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twisted smooth cohomology in string theory

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This entry contains lecture notes on

formulated in cohesive higher differential geometry.

Contents

A) Examples of twisted smooth cohomology in string theory

This section originates in notes prepared along with a lecture series (Schreiber ESI 2012) at the Erwin-Schrödinger institute in 2012.

This section discusses four classes of examples of twisted smooth cohomology in string theory

  1. Geometric structure (generalized, exceptional)

  2. Spin-, String-, and Fivebrane structure

  3. Spin cSpin^c-, String cString^c-, Fivebrane cFivebrane^c-structure

  4. Higher orientifold structure

Idea

The background gauge fields on spacetime appearing in string theory are mathematically described by cocycles in twisted and differential refinements of smooth cohomology. A famous example is the twisted K-theory that describes the B-field-twisted Yang-Mills fields over D-branes in type II string theory. But there are many more classes of examples of twisted / smooth / differential cohomology appearing throughout string theory.

This page provides a survey of and introduction to such examples, organized along a Table of twists, that indicates how all of these are instances a single pattern. For further reading and more details see the list of references below.

We start with an introduction to the general notion of twisted smooth cohomology by way of the simple but instructive class of examples of

via reduction of structure groups, which, simple as it is, serves as a blueprint for all of the examples to follow, and which we use to introduce the general machinery. It also serves to highlight the need and use of smooth cohomology in addition to both ordinary topological/homotopical as well as differential cohomology.

(The mathematically inclined reader wishing to see a more formal development of the general theory behind the discussion here should look at the section General theory below for pointers.)

Then we proceed in direct analogy, but now with ordinary gauge fields generalized to the genuine higher gauge fields of string theory, and discuss aspects of the main classes of examples of twisted smooth cohomology appearing there. First we indicate how higher spin structures as such lead to higher smooth homotopy theory:

Then we roughly indicate the relation between higher gauge fields and quantum anomalies:

Finally we put the pieces together and scan through various situations appearing in string theory with their anomaly structure and discuss the smooth moduli \infty-stacks of anomaly-free field configurations / of twisted smooth cocycles:

There are various further examples. As an outlook we indicate aspects of

Overview: The Table of Twists

The following sections discuss classes of examples of twisted smooth structures in string theory. All these examples are governed by the the same general pattern of twisted cohomology refined to smooth cohomology (a gentle explanation/example follows in a moment, for formal details see further below). They are specified by a universal local coefficient bundle

F E c BG \array{ F &\to& E \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }

which exists in a context of geometric homotopy types: an higher topos to be denoted H\mathbf{H} \coloneqq Smooth∞Grpd, of smooth ∞-groupoids/smooth ∞-stacks.

Here

Given such, and given a spacetime/target space XX, we have:

  • a morphism ϕ:XBG\phi : X \to \mathbf{B}G determines a twisting bundle or twisting background gauge field on XX;

  • a lift ϕ^\hat \phi in

    X ϕ^ E ϕ c BG \array{ X &&\stackrel{\hat \phi}{\to}&& E \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }

    is a ϕ\phi-twisted bundle, or ϕ\phi-twisted background gauge field.

The following table lists examples of such local coefficient bundles and tabulates the correspondings twisting fields and twisted fields. This is to be read as an extended table of contents. Explanations are in the sections to follow.

class of examplesuniversal local coefficient bundletwisting bundletwisting fieldtwisted bundletwisted field
0) general pattern
F E c BG\array{ F &\to& E \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }GG-principal bundleGG-gauge field instanton-sectorsection of ρ\rho-associated FF-bundletwisted ΩF\Omega F-gauge field instanton sector
B nU(1) B nU(1) curv dRB n+1U(1)\array{ \flat \mathbf{B}^n U(1) &\to& \mathbf{B}^n U(1) \\ && \downarrow^{\mathbf{curv}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1) }de Rham hypercohomologycircle n-bundle with connectionhigher abelian gauge field
F conn E conn c^ BG conn\array{ F_{conn} &\to& E_{conn} \\ && \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ && \mathbf{B}G_{conn} }GG-connectionGG-gauge fieldsection of ρ\rho-associated FF-bundletwisted ΩF\Omega F-gauge field
I) reduction of structure group/geometric structure
GL(n)/O(n) BO(n) BGL(n)\array{ GL(n)/O(n) &\to& \mathbf{B} O(n) \\ && \downarrow \\ && \mathbf{B} GL(n) }manifold structure / tangent bundlevielbeingravity
GL(n)/O(n) BO(n) conn BGL(n) conn\array{ GL(n)/O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow \\ && \mathbf{B} GL(n)_{conn} }affine connectionspin connectiongravity
O(2d,2d)/SU(d,d) BSU(d,d) BO(2d,2d)\array{ O(2d,2d)/SU(d,d) &\to& \mathbf{B} SU(d,d) \\ && \downarrow \\ && \mathbf{B} O(2d,2d) }generalized tangent bundlegeneralized Calabi-Yau manifolds
O(n)\O(n,n)/O(n) BO(n)×O(n) BO(n,n)\array{ O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B} O(n) \times O(n) \\ && \downarrow \\ && \mathbf{B} O(n,n) }generalized tangent bundletype II geometryDFT type II supergravity
SU(3)\O(6,6)/SU(3) BSU(3) BO(6,6)\array{ SU(3)\backslash O(6,6) / SU(3) &\to& \mathbf{B} SU(3) \\ && \downarrow \\ && \mathbf{B} O(6,6) }generalized tangent bundletype II geometryd=6d = 6, N=1N=1 type II supergravity comactifications
E 7(7)/SU(7) BSU(7) BE 7(7)\array{ E_{7(7)}/SU(7) &\to& \mathbf{B} SU(7) \\ && \downarrow \\ && \mathbf{B}E_{7(7)} }exceptional tangent bundleE7-U-duality moduli (split real form)d=7d = 7, N=1N=1 11d supergravity compactifications
E n(n)/H n BH n BE n(n)\array{ E_{n(n)}/H_n &\to& \mathbf{B}H_n \\ && \downarrow \\ && \mathbf{B}E_{n(n)} }exceptional tangent bundleU-duality moduliU-duality exceptional generalized geometrycompactification of 11d supergravity to d=11nd = 11-n
BU BPU dd B 2U(1)\array{ \mathbf{B} U &\to& \mathbf{B} P U \\ && \downarrow^{\mathbf{dd}} \\ && \mathbf{B}^2 U(1) }circle 2-bundle/U(1)U(1)-bundle gerbeB-fieldtwisted unitary bundleYang-Mills field
BString(E 8) BE 8 a B 3U(1)\array{ \mathbf{B}String(E_8) &\to& \mathbf{B} E_8 \\ && \downarrow^{\mathrlap{\mathbf{a}}} \\ && \mathbf{B}^3 U(1)} circle 3-bundle / bundle 2-gerbesupergravity C-fieldtwisted String(E8)-2-form gauge field
II) higher spin structures
BSpin BSO w 2 B 2 2\array{ \mathbf{B} Spin &\to& \mathbf{B} SO \\ && \downarrow^{\mathrlap{\mathbf{w}_2}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }second Stiefel-Whitney classtwisted spin structure
BString BSpin 12p 1 B 3U(1)\array{ \mathbf{B}String &\to& \mathbf{B} Spin \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1}} \\ && \mathbf{B}^3 U(1) }circle 3-bundle/U(1)U(1)-bundle 2-gerbeNS5-brane magnetic chargetwisted smooth string structure
BString conn BSpin conn 12p^ 1 B 3U(1) conn\array{ \mathbf{B}String_{conn} &\to& \mathbf{B} Spin_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\hat \mathbf{p}_1}} \\ && \mathbf{B}^3 U(1)_{conn} }circle 3-bundle with connectionNS5-brane magnetic currenttwisted differential string structureGreen-Schwarz mechanism gravity+B-field
BFivebrane BString 16p 2 B 7U(1)\array{ \mathbf{B}Fivebrane &\to& \mathbf{B} String \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\mathbf{p}_2}} \\ && \mathbf{B}^7 U(1) }circle 7-bundlestring electric chargetwisted smooth fivebrane structure
BFivebrane conn BString conn 16p^ 2 B 7U(1) conn\array{ \mathbf{B}Fivebrane_{conn} &\to& \mathbf{B} String_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\hat \mathbf{p}_2}} \\ && \mathbf{B}^7 U(1)_{conn} }circle 7-bundle with connectionstring electric currenttwisted differential fivebrane structuredual Green-Schwarz mechanism gravity+B6-field
III) higher spin^c-structures
BSpin c B(SO×U(1)) w 2c 1 B 2 2\array{ \mathbf{B}Spin^c &\to& \mathbf{B} (SO \times U(1)) \\ && \downarrow^{\mathrlap{ w_2 - c_1}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }twisted spin^c structure
B(Spin c) dd B(PU×SO) ddW 3 B 2U(1)\array{ \mathbf{B}(Spin^c)^{\mathbf{dd}} &\to& \mathbf{B} (PU \times SO) \\ && \downarrow^{\mathrlap{ \mathbf{dd} - \mathbf{W}_3 }} \\ && \mathbf{B}^2 U(1) }circle 2-bundleB-fieldtwisted spin^c structureFreed-Witten anomaly for type II superstring on D-brane
BString a BSpin×E 8 12p 12a B 3U(1)\array{ \mathbf{B}String^{\mathbf{a}} &\to& \mathbf{B} Spin \times E_8 \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - 2 \mathbf{a}}} \\ && \mathbf{B}^3 U(1) }circle 3-bundle/U(1)U(1)-bundle 2-gerbesupergravity C-fieldtwisted smooth string^c structuregravity+B-field+E8-gauge field
IV) Giraud-∞-gerbes
B 2U(1) BAut(BU(1)) B 2\array{ \mathbf{B}^2 U(1) &\to& \mathbf{B}Aut(\mathbf{B}U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }double coverJandl gerbeorientifold B-field
B 3U(1) BAut(B 2U(1)) B 2\array{ \mathbf{B}^3 U(1) &\to& \mathbf{B}Aut(\mathbf{B}^2 U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }double coverHořava-Witten orientifold

I) Geometric structure (generalized, exceptional)

The ordinary notion of vielbein in differential geometry (equivalently: soldering form or orthogonal structure) turns out to be a simple special case of the general notion of twisted smooth cohomology that we are concerned with here. Viewed from this perspective it already contains the seeds of all of the more sophisticated examples to be considered below. Therefore we discuss this case here as a warmup, such as to introduce the general theory by way of example.

The class of the tangent bundle

Let XX be a smooth manifold of dimension nn.

By definition this means that there is an atlas

{ nϕ i 1U iX} \{ \mathbb{R}^n \underoverset{\simeq}{\phi_i^{-1}}{\to} U_i \hookrightarrow X\}

of coordinate charts. On each overlap U iU jU_i \cap U_j of two charts, the partial derivatives of the corresponding coordinate transformations

ϕ jϕ i 1:U iU j n n \phi_j\circ \phi_i^{-1} : U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n

form the Jacobian matrix of smooth functions

((λ ij) μ μ)[ddx νϕ jϕ i 1(x μ)]:U iU jGL n ((\lambda_{i j})^{\mu}{}_{\mu}) \coloneqq \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu) \right] : U_i \cap U_j \to GL_n

with values in invertible matrices, hence in the general linear group GL(n)GL(n). By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations

(λ ij) μ λ(λ jk) λ ν=(λ ik) μ ν, (\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} = (\lambda_{i k})^\mu{}_{\nu} \,,

(here and in the following sums over an index appearing upstairs and downstairs are explicit)

hence the equation

λ ijλ jk=λ ik \lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}

in the group C (U iU jU k,GL(n))C^\infty(U_i \cap U_j \cap U_k, GL(n)) of smooth GL(n)GL(n)-valued functions on the chart overlaps.

This is the cocycle condition for a smooth Cech cocycle in degree 1 with coefficients in GL(n)GL(n) (precisely: with coefficients in the sheaf of smooth functions with values in GL(n)GL(n) ). We write

[(λ ij)]H smooth 1(X,GL n). [(\lambda_{i j})] \in H^1_{smooth}(X, GL_n) \,.

It is useful to reformulate this statement in the language of Lie groupoids/differentiable stacks.

  • XX itself is a Lie groupoid (XX)(X \stackrel{\to}{\to} X) with trivial morphism structure;

  • from the atlas {U iX}\{U_i \to X\} we get the corresponding Cech groupoid

    C({U i})=( i,jU iU j iU i)={ (x,j) = (x,i) (x,k)forxU iU jU k}, C(\{U_i\}) = (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i) = \left\{ \array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \;\;\; for\, x \in U_i \cap U_j \cap U_k \right\} \,,

    whose objects are the points in the atlas, with morphisms identifying lifts of a point in XX to different charts of the atlas;

  • any Lie group GG induces its delooping Lie groupoid

    BG=(G*). \mathbf{B}G = \left( G \stackrel{\to}{\to} * \right) \,.

The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form

C({U i}) λ BGL(n). X, \array{ C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n). \\ {}^{\mathllap{\simeq}}\downarrow \\ X } \,,

where

  • the left morphism is stalk-wisse (around small enough neighbourhoods of each point) an equivalence of groupoids (we make this more precise in a moment);

  • the horizontal functor has as components the functions λ ij\lambda_{i j} and its functoriality is the cocycle condition λ ijλ jk=λ ik\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}.

A transformation of smooth functors λ 1λ 2:C({U i})BGL(n)\lambda_1 \Rightarrow \lambda_2 : C(\{U_i\}) \to \mathbf{B} GL(n) is precisely a coboundary between two such cocycles.

Smooth moduli stacks

We want to think of such a diagram as being directly a morphism of smooth groupoids

TX:XBGL(n)H T X \; : \; X \to \mathbf{B} GL(n) \;\; \in \mathbf{H}

in a suitable context H\mathbf{H}, such that this may be regarded as a smooth refinement of the underlying homotopy class of a map into the classifying space BGL(n)B GL(n)

XBGL(n)Top. X \to B GL(n) \in Top \,.

Evidently, for this we need to turn the stalk-wise homotopy equivalence C({U i})XC(\{U_i\}) \to X into an actual homotopy equivalence. This is a non-abelian/non-stable generalization of what happens in the construction of a derived category, for instance in the theory of topological branes.

To make this precise, first notice that every Lie groupoid A=(A 1A 0)A = (A_1 \stackrel{\to}{\to} A_0) yields on each smooth manifold UU a groupoid of maps from UU into AA

A:U(C (U,A 1)C (U,A 0)), A : U \mapsto (C^\infty(U,A_1) \stackrel{\to}{\to} C^\infty(U,A_0)) \,,

the groupoid of smooth UU-families of elements of AA.

Moreover, for every smooth function U 1U 2U_1 \to U_2 there is an evident restriction map A(U 2)A(U 1)A(U_2) \to A(U_1) and so this yields a presheaf of groupoids, hence a functor AFunc(SmthMfd op,Grpd)A \in Func(SmthMfd^{op}, Grpd). The Yoneda lemma says that thinking of Lie groupoids as presheaves of ordinary groupoids this way does not lose information — and topos theory say that it is generally a good idea.

Let therefore

HFunc(SmthMfd op,Grpd)[{stalkwiseh.e} 1] \mathbf{H} \coloneqq Func(SmthMfd^{op}, Grpd)[\{stalkwise\, h.e\}^{-1}]

be the localization of groupoid-valued presheaves that universally turns stalkwise homotopy equivalences into actual homotopy equivalences: if a natural transformation η:AB\eta : A \to B in Func(SmthMfd op,Grpd)Func(SmthMfd^{op}, Grpd) is such that for each UU \in SmthMfd and each xUx \in U there is a neighbourhood U xUU_x \subset U of xx such that η(U x):A(U x)B(U x)\eta(U_x) : A(U_x) \to B(U_x) is an equivalence of groupoids, then η\eta has a homotopy inverse in H\mathbf{H}.

We call this H\mathbf{H} the (2,1)-topos of smooth groupoids or of smooth stacks.

Discussed there are tools for describing H\mathbf{H} concretely. For the moment we only need to know that

  1. the Cech nerve projection C({U i})XC(\{U_i\}) \to X of every open cover has a homotopy inverse in H\mathbf{H}, as already used;

  2. if the cover is good and GG is a Lie group, then every morphism XBGX \to \mathbf{B}G in H\mathbf{H} is presented by a zig-zag of the form XC({U i})BGX \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}G.

Then we have

  1. a morphism XBGL(n)X \to \mathbf{B} GL(n) in H\mathbf{H} is precisely a smooth real vector bundle on XX;

  2. a homotopy between two such morphisms in H\mathbf{H} is precisely a smooth GL(n)GL(n)-gauge transformation between the two vector bundles.

Therefore BGL(n)\mathbf{B}GL(n) regarded as an object of H\mathbf{H} is the moduli stack of real vector bundles.

Of course there is a “smaller” Lie groupoid that also classifies real vector bundles, but whose gauge transformations are restricted to be orthogonal group-valued functions. Passing to this “smaller” Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.

Reduction of the structure group

Consider the defining inclusion of the orthogonal group into the general linear group

O(n)GL(n). O(n) \hookrightarrow GL(n) \,.

We may understand this inclusion geometrically in terms of the canonical metric on n\mathbb{R}^n. We may also understand it purely Lie theoretically as the inclusion of the maximal compact subgroup of GL(n)GL(n). This makes it manifest that the inclusion is trivial at the level of homotopy theory (it is a homotopy equivalence of the underlying topological spaces) and hence only encodes geometric information.

The inclusion induces a corresponding inclusion (0-truncated morphism) of moduli stacks

Orth:BO(n)BGL(n)H \mathbf{Orth} : \mathbf{B} O(n) \to \mathbf{B} GL(n) \;\;\; \in \mathbf{H}

simply by regarding it as a morphism of Lie groupoids

(O(n)*)(GL(n)*) (O(n) \stackrel{\to}{\to} * ) \to (GL(n) \stackrel{\to}{\to} * )

in the evident way.

Now we can say what a Riemannian metric/orthogonal structure on XX is:

A choice of orthogonal structure on TXT X is a factorization of the above GL(n)GL(n)-valued cocycle λ\lambda through Orth\mathbf{Orth}, up to a smooth homotopy EE in H\mathbf{H}, hence a diagram

X h BO(n) λ E 1 Orth BGL(n) \array{ X &&\stackrel{h}{\to}&& \mathbf{B} O(n) \\ & {}_{\mathllap{\lambda}}\searrow &\swArrow_{\mathrlap{E^{-1}}}& \swarrow_{\mathrlap{\mathbf{Orth}}} \\ && \mathbf{B}GL(n) }

in H\mathbf{H}.

This consists of two pieces of data

  • the morphism hh is (by the same reasoning as for λ\lambda above) a O(n)O(n)-valued 1-cocycle – a collection of orthogonal transition functions – hence on each overlap of coordinate patches a smooth function

    ((h ij) a b):U iU jO(n) ((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n)

    such that

    h ijh jk=h ik h_{i j} \cdot h_{j k} = h_{i k}

    on all triple overlaps of coordinate charts U iU jU kU_i \cap U_j \cap U_k;

  • the homotopy EE is on each chart a function

    E i=((E i) a μ):U iGL(n) E_i = ((E_i)^a{}_\mu) : U_i \to GL(n)
    • such that on each overlap of coordinate charts it intertwines the transition functions λ\lambda of the tangent bundle with the new orthogonal transition functions, meaning that the equation

      (E i) a μ(λ ij) μ ν=(h ij) a b(E j) b ν (E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu = (h_{i j})^a{}_b (E_j)^b{}_\nu

      holds. This exhibits the naturality diagram of EE:

    * E i * λ ij h ij * E j * \array{ * &\stackrel{E_i}{\to}& * \\ {}^{\mathllap{\lambda_{i j}}}\downarrow && \downarrow^{\mathrlap{h_{i j}}} \\ * &\stackrel{E_j}{\to}& * }

The component hh defines an O(n)O(n)-principal bundle on XX, or its associated vector bundle. The component EE is the corresponding vielbein. It exhibits an isomorphism

E:TXV E : T X \stackrel{\simeq}{\to} V

between a vector bundle VXV \to X with structure group explicitly being the orthogonal group, and the tangent bundle itself, hence it exhibits the reduction of the structure group of TXT X from GL(n)GL(n) to O(n)O(n).

Moduli space of orthogonal structures: twisted cohomology

We consider now the space of choices of vielbein fields on a given tangent bundle, hence the moduli space or moduli stack of orthogonal structures/Riemannian metrics on XX.

This is usefully discussed in terms of the homotopy fiber of the morphism c:BO(n)BGL(n)\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n). One finds that the homotopy fiber is the coset O(n)\GL(n)O(n) \backslash GL(n).

This means that there is a diagram

GL(n)/O(n) BO(n) * BGL(n) \array{ GL(n)/O(n) &\to& \mathbf{B}O(n) \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& \mathbf{B} GL(n) }

in H\mathbf{H}, and that GL(n)/O(n)GL(n)/O(n) is universal with the property of sitting in such a diagram.

We may think of this fiber sequence as being a bundle in H\mathbf{H} over the moduli stack BGL(n)\mathbf{B}GL(n) with typical fiber GL(n)/O(n)GL(n)/O(n). As such, it is the smooth associated bundle to the smooth universal GL(n)-bundle induced by the canonical action of GL(n)GL(n) on O(n)\GL(n)O(n)\backslash GL(n).

One basic properties of homotopy pullbacks is that they are preserved by forming derived hom-spaces H(X,)\mathbf{H}(X,-) out of any other object XX. This means that also

C (X,GL(n)/O(n)) H(X,BO(n)) H(X,BGL(n)) \array{ C^\infty(X,GL(n)/O(n)) &\to& \mathbf{H}(X,\mathbf{B}O(n)) &\to& \mathbf{H}(X, \mathbf{B}GL(n)) }

is a fiber sequence. This in turn says that orthogonal structures on XX such that the underlying tangent bundle is trivializable, are given by smooth functions into GL(n)/O(n)GL(n)/O(n).

This means that if the tangent bundle TXT X is trivializable, then the coset space O(n)\GL(n)O(n)\backslash GL(n) is the moduli space for vielbein fields on TXT X:

(TXtrivializable)SpaceOfVielbeinFieldsOn(TX)=H(X,O(n)\GL(n))=C (X,O(n)\GL(n)). (T X \; trivializable) \Rightarrow SpaceOfVielbeinFieldsOn(T X) = \mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslash GL(n)) \,.

However, if TXT X is not trivial, then this is true only locally: there is then an atlas {U iX}\{U_i \to X\} such that restricted to each U iU_i the moduli space of vielbein fields is C (U i,GL(n)/O(n))C^\infty(U_i, GL(n)/ O(n)), but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that

the tangent bundle twists the functions XGL(n)/O(n)X \to GL(n)/O(n). If we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in TXT X-_twisted cohomology_ relative to the twisting local coefficient bundle c\mathbf{c}.

We can make this more manifest by writing equivalently

O(n)\GL(n) (O(n)\GL(n))//GL(n) BGL(n), \array{ O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n) \\ && \downarrow \\ && \mathbf{B}GL(n) } \,,

where now on the right we have inserted the fibration resolution of the morphism c\mathbf{c} as provided by the factorization lemma: this is the morphism out of the action groupoid of the action of GL(n)GL(n) on O(n)\GL(n)O(n)\backslash GL(n).

The pullback

TX× GL(n)(O(n)\GL(n)) O(n)\GL(n)//GL(n) X TX BGL(n) \array{ T X \times_{GL(n)} (O(n)\backslash GL(n)) &\to& O(n)\backslash GL(n) // GL(n) \\ \downarrow && \downarrow \\ X &\stackrel{T X}{\to}& \mathbf{B}GL(n) }

gives the non-linear TXT X-associated bundle whose space of sections is the “twisted O(n)\GL(n)O(n)\backslash GL(n)-0-cohomology”, hence the space of inequivalent vielbein fields.

Moduli stack of orthogonal structures

The above says that the space of vielbein fields is the cohomology of TXT X in the slice (2,1)-topos H /BGL(n)\mathbf{H}_{/\mathbf{B}GL(n)} with coefficients in Orth:BOBGL(n)\mathbf{Orth} : \mathbf{B}O \to \mathbf{B}GL(n)

OrthStruc TX(X)Orth(TX)H /BGL(n)(TX,Orth). \mathbf{Orth} Struc_{TX}(X) \coloneqq \mathbf{Orth}(T X) \coloneqq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{Orth}) \,.

But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth moduli stack. This is obtained by forming the internal hom in the slice over BGL(n)\mathbf{B}GL(n) of the locally cartesian closed (2,1)-category H\mathbf{H}.

OrthStruc TX(X)[TX,Orth] BGL(n). \mathbf{OrthStruc}_{T X}(X) \coloneqq [T X, \mathbf{Orth}]_{\mathbf{B}GL(n)} \,.

For more on this see also the discussion at general covariance.

Differential refinement: Spin connection

We may further refine this discussion to differential cohomology to get genuine differential TXT X-twisted c\mathbf{c}-structures.

Recall that the moduli stack BG\mathbf{B}G is presented in H\mathbf{H} by the presheaf of groupoids

BG:U(C (U,G)*). \mathbf{B} G : U \mapsto (C^\infty(U,G) \stackrel{\to}{\to} *) \,.

We may think of this for each UU as being the groupoid of GG-gauge transformations acting on the trivial GG-bundle over UU. A connection on the trivial GG-bundle is a Lie algebra valued form AΩ 1(U,𝔤)A \in \Omega^1(U, \mathfrak{g}). Accordingly, the presheaf of groupoids

BG conn:U(C (U,G)×Ω 1(U,𝔤)Ω 1(U,𝔤)) \mathbf{B}G_{conn} : U \mapsto (C^\infty(U,G) \times \Omega^1(U, \mathfrak{g}) \stackrel{\to}{\to} \Omega^1(U, \mathfrak{g}) )

is that of GG-connections and gauge transformations between them: the groupoid of Lie-algebra valued forms over UU. As an object of H=\mathbf{H} = SmoothGrpd this the moduli stack of GG-connections:

H(X,BG conn)GBund (X). \mathbf{H}(X, \mathbf{B}G_{conn}) \simeq G Bund_\nabla(X) \,.

The morphism Orth\mathbf{Orth} has an evident differential refinement to a morphism between such differentially refined moduli stacks

Orth conn:BO(n) connBGL(n) conn \mathbf{Orth}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn}

by acting on the differential forms with the induced inclusion of the orthogonal Lie algebra into the general linear Lie algebra 𝔬(n)𝔤𝔩(n)\mathfrak{o}(n) \hookrightarrow \mathfrak{gl}(n).

The homotopy fiber of this differential refinement turns out to be the same moduli space as before

GL(n)/O(n) BO(n) conn Orth conn BGL(n) conn, \array{ GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow^{\mathrlap{\mathbf{Orth}_{conn}}} \\ && \mathbf{B} GL(n)_{conn} } \,,

so that the moduli space of “differential vielbein fields” is the same as that of plain vielbein fields. But we nevertheless do gain differential information: consider an affine connection on the tangent bundle, which is now given by a morphism from XX to the moduli stack

TX:XBGL(n). \nabla_{T X} : X \to \mathbf{B}GL(n) \,.

This is a GL(n)GL(n)-principal connection which locally on an atlas is given by the Christoffel symbols

Γ i=((Γ i) μ α β)Ω 1(U i,𝔤𝔩(n)). \Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta) \in \Omega^1(U_i, \mathfrak{gl}(n)) \,.

A TX\nabla_{T X}-twisted differential cocycle is now a diagram

X V BO conn TX E 1 Orth^ BGL(n) conn. \array{ X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn} \\ & {}_{\mathllap{\nabla_{T X}}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\hat {\mathbf{Orth}}} \\ && \mathbf{B}GL(n)_{conn} } \,.

In components over the atlas, V\nabla_V is a “spin connection” given by local 1-forms {ω iΩ 1(U i,𝔬(n))}\{\omega_i \in \Omega^1(U_i, \mathfrak{o}(n))\}

ω i=E id dRE i 1+E iΓE i 1 \omega_i = E_i d_{dR} E_i^{-1} + E_i \Gamma E_i^{-1}

and the vielbeing EE now exhibits on each chart U iU_i the familiar relation between the components of the spin connection and the Christoffel-symbols:

ω a b=E i a νd dRE i ν b+E i a νΓ i ν λE i λ b. \omega{}^a{}_b = E_i^a{}_\nu d_{dR} E_i^\nu{}_b + E_i^a{}_\nu \Gamma_i {}^\nu{}_\lambda E_i^\lambda{}_b \,.

Pullback of orthogonal structures

It is a familiar fact that many fields in physics “naturally pull back”. For instance a scalar field on a spacetime XX is a function ϕ:X\phi : X \to \mathbb{C}, and for f:YXf : Y \to X any smooth function between spactimes, there is the corresponding pullback function/field f *ϕ:Yf^* \phi : Y \to \mathbb{C}.

Similarly for ϕ:XBG conn\phi : X \to \mathbf{B}G_{conn} a gauge field, as discussed above, it has naturally a pullback along ff given simply by forming the composite f *ϕ:YfXϕBG connf^* \phi : Y \stackrel{f}{\to} X \stackrel{\phi}{\to} \mathbf{B}G_{conn}.

But the situation is a little different for twisted fields such as orthogonal structures/Riemannian metrics. If we think of a Riemannian metric as given by a non-degenerate rank-2 tensor on XX, then the problem is that, while its pullback along ff will always be a rank-2 tensor, it is not in general non-degenerate anymore – unless ff is a local diffeomorphism.

This is also nicely formulated in the language used above, in a way that has a useful generalization when we come to higher twisted structures below: since the metric is encoded not just in a plain morphism h:XBOh : X \to \mathbf{B}O, but one that fits into a triangle

X h BO TX E orth BGL \array{ X && \stackrel{h}{\to} && \mathbf{B} O \\ & {}_{\mathllap{T X}}\searrow & \swArrow_{E}& \swarrow_{\mathrlap{orth}} \\ && \mathbf{B} GL }

a simple precomposition with just a morphism f:YXf : Y \to X is not the right operation to send this triangle based on XX to one based on YY.

But since this triangle is a morphism (h,E):TXorth(h,E) : T X \to \mathbf{orth} in the slice topos H /BGL\mathbf{H}_{/\mathbf{B}GL}, it is clear that it does pull back precisely along refinements of f:YXf : Y \to X to a morphism in H /BGL\mathbf{H}_{/\mathbf{B}GL}.

Such a refinement is a commuting triangle of the form

Y f X TY TX BGL \array{ Y &&\stackrel{f}{\to}&& X \\ & {}_{\mathllap{T Y}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{T X}} \\ && \mathbf{B} GL }

in H\mathbf{H}. But this is evidently the same as an isomorphism TYf *TXT Y \simeq f^* T X between the tangent bundle of YY and the pullback of the tangent bundle on XX. And this exhibits ff as a local diffeomorphism.

Generalized vielbein fields: type II geometry, generalized CY and U-duality

The above discussion of ordinary vielbein fields is just a special case of an analogous discussion for general reduction of structure groups, giving rise to generalized vielbein fields. Many geometric structures in string theory arise in this way, as indicated in the table of twists.

As one more out of these examples, we discuss in the above language of twisted smooth cohomology how a type II geometry of type II supergravity is the reduction of the structure group of the generalized tangent bundle along the inclusion O(d)×O(d)O(d,d)O(d) \times O(d) \to O(d,d).

Consider the Lie group inclusion

O(d)×O(d)O(d,d) \mathrm{O}(d) \times \mathrm{O}(d) \to \mathrm{O}(d,d)

of those orthogonal transformations, that preserve the positive definite part or the negative definite part of the bilinear form of signature (d,d)(d,d), respectively.

If O(d,d)\mathrm{O}(d,d) is presented as the group of 2d×2d2d \times 2d-matrices that preserve the bilinear form given by the 2d×2d2d \times 2d-matrix

η(0 id d id d 0) \eta \coloneqq \left( \array{ 0 & \mathrm{id}_d \\ \mathrm{id}_d & 0 } \right)

then this inclusion sends a pair (A +,A )(A_+, A_-) of orthogonal n×nn \times n-matrices to the matrix

(A +,A )12(A ++A A +A A +A A ++A ). (A_+ , A_-) \mapsto \frac{1}{\sqrt{2}} \left( \array{ A_+ + A_- & A_+ - A_- \\ A_+ - A_- & A_+ + A_- } \right) \,.

This induces the corresponding morphism of smooth moduli stacks, which we denote

TypeII:B(O(d)×O(d))BO(d,d). \mathbf{TypeII} : \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \to \mathbf{B} \mathrm{O}(d,d) \,.

Forming the homotopy fiber now yields the local coefficient bundle

O(d)\O(d,d)/O(d) B(O(d)×O(d)) TypeII BO(d,d), \array{ O(d) \backslash O(d,d) / O(d) &\to& \mathbf{B}(\mathrm{O}(d) \times \mathrm{O}(d)) \\ && \downarrow^{\mathrlap{\mathbf{TypeII}}} \\ && \mathbf{B} \mathrm{O}(d,d) } \,,

There is also a canonical embedding

GL(d)O(d,d) \mathrm{GL}(d) \hookrightarrow \mathrm{O}(d,d)

of the general linear group.

In the above matrix presentation this is given by sending

a(a 0 0 a T), a \mapsto \left( \array{ a & 0 \\ 0 & a^{-T} } \right) \,,

where in the bottom right corner we have the transpose of the inverse matrix of the invertble matrix aa.

Under this inclusion, the tangent bundle of a dd-dimensional manifold XX defines an O(d,d)\mathrm{O}(d,d)-cocycle

TXT *X:XTXBGL(d)BO(d,d). T X \oplus T^* X : X \stackrel{T X}{\to} \mathbf{B}\mathrm{GL}(d) \stackrel{}{\to} \mathbf{B} \mathrm{O}(d,d) \,.

The vector bundle canonically associated to this composite cocycle may canonically be identified with the direct sum vector bundle TXT *XT X \oplus T^* X, and so we will refer to this cocycle by these symbols, as indicated. This is also called the generalized tangent bundle of XX.

Therefore we may canonically consider the groupoid of TXT *XT X \oplus T^* X-twisted TypeII\mathbf{TypeII}-structures, according to the general notion of twisted differential c-structures.

A type II generalized vielbein on a smooth manifold XX is a diagram

X (˜TXT *X) B(O(n)×O(n)) TXT *X E TypeII BO(n,n) \array{ X &&\stackrel{\widetilde(T X \oplus T^* X)}{\to}&& \mathbf{B}(O(n) \times O(n)) \\ & {}_{\mathllap{T X \oplus T^* X}}\searrow &\swArrow_{E}& \swarrow_{\mathrlap{\mathbf{TypeII}}} \\ && \mathbf{B} O(n,n) }

in H\mathbf{H}, hence a cocycle in the smooth twisted cohomology

ETypeIIStruc(X)H /BO(n,n)(TXT *X,TypeII). E \in \mathbf{TypeII}Struc(X) \coloneqq \mathbf{H}_{/\mathbf{B} O(n,n)}(T X \oplus T^* X, \mathbf{TypeII}) \,.
Proposition / Definition

The groupoid TypeIIStruc(X)\mathbf{TypeII}\mathrm{Struc}(X) is that of “generalized vielbein fields” on XX, as considered for instance around equation (2.24) of (GMPW) (there only locally, though).

In particular, its set of equivalence classes is the set of type-II generalized geometry structures on XX.

Proof

Over a local coordinate chart dU iX\mathbb{R}^d \simeq U_i \hookrightarrow X, the most general such generalized vielbein (hence the most general O(d,d)\mathrm{O}(d,d)-valued function) may be parameterized as

E=12((e ++e )+(e + Te T)B (e + Te T) (e +e )(e + T+e T)B (e + T+e T)), E = \frac{1}{2} \left( \array{ (e_+ + e_-) + (e_+^{-T} - e_-^{-T})B & (e_+^{-T} - e_-^{-T}) \\ (e_+ - e_-) - (e_+^{-T} + e_-^{-T})B & (e_+^{-T} + e_-^{-T}) } \right) \,,

where e +,e C (U i,O(d))e_+, e_- \in C^\infty(U_i, \mathrm{O}(d)) are thought of as two ordinary vielbein fields, and where BB is any smooth skew-symmetric n×nn \times n-matrix valued function on dU i\mathbb{R}^d \simeq U_i.

By an O(d)×O(d)\mathrm{O}(d) \times \mathrm{O}(d)-gauge transformation this can always be brought into a form where e +=e =:12ee_+ = e_- =: \tfrac{1}{2}e such that

E=(e 0 e TB e T). E = \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) \,.

The corresponding “generalized metric” over U iU_i is

E TE=(e T Be 1 0 e 1)(e 0 e TB e T)=(gBg 1B Bg 1 g 1B g 1), E^T E = \left( \array{ e^T & B e^{-1} \\ 0 & e^{-1} } \right) \left( \array{ e & 0 \\ - e^{-T}B & e^{-T} } \right) = \left( \array{ g - B g^{-1} B & B g^{-1} \\ - g^{-1} B & g^{-1} } \right) \,,

where

ge Te g \coloneqq e^T e

is the metric (over qU i\mathbb{R}^q \simeq U_i a smooth function with values in symmetric n×nn \times n-matrices) given by the ordinary vielbein ee.

II) Spin-, String- and Fivebrane-structure

Above we have seen (pseudo-)Riemannian structure given by lifts through the inclusion BO(n)BGL(n)\mathbf{B} O(n) \to \mathbf{B} GL(n). Now we consider further lifts, through the Whitehead tower of BO\mathbf{B}O. This encodes higher spin structures.

Where a spin structure on spacetime is necessary to cancel a quantum anomaly of the spinning particle/superparticle sigma-model, so the heterotic superstring requires, in the absence of a gauge field, a “higher spin structure”, called a string structure. Further up in dimension, dual heterotic string theory in the absence of the gauge field involves a fivebrane structure.

In the presence of a nontrivial gauge fields, string structures are generalized to twisted string structure, which in heterotic string theory are part of the Green-Schwarz mechanism. These we discuss below.

Whitehead tower

The nature of higher spin structures is governed by what is called the Whitehead tower of the homotopy type of the classifying space BOB O of the orthogonal group, where in each stage a homotopy group is removed from below. This is dual to the Postnikov tower, where in each stage a homotopy group is added from above.

The homotopy groups of BOB O start out as

k=k =012345678
π k(BO)=\pi_k(B O) = * 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}

The Whitehead tower of BOB O starts out as

Whiteheadtower BFivebrane * secondfracPontr.class BString 16p 2 B 8 * firstfracPontr.class BSpin 12p 1 B 4 * secondSWclass BSO w 2 B 2 2 * firstSWclass BO τ 8BO τ 4BO τ 2BO w 1 τ 1BOB 2 Postnikovtower BGL(n) \array{ & Whitehead tower \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ second frac Pontr. class & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ first frac Pontr. class & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ second SW class & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ first SW class & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & Postnikov tower \\ & \downarrow \\ & B GL(n) }

where

For instance w 2w_2 can be identified as such by representing BOτ 2BOBO/ nB O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n} by a Kan fibration (see at Postnikov tower) between Kan complexes so that then the homotopy pullback (as discussed there) is given by an ordinary pullback. Since sSetsSet is a simplicial model category, sSet(S 2,)sSet(S^2,-) can be applied and preserves the pullback as well as the homotopy pullback, hence sends BOτ 2BO B O \to \tau_{\leq 2} B O to an isomorphism on connected components. This identifies BSOB 2B SO \to B^2 \mathbb{Z} as being an isomorphism on the second homotopy group. Therefore, by the Hurewicz theorem, it is also an isomorphism on the cohomology group H 2(, 2)H^2(-,\mathbb{Z}_2). Analogously for the other characteristic maps.

In summary, more concisely, the tower is

BFivebrane BString 16p 2 B 7U(1) B 8 BSpin 12p 1 B 3U(1) B 4 BSO w 2 B 2 2 BO w 1 B 2 BGL, \array{ \vdots \\ \downarrow \\ B Fivebrane \\ \downarrow \\ B String &\stackrel{\tfrac{1}{6}p_2}{\to}& B^7 U(1) & \simeq B^8 \mathbb{Z} \\ \downarrow \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& B^3 U(1) & \simeq B^4 \mathbb{Z} \\ \downarrow \\ B SO &\stackrel{w_2}{\to}& B^2 \mathbb{Z}_2 \\ \downarrow \\ B O &\stackrel{w_1}{\to}& B \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ B GL } \,,

where each “hook” is a fiber sequence.

The universal property of homotopy pullbacks says that

Necessity of smooth refinement

We will below consider a smooth refinement of the above Whitehead tower. Before we do so, here a few words on why we need to do this.

One way to state the general problem is:

  1. The classifying space of, say, the spin group is not afine moduli space_.

    Because, while homotopy classes Maps(X,BSpin) Maps(X, B Spin)_\sim of maps XBSpinX \to B Spin are in bijection with equivalence classes of spin bundles on XX, the homotopy classes Maps(X,ΩBSpin) =Maps(X,Spin) Maps(X, \Omega B Spin)_\sim = Maps(X, Spin)_\sim of homotopies from the trivial map X*BSpinX \to * \to B Spin are not in general in bijection with the gauge transformations of the trivial spin bundle: the latter form the set of smooth functions C (X,Spin)C^\infty(X,Spin), not just the homotopy classes of these.

  2. Maps(X,B()) Maps(X, B(-))_\sim does not give the right BRST-complex; hence speaking about gauge theory in terms of just bare (as opposed to geometric) homotopy theory does not yield an admissible starting point for quantization (by BV-BRST formalism).

  3. Maps(X,B()) Maps(X, B(-))_\sim cannot distinguish a group from its maximal compact subgroup, such as OGL nO \hookrightarrow GL_n, and hence cannot see vielbeins, not generalized vielbeins, not exceptional generalized geometry:

in terms of classifying spaces the entire discussion of vielbein fields above would collapse;

higher analogs of this problem include for instance that Maps(X,B()) Maps(X, B(-))_\sim cannot distinguish over 10-dimensional spacetime XX an E8-gauge field from a NS5-brane magnetic charge;

  1. eventually an action functional on the space of fields is to be constructed as a smooth function, or in fact as a smooth flat section of a smooth circle bundle with connection on the space of fields – which requires some smooth structure on that space.

These problems are all fixed by refining classifying spaces such as BSpinGrpdB Spin \in \infty Grpd to smooth moduli stacks such as BSpinSmoothGrpd\mathbf{B} Spin \in Smooth \infty Grpd.

Smooth refinement

We considered above the smooth refinement of the classifying space BGB G for GG a Lie group to a smooth moduli stack BG\mathbf{B}G. While that works well, one can see on general grounds that this cannot provide a smooth refinement of the higher stages of the Whitehead tower, if one asks the refinement to preserve obstruction theory. The problem is that a smooth stack is necessarily a smooth homotopy 1-type (even if its geometric realization is a higher type! see below), while the higher stages of the smooth Whitehead tower need to be smooth homotopy n-types/n-groupoids for higher nn.

But there is an evident refinement of the above discussion to such smooth nn-types.

To that end we first need a good model for bare homotopy types. One observes that the nerve functor embeds groupoids into Kan simplicial sets, as precisley those which are 2-coskeletal, meaning that only their 0-cells and 1-cells are non-trivial. Accordingly, a Kan complex which is (n+1)-coskeletal may be regarded as an n-groupoid modelling a homotopy n-type, and hence a general Kan complex as an ∞-groupoid.

Groupoids N Categories KanComplexes N QuasiCategories SimplicialSets. \array{ && Groupoids \\ & \swarrow && \searrow^{\mathrlap{N}} \\ Categories &&& & KanComplexes \\ & {}_{\mathllap{N}}\searrow && \swarrow \\ && QuasiCategories \\ && \downarrow \\ && SimplicialSets } \,.

A morphism between groupoids XYX \to Y is an equivalence of groupoids precisely if it is an essentially surjective functor and a full and faithful functor. This is equivalent to it inducing an isomorphism on isomorphism classes / connected components, and on automorphism groups. This in turn is equivalent to it inducing an isomorphism on the 0th and the first homotopy groups π 0\pi_0 and π 1\pi_1.

Accordingly, we say that a homotopy equivalence between Kan complexes is a morphism XYX \to Y which induces an isomorphism on all homotopy groups. These can be defined for general simplicial sets, and we say a morphism between these is a weak homotopy equivalence if it induces such isomorphisms.

Write

GroupoidsSimplicialSets[{weakhomotopyequivalences} 1] \infty Groupoids \simeq SimplicialSets[ \{ weak\;homotopy\;equivalences\}^{-1} ]

for the homotopy theory obtained by localization at the weak homotopy equivalences: ∞-groupoids.

A topological space XX defines a Kan complex/ ∞-groupoid by the singular simplicial complex construction SingXSing X, and this establishes an equivalence between the homotopy theory of topological spaces and simplicial sets

TopSing||sSet. Top \stackrel{\stackrel{{\vert-\vert}}{\leftarrow}}{ \underoverset{\simeq}{Sing}{\to} } sSet \,.

In view of this, the above Whitehead tower can be understood entirely as taking place in Kan complexes.

For instance for AA an abelian group, the 2-groupoid

B 2U(1) disc={ * aA * } \mathbf{B}^2 U(1)_{disc} = \left\{ \array{ && \to \\ & \nearrow && \searrow \\ * &&\Downarrow^{a \in A}&& * \\ & \searrow && \nearrow \\ && \to } \right\}

corresponds to the Eilenberg-MacLane space K(A,2)K(A,2).

More generally, for each chain complex A A_\bullet of abelian groups the Dold-Kan correspondence provides a Kan complex Ξ(A )\Xi(A_\bullet) whose simplicial homotopy groups are the chain homology groups of A A_\bullet. A quasi-isomorphism A B A_\bullet \to B_\bullet is sent to a weak homotopy equivalence Ξ(A )Ξ(B )\Xi(A_\bullet) \to \Xi(B_\bullet). In this sense ∞Grpd is a non-abelian generalization of chain complexes with quasi-isomorphisms inverted.

Using this, we can easily state the generalization of the definition of smooth stacks from above: we obtain a homotopy theory of smooth ∞-stacks H\mathbf{H} \coloneqq Smooth∞Grpd by considering simplicial sets parameterized over smooth manifolds and forcing stalkwise weak homotopy equivalences to become homotopy equivalences

HFunc(SmthMfd op,sSet)[{stalkwisew.h.e.} 1]. \mathbf{H} \coloneqq Func(SmthMfd^{op}, sSet)[\{stalkwise\, w.h.e.\}^{-1}] \,.

For instance there is the smooth 2-stack

B 2U(1)H \mathbf{B}^2 U(1) \in \mathbf{H}

given by assigning to each test space UU the Eilenberg-MacLane space on the (discrete) abelian group of smooth functions UU(1)U \to U(1)

B 2U(1):UK(C (U,U(1)),2)={ * cC (U,U(1)) * }. \mathbf{B}^2 U(1) : U \mapsto K( C^\infty(U, U(1)), 2) = \left\{ \array{ && \to \\ & \nearrow && \searrow \\ * &&\Downarrow^{c \in C^\infty(U,U(1))}&& * \\ & \searrow && \nearrow \\ && \to } \right\} \,.

A morphism XB 2U(1)X \to \mathbf{B}^2 U(1) in H\mathbf{H} is equivalently a zig-zag XC({U i})λB 2U(1)X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \stackrel{\lambda}{\to} \mathbf{B}^2 U(1) through the Cech nerve. This now defines in degree 2

(x,j) (x,1) (x,k) * λ ijk)(x * * \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,1) &&\to&& (x,k) } \;\;\; \mapsto \;\;\; \array{ && * \\ & \nearrow &\Downarrow^{\lambda_{i j k})(x}& \searrow \\ * &&\to&& * }

smooth functions λ\lambda on triple overlaps

λ ijk:U iU jU kU(1) \lambda_{i j k } : U_i \cap U_j \cap U_k \to U(1)

and the condition on quadruple overlaps U iU jU kU lU_i \cap U_j \cap U_k \cap U_l says that they satisfy

λ ijkλ ikl=λ jklλ ijk. \lambda_{i j k} \lambda_{i k l} = \lambda_{j k l} \lambda_{i j k} \,.

This now identifies (lamda ijk)(\lamda_{i j k}) with a cocycle in degree-2 Cech cohomology

[(λ ijk)]H smooth 2(X,U(1)). [(\lambda_{i j k})] \in H^2_{smooth}(X, U(1)) \,.

This classifies a smooth circle 2-bundle / bundle gerbe.

Generally we have

π 0H(X,B nU(1))H smooth n(X,U(1)) \pi_0 \mathbf{H}(X, \mathbf{B}^n U(1)) \simeq H^n_{smooth}(X, U(1))

and for XX a smooth manifold

H n+1(X,). \cdots \simeq H^{n+1}(X, \mathbb{Z}) \,.

So apparently the smooth nn-stack B nU(1)\mathbf{B}^n U(1) is a smooth refinement of the Eilenberg-MacLane space K(,n+1)K(\mathbb{Z},n+1).

This is made precise as follows.

Theorem There is an (∞,1)-functor

||:SmoothGrpdΠGrpd||Top {\vert-\vert} : Smooth\infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{{\vert-\vert}}{\to} Top

which is left adjoint to the functor that assigns constant ∞-stacks. We call this the geometric realization of smooth ∞-groupoids.

We say that a choice of lift of a diagram of bare homotopy types through geometric realization is a smooth geomtric refinement.

For instance one finds

|B nU(1)|B nU(1)B n+1K(,n+1) {\vert \mathbf{B}^n U(1) \vert} \simeq B^n U(1)\simeq B^{n+1}\mathbb{Z} \simeq K(\mathbb{Z}, n+1)

and hence B nU(1)\mathbf{B}^n U(1) is a smooth geometric refinement of K(,n+1)K(\mathbb{Z}, n+1).

We now apply this to the above Whitehead tower.

Smooth Whitehead tower

We state the smooth refinement of the above Whitehead tower and then explain some aspects of how it is constructed.

Theorem There is a smooth geometric refinement of the above Whitehead tower of bare homotopy types to a tower of smooth homotopy types/smooth ∞-stacks of the form

BFivebrane BString 16p 2 B 7U(1) BSpin 12p 1 B 3U(1) BSO w 2 B 2 2 BO w 1 B 2 BGL. \array{ \vdots \\ \downarrow \\ \mathbf{B} Fivebrane \\ \downarrow \\ \mathbf{B} String &\stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) \\ \downarrow \\ \mathbf{B} Spin &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} GL } \,.

where

and where

  • 12p 1\tfrac{1}{2} \mathbf{p}_1 classifies the universal Chern-Simons circle 3-bundle and hence identifies it with BStringBSpin\mathbf{B}String \to \mathbf{B}Spin;

  • 16p 2\tfrac{1}{6} \mathbf{p}_2 classifies the universal Chern-Simons circle 7-bundle and hence identifies it with BFivebraneBString\mathbf{B}Fivebrane \to \mathbf{B}String.

This is constructed using essentially the following three tools for presenting presheaves of higher groupoids:

  1. The Dold-Kan correspondence

    Ξ:Ch 0sAbsSet \Xi : Ch_{\geq 0} \stackrel{\simeq}{\to} sAb \to sSet

    and its prolongation to presheaves

    Ξ:[SmthMfd op,Ch 0)[SmthMfd op,sAb][SmthMfd op,sSet] \Xi : [SmthMfd^{op}, Ch_{\geq 0}) \stackrel{\simeq}{\to} [SmthMfd^{op},sAb] \to [SmthMfd^{op},sSet]

    allows to use presheaves of chain complexes of abelian groups to present presheaves of strict \infty-groupoids with strict abelian group structure.

    For instance

    B nU(1)Ξ(C (,U(1))[n]) \mathbf{B}^n U(1) \simeq \Xi( C^\infty(-,U(1))[n] )

    is equivalent to the image under the DK correspondence of the sheaf of chain complexes which is concentrated in degree nn on the group of U(1)U(1)-valued functions.

  2. Some nonabelian generalizations of the Dold-Kan correspondence allow to use chain complexes of not entirely abelian groups – crossed complexes – to present a few more classes of \infty-groupoids. Notably nonabelian 2-term chain complexes,

    G 1δG 0 G_1 \stackrel{\delta}{\to} G_0

    called crossed modules, due to them being equipped with a compatible action G 0Aut(G 1)G_0 \to Aut(G_1), serve to equivalently present strict 2-groups.

    For instance, one way to construct the string 2-group StringString above is via the crossed module (Ω^ *SpinP *Spin)(\hat \Omega_* Spin \to P_* Spin) induced from the Kac-Moody central extension of the loop group of SpinSpin.

    For a given crossed module, the corresponding moduli 2-stack B(G 1δG 0)\mathbf{B}(G_1 \stackrel{\delta}{\to} G_0) has 2-cells that look like

    B(G 1G 0)={ * g 1 h g 2 * δ(h)g 2g 1 |g 1,g 2G 0,hG 1}. \mathbf{B}(G_1 \to G_0) = \left\{ \array{ && * \\ & {}^{\mathllap{g_1}}\nearrow &\Downarrow^{\mathrlap{h}}& \searrow^{\mathrlap{g_2}} \\ * &&\underset{\delta(h) g_2 g_1}{\to}&& } \;\; | \;\; g_1,g_2 \in G_0, h \in G_1 \right\} \,.
  3. The Lie integration τ nexp(𝔤)\tau_{\leq n} \exp(\mathfrak{g}) of an L-∞ algebra 𝔤\mathfrak{g} yields the corresponding smooth ∞-group GG. For instance the string 2-group StringString above is also equivalently given by BStringτ 2exp(𝔰𝔱𝔯𝔦𝔫𝔤)\mathbf{B} String \simeq \tau_{\leq 2} \exp(\mathfrak{string}), where 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} is the string Lie 2-algebra.

Using these tools, the stages of the above smooth Whitehead tower are constructed as follows:

  • the morphism w 1:BOB 2\mathbf{w}_1 : \mathbf{B}O \to \mathbf{B} \mathbb{Z}_2 is directly induced from the canonical Lie group homomorphism O 2O \to \mathbb{Z}_2.

  • the morphism w 2:BSOB 2 2\mathbf{w}_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2 in H\mathbf{H} is presented by the zig-zag of crossed modules

    ( 2Spin) ( 21) (1SO) \array{ (\mathbb{Z}_2 \to Spin) &\stackrel{}{\to}& (\mathbb{Z}_2 \to 1) \\ {}^{\mathllap{\simeq}}\downarrow \\ (1 \to SO) }
  • the morphism 12p 1:BSpinB 3U(1)\tfrac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1) is constructed (FSSa) as the Lie integration of the canonical L-∞ 3-cocylce Bμ 3:B𝔰𝔬B 3\mathbf{B}\mu_3 : \mathbf{B}\mathfrak{so} \to \mathbf{B}^3 \mathbb{R}.

  • similarly 16p 2exp(μ 7)\tfrac{1}{6}\mathbf{p}_2 \simeq \exp(\mu_7) is the Lie integration of a canonical 7-cocycle Bμ 7:B𝔰𝔱𝔯𝔦𝔫𝔤B 7\mathbf{B}\mu_7 : \mathbf{B}\mathfrak{string} \to \mathbf{B}^7 \mathbb{R} (FSSa).

Differential refinement

While the above smooth refinement of the Whitehead tower already improves on the bare Whitehead tower by remembering the correct spaces of gauge transformations, it still only sees “instanton sectors” of gauge fields and higher gauge fields, namely the underlying principal ∞-bundles. We add now the refinement from smooth cohomology to differential cohomology such as to encode the actual higher gauge fields themselves. This differential cohomology in turn is naturally available in terms of curvature twisted flat cohomology or equivalently curvature-twisted local systems.

\,

In order to get a feeling for what differential refinements of higher moduli stacks are going to be like, recall two structures that we have already seen above:

  1. For GG a Lie group the smooth moduli stack of smooth GG-principal connections from above is presented by

    BG conn=(C (,G)×Ω 1(,𝔤)Ω 1(,𝔤))={Ag(g 1Ag+g 1dg)|AΩ 1(,𝔤),gC (,G)}. \mathbf{B}G_{conn} = (C^\infty(-,G) \times \Omega^1(-, \mathfrak{g}) \stackrel{\overset{}{\to}}{\underset{}{\to}} \Omega^1(-, \mathfrak{g})) = \left\{ A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g) | A \in \Omega^1(-,\mathfrak{g}), g \in C^\infty(-,G) \right\} \,.

    In the special case that G=U(1)G = U(1) is abelian, this is the image under the Dold-Kan correspondence of the length 1 complex of sheaves of abelian groups

    BU(1) conn=[C ()d dRΩ 1()]. \mathbf{B}U(1)_{conn} = [C^\infty(-) \stackrel{d_{dR}}{\to} \Omega^1(-)] \,.
  2. The smooth nn-stack B nU(1)\mathbf{B}^n U(1) is realized as the image under the Dold-Kan correspondence by the chain complex of sheaves C (,U(1))C^\infty(-,U(1))

    B nU(1)[C (,U(1))00]. \mathbf{B}^n U(1) \simeq [ C^\infty(-,U(1)) \to 0 \to \cdots \to 0 ] \,.

From the look of these expressions there is already a plausible candidate for the differential refinement B nU(1) conn\mathbf{B}^n U(1)_{conn}, the moduli nn-stack of circle n-bundles with connection – it should be the Deligne complex:

B nU(1) conn=[C (,U(1))d dRlogΩ 1()d dRΩ 2()d dRd dRΩ n()]. \mathbf{B}^{n} U(1)_{conn} = \left[ C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(- ) \right] \,.

For instance a cocycle XB 2U(1) connX \to \mathbf{B}^2 U(1)_{conn} in H\mathbf{H} is in Funct(SmthMfd op,sSet)Funct(SmthMfd^{op}, sSet) and relative to a good open cover given by a morphism C({U i})B 2U(1) connC(\{U_i\}) \to \mathbf{B}^2 U(1)_{conn}, which is

  • on each U iU_i a connection 2-form B iΩ 2(U i)B_i \in \Omega^2(U_i);

  • on each U iU jU_i \cap U_j a 1-form A ijΩ 1(U iU j)A_{i j} \in \Omega^1(U_i \cap U_j) such that B jB i=d dRA ijB_j - B_i = d_{dR} A_{i j} ;

  • on each U iU jU kU_i \cap U_j \cap U_k a smooth functor ϕ ijkC (U iU jU k,U(1))\phi_{i j k} \in C^\infty(U_i \cap U_j \cap U_k, U(1)) such that A ij+A jkA ik=d dRlogϕ ijkA_{i j} + A_{j k} - A_{i k} = d_{dR} log \phi_{i j k} and such that on each U iU jU kU lU_i \cap U_j \cap U_k \cap U_l the equation ϕ ijkϕ ikl=ϕ ijlϕ jkl\phi_{i j k} \phi_{i k l} = \phi_{i j l} \phi_{j k l} holds.

Tthe B-field on spacetime is (in the absence of various possible twists, to be discussed), such a cocycle XB 2U(1) connX \to \mathbf{B}^2 U(1)_{conn}; and the C-field (similarly in the absence of possible twists, to be discussed below) is given by a morphism XB 3U(1) connX \to \mathbf{B}^3 U(1)_{conn}.

Such cocycles in Deligne hypercohomology define classes in ordinary differential cohomology H diff n+1(X)H_{diff}^{n+1}(X):

H diff n+1(X)=π 0H(X,B nU(1) conn). H_{diff}^{n+1}(X) = \pi_0 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,.

There is an evident morphism

B nU(1) connB nU(1) \mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)

which forgets the connection data. We say that B nU(1) conn\mathbf{B}^n U(1)_{conn} is a differential refinement of B nU(1)\mathbf{B}^n U(1).

We now want to construct a differential refinement of the above smooth Whitehead tower, hence of the smooth universal characteristic classes appearing in it. To do so, we now first provide a more conceptual way to think of B nU(1) conn\mathbf{B}^n U(1)_{conn}, a way to obtain this more abstractly from fundamental principles.

The key is that the (∞,1)-topos H\mathbf{H} of smooth ∞-stacks comes with a canonical notion of local system or flat ∞-connection, and that we can twist this to find a notion of curvature-twisted and hence non-flat ∞-connection. The notion of local systems is induced from two basic derived adjoint functors that exist on H\mathbf{H}

HΓDiscGrpd, \mathbf{H} \stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \,,

where Γ\Gamma evaluates a presheaf on the point, and where DiscDisc sends an \infty-groupoid to the presheaf constant on that value. We form the composite

:HΓGrpdDiscH, \flat : \mathbf{H} \stackrel{\Gamma}{\to} \infty Grpd \stackrel{Disc}{\to} \mathbf{H} \,,

to be pronounced “flat”: for AHA \in \mathbf{H} a smooth homotopy type, we call A\flat A the corresponding flat local coefficient object.

For instance if GG is a Lie group, then ΓBGB(G disc)=K(G disc,1)\Gamma \mathbf{B}G \simeq B (G_{disc}) = K(G_{disc}, 1), and so a morphism XBGX \to \flat \mathbf{B}G is equivalently a cocycle XB(G disc)X \to \mathbf{B} (G_{disc}), hence a G discG_{disc}-covering space, hence a flat GG-principal connection.

Generally, we say that a morphism

XA X \to \flat A

is an AA-local system or AA-valued flat ∞-connection on XX.

There is a canonical forgetful morphism u:AAu : \flat A \to A which forgets the flat connection: this is the (DiscΓ)(\Disc \dashv \Gamma)-counit. Consider the coefficient object of those flat GG-connections whose underlying BG\mathbf{B}G-principal ∞-bundle is trivial

dRBG{BG|(u()*)}. \flat_{dR} \mathbf{B}G \coloneqq \left\{ \nabla \in \flat \mathbf{B}G | (u(\nabla) \simeq * ) \right\} \,.

From the example of ordinary principal connections it is familiar that flat GG-connections on trivial GG-principal bundles are equivalently flat Lie algebra valued differential forms. Below we will see that for general smooth ∞-groups GG, morphisms X dRBGX \to \flat_{dR} \mathbf{B}G are 𝔤\mathfrak{g}-∞-Lie algebra valued differential forms on XX.

Reading the above expression in homotopy type theory, its categorical semantics is the homotopy fiber of the counit

dRBG*× BGBG. \flat_{dR} \mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G \,.

By this construction and applying the pasting law, there is a canonical morphism θ:G dRBG\theta : G \to \flat_{dR} \mathbf{B}G, hence a canonical 𝔤\mathfrak{g}-valued form on any cohesive ∞-group GG: this identifies as the canonical Maurer-Cartan form on the ∞-group GG.

G * θ dRBG BG u * BG. \array{ G &\to& * \\ {}^{\mathllap{\theta}}\downarrow && \downarrow \\ \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{u}} \\ * &\to& \mathbf{B}G } \,.

For the special class of cases G=B nU(1)G = \mathbf{B}^n U(1) the circle (n+1)-group we call curv B nU(1)B nU(1) dRB n+1U(1)curv_{\mathbf{B}^n U(1)} \coloneqq \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1) the universal curvature class in degree (n+1)(n+1).

Due to the existence of the further functor Π:HGrpd\Pi : \mathbf{H} \to \infty Grpd discussed above it follows that :HGrpd\flat : \mathbf{H} \to \infty Grpd is a right adjoint and hence commutes with homotopy pullback. This in turn implies that by forming one more homotopy fiber above, we obtain the following differential version of a universal local coefficient bundle:

B nU(1) B nU(1) curv dRB n+1U(1). \array{ \flat \mathbf{B}^n U(1) &\to& \mathbf{B}^n U(1) \\ && \downarrow^{\mathrlap{curv}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1) } \,.

By the general concept of twisted cohomology, we see that this defines a notion of curvature twisted flat differential cohomology – hence of differential cohomology.

Specifically, for F X:XΩ cl n+1()F_X : X \to \Omega^{n+1}_{cl}(-) a closed differential form on XX, a cocycle in F XF_X-twisted curv\mathbf{curv}-cohomology is equivalently a circle n-bundle with connection with that curvature

X B nU(1) F curv dRB n+1U(1) connH B n+1U(1)(F X,curv). \array{ X &&\stackrel{}{\to}&& \mathbf{B}^n U(1) \\ & {}_{\mathllap{F}}\searrow &\swArrow_{\nabla}& \swarrow_{\mathrlap{\mathbf{curv}}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1)_{conn} } \;\; \in \mathbf{H}_{\flat \mathbf{B}^{n+1}U(1)}(F_X, \mathbf{curv}) \,.

For varying FF, the curv\mathbf{curv}-twisted cohomology in H\mathbf{H} identifies with ordinary differential cohomology: the homotopy pullback B nU(1) conn\mathbf{B}^n U(1)_{conn} in

B nU(1) conn Ω cl n+1() B nU(1) curv B n+1U(1) conn \array{ \mathbf{B}^n U(1)_{conn} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{curv}{\to}& \mathbf{B}^{n+1} U(1)_{conn} }

is presented, under the Dold-Kan correspondence, by the Deligne complex, discussed above. This exhibits ordinary differential cohomology as the curvature-twisted flat cohomology

H diff(X,B nU(1))=curvStruc Ω n+1(X). \mathbf{H}_{diff}(X, \mathbf{B}^n U(1)) = \mathbf{curv}Struc_{\Omega^{n+1}}(X) \,.

Using this geometric-homotopy-type theoretic description of ordinary differential cohomology, we obtain now a natural notion of differential refinement of smooth universal characteristic classes c:BGB n+1U(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^{n+1} U(1). We say that a differential refinement of c\mathbf{c} is a morphism c^\hat \mathbf{c} fitting into a diagram

BG c B n+1U(1) BG conn c^ B n+1U(1) conn curv BG c B n+1U(1) \array{ \flat \mathbf{B}G &\stackrel{\flat \mathbf{c}}{\to}& \flat \mathbf{B}^{n+1} U(1) \\ \downarrow && \downarrow \\ \mathbf{B}G_{conn} &\stackrel{\hat \mathbf{c}}{\to}& \mathbf{B}^{n+1} U(1)_{conn} \\ \downarrow && \downarrow^{\mathrlap{curv}} \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^{n+1} U(1) }

that factors the naturality square of \flat on c\mathbf{c}.

Differential Whitehead tower

Theorem (SSSa, FSSa) There exists a smooth differential refinement of the Whitehead tower of BO as follows:

BFivebrane conn BString conn 16p^ 2 B 7U(1) conn BSpin conn 12p^ 1 B 3U(1) conn BSO conn w 2 B 2 2 BO conn w 1 B 2 BGL conn. \array{ \vdots \\ \downarrow \\ \mathbf{B} Fivebrane_{conn} \\ \downarrow \\ \mathbf{B} String_{conn} &\stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow \\ \mathbf{B} Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow \\ \mathbf{B} S O_{conn} &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow \\ \mathbf{B} O_{conn} &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2 \\ \downarrow^{} \\ \mathbf{B} GL_{conn} } \,.

This construction is a joint generalization of Chern-Weil theory and Chern-Simons theory to ∞-Chern-Weil theory and ∞-Chern-Simons theory

For instance

  • the differential refinement of the first fractional Pontryagin class above yields the action functional

    exp(iS 12p 1):[Σ,BSpin conn]12p^ 1[Σ,B 3U(1) conn]exp(i Σ())U(1) \exp(i S_{\tfrac{1}{2}\mathbf{p}_1}) : [\Sigma, \mathbf{B}Spin_{conn}] \stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to} [\Sigma, \mathbf{B}^{3}U(1)_{conn}] \stackrel{\exp(i \int_\Sigma(-))}{\to} U(1)

    of Spin-Chern-Simons theory, refined to the integrated off-shell BRST-complex of the theory;

  • the differential refinement of the second fractional Pontryagin class above yields the action functional

    exp(iS 16p 2):[Σ,BString conn]16p^ 2[Σ,B 7U(1) conn]exp(i Σ())U(1) \exp(i S_{\tfrac{1}{6}\mathbf{p}_2}) : [\Sigma, \mathbf{B}String_{conn}] \stackrel{\tfrac{1}{6}\hat \mathbf{p}_2}{\to} [\Sigma, \mathbf{B}^{7}U(1)_{conn}] \stackrel{\exp(i \int_\Sigma(-))}{\to} U(1)

    of 7-dimensional Chern-Simons theory on nonabelian String 2-form fields (FSSb)

We indicate briefly how this is constructed.

(…)

Interlude) Anomaly line bundle on smooth moduli stacks of fields

Before coming to the description in smooth moduli ∞-stacks below, we make some introductory comments on the general origin of twisted differential structures in higher gauge theory, following (Freed). We add some stacky aspects to that and explain why.

In summary, we discuss how the action functional of higher gauge theory in the presence of electric and magnetic charge is a section of a circle bundle with connection highergaugeanomaly\nabla_{higher\;gauge\;anomaly} on the smooth ∞-stack [X,Fields][X, \mathbf{Fields}] of field configurations on a given spacetime XX, exhibited by a morphism

highergaugeanomalyexp(2πi Xc^ elc^ mag):[X,Fields]BU(1) conn, \nabla_{higher\;gauge\;anomaly} \coloneqq \exp(2 \pi i \int_X \hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} ) : [X, \mathbf{Fields}] \stackrel{}{\to} \mathbf{B} U(1)_{conn} \,,

where c^ elc^ mag:FieldsB dimX+2U(1) conn\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} : \mathbf{Fields} \to \mathbf{B}^{dim X +2} U(1)_{conn} is the differential characteristic morphism induced by the differential cup product (FSSd) of universal electric and magnetic currents, and where exp(2πi X())\exp(2\pi i \int_X(-)) is fiber integration in ordinary differential cohomology refined to smooth \infty-stacks (this is the “(dimX)+1(dim X)+1-dimensional infinity-Chern-Simons theory” of c^ elc^ mag\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} in codimension 1).

Higher gauge fields in the presence of magnetic charge current

Gauge theory starts maybe with Maxwell around 1850, who discovered, in modern language, that the field strength of the electromagnetic field on spacetime is encoded in a closed differential 2-form FΩ cl 2(X)F \in \Omega^2_{cl}(X). Then in the 1930s Dirac’s famous argument showed that more precisely – in the absence of, or outside of the support of magnetic charge current – this 2-form is the curvature of a circle group-principal bundle with connection, a 2-cocycle F^\hat F in ordinary differential cohomology.

In view of this the gauge equivalence classes of configurations of the electromagnetic field on XX form the set

H diff 2(X)τ 0H(X,BU(1) conn) H^2_{diff}(X) \coloneqq \tau_0 \mathbf{H}(X, \mathbf{B}U(1)_{conn})

of differential cohomology classes in degree 2 on XX.

Dirac’s argument works outside the support of the magnetic current, where the situation is comparatively easier to handle. But the twists and anomalies that we are concerned with here arise when one completes Dirac’s argument, and generalizes the model of the electromagnetic field to exist also over parts of spacetime where the magnetic current is non-trivial (Freed). Among other things the following shows that twists by higher bundles and differential cohomology is not just something that arises in string theory, but is already present in dear-old electromagnetism.

To see what happens in that general case, notice that the original Maxwell equations on the field strength/curvature 2-form of the electromagnetic field are:

  1. d dRF=J magd_{dR} F = J_{mag} (magnetic charge current)

  2. d dRF=J eld_{dR} \star F = J_{el} (electric charge current)

where \star is the Hodge star operator for the given pseudo Riemannian metric (the field of gravity) on XX.

  1. The first one is kinematics. For J mag=0J_{mag} = 0 it just expresses that the cuvature 2-form is closed, which is part of the fact that F^\hat F is a differential cocycle, so it is satisfied by all kinematic field configurations, meaning: all elements of H diff n+1(X)H^{n+1}_{diff}(X).

  2. The second is dynamics, being the equations of motion of the system. The configurations that satisfy this form the covariant phase space (BV-BRST complex) P[X,BU(1) conn]P \hookrightarrow [X, \mathbf{B}U(1)_{conn}] of the theory. For our purposes here this will not concern us, since the anomalies and twists are kinematic in nature, we work “off-shell”.

While for experimentally observed electromagnetism it is consistent to assume that J mag=0J_{mag} = 0, this is not the case for general gauge theories, notably not for heterotic supergravity, as we discuss in a moment. There the gauge field and the field of gravity induce a non-vanishing “fivebrane magnetic current”

  • J mag NS5(ϕ gr,ϕ ga)=F ωF ωF AF AJ^{NS5}_{mag}(\phi_{gr}, \phi_{ga}) = \langle F_\omega \wedge F_\omega\rangle -\langle F_A \wedge F_A\rangle

But for a circle n-bundle with connection in H diff n+1(X)H^{n+1}_{diff}(X), the curvature is necessarily closed, d dRF=0d_{dR} F = 0. So there must be another way to refine dF=J magd F = J_{mag} to differential cohomology.

(Notice for later that the natural home of J magJ_{mag} is not plain de Rham cohomology, but compactly supported cohomology. The equation d dRF=J magd_{dR} F = J_{mag} is a trivialization of the image of J magJ_{mag} in de Rham cohomology, but not in general a trivialization of the magnetic current as an entity living in compactly supported cohomology.)

Consider therefore now the groupoid

diff n+1(X)τ 1H(X,B nU(1) conn), \mathcal{H}^{n+1}_{diff}(X) \coloneqq \tau_1 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,,

whose

This “categorifies” the cohomology set H diff n+1(X)H^{n+1}_{diff}(X) in that the letter is its decategorification: the set of isomorphism classes of objects.

For instance if differential cohomology is modeled by the Deligne complex with differential D=d dR±δD = d_{dR} \pm \delta, then a morphism α^:F^ 1F^ 2\hat \alpha : \hat F_1 \to \hat F_2 in diff n+1(X)\mathcal{H}_{diff}^{n+1}(X) is a Deligne coboundary Dα^=F^ 2F^ 1D \hat \alpha = \hat F_2 - \hat F_1.

Or in terms of our smooth moduli stacks, this is a homotopy

F^ 1 X α^ B nU(1) conn F^ 2. \array{ & \nearrow \searrow^{\mathrlap{\hat F_1}} \\ X &\Downarrow^{ \hat \alpha }& \mathbf{B}^n U(1)_{conn} \\ & \searrow \nearrow_{ \mathrlap{\hat F_2} } } \,.

Notice that, since morphisms in diff n+1(X)\mathcal{H}^{n+1}_{diff}(X) preserve the higher connection, a morphism

0F^ 0 \to \hat F

in diff n+1(X)\mathcal{H}^{n+1}_{diff}(X) is a flat section of the corresponding circle nn-bundle, while a morphim

BF^ B \to \hat F

for some BΩ n(X) diff n+1(X)B \in \Omega^n(X) \hookrightarrow \mathcal{H}^{n+1}_{diff}(X) is a possibly non-flat section, hence a section just of the underlying circle n-group-principal ∞-bundle: it exhibits the fact that if the underlying bundle has a section, then the connection is equivalently given by a globally defined nn-form BB.

(Beware of this subtlety when comparing with (Freed): a differential as on the fifth line of p. 8 there may change the curvature by an exact term, hence may not preserve the connection, in contrast to the coboundaries further below on that page and on p. 9, which are the ones we are considering here.)

(Another reason for considering the groupoid diff n+1(X)\mathcal{H}^{n+1}_{diff}(X) is that it is needed in order to construct the quadratic refinement of the secondary intersection pairing that defines the partition function of self-dual higher gauge theory (Hopkins-Singer). This underlies the discussion of flux quantization below.)

Using this, we may improve the definition of the electromagnetic field on XX: take it to be a non-flat section

c^F^c mag. \hat \mathbf{c} \stackrel{\hat F}{\to} c_{mag} \,.

of a magnetic charge circle 2-bundle with connection c^ diff 3(X)\hat \mathbf{c} \in \mathcal{H}^{3}_{diff}(X). Equivalently, in terms of the corresponding classifying morphisms in H\mathbf{H} this is a homotopy in a diagram of the form

X c mag F^ c^ Ω 2() B 2U(1) conn. \array{ X &\to& \\ {}^{\mathllap{c_{mag}}}\downarrow &\swArrow_{\hat F}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\hookrightarrow}& \mathbf{B}^2 U(1)_{conn} } \,.

If F^\hat F is given by a Deligne cochain (g ij,A i)(g_{i j}, A_i), c^\hat \mathbf{c} by a cochain (c ijk,γ ij,β i)(c_{i j k}, \gamma_{i j}, \beta_i) then this means that

D(g ij,A i) ((δg) ijk,A jA i+d dRlogg ij,d dRA i) =(c ijk 1,γ ij,c magβ i) \begin{aligned} D (g_{i j}, A_i) & \coloneqq ((\delta g)_{i j k},\; A_j - A_i + d_{dR} log g_{i j},\; d_{dR} A_i) \\ & = ( c_{i j k}^{-1},\; -\gamma_{i j}, \; c_{mag} - \beta_i ) \end{aligned}

We say that F^\hat F is a c^\hat \mathbf{c}-twisted bundle with twisted curvature being

FdA i+β i. F \coloneqq d A_i + \beta_i \,.

This now correspondingly has a twisted Bianchi identity, which is precisely so that it solves the first Maxwell equation in the presence of magnetic current: d dRF=J magd_{dR} F = J_{mag}.

While we have been discussing this here for ordinary electromagnetism, this is precisely the mechanism by which also the higher cases will work: for the heterotic Green-Schwarz mechanism the analogy is

twistedcurvature d dR(gaugepotential) twist F = d dRA i + β i generalcase H i = d dRB i + CS(ω i)CS(A i) heteroticsugra. \array{ twisted\;curvature & & d_{dR}(gauge\;potential) && twist \\ F &=& d_{dR} A_i &+& \beta_i && general case \\ \\ H_i &=& d_{dR} B_i &+& CS(\omega_i) - CS(A_i) && heterotic\;sugra } \,.

Before we get there, we need to observe that working with the 1-groupoid diff n+1(X)\mathcal{H}^{n+1}_{diff}(X) is not sufficient. We discuss now that we necessarily need the full n-groupoid and moreover its smooth refinement to the full smooth n-stack [X,B nU(1) conn][X, \mathbf{B}^n U(1)_{conn}] in order to capture the physics situation.

Gauge transformations

To see that we need the full higher groupoid, just consider the question: what is a gauge transformation between twisted electromagnetic fields, that are now identified with morphisms c^F^c mag\hat \mathbf{c} \stackrel{\hat F}{\to} c_{mag} as above? Clearly, for this we need the 2-groupoid of differential cocycles τ 2H(X,B nU(1) conn)\tau_2 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})

to next say that the equivalenc class of a gauge transformation of twisted fields a^:F^ 1F^ 2\hat a : \hat F_1 \to \hat F_2 is a 2-morphism

F^ 1 c^ α^ J mag F^ 2τ 2H(X,B nU(1) conn). \array{ & \nearrow \searrow^{\mathrlap{\hat F_1}} \\ \hat \mathbf{c} &\Downarrow^{\hat \alpha}& J_{mag} \\ & \searrow \nearrow_{\mathrlap{\hat F_2}} } \;\;\; \in \tau_{2} \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \,.

That we moreover need the full smooth n-groupoid [X,B nU(1) conn][X, \mathbf{B}^n U(1)_{conn}] has several reasons, we discuss three. The third one of these is related to the higher gauge anomalies proper.

Magnetic current induced by background fields

The magnetic twist c^\hat \mathbf{c} will depend on other field configurations that induce magnetic charge. So it is not a constant, but varies with the fields.

(In fact, only this way is it a non-trivial stricture, for if both c^\hat c and c magc_{mag} are independent of the fields, the above definition of the groupoid of F^\hat Fs is equivalent to that of untwisted electromagnetic fields: because homotopy fibers only depend on the connected component of the base points, up to equivalence.)

Let therefore G bgG^{bg} be the gauge group of another background gauge field, and let BG conn bg\mathbf{B}G^{bg}_{conn} be its moduli stack of gauge field configurations. If a G bgG^{bg}-field induces magnetic current, then c^\hat \mathbf{c} must depend on the fields, hence it should be a map

[X,BG conn bg][X,B 2U(1) conn] [X, \mathbf{B}G^{bg}_{conn}] \to [X, \mathbf{B}^2 U(1)_{conn}]

between these spaces of fields, and it should be a smooth such map. Moreover, in general this is not expected to depend specifically the specific choice of XX, but just on the notion of G bgG^{bg}-fields in general, so it should be given just by postcompositon

c^:BG conn bgB 2U(1) conn \hat \mathbf{c} : \mathbf{B}G^{bg}_{conn} \to \mathbf{B}^2 U(1)_{conn}

with a universal smooth map on moduli. With that given, the above picture for the c^\hat \mathbf{c}-twisted higher electric field becomes

X ϕ bg BG conn bg F^ c^ Ω 2() B nU(1) conn. \array{ X &\stackrel{\phi_{bg}}{\to}& \mathbf{B}G^{bg}_{conn} \\ \downarrow &\swArrow_{\hat F}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\to}& \mathbf{B}^n U(1)_{conn} } \,.

We can then subsume all this and consider the smooth collection of all such twisting background fields ϕ bg\phi^{bg} and twisted gauge fields F^\hat F. By general reasoning, this is given by the homotopy pullback that universally completes the above diagram

Fields ϕ bg BG conn bg univ F^ univ c^ Ω 2() B nU(1) conn \array{ \mathbf{Fields} &\stackrel{\phi_{bg}}{\to}& \mathbf{B}G^{bg}_{conn}{}^{univ} \\ \downarrow &\swArrow_{\hat F^{univ}}& \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ \Omega^2(-) &\stackrel{}{\to}& \mathbf{B}^n U(1)_{conn} }

Infinitesimal moduli stacks: BRST complexes

So far we are talking about gauge fields and higher gauge fields on which we are evaluating an action functional (see below). Eventually one wants to quantize such a setup. There are two issues with this: first of all the action functional needs to be well-defined in the first place, we get to in the next point. But second, once we have a well-defined action functional on gauge fields, the only way to quantize this is to invove BV-BRST formalism: we need the BRST complex of the gauge fields.

or ordinary gauge theory this is the Lie algebroid of the smooth version [X,BG conn][X, \mathbf{B}G_{conn}]

BRST=C Lie([X,BG conn]). BRST = C^\infty Lie([X, \mathbf{B}G_{conn}]) \,.

Similarly for higher gauge theory it is the L-infinity algebroid.

Extended higher Chern-Simons-type functionals

The anomaly line bundle to be discussed in a moment below is a special case of a general construction in extended “∞-Chern-Simons theory”. So before getting to that special case, we indicate here the general pattern.

The action functional of ordinary Chern-Simons theory is traditionally taken to be simply a function, for a gven compact 3-manifold Σ 3\Sigma_3,

exp(iS cs):GBund (Σ 3)U(1) \exp( i S_{cs}) : G Bund_\nabla(\Sigma_3) \to U(1)

on GG-principal connections over Σ 3\Sigma_3. This perspective can be refined.

First of all, since this function is gauge invariant we may think of it as being defined on the full moduli stack

exp(iS cs):[Σ 3,BG conn]U(1). \exp( i S_{cs}) : [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,.

This also exhibits the smoothness of the action.

An important construction in Chern-Simons theory is the geometric quantization of this action functional in the case that Σ 3=Σ 2×Interval\Sigma_3 = \Sigma_2 \times Interval , which yields a holomorphic line bundle with connection on the covariant phase space of the theory, which may be identified with the space of flat connections over a 2-dimensional Σ 2\Sigma_2. Generally, one expects to see a circle k-bundle with connection assigned to a Σ\Sigma of codimension kk.

Above we had already seen such a structure in top codimension: if Σ=*\Sigma = * is the point, then [Σ,BG conn][*,BG conn]BG conn[\Sigma, \mathbf{B}G_{conn}] \simeq [*, \mathbf{B}G_{conn}] \simeq \mathbf{B}G_{conn}. So there should be a cricle 3-bundle with connection on this moduli stack. Taking G=SpinG = Spin, for definiteness, then the differential first fractional Pontryagin class from above is precisely of this form:

12p^ 1:BSpinn connB 3U(1). \tfrac{1}{2} \hat \mathbf{p}_1 : \mathbf{B} Spinn_{conn} \to \mathbf{B}^3 U(1) \,.

And indeed, as stated there, this induces the Chern-Simons action functional itself. Indeed, it induces a whole tower of higher circle bundles, in each codimension:

The operation of fiber integration of differential forms extends to an operation of fiber integration in ordinary differential cohomology, which in turn, as discussed there, extends to a morphism of smooth moduli stacks of the form

exp(2πi Σ k()):[Σ k,B nU(1) conn]B nkU(1) conn. \exp(2 \pi i \int_{\Sigma_k}(-)) : [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k} U(1)_{conn} \,.

If now

c^:BG connB nU(1) conn \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}

is any universal differential characteristic map, and Σ k\Sigma_k is compact closed of dimension kk, then the composite

exp(2πi Σ kc^):[Σ k,BG conn][Σ k,c^][Σ k,B nU(1) conn]exp(2πi Σ k())B nkU(1) conn \exp(2 \pi i \int_{\Sigma_k} \hat \mathbf{c}) : [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_k} (-))}{\to} \mathbf{B}^{n-k} U(1)_{conn}

is a “kk-extended action functional”.

An important class of ∞-Chern-Simons theories arising this way come from c^\hat \mathbf{c} that are cup products of two other differential classes (FSSd). For instance in ordinary abelian higher dimensional Chern-Simons theory one starts with the tautological differential class

DD^:B 2k+1U(1) connB 2k+1U(1) conn \hat \mathbf{DD} : \mathbf{B}^{2k+1} U(1)_{conn} \to \mathbf{B}^{2k+1} U(1)_{conn}

and then forms its differential cup product (FSSd)

c^B 2k+1U(1) connB 2k+1U(1) conn×B 2k+1U(1) connDD^DD^B 4k+3U(1) conn. \hat \mathbf{c} \coloneqq \mathbf{B}^{2k+1} U(1)_{conn} \stackrel{}{\to} \mathbf{B}^{2k+1} U(1)_{conn} \times \mathbf{B}^{2k+1} U(1)_{conn} \stackrel{\hat \mathbf{DD} \cup \hat \mathbf{DD}}{\to} \mathbf{B}^{4k + 3} U(1)_{conn} \,.

The action functional induced by this is that of (4k+3)(4k+3)-dimensional higher dimensional Chern-Simons theory which sends those (2k+1)(2k+1)-form fields CC whose underlying bundle happens to be trivial to exp(2πi Σ 4k+3Cd dRC)\exp(2 \pi i\int_{\Sigma_{4k+3}} C \wedge d_{dR} C).

The anomaly line bundle which we now turn to arises in this kind of way, only for the slightly more general case that the ∞-Chern-Simons theory involved is not given by a differential square, but by a genuine differential cup of two different cocycles: the electric and the magnetic differential cocycles.

Gauge interaction and the charge anomaly

We discuss now how the action functional of the (higher) gauge theory in the presence of electric charge current and magnetic charge current has in general an anomaly, but this anomaly exhibits itself, in traditional language, as something living over families of gauge fields. But by the formula for the internal hom of sheaves/stacks

[X,Fields]:UH(X×U,BG conn bg)={backgroundgaugefieldsonX×U}={UparameterizedfamlilyofgaugefieldsonX} [X, \mathbf{Fields}] : U \mapsto \mathbf{H}(X \times U, \mathbf{B}G^{bg}_{conn}) = \{ background\;gauge\;fields\; on\; X \times U \} = \{ U-parameterized\;famlily\;of\;gauge\;fields\; on\; X \}

this means effectively to work over the smooth moduli stack of fields itself. Notably, the anomaly is going to be (the non-triviality of) a circle bundle with connection on “the space of all fields”, so we certainly need a smooth structure on that space. We indicate now how that line bundle

anomaly:[X,Fields]BU(1) conn \nabla_{anomaly} : [X, \mathbf{Fields}] \to \mathbf{B}U(1)_{conn}

appears.

First, the kinetic piece of the action functional is simply exp(iS kin(F^))=exp(i XF*F)\exp(i S_{kin}(\hat F)) = \exp(i \int_X F \wedge \ast F).

But suppose there is a charged particle with trajectory γ:S 1X\gamma : S^1 \to X. Then there is an interaction term exp(2πi S 1γ *A)\exp(2\pi i \int_{S^1} \gamma^* A). Let J elJ_{el} be the Poincare dual form. Then =exp(i XAJ el)\cdots = \exp(i \int_X A \wedge J_{el}).

This may be expressed using the Beilinson-Deligne cup product and the fiber integration in ordinary differential cohomology as exp(2πi Xc^ elF^)\exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat F). (Here is where we need J elJ_{el} to have compact support.) This is a differential 2-cocycle on the moduli stack of field configurations: by the formula for the internal hom and for forms on a stack, we are evaluating for each test manifold UU on families of fields over X×UX \times U and then integrate out over XX.

But in the case where there is non-trivial c^ mag\hat \mathbf{c}_{mag} this is no longer the case, there instead this is a trivialization of a twist

ωexp(2πi Xc^ elF^)exp(2πi Xc^ elc^ mag). \omega \stackrel{\exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat F)}{\to} \exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}) \,.

Moreover, this twist matters in compactly supported cohomology (this is what fiber integration in ordinary differential cohomology sees), where it is in general not trivialized. So the action functional is not a function, but a section of a line bundle. Its first Chern class is the 2-class of

highergaugeanomaly:[X,Fields]exp(2πi Xc^ elc^ mag)BU(1) conn \nabla_{higher\;gauge\;anomaly} : [X, \mathbf{Fields}] \stackrel{ \exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}) } { \to } \mathbf{B} U(1)_{conn}

This is the anomaly line bundle with connection on the moduli stack of fields.

For this to cancel, there needs to be a fermionic anomaly – the Pfaffian line bundle of the Dirac operators on the fermions in the theory – of the same structure.

fermionanomaly:[X,Fields]BU(1) conn. \nabla_{fermion\;anomaly} : [X, \mathbf{Fields}] \to \mathbf{B} U(1)_{conn} \,.

The total action functional (higher gauge fields and fermions) is a section of the tensor product of these two

totalanomaly= highergaugeanomaly fermionanomaly:[X,Fields]BU(1) conn. \nabla_{total\;anomaly} = \nabla_{higher\;gauge\;anomaly} \otimes \nabla_{fermion\;anomaly} : [X, \mathbf{Fields}] \to \mathbf{B} U(1)_{conn} \,.

The action functional needs to be a flat section of totalanomaly\nabla_{total\;anomaly}. Hence the two line bundles need to be inverse to each other. This condition is the Green-Schwarz mechanism.

III) Spin cSpin^c-, String cString^c-, Fivebrane cFivebrane^c-structure

In the previous section we have considered higher differential structures originating in the orthogonal group. In applications to string theory these structures receive twists originating in the unitary group (or representations through the unitary group of groups like E8). (The orthogonal structures correspond to the field of gravity, while the unitary structures correspond to gauge fields.)

Accordingly, the above Whitehead tower of BO\mathbf{B}O has stage-wise unitary twistings. In the first stage this is given by the familiar spin^c-group, then there is a String^c 2-group, etc.

After discussing some generalities of these higher unitary-twisted connected covers of the orthogonal group below we then turn to discussing a list of twisted structures and their appearance in string theory:

unitary-twisted higher orthogonal structurerole in string theory
twisted differential spin^c structureFreed-Witten anomaly cancellation for type II strings on D-branes
twisted differential String^c-structureflux quantization in 11d sugra/M-theory with M5-branes
twisted differential string structureGreen-Schwarz mechanism in heterotic string theory
twisted differential fivebrane structureGreen-Schwarz mechanism in dual heterotic string theory

Higher unitary-twisted covers of OO

The Lie group spin^c is traditionally defined by the formula

Spin cSpin× 2U(1)=(Spin×U(1))/ 2, Spin^c \coloneqq Spin \times_{\mathbb{Z}_2} U(1) = (Spin \times U(1))/{\mathbb{Z}_2} \,,

which denotes the quotient of the product Spin×U(1)Spin \times U(1) by the diagonal action induced by the common canonical subgroup of order 2.

For our purposes it is useful to think of this as follows. We have the B 2\mathbf{B}\mathbb{Z}_2-higher fiber bundle classified by the smooth second Stiefel-Whitney class

B 2 BSpin BSO w 2 B 2 \array{ \mathbf{B} \mathbb{Z}_2 &\to& \mathbf{B} Spin \\ && \downarrow \\ && \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z} }

and we also have the higher fiber bundle

BU(1) 2 BU(1) c 1mod2 B 2 \array{ \mathbf{B} U(1) &\stackrel{\cdot 2}{\to}& \mathbf{B} U(1) \\ && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ && \mathbf{B}\mathbb{Z}_2 }

which is 2\mathbb{Z}_2-associated to the universal principal bundle universal 2\mathbb{Z}_2-bundle.

One finds, using the presentation of these maps as discussed above, that BSpin c\mathbf{B}Spin^c is the corresponding associated bundle, namely the homotopy pullback

BSpin c BU(1) c 1mod2 BSO w 2 B 2 2. \array{ \mathbf{B}Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.

Equivalently we may write this as a Mayer-Vietoris sequence and thus obtain the universal local BSpin c\mathbf{B}Spin^c-coefficient bundle over B 2\mathbf{B}^2 \mathbb{Z}

BSpin c B(SO×U(1)) w 2c 1mod2 B 2 2 \array{ \mathbf{B}Spin^c &\to& \mathbf{B} (SO \times U(1)) \\ && \downarrow^{\mathrlap{\mathbf{w}_2 - \mathbf{c}_1 mod 2}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }

We may read this as saying:

Where the moduli stack BSpin\mathbf{B}Spin is the homotopy fiber of the smooth second Stiefel-Whitney class w 2\mathbf{w}_2, the moduli stack BSpin c\mathbf{B}Spin^c is that homotopy fiber universally twisted by the smooth first Chern class c 1mod2\mathbf{c}_1 mod 2._

In the following we consider higher analogs of this, where homotopy fibers of “orthogonal classes” are twisted by “unitary classes”.

In particular, one step higher in the Whitehead tower of BO, we can twist the smooth first fractional Pontryagin class with the smooth second Chern class to obtain the delooping of the smooth String^c 2-group

BString c 2 B(Spin×SU) 12p 1c 2 B 3U(1). \array{ \mathbf{B} String^{\mathbf{c}_2} &\to& \mathbf{B}(Spin \times SU) \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - \mathbf{c}_2 }} \\ && \mathbf{B}^3 U(1) } \,.

This controls the supergravity C-field in M-theory/11-dimensional supergravity as well as the Green-Schwarz mechanism in heterotic string theory, discussed below.

Before we come to that we consider another variant, since that leads to the most familiar twisting, that of twisted K-theory.

One finds that there is also a universal local BSpin c\mathbf{B}Spin^c-coefficient bundle over B 2U(1)\mathbf{B}^2 U(1), and this is given by the smooth third integral Stiefel-Whitney class:

Spin c BSO W 3 B 2U(1). \array{ Spin^c &\to& \mathbf{B} SO \\ && \downarrow^{\mathrlap{\mathbf{W}_3}} \\ && \mathbf{B}^2 U(1) } \,.

Since that now lands in B 2U(1)\mathbf{B}^2 U(1), we can apply one more unitary twist by a corresponding class. The canonical such class is the universal Dixmier-Douady class dd\mathbf{dd} of (stable) projective unitary bundles

B(Spin c) dd B(SO×PU) W 3dd B 2U(1). \array{ \mathbf{B}(Spin^c)^{\mathbf{dd}} &\to& \mathbf{B} ( SO \times PU ) \\ && \downarrow^{\mathrlap{\mathbf{W}_3 - \mathbf{dd}}} \\ && \mathbf{B}^2 U(1) } \,.

This universal local coefficient bundle controls the Freed-Witten anomaly cancellation in type II string theory. To which we now turn.

a) Twisted differential Spin cSpin^c-structure: Freed-Witten mechanism

We discuss aspects of the twisted smooth cohomology involved over D-branes in type II string theory: the Freed-Witten anomaly cancellation mechanism in terms of twisted K-theory.

For each nn, the central extension of Lie groups

U(1)U(n)PU(n) U(1) \to U(n) \to PU(n)

that exhibits the unitary group as a circle group-extension of the projective unitary group induces the corresponding morphism of smooth moduli stacks

BU(1)BU(n)BPU(n). \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \,.

This is part of a long fiber sequence which continues to the right by a connecting homomorphism dd n\mathbf{dd}_n

BU(1)BU(n)BPU(n)dd nB 2U(1) \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1)

in H\mathbf{H}. Here the last morphism is presented in simplicial presheaves by the zig-zag of sheaves of crossed modules

[U(1)U(n)] [U(1)1] PU(n). \array{ [U(1) \to U(n)] &\to& [U(1) \to 1] \\ {}^{\mathllap{\simeq}}\downarrow \\ PU(n) } \,.

We have seen above that a morphism ϕ:XB 2U(1)\phi : X \to \mathbf{B}^2 U(1) classifies a circle 2-bundle encoded by a Cech 2-cocycle (ϕ ijk:U iU jU kU(1))(\phi_{i j k} : U_i \cap U_j \cap U_k \to U(1)) . This means that the universal local coefficient bundle

BU(n) BPU(n) dd n B 2U(1) \array{ \mathbf{B} U(n) &\to& \mathbf{B} PU(n) \\ && \downarrow^{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) }

induces a notion of unitary bundles that are twisted by a 2-bundle.

Indeed, unwinding the definition one finds that a ϕ\phi-twisted dd n\mathbf{dd}_n cocycle

X h BPU(n) ϕ dd n B 2U(1) \array{ X &&\stackrel{h}{\to}&& \mathbf{B} PU(n) \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) }

is, with respect to a resolution of XX a good open cover {U iX}\{U_i \to X\}, given by maps C({U i})B(U(1)U(n))C(\{U_i\}) \to \mathbf{B}(U(1) \to U(n)) whose components read

( (x,j) (x,i) (x,k))( * h ij(x) ϕ ijk(x) h jk * h ik *)B(U(1)U(n)). \left( \array{ && (x,j) \\ & \nearrow &\Downarrow& \searrow \\ (x,i) &&\to&& (x,k) } \right) \;\;\; \mapsto \;\;\; \left( \array{ && * \\ & {}^{\mathllap{h_{i j}(x)}}\nearrow &\Downarrow^{\phi_{i j k}(x)}& \searrow^{\mathrlap{h_{j k}}} \\ * &&\underset{h_{i k}}{\to}&& * } \right) \in \mathbf{B}(U(1) \to U(n)) \,.

Hence these are collections of smooth U(n)U(n)-valued functions

(h ij:U iU jU(n)) (h_{i j} : U_i \cap U_j \to U(n))

which satisfy on triple overlaps the equation

h ijh jk=h ikϕ ijk. h_{i j} h_{j k} = h_{i k} \phi_{i j k} \,.

If ϕ ijk\phi_{i j k} happens to be constant on the neutral element, then this is the condition for a cocycle in H smooth 1(X,U(n))H^1_{smooth}(X, U(n)). So in general we say it is a ϕ\phi-twisted such cocycle. And that (h ij)(h_{i j}) classifies a ϕ\phi-twisted unitary bundle.

In generalization of how unitary bundles constitute cocycles for K-theory, these ϕ\phi-twisted unitary bundles constitute cocycles for twisted K-theory.

For each nn \in \mathbb{N} there is a canonical inclusion BU(n)BU(n+1)\mathbf{B} U(n) \to \mathbf{B} U(n+1), exhibiting the fact that a rank-nn complex vector bundle canonically induces a rank-(n+1)(n+1)-bundle by added a trivial line bundle.

To get rid of the dependence on the rank nn – to stabilize the rank – we may form the directed colimit of smooth moduli stacks

BUlim nBU(n) \mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)

Proposition The smooth stack BU\mathbf{B} U is a smooth refinement of the classifying space BUB U of reduced K-theory. Also, for XX a compact smooth manifold smooth UU-principal bundles and smooth UU-gauge transformations on XX are represented by ordinary U(n)U(n)-bundles for some finite nn.

H(X,BU)lim nH(X,BU(n)). \mathbf{H}(X, \mathbf{B}U) \simeq \underset{\rightarrow_n}{\lim} \mathbf{H}(X, \mathbf{B} U(n)) \,.

Now we think of the manifold XX as a target space for the type II superstring, hence assume it to be orientable and spin: w 1(X)=0w_1(X) = 0 and w 2(X)=0w_2(X) = 0. Consider moreover a submanifold

ι:QX, \iota : Q \hookrightarrow X \,,

to be thought of as the worldvolume of a D-brane, which is also or orientable and spin.

Assume first that QQ also admits a spin^c structure, hence that also the third integral Stiefel-Whitney class vanishes W 3(Q)=0W_3(Q) = 0.

We can consider then a cocycle in the ι\iota-relative cohomology with coefficients in dd\mathbf{dd}, namely a diagram

Q ϕ ga BPU(n) i dd n X ϕ B B 2U(1)Q ϕ ga BPU(n) ϕ B| Q dd B 2U(1) \array{ Q &\stackrel{\phi_{ga}}{\to}& \mathbf{B} P U(n) \\ {}^{\mathllap{i}}\downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\mathbf{dd}_n}} \\ X &\stackrel{\phi_B}{\to}& \mathbf{B}^2 U(1) } \;\;\;\; \leftrightarrow \;\;\; \array{ Q &&\stackrel{\phi_{ga}}{\to}&& \mathbf{B} P U(n) \\ & {}_{\mathllap{\phi_B|_Q}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{dd}}} \\ && \mathbf{B}^2 U(1) }

in H\mathbf{H}.

This is

  1. a circle 2-bundle ϕ B:XB 2U(1)\phi_B : X \to \mathbf{B}^2 U(1) on spacetime XX: the underlying bundle of the B-field;

  2. a projective unitary bundle ϕ ga\phi_{ga} on QQ, a Chan-Paton bundle on the D-brane;

  • together with an identification of the restriction ϕ B| Q\phi_{B}|_Q of the BB-field to the DD-brane with the obstruction dd(ϕ ga)\mathbf{dd}(\phi_{ga}) to lift ϕ ga\phi_{ga} to a genuine unitary bundle.

In cohomology this says that

[dd(ϕ ga)]=[ϕ B| Q]H 3(Q). [\mathbf{dd}(\phi_{ga})] = [\phi_B|_Q] \;\; \in H^3(Q) \,.

This is the Freed-Witten anomaly cancellation condition for DD-branes with spin^c structure.

More generally, if QQ does not necessarily have Spin cSpin^c-structure, we consider ι\iota-relative cohomology with coefficients in the universal local coefficient bundle ddW 3\mathbf{dd} - \mathbf{W}_3:

Q (TX,ϕ ga) B(SO×PU(n)) i dd nW 3 X ϕ B B 2U(1)Q (TX,ϕ ga) B(SO×PU(n)) ϕ B| Q dd nW 3 B 2U(1). \array{ Q &\stackrel{(T X, \phi_{ga})}{\to}& \mathbf{B} ( SO \times P U(n)) \\ {}^{\mathllap{i}}\downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\mathbf{dd}_n - \mathbf{W}_3}} \\ X &\stackrel{\phi_B}{\to}& \mathbf{B}^2 U(1) } \;\;\;\; \leftrightarrow \;\;\; \array{ Q &&\stackrel{(T X, \phi_{ga})}{\to}&& \mathbf{B} (SO \times P U(n)) \\ & {}_{\mathllap{\phi_B|_Q}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{dd}_n-\mathbf{W}_3}} \\ && \mathbf{B}^2 U(1) } \,.

This now is equivalently a twisted Chan-Paton bundle and a BB-field such that in cohomology

[dd(ϕ ga)]+[W 3(Q)]=[ϕ B| Q]H 3(X,). [\mathbf{dd}(\phi_{ga})] + [W_3(Q)] = [\phi_B|_Q] \;\;\; \in H^3(X, \mathbb{Z}) \,.

This is the Freed-Witten anomaly cancellation condition for general QQ.

b) Twisted differential String c 2String^{\mathbf{c}_2}-structure: M-theory flux quantization

We discuss the twisted smooth cohomology of the supergravity C-field in 11-dimensional supergravity/M-theory. With the smooth and differential refinement of the Whitehead tower in hand, this proceeds essentially in higher analogy to the previous example.

From the effective QFT of 11-dimensional supergravity the bosonic massless field content consists locally of the graviton and a 3-form CC. We have the following information on how a model of this field content must behave globally

  1. Due to the existence of spinors, the graviton must be part of a spin connection:

    ϕ gr:XBSpin conn. \phi_{gr} : X \to \mathbf{B}Spin_{conn} \,.
  2. Due to the coupling to the M2-brane the 3-form must lift to a well-dfined 3-holonomy and hence must globally be a circle 3-bundle with connection

    ϕ C:XB 3U(1) conn. \phi_C : X \to \mathbf{B}^3 U(1)_{conn} \,.
  3. Due to the coupling to the M5-brane, there is an auxiliary E8-bundle

    ϕ ga:XBE 8 \phi_{ga} : X \to \mathbf{B} E_8

    and these fields must satisfy what in string theory literature is called the flux quantization condition, and what in Hopkins-Singer 05 is called an differential integral Wu structure, meaning that in cohomology

    [ϕ C]=[12p 1(ϕ gr)]+[2a(ϕ ga)]H 4(X,). [\phi_C] = [\frac{1}{2}p_1(\phi_{gr})] + [2\mathbf{a}(\phi_{ga})] \;\; \in H^4(X, \mathbb{Z}) \,.

(Depending on convention one may write “2ϕ C2 \phi_C” for “ϕ C\phi_C” here, regarding the physical CC-field as being “one half” of the differential cocycle XB 3U(1) connX \to \mathbf{B}^3 U(1)_conn above, see the remark below (1.2) in Witten’s arXiv:hep-th/9609122).

A discrete 1-groupoid model satisfying these points has been by Freed-Moore and others (see the references here). Using cohesive homotopy type theory and following (FSSc) we now obtain naturally a genuine smooth moduli 3-stack of such field configurations: the interpretation of the evident expression

CField(X){ϕ gr,ϕ ga,ϕ C|12p 1(ϕ gr)+2a(ϕ ga)ϕ C} \mathbf{CField}(X) \coloneqq \left\{ \phi_{gr}, \phi_{ga}, \phi_{C} | \frac{1}{2}\mathbf{p}_1(\phi_{gr}) + 2\mathbf{a}(\phi_{ga}) \simeq \phi_C \right\}

in homotopy type theory is (see HoTT methods for homotopy theorists for how this works) the smooth \infty-stack CFieldH\mathbf{CField} \in \mathbf{H} given as the homotopy pullback

CField BSpin conn×BE 8 12p 1+2a B 3U(1) conn B 3U(1). \array{ \mathbf{CField} &\to& \mathbf{B} Spin_{conn} \times \mathbf{B}E_8 \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\tfrac{1}{2}\mathbf{p}_1 + 2\mathbf{a}} \\ \mathbf{B}^3 U(1)_{conn} &\stackrel{}{\to}& \mathbf{B}^3 U(1) } \,.

On the right this has the universal local coefficient bundle for String 2a\mathbf{String}^{2\mathbf{a}} from above, and hence this identifies a gravity-C-field configuration as being (a partial differential refinement of) a [ϕ C][\phi_C]-twisted String 2aString^{2\mathbf{a}}-structure.

c) Twisted differential String-structure – Green-Schwarz mechanism

By Hořava-Witten theory, the 10-dimensional target spacetime of the heterotic string may be understood as being a boundary (or rather 2\mathbb{Z}_2-orbifold fixed points, see below) of the 11-dimensional spacetime of 11d SuGra. Over this boundary

  1. the curvature 4-form G 4(ϕ C)G_4(\phi_C) vanishes;

  2. and the E 8E_8-principal bundle picks up a connection.

This means that the above defining homotopy pullback for CField\mathbf{CField} goes over into the one that defines the differential refinement String conn aString^{\mathbf{a}}_{conn'} of String aString^{\mathbf{a}}:

BString conn a BSpin conn×B(E 8×E 8) conn ϕ B 12p^ 1+a^ Ω cl 3() ϕ C B 3U(1) conn. \array{ \mathbf{B}String^{\mathbf{a}}_{conn'} &\to& \mathbf{B} Spin_{conn} \times \mathbf{B}(E_8 \times E_8)_{conn} \\ \downarrow &{}^{\mathllap{\phi_B}}\swArrow_{\simeq}& \downarrow^{\tfrac{1}{2}\hat \mathbf{p}_1 + \hat \mathbf{a}} \\ \Omega^3_{cl}(-) & \stackrel{\phi_C}{\hookrightarrow} & \mathbf{B}^3 U(1)_{conn} } \,.

On cohomology classes this means that

[12p 1(ϕ gr)]=[a(ϕ ga)]H 4(X,). [\tfrac{1}{2}p_1(\phi_{gr})] = [a(\phi_{ga})] \;\; \in H^4(X, \mathbb{Z}) \,.

This is the integral part of the Green-Schwarz mechanism for the heterotic string.

Since this is now refined not just to cocycles, but to differential cocycles – to String a\mathrm{String}^{\mathbf{a}}-2-connections, there is, locally over a cover {U iX}\{U_i \to X\}, also an equation of differential forms that exhibits this in de Rham cohomology.

A morphism XString conn 2aX \to String^{2\mathbf{a}}_{conn} classifies field content that is expressed with respect to a good open cover {U iX}\{U_i \to X\} in particular over single patches U iU_i by (SSSa, FSSa)

which come with curvature/field strength forms

  • F A i=d dRA i+12[A iA i]F_{A_i} = d_{dR} A_i + \tfrac{1}{2}[A_i \wedge A_i]

  • F ω i=d dRω i+12[ω iω i]F_{\omega_i} = d_{dR} \omega_i + \tfrac{1}{2}[\omega_i \wedge \omega_i];

  • H i=dB i+CS(ω i)CS(A i)H_i = d B_i + CS(\omega_i) - CS(A_i) (BB-field strength shifted by the difference of the Chern-Simons forms of A iA_i and ω i\omega_i);

satisfying the (twisted) Bianchi identities (SSSa))

  • d dRF A i+[A iF A i]=0d_{dR} F_{A_i} + [A_i \wedge F_{A_i}] = 0;

  • d dRF ω i+[ω iF ω i]=0d_{dR} F_{\omega_i} + [\omega_i \wedge F_{\omega_i}] = 0;

  • d dRH=F ωF ωF AF Ad_{dR} H = \langle F_\omega \wedge F_\omega \rangle - \langle F_A \wedge F_A \rangle

(together with more local cocycle components on higher overlaps). Notably the twisted Bianchi identity of HH exhibits the above cohomological identity in de Rham cocycles.

These formulas characterize the Green-Schwarz anomaly cancellation conditions on the background gauge field content, that makes the heterotic string be well defined. Accordingly, String conn aString^{\mathbf{a}}_{conn} is the smooth moduli 2-stack of anomaly free heterotic background fields (in the massless bosonic sector).

Notice that if the twist 12p 1^(ϕ gr)a^(ϕ gau)\tfrac{1}{2}\hat \mathbf{p_1}(\phi_{gr}) - \hat\mathbf{a}(\phi_{gau}) happen to vanish (say because both the field of gravity and the gauge field are trivial), then the above homotopy pullback reduces to

B 2U(1) conn * ϕ B Ω cl 3() ϕ C B 3U(1) conn \array{ \mathbf{B}^2 U(1)_{conn} &\to& * \\ \downarrow &{}^{\mathllap{\phi_B}}\swArrow_{\simeq}& \downarrow^{} \\ \Omega^3_{cl}(-) & \stackrel{\phi_C}{\hookrightarrow} & \mathbf{B}^3 U(1)_{conn} }

and exhibits ϕ B\phi_B as a genuine circle 2-bundle with connection (and its 3-form curvature HH with ϕ C\phi_C.). Conversely, this shows how in the general situation ϕ B\phi_B is a twisted circle 2-bundle, with the twist given by the “magnetic fivebrane current” 12p^ 1(ϕ gr)a^(ϕ ga)\tfrac{1}{2}\hat \mathbf{p}_1(\phi_{gr}) - \hat \mathbf{a}(\phi_{ga}).

d) Twisted differential Fivebrane structure – dual Green-Schwarz mechanism

(…)

IV) Higher orientifold structure

Orientifolds

An orientifold target space for the bosonic string is a smooth manifold or more generally orbifold XX equipped with

  • a double cover w X:XB 2w_X : X \to \mathbf{B} \mathbb{Z}_2;

  • a twisted 2\mathbb{Z}_2-equivariant circle 2-bundle, given by a morphism ϕ B:XBAut(BU(1))\phi_B : X \to \mathbf{B}Aut(\mathbf{B}U(1)) whose underlying double cover is ww.

This means that this background is a cocycle in ww-twisted cohomology for the local coefficient bundle

BU(1) BAut(BU(1)) J B 2. \array{ \mathbf{B}U(1) &\to& \mathbf{B}Aut(\mathbf{B}U(1)) \\ && \downarrow^{\mathbf{J}} \\ && \mathbf{B}\mathbb{Z}_2 } \,.

Hence the B-field is now a cocycle in ww-twisted cohomology

ϕ bH /B 2(w X,J), \phi_b \in \mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}( w_X , \mathbf{J}) \,,

or rather a differential refinement thereof. For Σ\Sigma the worldvolume of a string, an orientifold string configuration is a cocycle

(φ,ν)H /B 2(w 1(Σ),w X), (\varphi, \nu) \in \mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}(\mathbf{w}_1(\Sigma), w_X) \,,

given in H\mathbf{H} by a diagram

Σ φ X w 1 ν w X B 2 \array{ \Sigma &&\stackrel{\varphi}{\to}&& X \\ & {}_{\mathllap{\mathbf{w}_1}}\searrow &{}^{\nu}\swArrow_{\simeq}& \swarrow_{\mathrlap{w_X}} \\ && \mathbf{B}\mathbb{Z}_2 }

consisting of

  • a map φ:ΣX\varphi : \Sigma \to X;

  • an isomorphism ν:φ *w Xw 1(Σ)\nu : \varphi^* w_X \to \mathbf{w}_1(\Sigma).

Hořava-Witten compactifications

In Hořava-Witten theory there is similarly a twisted 2\mathbb{Z}_2-action on the supergravity C-field, exhibited by a local coefficient bundle

B 2U(1) BAut(B 2U(1)) J B 2. \array{ \mathbf{B}^2U(1) &\to& \mathbf{B}Aut(\mathbf{B}^2U(1)) \\ && \downarrow^{\mathbf{J}} \\ && \mathbf{B}\mathbb{Z}_2 } \,.

Further twists

There are various further twisted cohomological structures in string theory known or conjectured (for some of which possibly no smooth refinement has been constructed yet). We briefly list some of them.

Twisted super bundle

In work like Loop Groups and Twisted K-Theory the following structure plays a role:

for GG a group, let

ϵ:G 2 \epsilon : G \to \mathbb{Z}_2

be a fixed group homomorphism. Then for E=E 0E 1XE = E_0 \oplus E_1 \to X a super vector bundle, an “ϵ\epsilon-twist” of EE is an action of GG on EE such that an element gGg \in G acts by an even automorphism if ϵ(g)\epsilon(g) is even, and by an odd automorphism if ϵ(g)\epsilon(g) is odd (Freed ESI lecture, (1.13)).

This is a special case of the general notion of twist discussed here by considering the canonical morphism

BAut(E) e B 2 \array{ \mathbf{B}Aut(E) \\ \downarrow^{\mathbf{e}} \\ \mathbf{B} \mathbb{Z}_2 }

as the local coefficient bundle, and considering ϵ\epsilon as the twist: then an ϵ\epsilon-twisting as above is a cocycle in the twisted cohomology H /B 2(Bϵ,e)\mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}(\mathbf{B}\epsilon, \mathbf{e}) given by a commuting triangle

BG BAut(E) Bϵ e B 2. \array{ \mathbf{B}G &&\to&& \mathbf{B}Aut(E) \\ & {}_{\mathllap{\mathbf{B}\epsilon}}\searrow && \swarrow_{\mathrlap{\mathbf{e}}} \\ && \mathbf{B}\mathbb{Z}_2 } \,.

Relative fields

Meanwhile in (Freed-Teleman 2012) special cases of the general notion of twisted fields above are being called relative fields. We briefly spell out how the definitions considered in that article are examples of the general notion above.

For πGrp(Set)Grp(SmoothGrpd)\pi \in Grp(Set) \hookrightarrow Grp(Smooth\infty Grpd) a discrete group and for X¯H\overline{X} \in \mathbf{H} \coloneqq Smooth∞Grpd any object (for instance a smooth manifold) a morphism ϕ X¯:X¯Bπ\phi_{\overline{X}} \colon \overline{X} \to \mathbf{B}\pi modulates a π\pi-principal bundle XX¯X \to \overline{X} over XX, hence a free π\pi-action on XX such that XX¯X \to \overline{X} is the quotient map.

Then the corresponding twisted cohomology H /Bπ(,ϕ X¯)\mathbf{H}_{/\mathbf{B}\pi}(-, \phi_{\overline{X}}) has

  1. domains are objects Σ\Sigma equipped with a π\pi-principal bundle, modulated by a morphism ϕ Σ:ΣBπ\phi_\Sigma \colon \Sigma \to \mathbf{B}\pi;

  2. cocycles are morphism ϕ Σϕ X\phi_\Sigma \to \phi_X in the slice (∞,1)-topos H /Bπ\mathbf{H}_{/\mathbf{B}\pi}, hence diagrams of the form

    Σ f X¯ ϕ Σ θ ϕ X Bπ \array{ \Sigma &&\stackrel{f}{\to}&& \overline{X} \\ & {}_{\mathllap{\phi_\Sigma}}\searrow &\swArrow_{\theta}& \swarrow_{\mathrlap{\phi_X}} \\ && \mathbf{B}\pi }

    in H\mathbf{H}, hence maps f:ΣX¯f \colon \Sigma \to \overline{X} equipped with equivalences

    θ:f *ϕ Xϕ Σ. \theta \colon f^* \phi_X \stackrel{\simeq}{\to} \phi_\Sigma \,.

This class of examples is what appears as def. 3.4 in (Freed-Teleman). It contains in particular the above examples of Reduction of structure group and its differential refinement.

Next, consider a compact Lie group G¯\overline{G} and a central group extension πGG¯\pi \to G \to \overline{G}. This is classified by a cocycle

c:BG¯B 2π. \mathbf{c} \colon \mathbf{B}\overline{G} \to \mathbf{B}^2 \pi \,.

Then the corresponding twisted cohomology H /B 2π(,c)\mathbf{H}_{/\mathbf{B}^2 \pi}(-, \mathbf{c}) has

  1. domains are objects Σ\Sigma equipped with a Bπ\mathbf{B}\pi-principal 2-bundle/bundle gerbe modulated by a morphism ϕ Σ:ΣB 2π\phi_\Sigma \colon \Sigma \to \mathbf{B}^2 \pi;

  2. cocycles are morphism ϕ Σc\phi_\Sigma \to \mathbf{c} in the slice (∞,1)-topos H /B 2π\mathbf{H}_{/\mathbf{B}^2\pi}, hence diagrams of the form

    Σ f X¯ ϕ Σ θ c B 2π \array{ \Sigma &&\stackrel{f}{\to}&& \overline{X} \\ & {}_{\mathllap{\phi_\Sigma}}\searrow &\swArrow_{\theta}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}^2\pi }

    in H\mathbf{H}, hence maps f:ΣX¯f \colon \Sigma \to \overline{X} equipped with equivalences

    θ:c(f)ϕ Σ \theta \colon \mathbf{c}(f) \stackrel{\simeq}{\to} \phi_\Sigma

    between the principal 2-bundle/bundle gerbe c(f)\mathbf{c}(f) induced by the G¯\overline{G}-principal bundl modulated by ff, and the one modulated by ϕ Σ\phi_\Sigma.

Alternatively one can use here the differential refinement of BG¯\mathbf{B}\overline{G} to the moduli stack BG¯ conn\mathbf{B}\overline{G}_{conn} of G¯\overline{G}-principal connections.

Examples and further details are discussed in Schreiber, section 4. In (Freed-Teleman) this example appears as def. 4.6.

Twisted tmf

Twisted Morava K-theory

B) Local boundary prequantum field theory from twisted smooth cohomology

This section originates in some talk notes (Schreiber, Twists 2013).

We indicate now how the twisted smooth cohomology data as in the examples above induces and in fact corresponds to data for prequantum field theory sigma-models localized (in the sense of the cobordism hypothesis) to local prequantum field theory (lpqft).

In order to formalize this accurately, we first talk a bit more about the relevant cohesion, differential cohesion and tangent cohesion. The way to understand this is as follows:

In full generality, cohomology (see there for details) is what is given by the (∞,1)-categorical hom-spaces in some ∞-topos: for X,AHX,A \in \mathbf{H} any two objects, then

H(X,A)π 0H(X,A) H(X,A) \coloneqq \pi_0 \mathbf{H}(X,A)

is the cohomology of XX with coefficients in AA.

Therefore it is natural to ask: What is an (∞,1)-topos to be like whose intrinsic cohomology is equivariant twisted stable differential cohomology?

And the answer we find is: it is to be the tangent (∞,1)-topos THT \mathbf{H} of a cohesive (∞,1)-topos H\mathbf{H}.

For more on this see at tangent cohesion.

Cohesive contexts for equivariant differential twisted cohomology

After the discovery of the role of topological K-theory in D-brane phenomena in string theory, the observation that more generally M-theory involves various other twisted cohomology theories such as tmf and Morava K-theory, has notably been highlighted by Hisham Sati, surveyed in (Sati 10).

The suggestion that the right context for formulating the smooth and differential refinement of these twisted cohomology theories is the (∞,1)-topos We

H =Sh (SmthMfd) Funct(SmoothMfd op,KanCplx)[{stalkwisehomotopyequivalences} 1] \begin{aligned} \mathbf{H} &= Sh_\infty(SmthMfd) \\ & \simeq Funct(SmoothMfd^{op}, KanCplx)[\{stalkwise\;homotopy\;equivalences\}^{-1}] \end{aligned}

of ∞-stacks over the site of smooth manifolds (the result of universally turning stalkwise homotopy equivalences on presheaves of Kan complexes into actual homotopy equivalences) is due to (Schreiber 09).

A full formalization of the classification of principal ∞-bundles in such (∞,1)-topos, and their classification by nonabelian cohomology was later given in (Nikolaus-Schreiber-Stevenson 12). There it is shown that for GGrp(H)G \in Grp(\mathbf{H}) an ∞-group of twists, the corresponding twisted cohomology is the plain cohomology of the slice (∞,1)-topos

H /BG(,1)Topos. \mathbf{H}_{/\mathbf{B}G} \;\; \in (\infty,1)Topos \,.

Precisely, for ρH BG\rho \in \mathbf{H}_{\mathbf{B}G} a GG-∞-action on some VV incarnated as its universal associated ∞-bundle (the local coefficient ∞-bundle) and for χ:XBG\chi \colon X \longrightarrow \mathbf{B}G a twist, then the χ\chi-twisted cohomology with local coefficients in VV is

H BG(χ X,ρ)={X ϕ V//G χ ρ BG}. \mathbf{H}_{\mathbf{B}G}(\chi_X,\rho) = \left\{ \array{ X && \stackrel{\phi}{\longrightarrow} && V//G \\ & {}_{\mathllap{\chi}}\searrow && \swarrow_{\mathrlap{\rho}} \\ && \mathbf{B}G } \right\} \,.

The observation that for the differential cohomology-refinement of this twisted geometric cohomology it is the adjoint quadruple of (∞,1)-functors between H\mathbf{H} to the base (∞,1)-topos (“cohesion”)

HcoDiscΓDiscΠGrpd \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd

(with Disc and coDisc full and faithful (∞,1)-functors and the fundamental ∞-groupoid/geometric realization Π\Pi finite product-preserving) which governs all the theory was observed first in (Sati-Schreiber-Stasheff 09 (11))

There it was shown that this serves to construct and characterize the twisted differential string structures and twisted differential Fivebrane structures in (dual) heterotic string theory.

A comprehensive theory of (twisted equivariant) differential cohomology formulated by just this axiom of “cohesion” was then laid out in the thesis (Schreiber 11). Parts of this appear in various articles, such as (Fiorenza-Schreiber-Rogers 13). See (Schreiber Synthetic 13eiberSynthetic)) for a fairly comprehensive survey.

Notice that such cohesion is a very special property of some (∞,1)-toposes, not a generic property. In particular the existence of Π\Pi means that H\mathbf{H} is a locally ∞-connected (∞,1)-topos and a globally ∞-connected (∞,1)-topos, and the existence of coDisccoDisc means that it is a local (∞,1)-topos.

Examples of cohesive higher geometry established and discussed in (Schreiber 11) include

Clearly these examples are all of a similar kind (modeled on variants of manifolds). There are also some diagram categories which are usefully cohesive, for instance

In October 2013 Charles Rezk announced a new kind of cohesion, namely:

But of course a central feature desireable for cohomology theory is stabilization of cohesion/cohesive spectrum objects.

In (Bunke-Nikolaus-Völkl 13 (14?)) is considered the stabilization of smooth cohesion, hence the stable (∞,1)-category Stab(Sh (SmthMfd))Stab(Sh_\infty(SmthMfd)) of spectrum object in smooth ∞-groupoids, which carries an analogous adjoint quadruple over the (∞,1)-category of spectra Stab(Grpd)SpectraStab(\infty Grpd) \simeq Spectra

Stab(Sh (SmthMfd))coDisc stabΓ stabDisc stabΠ stabSpectra. Stab(Sh_\infty(SmthMfd)) \stackrel{\overset{\Pi^{stab}}{\longrightarrow}}{\stackrel{\overset{Disc^{stab}}{\leftarrow}}{\stackrel{\overset{\Gamma^{stab}}{\longrightarrow}}{\underset{coDisc^{stab}}{\leftarrow}}}} Spectra \,.

But this stable aspect is unified with the unstable cohesion by the notion of “tangent cohesion”. This we turn to now.

Cohesive contexts for stable twisted cohomology

We first discuss generally how the tangent (∞,1)-category THT \mathbf{H} of an (∞,1)-topos H\mathbf{H} is itself an (∞,1)-topos over the tangent \infty-category of the original base (∞,1)-topos (Joyal 08). Then we observe that THT \mathbf{H} is cohesive if H\mathbf{H} is and is in fact an extension of the latter by its stabilization. a choice

Tangent \infty-topos

Definition

Let seqseq be the diagram category as follows:

seq{ x n1 p n1 * p n1 i n id * i n x n p n * id p n i n+1 * i n+1 x n+1 } n. seq \coloneqq \left\{ \array{ && \vdots && \vdots \\ && \downarrow && \\ \cdots &\to& x_{n-1} &\stackrel{p_{n-1}}{\longrightarrow}& \ast \\ &&{}^{\mathllap{p_{n-1}}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_n}} & \searrow^{\mathrm{id}} \\ &&\ast &\underset{i_n}{\longrightarrow}& x_n &\stackrel{p_n}{\longrightarrow}& \ast \\ && &{}_{\mathllap{id}}\searrow& {}^{\mathllap{p_n}}\downarrow &\swArrow& \downarrow^{\mathrlap{i_{n+1}}} \\ && && \ast &\stackrel{i_{n+1}}{\longrightarrow}& x_{n+1} &\to& \cdots \\ && && && \downarrow \\ && && && \vdots } \right\}_{n \in \mathbb{Z}} \,.

(Joyal 08, section 35.5)

Remark

Given an (∞,1)-topos H\mathbf{H}, an (∞,1)-functor

X :seqH X_\bullet \;\colon\; seq \longrightarrow \mathbf{H}

is equivalently

  1. a choice of object BHB \in \mathbf{H} (the image of *inseq\ast in seq]);

  2. a sequence of objects {X n}H /B\{X_n\} \in \mathbf{H}_{/B} in the slice (∞,1)-topos over BB;a choi

  3. a sequence of morphisms X nΩ BX n+1X_n \longrightarrow \Omega_B X_{n+1} from X nX_n into the loop space object of X n+1X_{n+1} in the slice.

This is a prespectrum object in the slice (∞,1)-topos H /B\mathbf{H}_{/B}.

A natural transformation f:X Y f \;\colon \;X_\bullet \to Y_\bullet between two such functors with components

{X n f n Y n p n X p n Y B 1 f b B 2} \left\{ \array{ X_n &\stackrel{f_n}{\longrightarrow}& Y_n \\ \downarrow^{\mathrlap{p_n^X}} && \downarrow^{\mathrlap{p_n^Y}} \\ B_1 &\stackrel{f_b}{\longrightarrow}& B_2 } \right\}

is equivalently a morphism of base objects f b:B 1B 2f_b \;\colon\; B_1 \longrightarrow B_2 in H\mathbf{H} together with morphisms X nf b *Y nX_n \longrightarrow f_b^\ast Y_n into the (∞,1)-pullback of the components of Y Y_\bullet along f bf_b.

Therefore the (∞,1)-presheaf (∞,1)-topos

H seqFunc(seq,H) \mathbf{H}^{seq} \coloneqq Func(seq, \mathbf{H})

is the codomain fibration of H\mathbf{H} with “fiberwise pre-stabilization”.

A genuine spectrum object is a prespectrum object for which all the structure maps X nΩ BX n+1X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1} are equivalences. The full sub-(∞,1)-category

THH seq T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}

on the genuine spectrum objects is therefore the “fiberwise stabilization” of the codomain fibration, hence the tangent (,1)(\infty,1)-category.

Lemma

(spectrification is left exact reflective)

The inclusion of spectrum objects into H\mathbf{H} is left reflective, hence it has a left adjoint (∞,1)-functor LL which preserves finite (∞,1)-limits.

THL lexH seq. T \mathbf{H} \stackrel{\overset{L_{lex}}{\leftarrow}}{\hookrightarrow} \mathbf{H}^{seq} \,.

(Joyal 08, section 35.1)

Proof

Forming degreewise loop space objects constitutes an (∞,1)-functor Ω:H seqH seq\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq} and by definition of seqseq this comes with a natural transformation out of the identity

θ:idΩ. \theta \;\colon\; id \longrightarrow \Omega \,.

This in turn is compatible with Ω\Omega in that

θΩΩθ:ΩΩΩ=Ω 2. \theta \circ \Omega \simeq \Omega \circ \theta \;\colon\; \Omega \longrightarrow \Omega \circ \Omega = \Omega^2 \,.

Consider then a sufficiently deep transfinite composition ρ tf\rho^{tf}. By the small object argument available in the presentable (∞,1)-category H\mathbf{H} this stabilizes, and hence provides a reflection L:H seqTHL \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}.

Since transfinite composition is a filtered (∞,1)-colimit and since in an (∞,1)-topos these commute with finite (∞,1)-limits, it follows that spectrum objects are an left exact reflective sub-(∞,1)-category.

Proposition

For H\mathbf{H} an (∞,1)-topos over the base (∞,1)-topos Grpd\infty Grpd, its tangent (∞,1)-category THT \mathbf{H} is an (∞,1)-topos over the base TGrpdT \infty Grpd (and hence in particular also over Grpd\infty Grpd itself).

(Joyal 08, section 35.5)

Proof

By the the spectrification lemma 1 THT \mathbf{H} has a geometric embedding into the (∞,1)-presheaf (∞,1)-topos H seq\mathbf{H}^{seq}, and this implies that it is an (∞,1)-topos (by the discussion there).

Moreover, since both adjoint (∞,1)-functor in the global section geometric morphism HΓΔGrpd\mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd preserve finite (∞,1)-limits they preserve spectrum objects and hence their immediate (∞,1)-presheaf prolongation immediately restricts to the inclusion of spectrum objects

TH TΓTΔ TGrpd incl incl H ΓΔ Grpd. \array{ T \mathbf{H} &\stackrel{\overset{T \Delta}{\leftarrow}}{\underset{T \Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} & \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} & \infty Grpd } \,.
Remark

We may think of the tangent (∞,1)-topos THT \mathbf{H} as being an extension of H\mathbf{H} by its stabilization Stab(H)T *HStab(\mathbf{H}) \simeq T_\ast \mathbf{H}:

Stab(H) Stab(Γ)Stab(Δ) Spectra incl incl TH TΓTΔ TGrpd base base H ΓΔ Grpd. \array{ Stab(\mathbf{H}) &\stackrel{\overset{Stab(\Delta)}{\leftarrow}}{\underset{Stab(\Gamma)}{\longrightarrow}}& Spectra \\ \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ T\mathbf{H} &\stackrel{\overset{T\Delta}{\leftarrow}}{\underset{T\Gamma}{\longrightarrow}}& T \infty Grpd \\ \downarrow^{\mathrlap{base}} && \downarrow^{\mathrlap{base}} \\ \mathbf{H} &\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}}& \infty Grpd } \,.

Crucial for the internal interpretation in homotopy type theory is that the homotopy types in T *HTHT_\ast \mathbf{H} \hookrightarrow T \mathbf{H} are stable homotopy types.

Tangent cohesive homotopy theory

Now consider the case that H\mathbf{H} is a cohesive (∞,1)-topos over ∞Grpd, in that there is an adjoint quadruple

HcoDiscΓDiscΠGrpd \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd

with Disc,coDiscDisc, coDisc being full and faithful (∞,1)-functors and Π\Pi preserving finite (∞,1)-products.

Since (∞,1)-limits and (∞,1)-colimits in an (∞,1)-presheaf (∞,1)-topos are computed objectwise, this adjoint quadruple immediately prolongs to H seq\mathbf{H}^{seq}

H seqcoDisc seqΓ seqDisc seqΠ seqGrpd seq. \mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} \infty Grpd^{seq} \,.

Moreover, all three right adjoints preserves the (∞,1)-pullbacks involved in the characterization of spectrum objects and hence restrict to THT \mathbf{H}

THcoDisc seqΓ seqDisc seqTGrpd. T\mathbf{H} \stackrel{}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} T\infty Grpd \,.

But then we have a further left adjoint given as the composite

THH seqDisc seqΠ seqGrpd seqLTGrpd. T\mathbf{H} \hookrightarrow \mathbf{H}^{seq} \stackrel{\overset{\Pi^{seq}}{\longrightarrow}}{\underset{Disc^{seq}}{\leftarrow}} \infty Grpd^{seq} \stackrel{\overset{L}{\longrightarrow}}{\underset{}{\leftarrow}} T \infty Grpd \,.

Again since LL is a left exact (∞,1)-functor this composite LΠL \Pi preserves finite (∞,1)-products.

So it follows in conclusion that if H\mathbf{H} is a cohesive (∞,1)-topos then its tangent (,1)(\infty,1)-category THT \mathbf{H} is itself a cohesive (∞,1)-topos over the tangent (,1)(\infty,1)-category TGrpdT \infty Grpd of the base (∞,1)-topos, which is an extension of the cohesion of the \infty-topos H\mathbf{H} over Grpd\infty Grpd by the cohesion of the stable \infty-category Stab(H)Stab(\mathbf{H}) over Stab(Grpd)SpecStab(\infty Grpd) \simeq Spec:

Stab(H) coDisc seqΓ seqDisc seqLΠ seq Stab(Grpd)Spectra T *H T *Grpd incl incl H Ω totd TH coDisc seqΓ seqDisc seqLΠ seq TGrpd base 0 base 0 H coDiscΓDiscΠ Grpd. \array{ && Stab(\mathbf{H}) & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & Stab(\infty Grpd) \simeq Spectra \\ && \simeq && \simeq \\ && T_\ast \mathbf{H} && T_\ast \infty Grpd \\ && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}} \\ \mathbf{H} &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}& T \mathbf{H} & \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}} & T \infty Grpd \\ && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} && {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} \\ && \mathbf{H} & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd } \,.

Here

  • Ω tot:THH\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H} assigns the total space of a spectrum bundle;

    its left adjoint is the tangent complex functor;

  • base:THHbase \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H} assigns the base space of a spectrum bundle;

    its left adjoint produces the 0-bundle.

Where the (∞,1)-categorical hom-space in a general (∞,1)-topos constitute a notion of cohomology, those of a tangent (∞,1)-topos specifically constitute twisted generalized cohomology, in fact twisted bivariant cohomology.

For consider a spectrum object ET *HE \in T_\ast \mathbf{H} and write GL 1(E)Grp(H)GL_1(E) \in Grp(\mathbf{H}) for its ∞-group of units. Then the ∞-action of this on EE is (by the discussion there) exhibited by an object

(E//GL 1(E) BGL 1(E))T BGL 1(E)HTH. \left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right) \;\;\; \in \;\;\; T_{\mathbf{B}GL_1(E)}\mathbf{H} \hookrightarrow T\mathbf{H} \,.

More generally, for Pic(E)HPic(E) \in \mathbf{H} the Picard ∞-groupoid of EE there is the universal (∞,1)-line bundle

(Pic(E)^Pic(E))TH. (\widehat{Pic(E)} \to Pic(E)) \in T \mathbf{H} \,.

Now for any object XHX \in \mathbf{H} we have

X×E(E×X X)T XHTH, X \times E \simeq \left( \array{ E \times X \\ \downarrow \\ X } \right) \;\;\; \in \;\;\; T_{X}\mathbf{H} \hookrightarrow T\mathbf{H} \,,

then morphisms in THT \mathbf{H} from the latter to the former

(E×X X)(E//GL 1(E) BGL 1(E)) \left( \array{ E \times X \\ \downarrow \\ X } \right) \longrightarrow \left( \array{ E//GL_1(E) \\ \downarrow \\ \mathbf{B}GL_1(E) } \right)

are equivalently

  1. a choice of twist of E-cohomology χ:XBGL 1(E)\chi \;\colon \; X \longrightarrow \mathbf{B}GL_1(E);

  2. an element in the χ\chi-twisted EE-cohomology of XX, hence cE (X,E)c \in E^{\bullet}(X,E).

If we consider the internal hom then we can use just XX instead of X×EX \times E:

Proposition

For XH0THX \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H} a geometric homotopy type and EStab(H)T *HTHE \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H} a spectrum object, then the internal hom/mapping stack

[X,E] THTH [X,E]_{T \mathbf{H}} \in T \mathbf{H}

(with respect to the Cartesian closed monoidal (∞,1)-category structure on the (∞,1)-topos is equivalently the mapping spectrum

[Σ X,E] Stab(H)Stab(H)TH, [\Sigma^\infty X, E]_{Stab(\mathbf{H})} \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H} \,,

in that

[X,E] TH[Σ X,E] Stab(H). [X,E]_{T \mathbf{H}} \simeq [\Sigma^\infty X,E]_{Stab(\mathbf{H})} \,.
Proof

Notice that as an object of THH seqT \mathbf{H} \hookrightarrow \mathbf{H}^{seq}, the object XX is the constant (∞,1)-presheaf on seqseq. By the formula for the internal hom in an (∞,1)-category of (∞,1)-presheaves we have

[X,E] H seq(X×,E). [X,E]_\bullet \simeq \mathbf{H}^{seq}(X \times \bullet, E) \,.

But since XX is constant the object X×X \times \bullet is for each object of seqseq the presheaf represented by that object. Therefore by the (∞,1)-Yoneda lemma it follows that

[X,E] [X,E ]. [X,E]_\bullet \simeq [X,E_\bullet] \,.

This is manifestly the same formula as for the mapping spectrum out of Σ X\Sigma^\infty X.

By the same kind of argument we have the following more general statement.

Proposition

For XH0THX \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H} a geometric homotopy type, for EE (H)E \in E_\infty(\mathbf{H}) an E-∞ ring with (Pic(E)^Pic(E))TH(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H} its universal (∞,1)-line bundle over its Picard ∞-groupoid, then the internal hom/mapping stack

[X,Pic(E)^] THTH [X,\widehat{Pic(E)}]_{T \mathbf{H}} \in T \mathbf{H}

is the object whose

Cohesive and differential refinement in tangent cohesion

Let THT\mathbf{H} be a tangent cohesive (,1)(\infty,1)-topos and write T *HT_\ast \mathbf{H} for the stable (∞,1)-category of spectrum objects inside it.

Proposition

For every AT *HA \in T_\ast \mathbf{H} the naturality square

A A/A (pb) Π(A) Π(A/A) \array{ A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }

(of the shape modality applied to the homotopy cofiber of the counit of the flat modality) is an (∞,1)-pullback square.

This was observed in (Bunke-Nikolaus-Völkl 13). It is an incarnation of a fracture theorem.

Proof

By cohesion and stability we have the diagram

A A A/A Π(A) Π(A) Π(A/A) \array{ \flat A &\longrightarrow & A &\stackrel{}{\longrightarrow}& A/\flat A \\ \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow \\ \Pi(\flat A) &\longrightarrow& \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }

where both rows are homotopy fiber sequences. By cohesion the left vertical map is an equivalence. The claim now follows with the homotopy fiber characterization of homotopy pullbacks.

Remark

This means that in stable cohesion every cohesive stable homotopy type is in controled sense a cohesive extension/refinement of its geometric realization geometrically discrete (“bare”) stable homotopy type by the non-discrete part of its cohesive structure;

In particular, A/AA/\flat A may be identified with differential cycle data. Indeed, by stability and cohesion it is the flat de Rham coefficient object

A/A= dRΣA A/\flat A = \flat_{dR}\Sigma A

of the suspension of AA. So

A θ A dRΣA (pb) Π(A) Π(A/A) \array{ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) }

exhibits AA as a differential cohomology-coefficient of the generalized cohomology theory Π(A)\Pi(A) (Bunke-Nikolaus-Völkl 13).

It follows by the discussion at differential cohomology in a cohesive topos that the further differential refinement A^\widehat{A} of AA should be given by a further homotopy pullback

A^ Ω 1(,Lie(A)) (pb) A θ A dRΣA (pb) Π(A) Π(A/A). \array{ \widehat{A} &\longrightarrow& \Omega^1(-,Lie(A)) \\ \downarrow &{}^{(pb)}& \downarrow \\ A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A) } \,.

Local prequantum field theory

We describe the formulation of local prequantum field theory in a cohesive (∞,1)-topos H\mathbf{H} (lpqft).

A classical field theory/prequantum field theory is traditionally defined by an action functional: given a smooth space Fields traj\mathbf{Fields}_{traj} “of trajectories” of a given physical system, then the action functional is a smooth function

Fields traj exp(iS) U(1) \array{ \mathbf{Fields}_{traj} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ U(1) }

to the circle group. The idea of producing a quantum field theory from this is to

  1. choose a linearization in the form of the group homomorphism U(1)GL 1()U(1) \longrightarrow GL_1(\mathbb{C}) to the group of units of the complex numbers,

  2. choose a measure dμd\mu on Fields traj\mathbf{Fields}_{traj}

and then declare that the integral (“path integral”)

ϕFields trajexp(iS(ϕ))dμ \underset{\phi \in \mathbf{Fields}_{traj}}{\int} \exp(i S(\phi))\, d\mu \in \mathbb{C}

is the partition function of the theory a kind of expectation value with probabilities replaced by probability amplitudes.

In order to make sense of this (for a full discussion of “motivic quantization” in this sense see (Nuiten 13), here we concentrate on the pre-quantum aspects), it is useful to allow some more conceptual wiggling room by passing to higher differential geometry. Notice that if we write BU(1)\mathbf{B}U(1) for the smooth universal moduli stack of circle group-principal bundles, then an action functional as above is equivalently a homotopy of the form

Fields traj * * 0 0 BU(1) Fields traj exp(iS) * U(1) * 0 (pb) 0 BU(1), \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & {}_{\mathllap{0}}\searrow && \swarrow_{\mathrlap{0}} \\ && \mathbf{B}U(1) } \;\;\;\; \simeq \;\;\;\; \array{ && \mathbf{Fields}_{traj} \\ & \swarrow &\downarrow^{\mathrlap{\exp(i S)}}& \searrow \\ \ast &\leftarrow& U(1) &\rightarrow& \ast \\ & {}_{\mathllap{0}}\searrow &{}^{(pb)}& \swarrow_{\mathrlap{0}} \\ && \mathbf{B}U(1) } \,,

where on the right we used the universal property of the homotopy pullback diagram which exhibits the smooth circle group U(1)U(1) as the loop space object of BU(1)\mathbf{B}U(1).

For instance for XX a smooth manifold (“spacetime”) and :XBU(1) conn\nabla \;\colon\; X \longrightarrow \mathbf{B}U(1)_{conn} a circle group-principal connection (“electromagnetic field on spacetime”) then for trajectories in XX of shape the circle, the canonical action functional (“Lorentz force gauge interaction”) is the holonomy functional

exp(iS Lor)exp(i S 1[S 1,]):[S 1,X][S 1,X][S 1,BU(1) conn]exp(i S 1())U(1). \exp(i S_{Lor}) \coloneqq \exp(i \int_{S^1} [S^1, \nabla]) \;\colon\; [S^1, X] \stackrel{[S^1, X]}{\longrightarrow} [S^1, \mathbf{B}U(1)_{conn}] \stackrel{\exp(i \int_{S^1}(-))}{\longrightarrow} U(1) \,.

But more generally, if the trajectories have a boundary, hence if they are of the shape of an interval I[0,1]I \coloneqq [0,1], then the holonomy functional on smooth loop space [S 1,X][S^1, X] generalizes to the parallel transport on the path space [I,X][I,X] and there it is no longer a function, but exists only as a homotopy of the form

[I,X] ()| 0 ()| 1 X exp(i I[I,]) X χ χ BU(1). \array{ && [I,X] \\ & {}^{(-)|_0}\swarrow && \searrow^{\mathrlap{(-)|_1}} \\ X && \swArrow_{\exp(i \int_{I}[I,\nabla])} && X \\ & {}_{\mathllap{\chi_\nabla}}\searrow && \swarrow_{\mathrlap{\chi_\nabla}} \\ && \mathbf{B}U(1) } \,.

Notice that this is a “local” description of the action functional: the data that determines it is the boundary

X BU(1) conn \array{ X \\ \downarrow^{\mathrlap{\nabla}} \\ \mathbf{B}U(1)_{conn} }

and from this the rest is induced by transgression.

A related class of examples are prequantized Lagrangian correspondences: Let

X ω Ω 2 \array{ X \\ \downarrow^{\mathrlap{\omega}} \\ \mathbf{\Omega}^2 }

be a symplectic manifold. Then a symplectomorphism f:XXf \;\colon\; X \longrightarrow X is a correspondence of the form

graph(f) X X ω ω Ω 2. \array{ && graph(f) \\ & \swarrow && \searrow \\ X && && X \\ & {}_{\mathllap{\omega}}\searrow && \swarrow_{\mathrlap{\omega}} \\ && \mathbf{\Omega}^2 } \,.

A prequantization of (X,ω)(X,\omega) is a lift \nabla in

X BU(1) conn F () Ω 2 \array{ X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ & \searrow & \downarrow^{\mathrlap{F_{(-)}}} \\ && \mathbf{\Omega}^2 }

and so a prequantized Lagrangian correspondence is

graph(f) X X BU(1) conn. \array{ && graph(f) \\ & \swarrow && \searrow \\ X && \swArrow && X \\ & _{\mathllap{\nabla}}\searrow && \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}U(1)_{conn} } \,.

To conceptualize all this, write

HSmoothGrpdFunc(SmoothMfd op,Grpd)[{stalkwiseequivalences} 1] \mathbf{H} \coloneqq SmoothGrpd \coloneqq Func(SmoothMfd^{op}, Grpd)[\{stalkwise\;equivalences\}^{-1}]

for the homotopy theory obtained from the category of groupoid-valued presheaves on the category of all smooth manifolds by universally turning stalkwise equivalences of groupoids into genuine homotopy equivalences (“simplicial localization”).

This is the (2,1)-topos of smooth groupoids/smooth (moduli) stacks.

Write

Corr 1(H)(2,1)Cat Corr_1(\mathbf{H}) \in (2,1)Cat

for the (2,1)-category of correspondences in H\mathbf{H}. Write H /BU(1) conn\mathbf{H}_{/\mathbf{B}U(1)_{conn}} for the slice (2,1)-topos over the smooth moduli stack of circle bundles with connection. Then the abovve diagrams are morphisms in Corr 1(H /BU(1) conn)Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}}).

Proposition

The automorphism group of Corr 1(H /BU(1) conn)\nabla \in Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}}) is the quantomorphism group of (X,ω)(X,\omega), hence the smooth group which is the Lie integration of the Poisson bracket Lie algebra of (X,ω)(X,\omega).

A concrete smooth 1-parameter subgroup

BBAut /BU(1) conn()H /BU(1) conn \mathbf{B}\mathbb{R} \longrightarrow \mathbf{B}\mathbf{Aut}_{/\mathbf{B}U(1)_{conn}}(\nabla) \hookrightarrow \mathbf{H}_{/\mathbf{B}U(1)_{conn}}

is equivalently a choice HC (X)H \in C^\infty(X) of a smooth function and sends

t(X exp(t{H,}) X exp(iS t) BU(1) conn), t \;\; \mapsto \;\; \left( \array{ X &&\stackrel{\exp(t \{H,-\})}{\longrightarrow}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\mathrlap{\exp(i S_t)}}& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}U(1)_{conn} } \right) \,,

where

  1. exp(t{H,})\exp(t \{H,-\}) is the Hamiltonian flow induced by HH;

  2. S t= 0 tLS_t = \int_0^t L is the Hamilton-Jacobi action functional, the integral of the Lagrangian of HH, hence of its Legendre transform.

(see Schreiber 13).

It is now clear how to pass from this to local prequantum field theory of higher dimension.

Let now more generally

HSmoothGrpdFunc(SuperMfd op,KanCplx)[{stalkwisehomotopyequivalences} 1] \mathbf{H} \coloneqq Smooth\infty Grpd \coloneqq Func(SuperMfd^{op}, KanCplx)[\{stalkwise\;homotopy\;equivalences\}^{-1}]

be the homotopy theory obtained from the category of Kan complex-valued presheaves on the category of all supermanifolds by universally turning stalkwise homotopy equivalences into actual homotopy equivalences.

We say that this is the (∞,1)-topos of smooth super ∞-groupoids/_supergeometric moduli ∞-stacks_.

Let

Corr n(H)(,n)Cat Corr_n(\mathbf{H}) \in (\infty,n)Cat

be the (∞,n)-category of nn-fold correspondences in H\mathbf{H}. This is a symmetric monoidal (∞,n)-category under the objectwise Cartesian product in H\mathbf{H}.

Smooth∞Grpd has the special property that it is cohesive in that it is equipped with an adjoint quadruple of adjoint (∞,1)-functors

HcoDiscΓDiscΠGrpd \mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd

which induce an adjoint triple of idempotent (∞,1)-monads/comonads

(Π):HH \left( \Pi \dashv \flat \dashv \sharp \right) \;\colon\; \mathbf{H} \longrightarrow \mathbf{H}

with Π\Pi product-preserving, called

Here the shape modality Π\Pi sends a simplicial manifold to the homotopy type of the fat geometric realization of the underlying simplicial topological space, hence in particular sends a smooth manifold to its homotopy type.

Write Bord nBord_n for the (∞,n)-category of framed n-dimensional cobordisms.

Proposition

A monoidal (∞,n)-functor

Fields:Bord nCorr n(H) \mathbf{Fields} \;\colon\; Bord_n \longrightarrow Corr_n(\mathbf{H})

is equivalently a choice of object FieldsH\mathbf{Fields} \in \mathbf{H}. It sends a cobordism Σ\Sigma to the internal hom of its shape into the higher moduli stack Fields\mathbf{Fields}:

( Σ Σ in Σ out)( [ΠΣ,Fields] ()| Σ in ()| Σ out [Π(Σ in),Fields] [Π(Σ out),Fields]). \left( \array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} && && \Sigma_{out} } \right) \;\; \mapsto \;\; \left( \array{ && [\Pi\Sigma, \mathbf{Fields}] \\ & {}^{(-)|_{\Sigma_{in}}}\swarrow && \searrow^{(-)|_{\Sigma_{out}}} \\ [\Pi(\Sigma_{in}), \mathbf{Fields}] && && [\Pi(\Sigma_{out}), \mathbf{Fields}] } \right) \,.

(lpqft)

Proposition

Under the Dold-Kan correspondence

DK:ChainComplexesSimplicialAbelianGroupsforgetKanComplexes DK \colon ChainComplexes \stackrel{\simeq}{\longrightarrow} SimplicialAbelianGroups \stackrel{forget}{\longrightarrow} KanComplexes

we have for all nn \in \mathbb{N} an equivalence

B n+1U(1)DK(U̲(1)dΩ 1dΩ 2ddΩ cl n+1) \flat \mathbf{B}^{n+1}U(1) \simeq DK \left( \underline{U}(1) \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^2 \stackrel{\mathbf{d}}{\longrightarrow} \cdots \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{n+1}_{cl} \right)

in H\mathbf{H}.

Example

Consider the induced canonical inclusion

Ω n+1B n+1U(1). \mathbf{\Omega}^{n+1} \longrightarrow \flat \mathbf{B}^{n+1}U(1) \,.

By the above we may regard this as an action functional for an (n+1)(n+1)-dimensional prequantum field theory with moduli stack of fields being Ω cl n+1\mathbf{\Omega}^{n+1}_{cl}. As such we denote it

Ω cl n+1 exp(iS tYM) B n+1U(1), \array{ \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \flat \mathbf{B}^{n+1}U(1) } \,,

where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.

Proposition

monoidal (∞,n)-functors

Bord n exp(iS) Corr n(H /B n+1U(1)) Fields Corr n(H) \array{ Bord_n &\stackrel{\exp(i S)}{\longrightarrow}& Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \\ & {}_{\mathbf{Fields}} \searrow & \downarrow \\ && Corr_n(\mathbf{H}) }

are equivalent to objects

(Fields exp(iS) B n+1U(1))H /B n+1U(1)Corr n(H /B n+1U(1)). \left( \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \flat \mathbf{B}^{n+1}U(1) } \right) \;\;\; \in \;\;\; \mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)} \hookrightarrow Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \,.

This sends the dual point to exp(iS)\exp(- i S) and sends the kk-sphere to the transgression of exp(iS)\exp(i S) to the mapping space [S k,Fields][S^k , \mathbf{Fields}].

(lpqft)

Example

Consider the induced canonical inclusion

Ω n+1B n+1U(1). \mathbf{\Omega}^{n+1} \longrightarrow \flat \mathbf{B}^{n+1}U(1) \,.

By the above we may regard this as an action functional for an (n+1)(n+1)-dimensional prequantum field theory with moduli stack of fields being Ω cl n+1\mathbf{\Omega}^{n+1}_{cl}. As such we denote it

Ω cl n+1 exp(iS tYM) B n+1U(1), \array{ \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow^{\mathrlap{\exp(i S_{tYM})}} \\ \flat \mathbf{B}^{n+1}U(1) } \,,

where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.

Observe that by the cobordism hypothesis Bord nBord_n is the free symmetric monoidal (∞,n)-category with fully dualizable objects generated from a single object *\ast.

Bord nFreeSMwD({*}). Bord_n \simeq FreeSMwD(\{\ast\}) \,.

Let then

Bord n FreeSMwD({*}) Bord_n^{\partial} \coloneqq FreeSMwD(\{\emptyset \longrightarrow \ast\})

the free symmetric monoidal (∞,n)-category with fully dualizable objects generated from a single object *\ast and a single morphism *\emptyset \longrightarrow \ast from the tensor unit to the generating object. By the boundary field theory/defect version of the cobordism hypothesis, this is equivalently the (∞,n)-category of cobordisms with possibly a boundary component of codimension (n1)(n-1).

Hence a boundary field theory is

Bord n exp(iS) Corr n(H /B n+1U(1)) Fields Corr n(H) \array{ Bord_n^\partial &\stackrel{\exp(i S)}{\longrightarrow}& Corr_n(\mathbf{H}_{/\flat \mathbf{B}^{n+1}U(1)}) \\ & {}_{\mathbf{Fields}^\partial} \searrow & \downarrow \\ && Corr_n(\mathbf{H}) }
Remark

A boundary field theory as above is equivalently a diagram in H\mathbf{H} of the form

Fields bdr * Fields exp(iS) B n+1U(1). \array{ && \mathbf{Fields}_{bdr} \\ & \swarrow && \searrow \\ \ast && \swArrow && \mathbf{Fields} \\ & \searrow && \swarrow_{\mathrlap{\exp(i S)}} \\ && \flat \mathbf{B}^{n+1}U(1) } \,.
Proposition

The universal boundary condition for the universal higher topological Yang-Mills theory of example 1 is the higher moduli stack B nU(1) conn\mathbf{B}^n U(1)_{conn} of circle n-bundle with connection, hence a general boundary condition for this higher topological Yang-Mills theory is a ∞-Chern-Simons theory].

The ∞-Wess-Zumino-Witten theory that we are after are boundaries of these boundary field theories, hence “corner field theories” (Sati 11, lpqft) of the higher universal topological Yang-Mills theory. This we turn to now.

Remark

The earliest and the only rigorously understood example of the holographic principle is the AdS3-CFT2 and CS-WZW correspondence between the WZW model on a Lie group GG and 3d GG-Chern-Simons theory.

In (Witten 98) it is argued that all examples of the AdS-CFT duality are governed by the higher Chern-Simons theory terms in the supergravity Lagrangian on one side of the correspondence, hence that the corresponding [[conformal field theories] are higher dimensional analogs of the traditional WZW model: that they are “∞-Wess-Zumino-Witten theory”-type models.

In particular for AdS7-CFT6 this means that the 6d (2,0)-superconformal QFT on the M5-brane worldvolume should be a 6d-dimensional WZW model holographically related to the 7d Chern-Simons theory which appears when 11-dimensional supergravity is KK-reduced on a 4-sphere:

∞-Chern-Simons theory\leftarrowholographic principle\rightarrow∞-Wess-Zumino-Witten theory
3d Chern-Simons theory2d Wess-Zumino-Witten model
7d Chern-Simons theory from 11-dimensional supergravity6d (2,0)-superconformal QFT on M5-brane

In (Witten 96) this is argued, by geometric quantization after transgression to codimension 1, for the bosonic and abelian contribution in 7d Chern-Simons theory. (The subtle theta characteristic involved was later formalized in Hopkins-Singer 02.)

In order to formalize this in generality, one needs a general formalization of holography for local prequantum field theory as these. How are ∞-Wess-Zumino-Witten theory-models higher holographic boundaries of ∞-Chern-Simons theory? This we are dealing with at Super Gerbes.

Motivic quantization of twisted fields

So far we have considered configuration spaces of fields, refined to smooth moduli ∞-stacks. The next step is to consider aspects of the quantization of these fields, at least as an effective quantum field theory (the full string theory being the corresponding UV-completion).

By the holographic principle and specifically by AdS-CFT duality, various of the twisted field configurations considered above participate either in higher dimensional Chern-Simons theory or in the corresponding self-dual higher gauge theory.

For instance the supergravity C-field, after compactification to dimension 7 in the context of AdS7-CFT6, has a topological action functional given by the secondary intersection pairing 7d Chern-Simons theory (or in fact, if quantum corrections are taken into account, a generalization of that to StringString-2-form fields FSSb).

The geometric quantization of these higher CS theories yields canonical states in codimension 1, which by AdS-CFT are interpreted as parts of the partition function of self-dual higher gauge fields.

This is described at motivic quantization (Nuiten 13).

Linearization by twisted EE-cohomology spectra

Before actually quantizing a local prequantum field theory [Fields exp(iS) B nU(1) conn]\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \mathbf{B}^n U(1)_{conn}}\right] as above, we choose linear coefficients, given by

  1. a choice of ground E-∞ ring EE

    (playing the role of the complex numbers in plain quantum mechanics);

  2. a choice of ∞-group homomorphism

    ρ:B n1U(1)GL 1(E) \rho \;\colon\; \mathbf{B}^{n-1}U(1) \longrightarrow GL_1(E)

    from the ∞-group of phases to the ∞-group of units of EE, hence an ∞-representation of the circle n-group on EE

    (playing the role of the canonical U(1) ×U(1) \hookrightarrow \mathbb{C}^\times in plain quantum mechanics).

Then for XB nU(1)X \longrightarrow \mathbf{B}^n U(1) modulating a circle n-bundle on XX, the composite

XB nU(1)ρBGL 1(E)EMod X \longrightarrow \mathbf{B}^n U(1) \stackrel{\rho}{\longrightarrow} B GL_1(E) \longrightarrow E Mod

modulates the associated ∞-bundle, which is an EE-(∞,1)-module bundle.

Specifically, given the higher prequantum bundle exp(iS):FieldsB nU(1) conn\exp(i S) \;\colon\; \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn} as above, the composite

χ:Fieldsexp(iS)B nU(1)ρBGL 1(E)Pic(E)EMod \chi \; \colon \; \mathbf{Fields} \stackrel{\exp(i S)}{\longrightarrow} \mathbf{B}^n U(1) \stackrel{\rho}{\longrightarrow} \mathbf{B} GL_1(E) \stackrel{}{\longrightarrow} Pic(E) \longrightarrow E Mod

modulates the associated higher prequantum E-line bundle.

A section of χ\chi is a higher wavefunction, hence a higher quantum state.

(At this point this looks un-polarized, but in fact we will see in the next section that the notion of polarization in higher prequantum geometry is automatic, but appears in a holographic/boundary field theory way in codimension (n1)(n-1) instead here in codimension nn.)

Accordingly, the space of sections of χ\chi is the higher space of quantum states in codimension 0.

If XX is a discrete ∞-groupoid then the space of sections has a particularly nice description, on which we focus for a bit:

The space of co-sections is the (∞,1)-colimit

E +χ(X)limχEMod. E_{\bullet + \chi}(X) \; \coloneqq\; \underset{\to}{\lim} \chi \; \in E Mod \,.

This is also known as the χ\chi-twisted EE-Thom spectrum of XX (Ando-Blumberg-Gepner 10).

  • a map EE +χ(X)E \to E_{\bullet + \chi}(X) is a cycle in χ\chi-twisted EE-generalized homology of XX;

  • a map E +χ(X)EE_{\bullet + \chi}(X) \to E is a cocycle in χ\chi-twisted EE-generalized cohomology of XX

    Hence we write

    E +χ(X)[E +χ(X),E]. E^{\bullet + \chi}\left(X\right) \coloneqq \left[ E_{\bullet + \chi}\left(X\right), \, E \right] \,.

Generally, for χ i:X iEMod\chi_i \colon X_i \to E Mod two EE-(∞,1)-module bundles over two spaces, a map

E +χ 1(X 1)E +χ 2(X 2) E_{\bullet + \chi_1}(X_1) \longrightarrow E_{\bullet + \chi_2}(X_2)

is a cocycle in (χ 1,χ 2)(\chi_1, \chi_2)-twisted bivariant EE-cohomology.

Now given a local action functional on a space of trajectories, hence a correspondence as above, this induces an integral kernel for linear maps between sections of higher prequantum line bundles:

Fields traj Fields in ρ(ξ) Fields out χ in χ out EMod Fields traj Fields in ξ Fields out exp(iS in) exp(iS out) B nU(1) BGL 1(E) Pic(E) EMod \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \mathbf{Fields}_{in} && \swArrow_{\rho(\xi)} && \mathbf{Fields}_{out} \\ & {}_{\mathllap{\chi_{in}}}\searrow && \swarrow_{\mathrlap{\chi_{out}}} \\ && E Mod } \;\;\; \coloneqq \;\;\; \array{ && \mathbf{Fields}_{traj} \\ & \swarrow && \searrow \\ \mathbf{Fields}_{in} && \swArrow_\xi && \mathbf{Fields}_{out} \\ & {}_{\mathllap{\exp(i S_{in})}}\searrow && \swarrow_{\mathrlap{\exp(i S_{out})}} \\ && \mathbf{B}^n U(1) \\ && \downarrow \\ && B GL_1(E) \\ && \downarrow \\ && Pic(E) \\ && \downarrow \\ && E Mod }

This is the integral kernel induced by the action functional, and acting on spaces of sections of the higher prequantum line bundle.

The linear map induced by these higher integral kernels is to be the quantum propagator. This we come to in the next section.

Notice that forming co-sections constitutes an (∞,1)-functor

E +()():H /B nU(1) connEMod. E_{\bullet + (-)}(-) \;\colon\; \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \longrightarrow E Mod \,.

Therefore forming co-sections sends an integral kernel as above to a correspondence of EE-(∞,1)-modules:

Fields traj i in i out Fields in Fields out χ in χ out EMod E +χ in(Fields in) (i in) * E +i in *χ in(Fields traj) E +i out *χ out(Fields traj) (i out) * E +χ out(Fields out). \array{ && \mathbf{Fields}_{traj} \\ & {}^{i_{in}}\swarrow && \searrow^{\mathrlap{i_{out}}} \\ \mathbf{Fields}_{in} & & \swArrow_{\simeq} & & \mathbf{Fields}_{out} \\ & {}_{\mathllap{\chi_{in}}}\searrow && \swarrow_{\mathrlap{\chi_{out}}} \\ && E Mod \\ \, \\ \, \\ E_{\bullet + \chi_{in}}(\mathbf{Fields}_{in}) & \stackrel{(i_{in})_\ast}{\longleftarrow}& \array{ E_{\bullet + i_{in}^\ast\chi_{in}}(\mathbf{Fields}_{traj}) \\ \simeq \\ E_{\bullet + i_{out}^\ast\chi_{out}}(\mathbf{Fields}_{traj}) } & \stackrel{(i_{out})^\ast}{\longrightarrow}& E_{\bullet + \chi_{out}}(\mathbf{Fields}_{out}) } \,.

The actual quantization/path integral as a pull-push transform map now consists in forming dual morphism in EModE Mod such as to turn one of the projections of such a correspondence arround a produce a quantum propagator

E +χ in(Fields in)E +χ out(Fields out) E^{\bullet + \chi_{in}}(\mathbf{Fields}_{in}) \longrightarrow E^{\bullet + \chi_{out}}(\mathbf{Fields}_{out})

that maps the incoming quantum states/wavefunctions to the outgoing ones.

Cohomological quantization by pull-push

What we need now for quantization is a path integral map that adds up the values of the action functional over the space of trajectories, a functor of the form

():Corr n(H /B nU(1)) (EMod n) \int (-) \;\; \colon \;\; Corr_n\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to (E Mod^{\Box^n})^\otimes

As such this will in general only exist for ∞-Dijkgraaf-Witten theory where Fields\mathbf{Fields} is a discrete ∞-groupoid and hence has a “counting measure”. This case has been considered in (Freed-Hopkins-Lurie-Teleman 09, Morton 10).

In the general case the path integral requires that we choose a suitable measure/orientation on the spaces of fields. We see below what this means, for the moment we just write

Corr n or(H /B nU(1)) Corr^{or}_n(\mathbf{H}_{/\mathbf{B}^n U(1)})^\otimes

(i.e. with an () or{(-)}^{or}-superscript) as a mnemonic for a suitable (∞,n)-category of suitably oriented/measured spaces of fields with action functional. Then we may consider lifts of the action functional to measure-valued action functionals

exp(iS)dμ:Corr n or(H /B nU(1)) (EMod n) . \exp(i S) \, d\mu \;\colon\; Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to \left( E Mod^{\Box^n} \right)^\otimes \,.

A path integral is then to be a monoidal functor of the form

():Corr n or(H /B nU(1)) (EMod n) . \int(-) \;\colon\; Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes \to \left( E Mod^{\Box^n} \right)^\otimes \,.

This we discuss now below. Once we have such a path integral functor, the quantization process is its composition with the given prequantum field theory exp(iS)dμ\exp(i S) \, d \mu to obtain the genuine quantized quantum field theory:

ϕFieldsexp(iS(ϕ))dμ(ϕ) : Bord n exp(iS)dμ Corr n or(H /B nU(1)) () (EMod n) Fields Corr n(H) . \array{ \underset{\phi \in \mathbf{Fields}}{\int} \exp(i S(\phi)) \, d \mu(\phi) & \colon & Bord_n^\otimes &\stackrel{\exp(i S)\, d\mu}{\to}& Corr_n^{or}\left(\mathbf{H}_{/\mathbf{B}^n U(1)}\right)^\otimes &\stackrel{\int (-) }{\to}& \left( E Mod^{\Box^n} \right)^\otimes \\ && & {}_{\mathllap{Fields}}\searrow & \downarrow \\ && && Corr_n\left(\mathbf{H}\right)^\otimes } \,.

We realize this now by fiber integration in generalized cohomology.

While traditionally the definition of path integral is notoriously elusive, here we make use of general abstract but basic facts of higher linear algebra in a tensor (∞,1)-category (a stable and symmetric monoidal (∞,1)-category): the simple basic idea is that

Cohomological integration

  1. Fiber integration of EE-modules along a map is forming the dual morphisms of pulling back EE-modules.

  2. The choice of measure against which one integrates is the choice of identification of dual objects.

More in detail, given a monoidal category 𝒞 \mathcal{C}^\otimes and given a morphism

f:V 1V 2 f \;\colon\; V_1 \to V_2

in 𝒞\mathcal{C}, an fiber integration/push-forward/index map is just

This allows in total to have a morphism between the same objects, but in the opposite direction

f !:V 2V 2 f V 1 V 1. f^! \;\colon\; V_2 \stackrel{\;\simeq\;}{\to} V_2^\vee \stackrel{\; f^\vee \;}{\to} V_1^\vee \stackrel{\; \simeq \;}{\to} V_1 \,.

That this is also the mechanism of fiber integration in generalized cohomology is almost explicit in the literature (Alexander-Whitehead-Atiyah duality), if maybe not fully clearly so. The statement is discussed explicitly in (Nuiten 13, section 4.1).

First, the basic example to keep in mind of is integration in ordinary cohomology. Write E=HR=HE = H R = H \mathbb{C} for the Eilenberg-MacLane spectrum of the complex numbers. Then for XX a manifold, the mapping spectrum

HR (X)[X,HR] H R^\bullet(X) \coloneqq [X,H R]

is the ordinary cohomology of XX, its dual the ordinary homology, with coefficients in RR.

For XX a closed manifold, Poincaré duality asserts that HR (X)HRModH R^\bullet(X) \in H R Mod is essentially a self-dual object, except for a shift in degree: a choice of orientation of XX induces an equivalence

HR (X)PD X(HR +dim(X)(X)) HR +dim(X)(X). H R^\bullet\left(X\right) \underoverset{\simeq}{\;\; PD_X\;\;}{\to} \left( H R^{\bullet+ dim(X)}\left(X\right)\right)^\vee \simeq H R_{\bullet + dim(X)}\left(X\right) \,.

Using this, for f:XYf \colon X \to Y a map of closed manifolds of dimension dd, a compatible choice of orientation of both XX and YY induces from the canonical push-forward map f *f_\ast on homology the Umkehr map/push-forward map on cohomology, by the composition

f != f:HR (X)PD XHR +dim(X)(X)f *HR +dim(X)(Y)PD Y 1HR n(dim(X)dim(Y))(Y). f^! = \int_f \;\colon\; H R^\bullet(X) \underoverset{\simeq}{\;PD_X\;}{\to} H R_{\bullet + dim(X)}(X) \stackrel{\; f_\ast \;}{\to} H R_{\bullet + dim(X)}(Y) \underoverset{\simeq}{\;PD_Y^{-1}\;}{\to} H R^{n-(dim(X)-dim(Y))}(Y) \,.

This is ordinary integration: if XX and YY are smooth manifolds, then H (X)H \mathbb{R}^\bullet(X) is modeled by differential forms on XX, PD XPD_X is given by a choice of volume form and f != ff^! = \int_{f} is ordinary integration of differential forms.

The shift in degree here seems to somewhat break the simple pattern. In fact this is not so, if only we realize that since we are working over spaces XX, we should use a relative/fiberwise point of view and regard not duality in EModE Mod itself, but in the functor categories Func(X,EMod)Func(X, E Mod), which is fiberwise duality in EModE Mod.

Accordingly, given an EE-(∞,1)-module bundle

χ:XEMod \chi \;\colon \; X \to E Mod

we form not just the mapping space E (X)=[X,E]E^\bullet(X) = [X, E] as above, but form the space of sections of this bundle, which we write:

E +χ(X)Γ X(χ) E^{\bullet + \chi}(X) \coloneqq \Gamma_X(\chi)

Here for XX a discrete ∞-groupoid

Γ X(χ)[limχ,E]. \Gamma_X(\chi) \coloneqq [\underset{\to}{\lim} \chi, E] \,.

Consider now a morphism

X f Y f *χ χ EMod \array{ X && \stackrel{f}{\to} && Y \\ & {}_{\mathllap{f^\ast \chi}}\searrow &\swArrow& \swarrow_{\mathrlap{\chi}} \\ && E Mod }

along which we want to integrate, whith χ\chi invertible in Func(Y,EMod)Func(Y, E Mod): (χ ) χ\left(\chi^\vee\right)^\vee \simeq \chi. (χ ) χ\left(\chi^\vee\right)^\vee \simeq \chi.

Observe that we have the pair of adjoint triples of left/right Kan extensions and colimits/limits

X f Y p * Func(X,EMod) f *f *f ! Func(Y,EMod) p *p *p ! EMod. \array{ X &\stackrel{f}{\to}& Y &\stackrel{p}{\to}& \ast \\ \\ Func(X, E Mod) &\stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\to}}} & Func(Y, E Mod) &\stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^\ast}{\leftarrow}}{\underset{p_\ast}{\to}}} & E Mod } \,.

Notice that f *f^\ast preserves duals, but f !f_! may not.

If f !f *χ f_! f^\ast \chi^\vee is a dualizable object, say that a choice of twisted orientation of ff in χ\chi-twisted cohomology is a choice of β:XEMod\beta \colon X \to E Mod together with a choice of a equivalence (if such exists) of the form

PD:(f !f *χ ) f !(f *χ+β), PD \;\colon \; \left( f_! f^\ast \chi^\vee \right)^\vee \simeq f_!\left( f^\ast \chi + \beta \right) \,,

hence a choice of correction of f !f_! preserving the duality of f *χf^\ast \chi.

Then the (f !f *)(f_! \dashv f^\ast) counit

f !f *χ χ f_! f^\ast \chi^\vee \to \chi^\vee

induces the dual morphism

χ(f !f *χ ) PDf !(f *χ+β) \chi \longrightarrow \left(f_! f^\ast \chi^\vee \right)^\vee \underoverset{\simeq}{PD}{\longrightarrow} f_!\left( f^\ast \chi + \beta \right)

and under [p !(),E]\left[ p_! \left( - \right), E \right] this becomes

[p !f !(f *χ+β),E][p !χ,E] \left[p_! f_! \left(f^\ast \chi + \beta\right), E\right] \longrightarrow \left[p_! \chi , E\right]

which is

f:E +f *χ+β(X)E +χ(Y). \int_f \;\colon \; E^{\bullet + f^\ast \chi + \beta}(X) \longrightarrow E^{\bullet + \chi}(Y) \,.

This we may call the the twisted fiber integration along ff in EE-cohomology, or the twisted EE-index map of ff, induced by (β,PD)(\beta, PD). If β=0\beta = 0 then we call PDPD anorientation_ of ff in χ\chi-twisted cohomology.

Notice that

General theory

We survey here some key aspects of a general theory of geometric twisted differential cohomology, following (DCCT), in which the above examples find a formal home. This is meant as a reference for readers of the Examples-section who wish to see pointers to formal details.

\,

We base the formulation of physics/string theory on the foundations of homotopy type theory, interpreted in (∞,1)-toposes. This provides a nicely natural and expressive language for the purpose of twisted smooth cohomology in string theory.

The following table indicates the hierarchy of axioms? that we invoke, the fragments of theory that can be interpreted with these and the models that we need. Essentially all of the above discussion works in the model Smooth∞Grpd. A more encompassing treatment uses supergeometry and works in the model SmoothSuper∞Grpd.

axioms?syntaxsemanticstypical modelsexpressiveness
bare minimumhomotopy type theory(∞,1)-topos theory∞Grpd, Super∞Grpdcohomology, principal bundles, twisted cohomology, associated and twisted bundles
+locality+∞-connectednesscohesive homotopy type theorycohesive (∞,1)-toposSmooth∞Grpd, SmoothSuper∞Grpdgeometric realization, differential cohomology, Chern-Weil theory Chern-Simons theory, geometric quantization
+infinitesimalsdifferential homotopy type theorydifferential (∞,1)-toposSynthDiff∞Grpdde Rham space, jet bundle, étale groupoid, orbifold

Homotopy type theory

Traditionally a homotopy type is a topological space regarded up to weak homotopy equivalence, hence equivalently an ∞-groupoid. More generally, we think of parameterized homotopy types – of ∞-stacks or (∞,1)-sheaves – as geometric homotopy types. The collection of such forms an (∞,1)-topos H\mathbf{H}. One regards (∞,1)-topos theory as part of homotopy theory, and, more specifically, the internal language of H\mathbf{H} is a homotopy-type theory.

We discuss now some basic structures that are expressible in such bare homotopy-type theory. (The fundamentals are due to Rezk and Lurie, see Higher Topos Theory. We point out the perspective of twisted cohomology in slices and add some aspects about higher bundle theory from (NSS)).

Where useful, we indicate some of the discussion in formal homotopy type theory syntax, see HoTT methods for homotopy theorists for more along such lines.

Cohomology

A group object in an (∞,1)-topos is a groupal A-∞-homotopy type: an ∞-group.

By looping and delooping there is an equivalence

Grp(H)BΩH 1 */ Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}_{\geq 1}^{*/}

between group objects and pointed connected homotopy types.

Remark

If pt:BGpt : \mathbf{B}G is the essentially uniqe point of the connected type BG\mathbf{B}G, then the group type itself is simply

G(ptpt):Type, G \coloneqq \vdash (pt \simeq pt) : Type \,,

the type of auto-equivalences of ptpt in BG\mathbf{B}G.

For AA a group object which admits an nn-fold delooping B nA\mathbf{B}^n A and XHX \in \mathbf{H} any object, we write

  • H(X,B nA)\mathbf{H}(X, \mathbf{B}^n A) for the space of degree-nn AA-cocycles c:XAc : X \to A;

  • H n(X,A)π 0H(X,B nA)H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X, \mathbf{B}^n A) for the degree-nn AA-cohomology of XX.

Principal bundles

For GGrp(H)G \in Grp(\mathbf{H}), GG-cocycles on XX have an equivalent geometric interpretation as GG-principal ∞-bundles: these are types

P X \array{ P \\ \downarrow \\ X }

over XX equipped with a GG-action ρ:P×GP\rho : P \times G \to P over XX such that the ∞-quotient of the action is XX.

Theorem There is an equivalence between the ∞-groupoid of GG-principal bundles over XX, and the cocycle \infty-groupoid of GG-cohomology over XX.

H(X,BG)fiblimGBund(X) \mathbf{H}(X, \mathbf{B}G) \stackrel{\overset{\underset{\to}{\lim}}{\leftarrow}}{\underoverset{fib}{\simeq}{\to}} G Bund(X)

From left to righr the equivalence is established by sending a cocycle XBGX\to \mathbf{B}G to its homotopy fiber.

(NSS)

Remark

With GG the type as in remark 6, the principal bundle corresponding to a coycle ϕ:XBG\phi : X \to \mathbf{B}G is

Px:X(ϕ(x)pt):Type P \; \coloneqq \; x : X \vdash ( \phi(x) \simeq pt ) : Type

And the action is

ρ:P×GG ((x:X;p:(ϕ(x)pt)),g:(ptpt))(x:X;gϕ) \begin{aligned} & \rho : P \times G \to G \\ & \;\;\coloneqq (( x : X; p : (\phi(x) \simeq pt)), g : (pt \simeq pt) ) \mapsto (x : X; g \circ \phi) \end{aligned}

Twisted cohomology

For H\mathbf{H} an (∞,1)-topos and BGH\mathbf{B}G \in \mathbf{H}, the collection of morphisms into BG\mathbf{B}G is the slice (∞,1)-topos H /BG\mathbf{H}_{/ \mathbf{B}G}.

The cohomology in H /BG\mathbf{H}_{/\mathbf{B}G} is naturally interpreted as follows

  1. a local coefficient object c:EBG\mathbf{c} : E \to \mathbf{B}G is a universal bundle of local coefficients;

  2. a domain object ϕ:XBG\phi : X \to \mathbf{B}G is a twisting bundle;

  3. a cocycle

    X σ E ϕ c BG \array{ X &&\stackrel{\sigma}{\to}&& E \\ & {}_{\mathllap{\phi}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }

    is equivalently a section of the (∞,1)-pullback bundle ϕ *EX\phi^* E \to X;

    ϕ *E E σ c X id X ϕ BG \array{ && \phi^* E &\to & E \\ &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow^{\mathbf{c}} \\ X &\stackrel{id}{\to}& X &\stackrel{\phi}{\to}& \mathbf{B}G }
  • the cohomology

    cStruc ϕ(X)c(ϕ)H /BG(ϕ,c) \mathbf{c}Struc_{\phi}(X) \coloneqq \mathbf{c}(\phi) \coloneqq \mathbf{H}_{/\mathbf{B}G}(\phi, \mathbf{c})

    is the ϕ\phi-twisted cohomology of XX (with local coefficients in the homotopy fiber FF of c\mathbf{c}).

Remark

In the syntax of the theory the type of twisted cocycles is

[ϕ,c] bBG(X(b)E(b)):Type= x:XE(ϕ(x)):Type [\phi, \mathbf{c}] \coloneqq \vdash \prod_{b \in \mathbf{B}G} (X(b) \to E(b)) : Type \;\; = \;\; \vdash \prod_{x : X} E(\phi(x)) : Type

(see here).

While on the right this expresses the collection of sections of the pullback bundle, the left hand side expresses explicitly a BG\mathbf{B}G-parameterized collection of cocycles X(b)E(b)X(b) \to E(b).

Remark

Cocycles in twisted cohomology relative to a local coefficient bundle c:EBG\mathbf{c} : E \to \mathbf{B}G do not pull back along morphisms in H\mathbf{H} (unless GG is trivial), but do pull back along morphisms in H\mathbf{H} that are lifted to morphisms in the slice H /BG\mathbf{H}_{/ \mathbf{B}G}.

Associated and twisted bundles

For

BA BG^ c BG \array{ \mathbf{B}A &\to& \mathbf{B}\hat G \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }

a universal local coefficient bundle in H\mathbf{H} and ϕ:XBG\phi : X \to \mathbf{B}G a twist and σ:XBG^\sigma : X \to \mathbf{B}\hat G a section, hence a cocycle in ϕ\phi-twisted cohomology, the corresponding geometric object is the twisted ∞-bundle σ˜\tilde \sigma on the total space PP of ϕ\phi

P˜ * P σ˜ BA * X σ BG^ c BG. \array{ \tilde P &\to& * \\ \downarrow && \downarrow \\ P &\stackrel{\tilde \sigma}{\to}& \mathbf{B}A &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{\sigma}{\to}& \mathbf{B}\hat G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G } \,.

(NSS)

(…)

Cohesive homotopy type theory

In general the geometry encoded by an (∞,1)-topos can be exotic. Two extra axioms ensure that it is modeled locally on \infty-connected geometrical archetypes, such as for instance on open disks for Euclidean-topological geometry and smooth open disks for smooth geometry. Following Lawvere, we call this refinement cohesive homotopy type theory interpreted in cohesive (∞,1)-toposes.

Geometric realization

A cohesive (∞,1)-topos is in particular equipped with a left derived adjoint Π\Pi to the locally constant ∞-stack-functor DiscDisc

(ΠDisc):HDiscΠGrpd. (\Pi \dashv Disc) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\underset{Disc}{\hookleftarrow}} \infty Grpd \,.

We may think of Π\Pi as being the functor that sends a smooth ∞-groupoid XX to its fundamental path ∞-groupoid

Under the identification (see homotopy hypothesis theorem)

Grpd||Top \infty Grpd \stackrel{{\vert-\vert}}{\to} Top

this identifies with geometric realization of geometric homotopy types / ∞-stacks.

We say that a lift of a diagram in ∞Grpd through Π\Pi is a geometric refinement of the diagram.

Differential cohomology

Write ΠDiscΠ\mathbf{\Pi} \coloneqq Disc \circ \Pi; and DiscΓ\flat \coloneqq Disc \circ \Gamma.

For GG an ∞-group, write dRBG*× BGBG\flat_{dR} \mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G: the de Rham coefficient object for BG\mathbf{B}G.

There is a canonical morphism θ:G dRBG\theta : G \to \flat_{dR} \mathbf{B}G: this identifies as the canonical Maurer-Cartan form on the ∞-group GG.

For G=B nU(1)G = \mathbf{B}^n U(1) the circle (n+1)-group we call curv B nU(1):B nU(1) dRB n+1U(1)curv \coloneqq_{\mathbf{B}^n U(1)} : \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1) the universal curvature class in degree (n+1)(n+1).

The curvcurv-twisted cohomology in H\mathbf{H} identifies with ordinary differential cohomology.

B nU(1) conn Ω cl n+1() B nU(1) curv B n+1U(1) conn \array{ \mathbf{B}^n U(1)_{conn} &\to& \Omega^{n+1}_{cl}(-) \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{curv}{\to}& \mathbf{B}^{n+1} U(1)_{conn} }

Chern-Weil theory

(…)

∞-Chern-Weil theory introduction

Differential homotopy-type theory

(…)

infinitesimals

de Rham stack

(…)

Étale stacks / orbifolds

(…)

Related entries

An entry with discussion of and list of examples of twisted differential structures is

References

The string-theoretic aspects of the above discussion owe a lot to Hisham Sati, who has pointed out the appearance of various twisted structures in string theory, notably in the series of articles

The notion of twisted cohomology by sections of twisting coeffcient \infty-bundles used here is similar to that in

but considered in the non-stable context of nonabelian cohomology and refined from bare homotopy types to geometric homotopy types.

The fundamental observation that background gauge fields in string theory are modeled by (twisted) differential cohomology goes back to

and literature referenced there. For this classical literature, notably on the example of twisted and differential K-theory, as well as on orientifolds, see the lists of references provided at these entries, notably

The 7d Chern-Simons theory that the supergravity C-field participates in, the relation of the flux quantization to the corresponding holographic description of the self-dual field on the M5-brane has been discussed in

A precise mathematical formulation of the proposal made there is given in

in terms of quadratic refinement of secondary intersection pairing via differential integral Wu structures. This also lays the mathematical foundation of much of differential cohomology.

The suggestion that it is the ∞-stack (∞,1)-topos over the site of smooth manifolds which is the right context for studying the twisted differential smooth cohomology in string theory was made in

The smooth \infty-stack refinements of these structures, as discussed above, have been developed in articles including the following

A general theory of such smooth homotopy-types is laid out in

The observation of the tangent (infinity,1)-topos is due to

\,

Section A) above originates in notes for an introductory lecture:

Closely related lectures at the same program included

Later, some special cases of the general notion of twisted fields considered above are being called relative fields in

as discussed above in the section Relative fields .

Section B) originates in notes for a talk

Revised on October 14, 2013 21:18:49 by Urs Schreiber (80.237.234.148)