Schreiber Background fields in twisted differential nonabelian cohomology

The following are notes from an early talk on the material developed at differential cohomology in a cohesive topos. Related later lectures notes include twisted smooth cohomology in string theory and geometry of physics.

---

This are notes for a talk given at Oberwolfach Workshop, June 2009 – Strings, Fields, Topology

• a version appeared as

  Oberwolfach Report No.28/2009 (pdf)

• based on joint work with John Baez, Thomas Nikolaus, Hisham Sati, Zoran Škoda, Jim Stasheff, Danny Stevenson, Konrad Waldorf

• the Oberwolfach “wall” announcement of the talk: Background fields in differential twisted nonabelian cohomology

• for more background, and more references see

  • Differential Nonabelian Cohomology

• for the following, recall

  • in particular the previous talk by Dan Freed and in general the article

    • Dan Freed, Dirac charge quantization and Generalized Differential Cohomology (arXiv)

Plan

• 1) Motivation

  • extended QFT wants functorial description of sigma-model background fields

  • (QFT as $n$-functor) $\stackrel{quantize}{\leftarrow|}$ (background field/classical action as $n$-functor)

  • recently: huge progress on formalizing the left hand side:

    • see (Mike Hopkins and) Jacob Lurie: On the Classification of Topological Field Theories

  • goal: formalize the right side! – that’s the topic here

  • second goal: formalize the quantization step in the functorial QFT picture – this step we shall not be concerned with here, for more on that see instead

    • Quantization as a Kan extension (by Johan Alm)

    • Nonabelian cocycles and their sigma model QFTs

    • and also section 3 of

      • Freed, Hopkins, Lurie, Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv)

• 2) Smooth nonabelian cohomology

  • for our task, the required generalized smooth spaces are

    • “sufficiently general smooth spaces”,

    • namely smooth infinity-groupoids modeled as infinity-stacks on $Diff$

    • think: $(A \in smooth \infty Grpd, X \in Diff) \Rightarrow (A(X) := Hom(X,A) \in \infty Grpd)$

• 3) Differential nonabelian cohomology

  • to every smooth $\infty$-groupoid $X$ is associated a smooth $\infty$-groupoid $\Pi(X)$ of $k$-disk-shaped smooth paths of sorts in $X$, for all $k$

  • differential cohomology of $X$ is essentially (up to a twist, to be discussed) cohomology of $\Pi(X)$

  • “$\mathbf{H}_{diff}(X,A) = H(\Pi(X),A)$”

• 4) Twisted nonabelian cohomology

  • for $(A \to \hat B \to B)$ a fibration sequence of smooth $\infty$-groupoids and $c \in \mathbf{H}(X,B)$ there exists a natural notion of

  • $c$-twisted $A$-cohomology $\mathbf{H}^c(X,A)$

• 5) Examples

  • there are a bunch of examples that physicists have been thinking about for a long time, which still are awaiting a proper formalization

    • compare Freed/Distler/Moore’s recent work on understanding what a background for 2d SCFT really is;

  • here is a list of some examples that we shall have something to say about:

  • purely topological QFTs

    • $n$-dimensional Dijkgraaf-Witten theory : target space is the groupoid incarnation $\mathbf{B}G$ of a finite group $G$ and the background field $\nabla : \mathbf{B}G \to \mathbf{B}^n U(1)$ is a $U(1)$-valued group $n$-cocycle on $G$

    • $n$-dimensional Chern-Simons theory, which is “as above”, but with $G$ replaced by a Lie group, so that the background field becomes an $(n-1)$-bundle gerbe with connection

  • topological parts of “physical” QFTs

    • Yang-Mills gauge field

      • vector bundle with connection

      • to which the charged particle couples (electrons, quarks);

    • Kalb-Ramond field

      • bundle gerbe with connection

      • to which the string couples;

    • twisted Green-Schwarz Kalb-Ramond field

      • twisted nonabelian String-gerbe with connection

      • to which the heterotic string couples;

    • orientifold Kalb-Ramond field

      • twisted gerbe with connection

      • to which the type II string couples;

    • twisted dual Green-Schwarz Kalb-Ramond field

      • twisted nonabelian Fivebrane 5-gerbe with connection

      • to which the fundamental 5-brane (the magnetic dual of the string) couples

1) Motivation

• understand background fields for sigma-model QFTs structurally:

  • (here a “sigma-model” is a $(d+1)$-dimensional quantum field theory describing the dynamics of a $d$-dimensional object propagating through a target space $X$ and subject to forces exerted by a background field (like an electromagnetic field) )

• such that path integral quantization to extended QFTs can be understood structurally by extension

by this we mean: we want to make sense of diagrams roughly of the form

$$\array{ \Pi_n(X) &&\stackrel{\nabla}{\to}&& A \\ \downarrow &&{}^{\exp(i S(-))}\nearrow && \\ Bord_n(X) \\ \downarrow &&& \nearrow_{\int \exp(i S(-))} \\ Bord_n }$$

(notice we don’t try to make precise or even correct this diagram here, that’s another topic, here it just serves to motivate why we are first of all interested in making precise and correct the top horizontal morphism data, which is the topic here)

where the objects and arrows appearing here indicate the following structures

$$\array{ n-dim paths in X &&\stackrel{\nabla}{\to}&& coefficient such as nVect \\ \downarrow &&{}^{\exp(i S(-))}\nearrow && \\ n-dim bordisms in X \\ \downarrow &&& \nearrow_{\int \exp(i S(-))} \\ abstract n-dim bordisms }$$

and are supposed to have the following interpretation

$$\array{ target space &&\stackrel{\stackrel{diff. cocycle}{background field}}{\to}&& A \\ \downarrow &&{}^{action}\nearrow && \\ field space \\ \downarrow &&& \nearrow_{QFT} \\ worldvolume }$$

• the top horizontal morphism: parallel transport of something like a higher connection along disks in target space $X$;

  • $\nabla : (x \stackrel{\gamma}{\to} y) \mapsto (E_x \stackrel{tra(\gamma)}{\to} E_y)$

• the middle morphism: an extension of that transport over disks in $X$ to entire cobordisms in $X$; amounting to equipping the connection with a notion of higher traces such as to yield holonomy. This holonomy is the action functional of a topological sigma-model QFT

• the lowest morphism: some extension of the action functor from bordisms in $X$ to abstract bordisms, where it represents an FQFT.

goal

• identify where this diagram lives;

• work out the above examples

2) Smooth nonabelian cohomology

so now: work out where this diagram lives

2.1) toy example: topological cohomology

• first toy case of Dijkgraaf-Witten theory ($n$-dimensional)

• here background field is just a continuous map

  $$g : B G \to K(\mathbb{Z},n+1)$$

  in Top

• two background fields are gauge equivalent if these maps are homotopic

  $$([g] = [g']) \Leftrightarrow \exist \left( \array{ & \nearrow \searrow^{g} \\ B G &\Downarrow& K(\mathbb{Z}, n+1) \\ & \searrow \nearrow^{g} } \right)$$

• even simpler by passing to a combinatorial model for topological spaces: infinity-groupoids (Kan complexes)

  $$g : \mathbf{B} G \to \mathbf{B}^n U(1)$$

  • here

    • $\mathbf{B}G = \{\bullet \stackrel{g}{\to} \bullet|g \in G\}$ is groupoid with one object and the group $G$ as the set of morphisms

    • $\mathbf{B}^n U(1)$ is similarly the $n$-gtoupoid with nontrivial morphisms in degree $n$ given by the group $U(1)$

• in either case there is an (infinity,1)-category – an $\infty Grpd$ enriched category

  $$\mathbf{H} = Top$$

  (compactly generated weakly Hausdorff topological spaces) or equivalently

  $$\mathbf{H} = \infty Grpd$$

• and we simply have $\mathbf{H}(X,A) := \{ A-valued background fields on X \} = \{ A-valued cocycles on X \}$

• and $H(X,A) := \pi_0 \mathbf{H}(X,A) = \{ gauge equivalence classes of A-valued background fields on X \} = \{ cohomology classes of A-valued cocycles on X \}$

2.2) generalization: $(\infty,1)$-topos of $\infty$-stacks

• generalize to smooth cocycles

• $\Rightarrow$ choose an $(\infty,1)$-category $\mathbf{H}$ that behaves essentially like $Top$…

  • means: $\mathbf{H}$ is an (infinity,1)-topos

• but contains smooth generalized spaces

  • means: topological spaces/$\infty$-groupoids parameterized over $Diff$, i.e. (infinity,1)-sheaves on $Diff$, aka infinity-stacks

• think of an $\infty$-stack on $Diff$ as a smooth $\infty$-groupoid

  • (for supposedly useful background ideas on that see: motivation for sheaves, cohomology and higher stacks)

2.3) model: $\infty$-groupoid valued sheaves

• to actually work with this, we chose a convenient model that presents $\infty$-stacks on $Diff$.

• there is an old construction, dating back to the remarkable

  • Kenneth Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology (1973)

• to model $\infty$-stacks as ordinary sheaves

  $$A : Diff^{op} \to \infty Grpd$$

  with values in $\infty$-groupoids, using

  • a certain model structure on simplicial presheaves

  • or, more lightweight, the structure of a Brown category

• this essentially amounts to remembering those morphisms of smooth $\infty$-groupoids that behave like surjective equivalences

  • ($f: Y \to X$) is surjective equivalence of smooth $\infty$-groupoids precisely if when regarded as a morphism of sheaves it it restricts to a surjective equivalence of ordinary $\infty$-groupoids locally (“stalkwise”)

  • these are also called

    • acyclic fibrations (generally)

    • hypercovers (the usefully suggestive terminology in our context)

  • the $\infty$-morphisms then are modeled by anafunctors, i.e. morphisms out of surjective equivalences $((X \to A)\in \mathbf{H}(X,A)) := \left[ \array{ Y &\to & A \\ \downarrow \\ X } \right]$

  • and the full hom-$\infty$-groupoid is obtained by doing this for all possible hypercovers

    • $\mathbf{H}(X,A) \simeq \colim_{Y \to X} Sh(Y,A)$

• this procedure amounts to using rectified $\infty$-stacks: those which as functors on the site are ordinary strict functors

• Jacob Lurie in HTT shows in particular that rectified $\infty$-stacks are sufficient :

  • this model structure on simplicial presheaves is a presentation of the (hypercompletion of, but nerver mind) the (infinity,1)-category of infinity-stacks

  • one useful technical upshot is:

    • we can implicitly work with $\infty$-stacks while in practice just handling $\infty$-groupoid valued sheaves as in (BaSc ScWa I ScWa II, ScWa III MaPi I)

  • finding the full weakly equivalent infinity-stackification is not in general necessary

2.3) cocycles and cohomology in this context

• unwraping what the above anafunctors mean gives all the cocycle equations that you’ll ever see, in particular, group cocycles, bundle gerbes, etc.

• in particular it reproduces abelian sheaf cohomology as a special case

• notice that there are the following simpler structures inside $\infty Grpds$

  $$\array{ Ch_+(Ab) &\hookrightarrow& CrsCh &\simeq& Str \infty Grpd &\hookrightarrow& \infty Grpd }$$

  • chain complexes of abelian groups in positive degree

  • inside crossed complexes of groupoids

  • which are equivalent to strict infinity-groupoids

  • which map under the omega-nerve into Kan complexes (as complicial sets)

Theorem (Kenneth Brown (1973))

• for the site being the category of open subsets of a space $X$

• $F$ a sheaf of abelian groups

• $A_F$ its corresponding $\infty$-groupoid valued sheaf under the above inclusion$Ch(Ab) \hookrightarrow \infty Grpd$

sheaf cohomology in degree $n$ with values in $F$ is cohomology of $X$ with coefficients in $\mathbf{B}^n A_F$

$$H^n(X,F) \simeq H(X,\mathbf{B}^n A_F)$$

(on the left ordinary sheaf cohomology, on the right nonabelian cohomology from above)

• so: abelian sheaf cohomology is really a way to compute the $\infty$-stackificatin of an $\infty$-prestack that happens to factor through $Ch_+(Ab) \hookrightarrow \infty Grpd$

  • the usual prescription (by right derived section functor) looks comparatively more unintuitive because $X$ itself does not represent an abelian sheaf, but a nonabelian sheaf. When regarded in nonabelian cohomology sheaf cohomology looksjust as cohomology as homotopy class of maps in $Top$ does.

3) Differential nonabelian cohomology

• we are already $\infty$-smooth, now we want $\infty$-connections

• main point of connections in physics: yield parallel transport and then from that actionafunctionals

• so consider: for each smooth $\infty$-groupoid $X$ there is a smooth path $\infty$-groupoid $P_n(X)$ whose

  • k-morphisms are smooth $k$-dimensional images of $k$-disks in $X$.

• idea: differential cohomology on $X$ is cohomology on $P_n(X)$: the cocycle $P_n(X) \to A$ is the parallel transport of a higher connection over $k$-volumes;

• this is a a functorial assignment

  $$P_n : Diff \to Sh(Diff, \infty Grpd)$$

  of smooth $n$-path groupoids to smooth spaces (BaSc, ScWa I ScWa II MaPi I)

• so we can define for each smooth coefficent $\infty$-groupoid $A$ its differential refinement $A^{P_n}$, which as a sheaf is

$$A^{P_n} := Sh(P_n(-), A) : Diff^{op} \to \infty Grpd$$

Theorem (BaSc ScWa I ScWa II, ScWa III MaPi I))

• $\mathbf{B}G^{P_1} = \Omega^1(-, Lie(G))$

  • means: pre $\infty$-stack $\mathbf{B}G^{P_1}$ is that of $Lie(G)$-valued differential forms and gauge transformations between them

• $\mathbf{B}^n U(1)^{P_n} = \mathbb{Z}(n+1)^\infty_D$

  • means: pre $\infty$-stack $\mathbf{B}^n U(1)^{P_n}$ is the Deligne complex of abelian sheaves

• $\mathbf{H}(X,\mathbf{B}G^{P_1}) \simeq G Bund_\nabla(X)$

  • means: $\infty$-stackification of $\mathbf{B}G^{P_1}$ is smooth $G$-bundles with connection

• $\mathbf{H}(X,\mathbf{B}^n U(1)^{P_n}) \simeq U(1) (n-1) BundGrb_\nabla(X)$

  • means: $\infty$-stackification of $\mathbf{B}^n U(1)^{P_n}$ is smooth $U(1)$ $(n-1)$-bundle gerbes with connection;

• given this consistency check with familiar structures, we get now much more:

  • we can now define higher nonabelian differential cohomology with parallel transport with coefficients in any smooth $\infty$-groupoid

  • for instance for general strict 2-group $G$ ((BaSc ScWa MaPi II), the differential nonabelian cohomolog $\mathbf{H}(X, \mathbf{G}G^{P_n})$ combines

    • the first nonabelian degree 1 example

    • with something abelian in higher degree

    • and with action of degree 1 group on the rest by automorphisms

• this appears in the twisting examples below:

Principle

• Higher nonabelian cohomology disguises as twisted higher abelian cohomology .

• conversely: twisted higher abelian cohomology is really nonabelian cohomology

• moreover

  • notice that for general coefficients the above notion of differential cohomology is too restrictive on curvature: curvature will only be allowed to be non-vanishing in higher degree (“fake flatness”)

  • real answer is: non-flat differential cocycle is (curvature characteristic form)-twisted flat differential cohomology

• so pass now to twisted cohomology

4) Twisted nonabelian cohomology

we

• recall the physical motivation of twisted cocycles

• present a formalization in smooth $\infty Grpds$

• demonstrates how this captures the two kinds of twists

  • smooth twist of cocycle by magnetic charges;

  • differential twist giving rise to curvature and characteristic forms for differential cocycles.

4.1) twists of background fields by charges

(Fr)

• recall twisting of electromagnetic field $\nabla$ by magnetic current $J_{magnetic}$ in Maxwell’s equations:

  $$d F_\nabla = J_{magnetic}$$

• this is called the twisted Bianchi identity

• so $\nabla$ here cannot be the connection on a bundle

• it is a connection on a twisted bundle. This generalizes to higher connections twisted by higher magnetic charges.

4.2) formalization

• since smooth $\infty$-groupoids live in an (infinity,1)-topos it makes sense to apply all the usual operations as used to from topological spaces;

• so we say a sequence $A \to \hat B \to B$ of smooth
  $\infty$-groupoids is a fibration sequence if

  $$\array{ A &\to& * \\ \downarrow && \downarrow \\ \hat B &\to & B }$$

  is homotopy pullback

• crucial property of homotopy pullback: preserved by Hom, so we get a homotopy pullback of cocycle $\infty$-groupoids

  $$\array{ \mathbf{H}(X,A) &\to& * \\ \downarrow && \downarrow^{*} \\ \mathbf{H}(X,\hat B) &\to & \mathbf{H}(X,B) }$$

• this says $A$-cocycles are precisely those $\hat B$-cocycles whose underlying $B$-cocycle is trivializable

• conversely: the obstruction to lifting a $\hat B$-cocycle to an $A$-cocycle is precisely its image as a $B$-cocycle

• now just tweak this situation a little

Definition: twisted cohomology (SaScSt III)

For $A \to \hat B \to B$ a fibration sequence and $c \in \mathbf{H}(X,B)$ a $B$-cocycle, the $c$-twisted $A$-cohomology $\mathbf{H}^c(X,A)$ is the homotopy pullback

$$\array{ \mathbf{H}^c(X,A) &\to& * \\ \downarrow && \downarrow^{c} \\ \mathbf{H}(X,\hat B) &\to & \mathbf{H}(X,B) }$$

• notice that what used to be the trivial cocycle in the right vertical morphism is now replaced by $c$

• claim this simple and systematic general nonsense definition reproduces the twists by charges one sees “in nature”

4.3) Examples and applications

in applications in physics there are two types of twists simultaneously

• the smooth twist that comes from charges

• the differential twist by characteristic forms that makes the cocycle non-flat

4.3.1) curvature characteristic forms of differential cocycles

we just briefly indicate the idea behind obtaining a Chern-Weil theory for higher differential cohomology in terms of curvature-twisted flat differential cohomology SaScSt III)

• find obstruction to extension problem

  $$\left( \array{ X &\to& A \\ \downarrow & \\ \Pi(X) } \right) \mapsto \left( \array{ X &\to& A \\ \downarrow & \nearrow \\ \Pi(X) } \right)$$

  that equips a cocycle with a flat connection

• solution by above twisting method:

  • lift relative cohomology on $X \hookrightarrow \Pi(X)$ through the reltive fibration sequence

    $$\array{ A &\to& A &\to& \mathbf{E}A \\ \downarrow && \downarrow && \downarrow \\ A &\to& \mathbf{E}A &\to& \mathbf{B} A }$$

  • turning the crank, one finds that the corresponding curvature-twisted cocycles are given by squares

    $$\array{ X &\stackrel{g}{\to}& A && underlying cocycle \\ \downarrow && \downarrow &&& first Ehresmann condition \\ \Pi(X) &\stackrel{\nabla}{\to}& \mathbf{E}A && connection \\ \downarrow && \downarrow &&& second Ehresmann condition \\ \Pi(X) &\stackrel{P}{\to}& \mathbf{B}A && char. forms }$$

    with lowest morphism trivializing when pulled back along $X \hookrightarrow \Pi(X)$ .

• obstruction to having flat connection is nontriviality of $P$: characteristic forms

• detailed examples in

  • Sati, Schreiber, Stasheff, $L_\infty$ connections (arXiv)

• abstract nonsense and more details on detailed examples:

  • Sati, Schreiber, Stasheff, Twisted differential String and Fivebrane structures (pdf)

• this produces the twisted Bianchi identities appearing in the following examples

4.3.2) charge twisted cocycles

we now

• list fibration sequence of smooth $\infty$-groupoids

• and indicate properties of the corresponding differential twisted nonabelian cohomology

Examples / Claim

• fibration sequence: $\mathbf{B}U(n) \to \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)$

  • twisting cocycle: lifting gerbe;

  • twisted cocycle: twisted bundles / gerbe modules

  • twisted Bianchi identity: $d F_\nabla = H_3$

  • occurence: Freed-Witten anomaly cancellation on D-brane

• fibration sequence: $\mathbf{B}String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$

  • twisting cocycle: Chern-Simons 2-gerbe;

  • twisted cocycle: twisted nonabelian String-gerbe with conection

  • twisted Bianchi identity: $d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle$

  • occurence: Green-Schwarz anomaly cancellation

  • Proof.

    • (with Danny Stevenson and Christoph Wockel: (SSSS)) use BCSS model (BCSS) of $String(n)$ with Brylinski-McLaughlin construction of $\frac{1}{2}p_1$

    • (using (SaScSt I, SaScSt III):) compute local differential form data after differentiating smooth $\infty$-groupoids to L-infinity algebroids using the formalism of (SaScSt I)

  • for aspects of the twisetd case see also

    • AsJu

    • for aspects of the untwisted case see also (Wa)

• fibration sequence: $\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)$

  • twisting cocycle: Chern-Simons 6-gerbe;

  • twisted cocycle: twisted nonabelian Fivebrane-gerbe with connection

  • occurence: dual Green-Schwarz anomaly cancellation for NS 5-brane magnetic dual to string

  • (SaScSt II SaScSt III)

• fibration sequence: $\mathbf{B}^2 U(1) \to \mathbf{B} (U(1) \to \mathbb{Z}_2) \stackrel{}{\to} \mathbf{B} \mathbb{Z}_2$

  • twisting cocycle: $\mathbb{Z}_2$-orbifold;

  • twisted cocycle: orientifold gerbe / Jandl gerbe with connection

  • occurence: unoriented string

  • unwrap the above abstract nonsense and use the above results to find SchrSchwWal and the bosonic part of DiFrMo