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

  • 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)


  • 1) Motivation

  • 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 DiffDiff

      • think: (AsmoothGrpd,XDiff)(A(X):=Hom(X,A)Grpd)(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 XX is associated a smooth \infty-groupoid Π(X)\Pi(X) of kk-disk-shaped smooth paths of sorts in XX, for all kk

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

    • H diff(X,A)=H(Π(X),A)\mathbf{H}_{diff}(X,A) = H(\Pi(X),A)

  • 4) Twisted nonabelian cohomology

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

    • cc-twisted AA-cohomology H c(X,A)\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

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

      • nn-dimensional Chern-Simons theory, which is “as above”, but with GG replaced by a Lie group, so that the background field becomes an (n1)(n-1)-bundle gerbe with connection

    • topological parts of “physical” QFTs

      • Yang-Mills gauge field

      • Kalb-Ramond field

      • 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)(d+1)-dimensional quantum field theory describing the dynamics of a dd-dimensional object propagating through a target space XX 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

Π n(X) A exp(iS()) Bord n(X) exp(iS()) Bord n \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

ndimpathsinX coefficientsuchasnVect exp(iS()) ndimbordismsinX exp(iS()) abstractndimbordisms \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

targetspace backgroundfielddiff.cocycle A action fieldspace QFT worldvolume \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 XX;

    • :(xγy)(E xtra(γ)E y)\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 XX to entire cobordisms in XX; 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 XX to abstract bordisms, where it represents an FQFT.


  • 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 (nn-dimensional)

  • here background field is just a continuous map

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

    in Top

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

    ([g]=[g])exist( g BG K(,n+1) g) ([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:BGB nU(1) g : \mathbf{B} G \to \mathbf{B}^n U(1)
    • here

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

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

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

    H=Top \mathbf{H} = Top

    (compactly generated weakly Hausdorff topological spaces) or equivalently

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

  • and H(X,A):=π 0H(X,A)={gaugeequivalenceclassesofAvaluedbackgroundfieldsonX}={cohomologyclassesofAvaluedcocyclesonX} 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: (,1)(\infty,1)-topos of \infty-stacks

  • generalize to smooth cocycles

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

  • but contains smooth generalized spaces

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

2.3) model: \infty-groupoid valued sheaves

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

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

  • to model \infty-stacks as ordinary sheaves

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

    with values in \infty-groupoids, using

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

    • (f:YXf: 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 ((XA)H(X,A)):=[Y A X] ((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

      • H(X,A)colim YXSh(Y,A)\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 :

2.3) cocycles and cohomology in this context

Theorem (Kenneth Brown (1973))

  • for the site being the category of open subsets of a space XX

  • FF a sheaf of abelian groups

  • A FA_F its corresponding \infty-groupoid valued sheaf under the above inclusionCh(Ab)GrpdCh(Ab) \hookrightarrow \infty Grpd

sheaf cohomology in degree nn with values in FF is cohomology of XX with coefficients in B nA F\mathbf{B}^n A_F

H n(X,F)H(X,B nA 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)GrpdCh_+(Ab) \hookrightarrow \infty Grpd

    • the usual prescription (by right derived section functor) looks comparatively more unintuitive because XX 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 TopTop 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 XX there is a smooth path \infty-groupoid P n(X)P_n(X) whose

    • k-morphisms are smooth kk-dimensional images of kk-disks in XX.
  • idea: differential cohomology on XX is cohomology on P n(X)P_n(X): the cocycle P n(X)AP_n(X) \to A is the parallel transport of a higher connection over kk-volumes;

  • this is a a functorial assignment

    P n:DiffSh(Diff,Grpd) P_n : Diff \to Sh(Diff, \infty Grpd)

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

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

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

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

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

    • means: pre \infty-stack BG P 1\mathbf{B}G^{P_1} is that of Lie(G)Lie(G)-valued differential forms and gauge transformations between them
  • B nU(1) P n=(n+1) D \mathbf{B}^n U(1)^{P_n} = \mathbb{Z}(n+1)^\infty_D

    • means: pre \infty-stack B nU(1) P n\mathbf{B}^n U(1)^{P_n} is the Deligne complex of abelian sheaves
  • H(X,BG P 1)GBund (X)\mathbf{H}(X,\mathbf{B}G^{P_1}) \simeq G Bund_\nabla(X)

    • means: \infty-stackification of BG P 1\mathbf{B}G^{P_1} is smooth GG-bundles with connection
  • H(X,B nU(1) P n)U(1)(n1)BundGrb (X)\mathbf{H}(X,\mathbf{B}^n U(1)^{P_n}) \simeq U(1) (n-1) BundGrb_\nabla(X)

    • means: \infty-stackification of B nU(1) P n\mathbf{B}^n U(1)^{P_n} is smooth U(1)U(1) (n1)(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 GG ((BaSc ScWa MaPi II), the differential nonabelian cohomolog H(X,GG P n)\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:


  • 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


  • recall the physical motivation of twisted cocycles

  • present a formalization in smooth Grpds\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


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

    dF =J magnetic 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 AB^BA \to \hat B \to B of smooth
    \infty-groupoids is a fibration sequence if

    A * B^ B \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

    H(X,A) * * H(X,B^) H(X,B) \array{ \mathbf{H}(X,A) &\to& * \\ \downarrow && \downarrow^{*} \\ \mathbf{H}(X,\hat B) &\to & \mathbf{H}(X,B) }
  • this says AA-cocycles are precisely those B^\hat B-cocycles whose underlying BB-cocycle is trivializable

  • conversely: the obstruction to lifting a B^\hat B-cocycle to an AA-cocycle is precisely its image as a BB-cocycle

  • now just tweak this situation a little

Definition: twisted cohomology (SaScSt III)

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

H c(X,A) * c H(X,B^) H(X,B) \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 cc

  • 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

    (X A Π(X))(X A Π(X)) \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 XΠ(X)X \hookrightarrow \Pi(X) through the reltive fibration sequence

      A A EA A EA BA \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

      X g A underlyingcocycle firstEhresmanncondition Π(X) EA connection secondEhresmanncondition Π(X) P BA char.forms \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 XΠ(X)X \hookrightarrow \Pi(X) .

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

  • detailed examples in

    • Sati, Schreiber, Stasheff, L 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: BU(n)BPU(n)B 2U(1)\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: dF =H 3d F_\nabla = H_3

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

  • fibration sequence: BString(n)BSpin(n)12p 1B 3U(1)\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: dH 3F F d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle

    • occurence: Green-Schwarz anomaly cancellation

    • Proof.

    • for aspects of the twisetd case see also

      • AsJu

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

  • fibration sequence: BFivebrane(n)BString(n)16p 2B 7U(1)\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)

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

    • twisting cocycle: 2\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

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