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
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
1) Motivation
extended QFT wants functorial description of sigma-model background fields
(QFT as $n$-functor) $\stackrel{\mathrm{quantize}}{\leftarrow \mid}$ (background field/classical action as $n$-functor)
recently: huge progress on formalizing the left hand side:
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
and also section 3 of
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 $\mathrm{Diff}$
think: $(A\in \mathrm{smooth}\mathrm{\infty}\mathrm{Grpd},X\in \mathrm{Diff})\Rightarrow (A(X):=\mathrm{Hom}(X,A)\in \mathrm{\infty}\mathrm{Grpd})$
3) Differential nonabelian cohomology
to every smooth $\mathrm{\infty}$-groupoid $X$ is associated a smooth $\mathrm{\infty}$-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 $\Pi (X)$
”${H}_{\mathrm{diff}}(X,A)=H(\Pi (X),A)$”
4) Twisted nonabelian cohomology
for $(A\to \hat{B}\to B)$ a fibration sequence of smooth $\mathrm{\infty}$-groupoids and $c\in H(X,B)$ there exists a natural notion of
$c$-twisted $A$-cohomology ${H}^{c}(X,A)$
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
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 $BG$ of a finite group $G$ and the background field $\nabla :BG\to {B}^{n}U(1)$ 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
understand background fields for sigma-model QFTs structurally:
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
(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
and are supposed to have the following interpretation
the top horizontal morphism: parallel transport of something like a higher connection along disks in target space $X$;
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
so now: work out where this diagram lives
first toy case of Dijkgraaf-Witten theory ($n$-dimensional)
here background field is just a continuous map
in Top
two background fields are gauge equivalent if these maps are homotopic
even simpler by passing to a combinatorial model for topological spaces: infinity-groupoids (Kan complexes)
here
$BG=\{\u2022\stackrel{g}{\to}\u2022\mid g\in G\}$ is groupoid with one object and the group $G$ as the set of morphisms
${B}^{n}U(1)$ 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 $\mathrm{\infty}\mathrm{Grpd}$ enriched category
(compactly generated weakly Hausdorff topological spaces) or equivalently
and we simply have $H(X,A):=\{A-\mathrm{valued}\mathrm{background}\mathrm{fields}\mathrm{on}X\}=\{A-\mathrm{valued}\mathrm{cocycles}\mathrm{on}X\}$
and $H(X,A):={\pi}_{0}H(X,A)=\{\mathrm{gauge}\mathrm{equivalence}\mathrm{classes}\mathrm{of}A-\mathrm{valued}\mathrm{background}\mathrm{fields}\mathrm{on}X\}=\{\mathrm{cohomology}\mathrm{classes}\mathrm{of}A-\mathrm{valued}\mathrm{cocycles}\mathrm{on}X\}$
generalize to smooth cocycles
$\Rightarrow $ choose an $(\mathrm{\infty},1)$-category $H$ that behaves essentially like $\mathrm{Top}$…
but contains smooth generalized spaces
think of an $\mathrm{\infty}$-stack on $\mathrm{Diff}$ as a smooth $\mathrm{\infty}$-groupoid
to actually work with this, we chose a convenient model that presents $\mathrm{\infty}$-stacks on $\mathrm{Diff}$.
there is an old construction, dating back to the remarkable
to model $\mathrm{\infty}$-stacks as ordinary sheaves
with values in $\mathrm{\infty}$-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 $\mathrm{\infty}$-groupoids that behave like surjective equivalences
($f:Y\to X$) is surjective equivalence of smooth $\mathrm{\infty}$-groupoids precisely if when regarded as a morphism of sheaves it it restricts to a surjective equivalence of ordinary $\mathrm{\infty}$-groupoids locally (“stalkwise”)
these are also called
acyclic fibrations (generally)
hypercovers (the usefully suggestive terminology in our context)
the $\mathrm{\infty}$-morphisms then are modeled by anafunctors, i.e. morphisms out of surjective equivalences $((X\to A)\in H(X,A)):=\left[\begin{array}{ccc}Y& \to & A\\ \downarrow \\ X\end{array}\right]$
and the full hom-$\mathrm{\infty}$-groupoid is obtained by doing this for all possible hypercovers
this procedure amounts to using rectified $\mathrm{\infty}$-stacks: those which as functors on the site are ordinary strict functors
Jacob Lurie in HTT shows in particular that rectified $\mathrm{\infty}$-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:
finding the full weakly equivalent infinity-stackification is not in general necessary
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 $\mathrm{\infty}\mathrm{Grpds}$
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 $\mathrm{\infty}$-groupoid valued sheaf under the above inclusion$\mathrm{Ch}(\mathrm{Ab})\hookrightarrow \mathrm{\infty}\mathrm{Grpd}$
sheaf cohomology in degree $n$ with values in $F$ is cohomology of $X$ with coefficients in ${B}^{n}{A}_{F}$
(on the left ordinary sheaf cohomology, on the right nonabelian cohomology from above)
so: abelian sheaf cohomology is really a way to compute the $\mathrm{\infty}$-stackificatin of an $\mathrm{\infty}$-prestack that happens to factor through ${\mathrm{Ch}}_{+}(\mathrm{Ab})\hookrightarrow \mathrm{\infty}\mathrm{Grpd}$
we are already $\mathrm{\infty}$-smooth, now we want $\mathrm{\infty}$-connections
main point of connections in physics: yield parallel transport and then from that actionafunctionals
so consider: for each smooth $\mathrm{\infty}$-groupoid $X$ there is a smooth path $\mathrm{\infty}$-groupoid ${P}_{n}(X)$ whose
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
of smooth $n$-path groupoids to smooth spaces (BaSc, ScWa I ScWa II MaPi I)
so we can define for each smooth coefficent $\mathrm{\infty}$-groupoid $A$ its differential refinement ${A}^{{P}_{n}}$, which as a sheaf is
Theorem (BaSc ScWa I ScWa II, ScWa III MaPi I))
$B{G}^{{P}_{1}}={\Omega}^{1}(-,\mathrm{Lie}(G))$
${B}^{n}U(1{)}^{{P}_{n}}=\mathbb{Z}(n+1{)}_{D}^{\mathrm{\infty}}$
$H(X,B{G}^{{P}_{1}})\simeq G{\mathrm{Bund}}_{\nabla}(X)$
$H(X,{B}^{n}U(1{)}^{{P}_{n}})\simeq U(1)(n-1){\mathrm{BundGrb}}_{\nabla}(X)$
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 $\mathrm{\infty}$-groupoid
for instance for general strict 2-group $G$ ((BaSc ScWa MaPi II), the differential nonabelian cohomolog $H(X,G{G}^{{P}_{n}})$ 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
we
recall the physical motivation of twisted cocycles
present a formalization in smooth $\mathrm{\infty}\mathrm{Grpds}$
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.
(Fr)
recall twisting of electromagnetic field $\nabla $ by magnetic current ${J}_{\mathrm{magnetic}}$ in Maxwell’s equations:
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.
since smooth $\mathrm{\infty}$-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
$\mathrm{\infty}$-groupoids is a fibration sequence if
crucial property of homotopy pullback: preserved by Hom, so we get a homotopy pullback of cocycle $\mathrm{\infty}$-groupoids
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 $c\in H(X,B)$ a $B$-cocycle, the $c$-twisted $A$-cohomology ${H}^{c}(X,A)$ is the homotopy pullback
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”
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
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
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
turning the crank, one finds that the corresponding curvature-twisted cocycles are given by squares
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
abstract nonsense and more details on detailed examples:
this produces the twisted Bianchi identities appearing in the following examples
we now
list fibration sequence of smooth $\mathrm{\infty}$-groupoids
and indicate properties of the corresponding differential twisted nonabelian cohomology
Examples / Claim
fibration sequence: $BU(n)\to B\mathrm{PU}(n)\to {B}^{2}U(1)$
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: $B\mathrm{String}(n)\to B\mathrm{Spin}(n)\stackrel{\frac{1}{2}{p}_{1}}{\to}{B}^{3}U(1)$
twisting cocycle: Chern-Simons 2-gerbe;
twisted cocycle: twisted nonabelian String-gerbe with conection
twisted Bianchi identity: $d{H}_{3}\propto \u27e8{F}_{\nabla}\wedge {F}_{\nabla}\u27e9$
occurence: Green-Schwarz anomaly cancellation
Proof.
(with Danny Stevenson and Christoph Wockel: (SSSS)) use BCSS model (BCSS) of $\mathrm{String}(n)$ with Brylinski-McLaughlin construction of $\frac{1}{2}{p}_{1}$
(using (SaScSt I, SaScSt III):) compute local differential form data after differentiating smooth $\mathrm{\infty}$-groupoids to L-infinity algebroids using the formalism of (SaScSt I)
for aspects of the twisetd case see also
fibration sequence: $B\mathrm{Fivebrane}(n)\to B\mathrm{String}(n)\stackrel{\frac{1}{6}{p}_{2}}{\to}{B}^{7}U(1)$
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
fibration sequence: ${B}^{2}U(1)\to B(U(1)\to {\mathbb{Z}}_{2})\stackrel{}{\to}B{\mathbb{Z}}_{2}$
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