group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
This entry contains lecture notes on
A) twisted smooth cohomology structures appearing in string theory/M-theory;
B) the corresponding sigma model local prequantum field theory
formulated in cohesive higher differential geometry.
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
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
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
which exists in a context of geometric homotopy types: an higher topos to be denoted $\mathbf{H} \coloneqq$ Smooth∞Grpd, of smooth ∞-groupoids/smooth ∞-stacks.
Here
$G$ is a smooth ∞-group – a higher gauge group;
$\mathbf{B}G$ is the smooth moduli ∞-stack of $G$-principal ∞-bundles;
$F$ is a coefficient object equipped with a $G$-action;
$E \to \mathbf{B}G$ is the associated bundle to the $G$-universal principal ∞-bundle over the moduli ∞-stack $\mathbf{B}G$ of $G$-principal ∞-bundles.
Given such, and given a spacetime/target space $X$, we have:
a morphism $\phi : X \to \mathbf{B}G$ determines a twisting bundle or twisting background gauge field on $X$;
a lift $\hat \phi$ in
is a $\phi$-twisted bundle, or $\phi$-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 examples | universal local coefficient bundle | twisting bundle | twisting field | twisted bundle | twisted field |
---|---|---|---|---|---|
0) general pattern | |||||
$\array{ F &\to& E \\ && \downarrow^{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}G }$ | $G$-principal bundle | $G$-gauge field instanton-sector | section of $\rho$-associated $F$-bundle | twisted $\Omega F$-gauge field instanton sector | |
$\array{ \flat \mathbf{B}^n U(1) &\to& \mathbf{B}^n U(1) \\ && \downarrow^{\mathbf{curv}} \\ && \flat_{dR} \mathbf{B}^{n+1} U(1) }$ | de Rham hypercohomology | circle n-bundle with connection | higher abelian gauge field | ||
$\array{ F_{conn} &\to& E_{conn} \\ && \downarrow^{\mathrlap{\hat \mathbf{c}}} \\ && \mathbf{B}G_{conn} }$ | $G$-connection | $G$-gauge field | section of $\rho$-associated $F$-bundle | twisted $\Omega F$-gauge field | |
I) reduction of structure group/geometric structure | |||||
$\array{ GL(n)/O(n) &\to& \mathbf{B} O(n) \\ && \downarrow \\ && \mathbf{B} GL(n) }$ | manifold structure / tangent bundle | vielbein | gravity | ||
$\array{ GL(n)/O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow \\ && \mathbf{B} GL(n)_{conn} }$ | affine connection | spin connection | gravity | ||
$\array{ O(2d,2d)/SU(d,d) &\to& \mathbf{B} SU(d,d) \\ && \downarrow \\ && \mathbf{B} O(2d,2d) }$ | generalized tangent bundle | generalized Calabi-Yau manifolds | |||
$\array{ O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B} O(n) \times O(n) \\ && \downarrow \\ && \mathbf{B} O(n,n) }$ | generalized tangent bundle | type II geometry | DFT type II supergravity | ||
$\array{ SU(3)\backslash O(6,6) / SU(3) &\to& \mathbf{B} SU(3) \\ && \downarrow \\ && \mathbf{B} O(6,6) }$ | generalized tangent bundle | type II geometry | $d = 6$, $N=1$ type II supergravity comactifications | ||
$\array{ E_{7(7)}/SU(7) &\to& \mathbf{B} SU(7) \\ && \downarrow \\ && \mathbf{B}E_{7(7)} }$ | exceptional tangent bundle | E7-U-duality moduli (split real form) | $d = 7$, $N=1$ 11d supergravity compactifications | ||
$\array{ E_{n(n)}/H_n &\to& \mathbf{B}H_n \\ && \downarrow \\ && \mathbf{B}E_{n(n)} }$ | exceptional tangent bundle | U-duality moduli | U-duality exceptional generalized geometry | compactification of 11d supergravity to $d = 11-n$ | |
$\array{ \mathbf{B} U &\to& \mathbf{B} P U \\ && \downarrow^{\mathbf{dd}} \\ && \mathbf{B}^2 U(1) }$ | circle 2-bundle/$U(1)$-bundle gerbe | B-field | twisted unitary bundle | Yang-Mills field | |
$\array{ \mathbf{B}String(E_8) &\to& \mathbf{B} E_8 \\ && \downarrow^{\mathrlap{\mathbf{a}}} \\ && \mathbf{B}^3 U(1)}$ | circle 3-bundle / bundle 2-gerbe | supergravity C-field | twisted String(E8)-2-form gauge field | ||
II) higher spin structures | |||||
$\array{ \mathbf{B} Spin &\to& \mathbf{B} SO \\ && \downarrow^{\mathrlap{\mathbf{w}_2}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }$ | second Stiefel-Whitney class | twisted spin structure | |||
$\array{ \mathbf{B}String &\to& \mathbf{B} Spin \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1}} \\ && \mathbf{B}^3 U(1) }$ | circle 3-bundle/$U(1)$-bundle 2-gerbe | NS5-brane magnetic charge | twisted smooth string structure | ||
$\array{ \mathbf{B}String_{conn} &\to& \mathbf{B} Spin_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\hat \mathbf{p}_1}} \\ && \mathbf{B}^3 U(1)_{conn} }$ | circle 3-bundle with connection | NS5-brane magnetic current | twisted differential string structure | Green-Schwarz mechanism gravity+B-field | |
$\array{ \mathbf{B}Fivebrane &\to& \mathbf{B} String \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\mathbf{p}_2}} \\ && \mathbf{B}^7 U(1) }$ | circle 7-bundle | string electric charge | twisted smooth fivebrane structure | ||
$\array{ \mathbf{B}Fivebrane_{conn} &\to& \mathbf{B} String_{conn} \\ && \downarrow^{\mathrlap{\tfrac{1}{6}\hat \mathbf{p}_2}} \\ && \mathbf{B}^7 U(1)_{conn} }$ | circle 7-bundle with connection | string electric current | twisted differential fivebrane structure | dual Green-Schwarz mechanism gravity+B6-field | |
III) higher spin^c-structures | |||||
$\array{ \mathbf{B}Spin^c &\to& \mathbf{B} (SO \times U(1)) \\ && \downarrow^{\mathrlap{ w_2 - c_1}} \\ && \mathbf{B}^2 \mathbb{Z}_2 }$ | twisted spin^c structure | ||||
$\array{ \mathbf{B}(Spin^c)^{\mathbf{dd}} &\to& \mathbf{B} (PU \times SO) \\ && \downarrow^{\mathrlap{ \mathbf{dd} - \mathbf{W}_3 }} \\ && \mathbf{B}^2 U(1) }$ | circle 2-bundle | B-field | twisted spin^c structure | Freed-Witten anomaly for type II superstring on D-brane | |
$\array{ \mathbf{B}String^{\mathbf{a}} &\to& \mathbf{B} Spin \times E_8 \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - 2 \mathbf{a}}} \\ && \mathbf{B}^3 U(1) }$ | circle 3-bundle/$U(1)$-bundle 2-gerbe | supergravity C-field | twisted smooth string^c structure | gravity+B-field+E8-gauge field | |
IV) Giraud-∞-gerbes | |||||
$\array{ \mathbf{B}^2 U(1) &\to& \mathbf{B}Aut(\mathbf{B}U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }$ | double cover | Jandl gerbe | orientifold B-field | ||
$\array{ \mathbf{B}^3 U(1) &\to& \mathbf{B}Aut(\mathbf{B}^2 U(1)) \\ && \downarrow \\ && \mathbf{B}\mathbb{Z}_2 }$ | double cover | Hořava-Witten orientifold |
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.
Let $X$ be a smooth manifold of dimension $n$.
By definition this means that there is an atlas
of coordinate charts. On each overlap $U_i \cap U_j$ of two charts, the partial derivatives of the corresponding coordinate transformations
form the Jacobian matrix of smooth functions
with values in invertible matrices, hence in the general linear group $GL(n)$. By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations
(here and in the following sums over an index appearing upstairs and downstairs are explicit)
hence the equation
in the group $C^\infty(U_i \cap U_j \cap U_k, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlaps.
This is the cocycle condition for a smooth Cech cocycle in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the sheaf of smooth functions with values in $GL(n)$ ). We write
It is useful to reformulate this statement in the language of Lie groupoids/differentiable stacks.
$X$ itself is a Lie groupoid $(X \stackrel{\to}{\to} X)$ with trivial morphism structure;
from the atlas $\{U_i \to X\}$ we get the corresponding Cech groupoid
whose objects are the points in the atlas, with morphisms identifying lifts of a point in $X$ to different charts of the atlas;
any Lie group $G$ induces its delooping Lie groupoid
The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form
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 $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.
A transformation of smooth functors $\lambda_1 \Rightarrow \lambda_2 : C(\{U_i\}) \to \mathbf{B} GL(n)$ is precisely a coboundary between two such cocycles.
We want to think of such a diagram as being directly a morphism of smooth groupoids
in a suitable context $\mathbf{H}$, such that this may be regarded as a smooth refinement of the underlying homotopy class of a map into the classifying space $B GL(n)$
Evidently, for this we need to turn the stalk-wise homotopy equivalence $C(\{U_i\}) \to X$ 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 $A = (A_1 \stackrel{\to}{\to} A_0)$ yields on each smooth manifold $U$ a groupoid of maps from $U$ into $A$
the groupoid of smooth $U$-families of elements of $A$.
Moreover, for every smooth function $U_1 \to U_2$ there is an evident restriction map $A(U_2) \to A(U_1)$ and so this yields a presheaf of groupoids, hence a functor $A \in Func(SmthMfd^{op}, Grpd)$. 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
be the localization of groupoid-valued presheaves that universally turns stalkwise homotopy equivalences into actual homotopy equivalences: if a natural transformation $\eta : A \to B$ in $Func(SmthMfd^{op}, Grpd)$ is such that for each $U \in$ SmthMfd and each $x \in U$ there is a neighbourhood $U_x \subset U$ of $x$ such that $\eta(U_x) : A(U_x) \to B(U_x)$ is an equivalence of groupoids, then $\eta$ has a homotopy inverse in $\mathbf{H}$.
We call this $\mathbf{H}$ the (2,1)-topos of smooth groupoids or of smooth stacks.
Discussed there are tools for describing $\mathbf{H}$ concretely. For the moment we only need to know that
the Cech nerve projection $C(\{U_i\}) \to X$ of every open cover has a homotopy inverse in $\mathbf{H}$, as already used;
if the cover is good and $G$ is a Lie group, then every morphism $X \to \mathbf{B}G$ in $\mathbf{H}$ is presented by a zig-zag of the form $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}G$.
Then we have
a morphism $X \to \mathbf{B} GL(n)$ in $\mathbf{H}$ is precisely a smooth real vector bundle on $X$;
a homotopy between two such morphisms in $\mathbf{H}$ is precisely a smooth $GL(n)$-gauge transformation between the two vector bundles.
Therefore $\mathbf{B}GL(n)$ regarded as an object of $\mathbf{H}$ 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.
Consider the defining inclusion of the orthogonal group into the general linear group
We may understand this inclusion geometrically in terms of the canonical metric on $\mathbb{R}^n$. We may also understand it purely Lie theoretically as the inclusion of the maximal compact subgroup of $GL(n)$. 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
simply by regarding it as a morphism of Lie groupoids
in the evident way.
Now we can say what a Riemannian metric/orthogonal structure on $X$ is:
A choice of orthogonal structure on $T X$ is a factorization of the above $GL(n)$-valued cocycle $\lambda$ through $\mathbf{Orth}$, up to a smooth homotopy $E$ in $\mathbf{H}$, hence a diagram
in $\mathbf{H}$.
This consists of two pieces of data
the morphism $h$ is (by the same reasoning as for $\lambda$ above) a $O(n)$-valued 1-cocycle – a collection of orthogonal transition functions – hence on each overlap of coordinate patches a smooth function
such that
on all triple overlaps of coordinate charts $U_i \cap U_j \cap U_k$;
the homotopy $E$ is on each chart a function
such that on each overlap of coordinate charts it intertwines the transition functions $\lambda$ of the tangent bundle with the new orthogonal transition functions, meaning that the equation
holds. This exhibits the naturality diagram of $E$:
The component $h$ defines an $O(n)$-principal bundle on $X$, or its associated vector bundle. The component $E$ is the corresponding vielbein. It exhibits an isomorphism
between a vector bundle $V \to X$ with structure group explicitly being the orthogonal group, and the tangent bundle itself, hence it exhibits the reduction of the structure group of $T X$ from $GL(n)$ to $O(n)$.
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 $X$.
This is usefully discussed in terms of the homotopy fiber of the morphism $\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)$. One finds that the homotopy fiber is the coset $O(n) \backslash GL(n)$.
This means that there is a diagram
in $\mathbf{H}$, and that $GL(n)/O(n)$ is universal with the property of sitting in such a diagram.
We may think of this fiber sequence as being a bundle in $\mathbf{H}$ over the moduli stack $\mathbf{B}GL(n)$ with typical fiber $GL(n)/O(n)$. As such, it is the smooth associated bundle to the smooth universal GL(n)-bundle induced by the canonical action of $GL(n)$ on $O(n)\backslash GL(n)$.
One basic properties of homotopy pullbacks is that they are preserved by forming derived hom-spaces $\mathbf{H}(X,-)$ out of any other object $X$. This means that also
is a fiber sequence. This in turn says that orthogonal structures on $X$ such that the underlying tangent bundle is trivializable, are given by smooth functions into $GL(n)/O(n)$.
This means that if the tangent bundle $T X$ is trivializable, then the coset space $O(n)\backslash GL(n)$ is the moduli space for vielbein fields on $T X$:
However, if $T X$ is not trivial, then this is true only locally: there is then an atlas $\{U_i \to X\}$ such that restricted to each $U_i$ the moduli space of vielbein fields is $C^\infty(U_i, GL(n)/ O(n))$, 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 $X \to GL(n)/O(n)$. 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 $T X$-_twisted cohomology_ relative to the twisting local coefficient bundle $\mathbf{c}$.
We can make this more manifest by writing equivalently
where now on the right we have inserted the fibration resolution of the morphism $\mathbf{c}$ as provided by the factorization lemma: this is the morphism out of the action groupoid of the action of $GL(n)$ on $O(n)\backslash GL(n)$.
The pullback
gives the non-linear $T X$-associated bundle whose space of sections is the “twisted $O(n)\backslash GL(n)$-0-cohomology”, hence the space of inequivalent vielbein fields.
The above says that the space of vielbein fields is the cohomology of $T X$ in the slice (2,1)-topos $\mathbf{H}_{/\mathbf{B}GL(n)}$ with coefficients in $\mathbf{Orth} : \mathbf{B}O \to \mathbf{B}GL(n)$
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 $\mathbf{B}GL(n)$ of the locally cartesian closed (2,1)-category $\mathbf{H}$.
For more on this see also the discussion at general covariance.
We may further refine this discussion to differential cohomology to get genuine differential $T X$-twisted $\mathbf{c}$-structures.
Recall that the moduli stack $\mathbf{B}G$ is presented in $\mathbf{H}$ by the presheaf of groupoids
We may think of this for each $U$ as being the groupoid of $G$-gauge transformations acting on the trivial $G$-bundle over $U$. A connection on the trivial $G$-bundle is a Lie algebra valued form $A \in \Omega^1(U, \mathfrak{g})$. Accordingly, the presheaf of groupoids
is that of $G$-connections and gauge transformations between them: the groupoid of Lie-algebra valued forms over $U$. As an object of $\mathbf{H} =$ SmoothGrpd this the moduli stack of $G$-connections:
The morphism $\mathbf{Orth}$ has an evident differential refinement to a morphism between such differentially refined moduli stacks
by acting on the differential forms with the induced inclusion of the orthogonal Lie algebra into the general linear Lie algebra $\mathfrak{o}(n) \hookrightarrow \mathfrak{gl}(n)$.
The homotopy fiber of this differential refinement turns out to be the same moduli space as before
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 $X$ to the moduli stack
This is a $GL(n)$-principal connection which locally on an atlas is given by the Christoffel symbols
A $\nabla_{T X}$-twisted differential cocycle is now a diagram
In components over the atlas, $\nabla_V$ is a “spin connection” given by local 1-forms $\{\omega_i \in \Omega^1(U_i, \mathfrak{o}(n))\}$
and the vielbeing $E$ now exhibits on each chart $U_i$ the familiar relation between the components of the spin connection and the Christoffel-symbols:
It is a familiar fact that many fields in physics “naturally pull back”. For instance a scalar field on a spacetime $X$ is a function $\phi : X \to \mathbb{C}$, and for $f : Y \to X$ any smooth function between spactimes, there is the corresponding pullback function/field $f^* \phi : Y \to \mathbb{C}$.
Similarly for $\phi : X \to \mathbf{B}G_{conn}$ a gauge field, as discussed above, it has naturally a pullback along $f$ given simply by forming the composite $f^* \phi : Y \stackrel{f}{\to} X \stackrel{\phi}{\to} \mathbf{B}G_{conn}$.
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 $X$, then the problem is that, while its pullback along $f$ will always be a rank-2 tensor, it is not in general non-degenerate anymore – unless $f$ 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 $h : X \to \mathbf{B}O$, but one that fits into a triangle
a simple precomposition with just a morphism $f : Y \to X$ is not the right operation to send this triangle based on $X$ to one based on $Y$.
But since this triangle is a morphism $(h,E) : T X \to \mathbf{orth}$ in the slice topos $\mathbf{H}_{/\mathbf{B}GL}$, it is clear that it does pull back precisely along refinements of $f : Y \to X$ to a morphism in $\mathbf{H}_{/\mathbf{B}GL}$.
Such a refinement is a commuting triangle of the form
in $\mathbf{H}$. But this is evidently the same as an isomorphism $T Y \simeq f^* T X$ between the tangent bundle of $Y$ and the pullback of the tangent bundle on $X$. And this exhibits $f$ as a local diffeomorphism.
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 $O(d) \times O(d) \to O(d,d)$.
Consider the Lie group inclusion
of those orthogonal transformations, that preserve the positive definite part or the negative definite part of the bilinear form of signature $(d,d)$, respectively.
If $\mathrm{O}(d,d)$ is presented as the group of $2d \times 2d$-matrices that preserve the bilinear form given by the $2d \times 2d$-matrix
then this inclusion sends a pair $(A_+, A_-)$ of orthogonal $n \times n$-matrices to the matrix
This induces the corresponding morphism of smooth moduli stacks, which we denote
Forming the homotopy fiber now yields the local coefficient bundle
There is also a canonical embedding
of the general linear group.
In the above matrix presentation this is given by sending
where in the bottom right corner we have the transpose of the inverse matrix of the invertble matrix $a$.
Under this inclusion, the tangent bundle of a $d$-dimensional manifold $X$ defines an $\mathrm{O}(d,d)$-cocycle
The vector bundle canonically associated to this composite cocycle may canonically be identified with the direct sum vector bundle $T X \oplus T^* X$, and so we will refer to this cocycle by these symbols, as indicated. This is also called the generalized tangent bundle of $X$.
Therefore we may canonically consider the groupoid of $T X \oplus T^* X$-twisted $\mathbf{TypeII}$-structures, according to the general notion of twisted differential c-structures.
A type II generalized vielbein on a smooth manifold $X$ is a diagram
in $\mathbf{H}$, hence a cocycle in the smooth twisted cohomology
The groupoid $\mathbf{TypeII}\mathrm{Struc}(X)$ is that of “generalized vielbein fields” on $X$, 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 $X$.
Over a local coordinate chart $\mathbb{R}^d \simeq U_i \hookrightarrow X$, the most general such generalized vielbein (hence the most general $\mathrm{O}(d,d)$-valued function) may be parameterized as
where $e_+, e_- \in C^\infty(U_i, \mathrm{O}(d))$ are thought of as two ordinary vielbein fields, and where $B$ is any smooth skew-symmetric $n \times n$-matrix valued function on $\mathbb{R}^d \simeq U_i$.
By an $\mathrm{O}(d) \times \mathrm{O}(d)$-gauge transformation this can always be brought into a form where $e_+ = e_- =: \tfrac{1}{2}e$ such that
The corresponding “generalized metric” over $U_i$ is
where
is the metric (over $\mathbb{R}^q \simeq U_i$ a smooth function with values in symmetric $n \times n$-matrices) given by the ordinary vielbein $e$.
Above we have seen (pseudo-)Riemannian structure given by lifts through the inclusion $\mathbf{B} O(n) \to \mathbf{B} GL(n)$. Now we consider further lifts, through the Whitehead tower of $\mathbf{B}O$. 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.
The nature of higher spin structures is governed by what is called the Whitehead tower of the homotopy type of the classifying space $B O$ 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 $B O$ start out as
$k =$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|
$\pi_k(B O) =$ | * | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |
The Whitehead tower of $B O$ starts out as
where
the stages are the deloopings of
… $\to$ fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group
the obstruction classes are the universal characteristic classes
first fractional Pontryagin class $\tfrac{1}{2}p_1$
second fractional Pontryagin class $\tfrac{1}{6}p_2$
every possible square in the above is a homotopy pullback square (using the pasting law).
For instance $w_2$ can be identified as such by representing $B O \to \tau_{\leq 2} B O \simeq BO/_{\sim_n}$ 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 $sSet$ is a simplicial model category, $sSet(S^2,-)$ can be applied and preserves the pullback as well as the homotopy pullback, hence sends $B O \to \tau_{\leq 2} B O$ to an isomorphism on connected components. This identifies $B SO \to B^2 \mathbb{Z}$ as being an isomorphism on the second homotopy group. Therefore, by the Hurewicz theorem, it is also an isomorphism on the cohomology group $H^2(-,\mathbb{Z}_2)$. Analogously for the other characteristic maps.
In summary, more concisely, the tower is
where each “hook” is a fiber sequence.
The universal property of homotopy pullbacks says that
the obstruction to lifting an orthogonal structure $T X : X \to B O$ to an orientation structure is the homotopy class $[w_1(T X)] \in H^1(B O, \mathbb{Z}_2)$ of $w_1(T X) : X \stackrel{T X}{\to} B O \stackrel{w_1}{\to} B \mathbb{Z}_2$, the first Stiefel-Whitney class;
the obstruction to lifting an orientation structure $T X : X \to B SO$ to an spin structure is the second Stiefel-Whitney class $w_2(T X) : X \stackrel{}{\to} B S \stackrel{w_2}{\to} B^2 \mathbb{Z}_2$;
the obstruction to lifting a spin structure $X \to B Spin$ to an string structure is the first fractional Pontryagin class;
the obstruction to lifting a string structure $X \to B String$ to a fivebrane structure is the second fractional Pontryagin class.
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:
The classifying space of, say, the spin group is not afine moduli space_.
Because, while homotopy classes $Maps(X, B Spin)_\sim$ of maps $X \to B Spin$ are in bijection with equivalence classes of spin bundles on $X$, the homotopy classes $Maps(X, \Omega B Spin)_\sim = Maps(X, Spin)_\sim$ of homotopies from the trivial map $X \to * \to B Spin$ are not in general in bijection with the gauge transformations of the trivial spin bundle: the latter form the set of smooth functions $C^\infty(X,Spin)$, not just the homotopy classes of these.
$Maps(X, B(-))_\sim$ 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).
$Maps(X, B(-))_\sim$ cannot distinguish a group from its maximal compact subgroup, such as $O \hookrightarrow GL_n$, 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 $Maps(X, B(-))_\sim$ cannot distinguish over 10-dimensional spacetime $X$ an E8-gauge field from a NS5-brane magnetic charge;
These problems are all fixed by refining classifying spaces such as $B Spin \in \infty Grpd$ to smooth moduli stacks such as $\mathbf{B} Spin \in Smooth \infty Grpd$.
We considered above the smooth refinement of the classifying space $B G$ for $G$ a Lie group to a smooth moduli stack $\mathbf{B}G$. 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 $n$.
But there is an evident refinement of the above discussion to such smooth $n$-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.
A morphism between groupoids $X \to Y$ 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 $\pi_0$ and $\pi_1$.
Accordingly, we say that a homotopy equivalence between Kan complexes is a morphism $X \to Y$ 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
for the homotopy theory obtained by localization at the weak homotopy equivalences: ∞-groupoids.
A topological space $X$ defines a Kan complex/ ∞-groupoid by the singular simplicial complex construction $Sing X$, and this establishes an equivalence between the homotopy theory of topological spaces and simplicial sets
In view of this, the above Whitehead tower can be understood entirely as taking place in Kan complexes.
For instance for $A$ an abelian group, the 2-groupoid
corresponds to the Eilenberg-MacLane space $K(A,2)$.
More generally, for each chain complex $A_\bullet$ of abelian groups the Dold-Kan correspondence provides a Kan complex $\Xi(A_\bullet)$ whose simplicial homotopy groups are the chain homology groups of $A_\bullet$. A quasi-isomorphism $A_\bullet \to B_\bullet$ is sent to a weak homotopy equivalence $\Xi(A_\bullet) \to \Xi(B_\bullet)$. 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 $\mathbf{H} \coloneqq$ Smooth∞Grpd by considering simplicial sets parameterized over smooth manifolds and forcing stalkwise weak homotopy equivalences to become homotopy equivalences
For instance there is the smooth 2-stack
given by assigning to each test space $U$ the Eilenberg-MacLane space on the (discrete) abelian group of smooth functions $U \to U(1)$
A morphism $X \to \mathbf{B}^2 U(1)$ in $\mathbf{H}$ is equivalently a zig-zag $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \stackrel{\lambda}{\to} \mathbf{B}^2 U(1)$ through the Cech nerve. This now defines in degree 2
smooth functions $\lambda$ on triple overlaps
and the condition on quadruple overlaps $U_i \cap U_j \cap U_k \cap U_l$ says that they satisfy
This now identifies $(\lamda_{i j k})$ with a cocycle in degree-2 Cech cohomology
This classifies a smooth circle 2-bundle / bundle gerbe.
Generally we have
and for $X$ a smooth manifold
So apparently the smooth $n$-stack $\mathbf{B}^n U(1)$ is a smooth refinement of the Eilenberg-MacLane space $K(\mathbb{Z},n+1)$.
This is made precise as follows.
Theorem There is an (∞,1)-functor
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
and hence $\mathbf{B}^n U(1)$ is a smooth geometric refinement of $K(\mathbb{Z}, n+1)$.
We now apply this to the above 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
where
$\mathbf{B}^{n+1} U(1)$ is the delooping of the smooth circle n-group, the smooth moduli $(n+1)$-stack of smooth circle n-bundles;
$\mathbf{B} String$ is the delooping of the smooth string 2-group, the moduli 2-stack of smooth $String$-principal 2-bundles
$\mathbf{B}Fivebrane$ is the delooping of the smooth fivebrane 6-group, the smooth moduli 6-stack of smooth fivebrane-principal 6-bundles.
and where
$\tfrac{1}{2} \mathbf{p}_1$ classifies the universal Chern-Simons circle 3-bundle and hence identifies it with $\mathbf{B}String \to \mathbf{B}Spin$;
$\tfrac{1}{6} \mathbf{p}_2$ classifies the universal Chern-Simons circle 7-bundle and hence identifies it with $\mathbf{B}Fivebrane \to \mathbf{B}String$.
This is constructed using essentially the following three tools for presenting presheaves of higher groupoids:
and its prolongation to presheaves
allows to use presheaves of chain complexes of abelian groups to present presheaves of strict $\infty$-groupoids with strict abelian group structure.
For instance
is equivalent to the image under the DK correspondence of the sheaf of chain complexes which is concentrated in degree $n$ on the group of $U(1)$-valued functions.
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,
called crossed modules, due to them being equipped with a compatible action $G_0 \to Aut(G_1)$, serve to equivalently present strict 2-groups.
For instance, one way to construct the string 2-group $String$ above is via the crossed module $(\hat \Omega_* Spin \to P_* Spin)$ induced from the Kac-Moody central extension of the loop group of $Spin$.
For a given crossed module, the corresponding moduli 2-stack $\mathbf{B}(G_1 \stackrel{\delta}{\to} G_0)$ has 2-cells that look like
The Lie integration $\tau_{\leq n} \exp(\mathfrak{g})$ of an L-∞ algebra $\mathfrak{g}$ yields the corresponding smooth ∞-group $G$. For instance the string 2-group $String$ above is also equivalently given by $\mathbf{B} String \simeq \tau_{\leq 2} \exp(\mathfrak{string})$, where $\mathfrak{string}$ is the string Lie 2-algebra.
Using these tools, the stages of the above smooth Whitehead tower are constructed as follows:
the morphism $\mathbf{w}_1 : \mathbf{B}O \to \mathbf{B} \mathbb{Z}_2$ is directly induced from the canonical Lie group homomorphism $O \to \mathbb{Z}_2$.
the morphism $\mathbf{w}_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2$ in $\mathbf{H}$ is presented by the zig-zag of crossed modules
the morphism $\tfrac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)$ is constructed (FSSa) as the Lie integration of the canonical L-∞ 3-cocylce $\mathbf{B}\mu_3 : \mathbf{B}\mathfrak{so} \to \mathbf{B}^3 \mathbb{R}$.
similarly $\tfrac{1}{6}\mathbf{p}_2 \simeq \exp(\mu_7)$ is the Lie integration of a canonical 7-cocycle $\mathbf{B}\mu_7 : \mathbf{B}\mathfrak{string} \to \mathbf{B}^7 \mathbb{R}$ (FSSa).
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:
For $G$ a Lie group the smooth moduli stack of smooth $G$-principal connections from above is presented by
In the special case that $G = U(1)$ is abelian, this is the image under the Dold-Kan correspondence of the length 1 complex of sheaves of abelian groups
The smooth $n$-stack $\mathbf{B}^n U(1)$ is realized as the image under the Dold-Kan correspondence by the chain complex of sheaves $C^\infty(-,U(1))$
From the look of these expressions there is already a plausible candidate for the differential refinement $\mathbf{B}^n U(1)_{conn}$, the moduli $n$-stack of circle n-bundles with connection – it should be the Deligne complex:
For instance a cocycle $X \to \mathbf{B}^2 U(1)_{conn}$ in $\mathbf{H}$ is in $Funct(SmthMfd^{op}, sSet)$ and relative to a good open cover given by a morphism $C(\{U_i\}) \to \mathbf{B}^2 U(1)_{conn}$, which is
on each $U_i$ a connection 2-form $B_i \in \Omega^2(U_i)$;
on each $U_i \cap U_j$ a 1-form $A_{i j} \in \Omega^1(U_i \cap U_j)$ such that $B_j - B_i = d_{dR} A_{i j}$;
on each $U_i \cap U_j \cap U_k$ a smooth functor $\phi_{i j k} \in C^\infty(U_i \cap U_j \cap U_k, U(1))$ such that $A_{i j} + A_{j k} - A_{i k} = d_{dR} log \phi_{i j k}$ and such that on each $U_i \cap U_j \cap U_k \cap U_l$ the equation $\phi_{i j k} \phi_{i k l} = \phi_{i j l} \phi_{j k l}$ holds.
Tthe B-field on spacetime is (in the absence of various possible twists, to be discussed), such a cocycle $X \to \mathbf{B}^2 U(1)_{conn}$; and the C-field (similarly in the absence of possible twists, to be discussed below) is given by a morphism $X \to \mathbf{B}^3 U(1)_{conn}$.
Such cocycles in Deligne hypercohomology define classes in ordinary differential cohomology $H_{diff}^{n+1}(X)$:
There is an evident morphism
which forgets the connection data. We say that $\mathbf{B}^n U(1)_{conn}$ is a differential refinement of $\mathbf{B}^n U(1)$.
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 $\mathbf{B}^n U(1)_{conn}$, a way to obtain this more abstractly from fundamental principles.
The key is that the (∞,1)-topos $\mathbf{H}$ 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 $\mathbf{H}$
where $\Gamma$ evaluates a presheaf on the point, and where $Disc$ sends an $\infty$-groupoid to the presheaf constant on that value. We form the composite
to be pronounced “flat”: for $A \in \mathbf{H}$ a smooth homotopy type, we call $\flat A$ the corresponding flat local coefficient object.
For instance if $G$ is a Lie group, then $\Gamma \mathbf{B}G \simeq B (G_{disc}) = K(G_{disc}, 1)$, and so a morphism $X \to \flat \mathbf{B}G$ is equivalently a cocycle $X \to \mathbf{B} (G_{disc})$, hence a $G_{disc}$-covering space, hence a flat $G$-principal connection.
Generally, we say that a morphism
is an $A$-local system or $A$-valued flat ∞-connection on $X$.
There is a canonical forgetful morphism $u : \flat A \to A$ which forgets the flat connection: this is the $(\Disc \dashv \Gamma)$-counit. Consider the coefficient object of those flat $G$-connections whose underlying $\mathbf{B}G$-principal ∞-bundle is trivial
From the example of ordinary principal connections it is familiar that flat $G$-connections on trivial $G$-principal bundles are equivalently flat Lie algebra valued differential forms. Below we will see that for general smooth ∞-groups $G$, morphisms $X \to \flat_{dR} \mathbf{B}G$ are $\mathfrak{g}$-∞-Lie algebra valued differential forms on $X$.
Reading the above expression in homotopy type theory, its categorical semantics is the homotopy fiber of the counit
By this construction and applying the pasting law, there is a canonical morphism $\theta : G \to \flat_{dR} \mathbf{B}G$, hence a canonical $\mathfrak{g}$-valued form on any cohesive ∞-group $G$: this identifies as the canonical Maurer-Cartan form on the ∞-group $G$.
For the special class of cases $G = \mathbf{B}^n U(1)$ the circle (n+1)-group we call $curv_{\mathbf{B}^n U(1)} \coloneqq \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1)$ the universal curvature class in degree $(n+1)$.
Due to the existence of the further functor $\Pi : \mathbf{H} \to \infty Grpd$ discussed above it follows that $\flat : \mathbf{H} \to \infty Grpd$ 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:
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 $F_X : X \to \Omega^{n+1}_{cl}(-)$ a closed differential form on $X$, a cocycle in $F_X$-twisted $\mathbf{curv}$-cohomology is equivalently a circle n-bundle with connection with that curvature
For varying $F$, the $\mathbf{curv}$-twisted cohomology in $\mathbf{H}$ identifies with ordinary differential cohomology: the homotopy pullback $\mathbf{B}^n U(1)_{conn}$ in
is presented, under the Dold-Kan correspondence, by the Deligne complex, discussed above. This exhibits ordinary differential cohomology as the curvature-twisted flat cohomology
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 $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^{n+1} U(1)$. We say that a differential refinement of $\mathbf{c}$ is a morphism $\hat \mathbf{c}$ fitting into a diagram
that factors the naturality square of $\flat$ on $\mathbf{c}$.
Theorem (SSSa, FSSa) There exists a smooth differential refinement of the Whitehead tower of BO as follows:
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
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
of 7-dimensional Chern-Simons theory on nonabelian String 2-form fields (FSSb)
We indicate briefly how this is constructed.
(…)
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 $\nabla_{higher\;gauge\;anomaly}$ on the smooth ∞-stack $[X, \mathbf{Fields}]$ of field configurations on a given spacetime $X$, exhibited by a morphism
where $\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag} : \mathbf{Fields} \to \mathbf{B}^{dim X +2} U(1)_{conn}$ is the differential characteristic morphism induced by the differential cup product (FSSd) of universal electric and magnetic currents, and where $\exp(2\pi i \int_X(-))$ is fiber integration in ordinary differential cohomology refined to smooth $\infty$-stacks (this is the “$(dim X)+1$-dimensional infinity-Chern-Simons theory” of $\hat\mathbf{c}_{el} \cup \hat \mathbf{c}_{mag}$ in codimension 1).
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 $F \in \Omega^2_{cl}(X)$. 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 $\hat F$ in ordinary differential cohomology.
In view of this the gauge equivalence classes of configurations of the electromagnetic field on $X$ form the set
of differential cohomology classes in degree 2 on $X$.
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:
$d_{dR} F = J_{mag}$ (magnetic charge current)
$d_{dR} \star F = J_{el}$ (electric charge current)
where $\star$ is the Hodge star operator for the given pseudo Riemannian metric (the field of gravity) on $X$.
The first one is kinematics. For $J_{mag} = 0$ it just expresses that the cuvature 2-form is closed, which is part of the fact that $\hat F$ is a differential cocycle, so it is satisfied by all kinematic field configurations, meaning: all elements of $H^{n+1}_{diff}(X)$.
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) $P \hookrightarrow [X, \mathbf{B}U(1)_{conn}]$ 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 $J_{mag} = 0$, 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”
But for a circle n-bundle with connection in $H^{n+1}_{diff}(X)$, the curvature is necessarily closed, $d_{dR} F = 0$. So there must be another way to refine $d F = J_{mag}$ to differential cohomology.
(Notice for later that the natural home of $J_{mag}$ is not plain de Rham cohomology, but compactly supported cohomology. The equation $d_{dR} F = J_{mag}$ is a trivialization of the image of $J_{mag}$ 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
whose
objects are cocycles in degree-$(n+1)$ differential cohomology: circle n-group-circle n-bundles with connection;
morphisms are equivalence classes of gauge transformations between these, hence equivalence classes of morphisms of higher bundles with connection.
This “categorifies” the cohomology set $H^{n+1}_{diff}(X)$ 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 $D = d_{dR} \pm \delta$, then a morphism $\hat \alpha : \hat F_1 \to \hat F_2$ in $\mathcal{H}_{diff}^{n+1}(X)$ is a Deligne coboundary $D \hat \alpha = \hat F_2 - \hat F_1$.
Or in terms of our smooth moduli stacks, this is a homotopy
Notice that, since morphisms in $\mathcal{H}^{n+1}_{diff}(X)$ preserve the higher connection, a morphism
in $\mathcal{H}^{n+1}_{diff}(X)$ is a flat section of the corresponding circle $n$-bundle, while a morphim
for some $B \in \Omega^n(X) \hookrightarrow \mathcal{H}^{n+1}_{diff}(X)$ 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 $n$-form $B$.
(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 $\mathcal{H}^{n+1}_{diff}(X)$ 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 $X$: take it to be a non-flat section
of a magnetic charge circle 2-bundle with connection $\hat \mathbf{c} \in \mathcal{H}^{3}_{diff}(X)$. Equivalently, in terms of the corresponding classifying morphisms in $\mathbf{H}$ this is a homotopy in a diagram of the form
If $\hat F$ is given by a Deligne cochain $(g_{i j}, A_i)$, $\hat \mathbf{c}$ by a cochain $(c_{i j k}, \gamma_{i j}, \beta_i)$ then this means that
We say that $\hat F$ is a $\hat \mathbf{c}$-twisted bundle with twisted curvature being
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: $d_{dR} F = J_{mag}$.
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
Before we get there, we need to observe that working with the 1-groupoid $\mathcal{H}^{n+1}_{diff}(X)$ 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 $[X, \mathbf{B}^n U(1)_{conn}]$ in order to capture the physics situation.
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 $\hat \mathbf{c} \stackrel{\hat F}{\to} c_{mag}$ as above? Clearly, for this we need the 2-groupoid of differential cocycles $\tau_2 \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$
to next say that the equivalenc class of a gauge transformation of twisted fields $\hat a : \hat F_1 \to \hat F_2$ is a 2-morphism
That we moreover need the full smooth n-groupoid $[X, \mathbf{B}^n U(1)_{conn}]$ has several reasons, we discuss three. The third one of these is related to the higher gauge anomalies proper.
The magnetic twist $\hat \mathbf{c}$ 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 $\hat c$ and $c_{mag}$ are independent of the fields, the above definition of the groupoid of $\hat F$s 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 $G^{bg}$ be the gauge group of another background gauge field, and let $\mathbf{B}G^{bg}_{conn}$ be its moduli stack of gauge field configurations. If a $G^{bg}$-field induces magnetic current, then $\hat \mathbf{c}$ must depend on the fields, hence it should be a map
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 $X$, but just on the notion of $G^{bg}$-fields in general, so it should be given just by postcompositon
with a universal smooth map on moduli. With that given, the above picture for the $\hat \mathbf{c}$-twisted higher electric field becomes
We can then subsume all this and consider the smooth collection of all such twisting background fields $\phi^{bg}$ and twisted gauge fields $\hat F$. By general reasoning, this is given by the homotopy pullback that universally completes the above diagram
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 $[X, \mathbf{B}G_{conn}]$
Similarly for higher gauge theory it is the L-infinity algebroid.
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 $\Sigma_3$,
on $G$-principal connections over $\Sigma_3$. 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
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 $\Sigma_3 = \Sigma_2 \times Interval$ , 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 $\Sigma_2$. Generally, one expects to see a circle k-bundle with connection assigned to a $\Sigma$ of codimension $k$.
Above we had already seen such a structure in top codimension: if $\Sigma = *$ is the point, then $[\Sigma, \mathbf{B}G_{conn}] \simeq [*, \mathbf{B}G_{conn}] \simeq \mathbf{B}G_{conn}$. So there should be a cricle 3-bundle with connection on this moduli stack. Taking $G = Spin$, for definiteness, then the differential first fractional Pontryagin class from above is precisely of this form:
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
If now
is any universal differential characteristic map, and $\Sigma_k$ is compact closed of dimension $k$, then the composite
is a “$k$-extended action functional”.
An important class of ∞-Chern-Simons theories arising this way come from $\hat \mathbf{c}$ 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
and then forms its differential cup product (FSSd)
The action functional induced by this is that of $(4k+3)$-dimensional higher dimensional Chern-Simons theory which sends those $(2k+1)$-form fields $C$ whose underlying bundle happens to be trivial to $\exp(2 \pi i\int_{\Sigma_{4k+3}} C \wedge d_{dR} C)$.
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.
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
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
appears.
First, the kinetic piece of the action functional is simply $\exp(i S_{kin}(\hat F)) = \exp(i \int_X F \wedge \ast F)$.
But suppose there is a charged particle with trajectory $\gamma : S^1 \to X$. Then there is an interaction term $\exp(2\pi i \int_{S^1} \gamma^* A)$. Let $J_{el}$ be the Poincare dual form. Then $\cdots = \exp(i \int_X A \wedge J_{el})$.
This may be expressed using the Beilinson-Deligne cup product and the fiber integration in ordinary differential cohomology as $\exp(2 \pi i \int_X \hat \mathbf{c}_{el} \cup \hat F)$. (Here is where we need $J_{el}$ 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 $U$ on families of fields over $X \times U$ and then integrate out over $X$.
But in the case where there is non-trivial $\hat \mathbf{c}_{mag}$ this is no longer the case, there instead this is a trivialization of a twist
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
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.
The total action functional (higher gauge fields and fermions) is a section of the tensor product of these two
The action functional needs to be a flat section of $\nabla_{total\;anomaly}$. Hence the two line bundles need to be inverse to each other. This condition is the Green-Schwarz mechanism.
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 $\mathbf{B}O$ 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 structure | role in string theory |
---|---|
twisted differential spin^c structure | Freed-Witten anomaly cancellation for type II strings on D-branes |
twisted differential String^c-structure | flux quantization in 11d sugra/M-theory with M5-branes |
twisted differential string structure | Green-Schwarz mechanism in heterotic string theory |
twisted differential fivebrane structure | Green-Schwarz mechanism in dual heterotic string theory |
The Lie group spin^c is traditionally defined by the formula
which denotes the quotient of the product $Spin \times U(1)$ 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 $\mathbf{B}\mathbb{Z}_2$-higher fiber bundle classified by the smooth second Stiefel-Whitney class
and we also have the higher fiber bundle
which is $\mathbb{Z}_2$-associated to the universal principal bundle universal $\mathbb{Z}_2$-bundle.
One finds, using the presentation of these maps as discussed above, that $\mathbf{B}Spin^c$ is the corresponding associated bundle, namely the homotopy pullback
Equivalently we may write this as a Mayer-Vietoris sequence and thus obtain the universal local $\mathbf{B}Spin^c$-coefficient bundle over $\mathbf{B}^2 \mathbb{Z}$
We may read this as saying:
Where the moduli stack $\mathbf{B}Spin$ is the homotopy fiber of the smooth second Stiefel-Whitney class $\mathbf{w}_2$, the moduli stack $\mathbf{B}Spin^c$ is that homotopy fiber universally twisted by the smooth first Chern class $\mathbf{c}_1 mod 2$._
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
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 $\mathbf{B}Spin^c$-coefficient bundle over $\mathbf{B}^2 U(1)$, and this is given by the smooth third integral Stiefel-Whitney class:
Since that now lands in $\mathbf{B}^2 U(1)$, we can apply one more unitary twist by a corresponding class. The canonical such class is the universal Dixmier-Douady class $\mathbf{dd}$ of (stable) projective unitary bundles
This universal local coefficient bundle controls the Freed-Witten anomaly cancellation in type II string theory. To which we now turn.
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 $n$, the central extension of Lie groups
that exhibits the unitary group as a circle group-extension of the projective unitary group induces the corresponding morphism of smooth moduli stacks
This is part of a long fiber sequence which continues to the right by a connecting homomorphism $\mathbf{dd}_n$
in $\mathbf{H}$. Here the last morphism is presented in simplicial presheaves by the zig-zag of sheaves of crossed modules
We have seen above that a morphism $\phi : X \to \mathbf{B}^2 U(1)$ classifies a circle 2-bundle encoded by a Cech 2-cocycle $(\phi_{i j k} : U_i \cap U_j \cap U_k \to U(1))$ . This means that the universal local coefficient bundle
induces a notion of unitary bundles that are twisted by a 2-bundle.
Indeed, unwinding the definition one finds that a $\phi$-twisted $\mathbf{dd}_n$ cocycle
is, with respect to a resolution of $X$ a good open cover $\{U_i \to X\}$, given by maps $C(\{U_i\}) \to \mathbf{B}(U(1) \to U(n))$ whose components read
Hence these are collections of smooth $U(n)$-valued functions
which satisfy on triple overlaps the equation
If $\phi_{i j k}$ happens to be constant on the neutral element, then this is the condition for a cocycle in $H^1_{smooth}(X, U(n))$. So in general we say it is a $\phi$-twisted such cocycle. And that $(h_{i j})$ classifies a $\phi$-twisted unitary bundle.
In generalization of how unitary bundles constitute cocycles for K-theory, these $\phi$-twisted unitary bundles constitute cocycles for twisted K-theory.
For each $n \in \mathbb{N}$ there is a canonical inclusion $\mathbf{B} U(n) \to \mathbf{B} U(n+1)$, exhibiting the fact that a rank-$n$ complex vector bundle canonically induces a rank-$(n+1)$-bundle by added a trivial line bundle.
To get rid of the dependence on the rank $n$ – to stabilize the rank – we may form the directed colimit of smooth moduli stacks
Proposition The smooth stack $\mathbf{B} U$ is a smooth refinement of the classifying space $B U$ of reduced K-theory. Also, for $X$ a compact smooth manifold smooth $U$-principal bundles and smooth $U$-gauge transformations on $X$ are represented by ordinary $U(n)$-bundles for some finite $n$.
Now we think of the manifold $X$ as a target space for the type II superstring, hence assume it to be orientable and spin: $w_1(X) = 0$ and $w_2(X) = 0$. Consider moreover a submanifold
to be thought of as the worldvolume of a D-brane, which is also or orientable and spin.
Assume first that $Q$ also admits a spin^c structure, hence that also the third integral Stiefel-Whitney class vanishes $W_3(Q) = 0$.
We can consider then a cocycle in the $\iota$-relative cohomology with coefficients in $\mathbf{dd}$, namely a diagram
in $\mathbf{H}$.
This is
a circle 2-bundle $\phi_B : X \to \mathbf{B}^2 U(1)$ on spacetime $X$: the underlying bundle of the B-field;
a projective unitary bundle $\phi_{ga}$ on $Q$, a Chan-Paton bundle on the D-brane;
In cohomology this says that
This is the Freed-Witten anomaly cancellation condition for $D$-branes with spin^c structure.
More generally, if $Q$ does not necessarily have $Spin^c$-structure, we consider $\iota$-relative cohomology with coefficients in the universal local coefficient bundle $\mathbf{dd} - \mathbf{W}_3$:
This now is equivalently a twisted Chan-Paton bundle and a $B$-field such that in cohomology
This is the Freed-Witten anomaly cancellation condition for general $Q$.
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 $C$. We have the following information on how a model of this field content must behave globally
Due to the existence of spinors, the graviton must be part of a spin connection:
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
Due to the coupling to the M5-brane, there is an auxiliary E8-bundle
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
(Depending on convention one may write “$2 \phi_C$” for “$\phi_C$” here, regarding the physical $C$-field as being “one half” of the differential cocycle $X \to \mathbf{B}^3 U(1)_conn$ 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
in homotopy type theory is (see HoTT methods for homotopy theorists for how this works) the smooth $\infty$-stack $\mathbf{CField} \in \mathbf{H}$ given as the homotopy pullback
On the right this has the universal local coefficient bundle for $\mathbf{String}^{2\mathbf{a}}$ from above, and hence this identifies a gravity-C-field configuration as being (a partial differential refinement of) a $[\phi_C]$-twisted $String^{2\mathbf{a}}$-structure.
By Hořava-Witten theory, the 10-dimensional target spacetime of the heterotic string may be understood as being a boundary (or rather $\mathbb{Z}_2$-orbifold fixed points, see below) of the 11-dimensional spacetime of 11d SuGra. Over this boundary
the curvature 4-form $G_4(\phi_C)$ vanishes;
and the $E_8$-principal bundle picks up a connection.
This means that the above defining homotopy pullback for $\mathbf{CField}$ goes over into the one that defines the differential refinement $String^{\mathbf{a}}_{conn'}$ of $String^{\mathbf{a}}$:
On cohomology classes this means that
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 $\mathrm{String}^{\mathbf{a}}$-2-connections, there is, locally over a cover $\{U_i \to X\}$, also an equation of differential forms that exhibits this in de Rham cohomology.
A morphism $X \to String^{2\mathbf{a}}_{conn}$ classifies field content that is expressed with respect to a good open cover $\{U_i \to X\}$ in particular over single patches $U_i$ by (SSSa, FSSa)
the gauge connection $A_i \in \Omega^1(U_i, \mathfrak{e}_8 \oplus \mathfrak{e}_8)$;
the spin connection $\omega_i \in \Omega^1(U_i, \mathfrak{so})$ (the field of gravity in first order formulation of gravity);
the B-field $\phi_B \in \Omega^2(U_i)$;
which come with curvature/field strength forms
$F_{A_i} = d_{dR} A_i + \tfrac{1}{2}[A_i \wedge A_i]$
$F_{\omega_i} = d_{dR} \omega_i + \tfrac{1}{2}[\omega_i \wedge \omega_i]$;
$H_i = d B_i + CS(\omega_i) - CS(A_i)$ ($B$-field strength shifted by the difference of the Chern-Simons forms of $A_i$ and $\omega_i$);
satisfying the (twisted) Bianchi identities (SSSa))
$d_{dR} F_{A_i} + [A_i \wedge F_{A_i}] = 0$;
$d_{dR} F_{\omega_i} + [\omega_i \wedge F_{\omega_i}] = 0$;
$d_{dR} H = \langle F_\omega \wedge F_\omega \rangle - \langle F_A \wedge F_A \rangle$
(together with more local cocycle components on higher overlaps). Notably the twisted Bianchi identity of $H$ 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, $String^{\mathbf{a}}_{conn}$ is the smooth moduli 2-stack of anomaly free heterotic background fields (in the massless bosonic sector).
Notice that if the twist $\tfrac{1}{2}\hat \mathbf{p_1}(\phi_{gr}) - \hat\mathbf{a}(\phi_{gau})$ happen to vanish (say because both the field of gravity and the gauge field are trivial), then the above homotopy pullback reduces to
and exhibits $\phi_B$ as a genuine circle 2-bundle with connection (and its 3-form curvature $H$ with $\phi_C$.). Conversely, this shows how in the general situation $\phi_B$ is a twisted circle 2-bundle, with the twist given by the “magnetic fivebrane current” $\tfrac{1}{2}\hat \mathbf{p}_1(\phi_{gr}) - \hat \mathbf{a}(\phi_{ga})$.
(…)
An orientifold target space for the bosonic string is a smooth manifold or more generally orbifold $X$ equipped with
a double cover $w_X : X \to \mathbf{B} \mathbb{Z}_2$;
a twisted $\mathbb{Z}_2$-equivariant circle 2-bundle, given by a morphism $\phi_B : X \to \mathbf{B}Aut(\mathbf{B}U(1))$ whose underlying double cover is $w$.
This means that this background is a cocycle in $w$-twisted cohomology for the local coefficient bundle
Hence the B-field is now a cocycle in $w$-twisted cohomology
or rather a differential refinement thereof. For $\Sigma$ the worldvolume of a string, an orientifold string configuration is a cocycle
given in $\mathbf{H}$ by a diagram
consisting of
a map $\varphi : \Sigma \to X$;
an isomorphism $\nu : \varphi^* w_X \to \mathbf{w}_1(\Sigma)$.
In Hořava-Witten theory there is similarly a twisted $\mathbb{Z}_2$-action on the supergravity C-field, exhibited by a local coefficient bundle
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.
In work like Loop Groups and Twisted K-Theory the following structure plays a role:
for $G$ a group, let
be a fixed group homomorphism. Then for $E = E_0 \oplus E_1 \to X$ a super vector bundle, an “$\epsilon$-twist” of $E$ is an action of $G$ on $E$ such that an element $g \in G$ acts by an even automorphism if $\epsilon(g)$ is even, and by an odd automorphism if $\epsilon(g)$ 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
as the local coefficient bundle, and considering $\epsilon$ as the twist: then an $\epsilon$-twisting as above is a cocycle in the twisted cohomology $\mathbf{H}_{/\mathbf{B}\mathbb{Z}_2}(\mathbf{B}\epsilon, \mathbf{e})$ given by a commuting triangle
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 $\pi \in Grp(Set) \hookrightarrow Grp(Smooth\infty Grpd)$ a discrete group and for $\overline{X} \in \mathbf{H} \coloneqq$ Smooth∞Grpd any object (for instance a smooth manifold) a morphism $\phi_{\overline{X}} \colon \overline{X} \to \mathbf{B}\pi$ modulates a $\pi$-principal bundle $X \to \overline{X}$ over $X$, hence a free $\pi$-action on $X$ such that $X \to \overline{X}$ is the quotient map.
Then the corresponding twisted cohomology $\mathbf{H}_{/\mathbf{B}\pi}(-, \phi_{\overline{X}})$ has
domains are objects $\Sigma$ equipped with a $\pi$-principal bundle, modulated by a morphism $\phi_\Sigma \colon \Sigma \to \mathbf{B}\pi$;
cocycles are morphism $\phi_\Sigma \to \phi_X$ in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}\pi}$, hence diagrams of the form
in $\mathbf{H}$, hence maps $f \colon \Sigma \to \overline{X}$ equipped with equivalences
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 $\overline{G}$ and a central group extension $\pi \to G \to \overline{G}$. This is classified by a cocycle
Then the corresponding twisted cohomology $\mathbf{H}_{/\mathbf{B}^2 \pi}(-, \mathbf{c})$ has
domains are objects $\Sigma$ equipped with a $\mathbf{B}\pi$-principal 2-bundle/bundle gerbe modulated by a morphism $\phi_\Sigma \colon \Sigma \to \mathbf{B}^2 \pi$;
cocycles are morphism $\phi_\Sigma \to \mathbf{c}$ in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^2\pi}$, hence diagrams of the form
in $\mathbf{H}$, hence maps $f \colon \Sigma \to \overline{X}$ equipped with equivalences
between the principal 2-bundle/bundle gerbe $\mathbf{c}(f)$ induced by the $\overline{G}$-principal bundl modulated by $f$, and the one modulated by $\phi_\Sigma$.
Alternatively one can use here the differential refinement of $\mathbf{B}\overline{G}$ to the moduli stack $\mathbf{B}\overline{G}_{conn}$ of $\overline{G}$-principal connections.
Examples and further details are discussed in Schreiber, section 4. In (Freed-Teleman) this example appears as def. 4.6.
twisted Morava K-theory (Sati-Westerland).
(…)
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 $X,A \in \mathbf{H}$ any two objects, then
is the cohomology of $X$ with coefficients in $A$.
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 $T \mathbf{H}$ of a cohesive (∞,1)-topos $\mathbf{H}$.
For more on this see at tangent cohesion.
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
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 $G \in Grp(\mathbf{H})$ an ∞-group of twists, the corresponding twisted cohomology is the plain cohomology of the slice (∞,1)-topos
Precisely, for $\rho \in \mathbf{H}_{\mathbf{B}G}$ a $G$-∞-action on some $V$ incarnated as its universal associated ∞-bundle (the local coefficient ∞-bundle) and for $\chi \colon X \longrightarrow \mathbf{B}G$ a twist, then the $\chi$-twisted cohomology with local coefficients in $V$ is
The observation that for the differential cohomology-refinement of this twisted geometric cohomology it is the adjoint quadruple of (∞,1)-functors between $\mathbf{H}$ to the base (∞,1)-topos (“cohesion”)
(with Disc and coDisc full and faithful (∞,1)-functors and the fundamental ∞-groupoid/geometric realization $\Pi$ 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 $\Pi$ means that $\mathbf{H}$ is a locally ∞-connected (∞,1)-topos and a globally ∞-connected (∞,1)-topos, and the existence of $coDisc$ means that it is a local (∞,1)-topos.
Examples of cohesive higher geometry established and discussed in (Schreiber 11) include
Euclidean topological ∞-groupoids $Sh_\infty(TopMfd)$ is cohesive
smooth ∞-groupoids $Sh_\infty(SmoothMfd)$ is cohesive
super ∞-groupoids $Sh_\infty(SuperPoints)$ is cohesive;
smooth super ∞-groupoids $Sh_\infty(SuperMfd)$ is cohesive over super ∞-groupoids;
synthetic differential ∞-groupoids $Sh_\infty(FormalMfds)$ is even “differentially cohesive”, which allows to axiomatize also the notions of manifold, jet bundle etc.;
synthetic differential super ∞-groupoids $Sh_\infty(FormalSuperMfds)$ is differentially cohesive over super ∞-groupoids.
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 $Stab(Sh_\infty(SmthMfd))$ of spectrum object in smooth ∞-groupoids, which carries an analogous adjoint quadruple over the (∞,1)-category of spectra $Stab(\infty Grpd) \simeq Spectra$
But this stable aspect is unified with the unstable cohesion by the notion of “tangent cohesion”. This we turn to now.
We first discuss generally how the tangent (∞,1)-category $T \mathbf{H}$ of an (∞,1)-topos $\mathbf{H}$ is itself an (∞,1)-topos over the tangent $\infty$-category of the original base (∞,1)-topos (Joyal 08). Then we observe that $T \mathbf{H}$ is cohesive if $\mathbf{H}$ is and is in fact an extension of the latter by its stabilization. a choice
Let $seq$ be the diagram category as follows:
Given an (∞,1)-topos $\mathbf{H}$, an (∞,1)-functor
is equivalently
a choice of object $B \in \mathbf{H}$ (the image of $\ast in seq$]);
a sequence of objects $\{X_n\} \in \mathbf{H}_{/B}$ in the slice (∞,1)-topos over $B$;a choi
a sequence of morphisms $X_n \longrightarrow \Omega_B X_{n+1}$ from $X_n$ into the loop space object of $X_{n+1}$ in the slice.
This is a prespectrum object in the slice (∞,1)-topos $\mathbf{H}_{/B}$.
A natural transformation $f \;\colon \;X_\bullet \to Y_\bullet$ between two such functors with components
is equivalently a morphism of base objects $f_b \;\colon\; B_1 \longrightarrow B_2$ in $\mathbf{H}$ together with morphisms $X_n \longrightarrow f_b^\ast Y_n$ into the (∞,1)-pullback of the components of $Y_\bullet$ along $f_b$.
Therefore the (∞,1)-presheaf (∞,1)-topos
is the codomain fibration of $\mathbf{H}$ with “fiberwise pre-stabilization”.
A genuine spectrum object is a prespectrum object for which all the structure maps $X_n \stackrel{\simeq}{\longrightarrow} \Omega_B X_{n+1}$ are equivalences. The full sub-(∞,1)-category
on the genuine spectrum objects is therefore the “fiberwise stabilization” of the codomain fibration, hence the tangent $(\infty,1)$-category.
(spectrification is left exact reflective)
The inclusion of spectrum objects into $\mathbf{H}$ is left reflective, hence it has a left adjoint (∞,1)-functor $L$ which preserves finite (∞,1)-limits.
Forming degreewise loop space objects constitutes an (∞,1)-functor $\Omega \colon \mathbf{H}^{seq} \longrightarrow \mathbf{H}^{seq}$ and by definition of $seq$ this comes with a natural transformation out of the identity
This in turn is compatible with $\Omega$ in that
Consider then a sufficiently deep transfinite composition $\rho^{tf}$. By the small object argument available in the presentable (∞,1)-category $\mathbf{H}$ this stabilizes, and hence provides a reflection $L \;\colon\; \mathbf{H}^{seq} \longrightarrow T \mathbf{H}$.
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.
For $\mathbf{H}$ an (∞,1)-topos over the base (∞,1)-topos $\infty Grpd$, its tangent (∞,1)-category $T \mathbf{H}$ is an (∞,1)-topos over the base $T \infty Grpd$ (and hence in particular also over $\infty Grpd$ itself).
By the the spectrification lemma 1 $T \mathbf{H}$ has a geometric embedding into the (∞,1)-presheaf (∞,1)-topos $\mathbf{H}^{seq}$, 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 $\mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd$ preserve finite (∞,1)-limits they preserve spectrum objects and hence their immediate (∞,1)-presheaf prolongation immediately restricts to the inclusion of spectrum objects
We may think of the tangent (∞,1)-topos $T \mathbf{H}$ as being an extension of $\mathbf{H}$ by its stabilization $Stab(\mathbf{H}) \simeq T_\ast \mathbf{H}$:
Crucial for the internal interpretation in homotopy type theory is that the homotopy types in $T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ are stable homotopy types.
Now consider the case that $\mathbf{H}$ is a cohesive (∞,1)-topos over ∞Grpd, in that there is an adjoint quadruple
with $Disc, coDisc$ being full and faithful (∞,1)-functors and $\Pi$ 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 $\mathbf{H}^{seq}$
Moreover, all three right adjoints preserves the (∞,1)-pullbacks involved in the characterization of spectrum objects and hence restrict to $T \mathbf{H}$
But then we have a further left adjoint given as the composite
Again since $L$ is a left exact (∞,1)-functor this composite $L \Pi$ preserves finite (∞,1)-products.
So it follows in conclusion that if $\mathbf{H}$ is a cohesive (∞,1)-topos then its tangent $(\infty,1)$-category $T \mathbf{H}$ is itself a cohesive (∞,1)-topos over the tangent $(\infty,1)$-category $T \infty Grpd$ of the base (∞,1)-topos, which is an extension of the cohesion of the $\infty$-topos $\mathbf{H}$ over $\infty Grpd$ by the cohesion of the stable $\infty$-category $Stab(\mathbf{H})$ over $Stab(\infty Grpd) \simeq Spec$:
Here
$\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the total space of a spectrum bundle;
its left adjoint is the tangent complex functor;
$base \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ 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 $E \in T_\ast \mathbf{H}$ and write $GL_1(E) \in Grp(\mathbf{H})$ for its ∞-group of units. Then the ∞-action of this on $E$ is (by the discussion there) exhibited by an object
More generally, for $Pic(E) \in \mathbf{H}$ the Picard ∞-groupoid of $E$ there is the universal (∞,1)-line bundle
Now for any object $X \in \mathbf{H}$ we have
then morphisms in $T \mathbf{H}$ from the latter to the former
are equivalently
a choice of twist of E-cohomology $\chi \;\colon \; X \longrightarrow \mathbf{B}GL_1(E)$;
an element in the $\chi$-twisted $E$-cohomology of $X$, hence $c \in E^{\bullet}(X,E)$.
If we consider the internal hom then we can use just $X$ instead of $X \times E$:
For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a geometric homotopy type and $E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ a spectrum object, then the internal hom/mapping stack
(with respect to the Cartesian closed monoidal (∞,1)-category structure on the (∞,1)-topos is equivalently the mapping spectrum
in that
Notice that as an object of $T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$, the object $X$ is the constant (∞,1)-presheaf on $seq$. By the formula for the internal hom in an (∞,1)-category of (∞,1)-presheaves we have
But since $X$ is constant the object $X \times \bullet$ is for each object of $seq$ the presheaf represented by that object. Therefore by the (∞,1)-Yoneda lemma it follows that
This is manifestly the same formula as for the mapping spectrum out of $\Sigma^\infty X$.
By the same kind of argument we have the following more general statement.
For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a geometric homotopy type, for $E \in E_\infty(\mathbf{H})$ an E-∞ ring with $(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H}$ its universal (∞,1)-line bundle over its Picard ∞-groupoid, then the internal hom/mapping stack
is the object whose
base homotopy type is the E-∞ ring $[X, Pic(E)]$ of $E$-twist on $X$;
whose spectrum bundle is the collection of $\chi$-twisted E-cohomology spectra for all twists $\chi$.
Let $T\mathbf{H}$ be a tangent cohesive $(\infty,1)$-topos and write $T_\ast \mathbf{H}$ for the stable (∞,1)-category of spectrum objects inside it.
For every $A \in T_\ast \mathbf{H}$ the naturality square
(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.
By cohesion and stability we have the diagram
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.
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, $A/\flat A$ may be identified with differential cycle data. Indeed, by stability and cohesion it is the flat de Rham coefficient object
of the suspension of $A$. So
exhibits $A$ as a differential cohomology-coefficient of the generalized cohomology theory $\Pi(A)$ (Bunke-Nikolaus-Völkl 13).
It follows by the discussion at differential cohomology in a cohesive topos that the further differential refinement $\widehat{A}$ of $A$ should be given by a further homotopy pullback
We describe the formulation of local prequantum field theory in a cohesive (∞,1)-topos $\mathbf{H}$ (lpqft).
A classical field theory/prequantum field theory is traditionally defined by an action functional: given a smooth space $\mathbf{Fields}_{traj}$ “of trajectories” of a given physical system, then the action functional is a smooth function
to the circle group. The idea of producing a quantum field theory from this is to
choose a linearization in the form of the group homomorphism $U(1) \longrightarrow GL_1(\mathbb{C})$ to the group of units of the complex numbers,
choose a measure $d\mu$ on $\mathbf{Fields}_{traj}$
and then declare that the integral (“path integral”)
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 $\mathbf{B}U(1)$ for the smooth universal moduli stack of circle group-principal bundles, then an action functional as above is equivalently a homotopy of the form
where on the right we used the universal property of the homotopy pullback diagram which exhibits the smooth circle group $U(1)$ as the loop space object of $\mathbf{B}U(1)$.
For instance for $X$ a smooth manifold (“spacetime”) and $\nabla \;\colon\; X \longrightarrow \mathbf{B}U(1)_{conn}$ a circle group-principal connection (“electromagnetic field on spacetime”) then for trajectories in $X$ of shape the circle, the canonical action functional (“Lorentz force gauge interaction”) is the holonomy functional
But more generally, if the trajectories have a boundary, hence if they are of the shape of an interval $I \coloneqq [0,1]$, then the holonomy functional on smooth loop space $[S^1, X]$ generalizes to the parallel transport on the path space $[I,X]$ and there it is no longer a function, but exists only as a homotopy of the form
Notice that this is a “local” description of the action functional: the data that determines it is the boundary
and from this the rest is induced by transgression.
A related class of examples are prequantized Lagrangian correspondences: Let
be a symplectic manifold. Then a symplectomorphism $f \;\colon\; X \longrightarrow X$ is a correspondence of the form
A prequantization of $(X,\omega)$ is a lift $\nabla$ in
and so a prequantized Lagrangian correspondence is
To conceptualize all this, write
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
for the (2,1)-category of correspondences in $\mathbf{H}$. Write $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$ for the slice (2,1)-topos over the smooth moduli stack of circle bundles with connection. Then the abovve diagrams are morphisms in $Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})$.
The automorphism group of $\nabla \in Corr_1(\mathbf{H}_{/\mathbf{B}U(1)_{conn}})$ is the quantomorphism group of $(X,\omega)$, hence the smooth group which is the Lie integration of the Poisson bracket Lie algebra of $(X,\omega)$.
A concrete smooth 1-parameter subgroup
is equivalently a choice $H \in C^\infty(X)$ of a smooth function and sends
where
$\exp(t \{H,-\})$ is the Hamiltonian flow induced by $H$;
$S_t = \int_0^t L$ is the Hamilton-Jacobi action functional, the integral of the Lagrangian of $H$, 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
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
be the (∞,n)-category of $n$-fold correspondences in $\mathbf{H}$. This is a symmetric monoidal (∞,n)-category under the objectwise Cartesian product in $\mathbf{H}$.
Smooth∞Grpd has the special property that it is cohesive in that it is equipped with an adjoint quadruple of adjoint (∞,1)-functors
which induce an adjoint triple of idempotent (∞,1)-monads/comonads
with $\Pi$ product-preserving, called
Here the shape modality $\Pi$ 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 $Bord_n$ for the (∞,n)-category of framed n-dimensional cobordisms.
is equivalently a choice of object $\mathbf{Fields} \in \mathbf{H}$. It sends a cobordism $\Sigma$ to the internal hom of its shape into the higher moduli stack $\mathbf{Fields}$:
(lpqft)
Under the Dold-Kan correspondence
we have for all $n \in \mathbb{N}$ an equivalence
in $\mathbf{H}$.
Consider the induced canonical inclusion
By the above we may regard this as an action functional for an $(n+1)$-dimensional prequantum field theory with moduli stack of fields being $\mathbf{\Omega}^{n+1}_{cl}$. As such we denote it
where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.
are equivalent to objects
This sends the dual point to $\exp(- i S)$ and sends the $k$-sphere to the transgression of $\exp(i S)$ to the mapping space $[S^k , \mathbf{Fields}]$.
(lpqft)
Consider the induced canonical inclusion
By the above we may regard this as an action functional for an $(n+1)$-dimensional prequantum field theory with moduli stack of fields being $\mathbf{\Omega}^{n+1}_{cl}$. As such we denote it
where the subscript is supposed to refer to “universal higher topological Yang-Mills theory”.
Observe that by the cobordism hypothesis $Bord_n$ is the free symmetric monoidal (∞,n)-category with fully dualizable objects generated from a single object $\ast$.
Let then
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 $(n-1)$.
Hence a boundary field theory is
A boundary field theory as above is equivalently a diagram in $\mathbf{H}$ of the form
The universal boundary condition for the universal higher topological Yang-Mills theory of example 1 is the higher moduli stack $\mathbf{B}^n U(1)_{conn}$ 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.
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 $G$ and 3d $G$-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:
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.
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 $String$-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).
Before actually quantizing a local prequantum field theory $\left[ \array{ \mathbf{Fields} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \mathbf{B}^n U(1)_{conn}}\right]$ as above, we choose linear coefficients, given by
a choice of ground E-∞ ring $E$
(playing the role of the complex numbers in plain quantum mechanics);
a choice of ∞-group homomorphism
from the ∞-group of phases to the ∞-group of units of $E$, hence an ∞-representation of the circle n-group on $E$
(playing the role of the canonical $U(1) \hookrightarrow \mathbb{C}^\times$ in plain quantum mechanics).
Then for $X \longrightarrow \mathbf{B}^n U(1)$ modulating a circle n-bundle on $X$, the composite
modulates the associated ∞-bundle, which is an $E$-(∞,1)-module bundle.
Specifically, given the higher prequantum bundle $\exp(i S) \;\colon\; \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn}$ as above, the composite
modulates the associated higher prequantum E-line bundle.
A section of $\chi$ 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 $(n-1)$ instead here in codimension $n$.)
Accordingly, the space of sections of $\chi$ is the higher space of quantum states in codimension 0.
If $X$ 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
This is also known as the $\chi$-twisted $E$-Thom spectrum of $X$ (Ando-Blumberg-Gepner 10).
a map $E \to E_{\bullet + \chi}(X)$ is a cycle in $\chi$-twisted $E$-generalized homology of $X$;
a map $E_{\bullet + \chi}(X) \to E$ is a cocycle in $\chi$-twisted $E$-generalized cohomology of $X$
Hence we write
Generally, for $\chi_i \colon X_i \to E Mod$ two $E$-(∞,1)-module bundles over two spaces, a map
is a cocycle in $(\chi_1, \chi_2)$-twisted bivariant $E$-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:
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
Therefore forming co-sections sends an integral kernel as above to a correspondence of $E$-(∞,1)-modules:
The actual quantization/path integral as a pull-push transform map now consists in forming dual morphism in $E Mod$ such as to turn one of the projections of such a correspondence arround a produce a quantum propagator
that maps the incoming quantum states/wavefunctions to the outgoing ones.
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
As such this will in general only exist for ∞-Dijkgraaf-Witten theory where $\mathbf{Fields}$ 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
(i.e. with an ${(-)}^{or}$-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
A path integral is then to be a monoidal functor of the form
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 $\exp(i S) \, d \mu$ to obtain the genuine quantized quantum field theory:
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
Fiber integration of $E$-modules along a map is forming the dual morphisms of pulling back $E$-modules.
The choice of measure against which one integrates is the choice of identification of dual objects.
More in detail, given a monoidal category $\mathcal{C}^\otimes$ and given a morphism
in $\mathcal{C}$, an fiber integration/push-forward/index map is just
forming the dual morphism $f^\vee \colon V_2^\vee \to V_1^\vee$;
such that equivalences $V_i^\vee \simeq V_i$ exhbiting self-dual objects exist (Poincaré duality) and have been chosen (orientation).
This allows in total to have a morphism between the same objects, but in the opposite direction
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 $E = H R = H \mathbb{C}$ for the Eilenberg-MacLane spectrum of the complex numbers. Then for $X$ a manifold, the mapping spectrum
is the ordinary cohomology of $X$, its dual the ordinary homology, with coefficients in $R$.
For $X$ a closed manifold, Poincaré duality asserts that $H R^\bullet(X) \in H R Mod$ is essentially a self-dual object, except for a shift in degree: a choice of orientation of $X$ induces an equivalence
Using this, for $f \colon X \to Y$ a map of closed manifolds of dimension $d$, a compatible choice of orientation of both $X$ and $Y$ induces from the canonical push-forward map $f_\ast$ on homology the Umkehr map/push-forward map on cohomology, by the composition
This is ordinary integration: if $X$ and $Y$ are smooth manifolds, then $H \mathbb{R}^\bullet(X)$ is modeled by differential forms on $X$, $PD_X$ is given by a choice of volume form and $f^! = \int_{f}$ 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 $X$, we should use a relative/fiberwise point of view and regard not duality in $E Mod$ itself, but in the functor categories $Func(X, E Mod)$, which is fiberwise duality in $E Mod$.
Accordingly, given an $E$-(∞,1)-module bundle
we form not just the mapping space $E^\bullet(X) = [X, E]$ as above, but form the space of sections of this bundle, which we write:
Here for $X$ a discrete ∞-groupoid
Consider now a morphism
along which we want to integrate, whith $\chi$ invertible in $Func(Y, E Mod)$: $\left(\chi^\vee\right)^\vee \simeq \chi$. $\left(\chi^\vee\right)^\vee \simeq \chi$.
Observe that we have the pair of adjoint triples of left/right Kan extensions and colimits/limits
Notice that $f^\ast$ preserves duals, but $f_!$ may not.
If $f_! f^\ast \chi^\vee$ is a dualizable object, say that a choice of twisted orientation of $f$ in $\chi$-twisted cohomology is a choice of $\beta \colon X \to E Mod$ together with a choice of a equivalence (if such exists) of the form
hence a choice of correction of $f_!$ preserving the duality of $f^\ast \chi$.
Then the $(f_! \dashv f^\ast)$ counit
induces the dual morphism
and under $\left[ p_! \left( - \right), E \right]$ this becomes
which is
This we may call the the twisted fiber integration along $f$ in $E$-cohomology, or the twisted $E$-index map of $f$, induced by $(\beta, PD)$. If $\beta = 0$ then we call $PD$ anorientation_ of $f$ in $\chi$-twisted cohomology.
Notice that
Under fiber integration in twisted cohomology, the twist may change.
Grading in cohomology is just one incarnation of twist. Hence the fact that the twist changes under duality was already seen above in the ordinary case of Poincaré duality in ordinary cohomology.
For the special case that $X$ is a manifold, Atiyah duality identifies the dual cohomology spectrum with the Thom space cohomology spectrum. Then a choice of orientation amounts to a choice of Thom isomorphism, as traditionally considered.
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.
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 $\mathbf{H}$. One regards (∞,1)-topos theory as part of homotopy theory, and, more specifically, the internal language of $\mathbf{H}$ 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.
A group object in an (∞,1)-topos is a groupal A-∞-homotopy type: an ∞-group.
By looping and delooping there is an equivalence
between group objects and pointed connected homotopy types.
If $pt : \mathbf{B}G$ is the essentially uniqe point of the connected type $\mathbf{B}G$, then the group type itself is simply
the type of auto-equivalences of $pt$ in $\mathbf{B}G$.
For $A$ a group object which admits an $n$-fold delooping $\mathbf{B}^n A$ and $X \in \mathbf{H}$ any object, we write
$\mathbf{H}(X, \mathbf{B}^n A)$ for the space of degree-$n$ $A$-cocycles $c : X \to A$;
$H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X, \mathbf{B}^n A)$ for the degree-$n$ $A$-cohomology of $X$.
For $G \in Grp(\mathbf{H})$, $G$-cocycles on $X$ have an equivalent geometric interpretation as $G$-principal ∞-bundles: these are types
over $X$ equipped with a $G$-action $\rho : P \times G \to P$ over $X$ such that the ∞-quotient of the action is $X$.
Theorem There is an equivalence between the ∞-groupoid of $G$-principal bundles over $X$, and the cocycle $\infty$-groupoid of $G$-cohomology over $X$.
From left to righr the equivalence is established by sending a cocycle $X\to \mathbf{B}G$ to its homotopy fiber.
(NSS)
With $G$ the type as in remark 6, the principal bundle corresponding to a coycle $\phi : X \to \mathbf{B}G$ is
And the action is
For $\mathbf{H}$ an (∞,1)-topos and $\mathbf{B}G \in \mathbf{H}$, the collection of morphisms into $\mathbf{B}G$ is the slice (∞,1)-topos $\mathbf{H}_{/ \mathbf{B}G}$.
The cohomology in $\mathbf{H}_{/\mathbf{B}G}$ is naturally interpreted as follows
a local coefficient object $\mathbf{c} : E \to \mathbf{B}G$ is a universal bundle of local coefficients;
a domain object $\phi : X \to \mathbf{B}G$ is a twisting bundle;
a cocycle
is equivalently a section of the (∞,1)-pullback bundle $\phi^* E \to X$;
the cohomology
is the $\phi$-twisted cohomology of $X$ (with local coefficients in the homotopy fiber $F$ of $\mathbf{c}$).
In the syntax of the theory the type of twisted cocycles is
(see here).
While on the right this expresses the collection of sections of the pullback bundle, the left hand side expresses explicitly a $\mathbf{B}G$-parameterized collection of cocycles $X(b) \to E(b)$.
Cocycles in twisted cohomology relative to a local coefficient bundle $\mathbf{c} : E \to \mathbf{B}G$ do not pull back along morphisms in $\mathbf{H}$ (unless $G$ is trivial), but do pull back along morphisms in $\mathbf{H}$ that are lifted to morphisms in the slice $\mathbf{H}_{/ \mathbf{B}G}$.
For
a universal local coefficient bundle in $\mathbf{H}$ and $\phi : X \to \mathbf{B}G$ a twist and $\sigma : X \to \mathbf{B}\hat G$ a section, hence a cocycle in $\phi$-twisted cohomology, the corresponding geometric object is the twisted ∞-bundle $\tilde \sigma$ on the total space $P$ of $\phi$
(NSS)
(…)
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.
A cohesive (∞,1)-topos is in particular equipped with a left derived adjoint $\Pi$ to the locally constant ∞-stack-functor $Disc$
We may think of $\Pi$ as being the functor that sends a smooth ∞-groupoid $X$ to its fundamental path ∞-groupoid
Under the identification (see homotopy hypothesis theorem)
this identifies with geometric realization of geometric homotopy types / ∞-stacks.
We say that a lift of a diagram in ∞Grpd through $\Pi$ is a geometric refinement of the diagram.
Write $\mathbf{\Pi} \coloneqq Disc \circ \Pi$; and $\flat \coloneqq Disc \circ \Gamma$.
For $G$ an ∞-group, write $\flat_{dR} \mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G$: the de Rham coefficient object for $\mathbf{B}G$.
There is a canonical morphism $\theta : G \to \flat_{dR} \mathbf{B}G$: this identifies as the canonical Maurer-Cartan form on the ∞-group $G$.
For $G = \mathbf{B}^n U(1)$ the circle (n+1)-group we call $curv \coloneqq_{\mathbf{B}^n U(1)} : \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1} U(1)$ the universal curvature class in degree $(n+1)$.
The $curv$-twisted cohomology in $\mathbf{H}$ identifies with ordinary differential cohomology.
(…)
∞-Chern-Weil theory introduction
(…)
(…)
(…)
An entry with discussion of and list of examples of twisted differential structures is
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
Domenico Fiorenza Chris Rogers, Urs Schreiber, Higher geometric prequantum theory, 2013 (arXiv:1304.0236)
Urs Schreiber, Classical field theory via Cohesive homotopy types
Joost Nuiten, Cohomological quantization of local boundary prequantum field theory, Master thesis Utrecht 2013
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