This entry is about the article
We generalize the topological notion of string structure
to that of higher string structures such as Fivebrane structure;
and to the smooth context.
Then we refine the notion of higher smooth String structure from smooth nonabelian cohomology to
and
We show
that the data known in quantum field theory as a Green-Schwarz mechanism encodes a cocycle in differential twisted cohomology. Concretely, we identitfy the Green-Schwarz mechanism in heterotic string theory with the existence of differential twisted String-structures
that the dual Green-Schwarz mechanism in dual heterotic string theory with differential twisted Fivebrane-structures.
Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in String theory, such as notably the Green-Schwarz mechanism. On the other hand, this mechanism, as well as the Freed- Witten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian $O(n)$- and $U(n)$-principal bundles underlying the tangent bundle and the gauge bundle on target space.
We unify the picture by describing differential nonabelian cohomology and differential twisted cohomology and apply it to these situations. We demonstrate that the Freed-Witten mechanism for the Kalb-Ramond field, the Green-Schwarz mechanism for the Kalb-Ramond field, as well as its magnetic dual version for the $H_7$-field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted $Spin(n)$-, twisted String structure - and twisted $Fivebrane(n)$- structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of $U(n)$. We work out the (nonabelian) ∞-Lie algebroid valued differential forms data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the string theory literature. The treatment for M-theory leads to models for the $C$-field and its dual in differential nonabelian cohomology.
Contents
1) Plan
this is the page seen here
2) Differential nonabelian cohomology
this is the section at differential nonabelian cohomology
3) Twisted topological structures
this is essentially the material at twisted cohomology
4) $L_\infty$-connections
this is essentially the material at Cartan-Ehresmann ∞-connection
5) Twisted differential structures
this is essentially the material at differential twisted cohomology
An interesting supply of motivations for and applications of generalized notions of cohomology arises in formal higher energy physics in the context of theories that combine and generalize Maxwell’s theory of the electromagnetic field, Einstein’s theory of the gravitational field and Yang-Mills' theory of general gauge fields.
In order to formalize and study certain phenomena exhibited by such higher background fields – such as
the Kalb-Ramond field in heterotic or type II supergravity
the RR field in type II supergravity
the supergravity C-field maximal supergravity –
Hopkins and Singer in their seminal work developed the general theory of differential refinements of generalized (Eilenberg-Steenrod) cohomology. Based on this, Freed explained certain subtle effects, previously observed semi-rigorously by physicists, systematically as phenomena exhibited by cocycles in generalized abelian differential cohomology.
This involves notably various twists of one kind of cohomology by another. The most familiar example is twisted K-theory, the cohomology theory that in formal high energy physics describes the RR field.
But the physical applications indicate that this is only the simplest example in a more general theory of twisted generalized cohomology. The next example is the famous Green-Schwarz mechanism in heterotic supergravity, which, as Freed explained, amounts to asserting a kind of twist of a higher differential cohomology class.
While this clarifies some of the structure, it remains noteworthy that the twist in the Green-Schwarz mechanism is fundamentally encoded not in abelian generalized coholomology, but by the class of a $G$-principal bundle for a nonabelian group $G$, hence by a class in nonabelian cohomology.
The coefficients of ordinary generalized cohomology are spectra. If connective, these are maximally abelian namely symmetric monoidal $\infty$-groupoids or equivalently – by the homotopy hypothesis – topological spaces. In contrast to that, degree $n$-nonabelian cohomology allows as coefficients arbitray n-groupoids, i.e. homotopy n-types. Such nonabelian cohomology is traditionally most familiar in the study of 1- and 2-gerbes as well as in the higher Schreier theory of nonabelian group extensions. Here $n$ may be $n = \infty$.
Using the language of Quillen model categories it was already realized in BrownAHT that generalized (Eilenberg-Steenrod) cohomology, abelian sheaf cohomology as well as nonabelian cohomology all describe hom-sets in homotopy categories of (pre)sheaves with values in infinity-groupoids – called simplicial presheaves
This perspective was later refined by Joyal and Jardine’s study of the model structure on simplicial presheaves. By the recent result of Lurie HTT we know that these constructions model precisely the theory of ∞-stacks in that they are (hypercomplete) models for ∞-stack (∞,1)-toposes.
This leads one to expect that a general theory of smooth cohomology that encompasses abelian as well as nonabelian phenomena concerns a smooth (∞,1)-topos of ∞-stacks on a site such as that of smooth loci: these may be regarded as ∞-Lie groupoids. This perspective on $\infty$-stacks as the truly general notion of cohomology is implied by Lurie’s very notion of $(\infty,1)$-topos as a context that “behaves like topological spaces” . It can be found made explicit for instance in Dug.
We therefore place ourselves in a general context in which all the familiar operations of the homotopy theory of topological spaces – such as notably fibration sequences – make sense and work as expected, but where these spaces may be something more general and/or richer than topological spaces: Specifically, for applications to physics in general and for our applications to differential cohomology in higher gauge theory in particular they may be smooth and wildly infinite-dimensional spaces. The general tool for describing contexts that behave like the category Top of topological spaces but are richer are (∞,1)-toposes of infinity-stacks.
We pick a well-adapted smooth (∞,1)-topos $\mathbf{H}$ of ∞-Lie groupoids as our context and then study the following questions:
What is the general notion of differential cohomology in $\mathbf{H}$?
What is the general notion of twisted cohomology in $\mathbf{H}$?
How does this describe phenomena exhibited by background fields in string theory?
How does this induce the corresponding quantum field theories of objects charged under these background fields?
The last of these four questions is our main motivation. Here, however, we don’t go into this last question except that it shall serve to motivate our answer to the first question:
In the context of general (nonabelian) cohomology we set up notions of
First notice how the (∞,1)-topos $\mathbf{H}$ comes with its notion of
Before stating our formulation of differential cohomology the reader familiar with functorial quantum field theory may find it helpful to consider the following motivation. Other readers should skip this motivation.
Recall that, as finally fully formalized in TFTClass, an $n$-dimensional topological quantum field theory is an $(\infty,n)$-functor
on the (∞,n)-category of cobordisms with values in an (∞,n)-category of something like $n$-vector spaces.
On the other hand, when we have on a target space object $X$ a background field under which an $n$-dimensional object is charged, we expect a notion of parallel transport and holonomy encoded by an $(\infty,n)$-functor
from bordisms equipped with maps into $X$, that assigns to an $n$-dimensional bordism its parallel transport or holonomy, as a morphism in $\mathcal{V}$.
The above suggests that those QFTs $Z$ that arise as sigma models in that they are encoded by a background field $\nabla$ are, in some sense, \emph{extensions}
along the obvious forgetful $(\infty,n)$-functor $\mathrm{Bord}_n(X) \to \mathrm{Bord}(X)$.
Whatever this extension procedure may be in detail, a necessary prerequisite for studying it is a good grasp of how to encode differential cocycles on $X$ in terms of functors on $\mathrm{Bord}_n(X)$.
In a series of articles BaezSchr,SWI, SWII, SWIII it was shown that gerbes with connection and more generally higher principal bundles with connection are indeed encoded as morphisms in $\mathbf{H}$ of the form
where the $\mathcal{P}_n(X)$ is essentially the subobject of $\mathrm{Bord}_n(X)$ consisting only of bordisms in $X$ that are homeomorphic to a disk: the path n-groupoid of $X$.
Let $G$ be a Lie group, $\mathbf{B}G$ the corresponding delooped one-object smooth groupoid in $\mathbf{H}$ – this is a smooth version of the classifying space of $G$ – and similarly $\mathbf{B}\mathrm{AUT}(G)$ the one-object 2-groupoid coming from the automorphism 2-group of $G$. Let $X$ be a manifold, then
Jim Stasheff: SUGGEST SPLITTING THAT IN TWO SINCE THE FIRST HALF NEEDS ONLY AN ORDINARY GROUP WHEREAS THE SECOND NEEDS 2-GROUPS, GERBES, FAKE-FLAT ETC ETC WHICH WOULD BE HANDLED BY AN INSERT
Urs Schreiber: yes, that needs more detail and polishing – will try to get to that soon
In particular let $\mathbf{B}\mathbf{B} U(1)$ be the smooth 2-groupoid with the Lie group $U(1)$ in degree 2, then
Here the notation means the following:
$\mathbf{B} U(1)$ is a smooth realization in our smooth (∞,1)-topos $\mathbf{H}$ of the familiar classifying space for $U(1)$-principal bundles. Similarly, $\mathbf{B}^n U(1)$ is a smooth model the iterated classifying space $\mathcal{B}^n U(1)$.
For $X$ and $A$ objects in $\mathbf{H}_{smooth}$, the notation $\mathbf{H}_{smooth}(X,A)$ or $\mathbf{H}(X,A)$ for short denotes the topological space or infinity-groupoid of maps from $X$ to $A$, homotopies between maps, homotopies between homotopies, etc.
In particular, the cohomology itself on $X$ with coefficients in $A$ is the the set of homotopy classes of such maps
Finally, fake flatness is a certain condition on differential structures that is invisible in ordinary connections on a bundle but appears in the generalized context considered here: for $A$ an n-truncated object the curvature? of a differential cocycle is a list of differential forms in degree 1 through $(n+1)$. For $n=2$ the 2-form curvature is sometimes called the “fake” curvature, whereas the “true” one is the 3-form curvature.
Also in the case of abelian coefficients $\mathbf{B}^n U(1)$ the fake-flatness condition is empty and plays no role. But in the nonabelian case, as well as in the abelian equivariant case, fake flatness is more restrictive than what one might expect. For full nonabelian differential cohomology the above is slightly too naive and replaced by the following.
For every object $X \in \mathbf{X}$ there is an object $\Pi(X) = colim \mathcal{P}_n(X)$, the path infinity-groupoid? of $X$. The $k$-cells of $\Pi(X)$ are generated from the existing $k$-cells of $X$ and from $k$-dimensional smooth paths (disk-shaped cobordisms) in $X$. See path infinity-groupoid? for details.
For any coefficient object $A \in \mathbf{H}$ we call $\mathbf{H}(\Pi(X), A)$ the flat differential $A$-cohomology of $X$ or the $A$-local systems on $X$.
For $A$ once deloopable there is a morphism
whose image $P(c)$ of an $A$-cocycle $c$ we call the characteristic curvature class of $c$.
Given a class $P \in \mathbf{H}(\Pi(X),\mathbf{B}A)$ we call the corresponding $P$-twisted flat differential $A$-cohomology $\mathbf{H}^{P}(\Pi(X),A)$ the differential A-valued cohomology with curvature $P$.
The last clause uses the following definition of twisted cohomology.
Recall that the main point of having an (infinity,1)-topos $\mathbf{H}$ that replaces the ordinary (infinity,1)-category Top is that familiar notions of homotopy theory all apply as expected in $\mathbf{H}$. In particular, there is a notion of homotopy pullback and fibration sequence in $\mathbf{H}_{synth}$.
Let $F \to G \to A$ be such a fibration sequence in $\mathbf{H}$. We have left-exactness of the Hom for any object $X \in \mathbf{H}$ we obtain the fibration sequence
of $\infty$-groupoids or topological spaces, that characterizes $F$-cocycles as those $G$-cocycles whose obstructing $A$-cocycle is trivializable.
For $c \in \mathbf{H}(X,A)$ any possibly nontrivial $A$-cocycle on $X$, define the $c$-twisted $F$-cohomology $\mathbf{H}^c(X,A)$ to be the homotopy pullback
We discuss the following exampleS and applicationS of twisted differential cohomology .
Jim I TAKE THIS AS AN OUTLINE
Urs yes, I will work on this
fibration sequence: $\mathbf{B}U(n) \to \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)$
twisting cocycle: lifting gerbe;
twisted cocycle: twisted bundles / gerbe modules
twisted Bianchi identity: $d F_\nabla = H_3$
occurence: Freed-Witten anomaly cancellation on D-brane
fibration sequence: $\mathbf{B}String(n) \to \mathbf{B} Spin(n) \stackrel{\frac{1}{2}p_1}{\to} \mathbf{B}^3 U(1)$
twisting cocycle: Chern-Simons 2-gerbe;
twisted cocycle: twisted nonabelian String-gerbe with conection
twisted Bianchi identity: $d H_3 \propto \langle F_\nabla \wedge F_\nabla \rangle$
occurence: Green-Schwarz anomaly cancellation
Proof.
(with Danny Stevenson and Christoph Wockel: (SSSS)) use BCSS model (BCSS) of $String(n)$ with Brylinski-McLaughlin construction of $\frac{1}{2}p_1$
(using (SaScSt I, SaScSt III):) compute local differential form data after differentiating smooth $\infty$-groupoids to L-infinity algebroids using the formalism of (SaScSt I)
for aspects of the twisted case see also
fibration sequence: $\mathbf{B}Fivebrane(n) \to \mathbf{B} String(n) \stackrel{\frac{1}{6}p_2}{\to} \mathbf{B}^7 U(1)$
twisting cocycle: Chern-Simons 6-gerbe;
twisted cocycle: twisted nonabelian Fivebrane-gerbe with connection
occurence: dual Green-Schwarz anomaly cancellation for NS 5-brane magnetic dual to string
fibration sequence: $\mathbf{B}^2 U(1) \to \mathbf{B} (U(1) \to \mathbb{Z}_2) \stackrel{}{\to} \mathbf{B} \mathbb{Z}_2$
twisting cocycle: $\mathbb{Z}_2$-orbifold;
twisted cocycle: orientifold gerbe / Jandl gerbe with connection
occurence: unoriented string
unwrap the above abstract nonsense and use the above results to find SchrSchwWal and the bosonic part of DiFrMo
this is the end of section 1) Plan