### cohomology

To continue from the previous post

Twisted Differential Cohomology

Ulrich Bunke gave a talk introducing differential cohomology theories, and Thomas Nikolaus gave one about a twisted version of such theories (unfortunately, perhaps in the wrong order). The idea here is that cohomology can give a classification of field theories, and if we don’t want the theories to be purely topological, we would need to refine this. A cohomology theory is a (contravariant) functorial way of assigning to any space $X$, which we take to be a manifold, a $\mathbb{Z}$-graded group: that is, a tower of groups of “cocycles”, one group for each $n$, with some coboundary maps linking them. (In some cases, the groups are also rings) For example, the group of differential forms, graded by degree.

Cohomology theories satisfy some axioms – for example, the Mayer-Vietoris sequence has to apply whenever you cut a manifold into parts. Differential cohomology relaxes one axiom, the requirement that cohomology be a homotopy invariant of $X$. Given a differential cohomology theory, one can impose equivalence relations on the differential cocycles to get a theory that does satisfy this axiom – so we say the finer theory is a “differential refinement” of the coarser. So, in particular, ordinary cohomology theories are classified by spectra (this is related to the Brown representability theorem), whereas the differential ones are represented by sheaves of spectra – where the constant sheaves represent the cohomology theories which happen to be homotopy invariants.

The “twisting” part of this story can be applied to either an ordinary cohomology theory, or a differential refinement of one (though this needs similarly refined “twisting” data). The idea is that, if $R$ is a cohomology theory, it can be “twisted” over $X$ by a map $\tau: X \rightarrow Pic_R$ into the “Picard group” of $R$. This is the group of invertible $R$-modules (where an $R$-module means a module for the cohomology ring assigned to $X$) – essentially, tensoring with these modules is what defines the “twisting” of a cohomology element.

An example of all this is twisted differential K-theory. Here the groups are of isomorphism classes of certain vector bundles over $X$, and the twisting is particularly simple (the Picard group in the topological case is just $\mathbb{Z}_2$). The main result is that, while topological twists are classified by appropriate gerbes on $X$ (for K-theory, $U(1)$-gerbes), the differential ones are classified by gerbes with connection.

Fusion Categories

Scott Morrison gave a talk about Classifying Fusion Categories, the point of which was just to collect together a bunch of results constructing particular examples. The talk opens with a quote by Rutherford: “All science is either physics or stamp collecting” – that is, either about systematizing data and finding simple principles which explain it, or about collecting lots of data. This talk was unabashed stamp-collecting, on the grounds that we just don’t have a lot of data to systematically understand yet – and for that very reason I won’t try to summarize all the results, but the slides are well worth a look-over. The point is that fusion categories are very useful in constructing TQFT’s, and there are several different constructions that begin “given a fusion category $\mathcal{C}$“… and yet there aren’t all that many examples, and very few large ones, known.

Scott also makes the analogy that fusion categories are “noncommutative finite groups” – which is a little confusing, since not all finite groups are commutative anyway – but the idea is that the symmetric fusion categories are exactly the representation categories of finite groups. So general fusion categories are a non-symmetric generalization of such groups. Since classifying finite groups turned out to be difficult, and involve a laundry-list of sporadic groups, it shouldn’t be too surprising that understanding fusion categories (which, for the symmetric case, include the representation categories of all these examples) should be correspondingly tricky. Since, as he points out, we don’t have very many non-symmetric examples beyond rank 12 (analogous to knowing only finite groups with at most 12 elements), it’s likely that we don’t have a very good understanding of these categories in general yet.

There were a couple of talks – one during the workshop by Sonia Natale, and one the previous week by Sebastian Burciu, whom I also had the chance to talk with that week – about “Equivariantization” of fusion categories, and some fairly detailed descriptions of what results. The two of them have a paper on this which gives more details, which I won’t summarize – but I will say a bit about the construction.

An “equivariantization” of a category $C$ acted on by a group $G$ is supposed to be a generalization of the notion of the set of fixed points for a group acting on a set.  The category $C^G$ has objects which consist of an object $x \in C$ which is fixed by the action of $G$, together with an isomorphism $\mu_g : x \rightarrow x$ for each $g \in G$, satisfying a bunch of unsurprising conditions like being compatible with the group operation. The morphisms are maps in $C$ between the objects, which form commuting squares for each $g \in G$. Their paper, and the talks, described how this works when $C$ is a fusion category – namely, $C^G$ is also a fusion category, and one can work out its fusion rules (i.e. monoidal structure). In some cases, it’s a “group theoretical” fusion category (it looks like $Rep(H)$ for some group $H$) – or a weakened version of such a thing (it’s Morita equivalent to ).

A nice special case of this is if the group action happens to be trivial, so that every object of $C$ is a fixed point. In this case, $C^G$ is just the category of objects of $C$ equipped with a $G$-action, and the intertwining maps between these. For example, if $C = Vect$, then $C^G = Rep(G)$ (in particular, a “group-theoretical fusion category”). What’s more, this construction is functorial in $G$ itself: given a subgroup $H \subset G$, we get an adjoint pair of functors between $C^G$ and $C^H$, which in our special case are just the induced-representation and restricted-representation functors for that subgroup inclusion. That is, we have a Mackey functor here. These generalize, however, to any fusion category $C$, and to nontrivial actions of $G$ on $C$. The point of their paper, then, is to give a good characterization of the categories that come out of these constructions.

Quantizing with Higher Categories

The last talk I’d like to describe was by Urs Schreiber, called Linear Homotopy Type Theory for Quantization. Urs has been giving evolving talks on this topic for some time, and it’s quite a big subject (see the long version of the notes above if there’s any doubt). However, I always try to get a handle on these talks, because it seems to be describing the most general framework that fits the general approach I use in my own work. This particular one borrows a lot from the language of logic (the “linear” in the title alludes to linear logic).

Basically, Urs’ motivation is to describe a good mathematical setting in which to construct field theories using ingredients familiar to the physics approach to “field theory”, namely… fields. (See the description of Kevin Walker’s talk.) Also, Lagrangian functionals – that is, the notion of a physical action. Constructing TQFT from modular tensor categories, for instance, is great, but the fields and the action seem to be hiding in this picture. There are many conceptual problems with field theories – like the mathematical meaning of path integrals, for instance. Part of the approach here is to find a good setting in which to locate the moduli spaces of fields (and the spaces in which path integrals are done). Then, one has to come up with a notion of quantization that makes sense in that context.

The first claim is that the category of such spaces should form a differentially cohesive infinity-topos which we’ll call $\mathbb{H}$. The “infinity” part means we allow morphisms between field configurations of all orders (2-morphisms, 3-morphisms, etc.). The “topos” part means that all sorts of reasonable constructions can be done – for example, pullbacks. The “differentially cohesive” part captures the sort of structure that ensures we can really treat these as spaces of the suitable kind: “cohesive” means that we have a notion of connected components around (it’s implemented by having a bunch of adjoint functors between spaces and points). The “differential” part is meant to allow for the sort of structures discussed above under “differential cohomology” – really, that we can capture geometric structure, as in gauge theories, and not just topological structure.

In this case, we take $\mathbb{H}$ to have objects which are spectral-valued infinity-stacks on manifolds. This may be unfamiliar, but the main point is that it’s a kind of generalization of a space. Now, the sort of situation where quantization makes sense is: we have a space (i.e. $\mathbb{H}$-object) of field configurations to start, then a space of paths (this is WHERE “path-integrals” are defined), and a space of field configurations in the final system where we observe the result. There are maps from the space of paths to identify starting and ending points. That is, we have a span:

$A \leftarrow X \rightarrow B$

Now, in fact, these may all lie over some manifold, such as $B^n(U(1))$, the classifying space for $U(1)$ $(n-1)$-gerbes. That is, we don’t just have these “spaces”, but these spaces equipped with one of those pieces of cohomological twisting data discussed up above. That enters the quantization like an action (it’s WHAT you integrate in a path integral).

Aside: To continue the parallel, quantization is playing the role of a cohomology theory, and the action is the twist. I really need to come back and complete an old post about motives, because there’s a close analogy here. If quantization is a cohomology theory, it should come by factoring through a universal one. In the world of motives, where “space” now means something like “scheme”, the target of this universal cohomology theory is a mild variation on just the category of spans I just alluded to. Then all others come from some functor out of it.

Then the issue is what quantization looks like on this sort of scenario. The Atiyah-Singer viewpoint on TQFT isn’t completely lost here: quantization should be a functor into some monoidal category. This target needs properties which allow it to capture the basic “quantum” phenomena of superposition (i.e. some additivity property), and interference (some actual linearity over $\mathbb{C}$). The target category Urs talked about was the category of $E_{\infty}$-rings. The point is that these are just algebras that live in the world of spectra, which is where our spaces already lived. The appropriate target will depend on exactly what $\mathbb{H}$ is.

But what Urs did do was give a characterization of what the target category should be LIKE for a certain construction to work. It’s a “pull-push” construction: see the link way above on Mackey functors – restriction and induction of representations are an example . It’s what he calls a “(2-monoidal, Beck-Chevalley) Linear Homotopy-Type Theory”. Essentially, this is a list of conditions which ensure that, for the two morphisms in the span above, we have a “pull” operation for some and left and right adjoints to it (which need to be related in a nice way – the jargon here is that we must be in a Wirthmuller context), satisfying some nice relations, and that everything is functorial.

The intuition is that if we have some way of getting a “linear gadget” out of one of our configuration spaces of fields (analogous to constructing a space of functions when we do canonical quantization over, let’s say, a symplectic manifold), then we should be able to lift it (the “pull” operation) to the space of paths. Then the “push” part of the operation is where the “path integral” part comes in: many paths might contribute to the value of a function (or functor, or whatever it may be) at the end-point of those paths, because there are many ways to get from A to B, and all of them contribute in a linear way.

So, if this all seems rather abstract, that’s because the point of it is to characterize very generally what has to be available for the ideas that appear in physics notions of path-integral quantization to make sense. Many of the particulars – spectra, $E_{\infty}$-rings, infinity-stacks, and so on – which showed up in the example are in a sense just placeholders for anything with the right formal properties. So at the same time as it moves into seemingly very abstract terrain, this approach is also supposed to get out of the toy-model realm of TQFT, and really address the trouble in rigorously defining what’s meant by some of the standard practice of physics in field theory by analyzing the logical structure of what this practice is really saying. If it turns out to involve some unexpected math – well, given the underlying issues, it would have been more surprising if it didn’t.

It’s not clear to me how far along this road this program gets us, as far as dealing with questions an actual physicist would like to ask (for the most part, if the standard practice works as an algorithm to produce results, physicists seldom need to ask what it means in rigorous math language), but it does seem like an interesting question.

This is the 100th entry on this blog! It’s taken a while, but we’ve arrived at a meaningless but convenient milestone. This post constitutes Part III of the posts on the topics course which I shared with Susama Agarwala. In the first, I summarized the core idea in the series of lectures I did, which introduced toposes and sheaves, and explained how, at least for appropriate sites, sheaves can be thought of as generalized spaces. In the second, I described the guest lecture by John Huerta which described how supermanifolds can be seen as an example of that notion.

In this post, I’ll describe the machinery I set up as part of the context for Susama’s talks. The connections are a bit tangential, but it gives some helpful context for what’s to come. Namely, my last couple of lectures were on sheaves with structure, and derived categories. In algebraic geometry and elsewhere, derived categories are a common tool for studying spaces. They have a cohomological flavour, because they involve sheaves of complexes (or complexes of sheaves) of abelian groups. Having talked about the background of sheaves in Part I, let’s consider how these categories arise.

### Structured Sheaves and Internal Constructions in Toposes

The definition of a (pre)sheaf as a functor valued in $Sets$ is the basic one, but there are parallel notions for presheaves valued in categories other than $Sets$ – for instance, in Abelian groups, rings, simplicial sets, complexes etc. Abelian groups are particularly important for geometry/cohomology.

But for the most part, as long as the target category can be defined in terms of sets and structure maps (such as the multiplication map for groups, face maps for simplicial sets, or boundary maps in complexes), we can just think of these in terms of objects “internal to a category of sheaves”. That is, we have a definition of “abelian group object” in any reasonably nice category – in particular, any topos. Then the category of “abelian group objects in $Sh(\mathcal{T})$” is equivalent to a category of “abelian-group-valued sheaves on $\mathcal{T}$“, denoted $Sh((\mathcal{T},J),\mathbf{AbGrp})$. (As usual, I’ll omit the Grothendieck topology $J$ in the notation from now on, though it’s important that it is still there.)

Sheaves of abelian groups are supposed to generalize the prototypical example, namely sheaves of functions valued in abelian groups, (indeed, rings) such as $\mathbb{Z}$, $\mathbb{R}$, or $\mathbb{C}$.

To begin with, we look at the category $Sh(\mathcal{T},\mathbf{AbGrp})$, which amounts to the same as the category of abelian group objects in  $Sh(\mathcal{T})$. This inherits several properties from $\mathbf{AbGrp}$ itself. In particular, it’s an abelian category: this gives us that there is a direct sum for objects, a zero object, exact sequences split, all morphisms have kernels and cokernels, and so forth. These useful properties all hold because at each $U \in \mathcal{T}$, the direct sum of sheaves of abelian group just gives $(A \oplus A')(U) = A(U) \oplus A'(U)$, and all the properties hold locally at each $U$.

So, sheaves of abelian groups can be seen as abelian groups in a topos of sheaves $Sh(\mathcal{T})$. In the same way, other kinds of structures can be built up inside the topos of sheaves, and there are corresponding “external” point of view. One good example would be simplicial objects: one can talk about the simplicial objects in $Sh(\mathcal{T},\mathbf{Set})$, or sheaves of simplicial sets, $Sh(\mathcal{T},\mathbf{sSet})$. (Though it’s worth noting that since simplicial sets model infinity-groupoids, there are more sophisticated forms of the sheaf condition which can be applied here. But for now, this isn’t what we need.)

Recall that simplicial objects in a category $\mathcal{C}$ are functors $S \in Fun(\Delta^{op},\mathcal{C})$ – that is, $\mathcal{C}$-valued presheaves on $\Delta$, the simplex category. This $\Delta$ has nonnegative integers as its objects, and the morphisms from $n$ to $m$ are the order-preserving functions from $\{ 1, 2, \dots, n \}$ to $\{ 1, 2, \dots, m \}$. If $\mathcal{C} = \mathbf{Sets}$, we get “simplicial sets”, where $S(n)$ is the “set of $n$-dimensional simplices”. The various morphisms in $\Delta$ turn into (composites of) the face and degeneracy maps. Simplicial sets are useful because they are a good model for “spaces”.

Just as with abelian groups, simplicial objects in $Sh(\mathcal{T})$ can also be seen as sheaves on $\mathcal{T}$ valued in the category $\mathbf{sSet}$ of simplicial sets, i.e. objects of $Sh(\mathcal{T},\mathbf{sSet})$. These things are called, naturally, “simplicial sheaves”, and there is a rather extensive body of work on them. (See, for instance, the canonical book by Goerss and Jardine.)

This correspondence is just because there is a fairly obvious bunch of isomorphisms turning functors with two inputs into functors with one input returning another functor with one input:

$Fun(\Delta^{op} \times \mathcal{T}^{op},\mathbf{Sets}) \cong Fun(\Delta^{op}, Fun(\mathcal{T}^{op}, \mathbf{Sets}))$

and

$Fun(\Delta^{op} \times \mathcal{T}^{op},\mathbf{Sets}) \cong Fun(\mathcal{T}^{op},Fun(\Delta^{op},\mathbf{Sets})$

(These are all presheaf categories – if we put a trivial topology on $\Delta$, we can refine this to consider only those functors which are sheaves in every position, where we use a certain product topology on $\Delta \times \mathcal{T}$.)

Another relevant example would be complexes. This word is a bit overloaded, but here I’m referring to the sort of complexes appearing in cohomology, such as the de Rahm complex, where the terms of the complex are the sheaves of differential forms on a space, linked by the exterior derivative. A complex $X^{\bullet}$ is a sequence of Abelian groups with boundary maps $\partial^i : X^i \rightarrow X^{i+1}$ (or just $\partial$ for short), like so:

$\dots \rightarrow^{\partial} X^0 \rightarrow^{\partial} X^1 \rightarrow^{\partial} X^2 \rightarrow^{\partial} \dots$

with the property that $\partial^{i+1} \circ \partial^i = 0$. Morphisms between these are sequences of morphisms between the terms of the complexes $(\dots,f_0,f_1,f_2,\dots)$ where each $f_i : X^i \rightarrow Y^i$ which commute with all the boundary maps. These all assemble into a category of complexes $C^{\bullet}(\mathbf{AbGrp})$. We also have $C^{\bullet}_+$ and $C^{\bullet}_-$, the (full) subcategories of complexes where all the negative (respectively, positive) terms are trivial.

One can generalize this to replace $\mathbf{AbGrp}$ by any category enriched in abelian groups, which we need to make sense of the requirement that a morphism is zero. In particular, one can generalize it to sheaves of abelian groups. This is an example where the above discussion about internalization can be extended to more than one structure at a time: “sheaves-of-(complexes-of-abelian-groups)” is equivalent to “complexes-of-(sheaves-of-abelian-groups)”.

This brings us to the next point, which is that, within $Sh(\mathcal{T},\mathbf{AbGrp})$, the last two examples, simplicial objects and complexes, are secretly the same thing.

### Dold-Puppe Correspondence

The fact I just alluded to is a special case of the Dold-Puppe correspondence, which says:

Theorem: In any abelian category $\mathcal{A}$, the category of simplicial objects $Fun(\Delta^{op},\mathcal{A})$ is equivalent to the category of positive chain complexes $C^{\bullet}_+(\mathcal{A})$.

The better-known name “Dold-Kan Theorem” refers to the case where $\mathcal{A} = \mathbf{AbGrp}$. If $\mathcal{A}$ is a category of $\mathbf{AbGrp}$-valued sheaves, the Dold-Puppe correspondence amounts to using Dold-Kan at each $U$.

The point is that complexes have only coboundary maps, rather than a plethora of many different face and boundary maps, so we gain some convenience when we’re looking at, for instance, abelian groups in our category of spaces, by passing to this equivalent description.

The correspondence works by way of two maps (for more details, see the book by Goerss and Jardine linked above, or see the summary here). The easy direction is the Moore complex functor, $N : Fun(\Delta^{op},\mathcal{A} \rightarrow C^{\bullet}_+(\mathcal{A})$. On objects, it gives the intersection of all the kernels of the face maps:

$(NS)_k = \bigcap_{j=1}^{k-1} ker(d_i)$

The boundary map from this is then just $\partial_n = (-1)^n d_n$. This ends up satisfying the “boundary-squared is zero” condition because of the identities for the face maps.

The other direction is a little more complicated, so for current purposes, I’ll leave you to follow the references above, except to say that the functor $\Gamma$ from complexes to simplicial objects in $\mathcal{A}$ is defined so as to be adjoint to $N$. Indeed, $N$ and $\Gamma$ together form an adjoint equivalence of the categories.

### Chain Homotopies and Quasi-Isomorphisms

One source of complexes in mathematics is in cohomology theories. So, for example, there is de Rahm cohomology, where one starts with the complex with $\Omega^n(M)$ the space of smooth differential $n$-forms on some smooth manifold $M$, with the exterior derivatives as the coboundary maps. But no matter which complex you start with, there is a sequence of cohomology groups, because we have a sequence of cohomology functors:

$H^k : C^{\bullet}(\mathcal{A}) \rightarrow \mathcal{A}$

given by the quotients

$H^k(A^{\bullet}) = Ker(\partial_k) / Im(\partial_{k-1})$

That is, it’s the cocycles (things whose coboundary is zero), up to equivalence where cocycles are considered equivalent if their difference is a coboundary (i.e. something which is itself the coboundary of something else). In fact, these assemble into a functor $H^{\bullet} : C^{\bullet}(\mathcal{A}) \rightarrow C^{\bullet}(\mathcal{A})$, since there are natural transformations between these functors

$\delta^k(A^{\bullet}) : H^k(A^{\bullet} \rightarrow H^{k+1}(A^{\bullet})$

which just come from the restrictions of the $\partial^k$ to the kernel $Ker(\partial^k)$. (In fact, this makes the maps trivial – but the main point is that this restriction is well-defined on equivalence classes, and so we get an actual complex again.) The fact that we get a functor means that any chain map $f^{\bullet} : A^{\bullet} \rightarrow B^{\bullet}$ gives a corresponding $H^{\bullet}(f^{\bullet}) : H^{\bullet}(A^{\bullet}) \rightarrow H^{\bullet}(B^{\bullet})$.

Now, the original motivation of cohomology for a space, like the de Rahm cohomology of a manifold $M$, is to measure something about the topology of $M$. If $M$ is trivial (say, a contractible space), then its cohomology groups are all trivial. In the general setting, we say that $A^{\bullet}$ is acyclic if all the $H^k(A^{\bullet}) = 0$. But of course, this doesn’t mean that the chain itself is zero.

More generally, just because two complexes have isomorphic cohomology, doesn’t mean they are themselves isomorphic, but we say that $f^{\bullet}$ is a quasi-isomorphism if $H^{\bullet}(f^{\bullet})$ is an isomorphism. The idea is that, as far as we can tell from the information that coholomology detects, it might as well be an isomorphism.

Now, for spaces, as represented by simplicial sets, we have a similar notion: a map between spaces is a quasi-isomorphism if it induces an isomorphism on cohomology. Then the key thing is the Whitehead Theorem (viz), which in this language says:

Theorem: If $f : X \rightarrow Y$ is a quasi-isomorphism, it is a homotopy equivalence.

That is, it has a homotopy inverse $f' : Y \rightarrow X$, which means there is a homotopy $h : f' \circ f \rightarrow Id$.

What about for complexes? We said that in an abelian category, simplicial objects and complexes are equivalent constructions by the Dold-Puppe correspondence. However, the question of what is homotopy equivalent to what is a bit more complicated in the world of complexes. The convenience we gain when passing from simplicial objects to the simpler structure of complexes must be paid for it with a little extra complexity in describing what corresponds to homotopy equivalences.

The usual notion of a chain homotopy between two maps $f^{\bullet}, g^{\bullet} : A^{\bullet} \rightarrow B^{\bullet}$ is a collection of maps which shift degrees, $h^k : A^k \rightarrow B^{k-1}$, such that $f-g = \partial \circ h$. That is, the coboundary of $h$ is the difference between $f$ and $g$. (The “co” version of the usual intuition of a homotopy, whose ingoing and outgoing boundaries are the things which are supposed to be homotopic).

The Whitehead theorem doesn’t work for chain complexes: the usual “naive” notion of chain homotopy isn’t quite good enough to correspond to the notion of homotopy in spaces. (There is some discussion of this in the nLab article on the subject. That is the reason for…

### Derived Categories

Taking “derived categories” for some abelian category can be thought of as analogous, for complexes, to finding the homotopy category for simplicial objects. It compensates for the fact that taking a quotient by chain homotopy doesn’t give the same “homotopy classes” of maps of complexes as the corresponding operation over in spaces.

That is, simplicial sets, as a model category, know everything about the homotopy type of spaces: so taking simplicial objects in $\mathcal{C}$ is like internalizing the homotopy theory of spaces in a category $\mathcal{C}$. So, if what we’re interested in are the homotopical properties of spaces described as simplicial sets, we want to “mod out” by homotopy equivalences. However, we have two notions which are easy to describe in the world of complexes, which between them capture the notion “homotopy” in simplicial sets. There are chain homotopies and quasi-isomorphisms. So, naturally, we mod out by both notions.

So, suppose we have an abelian category $\mathcal{A}$. In the background, keep in mind the typical example where $\mathcal{A} = Sh( (\mathcal{T},J), \mathbf{AbGrp} )$, and even where $\mathcal{T} = TOP(X)$ for some reasonably nice space $X$, if it helps to picture things. Then the derived category of $\mathcal{A}$ is built up in a few steps:

1. Take the category $C^{\bullet} ( \mathcal{A} )$ of complexes. (This stands in for “spaces in $\mathcal{A}$” as above, although we’ve dropped the “$+$“, so the correct analogy is really with spectra. This is a bit too far afield to get into here, though, so for now let’s just ignore it.)
2. Take morphisms only up to homotopy equivalence. That is, define the equivalence relation with $f \sim g$ whenever there is a homotopy $h$ with $f-g = \partial \circ h$.  Then $K^{\bullet}(\mathcal{A}) = C^{\bullet}(\mathcal{A})/ \sim$ is the quotient by this relation.
3. Localize at quasi-isomorphisms. That is, formally throw in inverses for all quasi-isomorphisms $f$, to turn them into actual isomorphisms. The result is $D^{\bullet}(\mathcal{A})$.

(Since we have direct sums of complexes (componentwise), it’s also possible to think of the last step as defining $D^{\bullet}(\mathcal{A}) = K^{\bullet}(\mathcal{A})/N^{\bullet}(\mathcal{A})$, where $N^{\bullet}(\mathcal{A})$ is the category of acyclic complexes – the ones whose cohomology complexes are zero.)

Explicitly, the morphisms of $D^{\bullet}(\mathcal{A})$ can be thought of as “zig-zags” in $K^{\bullet}(\mathcal{A})$,

$X^{\bullet}_0 \leftarrow X^{\bullet}_1 \rightarrow X^{\bullet}_2 \leftarrow \dots \rightarrow X^{\bullet}_n$

where all the left-pointing arrows are quasi-isomorphisms. (The left-pointing arrows are standing in for their new inverses in $D^{\bullet}(\mathcal{A})$, pointing right.) This relates to the notion of a category of spans: in a reasonably nice category, we can always compose these zig-zags to get one of length two, with one leftward and one rightward arrow. In general, though, this might not happen.

Now, the point here is that this is a way of extracting “homotopical” or “cohomological” information about $\mathcal{A}$, and hence about $X$ if $\mathcal{A} = Sh(TOP(X),\mathbf{AbGrp})$ or something similar. In the next post, I’ll talk about Susama’s series of lectures, on the subject of motives. This uses some of the same technology described above, in the specific context of schemes (which introduces some extra considerations specific to that world). It’s aim is to produce a category (and a functor into it) which captures all the cohomological information about spaces – in some sense a universal cohomology theory from which any other can be found.

Since the last post, I’ve been busily attending some conferences, as well as moving to my new job at the University of Hamburg, in the Graduiertenkolleg 1670, “Mathematics Inspired by String Theory and Quantum Field Theory”.  The week before I started, I was already here in Hamburg, at the conference they were organizing “New Perspectives in Topological Quantum Field Theory“.  But since I last posted, I was also at the 20th Oporto Meeting on Geometry, Topology, and Physics, as well as the third Higher Structures in China workshop, at Jilin University in Changchun.  Right now, I’d like to say a few things about some of the highlights of that workshop.

Higher Structures in China III

So last year I had a bunch of discussions I had with Chenchang Zhu and Weiwei Pan, who at the time were both in Göttingen, about my work with Jamie Vicary, which I wrote about last time when the paper was posted to the arXiv.  In that, we showed how the Baez-Dolan groupoidification of the Heisenberg algebra can be seen as a representation of Khovanov’s categorification.  Chenchang and Weiwei and I had been talking about how these ideas might extend to other examples, in particular to give nice groupoidifications of categorified Lie algebras and quantum groups.

That is still under development, but I was invited to give a couple of talks on the subject at the workshop.  It was a long trip: from Lisbon, the farthest-west of the main cities of (continental) Eurasia all the way to one of the furthest-East.   (Not quite the furthest, but Changchun is in the northeast of China, just a few hours north of Korea, and it took just about exactly 24 hours including stopovers to get there).  It was a long way to go for a three day workshop, but as there were also three days of a big excursion to Changbai Mountain, just on the border with North Korea, for hiking and general touring around.  So that was a sort of holiday, with 11 other mathematicians.  Here is me with Dany Majard, in a national park along the way to the mountains:

Here’s me with Alex Hoffnung, on Changbai Mountain (in the background is China):

And finally, here’s me a little to the left of the previous picture, where you can see into the volcanic crater.  The lake at the bottom is cut out of the picture, but you can see the crater rim, of which this particular part is in North Korea, as seen from China:

Well, that was fun!

Anyway, the format of the workshop involved some talks from foreigners and some from locals, with a fairly big local audience including a good many graduate students from Jilin University.  So they got a chance to see some new work being done elsewhere – mostly in categorification of one kind or another.  We got a chance to see a little of what’s being done in China, although not as much as we might have. I gather that not much is being done yet that fit the theme of the workshop, which was part of the reason to organize the workshop, and especially for having a session aimed specially at the graduate students.

### Categorified Algebra

This is a sort of broad term, but certainly would include my own talk.  The essential point is to show how the groupoidification of the Heisenberg algebra is a representation of Khovanov’s categorification of the same algebra, in a particular 2-category.  The emphasis here is on the fact that it’s a representation in a 2-category whose objects are groupoids, but whose morphisms aren’t just functors, but spans of functors – that is, composites of functors and co-functors.  This is a pretty conservative weakening of “representations on categories” – but it lets one build really simple combinatorial examples.  I’ve discussed this general subject in recent posts, so I won’t elaborate too much.  The lecture notes are here, if you like, though – they have more detail than my previous post, but are less technical than the paper with Jamie Vicary.

Aaron Lauda gave a nice introduction to the program of categorifying quantum groups, mainly through the example of the special case $U_q(sl_2)$, somewhat along the same lines as in his introductory paper on the subject.  The story which gives the motivation is nice: one has knot invariants such as the Jones polynomial, based on representations of groups and quantum groups.  The Jones polynomial can be categorified to give Khovanov homology (which assigns a complex to a knot, whose graded Euler characteristic is the Jones polynomial) – but also assigns maps of complexes to cobordisms of knots.  One then wants to categorify the representation theory behind it – to describe actions of, for instance, quantum $sl_2$ on categories.  This starting point is nice, because it can work by just mimicking the construction of $sl_2$ and $U_q(sl_2)$ representations in terms of weight spaces: one gets categories $V_{-N}, \dots, V_N$ which correspond to the “weight spaces” (usually just vector spaces), and the $E$ and $F$ operators give functors between them, and so forth.

Finding examples of categories and functors with this structure, and satisfying the right relations, gives “categorified representations” of the algebra – the monoidal categories of diagrams which are the “categorifications of the algebra” then are seen as the abstraction of exactly which relations these are supposed to satisfy.  One such example involves flag varieties.  A flag, as one might eventually guess from the name, is a nested collection of subspaces in some $n$-dimensional space.  A simple example is the Grassmannian $Gr(1,V)$, which is the space of all 1-dimensional subspaces of $V$ (i.e. the projective space $P(V)$), which is of course an algebraic variety.  Likewise, $Gr(k,V)$, the space of all $k$-dimensional subspaces of $V$ is a variety.  The flag variety $Fl(k,k+1,V)$ consists of all pairs $W_k \subset W_{k+1}$, of a $k$-dimensional subspace of $V$, inside a $(k+1)$-dimensional subspace (the case $k=2$ calls to mind the reason for the name: a plane intersecting a given line resembles a flag stuck to a flagpole).  This collection is again a variety.  One can go all the way up to the variety of “complete flags”, $Fl(1,2,\dots,n,V)$ (where $V$ is $n$-dimenisonal), any point of which picks out a subspace of each dimension, each inside the next.

The way this relates to representations is by way of geometric representation theory. One can see those flag varieties of the form $Fl(k,k+1,V)$ as relating the Grassmanians: there are projections $Fl(k,k+1,V) \rightarrow Gr(k,V)$ and $Fl(k,k+1,V) \rightarrow Gr(k+1,V)$, which act by just ignoring one or the other of the two subspaces of a flag.  This pair of maps, by way of pulling-back and pushing-forward functions, gives maps between the cohomology rings of these spaces.  So one gets a sequence $H_0, H_1, \dots, H_n$, and maps between the adjacent ones.  This becomes a representation of the Lie algebra.  Categorifying this, one replaces the cohomology rings with derived categories of sheaves on the flag varieties – then the same sort of “pull-push” operation through (derived categories of sheaves on) the flag varieties defines functors between those categories.  So one gets a categorified representation.

Heather Russell‘s talk, based on this paper with Aaron Lauda, built on the idea that categorified algebras were motivated by Khovanov homology.  The point is that there are really two different kinds of Khovanov homology – the usual kind, and an Odd Khovanov Homology, which is mainly different in that the role played in Khovanov homology by a symmetric algebra is instead played by an exterior (antisymmetric) algebra.  The two look the same over a field of characteristic 2, but otherwise different.  The idea is then that there should be “odd” versions of various structures that show up in the categorifications of $U_q(sl_2)$ (and other algebras) mentioned above.

One example is the fact that, in the “even” form of those categorifications, there is a natural action of the Nil Hecke algebra on composites of the generators.  This is an algebra which can be seen to act on the space of polynomials in $n$ commuting variables, $\mathbb{C}[x_1,\dots,x_n]$, generated by the multiplication operators $x_i$, and the “divided difference operators” based on the swapping of two adjacent variables.  The Hecke algebra is defined in terms of “swap” generators, which satisfy some $q$-deformed variation of the relations that define the symmetric group (and hence its group algebra).   The Nil Hecke algebra is so called since the “swap” (i.e. the divided difference) is nilpotent: the square of the swap is zero.  The way this acts on the objects of the diagrammatic category is reflected by morphisms drawn as crossings of strands, which are then formally forced to satisfy the relations of the Nil Hecke algebra.

The ODD Nil Hecke algebra, on the other hand, is an analogue of this, but the $x_i$ are anti-commuting, and one has different relations satisfied by the generators (they differ by a sign, because of the anti-commutation).  This sort of “oddification” is then supposed to happen all over.  The main point of the talk was to to describe the “odd” version of the categorified representation defined using flag varieties.  Then the odd Nil Hecke algebra acts on that, analogously to the even case above.

Marco Mackaay gave a couple of talks about the $sl_3$ web algebra, describing the results of this paper with Weiwei Pan and Daniel Tubbenhauer.  This is the analog of the above, for $U_q(sl_3)$, describing a diagram calculus which accounts for representations of the quantum group.  The “web algebra” was introduced by Greg Kuperberg – it’s an algebra built from diagrams which can now include some trivalent vertices, along with rules imposing relations on these.  When categorifying, one gets a calculus of “foams” between such diagrams.  Since this is obviously fairly diagram-heavy, I won’t try here to reproduce what’s in the paper – but an important part of is the correspondence between webs and Young Tableaux, since these are labels in the representation theory of the quantum group – so there is some interesting combinatorics here as well.

### Algebraic Structures

Some of the talks were about structures in algebra in a more conventional sense.

Jiang-Hua Lu: On a class of iterated Poisson polynomial algebras.  The starting point of this talk was to look at Poisson brackets on certain spaces and see that they can be found in terms of “semiclassical limits” of some associative product.  That is, the associative product of two elements gives a power series in some parameter $h$ (which one should think of as something like Planck’s constant in a quantum setting).  The “classical” limit is the constant term of the power series, and the “semiclassical” limit is the first-order term.  This gives a Poisson bracket (or rather, the commutator of the associative product does).  In the examples, the spaces where these things are defined are all spaces of polynomials (which makes a lot of explicit computer-driven calculations more convenient). The talk gives a way of constructing a big class of Poisson brackets (having some nice properties: they are “iterated Poisson brackets”) coming from quantum groups as semiclassical limits.  The construction uses words in the generating reflections for the Weyl group of a Lie group $G$.

Li Guo: Successors and Duplicators of Operads – first described a whole range of different algebra-like structures which have come up in various settings, from physics and dynamical systems, through quantum field theory, to Hopf algebras, combinatorics, and so on.  Each of them is some sort of set (or vector space, etc.) with some number of operations satisfying some conditions – in some cases, lots of operations, and even more conditions.  In the slides you can find several examples – pre-Lie and post-Lie algebras, dendriform algebras, quadri- and octo-algebras, etc. etc.  Taken as a big pile of definitions of complicated structures, this seems like a terrible mess.  The point of the talk is to point out that it’s less messy than it appears: first, each definition of an algebra-like structure comes from an operad, which is a formal way of summing up a collection of operations with various “arities” (number of inputs), and relations that have to hold.  The second point is that there are some operations, “successor” and “duplicator”, which take one operad and give another, and that many of these complicated structures can be generated from simple structures by just these two operations.  The “successor” operation for an operad introduces a new product related to old ones – for example, the way one can get a Lie bracket from an associative product by taking the commutator.  The “duplicator” operation takes existing products and introduces two new products, whose sum is the previous one, and which satisfy various nice relations.  Combining these two operations in various ways to various starting points yields up a plethora of apparently complicated structures.

Dany Majard gave a talk about algebraic structures which are related to double groupoids, namely double categories where all the morphisms are invertible.  The first part just defined double categories: graphically, one has horizontal and vertical 1-morphisms, and square 2-morphsims, which compose in both directions.  Then there are several special degenerate cases, in the same way that categories have as degenerate cases (a) sets, seen as categories with only identity morphisms, and (b) monoids, seen as one-object categories.  Double categories have ordinary categories (and hence monoids and sets) as degenerate cases.  Other degenerate cases are 2-categories (horizontal and vertical morphisms are the same thing), and therefore their own special cases, monoidal categories and symmetric monoids.  There is also the special degenerate case of a double monoid (and the extra-special case of a double group).  (The slides have nice pictures showing how they’re all degenerate cases).  Dany then talked about some structure of double group(oids) – and gave a list of properties for double groupoids, (such as being “slim” – having at most one 2-cell per boundary configuration – as well as two others) which ensure that they’re equivalent to the semidirect product of an abelian group with the “bicrossed product”  $H \bowtie K$ of two groups $H$ and $K$ (each of which has to act on the other for this to make sense).  He gave the example of the Poincare double group, which breaks down as a triple bicrossed product by the Iwasawa decomposition:

$Poinc = (SO(3) \bowtie (SO(1; 1) \bowtie N)) \ltimes \mathbb{R}_4$

($N$ is certain group of matrices).  So there’s a unique double group which corresponds to it – it has squares labelled by $\mathbb{R}_4$, and the horizontial and vertical morphisms by elements of $SO(3)$ and $N$ respectively.  Dany finished by explaining that there are higher-dimensional analogs of all this – $n$-tuple categories can be defined recursively by internalization (“internal categories in $(n-1)$-tuple-Cat”).  There are somewhat more sophisticated versions of the same kind of structure, and finally leading up to a special class of $n$-tuple groups.  The analogous theorem says that a special class of them is just the same as the semidirect product of an abelian group with an $n$-fold iterated bicrossed product of groups.

Also in this category, Alex Hoffnung talked about deformation of formal group laws (based on this paper with various collaborators).  FGL’s are are structures with an algebraic operation which satisfies axioms similar to a group, but which can be expressed in terms of power series.  (So, in particular they have an underlying ring, for this to make sense).  In particular, the talk was about formal group algebras – essentially, parametrized deformations of group algebras – and in particular for Hecke Algebras.  Unfortunately, my notes on this talk are mangled, so I’ll just refer to the paper.

### Physics

I’m using the subject-header “physics” to refer to those talks which are most directly inspired by physical ideas, though in fact the talks themselves were mathematical in nature.

Fei Han gave a series of overview talks intorducing “Equivariant Cohomology via Gauged Supersymmetric Field Theory”, explaining the Stolz-Teichner program.  There is more, using tools from differential geometry and cohomology to dig into these theories, but for now a summary will do.  Essentially, the point is that one can look at “fields” as sections of various bundles on manifolds, and these fields are related to cohomology theories.  For instance, the usual cohomology of a space $X$ is a quotient of the space of closed forms (so the $k^{th}$ cohomology, $H^{k}(X) = \Omega^{k}$, is a quotient of the space of closed $k$-forms – the quotient being that forms differing by a coboundary are considered the same).  There’s a similar construction for the $K$-theory $K(X)$, which can be modelled as a quotient of the space of vector bundles over $X$.  Fei Han mentioned topological modular forms, modelled by a quotient of the space of “Fredholm bundles” – bundles of Banach spaces with a Fredholm operator around.

The first two of these examples are known to be related to certain supersymmetric topological quantum field theories.  Now, a TFT is a functor into some kind of vector spaces from a category of $(n-1)$-dimensional manifolds and $n$-dimensional cobordisms

$Z : d-Bord \rightarrow Vect$

Intuitively, it gives a vector space of possible fields on the given space and a linear map on a given spacetime.  A supersymmetric field theory is likewise a functor, but one changes the category of “spacetimes” to have both bosonic and fermionic dimension.  A normal smooth manifold is a ringed space $(M,\mathcal{O})$, since it comes equipped with a sheaf of rings (each open set has an associated ring of smooth functions, and these glue together nicely).  Supersymmetric theories work with manifolds which change this sheaf – so a $d|\delta$-dimensional space has the sheaf of rings where one introduces some new antisymmetric coordinate functions $\theta_i$, the “fermionic dimensions”:

$\mathcal{O}(U) = C^{\infty}(U) \otimes \bigwedge^{\ast}[\theta_1,\dots,\theta_{\delta}]$

Then a supersymmetric TFT is a functor:

$E : (d|\delta)-Bord \rightarrow STV$

(where $STV$ is the category of supersymmetric topological vector spaces – defined similarly).  The connection to cohomology theories is that the classes of such field theories, up to a notion of equivalence called “concordance”, are classified by various cohomology theories.  Ordinary cohomology corresponds then to $0|1$-dimensional extended TFT (that is, with 0 bosonic and 1 fermionic dimension), and $K$-theory to a $1|1$-dimensional extended TFT.  The Stoltz-Teichner Conjecture is that the third example (topological modular forms) is related in the same way to a $2_1$-dimensional extended TFT – so these are the start of a series of cohomology theories related to various-dimension TFT’s.

Last but not least, Chris Rogers spoke about his ideas on “Higher Geometric Quantization”, on which he’s written a number of papers.  This is intended as a sort of categorification of the usual ways of quantizing symplectic manifolds.  I am still trying to catch up on some of the geometry This is rooted in some ideas that have been discussed by Brylinski, for example.  Roughly, the message here is that “categorification” of a space can be thought of as a way of acting on the loop space of a space.  The point is that, if points in a space are objects and paths are morphisms, then a loop space $L(X)$ shifts things by one categorical level: its points are loops in $X$, and its paths are therefore certain 2-morphisms of $X$.  In particular, there is a parallel to the fact that a bundle with connection on a loop space can be thought of as a gerbe on the base space.  Intuitively, one can “parallel transport” things along a path in the loop space, which is a surface given by a path of loops in the original space.  The local description of this situation says that a 1-form (which can give transport along a curve, by integration) on the loop space is associated with a 2-form (giving transport along a surface) on the original space.

Then the idea is that geometric quantization of loop spaces is a sort of higher version of quantization of the original space. This “higher” version is associated with a form of higher degree than the symplectic (2-)form used in geometric quantization of $X$.   The general notion of n-plectic geometry, where the usual symplectic geometry is the case $n=1$, involves a $(n+1)$-form analogous to the usual symplectic form.  Now, there’s a lot more to say here than I properly understand, much less can summarize in a couple of paragraphs.  But the main theorem of the talk gives a relation between n-plectic manifolds (i.e. ones endowed with the right kind of form) and Lie n-algebras built from the complex of forms on the manifold.  An important example (a theorem of Chris’ and John Baez) is that one has a natural example of a 2-plectic manifold in any compact simple Lie group $G$ together with a 3-form naturally constructed from its Maurer-Cartan form.

At any rate, this workshop had a great proportion of interesting talks, and overall, including the chance to see a little more of China, was a great experience!

I’ve written here before about building topological quantum field theories using groupoidification, but I haven’t yet gotten around to discussing a refinement of this idea, which is in the most recent version of my paper on the subject.  I also gave a talk about this last year in Erlangen. The main point of the paper is to pull apart some constructions which are already fairly well known into two parts, as part of setting up a category which is nice for supporting models of fairly general physical systems, using an extension of the  concept of groupoidification. So here’s a somewhat lengthy post which tries to unpack this stuff a bit.

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

$Z : 3Cob \rightarrow Vect$

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

$A^{\omega} : nCob_2 \rightarrow Span_2(Gpd)^{U(1)}$

This depends on a cohomology class $[\omega] \in H^3(G,U(1))$. The “quantization functor” (which in this case I call “2-linearization”):

$\Lambda^{U(1)} : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

The middle stage involves the monoidal 2-category I call $Span_2(Gpd)^{U(1)}$.  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through $Span_2(Gpd)$.

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both $Span_2(Gpd)^{U(1)}$ and $\Lambda^{U(1)}$.

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category $Span(Gpd)$, whose objects are groupoids ($A$, $B$, etc.), whose morphisms are spans of groupoids:

$A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B$

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps $s,t$ pick out its starting and ending configuration.  That is, $A = A_G(S)$ is the groupoid of flat $G$-connections on a manifold $S$, and $X = A_G(\Sigma)$ is the groupoid of flat $G$-connections on some cobordism $\Sigma$, of which $S$ is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is $s$.

Now 2-linearization is a 2-functor:

$\Lambda : Span_2(Gpd)^{U(1)} \rightarrow 2Vect$

It gives a 2-vector space (a nice kind of category) for each groupoid $G$.  Specifically, the category of its representations, $Rep(G)$.  Then a span turns into a functor which comes from “pulling” back along $s$ (the restricted representation where $X$ acts by first applying $s$ then the representation), then “pushing” forward along $t$ (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of $Y$), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if $S=0$, so that what’s integrated over paths (“histories”) is just $e^{iS}=1$. So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid $G$ (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an $n$-cocycle is some function of $n$ composable morphisms of $G$ (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always $U(1)$

The trivial case where $n=0$ is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case $n=1$ is maybe a little more obvious. A 1-cochain $f \in Z^1_{gpd}(G,U(1))$ can measure how a function $h$ on objects might fail to be a 0-cocycle. It is a $U(1)$-valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

$f(g_1 \circ g_2) = f(g_1) f(g_2)$

This condition means that a cochain $f$ is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in $U(1)$. It will automatically by satisfied for a coboundary (i.e. if $f$ comes from a function $h$ on objects as $f(g) = \delta h (g) = h(t(g)) - h(s(g))$). But not every cocycle is a coboundary: the first cohomology $H^1(G,U(1))$ is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges $g_1$, $g_2$, and $g_1 \circ g_2$.  It says that if we go from the source to the target of the sequence $(g_1, g_2)$ with or without composing, and accumulate $f$-values, our $f$ gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an $n$-simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation $R$ of a group is not a representation in the strict sense. That would be a map into $End(V)$, such that $R(g_1) \circ R(g_2) = R(g_1 \circ g_2)$.  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space $V$.  A twisted representation $\rho$ only satisfies this up to a phase:

$\rho(g_1) \circ \rho(g_2) = \theta(g_1,g_2) \rho(g_1 \circ g_2)$

where $\theta : G^2 \rightarrow U(1)$ is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: $\theta$ is a “cocycle” exactly when this twisting still makes $\rho$ respect the associative law:

$\rho(g_1) \rho( g_2 \circ g_3) = \rho( g_1 \circ g_2) \circ \rho( g_3)$

Working out what this says in terms of $\theta$, the cocycle condition says that for any composable triple $(g_1, g_2, g_3)$ we have:

$\theta( g_1, g_2 \circ g_3) \theta (g_2,g_3) = \theta(g_1,g_2) \theta(g_1 \circ g_2, g_3)$

So $H^2_{grp}(G,U(1))$ – the second group-cohomology group of $G$ – consists of exactly these $\theta$ which satisfy this condition, which ensures we have associativity.

Given one of these $\theta$ maps, we get a category $Rep^{\theta}(G)$ of all the $\theta$-twisted representations of $G$. It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of $G$, called $C^{\theta}(G)$. The point is, we can use $\theta$ to twist the convolution product for functions on $G$, and this is still an associative algebra just because $\theta$ satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category $Span(Gpd)^{U(1)}$ is that it contains $Span(Gpd)$, but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization $\Lambda$.  So in particular, what we have is a (symmetric monoidal) 2-category where:

• Objects consist of $(A, \theta)$, where $A$ is a groupoid and $\theta \in Z^2(A,U(1))$
• Morphisms from $A$ to $B$ consist of a span $(X,s,t)$ from $A$ to $B$, together with $\alpha \in Z^1(X,U(1))$
• 2-Morphisms from $X_1$ to $X_2$ consist of a span $(Y,\sigma,\tau)$ from $X$, together with $\beta \in Z^0(Y,U(1))$

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that $Span(Gpd)$ is a subcategory of $Span(Gpd)^{U(1)}$. The 2-linearization functor extends to $\Lambda^{U(1)} : Span(Gpd)^{U(1)} \rightarrow 2Vect$:

• On Objects: $\Lambda^{U(1)} (A, \theta) = Rep^{\theta}(A)$, the category of $\theta$-twisted representation of $A$
• On Morphisms: $\Lambda^{U(1)} ( (X,s,t) , \alpha )$ comes by pulling back a twisted representation in $Rep^{\theta_A}(A)$ to one in $Rep^{s^{\ast}\theta_A}(X)$, pulling it through the algebra map “multiplication by $\alpha$“, and pushing forward to $Rep^{\theta_B}(B)$
• On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects $y \in Y$ picks up a weight of $\beta(y)$ at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that $\Lambda^{U(1)}$ is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class $[\omega] \in H^3_{grp}(G,U(1))$. But this is the same thing as a class in the cohomology of the classifying space: $[\omega] \in H^3(BG,U(1))$. This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) $S$, which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in $S$:

$A : \pi_1(M) \rightarrow G$

But this amounts to the same thing as a map into the classifying space of $G$:

$f_A : M \rightarrow BG$

This is essentially the definition of $BG$, and it implies various things, such as the fact that $BG$ is a space whose fundamental group is $G$, and has all other homotopy groups trivial. That is, $BG$ is the Eilenberg-MacLane space $K(G,1)$. But the point is that the groupoid of connections and gauge transformations on $S$ just corresponds to the mapping space $Maps(S,BG)$. So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given $[\omega] \in H^3(BG,U(1))$, then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle $\omega$ (representing the class $[\omega]$) amounts to a nice 3-form on $BG$ which we can integrate over a 3-dimentional submanifold to get a number. For a $d$-dimensional $S$, we get such a 3-manifold from a $(3-d)$-dimensional submanifold of $Maps(S,BG)$: each point gives a copy of $S$ in $BG$. Then we get a $(3-d)$-cocycle on $Maps(S,BG)$ whose values come from integrating $\omega$ over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of $3Cob_2$, 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in $BG$, these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of $Span^{U(1)}(Gpd)$. This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of $\omega$ as seen by transgression: pulling back along the evaluation map $ev$ from the picture. Then the way you use it in the quantization is described exactly by $\Lambda^{U(1)}$.

What I like about this is that $\Lambda^{U(1)}$ is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of $BG$ which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

As usual, this write-up process has been taking a while since life does intrude into blogging for some reason.  In this case, because for a little less than a week, my wife and I have been on our honeymoon, which was delayed by our moving to Lisbon.  We went to the Azores, or rather to São Miguel, the largest of the nine islands.  We had a good time, roughly like so:

Now that we’re back, I’ll attempt to wrap up with the summaries of things discussed at the workshop on Higher Gauge Theory, TQFT, and Quantum Gravity.  In the previous post I described talks which I roughly gathered under TQFT and Higher Gauge Theory, but the latter really ramifies out in a few different ways.  As began to be clear before, higher bundles are classified by higher cohomology of manifolds, and so are gerbes – so in fact these are two slightly different ways of talking about the same thing.  I also remarked, in the summary of Konrad Waldorf’s talk, the idea that the theory of gerbes on a manifold is equivalent to ordinary gauge theory on its loop space – which is one way to make explicit the idea that categorification “raises dimension”, in this case from parallel transport of points to that of 1-dimensional loops.  Next we’ll expand on that theme, and then finally reach the “Quantum Gravity” part, and draw the connection between this and higher gauge theory toward the end.

## Gerbes and Cohomology

The very first workshop speaker, in fact, was Paolo Aschieri, who has done a lot of work relating noncommutative geometry and gravity.  In this case, though, he was talking about noncommutative gerbes, and specifically referred to this work with some of the other speakers.  To be clear, this isn’t about gerbes with noncommutative group $G$, but about gerbes on noncommutative spaces.  To begin with, it’s useful to express gerbes in the usual sense in the right language.  In particular, he explain what a gerbe on a manifold $X$ is in concrete terms, giving Hitchin’s definition (viz).  A $U(1)$ gerbe can be described as “a cohomology class” but it’s more concrete to present it as:

• a collection of line bundles $L_{\alpha \beta}$ associated with double overlaps $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$.  Note this gets an algebraic structure (multiplication $\star$ of bundles is pointwise $\otimes$, with an inverse given by the dual, $L^{-1} = L^*$, so we can require…
• $L_{\alpha \beta}^{-1} \cong L_{\beta \alpha}$, which helps define…
• transition functions $\lambda _{\alpha \beta \gamma}$ on triple overlaps $U_{\alpha \beta \gamma}$, which are sections of $L_{\alpha \beta \gamma} = L_{\alpha \beta} \star L_{\beta \gamma} \star L_{\gamma \alpha}$.  If this product is trivial, there’d be a 1-cocycle condition here, but we only insist on the 2-cocycle condition…
• $\lambda_{\beta \gamma \delta} \lambda_{\alpha \gamma \delta}^{-1} \lambda_{\alpha \beta \delta} \lambda_{\alpha \beta \gamma}^{-1} = 1$

This is a $U(1)$-gerbe on a commutative space.  The point is that one can make a similar definition for a noncommutative space.  If the space $X$ is associated with the algebra $A=C^{\infty}(X)$ of smooth functions, then a line bundle is a module for $A$, so if $A$ is noncommutative (thought of as a “space” $X$), a “bundle over $X$ is just defined to be an $A$-module.  One also has to define an appropriate “covariant derivative” operator $D$ on this module, and the $\star$-product must be defined as well, and will be noncommutative (we can think of it as a deformation of the $\star$ above).  The transition functions are sections: that is, elements of the modules in question.  his means we can describe a gerbe in terms of a big stack of modules, with a chosen algebraic structure, together with some elements.  The idea then is that gerbes can give an interpretation of cohomology of noncommutative spaces as well as commutative ones.

Mauro Spera spoke about a point of view of gerbes based on “transgressions”.  The essential point is that an $n$-gerbe on a space $X$ can be seen as the obstruction to patching together a family of  $(n-1)$-gerbes.  Thus, for instance, a $U(1)$ 0-gerbe is a $U(1)$-bundle, which is to say a complex line bundle.  As described above, a 1-gerbe can be understood as describing the obstacle to patching together a bunch of line bundles, and the obstacle is the ability to find a cocycle $\lambda$ satisfying the requisite conditions.  This obstacle is measured by the cohomology of the space.  Saying we want to patch together $(n-1)$-gerbes on the fibre.  He went on to discuss how this manifests in terms of obstructions to string structures on manifolds (already discussed at some length in the post on Hisham Sati’s school talk, so I won’t duplicate here).

A talk by Igor Bakovic, “Stacks, Gerbes and Etale Groupoids”, gave a way of looking at gerbes via stacks (see this for instance).  The organizing principle is the classification of bundles by the space maps into a classifying space – or, to get the category of principal $G$-bundles on, the category $Top(Sh(X),BG)$, where $Sh(X)$ is the category of sheaves on $X$ and $BG$ is the classifying topos of $G$-sets.  (So we have geometric morphisms between the toposes as the objects.)  Now, to get further into this, we use that $Sh(X)$ is equivalent to the category of Étale spaces over $X$ – this is a refinement of the equivalence between bundles and presheaves.  Taking stalks of a presheaf gives a bundle, and taking sections of a bundle gives a presheaf – and these operations are adjoint.

The issue at hand is how to categorify this framework to talk about 2-bundles, and the answer is there’s a 2-adjunction between the 2-category $2-Bun(X)$ of such things, and $Fib(X) = [\mathcal{O}(X)^{op},Cat]$, the 2-category of fibred categories over $X$.  (That is, instead of looking at “sheaves of sets”, we look at “sheaves of categories” here.)  The adjunction, again, involves talking stalks one way, and taking sections the other way.  One hard part of this is getting a nice definition of “stalk” for stacks (i.e. for the “sheaves of categories”), and a good part of the talk focused on explaining how to get a nice tractable definition which is (fibre-wise) equivalent to the more natural one.

Bakovic did a bunch of this work with Branislav Jurco, who was also there, and spoke about “Nonabelian Bundle 2-Gerbes“.  The paper behind that link has more details, which I’ve yet to entirely absorb, but the essential point appears to be to extend the description of “bundle gerbes” associated to crossed modules up to 2-crossed modules.  Bundles, with a structure-group $G$, are classified by the cohomology $H^1(X,G)$ with coefficients in $G$; and whereas “bundle-gerbes” with a structure-crossed-module $H \rightarrow G$ can likewise be described by cohomology $H^1(X,H \rightarrow G)$.  Notice this is a bit different from the description in terms of higher cohomology $H^2(X,G)$ for a $G$-gerbe, which can be understood as a bundle-gerbe using the shifted crossed module $G \rightarrow 1$ (when $G$ is abelian.  The goal here is to generalize this part to nonabelian groups, and also pass up to “bundle 2-gerbes” based on a 2-crossed module, or crossed complex of length 2, $L \rightarrow H \rightarrow G$ as I described previously for Joao Martins’ talk.  This would be classified in terms of cohomology valued in the 2-crossed module.  The point is that one can describe such a thing as a bundle over a fibre product, which (I think – I’m not so clear on this part) deals with the same structure of overlaps as the higher cohomology in the other way of describing things.

Finally,  a talk that’s a little harder to classify than most, but which I’ve put here with things somewhat related to string theory, was Alexander Kahle‘s on “T-Duality and Differential K-Theory”, based on work with Alessandro Valentino.  This uses the idea of the differential refinement of cohomology theories – in this case, K-theory, which is a generalized cohomology theory, which is to say that K-theory satisfies the Eilenberg-Steenrod axioms (with the dimension axiom relaxed, hence “generalized”).  Cohomology theories, including generalized ones, can have differential refinements, which pass from giving topological to geometrical information about a space.  So, while K-theory assigns to a space the Grothendieck ring of the category of vector bundles over it, the differential refinement of K-theory does the same with the category of vector bundles with connection.  This captures both local and global structures, which turns out to be necessary to describe fields in string theory – specifically, Ramond-Ramond fields.  The point of this talk was to describe what happens to these fields under T-duality.  This is a kind of duality in string theory between a theory with large strings and small strings.  The talk describes how this works, where we have a manifold with fibres at each point $M\times S^1_r$ with fibres strings of radius $r$ and $M \times S^1_{1/r}$ with radius $1/r$.  There’s a correspondence space $M \times S^1_r \times S^1_{1/r}$, which has projection maps down into the two situations.  Fields, being forms on such a fibration, can be “transferred” through this correspondence space by a “pull-back and push-forward” (with, in the middle, a wedge with a form that mixes the two directions, $exp( d \theta_r + d \theta_{1/r})$).  But to be physically the right kind of field, these “forms” actually need to be representing cohomology classes in the differential refinement of K-theory.

## Quantum Gravity etc.

Now, part of the point of this workshop was to try to build, or anyway maintain, some bridges between the kind of work in geometry and topology which I’ve been describing and the world of physics.  There are some particular versions of physical theories where these ideas have come up.  I’ve already touched on string theory along the way (there weren’t many talks about it from a physicist’s point of view), so this will mostly be about a different sort of approach.

Benjamin Bahr gave a talk outlining this approach for our mathematician-heavy audience, with his talk on “Spin Foam Operators” (see also for instance this paper).  The point is that one approach to quantum gravity has a theory whose “kinematics” (the description of the state of a system at a given time) is described by “spin networks” (based on $SU(2)$ gauge theory), as described back in the pre-school post.  These span a Hilbert space, so the “dynamical” issue of such models is how to get operators between Hilbert spaces from “foams” that interpolate between such networks – that is, what kind of extra data they might need, and how to assign amplitudes to faces and edges etc. to define an operator, which (assuming a “local” theory where distant parts of the foam affect the result independently) will be of the form:

$Z(K,\rho,P) = (\prod_f A_f) \prod_v Tr_v(\otimes P_e)$

where $K$ is a particular complex (foam), $\rho$ is a way of assigning irreps to faces of the foam, and $P$ is the assignment of intertwiners to edges.  Later on, one can take a discrete version of a path integral by summing over all these $(K, \rho, P)$.  Here we have a product over faces and one over vertices, with an amplitude $A_f$ assigned (somehow – this is the issue) to faces.  The trace is over all the representation spaces assigned to the edges that are incident to a vertex (this is essentially the only consistent way to assign an amplitude to a vertex).  If we also consider spacetimes with boundary, we need some amplitudes $B_e$ at the boundary edges, as well.  A big part of the work with such models is finding such amplitudes that meet some nice conditions.

Some of these conditions are inherently necessary – to ensure the theory is invariant under gauge transformations, or (formally) changing orientations of faces.  Others are considered optional, though to me “functoriality” (that the way of deriving operators respects the gluing-together of foams) seems unavoidable – it imposes that the boundary amplitudes have to be found from the $A_f$ in one specific way.  Some other nice conditions might be: that $Z(K, \rho, P)$ depends only on the topology of $K$ (which demands that the $P$ operators be projections); that $Z$ is invariant under subdivision of the foam (which implies the amplitudes have to be $A_f = dim(\rho_f)$).

Assuming all these means the only choice is exactly which sub-projection $P_e$ is of the projection onto the gauge-invariant part of the representation space for the faces attached to edge $e$.  The rest of the talk discussed this, including some examples (models for BF-theory, the Barrett-Crane model and the more recent EPRL/FK model), and finished up by discussing issues about getting a nice continuum limit by way of “coarse graining”.

On a related subject, Bianca Dittrich spoke about “Dynamics and Diffeomorphism Symmetry in Discrete Quantum Gravity”, which explained the nature of some of the hard problems with this sort of discrete model of quantum gravity.  She began by asking what sort of models (i.e. which choices of amplitudes) in such discrete models would actually produce a nice continuum theory – since gravity, classically, is described in terms of spacetimes which are continua, and the quantum theory must look like this in some approximation.  The point is to think of these as “coarse-graining” of a very fine (perfect, in the limit) approximation to the continuum by a triangulation with a very short length-scale for the edges.  Coarse graining means discarding some of the edges to get a coarser approximation (perhaps repeatedly).  If the $Z$ happens to be triangulation-independent, then coarse graining makes no difference to the result, nor does the converse process of refining the triangulation.  So one question is:  if we expect the continuum limit to be diffeomorphism invariant (as is General Relativity), what does this say at the discrete level?  The relation between diffeomorphism invariance and triangulation invariance has been described by Hendryk Pfeiffer, and in the reverse direction by Dittrich et al.

Actually constructing the dynamics for a system like this in a nice way (“canonical dynamics with anomaly-free constraints”) is still a big problem, which Bianca suggested might be approached by this coarse-graining idea.  Now, if a theory is topological (here we get the link to TQFT), such as electromagnetism in 2D, or (linearized) gravity in 3D, coarse graining doesn’t change much.  But otherwise, changing the length scale means changing the action for the continuum limit of the theory.  This is related to renormalization: one starts with a “naive” guess at a theory, then refines it (in this case, by the coarse-graining process), which changes the action for the theory, until arriving at (or approximating to) a fixed point.  Bianca showed an example, which produces a really huge, horrible action full of very complicated terms, which seems rather dissatisfying.  What’s more, she pointed out that, unless the theory is topological, this always produces an action which is non-local – unlike the “naive” discrete theory.  That is, the action can’t be described in terms of a bunch of non-interacting contributions from the field at individual points – instead, it’s some function which couples the field values at distant points (albeit in a way that falls off exponentially as the points get further apart).

In a more specific talk, Aleksandr Mikovic discussed “Finiteness and Semiclassical Limit of EPRL-FK Spin Foam Models”, looking at a particular example of such models which is the (relatively) new-and-improved candidate for quantum gravity mentioned above.  This was a somewhat technical talk, which I didn’t entirely follow, but  roughly, the way he went at this was through the techniques of perturbative QFT.  That is, by looking at the theory in terms of an “effective action”, instead of some path integral over histories $\phi$ with action $S(\phi)$ – which looks like $\int d\phi e^{iS(\phi)}$.  Starting with some classical history $\bar{\phi}$ – a stationary point of the action $S$ – the effective action $\Gamma(\bar{\phi})$ is an integral over small fluctuations $\phi$ around it of $e^{iS(\bar{\phi} + \phi)}$.

He commented more on the distinction between the question of triangulation independence (which is crucial for using spin foams to give invariants of manifolds) and the question of whether the theory gives a good quantum theory of gravity – that’s the “semiclassical limit” part.  (In light of the above, this seems to amount to asking if “diffeomorphism invariance” really extends through to the full theory, or is only approximately true, in the limiting case).  Then the “finiteness” part has to do with the question of getting decent asymptotic behaviour for some of those weights mentioned above so as to give a nice effective action (if not necessarily triangulation independence).  So, for instance, in the Ponzano-Regge model (which gives a nice invariant for manifolds), the vertex amplitudes $A_v$ are found by the 6j-symbols of representations.  The asymptotics of the 6j symbols then becomes an issue – Alekandr noted that to get a theory with a nice effective action, those 6j-symbols need to be scaled by a certain factor.  This breaks triangulation independence (hence means we don’t have a good manifold invariant), but gives a physically nicer theory.  In the case of 3D gravity, this is not what we want, but as he said, there isn’t a good a-priori reason to think it can’t give a good theory of 4D gravity.

Now, making a connection between these sorts of models and higher gauge theory, Aristide Baratin spoke about “2-Group Representations for State Sum Models”.  This is a project Baez, Freidel, and Wise, building on work by Crane and Sheppard (see my previous post, where Derek described the geometry of the representation theory for some 2-groups).  The idea is to construct state-sum models where, at the kinematical level, edges are labelled by 2-group representations, faces by intertwiners, and tetrahedra by 2-intertwiners.  (This assumes the foam is a triangulation – there’s a certain amount of back-and-forth in this area between this, and the Poincaré dual picture where we have 4-valent vertices).  He discussed this in a couple of related cases – the Euclidean and Poincaré 2-groups, which are described by crossed modules with base groups $SO(4)$ or $SO(3,1)$ respectively, acting on the abelian group (of automorphisms of the identity) $R^4$ in the obvious way.  Then the analogy of the 6j symbols above, which are assigned to tetrahedra (or dually, vertices in a foam interpolating two kinematical states), are now 10j symbols assigned to 4-simplexes (or dually, vertices in the foam).

One nice thing about this setup is that there’s a good geometric interpretation of the kinematics – irreducible representations of these 2-groups pick out orbits of the action of the relevant $SO$ on $R^4$.  These are “mass shells” – radii of spheres in the Euclidean case, or proper length/time values that pick out hyperboloids in the Lorentzian case of $SO(3,1)$.  Assigning these to edges has an obvious geometric meaning (as a proper length of the edge), which thus has a continuous spectrum.  The areas and volumes interpreting the intertwiners and 2-intertwiners start to exhibit more of the discreteness you see in the usual formulation with representations of the $SO$ groups themselves.  Finally, Aristide pointed out that this model originally arose not from an attempt to make a quantum gravity model, but from looking at Feynman diagrams in flat space (a sort of “quantum flat space” model), which is suggestively interesting, if not really conclusively proving anything.

Finally, Laurent Freidel gave a talk, “Classical Geometry of Spin Network States” which was a way of challenging the idea that these states are exclusively about “quantum geometries”, and tried to give an account of how to interpret them as discrete, but classical.  That is, the quantization of the classical phase space $T^*(A/G)$ (the cotangent bundle of connections-mod-gauge) involves first a discretization to a spin-network phase space $\mathcal{P}_{\Gamma}$, and then a quantization to get a Hilbert space $H_{\Gamma}$, and the hard part is the first step.  The point is to see what the classical phase space is, and he describes it as a (symplectic) quotient $T^*(SU(2)^E)//SU(2)^V$, which starts by assigning $T^*(SU(2))$ to each edge, then reduced by gauge transformations.  The puzzle is to interpret the states as geometries with some discrete aspect.

The answer is that one thinks of edges as describing (dual) faces, and vertices as describing some polytopes.  For each $p$, there’s a $2(p-3)$-dimensional “shape space” of convex polytopes with $p$-faces and a given fixed area $j$.  This has a canonical symplectic structure, where lengths and interior angles at an edge are the canonically conjugate variables.  Then the whole phase space describes ways of building geometries by gluing these things (associated to vertices) together at the corresponding faces whenever the two vertices are joined by an edge.  Notice this is a bit strange, since there’s no particular reason the faces being glued will have the same shape: just the same area.  An area-1 pentagon and an area-1 square associated to the same edge could be glued just fine.  Then the classical geometry for one of these configurations is build of a bunch of flat polyhedra (i.e. with a flat metric and connection on them).  Measuring distance across a face in this geometry is a little strange.  Given two points inside adjacent cells, you measure orthogonal distance to the matched faces, and add in the distance between the points you arrive at (orthogonally) – assuming you glued the faces at the centre.  This is a rather ugly-seeming geometry, but it’s symplectically isomorphic to the phase space of spin network states – so it’s these classical geometries that spin-foam QG is a quantization of.  Maybe the ugliness should count against this model of quantum gravity – or maybe my aesthetic sense just needs work.

(Laurent also gave another talk, which was originally scheduled as one of the school talks, but ended up being a very interesting exposition of the principle of “Relativity of Localization”, which is hard to shoehorn into the themes I’ve used here, and was anyway interesting enough that I’ll devote a separate post to it.)

Continuing from the previous post, there are a few more lecture series from the school to talk about.

## Higher Gauge Theory

The next was John Huerta’s series on Higher Gauge Theory from the point of view of 2-groups.  John set this in the context of “categorification”, a slightly vague program of replacing set-based mathematical ideas with category-based mathematical ideas.  The general reason for this is to get an extra layer of “maps between things”, or “relations between relations”, etc. which tend to be expressed by natural transformations.  There are various ways to go about this, but one is internalization: given some sort of structure, the relevant example in this case being “groups”, one has a category ${Groups}$, and can define a 2-group as a “category internal to ${Groups}$“.  So a 2-group has a group of objects, a group of morphisms, and all the usual maps (source and target for morphisms, composition, etc.) which all have to be group homomorphisms.  It should be said that this all produces a “strict 2-group”, since the objects $G$ necessarily form a group here.  In particular, $m : G \times G \rightarrow G$ satisfies group axioms “on the nose” – which is the only way to satisfy them for a group made of the elements of a set, but for a group made of the elements of a category, one might require only that it commute up to isomorphism.  A weak 2-group might then be described as a “weak model” of the theory of groups in $Cat$, but this whole approach is much less well-understood than the strict version as one goes to general n-groups.

Now, as mentioned in the previous post, there is a 1-1 correspondence between 2-groups and crossed modules (up to equivalence): given a crossed module $(G,H,\partial,\rhd)$, there’s a 2-group $\mathcal{G}$ whose objects are $G$ and whose morphisms are $G \ltimes H$; given a 2-group $\mathcal{G}$ with objects $G$, there’s a crossed module $(G, Aut(1_G),1,m)$.  (The action $m$ acts on a morphism in such as way as to act by multiplication on its source and target).  Then, for instance, the Peiffer identity for crossed modules (see previous post) is a consequence of the fact that composition of morphisms is supposed to be a group homomorphism.

Looking at internal categories in [your favourite setting here] isn’t the only way to do categorification, but it does produce some interesting examples.  Baez-Crans 2-vector spaces are defined this way (in $Vect$), and built using these are Lie 2-algebras.  Looking for a way to integrate Lie 2-algebras up to Lie 2-groups (which are internal categories in Lie groups) brings us back to the current main point.  This is the use of 2-groups to do higher gauge theory.  This requires the use of “2-bundles”.  To explain these, we can say first of all that a “2-space” is an internal category in $Spaces$ (whether that be manifolds, or topological spaces, or what-have-you), and that a (locally trivial) 2-bundle should have a total 2-space $E$, a base 2-space $M$, and a (functorial) projection map $p : E \rightarrow M$, such that there’s some open cover of $M$ by neighborhoods $U_i$ where locally the bundle “looks like” $\pi_i : U_i \times F \rightarrow U_i$, where $F$ is the fibre of the bundle.  In the bundle setting, “looks like” means “is isomorphic to” by means of isomorphisms $f_i : E_{U_i} \rightarrow U_i \times F$.  With 2-bundles, it’s interpreted as “is equivalent to” in the categorical sense, likewise by maps $f_i$.

Actually making this precise is a lot of work when $M$ is a general 2-space – even defining open covers and setting up all the machinery properly is quite hard.  This has been done, by Toby Bartels in his thesis, but to keep things simple, John restricted his talk to the case where $M$ is just an ordinary manifold (thought of as a 2-space which has only identity morphisms).   Then a key point is that there’s an analog to how (principal) $G$-bundles (where $F \cong G$ as a $G$-set) are classified up to isomorphism by the first Cech cohomology of the manifold, $\check{H}^1(M,G)$.  This works because one can define transition functions on double overlaps $U_{ij} := U_i \cap U_j$, by $g_{ij} = f_i f_j^{-1}$.  Then these $g_{ij}$ will automatically satisfy the 1-cocycle condidion ($g_{ij} g_{jk} = g_{ik}$ on the triple overlap $U_{ijk}$) which means they represent a cohomology class $[g] = \in \check{H}^1(M,G)$.

A comparable thing can be said for the “transition functors” for a 2-bundle – they’re defined superficially just as above, except that being functors, we can now say there’s a natural isomorphism $h_{ijk} : g_{ij}g_{jk} \rightarrow g_{ik}$, and it’s these $h_{ijk}$, defined on triple overlaps, which satisfy a 2-cocycle condition on 4-fold intersections (essentially, the two ways to compose them to collapse $g_{ij} g_{jk} g_{kl}$ into $g_{il}$ agree).  That is, we have $g_{ij} : U_{ij} \rightarrow Ob(\mathcal{G})$ and $h_{ijk} : U_{ijk} \rightarrow Mor(\mathcal{G})$ which fit together nicely.  In particular, we have an element $[h] \in \check{H}^2(M,G)$ of the second Cech cohomology of $M$: “principal $\mathcal{G}$-bundles are classified by second Cech cohomology of $M$“.  This sort of thing ties in to an ongoing theme of the later talks, the relationship between gerbes and higher cohomology – a 2-bundle corresponds to a “gerbe”, or rather a “1-gerbe”.  (The consistent terminology would have called a bundle a “0-gerbe”, but as usual, terminology got settled before the general pattern was understood).

Finally, having defined bundles, one usually defines connections, and so we do the same with 2-bundles.  A connection on a bundle gives a parallel transport operation for paths $\gamma$ in $M$, telling how to identify the fibres at points along $\gamma$ by means of a functor $hol : P_1(M) \rightarrow G$, thinking of $G$ as a category with one object, and where $P_1(M)$ is the path groupoid whose objects are points in $M$ and whose morphisms are paths (up to “thin” homotopy). At least, it does so once we trivialize the bundle around $\gamma$, anyway, to think of it as $M \times G$ locally – in general we need to get the transition functions involved when we pass into some other local neighborhood.  A connection on a 2-bundle is similar, but tells how to parallel transport fibres not only along paths, but along homotopies of paths, by means of $hol : P_2(M) \rightarrow \mathcal{G}$, where $\mathcal{G}$ is seen as a 2-category with one object, and $P_2(M)$ now has 2-morphisms which are (essentially) homotopies of paths.

Just as connections can be described by 1-forms $A$ valued in $Lie(G)$, which give $hol$ by integrating, a similar story exists for 2-connections: now we need a 1-form $A$ valued in $Lie(G)$ and a 2-form $B$ valued in $Lie(H)$.  These need to satisfy some relations, essentially that the curvature of $A$ has to be controlled by $B$.   Moreover, that $B$ is related to the $B$-field of string theory, as I mentioned in the post on the pre-school… But really, this is telling us about the Lie 2-algebra associated to $\mathcal{G}$, and how to integrate it up to the group!

## Infinite Dimensional Lie Theory and Higher Gauge Theory

This series of talks by Christoph Wockel returns us to the question of “integrating up” to a Lie group $G$ from a Lie algebra $\mathfrak{g} = Lie(G)$, which is seen as the tangent space of $G$ at the identity.  This is a well-understood, well-behaved phenomenon when the Lie algebras happen to be finite dimensional.  Indeed the classification theorem for the classical Lie groups can be got at in just this way: a combinatorial way to characterize Lie algebras using Dynkin diagrams (which describe the structure of some weight lattice), followed by a correspondence between Lie algebras and Lie groups.  But when the Lie algebras are infinite dimensional, this just doesn’t have to work.  It may be impossible to integrate a Lie algebra up to a full Lie group: instead, one can only get a little neighborhood of the identity.  The point of such infinite-dimensional groups, and ultimately their representation theory, is to deal with string groups that have to do with motions of extended objects.  Christoph Wockel was describing a result which says that, going to 2-groups, this problem can be overcome.  (See the relevant paper here.)

The first lecture in the series presented some background on a setting for infinite dimensional manifolds.  There are various approaches, a popular one being Frechet manifolds, but in this context, the somewhat weaker notion of locally convex spaces is sufficient.  These are “locally modelled” by (infinite dimensional) locally convex vector spaces, the way finite dimensonal manifolds are locally modelled by Euclidean space.  Being locally convex is enough to allow them to support a lot of differential calculus: one can find straight-line paths, locally, to define a notion of directional derivative in the direction of a general vector.  Using this, one can build up definitions of differentiable and smooth functions, derivatives, and integrals, just by looking at the restrictions to all such directions.  Then there’s a fundamental theorem of calculus, a chain rule, and so on.

At this point, one has plenty of differential calculus, and it becomes interesting to bring in Lie theory.  A Lie group is defined as a group object in the category of manifolds and smooth maps, just as in the finite-dimensional case.  Some infinite-dimensional Lie groups of interest would include: $G = Diff(M)$, the group of diffeomorphisms of some compact manifold $M$; and the group of smooth functions $G = C^{\infty}(M,K)$ from $M$ into some (finite-dimensional) Lie group $K$ (perhaps just $\mathbb{R}$), with the usual pointwise multiplication.  These are certainly groups, and one handy fact about such groups is that, if they have a manifold structure near the identity, on some subset that generates $G$ as a group in a nice way, you can extend the manifold structure to the whole group.  And indeed, that happens in these examples.

Well, next we’d like to know if we can, given an infinite dimensional Lie algebra $X$, “integrate up” to a Lie group – that is, find a Lie group $G$ for which $X \cong T_eG$ is the “infinitesimal” version of $G$.  One way this arises is from central extensions.  A central extension of Lie group $G$ by $Z$ is an exact sequence $Z \hookrightarrow \hat{G} \twoheadrightarrow G$ where (the image of) $Z$ is in the centre of $\hat{G}$.  The point here is that $\hat{G}$ extends $G$.  This setup makes $\hat{G}$ is a principal $Z$-bundle over $G$.

Now, finding central extensions of Lie algebras is comparatively easy, and given a central extension of Lie groups, one always falls out of the induced maps.  There will be an exact sequence of Lie algebras, and now the special condition is that there must exist a continuous section of the second map.  The question is to go the other way: given one of these, get back to an extension of Lie groups.  The problem of finding extensions of $G$ by $Z$, in particular as a problem of finding a bundle with connection having specified curvature, which brings us back to gauge theory.  One type of extension is the universal cover of $G$, which appears as $\pi_1(G) \hookrightarrow \hat{G} \twoheadrightarrow G$, so that the fibre is $\pi_1(G)$.

In general, whether an extension can exist comes down to a question about a cocycle: that is, if there’s a function $f : G \times G \rightarrow Z$ which is locally smooth (i.e. in some neighborhood in $G$), and is a cocyle (so that $f(g,h) + f(gh,k) = f(g,hk) + f(h,k)$), by the same sorts of arguments we’ve already seen a bit of.  For this reason, central extensions are classified by the cohomology group $H^2(G,Z)$.  The cocycle enables a “twisting” of the multiplication associated to a nontrivial loop in $G$, and is used to construct $\hat{G}$ (by specifying how multiplication on $G$ lifts to $\hat{G}$).  Given a  2-cocycle $\omega$ at the Lie algebra level (easier to do), one would like to lift that up the Lie group.  It turns out this is possible if the period homomorphism $per_{\omega} : \Pi_2(G) \rightarrow Z$ – which takes a chain $[\sigma]$ (with $\sigma : S^2 \rightarrow G$) to the integral of the original cocycle on it, $\int_{\sigma} \omega$ – lands in a discrete subgroup of $Z$. A popular example of this is when $Z$ is just $\mathbb{R}$, and the discrete subgroup is $\mathbb{Z}$ (or, similarly, $U(1)$ and $1$ respectively).  This business of requiring a cocycle to be integral in this way is sometimes called a “prequantization” problem.

So suppose we wanted to make the “2-connected cover” $\pi_2(G) \hookrightarrow \pi_2(G) \times_{\gamma} G \twoheadrightarrow G$ as a central extension: since $\pi_2(G)$ will be abelian, this is conceivable.  If the dimension of $G$ is finite, this is trivial (since $\pi_2(G) = 0$ in finite dimensions), which is why we need some theory  of infinite-dimensional manifolds.  Moreover, though, this may not work in the context of groups: the $\gamma$ in the extension $\pi_2(G) \times_{\gamma} G$ above needs to be a “twisting” of associativity, not multiplication, being lifted from $G$.  Such twistings come from the THIRD cohomology of $G$ (see here, e.g.), and describe the structure of 2-groups (or crossed modules, whichever you like).  In fact, the solution (go read the paper for more if you like) to define a notion of central extension for 2-groups (essentially the same as the usual definition, but with maps of 2-groups, or crossed modules, everywhere).  Since a group is a trivial kind of 2-group (with only trivial automorphisms of any element), the usual notion of central extension turns out to be a special case.  Then by thinking of $\pi_2(G)$ and $G$ as crossed modules, one can find a central extension which is like the 2-connected cover we wanted – though it doesn’t work as an extension of groups because we think of $G$ as the base group of the crossed module, and $\pi_2(G)$ as the second group in the tower.

The pattern of moving to higher group-like structures, higher cohomology, and obstructions to various constructions ran all through the workshop, and carried on in the next school session…

## Higher Spin Structures in String Theory

Hisham Sati gave just one school-lecture in addition to his workshop talk, but it was packed with a lot of material.  This is essentially about cohomology and the structures on manifolds to which cohomology groups describe the obstructions.  The background part of the lecture referenced this book by Fridrich, and the newer parts were describing some of Sati’s own work, in particular a couple of papers with Schreiber and Stasheff (also see this one).

The basic point here is that, for physical reasons, we’re often interested in putting some sort of structure on a manifold, which is really best described in terms of a bundle.  For instance, a connection or spin connection on spacetime lets us transport vectors or spinors, respectively, along paths, which in turn lets us define derivatives.  These two structures really belong on vector bundles or spinor bundles.  Now, if these bundles are trivial, then one can make the connections on them trivial as well by gauge transformation.  So having nontrivial bundles really makes this all more interesting.  However, this isn’t always possible, and so one wants to the obstruction to being able to do it.  This is typically a class in one of the cohomology groups of the manifold – a characteristic class.  There are various examples: Chern classes, Pontrjagin classes, Steifel-Whitney classes, and so on, each of which comes in various degrees $i$.  Each one corresponds to a different coefficient group for the cohomology groups – in these examples, the groups $U$ and $O$ which are the limits of the unitary and orthogonal groups (such as $O := O(\infty) \supset \dots \supset O(2) \supset O(1)$)

The point is that these classes are obstructions to building certain structures on the manifold $X$ – which amounts to finding sections of a bundle.  So for instance, the first Steifel-Whitney classes, $w_1(E)$ of a bundle $E$ are related to orientations, coming from cohomology with coefficients in $O(n)$.  Orientations for the manifold $X$ can be described in terms of its tangent bundle, which is an $O(n)$-bundle (tangent spaces carry an action of the rotation group).  Consider $X = S^1$, where we have actually $O(1) \simeq \mathbb{Z}_2$.  The group $H^1(S^1, \mathbb{Z}_2)$ has two elements, and there are two types of line bundle on the circle $S^1$: ones with a nowhere-zero section, like the trivial bundle; and ones without, like the Moebius strip.  The circle is orientable, because its tangent bundle is of the first sort.

Generally, an orientation can be put on $X$ if the tangent bundle, as a map $f : X \rightarrow B(O(n))$, can be lifted to a map $\tilde{f} : X \rightarrow B(SO(n))$ – that is, it’s “secretly” an $SO(n)$-bundle – the special orthogonal group respects orientation, which is what the determinant measures.  Its two values, $\pm 1$, are what’s behind the two classes of bundles.  (In short, this story relates to the exact sequence $1 \rightarrow SO(n) \rightarrow O(n) \stackrel{det}{\rightarrow} O(1) = \mathbb{Z}_2 \rightarrow 1$; in just the same way we have big groups $SO$, $Spin$, and so forth.)

So spin structures have a story much like the above, but where the exact sequence $1 \rightarrow \mathbb{Z}_2 \rightarrow Spin(n) \rightarrow SO(n) \rightarrow 1$ plays a role – the spin groups are the universal covers (which are all double-sheeted covers) of the special rotation groups.  A spin structure on some $SO(n)$ bundle $E$, let’s say represented by $f : X \rightarrow B(SO(n))$ is thus, again, a lifting to $\tilde{f} : X \rightarrow B(Spin(n))$.  The obstruction to doing this (the thing which must be zero for the lifting to exist) is the second Stiefel-Whitney class, $w_2(E)$.  Hisham Sati also explained the example of “generalized” spin structures in these terms.  But the main theme is an analogous, but much more general, story for other cohomology groups as obstructions to liftings of some sort of structures on manifolds.  These may be bundles, for the lower-degree cohomology, or they may be gerbes or n-bundles, for higher-degree, but the setup is roughly the same.

The title’s term “higher spin structures” comes from the fact that we’ve so far had a tower of classifying spaces (or groups), $B(O) \leftarrow B(SO) \leftarrow B(Spin)$, and so on.  Then the problem of putting various sorts of structures on $X$ has been turned into the problem of lifting a map $f : X \rightarrow S(O)$ up this tower.  At each point, the obstruction to lifting is some cohomology class with coefficients in the groups ($O$, $SO$, etc.)  So when are these structures interesting?

This turns out to bring up another theme, which is that of special dimensions – it’s just not true that the same phenomena happen in every dimension.  In this case, this has to do with the homotopy groups  – of $O$ and its cousins.  So it turns out that the homotopy group $\pi_k(O)$ (which is the same as $\pi_k(O_n)$ as long as $n$ is bigger than $k$) follows a pattern, where $\pi_k(O) = \mathbb{Z}_2$ if $k = 0,1 (mod 8)$, and $\pi_k(O) = \mathbb{Z}$ if $k = 3,7 (mod 8)$.  The fact that this pattern repeats mod-8 is one form of the (real) Bott Periodicity theorem.  These homotopy groups reflect that, wherever there’s nontrivial homotopy in some dimension, there’s an obstruction to contracting maps into $O$ from such a sphere.

All of this plays into the question of what kinds of nontrivial structures can be put on orthogonal bundles on manifolds of various dimensions.  In the dimensions where these homotopy groups are non-trivial, there’s an obstruction to the lifting, and therefore some interesting structure one can put on $X$ which may or may not exist.  Hisham Sati spoke of “killing” various homotopy groups – meaning, as far as I can tell, imposing conditions which get past these obstructions.  In string theory, his application of interest, one talks of “anomaly cancellation” – an anomaly being the obstruction to making these structures.  The first part of the punchline is that, since these are related to nontrivial cohomology groups, we can think of them in terms of defining structures on n-bundles or gerbes.  These structures are, essentially, connections – they tell us how to parallel-transport objects of various dimensions.  It turns out that the $\pi_k$ homotopy group is related to parallel transport along $(k-1)$-dimensional surfaces in $X$, which can be thought of as the world-sheets of $(k-2)$-dimensional “particles” (or rather, “branes”).

So, for instance, the fact that $\pi_1(O)$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^2(X,\mathbb{Z})$, which has to do with spin structure – as above.  “Cancelling” this “anomaly” means that for a theory involving such a spin structure to be well-defined, then this characteristic class for $X$ must be zero.  The fact that $\pi_3(O) = \mathbb{Z}$ is nontrivial means there’s an obstruction to a lifting in the form of a class in $H^4(X, \mathbb{Z})$.  This has to do with “string bundles”, where the string group is a higher analog of $Spin$ in exactly the sense we’ve just described.  If such a lifting exists, then there’s a “string-structure” on $X$ which is compatible with the spin structure we lifted (and with the orientation a level below that).  Similarly, $\pi_7(O) = \mathbb{Z}$ being nontrivial, by way of an obstruction in $H^8$, means there’s an interesting notion of “five-brane” structure, and a $Fivebrane$ group, and so on.  Personally, I think of these as giving a geometric interpretation for what the higher cohomology groups actually mean.

A slight refinement of the above, and actually more directly related to “cancellation” of the anomalies, is that these structures can be defined in a “twisted” way: given a cocycle in the appropriate cohomology group, we can ask that a lifting exist, not on the nose, but as a diagram commuting only up to a higher cell, which is exactly given by the cocycle.  I mentioned, in the previous section, a situation where the cocycle gives an associator, so that instead of being exactly associative, a structure has a “twisted” associativity.  This is similar, except we’re twisting the condition that makes a spin structure (or higher spin structure) well-defined.  So if $X$ has the wrong characteristic class, we can only define one of these twisted structures at that level.

This theme of higher cohomology and gerbes, and their geometric interpretation, was another one that turned up throughout the talks in the workshop…

And speaking of that: coming up soon, some descriptions of the actual workshop.

So this is a couple of weeks backdated.  I’ve had a pretty serious cold for a while – either it was bad in its own right, or this was just a case of the difference in native viruses between two different continents that my immune system wasn’t prepared for.  Then, too, last week was Republic Day – the 100th anniversary of the middle of three revolutions (the Liberal, the Republican, and the Carnation revolution that ousted the dictatorship regime in 1974 – and let me say that it’s refreshing for a North American to be reminded that Republicanism is a refinement of Liberalism, though how the flowers fit into it is less straightforward).  So my family and I went to attend some of the celebrations downtown, which were impressive.

Anyway, with the TQFT club seminars starting up very shortly, I wanted to finish this post on the first talks I got to see here at IST, which were on pretty widely different topics.  The first was by Ivan Smith, entitled “Quadrics, 3-Manifolds and Floer Cohomology”.  The second was a recorded video talk arranged by the string theory group.  This was a recording of a talk given by Kostas Skenderis a couple of years ago, entitled “The Fuzzball Proposal for Black Holes”.

## Ivan Smith – Quadrics, 3-Manfolds and Floer Cohomology

Ivan Smith’s talk began with some motivating questions from topology, symplectic geometry, and from the study of moduli spaces.  The topological question talks about 3-manifolds $Y$ and the space of representations $Hom(\pi_1(Y),G)$ of its fundamental group into a compact Lie group $G$, which was generally $SO(3)$ or $SU(2)$.  Specifically, the question is how this space is affected by operations on $Y$ such as surgery, taking covering spaces, etc.  The symplectic geometry question asks, for a symplectic manifold $(X,\omega)$, what the “mapping class group” of symplectic transformations – that is, the group $\pi_0(Symp(X))$ of connected components of symplectomorphisms from $X$ to itself – in a sense, this is asking how much of the geometry is seen by the symplectic situation.  The question about moduli spaces asks to characterize the action of the (again, mapping class group of) diffeomorphisms of a Riemann surface on the moduli space of bundles on it.  (This space, for  $\Sigma$ with genus $g \geq 2$, look like $M_g \simeq Hom(\pi_1(\Sigma),SU(2))$ modulo conjugation.  It is the complex-manifold version of the space of flat connections which I’ve been quite interested in for purposes of TQFT, though this is a coarse quotient, not a stack-like quotient.  Lots of people are interested in this space in its various hats.)

The point of the talk being to elucidate how these all fit together.  The first part of the title, “Quadrics”, referred to the fact that, when $\Sigma$ has genus 2, the moduli space we’ll be looking at can be described as an intersection of some varieties (defined by quadric equations) in the projective space $\mathbb{CP}^5$.  Knowing this, one can describe some of its properties just by looking at intersections of curves.

In general we’re talking about complex manifolds, here.  To start with, for Riemann surfaces (one-dimensional complex manifolds), he pointed out that there is an isomorphism between the mapping class groups of symplectomorphisms and diffeomorphisms: $\pi_0(Symp(\Sigma)) \simeq \pi_0(Diff(\Sigma))$.  But in general, for example, for 3-dimensional manifolds, there is structure in the symplectic maps which is forgotten by the smooth ones – there’s still a map $\pi_0(Symp(\Sigma)) \rightarrow \pi_0(Diff(\Sigma))$, but it has a kernel – there are distinct symplectic maps that all look like the identity up to smooth deformation.

Now, our original question was what the action of the diffeomorphisms of on the moduli space $M_g$ of bundles over $\Sigma$.  An element $h$ of $\pi_0(Diff(\Sigma))$ acts (by symplectic map) on it.  The discrepancy we mentioned is that the map corresponding to $h$ will always have fixed points, but be smoothly equivalent to one that doesn’t.  So the smooth mapping class group can’t detect the property of having fixed points.  What it CAN detect, however, is information about intersections.  In particular,   as mentioned above, the moduli space of bundles over a genus 2 surface is an intersection; in this situation, there is an injective map back from the smooth mapping class group into the group of classes of symplectic maps.  So looking symplectically loses nothing from the smooth case.

Now, these symplectic maps tie into the third part of the title, “Floer Homology”, as follows.  Given a symplectic map $\phi : (X,\omega) \rightarrow (X,\omega)$, one can define a complex of vector spaces $HF(\phi)$ which is the usual cohomology of a chain complex generated by fixed points of the map $\phi$, and with a differential $\partial$ which is defined by counting certain curves.  The way this is set up, if $\phi$ is the identity so that all points are fixed points, one gets the usual cohomology of the space $X$ – except that it’s defined so as to be the quantum cohomology of $X$ (for more, check out this tutorial by Givental).  This has the same complex as the usual cohomology, but with the cup product replaced by a deformed product.  It’s an older theorem (due to Donaldson) that, at least for genus 2, the quantum cohomology of the moduli space of bundles over $\Sigma$ splits into a direct sum of rings:

$QH^*(M_2) \cong \mathbb{C} \oplus QH^*(\Sigma_2) \oplus \mathbb{C}$

So one of the key facts is that this works also with Floer homology for other maps than the identity (so this becomes a special case).  So replacing $QH^*$ in the above with $HF^*(\phi)$ for any $\phi$ (acting either on the surface $\Sigma$, or the induced action on the moduli space) still gives a true statement.  Note that this actually implies the theorem that there are fixed points in the space of bundles, since the right hand side is always nontrivial.

So at this point we have some idea of how Floer cohomology is part of what ties the original three questions together.  To take a further look at these we can start to build a category combining much of the same information.  This is the (derived) Fukaya category.  The objects are Lagrangian submanifolds of a symplectic manifold $(X,\omega)$ – ones where the symplectic form vanishes.  To start building the category, consider what we can build from pairs of such objects $(L_1,L_2)$.  This is rather like the above – we define a complex of vector spaces, which is the cohomology of another complex.  Instead of being the complex freely generated by fixed points, though, it’s generated by intersection points of $L_1$ and $L_2$.  This automatically becomes a module over $QH^*(X)$, so the category we’re building is enriched over these.

Defining the structure of this category is apparently a little bit complicated – in particular, there is a composition product $HF(L_1,L_2) \otimes HF(L_2,L_3) \rightarrow HF(L_1,L_3)$ in the form of a cohomology operation.  Furthermore, which Ivan Smith didn’t have time to describe in detail, there are other “higher” products.  These are Massey type products, which is to say higher-order cohomology operations, which involve more than two inputs.  These give the whole structure (where one takes the direct sum of all those hom-modules $HF(L_i,L_j)$ to get one big module) the structure of an $A_{\infty}$-algebra (so the Fukaya category is an $A_{\infty}$-category, I suppose).  This is one way of talking about weak higher categories (the higher products give the associator for composition, and its higher analogs), so in fact this is a pretty complex structure, which the talk didn’t dwell on in detail.  But in any case, the point is that the operations in the category correspond to cohomology operations.

Then one deals with the “derived” Fukaya category $\mathcal{DF}(X)$.  I understand derived categories to be (at least among other examples) a way of taking categories of complexes “up to homotopy”, perhaps as a way of getting rid of some of this complication.  Again, the talk didn’t elaborate too much on this.  However, the fundamental theorem about this category is a generalization of the theorem above above quantum cohomology:

$\mathcal{DF}(M_2) \cong \mathcal{DF}(pt) \oplus \mathcal{DF}(\Sigma_2) \oplus \mathcal{DF}(pt)$

That is, the derived Fukaya category for the moduli space of bundles over $\Sigma_2$ is the category for the Riemann surface itself, summed with two copies of the category for a single point (which is replacing the two copies of $\mathbb{C}$).  This reduces to the previous theorem when we’re looking at the map $\phi = id$, just as before.

So the last question Ivan Smith addressed about this is the fact that these sorts of categories are often hard to calculate explicitly, but they can be described in terms of some easily-described data.  He gave the analogy of periodic functions – which may be quite complicated, but by means of Fourier decompositions, can be easily described in terms of sines and cosines, which are easy to analyze.  In the same way, although the Fukaya categories for particular spaces might be complicated, they can be described in terms of the (derived) category of modules over the $A_{\infty}$-algebras.  In particular, every category $\mathcal{DF}(X)$ embeds in a generic example $\mathcal{D}(mod-A_{\infty}-alg)$.  So by understanding categories like this, one can understand a lot about the categories that come from spaces, which generalize quantum cohomology as described above.

I like this punchline of the analogy with Fourier analysis, as imprecise as it might be, because it suggests a nice way to approach complex entities by finding out the parts that can generate them, or simple but large things you might discover them inside.

## Fuzzballs

The Skenderis talk about black holes was interesting, in that it was a recorded version of a talk given somewhere else – I haven’t seen this done before, but apparently the String Theory group does it pretty regularly.  This has some obvious advantages – they can get a wider range of talks by many different speakers.  There was some technical problem – I suppose due to the way the video was encoded – that meant the slides were sometimes unreadably blurry, but that’s still better than not getting the speaker at all.  I don’t have the background in string theory to be able to really get at the meat of the talk, though it did involve the AdS/CFT correspondence.  However, I can at least say a few concrete things about the motivation.  First, the “fuzzball” proposal is a more-or-less specific proposal to deal with the problem of black hole entropy.

The problem, basically, is that it’s known that the thermodynamic entropy associated to a black hole – which can be computed in completely macroscopic terms – is proportional to the area of its horizon.  On the other hand, in essentially every other setting, entropy has an interpretation in terms of counting microstates, so that the entropy of a “macrostate” is proportional to the logarithm of the number of microstates.  (Or, in a thermal state, which is a statistical distribution, this is weighted by the probability of the microstate).  So, for example, with a gas in a box, there are many macrostates that correspond to a relatively even distribution of position and momentum among the molecules, and relatively few in which all molecules are all in one small corner of the box.

The reason this is a problem is that, classically, the state of a black hole is characterized by very few numbers: the mass, angular momentum, and electric charge.   There doesn’t seem to be room for “microstates” in a classical black hole.  So the overall point of the proposal is to describe what microstates would be.  The specific way this is done with “fuzzballs” is somewhat mysterious to me, but the overall idea makes sense.  One interesting consequence of this approach is that event horizons would be strictly a property of thermal states, in whatever underlying theory one takes to be the quantum theory behind classical gravity (here assumed to be some specific form of string theory – the example he was using is something called the B1-B5 black hole, which I know nothing about).  That’s because a pure state would have a single microstate, hence have zero entropy, hence no horizon.

Now, what little I do understand about the particular model relies on the fact that near a (classical) event horizon, the background metric has a component that looks like anti-deSitter space – a vacuum solution to the Einstein equations with a negative cosmological constant.  (This part isn’t so hard to see – AdS space has that “saddle-shaped” appearance of a hyperbolic surface, and so does the area around a horizon, even when you draw it like this.)  But then, there is the AdS/CFT correspondence that says states for a gravitational field in (asymptotically) anti-deSitter space correspond to states for a conformal field theory (CFT) at the boundary.  So the way to get microstates, in the “fuzzball” proposal, is to look at this CFT, and find geometries that correspond to them.  Some would be well-approximated by the classical, horizon-ridden geometry, but others would be different.  The fact that this CFT is defined at the boundary explains why entropy would be proportional to area, not volume, of the black hole – this being a manifestation of the so-called “holographic principle”.  The “fuzziness” that one throws away by reducing a thermal state that combines these many geometries to the classical “no-hair” black hole determined by just three numbers is exactly the information described by the entropy.

I couldn’t follow some parts of it, not having much string-theory background – I don’t feel qualified to judge whether string theory makes sense as physics, but it isn’t an approach I’ve studied much.  Still, this talk did reinforce my feeling that the AdS/CFT correspondence, at the very least, is something well-worth learning about and important in its own right.

Coming soon: descriptions of the TQFT club seminars which are starting up at IST.

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