Now for a more sketchy bunch of summaries of some talks presented at the HGTQGR workshop.  I’ll organize this into a few themes which appeared repeatedly and which roughly line up with the topics in the title: in this post, variations on TQFT, plus 2-group and higher forms of gauge theory; in the next post, gerbes and cohomology, plus talks on discrete models of quantum gravity and suchlike physics.

## TQFT and Variations

I start here for no better reason than the personal one that it lets me put my talk first, so I’m on familiar ground to start with, for which reason also I’ll probably give more details here than later on.  So: a TQFT is a linear representation of the category of cobordisms – that is, a (symmetric monoidal) functor $nCob \rightarrow Vect$, in the notation I mentioned in the first school post.  An Extended TQFT is a higher functor $nCob_k \rightarrow k-Vect$, representing a category of cobordisms with corners into a higher category of k-Vector spaces (for some definition of same).  The essential point of my talk is that there’s a universal construction that can be used to build one of these at $k=2$, which relies on some way of representing $nCob_2$ into $Span(Gpd)$, whose objects are groupoids, and whose morphisms in $Hom(A,B)$ are pairs of groupoid homomorphisms $A \leftarrow X \rightarrow B$.  The 2-morphisms have an analogous structure.  The point is that there’s a 2-functor $\Lambda : Span(Gpd) \rightarrow 2Vect$ which is takes representations of groupoids, at the level of objects; for morphisms, there is a “pull-push” operation that just uses the restricted and induced representation functors to move a representation across a span; the non-trivial (but still universal) bit is the 2-morphism map, which uses the fact that the restriction and induction functors are bi-ajdoint, so there are units and counits to use.  A construction using gauge theory gives groupoids of connections and gauge transformations for each manifold or cobordism.  This recovers a form of the Dijkgraaf-Witten model.  In principle, though, any way of getting a groupoid (really, a stack) associated to a space functorially will give an ETQFT this way.  I finished up by suggesting what would need to be done to extend this up to higher codimension.  To go to codimension 3, one would assign an object (codimension-3 manifold) a 3-vector space which is a representation 2-category of 2-groupoids of connections valued in 2-groups, and so on.  There are some theorems about representations of n-groupoids which would need to be proved to make this work.

The fact that different constructions can give groupoids for spaces was used by the next speaker, Thomas Nicklaus, whose talk described another construction that uses the $\Lambda$ I mentioned above.  This one produces “Equivariant Dijkgraaf-Witten Theory”.  The point is that one gets groupoids for spaces in a new way.  Before, we had, for a space $M$ a groupoid $\mathcal{A}_G(M)$ whose objects are $G$-connections (or, put another way, bundles-with-connection) and whose morphisms are gauge transformations.  Now we suppose that there’s some group $J$ which acts weakly (i.e. an action defined up to isomorphism) on $\mathcal{A}_G(M)$.  We think of this as describing “twisted bundles” over $M$.  This is described by a quotient stack $\mathcal{A}_G // J$ (which, as a groupoid, gets some extra isomorphisms showing where two objects are related by the $J$-action).  So this gives a new map $nCob \rightarrow Span(Gpd)$, and applying $\Lambda$ gives a TQFT.  The generating objects for the resulting 2-vector space are “twisted sectors” of the equivariant DW model.  There was some more to the talk, including a description of how the DW model can be further mutated using a cocycle in the group cohomology of $G$, but I’ll let you look at the slides for that.

Next up was Jamie Vicary, who was talking about “(1,2,3)-TQFT”, which is another term for what I called “Extended” TQFT above, but specifying that the objects are 1-manifolds, the morphisms 2-manifolds, and the 2-morphisms are 3-manifolds.  He was talking about a theorem that identifies oriented TQFT’s of this sort with “anomaly-free modular tensor categories” – which is widely believed, but in fact harder than commonly thought.  It’s easy enough that such a TQFT $Z$ corresponds to a MTC – it’s the category $Z(S^1)$ assigned to the circle.  What’s harder is showing that the TQFT’s are equivalent functors iff the categories are equivalent.  This boils down, historically, to the difficulty of showing the category is rigid.  Jamie was talking about a project with Bruce Bartlett and Chris Schommer-Pries, whose presentation of the cobordism category (described in the school post) was the basis of their proof.

Part of it amounts to giving a description of the TQFT in terms of certain string diagrams.  Jamie kindly credited me with describing this point of view to him: that the codimension-2 manifolds in a TQFT can be thought of as “boundaries in space” – codimension-1 manifolds are either time-evolving boundaries, or else slices of space in which the boundaries live; top-dimension cobordisms are then time-evolving slices of space-with-boundary.  (This should be only a heuristic way of thinking – certainly a generic TQFT has no literal notion of “time-evolution”, though in that (2+1) quantum gravity can be seen as a TQFT, there’s at least one case where this picture could be taken literally.)  Then part of their proof involves showing that the cobordisms can be characterized by taking vector spaces on the source and target manifolds spanned by the generating objects, and finding the functors assigned to cobordisms in terms of sums over all “string diagrams” (particle worldlines, if you like) bounded by the evolving boundaries.  Jamie described this as a “topological path integral”.  Then one has to describe the string diagram calculus – ridigidy follows from the “yanking” rule, for instance, and this follows from Morse theory as in Chris’ presentation of the cobordism category.

There was a little more discussion about what the various properties (proved in a similar way) imply.  One is “cloaking” – the fact that a 2-morphism which “creates a handle” is invisible to the string diagrams in the sense that it introduces a sum over all diagrams with a string “looped” around the new handle, but this sum gives a result that’s equal to the original map (in any “pivotal” tensor category, as here).

Chronologically before all these, one of the first talks on such a topic was by Rafael Diaz, on Homological Quantum Field Theory, or HLQFT for short, which is a rather different sort of construction.  Remember that Homotopy QFT, as described in my summary of Tim Porter’s school sessions, is about linear representations of what I’ll for now call $Cob(d,B)$, whose morphisms are $d$-dimensional cobordisms equipped with maps into a space $B$ up to homotopy.  HLQFT instead considers cobordisms equipped with maps taken up to homology.

Specifically, there’s some space $M$, say a manifold, with some distinguished submanifolds (possibly boundary components; possibly just embedded submanifolds; possibly even all of $M$ for a degenerate case).  Then we define $Cob_d^M$ to have objects which are $(d-1)$-manifolds equipped with maps into $M$ which land on the distinguished submanifolds (to make composition work nicely, we in fact assume they map to a single point).  Morphisms in $Cob_d^M$ are trickier, and look like $(N,\alpha, \xi)$: a cobordism $N$ in this category is likewise equipped with a map $\alpha$ from its boundary into $M$ which recovers the maps on its objects.  That $\xi$ is a homology class of maps from $N$ to $M$, which agrees with $\alpha$.  This forms a monoidal category as with standard cobordisms.  Then HLQFT is about representations of this category.  One simple case Rafael described is the dimension-1 case, where objects are (ordered sets of) points equipped with maps that pick out chosen submanifolds of $M$, and morphisms are just braids equipped with homology classes of “paths” joining up the source and target submanifolds.  Then a representation might, e.g., describe how to evolve a homology class on the starting manifold to one on the target by transporting along such a path-up-to-homology.  In higher dimensions, the evolution is naturally more complicated.

A slightly looser fit to this section is the talk by Thomas Krajewski, “Quasi-Quantum Groups from Strings” (see this) – he was talking about how certain algebraic structures arise from “string worldsheets”, which are another way to describe cobordisms.  This does somewhat resemble the way an algebraic structure (Frobenius algebra) is related to a 2D TQFT, but here the string worldsheets are interacting with 3-form field, $H$ (the curvature of that 2-form field $B$ of string theory) and things needn’t be topological, so the result is somewhat different.

Part of the point is that quantizing such a thing gives a higher version of what happens for quantizing a moving particle in a gauge field.  In the particle case, one comes up with a line bundle (of which sections form the Hilbert space) and in the string case one comes up with a gerbe; for the particle, this involves associated 2-cocycle, and for the string a 3-cocycle; for the particle, one ends up producing a twisted group algebra, and for the string, this is where one gets a “quasi-quantum group”.  The algebraic structures, as in the TQFT situation, come from, for instance, the “pants” cobordism which gives a multiplication and a comultiplication (by giving maps $H \otimes H \rightarrow H$ or the reverse, where $H$ is the object assigned to a circle).

There is some machinery along the way which I won’t describe in detail, except that it involves a tricomplex of forms – the gradings being form degree, the degree of a cocycle for group cohomology, and the number of overlaps.  As observed before, gerbes and their higher versions have transition functions on higher numbers of overlapping local neighborhoods than mere bundles.  (See the paper above for more)

## Higher Gauge Theory

The talks I’ll summarize here touch on various aspects of higher-categorical connections or 2-groups (though at least one I’ll put off until later).  The division between this and the section on gerbes is a little arbitrary, since of course they’re deeply connected, but I’m making some judgements about emphasis or P.O.V. here.

Apart from giving lectures in the school sessions, John Huerta also spoke on “Higher Supergroups for String Theory”, which brings “super” (i.e. $\mathbb{Z}_2$-graded) objects into higher gauge theory.  There are “super” versions of vector spaces and manifolds, which decompose into “even” and “odd” graded parts (a.k.a. “bosonic” and “fermionic” parts).  Thus there are “super” variants of Lie algebras and Lie groups, which are like the usual versions, except commutation properties have to take signs into account (e.g. a Lie superalgebra’s bracket is commutative if the product of the grades of two vectors is odd, anticommutative if it’s even).  Then there are Lie 2-algebras and 2-groups as well – categories internal to this setting.  The initial question has to do with whether one can integrate some Lie 2-algebra structures to Lie 2-group structures on a spacetime, which depends on the existence of some globally smooth cocycles.  The point is that when spacetime is of certain special dimensions, this can work, namely dimensions 3, 4, 6, and 10.  These are all 2 more than the real dimensions of the four real division algebras, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$.  It’s in these dimensions that Lie 2-superalgebras can be integrated to Lie 2-supergroups.  The essential reason is that a certain cocycle condition will hold because of the properties of a form on the Clifford algebras that are associated to the division algebras.  (John has some related material here and here, though not about the 2-group case.)

Since we’re talking about higher versions of Lie groups/algebras, an important bunch of concepts to categorify are those in representation theory.  Derek Wise spoke on “2-Group Representations and Geometry”, based on work with Baez, Baratin and Freidel, most fully developed here, but summarized here.  The point is to describe the representation theory of Lie 2-groups, in particular geometrically.  They’re to be represented on (in general, infinite-dimensional) 2-vector spaces of some sort, which is chosen to be a category of measurable fields of Hilbert spaces on some measure space, which is called $H^X$ (intended to resemble, but not exactly be the same as, $Hilb^X$, the space of “functors into $Hilb$ from the space $X$, the way Kapranov-Voevodsky 2-vector spaces can be described as $Vect^k$).  The first work on this was by Crane and Sheppeard, and also Yetter.  One point is that for 2-groups, we have not only representations and intertwiners between them, but 2-intertwiners between these.  One can describe these geometrically – part of which is a choice of that measure space $(X,\mu)$.

This done, we can say that a representation of a 2-group is a 2-functor $\mathcal{G} \rightarrow H^X$, where $\mathcal{G}$ is seen as a one-object 2-category.  Thinking about this geometrically, if we concretely describe $\mathcal{G}$ by the crossed module $(G,H,\rhd,\partial)$, defines an action of $G$ on $X$, and a map $X \rightarrow H^*$ into the character group, which thereby becomes a $G$-equivariant bundle.  One consequence of this description is that it becomes possible to distinguish not only irreducible representations (bundles over a single orbit) and indecomposible ones (where the fibres are particularly simple homogeneous spaces), but an intermediate notion called “irretractible” (though it’s not clear how much this provides).  An intertwining operator between reps over $X$ and $Y$ can be described in terms of a bundle of Hilbert spaces – which is itself defined over the pullback of $X$ and $Y$ seen as $G$-bundles over $H^*$.  A 2-intertwiner is a fibre-wise map between two such things.  This geometric picture specializes in various ways for particular examples of 2-groups.  A physically interesting one, which Crane and Sheppeard, and expanded on in that paper of [BBFW] up above, deals with the Poincaré 2-group, and where irreducible representations live over mass-shells in Minkowski space (or rather, the dual of $H \cong \mathbb{R}^{3,1}$).

Moving on from 2-group stuff, there were a few talks related to 3-groups and 3-groupoids.  There are some new complexities that enter here, because while (weak) 2-categories are all (bi)equivalent to strict 2-categories (where things like associativity and the interchange law for composing 2-cells hold exactly), this isn’t true for 3-categories.  The best strictification result is that any 3-category is (tri)equivalent to a Gray category – where all those properties hold exactly, except for the interchange law $(\alpha \circ \beta) \cdot (\alpha ' \circ \beta ') = (\alpha \cdot \alpha ') \circ (\beta \circ \beta ')$ for horizontal and vertical compositions of 2-cells, which is replaced by an “interchanger” isomorphism with some coherence properties.  John Barrett gave an introduction to this idea and spoke about “Diagrams for Gray Categories”, describing how to represent morphisms, 2-morphisms, and 3-morphisms in terms of higher versions of “string” diagrams involving (piecewise linear) surfaces satisfying some properties.  He also carefully explained how to reduce the dimensions in order to make them both clearer and easier to draw.  Bjorn Gohla spoke on “Mapping Spaces for Gray Categories”, but since it was essentially a shorter version of a talk I’ve already posted about, I’ll leave that for now, except to point out that it linked to the talk by Joao Faria Martins, “3D Holonomy” (though see also this paper with Roger Picken).

The point in Joao’s talk starts with the fact that we can describe holonomies for 3-connections on 3-bundles valued in Gray-groups (i.e. the maximally strict form of a general 3-group) in terms of Gray-functors $hol: \Pi_3(M) \rightarrow \mathcal{G}$.  Here, $\Pi_3(M)$ is the fundamental 3-groupoid of $M$, which turns points, paths, homotopies of paths, and homotopies of homotopies into a Gray groupoid (modulo some technicalities about “thin” or “laminated”  homotopies) and $\mathcal{G}$ is a gauge Gray-group.  Just as a 2-group can be represented by a crossed module, a Gray (3-)group can be represented by a “2-crossed module” (yes, the level shift in the terminology is occasionally confusing).  This is a chain of groups $L \stackrel{\delta}{\rightarrow} E \stackrel{\partial}{\rightarrow} G$, where $G$ acts on the other groups, together with some structure maps (for instance, the Peiffer commutator for a crossed module becomes a lifting $\{ ,\} : E \times E \rightarrow L$) which all fit together nicely.  Then a tri-connection can be given locally by forms valued in the Lie algebras of these groups: $(\omega , m ,\theta)$ in  $\Omega^1 (M,\mathfrak{g} ) \times \Omega^2 (M,\mathfrak{e}) \times \Omega^3(M,\mathfrak{l})$.  Relating the global description in terms of $hol$ and local description in terms of $(\omega, m, \theta)$ is a matter of integrating forms over paths, surfaces, or 3-volumes that give the various $j$-morphisms of $\Pi_3(M)$.  This sort of construction of parallel transport as functor has been developed in detail by Waldorf and Schreiber (viz. these slides, or the full paper), some time ago, which is why, thematically, they’re the next two speakers I’ll summarize.

Konrad Waldorf spoke about “Abelian Gauge Theories on Loop Spaces and their Regression”.  (For more, see two papers by Konrad on this)  The point here is that there is a relation between two kinds of theories – string theory (with $B$-field) on a manifold $M$, and ordinary $U(1)$ gauge theory on its loop space $LM$.  The relation between them goes by the name “regression” (passing from gauge theory on $LM$ to string theory on $M$), or “transgression”, going the other way.  This amounts to showing an equivalence of categories between [principal $U(1)$-bundles with connection on $LM$] and [$U(1)$-gerbes with connection on $M$].  This nicely gives a way of seeing how gerbes “categorify” bundles, since passing to the loop space – whose points are maps $S^1 \rightarrow M$ means a holonomy functor is now looking at objects (points in $LM$) which would be morphisms in the fundamental groupoid of $M$, and morphisms which are paths of loops (surfaces in $M$ which trace out homotopies).  So things are shifted by one level.  Anyway, Konrad explained how this works in more detail, and how it should be interpreted as relating connections on loop space to the $B$-field in string theory.

Urs Schreiber kicked the whole categorification program up a notch by talking about $\infty$-Connections and their Chern-Simons Functionals .  So now we’re getting up into $\infty$-categories, and particularly $\infty$-toposes (see Jacob Lurie’s paper, or even book if so inclined to find out what these are), and in particular a “cohesive topos”, where derived geometry can be developed (Urs suggested people look here, where a bunch of background is collected). The point is that $\infty$-topoi are good for talking about homotopy theory.  We want a setting which allows all that structure, but also allows us to do differential geometry and derived geometry.  So there’s a “cohesive” $\infty$-topos called $Smooth\infty Gpds$, of “sheaves” (in the $\infty$-topos sense) of $\infty$-groupoids on smooth manifolds.  This setting is the minimal common generalization of homotopy theory and differential geometry.

This is about a higher analog of this setup: since there’s a smooth classifying space (in fact, a Lie groupoid) for $G$-bundles, $BG$, there’s also an equivalence between categories $G-Bund$ of $G$-principal bundles, and $SmoothGpd(X,BG)$ (of functors into $BG$).  Moreover, there’s a similar setup with $BG_{conn}$ for bundles with connection.  This can be described topologically, or there’s also a “differential refinement” to talk about the smooth situation.  This equivalence lives within a category of (smooth) sheaves of groupoids.  For higher gauge theory, we want a higher version as in $Smooth \infty Gpds$ described above.  Then we should get an equivalence – in this cohesive topos – of $hom(X,B^n U(1))$ and a category of $U(1)$$(n-1)$-gerbes.

Then the part about the  “Chern-Simons functionals” refers to the fact that CS theory for a manifold (which is a kind of TQFT) is built using an action functional that is found as an integral of the forms that describe some $U(1)$-connection over the manifold.  (Then one does a path-integral of this functional over all connections to find partition functions etc.)  So the idea is that for these higher $U(1)$-gerbes, whose classifying spaces we’ve just described, there should be corresponding functionals.  This is why, as Urs remarked in wrapping up, this whole picture has an explicit presentation in terms of forms.  Actually, in terms of Cech-cocycles (due to the fact we’re talking about gerbes), whose coefficients are taken in sheaves of complexes (this is the derived geometry part) of differential forms whose coefficients are in $L_\infty$-algebroids (the $\infty$-groupoid version of Lie algebras, since in general we’re talking about a theory with gauge $\infty$-groupoids now).

Whew!  Okay, that’s enough for this post.  Next time, wrapping up blogging the workshop, finally.