The n-Category Café

Is there a path from your material to the other term in the conference title – automorphic forms?

Hmm, $n$Lab has a link to automorphic form, e.g. from Langlands program, but no entry yet.

But the conference description specifies topological automorphic forms, and we certainly don’t have anything on them. Topological automorphic forms by Behrens and Lawson seems to be the standard reference

We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type $U(1, n−1)$. These cohomology theories of topological automorphic forms (TAF) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves.

So does the topic arise at this conference through something like a parallel for TAF of Stolz-Teichner’s rendition of TMF in terms of supersymmetric Euclidean field theories?

Andrey Lazarev started by giving a review of basic facts of deformation theory along the lines of, say, Hinich’s DG coalgebras as formal stacks.

The central result that he discussed, joint with Jonathan Block he says, is a slight refinement of Sullivan’s classical statement of how automorphism $L_\infty$-algebras model classifying spaces of $V$-fibrations for rational $V$. He pointed out that Sullivan never, apparently, provided a proof of his claims and that his theorem with Jonathan Block is now apparently the first detailed proof.

I need to check if any writeup is available…

Henn is reviewing aspects of the work

• P. Goerss, H.-W. Henn, M. Mahowald, C. Rezk, A resolution of the K(2)-local sphere at the prime 3 (arXiv:0706.2175)

Abstract We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory $K(2)$. At the prime 3, we write the spectrum $L_{K(2)}S^0$ as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form $E_2^{hF}$ where $F$ is a finite subgroup of the Morava stabilizer group and $E_2$ is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case $n=2$ at $p=3$ represents the edge of our current knowledge: $n=1$ is classical and at $n=2$, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.

Now bell-show: 10+$\epsilon$-minute talks.

Lennart Meier states the following result and sketches the proof, based on results of Jacob Lurie:

Let $\mathcal{M}$ be the derived moduli stack of elliptic curves, i.e. the moduli stack of elliptic curves equipped with its structure $\infty$-sheaf of $E_\infty$-rings that over a given elliptic curve assigns the spectrum representing the corresponding elliptic cohomology theory. (See A survey of elliptic cohomology).

Let then $QC(\mathcal{M})$ be the $\infty$-category of quasicoherent sheaves on $\mathcal{M}$. The global sections of these take values in tmf-module spectra.

$$\Gamma : QC(\mathcal{M}) \to tmf Mod \,.$$

Claim: This is an equivalence of infinity-categories.

Nitu Kitchloo talks about the work

• Nitu Kitchloo, The Stable Symplectic Category and Geometric Quantization (arXiv:1204.5720)

He is showing slides and will make them online available, he says.

He starts by briefly reviewing the idea, due to Alan Weinstein, that geometric quantization should eventually give a representation of the “symplectic category” whose

• objects are symplectic manifolds $(X, \omega)$;

• morphisms $(X_1,\omega_1) \to (X_2,\omega_2)$ are Lagrangian correspondences: Lagrangian submanifolds of $(X_1, \omega_1) \times (X_2, -\omega_2)$;

and composition is given by taking fiber products of these Lagrangian submanifolds.

However, this is not actually quite a category, since composition is only well-defined when the intersection of $L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times X_3$ is transverse.

(This kind of issue is familiar from constructions of the Fukaya category.)

Now Nitu Kitchloo explains how he proposes to rectify this: he introduces a notion of stabilized Lagrangian correspondences. For any two symplecitc manifolds, the collection of all of them forms a spectrum, which is a module spectrum over the corresponding spectrum $\Omega$ where both these spaces are the point. $\Omega$ is a retract of the Thom spectrum $MO$. (details are all in the above article).

Then the statement is that there is now an honest $\Omega$-module-spectrum-enriched category whose objects are (compact maybe?) symplectic manifolds, and whose hom-objects are these spectra of stabilized Lagrangians.

There is a version of this construction with and without metaplectic correction. From p. 16 of his article I see that with metaplectic correction built in, he does obtain a representation of the symplectomorphism group of a given symplectic manifold on the “stable symplectic category”.

Now he gets back to geometric quantization. He wants to discuss a refinement of his “stable symplectic category” where now the objects are prequantized symplectic manifolds. However, what he shows in his slides are lifts from de Rham cohomology to integral cohomology instead of to differential cohomology as it should be for prequantization. (I asked about this, but currently it remains inconclusive to me what’s going on now.)

The next slide had something on the step from prequantization to actual geoemtric quantization. But I didn’t follow what he said there.

Vassily Gorbounov talked about

• V. Gorbounov, R. Rimanyi, V. Tarasov, A. Varchenko, Cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra (arXiv:1204.5138)

Abstract We interpret the equivariant cohomology algebra $H^\ast_{GL_n\times\C^\ast}(T^\ast F_\lambda ;\C)$ of the cotangent bundle of a partial flag variety $F_\lambda$ parametrizing chains of subspaces $0=F_0\subset F_1\subset \cdots\subset F_N =\C^n, \dim F_i/F_{i-1}=\lambda_i$, as the Yangian Bethe algebra of a $\mathfrak{gl}_N$-weight subspace of a $Y(gl_N)$-module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] if $F_\lambda$ is the full flag variety.

Chris Schommer-Pries talked about definition and construction of the String group and of its refinement to the smooth String 2-group.

Peter Teichner talks about that special case of extended functorial QFT which is the higher parallel transport of a circle n-bundle with connection.

Originally we did this for low $n$, see the references here. It was kind of clear early on that this higher parallel transport is to be thought of as the “classical version” (in the sense of the classical physics/quantum physics dichotomy) of $n$-dimensional quantum field theories formulated as extended FQFT (not-necessarily topological). I had amplified this relation for instance in

• Urs Schreiber, AQFT from n-functorial QFT (arXiv:0806.1079),

which observes that when a 2-dimensional non-topological QFT is realized as a parallel transport 2-functor on Minkowski spacetime, then then corresponding algebras of observables satisfy the basic axiom of AQFT: they form a local net. (This is the step from the Schrödinger picture to the Heisenberg picture in the context of extended QFT).

So for me an extended QFT was a “quantized higher parallel transport”.

On the other hand, Stefan Stolz and Peter Teichner in their program had started to study non-topological extended QFTs, and so they are now coming from the other direction: they think of higher parallel transport as being an “invertible quantum field theory”. Just two different ways to think of the same relation.

So for the case of higher parallel transport in circle $n$-bundles, the idea is that

• to each smooth manifold $X$ there is a smooth n-groupoid (an $n$-stack on $SmoothManifolds$) $\mathbf{P}_n(X)$ – the path n-groupoid;

• a circle $n$-bundle with connection is a morphism

  $$\nabla : \mathbf{P}_n(X) \to \mathbf{B}^n U(1) \,,$$

  where on the right we have the n-fold stacky delooping of $U(1)$.

I have more of a survey and exposition of this in the introduction section 1.3 of differential cohomology in a cohesive topos.

Back then we only formally proved the above higher parallel transport description of circle $n$-connections for low $n$, since we were then concentrating on its generalization to more general principal 2-connections and then eventually generalizing to connections on principal infinity-bundles by other means. But it seemed clear that the proofs should generalize to all $n$.

This is, essentially, what Peter Teichner says they have proven now. But let’s see, the talk is still running, so far it was all review of extended QFT. More in the next comment.

Friedrich Wagemann (replacing Christoph Wockel, who is apparently sick at home – hope you get well soon, Christoph!) speaks about

• Hossein Abbaspour, Friedrich Wagemann, On 2-Holonomy (arXiv:1202.2292)

This article suggests a higher Hochschild-cohomology-perspective of the nonabelian surface parallel transport of a connection on a principal 2-bundle in the form of a parallel transport along paths in path space in section 2.3.4 of

• John Baez, Urs Schreiber, 2-Connections on 2-bundles (arXiv:hep-th/0412325)

and section 2.3.1 of

• Urs Schreiber, Konrad Waldorf, Smooth functors vs. differential forms, Homology, Homotopy Appl., 13(1), 143-203 (2011) (arXiv:0802.0663)

by combining that with the constructions around example 2.3.2 of

• Gregory Ginot, Thomas Tradler, Mahmoud Zeinalian, A Chen model for mapping spaces and the surface product, Annales scientifiques de l’ENS, série 4 43, fascicule 5 (2010), 811-881(arXiv:0905.2231)

The idea behind this is that one can always use the construction of

• Friedrich Wagemann, On Lie algebra crossed modules, Comm. Algebra 34 (2006) no. 5, 1699-1722 (arXiv:math/0611375)

to replace the given crossed module of groups/Lie algebras by one whose top-degree component is abelian, which hence makes that non-abelian 1-form on path space be actually abelian.

I think, as I have discussed with Friedrich, that there is still an issue when the endpoints of the paths in the Hochschild construction are allowed to move, since to match that to the 2-holonomy as defined above requires, in general, a correction term by parallel transport along that string boundary.

Konrad Waldorf talks about a new result within his program of equipping smooth bundles on smooth loop spaces with enough extra structure such as to make the transgression map from 2-bundles on the base space to bundles on loop space be an equivalence, or at least an isomorphism on equivalence classes:

• Konrad Waldorf, Transgression to loop spaces and its inverse I - III (arXiv:0911.3212, arXiv:1004.0031, arXiv:1109.0480).

The new result concerns the very origin of string structures in string theory: originally Killingback and then Witten defined (see at the above link) a String structure on spacetime $X$ to be a Spin structure, in a certain sense, on loop space $L X$. But really the Spin structure on loop space is just the transgression of the String structure down on the base, and the transgression map can in general be neither injective or surjective. So considering bare Spin structures on the loop spaces misses out on crucial information.

But, since a spin structure is given by a Spin-principal bundle and, by the modern understanding, a String structure is given by a String 2-group-principal 2-bundle, this situation is an instance of what Konrad’s general theory applies to.

He has a nice writeup that should appear soon, and I won’t be able to do better here, so I’ll just be very brief:

a bundle on loop space that arises from transgression of a 2-bundle down on the base has one crucial bit of extra structure: it “is fusion” in that it is compatible with the overlapping-concatenation (fusion) of loops. If this structure is remembered, Konrad calls it a “fusion bundle”. Correspondingly there is a notion of “fusion spin structure”, etc.

The theorem then is: transgression is an isomorphism from string structures on $X$ to fusion spin structures on $L X$.

There is a simpler variant of this setup, where Spin-structures on spacetime are related to orientations of loop space. In this case, too, it is fusion orientations that one needs to make this be a bijection. This was shown in theorem 9 of:

• Stephan Stolz and Peter Teichner, The Spinor bundle on loop space (pdf).

Konrads construction and result is a generalization of this. But since an orientation corresponds to a $\mathbb{Z}_2$-bundle with discrete structure group, he needs a good bit more of technology to handle the more general case where the structure group of the bundle on loop space is a non-trivial Lie group.

André Henriques speaks the about the following question (my paraphrase):

What is a definition of 2-trace that can be applied to a suitable surface diagram of 2-vector spaces in analogy to how the ordinary trace can be applied to a suitable string diagram of (finite dimensional) vector spaces?

Or rather: given an evident guess for a definition, is it well defined?

This question arises for instance when you have a parallel 2-transport or extended 2d QFT with values in 2-vector spaces

$$P_2(X) \to 2 Vect$$

and want to compute its higher holonomy.

Since the 2-category $2 Vect$ over some field $k$ is equivalent to that of $k$-algebras, algebra bimodules between these as 1-morphisms, and bimodule homomorphisms between those as 2-morphisms, this question more concretely reads as follows:

suppose you have a surface, say a torus, tringaulated in an oriented way, or otherwise cell-decomposed, with

• an algebra $A_v$ assigned to each vertex $v$;

• an $A_{v_1}-A_{v_2}$-bimodule $N_e$ assigned to each edge $e$ between vertices $v_1$ and $v_2$;

• a bimodule homomorphism $\phi_\Sigma : N_{e_1} \to N_{e_2}$ for every 2-cell $\Sigma$ with incoming boundary $e_1$ and outgoing boundary $e_2$.

Then a double trace should send this to a number by producing tensor product of bimodules along vertices and traces of bimodule homomorphisms, at least provided that the bimodules involved are suitably “trace class”.

André gives a comparatively simple example of this for the case of a 2-torus.

The next question is how to formalize what it means for a morphism of 2-vector spaces, hence for a bimodule, to be “trace class”.

To address this, it should help to first see how “trace class” can be defined generally in monoidal categories. This questions was addressed and answered in

• Stephan Stolz, Peter Teichner, Traces in monoidal categories (arXiv:1010.4527)

André says he is generalizing the kind of argument given there now to the 2-dimensional case, terming the concept “double trace class” for the moment. This involves drawing some nice surface diagrams, which I can’t try to indicate here.

After this definition, André turns to the intended application to (extended) 2-dimensional super-conformal field theory. Here we think of the above bimodules as spaces of states of the theory, and the bimodule homomorphisms as correlators. The 2-trace of these on a given surface is going to be the partition function of the theory. Specifically for the heterotic string this is going to be a modular form on the moduli of the given torus.

So I asked how this relates to

• Kate Ponto, Michael Shulman, Shadows and traces in bicategories (arXiv:0910.1306)

André says its similar, but probably a little different. Certainly the “shadow” he uses is not defined globally on the 2-category as in Ponto-Shulman – that’s precisely what the “2-trace class” condition above is about. But to which extent the concepts coincides once this is taken care of seems to be unclear at the moment.

Aturo Prat-Waldron speaks about looking into geometric quantization of (1|1)-dimensional supersymmetric sigma-models, also known as the (worldline supersymmetric) superparticle on some Riemannian spin target space.

He says he is motivated by trying to understand in a precise fashion the famous but non-rigorous physics derivations of the Atiyah-Singer index theorem, based on this $\sigma$-model… and then, eventually, by trying to generalize this to the heterotic superstring, i.e. the (2|1)-dimensional sigma model (which, by analogy, would be expected to yield some tmf-index theorem, of sorts).

So he proceeds by writing out the standard action functional and determines the covariant phase space, all done cleanly in supergeometry.

Then he turns to the geometric quantization. Since he does not consider a background gauge field the phase space has an exact symplectic form with a symplectic potential caninically given by the variational principle, and so he can choose the trivial line bundle with that global 1-form as its connection to be the prequantum line bundle.

The super-geometric variant of geometric quantization is not so well-studied (see some references here). Aturo points out that take at face falues there will be no super-Lagrangian subspaces. He says, based on suggestions by Stephan Stolz and Peter Teichner, he fixes that by chaning the setup slightly: instead of taking the target space to be an ordinary manifold $X$, he replaces that by the supermanifold $X \times \mathbb{R}^{0|n}$, for some $n$.

Then Aturo continues going through the program, now discussing choices of polarization and metaplectic correction.

After the dust has settled he does find what one would hope for: the Hilbert space of states of the system is that of square integrable spinors on the target space: the superparticle is the spinning particle.

Finally his main theorem is to derive the Hamiltonian time evolution. Apparently it’s non-trivial and involves taking care of various subtle terms, but out comes luckily, as expected, the expected super-time evolution operator

$$U(t, \theta) = \exp(- t D^2 + \theta D) \,,$$

where $(t, \theta)$ is a super-time interval, and where $D$ is the Dirac operator on $X$ on the given spinor bundle.

So all is well and just as expected. But apparently this was not written out precisely before.

Any good discussion after your talk?