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August 13, 2012

Strings and Automorphic Forms in Topology

Posted by Urs Schreiber

This week takes place the conference

  • Strings and Automorphic Forms in Topology


intended as the pure-math satellite conference of the recent String-Math 2012 meeting :

The conference focuses on the topological aspects of the series of String-Math meetings. It aims to engage the growing number of mathematicians working on the new interactions of topology with string geometry in mathematical physics and algebraic geometry. Its main topics include topological automorphic forms, conformal field theories, the string group and topological algebraic geometry.

I have missed the first day, having only just returned from a fairly long and very off-line vacation.

Myself. I am speaking about this:

  • Urs Schreiber, Higher prequantum string geometry (pdf)

I’ll be reporting from events at the conference in the comment section below.

Posted at August 13, 2012 9:18 PM UTC

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Re: Strings and Automorphic Forms in Topology

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

Hmm, nnLab 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,n1)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?

Posted by: David Corfield on August 14, 2012 9:50 AM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

but the conference description specifies topological automorphic forms, and we certainly don’t have anything on them.

I have now at least created a stub with some references: nLab:autmorphic form.

I have to admit that I am not entirely sure how the conference organizers thought of the topic selection. Neither the conference abstract nor the talk schedule seems to be focusing much on TAF as opposed to TMF (unless “TBA” is now an abbreviation for “topologial bounded automorphic forms”, or something ;-).

But, yes, as you indicate, on general grounds the path between the terms in the title is this:

stringgeometryheteroticstring STconj.differentialtmftmftaf. string\,geometry \Rightarrow heterotic\,string \sim_{ST\,conj.} differential\,tmf \to tmf \hookrightarrow taf.

Posted by: Urs Schreiber on August 14, 2012 10:52 AM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

Along with the question raised here about

(n,1)(n,1)-dimensional Euclidean field theories and taf,

maybe there’s a (vaguer) question to be asked about the ways mathematical physics is seeing the Langlands Program. There is the S-duality – geometric Langlands link. Is something new expected to emerge here? After all, Shimura varieties figure prominently.

Posted by: David Corfield on August 15, 2012 9:12 AM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

All right, so I decided to abuse (sorry again) the end of the question period after Peter Teichner’s talk to ask (paraphrasing now):

Can anyone say anything about whether/how TAF is related to higher dimensional super-QFT in analogy to how KO is related to the superparticle and TMF to the heterotic superstring (given the table that serves to motivate TAF in the first place)?

Nobody seemed to know anything to that extent. But Chris Schommer-Pries kindly tells me that there is a result which implies that this can not quite work, at least not as naively as the analogy might suggest: it is known, somehow, that there cannot be a Fivebrane-orientation of any TAF theory, nor any further higher orientation.

(But we only had a few minutes to discuss, the next speaker already waiting to start.)

I hope I am reproducing this roughly accurately here. If not and, generally, if you are an expert on (parts of) this question, please drop a note!

Posted by: Urs Schreiber on August 16, 2012 9:46 AM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

Can one know anything about a potential genus for MFivebraneM Fivebrane?

Posted by: David Corfield on August 16, 2012 10:55 AM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

It is my understanding that for TAF of chromatic height greater than 2 (ie for things beyond K-theory and tmf) there can be no orientation from a bordism theory MO[n] for any n. I have heard this attributed to Hopkins and Behrens.

The argument I was told works by noting that TAF has a map to a certain special K(m)-local cohomology theory. This cohomology theory has the property that it detects certain elements in the stable homotopy groups of spheres and also that any orientation from MO[n] factors through an orientation from a lower dimensional MO[k] which does not detect these elements. This is a contradiction, so there cannot be such an orientation of TAF.

Posted by: Chris SP on August 16, 2012 2:33 PM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

Is there any more reason to hope that TAF has a relation to super-QFT than other spectra aiming at higher chromatic filtration, such as K3-spectra? By the way, the author of that, Markus Szymik, was speaking at your meeting on Monday.

Posted by: David Corfield on August 16, 2012 3:51 PM | Permalink | Reply to this

Lazarev: Derived deformation theory and models for classifying spaces.

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 L_\infty-algebras model classifying spaces of VV-fibrations for rational VV. 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…

Posted by: Urs Schreiber on August 14, 2012 10:13 AM | Permalink | Reply to this

Henn (no title)

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)K(2). At the prime 3, we write the spectrum L K(2)S 0L_{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 hFE_2^{hF} where FF is a finite subgroup of the Morava stabilizer group and E 2E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2n=2 at p=3p=3 represents the edge of our current knowledge: n=1n=1 is classical and at n=2n=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.

Posted by: Urs Schreiber on August 14, 2012 11:31 AM | Permalink | Reply to this

Re: Henn (no title)

This is part of the TAF program, isn’t it?

Posted by: David Corfield on August 14, 2012 12:53 PM | Permalink | Reply to this

Re: Henn (no title)

This is part of the TAF program, isn’t it?

Yes, thanks, I should have mentioned this:

for the role of the K(2)K(2)-local sphere in TAF see already the abstract of

  • Behrens, Lawson, Topological automorphic forms (pdf)
Posted by: Urs Schreiber on August 14, 2012 2:19 PM | Permalink | Reply to this

Meier (no title)

Now bell-show: 10+ϵ\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 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()QC(\mathcal{M}) be the \infty-category of quasicoherent sheaves on \mathcal{M}. The global sections of these take values in tmf-module spectra.

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

Claim: This is an equivalence of infinity-categories.

Posted by: Urs Schreiber on August 14, 2012 1:42 PM | Permalink | Reply to this

Re: Meier (no title)

That seems strange… it makes it seem like the moduli space is affine in the infinity setting; why should that be true? Or am I missing something and really we only need affineness of the diagonal map?

Posted by: Dylan Wilson on August 15, 2012 6:27 AM | Permalink | Reply to this

Re: Meier (no title)

Affine features are much more common in the derived world since the global sections see much more of the underlying stack. In some sense, TMFTMF knows about all the stackiness of the moduli stack of elliptic curves. There are currently two approaches to prove these things, one due to myself and due to Akhil Mathews (on which there will be papers online in the near future I guess). Both use the nilpotence theorem in a crucial way. So you see that it is not completely formal.

Posted by: Lennart Meier on August 21, 2013 4:23 PM | Permalink | Reply to this

Re: Meier (no title)

What did you make of the (10+ε)(10+\varepsilon)-minute format? I know that very short talks are common in large conferences in some of the life sciences, but I’ve never seen it done in (pureish) mathematics. I’m curious to know how well the format works.

Posted by: Tom Leinster on August 19, 2012 8:16 PM | Permalink | Reply to this

Kitchloo: The Stable Symplectic Category and derived Geometric Quantization.

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,ω)(X, \omega);

  • morphisms (X 1,ω 1)(X 2,ω 2)(X_1,\omega_1) \to (X_2,\omega_2) are Lagrangian correspondences: Lagrangian submanifolds of (X 1,ω 1)×(X 2,ω 2)(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×L 2X 1×Δ(X 2)×X 3L_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 Ω\Omega where both these spaces are the point. Ω\Omega is a retract of the Thom spectrum MOMO. (details are all in the above article).

Then the statement is that there is now an honest Ω\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.

Posted by: Urs Schreiber on August 15, 2012 9:07 AM | Permalink | Reply to this

Grobounov: Quantum cohomology and quantum groups

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 GL n×C * *(T *F λ;C)H^\ast_{GL_n\times\C^\ast}(T^\ast F_\lambda ;\C) of the cotangent bundle of a partial flag variety F λF_\lambda parametrizing chains of subspaces 0=F 0F 1F N=C n,dimF i/F i1=λ i0=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 𝔤𝔩 N\mathfrak{gl}_N-weight subspace of a Y(gl N)Y(gl_N)-module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] if F λF_\lambda is the full flag variety.

Posted by: Urs Schreiber on August 15, 2012 11:42 AM | Permalink | Reply to this

Schommer-Pries: A finite dimensional model of the string group.

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

Posted by: Urs Schreiber on August 15, 2012 12:22 PM | Permalink | Reply to this

Teichner: Invertible field theories and differential cohomology

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 nn, 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 nn-dimensional quantum field theories formulated as extended FQFT (not-necessarily topological). I had amplified this relation for instance in

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 nn-bundles, the idea is that

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

  • a circle nn-bundle with connection is a morphism

    :P n(X)B nU(1), \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)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 nn-connections for low nn, 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 nn.

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.

Posted by: Urs Schreiber on August 16, 2012 8:37 AM | Permalink | Reply to this

Re: Teichner: Invertible field theories and differential cohomology

After these preliminaries of mine, now the statement of the result that Peter Teichner presents:

It is due to Dmitri Pavlov, Stefan Stolz and Peter Teichner and and it says, as mentioned above, that equivalence classes of smooth invertible dd-dimensional TQFTs over a smooth manifold XX, hence (dropping back to a bit more of my notation to speed up my note-taking) morphisms of smooth \infty-stacks

Bord d(X)B dU(1) \mathbf{Bord}_d(X) \to \mathbf{B}^d U(1)

are in bijection with ordinary differential cohomology H diff d+1(X)H_{diff}^{d+1}(X).

The proof, Peter says, proceeds by adopting the setup of projective local simplicial presheaves over smooth manifolds as set up in

(and discussed more extensively in DCCT).

Then it proceeds by making use of the sequence U(1)\mathbb{Z} \to \mathbb{R} \to U(1) and thereby reducing the computation to that where the coefficients are either \mathbb{Z} (which is easy) and \mathbb{R} (which is where a main lemma will do the work, namely giving the equivalence classes of connection on trivial circle nn-bundles). Then one uses the curvature and characteristic class exact sequences of ordinary differential cohomology to deduce the result.

Posted by: Urs Schreiber on August 16, 2012 9:08 AM | Permalink | Reply to this

Wagemann: On 2-holonomy

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

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

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.

Posted by: Urs Schreiber on August 16, 2012 2:35 PM | Permalink | Reply to this

Waldorf: A characterization of string manifolds via spin structures on loop spaces

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:

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 XX to be a Spin structure, in a certain sense, on loop space LXL 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 XX to fusion spin structures on LXL 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 2\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.

Posted by: Urs Schreiber on August 16, 2012 3:56 PM | Permalink | Reply to this

Henriques: The double trace

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)2Vect P_2(X) \to 2 Vect

and want to compute its higher holonomy.

Since the 2-category 2Vect2 Vect over some field kk is equivalent to that of kk-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 vA_v assigned to each vertex vv;

  • an A v 1A v 2A_{v_1}-A_{v_2}-bimodule N eN_e assigned to each edge ee between vertices v 1v_1 and v 2v_2;

  • a bimodule homomorphism ϕ Σ:N e 1N e 2\phi_\Sigma : N_{e_1} \to N_{e_2} for every 2-cell Σ\Sigma with incoming boundary e 1e_1 and outgoing boundary e 2e_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.

Posted by: Urs Schreiber on August 17, 2012 10:22 AM | Permalink | Reply to this

Prat-Waldron: Geometric quantization of a non-linear supersymmetric sigma-model

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 σ\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 XX, he replaces that by the supermanifold X× 0|nX \times \mathbb{R}^{0|n}, for some nn.

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,θ)=exp(tD 2+θD), U(t, \theta) = \exp(- t D^2 + \theta D) \,,

where (t,θ)(t, \theta) is a super-time interval, and where DD is the Dirac operator on XX on the given spinor bundle.

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

Posted by: Urs Schreiber on August 17, 2012 12:03 PM | Permalink | Reply to this

Re: Strings and Automorphic Forms in Topology

Any good discussion after your talk?

Posted by: David Corfield on August 18, 2012 7:48 AM | Permalink | Reply to this

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