The n-Category Café

Edward Frenkel gave a survey of his joint project with A. Losev and Nikita Nekrasov that they have been pursuing for serveral years already.

This starts with the observation is that there are several quantum field theories whose action functional is the sum of a “kinetic” and a “topological” piece, in such a way that the prefactors of those two pieces are usefully regarded as the real and the imaginary part of a single but complex coupling constant. The most famous example is probably Yang-Mills theory (see here) in which case this complex parameter notably controls (or is controled by) S-duality. Another famous example, actually closely related, which they amplify is the $(2|2)$-supersymmetric sigma-model, the superstring in a background field of gravity and B-field. Finally, to bring out the underlying formal structure most clearly, Frenkel and his coauthors observe that one can go down even one more step and still observe this phenomenon in the toy model of supersymmetric quantum mechanics. There is a useful review of their considerations in that simple case by Jacques Distler here.

The content of the program now is to consider the theory in a certain limit of that complex parameter, which is such as to make the possible physical configurations restrict to (“localize” to) only very special “instanton” configurations, which no longer form a huge space, but just a finite-dimensional manifold, over which the path integral is well defined. A famous examples of this is the A-model variant of the above 2-dimensional $\Sigma$-model, where this path integral famously computes the Gromov-Witten invariants of its target space.

The goal of the Frenkel-Losev-Nekrasov program is to go beyond well-known examples such as this one, considering corrections to the full “topological limit”. For instance they get Gromov-Witten invariants next not on base space itself, but on its jet space, and not a topological field theory but a “logarithmic conformal field theory”.

An exposition of the general program is given in

• Frenkel, Losev, Nekrasov, Notes on instantons in topological field theory and beyond (arXiv:hep-th/0702137)

Since I see pretty much all the notes that I took in the talk today have their counterpart in that note, I’ll just point to that. A few more references and pointers are listed at the $n$Lab entry on instanton.

Maxim Kontsevich talks about new results and conjectures related to his wall crossing formula for BPS states in supersymmetric theories.

There is no point in me reproducing the notes taken from the talk here. But afterwards I looked around a bit and found something that is worthwhile to be linked to here, namely there is a nice set of relatively recent introductory lectures notes:

• Greg Moore, PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories (2010) (pdf)

which look pretty good for an introduction.

In that direction there is also

• Sergio Cecotti, Trieste lectures on wall-crossing invariants (2010) (pdf)

which however assumes that you know a bit of background. For more references see behind the above links.

Catharina Stroppel is surveying her theorems on categoriefied knot invariants and related, from her article with Frenkel and Sussan that was discussed here before and from joint work with Ben Webster.

The relevant references are at the end of this list at categorification in representation theory.

Tom Bridgeland outlines a proof, or the ideas going into it, that the moduli space of Bridgeland stability conditions on the derived category over a suitable complex 3d manifold is equivalent to that of quadratic differentials on that space (with simple 0s).

He advertizes this as a first little step in making mathematically precise the considerations in

• Davide Gaiotto, Gregory W. Moore, Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation (arXiv:0907.3987)

see also the talk by Kontsevich above.

(And now my battery dies…)

Greg Moore now gives what would have served as the pilot talk to the previous talks of Cecotti, Cordova, Kontsevich and Bridgeland: he surveys, as the title say, recent developments in understanding the vacuum moduli spaces, notably the spaces of BPS states, in N=2 D=4 super Yang-Mills theory.

There is no chance to take notes from the slides that are being flashed, but I notice that the following set of slides is very similar, and not too much out of date:

• Greg Moore, Surface Defects and the BPS Spectrum of 4d N=2 Theories, Solvay 2011 (pdf)

I skipped yesterday’s afternoon talks in order to discuss some math with Peter Teichner’s group over at the Max-Planck institute (after all the string-math ;-).

Now on to Wednesday’s parallel sessions, and so I have to make choices.

Hagen Triendl just talked about joint work with Mariana Graña on exceptional generalized geometry in supergravity and string theory. As usual here, this proceeds by slide flashing, and so I can’t provide much detailed notes. But it seems that many of the aspects mentioned in the talk appear in detail in

• Mariana Graña, Francesco Orsi, N=2 vacua in Generalized Geometry (arXiv:1207.3004)

The basic idea is this:

the U-duality-invariant formulation of, in particular, type II supergravity in the absence of branes (hence of charges) is naturally an “exceptional generalized geometry” – see here – given nicely by reduction of the structure group of an exceptional tangent bundle from (the split real form of) an exceptional Lie group to, typically, its maximal compact subgroup. This is in direct generalization of how an ordinary Riemannian metric on a smooth manifold is precisely the ( smooth ) reduction of the structure group of the ordinary tangent bundle along the inclusion $O(n) \hookrightarrow GL(n)$: a vielbein / orthogonal structure.

Now, in the presence of branes, hence of charges, these geometric structures are further twisted. For instance the B-field which, at least locally, is part of a type II geometry given by reduction of the generalized tangent bundle along $O(n)\times O(n) \hookrightarrow O(n,n)$, becomes a twisted B-field and it is not a-priori clear how that fits into the perspective of (exceptional) generalized geometry.

Now the idea of the research that Triendl indicated is to circumvent this problem by making use of the fact that several branes in type II backgrounds come from “pure geometry” under the compactfication from M-theory/11-dimensional supergravity. This means that there is a chance to describe their preimages there by exceptional generalized geometry, and then follow what happens to that geometry when put through the KK-reduction.

All this is done, in turn, for compactifications on 7-dimensional respectively 6-dimensional internal spaces, hence along a geometry that (locally) is of the form

$$\array{ S^1 \times Y_6 \times \mathbb{R}^{3,1} &\to& Y_6 \times \mathbb{R}^{3,1} &\to& \mathbb{R}^{3,1} \\ exceptional M-geometry && exceptional \,{type\,II\, geometry} && effective \, {4-d theory} } \,.$$

Accordingly, the work to be done now consists of taking the exceptional structure group $E_{7(7)}$ and decompose its representations according to this splitting, then tracking which kind of generalized geoemtry this yields in $6+4$ dimensions induced from exceptional generalized geometry in $1+6+4$-dimensions.

As I said, it seems that much of this appears already in the above article, but I’ll try to get hold of whatever further details there are available.

Axel Kleinschmidt surveyed his recent work:

• Philipp Fleig, Axel Kleinschmidt, Eisenstein series for infinite-dimensional U-duality groups (arXiv:1204.3043)

The phenomenon investigated here is that the graviton-scattering amplitudes in higher dimensional supergravity compactified to $(11-n)$-dimensions is invariant under the U-duality group $E_{n}(\mathbb{Z})$ and as such is given by a series of higher curvature corrections which is controled by a kind of Eisenstein series of functions on the moduli, for the corresponding U-duality group. The point of the work is to demonstrate that this makes sense and remains true even as $n \gt 8$ in which case one deals with the still rather mysterious Kac-Moody groups.

This is part of a big program that Kleinschmidt has been following with Hermann Nicolai for many years. See the references here for an impression.

Nils Carqueville reports on joint work with Ingo Runkel and Daniel Murfet building on

• Nils Carqueville, Ingo Runkel, Rigidity and defect actions in Landau-Ginzburg models (arXiv:1006.5609)

I arrived very late into this talk from lunch (or rather: from getting some work done), so I missed lots of information.

But the basic idea is to show how a 2d TQFT with defects is induced by a bicategory with all 1-adjoints, in particular so for Landau-Ginzburg models, where the adjoints of 1-morphism corresponds to reverseal of defects.

David Andriot is showing us these slides:

• Non-geometric fluxes in higher dimensions: I (pdf)

The basic idea seems to be to try to make sense of that fragment of type II geometry (see there) which is usually called “non-geometric”, namely where the transition functions of the generalized tangent bundle are allowed to involve those “$\beta$-transformations”, as in, for instance:

• Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, Generalized Geometry and Non-Geometric Backgrounds (arXiv:0807.4527)

Lara Anderson talks about a new strategy to stabilize moduli in compactifications of the heterotic string.

Two of the three arXiv numbers shown at the beginning lead to the following two articles with the details:

• Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut, Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua (arXiv:1102.0011)

• Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut, The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications (arXiv:1107.5076)

The context is compactifications of $E_8 \times E_8$ heterotic string backgrounds on a Calabi-Yau complex 3-fold with a holomorphic vector bundle with structure group $G \subset E_8$.

Recent years have seen lots of activity on the problem of moduli stabilization in type II string theory. These techniques can’t really be adapted to the heterotic case, and so one needs other tools here.

The idea studied here was introduced in

• Lara B. Anderson, James Gray, Andre Lukas, Burt Ovrut, Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua (arXiv:1010.0255)

and it consists of the observation that a holomorphic background gauge field on the internal Calabi-Yau may serve to constrain the moduli to vary only such that holomorphicity is preserved.

Hence one is led to solve the following mathematical question: given a 6d manifold $Y$, how to find families of complex bundles over $Y$ equipped with a CY-structure, such that the bundle is holomorphic only at isolated points of the CY moduli space?

Lara Anderson is showing slides that first decompose this problem into some homological algebra, expressed in terms of various Ext-groups of various sheaves. Then at some point a computer algebra program gets into the game and is used to scan the moduli spaces for the desired isolated regions.

And so forth. The computer apparently spits out 27 such regions under some assumptions, but most of them are “too singular”. So now we hear about how to blow up these singularities. Some of these are well-understood because they correspond to conifold transitions, so now we focus on these.

The talk by Albert Schwarz was impressive in its way, given who Albert Schwarz is. It was however hard to understand, even from the second row where I was sitting.

He started by highlighting his work with Movshev on deformations of super Yang-Mills theories. Then he amplified the observation that a Chern-Simons term can be written down as soon as one as an associative dg-algebra, a basic phenomenon underlying cubic string field theory and its various limiting cases. He closed by pointing to his recent work on the cohomology of super-Poincaré Lie algebras, which we have been discussing a while back here.

Stephan Stieberger reports on recent progress of understanding $n$-point tree-level superstring scattering amplitudes, first for the open but then also for the closed superstring, and drastically simplifying their expression in terms of motivic multi-zeta values. As a result there is apparently a graded Hopf algebra structure on the space of states, which turns out to be useful.

The details are in

• O. Schlotterer, S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes (arXiv.1205.1516)

I was late, tired and overworked when I arrived this morning. The last minutes of Don Zagier’s talk on Mock Modularity and Applications that I saw reminded me – positively but vividly – of a live blackboard version of Vi Hart’s math videos. In any case, I don’t have notes.

Then Jeff Harvey gave an exposition of work described in the article

• Miranda Cheng, John Duncan, Jeffrey Harvey, Umbral Moonshine (arXiv:1204.2779)

Ashoke Sen is showing these slides:

• Ashoke Sen, Black holes to quivers (pdf)

Ah, now a talk on a fundamental mathematical aspect of string theory.

Leonardo Rastelli is talking about Distances between Conformal Field Theories, following proposals by Maxim Kontsevich and Yan Soibelman as laid out and referenced notably in

• Yan Soibelman, Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry (PSPM 83)

Recall that the basic premise of perturbative string theory is that spacetime together with the background gauge fields on it is encoded in terms of a 2-dimensional CFT that is, or behaves as if it is, the sigma-model that describes the quantum mechanics of single strings propagating in this spacetime, charged under these background fields.

Hence it is of interest to get an idea of “the space of all such 2dCFTs”.

In its entirety, very little is known about this space. One tiny corner that has been described in full detail by mathematical classification theorems is rational 2d CFT (this was achieved by FRS formalism).

Another tiny corner, consisting of sigma-models on target spaces which are Calabi-Yau fibrations with various assumptions on the background fields, attracted some attention a few years back, because it was a little less tiny than some people had hoped: this has come to be called the landscape.

To appreciate the general problem, it is useful to consider its analog for particle phyiscs: we may decide that it is a good idea to describe a spacetime with its background fields in terms of the quantum mechanics of a single particle propagating on this manifold, and charged under these fields. That quantum mechanics is given by three pieces of data: a Hilbert space of states, an algebra (of smooth functions on spacetime) densely embedded in it, and a certain operator on that space.

Such a triple of data is called a spectral triple. The study of target space geometries in terms of their spectral triples is known, somewhat unfortunately, a Connes-style noncommutative Riemannian geometry (or some variant of these words). A better term would be spectral geometry.

In any case, there is a rather well developed theory of such spectral geometry. The starting point of perturbative string theory is entirely analogous to this, just “lifted in dimension” by one, in a way that is a lot like categorification in various ways: it is like studying the moduli space of 2-spectral triples.

In Yan Soibelman’s article mentioned above the strategy is to turn this around and see what one can say about the 1-spectral geometry that arises from the 2-spectral stringy/CFT geometry in the limit that we assume the string to have vanishing extension.

Now back to Rastelli’s talk, finally: he points out how we are thus looking for analogs and generalizations of distance on moduli spaces of Riemannian metrics. He is scanning through various proposal of what one could define to be a sensible distance between CFTs, and what one knows about each proposal (not much in general).

The main proposal that he says he wants to propose now involves a concept he calls “conformal interfaces”, which is formulated in terms of boundary states for open string CFTs. I should better listen now and stop typing…

Anastasia Volovich is talking about the recent drastic progress in organizing scattering amplitudes in N=4 D=4 super Yang-Mills theory in terms of motivic structures.

I am starting to collect her and other related references here (but eventually I may decide to move all that elsewhere).

We had already heard Stieberger talk about this here yesterday, and apparently we will hear more about this today and, I guess, on Saturday.

Matthias Gaberdiel talks about Mathieu moonshine (see there for more).

In the evening Christophe Grojean surveyed the state of the art of Higgs measurements at the LHC.

One little fact which I hadn’t heard before: the data already shows that whatever that particle is that is seen at 125 GeV, it does couple to the fermions in the standard model (as of course it better does if indeed it is the Higgs particle).

He had detailed slides on the subtlety of how to measure the sign whith which it does so, but had to skip these for time reasons.

Edward Witten in his talk made the point that despite the evident relevance of supergeometry in superstring perturbation theory, there are some technical aspects that remain unnecessarily unclear in the literature due to a lack of intrinsically supergeometric formulation.

Examples he amplified was

1. the natural characterization of Neuveu-Schwarz and of Ramond vertex operators on the superworldsheet in terms of divisors of this supermanifold and

2. in some detail, the interpretation of Friedan-Martinec-Shenker’s notion of picture sectors in 2d SCFT in terms simply of the degree of “integrable super-differential forms” on which the notion of integration over supermanifolds is based.

These aspects are “known, but not well-known”. The supergeometric interpretation of SCFT picture sectors as above is given in

• Alexander Belopolsky, Picture changing operators in supergeometry and superstring theory (arXiv:hep-th/9706033v2)

following work of E. and H. Verlinde in the late 1980s, but was not widely taken note of, apparently. (Interesting to check out the citations to it to see who did take note of it.)

Witten’s main point, however, was to amplify that a more intrinsically supergeometric approach should help make superstring perturbation theory be more tractable and more transparent at higher worldsheet genus. He recalled and emphasized that existing literature makes extensive use of accidental global splitting of the (complex) super-moduli super-space $\tilde M_{g,n_{NS}, n_R}$ of super-Riemann surfaces (of genus $g$ with $n_{NS}$ Neveu-Schwarz and $n_R$ Ramond-type punctures) into an ordinary (bosonic) manifold with a bundle of super-directions above it, which works for low enough $g,n$ only.

In particular the overall strategy of the celebrated computation of D’Hoker and Phong of $g = 2$ amplitudes,

• Eric D’Hoker, D.H. Phong, Lectures on Two-Loop Superstrings (arXiv:hep-th/0211111)

will not generalize to higher genus.

Jacques Distler once posted some nice discussion of related matters on his blog:

• Jacques Distler,

  Chiral superstring measure (blog)

  More D’Hoker and Phong (blog)

Right now Ron Donagi is speaking about joint work with Edward Witten related to this. They prove that

• $\tilde M_g$ is not split for $g \geq 5$,

• $\tilde M_{g,1}$ not split for $g \geq 2$.

Witten has been given talks with the same title as this one, but with non-identical content on other conferences recently. One occasion for which I see that the slides are online is

• Edward Witten, Superstring perturbation theory revisited, March 2012 (pdf)

This has emphasis on different aspects than the talk given today. But it seems to essentially verbatim overlap from page 22/slide55 to page 40/slice 116.

Nigel Hitchin talks about the idea first indicated in section 2.4 of

• David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry (arXiv:1101.0856)

about a generalization of type II geometry motivated by (and I think meant to be a toy example of?) the exceptional generalized geometry (as first seen by Chris Hull, see the references given behing that link) in 11-dimensional supergravity.

By the way, the Lie algebra structure in the Chern-Simons term of 11-dimensional supergravity that Baraglia observes in his section 2.3 is the ungraded shadow of the respective dg-/Lie algebra / $L_\infty$-algebra observed in section 4 The Gauge Algebra of Supergravity in 6k−1 Dimensions in

• Hisham Sati, Geometric and topological structures related to M-branes (arXiv:1001.5020)

There is more here than meets the eye.

Joel Kamnitzer gives an exposition of the basic idea and central results of his joint

project with Sabin Cautis and Anthony Licata

on the categorification of knot invariants (notably including and hence generalizing Khovanov homology) by means of geometric representation theory.

A little bit of introduction to this project is in section 1 of the older article

• Sabin Cautis, Joel Kamnitzer, Knot homology via derived categories of coherent sheaves I, sl(2) case (arXiv:0701194)