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July 16, 2012

Notes from String-Math 2012

Posted by Urs Schreiber

This week I am in Bonn at String-Math 2012, the second in a new annual series of conferences on mathematical aspects of string theory (the first was in Philadelphia, last year).

I will post notes about some talks below in the comment section.

They will be a bit more terse than I originally intended, because at the same time I have to be finalizing an article with Hisham and Domencio, on Extended higher cup-product Chern-Simons theories before I go on vacation next week.

We’ll see how it works out. Now on to Frenkel’s talk…

Posted at July 16, 2012 11:24 AM UTC

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Frenkel: Beyond topological field theory

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)(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 Σ\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 nnLab entry on instanton.

Posted by: Urs Schreiber on July 16, 2012 12:47 PM | Permalink | Reply to this

Re: Frenkel: Beyond topological field theory

I see from Jacques Distler’s post you link to that Morse theory is involved. We could do with a bright young person beefing up the nLab Morse Theory page. I have been surprised before that Morse theory didn’t have greater prominence.

Posted by: David Corfield on July 17, 2012 9:00 AM | Permalink | Reply to this

Kontsevich: Integrality for K_2-Symplectomorphisms

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=4d = 4, 𝒩=2\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.

Posted by: Urs Schreiber on July 17, 2012 8:53 AM | Permalink | Reply to this

Stroppel: Categorification and Fractional Euler Characteristics

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.

Posted by: Urs Schreiber on July 17, 2012 9:51 AM | Permalink | Reply to this

Bridgeland: Tom Bridgeland: Quadratic Differentials as Stability conditions

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…)

Posted by: Urs Schreiber on July 17, 2012 10:47 AM | Permalink | Reply to this

Moore: Progress in N=2 Field Theory

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)
Posted by: Urs Schreiber on July 17, 2012 11:24 AM | Permalink | Reply to this

Re: Moore: Progress in N=2 Field Theory

Much of the talk now is about the Spectral networks tool for studying BPS states.

Posted by: Urs Schreiber on July 17, 2012 12:04 PM | Permalink | Reply to this

Triendl: Generalized type IIA orientifold and M-theory compactifications

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)GL(n)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)×O(n)O(n,n)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

S 1×Y 6× 3,1 Y 6× 3,1 3,1 exceptionalMgeometry exceptionaltypeIIgeometry effective4dtheory. \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)E_{7(7)} and decompose its representations according to this splitting, then tracking which kind of generalized geoemtry this yields in 6+46+4 dimensions induced from exceptional generalized geometry in 1+6+41+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.

Posted by: Urs Schreiber on July 18, 2012 9:21 AM | Permalink | Reply to this

Kleinschmidt: Eisenstein series on Kac-Moody groups

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 (11n)(11-n)-dimensions is invariant under the U-duality group E n()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>8n \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.

Posted by: Urs Schreiber on July 18, 2012 11:11 AM | Permalink | Reply to this

Carqueville: Defects and adjunctions in Landau-Ginzburg models

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.

Posted by: Urs Schreiber on July 18, 2012 2:00 PM | Permalink | Reply to this

Re: Carqueville: Defects and adjunctions in Landau-Ginzburg models

Nils kindly points out by email that the slides for his talk are available online:

  • Nils Carqueville, Defects and adjunctions in Landau-Ginburg models (pdf)

Also, I should say that this is about two projects, one with Ingo Runkel on generalized orbifolds, the other with Daniel Murfet on adjunctions in the bicategory of Landau-Ginzburg models. The former uses results of the latter, which in turn is a big generalization of the one that I did link to above.

I hope I got this right. Otherwise, feel free to hit “reply to” and add more information.

Posted by: Urs Schreiber on July 19, 2012 6:15 AM | Permalink | Reply to this

Re: Carqueville: Defects and adjunctions in Landau-Ginzburg models

From the slides

2d TFTs with defects are naturally described in terms of bicategories with extra structure.

So this approach relates to Vaughan Jones’ planar algebras? Urs once wrote

If you’d ask me how I would summarize this in few words i’d say:

A planar algebra is a 2d genus-0 TFT with defect lines,

elaborated here.

But then should we not see some von Neumann algebra-like entities appear?

Posted by: David Corfield on July 19, 2012 9:25 AM | Permalink | Reply to this

Andriot: Non-geometric fluxes versus (non)-geometry

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 “β\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)
Posted by: Urs Schreiber on July 18, 2012 2:16 PM | Permalink | Reply to this

Re: Andriot: Non-geometric fluxes versus (non)-geometry

A review is in section 2 of

  • David Andriot, Magdalena Larfors, Dieter Lust, Peter Patalong, A ten-dimensional action for non-geometric fluxes (arXiv:1106.4015)

For the group’s work it seems this one here is the latest:

  • David Andriot, Olaf Hohm, Magdalena Larfors, Dieter Lust, Peter Patalong, Non-Geometric Fluxes in Supergravity and Double Field Theory (arXiv:1204.1979)
Posted by: Urs Schreiber on July 18, 2012 2:38 PM | Permalink | Reply to this

Re: Andriot: Non-geometric fluxes versus (non)-geometry

Thanks a lot for this post! For those interested, I guess a good summary of my talk can also be given by its abstract, that I copy here:

Non-geometry appeared initially through new types of string backgrounds, where stringy symmetries were allowed to serve as transition functions between patches. It was argued later on that some terms in the potential of four-dimensional gauged supergravities, generated by so-called non-geometric fluxes, should be obtained from a compactification on those backgrounds. In this talk, I will present recent results clarifying such a relation. Thanks to a field redefinition performed on an NSNS non-geometric configuration, one can restore a standard notion of geometry, and make the non-geometric fluxes appear in ten dimensions. A dimensional reduction then leads to the desired potential terms in four dimensions. Implementing this field redefinition in doubled field theory, one provides additionally the non-geometric fluxes with a geometrical role, in view of (doubled) diffeomorphisms. Finally, I will mention relations to non-commutative geometry.

All my best wishes!

Posted by: David Andriot on July 19, 2012 11:56 AM | Permalink | Reply to this

Re: Andriot: Non-geometric fluxes versus (non)-geometry

Can someone point me to a *definition* of geometric and non-?

Posted by: jim stasheff on July 19, 2012 1:47 PM | Permalink | Reply to this

Re: Andriot: Non-geometric fluxes versus (non)-geometry

Can someone point me to a definition of geometric and non-?

For instance in arXiv:0807.4527 right below equation (2.12) you see the subgroup

G geomO(n,n) G_{geom} \hookrightarrow O(n,n)

defined. A type II geometry is defined by the reduction of the structure group of an O(n,n)O(n,n)-principal bundle much like a Riemannian geometric is defined by reduction of a GL(n)GL(n)-principal bundle. If reduction lands in G geomG_{geom} we say we have a geometric type II background. Otherwise it is called “non-geometric”.

Posted by: Urs Schreiber on July 19, 2012 2:22 PM | Permalink | Reply to this

Re: Andriot: Non-geometric fluxes versus (non)-geometry

The slides Urs Schreiber indicated in his first message are actually those of a talk I gave recently at the conference String Phenomenology 2012. This talk was more physics oriented. The talk I gave in String Math 2012 was different, and more towards mathematical aspects of the work. I am sure the corresponding slides will be put online on the conference website, pretty soon. This could be the link for that:

Best wishes

Posted by: David Andriot on July 22, 2012 5:50 PM | Permalink | Reply to this

Anderson: Heterotic Vector Bundles, Deformations and Conifold Transitions

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×E 8E_8 \times E_8 heterotic string backgrounds on a Calabi-Yau complex 3-fold with a holomorphic vector bundle with structure group GE 8G \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 YY, how to find families of complex bundles over YY 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.

Posted by: Urs Schreiber on July 18, 2012 3:28 PM | Permalink | Reply to this

Schwarz: Generalized Chern-Simons theory for large N and SUSY deformations of maximally supersymmetric gauge theories

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.

Posted by: Urs Schreiber on July 18, 2012 4:47 PM | Permalink | Reply to this

Stieberger: Motivic Multiple Zeta Values and Superstring Amplitudes

Stephan Stieberger reports on recent progress of understanding nn-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)
Posted by: Urs Schreiber on July 18, 2012 5:00 PM | Permalink | Reply to this

Harvey: Umbral Moonshine

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)
Posted by: Urs Schreiber on July 19, 2012 11:38 AM | Permalink | Reply to this

Sen: Black holes to quivers

Ashoke Sen is showing these slides:

  • Ashoke Sen, Black holes to quivers (pdf)
Posted by: Urs Schreiber on July 19, 2012 11:43 AM | Permalink | Reply to this

Rastelli: Distances between Conformal Field Theories

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…

Posted by: Urs Schreiber on July 19, 2012 2:14 PM | Permalink | Reply to this

Volovich: Mathematical Structures of Scattering Amplitudess from String-Math 2012

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.

Posted by: Urs Schreiber on July 19, 2012 3:15 PM | Permalink | Reply to this

Gaberdiel: Mathieu moonshine

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

Posted by: Urs Schreiber on July 19, 2012 4:03 PM | Permalink | Reply to this

Grojean: Quo Vadis LHC?

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.

Posted by: Urs Schreiber on July 19, 2012 7:07 PM | Permalink | Reply to this

Witten: Superstring Perturbation Theory Revisited

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 M˜ g,n NS,n R\tilde M_{g,n_{NS}, n_R} of super-Riemann surfaces (of genus gg with n NSn_{NS} Neveu-Schwarz and n Rn_R Ramond-type punctures) into an ordinary (bosonic) manifold with a bundle of super-directions above it, which works for low enough g,ng,n only.

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

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

  • M˜ g\tilde M_g is not split for g5g \geq 5,

  • M˜ g,1\tilde M_{g,1} not split for g2g \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.

Posted by: Urs Schreiber on July 20, 2012 10:15 AM | Permalink | Reply to this

Re: Witten: Superstring Perturbation Theory Revisited

I guess I should point also to this earlier article here, relevant in the above context:

Abstract. We present a new geometrical approach to superstrings based on the geometrical theory of integration on supermanifolds. This approach provides an effective way to calculate multi-loop superstring amplitudes for arbitrary backgrounds. It makes possible to calculate amplitudes for the physical states defined as BRST cohomology classes using arbitrary representatives. Since the new formalism does not rely on the presence of primary representatives for the physical states it is particulary valuable for analyzing the discrete states for which no primary representatives are available. We show that the discrete states provide information about symmetries of the background including odd symmetries which mix Bose and Fermi states. The dilaton is an example of a non-discrete state which cannot be covariantly represented by a primary vertex. The new formalism allows to prove the dilaton theorem by a direct calculation.

Posted by: Urs Schreiber on July 20, 2012 11:04 AM | Permalink | Reply to this

Hitchin: Generalized Geometry of Type B_n

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 L_\infty-algebra observed in section 4 The Gauge Algebra of Supergravity in 6k−1 Dimensions in

There is more here than meets the eye.

Posted by: Urs Schreiber on July 20, 2012 11:19 AM | Permalink | Reply to this

Re: Hitchin: Generalized Geometry of Type B_n

Alexander Kahle kindly points out that slides containing a good bit of the material that Hitchin showed are in last year’s

  • Peter Bouwknegt, Courant algebroids and generalizations of geometry, String-Math 2011 (slides)

Particularly the material on “B nB_n-geometry” starts at slide 13 there.

(Hitchin showed a good bit further material, though.)

Posted by: Urs Schreiber on July 20, 2012 11:49 AM | Permalink | Reply to this

Kamnitzer: Categorification using the Affine Grassmannian

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)
Posted by: Urs Schreiber on July 20, 2012 2:04 PM | Permalink | Reply to this
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