# Schedule for String-Math 2012

(Conference videos and lecture notes are available for download below next to the abstracts.)

## Monday, July 16

 07:00 Registration 08:50 - 09:00 Opening 09:00 - 09:55 Andrei Okounkov: Quantum Groups and Quantum Cohomology 10:00 - 10:45 Samson Shatashvili: Gauge Theory Angle on Integrability 10:45 - 11:15 Coffee break 11:15 - 12:00 Joerg Teschner: SUSY Gauge Theories, Quantized Moduli Spaces of Flat Cconnections, and Liouville Theory 12:05 - 12:50 Edward Frenkel: Beyond Topological Field Theory 12:50 - 14:30 Lunch 14:30 - 15:15 Sergio Cecotti: N=2 Gauge Theories, Half-Hypers, and Quivers 15:20 - 16:05 Clay Cordova: Abelian Chern-Simons Theory and the M5-Brane 16:00 - 16:30 Coffee break 16:35 - 17:20 Sara Pasquetti: Holomorphic Blocks in 3d 17:25 - 18:10 Tudor Dimofte: Holomorphic Blocks and Stokes Phenomena 18:30 Reception at the Max Planck Institut for Mathematics, Vivatsgasse 7, Bonn

## Tuesday, July 17

 09:00 - 09:55 Maxim Kontsevich: Integrality for K_2-Symplectomorphisms 09:55 - 10:45 Catharina Stroppel: Categorification and Fractional Euler Characteristics 10:45 - 11:15 Coffee break 11:15 - 12:00 Tom Bridgeland: Quadratic Differentials as Stability conditions 12:05 - 12:50 Greg Moore: Progress in N=2 Field Theory 12:50 - 14:30 Lunch 14:30 - 15:15 Richard Thomas: Cubic Fourfolds and K3 Surfaces 15:20 - 16:05 Johannes Walcher: On the Arithmetic of D-brane Superpotentials 16:00 - 16:30 Coffee break 16:35 - 17:20 Nick Sheridan: Homological Mirror Symmetry for a Calabi-Yau hypersurface in Projective Space 17:25 - 18:10 Rahul Pandharipande: GW/Pairs Correspondence for the Quintic 3-fold

## Wednesday, July 18 (3 Parallel sessions)

 Venue: Großer Hörsaal Mathematik Kleiner Hörsaal Mathematik Seminarraum bctp 1 09:00 - 09:25 Motohico Mulase Raffaele Savelli Partha Mukhapadhyay 09:30 - 09:55 Vasily Pestun Hagen Triendl Alexandr Buryak 10:00 - 10:25 Sam Gunningham Ben Jurke Tadashi Okazaki 10:30 - 10:55 Ricardo Schiappa Andreas Deser Francesco Sala 11:00 - 11:30 Coffee break 11:30 - 11:55 Emanuel Scheidegger Susanne Reffert Axel Kleinschmidt 12:00 - 12:25 Minxin Huang Noppadol Mekareeya Calin Lazaroiu 12:25 - 14:00 Lunch 14:00 - 14:25 Daniel Persson Amir Kashani Poor Simon Wood 14:30 - 14:55 Boris Pioline Andy Royston Nils Carqueville 15:00 - 15:25 Piotr Sulkowski Anindya Dey David Andriot 15:30 - 16:00 Coffee break 16:00 - 16:25 Artan Sheshmani Junya Yagi Lara Anderson 16:30 - 16:55 Will Donovan Grigory Vartanov Ido Adam 17:00 - 17:25 Balazs Szendroi Albert Schwarz Jock McOrist 17:30 - 18:10 Yan Soibelman Stephan Stieberger Ilarion Melnikov 18:45 - 23:00 Dinner Hotel Königshof, Adenauerallee 9, Bonn

## Thursday, July 19

 09:00 - 09:55 Don Zagier: Mock Modularity and Applications 09:55 - 10:45 Hirosi Ooguri: Modular Constraints on Calabi-Yau Compactification 10:45 - 11:15 Coffee break 11:15 - 12:00 Jeff Harvey: Umbral Moonshine 12:05 - 12:50 Ashoke Sen: Black Holes to Quivers 12:50 - 14:30 Lunch 14:30 - 15:15 Leonardo Rastelli: Distances between Conformal Field Theories 15:20 - 16:05 Anastasia Volovich: Mathematical Structures of Scattering Amplitudes 16:00 - 16:30 Coffee break 16:35 - 17:20 Matthias Gaberdiel: Mathieu Moonshine 17:25 - 18:10 Katrin Wendland: Large Subgroups of M24 Form Overarching Symmetry Groups of K3 18:20 Christophe Grojean, "Quo Vadis LHC?", Wolfgang-Paul-Hörsaal, Kreuzbergweg 28

## Friday, July 20

 09:00 - 09:55 Edward Witten: Superstring Perturbation Theory Revisited 09:55 - 10:45 Ron Donagi: Non Splitness of Supermoduli Space 10:45 - 11:15 Coffee break 11:15 - 12:00 Nigel Hitchin: Generalized Geometry of Type B_n 12:05 - 12:50 Vivek Shende: Large N Duality, Homological Knot Invariants, and the Rational DAHA 12:50 - 14:30 Lunch 14:30 - 15:15 Joel Kamnitzer: Categorification using the Affine Grassmannian 15:20 - 16:05 Hiroyuki Fuji: On Asymptotic Behavior of the Colored Superpolynomials 16:00 - 16:30 Coffee break 16:35 - 17:20 Alessandro Torrielli: SecretSymmetries of AdS/CFT 17:25 - 18:10 Matthias Staudacher: Qoperators, Yangians, and AdS/CFT Integrability

## Saturday, July 21

 09:00 - 09:45 Lionel Mason: The S-Matrix/Wilson-Loop Duality in Twistor Space 09:50 - 10:35 Nima Arkani-Hamed: Scattering Amplitudes and the Positive Grassmannian 10:45 - 11:15 Coffee break 11:15 - 12:00 Summary talk 13:00 End

## Abstracts:

Ido Adam: On the marginal deformations of general (0,2) non-linear sigma-models

In this talk we explore the possible marginal deformationsof general (0,2) non-linear sigma-models, which arise asdescriptions of the weakly-coupled (large radius) limits offour-dimensional $cN = 1$ compactifications of the heterotic string, to lowest order in $alpha'$ and first order in conformal perturbation theory. The results shed light from the world-sheet perspective on the classical moduli space of such compactifications.

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Lara Anderson: Heterotic Vector Bundles, Deformations and Conifold Transitions

I will discuss recent work on moduli stabilization without Neveu-Schwarz three-form flux in Calabi-Yau compactifications of Heterotic M-theory. In particular, I discuss a systematic approach to stabilizing the complex structure moduli in heterotic vacua and how the holomorphic deformations of the gauge bundle can naturally lead to conifold transitions in the Calabi-Yau base geometry.

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

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.

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Nima Arkani-Hamed: Scattering Amplitudes and the Positive Grassmannian

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Tom Bridgeland: Quadratic Differentials as Stability conditions

I describe joint work with Ivan Smith in which we identify spaces of stability conditions on certain CY3 triangulated categories with spaces of meromorphic quadratic differentials on Riemann surfaces. This work is inspired by Gaiotto, Moore and Neitzke's paper Wall-crossing, Hitchin systems and the WKB approximation'.

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Alexandr Buryak: Integrals of psi-classes over double ramification cycles

Double ramification cycles are certain codimension g cycles in the moduli space M_{g,n} of stable genus g curves with n marked points. They have proved to be very useful in the study of the intersection theory of M_{g,n}. In my talk I will explain that integrals of arbitrary monomials in psi-classes over double ramification cycles have an elegant expression in terms of vacuum expectations of certain operators that act in the infinite wedge space.

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Cordova Clay: Abelian Chern-Simons Theory and the M5-Brane

We discuss reduction of the Abelian M5-brane theory on three-manifolds, as a means of geometrically studying Abelian Spin Chern-Simons theories. The construction makes manifest certain features of the effective theory such as quantum equivalences of classically distinct Lagrangians. We explore the relevant geometry and describe an interesting connection with knot theory.

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Nils Carqueville: Defects and adjunctions in Landau-Ginzburg models

Two-dimensional topological field theories with defects are in general expected to be described by certain bicategories with adjoints. This can be worked out very explicitly and elegantly in the case of Landau-Ginzburg models. I shall explain the construction (which is joint work with Daniel Murfet) and indicate some of its applications, including an astonishingly simple proof of the general Cardy condition.

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Sergio Cecotti: N=2 Gauge Theories, Half-Hypers, and Quiver

Via the Quiver formulation, the problem of determining the BPS mass spectrum of a 4d N=2 QFT is equivalent to a canoni al problem in Representation Theory, and all aspects of the field Theory may be' phrased and understood in Representation theoretical terms. After reviewing these aspects in standard theories, as SQCD with general gauge group and matter representations, showing how the Representation theoretical viewpoint magically realizes the expected physical properties, we address the subtler theories which container HALF hypermultiplets - the simplest case being E_7 with half a 56. Such theories live on the verte of inconsistency. We descrive explocitly their Quiver formulation, giving the Quiver and superpotential for all Such models, and show how the corresponding representation theory realizes the miracles which make the Theory to be' quantum consistent. For some models we give a complete description of the strong coupling regime.

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Andreas Deser: Quasi-Poisson structures, Courant algebroids and Bianchi identities for non-geometric fluxes

The notion of non-geometric flux was introduced in string theory to correctly describe the degrees of freedom in four-dimensional effective superpotentials of type II flux compactifications. Whereas the H-flux is simply the field strength of the NS-NS B-field and the f-flux can be interpreted as the structure constants of a non-holonomic basis of the tangent bundle, the physical and mathematical properties of the remaining Q- and R-fluxes are to a great extent unknown. In this talk, we will first use the notion of a (quasi-)Poisson structure to re-derive directly on the tangent bundle the commutation relations containing all four fluxes as structure constants. As a consequence, we get Bianchi-type identities for the fluxes by writing down the Jacobi-identities for the algebra. Using these Bianchi-identities and the theory of quasi Lie-algebroids, we are able to identify the associated Courant algebroid structure, which is essential for dealing with all fluxes being non-zero.

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Anindya Dey: Partition Functions on S^3 and Three Dimensional Mirror Symmetry

Mirror Symmetry for a large class of  three dimensional $mathcal{N}=4$ supersymmetric gauge theories has a natural explanation in terms of M-theory compactified on a product of $text{ALE}$ spaces. A pair of such mirror duals can be described as two different deformations of the eleven-dimensional supergravity background $mathcal{M}=mathbb{R}^{2,1} times text{ALE}_{1} times text{ALE}_{2}$, to which they flow in the deep IR. The $A-D-E$ classification of $text{ALE}$ spaces, therefore provides a neat way to catalogue a large class of dual quiver gauge theories . For a certain subset of dual theories which arise from the aforementioned M-theory background with an  $A$-type $text{ALE}_{1}$ and a $D$-type $text{ALE}_2$, we verify the duality explicitly by a computation of partition functions of the theories on $S^3$, using localization techniques . We derive the relevant mirror map and  discuss its agreement  with predictions from the Type IIB brane construction for these theories. Our computation automatically provides rules for implementing &quot;orbifold 5-plane/orientifold 5-plane&quot; S-duality at the level of the partition function.

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Tudor Dimofte: Holomorphic Blocks and Stokes Phenomena

Following on the presentation by S. Pasquetti, I will discuss one of the most interesting (and physically meaningful) aspects of holomorphic blocks for 3d N=2 theories: their Stokes phenomena as parameters of the theory are varied. This can be analyzed explicitly using a non-perturbative integral construction for the blocks. I will also relate the blocks of a 3-manifold theory T_M to non-perturbative partition functions in Chern-Simons theory on M itself (which also display Stokes phenomena).

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Ron Donagi: Non Splitness of Supermoduli Space

The simplest type of supermanifold S is obtained from a vector bundle V on an ordinary manifold M: S has even coordinates along M and odd ones along V. Such a supermanifold is called split, and is characterized by admitting a projection to its reduced space M. In joint work with Edward Witten, we show that the moduli space \S_g of super Riemann surfaces (of genus g>=5) is not split, so it does not map to its underlying reduced space \M_g, which is the moduli space of (ordinary) Riemann surfaces with a spin structure.

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Will Donovan: Window shifts, flop equivalences and Grassmannian twists

A long-standing question in the theory of derived categories of coherent sheaves is to establish the existence of derived equivalences corresponding to flops. We will discuss in particular the Grassmannian flop, and show how appropriate equivalences can be quickly established in this case using a non-unique &quot;grade restriction window&quot; inspired by the work of Herbst-Hori-Page on gauged linear sigma models. (This can be viewed as extension of their ideas to non-abelian gauge groups.) We also discuss autoequivalences produced by appropriate composition of equivalences associated to different windows, and give a description of these autoequivalences as twists of certain spherical functors. This is joint work with Ed Segal.

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Edward Frenkel: Beyond Topological Field Theory

In a large class of quantum field theories in dimensions 1,2, and 4 there is a particular small coupling limit in which the contribution of instantons is on par with the perturbative sector, and anti-tinstantons die out. As the result, all correlation functions (not just those of topological, or BPS observables, in theories with SUSY) localize on finite-dimensional moduli spaces of instantons (such as holomorphic maps in 2D and anti-self-dual connections in 4D). Thus, we obtain full-fledged models of QFT that include as a small subsector,and vastly extend, the familiar topological field theories.These models enjoy many of the properties of TFT, such as the fact that path integrals are finite-dimensional. In the special case of the purely bosonic 2D sigma model with the sphere as a target manifold, we find an affine Kac-Moody algebra symmetry at the critical level and using it, obtain in a novel way the fermionic formula by Fateev-Frolov-Schwarz for the partition function. This is a joint work with A. Losev and N. Nekrasov.

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Horoyuki Fuji: On Asymptotic Behavior of the Colored Superpolynomials

In this talk, a generalization of the volume conjecture to the homological knot invariant will be discussed. Originally, the volume conjecture is studied fort he colored Jones polynomials for knots, and the geometric invariants for the knot complements are extracted from the asymptotic expansions. One of such invariants is A-polynomial which is the characteristic polynomial for the knot. In this talk, I would like to discuss the asymptotic behavior for the colored superpolynomial and find the generalized characteristic polynomial which we call super-A-polynomial. This talk is based on the works in collaboration with Sergei Gukov and Piotr Sulkowski, arXiv:1203.2182[hep-th], arXiv:1205.1515[hep-th].

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Matthias Gaberdiel: Mathieu Moonshine

The proposal of Eguchi, Ooguri and Tachikawa for an M24 symmetry underlying the elliptic genus of K3 is reviewed, and it is shown that the construction of all twining genera effectively proves their conjecture. In order to understand the microscopic reason for the emergence of M24, the symmetries of arbitrary K3 sigma-models are classified. While not all symmetry generators can be embedded into M24, those that cannot seem to be closely related to K3 sigma-models that are torus orbifolds. [Based on joint work with (Stefan Hohenegger &) Roberto Volpato.]

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Sam Gunningham: Spin Hurwitz Numbers and TQFT

Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed $pm 1$ according to the Atiyah invariant (parity) of the covering surface.I will describe an extended spin TQFT whose value on surfaces encodes these numbers. Using this, and some representation theory, we can compute the spin Hurwitz numbers for any genus.

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Jeff Harvey: Umbral Moonshine

Generalizes and extends the connection between the elliptic genus of K3 and M24 discovered by Eguchi-Ooguri-Tachikawa.

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Nigel Hitchin: Generalized Geometry of Type B_n

Conventional generalized geometry uses the SO(n,n) structure of T+T^*, i.e. Dynkin type D_n. We shall consider the SO(n+1,n) structure of T+1+T^*, twisted versions, twisted de Rham cohomology, generalized metrics, connections and Ricci curvature.

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Minxin Huang: Refined topological string

We discuss refined topological string theory on Calabi-Yau manifolds, and application to Seiberg-Witten gauge theory and Nekrasov function. In particular we discuss the Nekrasov-Shatashvili limit of the refined amplitudes. We discuss the refined BPS invariants for local Calabi-Yau manifolds.

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Ben Jurke: Scanning for Swiss Cheese geometries (preliminary)

Moduli stabilization in the Large Volume Scenario requires a specific kind of six-dimensional Calabi-Yau geometry for the compactification procedure, which is being referred to as &quot;Swiss Cheese&quot; due to the presence of certain holes for instanton wrappings. Only a handful of such manifolds are known explicitly. I will give an overview of the Large Volume Scenario and present recent progress on a general scan of the Calabi-Yau threefold landscape for the Swiss Cheese property. (preliminary)

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Joel Kamnitzer: Categorification using the Affine Grassmannian

I will give an overview of recent work (both mine and others) on categorification of quantum group representations and the associated knot and tangle invariants. I will concentrate on geometric approaches to categorification using quiver varieties and affine Grassmannians. I will discuss both coherent sheaves and modules over quantizations.

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Amir Kashani Poor: AGT and the topological string

In the case of massless Seiberg-Witten theory at N_f=4, we compare results from conformal field theory, obtained via the AGT correspondence, to results obtained via application of the holomorphic anomaly equations of the topological string. CFT methods yield exact results in the string coupling g_s in an instanton expansion in q, while the holomorphic anomaly yields exact results in q in a g_s expansion. We identify how the IR coupling of SW arises in the CFT context. We comment on the role of modularity and peculiarities of the asymptotic expansion of the topological string that become visible in the juxtaposition with CFT.

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Axel Kleinschmidt: Eisenstein series on Kac-Moody groups

Infinite-dimensional Kac-Moody groups are thought to play an important role in understanding the structure of M theory. I will discuss recent progress in understanding the construction of Eisenstein series on such groups and their relation to string theory.

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Calin Lazaroiu: Revisiting eight-manifold compactifications of M-theory with two generalized Killing spinors

Motivated by open problems in F-theory, we reconsider warped   compactifications of M-theory on eight-manifolds to $\AdS_3$ spaces in the presence of non-trivial field strength of the M-theory 3-form,   studying the most general conditions under which such backgrounds   preserve $N=2$ supersymmetry in three dimensions. In contradistinction   with the well-known cases studied previously, we allow for the most   general pair of Majorana generalized Killing spinors on the internal eight-manifold, without imposing any chirality conditions on those  spinors. Similarly to what happens for $N=1$ backgrounds, we show that such compactifications need not be manifolds with $\SU(4)$ structure,   being instead described by a $G_2$-reduction of the structure group of the nine-dimensional metric cone over the compactification space. By   studying the associated Fierz bilinears in a geometric algebra/Clifford bundle formalism, we give a characterization of such eight-manifolds   through an associative algebra of differential forms satisfying a set of   first order differential constraints.

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Lionel Mason: The S-Matrix/Wilson-Loop Duality in Twistor Space

Thirty years ago, Atiyah introduced the concept of holomorphic linking for certain holomorphic curves in a complex three-fold by analogy with the corresponding topological story for real curves in a real three-space. Recently, a remarkable correspondence has emerged between scattering amplitudes for planar maximally supersymmetric Yang-Mills theory and Wilson loops around a null polygon. The proof of this correspondence is most directly seen from a reformulation in twistor space. This leads to holomorphic analogues of Chern-Simons link invariants associated to a holomorphic Wilson loop on a nodal holomorphic curve in twistor space and gives a natural nonlinear extension of Atiyah's idea. In twistor space the proof of the duality follows from the observation that the Feynman diagrams of the S-matrix are the planar duals of those of the holmorphic Wilson loop. This talk will give an introduction to the basic ideas.

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Jock McOrist: (0,2) Sigma Models and Heterotic Metrics

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Noppadol Mekareeya: Recent Progress on Instanton Partition Functions

In this talk, I will discuss the moduli spave of instantons on C^2 in details. For a classical gauge group, the ADHM constuction of the instanton solution is known and can be realised from the Lagrangian of a certain quiver gauge theory with 8 supercharges. Hoewever, such a construction is not available for instantons in exceptional gauge groups. Moreover, all known stringy constructions of the latter so far do not admit a perturbative Lagrangian description. Neverthelss, due to recent developments of computations of Hilbert series and superconformal indices, partition functions for one an two instantons can be computed exactly and explicitly for any simple group, regardless of the existence of a Lagrangian description. This talk is based on the following papers: [arXiv:1205.4741], [arXiv:1111.5624] and [arXiv:1005,3026].

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Ilarion Melnikov: Heterotic flux vacua and their IIA duals

I will give a world sheet non-linear sigma model perspective on heterotic vacua with 8 supercharges.  This structure motivates a proposal for a large class of type II/heterotic dual pairs where the Calabi-Yau manifold on the IIA side need not be elliptically fibered.

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Gregory Moore: Progress in N=2 Field Theory

We briefly describe some new results on the BPS spectra of pure SU(K) SYM. We then describe spectral networks. Physically, spectral networks are combinatorial objects associated with N=2 theories of class S which encode various BPS spectra of the theory. Mathematically, spectral networks are associated with a branched covering of Riemann surfaces S -> C. They provide an interesting „nonabelianization map“ allowing one to construct Darboux coordinates on moduli spaces of flat connections on C in terms of holonomies of abelian flat connections on S.

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Partha Mukhopadhyay: On a semi-classical limit of loop space quantum mechanics

We study a loop space description of strings in curved background ($M$).  In a semi-classical limit the string wavefunction is localized on the submanifold of vanishing loops in $LM$ which is isomorphic to $M$ itself. And the semi-classical expansion is related to the tubular expansion of the loop space quantum mechanics around this submanifold. We develop the mathematical framework required to compute the effective dynamics on the submanifold in the Born-Oppenheimer sense at leading order in $alpha'$ expansion. In particular, we show that the linearized tachyon effective equation is reproduced correctly with divergent terms all proportional to the Ricci scalar of $M$. The mathematical framework involved in this analysis is an explicit construction of the Fermi normal coordinates in the tubular neighborhood of $M$ in $LM$.

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Motohico Mulase: Eynard-Orantin recursion and enumerative geometry

The topological recursion formula of Eynard and Orantin provides an inductive mechanism to solve certain geometric enumeration problems. Although physical speculation suggests that this formula is valid for a vast class of interesting problems, so far only a handful cases have been mathematically proven. This talk is aimed at presenting a mathematical theory, using a recent result on double Hurwitz numbers as an example. The talk is based on my collaboration with Dumitrescu, Hernandez-Serrano, and Sorkin.

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Tadashi Okazaki: String Duality and Stringy Interpretation of Wall-Crossing Phenomena

We study the spectrum of BPS D5-D3-F1 states in type IIB theory, which are proposed to be dual to D4-D2-D0 states on the resolved conifold in type IIA theory. We evaluate the BPS partition functions for all values of the moduli parameter in the type IIB side, and find them completely agree with the results in the type IIA side which was obtained by using Kontsevich-Soibelman's wall-crossing formula. Our result is not only a quite strong evidence for string dualities on the conifold but also gives us stringy interpretation of wall-crossing phenomena.

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Horosi Ooguri: Modular Constraints on Calabi-Yau Compactification

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Sara Pasquetti: Holomorphic Blocks in 3d

I will show how ellipsoid partition functions and indices of 3d N=2 SCFT can be factorized into a sum of products of holomorphic blocks which are partition functions on twisted solid tori, labelled by massive vacua of the theory. If the 3d SCFT is engineered on a brane in a non-compact Calabi-Yau geometry, the blocks are identified as open topological string amplitudes and are labelled by possible brane placements. I will discuss some analytic properties of blocks and their behavior under mirror symmetry transformations. This is a joint work with C. Beem and T. Dimofte.

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Rahul Pandharipande: GW/Pairs Correspondence for the Quintic 3-fold

I will present a proof of the GW/Pairs correspondence for the quintic using degeneration, descendents, and an analysis of projective bundles over surfaces (joint work with A. Pixton)

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Daniel Persson: Wall-crossing, hyperholomorphic bundles and dilogarithm identities

I will discuss a recent construction of a hyperholomorphic line bundle over a hyperkähler manifold. The construction relies on a general duality between 4n-dimensional quaternion-Kähler and hyperkähler spaces with certain continuous isometries, and involves a lift of the Kontsevich-Soibelman wall-crossing formula to the total space of the line bundle. Physically, this allows to describe the wall-crossing behaviour of D-instantons in type II Calabi-Yau compactifications via techniques developed in field theory.

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Vasily Pestun: Seiberg-Witten curves for N=2 quiver gauge theories

Starting from instanton partition functions we will systematically derive Seiberg-Witten curves for all N=2 ADE quiver gauge theories (in collaboration with Nikita Nekrasov).

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Boris Pioline: D3-instantons, Mock Theta Series and Twistors

The D-instanton corrected hypermultiplet moduli space in type IIB Calabi-Yau string vacua is known to be determined in terms of the generalized Donaldson-Thomas invariants, via a twistorial construction. In the large volume limit where D5 and NS5-instantons can be ignored, S-duality requires that this moduli space admits an isometric action of $SL(2,\IZ)$. We show that D3-D1-D(-1) instanton corrections are consistent with this symmetry. The proof hinges on the relation between generalized DT invariants for D3-instantons and the elliptic genus of the MSW conformal field theory, and on a representation of the instanton corrected Darboux coordinates in terms of an Eichler integral of the Siegel-Narain theta series corresponding to the sum over D1 charges. The Darboux coordinates have a modular anomaly under S-duality, which can be cancelled by a contact transformation generated by a mock theta series. This ensures that the twistor space, hence the HM moduli space, is modular invariant.

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Susanne Reffert: 2d Gauge/Bethe correspondence from String Theory

The gauge/Bethe correspondence of Nekrasov/Shatashvili links the full spectrum of integrable spin chains to the ground states of N=(2,2) gauge theories in 2 dimensions with twisted masses and can be understood in terms of geometric representation theory. We realize these gauge theories via a brane construction in a deformed background. The symmetry group of the integrable system is manifest in this set-up as an enhanced symmetry. The twisted masses are inherited from this so-called fluxtrap background, which also provides a string theory realization of the Omega deformation.

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Andy Royston: Semiclassical Framed BPS States

We provide a definition of framed BPS states in N=2 SYM with line operator defects in the semiclassical approximation, in terms of the kernel of a twisted Dirac operator on the moduli space of certain singular monopole configurations.    Both framed and ordinary BPS states transform in representations of a global symmetry group that is the product of spatial rotations and SU(2) R-symmetry.  We describe the action of this group and define the protected spin character of Gaiotto-Moore-Neitzke in geometric language.  We reformulate the strong positivity conjecture of GMN as a condition on the kernel of the Dirac operator.

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Francesco Sala: Sheaf theory and instantons on ALE spaces

I describe toric orbifold compactifications of ALE spaces and framed sheaves on such orbifolds by using the theory of root stacks. I give an algebro-geometric description of the relation between framed instantons on ALE spaces and framed sheaves on orbifold compactifications. This is equivalent to a similar description provided by Nakajima in the setting of complex differential geometry. This is part of a joint project with Richard Szabo.

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Samson Shatashvili: Gauge Theory Angle on Integrability

I review recent results on relations between quantum integrability and supersymmetric vacua, and formulate some open questions/hypothesis. Based on joint work with N. Nekrasov

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Raffaele Savelli: Flux Quantization in F-theory and Freed-Witten Anomaly

Using M/F-theory duality, I will relate the quantization of the G-flux in M-theory to the one of the gauge flux on type IIB D7-branes prescribed by Freed-Witten anomaly cancellation. The correspondence is shown for all unitary and symplectic Kodaira singularities, by explicitly constructing appropriate 4-cycles in the M-theory elliptic fourfold, which are able to detect the global anomaly.

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Emanuel Scheidegger: Topological Strings and Elliptic Fibrations

We will explain a conjecture that expresses the Gromov-Witten invariants for elliptically fibered Calabi-Yau threefolds in terms of modular forms. In particular, there is a recursion relation which governs these modular forms. Evidence comes from the polynomial formulation of the higher genus topological string amplitudes with insertions.

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Ricardo Schiappa: Resurgent Analysis of Random Matrices at Large N

D-brane instantons are nonperturbative effects in string theory which control the large-order behavior of string perturbation theory, leading to the well-known (2g)! growth of the genus expansion. On the other hand, a complete analysis of the string theoretic asymptotic behavior, including both the perturbative series and multi-instanton sectors, needs the general framework of transseries and resurgent analysis. We show how to develop and apply these techniques within the contexts of matrix models and topological strings, and explicitly show how new nonperturbative sectors, beyond the standard D-brane sector, are needed in order to account for the full (resurgent) asymptotic behavior of all multi-instanton sectors. We also show how the transseries solution is a true nonperturbative solution, allowing to probe all possible (holographically dual) large N backgrounds which arise from a single matrix model once we allow the large N limit to be taken in any direction in the complex plane. Distinct backgrounds are related via Stokes phenomena and we explore how the aforementioned new nonperturbative sectors play a fundamental role within this context of large N background (in)dependence.

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

In the framework of BV formalism one can construct Chern-Simons action functional defined on NXN matrices with entries from differential associative algebra with trace. We are studying the properties of this action functional (observables, symmetries, deformations) for large N using methods of homological algebra and noncommutative geometry.We apply our results to the analysis of supersymmetric deformations of dimensional reductions of ten-dimensional super Yang-Mills theory; these results can be used to study BPS fields. The talk is based on joint work with M.Movshev.

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Ashoke Sen: Black Holes to Quivers

Using the connection between quiver quantum mechanics on the Coulomb branch and the quantum mechanics of multi-centered black holes, we propose a general algorithm for reconstructing the full moduli-dependent cohomology of the moduli space of an arbitrary quiver, in terms of the BPS invariants of the pure Higgs states. We analyze many examples of quivers with loops, including all cyclic Abelian quivers and several examples with two loops or non-Abelian gauge groups, and provide supporting evidence for this proposal.

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Vivek Shende: Large N Duality, Homological Knot Invariants, and the Rational DAHA

A couple years ago, Oblomkov and I conjectured a relation between Hilbert schemes of points on singular curves and the HOMFLY polynomials of their links. I will discuss recent developments: the derivation (with Diaconescu and Vafa) of the conjecture via large N duality; the promotion of the conjecture to a relation to Khovanov-Rozansky homology, and the reformulation (with Gorsky, Oblomkov, and Rasmussen) in terms of parabolic Hitchin fibers. In the latter context the torus knots give rise to representations of the rational DAHA, and the differentials relating HOMFLY homology to sl(n) homology appear to admit explicit expressions in terms of the DAHA action.

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Nick Sheridan: Homological Mirror Symmetry for a Calabi-Yau hypersurface in
Projective Space

We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. After stating the result carefully, we will describe some of the techniques use.

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Artan Shesmani: Nested Donaldson-Thomas invariants and the Elliptic genera in String theory

Gaiotto et al in (hep-th/0607010) discussed a strategy to compute a finite set of BPS invariants associated to a D4-D2-D0 system in an ambient Fermat quintic Calabi Yau threefold. Moreover, using the predictions about the modularity property of the associated generating series for the invariants, the authors provided a formula for the ("modified") elliptic genus of an M5-brane wrapping a hyperplane section of that threefold. These invariants, roughly speaking, compute the contribution of points and curves lying scheme theoretically on a moving Divisor embedded in an ambient Calabi Yau threefold. The computation of BPS states of D4-D2-D0 systems is extremely sensitive to how the underlying divisor deforms inside the threefold and in cases where the divisor becomes singular there is no rigorous treatment of how to compute these invariants in full generality. We introduce Nested Donaldson Thomas invariants (NDT) corresponding to invariants of torsion sheaves with 2 dimensional support in a Calabi Yau threefold. These are the mathematical counterpart of the physicists' invariants in this context. We compute the NDT invariants via constructing the virtual fundamental class over their moduli space in cases where the underlying threefold is smooth or it has certain singularities; In cases where the ambient threefold has singularities, we use Jun Li's degeneration techniques and certain Conifold transition formula developed for NDT invariants to compute a partition function for the invariants. We also show that these invariants, as predicted by string theory, satisfy nice modularity properties. Finally, we will

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Yan Soibelman: Donaldson-Thomas invariants and integrable systems

I am going to discuss the relationship between theory of Donaldson-Thomas invariants developed by Kontsevich and myself, complex integrable systems of Hitchin type and geometry of 3-dimensional Calabi-Yau manifolds.

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Matthias Staudacher: Qoperators, Yangians, and AdS/CFT Integrability

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Stephan Stieberger: Motivic Multiple Zeta Values and Superstring Amplitudes

The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows to disentangle its alpha'-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the alpha'-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the alpha'-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a transcendental function. In terms of elements of this Hopf algebra the alpha'-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

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Piotr Sulkowski: (Quantum) Super-A-polynomial

In this talk we introduce the "quantum super-A-polynomial", i.e. an object which encodes recursion relations for homological knot invariants (such as colored superpolynomials). Quantum super-A-polynomial is a 2-parameter deformation of an ordinary quantum A-polynomial, and in the classical limit it reduces to the (classical) super-A-polynomial, i.e. an algebraic curve which encodes asymptotics of colored superpolynomials. We present quantum super-A-polynomial in several interesting cases, including figure-8 knot and torus knots, and discuss its quantizability properties. We also discuss a physical interpretation of the super-A-polynomial, as describing the SUSY vacua of the dual 3d N=2 theory associated to the knot complement . These results are based on recent works in collaboration with Hiroyuki Fuji and Sergei Gukov, presented in arXiv: 1203.2182 [hep-th], 1205.1515 [hep-th].

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Balazs Szendroi: Refined DT theory and Nekrasov's formula

I revisit the identification of Nekrasov's partition function counting instantons on R^4, and the (refined) DT/GW series of the associated local Calabi-Yau threefold, mainly in the case of the conifold corresponding to the gauge group U(1). I will show how recent mathematical results confirm this identification, and speculate on how one could lift the equality of partition functions to an isomorphism of vector spaces.

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Jörg Teschner: SUSY Gauge Theories, Quantized Moduli Spaces of Flat Cconnections, and Liouville Theory

We will argue that the exact results on partition functions and loop operator expectation values in certain SUSY gauge theories obtained by Pestun, Alday-Gaiotto-Tachikawa (AGT) and others are best understood in terms of a reduction to an effective quantum mechanics that describes the gauge theory at low energies. The relevant quantum mechanics is obtained by quantizing moduli spaces of flat SL(2,R)-connections. Natural physical assumptions concerning the action of electric-magnetic dualities (exchange of Wilson- and 't Hooft loops) characterize the relevant wave functions uniquely. The conformal blocks of Liouville theory are the solutions. This line of reasoning characterizes the instanton partition functions in a way that is similar to the reasoning by which Seiberg and Witten determined the prepotential for such theories.

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Richard Thomas: Cubic Fourfolds and K3 Surfaces

I will start by reviewing, with examples, some of the amazing similarities between cubic 4-folds and K3 surfaces.
Hassett: the (interesting bit of the) Hodge diamond of a cubic 4-fold looks remarkably like that of a K3 surface; conjecturally it is exactly that of a K3 surface iff the cubic 4-fold is rational.
Kuznetsov: the (interesting bit of the) derived category of a cubic 4-fold looks remarkably like that of a K3 surface; conjecturally it is exactly that of a K3surface iff the cubic 4-fold is rational.
Then I will discuss joint work with Nick Addington. A cubic 4-fold is Hassett if it is Kuznetsov, and the converse is true over at least a Zariski open subset of (each irreducible component of) the moduli space of Hassett cubic 4-folds. If there's time I will review our attempts to close this statement up.

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We will review special quantum group symmetries underlying the integrability of the AdS/CFT spectral problem, with particular emphasis on the 'secret' or bonus' symmetry. This is a particular symmetry of Yangian type, however not accounted for by the standard Yangian. This symmetry has been observed not only in the spectrum, but also in the presence of D-branes and integrable boundaries, in scattering amplitudes, in the pure spinor formalism and, recently, in the quantum affine deformation.

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Hagen Triendl: Generalized type IIA orientifold and M-theory compactifications

We discuss general N=1 and N=2 compactifications of M-theory to four dimensions in the E7-covariant formalism of exceptional generalized geometry. We find that these backgrounds have a natural interpretation in terms of an eight-dimensional internal space of fixed volume. Furthermore, we make contact to type IIA compactifications and, more importantly, their O6-orientifolds and relate these backgrounds to their M-theory parents. We finally discuss possible resolutions of the problem of sources in generalized geometry.

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Grigory Vartanov: 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories

I wanted to talk about my recent work with J. Teschner "6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories" based on the paper in Arxiv arxiv.org/abs/1202.4698. In our paper we revisited the definition of the 6j-symbols from the modular double of U_q(sl(2,R)) (b-6j symbols) and, in particular, we showed the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We observed a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Also we found a new integral representation which indicates a possible interpretation of the b-6j symbols as partition functions of three-dimensional N=2 supersymmetric gauge theories with nonabelian gauge groups.

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Anastasia Volovich: Mathematical Structures of Scattering Amplitudes

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Johannes Walcher: On the Arithmetic of D-brane Superpotentials

Irrational invariants from D-brane superpotentials are pursued on the mirror quintic, systematically according to the degree of a representative curve. Lines are completely understood: the contribution from isolated lines vanishes. All other lines can be deformed holomorphically to the van Geemen lines, whose superpotential is determined via the associated inhomogeneous Picard-Fuchs equation. Substantial progress is made for conics: the families found by Mustata contain conics reducible to isolated lines, hence they have a vanishing superpotential. The search for all conics invariant under a residual Z2 symmetry reduces to an algebraic problem at the limit of our computational capabilities. The main results are of arithmetic flavor: the extension of the moduli space by the algebraic cycle splits in the large complex structure limit into groups each governed by an algebraic number field. The expansion coefficients of the superpotential around large volume remain irrational. The integrality of those coefficients is revealed by a new, arithmetic twist of the di-logarithm: the D-logarithm. There are several options for attempting to explain how these invariants could arise from the A-model perspective. A successful spacetime interpretation will require spaces of BPS states to carry number theoretic structures, such as an action.

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Katrin Wendland: Large Subgroups of M24 Form Overarching Symmetry Groups of K3

Byclassical results due to Nikulin, Mukai, Xiao and Kondo in the 1980's and 90's, the finite symplectic automorphism groups of K3 surfaces are always subgroups of the sporadic simple group known as the Mathieu group M24, which can be constructed as the automorphism group of the Golay code. There are eleven "maximal" subgroups of M24, which were also determined by Mukai, such that every finite symplectic automorphism group of a K3 surface is contained in one of these eleven groups as a subgroup. The group M24 is by a factor of about half a million larger than each of these eleven groups. Inspired by a recent observation by Eguchi, Ooguri and Tachikawa that the elliptic genus of K3 surfaces seems to exhibit a mysterious footprint of M24, we investigate the polarization preserving automorphisms of Kummer K3 surfaces in terms of automorphisms of the Golay code. We explain how several such automorphism groups can be combined to a subgroup of M24 which is by a factor of 100 larger than any of Mukai's eleven "maximal" groups.

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Edward Witten: Superstring Perturbation Theory Revisited

I will describe the natural formulation of superstring perturbation theory via integration over the moduli space of super Riemann surfaces, and some of the drawbacks of attempting to reduce all of the computations to the moduli space of ordinary Riemann surfaces. In particular, such a reduction makes spacetime supersymmetry and some of the delicate properties of the theory difficult to understand even when it works; and moreover it does not work in general.

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Simon Wood: Understanding logarithmic CFT

Log CFT generalises ordinary CFT, by allowing logarithmic singularities in addition to poles in correlators. A direct consequence of such logarithms is that the zero mode of the energy-momentum tensor acts non-diagonalisably on the space of states and that the representation theory is non-semi-simple (there exist reducible yet indecomposable representations). In analogy to non-log rational CFT, there are families of log CFTs, for which the number of irreducible representations is finite. I would like to present two such families (the W_p and the W_p,q models), explain how they differ from rational CFTs such as the minimal models and explain how they naturally generalise rational CFT.

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Junya Yagi: (2, 0) theory, cigars, and AGT

It has been argued that the AGT correspondence originates from two ways to compactify the (2,0) theory in six dimensions.  One is compactification on a punctured Riemann surface, which produces an N=2 supersymmetric gauge theory in four dimensions.  The other is compactification on the Ω-background.  I will discuss how the latter can lead to Liouville theory and its generalizations.  The basic strategy is to consider another kind of compactification using two cigar-like manifolds.

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Don Zagier: Mock Modularity and Applications

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