# Schedule of the Workshop: Integrability in topological field theory

## Monday, April 16

9:00 |
Registration |

9:30 - 10:30 |
Alexander Varchenko: Cohomology of the cotangent bundle of a flag variety as an XXX Bethe algebra |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Emanuel Scheidegger: Topological strings and elliptic fibrations |

12:00 - 13:00 |
Hans Jockers: Conifold Transitions in M-theory on Calabi-Yau fourfolds |

13:00 - 15:00 |
Lunch break |

15:00 - 16:00 |
Jan Manschot: BPS invariants of rational surfaces |

16:00 - 16:30 |
Tea break |

16:30 - 17:30 |
Oleg Sheinman: Finite-dimensional Lax integrable systems and 2D CFT |

## Tuesday, April 17

9:30 - 10:30 |
Nils Scheithauer: Modular forms for the Weil representation and Borcherds' conjecture |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Daniel Persson: Hyperholomorphic wall-crossing and geometric quantization of cluster varieties |

12:00 - 13:00 |
Dani Hernández Serrano: Topological recursion for Gromow-Witten invariants of BG |

13:00 - 15:00 |
Lunch break |

15:00 - 16:00 |
Motohico Mulase: Laplace transform of Catalan Numbers and Topological Recursion |

16:00 - 16:30 |
Tea break |

16:30 - 17:30 |
Gaёtan Borot: Topological recursion, theta functions, integrability, and a conjecture on asymptotics of the Jones polynomial |

## Wednesday, April 18

9:30 - 10:30 |
Sara Pasquetti: N=2 theories partition functions on the squashed 3-sphere |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Andrey Mironov: Semi-Hamiltonian systems for integrable geodesic flow on 2-torus |

12:00 - 13:00 |
Alexander Alexandrov: On a relation between two enumerative geometry tau-functions |

13:00 - 24:00 |
EXCURSION and CONFERENCE DINNER |

## Thursday, April 19

9:30 - 10:30 |
Marcos Marino: Torus knots and mirror symmetry |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Piotr Sulkowski: Volume conjecture: refined and categorified |

12:00 - 13:00 |
Amir-Kian Kashani-Poor: The holomorphic anomaly equations in Omega deformed backgrounds |

13:00 - 15:00 |
Lunch break |

15:00 - 16:00 |
Christian Korff: Small quantum cohomology as quantum integrable system |

16:00 - 16:30 |
Tea break |

16:30 - 17:30 |
Giulio Bonelli: On some extensions of AGT correspondence |

## Friday, April 20

9:30 - 10:30 |
Alexey Bondal: Poincare groupoids of graphs and Noncommutative Geometry |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Marta Mazzocco: Upper-triangular bilinear forms and braid group action |

12:00 - 13:00 |
Todor Milanov: Period integrals and twisted representations of vertex algebras |

## EXCURSION and CONFERENCE DINNER

The excursion will take us on a boat trip to the beautiful town of Königswinter. Information-Link

There, one can, for example, head off by foot or by cog-wheel train to the Drachenfels, or simply enjoy the town. Further information and sign-up for the excursion: At registrationon Monday, April 16, 9am.

Conference dinner: Wednesday, April 18, 7pm, Restaurant OPERA, Kapuzinerstr. 13, 53117 Bonn. Information-Link

## Abstracts:

Alexander Alexandrov: On a relation between two enumerative geometry tau-functions

In this talk we present our conjecture on the connection between the Kontsevich-Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the GL(∞) group element. The important feature of this group element is its simplicity: it consists of only generators of the Virasoro algebra. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich-Witten tau-function.

Alexey Bondal: Poincare groupoids of graphs and Noncommutative Geometry

TBA

Giulio Bonelli: On some extensions of AGT correspondence

In the first part of the talk I will present a relation between N=2 quiver gauge theories on the ALE space O_P^{^1}(-2) and correlators of N=1 super Liouville conformal field theory, provide checks in the case of punctured spheres and tori. The instanton part is matched with the conformal block while the one-loop contribution is matched with the three-point function. In the second part I will introduce N=2 supersymmetric SU(2) gauge theories on R^{^4} coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A_{1}(2,0) theory on Riemann surfaces with irregular punctures. These are associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces.

Gaetan Borot: Topological recursion, theta functions, integrability, and a conjecture on asymptotics of the Jones polynomia

The topological recursion takes as initial data a plane curve, and produces a tower of geometric invariants, namely numbers F_{h} and differentials forms in n variables W_{n}^{h} (for h integers, and n positive integers), which are solution to loop equations. It can be used to build a dispersionful integrable system S whose dispersionless limit S_{0} reproduces the algebro-geometric solutions of KP associated to the curve. The extra ingredient for this construction are the theta functions associated to the curve (and their derivatives): we have proposed explicit formulas for the tau function and the wave function of S as formal asymptotic expansions in hbar, and checked Hirota equation to the first non trivial order in hbar. When the plane curve is chosen to be the A-polynomial of a hyperbolic knot K, this asymptotic expansion has special properties, and we conjecture that the all-order asymptotics of the Jones polynomial of K can be extracted from the wave function. This conjecture is checked up to first orders for the figure eight-knot and the once-punctured torus bundle L^{2}R. Our proposal corrects and completes the conjecture of Dijkgraaf, Fuji and Manabe: we explain that the ad hoc renormalizations constants they needed to all orders to match the perturbative knot invariants with the topological recursion are essentially values of derivatives of theta functions. This talk is based on a joint work with Bertrand Eynard, and the aim is to describe algebraic geometry on the A-polynomial curve and our completed conjecture, with emphasis on the example of the figure-eight knot.

Hans Jockers: Conifold Transitions in M-theory on Calabi-Yau fourfolds

In this talk I consider topology changing transitions for M-theory compactifications on Calabi-Yau fourfolds. M-theory requires us to include background G-fluxes, which are represented by integral four-form cohomology classes. I discuss a set of canonical G-fluxes that fulfill the consistency conditions imposed by M-theory along Calabi-Yau fourfold transitions. The local geometry of the transition is generically a genus g curve of double points. In physics it engineers a certain three-dimensional gauge theory near the intersection of Coulomb and Higgs branches with interesting connections to the classical theory of Riemann surfaces.

Christian Korff: Small quantum cohomology as quantum integrable system

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder. These lattice paths form exactly solvable statistical mechanics or quantum integrable models and are obtained from solutions to the Yang-Baxter equation.

Amir-Kian Kashani-Poor: The holomorphic anomaly equations in Omega deformed backgrounds

We discuss how the holomorphic anomaly equations generalized to Omega deformed backgrounds can be used to obtain exact results for refined topological string amplitudes. The role of modularity and the application to conformal theories will be emphasized.

Jan Manschot: BPS invariants of rational surfaces

I'll discuss the partition function of topologically twisted N=4 U(r) Yang-Mills theory on rational surfaces. In particular, I'll explain how to compute its BPS invariants, which capture topological invariants of moduli spaces of semi-stable sheaves. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan filtrations and a blow-up formula.

Marcos Marino: Torus knots and mirror symmetry

A spectral curve is proposed and derived encoding all the colored HOMFLY invariants of torus knots, in the context of the type B open topological string. This curve can be derived by using a matrix model which computes this invariants, and solving it in the large N limit.

Marta Mazzocco: Upper-triangular bilinear forms and braid group action

In this talk we will discuss a quadratic Poisson algebra structure on the space of bilinear forms on C^{N} with the property that for any n, m ∈N such that n m = N, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size m x m is Poisson. We give some examples and classify all central elements. We also obtain the braid group action on the Poisson algebra. Finally, we discuss quantisation.

Todor Milanov: Period integrals and twisted representations of vertex algebras

This talk is based on my recent work with B. Bakalov. Using period integrals for hyper-surface singularities we managed to construct a representation of a certain vertex algebras and use it to obtain differential operator constraints for the so called FJRW invariants. The latter are certain intersection numbers on the moduli space of Riemann surfaces equipped with a spin structure. It is quite plausible that our construction can be applied to the Gromov-Witten theory of orbifold projective lines. I am planning to report on our progress in this direction as well.

Andrey Mironov: Semi-Hamiltonian systems for integrable geodesic flow on 2-torus

We prove that the question of existence of polynomial first integrals of geodesic flow on torus leads naturally to a remarkable system of quasi-linear equations which turns out to be a Semi-Hamiltonian system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Semi-Hamiltonian quasi-linear system.

Motohico Mulase: Laplace transform of Catalan Numbers and Topological Recursion

What is the mirror dual of the Catalan numbers? Starting with this question, the talk provides a mathematical introduction to the Eynard-Orantin topological recursion.

The topological recursion is an effective quantization mechanism. The physics predictions include spectacular applications of the recursion in many quantum invariants of geometry, such as Gromov-Witten invariants and 3-manifold invariants, including the hyperbolic volume. As of now still very little is mathematically established.

This talk is aimed at presenting a simple mathematical example of the Eynard-Orantin theory, identifying the recursive nature via the Laplace transform as mirror symmetry. The spectral curve appears as the mirror of the Catalan numbers.

Sara Pasquetti: N=2 theories partition functions on the squashed 3-sphere

We show that partition functions of N=2 theories on the squashed 3-sphere can be factorised into vortex theories defined on the two solid tori corresponding to the two degeneration limits of the ellipsoid. We comment on how the ellipsoid partition function appears to provide the modular invariant non-perturbative completion of the vortex theory. We conclude with the geometrical engineering of the vortex theory.

Daniel Persson: Hyperholomorphic wall-crossing and geometric quantization of cluster varieties

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. Finally, I will offer some speculations on an interesting relation with the geometric quantization of cluster varieties, which suggests a way to incorporate effects from NS5-branes.

Emanuel Scheidegger: Topological strings and elliptic fibrations

We will explain a conjecture that expresses the BPS invariants (Gopakumar-Vafa invariants) for elliptically fibered Calabi-Yau threefolds in terms of modular forms. Evidence comes from the polynomial formulation of the higher genus topological string amplitudes.

Nils Scheithauer: Modular forms for the Weil representation and Borcherds' conjecture

In 1995 Borcherds conjectured that the twisted denominator identities of the fake monster algebra under Conway's group are automorphic forms of singular weight on orthogonal groups. We describe the final steps in the proof.

Daniel Hernández Serrano: Topological recursion for Gromow-Witten invariants of BG

We show that intersection numbers of cotangent classes on the moduli space of stable maps to the classifying space of a finite group satisfy a Eynard-Orantin type topological recursion. Work in progress.

Oleg Sheinman: Finite-dimensional Lax integrable systems and 2D CFT.

In 2001 I. Krichever proposed a new notion of Lax operator with thespectral parameter on a Riemann surface. He has given a general and transparent treatment of Hamiltonian theory of the corresponding Lax equations. This work has led to the notion of Lax operator algebras (I. Krichever, O. Sheinman, 2007) and consequent generalization of the Krichever's approach on Lax operators taking values in the classical Lie algebras. The corresponding class of Lax integrable systems contains Hitchin systems and their analog for pointed Riemann surfaces, integrable gyroscopes and similar examples.

In the talk, given a Lax integrable system of the just mentioned type, we construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields. For the Lax equations in question, we propose a way to represent Hamiltonian vector fields by covariant derivatives with respect to the (high-genus) Knizhnik-Zamolodchikov connection. This provides a prequantization of the Lax system. The representation operators of Poisson commuting Hamiltonians of the Lax system projectively commute. If Hamiltonians depend only on the action variables then the corresponding operators commute. The idea of quantization of Hitchin systems by means of the Knizhnik-Zamolodchikov connection was addressed, or at least mentioned, many times in the theoretical physics literature (N. Hitchin, D. Ivanov, G. Felder and Ch. Wieczerkowski, M.A. Olshanetsky and A.M. Levin) but only the second order Hamiltonians were involved. On the level of prequantization, we have observed such relation for all Hamiltonians, and, moreover, for all observables of the Hamiltonian system given by the Lax equations in question.

A part of this talk, mainly concentrated on the Lax operator algebras, has been given at the first Workshop of the Program (Integrability - modern variations). At the current Workshop I would like to focus on the correspondense between a Lax integrable system and CFT.

Piotr Sulkowski: Volume conjecture: refined and categorified

The volume conjecture relates asymptotic behavior of the colored Jones polynomial or the Chern-Simons partition function to objects defined on the zero locus of A-polynomial, i.e. an algebraic curve A(x,y)=0. A related, so called AJ-conjecture, states that the Chern-Simons partition function is annihilated by a quantum counterpart of this curve, Â(x,y;q)=0. In this talk the refined, or categorified, versions of both conjectures will be presented. These new conjectures involve t-deformations of the above classical and quantum A-polynomials, and describe properties of homological knot invariants and refined BPS invariants. Explicit examples of classical and quantum t-deformed curves, relevant for colored knot homologies as well as refined BPS invariants, will be provided.

Alexander Varchenko: Cohomology of the cotangent bundle of a flag variety as an XXX Bethe algebra

I will interpret the equivariant cohomology of the cotangent bundle of a flag variety F as the XXX Bethe algebra of a suitable Yangian module. Under this identification the dynamical connection of the XXX model turns into the quantum connection of Braverman-Maulik-Okounkov. This gives a relation between the XXX integrable model and quantum cohomology, the relation discussed in physics literature by Nekrasov and Shatashvili.