This talk has four parts. Part one reviews the derivation of the Kontsevich-Soibelman wall-crossing formula for BPS degeneracies in four-dimensional theories with N=2 supersymmetry using framed BPS states. This follows the papers [ArXiv:1006.0146,1008.0030]. I might also mention briefly a possible generalization under discussion with T. Dimofte and D. Gaiotto. Motivated by this possible...

In the first part of the talk I’ll describe a joint work with Tomoki Nakanishi and explain how the Voros symbols in exact WKB analysis realize (generalized) cluster variables. In the second part I’ll generalize the notion of the Voros symbols to the Painlevé equations, and discuss their applications.

We discuss some aspects of the reduction process leading to a pre-building associated to an SL(3) spectral curve. This construction is related to harmonic maps and the WKB problem, and has potential applications to the construction of stability conditions. This is joint work with Katzarkov, Noll and Pandit [arXiv:1503.00989], see also [arXiv:1311.7101]

In this talk I will review recent developments on the relationship between meromorphic connections on the Riemann sphere and quivers. Such relationship was first found by Crawley-Boevey in the case of logarithmic connections. He used it to solve the additive Deligne-Simpson problem, a sort of existence problem on logarithmic connections. I will explain the generalization of Crawley-Boevey's...

Inclusion of surface operators leads to interesting generalisations of the correspondence discovered by Alday, Gaiotto and Tachikawa between four-dimensional N=2 SUSY field theories and conformal field theory. My goal will be to outline how this generalisation is related to the geometric Langlands correspondence and to a certain quantum generalisation of this correspondence. The resulting...

We will derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,C)$. The corresponding operator acts in the direct sum of $N( n-3) $ copies of $L^2(S^1)$. Its kernel is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems...

I explain an ansatz for the partition function of elliptically fibered Calabi-Yau threefolds in terms of Jacobi forms using a combination of B-model, homological mirror symmetry, and geometric techniques. This talk is based on joint work with Minxin Huang and Albrecht Klemm appearing in [arXiv:1501.04891] as well as work in progress.

Modular anomaly have been discovered in topological string theory on elliptic Calabi-Yau spaces. We extend the derivation of genus zero anomaly equation for non-compact cases in the literature to compact cases. For higher genus, we derive the modular anomaly equation from BCOV holomorphic anomaly equation. Based on [arXiv:1501.04891].

In [arXiv:1503.03676,1601.03586] (with Braverman and Finkelberg), I have proposed a mathematical approach to define Coulomb branches of 3d N=4 SUSY gauge theories. It is based on the homology group of a certain moduli space, and has a natural quantization by the equivariant homology group. For a quiver gauge theory, the quantized Coulomb branch has an embedding into the ring of difference...

The Hilbert series is a generating function that enumerates gauge invariant chiral operators of a supersymmetric field theory with four supercharges and an R-symmetry. In this talk I will explain how the counting of dressed ‘t Hooft monopole operators leads to a formula for the Hilbert series of a 3d N=2 gauge theory, which captures precious information about the chiral ring and the geometry...

We consider cohomological Hall algebras associated to quivers and their actions on the cohomology of Nakajima varieties; we relate these algebras with the Yangians constructed by Maulik and Okounov, and show that their Hilbert series are encoded by the Kac polynomials of the underlying quiver. For instance, for the 1-loop quiver, one obtains the Yangian of $\widehat{gl(1)}$ used relevant in...

I will talk about new example of W algebras depending on three integer numbers n,m,k. Category of representations of such algebras is equivalent (similar to Drinfeld–Kohno or Kazhdan–Lusztig theorem) to the category of representations of product of three quantum groups gl_{n|k}, gl_{k|m} and gl_{m|n}. Irreducible representations of these W algebras have a basis labeled by plane partition with...

I present a conjectural correspondence between topological string theory on toric Calabi-Yau manifolds, and the spectral theory of certain trace class operators on the real line, in the spirit of large N dualities. The operators are obtained by quantization of the algebraic curves which define the mirror manifolds to the Calabi-Yau's. This conjecture can be regarded as a non-perturbative...

Any four-dimensional N=2 superconformal field theory (SCFT) admits a subsector of operators and observables isomorphic to a vertex operator algebra. After reviewing this correspondence (first identified in arXiv:1312.5344), I will aim to characterize the relationship between the Higgs branch of the SCFT (as an algebraic geometric object) and the associated vertex operator algebra. Our proposal...

According the cobordism hypothesis (proposed by Baez-Dolan, and proved by Lurie), an extended topological quantum field theory is fully determined by its value on the point. A natural question is then: does this classification theorem apply to the topological quantum field theories of physical interest? And if yes, what is then the value of those theories on a point (the latter will then...

I will present a framework for computing correlators of three single trace operators in planar N=4 SYM theory that uses hexagonal patches as building blocks. This approach allows one to exploit the integrability of the theory and derive all loop predictions for its structure constants. After presenting the main ideas and results, I will discuss recent perturbative tests and open problems....

I will describe joint work with Yuuji Tanaka. We define a Vafa-Witten invariant for algebraic surfaces. For Fano and K3 surfaces, a standard vanishing theorem means it reduces to (roughly speaking) the Euler characteristic of the moduli space of sheaves on the surface. For general type surfaces there are other contributions, which we calculate.

I will discuss a new moduli space of holomorphic/meromorphic differentials on Riemann surfaces (joint work with G. Farkas) and propose connections between the fundamental class to Pixton's formulas and Witten's r-spin class (joint work with F. Janda, A. Pixton, and D. Zvonkine).

The Verlinde bundle, or the bundle of conformal blocks, is a vector bundle whose rank is given by the well-known Verlinde formula. We will explain how Teleman's classification of semi-simple cohomological field theories allow one to find the Chern character of this vector bundle.

We discuss the exact computation of correlation functions of local operators in the Coulomb branch in four-dimensional N=2 superconformal field theories.

I will explore the constraints imposed by conformal invariance on defects in a conformal field theory. Correlation function of a conformal defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. A correlator of two defect operators admits a...

In this talk I will discuss the moduli spaces of pointed curves with possibly non-nodal singularities such that the marked points form a nonspecial ample divisor. I will show that such curves have natural projective embeddings, with a canonical choice of homogenous coordinates up to rescaling. Using Groebner bases technique this leads to the identification of the moduli with the quotient of...

We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration, and

recalling how to realize Langlands duality through spectral data. We will then look at a natural construction of families of subspaces which give different types of branes, and explain how the topology of some of these branes can be described by considering the spectral...

Gaiotto's conjecture (2014) is a particular construction of opers from Higgs bundles in one Hitchin component. The conjecture has been recently solved by a joint paper of Dumitrescu, Fredrickson, Kydonakis, Mazzeo, Mulase, and Neitzke (2016). In this talk, I will present a holomorphic description of the limiting oper, and its geometry. The importance of this correspondence, in particular the...

I will present a new perspective on Riemann-Hilbert correspondence and wall-crossing based on the considerations of Fukaya categories associated with a holomorphic symplectic manifold and a possibly singular analytic Lagrangian subvariety. This framework includes holonomic D-modules (for the case of cotangent bundles) on the same footing as q-difference equations.

The rationale for interpreting the recently announced events

of the Laser Interferometer Gravitational-Wave Observatory (LIGO) as gravitational wave (GW) signals emitted during the coalescence of two black holes is the excellent match between these events and the corresponding theoretical predictions within General Relativity. We shall review the mix of analytical and numerical methods that...

I will give a review of Poisson-Lie T-duality, which is a non-Abelian generalization of T-duality, and explain it in terms of Chern-Simons theory and its generalizations (AKSZ models) with appropriate boundary conditions. Based on [arXiv:1602.05126]

I will describe two mathematical applications of little string theory. The first leads to a variant of AGT correspondence that relates q-deformed W-algebra conformal blocks to K-theoretic instanton counting. This correspondence can be proven for any simply laced Lie algebra. The second leads to a variant of quantum Langlands correspondence which relates q-deformed conformal blocks of an affine...

I will discuss the properties of boundary conditions of maximally supersymmetric Yang Mills theory compactified on a Riemann surface. Depending on the details of the compactification, this produces BAA branes (i.e. complex Lagrangian submanifolds) or BBB branes (i.e. hyper-holomorphic sheaves)

for the two-dimensional sigma model in the Hitchin moduli space. I will discuss the map from...