I would like to briefly review our recent works on an application of topological string theory to 2d electron system. I will explain how to generalize this idea to the case of honeycomb lattice.

I will review a recently discovered correspondence between

topological strings (and knots and their invariants in particular) and quiver representation theory.

Perturbation theory is generically divergent, leading to series with zero radius of convergence. When such asymptotic perturbative-series are resurgent, this problem can be tackled by extending the perturbative series into a non-perturbative trans-series, in a specified fashion. Resurgent trans-series may then be used to go beyond perturbation theory in generic problems across theoretical...

We describe four methods to solve the topological string on elliptic Calabi-Yau manifolds and its relation to 6d SCFT.

The existence, decay and formation of BPS bound-states in N=2 theories are controlled by the central charge function through the notion of Pi-stability. In this talk I will describe the variation of the central charge over the moduli space of a given theory introducing a notion of BPS variation of Hodge structure. This description leads to exact results, giving exact descriptions of the walls...

I discuss a class of little string theories (LSTs) with eight supercharges on the world-volume of N M5-branes probing a transverse Z_M orbifold. These M-brane configurations compactified on a circle are dual to M D5-branes intersecting N NS5-branes on T^2 x R^{7,1} as well as to F-theory compactified on a toric Calabi-Yau threefold X_{N,M}. I argue that the Kähler cone of X_{N,M} admits three...

In the last few years there have been many new results connecting (linear quiver) N=2 class S theories, and the topological strings that engineeer them, to the theory of isomonodromic deformations on the sphere and their q-deformations.

The aim of this talk is to show how this connection can be extended beyond the case of genus zero, which corresponds to circular quiver gauge theories,...

I give a very brief review of the main points of my recent work on the constructive field theory of the most general class of B-type Landau-Ginzburg models, stressing the conceptually important parts while avoiding most of the mathematical details.

In their recent work R. Panharipande, J. Solomon and R.

Tessler constructed intersection theory on the moduli spaces of

Riemann surfaces with boundaries. The goal of my talk is to describe

the generating function of this theory, which is a simplest example of

the open topological string model, in terms of matrix models and

integrable hierarchies. The geometrical description of the open...

I will discuss an approach to establishing the foundations of open Gromov-Witten theory based on bounding chains and Fukaya A-infinity algebras.

We exhibit a relation between the analytic continuation of the periods of the mirror quintic to the conifold and periods and quasiperiods of a certain weight 4 Hecke eigenform associated to the mirror quintic. To explain this, we review the theory of periods of modular forms and extend it to quasiperiods.

We show that there is a universal algebraic structure, closely related with that of the WDVV equation, governing quantum correlation functions of every quantum field theory based on the BV quantization scheme.

The sphere partition function of the gauged linear sigma model computes the exact metric on the Kaehler moduli space of a Calabi-Yau. We use this to test the refined swampland distance conjecture for examples of one-parameter Calabi-Yau threefolds with exotic hybrid points. This is joint work with David Erkinger.

We discuss the relation between certain auto-equivalences of the category of B-branes on elliptic Calabi-Yau threefolds and modular properties of the corresponding topological string partition function.

This suggests a geometric explanation and generalization of recent conjectures on the appearance of lattice Jacobi forms.

In particular, we will shed light on the special case where the...