Workshop on Gromov-Witten Theory and Integrable Hierarchies

In this talk, I'll introduce some recent results on peakon, Toda lattices, and orthogonal polynomials.

In 2012, Pantev, Toën, Vaquié and Vezzosi introduced the notion of shifted symplectic structures for derived stacks. This not only generalizes the classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on the various moduli spaces that mathematicians are now studying. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincare duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to non-commutative spaces, such as Calabi-Yau categories.

In this talk, we focus on a special class of non-commutative Calabi-Yau spaces, namely Calabi-Yau algebras, and show that for a Koszul Calabi-Yau algebra, there is a noncommutative shifted symplectic structure, which is also called the shifted bi-symplectic structure, on the cobar construction of its Koszul dual coalgebra adjoining a co-unit. Such shifted bi-symplectic structure naturally induces a shifted symplectic structure on the corresponding differential graded (DG) representation schemes, and also induces a shifted Poisson structure on the derived representation schemes of the Calabi-Yau algebra. Joint work with F. Eshmatov.

I will discuss the integrals on the T-dual branes in the Landau-Ginzburg mirror of toric varieties. The results are genus 0 Gromov-Witten descendant potentials. In particular, based on this result and with some input from homological mirror symmetry, I will sketch a proof of Gamma II conjecture, which concerns such potential's asymptotic behavior.

6d superconformal theories are constructed by the building blocks of a finite set of minimal theories. We develop method to compute the elliptic genus of tensionless strings in these minimal theories, using Weyl invariant Jacobi forms of simple Lie algebra. The elliptic genus is related to topological string partition functions and Gromov-Witten invariants of the corresponding Calabi-Yau three-folds.

We discuss the physics origin of a two-parameter quantum integrable hierarchies associated to Calabi-Yau geometries as a natural quantum gauge symmetry in B-twisted topological string field theory.

The stability of solitary waves (or solitons) is one of the fundamental qualitative properties of the solutions of nonlinear waves equations. In this talk, I will review some progress on the constructions of the orbital stability of the peaked solitary waves (or peakons) by a direct method of conservation laws; present some recent results for the higher-order Camassa-Holm-type dispersive equations; and investigate the effect on the issue of stability of higher-order nonlinearity.

Using parity and time reversal symmetries, some nonequivalent integrable nonlocal KdV (Alice-Bob KdV) systems are obtained from the well-known systems such as the Hirota-Satsuma system, the usual KdV equation with source, coupled KdV systems and the dispersive long wave equations. Same idea can be used to find other types of nonlinear nonlocal integral systems. For instance, the AB-Burgers, AB-NLS type, AB-DNLS systems can be obtained from the Kaup-Broer system, the Boussinesq system and the third CLL systems respectively.

We introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards. The talk is based on the joint paper with M. Bialy (Tel Aviv).

We consider integrable (by the Extended Hodograph Method) multi-component hydrodynamic type systems, which possess just one multiple root of characteristic polynomials of velocity matrices.

TBA

In the present work, we calculate spectral curves for quasi-rational solutions of equations from AKNS hierarchy. Also, we found spectral curves for the simplest trigonometric, hyperbolic and elliptic solutions.

TBA

Donaldson-Thomas invariants of Calabi-Yau threefolds are holomorphic analogue of Casson invariants of three-dimensional real manifolds. They are also closely related to Gromov-Witten invariants in various ways. I will explain briefly this analogue, the Gromov-Witten/Donaldson-Thomas correspondence. Then I will focus on the degeneration method applying to moduli problem, especially Hilbert schemes. As application, we have the degeneration formula of Donaldson-Thomas invariants. This is joint work with Jun Li.

We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. This is a collaboration with Mattia Cafasso.

I will talk about the correspondence between quantum integrable systems and algebro-geometric spectral data. The most important ingredient of these data are spectral sheaves --- torsion free sheaves of special kind on algebraic varieties. I will also discuss how explicit is this correspondence, in particular, could we produce integrable systems starting from spectral data, even if the spectral variety has dimension greater than one.

Kepler system is one of the first integrable systems. Modern study of the Kepler problem reveals some rich connections with other mathematical objects, for example hidden symmetry of the Kepler system and minimal coadjoint orbits, relationships with Penrose twistor theory and Howe duality, etc. There are also generalization of the Kepler problems, which are related to Jordan algebras and bounded symmetric domains. We will introduce a connection of the Kepler problem with the AdS/CFT correspondence. This leads us to hidden supersymmetry that unites all aspects of the Kepler problem.

In 2004 and 2005, the work of Bryan-Pandharipande and Okounkov-Pandharipande build the Gromov-Witten/Donaldson-Thomas correspondence for local curves. In this talk, I will discuss an ongoing project on the Gromov-Witten/Donaldson-Thomas correspondence for local gerby curves, which is an orbifold generalization of the above work. More concretely, we will consider a $\mathbb{Z}_{n+1}$-gerbe $C$ over a smooth genus $g$ curve. Let $L_1$ and $L_2$ be two orbifold line bundles over $C$ such that the isotropy group $\mathbb{Z}_{n+1}$ acts on them via fiberwisely multiplication by $\exp\left(\frac{2\pi i}{n + 1}\right)$ and $\exp\left(-\frac{2\pi i}{n + 1}\right)$ respectively. We will study the Gromov-Witten/Donaldson-Thomas correspondence for the total space of $X := L_1\oplus L_2$. The strategy of the proof is to apply degeneration formula to $X$ to reduce the problem to the case of cap, cylinder, and pair of pants. This work is joint with Zijun Zhou.