Research Interests

My research interests are in the intersection of three different areas: symplectic geometry, complex geometry, and representations of infinite-dimensional Lie algebras. The starting point of my research is the so-called Witten's conjecture proved by Kontsevich. It says that the full generating function of intersection numbers on the Delign--Mumford compactificatio of the moduli spaces ℳg,n of Riemann surfaces is a tau-function of the KdV hierarchy. It is well known (M. Sato, 1981) that the tau-functions of the KdV hierarchy can be described in terms of the so-called Hirota bilinear equations. Equivalently, the coefficients of the tau-functions satisfy an infinite system of quadratic equations which can be interpreted as the Plücker relations of the embedding of an infinite Grassmannian in a projective space. Following Givental we refer to this system of quadratic equations as Hirota quadratic equations (HQEs). It turns out that the HQEs of KdV can be interpreted as the regularity condition of a certain bilinear operator acting on the tensor square of the tau-function. The key observation, due to Givental (in 2003), is that the bilinear operator is defined in terms of vertex operators whose coefficients are periods of the A1 singularity (Givental, 2003). The project that I started working on was suggested by Givental. He asked me to investigate whether we can construct HQEs more generally in Gromov--Witten theory of a smooth projective variety X by using the period integrals associated with the mirror model of X. After the work of Dubrovin and Zhang, it became clear that we have to work with targets X, such that, their quantum cohomology is semi-simple. There are 3 major developments that happened since 2003. First, in a joint work (in 2013) with Bojko Bakalov, we discovered that the vertex operators with coefficients period integrals define a representation of the lattice vertex algebra associated to the Milnor lattice -- the lattice of vanishing cycles or corresponding Lefschetz thimbels. Second, Hiroshi Iritani found (in 2009) a remarkable conjecture about the Milnor lattice of the mirror model in terms of the topological K-ring of X. Finally, partially motivated by my joint work with Givental and Frenkel (in 2010), I was able to find certain connection formulas for the Operator Product Expansion (OPE) of the vertex operators. The project of constructing HQEs in GW theory of a smooth projective variety X can be formulated now in the language of the lattice vertex algebras associated with the lattice underlying the topological K-ring of X. Namely, one has to construct a state in the lattice vertex algebra satisfying certain screening equations. So far, the targets for which the construction was worked out are the Fano orbifold lines Pa,b,c.

The above paragraph describes the main focus of my research. During the years I got involved in various related projects, such as, the topological recursion of Eynard and Orantin, matrix models, mirror symmetry for manifolds with semi-simple quantum cohomology, and K-theoretic GW theory.

My work is available on the arXiv.

The journal references are available in my Publication List.


1. Chenghan Zha

Master Thesis: The period map of a two-dimensional semi-simple Frobenius manifold.

PhD Thesis: Integral structures in the local algebra of a singularity.

2. Xiaokun Xia

Master Thesis: Gamma integral structure for the blowup of Pn at a point.

Lecture notes

1. Gromov--Witten theory and representations of affine Lie algebras. Lectures that I gave in 2007 in Stanford.
2. Gromov--Witten theory and integrable hierarchies. Lectures that I gave in Beijing 2009.
3. IPMU seminar on Bolibrukh's counter example for the Riemann--Hilbert problem.
4. Primitive forms and vertex operators. The first draft of a book.