Huijun Fan

Peking University | PKU · School of Mathematical Sciences

For a nondegenerate homogeneous polynomial $f\in\mathbb{C}[z_0, \dots, z_{n+1}]$ with degree $n+2$, we can obtain a $tt^*$ structure from the Landau-Ginzburg model $(\C^{n+2}, f)$ and a (new) $tt^*$ structure on the Calabi-Yau hypersurface defined by the zero locus of $f$ in $\C P^{n+1}$. We can prove that the big residue map considered by Steenbri...

Inspired by the LG/CY correspondence, we study the local index theory of the Schrodinger operator associated to a singularity defined on Cn by a quasi-homogeneous polynomial f . Under some mild assumption to f , we show that the small time heat kernel expansion of the corresponding Schrodinger operator exists and is a series of fractional powers of...

The concept of $tt^*$ geometric structure was introduced by physicists (see \cite{CV1, BCOV} and references therein) , and then studied firstly in mathematics by C. Hertling \cite{Het1}. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of a two dimensional topological field theory. In this paper, we propos...

Inspired by the LG/CY correspondence, we study the local index theory of the Schr\"odinger operator associated to a singularity defined on ${\mathbb C}^n$ by a quasi-homogeneous polynomial $f$. Under some mild assumption on $f$, we show that the small time heat kernel expansion of the corresponding Schr\"odinger operator exists and is a series of f...

This is a survey article for the mathematical theory of Witten's Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper we focus on the description of the moduli.

The authors prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D 4 with symmetry group 〈J〉 and for D 4T with symmetry group G max, respectively, are both tau-functions of the D 4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in the article by Fan-Jarvis-Ruan (2013), of the Witt...

We construct a rigorous mathematical theory of Witten's Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with non-Abelian gauge group. Both the Gromov-Witten theory of a Calabi-Yau complete intersection X and the Landau-Ginzburg dual (FJRW-theory) of X can be expressed as gauged linear sigma mod...

A twisted $\bar{\partial}_f$-Neumann problem associated to a singularity $(\mathscr{O}_n,f)$ is established. By constructing the connection to the Koszul complex for toeplitz $n$-tuples $(f_1,\cdots,f_n)$ on Bergman spaces $B^0(D)$, we can solve this $\bar{\partial}_f$-Neumann problem. Moreover, we can compute the cohomology of the $L^2$ holomorphi...

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial X p + XY q (p≥2, q>1) with some admissible symmetry group G defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial X p Y+ Y q. We will construct an concrete isomorphism between them. The construction is a little bi...

This is the first of a series of papers to construct the deformation theory of the form Schr\"odinger equation, which is related to a section-bundle system $(M,g,f)$, where $(M,g)$ is a noncompact complete K\"ahler manifold with bounded geometry and $f$ is a holomorphic function defined on $M$. This work is also the first step attempting to underst...

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r-spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W-curves is canonically isomorphic to the stack of r-spin curves described by Abramovich and Jarvis. We further...

We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4 Kac-Wakimoto/Drinfeld-Sokolov hierarchy. This completes the proof, begun in [FJR], of the Witten Integrable Hierarchies Conjecture for all s...

In this paper, we will prove that the quantum ring of the quasi-homogeneous polynomial $X^{p}+XY^{q}(p\ge 2,q>1)$ with some admissible symmetry group $G$ defined by Fan-Jarvis-Ruan-Witten theory is isomorphic to the Milnor ring of its mirror dual polynomial $X^{p}Y+Y^{q}$. We will construct an concrete isomorphism between them. The construction is...

We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the...

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_{r-1}. W...

We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the Cauchy-Riemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, w...

The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one...

This paper discusses the Cauchy problem of the equation $$ u_{t} = \nabla \cdot {\left( {{\left| \nabla \right|}_{u} ^{m} \left| {^{{p - 2}} } \right.\nabla _{u} ^{m} } \right)} - u^{q} $$ (1) with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative generalized solution...

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carry...

The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley's theory for dynamical systems. We introduce the concept of a flow carrying a cocycle , (generalized) -flow for short, where is a continuous cocycle in bounded Alexander-Spanier cohomolo...

Introduction and the main result This paper can be seen as the nal remark of the previous paper written by the rst author [D]. We consider the existence of harmonic maps between two spheres, via Hopf constructions. Given a non trivial bi-eigenmap f : S p S q ! S n with bi-eigenvalue (; ) (; > 0) and a continuous function : [ 0, 2 ] ! [ 0, ] with (0...

The present paper contains an interpretation and generalization of Novikov's theory of Morse type inequalities for 1-forms in terms of Conley's theory for dynamical systems.

It is proved that the heat equations of harmonic maps have self-similar solutions satisfying certain energy condition.

We introduce the W -spin structures on a Riemann surface Σ and give a precise definition to the corresponding W -spin equations for W being a quasi-homogeneous polynomial. When W is the Ar-potential, then they correspond to the r-spin structures and the r-spin equations considered by E. Witten [W2]. If the number of the Ramond marked points on Σ is...