Gauged Linear σ-model (GLSM)
Around 1990s, physicists discovered a correspondence between conformal field theories: the nonlinear σ-model of a Calabi-Yau hypersurface defined by a quintic polynomial Q corresponds to the Landau-Ginzburg model of the singularity defined by the same Q. Later on Witten introduced the gauged linear σ-model (GLSM) ("Phases of N=2 theories in two dimensions", 1993), giving a geometric explanation of LG/CY correspondence. Both of the two sides of the correspondence have been constructed as rigorous mathematics, i.e., the Gromov-Witten theory and the Landau-Ginzburg A-model theory, and they have great impact on geometry.
Tian and I initiated the program on a mathematical construction of GLSM. It is based on extending the techniques of the symplectic vortex equation to the new situation where a superpotential comes into play. The new equation, which we call the gauged Witten equation, is the equation of motion in the corresponding physics theory.
- (with Gang Tian) Analysis of gauged Witten equation, to appear in Crelle's journal, arxiv link.
In this paper we set up the basic framework of the gauged Witten equation, for the superpotentials of Lagrange multiplier type. We provide a convenient perturbation scheme, proved the asymptotic behavior of finite energy solutions and studied the linear Fredholm theory. We also proved the compactness of solutions to the perturbed gauged Witten equation when the domain curve is fixed.
- (with Gang Tian) Correlation functions of gauged linear σ-model
In this paper we give a formal definition of the correlation function in GLSM over a fixed smooth domain curve, assuming the existence of a virtual fundamental cycle on the moduli space of solutions to the gauged Witten equation.
- U(1)-vortices and quantum Kirwan map, submitted,
In this paper we classified affine vortices in the Euclidean space with the standard U(1)-action. This generalizes the classical result of Taubes in 1980 on the classification of affine abelian vortices.
Gauged Hamiltonian Floer theory
Hamiltonian Floer homology was invented by Andreas Floer in 1980s. Formally it took Witten's point of view of Morse theory ("Supersymmetry and Morse theory, 1982") by studying the equation of negative gradient flow lines of certain action functional on the loop space of a symplectic manifold M, where the action functional is canonically associated with a Hamiltonian system. In addition, Floer's equation is a variant of the nonlinear Cauchy-Riemann equation, on which Gromov started his pioneering work ("Pseudoholomorphic curves in symplectic manifolds, 1985").
On a Hamiltonian G-manifold, the equivariant counter-part of pseudoholomorphic curves are vortices. Then for G-invariant Hamiltonian system, Cieliebak-Gaio-Salamon proposed a vortex version of Hamiltonian Floer homology.
- Gauged Hamiltonian Floer homology I: the definition of Floer homology groups, to appear on Trans. Amer. Math. Soc. arxiv link
In this paper we complete the construction of gauged Hamiltonian Floer homology groups proposed by Cieliebak-Gaio-Salamon. We assumed that the Hamiltonian G-manifold is aspherical, and achieved the transversality without using the virtual technique. Since the symplectic quotient may be non-aspherical, and this Floer homology gives a nontrivial alternative for the usual Hamiltonian Floer homology of the symplectic quotient.
- (with Stephen Schecter) Morse theory for Lagrange multipliers and adiabatic limits, J. Differential Equations, 257 (2014) 4277-4318.
In this paper we studied a finite-dimensional model of the gauged Hamiltonian Floer theory, i.e., the Morse homology theory of Lagrange multipliers. Namely, the Morse theory for the function f + tμ. By varying a parameter in this theory, we obtain two different limits: the first limit gives the Morse homology of the hypersurface μ=0 and the second limit is associated with the fast-slow dynamics in ODE theory, which provides a different perspective of Morse theory of hypersurfaces.
- (with Weiwei Wu) Vortex Floer homology and spectral invariants, preprint
The energy filtration in Hamiltonian Floer theory induces chain-level invariants in Hamiltonian Floer theory called the spectral invariants, introduced by Viterbo, Schwarz and Oh. They are used by Entov-Polterovich to construct quasi-morphisms and quasi-states in symplectic topology. In this paper we apply the same idea for vortex Hamiltonian Floer theory. Such construction leads to a reproduction of the results of Fukaya-Oh-Ohta-Ono on the relation between spectral invariants and Lagrangian Floer theory without using virtual technique. For example, we can show that certain toric fibres in toric manifolds are heavy.
Gauged Lagrangian Floer theory
- Moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness, Adv. Math. 242 (2013) 1-49
In this paper I studied the compactification of symplectic vortex equation with Lagrangian boundary condition for compact Hamiltonian U(1)-manifold and invariant Lagrangian submanifolds. This generalizes the compactness results of Mundet and Tian.
- (joint with Dongning Wang) Compactness in adiabatic limit of disk vortices, arxiv link.
Affine vortices are vortices over the complex plane. They leads to a relation between Hamiltonian Gromov-Witten invariants of a Hamiltonian G-manifold and the Gromov-Witten invariants of its symplectic reduction, conjectured by D. Salamon. In the corresponding open-string theory, in genus zero, it is supposed to intertwine A-infinity relations.
Master Thesis
- (with Huitao Feng and Weiping Zhang) Real embedding, η-invariant and Chern-Simons current, Pure Appl. Math. Q. 5 (2009), no. 3, Special Issue: In honor of Friedrich Hirzebruch. Part 2, 1113–1137. 58J28
In this paper we give a new, more elementary proof of Bismut-Zhang's formula for η-invariants of Dirac operators on odd-dimensional manifolds.
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