C. M. Liu

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Let (X, ω) be a symplectic manifold, J be an ω-tame almost complex structure, and L be a Lagrangian submanifold. The stable compactifi-cation of the moduli space of parametrized J-holomorphic curves in X with boundary in L (with prescribed topological data) is compact and Hausdorff in Gromov's C ∞-topology. We construct a Kuranishi structure with corners in(More)
We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three(More)
In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to(More)
We propose a conjectural formula expressing the generating series of some Hodge integrals in terms of representation theory of Kac-Moody algebras. Such generating series appear in calculations of Gromov-Witten in-variants by localization techniques. It generalizes a formula conjectured by Mariño and Vafa, recently proved in joint work with Chiu-Chu Melissa(More)
We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons.(More)