Chiu-Chu Melissa Liu

A mathematical theory of the topological vertex

We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 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 manifolds.

Positivity of quasilocal mass.

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.

A formula of two-partition Hodge integrals

JMg,n where tpi = ci(L?) is the first Chern class of L?, and Xj = Cj(E) is the j-th Chern class of the Hodge bundle. The study of Hodge integrals is an important part of intersection theory on Mg,n> Hodge integrals also naturally arise when one computes Gromov-Witten invariants by localization techniques. For example, the following generating series of Hodge integrals arises when one computes local invariants of a toric Fano surface in a Calabi-Yau 3-fold by virtual localization [33]:

The Eynard–Orantin recursion and equivariant mirror symmetry for the projective line

We study the equivariantly perturbed mirror Landau-Ginzburg model of the projective line. We show that the Eynard-Orantin recursion on this model encodes all genus all descendants equivariant Gromov-Witten invariants of the projective line. The non-equivariant limit of this result is the Norbury-Scott conjecture, while by taking large radius limit we recover the Bouchard-Marino conjecture on simple Hurwitz numbers.

Gromov-Witten Theory of Calabi-Yau 3-folds

We describe some recent progress and open problems in Gromov-Witten theory of Calabi-Yau 3-folds, focusing on the quintic 3-fold and toric Calabi-Yau 3folds.