Si Li

On the B-twisted topological sigma model and Calabi–Yau geometry

We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin-Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi-Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau-Ginzburg case, and show that the resulting topological correlations coincide with Vafas residue formula.

Batalin–Vilkovisky quantization and the algebraic index

Abstract Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosovs deformation quantization of a symplectic manifold X and the Batalin–Vilkovisky (BV) quantization of a one-dimensional sigma model with target X . This model is a quantum field theory of AKSZ type and is quantized rigorously using Costellos homotopic theory of effective renormalization. We show that Fedosovs Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.

A mirror theorem between Landau–Ginzburg models

Abstract We survey on the recent progress toward mirror symmetry between Landau–Ginzburg models.