Naichung Conan Leung

SYZ mirror symmetry for toric Calabi-Yau manifolds

We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric CalabiYau manifold X, we construct a complex manifold y X using Tduality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold y X, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series ex(

Mirror symmetry for toric Fano manifolds via SYZ transformations

Abstract We construct and apply Strominger–Yau–Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau–Ginzburg models.

Mirror maps equal SYZ maps for toric Calabi–Yau surfaces

We prove that the mirror map is the Strominger–Yau–Zaslow map for every toric Calabi–Yau surface. As a consequence, one obtains an enumerative meaning of the mirror map. This involves computing genus-0 open Gromov–Witten invariants, which is done by relating them with closed Gromov–Witten invariants via compactification and using an earlier computation by Bryan–Leung.

Orientability for gauge theories on Calabi–Yau manifolds

Abstract We study orientability issues of moduli spaces from gauge theories on Calabi–Yau manifolds. Our results generalize and strengthen those for Donaldson–Thomas theory on Calabi–Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing formula in DT theory.

BV Quantization of the Rozansky–Witten Model

We investigate the perturbative aspects of Rozansky–Witten’s 3d

Branes and toric geometry

{\sigma}

Mirror Symmetry Without Corrections

σ-model (Rozansky and Witten in Sel Math 3(3):401–458, 1997) using Costello’s approach to the Batalin–Vilkovisky (BV) formalism (Costello in Renormalization and effective field theory, American Mathematical Society, Providence, 2011). We show that the BV quantization (in Costello’s sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich (First European congress of mathematics, Paris, 1992) and Axelrod–Singer (J Differ Geom 39(1):173–213, 1994). We also study the factorization algebra structure of quantum observables following Costello–Gwilliam (Factorization algebras in quantum field theory, Cambridge University Press, Cambridge 2017). In particular, we show that the cohomology of local quantum observables on a genus g handle body is given by