Vertex Algebras, Mirror Symmetry, And D-Branes: The Case Of Complex Tori
Anton Kapustin, Dmitri Orlov
TL;DR
The paper develops a generalized vertex-algebra framework to study mirror symmetry for complex tori, constructing $N=2$ superconformal vertex algebras $Vert( ext{Γ},I,G,B)$ from torus data and giving explicit isomorphism and mirror criteria via lattice data. It shows that for algebraic tori with torsion B-fields, Kontsevich's Homological Mirror Symmetry holds after replacing $D^b(X)$ by $D^b(X, ext{B})$ and twisting the Fukaya category by the B-field, while for higher dimensions a full HMS requires Azumaya/Morita twists. The work connects to D-brane physics, suggesting that BPS branes of type B correspond to coherent-sheaf objects, while A-branes become twisted Fukaya-category objects, with the B-field mediating noncommutative geometry via Azumaya structures. Together, these results illuminate the role of the B-field in both mathematical HMS and physical D-brane configurations on Calabi–Yau manifolds, and provide a concrete torus setting in which to test and refine HMS under nontrivial B-field twists.
Abstract
A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds.