Kapustin, Anton; Orlov, Dmitri Vertex algebras, mirror symmetry, and D-branes: the case of complex tori. (English) Zbl 1051.17017 Commun. Math. Phys. 233, No. 1, 79-136 (2003). Summary: 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 Kähler 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 the 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 V. Golyshev, V. Lunts, and D. Orlov [J. Alg. Geom. 10, No. 3, 433–496 (2001; Zbl 1014.14020); see also Preprint math/9912003] 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. Cited in 1 ReviewCited in 34 Documents MSC: 17B69 Vertex operators; vertex operator algebras and related structures 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14J81 Relationships between surfaces, higher-dimensional varieties, and physics 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 14J32 Calabi-Yau manifolds (algebro-geometric aspects) Citations:Zbl 1014.14020 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link