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Genus two partition and correlation functions for fermionic vertex operator superalgebras. I. (English) Zbl 1254.17024
This paper is a very meaningful work for string and conformal field theory. Here the authors study the genus two partition and $$n$$-point correlation functions for vertex operator superalgebras on a genus two Riemann surface formed by sewing two tori together. A closed formula is obtained for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegö kernels. The authors prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. The authors compute and discuss the modular properties of the generating function for all $$n$$-point functions in terms of a genus two Szegö kernel determinant. The authors also show that the Virasoro vector one point function satisfies a genus two Ward identity.

MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 17B68 Virasoro and related algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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