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Torus chiral $$n$$-point functions for free boson and lattice vertex operator algebras. (English) Zbl 1020.17020
Given a vertex operator algebra (VOA) $$V$$ one may define chiral $$n$$-point functions at genus one following Y. Zhu [J. Am. Math. Soc. 9, 237-302 (1996; Zbl 0854.17034)] and then use various sewing procedures to define such functions at higher genera. However, these procedures require a detailed description of the genus one $$n$$-point functions. In the paper under review the authors obtain explicit expressions for all genus one $$n$$-point functions in two important cases, namely when $$V$$ is either a Heisenberg VOA (also called a free boson theory) or a lattice VOA, that is, a VOA associated to a positive-definite, even lattice. Roughly speaking, in a free boson theory the $$n$$-point functions are elliptic functions whose structure depends on certain combinatorial data determined by the states in $$V$$ under consideration. In the case of a lattice VOA, the function is the product of two pieces, one determined by the Heisenberg sub-VOA of $$V$$ and one which may be described in terms of the lattice itself and the genus one prime form. The paper concludes with a discussion of $$n$$-point functions from the point of view of their symmetry and elliptic properties.

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81R99 Groups and algebras in quantum theory
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