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Free bosonic vertex operator algebras on genus two Riemann surfaces. I. (English) Zbl 1226.17024
One feature of chiral conformal field theory is the natural occurrence of elliptic functions and modular forms in the $$n$$-point correlation functions. In string theory their occurrence has been known for a long time and has inspired mathematicians and physicists alike. In the present paper the authors define the partition function and the $$n$$-point functions for a vertex operator algebra (VOA) on a genus-2 Riemann surface obtained by sewing two genus-1 tori together. They compute closed formulas for the partition function for the rank one Heisenberg VOA and for any pair of simple Heisenberg modules. They prove that the partition function is holomorphic in the sewing parameters within some complex domain and describe its modular properties for the Heisenberg VOA and the lattice VOA. They also describe a continuous orbifolding of the fermion vertex operator super algebra of rank two, compute the $$n$$-point function, and show that the Virasoro vector 1-point function satisfies a genus-2 Ward identity.

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81T10 Model quantum field theories 81T45 Topological field theories in quantum mechanics 58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx) 57R56 Topological quantum field theories (aspects of differential topology) 53Z05 Applications of differential geometry to physics
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