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Current Algebra on the Torus
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We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant tau-dependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications.
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On theta function expressions of cyclic products of fermion correlation functions in genus two
Cyclic products of genus-two fermion correlators decompose so spin-structure dependence lives only in Pe functions at even half-periods, with explicit theta-function forms obtained for N=2, 3 and partially for N=4.
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