Abstract
The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.
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Acknowledgments
The research of ED is supported in part by NSF grants PHY-19-14412 and PHY-22-09700. The research of MH and OS is supported by the European Research Council under ERC-STG-804286 UNISCAMP. MH and OS are grateful to UCLA and the Mani Bhaumik Institute for kind hospitality and creating a stimulating atmosphere during initiation of this work. We gratefully acknowledge the hospitality of the KITP during early stages of this work, and support from National Science Foundation grant PHY-17-48958.
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D’Hoker, E., Hidding, M. & Schlotterer, O. Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation. J. High Energ. Phys. 2023, 73 (2023). https://doi.org/10.1007/JHEP05(2023)073
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DOI: https://doi.org/10.1007/JHEP05(2023)073