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Threshold corrections, generalised prepotentials and Eichler integrals. (English) Zbl 1329.81300

Summary: We continue our study of one-loop integrals associated to BPS-saturated amplitudes in \(\mathcal{N} = 2\) heterotic vacua. We compute their large-volume behaviour, and express them as Fourier series in the complexified volume, with Fourier coefficients given in terms of Niebur-Poincaré series in the complex structure modulus. The closure of Niebur-Poincaré series under modular derivatives implies that such integrals derive from holomorphic prepotentials \(f_n\), generalising the familiar prepotential of \(\mathcal{N} = 2\) supergravity. These holomorphic prepotentials transform anomalously under T-duality, in a way characteristic of Eichler integrals. We use this observation to compute their quantum monodromies under the duality group. We extend the analysis to modular integrals with respect to Hecke congruence subgroups, which naturally arise in compactifications on non-factorisable tori and freely-acting orbifolds. In this case, we derive new explicit results including closed-form expressions for integrals involving the \(\Gamma_0(N)\) Hauptmodul, a full characterisation of holomorphic prepotentials including their quantum monodromies, as well as concrete formulæ for holomorphic Yukawa couplings.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
83E50 Supergravity
57R18 Topology and geometry of orbifolds
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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References:

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