Abstract
We continue our study of one-loop integrals associated to BPS-saturated amplitudes in 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 fn, generalising the familiar prepotential of 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 Γ0(N) Hauptmodul, a full characterisation of holomorphic prepotentials including their quantum monodromies, as well as concrete formulæ for holomorphic Yukawa couplings.
Highlights
In a series of recent works [5,6,7], we proposed a new method, which assumes that Φ can be written as an absolutely convergent Poincaré series under the modular group
Which we refer to as the Narain lattice, is a sum over integers mi and ni identified with the Kaluza-Klein momentum and winding numbers along a two-dimensional torus
Using the behaviour (2.9) of Niebur-Poincaré series under the action of modular derivatives, we show that for integer s = 1 + n > 1, the modular integral (2.16) can be expressed in terms of a holomorphic function fn(T, U), which we refer to as the generalised prepotential8, I(1 + n)
Summary
We can proceed to the study of the integral (1.1). Since any weak almost holomorphic modular form Φ can be decomposed as in (2.11) it is sufficient to focus our attention on the basic regularised modular integral. For s = 1, the integral (2.16) develops a simple pole, which must be subtracted [6,8]. Eq (2.17) corresponds to the so-called Hamiltonian representation of the Narain lattice, and exhibits manifest invariance under the duality group O(2, 2; Z) ∼ SL(2; Z)T × SL(2; Z)U σT,U, where the Z2 generator σT,U exchanges the T and U moduli. Γ2,2(T, U; τ) is invariant under the action of SL(2; Z)τ × SL(2; Z)T × SL(2; Z)U στ,T,U, where στ,T,U permutes the τ, T and U moduli Where the source term is due to the subtraction of the simple pole in the integral (2.16) at s = 1.
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