A new look at one-loop integrals in string theory

TL;DR

The paper develops a modular-invariant framework to evaluate one-loop string theory integrals, keeping T-duality explicit by employing the Rankin-Selberg-Zagier method and modular regularisation. It shows that lattice partition functions can be renormalised and expressed as constrained Epstein zeta series tied to Langlands-Eisenstein objects, enabling exact results for general spacetime dimension $d$ and transparent low-dimensional cases ($d=1,2$). For integrals with unphysical tachyons, the authors introduce a non-holomorphic Poincaré-series unfolding that yields holomorphic limits linked to the Klein $j$-function and its Hecke transforms, culminating in shifted constrained Epstein zeta series representations. The work also discusses decompactification, alternate regulators, and the interplay between modular forms and automorphic objects, providing a robust, symmetry-preserving toolkit for one-loop amplitudes in string theory. These methods offer a modularly invariant path beyond the traditional orbit method, with potential extensions to higher genus and enhanced symmetry loci.

Abstract

We revisit the evaluation of one-loop modular integrals in string theory, employing new methods that, unlike the traditional 'orbit method', keep T-duality manifest throughout. In particular, we apply the Rankin-Selberg-Zagier approach to cases where the integrand function grows at most polynomially in the IR. Furthermore, we introduce new techniques in the case where `unphysical tachyons' contribute to the one-loop couplings. These methods can be viewed as a modular invariant version of dimensional regularisation. As an example, we treat one-loop BPS-saturated couplings involving the $d$-dimensional Narain lattice and the invariant Klein $j$-function, and relate them to (shifted) constrained Epstein Zeta series of O(d,d;Z). In particular, we recover the well-known results for d=2 in a few easy steps.