Two-Loop Superstrings I, Main Formulas

TL;DR

This work resolves long-standing ambiguities in two-loop superstring amplitudes by formulating a gauge-fixing scheme based on the super period matrix, yielding a slice-independent, modular-covariant measure for even spin structures at genus 2. The per-spin-structure measures are expressed via modular forms, enabling a unique GSO projection whose sum vanishes pointwise, hence a zero cosmological constant, and producing finite, well-defined 4-point amplitudes with controlled low-energy limits. The framework naturally extends to compactifications with ${\cal N}=1$ worldsheet supersymmetry and central charge $c=15$, providing a practical method to compute cosmological constants in a broad class of models. Overall, the paper establishes a consistent, first-principles construction of the two-loop superstring measure and scattering amplitudes, with robust gauge invariance and modular properties underpinning the vanishing cosmological constant and finite higher-point functions.

Abstract

An unambiguous and slice-independent formula for the two-loop superstring measure on moduli space for even spin structure is constructed from first principles. The construction uses the super-period matrix as moduli invariant under worldsheet supersymmetry. This produces new subtle contributions to the gauge-fixing process, which eliminate all the ambiguities plaguing earlier gauge-fixed formulas. The superstring measure can be computed explicitly and a simple expression in terms of modular forms is obtained. For fixed spin structure, the measure exhibits the expected behavior under degenerations of the surface. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons all vanish pointwise on moduli space without the appearance of boundary terms. A certain disconnected part of the 4-point function is shown to be given by a convergent, finite integral on moduli space. A general slice-independent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and N=1 worldsheet supersymmetry in terms of the data of the compactification conformal field theory. In this paper, a summary of the above results is presented with detailed constructions, derivations and proofs to be provided in a series of subsequent publications.