Two-Loop Superstrings I, Main Formulas

TL;DR

This paper derives an unambiguous, slice-independent formula for the two-loop superstring measure on moduli space, enabling explicit computation and showing the vanishing of certain amplitudes, advancing the understanding of superstring perturbation theory.

Contribution

It introduces a new superstring measure formula based on the super-period matrix, resolving ambiguities and enabling explicit, modular covariant calculations at two loops.

Findings

Explicit superstring measure in terms of modular forms

Vanishing of cosmological constant and low-point functions

Finite, convergent integral for certain four-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.