Two-Loop Superstrings II, The Chiral Measure on Moduli Space

TL;DR

This paper derives a precise, gauge-independent formula for the two-loop superstring chiral measure by analyzing supergeometries, super period matrices, and associated subtleties, ensuring the measure's consistency and independence from gauge choices.

Contribution

It provides the first unambiguous, slice-independent derivation of the two-loop superstring chiral measure from first principles, addressing subtleties and gauge invariance issues.

Findings

Derived a slice-independent formula for the two-loop superstring chiral measure.

Identified and incorporated necessary Jacobian superdeterminants and gauge choices.

Confirmed the gauge independence of the resulting measure.

Abstract

A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their super period matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, ``slice \hat\mu choice'', whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its super period matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.