Integrating out the fermions in AdS

TL;DR

This paper computes a three-point superstring amplitude in AdS3 using the hybrid formalism, showing agreement with RNS results and revealing how fermion integration encodes boundary conformal symmetry.

Contribution

It introduces a method to integrate out fermions in AdS backgrounds, connecting worldsheet fermions to target-space symmetries in superstring theory.

Findings

The three-point amplitude matches RNS computations.

Fermion integration yields the target-space vielbein.

Boundary conformal group is realized through fermion integration.

Abstract

Not much is known about superstring scattering amplitudes in curved backgrounds. Using the hybrid formalism in $\rm AdS_3 \times S^3$ with pure NS-NS three-form flux, we compute a $\rm PSU(1,1|2)$-covariant three-point amplitude for half-BPS vertex operators inserted on the $\rm AdS_3$ boundary and show that it agrees with the RNS computation. The zero-mode prescription for the fermions in $\rm AdS$ is defined in terms of the "standard" spacetime SUSY generator. It is found that integrating out the fermionic worldsheet fields in the path integral gives rise to the target-space vielbein, which explicitly encodes that the conformal group on the boundary is identified with the symmetry group of the $\rm AdS$ bulk.