A search for exact superstring vacua

TL;DR

This paper examines conditions under which supersymmetric 2D sigma-models with specific target space metrics can produce exact string vacua, revealing that higher-order quantum corrections complicate the derivation of such vacua.

Contribution

It establishes non-renormalization theorems relating the vector $M_{}$ to beta functions and analyzes higher-loop contributions that challenge the existence of exact superstring vacua.

Findings

Non-vanishing four-loop contributions to the beta function.

Higher-order corrections prevent closed-form superstring vacua.

Relation between $M_u$ and the beta function at one-loop level.

Abstract

We investigate $2d$ sigma-models with a $2+N$ dimensional Minkowski signature target space metric and Killing symmetry, specifically supersymmetrized, and see under which conditions they might lead to corresponding exact string vacua. It appears that the issue relies heavily on the properties of the vector $M_{\mu}$, a reparametrization term, which needs to possess a definite form for the Weyl invariance to be satisfied. We give, in the $n = 1$ supersymmetric case, two non-renormalization theorems from which we can relate the $u$ component of $M_{\mu}$ to the $\beta^G_{uu}$ function. We work out this $(u,u)$ component of the $\beta^G$ function and find a non-vanishing contribution at four loops. Therefore, it turns out that at order $\alpha^{\prime 4}$, there are in general non-vanishing contributions to $M_u$ that prevent us from deducing superstring vacua in closed form.