The Mechanism behind the Embeddings of String Theories

TL;DR

This paper investigates the mechanisms behind embeddings of different string theories, revealing how various theories' spectra and amplitudes relate through BRST cohomology and affine algebra embeddings, and demonstrating their deformations and isomorphisms.

Contribution

It provides a detailed analysis of the embedding mechanism between string theories via affine algebra and BRST cohomology, including proofs of isomorphism for specific classes of embeddings.

Findings

Embedding of affine algebras preserves BRST cohomology structures.

All similar embeddings are deformations of the simplest case.

Certain classes of embeddings have isomorphic cohomologies.

Abstract

It has been realised recently that there is no unique way to describe the physical states of a given string theory. In particular, it has been shown that any bosonic string theory can be embedded in a particular $N{=}1$ string background in such a way that the spectrum and the amplitudes of both theories agree. Similarly, it is also known that the amplitudes of any $N{=}1$ string theory can be obtained from a particular $N{=}2$ string background. When rephrased in the language of BRST cohomology, these results suggest a close connection to the theory of induced representations. The purpose of this note is to investigate this connection further and at the same time to reveal the mechanism behind these embeddings between string theories. We will first analyze the embedding of an affine algebra $\gghat$ in the $N{=}1$ affine algebra associated to $\gg$. Given any BRST cohomology theory for $\gghat$ we will be able to construct one for the $N{=}1$ affine algebra associated to $\gg$ such that the cohomologies agree as operator product algebras. This is proven in two different ways. This example is the simplest in its kind and, in a sense that is made precise in the paper, all other similar embeddings are deformations of this one. We conclude the paper with a brief treatment of the general case, where we prove that for a particular class of ``good'' embeddings, the cohomologies are again isomorphic.