Time-like T-duality algebra

TL;DR

This paper investigates the algebraic structure of low energy theories from string theory compactifications on tori with indefinite signatures, computing and classifying the relevant symmetry groups H_n for various theories and signatures.

Contribution

It provides explicit formulas and classifications for the groups H_n in indefinite signature compactifications, including new groups not previously documented.

Findings

Computed H_n groups for all compactifications of M-, M*-, M'-theories, and type II theories.

Identified new real forms of E_n and H_n relevant to timelike compactifications.

Compiled tables outlining dualities between different signatures and theories.

Abstract

When compactifying M- or type II string-theories on tori of indefinite space-time signature, their low energy theories involve sigma models on E_{n(n)}/H_n, where H_n is a not necessarily compact subgroup of E_{n(n)} whose complexification is identical to the complexification of the maximal compact subgroup of E_{n(n)}. We discuss how to compute the group H_n. For finite dimensional E_{n(n)}, a formula derived from the theory of real forms of E_n algebra's gives the possible groups immediately. A few groups that have not appeared in the literature are found. For n=9,10,11 we compute and describe the relevant real forms of E_n and H_n. A given H_n can correspond to multiple signatures for the compact torus. We compute the groups H_n for all compactifications of M-, M*-, and M'-theories, and type II-, II*- and II'-theories on tori of arbitrary signature, and collect them in tables that outline the dualities between them. In an appendix we list cosets G/H, with G split and H a subgroup of G, that are relevant to timelike toroidal compactifications and oxidation of theories with enhanced symmetries.