The many faces of OSp(1|32)

TL;DR

This paper demonstrates that various superalgebras underlying F-theories, M-theories, and type II string theories across different dimensions and signatures are unified as different covariant forms of the same OSp(1|32) algebra, with dualities interpreted as basis changes.

Contribution

It develops a unified framework using the complex form of OSp(1|32) to relate different signatures and theories through basis changes, dualities, and real forms, clarifying their algebraic structure.

Findings

Unified description of superalgebras across theories and signatures.

Demonstrated T-duality transformations involving spacelike and timelike directions.

Connected translation generators to central charges, leading to star algebras.

Abstract

We show that the complete superalgebra of symmetries, including central charges, that underlies F-theories, M-theories and type II string theories in dimensions 12, 11 and 10 of various signatures correspond to rewriting of the same OSp(1|32) algebra in different covariant ways. One only has to distinguish the complex and the unique real algebra. We develop a common framework to discuss all signatures theories by starting from the complex form of OSp(1|32). Theories are distinguished by the choice of basis for this algebra. We formulate dimensional reductions and dualities as changes of basis of the algebra. A second ingredient is the choice of a real form corresponding to a specific signature. The existence of the real form of the algebra selects preferred spacetime signatures. In particular, we show how the real d=10 IIA and IIB superalgebras for various signatures are related by generalized T-duality transformations that not only involve spacelike but also timelike directions. A third essential ingredient is that the translation generator in one theory plays the role of a central charge operator in the other theory. The identification of the translation generator in these algebras leads to the star algebras of Hull, which are characterized by the fact that the positive definite energy operator is not part of the translation generators. We apply our results to discuss different T-dual pictures of the D-instanton solution of Euclidean IIB supergravity.