On 2d Gauged Supergravity

Abstract

Supergravity theories are gravitational field theories with local supersymmetry. The highest dimensional supergravity theory is 11d supergravity, from which one obtains lower-dimensional ungauged maximal supergravity theories by dimensional reduction. The d dimensional ungauged theory has global e_11−d symmetry and local k(e_11−d) symmetry, where k(e_11−d) is the maximal compact subalgebra of e_11−d. These symmetries become infinite-dimensional in the reduction to 2d supergravity.

Other maximal supergravity theories in d dimensions can be obtained as gauge deformations of the ungauged theory, where a subalgebra of the global symmetry is promoted to a local symmetry. While the supergravity theories in dimensions above two were gauged, it remains open to gauge 2d supergravity. For the bosonic sector of 2d gauged supergravity, important results are in [3, 4]. However, all fermionic contributions, the supersymmetry variations, and the full scalar potential remained unknown. The main reason is that the bosons transform in e9 representations, while the fermions transform under k(e₉). To understand supersymmetry, we must understand how the e9 representations of the bosons ‘decompose’ under k(e₉).

In this thesis, we obtain the ‘decomposition’ of the e_g basic representation in k(e₉) representations. This result allows us to construct the fermionic sector and the supersymmetry variation of maximal 2d gauged supergravity. These are the main results of the thesis.

More generally, for every split real simply-laced affine Kac-Moody algebra with maximal compact subalgebra k, we provide infinitely many k-subrepresentations of the basic representation. For this infinite set of subrepresentations we prove a completeness statement.

These results allow for further extensions: The representation theoretical result is a starting point for understanding the ‘decomposition’ of further highest weight representations of affine algebras. Ultimately, this allows for studying the coset of a hyperbolic Kac-Moody algebra by its maximal compact subalgebra having important applications in physical models of emergent space. The results on 2d supergravity allow the determination of the scalar potential by supersymmetry, and this way, to obtain the complete 2d gauged supergravity theory. The theory also serves as a toy model of emergent space, and a particular 2d gauged supergravity theory allows the study of the dWHN-BFSS matrix model by holographic methods.