E 9 exceptional field theory. Part II. The complete dynamics

Abstract

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends $$ {\mathfrak{e}}_9 $$ e 9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

A master exceptional field theory

“…In [58] we computed the explicit components of η IJ in GL(11) level decomposition that are K(E 11 )-invariant for the first seven gl(11) levels. Considering the e 9 subalgebra of e 11 as in [30] establishes the existence of η IJ on an infinite-dimensional submodule, including the first three e 9 levels.…”

“…In a different, but not unrelated, strand of research, field theories with extended spacetime symmetries and exceptional symmetry groups have been constructed. These so-called exceptional field theories (ExFT) [21][22][23][24][25][26][27][28][29][30] possess fields in a non-linear realisation of E n (for n ≤ 9) and these fields depend on extended (internal) coordinates Y M involving representations of E n , see also [31][32][33]. Exceptional field theories have proved to be very powerful tools in analysing Kaluza-Klein reductions in supergravity [31,[34][35][36][37][38][39][40][41][42][43][44][45][46].…”

“…7 • The topological term L top generalises the topological term of other ExFTs that does not depend on the external metric. It depends on the E 11 coset fields only through the e 11 current, without the explicit appearance of the generalised metric M. It is defined as a rigid E 11 -invariant completion of the total derivative of a constrained field transforming in an indecomposable representation together with the e 11 current, as does the topological term in E 9 ExFT [30].…”

“…In the list (1.3), we have not included cases where the Levi subgroup is infinitedimensional. One of those cases is L 2 = SL(2) × E 9 whose ExFT version has been constructed recently [30] and has served as an inspiration for the present paper. The other Kac-Moody group is L 1 = GL(1) × E 10 and we find that the same decomposition (1.4) of our model applies to D = 1 for E 10 , although the content of the field strength representation is more conjectural.…”

“…For this we take inspiration from E 9 exceptional field theory [30] and look for an E 11 -invariant completion of the derivative of the constrained field χ Mα , i.e., we start from…”