Abstract
Abstract The $T$-duality of string theory can be extended to the Poisson–Lie $T$-duality when the target space has a generalized isometry group given by a Drinfel’d double. In M-theory, $T$-duality is understood as a subgroup of $U$-duality, but the non-Abelian extension of $U$-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal E_n$ algebra. This provides a natural setup to study non-Abelian $U$-duality because the $\mathcal E_n$ algebra has been proposed as a $U$-duality extension of the Drinfel’d double. We show that the standard treatment of Abelian $U$-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian $U$-duality still exists in the non-Abelian extension.
Highlights
Abelian T -duality is a symmetry of string theory when the target space has D commuting Killing vector fields
By requiring the target space to have a symmetry of the En algebra, we show that the generalized displacement satisfies the equation dP A
Similar to the case of the Drinfel’d double, the generalized frame fields EAI satisfy the relation (3.12) by means of the generalized Lie derivative in exceptional field theory (EFT), and we can show that the target geometry has the generalized isometry group, which is generated by the En algebra
Summary
Abelian T -duality is a symmetry of string theory when the target space has D commuting Killing vector fields. The MC equation does not depend on the choice of the generators TA in the En algebra, and it is manifestly covariant under the change of generators TA′ = CAB TB , where the constant matrix CAB is an element of the U -duality group E3 ≡ SL(2) × SL(3) This arbitrariness in the choice of generators is what we call non-Abelian U -duality. Our result is a natural non-Abelian extension of the standard Abelian U -duality, and the fact that the generalized displacement PA satisfies the MC equation is non-trivial.
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