Currents, charges and algebras in exceptional generalised geometry

Abstract

A classical Ed(d)-invariant Hamiltonian formulation of world-volume theories of half-BPS p-branes in type IIb and eleven-dimensional supergravity is proposed, extending known results to d ≤ 6. It consists of a Hamiltonian, characterised by a generalised metric, and a current algebra constructed s.t. it reproduces the Ed(d) generalised Lie derivative. Ed(d)-covariance necessitates the introduction of so-called charges, specifying the type of p-brane and the choice of section. For p > 2, currents of p-branes are generically non- geometric due to the imposition of U-duality, e.g. the M5-currents contain coordinates associated to the M2-momentum.A derivation of the Ed(d)-invariant current algebra from a canonical Poisson structure is in general not possible. At most, one can derive a current algebra associated to para-Hermitian exceptional geometry.The membrane in the SL(5)-theory is studied in detail. It is shown that in a generalised frame the current algebra is twisted by the generalised fluxes. As a consistency check, the double dimensional reduction from membranes in M-theory to strings in type IIa string theory is performed. Many features generalise to p-branes in SL(p + 3) generalised geometries that form building blocks for the Ed(d)-invariant currents.

The canonical formulation of E6(6) exceptional field theory

“…We should note that the time evolution of the one-forms (5.27) is most concisely written in terms of the modified momenta P n N (A) which may be seen as another argument in favour of the modified momenta. While the time evolution of the fields in E 6 (6) ExFT agrees in form with that of fivedimensional E 6(6) invariant supergravity [14] the time evolution of the canonical momenta is significantly more complicated in ExFT. Part of the complexity is expected because the analogous transformations are already relatively complicated in five-dimensional supergravity but the scalar potential and covariant derivatives lead to many additional terms.…”

“…In this work we focus on the E 6(6) exceptional field theory, which is formulated on an extended 5 + 27 dimensional extended geometry [3,11]. The E 6 (6) ExFT is, in a sense, the simplest case of the true exceptional groups n = 6, 7, 8, because there are no self-dual forms in the five external dimensions (unlike in the E 7 (7) ExFT where one has to consider a pseudo-action with an additional self-duality relation) and there are no constrained compensator fields (unlike the E 8(8) ExFT). For simplicity we furthermore focus on the bosonic sector of the E 6(6) ExFT as described in [3,11].…”

“…Some aspects of exceptional geometry, across different exceptional groups, can be phrased in terms of an object called the Y-tensor [4]. The Y-tensor of E 6 (6) , which follows from the projector (2.8), is given by (2.9).…”

“…So far the canonical formulation of exceptional field theory has not been investigated. In [5] the canonical formulation of O(n, n) double field theory has been discussed and very recently E n(n) covariant world-volume theories have been analysed canonically in [6].…”

Brane current algebras and generalised geometry from QP manifolds. Or, “when they go high, we go low”

“…We emphasise that in the QP constructions in this paper there appear no "extended coordinates" corresponding to brane winding modes, because there appears to be some tension between Q 2 = 0 and extended coordinates [16][17][18]. Works including extended coordinates in the hamiltonian formalism on the brane include Sakatani and Uehara [19,20], Linch and Siegel [21,22] and the very recent paper of Osten [23] that focusses on the realisation of exceptional generalised geometry on brane currents. Another perspective complementary to ours is by Strickland-Constable [24], where the brane equations of motion on phase space are interpreted as geodesic equations for (exceptional) generalised geometry (see also [25,26]).…”