Courant-like brackets and loop spaces

Abstract

We study the algebra of local functionals equipped with a Poisson bracket. We discuss the underlying algebraic structures related to a version of the Courant-Dorfman algebra. As a main illustration, we consider the functionals over the cotangent bundle of the superloop space over a smooth manifold. We present a number of examples of the Courant-like brackets arising from this analysis.Comment: 20 pages, the version published in JHE

p-brane actions and higher Roytenberg brackets

“…It was observed by Alekseev and Strobl in [13], that the current algebra of sigma models naturally involves the structures of generalized geometry [14,15], such as Dorfman bracket and Dirac structures. This was further developed by Ekstrand and Zabzine in [16] and Bonelli and Zabzine in [17]. Recently, D-branes have been identified with Dirac structures [18].…”

“…In fact, we shall consider more general charges, involving background fields Π and B. This can be done in a straightforward manner; however it is easier to use the results of [16]:…”

On the generalized geometry origin of noncommutative gauge theory

“…In [12], it has been observed that the current algebra of sigma models naturally involves structures of generalized geometry, such as the Dorfman bracket and Dirac structures. This was further developed in [13] and [14]. In [15], it was observed that in the first order (nontopological) Poisson sigma model characterized by a 2-form B and a bivector θ, a more general form of world-sheet currents appears.…”

Extended generalized geometry and a DBI-type effective action for branes ending on branes

“…To express this matrix in open variables we introduce the following notation: (57,58), respectively, into the picture. The determinant of the left-hand side of (61) seems to be a likely candidate to appear in the "commutative" membrane DBI action, whereas the determinant on the right-hand side of (61) seems to contain as a factor a likely candidate to appear in its "noncommutative" counterpart.…”

“…Generalized geometry (mostly Courant algebroid brackets) was also used in relation to worldsheet algebras and nongeometric backgrounds. See, for example, [56][57][58] and [59,60]. One should also mention the use of generalized geometry in the description of T-duality, see [61], or the lecture notes [62].…”