TOPOLOGICAL OPEN P-BRANES

Abstract

By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3-form C-field leads to deformations of the algebras of multi-vectors on the Dirichlet-brane world-volume as 2-algebras. This would shed some new light on geometry of M-theory 5-brane and associated decoupled theories. We show that, in general, topological open p-brane has a structure of (p + 1)-algebra in the bulk, while a structure of p-algebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of p-algebras. It also imply that the algebras of quantum observables of (p − 1)-brane are "close to" the algebras of its classical observables as p-algebras. We interpret above as deformation quantization of (p − 1)-brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to topological strings and conjecture that the homological mirror symmetry has further generalizations to the categories of p-algebras.

On dual pairs in Dirac geometry

“…Courant algebroids may be twisted by closed three-forms. Specifically, let φ ∈ Ω 3 (M ) be closed, and consider the φ-twisted Dorfman bracket Twisted Dirac structures go back, in one form or another, to [27,32,34,35]. A crucial example of such structures is given by Cartan-Dirac structures associated to nondegenerate, invariant inner products on the Lie algebra g of a Lie group G [35, Example 4.2], and whose presymplectic realizations correspond to the quasiHamiltonian g-spaces of [1] (see [10]).…”

A minimal BV action for Vasiliev’s four-dimensional higher spin gravity

“…therein); interestingly, the BF model without Poisson structure on a non-commutative manifold was studied in [20,21]. Further developments of the AKSZ formalism can be found in [22,23,24] and [25,26,27,28,29,30,31,32,33], and its close ties to unfolded dynamics have been stressed in [34,35,36,37,38,39]. For related treatments of more general dynamical systems, not necessarily based on differential algebras, see [40,41] and references therein.…”

“…In all types of generalized Poisson sigma models, whether on commutative or non-commutative base manifolds, the physical degrees of freedom are contained in boundary vertex operators [12,23]. The boundary lives in a graded target-space manifold equipped with a nilpotent vector field of degree 1, referred to as the Q-structure, and compatible poly-vector fields of suitable degrees depending on the dimension of the base manifold, whose mutual Schouten brackets vanish, thus defining a generalized Poisson structure referred to as a QP -structure in the bi-vector case 2 ; see [42] and refs.…”

AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories

“…Cattaneo-Felder [4] and Park [8] further refined the AKSZ procedure by generalizing it to the case of manifolds with boundary, and produced new examples: Cattaneo and Felder studied the Poisson sigma-model [11] on the disk [3] [4], while Park considered its higher-dimensional generalizations, the topological open p-branes.…”