Topological membranes, current algebras and H-flux-R-flux duality based on Courant algebroids

Abstract

We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold. Recently, there are further developments related to T-duality. Double field theory [4] is a manifestly O(d, d) covariant field theory which allows also for T-duality along non-isometry directions. Examples for other developments are the branes as sources for Q-and R-fluxes [5, 6] and the β-supergravity [7]. The topological T-duality [8, 9] is also proposed to analyze T-duality with flux. However, the background geometric structures for nongeometric fluxes are not well understood. A background geometry in string theory with NS H-flux [10] is known to be a Courant algebroid [11, 12], and the standard Courant algebroid of the generalized tangent bundle T M ⊕ T * M is of particular interest in the framework of generalized geometry [13, 14]. The T-duality on the H-flux is well understood as an automorphism on the standard Courant algebroid if ι X ι Y H = 0 [15]. However, we cannot simultaneously introduce all degrees of freedom of H-, F -, Q-, R-fluxes as deformation of the Courant algebroid. The only independent deformation in the exact Courant algebroid is a 3-form (H-flux) degree of freedom [16]. Recently, the Courant algebroid on a Poisson manifold, i.e. the Poisson Courant algebroid, has been introduced in [17] as a geometric object for a background with R-flux. It is shown that the nontrivial flux R of a 3-vector can be introduced consistently on a Poisson manifold as a deformation of the Courant algebroid. It is the 'contravariant object' [18] with respect to the standard Courant algebroid, which is the exchange of T * M with T M and H-flux with R-flux. The T-duality on the R-flux has also been analyzed and it has been shown that the duality of R-flux with Q-flux is also understood as an automorphism on the Poisson Courant algebroid [19].In this paper, we analyze the geometric structure of the Poisson Courant algebroid and a

Double Field Theory for the A/B‐Models and Topological S‐Duality in Generalized Geometry

“…The contravariant Courant sigma‐model was introduced in [] as the Courant sigma‐model corresponding to a Poisson Courant algebroid. It is defined by the AKSZ action In the absence of R ‐flux the master equation gives the Poisson condition for the bivector π, so we can expect that the contravariant Courant sigma‐model is closely related to the Poisson sigma‐model.…”

“…Based on our observation that the Poisson sigma‐model on doubled spaces captures both the A‐ and B‐models with different choices of the doubled Poisson structure, we propose an open AKSZ membrane sigma‐model, inspired by the approach of [] to T‐duality between geometric and non‐geometric fluxes, which gives back the doubled Poisson sigma‐model on the boundary in a specific gauge. Then we reduce the fields in a way which can be interpreted as the same reduction performed in [], where it was called a DFT projection.…”

“…In Figure we summarize the relations between the different AKSZ string and membrane sigma‐models appearing in this paper. In the following we also discuss some additional new results: We introduce a gauge fixing method which is a natural way to perform boundary reductions, and we show that the contravariant Courant sigma‐model of [] reduces to the Poisson sigma‐model in this gauge; we also relate its twisting by (geometric) R ‐flux to the membrane sigma‐model introduced in [] which describes the nonassociative deformations of (locally non‐geometric) R ‐flux backgrounds. We further show that the standard Courant sigma‐model twisted by H ‐flux is related to the B‐model in a particular gauge.…”

Higher gauge theories from Lie n-algebras and off-shell covariantization

“…Since the generator of the canonical transformation α is of degree n, it is degree-preserving. For details on the conventions related to QP-manifolds and graded differential calculus, we refer to [27].…”

Courant Sigma Model and ‐algebras

“…2 This Courant sigma model belongs to a general class of topological sigma models of AKSZ type [18] satisfying the classical master equation. The membrane sigma models were subsequently used for a systematic description of closed strings in non-geometric flux backgrounds [19][20][21][22][23]. In particular, the expression for the fluxes and their Bianchi identities coincide with the local form of the axioms of a Courant algebroid.…”