First Class Constrained Systems and Twisting of Courant Algebroids by a Closed 4-Form

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the corresponding operations in DFT. This describes precisely how the geometric structure of DFT lies in between two Courant algebroids and is reconciled with generalized geometry. We construct the membrane sigma-model that corresponds to DFT, and demonstrate how the standard T-duality orbit of geometric and non-geometric flux backgrounds is captured by its action functional in a unified way. This also clarifies the appearence of noncommutative and nonassociative deformations of geometry in non-geometric closed string theory. Gauge invariance of the DFT membrane sigma-model is compatible with the flux formulation of DFT and its strong constraint, whose geometric origin is explained. Our approach leads to a new generalization of a Courant algebroid, that we call a DFT algebroid and relate to other known generalizations, such as pre-Courant algebroids and symplectic nearly Lie 2-algebroids. We also describe the construction of a gauge-invariant doubled membrane sigma-model that does not require imposing the strong constraint. 1 a.chatzistavrakidis@gmail.com 2 larisa@irb.hr 3 Fech.Scen.Khoo@irb.hr 4 R.J.Szabo@hw.ac.uk 29)19 In this subsection we simplify the notation for the components of the map ρ + by denoting (ρ + ) I J as ρ I J .

Section: Summary Of Results and Outlinementioning

confidence: 98%

Section: Summary Of Results and Outlinementioning

confidence: 99%

Section: Gauge Invariance and Bianchi Identitiesmentioning

confidence: 99%

“…[26,38]. More notably, in [64] where sigmamodels were used to derive a definition of a Courant algebroid twisted by a closed 4-form, the fiber metric is also dynamical.…”

Section: Sigma-models With Dynamical Fiber Metricmentioning

confidence: 99%

“…In[64] this 4-form is denoted by H. Here we use a different notation in order to avoid confusion with the NS-NS 3-form flux H.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Double field theory and membrane sigma-models

Chatzistavrakidis

Jonke

Khoo

et al. 2018

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

Bessho

Heller

Ikeda

et al. 2016

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

Geometric BV for twisted Courant sigma models and the BRST power finesse

Chatzistavrakidis,

Ikeda,

Jonke

2024

We study twisted Courant sigma models, a class of topological field theories arising from the coupling of 3D 0-/2-form BF theory and Chern-Simons theory and containing a 4-form Wess-Zumino term. They are examples of theories featuring a nonlinearly open gauge algebra, where products of field equations appear in the commutator of gauge transformations, and they are reducible gauge systems. We determine the solution to the master equation using a technique, the BRST power finesse, that combines aspects of the AKSZ construction (which applies to the untwisted model) and the general BV-BRST formalism. This allows for a geometric interpretation of the BV coefficients in the interaction terms of the master action in terms of an induced generalised connection on a 4-form twisted (pre-)Courant algebroid, its Gualtieri torsion and the basic curvature tensor. It also produces a frame independent formulation of the model. We show, moreover, that the gauge fixed action is the sum of the classical one and a BRST commutator, as expected from a Schwarz type topological field theory.