Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

Ikeda, Noriaki; Xu, Xiaomeng

doi:10.1063/1.4900834

Cited by 13 publications

(14 citation statements)

References 56 publications

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“…The underlying structure of the target space in that case was found to be that of a Courant algebroid twisted by a 4-form. Moreover, a more general approach to topological field theories with Wess-Zumino term in the framework of a generalization of the AKSZ construction was presented in [22].…”

Section: Jhep09(2021)045mentioning

confidence: 99%

Topological field theories induced by twisted R-Poisson structure in any dimension

Chatzistavrakidis

2021

J. High Energ. Phys.

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We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.

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Section: Jhep09(2021)045mentioning

confidence: 99%

Topological field theories induced by twisted R-Poisson structure in any dimension

Chatzistavrakidis

2021

J. High Energ. Phys.

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“…Let us define an exponential adjoint operation e δα on a general QP-manifold M, 124) where α ∈ C ∞ (M). A function α with the property defined in Proposition 10.4 is called a Poisson function [141,97] or a canonical function [82]. The structures for general n have been analyzed in Ref.…”

Section: Canonical Transformation Of Q-structure Functionmentioning

confidence: 99%

Lectures on AKSZ Sigma Models for Physicists

Ikeda

2017

Noncommutative Geometry and Physics 4

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This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analysis of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional topological field theory and explain the ideas of the AKSZ sigma models. This construction is then generalized and leads to a mathematical formulation of a general topological sigma model. We review the mathematical objects, such as algebroids and supergeometry, that are used in the analysis of general gauge structures. The quantization of the Poisson sigma model is presented as an example of a quantization of an AKSZ sigma model.

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“…The condition is expressed on M as follows. [28,44] Let α ∈ C ∞ (M) of degree n be a canonical function with respect to L, i.e., e δα Θ| L = 0.…”

Section: Aksz Sigma Modelsmentioning

confidence: 99%

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

Bessho

Heller

Ikeda

et al. 2016

J. High Energ. Phys.

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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

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Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

Cited by 13 publications

References 56 publications

Topological field theories induced by twisted R-Poisson structure in any dimension

Topological field theories induced by twisted R-Poisson structure in any dimension

Lectures on AKSZ Sigma Models for Physicists

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

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