On the geometric quantization of Jacobi manifolds

Abstract We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

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Generalized operator-valuedpq-symbols and geometric quantization

Functional Analysis and Its Applications ◽

10.1007/bf01080015 ◽

1995 ◽

Vol 29 (2) ◽

pp. 133-135 ◽

Cited By ~ 1

Author(s):

A. V. Karabegov

Keyword(s):

Geometric Quantization ◽

Generalized Operator

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On twisted contact groupoids and on integration of twisted Jacobi manifolds

Bulletin des Sciences Mathématiques ◽

10.1016/j.bulsci.2010.01.002 ◽

2010 ◽

Vol 134 (5) ◽

pp. 475-500 ◽

Cited By ~ 2

Author(s):

Fani Petalidou

Keyword(s):

Jacobi Manifolds

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Topological field theories on manifolds with Wu structures

Reviews in Mathematical Physics ◽

10.1142/s0129055x17500155 ◽

2017 ◽

Vol 29 (05) ◽

pp. 1750015 ◽

Cited By ~ 6

Author(s):

Samuel Monnier

Keyword(s):

Topological Field Theories ◽

Geometric Quantization ◽

Discrete Symmetry ◽

Local System ◽

Field Theories ◽

Simons Theory ◽

General Point ◽

Topological Field ◽

Generalized Cohomology ◽

Chern Simons

We construct invertible field theories generalizing abelian prequantum spin Chern–Simons theory to manifolds of dimension [Formula: see text] endowed with a Wu structure of degree [Formula: see text]. After analyzing the anomalies of a certain discrete symmetry, we gauge it, producing topological field theories whose path integral reduces to a finite sum, akin to Dijkgraaf–Witten theories. We take a general point of view where the Chern–Simons gauge group and its couplings are encoded in a local system of integral lattices. The Lagrangian of these theories has to be interpreted as a class in a generalized cohomology theory in order to obtain a gauge invariant action. We develop a computationally friendly cochain model for this generalized cohomology and use it in a detailed study of the properties of the Wu Chern–Simons action. In the 3-dimensional spin case, the latter provides a definition of the “fermionic correction” introduced recently in the literature on fermionic symmetry protected topological phases. In order to construct the state space of the gauged theories, we develop an analogue of geometric quantization for finite abelian groups endowed with a skew-symmetric pairing. The physical motivation for this work comes from the fact that in the [Formula: see text] case, the gauged 7-dimensional topological field theories constructed here are essentially the anomaly field theories of the 6-dimensional conformal field theories with [Formula: see text] supersymmetry, as will be discussed elsewhere.

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