2d Gauge Theories and Generalized Geometry

Abstract

We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra $\mathfrak{g}$ leads naturally to the appearance of the "generalized tangent bundle" $\mathbb{T}M\equiv TM\oplus T^{\ast }M$ by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure $D\subset \mathbb{T}M$ (or, more generally, the choide of a "small Dirac-Rinehart sheaf" $\mathcal{D}$), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: A gauging of $\mathfrak{g}$ of a standard sigma model with Wess-Zumino term exists, \emph{iff} there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid $M\times \mathfrak{g}\to M$ into $D\to M$ (or the algebraic analogue of the morphism in the case of $\mathcal{D}$). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.

22 pages, 2 figures; To appear in Journal of High Energy Physics

References in corpus (14)

• Two-Dimensional Gravity and Nonlinear Gauge Theory
• Poisson Structure Induced (Topological) Field Theories
• WZW-Poisson manifolds
• Current Algebras and Differential Geometry
• Generalized Kahler manifolds and off-shell supersymmetry
• L-infinity algebras from multisymplectic geometry
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Topological Open P-Branes
• Lectures on Generalized Complex Geometry and Supersymmetry
• Dirac Sigma Models
• Gauging without Initial Symmetry
• Equivariant Cohomology and Gauged Bosonic sigma-Models
• Characteristic classes associated to Q-bundles
• Dirac Sigma Models from Gauging

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• Beyond the standard gauging: gauge symmetries of Dirac Sigma Models
• T-duality without isometry via extended gauge symmetries of 2D sigma models
• Strings in Singular Space-Times and their Universal Gauge Theory
• On the relation of Lie algebroids to constrained systems and their BV/BFV formulation
• Topological Dirac Sigma Models and the Classical Master Equation
• Graded geometry in gauge theories and beyond