2d Gauge Theories and Generalized Geometry

TL;DR

This paper demonstrates that in 2D sigma models, gauge symmetries naturally lead to generalized geometry structures, specifically Dirac structures, and introduces a universal Dirac sigma model framework for gauging symmetries.

Contribution

It reveals the natural emergence of generalized tangent bundles and Dirac structures in 2D gauge theories and establishes a universal Dirac sigma model formulation for gauging symmetries.

Findings

Gauge transformations follow the Courant bracket.

Gauging exists iff symmetry extends to a Lie algebroid morphism.

Gauged models derive from a pullback of the Dirac sigma model.

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" $\mathbb{T}M \equiv TM \oplus T^*M$ 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" $\cal{D}$), 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 $\cal{D}$). 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.