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 g leads naturally to the appearance of the “generalized tangent bundle” $$ \mathbb{T} $$ M≡TM⊕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 ⊂ $$ \mathbb{T} $$ 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 g of a standard sigma model with Wess-Zumino term exists, iff there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid M × $$ \mathfrak{g} $$ → M into D → 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.