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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.
Highlights
TM = T M ⊕ T ∗M is the vector bundle sum of the tangent and the cotangent bundle, which sometimes is called the “generalized tangent bundle”
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” TM ≡ T M ⊕ T ∗M by means of composite fields
The gauge theory takes the form of a 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 × g → M into D → M
Summary
The standard sigma model in defined as a functional of smooth maps X : Σ → M where (Σ, h) is some oriented Lorentzian signature pseudo-Riemannian d-manifold and (M, g) is a Riemannian or pseudo-Riemannian n-manifold. We can extend the above action by adding the pullback of a 2-form B on M assumed to be part of the given data on the target: S[X] =. The action (2.2) becomes invariant under the rigid symmetry group G. the action (2.2) becomes invariant under the rigid symmetry group G In formulas this implies that for any such a vector field v on M , an infinitesimal change of the fields induced by δXi := X∗vi leaves the action (2.2) invariant (up to a boundary term ∂Σ X∗β, that is usually considered irrelevant at this point). For H = dB and α = β − ιvB this reproduces the previous situation (the boundary of Σis assumed to be Σ and Xrestricts to X on the boundary by assumption), but the variational problem is well-defined more generally for H a closed d + 1-form.
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