Gauging as constraining: the universal generalised geometry action in two dimensions

Abstract

One of the central concepts in modern theoretical physics, gauge symmetry, is typically realised by lifting a finite-dimensional global symmetry group of a given functional to an infinite-dimensional local one by extending the functional to include gauge fields. In this contribution we review the construction of gauged actions for two-dimensional sigma models, considering a more general notion to be gauged, namely that of a (possibly singular) foliation. In particular, the original action does not need to have any global symmetry for this purpose. Moreover, reformulating the ungauged theory by means of auxiliary 1-form fields taking values in the generalised tangent bundle over the target, all possible such gauge theories result from restriction of these fields to take values in (possibly small) Dirac structures. This turns all the remaining 1-form fields into gauge fields and leads to the presence of a local symmetry. We recall all needed mathematical notions, those of (higher) Lie algebroids, Courant algebroids, and Dirac structures.

Lie algebroid gauging of non-linear sigma models

“…An answer to the existence question is given for special cases in [5,7]. The results of this paper give a more complete answer: Corollary 6.1 states that for any choice of G and H there exist (ρ a , α a , ω a µb , φ a µb ) which satisfy the constraints (2.14) for some U ⊂ M .…”

“…In [5] a particular solution for φ and ω was found in the special case that (α * + ρ) −1 exists. 7 It follows from Theorem 4.1 that a family of solutions exist in this case, with the given choice being a particular solution. No statement was made about the closure of the gauge algebroid for this choice, nor the existence of solutions when (α * + ρ) −1 does not exist.…”

Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

“…where in the second line we used the Cartan relation 42) and in the last line we also used the constraint (2.37a), which may be written explicitly as…”

Momentum Sections in Hamiltonian Mechanics and Sigma Models

“…[ρ(e a ), ρ(e b )] = ρ([e a , e b ]), (50) b We can consider a more general ansatz of a gauge transformation as δA a = Dǫ a +[A, ǫ] a = dǫ a +C a bc A b ǫ c + ∆A a , where ∆A a is a 1-form taking a value on a pullback of E, which is linear with respect to the infinitesimal parameter ǫ a . [8,9,10] where ∨ is a symmetric product of 1-forms. Equation (49) is equivalent to E ∇g = 0.…”

“…Recently, a two-dimensional gauged sigma model with a two-form b-field with three dimensional Wess-Zumino term [14] is analyzed related to T-duality in string theory [4,5,7,8,9,10]. For such an application, it is interesting to generalize a momentum section in a pre-multisymplectic manifold.…”