Jacobi sigma models

Francesco Bascone,Franco Pezzella,Patrizia Vitale

doi:10.1007/jhep03(2021)110

Abstract

We introduce a two-dimensional sigma model associated with a Jacobi manifold. The model is a generalisation of a Poisson sigma model providing a topological open string theory. In the Hamiltonian approach first class constraints are derived, which generate gauge invariance of the model under diffeomorphisms. The reduced phase space is finite-dimensional. By introducing a metric tensor on the target, a non-topological sigma model is obtained, yielding a Polyakov action with metric and B-field, whose target space is a Jacobi manifold.

Highlights

Path integral quantisation of the model furnishes a field-theoretical proof of Kontsevich star product quantisation of Poisson manifolds [4, 5]
On using a consistent definition of Hamiltonian vector fields for Jacobi manifolds, we show that the latter can be associated with gauge transformations and verify that they close under Lie bracket, generating space-time diffeomorphisms
We started from the concept of Poissonization of a Jacobi manifold, which consists in the construction of a homogeneous Poisson structure on the extended manifold M ×R from a Jacobi structure on M

Summary

Poisson sigma models

Let (M, Π) be a Poisson manifold, where Π ∈ Γ(∧2T M ) is a Poisson structure on the smooth m-dimensional manifold M , and Σ a 2-dimensional orientable smooth manifold, eventually with boundary. By indicating with f (u)∂u a generic space diffeomorphism, it is immediate to check that this is the generator of an infinitesimal symmetry for the model, it being the Hamiltonian vector field associated with Hβ, for β = f (u)ζ. This is a direct consequence of the invariance of the action under the exchange X ↔ Xand β ↔ −ζ. The model is invariant under space-time diffeomorphisms and the reduced phase space can be defined as G = C/Diff(Σ) It can be proven [3, 4] that the latter is a finite-dimensional, closed subspace of phase space, of dimension 2dim(M ), with a natural groupoid structure. We will project the obtained dynamics on the underlying Jacobi manifold and propose a consistent model, directly defined on the Jacobi manifold, whose dynamics is proven to coincide with the projected one

Jacobi brackets and Jacobi manifold

Action principle on the Jacobi manifold

Metric extension and Polyakov action

Findings

Conclusions and outlook