Higher dimensional Lie algebroid sigma model with WZ term

Abstract

We generalize the (n + 1)-dimensional twisted R-Poisson topological sigma model with flux on a target Poisson manifold to a Lie algebroid. Analyzing consistency of constraints in the Hamiltonian formalism and the gauge symmetry in the Lagrangian formalism, geometric conditions of the target space to make the topological sigma model consistent are identified. The geometric condition is an universal compatibility condition of a Lie algebroid with the multi-symplectic structure. This condition is a generalization of the momentum map theory of a Lie group and is regarded as a generalization of the momentum section condition of the Lie algebroid.

Homotopy momentum sections on multisymplectic manifolds

“…The twisted R-Poisson structure is also regarded as a special case of the homotopy momentum section. Refer to [28] for a generalization of the twisted R-Poisson structure to a general Lie algebroid and a topological sigma model. ¶ An n-multivector field is denoted by J in this paper though it is denoted by R in [15].…”

Topological Dirac Sigma Models and the Classical Master Equation

“…In a similar spirit to the transition from 2D Poisson to R-Poisson structures in any dimension [24], one could generalize Dirac sigma models in higher than 2 dimensions. Some considerations in this direction are found in [25].…”

“…Such a situation typically arises in presence of Wess-Zumino terms, which present obstructions to QP-ness, as e.g. in the H-twisted Poisson sigma model [13] and higher dimensional generalizations thereof [23][24][25]. The second and more radical reason is that the Q manifold at hand might not even admit a natural symplectic structure, for example when it is not a (graded) cotangent bundle.…”

Compatible E-Differential Forms on Lie Algebroids Over (Pre-)Multisymplectic Manifolds

“…If and only if a multivector field on the target space is satisfied a geometric condition, the topological sigma model is consistent. A generalization to general Lie algebroid setting is formualted in the paper [24]. Consistency conditions of these sigma models imposes a geometric condition of an E-differential form with a Lie algebroid structure and a (pre-)(multi)symplectic form..…”