Frobenius-Chern-Simons gauge theory

TL;DR

This paper develops a broad class of covariant gauge theories on noncommutative manifolds, generalizing the Frobenius-Chern-Simons model to include new algebraic structures and applications in higher spin gravity.

Contribution

It introduces a general framework for cubic covariant Hamiltonian actions controlled by Z_2-graded associative algebras, extending the Frobenius-Chern-Simons model with novel algebraic and geometric features.

Findings

Constructed a class of models generalizing FCS with new algebraic structures.

Presented a model based on twisting C[Z_2 x Z_4] leading to self-dual gauge fields on AdS_4.

Demonstrated consistent truncations for 3-graded FCS models, including connections to Vasiliev's higher spin theories.

Abstract

Given a set of differential forms on an odd-dimensional noncommutative manifold valued in an internal associative algebra H, we show that the most general cubic covariant Hamiltonian action, without mass terms, is controlled by an Z_2-graded associative algebra F with a graded symmetric nondegenerate bilinear form. The resulting class of models provide a natural generalization of the Frobenius-Chern-Simons model (FCS) that was proposed in arXiv:1505.04957 as an off-shell formulation of the minimal bosonic four-dimensional higher spin gravity theory. If F is unital and the Z_2-grading is induced from a Klein operator that is outer to a proper Frobenius subalgebra, then the action can be written on a form akin to topological open string field theory in terms of a superconnection valued in the direct product of H and F. We give a new model of this type based on a twisting of C[Z_2 x Z_4], which leads to self-dual complexified gauge fields on AdS_4. If F is 3-graded, the FCS model can be truncated consistently as to zero-form constraints on-shell. Two examples thereof are a twisting of C[(Z_2)^3] that yields the original model, and the Clifford algebra Cl_2n which provides an FCS formulation of the bosonic Konstein--Vasiliev model with gauge algebra hu(4^{n-1},0).