On the symmetries of BF models and their relation with gravity

Abstract

The perturbative finiteness of various topological models (e.g. BF models) has its origin in an extra symmetry of the gauge-fixed action, the so-called vector supersymmetry. Since an invariance of this type also exists for gravity and since gravity is closely related to certain BF models, vector supersymmetry should also be useful for tackling various aspects of quantum gravity. With this motivation and goal in mind, we first extend vector supersymmetry of BF models to generic manifolds by incorporating it into the BRST symmetry within the Batalin-Vilkovisky framework. Thereafter, we address the relationship between gravity and BF models, in particular for three-dimensional space-time.

Symmetries and observables in topological gravity

“…The action of the operator e i ξ can be factorized in all terms of equations (2.6) and (2.8) by virtue of the following operatorial relation [31,33] which results from equations (2.4):…”

“…Degenerate supervielbeins may well appear in a general gauge. This situation is somewhat reminiscent of the fact that the invertibility problem does not manifest itself in 3-dimensional quantum gravity if the latter is expressed as a topological theory of Chern-Simons or of BF type [40,33].…”

Covariant canonical formulations of classical field theories

“…We recall that the BV approach [18,39,[183][184][185][186][187] amounts to a symplectic-type reformulation and generalization of the usual BRST quantization procedure [188,189] for which the interplay between Lagrangian symmetries and equations of motion plays a central role. In the following, we outline some of its basic points for the case of an irreducible gauge theory [18] while taking pure YM-theory in four-dimensional Minkowski space-time as an explicit example.…”

“…All of our functional derivatives are left derivatives[185], but we note that the BV bracket is traditionally expressed in terms of both left and right derivatives which implies the presence of other sign factors.…”

Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity

“…Three-dimensional gravity is a special case of the topological BF theory [25,72] which can be defined in arbitrary dimensions. Though gravity and BF theory are not equivalent in dimension four and higher, since they differ by the presence and the lack of local degrees of freedom, it turns out that they are still related [41,58,60,72]. One of these relations consists in formulating gravity as a BF model supplemented with constraints, and this is the main road to the spin foam quantization attempts to quantum gravity.…”