Covariant momentum map for non-Abelian topological BF field theory

Abstract

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.

Covariant canonical formulations of classical field theories

“…• Non-relativistic mechanics [5,7,53] • Relativistic mechanics [3] • Hydrodynamics [60,61] • Gauge field theories [3-5, 48, 53, 61, 62] • Spinor fields [63] • String theory [53,61] • Gravity [5,64,65] • Massive spin-2 field [65] • Topological field theories [53,66] • Korteweg-De Vries equation [67] • WZW conformal field theory [68] • Rarita-Schwinger field [63] • Supersymmetric sigma-models in two dimensions [69] • Fronsdal theory for massless fields of arbitrary integer spin [65] •…”

“…Here, we wish to describe how the expressions and results of Dirac's procedure for the standard Hamiltonian approach (in particular the FCC's and the gauge transformations that they generate, as discussed for instance in section 5.3 of reference [19]) are encoded in the multisymplectic approach to free Maxwell theory in four space-time dimensions. In this respect, we note that the general mathematical formulation of YM-theories has been addressed in the unpublished third part of the Gimmsy papers [53] and that the case of non-Abelian topological BF-theory in four space-time dimensions has been investigated quite recently in reference [66].…”

“…a foliation of the space-time manifold into space-like hypersurfaces [53,77]. To do so, we consider 3-dimensional hypersurfaces Σ t in space-time corresponding to fixed values of time t. Restriction of the covariant momentum map J defined by (2.17) to such a hypersurface amounts to the restriction σ of fields to Σ t : this yields [53,66] a map J t whose action on ξ writes…”

“…see equation (5.15) of reference [19]. In summary, the restriction of the covariant momentum map (of the multisymplectic formulation), that is determined by the invariance of the theory under local gauge transformations, yields upon a (3 + 1)-decomposition the generator of (Lagrangian) gauge transformations of the standard Hamiltonian approach -see references [53,66] for other examples and for the mathematical underpinnings. As a matter of fact, the extended Hamiltonian of Dirac's formulation (e.g.…”

A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity

“…Such canonical form, known as the polysymplectic form, encodes the relevant physical data of a given classical field theory in order to construct a well-defined Poisson-Gerstenhaber bracket for a set of appropriately prescribed differential Hamiltonian forms, thus allowing us to analyze an arbitrary classical field theory in a covariant Poisson-Hamiltonian framework [21][22][23]. Some physically motivated examples for which the multisymplectic and polysymplectic formalisms have been applied may be encountered in references [5,8,10,21,[24][25][26][27][28][29][30][31][32][33][34][35]. Despite their mathematical elegance, from our point of view, the analysis of the gauge content for a given classical field theory from the perspective of such geometric formulations has been rarely exploited, especially when considering certain highly non-trivial gauge models associated with general relativity.…”