Topological field theories inn-dimensional spacetimes and Cartan’s equations

Abstract

Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos' recent paper where Cartan's first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional spacetimes. Dirac's canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing additional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations. .unam.mx end, four different types of auxiliary fields are introduced. The first one φ I is a (n − 2)-form that will be associated to the basis of vielbeins e I ; the second one φ IJ = −φ JI is also a (n − 2)-form that will be associated to the Lorentz connection ω IJ . The other two auxiliary fields ψ I and ψ IJ = −ψ JI are (n − 3)-forms that will be associated to the "torsion" T I and the "curvature" R IJ , respectively. Note that R I J is not the same as R I J [ω]: R IJ is a set of two-forms while R I J [ω] is the curvature of ω I J (an analog comment applies to T I , see Ref.[3] for more details). In this way, for n-dimensional spacetimes M n with n ≥ 3, the field theories studied in this paper are defined by the action principlewhere ω IJ = −ω JI is a Lorentz (or Euclidean) connection valued in the so(n − 1, 1) or so(n) Lie algebra, e I is a basis of one-forms, T I is a set of n two-forms, R IJ = −R JI is a set of n(n − 1)/2 two-forms. The indices I, J, K, . . . , are raised and lowered with the Minkowski (σ = −1) or Euclidean (σ = +1) metric (η IJ ) = diag(σ, +1, +1, . . . , +1) (see Ref. [4] for the canonical analysis of BF theory with structure group SO(3, 1), Refs. [5,6] for alternative action principles for SO(3, 1) BF theory, and Ref.[7] for the study of its symmetries). The equations of motion that follow from the variation of the action (1) with respect to the independent fields arewhere D is the covariant derivative computed with respect to the connection ω I J . In two-dimensional spacetimes M 2 , on the other hand, only Cartan's first and second structure equations are allowed because there is no room for the first and second Bianchi identities, i.e., the terms involving the ψ's in Eq.(1) are not allowed. Consequently, the natural action principle is given bywhich are totally antisymmetric in the free indices. The constraints (16) behave as constraints (10) do for BF theory, i.e., repeatedly applying the operator D a to Eqs. (16)...

Poincaré–Plebański formulation of GR and dual simplicity constraints

“…The main assumption made about the frame field is that it is non-degenerate, and the matrix e A a is invertible. In the form of the action we may recognize the special case of the theory [43,44], designed to obey the Cartan's (also Bianchi's) structure equations (which are the kinematical basis for Riemannian geometry). One could anticipate and announce (4.4) as the Poincaré BF theory -this assertion is made precise, as we develop in this section the structure of the theory in full detail.…”

BF gravity with Immirzi parameter and matter fields

“…Just to mention some of them, Cartan's equations in the framework of BF theories are analyzed in Refs. [3,4] while the relationship of general relativity to the Husain-Kuchar model in the framework of BF theory is analyzed in Refs. [5][6][7].…”

Bi-gravity with a single graviton

“…it is not a wedge product of two 1forms), it is quite involved to define the inverse of the algebraic operator acting on the connection and find the general solution. Although in principle such general solution is known [10] (see also [33]), it is a non-linear expression in B IJ constructed using the Urbantke metrics defined from a general 2-form. This solution does not appear to significantly simplify in the special case (2.9) of our current interest, and we leave investigations of this approach to future work.…”