Algebraic structure of gravity with torsion

Abstract

The BRS transformations for gravity with torsion are discussed by using the Maurer-Cartan horizontality conditions. With the help of an operator δ which allows to decompose the exterior space-time derivative as a BRS commutator we solve the Wess-Zumino consistency condition corresponding to invariant Lagrangians and anomalies. * Supported in part by the "Fonds zur Förderung der wissenschaftlichen Forschung" under Grant No. P9116-PHY.† Supported in part by the "Fonds zur Förderung der wissenschaftlichen Forschung", M008-Lise Meitner Fellowship.

Ghost equations and diffeomorphism-invariant theories

“…And, remarkably, it is linked to our vector supersymmetry, hence to the diffeomorphism ghost equation, in the following way. Considering the supersymmetry transformation rules (3.4) for a constant 13 vector field ε, we define the action of the operator δ as given by these transformations, with ε µ replaced by the differential dx µ : 14) which is the result of [15,16] -up to the action on the Lagrange multiplier fields B i , not considered there. We can easily check the commutation rule (3.13), and also that δ commutes with d. Note that, as in [16], we can write the first of eqs.…”

BRST Cohomology of the Superstring in Super-Beltrami Parametrization

“…The operator δ can be used to solve the descent equations (1.2). As it has been shown in [17,18,19] these solutions can be obtained from the equation…”

Algebraic structure of gravity in Ashtekar’s variables

“…The tetrad 1-form is related to the Ashtekar torsion 2-form by 15) where D = d + A is the Ashtekar covariant exterior derivative. This Ashtekar torsion 2form does not vanish even though the spin-connection ω ab , related to A ab by the equation…”

“…Besides, the Russian formula, in the form given for the Yang-Mills case [5], does not hold and we have to find out a new method for obtaining a generalization of it in this case. On the other hand, for the Ashtekar variables, as well as for the gravity with torsion [15] it is difficult to write down a transgression equation and to obtain the anomalies, the Schwinger terms and the invariant Lagrangians. All these quantities are BRST-invariants modulo d-exact terms and they can be obtained by using an operator δ which allows to express the exterior derivative d as a BRST commutator d = −[s, δ] .…”