Constraint Analysis and Quantization of Anomalous 2-D Thomas-Whitehead Gravity

Abstract

The two-dimensional effective Polyakov action is often realized as the anomalous contributions of string theories and fermions coupled to gravity in two dimensions. However, as a result of the reparametrization invariance, one finds that the effective action produces vanishing Hamiltonians as constraints even in disparate gauges such as the dynamical light-cone and the Arnowitt-Deser-Misner (ADM) formalism of the metric. On the other hand, two-dimensional gravitational theories naturally arise as geometric actions on the coadjoint orbits of the Virasoro algebra. The Thomas-Whitehead gravity formalism extends the effective Polyakov action in such a way that the defining coadjoint element for the orbit becomes a dynamical field, viz. the diffeomorphism field. In this work, we examine the role the diffeomorphism field plays through the well-understood quantization of the two-dimensional anomalous contributions to gravity. This is first done in the dynamical light-cone and then with the ADM formalism of the metric. To examine a dynamical diffeomorphism field, the constraint analysis is then repeated in a Minkowski background, where the dynamics of the diffeomorphism field arise from the Thomas-Whitehead action. Adding dynamics to the diffeomorphism field appears to remove the vanishing Hamiltonians; however, expressing the diffeomorphism in terms of the P-tensor recovers the Hamiltonian constraint. One can compare this investigation to that of a gauge Wess-Zumino-Witten action where the gauge field has become dynamical through the inclusion of a Yang-Mills term.