Vincent Rodgers | Searcharxiv

arXiv2026

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

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.

gr-qc↗

arXiv2020

General structure of Thomas$-$Whitehead gravity

Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity over a d-dimensional manifold that is intimately related to string theory through reparameterization invariance. Unparameterized geodesics are the ubiquitous structure that ties together string theory and higher dimensional gravitation. This is realized through the projective geometry of Tracy Thomas. The projective connection, due to Thomas and later Whitehead, admits a component that in one dimension is in one-to-one correspondence with the coadjoint elements of the Virasoro algebra. This component is called the diffeomorphism field $\mathcal{D}_{ab }$ in the literature. It also has been shown that in four dimensions, the TW\ action collapses to the Einstein-Hilbert action with cosmological constant when $\mathcal{D}_{ab}$ is proportional to the Einstein metric. These previous results have been restricted to either particular metrics, such as the Polyakov 2D\ metric, or were restricted to coordinates that were volume preserving. In this paper, we review TW gravity and derive the gauge invariant TW action that is explicitly projectively invariant and general coordinate invariant. We derive the covariant field equations for the TW action and show how fermionic fields couple to the gauge invariant theory. The independent fields are the metric tensor $g_{ab}$, the fundamental projective invariant $\Pi^{a}_{\,\,\,bc}$, and the diffeomorphism field $\mathcal D_{ab}$.

hep-th↗