Gauge-unfixed Hamiltonian Casimir and static torsionful sector in the Katanaev–Volovich model

Abstract

We give a gauge-unfixed Hamiltonian analysis of the first-order Katanaev–Volovich model of two-dimensional gravity with torsion. The first-order action already singles out natural canonical pairs: the auxiliary Lorentz scalars are conjugate to the spatial connection and zweibein. An extended Dirac–Bergmann embedding isolates an auxiliary second-class sector whose elimination recovers the canonical structure encoded in the first-order action and leaves the first-class constraints generating the local gauge symmetries. An independent Faddeev–Jackiw reduction yields the same reduced brackets. The Katanaev/Poisson–Sigma Casimir is then recovered, up to normalization, directly from the reduced Dirac–Bergmann first-class constraint ideal, before imposing any gauge condition. This identifies the Casimir as a global label of the reduced canonical sectors and supplies a convenient parameter for the torsionful branch. The independent constant normalization of the Killing time is displayed explicitly and then fixed to the local convention \(A=1\). In this convention the Casimir-normalized radial field does not coincide with the metric Killing norm: the diagonal representative satisfies \(NB=e^{4\beta r}\). Consequently, torsion modifies the Killing temperature in the chosen Killing-time normalization, while the horizon entropy retains its standard two-dimensional dilaton value and the Casimir-normalized first law holds.