2D gravitational Mabuchi action on Riemann surfaces with boundaries

Abstract

Abstract:We study the gravitational action induced by coupling two-dimensional nonconformal, massive matter to gravity on a Riemann surface with boundaries. A smallmass expansion gives back the Liouville action in the massless limit, while the first-order mass correction allows us to identify what should be the appropriate generalization of the Mabuchi action on a Riemann surface with boundaries. We provide a detailed study for the example of the cylinder. Contrary to the case of manifolds without boundary, we find that the gravitational Lagrangian explicitly depends on the space-point, via the geodesic distances to the boundaries, as well as on the modular parameter of the cylinder, through an elliptic θ-function.

Minisuperspace computation of the Mabuchi spectrum

“…The next step is to study the minisuperspace of the coupled Liouville-Mabuchi theory, and more particularly by taking into account the precise coefficients related to the matter content. To this aim, the gravitational action for a massive scalar field on a cylinder computed in [21] can be used.…”

Effective gravitational action for 2D massive fermions

“…The partition function of quantum gravity at fixed area, with a gravitational action being a combination of the Liouville and Mabuchi actions, has been studied at one loop in [7] and at two and three loops in [8]. The effective gravitational action 1 for such massive scalar fields including higher-order corrections in m 2 has been studied in [9], and on manifolds with boundaries in [10]. A rigourous mathematical construction of the functional integral based on the coupling of the Liouville and Mabuchi actions has since been obtained in ref [11] by means of probabilistic tools.…”

“…References [6,9,10] all studied the effective gravitational action of a two-dimensional massive scalar field, and a natural question to ask was what happens for two-dimensional massive fermions. As we will briefly discuss below, the most interesting case corresponds to a massive "Majorana" fermion.…”

“…Note that at present we call Sgrav[g, ĝ] what was called Sgrav[ĝ, g] in refs[6,[8][9][10] 2. This is a 2D fermion with a Majorana-type mass term.…”

BRST cohomology of timelike Liouville theory

“…Second, it directly generalizes to the case where the matter is not a CFT. In this case, the only known formulation is through the path integral, and the gravitational action is given by the Liouville action plus corrections which are not conformally invariant [7][8][9][10][11][12][13]. Third, most higher-dimensional theories of quantum gravity are defined from an action.…”