Gravitational actions in two dimensions and the Mabuchi functional

Abstract

The Mabuchi energy is an interesting geometric functional on the space of Kähler metrics that plays a crucial rôle in the study of the geometry of Kähler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.October 26, 2018 † On leave of absence from ITEP, Moscow, Russia.

Multi-loop zeta function regularization and spectral cutoff in curved spacetime

“…Similar transformation laws were studied in [7] in two dimensions and the above relations can be proved by using similar methods.…”

“…Recently, a non-perturbative definition of the path integral over the Kähler metrics on a compact complex manifold of arbitrary dimension was proposed in [6] (see also [7,8] for related works). The main ingredient in the construction of the path integral is to approximate the infinite dimensional space of Kähler metrics M at fixed Kähler class by finite dimensional subspaces M n of so-called Bergman metrics.…”

“…[7] and references therein. The scale µ has been introduced for dimensional reason and can be viewed as an arbitrary renormalization scale.…”

2D gravitational Mabuchi action on Riemann surfaces with boundaries

“…In the mathematical literature they appear in relation with the characterization of constant scalar curvature metrics. Their roles as two-dimensional gravitational actions in the sense of (1.1) have been put forward in [6]. In particuler, ref.…”

“…[6] has studied the metric dependence of the partition function of non-conformal massive matter on compact Riemann surfaces and shown that a gravitational action defined by (1.1) contains these Mabuchi and Aubin-Yau actions as first-order corrections (first order in m 2 A where m is the mass and A the area of the Riemann surface) to the Liouville action. The study of [6] was further confirmed and generalized in [7] where an exact formula for the gravitational action for any value of the mass was obtained, its expansion in m 2 A giving back the results of [6]. The Mabuchi action has drawn much attention in recent years.…”