2D quantum gravity on compact Riemann surfaces with non-conformal matter

Abstract

Abstract:We study the gravitational action induced by coupling two-dimensional nonconformal, massive matter to gravity on a compact Riemann surface. We express this gravitational action in terms of finite and well-defined quantities for any value of the mass. A small-mass expansion gives back the Liouville action in the massless limit, the Mabuchi and Aubin-Yau actions to first order, as well as an infinite series of higher-order contributions written in terms of purely geometric quantities.

2D gravitational Mabuchi action on Riemann surfaces with boundaries

“…[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.…”

“…As noted before, δλ 0 = 0 and, hence, there is no zero-mode contribution to the second term and one could just equally well rewrite the following results in terms of the ζ-functions defined by excluding the zero-mode [7]. Here, however, this is not of particular interest to us, and thus…”

“…In the massless case, this has to be slightly modified, see [6,7]. Of course, the determinant is ill-defined and needs to be regularized.…”

“…Formally, the ζ-functions are always defined by (2.3), but the properties of the manifold are encoded in the eigenvalues λ n that appear in the sum. The strategy of [6] and [7], that we will also follow here, was to determine the infinitesimal change of the ζ-functions from the infinitesimal change of the eigenvalues λ n under an infinitesimal change of the metric, and then to integrate this relation to get S grav . The change of the eigenvalues is obtained from (almost) standard quantum mechanical perturbation theory, as we discuss next.…”

“…Next, the variation of ℓ 2 (x, y) was given e.g. in [6,7,21]. In the limit x → y one simply has ℓ 2 (x, y) ≃ g ab dx a dx b = e 2σ(y) g (0) ab dx a dx b which shows that one has δℓ 2 (x, y) ≃ 2δσ(y) ℓ 2 (x, y) and, hence, lim x→y δ log ℓ 2 (x, y)µ 2 = 2 δσ(y) .…”