We argue that in addition to the Hall conductance and the non-dissipative component of the viscous tensor there exists a third independent transport coefficient, which is precisely quantized, taking on constant values along quantum Hall plateaus. Relying on the holomorphic properties of the quantum Hall states, we show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus two or higher and therefore can not change continuously along the plateau. As such it does not transpire on a sphere or a torus. In the linear response theory this coefficient determines intensive forces exerted on electronic fluid by adiabatic deformations of geometry and represents the effect of the gravitational anomaly. We also present the method of computing the transport coefficients for QH-states. Precise quantization in materials occurs when the transport is a non-dissipative adiabatic process. QHE is an example of a system where adiabatic conditions are in place. Namely, the low energy states are separated from the rest of the spectrum by a gap and adiabatic changes of parameters produce states with the flux. However, only adiabatic processes with the nontrivial first Chern class yield quantized transport coefficients.In this paper we show that apart from the Hall conductance there exist two more quantized transport coefficients, although at present only the former is experimentally accessible. One of these coefficients is the nondissipative component of viscous tensor introduced in [1][2][3]. Indications for the existence of another precise transport coefficient appeared recently in connection with the gravitational anomaly found in the context of QHE in Refs. [4][5][6][7][8][9][10][11][12], see also [13].Precise quantization on QH plateaus of the nondissipative transport coefficients can be explained from two points of view. The first, topological explanation is through their relation to topological invariants, such as Chern numbers of vector bundles over the appropriate parameter space [14][15][16]. For example, in the case of the Hall conductance, the parameter space is spanned by Aharonov-Bohm fluxes piercing through the handles of the Riemann surface. For the non-dissipative viscosity the relevant parameter space is the moduli space of complex structures on the torus [1][2][3]. In this paper we show that the third coefficient shows up, when the parameter space is the moduli space of complex structures for the surfaces of genus 2 and higher.For this reason, we discuss the precise transport for QH-states on compact Riemann surfaces. We develop a general method to compute all three transport coeffi-

We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix model free energy and its relation to gravitational effective actions in two dimensions. We compute the leading terms of the free energy expansion in the pure bulk case, and make some observations on the structure of the expansion to all orders. As an application of these results, we propose an asymptotic formula for the Liouville action, restricted to the space of the Bergman metrics.

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.

Abstract:We study the free energy of the Laughlin state on curved backgrounds, starting from the free field representation. A simple argument, based on the computation of the gravitational effective action from the transformation properties of Green functions under the change of the metric, allows to compute the first three terms of the expansion in large magnetic field. The leading and subleading contributions are given by the Aubin-Yau and Mabuchi functionals respectively, whereas the Liouville action appears at next-to-next-toleading order. We also derive a path integral representation for the remainder terms. They correspond to a large mass expansion for a related interacting scalar field theory and are thus given by local polynomials in curvature invariants.