Random normal matrices, Bergman kernel and projective embeddings

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.Comment: 16 pages, typos corrected, references adde

Laughlin states on higher genus Riemann surfaces

Considering quantum Hall states on geometric backgrounds has proved over the past few years to be a useful tool for uncovering their less evident properties, such as gravitational and electromagnetic responses, topological phases and novel geometric adiabatic transport coefficients. One of the transport coefficients, the central charge associated with the gravitational anomaly, appears as a Chern number for the adiabatic transport on the moduli spaces of higher genus Riemann surfaces. This calls for a better understanding of the QH states on these backgrounds. Here we present a rigorous definition and give a detailed account of the construction of Laughlin states on Riemann surfaces of genus ${\rm g}>1$. By the first principles construction we prove that the dimension of the vector space of Laughlin states is at least $\beta^{\rm g}$ for the filling fraction $\nu=1/\beta$. Then using the path integral for the 2d bosonic field compactified on a circle, we reproduce the conjectured $\beta^{\rm g}$-degeneracy as the number of independent holomorphic blocks. We also discuss the lowest Landau level, integer QH state and its relation to the bosonization formulas on higher genus Riemann surfaces.Comment: 33 pages, 2 figures, v2: Fay's conventions for the $\sigma$-differential and the Arakelov metric are adopted, resulting in slight modifications of the affected formulas. Several other cosmetic changes and fixed typos, v3: further corrections, version to appear in Commun. Math. Phy

Geometric adiabatic transport in quantum Hall states

We argue that in addition to the Hall conductance and the nondissipative component of the viscous tensor, there exists a third independent transport coefficient, which is precisely quantized. It takes constant values along quantum Hall plateaus. We show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus 2 or higher and therefore cannot 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 quantum Hall states.Comment: 6 pages, discussion of angular momentum formulas in sec. 7 is amende

FQHE on curved backgrounds, free fields and large N

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-to-leading 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.Comment: 14 pages; v3: conformal spin rescaled, minor change

Heat kernel measures on random surfaces

The heat kernel on the symmetric space of positive definite Hermitian matrices is used to endow the spaces of Bergman metrics of degree k on a Riemann surface M with a family of probability measures depending on a choice of the background metric. Under a certain matrix-metric correspondence, each positive definite Hermitian matrix corresponds to a Kahler metric on M. The one and two point functions of the random metric are calculated in a variety of limits as k and t tend to infinity. In the limit when the time t goes to infinity the fluctuations of the random metric around the background metric are the same as the fluctuations of random zeros of holomorphic sections. This is due to the fact that the random zeros form the boundary of the space of Bergman metrics.Comment: 20 pages, v2: minor correction

Stability and integration over Bergman metrics

We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges. It measures the minimal degree of stability of geodesic rays in the space the Bergman metrics, with respect to the action. We calculate this critical value when the action is the $\nu$-balancing energy, and show that $\gamma_k^{\rm crit} = k - h$ on a Riemann surface of genus $h$.Comment: 24 pages, 3 figure

Black holes and balanced metrics

We consider a probe in a BPS black hole in type II strings compactified on Calabi-Yau manifolds, and conjecture that its moduli space metric is the balanced metric.Comment: 15 page

Quantum Hall effect and Quillen metric

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soul\'e curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.Comment: 36 pages, v4: greatly expanded version, added: references, Sec. 1.1 and Appendix A with background material, examples in Sec. 2.3 and Sec. 4, Thm. 3 and Sec. 5 expanded with more details on the relation between the adiabatic phase, eta invariant and Chern-Simons functional. To appear in Commun. Math. Phy