310 research outputs found
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 . By the first principles construction we prove
that the dimension of the vector space of Laughlin states is at least
for the filling fraction . Then using the path
integral for the 2d bosonic field compactified on a circle, we reproduce the
conjectured -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
-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
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
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
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 -balancing energy, and show
that on a Riemann surface of genus .Comment: 24 pages, 3 figure
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
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 of the positive
Hermitian line bundle . 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 , at large . 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
- …