1,675 research outputs found
2D Gravity and Random Matrices
We review recent progress in 2D gravity coupled to conformal matter,
based on a representation of discrete gravity in terms of random matrices. We
discuss the saddle point approximation for these models, including a class of
related matrix models. For matter, the matrix problem can be
completely solved in many cases by the introduction of suitable orthogonal
polynomials. Alternatively, in the continuum limit the orthogonal polynomial
method can be shown to be equivalent to the construction of representations of
the canonical commutation relations in terms of differential operators. In the
case of pure gravity or discrete Ising--like matter, the sum over topologies is
reduced to the solution of non-linear differential equations (the Painlev\'e
equation in the pure gravity case) which can be shown to follow from an action
principle. In the case of pure gravity and more generally all unitary models,
the perturbation theory is not Borel summable and therefore alone does not
define a unique solution. In the non-Borel summable case, the matrix model does
not define the sum over topologies beyond perturbation theory. We also review
the computation of correlation functions directly in the continuum formulation
of matter coupled to 2D gravity, and compare with the matrix model results.
Finally, we review the relation between matrix models and topological gravity,
and as well the relation to intersection theory of the moduli space of
punctured Riemann surfaces.Comment: 190 pages (harvmac l mode), 400kb (don't even dream of requesting
hardcopy
Lectures on the topological recursion for Higgs bundles and quantum curves
© 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
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