9 research outputs found
How instanton combinatorics solves Painlev\'e VI, V and III's
We elaborate on a recently conjectured relation of Painlev\'e transcendents
and 2D CFT. General solutions of Painlev\'e VI, V and III are expressed in
terms of conformal blocks and their irregular limits, AGT-related to
instanton partition functions in supersymmetric gauge theories
with . Resulting combinatorial series representations of
Painlev\'e functions provide an efficient tool for their numerical computation
at finite values of the argument. The series involve sums over bipartitions
which in the simplest cases coincide with Gessel expansions of certain Toeplitz
determinants. Considered applications include Fredholm determinants of
classical integrable kernels, scaled gap probability in the bulk of the GUE,
and all-order conformal perturbation theory expansions of correlation functions
in the sine-Gordon field theory at the free-fermion point.Comment: 34 pages, 3 figures; v2: minor improvement
Conformal field theory of Painlev\'e VI
Generic Painlev\'e VI tau function \tau(t) can be interpreted as four-point
correlator of primary fields of arbitrary dimensions in 2D CFT with c=1. Using
AGT combinatorial representation of conformal blocks and determining the
corresponding structure constants, we obtain full and completely explicit
expansion of \tau(t) near the singular points. After a check of this expansion,
we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and
algebraic solutions of Painlev\'e VI.Comment: 24 pages, 1 figure; v3: added refs and minor clarifications, few
typos corrected; to appear in JHE
Tabulation of PVI Transcendents and Parametrization Formulas (August 17, 2011)
The critical and asymptotic behaviors of solutions of the sixth Painlev\'e
equation PVI, obtained in the framework of the monodromy preserving deformation
method, and their explicit parametrization in terms of monodromy data, are
tabulated.Comment: 30 pages, 1 figure; Nonlinearity 201