332 research outputs found
Shear coordinate description of the quantised versal unfolding of D_4 singularity
In this paper by using Teichmuller theory of a sphere with four
holes/orbifold points, we obtain a system of flat coordinates on the general
affine cubic surface having a D_4 singularity at the origin. We show that the
Goldman bracket on the geodesic functions on the four-holed/orbifold sphere
coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We
prove that this bracket is the image under the Riemann-Hilbert map of the
Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the
action of the mapping class group by the action of the braid group on the
geodesic functions . This action coincides with the procedure of analytic
continuation of solutions of the sixth Painlev\'e equation. Finally, we produce
the explicit quantisation of the Goldman bracket on the geodesic functions on
the four-holed/orbifold sphere and of the braid group action.Comment: 14 pages, 2 picture
Two-logarithm matrix model with an external field
We investigate the two-logarithm matrix model with the potential
related to an exactly solvable
Kazakov-Migdal model. In the proper normalization, using Virasoro constraints,
we prove the equivalence of this model and the Kontsevich-Penner matrix model
and construct the 1/N-expansion solution of this model.Comment: 15pp., LaTeX, no figures, reference adde
Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
We solve the loop equations of the -ensemble model analogously to the
solution found for the Hermitian matrices . For \beta=1y^2=U(x)\beta((\hbar\partial)^2-U(x))\psi(x)=0\hbar\propto
(\sqrt\beta-1/\sqrt\beta)/Ny^2-U(x)[y,x]=\hbarF_h-expansion at arbitrary . The set of "flat"
coordinates comprises the potential times and the occupation numbers
\widetilde{\epsilon}_\alpha\mathcal F_0\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure
Effective Action and Measure in Matrix Model of IIB Superstrings
We calculate an effective action and measure induced by the integration over
the auxiliary field in the matrix model recently proposed to describe IIB
superstrings. It is shown that the measure of integration over the auxiliary
matrix is uniquely determined by locality and reparametrization invariance of
the resulting effective action. The large-- limit of the induced measure for
string coordinates is discussed in detail. It is found to be ultralocal and,
thus, possibly is irrelevant in the continuum limit. The model of the GKM type
is considered in relation to the effective action problem.Comment: 9pp., Latex; v2: the discussion of the large N limit of the induced
measure is substantially expande
Cut-and-Join operator representation for Kontsevich-Witten tau-function
In this short note we construct a simple cut-and-join operator representation
for Kontsevich-Witten tau-function that is the partition function of the
two-dimensional topological gravity. Our derivation is based on the Virasoro
constraints. Possible applications of the obtained expression are discussed.Comment: 5 pages, minor correction
Instantons and Merons in Matrix Models
Various branches of matrix model partition function can be represented as
intertwined products of universal elementary constituents: Gaussian partition
functions Z_G and Kontsevich tau-functions Z_K. In physical terms, this
decomposition is the matrix-model version of multi-instanton and multi-meron
configurations in Yang-Mills theories. Technically, decomposition formulas are
related to representation theory of algebras of Krichever-Novikov type on
families of spectral curves with additional Seiberg-Witten structure.
Representations of these algebras are encoded in terms of "the global partition
functions". They interpolate between Z_G and Z_K associated with different
singularities on spectral Riemann surfaces. This construction is nothing but
M-theory-like unification of various matrix models with explicit and
representative realization of dualities.Comment: 54 page
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
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