85 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
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
Matrix and vector models in the strong coupling limit
In this paper we consider matrix and vector models in the large N limit ( matrices and vectors with N^{2} components). For the case of
zero-dimensional model (D=0) it is proved that in the strong coupling limit statistical sums of both models coincide up to a coefficient. This
is also true for D=1.Comment: 8 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
Right tail expansion of Tracy-Widom beta laws
Using loop equations, we compute the large deviation function of the maximum
eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles,
to all orders in 1/N. We then give a physical derivation of the all order
asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive
beta, by studying the double scaling limit.Comment: 23 page
Mumford dendrograms and discrete p-adic symmetries
In this article, we present an effective encoding of dendrograms by embedding
them into the Bruhat-Tits trees associated to -adic number fields. As an
application, we show how strings over a finite alphabet can be encoded in
cyclotomic extensions of and discuss -adic DNA encoding. The
application leads to fast -adic agglomerative hierarchic algorithms similar
to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint
of -adic geometry, to encode a dendrogram in a -adic field means
to fix a set of -rational punctures on the -adic projective line
. To is associated in a natural way a
subtree inside the Bruhat-Tits tree which recovers , a method first used by
F. Kato in 1999 in the classification of discrete subgroups of
.
Next, we show how the -adic moduli space of
with punctures can be applied to the study of time series of
dendrograms and those symmetries arising from hyperbolic actions on
. In this way, we can associate to certain classes of dynamical
systems a Mumford curve, i.e. a -adic algebraic curve with totally
degenerate reduction modulo .
Finally, we indicate some of our results in the study of general discrete
actions on , and their relation to -adic Hurwitz spaces.Comment: 14 pages, 6 figure
Double Scaling Limits in Gauge Theories and Matrix Models
We show that gauge theories with an adjoint chiral multiplet admit a
wide class of large-N double-scaling limits where is taken to infinity in a
way coordinated with a tuning of the bare superpotential. The tuning is such
that the theory is near an Argyres-Douglas-type singularity where a set of
non-local dibaryons becomes massless in conjunction with a set of confining
strings becoming tensionless. The doubly-scaled theory consists of two
decoupled sectors, one whose spectrum and interactions follow the usual large-N
scaling whilst the other has light states of fixed mass in the large-N limit
which subvert the usual large-N scaling and lead to an interacting theory in
the limit. -term properties of this interacting sector can be calculated
using a Dijkgraaf-Vafa matrix model and in this context the double-scaling
limit is precisely the kind investigated in the "old matrix model'' to describe
two-dimensional gravity coupled to conformal field theories. In
particular, the old matrix model double-scaling limit describes a sector of a
gauge theory with a mass gap and light meson-like composite states, the
approximate Goldstone boson of superconformal invariance, with a mass which is
fixed in the double-scaling limit. Consequently, the gravitational -terms in
these cases satisfy the string equation of the KdV hierarchy.Comment: 38 pages, 1 figure, reference adde
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