310 research outputs found
Matrix Models and Geometry of Moduli Spaces
We give the description of discretized moduli spaces (d.m.s.) \Mcdisc
introduced in \cite{Ch1} in terms of discrete de Rham cohomologies for moduli
spaces \Mgn. The generating function for intersection indices (cohomological
classes) of d.m.s. is found. Classes of highest degree coincide with the ones
for the continuum moduli space \Mc. To show it we use a matrix model
technique. The Kontsevich matrix model is the generating function in the
continuum case, and the matrix model with the potential N\alpha \tr {\bigl(-
\fr 14 \L X\L X -\fr12\log (1-X)-\fr12X\bigr)} is the one for d.m.s. In the
latest case the effects of Deligne--Mumford reductions become relevant, and we
use the stratification procedure in order to express integrals over open spaces
\Mdisc in terms of intersection indices, which are to be calculated on
compactified spaces \Mcdisc. We find and solve constraint equations on
partition function of our matrix model expressed in times for d.m.s.:
t^\pm_m=\tr \fr{\d^m}{\d\l^m}\fr1{\e^\l-1}. It appears that depends
only on even times and {\cal Z}[t^\pm_\cdot]=C(\aa N) \e^{\cal
A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}, where is a
logarithm of the partition function of the Kontsevich model, being a
quadratic differential operator in \dd{t^\pm_{2n}}.Comment: 40pp., LaTeX, no macros needed, 8 figures in tex
Spectral problem on graphs and L-functions
The scattering process on multiloop infinite p+1-valent graphs (generalized
trees) is studied. These graphs are discrete spaces being quotients of the
uniform tree over free acting discrete subgroups of the projective group
. As the homogeneous spaces, they are, in fact, identical to
p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the
finite subgraph-the reduced graph containing all loops of the generalized tree.
We study the spectral problem on these graphs, for which we introduce the
notion of spherical functions-eigenfunctions of a discrete Laplace operator
acting on the graph. We define the S-matrix and prove its unitarity. We present
a proof of the Hashimoto-Bass theorem expressing L-function of any finite
(reduced) graph via determinant of a local operator acting on this
graph and relate the S-matrix determinant to this L-function thus obtaining the
analogue of the Selberg trace formula. The discrete spectrum points are also
determined and classified by the L-function. Numerous examples of L-function
calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur
The NBI matrix model of IIB Superstrings
We investigate the NBI matrix model with the potential
recently proposed to describe IIB
superstrings. With the proper normalization, using Virasoro constraints, we
prove the equivalence of this model and the Kontsevich matrix model for
and find the explicit transformation between the two models.Comment: LaTeX, 11p
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
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