6,001 research outputs found
Matrix models and stochastic growth in Donaldson-Thomas theory
We show that the partition functions which enumerate Donaldson-Thomas
invariants of local toric Calabi-Yau threefolds without compact divisors can be
expressed in terms of specializations of the Schur measure. We also discuss the
relevance of the Hall-Littlewood and Jack measures in the context of BPS state
counting and study the partition functions at arbitrary points of the Kaehler
moduli space. This rewriting in terms of symmetric functions leads to a unitary
one-matrix model representation for Donaldson-Thomas theory. We describe
explicitly how this result is related to the unitary matrix model description
of Chern-Simons gauge theory. This representation is used to show that the
generating functions for Donaldson-Thomas invariants are related to
tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The
matrix model also leads to an interpretation of Donaldson-Thomas theory in
terms of non-intersecting paths in the lock-step model of vicious walkers. We
further show that these generating functions can be interpreted as
normalization constants of a corner growth/last-passage stochastic model.Comment: 31 pages; v2: comments and references added; v3: presentation
improved, comments added; final version to appear in Journal of Mathematical
Physic
Counting tropical elliptic plane curves with fixed j-invariant
In complex algebraic geometry, the problem of enumerating plane elliptic
curves of given degree with fixed complex structure has been solved by
R.Pandharipande using Gromov-Witten theory. In this article we treat the
tropical analogue of this problem, the determination of the number of tropical
elliptic plane curves of degree d and fixed ``tropical j-invariant''
interpolating an appropriate number of points in general position. We show that
this number is independent of the position of the points and the value of the
j-invariant and that it coincides with the number of complex elliptic curves.
The result can be used to simplify Mikhalkin's algorithm to count curves via
lattice paths in the case of rational plane curves.Comment: 34 pages; minor changes to match the published versio
The interaction of a gap with a free boundary in a two dimensional dimer system
Let be a fixed vertical lattice line of the unit triangular lattice in
the plane, and let \Cal H be the half plane to the left of . We
consider lozenge tilings of \Cal H that have a triangular gap of side-length
two and in which is a free boundary - i.e., tiles are allowed to
protrude out half-way across . We prove that the correlation function of
this gap near the free boundary has asymptotics ,
, where is the distance from the gap to the free boundary. This
parallels the electrostatic phenomenon by which the field of an electric charge
near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te
Counting interesting elections
We provide an elementary proof of a formula for the number of northeast
lattice paths that lie in a certain region of the plane. Equivalently, this
formula counts the lattice points inside the Pitman--Stanley polytope of an
n-tuple.Comment: 7 pages, 1 figure; published versio
Quivers, YBE and 3-manifolds
We study 4d superconformal indices for a large class of N=1 superconformal
quiver gauge theories realized combinatorially as a bipartite graph or a set of
"zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we
call a "double Yang-Baxter move", gives the Seiberg duality of the gauge
theory, and the invariance of the index under the duality is translated into
the Yang-Baxter-type equation of a spin system defined on a "Z-invariant"
lattice on T^2. When we compactify the gauge theory to 3d, Higgs the theory and
then compactify further to 2d, the superconformal index reduces to an integral
of quantum/classical dilogarithm functions. The saddle point of this integral
unexpectedly reproduces the hyperbolic volume of a hyperbolic 3-manifold. The
3-manifold is obtained by gluing hyperbolic ideal polyhedra in H^3, each of
which could be thought of as a 3d lift of the faces of the 2d bipartite
graph.The same quantity is also related with the thermodynamic limit of the BPS
partition function, or equivalently the genus 0 topological string partition
function, on a toric Calabi-Yau manifold dual to quiver gauge theories. We also
comment on brane realization of our theories. This paper is a companion to
another paper summarizing the results.Comment: 61 pages, 16 figures; v2: typos correcte
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