336 research outputs found
Computing topological invariants with one and two-matrix models
A generalization of the Kontsevich Airy-model allows one to compute the
intersection numbers of the moduli space of p-spin curves. These models are
deduced from averages of characteristic polynomials over Gaussian ensembles of
random matrices in an external matrix source. After use of a duality, and of an
appropriate tuning of the source, we obtain in a double scaling limit these
intersection numbers as polynomials in p. One can then take the limit p to -1
which yields a matrix model for orbifold Euler characteristics. The
generalization to a time-dependent matrix model, which is equivalent to a
two-matrix model, may be treated along the same lines ; it also yields a
logarithmic potential with additional vertices for general p.Comment: 30 pages, added references, changed conten
RECENT DEVELOPMENTS IN THE THEORY OF GENERAL HYPERGEOMETRIC FUNCTIONS(Special Differential Equations)
Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups
We introduce and study action of quantum groups on skew polynomial rings and
related rings of quotients. This leads to a ``q-deformation'' of the
Gel'fand-Kirillov conjecture which we partially prove. We propose a
construction of automorphisms of certain non-commutaive rings of quotients
coming from complex powers of quantum group generators; this is applied to
explicit calculation of singular vectors in Verma modules over
U_{q}(\gtsl_{n+1}).
We finally give a definition of a connection with coefficients in a ring
of skew polynomials and study the structure of quantum group modules twisted by
a connection.Comment: 25 page
Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation
Two families of solutions of a generalized non-Abelian Toda lattice are
considered. These solutions are expressed in terms of quasideterminants,
constructed by means of Darboux and binary Darboux transformations. As an
example of the application of these solutions, we consider the 2-periodic
reduction to a matrix sine-Gordon equation. In particular, we investigate the
interaction properties of polarized kink solutions.Comment: 14 pages; 4 picture
Ising model with a boundary magnetic field - an example of a boundary flow
In hep-th/0312197 a nonperturbative proof of the g-theorem of Affleck and
Ludwig was put forward. In this paper we illustrate how the proof of
hep-th/0312197 works on the example of the 2D Ising model at criticality
perturbed by a boundary magnetic field. For this model we present explicit
computations of all the quantities entering the proof including various contact
terms. A free massless boson with a boundary mass term is considered as a
warm-up example.Comment: 1+20 pages, Latex, 2 eps figures; v2: references adde
The mapping class group and the Meyer function for plane curves
For each d>=2, the mapping class group for plane curves of degree d will be
defined and it is proved that there exists uniquely the Meyer function on this
group. In the case of d=4, using our Meyer function, we can define the local
signature for 4-dimensional fiber spaces whose general fibers are
non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our
local signature will be given.Comment: 24 pages, typo adde
Boundary three-point function on AdS2 D-branes
Using the H3+-Liouville relation, I explicitly compute the boundary
three-point function on AdS2 D-branes in H3+, and check that it exhibits the
expected symmetry properties and has the correct geometrical limit. I then find
a simple relation between this boundary three-point function and certain fusing
matrix elements, which suggests a formal correspondence between the AdS2
D-branes and discrete representations of the symmetry group. Concluding
speculations deal with the fuzzy geometry of AdS2 D-branes, strings in the
Minkowskian AdS3, and the hypothetical existence of new D-branes in H3+.Comment: 27 pages, v2: significant clarifications added in sections 4.3 and
Modular Lie algebras and the Gelfand-Kirillov conjecture
Let g be a finite dimensional simple Lie algebra over an algebraically closed
field of characteristic zero. We show that if the Gelfand-Kirillov conjecture
holds for g, then g has type A_n, C_n or G_2.Comment: 20 page
The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average
We introduce an algorithm which solves mean payoff games in polynomial time
on average, assuming the distribution of the games satisfies a flip invariance
property on the set of actions associated with every state. The algorithm is a
tropical analogue of the shadow-vertex simplex algorithm, which solves mean
payoff games via linear feasibility problems over the tropical semiring
. The key ingredient in our approach is
that the shadow-vertex pivoting rule can be transferred to tropical polyhedra,
and that its computation reduces to optimal assignment problems through
Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP
2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part
Singularities of bi-Hamiltonian systems
We study the relationship between singularities of bi-Hamiltonian systems and
algebraic properties of compatible Poisson brackets. As the main tool, we
introduce the notion of linearization of a Poisson pencil. From the algebraic
viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with
a fixed 2-cocycle. In terms of such linearizations, we give a criterion for
non-degeneracy of singular points of bi-Hamiltonian systems and describe their
types
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