180 research outputs found
On action of the Virasoro algebra on the space of univalent functions
We obtain explicit expressions for differential operators defining the action
of the Virasoro algebra on the space of univalent functions. We also obtain an
explicit Taylor decomposition for Schwarzian derivative and a formula for the
Grunsky coefficients.Comment: 15
Noncommutative integrability, paths and quasi-determinants
In previous work, we showed that the solution of certain systems of discrete
integrable equations, notably and -systems, is given in terms of
partition functions of positively weighted paths, thereby proving the positive
Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of
solution is amenable to generalization to non-commutative weighted paths. Under
certain circumstances, these describe solutions of discrete evolution equations
in non-commutative variables: Examples are the corresponding quantum cluster
algebras [BZ], the Kontsevich evolution [DFK09b] and the -systems themselves
[DFK09a]. In this paper, we formulate certain non-commutative integrable
evolutions by considering paths with non-commutative weights, together with an
evolution of the weights that reduces to cluster algebra mutations in the
commutative limit. The general weights are expressed as Laurent monomials of
quasi-determinants of path partition functions, allowing for a non-commutative
version of the positive Laurent phenomenon. We apply this construction to the
known systems, and obtain Laurent positivity results for their solutions in
terms of initial data.Comment: 46 pages, minor typos correcte
K-matrices for 2D conformal field theories
In this paper we examine fermionic type characters (Universal Chiral
Partition Functions) for general 2D conformal field theories with a bilinear
form given by a matrix of the form K \oplus K^{-1}. We provide various
techniques for determining these K-matrices, and apply these to a variety of
examples including (higher level) WZW and coset conformal field theories.
Applications of our results to fractional quantum Hall systems and (level
restricted) Kostka polynomials are discussed.Comment: 59 pages, 2 figures, v2: note added, minor changes, references added,
v3: typos correcte
Boundary Ground Ring in Minimal String Theory
We obtain relations among boundary states in bosonic minimal open string
theory using the boundary ground ring. We also obtain a difference equation
that boundary correlators must satisfy.Comment: 28 pages, 1 figur
Discrete integrable systems, positivity, and continued fraction rearrangements
In this review article, we present a unified approach to solving discrete,
integrable, possibly non-commutative, dynamical systems, including the - and
-systems based on . The initial data of the systems are seen as cluster
variables in a suitable cluster algebra, and may evolve by local mutations. We
show that the solutions are always expressed as Laurent polynomials of the
initial data with non-negative integer coefficients. This is done by
reformulating the mutations of initial data as local rearrangements of
continued fractions generating some particular solutions, that preserve
manifest positivity. We also show how these techniques apply as well to
non-commutative settings.Comment: 24 pages, 2 figure
Fusion products, Kostka polynomials, and fermionic characters of su(r+1)_k
Using a form factor approach, we define and compute the character of the
fusion product of rectangular representations of \hat{su}(r+1). This character
decomposes into a sum of characters of irreducible representations, but with
q-dependent coefficients. We identify these coefficients as (generalized)
Kostka polynomials. Using this result, we obtain a formula for the characters
of arbitrary integrable highest-weight representations of \hat{su}(r+1) in
terms of the fermionic characters of the rectangular highest weight
representations.Comment: 21 pages; minor changes, typos correcte
The solution of the quantum T-system for arbitrary boundary
We solve the quantum version of the -system by use of quantum
networks. The system is interpreted as a particular set of mutations of a
suitable (infinite-rank) quantum cluster algebra, and Laurent positivity
follows from our solution. As an application we re-derive the corresponding
quantum network solution to the quantum -system and generalize it to
the fully non-commutative case. We give the relation between the quantum
-system and the quantum lattice Liouville equation, which is the quantized
-system.Comment: 24 pages, 18 figure
Parafermionic quasi-particle basis and fermionic-type characters
A new basis of states for highest-weight modules in \ZZ_k parafermionic
conformal theories is displayed. It is formulated in terms of an effective
exclusion principle constraining strings of fundamental parafermionic
modes. The states of a module are then built by a simple filling process, with
no singular-vector subtractions. That results in fermionic-sum representations
of the characters, which are exactly the Lepowsky-Primc expressions. We also
stress that the underlying combinatorics -- which is the one pertaining to the
Andrews-Gordon identities -- has a remarkably natural parafermionic
interpretation.Comment: minor modifications and proof in app. C completed; 34 pages (harvmac
b
Q-systems, Heaps, Paths and Cluster Positivity
We consider the cluster algebra associated to the -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the -system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the -system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure
Determination of quantum symmetries for higher ADE systems from the modular T matrix
We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram
(or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system)
is, in most cases, deduced from the structure of the modular T matrix in the A
series. We recover in this way the (known) quantum symmetries of su(2) diagrams
and illustrate our method by studying those associated with the three genuine
exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides
the shortest way to the determination of twisted partition functions in
boundary conformal field theory with defect lines.Comment: 30 pages, 16 figures. Several misprints have been corrected. We added
several references and the appendix has been enlarged (one section on
essential paths and one section devoted to open problems). This article will
appear in the Journal of Mathematical Physic
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