420 research outputs found
Local Operators in Massive Quantum Field Theories
Contribution to the proceedings of Schladming 1995. A review of the form
factor approach and its utilisation to determine the space of local operators
of integrable massive quantum theories is given. A few applications are
discussed.Comment: 6 pages, late
Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich
We prove a conjecture of Kontsevich regarding the solutions of rank two
recursion relations for non-commutative variables which, in the commutative
case, reduce to rank two cluster algebras of affine type. The conjecture states
that solutions are positive Laurent polynomials in the initial cluster
variables. We prove this by use of a non-commutative version of the path models
which we used for the commutative case.Comment: 17 pages, 2 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
A quasi-particle description of the M(3,p) models
The M(3,p) minimal models are reconsidered from the point of view of the
extended algebra whose generators are the energy-momentum tensor and the
primary field \phi_{2,1} of dimension . Within this framework, we
provide a quasi-particle description of these models, in which all states are
expressed solely in terms of the \phi_{2,1}-modes. More precisely, we show that
all the states can be written in terms of \phi_{2,1}-type highest-weight states
and their phi_{2,1}-descendants. We further demonstrate that the conformal
dimension of these highest-weight states can be calculated from the \phi_{2,1}
commutation relations, the highest-weight conditions and associativity. For the
simplest models (p=5,7), the full spectrum is explicitly reconstructed along
these lines. For odd, the commutation relations between the \phi_{2,1}
modes take the form of infinite sums, i.e., of generalized commutation
relations akin to parafermionic models. In that case, an unexpected operator,
generalizing the Witten index, is unravelled in the OPE of \phi_{2,1} with
itself. A quasi-particle basis formulated in terms of the sole \phi_{1,2} modes
is studied for all allowed values of p. We argue that it is governed by
jagged-type partitions further subject a difference 2 condition at distance 2.
We demonstrate the correctness of this basis by constructing its generating
function, from which the proper fermionic expression of the combination of the
Virasoro irreducible characters \chi_{1,s} and \chi_{1,p-s} (for 1\leq s\leq
[p/3]+1) are recovered. As an aside, a practical technique for implementing
associativity at the level of mode computations is presented, together with a
general discussion of the relation between associativity and the Jacobi
identities.Comment: 29 pages; revised version with two appendices adde
Fermionic characters for graded parafermions
Fermionic-type character formulae are presented for charged
irreduciblemodules of the graded parafermionic conformal field theory
associated to the coset . This is obtained by counting the
weakly ordered `partitions' subject to the graded exclusion principle.
The bosonic form of the characters is also presented.Comment: 24 p. This corrects typos (present even in the published version) in
eqs (4.4), (5.23), (5.24) and (C.4
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
Parafermionic character formulae
We study various aspects of parafermionic theories such as the precise field
content, a description of a basis of states (that is, the counting of
independent states in a freely generated highest-weight module) and the
explicit expression of the parafermionic singular vectors in completely
irreducible modules. This analysis culminates in the presentation of new
character formulae for the parafermionic primary fields. These characters
provide novel field theoretical expressions for \su(2) string functions.Comment: Harvmac (b mode : 37 p
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