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 $k$ 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 $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-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 $Q$-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 $A_r$ $Q$-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 $(p-2)/4$. 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 $p$ 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 $osp(1,2)_k/u(1)$. This is obtained by counting the weakly ordered `partitions' subject to the graded $Z_k$ 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 $Z_N$ parafermionic primary fields. These characters provide novel field theoretical expressions for \su(2) string functions.Comment: Harvmac (b mode : 37 p