Rinat Kedem

Fermionic sum representations for conformal field theory characters

Abstract We present sum representations for all characters of the unitary Virasoro minimal models. They can be viewed as fermionic companions of the Rocha-Caridi sum representations, the latter related to the (bosonic) Feigin-Fuchs-Felder construction. We also give fermionic representations for certain characters of the general ( G (1) ) k × ( G (1) ) l ( G (1) ) k+l coset conformal field theories, the non-unitary minimal models M (p, p+2) and M (p, kp+1) , the N = 2 superconformal series, and the Z N -parafermion theories, and relate the q →1 behaviour of all these fermionic sum representations to the thermodynamic Bethe ansatz.

XXZ chain with a boundary

The XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h ⩾ 0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary S-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the XXX model.

Difference equations in spin chains with a boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the eight-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary S-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal-Zamolodchikovs result. The axioms satisfied by the form factors in the boundary theory are formulated.

Q-systems as cluster algebras

Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211–21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac–Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497–529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation–theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119–44).

Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property

We define the cluster algebra associated with the Q-system for the Kirillov–Reshetikhin characters of the quantum affine algebra

Q-Systems, Heaps, Paths and Cluster Positivity

{U_q(\widehat{\mathfrak {g}})}

Quasi-Particles, Conformal Field Theory, and q-Series

for any simple Lie algebra