International Journal of Modern Physics A

Basing on the numerical observations of the eigenspectra of the SU(N) Perk-Schultz model at the special value of the anisotropy q=exp(iπ(N-1)/N), we formulate a set of conjectures concerning the existence of the free-fermion like eigenenergies. We prove analytically a part of these conjectures.

We construct integrable spin chains with the inhomogeneous periodic disposition of the anisotropy parameter. The periodicity holds for both auxiliary (space) and quantum (time) directions. The integrability of the model is based on a set of coupled Yang–Baxter equations. This construction yields the P-leg integrable ladder Hamiltonians. We analyse the corresponding quantum group symmetry.

The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, the program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.

We describe the center of the algebra of monodromy matrices connected with the R-matrix of the eight-vertex model in the case where the crossing parameter has rational values.

We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field. We are interested in a probability of a formation of a ferromagnetic string P(n) in the antiferromagnetic ground-state. We call it emptiness formation probability [EFP]. We suggest a new technique for computation of the EFP in the inhomogeneous case. It is based on the quantum Knizhnik-Zamolodchikov equation [qKZ]. We calculate EFP for n≤6 for the inhomogeneous case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations and number theory. We also make a conjecture about a structure of EFP for arbrary n.

Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type.

We present two applications of quantum integrable systems. First, we predict that it is possible to generate high harmonics from solid state devices by demonstrating that the emission spectrum of a minimally coupled laser field of frequency ω to an impurity system of a quantum wire, contains multiples of the incoming frequency. Second, by evaluating expressions for the conductance in the high temperature regime we show that multiples of the characteristic filling fractions of the Jain sequence, which occur in the fractional quantum Hall effect, can be obtained from quantum wires which are described by minimal affine Toda field theories.

A universal symmetric truncation of the bosonic string Hilbert space yields all known closed fermionic string theories in ten dimensions, their D-branes and their open descendants. We highlight the crucial role played by group theory and two-dimensional conformal field theory in the construction and emphasize the predictive power of the truncation. Such circumstantial evidence points towards the existence of a mechanism which generates space-time fermions out of bosons dynamically within the framework of bosonic string theory.

A unified analysis of the disorder operators for ghosts, complex boson and fermion fields is presented. Matrix elements on the asymptotic states of these operators can be exactly computed by solving the Form Factor functional equations. The two–point correlation functions of the disorder operators depend only on the statistics and can be expressed in terms of a solution of a non–linear differential equation of Painleve' type.

We consider spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra . The first one is generated by , and the second one is generated by , where P(t), are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and q-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion product of -modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.

By considering the continuum scaling limit of the A₄ RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit two parameters ξ and η which play the role of the perturbing boundary fields φ_1,3 and φ_1,2 and induce the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.

We present a procedure, allowing to derive the coefficients of the Seiberg Witten prepotential of the N=2 Super Yang-Mills theory in the framework of the instanton calculations.

This is a brief introduction to the subject of Conformal Field Theory on surfaces with boundaries and crosscaps, which describes the perturbative expansion of open string theory.

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator in the space of a triple Weyl algebra. is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for follows without further calculation. If the Weyl parameter is taken to be a root of unity, decomposes into a matrix conjugation operator R_1,2,3 and a c-number functional mapping . The operator R_1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R_1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.

In this paper we define a new class of the quantum integrable systems associated with the quantization of the cotangent bundle T*(GL(N)) to the Lie algebra . The construction is based on the Gelfand-Zetlin maximal commuting subalgebra in . We discuss the connection with the other known integrable systems based on T*GL(N). The construction of the spectral tower associated with the proposed integrable theory is given. This spectral tower appears as a generalization of the standard spectral curve for an integrable system.

We study the dyon electric charge of D6-branes as eleven dimensional KK-monopoles. We observe that the dyon charge is intimately related with the existence of gauge connections and antisymmetric fields on the brane world-volume.

The aim of the article is to derive in the explicit form the radial components of Casimir elements of U_q(gl_n) corresponding to a quantum analogue of the homogeneous space GL(n)/SO(n). They coincide with the Macdonald–Ruijsenaars difference operators (MRDOs), if one starts from a special set of Casimir elements from the center of U_q(gl_n). The derivation is essentially based on Cherednik's approach to MRDOs by means of affine Hecke algebras. From the other side, MRDOs coincide with commuting Hamiltonians of quantum trigonometric n-particle Ruijsenaars model.

We continue our study of quantum Lie algebras, an important class of quadratic algebras arising in the Woronowicz calculus on a quantum group. Quantum Lie algebras are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit "quantum" analogues. In particular, there is a BRST operator Q(Q²=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given. Here we solve this recurrence relation and obtain an explicit formula for the BRST operator.

We review some recent advances in the computation of exact correlation functions of the XXZ-½ Heisenberg chain. We first give a general introduction to our method which is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem, leading in particular to multiple integral representations for the correlation functions. Then we describe recently obtained compact formulas for the spin-spin correlation functions of the XXZ-½ Heisenberg chain. We outline how this leads to several explicit results including the known two point functions in the limit of free fermions, the so-called emptiness formation probability at anisotropy Δ=1/2 and its large distance asymptotic behaviour in the massless phase of the model.

In this note, I review some recent results concerning spin-spin Ising correlators on a cylinder: lattice form factor expansions, a scaling limit and nonlinear differential equations. In particular, the pair correlation function is shown to satisfy the same three PDEs of the fourth order in the paramagnetic and ferromagnetic phase. I derive determinant representations of these equations and their multilinear Hirota form. I also conjecture an overdetermined system of PDEs, related to the isomonodromy problem for the Dirac equation on the cylinder, for which n-point functions of the Ising model should give a solution.

We generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals (the Gross-Witten integral, the integrals over complex matrices and the integrals over rectangle matrices) using a notion of the tau function of the matrix argument. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions. We also consider a generalization of the Kontsevich integral.

The approach to construction of D-branes in the N=2 superconformal minimal models based on a free-field realization of the N=2 super-Virasoro algebra unitary modules is represented.

An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.

Three explicit and equivalent representations for the monodromy of the conformal blocks in the non compact SL(2)_k WZNW model are proposed in terms of the same quantity computed in Liouville field theory.

Modular theory of operator algebras and the associated KMS property are used to obtain a unified description for the thermal aspects of the standard heat bath situation and those caused by quantum vacuum fluctuations from localization. An algebraic variant of lightfront holography reveals that the vacuum polarization on wedge horizons is compressed into the lightray direction. Their absence in the transverse direction is the prerequisite to an area (generalized Bekenstein-) behaviour of entropy-like measures which reveal the loss of purity due to restrictions to wedges and their horizons. Besides the well-known fact that localization-induced (generalized Hawking-) temperature is fixed by the geometric aspects, this area behaviour (versus the standard volume dependence) constitutes the main difference between localization-caused and standard thermal behaviour.

Three examples of free field constructions for the vertex operators of the elliptic quantum group are obtained. Two of these (for p^1/2=±q^3/2, p^1/2=-q²) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 Z-algebra of Lepowsky and Wilson. The third one (p^1/2=q³) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at (p^1/2=q³), however, is not related to the deformed Virasoro algebra. Using these free field constructions, an integral formula for the correlation functions of Baxter's eight-vertex model is obtained. This formula shows different structure compared with the one obtained by Lashkevich and Pugai.

We summarize previous works on the exact ground state and the elementary excitations of the exactly solvable BCS model in the canonical ensemble. The BCS model is solved by Richardson equations, and, in the large coupling limit, by Gaudin equations. The relationship between this two kinds of solutions is used to classify the excitations.

In this paper the proofs are given of important properties of deformed Abelian differentials introduced earlier in connection with quantum integrable systems. The starting point of the construction is the Baxter equation. In particular, we prove the Riemann bilinear relation. The duality plays an important role in our consideration. The classical limit is considered in details.

Trigonometric degeneration of the Baxter-Belavin elliptic r matrix is described by the degeneration of the twisted WZW model on elliptic curves. The spaces of conformal blocks and conformal coinvariants of the degenerate model are factorised into those of the orbifold WZW model.

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov.

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.

We present face-type elliptic solutions to the Yang-Baxter equation. They have 2N-2 real parameters. When specializing them to definite values, we recover the various models so far known. The intertwining relation between the face models above and the Z_N-symmetric vertex model of Belavin is also given.

An infinite set of the operator-valued relations in the Liouville field theory is established. These relations are enumerated by a pair of positive integers (m, n), the first (1,1) representative being the usual Liouville equation of motion. The relations are proven in the framework of conformal field theory on a basis of the exact structure constants in the Liouville operator product expansions. Possible applications in 2D gravity are discussed.