Extended Complex Trigonometry in Relation to Integrable 2D-Quantum Field Theories and Duality

Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current conservation. In two dimensions for n < 6 known integrable models (deformed Toda and non-linear sigma, pure affine Toda...) with dual counterparts are obtained in this way from the multicomplex space MC itself and from the natural embedding \MC_n \subset \MMC_m, n < m. For $n \ge 6$ a generic constraint on the space of parametersis obtained from current conservation at first order in the interaction Lagragien.Comment: 11 pages, no figure, LaTex with amsmath accepted by Phys. Lett.

Vertex operator approach to semi-infinite spin chain : recent progress

Vertex operator approach is a powerful method to study exactly solvable models. We review recent progress of vertex operator approach to semi-infinite spin chain. (1) The first progress is a generalization of boundary condition. We study $U_q(\widehat{sl}(2))$ spin chain with a triangular boundary, which gives a generalization of diagonal boundary [Baseilhac and Belliard 2013, Baseilhac and Kojima 2014]. We give a bosonization of the boundary vacuum state. As an application, we derive a summation formulae of boundary magnetization. (2) The second progress is a generalization of hidden symmetry. We study supersymmetry $U_q(\widehat{sl}(M|N))$ spin chain with a diagonal boundary [Kojima 2013]. By now we have studied spin chain with a boundary, associated with symmetry $U_q(\widehat{sl}(N))$, $U_q(A_2^{(2)})$ and $U_{q,p}(\widehat{sl}(N))$ [Furutsu-Kojima 2000, Yang-Zhang 2001, Kojima 2011, Miwa-Weston 1997, Kojima 2011], where bosonizations of vertex operators are realized by "monomial" . However the vertex operator for $U_q(\widehat{sl}(M|N))$ is realized by "sum", a bosonization of boundary vacuum state is realized by "monomial".Comment: Proceedings of 10-th Lie Theory and its Applications in Physics, LaTEX, 10 page

A note on the Oq(sl2^)O_q(\hat{sl_2}) algebra

An explicit homomorphism that relates the elements of the infinite dimensional non-Abelian algebra generating $O_q(\hat{sl_2})$ currents and the standard generators of the $q-$Onsager algebra is proposed. Two straightforward applications of the result are then considered: First, for the class of quantum integrable models which integrability condition originates in the $q-$Onsager spectrum generating algebra, the infinite $q-$deformed Dolan-Grady hierarchy is derived - bypassing the transfer matrix formalism. Secondly, higher Askey-Wilson relations that arise in the study of symmetric special functions generalizing the Askey-Wilson $q-$orthogonal polynomials are proposed.Comment: 11 page

Form factors of the half-infinite XXZ spin chain with a triangular boundary

The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of form factors of local operators are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. The expressions are compared with known results upon specializations.Using the spin-reversal property which relates the Hamiltonian with upper and lower triangular boundary conditions, new identities between multiple integrals of infinite products are extracted.Comment: LaTEX, 31 page

Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory

The transfer matrix of the XXZ open spin-1/2 chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q|=1) is diagonalized using the representation theory of the q-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d=2^N. The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results by Nepomechie et al. and Cao et al.. In the latter case the complete family of eigenvalues and eigenstates splits in two sets, each associated with a characteristic polynomial of degree $d< 2^N$. Numerical checks performed for small values of $N$ support the analysis.Comment: 21 pages; LaTeX file with amssymb; v2: typos corrected, references added, more details, to appear in JSTA

The half-infinite XXZ chain in Onsager's approach

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed `Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime $-1<q<0$, level one infinite dimensional representation ($q-$vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo {\it et al.} are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using $q-$bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for $q$ generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the $q-$Onsager algebra (generic non-diagonal case) or the augmented $q-$Onsager algebra (generic diagonal case).Comment: 28 pages; Presentation improved; misprints corrected; to appear in Nucl. Phys.

Analogues of Lusztig's higher order relations for the q-Onsager algebra

Let $A,A^*$ be the generators of the $q-$Onsager algebra. Analogues of Lusztig's $r-th$ higher order relations are proposed. In a first part, based on the properties of tridiagonal pairs of $q-$Racah type which satisfy the defining relations of the $q-$Onsager algebra, higher order relations are derived for $r$ generic. The coefficients entering in the relations are determined from a two-variable polynomial generating function. In a second part, it is conjectured that $A,A^*$ satisfy the higher order relations previously obtained. The conjecture is proven for $r=2,3$. For $r$ generic, using an inductive argument recursive formulae for the coefficients are derived. The conjecture is checked for several values of $r\geq 4$. Consequences for coideal subalgebras and integrable systems with boundaries at $q$ a root of unity are pointed out.Comment: 19 pages. v2: Some basic material in subsections 2.1,2.2,2.3 of pages 3-4 (Definitions 2.1,2.2, Lemma 2.2, Theorem 1) from Terwilliger's and coauthors works (see also arXiv:1307.7410); Missprints corrected; Minor changes in the text; References adde

On the third level descendent fields in the Bullough-Dodd model and its reductions

Exact vacuum expectation values of the third level descendent fields $$ in the Bullough-Dodd model are proposed. By performing quantum group restrictions, we obtain $<L_{-3}{\bar L}_{-3}{\Phi}_{lk}>$ in perturbed minimal conformal field theories.Comment: 7 pages, LaTeX file with amssymb; to appear in Phys. Lett.