18 research outputs found
Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
We compute explicitly the two-dimensional version of Basso-Dixon type
integrals for the planar four-point correlation functions given by conformal
fishnet Feynman graphs. These diagrams are represented by a fragment of a
regular square lattice of power-like propagators, arising in the recently
proposed integrable bi-scalar fishnet CFT. The formula is derived from first
principles, using the formalism of separated variables in integrable SL(2,C)
spin chain. It is generalized to anisotropic fishnet, with different powers for
propagators in two directions of the lattice.Comment: 30 pages, 13 figures, v2: improved formulas, typos correcte
R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain
The problem of constructing the invariant solutions to the
Yang-Baxter equation is considered. The solutions (-operators) for
arbitrarily principal series representations of are obtained
in an explicit form. We construct the commutative family of the operators
which can be identified with the Baxter operators for the
noncompact spin magnet.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
We start from known solutions of the Yang-Baxter equation with a spectral
parameter defined on the tensor product of two infinite-dimensional principal
series representations of the group or Faddeev's
modular double. Then we describe its restriction to an irreducible
finite-dimensional representation in one or both spaces. In this way we obtain
very simple explicit formulas embracing rational and trigonometric
finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct
these finite-dimensional solutions by means of the fusion procedure and find a
nice agreement between two approaches
Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
We construct the Baxter Q-operator and the representation of the Separated
Variables (SoV) for the homogeneous open SL(2,R) spin chain. Applying the
diagrammatical approach, we calculate Sklyanin's integration measure in the
separated variables and obtain the solution to the spectral problem for the
model in terms of the eigenvalues of the Q-operator. We show that the
transition kernel to the SoV representation is factorized into the product of
certain operators each depending on a single separated variable. As a
consequence, it has a universal pyramid-like form that has been already
observed for various quantum integrable models such as periodic Toda chain,
closed SL(2,R) and SL(2,C) spin chains.Comment: 29 pages, 9 figures, Latex styl
Separation of variables for the quantum SL(2,R) spin chain
We construct representation of the Separated Variables (SoV) for the quantum
SL(2,R) Heisenberg closed spin chain and obtain the integral representation for
the eigenfunctions of the model. We calculate explicitly the Sklyanin measure
defining the scalar product in the SoV representation and demonstrate that the
language of Feynman diagrams is extremely useful in establishing various
properties of the model. The kernel of the unitary transformation to the SoV
representation is described by the same "pyramid diagram" as appeared before in
the SoV representation for the SL(2,C) spin magnet. We argue that this kernel
is given by the product of the Baxter Q-operators projected onto a special
reference state.Comment: 26 pages, Latex style, 9 figures. References corrected, minor
stylistic changes, version to be publishe
Mirror channel eigenvectors of the d-dimensional fishnets
International audienceWe present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d-dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers (u, l ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in (1 + 1)-dimensions with O(d) internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation, new to our knowledge, of the matrix elements of the fused R-matrix with O(d) symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions
On Complex Gamma-Function Integrals
It was observed recently that relations between matrix elements of certain operators in the spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with symmetry group and as a local Hilbert space give rise to a new type of -function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals