55 research outputs found
Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces
Using Polyakov's functional integral approach with the Liouville action
functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville
theory on a compact Riemann surface X of genus g > 1. For the partition
function and for the correlation functions with the stress-energy tensor
components , we
describe Feynman rules in the background field formalism by expanding
corresponding functional integrals around a classical solution - the hyperbolic
metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define the
regularization scheme for any choice of global coordinate on X, and for
Schottky and quasi-Fuchsian global coordinates we rigorously prove that one-
and two-point correlation functions satisfy conformal Ward identities in all
orders of the perturbation theory. Obtained results are interpreted in terms of
complex geometry of the projective line bundle \cE_{c}=\lambda_{H}^{c/2} over
the moduli space , where c is the central charge and
is the Hodge line bundle, and provide Friedan-Shenker \cite{FS}
complex geometry approach to CFT with the first non-trivial example besides
rational models.Comment: 67 pages, 4 figures (Typos corrected as in the published version
Semi-classical scalar products in the generalised SU(2) model
In these notes we review the field-theoretical approach to the computation of
the scalar product of multi-magnon states in the Sutherland limit where the
magnon rapidities condense into one or several macroscopic arrays. We formulate
a systematic procedure for computing the 1/M expansion of the
on-shell/off-shell scalar product of M-magnon states in the generalised
integrable model with SU(2)-invariant rational R-matrix. The coefficients of
the expansion are obtained as multiple contour integrals in the rapidity plane.Comment: 13 pages, 3 figures. Based on a talk delivered at the X.
International Workshop "Lie Theory and Its Applications in Physics", (LT-10),
Varna, Bulgaria, 17-23 June 201
A system of difference equations with elliptic coefficients and Bethe vectors
An elliptic analogue of the deformed Knizhnik-Zamolodchikov equations is
introduced. A solution is given in the form of a Jackson-type integral of Bethe
vectors of the XYZ-type spin chains.Comment: 20 pages, AMS-LaTeX ver.1.1 (amssymb), 15 figures in LaTeX picture
environment
Separation of Variables in the Classical Integrable SL(3) Magnetic Chain
There are two fundamental problems studied by the theory of hamiltonian
integrable systems: integration of equations of motion, and construction of
action-angle variables. The third problem, however, should be added to the
list: separation of variables. Though much simpler than two others, it has
important relations to the quantum integrability. Separation of variables is
constructed for the magnetic chain --- an example of integrable model
associated to a nonhyperelliptic algebraic curve.Comment: 13 page
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
Minimal surfaces and particles in 3-manifolds
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic,
anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these
manifolds admit ``nice'' foliations and explicit metrics, and whether the space
of these metrics has a simple description in terms of Teichm\"uller theory. In
the hyperbolic settings both questions have positive answers for a certain
subset of the quasi-Fuchsian manifolds: those containing a closed surface with
principal curvatures at most 1. We show that this subset is parameterized by an
open domain of the cotangent bundle of Teichm\"uller space. These results are
extended to ``quasi-Fuchsian'' manifolds with conical singularities along
infinite lines, known in the physics literature as ``massive, spin-less
particles''.
Things work better for globally hyperbolic anti-de Sitter manifolds: the
parameterization by the cotangent of Teichm\"uller space works for all
manifolds. There is another description of this moduli space as the product two
copies of Teichm\"uller space due to Mess. Using the maximal surface
description, we propose a new parameterization by two copies of Teichm\"uller
space, alternative to that of Mess, and extend all the results to manifolds
with conical singularities along time-like lines. Similar results are obtained
for de Sitter or Minkowski manifolds.
Finally, for all four settings, we show that the symplectic form on the
moduli space of 3-manifolds that comes from parameterization by the cotangent
bundle of Teichm\"uller space is the same as the 3-dimensional gravity one.Comment: 53 pages, no figure. v2: typos corrected and refs adde
Noncompact SL(2,R) spin chain
We consider the integrable spin chain model - the noncompact SL(2,R) spin
magnet. The spin operators are realized as the generators of the unitary
principal series representation of the SL(2,R) group. In an explicit form, we
construct R-matrix, the Baxter Q-operator and the transition kernel to the
representation of the Separated Variables (SoV). The expressions for the energy
and quasimomentum of the eigenstates in terms of the Baxter Q-operator are
derived. The analytic properties of the eigenvalues of the Baxter operator as a
function of the spectral parameter are established. Applying the diagrammatic
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 a product of certain operators
each depending on a single separated variable.Comment: 29 pages, 12 figure
Liouville field theory with heavy charges. I. The pseudosphere
We work out the perturbative expansion of quantum Liouville theory on the pseudosphere starting from the semiclassical limit of a background generated by heavy charges. By solving perturbatively the Riemann-Hilbert problem for the Poincar\ue9 accessory parameters, we give in closed form the exact Green function on the background generated by one finite charge. Such a Green function is used to compute the quantum determinants i.e. the one loop corrections to known semiclassical limits thus providing the resummation of infinite classes of standard perturbative graphs. The results obtained for the one point function are compared with the bootstrap formula while those for the two point function are compared with the existing double perturbative expansion and with a degenerate case, finding complete agreement. \ua9 SISSA 2006
Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets
We consider correlation functions of the spin-\half XXX and XXZ Heisenberg
chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive
representations for various correlation functions in terms of determinants of
Fredholm integral operators.Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos correcte
Differential Calculus on the Quantum Superspace and Deformation of Phase Space
We investigate non-commutative differential calculus on the supersymmetric
version of quantum space where the non-commuting super-coordinates consist of
bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum
deformation of the general linear supergroup, , is studied and the
explicit form for the -matrix, which is the solution of the
Yang-Baxter equation, is presented. We derive the quantum-matrix commutation
relation of and the quantum superdeterminant. We apply these
results for the to the deformed phase-space of supercoordinates and
their momenta, from which we construct the -matrix of q-deformed
orthosymplectic group and calculate its -matrix. Some
detailed argument for quantum super-Clifford algebras and the explict
expression of the -matrix will be presented for the case of
.Comment: 17 pages, KUCP-4
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