18 research outputs found
R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
The problem of constructing the SL(N,C) invariant solutions to the Yang-Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N,C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be identified with the Baxter operators for the noncompact SL(N,C) spin magnet
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
The spectrum of the anomalous dimensions of the composite operators in the - expansion in the scalar - field theory
The spectrum of the anomalous dimensions of the composite operators (with
arbitrary number of fields and derivatives ) in the scalar -
theory in the first order of the -expansion is investigated. The
exact solution for the operators with number of fields is presented.
The behaviour of the anomalous dimensions in the large limit has been
analyzed. It is given the qualitative description of the %structure of the
spectrum for the arbitrary .Comment: 25 pages, latex, a few changes in latex command
Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
We analyze a completely integrable two-dimensional quantum-mechanical model
that emerged in the recent studies of the compound gluonic states in
multi-color QCD at high energy. The model represents a generalization of the
well-known homogenous Heisenberg spin magnet to infinite-dimensional
representations of the SL(2,C) group and can be reformulated within the Quantum
Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the
R-matrix for the SL(2,C) representations of the principal series and discuss
its properties. We explicitly construct the Baxter Q-operator for this model
and show how it can be used to determine the energy spectrum. We apply
Sklyanin's method of the Separated Variables to obtain an integral
representation for the eigenfunctions of the Hamiltonian. We demonstrate that
the language of Feynman diagrams supplemented with the method of uniqueness
provide a powerful technique for analyzing the properties of the model.Comment: 61 pages, 19 figures; version to appear in Nucl.Phys.
Quantum integrability in (super) Yang-Mills theory on the light-cone
We employ the light-cone formalism to construct in the (super) Yang-Mills
theories in the multi-color limit the one-loop dilatation operator acting on
single trace products of chiral superfields separated by light-like distances.
In the N=4 Yang-Mills theory it exhausts all Wilson operators of the maximal
Lorentz spin while in nonsupersymmetric Yang-Mills theory it is restricted to
the sector of maximal helicity gluonic operators. We show that the dilatation
operator in all N-extended super Yang-Mills theories is given by the same
integral operator which acts on the (N+1)-dimensional superspace and is
invariant under the SL(2|N) superconformal transformations. We construct the
R-matrix on this space and identify the dilatation operator as the Hamiltonian
of the Heisenberg SL(2|N) spin chain.Comment: 18 pages, 1 figure; replaced with correct revised versio
Superconformal operators in Yang-Mills theories on the light-cone
We employ the light-cone superspace formalism to develop an efficient
approach to constructing superconformal operators of twist two in Yang-Mills
theories with N=1,2,4 supercharges. These operators have an autonomous scale
dependence to one-loop order and determine the eigenfunctions of the dilatation
operator in the underlying gauge theory. We demonstrate that for arbitrary N
the superconformal operators are given by remarkably simple, universal
expressions involving the light-cone superfields. When written in components
field, they coincide with the known results obtained by conventional
techniques.Comment: 29 pages, Late
Computation of quark mass anomalous dimension at O(1/N_f^2) in quantum chromodynamics
We present the formalism to calculate d-dimensional critical exponents in QCD
in the large N_f expansion where N_f is the number of quark flavours. It relies
in part on demonstrating that at the d-dimensional fixed point of QCD the
critical theory is equivalent to a non-abelian version of the Thirring model.
We describe the techniques used to compute critical two and three loop Feynman
diagrams and as an application determine the quark wave function, eta, and mass
renormalization critical exponents at O(1/N_f^2) in d-dimensions. Their values
when expressed in relation to four dimensional perturbation theory are in exact
agreement with the known four loop MSbar results. Moreover, new coefficients in
these renormalization group functions are determined to six loops and
O(1/N_f^2). The computation of the exponents in the Schwinger Dyson approach is
also provided and an expression for eta in arbitrary covariant gauge is given.Comment: 41 latex pages, 17 postscript figure
Dilatation operator in (super-)Yang-Mills theories on the light-cone
The gauge/string correspondence hints that the dilatation operator in gauge
theories with the superconformal SU(2,2|N) symmetry should possess universal
integrability properties for different N. We provide further support for this
conjecture by computing a one-loop dilatation operator in all (super)symmetric
Yang-Mills theories on the light-cone ranging from gluodynamics all the way to
the maximally supersymmetric N=4 theory. We demonstrate that the dilatation
operator takes a remarkably simple form when realized in the space spanned by
single-trace products of superfields separated by light-like distances. The
latter operators serve as generating functions for Wilson operators of the
maximal Lorentz spin and the scale dependence of the two are in the one-to-one
correspondence with each other. In the maximally supersymmetric, N=4 theory all
nonlocal light-cone operators are built from a single CPT self-conjugated
superfield while for N=0,1,2 one has to deal with two distinct superfields and
distinguish three different types of such operators. We find that for the
light-cone operators built from only one species of superfields, the one-loop
dilatation operator takes the same, universal form in all SYM theories and it
can be mapped in the multi-color limit into a Hamiltonian of the SL(2|N)
Heisenberg (super)spin chain of length equal to the number of superfields
involved. For "mixed'' light-cone operators involving both superfields the
dilatation operator for N<=2 receives an additional contribution from the
exchange interaction between superfields on the light-cone which breaks its
integrability symmetry and creates a mass gap in the spectrum of anomalous
dimensions.Comment: 70 pages, 3 figures; minor changes, references 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
The simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion
The simple method for the calculating of the anomalous dimensions of the
composite operators up to 1/N^2 order is developed. We demonstrate the
effectiveness of this approach by computing the critical exponents of the
and
operators in the 1/N^2 order in the nonlinear sigma model. The special
simplifications due to the conformal invariance of the model are discussed.Comment: 20 pages, Latex, uses Feynman.st