2,338 research outputs found
Cayley--Klein Contractions of Quantum Orthogonal Groups in Cartesian Basis
Spaces of constant curvature and their motion groups are described most
naturally in Cartesian basis. All these motion groups also known as CK groups
are obtained from orthogonal group by contractions and analytical
continuations. On the other hand quantum deformation of orthogonal group is most easily performed in so-called symplectic basis. We reformulate its
standard quantum deformation to Cartesian basis and obtain all possible
contractions of quantum orthogonal group both for untouched and
transformed deformation parameter. It turned out, that similar to undeformed
case all CK contractions of are realized. An algorithm for obtaining
nonequivalent (as Hopf algebra) contracted quantum groups is suggested.
Contractions of are regarded as an examples.Comment: The statement of the basic theorem have correct. 30 pages, Latex.
Report given at X International Conference on Symmetry Methods in Physics,
August 13-19, 2003, Yerevan, Armenia. Submitted in Journal Physics of Atomic
Nucle
Analytic Solution of Bremsstrahlung TBA
We consider the quark--anti-quark potential on the three sphere or the
generalized cusp anomalous dimension in planar N=4 SYM. We concentrate on the
vacuum potential in the near BPS limit with units of R-charge.
Equivalently, we study the anomalous dimension of a super-Wilson loop with L
local fields inserted at a cusp. The system is described by a recently proposed
infinite set of non-linear integral equations of the Thermodynamic Bethe Ansatz
(TBA) type. That system of TBA equations is very similar to the one of the
spectral problem but simplifies a bit in the near BPS limit. Using techniques
based on the Y-system of functional equations we first reduced the infinite
system of TBA equations to a Finite set of Nonlinear Integral Equations
(FiNLIE). Then we solve the FiNLIE system analytically, obtaining a simple
analytic result for the potential! Surprisingly, we find that the system has
equivalent descriptions in terms of an effective Baxter equation and in terms
of a matrix model. At L=0, our result matches the one obtained before using
localization techniques. At all other L's, the result is new. Having a new
parameter, L, allows us to take the large L classical limit. We use the matrix
model description to solve the classical limit and match the result with a
string theory computation. Moreover, we find that the classical string
algebraic curve matches the algebraic curve arising from the matrix model.Comment: 50 pages, 5 figures. v2: references added, JHEP versio
Quark--anti-quark potential in N=4 SYM
We construct a closed system of equations describing the quark--anti-quark
potential at any coupling in planar N=4 supersymmetric Yang-Mills theory. It is
based on the Quantum Spectral Curve method supplemented with a novel type of
asymptotics. We present a high precision numerical solution reproducing the
classical and one-loop string predictions very accurately. We also analytically
compute the first 7 nontrivial orders of the weak coupling expansion.
Moreover, we study analytically the generalized quark--anti-quark potential
in the limit of large imaginary twist to all orders in perturbation theory. We
demonstrate how the QSC reduces in this case to a one-dimensional Schrodinger
equation. In the process we establish a link between the Q-functions and the
solution of the Bethe-Salpeter equation.Comment: 31 pages, 1 figure; v2: minor correcton
Numerical results for the exact spectrum of planar AdS4/CFT3
We compute the anomalous dimension for a short single-trace operator in
planar ABJM theory at intermediate coupling. This is done by solving
numerically the set of Thermodynamic Bethe Ansatz equations which are expected
to describe the exact spectrum of the theory. We implement a truncation method
which significantly reduces the number of integral equations to be solved and
improves numerical efficiency. Results are obtained for a range of 't Hooft
coupling lambda corresponding to , where h(lambda) is
the interpolating function of the AdS4/CFT3 Bethe equations.Comment: v3: corrected Acknowledgements section; v4: minor changes, published
version; v5: fixed typos in Eq. (3.9
Wrapping corrections, reciprocity and BFKL beyond the sl(2) subsector in N=4 SYM
We consider N=4 SYM and a class of spin N, length-3, twist operators beyond
the well studied sl(2) subsector. They can be identified at one-loop with three
gluon operators. At strong coupling, they are associated with spinning strings
with two spins in AdS5. We exploit the Y-system to compute the leading
weak-coupling four loop wrapping correction to their anomalous dimension. The
result is written in closed form as a function of the spin N. We combine the
wrapping correction with the known four-loop asymptotic Bethe Ansatz
contribution and analyze special limits in the spin N. In particular, at large
N, we prove that a generalized Gribov-Lipatov reciprocity holds. At negative
unphysical spin, we present a simple BFKL-like equation predicting the
rightmost leading poles.Comment: 18 page
Quantum Spectral Curve at Work: From Small Spin to Strong Coupling in N=4 SYM
We apply the recently proposed quantum spectral curve technique to the study
of twist operators in planar N=4 SYM theory. We focus on the small spin
expansion of anomalous dimensions in the sl(2) sector and compute its first two
orders exactly for any value of the 't Hooft coupling. At leading order in the
spin S we reproduced Basso's slope function. The next term of order S^2
structurally resembles the Beisert-Eden-Staudacher dressing phase and takes
into account wrapping contributions. This expansion contains rich information
about the spectrum of local operators at strong coupling. In particular, we
found a new coefficient in the strong coupling expansion of the Konishi
operator dimension and confirmed several previously known terms. We also
obtained several new orders of the strong coupling expansion of the BFKL
pomeron intercept. As a by-product we formulated a prescription for the correct
analytical continuation in S which opens a way for deriving the BFKL regime of
twist two anomalous dimensions from AdS/CFT integrability.Comment: 53 pages, references added; v3: due to a typo in the coefficients C_2
and D_2 on page 29 we corrected the rational part of the strong coupling
predictions in equations (1.5-6), (6.22-24), (6.27-30) and in Table
Noncommutative space-time models
The FRT quantum Euclidean spaces are formulated in terms of Cartesian
generators. The quantum analogs of N-dimensional Cayley-Klein spaces are
obtained by contractions and analytical continuations. Noncommutative constant
curvature spaces are introduced as a spheres in the quantum Cayley-Klein
spaces. For N=5 part of them are interpreted as the noncommutative analogs of
(1+3) space-time models. As a result the quantum (anti) de Sitter, Newton,
Galilei kinematics with the fundamental length and the fundamental time are
suggested.Comment: 8 pages; talk given at XIV International Colloquium of Integrable
Systems, Prague, June 16-18, 200
On contractions of classical basic superalgebras
We define a class of orthosymplectic and unitary
superalgebras which may be obtained from and
by contractions and analytic continuations in a similar way as the
special linear, orthogonal and the symplectic Cayley-Klein algebras are
obtained from the corresponding classical ones. Casimir operators of
Cayley-Klein superalgebras are obtained from the corresponding operators of the
basic superalgebras. Contractions of and are regarded as
an examples.Comment: 15 pages, Late
On the Fermionic Frequencies of Circular Strings
We revisit the semiclassical computation of the fluctuation spectrum around
different circular string solutions in AdS_5xS^5 and AdS_4xCP^3, starting from
the Green-Schwarz action. It has been known that the results for these
frequencies obtained from the algebraic curve and from the worldsheet
computations sometimes do not agree. In particular, different methods give
different results for the half-integer shifts in the mode numbers of the
frequencies. We find that these discrepancies can be removed if one carefully
takes into account the transition matrices in the spin bundle over the target
space.Comment: 13 pages, 1 figur
Possible contractions of quantum orthogonal groups
Possible contractions of quantum orthogonal groups which correspond to
different choices of primitive elements of Hopf algebra are considered and all
allowed contractions in Cayley--Klein scheme are obtained. Quantum deformations
of kinematical groups have been investigated and have shown that quantum analog
of (complex) Galilei group G(1,3) do not exist in our scheme.Comment: 10 pages, Latex. Report given at XXIII Int. Colloquium on Group
Theoretical Methods in Physics, July 31- August 5, 2000, Dubna (Russia
- …