arXiv subjects

C. A. S. Young

Publications and source records attributed to C. A. S. Young.

At least 19 recordsLinked to original sources

arXiv2014

Quantum loop algebras and l-root operators

Let g be a simple Lie algebra and q transcendental. We consider the category C_P of finite-dimensional representations of the quantum loop algebra Uq(Lg) in which the poles of all l-weights belong to specified finite sets P. Given the data (g,q,P), we define an algebra A whose raising/lowering operators are constructed to act with definite l-weight (unlike those of Uq(Lg) itself). It is shown that there is a homomorphism Uq(Lg) -> A such that every representation V in C_P is the pull-back of a representation of A.

math.QA↗

arXiv2012

Affinization of category O for quantum groups

Let g be a simple Lie algebra. We consider the category O-hat of those modules over the affine quantum group Uq(g-hat) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category O-hat. In particular, we develop the theory of q-characters and define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.

math.QA↗

arXiv2012

Extended T-systems

We use the theory of q-characters to establish a number of short exact sequences in the category of finite-dimensional representations of the quantum affine groups of types A and B. That allows us to introduce a set of 3-term recurrence relations which contains the celebrated T-system as a special case.

math.QA↗

arXiv2011

Reflecting magnons from D7 and D5 branes

We obtain the reflection matrices for the scattering of elementary magnons from certain open boundaries, corresponding to open strings ending on D7 and D5 branes in $AdS_5\times S^5$. In each case we consider two possible orientations for the vacuum state. We show that symmetry arguments are sufficient to determine the reflection matrices up to at most two unknown functions. The D7 reflection matrices obey the boundary Yang Baxter-Equation. This is automatic for one vacuum orientation, and requires a natural choice of ratio between two unknowns for the other. In contrast, the D5 reflection matrices do not obey the boundary Yang Baxter-Equation. In both cases we show consistency with the existent weak and strong coupling results.

hep-th↗

arXiv2011

Path description of type B q-characters

We give a set of sufficient conditions for a Laurent polynomial to be the q-character of a finite-dimensional irreducible representation of a quantum affine group. We use this result to obtain an explicit path description of q-characters for a class of modules in type B. In particular, this proves a conjecture of Kuniba-Ohta-Suzuki.

math.QA↗

arXiv2010

Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let Uq(ghat) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey's rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations V_{i,a} of Uq(ghat). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.

math.QA↗

arXiv2010

Sutherland Models for Complex Reflection Groups

There are known to be integrable Sutherland models associated to every real root system -- or, which is almost equivalent, to every real reflection group. Real reflection groups are special cases of complex reflection groups. In this paper we associate certain integrable Sutherland models to the classical family of complex reflection groups. Internal degrees of freedom are introduced, defining dynamical spin chains, and the freezing limit taken to obtain static chains of Haldane-Shastry type. By considering the relation of these models to the usual BC_N case, we are led to systems with both real and complex reflection groups as symmetries. We demonstrate their integrability by means of new Dunkl operators, associated to wreath products of dihedral groups.

math-ph↗

arXiv2010

Asymptotic Bethe equations for open boundaries in planar AdS/CFT

We solve, by means of a nested coordinate Bethe ansatz, the open-boundaries scattering theory describing the excitations of a free open string propagating in $AdS_5\times S^5$, carrying large angular momentum $J=J_{56}$, and ending on a maximal giant graviton whose angular momentum is in the same plane. We thus obtain the all-loop Bethe equations describing the spectrum, for $J$ finite but large, of the energies of such strings, or equivalently, on the gauge side of the AdS/CFT correspondence, the anomalous dimensions of certain operators built using the epsilon tensor of SU(N). We also give the Bethe equations for strings ending on a probe D7-brane, corresponding to meson-like operators in an $\mathcal N=2$ gauge theory with fundamental matter.

hep-th↗

arXiv2009

Finite size corrections for open strings/open chains in planar AdS/CFT

We identify the leading finite-size (Luscher-type) correction to the energy of open strings ending on maximal giant gravitons. In particular we obtain the leading finite size correction at weak 't Hooft coupling and in the planar limit to the energy of very short vacuum states. These results are shown to agree with certain 1, 2, 3 and 4-loop dual gauge theory perturbative calculations, which we also perform.

hep-th↗

arXiv2008

Triangular quasi-Hopf algebra structures on certain non-semisimple quantum groups

One way to obtain Quantized Universal Enveloping Algebras (QUEAs) of non-semisimple Lie algebras is by contracting QUEAs of semisimple Lie algebras. We prove that every contracted QUEA in a certain class is a cochain twist of the corresponding undeformed universal envelope. Consequently, these contracted QUEAs possess a triangular quasi-Hopf algebra structure. As examples, we consider kappa-Poincare in 3 and 4 spacetime dimensions.

math.QA↗

arXiv2008

On kappa-deformation and triangular quasibialgebra structure

We show that, up to terms of order 1/kappa^5, the kappa-deformed Poincare algebra can be endowed with a triangular quasibialgebra structure. The universal R matrix and coassociator are given explicitly to the first few orders. In the context of kappa-deformed quantum field theory, we argue that this structure, assuming it exists to all orders, ensures that states of any number of identical particles, in any representation, can be defined in a kappa-covariant fashion.

hep-th↗

arXiv2008

Covariant particle exchange for kappa-deformed theories in 1+1 dimensions

We consider the exchange of identical scalar particles in theories with kappa-deformed Poincare symmetry. We argue that, at least in 1+1 dimensions, the symmetric group S_N can be realized on the space of N-particle states in a kappa-covariant fashion. For the case of two particles this realization is unique: we show that there is only one non-trivial intertwiner, which automatically squares to the identity.

hep-th↗

arXiv2007

Covariant particle statistics and intertwiners of the kappa-deformed Poincare algebra

To speak about identical particles - bosons or fermions - in quantum field theories with kappa-deformed Poincare symmetry, one must have a kappa-covariant notion of particle exchange. This means constructing intertwiners of the relevant representations of kappa-Poincare. We show, in the simple case of spinless particles, that intertwiners exist, and, supported by a perturbative calculation to third order in 1/kappa, make a conjecture about the existence and uniqueness of a certain preferred intertwiner defining particle exchange in kappa-deformed theories.

hep-th↗

arXiv2007

q-Deformed Supersymmetry and Dynamic Magnon Representations

It was recently noted that the dispersion relation for the magnons of planar N=4 SYM can be identified with the Casimir of a certain deformation of the Poincare algebra, in which the energy and momentum operators are supplemented by a boost generator J. By considering the relationship between J and su(2|2) x R^2, we derive a q-deformed super-Poincare symmetry algebra of the kinematics. Using this, we show that the dynamic magnon representations may be obtained by boosting from a fixed rest-frame representation. We comment on aspects of the coalgebra structure and some implications for the question of boost-covariance of the S-matrix.

hep-th↗

arXiv2005

Bethe Equations for a g_2 Model

We prove, using the coordinate Bethe ansatz, the exact solvability of a model of three particles whose point-like interactions are determined by the root system of g_2. The statistics of the wavefunction are left unspecified. Using the properties of the Weyl group, we are also able to find Bethe equations. It is notable that the method relies on a certain generalized version of the well-known Yang-Baxter equation. A particular class of non-trivial solutions to this equation emerges naturally.

math-ph↗

arXiv2005

Quantum, higher-spin, local charges in symmetric space sigma models

Potential anomalies are analysed for the local spin-3 and spin-4 classically conserved currents in any two-dimensional sigma model on a compact symmetric space $G/H$, with $G$ and $H$ classical groups. Quantum local conserved charges are shown to exist in exactly those models which also possess quantum non-local (Yangian) charges. The possibility of larger sets of quantum local charges is discussed and shown to be consistent with known S-matrix results and the behaviour of the corresponding Yangian representations.

hep-th↗

arXiv2005

Higher-spin conserved currents in supersymmetric sigma models on symmetric spaces

Local higher-spin conserved currents are constructed in the supersymmetric sigma models with target manifolds symmetric spaces $G/H$. One class of currents is based on generators of the de Rham cohomology ring of $G/H$; a second class of currents are higher-spin generalizations of the (super)energy-momentum tensor. A comprehensive analysis of the invariant tensors required to construct these currents is given from two complimentary points of view, and sets of primitive currents are identified from which all others can be constructed as differential polynomials. The Poisson bracket algebra of the top component charges of the primitive currents is calculated. It is shown that one can choose the primitive currents so that the bosonic charges all Poisson-commute, while the fermionic charges obey an algebra which is a form of higher-spin generalization of supersymmetry. Brief comments are made on some implications for the quantized theories.

hep-th↗

arXiv2004

Non-local charges and quantum integrability of sigma models on the symmetric spaces SO(2n)/SO(n)xSO(n) and Sp(2n)/Sp(n)xSp(n)

Non-local conserved charges in two-dimensional sigma models with target spaces $SO(2n)/SO(n){\times}SO(n)$ and $Sp(2n)/Sp(n){\times}Sp(n)$ are shown to survive quantization, unspoiled by anomalies; these theories are therefore integrable at the quantum level. Local, higher-spin, conserved charges are also shown to survive quantization in the $SO(2n)/SO(n){\times}SO(n)$ models.

hep-th↗