22 research outputs found
Vertex Lie algebras and cyclotomic coinvariants
Electronic version of an article published as BenoĂźt Vicedo and Charles Young, Commun. Contemp. Math. 0, 1650015 (2016) [62 pages] DOI: http://dx.doi.org/10.1142/S0219199716500152 Vertex Lie algebras and cyclotomic coinvariants.Given a vertex Lie algebra equipped with an action by automorphisms of a cyclic group , we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over `local' Lie algebras assigned to marked points , by the action of a `global' Lie algebra of -equivariant functions. On the other hand, the universal enveloping vertex algebra of is itself a vertex Lie algebra with an induced action of . This gives `big' analogs of the Lie algebras above. From these we construct the space of `big' cyclotomic coinvariants, i.e. coinvariants with respect to . We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in arXiv:1409.6937. At the origin, which is fixed by , one must assign a module over the stable subalgebra of . This module becomes a -quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.Peer reviewe
A Symplectic Structure for String Theory on Integrable Backgrounds
We define regularised Poisson brackets for the monodromy matrix of classical
string theory on R x S^3. The ambiguities associated with Non-Ultra Locality
are resolved using the symmetrisation prescription of Maillet. The resulting
brackets lead to an infinite tower of Poisson-commuting conserved charges as
expected in an integrable system. The brackets are also used to obtain the
correct symplectic structure on the moduli space of finite-gap solutions and to
define the corresponding action-angle variables. The canonically-normalised
action variables are the filling fractions associated with each cut in the
finite-gap construction. Our results are relevant for the leading-order
semiclassical quantisation of string theory on AdS_5 x S^5 and lead to
integer-valued filling fractions in this context.Comment: 41 pages, 2 figures; added references, corrected typos, improved
discussion of Hamiltonian constraint
Giant Magnons and Singular Curves
We obtain the giant magnon of Hofman-Maldacena and its dyonic generalisation
on R x S^3 < AdS_5 x S^5 from the general elliptic finite-gap solution by
degenerating its elliptic spectral curve into a singular curve. This alternate
description of giant magnons as finite-gap solutions associated to singular
curves is related through a symplectic transformation to their already
established description in terms of condensate cuts developed in
hep-th/0606145.Comment: 34 pages, 17 figures, minor change in abstrac
Gaudin models for gl(m|n)
Date of Acceptance: 16/04/2015We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.Peer reviewedFinal Accepted Versio
Deformed integrable Ï-models, classical R-matrices and classical exchange algebra on Drinfelâd doubles
We describe a unifying framework for the systematic construction of integrable deformations of integrable Ï-models within the Hamiltonian formalism. It applies equally to both the 'YangâBaxter' type as well as 'gauged WZW' type deformations which were considered recently in the literature. As a byproduct, these two families of integrable deformations are shown to be PoissonâLie T-dual of one anotherPeer reviewedFinal Accepted Versio
The classical R-matrix of AdS/CFT and its Lie dialgebra structure
The classical integrable structure of Z_4-graded supercoset sigma-models,
arising in the AdS/CFT correspondence, is formulated within the R-matrix
approach. The central object in this construction is the standard R-matrix of
the Z_4-twisted loop algebra. However, in order to correctly describe the Lax
matrix within this formalism, the standard inner product on this twisted loop
algebra requires a further twist induced by the Zhukovsky map, which also plays
a key role in the AdS/CFT correspondence. The non-ultralocality of the
sigma-model can be understood as stemming from this latter twist since it leads
to a non skew-symmetric R-matrix.Comment: 22 pages, 2 figure
Cyclotomic Gaudin models, Miura opers and flag varieties
Let g be a semisimple Lie algebra over C. Let ÎœâAutg be a diagram automorphism whose order divides TâZâ„1. We define cyclotomic g-opers over the Riemann sphere P1 as gauge equivalence classes of g-valued connections of a certain form, equivariant under actions of the cyclic group Z/TZ on g and P1. It reduces to the usual notion of g-opers when T=1. We also extend the notion of Miura g-opers to the cyclotomic setting. To any cyclotomic Miura g-oper â we associate a corresponding cyclotomic g-oper. Let â have residue at the origin given by a Îœ-invariant rational dominant coweight λË0 and be monodromy-free on a cover of P1. We prove that the subset of all cyclotomic Miura g-opers associated with the same cyclotomic g-oper as â is isomorphic to the Ï-invariant subset of the full flag variety of the adjoint group G of g, where the automorphism Ï depends on Îœ, T and λË0. The big cell of the latter is isomorphic to NÏ, the Ï-invariant subgroup of the unipotent subgroup NâG, which we identify with those cyclotomic Miura g-opers whose residue at the origin is the same as that of â. In particular, the cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is interpreted as taking â to other cyclotomic Miura g-opers corresponding to elements of NÏ associated with simple root generators. We motivate the introduction of cyclotomic g-opers by formulating two conjectures which relate them to the cyclotomic Gaudin model of [arXiv:1409.6937]
Cyclotomic Gaudin models: construction and Bethe ansatz
This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra and automorphism we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case . We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe