Searcharxiv⌕ Search

arXiv subjects

Benoit Vicedo

Publications and source records attributed to Benoit Vicedo.

54 records · Page 3Linked to original sources
arXiv2014

Cyclotomic Gaudin models: construction and Bethe ansatz

To any simple Lie algebra $\mathfrak g$ and automorphism $σ:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $σ= \text{id}$. 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.

math.QA↗
arXiv2014

Derivation of the action and symmetries of the q-deformed AdS_5 x S^5 superstring

We recently proposed an integrable q-deformation of the AdS_5 x S^5 superstring action. Here we give details on the hamiltonian origin and construction of this deformation. The procedure is a generalization of the one previously developed for deforming principal chiral and symmetric space sigma-models. We also show that the original psu(2,2|4) symmetry is replaced in the deformed theory by a classical analog of the quantum group U_q(psu(2,2|4)) with q real. The relation between q and the deformation parameter entering the action is given. The framework used to derive the deformation also enables to prove that at the hamiltonian level, the "maximal deformation" limit corresponds to an undeformed semi-symmetric space sigma-model with bosonic part dS_5 x H^5. Finally, we discuss the various freedoms in the construction.

hep-th↗
arXiv2014

Gaudin models for gl(m|n)

We 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.

math.QA↗
arXiv2013

On classical q-deformations of integrable sigma-models

A procedure is developed for constructing deformations of integrable sigma-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter sigma-model introduced a few years ago by C. Klimcik. In the case of the symmetric space sigma-model on F/G we obtain a new one-parameter family of integrable sigma-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset sigma-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset sigma-model which interpolates all the way to the SU(1,1)/U(1) coset sigma-model.

hep-th↗
arXiv2012

Generalized sine-Gordon models and quantum braided groups

We determine the quantized function algebras associated with various examples of generalized sine-Gordon models. These are quadratic algebras of the general Freidel-Maillet type, the classical limits of which reproduce the lattice Poisson algebra recently obtained for these models defined by a gauged Wess-Zumino-Witten action plus an integrable potential. More specifically, we argue based on these examples that the natural framework for constructing quantum lattice integrable versions of generalized sine-Gordon models is that of affine quantum braided groups.

math-ph↗
arXiv2012

Alleviating the non-ultralocality of coset sigma models through a generalized Faddeev-Reshetikhin procedure

The Faddeev-Reshetikhin procedure corresponds to a removal of the non-ultralocality of the classical SU(2) principal chiral model. It is realized by defining another field theory, which has the same Lax pair and equations of motion but a different Poisson structure and Hamiltonian. Following earlier work of M. Semenov-Tian-Shansky and A. Sevostyanov, we show how it is possible to alleviate in a similar way the non-ultralocality of symmetric space sigma models. The equivalence of the equations of motion holds only at the level of the Pohlmeyer reduction of these models, which corresponds to symmetric space sine-Gordon models. This work therefore shows indirectly that symmetric space sine-Gordon models, defined by a gauged Wess-Zumino-Witten action with an integrable potential, have a mild non-ultralocality. The first step needed to construct an integrable discretization of these models is performed by determining the discrete analogue of the Poisson algebra of their Lax matrices.

hep-th↗
arXiv2012

Alleviating the non-ultralocality of the AdS_5 x S^5 superstring

We generalize the initial steps of the Faddeev-Reshetikhin procedure to the AdS_5 x S^5 superstring theory. Specifically, we propose a modification of the Poisson bracket whose alleviated non-ultralocality enables to write down a lattice algebra for the Lax matrix. We then show that the dynamics of the Pohlmeyer reduction of the AdS_5 x S^5 superstring can be naturally reproduced with respect to this modified Poisson bracket. This work generalizes the alleviation procedure recently developed for symmetric space sigma-models. It also shows that the lattice algebra recently obtained for the AdS_5 x S^5 semi-symmetric space sine-Gordon theory coincides with the one obtained by the alleviation procedure.

hep-th↗
arXiv2011

Splitting strings on integrable backgrounds

We use integrability to construct the general classical splitting string solution on R x S^3. Namely, given any incoming string solution satisfying a necessary self-intersection property at some given instant in time, we use the integrability of the worldsheet sigma-model to construct the pair of outgoing strings resulting from a split. The solution for each outgoing string is expressed recursively through a sequence of dressing transformations, the parameters of which are determined by the solutions to Birkhoff factorization problems in an appropriate real form of the loop group of SL(2,C).

hep-th↗
arXiv2010

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.

hep-th↗
arXiv2009

Hamiltonian dynamics and the hidden symmetries of the AdS_5 x S^5 superstring

We construct the Lax connection of the Green-Schwarz superstring in AdS_5 x S^5 within the Hamiltonian formalism and obtain precisely that used in 0810.4136. It differs in a crucial way from the Bena-Polchinski-Roiban connection by terms proportional to the Hamiltonian constraints. These extra terms ensure firstly that the integrals of motion are all first class and secondly that the Lax connection is flat in the strong sense.

hep-th↗
arXiv2008

Finite-g Strings

In view of one day proving the AdS/CFT correspondence, a deeper understanding of string theory on certain curved backgrounds such as AdS_5xS^5 is required. In this dissertation we make a step in this direction by focusing on RxS^3. It was discovered in recent years that string theory on AdS_5xS^5 admits a Lax formulation. However, the complete statement of integrability requires not only the existence of a Lax formulation, but also that the resulting integrals of motion are in pairwise involution. This idea is central to the first part of this thesis. Exploiting this integrability we apply algebro-geometric methods to string theory on RxS^3 and obtain the general finite-gap solution. The construction is based on an invariant algebraic curve previously found in the AdS_5xS^5 case. However, encoding the dynamics of the solution requires specification of additional marked points. By restricting the symplectic structure of the string to this algebro-geometric data we derive the action-angle variables of the system. We then perform a first-principle semiclassical quantisation of string theory on RxS^3 as a toy model for strings on AdS_5xS^5. The result is exactly what one expects from the dual gauge theory perspective, namely the underlying algebraic curve discretises in a natural way. We also derive a general formula for the fluctuation energies around the generic finite-gap solution. The ideas used can be generalised to AdS_5xS^5.

hep-th↗
arXiv2008

Semiclassical Quantisation of Finite-Gap Strings

We perform a first principle semiclassical quantisation of the general finite-gap solution to the equations of a string moving on R x S^3. The derivation is only formal as we do not regularise divergent sums over stability angles. Moreover, with regards to the AdS/CFT correspondence the result is incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are not taken into account. Nevertheless, the calculation serves the purpose of understanding how the moduli of the algebraic curve gets quantised semiclassically, purely from the point of view of finite-gap integration and with no input from the gauge theory side. Our result is expressed in a very compact and simple formula which encodes the infinite sum over stability angles in a succinct way and reproduces exactly what one expects from knowledge of the dual gauge theory. Namely, at tree level the filling fractions of the algebraic curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At 1-loop order the filling fractions receive Maslov index corrections of hbar/2 and all the singular points of the spectral curve become filled with small half-integer multiples of hbar. For the subsector in question this is in agreement with the previously obtained results for the semiclassical energy spectrum of the string using the method proposed in hep-th/0703191. Along the way we derive the complete hierarchy of commuting flows for the string in the R x S^3 subsector. Moreover, we also derive a very general and simple formula for the stability angles around a generic finite-gap solution. We also stress the issue of quantum operator orderings since this problem already crops up at 1-loop in the form of the subprincipal symbol.

hep-th↗
arXiv2007

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.

hep-th↗
arXiv2007

Large Winding Sector of AdS/CFT

We study a family of classical strings on R x S^3 subspace of the AdS_5 x S^5 background that interpolates between pulsating strings and single-spike strings. They are obtained from the helical strings of hep-th/0609026 by interchanging worldsheet time and space coordinates, which maps rotating/spinning string states with large spins to oscillating states with large winding numbers. From a finite-gap perspective, this transformation is realised as an interchange of quasi-momentum and quasi-energy defined for the algebraic curve. The gauge theory duals are also discussed, and are identified with operators in the non-holomorphic sector of N=4 super Yang-Mills. They can be viewed as excited states above the ``antiferromagnetic'' state, which is ``the farthest from BPS'' in the spin-chain spectrum. Furthermore, we investigate helical strings on AdS_3 x S^1 in an appendix.

hep-th↗
arXiv2007

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.

hep-th↗
arXiv2006

On the Dynamics of Finite-Gap Solutions in Classical String Theory

We study the dynamics of finite-gap solutions in classical string theory on R x S^3. Each solution is characterised by a spectral curve, Σ, of genus g and a divisor, γ, of degree g on the curve. We present a complete reconstruction of the general solution and identify the corresponding moduli-space, M^(2g)_R, as a real symplectic manifold of dimension 2g. The dynamics of the general solution is shown to be equivalent to a specific Hamiltonian integrable system with phase-space M^(2g)_R. The resulting description resembles the free motion of a rigid string on the Jacobian torus J(Σ). Interestingly, the canonically-normalised action variables of the integrable system are identified with certain filling fractions which play an important role in the context of the AdS/CFT correspondence.

hep-th↗