arXiv subjects
Publications and source records attributed to Rafael Hernandez.
We consider the quantization of the non-relativistic limit of electrodynamics called Galilean electrodynamics. To that end, we apply the Dirac bracket formalism to understand the constraints and dynamics of the theory and analyze its gauge structure. We show that the theory is fully constrained, which eliminates all the degrees of freedom from the path integral.
The Bethe ansatz, both in its coordinate and its algebraic version, is an exceptional method to compute the eigenvectors and eigenvalues of integrable systems. However, computing correlation functions using the eigenvectors thus constructed complicates rather fast. In this article, we will compute some simple correlation functions for the isotropic Heisenberg spin chain to highlight the shortcomings of both Bethe ansätze. In order to compare the results obtained from each approach, a discussion on the normalization of states in each ansatz will be required. We will show that the analysis can be extended to the long-range spin chain governing the spectrum of anomalous dimensions of single trace operators in four-dimensional Yang-Mills with maximal supersymmetry.
The simplest example of the $λ$-deformation connects the SU(2) Wess-Zumino-Witten model with the non-Abelian T-dual (NATD) of the SU(2) principal chiral model. We analyze spinning strings with one spin propagating through the $λ$-deformation of the target space of the interpolation. We show that the situation apart from the NATD limit parallels the undeformed case. We demonstrate that regular spinning strings are either folded or circular, and that nearly degenerate spinning strings are either nearly point-like, fast, or slow. The effects of the $λ$-deformation are both the overall increment of the energy of spinning strings and the enlargement of the gap between the energies of folded and circular strings. In the NATD limit, we prove that circular strings disappear and that fast strings realize the dispersion relation of Gubser-Klebanov-Polyakov strings.
We study the reduction of classical strings rotating in the deformed three-sphere truncation of the double Yang-Baxter deformation of the $\hbox{AdS}_3 \times \hbox{S}^3 \times \hbox{T}^4$ background to an integrable mechanical model. The use of the generalized spinning-string ansatz leads to an integrable deformation of the Neumann-Rosochatius system. Integrability of this system follows from the fact that the usual constraints for the Uhlenbeck constants apply to any deformation that respects the isometric coordinates of the three-sphere. We construct solutions to the system in terms of the underlying ellipsoidal coordinate. The solutions depend on the domain of the deformation parameters and the reality conditions of the roots of a fourth order polynomial. We obtain constant-radii, giant-magnon and trigonometric solutions when the roots degenerate, and analyze the possible solutions in the undeformed limit. In the case where the deformation parameters are purely imaginary and the polynomial involves two complex-conjugated roots, we find a new class of solutions. The new class is connected with twofold giant-magnon solutions in the degenerate limit of infinite period.
Classical strings propagating in $AdS_{3}\times S^{3}\times T^{4}$ supported with Neveu-Schwarz-Neveu-Schwarz flux are described by a Wess-Zumino-Witten model. In this note, we study the emergence of their semiclassical $SU(2)$ spectrally flowed sectors as the Landau-Lifshitz limit of the underlying quantum spin chain. We consider the propagator in the coherent state picture, and find that the time interval is discretized proportionally to the lattice spacing. In the Landau-Lifshitz limit, where both time and space become continuous, we derive a path integral representation of the propagator for each spectrally flowed sector. We prove that the arbitrariness of the global phase of coherent states is mapped to the gauge freedom of the $B$-field in the classical action. We show that higher order corrections in the Landau-Lifshitz limit are suppressed by inverse powers of the 't Hooft coupling.
We obtain the ratio of semiclassical partition functions for the extension under mixed flux of the minimal surfaces subtending a circumference and a line in Euclidean $AdS_{3}\times S^{3}\times T^{4}$. We reduce the problem to the computation of a set of functional determinants. If the Ramond-Ramond flux does not vanish, we find that the contribution of the $B$-field is comprised in the conformal anomaly. In this case, we successively apply the Gel'fand-Yaglom method and the Abel-Plana formula to the flat-measure determinants. To cancel the resultant infrared divergences, we shift the regularization of the sum over half-integers depending on whether it corresponds to massive or massless fermionic modes. We show that the result is compatible with the zeta-function regularization approach. In the limit of pure Neveu-Schwarz-Neveu-Schwarz flux we argue that the computation trivializes. We extend the reasoning to other surfaces with the same behavior in this regime.
We study minimal area world sheets ending on two concentric circumferences on the boundary of Euclidean $AdS_{3}$ with mixed R-R and NS-NS three-form fluxes. We solve the problem by reducing the system to a one-dimensional integrable model. We find that the NS-NS flux term either brings the surface near to the boundary or separates the circumferences. In the limit of pure NS-NS flux the solution adheres to the boundary in the former case and the outer radius diverges in the latter. We further construct the underlying elliptic spectral curve, which allows us to analyze the deformation of other related minimal surfaces. We show that in the regime of pure NS-NS flux the elliptic curve degenerates.
Circular strings pulsating in $AdS_3 \times S^3 \times T^4$ with mixed R-R and NS-NS three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in $AdS_{3}$ with pure NS-NS three-form flux, where the action reduces to the $SL(2,\mathbb{R})$ WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in $AdS_{3}$ to extend the dispersion relation beyond the limit of pure NS-NS flux.
The sigma-model of closed strings spinning in the $η$-deformation of $AdS_{5} \times S^{5}$ leads to an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical system. In this article we construct general solutions to this system that can be written in terms of elliptic functions. The solutions correspond to closed strings with non-constant radii rotating with two different angular momenta in an $η$-deformed three-sphere. We analyse the reduction of the elliptic solutions for some limiting values of the deformation parameter. For the case of solutions with constant radii we find the dependence of the classical energy of the string on the angular momenta as an expansion in the 't Hooft coupling.
Closed strings spinning in AdS_3 x S^3 x T^4 with mixed R-R and NS-NS three-form fluxes are described by a deformation of the one-dimensional Neumann-Rosochatius integrable system. In this article we find general solutions to this system that can be expressed in terms of elliptic functions. We consider closed strings rotating either in S^3 with two different angular momenta or in AdS_3 with one spin. In order to find the solutions we will need to extend the Uhlenbeck integrals of motion of the Neumann-Rosochatius system to include the contribution from the flux. In the limit of pure NS-NS flux, where the problem can be described by a supersymmetric WZW model, we find exact expressions for the classical energy in terms of the spin and the angular momenta of the spinning string.
The sigma model describing closed strings rotating in AdS_3 x S^3 is known to reduce to the one-dimensional Neumann-Rosochatius integrable system. In this article we show that closed spinning strings in AdS_3 x S^3 x T^4 in the presence of NS-NS three-form flux can be described by an extension of the Neumann-Rosochatius system. We consider closed strings rotating with one spin in AdS_3 and two different angular momenta in S^3. For a class of solutions with constant radii we find the dependence of the classical energy on the spin and the angular momenta as an expansion in the square of the 't Hooft coupling of the theory.
We consider the ratio of the correlation function of an hexagon light-like Wilson loop with one local operator over the expectation value of the Wilson loop within the strong-coupling regime of the AdS/CFT correspondence. We choose the hexagon Wilson loop within a class of minimal solutions obtained by cutting and gluing light-like quadrangle loops. These surfaces do not have an interpretation in terms of dual scattering amplitudes but they still exhibit general features of the mixed correlation function. In the case of a regular null hexagon conformal symmetry constrains the space-time dependence of the correlator up to a function of three conformal cross-ratios. We obtain the leading-order contribution to the correlation function in the semiclassical approximation of large string tension, and express the result in terms of three conformal ratios in the case where the local operator is taken to be the dilaton. We include the analysis of an irregular Wilson loop obtained after a boost of the regular hexagon.
We analyze correlation functions of Wilson loop observables and local vertex operators within the strong-coupling regime of the AdS/CFT correspondence. When the local operator corresponds to a light string state with finite conserved charges the correlation function can be evaluated in the semiclassical approximation of large string tension, where the contribution from the light vertex can be neglected. We consider the cases where the Wilson loops are described by two concentric surfaces and the local vertices are the superconformal chiral primary scalar or a singlet massive scalar operator.
Three-point correlation functions in the strong-coupling regime of the AdS/CFT correspondence can be analyzed within a semiclassical approximation when two of the vertex operators correspond to heavy string states having large quantum numbers while the third vertex corresponds to a light state with fixed charges. We consider the case where the heavy string states are chosen to be giant magnon solitons with either a single or two different angular momenta, for various different choices of light string states.
The strong-coupling limit of three-point correlation functions of local operators can be analyzed beyond the supergravity regime using vertex operators representing spinning string states. When two of the vertex operators correspond to heavy string states having large quantum numbers, while the third operator corresponds to a light state with fixed charges, the correlator can be computed in the large string tension limit by means of a semiclassical approximation. We study the case when the heavy string states are circular string solutions with one AdS_5 spin and three different angular momenta along S^5, for several choices of the light string state.
We extract from the double logarithmic contributions to DGLAP anomalous dimensions for twist-two operators up to three-loops the magnon dispersion relation for planar N=4 supersymmetric Yang-Mills. Perturbatively the magnon dispersion relation agrees with the expansion of the anomalous dimension for spin-one as well as with the non-collinear double logarithmic contributions to the BFKL anomalous dimensions analytically extended to negative spin. The all-loop expression for the magnon dispersion relation is determined by the double logarithmic resummation of the corresponding Bethe-Salpeter equation. A potential map relating the spin chain magnon to BFKL eigenfunctions in the double logarithm approximation is suggested.
The scattering-matrix for planar Yang-Mills with N=4 supersymmetry relies on the assumption that integrability holds to all orders in perturbation theory. In this note we define a map from the spectral variables x^{\pm}, parameterizing the long-range magnon momenta, to couplings in a two-dimensional Ising model. Under this map integrability of planar N=4 Yang-Mills becomes equivalent to the Yang-Baxter equation for the two-dimensional Ising model, and the long-range variables x^{\pm} translate into the entries of the Ising transfer matrices. We explore the Ising correlation length which equals the inverse magnon momentum in the small momentum limit. The critical regime is thus reached for vanishing magnon momentum. We also discuss the meaning of the Kramers-Wannier duality transformation on the gauge theory, together with that of the Ising model critical points.
The dispersion relation for planar N=4 supersymmetric Yang-Mills is identified with the Casimir of a quantum deformed two-dimensional kinematical symmetry, E_q(1,1). The quantum deformed symmetry algebra is generated by the momentum, energy and boost, with deformation parameter q=e^{2πi/λ}. Representing the boost as the infinitesimal generator for translations on the rapidity space leads to an elliptic uniformization with crossing transformations implemented through translations by the elliptic half-periods. This quantum deformed algebra can be interpreted as the kinematical symmetry of a discrete integrable model with lattice spacing given by the BMN length a=2π/\sqrtλ. The interpretation of the boost generator as the corner transfer matrix is briefly discussed.