Motivated by the recent proposal for the S-matrix in AdS 3 × S 3 with mixed three form fluxes, we study classical folded string spinning in AdS 3 with both Ramond and Neveu-Schwarz three form fluxes. We solve the equations of motion of these strings and obtain their dispersion relation to the leading order in the Neveu-Schwarz flux b. We show that dispersion relation for the spinning strings with large spin S acquires a term given by − √ λ 2π b 2 log 2 S in addition to the usual √ λ π log S term where √ λ is proportional to the square of the radius of AdS 3 . Using SO(2, 2) transformations and re-parmetrizations we show that these spinning strings can be related to light like Wilson loops in AdS 3 with Neveu-Schwarz flux b. We observe that the logarithmic divergence in the area of the light like Wilson loop is also deformed by precisely the same coefficient of the b 2 log 2 S term in the dispersion relation of the spinning string. This result indicates that the coefficient of b 2 log 2 S has a property similar to the coefficient of the log S term, known as cusp-anomalous dimension, and can possibly be determined to all orders in the coupling λ using the recent proposal for the S-matrix.
Using the fact the BTZ black hole is a quotient of AdS_3 we show that classical string propagation in the BTZ background is integrable. We construct the flat connection and its monodromy matrix which generates the non-local charges. From examining the general behaviour of the eigen values of the monodromy matrix we determine the set of integral equations which constrain them. These equations imply that each classical solution is characterized by a density function in the complex plane. For classical solutions which correspond to geodesics and winding strings we solve for the eigen values of the monodromy matrix explicitly and show that geodesics correspond to zero density in the complex plane. We solve the integral equations for BMN and magnon like solutions and obtain their dispersion relation. Finally we show that the set of integral equations which constrain the eigen values of the monodromy matrix can be identified with the continuum limit of the Bethe equations of a twisted SL(2, R) spin chain at one loop.Comment: 45 pages, Reference added, typos corrected, discussion on geodesics improved to include all geodesic
The string sigma model in AdS 3 ×S 3 supported by mixed three-form fluxes has recently been proved to be integrable which led to a plethora of work in this background including proposals for S matrix and study of semiclassical string profiles. Motivated by this, in this paper, we present a study of 'spiky' strings in this background. We analyze the string profiles in detail and also find the dispersion relation between the charges in the 'long' string limit after solving the equations of motion perturbatively upto the leading order in the Neveu-Schwarz flux b. We find that the dispersion for 2 spikes gets corrected by the term − b 2 2 log S. We also discuss the fate of the solution in the limit of pure NS-NS flux.
We study the structure constants of the N = 1 beta deformed theory perturbatively and at strong coupling. We show that the planar one loop corrections to the structure constants of single trace gauge invariant operators in the scalar sector is determined by the anomalous dimension Hamiltonian. This result implies that 3 point functions of the chiral primaries of the theory do not receive corrections at one loop. We then study the structure constants at strong coupling using the Lunin-Maldacena geometry. We explicitly construct the supergravity mode dual to the chiral primary with three equal U(1) R-charges in the Lunin-Maldacena geometry. We show that the 3 point function of this supergravity mode with semi-classical states representing two other similar chiral primary states but with large U(1) charges to be independent of the beta deformation and identical to that found in the AdS 5 × S 5 geometry. This together with the one-loop result indicate that these structure constants are protected by a non-renormalization theorem. We also show that three point function of U(1) R-currents with classical massive strings is proportional to the R-charge carried by the string solution. This is in accordance with the prediction of the R-symmetry Ward identity.1 A perturbative study of BPS operators of this theory was also done in [29,30]. 12 . (5.4) Comparing (5.3) and (5.4) results in C J (λ) = − J 2π 2 . (5.5)This is an all-loop prediction for the structure constant C J (λ). In the next section we will verify this at strong coupling in the Lunin-Maldacena geometry. The method
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