Spinning strings and minimal surfaces in $Ad{S}_3$ with mixed 3-form fluxes

Abstract

Motivated by the recent proposal for the S-matrix in $Ad{S}_3\times S^3$ with mixed three form fluxes, we study classical folded string spinning in $Ad{S}_3$ 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 $\mathcal{S}$ acquires a term given by $-\frac{\sqrt{\lambda }}{2\pi }{b}^2{\log }^2\mathcal{S}$ in addition to the usual $\frac{\sqrt{\lambda }}{\pi }\log \mathcal{S}$ term where $\sqrt{\lambda }$ is proportional to the square of the radius of $Ad{S}_3$. Using SO(2,2) transformations and re-parmetrizations we show that these spinning strings can be related to light like Wilson loops in $Ad{S}_3$ 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 $b^2{\log }^2\mathcal{S}$ term in the dispersion relation of the spinning string. This result indicates that the coefficient of $b^2{\log }^2\mathcal{S}$ has a property similar to the coefficient of the $\log \mathcal{S}$ term, known as cusp-anomalous dimension, and can possibly be determined to all orders in the coupling $\lambda$ using the recent proposal for the S-matrix.

arXiv comment34 pages, Accepted for publication in JHEP

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