TheAdS5×S5superstring worldsheetSmatrix and crossing symmetry

An $S$ matrix satisfying the Yang-Baxter equation with symmetries relevant to the ${\mathrm{AdS}}_5\times S^5$ superstring recently has been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations; however, due to the lack of conventional relativistic invariance, in this case its determination remained an open problem. In this paper we propose an algebraic way to implement crossing relations for the ${\mathrm{AdS}}_5\times S^5$ superstring worldsheet $S$ matrix. We base our construction on a Hopf-algebraic formulation of crossing in terms of the antipode and introduce generalized rapidities living on the universal cover of the parameter space which is constructed through an auxillary, coupling-constant dependent, elliptic curve. We determine the crossing transformation and write functional equations for the scalar factor of the $S$ matrix in the generalized rapidity plane.