Classical integrability and super Yangian of superstring on AdS$_5\times {\rm S}^5$

Abstract

We discuss a classical integrability in the type IIB string theory on the AdS 5 × S 5 background. By using the Roiban-Siegel formulation of the superstring on the AdS 5 ×S 5 , we carefully treat the Wess-Zumino term and the constraint conditions intrinsic to the supersymmetric case, and construct explicitly non-local charges for a hidden infinite-dimensional symmetry. The existence of the symmetry is shown by Bena-Polchinski-Roiban. Then the super Yangian algebra is calculated. We also show the Serre relation ensuring the structure of the Hopf algebra.In addition, the classical integrability is discussed by constructing the Lax pair and the transfer matrix.

A Yangian double for the AdS/CFT classical r-matrix

“…α with the central element C, and their infinite Yangian partners labeled by an integer n, whose fundamental representation for n = 0 is given by 10) and 12) as well as 13) with a, b, c, d defined by the limit ζ → 0 of the corresponding variables introduces in [4] (see also [18])…”

“…The additional scalar (dressing) factor [6] is constrained by the Hopf-algebraic analog of crossing symmetry [7]. Remarkable advances in the exact determination of this phase factor have been recently made [8,9], as well as progress in understanding the nature of the infinite dimensional symmetry algebra underlying the planar integrability of the model [10,11,12]. In [13], in particular, a Yangian symmetry has been advocated, and the hope for the existence of a universal R-matrix which could reproduce the AdS/CFT scattering matrix upon choosing an appropriate representation, has been put on somewhat firmer grounds.…”

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

“…It has applications in mathematical physics because it describes symmetry in integrable models of Calogero-Sutherland systems [1,12], superstrings in AdS 5 × S 5 [11], and in the hierarchy of a form of the non-linear super-Schrödinger equation with m bosons and n fermions [4]. The centre of the Yangian Y (gl m|n ) is conveniently described using a formal power series called the quantum Berezinian (see Section 7).…”

Ramond-Ramond gauge fields in superspace with manifest T-duality

“…space Poincaré/Lorentz, S mn = S m ′ n ′ = 0 are coset constraints andΣ mn andΣ m ′ n ′ are also constrained as gauge fields [42][43][44]. From the algebra (2.6) both S mn = 0 and Σ mn = 0 are not first class constraints, but S mn = 0 and the symmetry generatorΣ µν ′ = 0 are first class constraints where symmetry generators commute with covariant derivatives.…”