The AdS₅ × S⁵ semi-symmetric space sine-Gordon theory

Abstract

The generalized symmetric space sine-Gordon theories are a series of 1 + 1-integrable field theories that are classically equivalent to superstrings on symmetric space spacetimes F/G. They are formulated in terms of a semi-symmetric space as a gauged WZW model with fermions and a potential term to deform it away from the conformal fixed point. We consider in particular the case of PSU(2, 2–4)/Sp(2, 2) × Sp(4) which corresponds to AdS₅ × S⁵. We argue that the infinite tower of conserved charges of these theories includes an exotic $\mathcal{N} = \left( {8,8} \right)$ supersymmetry that is realized in a mildy non-local way at the Lagrangian level. The supersymmetry is associated to a double central extension of the superalgebra $\mathfrak{p}\mathfrak{s}\mathfrak{u}\left( {2\left| 2 \right.} \right) \oplus \mathfrak{p}\mathfrak{s}\mathfrak{u}\left( {2\left| 2 \right.} \right)$ and includes a non-trivial R symmetry algebra corresponding to global gauge transformations, as well as 2-dimensional spacetime translations. We then explicitly construct soliton solutions and show that they carry an internal moduli superspace $\mathbb{C}{P^{2\left| 1 \right.}} \times \mathbb{C}{P^{2\left| 1 \right.}}$ with both bosonic and Grassmann collective coordinates. We show how to semi-classical quantize the solitons by writing an effective quantum mechanical system on the moduli space which takes the form of a co-adjoint orbit of SU(2–2)^×2. The spectrum consists of a tower of massive states in the short, or atypical, symmetric representations, just as the giant magnon states of the string world sheet theory, although here the tower is truncated.