Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries

Abstract

Abstract:We review the role of integrability in the planar spectral problem of fourdimensional superconformal gauge theories besides N = 4 SYM. The cases considered include the Leigh-Strassler marginal deformations of N = 4 SYM, quiver theories which arise as orbifolds of AdS 5 ×S 5 on the dual gravity side, as well as various theories involving open spin chains.

The AdS3× S3× S3× S1worldsheetSmatrix

“…All the remaining ones can be grouped together, as the representations for masses |m| = 0, 1 −α, α have the same grading. We read off two irreducible representations for each mass, each labelled by left or right, 14 for a total of eight two-dimensional irreducible representations. They are all short representations, satisfying the shortening condition…”

“…The best understood AdS/CFT dual pairs are given by type IIB strings on AdS 5 × S 5 and the dual N = 4 Supersymmetric Yang-Mills (SYM) theory, and its close relative type IIA string theory on AdS 4 × CP 3 [6][7][8] and the dual ABJM Chern-Simons theory [9,10], see references [11][12][13]] for reviews and a more complete list of references. Integrability seems to be quite a robust feature of such backgrounds, as it persists for their orbifolds, orientifolds as well as for certain deformations [14,15]. It is natural to wonder if integrability underlies other instances of AdS/CFT, and in particular whether AdS 3 /CFT 2 enjoys such hidden symmetries.…”

Integrable Deformation of theAdS5×S5Superstring Action

“…An important example of such an integrable deformation is the so called β-deformation associated with strings on the Lunin-Maldacena background [5]. The integrability of this model was shown in [6,7] (see also the review [8] and references therein). Here we shall take a more systematic approach to the construction of integrable deformations by demanding the deformed theory to be integrable from the very outset.…”

The dual string σ-model of the SU q (3) sector

“…However, no comparison could be made there between gauge and string theory, since no Bethe equations were formulated for the SU(3) sector for complex β. 6 For more details, we refer the reader to the recent review [34] and references therein.…”