Abstract
One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, that is, the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two‐dimensional σ‐models and in the AdS/CFT context. The first part is focused on the AdS5/CFT4 duality, especially the classical and quantum integrability of the type IIB superstring on AdS5 × S5 which is discussed in both pure spinor and Green‐Schwarz formulations. The second part is dedicated to the AdS4/CFT3 duality with particular attention to the type IIA superstring on AdS4 × ℂP3 and its integrability. This review is based on the author′s PhD thesis discussed at Uppsala University the 21st September 2009.
Highlights
Motivations, Overview, and OutlineIn 1997, Maldacena conjectured that type IIB superstrings on AdS5 × S5 describe the same physics of the supersymmetric SU N Yang-Mills theory in four dimensions 1 AdS5/CFT4
On the gauge theory side, this is due to the correspondence between the N 4 SYM theory and the one-dimensional spin chain, in particular, it follows from the identification between the dilatation operator and the spin chain Hamiltonian, cf
The integrable structures which emerge on both sides of the AdS5/CFT4 correspondence, manifest themselves with an infinite set of conserved charges
Summary
The AdS4/CFT3 states a duality between a three-dimensional conformal field theory and an M-theory on eleven dimensions. The theory is conformally invariant at classical and quantum level and it possesses N 6 supersymmetries It contains two parameters:[83] the gauge group rank, N, and the level of the algebra k. The circle radius is given by RS1 ∼ Nk 1/6/k When such radius is very large, namely, when N k5, the theory is strongly coupled and the proper description is in terms of the M-theory. We are going to consider only a specific region for the gravity side of the correspondence: the string regime This means that for us N and k are very large and in particular are such that N k5 or 1 λ k4.
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