[Paper Review] Integrable strings for AdS/CFT

Riccardo Borsato|arXiv (Cornell University)|May 10, 2016

Black Holes and Theoretical Physics175 references17 citations

TL;DR

This PhD thesis presents exact S-matrix solutions for string theory on AdS₃×S³×T⁴ with pure Ramond-Ramond flux and the η-deformed AdS₅×S⁵ background, using symmetry-based methods in light-cone gauge. It derives all-loop Bethe-Yang equations, constructs spin-chain descriptions, and solves crossing equations for dressing phases, providing a complete integrable framework for both massive and massless sectors.

ABSTRACT

In this PhD thesis we review some aspects of integrable models related to string backgrounds or their deformations. In the first part we develop methods to obtain exact results in the AdS3/CFT2 correspondence. We consider the AdS_3 x S^3 x T^4 background with pure Ramond-Ramond flux and we find the all-loop worldsheet S-matrix by exploiting the symmetries of the model in light-cone gauge. As we naturally include the massless modes on the worldsheet, we derive the full set of Bethe-Yang equations. In the massive sector we give also a spin-chain description and we write down solutions compatible with crossing for the scalar factors which are not constrained by simmetries. In the second part of the thesis we consider the so-called "eta-deformation" of the superstring on AdS_5 x S^5. We first discuss the effects of the deformation at the level of the bosonic sigma-model, and we match the tree-level worldsheet scattering processes to the expansion of the q-deformed S-matrix. To identify the missing Ramond-Ramond fields we then compute the action quadratic in fermions, and we provide an alternative derivation by looking at the kappa-symmetry variations. The resulting background fields do not solve the equations of motion of type IIB supergravity and we comment on this.

Motivation & Objective

• To develop exact methods for the AdS₃/CFT₂ correspondence using integrability.
• To derive the all-loop worldsheet S-matrix for AdS₃×S³×T⁴ with pure Ramond-Ramond flux by exploiting light-cone gauge symmetries.
• To construct a complete set of Bethe-Yang equations including massless modes and provide a spin-chain description for the massive sector.
• To investigate the η-deformation of AdS₅×S⁵ at the bosonic and fermionic levels, matching to q-deformed S-matrices.
• To analyze the consistency of the deformed background with type IIB supergravity equations of motion.

Proposed method

• Utilizes light-cone gauge quantization to derive the worldsheet S-matrix for AdS₃×S³×T⁴, preserving all symmetries of the model.
• Applies the off-shell symmetry algebra 𝔰𝔲(1|1)²₊ to construct S-matrix elements invariant under the central charge extension.
• Derives Bethe-Yang equations from the S-matrix using the Yang-Baxter equation and crossing symmetry constraints.
• Constructs a spin-chain realization of the massive sector using the 𝔰𝔭𝔰𝔲(1,1|2)² algebra and computes central extensions.
• Solves the crossing equations for dressing factors using functional equations and bound-state structures.
• Performs a perturbative expansion of the fermionic action in the η-deformed model and verifies consistency with the q-deformed S-matrix at tree level.

Experimental results

Research questions

• RQ1 How can the all-loop S-matrix be constructed for AdS₃×S³×T⁴ with pure Ramond-Ramond flux using symmetry-based methods?
• RQ2 What is the complete set of Bethe-Yang equations including massless modes in the AdS₃×S³×T⁴ background?
• RQ3 How do the dressing factors in the S-matrix satisfy crossing symmetry and unitarity in the massive sector?
• RQ4 What is the structure of the η-deformed bosonic and fermionic worldsheet S-matrix, and how does it match the q-deformed S-matrix?
• RQ5 Do the background fields resulting from the η-deformation satisfy the equations of motion of type IIB supergravity?

Key findings

• The all-loop worldsheet S-matrix for AdS₃×S³×T⁴ is derived as a tensor product of 𝔰𝔲(1|1)²₊-invariant S-matrices, with full crossing invariance and unitarity satisfied.
• The full set of Bethe-Yang equations is derived, including contributions from massive, mixed-mass, and massless modes, consistent with the worldsheet spectrum.
• A spin-chain description is constructed for the massive sector using the 𝔰𝔭𝔰𝔲(1,1|2)² algebra, with central charges correctly realized.
• The dressing factors are solved via crossing equations, with explicit solutions for the scalar factors not fixed by symmetry.
• The η-deformation of the AdS₅×S⁵ background leads to a q-deformed S-matrix at tree level, confirmed by matching the quartic action and scattering amplitudes.
• The resulting deformed background fields do not solve the type IIB supergravity equations of motion, indicating a breakdown of the supergravity approximation in the η-deformed case.