Integrable effective field theories and their holographic descriptions

Two-dimensional quantum field theory deformed by Zamolodchikov's TTbar operator has recently attracted the attention of theoretical physicists due to the many links with string theory and AdS/CFT. In this talk, I will review some interesting facts about the TTbar-like deformations of simple systems in one, two and four space-time dimensions.

Among many interesting features, the TTbar deformation preserves the integrability of the original theory. In the first part of the talk, I will present an efficient method to explicitly compute the local conserved currents of a TTbar-deformed integrable field theory (IFT), using a field-dependent coordinate transformation involving the stress-energy tensor. In the second part, I consider an infinite family of coordinate transformations built out of generic conserved currents of the original IFT and I study the integrable structure of the corresponding deformed theories for a simple model. Quite remarkably, this analysis suggests that the finite-size spectrum of these perturbed systems is obtained by introducing Lorentz-breaking scattering phase factors in the S-matrix of the original theory.

I will argue that the natural gravitational picture of the TTbar deformation is ghost-free massive gravity, showing that massive gravity naturally emerges as an appropriately defined version of the random geometries framework. The important role of field dependent diffs is just a straightforward consequence of the Stuckelberg formalism of massive gravity. I will discuss these various connections, how (non)-critical strings arise naturally as the deformation of CFTs, and how to give path integral definitions for more general deformations.

Starting from the definition of the TTbar deformation as a CDD factor in the factorised S-matrix, I will show that one can derive generalised flow equations for two-dimensional Quantum Field Theories deformed by generalised current-current operators. I will also argue that a similar picture holds for integrable spin chains, where the CDD factor is taken as a deformation of the Bethe ansatz equations. This talk will be based on arXiv:1911.12233 and 1911.12315.

We show that the T-Tbar - deformation of integrable QFT has a natural counterpart in integrable spin chains: it corresponds to a certain kind of long range deformation which had already been studied in the literature. However, the connection between the two deformations was not yet noticed. We show that this connection can be understood through the current and generalized current operators. As a by-product we derive a rather general formula for the mean values of current operators in Bethe Ansatz solvable models.

Integrable quantum field theories (IQFT) can be regularized on the lattice while preserving integrability. The resulting lattice model is also integrable. We propose an integrable deformation of the lattice model such that in the continuum limit we obtain the TTbar deformed IQFT.

I will review various aspects of Yang-Baxter deformations, keeping as a common theme their interpretation as marginal deformations. I will also remind how a subclass of Yang-Baxter (the TsT transformations) provide the holographic analog of the single-trace TTbar deformation, and of its generalisations.

The study of integrable eta-deformations of AdS superstrings has led to a number of puzzles, in particular concerning the status of Weyl invariance in the deformed theories. In this talk, I will review the construction of the eta-models and discuss when the deformed backgrounds solve the standard type IIB supergravity equations of motion. While this resolves one of the main puzzles of eta-deformed superstrings, others remain including the behavior in the maximal deformation limit.

I will briefly review integrable $\lambda$-deformations of worldsheet models and their relation to T-dualities. I will discuss how the construction of the deformation can be generalised in various ways—while keeping (classical) integrability—focusing on recent work where the theory deforms an exact worldsheet CFT realised as an asymmetrically gauged Wess-Zumino-Witten model. As a low-dimensional example, I will apply this deformation to the Euclidean d = 2 ‘cigar’ black hole, corresponding to the axially gauged SL(2,R)/U(1) WZW theory, and I will argue the interest of this example as an integrable deformation of its Sine-Liouville holographic dual.