We construct two-parameter families of integrable λ-deformations of two-dimensional field theories. These interpolate between a CFT (a WZW/gauged WZW model) and the non-Abelian T-dual of a principal chiral model on a group/symmetric coset space. In examples based on the SU(2) WZW model and the SU(2)/U(1) exact coset CFT, we show that these deformations are related to bi-Yang-Baxter generalisations of ηdeformations via Poisson-Lie T-duality and analytic continuation. We illustrate the quantum behaviour of our models under RG flow. As a byproduct we demonstrate that the bi-Yang-Baxter σ-model for a general group is one-loop renormalisable. Introduction and motivationOne of the most powerful tools available to the modern holographic practitioner is integrability. Most famously, the problem of determining the anomalous dimensions of single trace operators in the planar limit of N = 4 supersymmetric Yang-Mills gauge theory with gauge group SU(N) can be mapped to the problem of determining eigenvalues of an integrable spin-chain Hamiltonian [1]. On the other side of the AdS/CFT conjecture, the AdS 5 × S 5 string σ-model is, classically at least, integrable. The reason for this is that the σ-model's target space is exceptionally symmetric; the world sheet theory takes the form of a σ-model on a semi-symmetric space PSU(2, 2|4)/SO(4, 1) × SO(5) [2]. The two-dimensional σ-model admits a Lax pair formulation from which an infinite tower of conserved quantities can be deduced [3]. Given this success, one would hope to find ways in which the AdS/CFT correspondence can be generalised from the AdS 5 × S 5 setting whilst still maintaining the properties of integrability. Two novel and related classes of two-dimensional σ-models, that we shall refer to as η-and λ-deformations, have recently been developed and provide a new perspective on this challenge. The η-deformation of the AdS 5 × S 5 superstring proposed by Delduc, Magro and Vicedo [4, 5] is a generalisation of the Yang-Baxter (YB) deformations introduced by Klimčík in [6]. A central rôle in the construction of such YB deformations is played by the antisymmetric R-matrix; an endomorphism of a Lie-algebra g that obeys a modified YB (mYB) equation [RA, RB] − R([RA, B] + [A, RB]) = −c 2 [A, B] , ∀A, B ∈ g , c ∈ C .(1.1)There are three distinct choices for the parameter c; c 2 > 0, c 2 < 0 and c 2 = 0 and the corresponding solutions of the mYB are referred to as being, respectively, on the real, complex and classical branch. 1 The complex branch, c 2 < 0, is the setting for the η-deformations. Using such an R-matrix one can construct a one-parameter family of deformations of the principal chiral model on a group G which were shown 1 Contracting (A.4), equivalent form of (1.1), with f abc and using the Jacobi identity we easily find that c G c 2 dimG + 3 2 ||ξ|| 2 = 0 , ||ξ|| 2 = δ ab ξ a ξ b , ξ a = f abc R bc , which has no solution for compact groups and c 2 = 1, referred to as the real branch.
We provide the set of equations for non-relativistic fluid dynamics on arbitrary, possibly time-dependent spaces, in general coordinates. These equations are fully covariant under either local Galilean or local Carrollian transformations, and are obtained from standard relativistic hydrodynamics in the limit of infinite or vanishing velocity of light. All dissipative phenomena such as friction and heat conduction are included in our description. Part of our work consists in designing the appropriate coordinate frames for relativistic spacetimes, invariant under Galilean or Carrollian diffeomorphisms. The guide for the former is the dynamics of relativistic point particles, and leads to the Zermelo frame. For the latter, the relevant objects are relativistic instantonic space-filling branes in Randers-Papapetrou backgrounds. We apply our results for obtaining the general first-derivative-order Galilean fluid equations, in particular for incompressible fluids (Navier-Stokes equations) and further illustrate our findings with two applications: Galilean fluids in rotating frames or inflating surfaces and Carrollian conformal fluids on two-dimensional time-dependent geometries. The first is useful in atmospheric physics, while the dynamics emerging in the second is governed by the Robinson-Trautman equation, describing a Calabi flow on the surface, and known to appear when solving Einstein's equations for algebraically special Ricci-flat or Einstein spacetimes.
We show that a holographic description of four-dimensional asymptotically locally flat spacetimes is reached smoothly from the zero-cosmological-constant limit of anti-de Sitter holography. To this end, we use the derivative expansion of fluid/gravity correspondence. From the boundary perspective, the vanishing of the bulk cosmological constant appears as the zero velocity of light limit. This sets how Carrollian geometry emerges in flat holography. The new boundary data are a two-dimensional spatial surface, identified with the null infinity of the bulk Ricci-flat spacetime, accompanied with a Carrollian time and equipped with a Carrollian structure, plus the dynamical observables of a conformal Carrollian fluid. These are the energy, the viscous stress tensors and the heat currents, whereas the Carrollian geometry is gathered by a two-dimensional spatial metric, a frame connection and a scale factor. The reconstruction of Ricci-flat spacetimes from Carrollian boundary data is conducted with a flat derivative expansion, resummed in a closed form in Eddington-Finkelstein gauge under further integrability conditions inherited from the ancestor anti-de Sitter set-up. These conditions are hinged on a duality relationship among fluid friction tensors and Cotton-like geometric data. We illustrate these results in the case of conformal Carrollian perfect fluids and Robinson-Trautman viscous hydrodynamics. The former are dual to the asymptotically flat Kerr-Taub-NUT family, while the latter leads to the homonymous class of algebraically special Ricci-flat spacetimes.
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