<p>In a flat background, the canonical energy momentum tensor of Lorentz and conformally invariant matter field theories can be improved to a symmetric and traceless tensor that gives the same conserved charges. We argue that the geometric origin of this improvement process is unveiled when the matter theory is coupled to metric-affine gravity. In particular, we show that the Belinfante-Rosenfeld improvement terms correspond to the matter theory’s hypermomentum. The improvement terms in conformally invariant matter theories are also related to the hypermomentum; however, a general proof would require an extended investigation. We demonstrate our results through various examples, such as the free massless scalar, the Maxwell field, Abelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>-forms, the Dirac field, and a nonunitary massless scalar field. Possible applications of our method for theories that break Lorentz or special conformal invariance are briefly discussed.</p>

Estonia

Damianos Iosifidis

United States

Manthos Karydas

Greece

Anastasios Petkou

Konstantinos Siampos

Geometric origin of the energy-momentum tensor improvement terms

We construct a generalisation of the -deformation of the Principal Chiral Model (PCM) where we deform just a subgroup of the full symmetry group . We find that demanding Lax integrability imposes a crucial restriction, namely that the coset must be symmetric. Surprisingly, we also find that (when is non-abelian) integrability requires that the term in the action involving only the spectator fields should have a specific -dependence, which is a curious modification of the procedure expected from the known = case. The resulting Lax connection has a novel analytical structure, with four single poles as opposed to the two poles of the cases of the PCM and of the standard -deformation. We also explicitly work out the example of = SU(2) and = U(1), discussing its renormalisation group flow to two loops.

Spain

Riccardo Borsato

Germany

Georgios Itsios

J. Miramontes

Integrability of the

We provide the first supersymmetric embedding of an integrable λ-deformation to type-II supergravity. Specifically, that of the near horizon of the NS1-NS5 brane intersection, geometrically corresponding to AdS 3 × S 3 × T 4. We show that the deformed background preserves 1/4 of the maximal supersymmetry. In the Penrose limit we show that it preserves no-more than one half of the maximal supersymmetry.

Konstantinos Sfetsos

Supersymmetric backgrounds from λ-deformations

In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as the variation of the gravitational Chern-Simons action with respect to the metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved. Performing a reduction of the Cotton tensor with respect to Carrollian diffeomorphisms in a suitable frame, one discloses four sets of Cotton Carrollian relatives, which are conformal and obey Carrollian conservation equations. Each set of Carrollian Cotton tensors is alternatively obtained as the variation of a distinct Carroll-Chern-Simons action with respect to the degenerate metric and the clock form of a strong Carroll structure. The four Carroll-Chern-Simons actions emerge in the Carrollian reduction of the original Chern-Simons ascendant. They inherit its anomalous behaviour under diffeomorphisms and Weyl transformations. The extremums of these Carrollian actions are commented and illustrated.

Chile

Olivera Mišković

Rodrigo Olea

France

P. Petropoulos

David Rivera-Betancour

Chern-Simons action and the Carrollian Cotton tensors

We initiate the construction of integrable λ-deformed WZW models based on non-semisimple groups. We focus on the four-dimensional case whose underlying symmetries are based on the non-semisimple group   <math> <msubsup> <mi>E</mi> <mn>2</mn> <mi>c</mi> </msubsup> </math>  $$ {E}_2^c $$ . The corresponding gravitational backgrounds of Lorentzian signature are plane waves which can be obtained as Penrose limits of the λ-deformed SU(2) background times a timelike coordinate for appropriate choices of the λ-matrix. We construct two such deformations which we demonstrate to be integrable. They both deform the Nappi-Witten plane wave and are inequivalent. Nevertheless, they have the same underlying symmetry algebra which is a Saletan-type contraction of that for the λ-deformed SU(2) background with a timelike direction. We also construct a plane wave from the Penrose limit of the λ-deformation of the   <math> <mfrac> <mrow> <mi>SU</mi> <mfenced> <mn>2</mn> </mfenced> </mrow> <mrow> <mi>U</mi> <mfenced> <mn>1</mn> </mfenced> </mrow> </mfrac> </math>  $$ \frac{\textrm{SU}(2)}{\textrm{U}(1)} $$  coset CFT times a timelike coordinate which represents the deformation of a logarithmic CFT constructed in the past. Finally, we briefly consider contractions based on the simplest Yang-baxter σ-models.

Integrable models based on non-semi-simple groups and plane wave target spacetimes

We comprehensively study Galilean and Carrollian hydrodynamics on arbitrary backgrounds, in the presence of a matter/charge conserved current. For this purpose, we follow two distinct and complementary paths. The first is based on local invariance, be it Galilean or Carrollian diffeomorphism invariance, possibly accompanied by Weyl invariance. The second consists in analyzing the relativistic fluid equations at large or small speed of light, after choosing an adapted gauge, Arnowitt-Deser-Misner-Zermelo for the former and Papapetrou-Randers for the latter. Unsurprisingly, the results agree, but the second approach is superior as it effortlessly captures more elaborate situations with multiple degrees of freedom. It furthermore allows to investigate the fate of hydrodynamic-frame invariance in the two limits at hand, and conclude that its breaking (in the Galilean) or its preservation (in the Carrollian) are fragile consequences of the behaviour of transport attributes at large or small c. Both methods do also agree on the doom of Nœtherian currents generated in the relativistic theory by isometries: conserved currents are not always guaranteed in Newton-Cartan or Carroll spacetimes as a consequence of Galilean or Carrollian isometries. Comparison of Galilean and Carrollian fluid equations exhibits a striking but often superficial resemblance, which we comment in relation to black-hole horizon dynamics, awkwardly akin to Navier-Stokes equations. This congruity is authentic in one instance though and turns out then to describe Aristotelian dynamics, which is the last item in our agenda.

Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars

Kerr–Schild perturbations in General Relativity provide a fruitful way of constructing new exact solutions starting from known ones, elucidating also the structure of the spacetimes. We initiate such a study in the context of string theory and supergravity. Specifically, we explicitly construct Kerr–Schild perturbations of coset CFTs based on low dimensionality orthogonal groups. We show that these give rise to scale, but not Weyl, invariant integrable σ-models. We explicitly demonstrate that these models can also be derived from a particular limiting procedure of λ-deformed coset CFTs based on non-compact groups. The target space of the simplest σ-model describes a two-dimensional scale invariant black hole for which we also provide two different embeddings to type-II supergravity.

Kerr–Schild perturbations of coset CFTs as scale invariant integrable -models

We formulate λ-deformed σ-models as QFTs in the upper-half plane. For different boundary conditions we compute correlation functions of currents and primary operators, exactly in the deformation parameter λ and for large values of the level k of the underlying WZW model. To perform our computations we use either conformal perturbation theory in association with Cardy's doubling trick, as well as meromorphicity arguments and a non-perturbative symmetry in the parameter space <math><mo>(</mo><mi>λ</mi><mo>,</mo><mi>k</mi><mo>)</mo></math>, or standard QFT techniques based on the free field expansion of the σ-model action, with the free fields obeying appropriate boundary conditions. Both methods have their own advantages yielding consistent and rich, compared to those in the absence of a boundary, complementary results. We pay particular attention, albeit not exclusively, to integrability preserving boundary conditions.

-Deformations in the upper-half plane

A novel class of integrable σ-models interpolating between exact coset conformal field theories in the IR and hyperbolic spaces in the UV is constructed. We demonstrate the relation to the asymptotic limit of λ-deformed models for cosets of non-compact groups. An integrable model interpolating between two spacetimes with cosmological and black hole interpretations and exact conformal field theory descriptions is also provided. In the process of our work, a new zoom-in limit, distinct from the well known non-Abelian T-duality limit, is found.

Novel integrable interpolations

In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.