# Integrability, Dualities and Deformations

Aug 1 – 5, 2022
Humboldt Universität zu Berlin, IRIS building, Adlershof
Europe/Zurich timezone

## Titles and Abstracts

Chris Blair (Vrije U., Brussels and Intl. Solvay Inst.)

Generalised U-dual solutions in supergravity

I will discuss some notions and examples of generalised U-duality, viewed as a solution generating mechanism in (10- and 11-dimensional) supergravity.

Topological defects and fusion for generalised T-duality

Topological defects have long been known to encode symmetries and dualities between physical systems. In this talk, I will first explain how Poisson-Lie T-duality can be realised as a topological defect at the level of the target space. A fundamental property of topological defect is that they can be fused together or with boundary conditions. I will sketch how this topological defect consistently leads to the known generalised duality transformations for boundary conditions of open strings. Finally, I will discuss how the language of Dirac geometry can be used as a universal tool to describe defects, boundary conditions and their corresponding fusion. This work is in collaboration with Thomas Raml.

Sibylle Driezen (IGFAE Santiago de Compostela)

Integrable deformations and dualities of string sigma-models

In this talk, I will review recent work on deformations and generalised T-dualities which preserve the integrability of string sigma-models. As a guiding exemplar, I will focus on the so-called Homogeneous Yang-Baxter deformations, which are a class of deformations that interpolate between a sigma-model and its non-abelian T-dual. As a special case they capture the well-known TsT-transformations involving standard abelian T-duality. At the level of the low-energy string as well as worldsheet integrability it has been understood that many of the attractive features of TsT carry over to the generalised Yang-Baxter case. Using the framework of Double Field Theory, I will first show how one can easily understand Yang-Baxter deformations as solution-generating techniques of supergravity, and I will argue how a large web of transformations with the same feature can be more generally classified. At the same time, these transformations can be understood as canonical transformations of the worldsheet. I will show in the Yang-Baxter case how this implies that they can be equivalently described on-shell in terms of a twisting of the worldsheet boundary conditions of the undeformed model. This feature makes it particularly efficient to employ integrable methods such as the classical spectral curve and its semi-classical quantisation to obtain energy corrections, and thus opens a route to study integrable deformations of AdS/CFT. I will report on progress obtained for a Homogeneous Yang-Baxter deformation of the AdS5 x S5 superstring.

Sergey Frolov (Hamilton Math. Inst., Dublin and Trinity Coll., Dublin)

Dressing Factors and TBA for AdS3 x S3 x T4

New dressing factors for the world-sheet S-matrix of the RR superstrings on AdS3 x S3 x T4 are proposed. The resulting S-matrix is continued to the mirror region, and used to derive the mirror TBA equations describing the spectrum of the superstring theory in the zero winding sector.

Thomas Grimm (Utrecht U.)

Tame geometry in QFTs and Quantum Gravity

In this talk I will introduce a generalized notion of finiteness and argue that it appears in QFTs and in all well-understood string theory effective theories. The underlying mathematical foundation is described by the tame geometry that is built from o-minimal structures. These remarkable structures originated in logic and have recently been used to prove many longstanding mathematics conjectures. I will argue that tameness appears in many physical theories and describe how it constrains coupling functions, field spaces, and amplitudes. If time permits, I will discuss the tameness in lambda-deformed WZW models.

Superintegrability, Killing symmetry and point particle T-duality

We report on a recent progress in the point particle T-duality story. We argue that the ideas  related to Killing symmetry, integrability and deformation play an important role not only in a string T-duality story but also in its point particle counterpart. Applying those ideas we find, quite remarkably, that the  T-duality seems to be a more widespread phenomenon in the context of the point particle dynamics  than in the string one. In this talk, we focus on superintegrable spherically symmetric electro-gravitational backgrounds in n dimensions where we can produce explicit new examples of point particle T-duality having an interesting physical consequences. In particular, the dynamics of a charged particle scattered by a repulsive electric potential in a flat space is T-dual to the dynamics of scattering in the space of constant negative curvature. Thus knowing just the exact Hamiltonian dynamics of the scattered particle cannot give us an information about the curvature of the space in which the particle moves.

Sylvain Lacroix (ETH Zurich)

Integrable sigma-models at RG fixed points: quantisation as affine Gaudin models

In this talk, I will present first steps towards the quantisation of integrable sigma-models using the formalism of affine Gaudin models, approaching these theories through their conformal limits. The talk will mostly focus on a specific example called the Klimčík (or bi-Yang-Baxter) model. After recalling the relation between this theory and affine Gaudin models at the classical level, I will explain how its integrable structure splits into two decoupled chiral parts in the conformal limit, built respectively from left-moving and right-moving degrees of freedom. Finally, I will briefly sketch how the quantisation of these chiral integrable structures can be studied using the language of affine Gaudin models and vertex operator algebras. This is based on joint work with G. Kotousov and J. Teschner.

Jules Lamers (IPhT Saclay)

Long-range spin chains - no strings attached

I will give an introduction to quantum-integrable spin chains with long-range interactions (inhomogeneous Heisenberg, Inozemtsev, Haldane--Shastry) and discuss their intdualdef. Based on joint works with R. Klabbers, D. Serban and V. Pasquier, and work in progress.

André LeClair (Cornell U.)

On the classification of UV completions of integrable TTbar deformations of CFT

It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant TTbar perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher TTbar_{s>1} irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge c_UV > c_IR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by TTbar_s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with c_IR = 1/2 we find 3 UV completions based on a single massless Majorana fermion description with c_UV = 7/10 and 3/2, which both have N=1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E_8 appropriate to a magnetic perturbation with c_UV = 21/22, 15/2 and 31/2. We argue that it is likely there are more cases for this E_8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infra-red (IR) theory is the 3-state Potts model with c_IR = 4/5 and we find 3 completions with 4/5 < c_UV ≤ 16/5. For the su(4) case, which has 3 particles and c_IR=1, and we find 11 UV completions with 1 < c_UV ≤ 5, most of which were previously unknown.

George Linardopoulos (Wigner Research Centre for Physics)

String integrability in brane-deformed holography

The D3-D5 and D2-D4 probe-brane systems with nonzero worldvolume flux are holographically dual to N=4 super Yang-Mills and ABJM theory in the presence of half-BPS domain walls. The two domain wall systems are thought to be integrable; the evidence comes mainly from the study of correlation functions at weak coupling. In the present talk we show that the string theory duals of these systems are classically integrable. In other words, the string boundary conditions on the probe branes preserve the integrability of the corresponding Green-Schwarz sigma models. This finding suggests that the dual domain wall systems are integrable to all loop orders and for any value of the bond dimension.

Jeremy Mann (DESY)

Conformal blocks in d dimensions from Gaudin integrable models and applications

The decomposition of functions with Lie group symmetry G into a basis of G-invariant functions is an important problem in many areas of theoretical physics and mathematics. In conformal field theory, this conformal block decomposition of correlation functions is a crucial tool for many modern approaches to computing and constraining the physical data. In this talk, I will review the construction of conformal blocks as integrable systems and discuss some applications to the conformal bootstrap program. Our results also demonstrate a number of precise relations between different types of Gaudin and Calogero-Moser-Sutherland integrable models.

Tim Meier (Humboldt U. Berlin)

Noncommutative N = 4 SYM via Drinfel’d twists

In the realm of the AdS/CFT correspondence, integrable deformations open up a field of new integrable theories. A well-studied group of such deformations is spanned by the homogeneous Yang-Baxter deformations of the AdS5xS5 string, which result in Drinfel’d twists of the underlying symmetry algebra. The CFT dual to the AdS5xS5 string is believed to be N=4 SYM. Here, the underlying symmetry algebra include spacetime symmetries. Drinfel’d twists of spacetime symmetries lead to so-called non-commutative field theories in which the notion of gauge symmetry changes drastically. In fact, the construction of noncommutative gauge theory is explicitly known only for the simplest, so-called canonical deformation. I will present an approach which can go beyond the canonical case, and derive gauge invariant actions for what can be called the Lorentz deformation as an example of a broader class of deformations. The resulting theories could give possible dual models to the Yang Baxter deformed string.

Elli Pomoni (DESY)

Quantum symmetries in N = 2 SCFTs

Fiona Seibold (Imperial College London)

I will present a family of type IIB supergravity backgrounds that are deformations of AdS3xS3xT4 with squashed metric, a combination of NSNS and RR fluxes, constant dilaton and preserving 8 supercharges. On the one hand they can be obtained from AdS3xS3xT4 through T- and S-dualities. On the other hand, their classical integrability follows from their construction, up to T-dualities, as an inhomogenenous Yang-Baxter deformation of AdS3xS3xT4. Based on 2206.12347 with Ben Hoare and Arkady Tseytlin.

Daniel Thompson (Swansea U.)

Aspects of chiral dynamics and resurgence in E-models

This is a talk in two parts. A variety of integrable sigma models can be interpreted as ${\cal E}$ models; a first order description with chiral currents defined on a group manifold of double the dimension of the sigma target space. I will outline some recent developments of topological aspects of these the theories.  I will further move on to review Severa's construction of these ${\cal E}$ models as boundary dynamics of 3d Chern Simons which will allow us to make some novel remarks on the treatment of Lorentz invariant formulations of chiral dynamics.    In the second part of the talk, I will describe recent emerging findings of the resurgent structure of lambda deformations giving a snapshop into the asymptotic behaviour of an asymptotic CFT.

Alessandro Torrielli (Surrey U.)

Tools for massless form factors in AdS3

We present a study of form factors in the relativistic limit of a massless scattering problem connected with AdS3 integrable superstrings. After briefly summarising the relevant spectrum and S-matrix theory, we describe minimal solutions for the two- and three-point form factors, outline the proofs of the form-factor axioms, and discuss the general features, difficulties and complications which this model presents with respect to the very-closely related, and yet rather simpler, Sine-Gordon soliton-antisoliton problem.