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Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterising the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly-integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have non-monotonic behaviour as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution., Comment: 43 pages, 19 figures; v2: references and minor comments added. Also, a new result in the 1/q corrections

We review different exact approaches to string theory. In the context of the Green-Schwarz superstring, we discuss the action in curved backgrounds and its supercoset formulation, with particular attention to superstring backgrounds of the $AdS_3$ type supported by both Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. This is the basis for the discussion of classical integrability, of worldsheet-scattering factorisation in the uniform lightcone gauge, and eventually of the string spectrum through the mirror thermodynamic Bethe ansatz, which for $AdS_3$ backgrounds was only derived and analysed very recently. We then illustrate some aspects of the Ramond-Neveu-Schwarz string, and introduce the formalism of Berkovits-Vafa-Witten, which has seen very recent applications to $AdS_3$ physics, which we also briefly review. Finally, we present the relation between M-theory in the discrete lightcone quantisation and decoupling limits of string theory that exhibit non-relativistic behaviours, highlighting the connection with integrable $T\bar{T}$ deformations, as well as the relation between spin-matrix theory and Landau-Lifshitz models. This review is based on lectures given at the Young Researchers Integrability School and Workshop 2022 "Taming the string worldsheet" at NORDITA, Stockholm., Comment: 283 pages; v2: references added

The distance conjecture diagnoses viable low-energy effective realisation of consistent theories of quantum gravity by examining their breakdown at infinite distance in their parameter space. At the same time, infinite distance points in parameter space are naturally intertwined with string dualities. We explore the implications of the distance conjecture when T-duality is applied to curved compact manifolds and in presence of (non-)geometric fluxes. We provide evidence of how divergent potentials signal pathological infinite distance points in the scalar field space where towers of light states cannot be sustained by the curved background. This leads us to suggest an extension to the current statement of the Swampland distance conjecture in curved spaces or in presence of non-trivial fluxes supporting the background., Comment: 66 pages, v3: published version for JHEP

Topological defects have long been known to encode symmetries and dualities between physical systems. In the context of string theory, defects have been intensively studied at the level of the worldsheet. Although marked by a number of pioneering milestones, the target space picture of defects is much less understood. In this paper, we show, at the level of the target space, that Poisson-Lie T-duality can be encoded as a topological defect. With this result at hand, we can postulate the kernel capturing the Fourier-Mukai transform associated to the action of Poisson-Lie T-duality on the RR-sector. Topological defects have the remarkable property that they can be fused together or, alternatively, with worldsheet boundary conditions. We study how fusion of the proposed generalised T-duality topological defect consistently leads to the known duality transformations for boundary conditions. Finally, taking a step back from generalised T-duality, we tackle the general problem of understanding the effect of fusion at the level of the target space. We propose to use the framework of Dirac geometry and formulate the fusion of topological defects and D-branes in this language., Comment: 33 pages, 1 figure

We consider the setup of an evaporating black hole in AdS$_{4}$ coupled to an external bath, embedded in type IIB string theory. We study quantum extremal islands in these backgrounds, in relation to the existence of a massive graviton. Using explicit results of the microscopic embedding of AdS$_{4}$ massive gravity in string theory, we investigate whether it is possible to achieve backgrounds with extremal islands, in which the lowest lying graviton is only slightly massive. For certain regions of the microscopic parameters, the graviton mass can be computed explicitly, and we explain how it directly affects the existence and the properties of the islands. We also show that islands can in principle exist within the regime of validity of the massive gravity effective field theory. However we see via numerical computations that the existence of quantum extremal islands at zero temperature is highly constrained, also when the dilaton is allowed to vary, so that the mass of the graviton cannot be made arbitrarily light. At finite temperature, we also identify a critical parameter, above and below which islands still exist but exhibit a different behavior. Our work supports recent proposals that the unitary evolution of black holes in higher dimensions, and more precisely their Page curve, strongly relies on the presence of a massive graviton in the effective theory., Comment: Updated plots, corrected typos, 1 ancillary notebook

We explore two distinct methods to introduce integrable defects in a family of integrable sigma-models known as Yang-Baxter models. The first method invokes a modified monodromy matrix encoding an integrable defect separating two integrable systems. As an example we construct integrable defects in the ultralocal version of the $S^2$ Yang-Baxter model or 2d Fateev sausage model. The second method is based on the so-called "frozen" B\"acklund transformations. Lifting the construction to the Drinfel'd double, we show how defect matrices can be constructed for inhomogeneous Yang-Baxter models. We provide explicit expressions for the $SU(2)$ non-split Yang-Baxter model for this class of integrable defects., Comment: 22 pages

We build on the results of arXiv:1912.11036 for generalised frame fields on generalised quotient spaces and study integrable deformations for $\mathbb{CP}^n$. In particular we show how, when the target space of the Principal Chiral Model is a complex projective space, a two-parameter deformation can be introduced in principle. The second parameter can however be removed via a diffeomorphism, which we construct explicitly, in accordance with the results stemming from a thorough integrability analysis we carry out. We also elucidate how the deformed target space can be seen as an instance of generalised Kaehler, or equivalently bi-Hermitian, geometry. In this respect, we find the generic form of the pure spinors for $\mathbb{CP}^n$ and the explicit expression for the generalised Kaehler potential for $n=1,2$., Comment: 25 pages

Recent work has shown that two-dimensional non-linear $\sigma$-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces. Here we extend this idea to target spaces constructed as double cosets $M=\widetilde{G} \backslash D / H$. Mirroring conventional coset geometries, we show that on $M$ one can construct a generalised frame field and a $H$-valued generalised spin connection that together furnish an algebra under the generalised Lie derivative. This results naturally in a generalised covariant derivative with a (covariantly) constant generalised intrinsic torsion, lending itself to the construction of consistent truncations of 10-dimensional supergravity compactified on $M$. An important feature is that $M$ can admit distinguished points, around which the generalised tangent bundle should be augmented by localised vector multiplets. We illustrate these ideas with explicit examples of two-dimensional parafermionic theories and NS5-branes on a circle., Comment: 24 pages, Merry Christmas!

Poisson-Lie (PL) T-duality has received much attention over the last five years in connection with integrable string worldsheet theories. At the level of the worldsheet, the algebraic structure underpinning these connections is made manifest with the $\mathcal{E}$-model, a first order Hamiltonian description of the string. The $\mathcal{E}$-model shares many similarities with Double Field Theory (DFT). We report on recent progress in establishing a precise linkage with DFT as the target space description of the $\mathcal{E}$-model. There are three important outcomes of this endeavor: 1) PL symmetry is made manifest at the level of (generalized) supergravity in DFT. 2) PL symmetric target spaces are described by a set of generalized frame fields that encode consistent truncations of supergravity. 3) PL dualisation rules are made explicit and are readily extended to include the R/R sector of the type II theory. These general results are put into context with their application to the the integrable Yang-Baxter model ($\eta$-deformation). This extended proceedings provides some introductory review of PL and an orientation to the results of arXiv:1707.08624 and arXiv:1810.11446., Comment: 24 pages, 3 figures, contribution to the proceedings of the workshop "Dualities and Generalized Geometries", Corfu Summer Institute 2018

In this short proceedings we discuss some of the results obtained in [1]. Integrable deformations enlarge the landscape and understanding of integrable models and its algebraic structures like quantum groups. In this short proceedings, we will review the one-loop renormalisation group analysis of an integrable deformation known as the Yang-Baxter Wess-Zumino model. This classically integrable model shows a striking stability under one-loop renormalisation. In addition, we show how Poisson-Lie T-duality, a generalisation of T-duality that is closely intertwined with integrable deformations, is particularly simple and elegant for the Yang-Baxter Wess-Zumino model., Comment: 13 pages, 3 figures, Submitted as proceedings for the 2018 conference "School and Workshops on Elementary Particle Physics and Gravity" held at the Corfu Summer Institute

The worldsheet theories that describe Poisson-Lie T-dualisable $\sigma$-models on group manifolds as well as integrable $\eta$, $\lambda$ and $\beta$-deformations provide examples of ${\cal E}$-models. Here we show how such ${\cal E}$-models can be given an elegant target space description within Double Field Theory by specifying explicitly generalised frame fields forming an algebra under the generalised Lie derivative. With this framework we can extract simple criteria for the R/R fields and the dilaton that extend the ${\cal E}$-model conditions to type II backgrounds. In particular this gives conditions for a type II background to be Poisson-Lie T-dualisable. Our approach gives rise to algebraic field equations for Poisson-Lie symmetric spacetimes and provides an effective tool for their study., Comment: 55 pages

We investigate the integrable Yang-Baxter deformation of the 2d Principal Chiral Model with a Wess-Zumino term. For arbitrary groups, the one-loop beta functions are calculated and display a surprising connection between classical and quantum physics: the classical integrability condition is necessary to prevent new couplings being generated by renormalisation. We show these theories admit an elegant realisation of Poisson-Lie T-duality acting as a simple inversion of coupling constants. The self-dual point corresponds to the Wess-Zumino-Witten model and is the IR fixed point under RG. We address the possibility of having supersymmetric extensions of these models showing that extended supersymmetry is not possible in general., Comment: 44 pages; 4 figures; v2: references added; v3: published version, text edited and restructured, result added on CFT operators driving the flow

We study the $SU(2)$ Principal Chiral Model (PCM) in the presence of an integrable $\eta$-deformation. We put the theory on $\mathbb{R}\times S^1$ with twisted boundary conditions and then reduce the circle to obtain an effective quantum mechanics associated with the Whittaker-Hill equation. Using resurgent analysis we study the large order behaviour of perturbation theory and recover the fracton events responsible for IR renormalons. The fractons are modified from the standard PCM due to the presence of this $\eta$-deformation but they are still the constituents of uniton-like solutions in the deformed quantum field theory. We also find novel $SL(2,\mathbb{C})$ saddles, thus strengthening the conjecture that the semi-classical expansion of the path integral gives rise to a resurgent transseries once written as a sum over Lefschetz thimbles living in a complexification of the field space. We conclude by connecting our quantum mechanics to a massive deformation of the $\mathcal{N}=2~$ $4$-d gauge theory with gauge group $SU(2)$ and $N_f=2$., Comment: 56 pages, 19 figures; v2: references amended and small clarifications made in the text

We examine integrable $\lambda$-deformations of $SO(n+1)/SO(n)$ coset CFTs and their analytic continuations. We provide an interpretation of the deformation as a squashing of the corresponding coset $\sigma$-model's target space. We realise the $\lambda$-deformation for $n=5$ case as a solution to supergravity supported by non-vanishing five-form and dilaton. This interpolates between the coset CFT $SO(4,2)/SO(4,1)\times SO(6)/SO(5)$ constructed as a gauged WZW model and the non-Abelian T-dual of the $AdS_5\times S^5$ spacetime., Comment: 30 pages; v2 journal version with note added

This study examined how existing literacy practices in native language and English affect the ways parents manage their own lives and support emergent literacy in their children. Participating in the study were non-English speaking parents of successive cohorts of children attending a preschool in Arlington, Virginia. Data collection methods included focus group and individual telephone interviews with parents (mostly mothers) regarding family literacy practices in Spanish and English. Findings revealed complex differences in literacy practices among and within families, with the typical family having mothers who had some literacy in their native language and fathers with at least minimal English literacy. Parents, especially mothers, were very aware of the consequences of their not having fluency in English, including power relations within families, ability to support children's literacy, mothers' employment opportunities, and ability to interact in the larger society. Some families support the developing literacy of their children in ways that schools may not recognize, such as reading to their children in another language. Mothers expressed the desire and intention to learn English, with a number registered for English classes offered through the preschool parent support program. (Contains 17 references.) (KB)

This paper uses teachers' voices to explore transformation processes in teachers as a result of participation in an innovative school-based Master's degree program at the Institute for Educational Transformation. The program is built around a set of beliefs and principles that enunciate a philosophy of teaching and learning emphasizing work within teams, developing reflective practice, school-based inquiry, and continuous improvement. Data for this paper come from reflective essays written by 103, 1997 graduates of the program seeking to understand their own processes of transformation over its duration. They were asked to think about and document in a portfolio how they had changed over the course of approximately 2 years. Their writings described important changes they could see in their professional and personal lives. The 1997 graduates described the ability to see children and classrooms through new perspectives, such that they felt the program helped them develop stronger professional voices, more confidence in their professional judgments, and more self-efficacy and power. Teachers also reported: a greater knowledge and sense of self; increased technology proficiency; improved understanding of the writing process and improved writing style; improved teaching practices; changed educational philosophy; improved professional and personal relationships; and broader mentoring and modeling for others. (Contains 13 references.) (SM)