A quantum integrability index was proposed in \cite{KMS}. It systematizes the Goldschmidt and Witten's operator counting argument \cite{GW} by using the conformal symmetry. In this work we compute the quantum integrability indexes for the symmetric coset models ${SU(N)}/{SO(N)}$ and $SO(2N)/{SO(N)\times SO(N)}$. The indexes of these theories are all non-positive except for the case of ${SO(4)}/{SO(2)\times SO(2)}$. Moreover we extend the analysis to the theories with fermions and consider a concrete theory: the $\mathbb{CP}^N$ model coupled with a massless Dirac fermion. We find that the indexes for this class of models are non-positive as well.
We define non-relativistic string in the background with non-trivial metric. Then we analyze its properties when we focus on integrability of this string in AdS(5)xS(5) background.
This short note is devoted to the study of the integrability of the bosonic string on AdS(5)xS(5) in the uniform light-cone gauge. We construct Lax connection for gauge fixed theory and we argue that it is flat.
A wide class of models, built of the three component unit vector field living in the (3+1) Minkowski space-time, which break explicitly global O(3) symmetry are discussed. The symmetry breaking occurs due to the so-called dielectric function multiplying a standard symmetric term. Integrability conditions are found. Moreover, for some particular forms of the Lagrangian exact toroidal solutions with any Hopf index are obtained. It is proved that such symmetry breaking influences the shape of the solitons whereas the energy as well as the Hopf index remain unchanged.
These are notes based on a lecture given at the Cargese summer school 1995. I describe evidence that the (two-dimensional) integrable chiral Gross-Neveu model might remain integrable when coupled to gravity. The results presented here were obtained in collaboration with Ian Kogan.
We study the anomalous dimensions of operators in the scalar sector of β-deformed ABJ(M) theories. We show that the anomalous dimension matrix at two-loop order gives an integrable Hamiltonian acting on an alternating SU(4) spin chain with the spins at odd lattice sides in the fundamental representation and the spins at even lattices in the anti-fundamental representation. We get a set of $β$-deformed Bethe ansatz equations which give the eigenvalues of Hamiltonian of this deformed spin chain system. Based on our computations, we also extend our study to non-supersymmetric three-parameter $γ$-…
These notes provide an introduction toward Wilson loops in N=4 supersymmetric Yang-Mills theory with a focus toward their integrability properties. In addition to a brief discussion of exact results for the circular Wilson loop and the cusp anomalous dimension, the notes focus on non-local symmetries, utilizing the integrability of the minimal surface problem that appears at strong coupling. This work is based on lectures given at the Young Researchers Integrability School and Workshop 2018. To appear in a special issue of J. Phys. A.
We consider non-Hermitian but PT-symmetric extensions of Calogero models, which have been proposed by Basu-Mallick and Kundu for two types of Lie algebras. We address the question of whether these extensions are meaningful for all remaining Lie algebras (Coxeter groups) and if in addition one may extend the models beyond the rational case to trigonometric, hyperbolic and elliptic models. We find that all these new models remain integrable, albeit for the non-rational potentials one requires additional terms in the extension in order to compensate for the breaking of integrability.
In the first half of this note, after briefly motivating and reviewing membrane field theories, we consider their BPS funnel solutions. We discuss some aspects of embedding M-theory fuzzy funnels in these theories. In the second half, we focus on ABJM theory and explain a test of AdS4/CFT3 based on integrability. We discuss a numerical mismatch at one loop in worldsheet perturbation theory and its possible resolutions.
In this note we prove that the Hamilton-Jacobi equation for a particle in the five dimensional Kerr-(A)dS black hole is separable, for arbitrary rotation parameters. As a result we find an irreducible Killing tensor. We also consider the Klein-Gordon equation in this background and show that this is also separable. Finally we comment on extensions and implications of these results.
In this note, we study the action of $O(d,d)$ transformations on the integrable structure of two-dimensional non-linear sigma models via the doubled formalism. We construct the Lax pairs associated with the $O(d,d)$-transformed model and find that they are in general non-local because they depend on the winding modes. We conclude that every $O(d,d;\mathbb{R})$ deformation preserves integrability. As an application we compute the Lax pairs for continuous families of deformations, such as $J\bar{J}$ marginal deformations and TsT transformations of the three-sphere with $H$-flux.
The existence of higher-spin quantum conserved currents in two dimensions guarantees quantum integrability. We revisit the question of whether classically-conserved local higher-spin currents in two-dimensional sigma models survive quantization. We define an integrability index $\mathcal{I}(J)$ for each spin $J$, with the property that $\mathcal{I}(J)$ is a lower bound on the number of quantum conserved currents of spin $J$. In particular, a positive value for the index establishes the existence of quantum conserved currents. For a general coset model, with or without extra discrete symmetries…
The main result of this note is the proof of degenerate quantum integrability of quantum spin Calogero--Moser systems and the description of the spectrum of quantum Hamiltonians in terms of the decomposition of tensor products of irreducible representations of corresponding Lie algebra.
These lecture notes are based on a blackboard course given at the XVII Modave Summer School in Mathematical Physics held from 13 -- 17 September 2021 in Brussels (Belgium), and aimed at Ph.D. students in High Energy Theoretical Physics. We start with introducing classical integrability in finite-dimensional systems to set the stage for our main purpose: introducing two-dimensional classical field theories which are integrable. We focus on their zero-curvature formulation through the so-called Lax connection, which ensures the existence of an infinite tower of conserved charges. We then move on…
This short note is devoted to the study of the Hamiltonian formalism and the integrability of the bosonic model introduced in [hep-th/0612079]. We calculate Poisson bracket of spatial components of Lax connection and we argue that its structure implies classical integrability of the theory.
We study integrable models in the context of the recently discovered Gauge/YBE correspondence, where the Yang-Baxter equation is promoted to a duality between two supersymmetric gauge theories. We study flavored elliptic genus of 2d $\mathcal{N}=(2,2)$ quiver gauge theories, which theories are defined from statistical lattices regarded as quiver diagrams. Our R-matrices are written in terms of theta functions, and simplifies considerably when the gauge groups at the quiver nodes are Abelian. We also discuss the modularity properties of the R-matrix, reduction of 2d index to 1d Witten index, an…
We study topological vortex solutions in a generalized Abelian Higgs model with non-polynomial dielectric and potential functions. These quantities are chosen by requiring integrability of the self-dual limit of the theory for all values of the magnetic flux. All the vortex profiles are described by exact analytical expressions that solve the self-dual vortex equations. There is only a symmetry-breaking superconducting phase and the model sustains regular phenomenology.
In these notes we review the S-matrix theory in (1+1)-dimensional integrable models, focusing mainly on the relativistic case. Once the main definitions and physical properties are introduced, we discuss the factorization of scattering processes due to integrability. We then focus on the analytic properties of the two-particle scattering amplitude and illustrate the derivation of the S-matrices for all the possible bound states using the so-called bootstrap principle. General algebraic structures underlying the S-matrix theory and its relation with the form factors axioms are briefly mentioned…
The UV fixed point theory of SU(2) gauge theory with N_f = 0,1,...,7 flavors is believed to have the enlarged E_{N_f +1} flavor symmetry. Actually it is not easy to check this conjecture because the UV theory is strongly-coupled, however, computation of certain SUSY protected quantities provides strong evidence for the enhancement of flavor symmetry. We study the superconformal index for SU(2) gauge theory with N_f = 0, 1 flavors in details, and we give a support for the enhancement by studying combinatorial structure of the superconformal indexes of these theories. We also give a nontrivial e…
In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the Sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-ma…
These notes accompany an introductory lecture course on the twistor approach to supersymmetric gauge theories aimed at early-stage PhD students. It was held by the author at the University of Cambridge during the Michaelmas term in 2009. The lectures assume a working knowledge of differential geometry and quantum field theory. No prior knowledge of twistor theory is required.
We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples (harmonic oscillator, Kepler problem); 3. Algebraic tools: Lax pairs, monodromy and transfer matrices, classical r-matrices and exchange relations, non-ultralocal Poisson brackets, with examples (non-linear Schroedinger model, principal chiral field); 4. Features of classical r-matrices: Belavin-Drinfeld theorems, analyticity properties, and lift of the classi…
Partial symmetries are described by generalized group structures known as symmetric inverse semigroups. We use the algebras arising from these structures to realize supersymmetry in (0+1) dimensions and to build many-body quantum systems on a chain. This construction consists in associating appropriate supercharges to chain sites, in analogy to what is done in spin chains. For simple enough choices of supercharges, we show that the resulting states have a finite non-zero Witten index, which is invariant under perturbations, therefore defining supersymmetric phases of matter protected by the in…
It is noted that the Schrodinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order integrals of motion are presented. That does not mean that we can explicitly compute the time dependence for expectation value of any quantum observable. A similar result is indicated for classical dynamical systems in terms of Koopman's approach. Explicit transformations of quantum and classical dynamics to the free evolution by using direct methods of scatterin…
Donaldson has shown that the moduli space of monopoles $M_k$ is diffeomorphic to the space $\Rat_k$ of based rational maps from the two-sphere to itself. We use this diffeomorphism to give an explicit description of the bundle on $\Rat_k$ obtained by pushing out the index bundle from $M_k$. This gives an alternative and more explicit proof of some earlier results of Cohen and Jones.
These are lecture notes of an introduction to quantum integrability given at the Tenth Modave Summer School in Mathematical Physics, 2014, aimed at PhD candidates and junior researchers in theoretical physics. We introduce spin chains and discuss the coordinate Bethe ansatz (CBA) for a representative example: the Heisenberg XXZ model. The focus lies on the structure of the CBA and on its main results, deferring a detailed treatment of the CBA for the general $M$-particle sector of the XXZ model to an appendix. Subsequently the transfer-matrix method is discussed for the six-vertex model, uncov…
We discuss mock automorphic forms from the point of view of representation theory, that is, obtained from weak harmonic Maass forms give rise to nontrivial $(\mathfrak{g},K)$-cohomology. We consider the possibility of replacing the `holomorphic' condition with `cohomological' when generalizing to general reductive groups. Such a candidate allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic for…
In this note, we study the integrability of geodesic flow in the background of a very general class of spacetimes with NUT-charge(s) in higher dimensions. This broad set encompasses multiply NUT-charged solutions, electrically and magnetically charged solutions, solutions with a cosmological constant, and time dependant bubble-like solutions. We also derive first-order equations of motion for particles in these backgrounds. Separability turns out to be possible due to the existence of non-trivial irreducible Killing tensors. Finally, we also examine the Klein-Gordon equation for a scalar field…
