Narrow Results

The parametric families of integrable boundary affine Toda theories are considered. We calculate boundary one-point functions and propose boundary S-matrices in these theories. We use boundary one-point functions and S-matrix amplitudes to derive boundary ground state energies and exact solutions describing classical vacuum configurations.

Yang–Mills theories in four space–time dimensions possess a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. It turns out that the effective Yang–Mills dynamics in several important limits is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations. In this review we explain the general phenomenon of complete integrability and its realization in several different situations. As a prime example, we consider in some detail the scale dependence of composite (Wilson) operators in QCD and super-Yang–Mills (SYM) theories. High-energy (Regge) behavior of scattering amplitudes in QCD is also discussed and provides one with another realization of the same phenomenon that differs, however, from the first example in essential details. As the third example, we address the low-energy effective action in a SYM theory which, contrary to the previous two cases, corresponds to a classical integrable model. Finally, we include a short overview of recent attempts to use gauge/string duality in order to relate integrability of Yang–Mills dynamics with the hidden symmetry of a string theory on a curved background.

We study classical integrability of the supersymmetric U(N) σ model with the Wess–Zumino–Witten term on infinite and half-plane. We demonstrate the existence of nonlocal conserved currents of the model and derive general recursion relations for the infinite number of the corresponding charges in a superfield framework. The explicit forms of the first few supersymmetric charges are constructed. We show that the considered model is integrable on infinite plane as a consequence of the conservation of the supersymmetric charges. Also, we study the model on half-plane with free boundary, and examine the conservation of the supersymmetric charges on half-plane and find that they are conserved as a result of the equations of motion and the free boundary condition. As a result, the model on half-plane with free boundary is integrable. Finally, we conclude the paper and some features and comments are presented.

A large class of integrable cosmological models with two matter components is presented. The D-dimensional models on the space-time manifold are studied in the presence of 2 separately conserved barotropic perfect fluids. Such model are reducible to pseudo-Euclidean Toda-like system and integrable when their barotropic parameters satisfy some algebraic relations. Methods for integrating of pseudo-Euclidean Toda-like systems are based on the Minkowski-like geometry for characteristic vectors composed from the barotropic parameters. We also apply the methods for the spatially flat Friedmann-Robertson-Walker universe containing a perfect fluid and a minimally coupled self-interacting scalar field with a potential comprised of two exponentials.

The geometry of five-dimensional Kerr black holes is discussed based on geodesics and Weyl curvatures. Kerr-Star space, Star-Kerr space and Kruskal space are naturally introduced by using special null geodesics. We show that the geodesics of AdS Kerr black hole are integrable, which generalizes the result of Frolov and Stojkovic. We also show that five-dimensional AdS Kerr black holes are isospectrum deformations of Ricci-flat Kerr black holes in the sense that the eigenvalues of the Weyl curvature are preserved.

We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M, with volume form μ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M. Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang–Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of μ. The magnetic field evolves by a quadratically nonlinear "Euler" equation, which may also be regarded as describing geodesic flow on SDiff(M, μ). Static solutions are obtained. For uniform μ, an infinite sequence of local conserved charges beginning with the Hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang–Mills theory and matrix field theories, whose gauge-invariant phase-space is conjectured to be a coadjoint orbit of the diffeomorphism group of a noncommutative space.

We find a new family of classical oscillating strings with large winding numbers, which are constructed by interchanging τ and σ of helical strings' solutions. We also give a finite-gap interpretation of interchanging τ and σ. Gauge dual operators are discussed at the end.

We find a new quantum Yangian symmetry of AdS/CFT S-matrix, which does not have the original Lie algebra 𝔰𝔲(2|2) analogue, and persue the origin of this symmetry in exceptional superalgebra 𝔡(2, 1; ε) , which recovers 𝔰𝔲(2|2) when the paramater ε taken to zero.

We reconsider the perturbative expansion of the Wilson loop in 2D noncommutative gauge theories, using an improved integration method. For the class of maximally crossed diagrams in the θ → ∞ limit we find an intriguing formula, easily generalizable to all orders in perturbation theory.

The existence of crystalline condensates in the temperature and chemical potential phase diagram of the Gross-Neveu models can be traced to intricate symmetries of the associated inhomogeneous gap equation. The gap equation based on the Ginzburg-Landau expansion is precisely the mKdV or AKNS hierarchy of integrable nonlinear equations for the Gross-Neveu model with discrete or continuous chiral symmetry, respectively. The former model also has a dense-dilute symmetry that is due to the energy-reflection duality of the underlying quasi-exactly soluble spectral operators.

The quantum H₃ integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H₃. It is shown that the gauge-rotated H₃ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H₃, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector . One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H₃ Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H₃-invariants "polynomially"-isospectral to the quantum H₃ model is defined.

We solve exactly the scalar box integral using the Mellin–Barnes representation. First we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we perform an analytic continuation before applying the Laurent expansion in ϵ = (d-4)/2 of the result.

We directly compute Wilson loop/Wilson loop correlators on ℝ ×S³ in AdS/CFT by constructing spacelike minimal surfaces that connect two spacelike circular contours on the boundary of global AdS that are separated by a spacelike interval. We compare these minimal surfaces to the disconnected "double cap" solutions both to regulate the area, and show when the connected/disconnected solution is preferred. We find that for sufficiently large Wilson loops no transition occurs because the Wilson loops cannot be sufficiently separated on the sphere. This may be considered an effect similar to the Hawking–Page transition: the size of the sphere introduces a new scale into the problem, and so one can expect phase transitions to depend on this data. To construct the minimal area solutions, we employ a reduction a la Arutyunov–Russo–Tseytlin (used by them for spinning strings), and rely on the integrability of the reduced set of equations to write explicit results.

Basing upon experience from performing double-dimensional reduction of the D = 11 supermembrane on AdS₄ × S⁷ background to Type IIA superstring on AdS₄ × ℂℙ³ we introduce Kaluza–Klein (partial) κ-symmetry gauge as a vanishing condition of the contributions to the D = 11 supervielbein components tangent to D = 10 space–time that are proportional to the differential of the coordinate parametrizing compact 11th space–time dimension identified with the supermembrane worldvolume compact dimension. For AdS₄ × S⁷ supermembrane Kaluza–Klein gauge removes half Grassmann coordinates associated with eight space–time supersymmetries, broken by the AdS₄ × ℂℙ³ superbackground, by imposing D = 3 (anti-)Majorana condition on them. The consideration relies on the realization of osp(4|8) isometry superalgebra of the AdS₄ × S⁷ superbackground as superconformal algebra. Requiring further vanishing of the D = 10 dilaton leaves in the sector of broken supersymmetries just two Grassmann coordinates organized into D = 3 (anti-)Majorana spinor that defines minimal SL(2, ℝ)-covariant extension of the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model. Among four possibilities of such a minimal extension we consider in detail one, that corresponds to picking out D = 3 Majorana coordinate related to broken Poincaré supersymmetry, and show that the AdS₄ × ℂℙ³ superstring equations of motion in this partial κ-symmetry gauge are integrable. Also the relation between the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model and the AdS₄ × ℂℙ³ superstring is revisited.

In this paper, we study the action of the nonplanar two-loop dilatation operator in an SU(2)×SU(2) subsector of the ABJ Chern–Simons-matter theory. The gauge invariant operators we consider are the restricted Schur polynomials. As in ABJM theory, there is a limit in which the spectrum reduces to a set of decoupled harmonic oscillators, indicating integrability in the large M and N double limit of the theory. We then consider parity transformations on the gauge invariant operators. In this case the nonplanar anomalous dimensions break parity invariance. Our analysis shows that (M-N) is related to the holonomy in the string theory, confirming one of the main features of the theory and its string dual. Furthermore, in the limit where ABJ theory reduces to ABJM theory, parity invariance is restored.

We analyze the constraints for a system of antiself-dual Yang–Mills (ASDYM) equations by means of the modified Faddeev–Jackiw method in $K$ and $J$ gauges à la Yang. We also establish the Hamiltonian flow for ASDYM system through the hidden BRS invariance in both the gauges. Finally, we remark on the bi-Hamiltonian nature of ASDYM and the compatibility of the symplectic structures therein.