Spectral Duality in Integrable Systems from AGT Conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N) Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrodinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2x2 and NxN representations of the Toda chain and the famous AHH duality

On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system

A simple derivation of the spectra of the action-variables of the quantized compactified Ruijsenaars-Schneider system is presented. The spectra are obtained by combining Kahler quantization with the identification of the classical action-variables as a standard toric moment map on the complex projective space. The result is consistent with the Schrodinger quantization of the system worked out previously by van Diejen and Vinet.Comment: Based on talk at the workshop CQIS-2011 (Protvino, Russia, January 2011), 12 page

Angular Momentum and Gravimagnetization of the N=2{\cal N}=2 SYM vacuum

In this note we discuss the gravimagnetization of the ${\cal N}=2$ SYM vacuum in the $\Omega$-background. It is argued that the Seiberg-Witten prepotential is related to the vacuum density of the angular momentum in the Euclidean $R^4$ space. The possible role of the dyonic instantons as the microscopic angular momentum carriers which could yield the spontaneous vacuum gravimagnetization is conjectured. We interpret the dyonic instanton as a kind of the Euclidean bounce in $R^4$ similar to one responsible for the Schwinger pair creation. The induced angular momentum in $R^4$ is also briefly considered in the dual Liouville formulation of $SU(2)$ theory via AGT relation.Comment: 20 page

Integrability in QCD and beyond

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 N=2 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.Comment: 87 pages, 4 figures; minor stylistic changes, references added. To be published in the memorial volume 'From Fields to Strings: Circumnavigating Theoretical Phyiscs', World Scientific, 2004. Dedicated to the memory of Ian Koga

Type I Non-Abelian Superconductors in Supersymmetric Gauge Theories

Non-BPS non-Abelian vortices with CP^1 internal moduli space are studied in an N=2 supersymmetric U(1) x SU(2) gauge theory with softly breaking adjoint mass terms. For generic internal orientations the classical force between two vortices can be attractive or repulsive. On the other hand, the mass of the scalars in the theory is always less than that of the vector bosons; also, the force between two vortices with the same CP^1 orientation is always attractive: for these reasons we interpret our model as a non-Abelian generalization of type I superconductors. We compute the effective potential in the limit of two well separated vortices. It is a function of the distance and of the relative colour-flavour orientation of the two vortices; in this limit we find an effective description in terms of two interacting CP^1 sigma models. In the limit of two coincident vortices we find two different solutions with the same topological winding and, for generic values of the parameters, different tensions. One of the two solutions is described by a CP^1 effective sigma model, while the other is just an Abelian vortex without internal degrees of freedom. For generic values of the parameters, one of the two solutions is metastable, while there are evidences that the other one is truly stable.Comment: 35 pages, 8 figures. v2: fixed typos and added small comments, v3 removed an unecessary figur

Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space

We discuss multiloop MHV amplitudes in the N=4 SYM theory in terms of effective gravity in the momentum space with IR regulator branes as degrees of freedom. Kinematical invariants of external particles yield the moduli spaces of complex or Kahler structures which are the playgrounds for the Kodaira-Spencer(KS) or Kahler type gravity. We suggest fermionic representation of the loop MHV amplitudes in the N=4 SYM theory assuming the identification of the IR regulator branes with KS fermions in the B model and Lagrangian branes in A model. The two-easy mass box diagram is related to the correlator of fermionic currents on the spectral curve in B model or hyperbolic volume in the A model and it plays the role of a building block in the whole picture. The BDS-like anzatz has the interpretation as the semiclassical limit of a fermionic correlator. It is argued that fermionic representation implies a kind of integrability on the moduli spaces. We conjecture the interpretation of the reggeon degrees of freedom in terms of the open strings stretched between the IR regulator branes.Comment: 39 pages, typos corrected, journal versio

On Integrable Systems and Supersymmetric Gauge Theories

The properties of the N=2 SUSY gauge theories underlying the Seiberg-Witten hypothesis are discussed. The main ingredients of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential are presented, the invariant sense of these definitions is illustrated. Recently found exact nonperturbative solutions to N=2 SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to integrable systems are discussed.Comment: LaTeX, 38 pages, no figures; based on the lecture given at INTAS School on Advances in Quantum Field Theory and Statistical Mechanics, Como, Italy, 1996; minor changes, few references adde

Dualities in Quantum Hall System and Noncommutative Chern-Simons Theory

We discuss different dualities of QHE in the framework of the noncommutative Chern-Simons theory. First, we consider the Morita or T-duality transformation on the torus which maps the abelian noncommutative CS description of QHE on the torus into the nonabelian commutative description on the dual torus. It is argued that the Ruijsenaars integrable many-body system provides the description of the QHE with finite amount of electrons on the torus. The new IIB brane picture for the QHE is suggested and applied to Jain and generalized hierarchies. This picture naturally links 2d $\sigma$-model and 3d CS description of the QHE. All duality transformations are identified in the brane setup and can be related with the mirror symmetry and S duality. We suggest a brane interpretation of the plateu transition in IQHE in which a critical point is naturally described by $SL(2,R)$ WZW model.Comment: 31 pages, 4 figure

Static Interactions of non-Abelian Vortices

Interactions between non-BPS non-Abelian vortices are studied in non-Abelian U(1) x SU(N) extensions of the Abelian-Higgs model in four dimensions. The distinctive feature of a non-Abelian vortex is the presence of an internal CP^{N-1} space of orientational degrees of freedom. For fine-tuned values of the couplings, the vortices are BPS and there is no net force between two static parallel vortices at arbitrary distance. On the other hand, for generic values of the couplings the interactions between two vortices depend non-trivially on their relative internal orientations. We discuss the problem both with a numerical approach (valid for small deviations from the BPS limit) and in a semi-analytical way (valid at large vortex separations). The interactions can be classified with respect to their asymptotic property at large vortex separation. In a simpler fine-tuned model, we find two regimes which are quite similar to the usual type I/II Abelian superconductors. In the generic model we find other two new regimes: type I*/II*. Unlike the type I (type II) case, where the interaction is always attractive (repulsive), the type I* and II* have both attractive and repulsive interactions depending on the relative orientation. We have found a rich variety of interactions at small vortex separations. For some values of the couplings, a bound state of two static vortices at a non-zero distance exists.Comment: 36 pages, 13 figures; v2 a small comment and a reference adde