Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of the flag variety itself as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.
From our viewpoint, the operators of quantum multiplication form a family (parametrized by @math ) of maximal abelian subalgebras in a certain geometrically constructed Yangian. Long before geometric representation theory, Yangians appeared in mathematical physics as symmetries of integrable models of quantum mechanics and quantum field theory. The maximal abelian subalgebra there is the algebra of quantum integrals of motion, that is, of the operators commuting with the Hamiltonian. A profound correspondence between quantum integrable systems and supersymmetric gauge theories was discovered by Nekrasov and Shatashvili, see @cite_41 . It connects the two appearances of Yangians and, for example, correctly predicts that the eigenvalues of the operators of quantum multiplication are given by solutions of certain Bethe equations (well-known in quantum integrable systems, but probably quite mysterious to geometers). The integrable and geometric viewpoints are complementary in many ways and, no doubt, will lead to new insights on both sides of the correspondence.
@cite_41
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We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system. 1. IntroductionIn this work we address to the problem of the membership to the ideal of r-nuclearoperators for integral operators of the form(1.1) Tf(x) =Z G(x)F(xy)H(y)f(y)dy;where Gand H are bounded functions and F is a smooth function admitting asuitable Taylor expansion. A such kind of operator arises in the theory of laser,particularly in the form of the Fox-Li operator which has attracted the interest ofresearchers in recent years(cf. [BBIN10], [BGI12], [BGHI12]). In those works, theauthors were focused in the investigation of the asymptotic distribution of the sin-gular values and eigenvalues. More recently, the r-nuclearity and its connection withGrothendieck-Lidskii’s formula on compact Lie groups and applications to some dif-ferential operators have been studied in [DR13]. Applications of nuclear operators tothe study of periodic pseudo-di erential operators have been established in [DW13].The nuclearity of operators on spaces of Bochner integrable functions(cf. [DE09])and the corresponding relation with the martingale vector-maximal function for thestudy of traces have been considered in [Del]. Recent progress on the nuclearity andsingular values can also be found in [FMO09] and [MO12].The Fox-Li operator was introduced in [FL61], A. G. Fox and T. Li considered therepeated reection of an electromagnetic wave between two plane-parallel rectangularmirrors. The current density at the surface of each reector satis es the eigenvalueintegral equationTf(x) = f(x):For a good account on the physics of such kind of operators we recommend the chapterXI, section 60 in the book [NE05] by Trefethen and Embree. The Fox-Li operator is We consider the perturbation of elliptic pseudodifferential operators @math with more than square integrable Green's functions by random, rapidly varying, sufficiently mixing, potentials of the form @math . We analyze the source and spectral problems associated with such operators and show that the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as @math as explicit Gaussian processes. Such results may be seen as central limit corrections to homogenization (law of large numbers). Similar results are derived for more general elliptic equations with random coefficients in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions. We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+?j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I?I?Rd,f?TdI?I?Rd,f?Td, p,q?Rnp,q?Rn, t?T1t?T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I?I?RdI?I?Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in and