Lattice QCD as a theory of interacting surfaces

Pure gauge lattice QCD at arbitrary D is considered. Exact integration over link variables in an arbitrary D-volume leads naturally to an appearance of a set of surfaces filling the volume and gives an exact expression for functional of their boundaries. The interaction between each two surfaces is proportional to their common area and is realized by a non-local matrix differential operator acting on their boundaries. The surface self-interaction is given by the QCD$_2$ functional of boundary. Partition functions and observables (Wilson loop averages) are written as an averages over all configurations of an integer-valued field living on a surfaces.Comment: TAUP-2204-94, 12pp., LaTe

Distributive Lattices, Affine Semigroups, and Branching Rules of the Classical Groups

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.Comment: 30 pages, extensively revise

Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an \mr{Spin}(2, 2n+2) dynamical symmetry which extends the manifest \mr{Spin}(2n+1) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight \mr{Spin}(2, 2n+2)-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight \mr{Spin}(2, 2n+2)-module.Comment: 27 pages, Refs. update

Born--Oppenheimer corrections to the effective zero-mode Hamiltonian in SYM theory

We calculate the subleading terms in the Born--Oppenheimer expansion for the effective zero-mode Hamiltonian of N = 1, d=4 supersymmetric Yang--Mills theory with any gauge group. The Hamiltonian depends on 3r abelian gauge potentials A_i, lying in the Cartan subalgebra, and their superpartners (r being the rank of the group). The Hamiltonian belongs to the class of N = 2 supersymmetric QM Hamiltonia constructed earlier by Ivanov and I. Its bosonic part describes the motion over the 3r--dimensional manifold with a special metric. The corrections explode when the root forms \alpha_j(A_i) vanish and the Born--Oppenheimer approximation breaks down.Comment: typos correcte

Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory

We obtain integral representations for the wave functions of Calogero-type systems,corresponding to the finite-dimentional Lie algebras,using exact evaluation of path integral.We generalize these systems to the case of the Kac-Moody algebras and observe the connection of them with the two dimensional Yang-Mills theory.We point out that Calogero-Moser model and the models of Calogero type like Sutherland one can be obtained either classically by some reduction from two dimensional Yang-Mills theory with appropriate sources or even at quantum level by taking some scaling limit.We investigate large k limit and observe a relation with Generalized Kontsevich Model.Comment: 34 pages,UUITP-6/93 and ITEP-20/9

Generalized Calogero-Moser-Sutherland models from geodesic motion on GL(n, R) group manifold

It is shown that geodesic motion on the GL(n, R) group manifold endowed with the bi-invariant metric d s^2 = tr(g^{-1} d g)^2 corresponds to a generalization of the hyperbolic n-particle Calogero-Moser-Sutherland model. In particular, considering the motion on Principal orbit stratum of the SO(n, R) group action, we arrive at dynamics of a generalized n-particle Calogero-Moser-Sutherland system with two types of internal degrees of freedom obeying SO(n, R) \bigoplus SO(n, R) algebra. For the Singular orbit strata of SO(n, R) group action the geodesic motion corresponds to certain deformations of the Calogero-Moser-Sutherland model in a sense of description of particles with different masses. The mass ratios depend on the type of Singular orbit stratum and are determined by its degeneracy. Using reduction due to discrete and continuous symmetries of the system a relation to II A_n Euler-Calogero-Moser-Sutherland model is demonstrated.Comment: 16 pages, LaTeX, no figures. V2: Typos corrected, two references added. V3: Abstract changed, typos corrected, a few formulas and references added. The presentation in the last section has been clarified and it was restricted to the case of GL(3, R) group, the analysis of GL(4, R) will be given elsewhere. V4: Minor corrections in the whole text, more formulas and references added, accepted for publication in PL

Finite Factorization equations and Sum Rules for BPS correlators in N=4 SYM theory

A class of exact non-renormalized extremal correlators of half-BPS operators in N=4 SYM, with U(N) gauge group, is shown to satisfy finite factorization equations reminiscent of topological gauge theories. The finite factorization equations can be generalized, beyond the extremal case, to a class of correlators involving observables with a simple pattern of SO(6) charges. The simple group theoretic form of the correlators allows equalities between ratios of correlators in N=4 SYM and Wilson loops in Chern-Simons theories at k=\infty, correlators of appropriate observables in topological G/G models and Wilson loops in two-dimensional Yang-Mills theories. The correlators also obey sum rules which can be generalized to off-extremal correlators. The simplest sum rules can be viewed as large k limits of the Verlinde formula using the Chern-Simons correspondence. For special classes of correlators, the saturation of the factorization equations by a small subset of the operators in the large N theory is related to the emergence of semiclassical objects like KK modes and giant gravitons in the dual ADS \times S background. We comment on an intriguing symmetry between KK modes and giant gravitons.Comment: 1+69 pages, harvmac, 38 figures; v2: references added, comment added on next-to-extremal correlator

Can fusion coefficients be calculated from the depth rule ?

The depth rule is a level truncation of tensor product coefficients expected to be sufficient for the evaluation of fusion coefficients. We reformulate the depth rule in a precise way, and show how, in principle, it can be used to calculate fusion coefficients. However, we argue that the computation of the depth itself, in terms of which the constraints on tensor product coefficients is formulated, is problematic. Indeed, the elements of the basis of states convenient for calculating tensor product coefficients do not have a well-defined depth! We proceed by showing how one can calculate the depth in an `approximate' way and derive accurate lower bounds for the minimum level at which a coupling appears. It turns out that this method yields exact results for $\widehat{su}(3)$ and constitutes an efficient and simple algorithm for computing $\widehat{su}(3)$ fusion coefficients.Comment: 27 page

Higgs Bundles, Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schr\"{o}dinger equation in the $N$-particle sector. This implies the full equivalence between the above gauge theory and the $N$-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the $G/G$ gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed.Comment: 48 pages, typos corrected, one reference adde