On two geometric constructions of U(sl_n) and its representations

Ginzburg and Nakajima have given two different geometric constructions of quotients of the universal enveloping algebra of sl_n and its irreducible finite-dimensional highest weight representations using the convolution product in the Borel-Moore homology of flag varieties and quiver varieties respectively. The purpose of this paper is to explain the precise relationship between the two constructions. In particular, we show that while the two yield different quotients of the universal enveloping algebra, they produce the same representations and the natural bases which arise in both constructions are the same. We also examine how this relationship can be used to translate the crystal structure on irreducible components of quiver varieties, defined by Kashiwara and Saito, to a crystal structure on the varieties appearing in Ginzburg's construction, thus recovering results of Malkin.Comment: 20 pages, 1 figure; v2: Minor changes to match published versio

The Landau gauge lattice QCD simulation and the gluon propagator

The gluon propagator in the Landau gauge lattice QCD simulation of quenched 8^3 x 16 lattice is measured. The data suggests the confinement mechanism of the Gribov-Zwanziger theory.Comment: LATTICE98(confine), 3 pages, 3 eps figure

Numerical studies of confinement in the lattice Landau gauge

Critical conjectures on confinement in the Landau gauge is numerically tested in focus to Gribov copy effects. One of the subjects is of the Kugo-Ojima confinement criterion and the other is of various viewpoints in the Gribov-Zwanziger theory. We use the smearing gauge as a reference gauge free of Gribov copy, and performed three types of simulations, log U, U-linear and log U in the smearing gauge. It is found that Gribov copy effect on the Kugo-Ojima parameter is small. log U and U-linear simulations yield only global scale factor difference in gluon propagator and in ghost propagator, and about 10% difference in Kugo-Ojima parameter. The horizon function defined by Zwanziger is evaluated in three types of gauge field and compared. All data show the negative horizon function as expected.Comment: 4 pages, 2 eps figures, espcrc2.sty included, Lattice 2000 contribution(Confinement and Strings

Existence of random gradient states

We consider two versions of random gradient models. In model A the interface feels a bulk term of random fields while in model B the disorder enters through the potential acting on the gradients. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are "gradient Gibbs measures" describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and K\"{u}lske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt $u\in \mathbb{R}^d$ for model A when $d\geq3$ and the disorder has mean zero, and for model B when $d\geq1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.Comment: Published in at http://dx.doi.org/10.1214/11-AAP808 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quiver varieties and fusion products for sl_2

We construct the fusion product of finite-dimensional sl_2-modules in the homology of (or in the space of constructible functions on) a certain subvariety L_l(w_1, ..., w_r) of Nakajima's tensor product variety L(w_1,..., w_r). We also give a combinatorial description of the irreducible components of this subvariety using the notions of graphical calculus and crossingless matches for sl_2.Comment: 11 pages, Latex. v2: Some minor typos correcte

Test of the Kugo-Ojima Confinement Criterion in the Lattice Landau Gauge

We present the first results of numerical test of the Kugo-Ojima confinement criterion in the lattice Landau gauge. The Kugo-Ojima criterion of colour confinement in the BRS formulation of the continuum gauge theory is given by $u_b^a(p^2)=-\delta_b^a$, where $u_b^a$ is defined by the two point function of the Faddeev-Popov ghost fields $c, \bar c$ and the gauge field $A_\mu$. We measured the lattice version of $u_b^a(0)$ in use of $1/(-\partial D(A))$ where $D_\mu(A)$ is a lattice covariant derivative in the new definition of the gauge fields as $U=e^A$. We obtained that $u_b^a(0)$ is consistent with $-c\delta_b^a, c=0.7$ in SU(3) quenched simulation data of $\beta=5.5$ on $8^4$ and $12^4$. We report the $\beta$ dependence and finite-size effect of c.Comment: 3 pages Latex, 2 eps figures, Talk given at Lattice99(confine), Pisa, Ital

A new algorithm of Langevin simulation and its application to the SU(2) and SU(3) lattice gauge

The 2nd order Runge-Kutta scheme Langevin simulation of unquenched QCD in pseudofermion method derived from our general theory shows a behaviour as a function of the Langevin step t better than the Fukugita,Oyanagi,Ukawa's scheme.Comment: Talk given at the ``XVth International Symposium on Lattice Field Theory'', Edinburgh (UK), July 22nd-26th 1997 (LATTICE 97); 3 pages, LaTeX file, one Latex picture, uses espcrc2.st

Dual F-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities

The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called "special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteiness.Comment: 14 pages, to appear in J. Commut. Algebra, v3: improved proofs of theorems, v2: minor change