Operator-Valued Norms

We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties.Comment: 8 page

A Family of N=1 SU(N)^k Theories from Branes at Singularities

We obtain N=1 SU(N)^k gauge theories with bifundamental matter and a quartic superpotential as the low energy theory on D3-branes at singular points. These theories generalize that on D3-branes at a conifold point, studied recently by Klebanov and Witten. For k=3 the defining equation of the singular point is that of an isolated D_4 singularity. For k>3 we obtain a family of multimodular singularities. The considered SU(N)^k theories flow in the infrared to a non-trivial fixed point. We analyze the AdS/CFT correspondence for our examples.Comment: 18 pages, 1 figure, TeX; v2 minor change

Artinian level algebras of codimension 3

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector \H cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function \H for some value of $h_{d+1}$.Comment: 15 page

Systematic study of autocorrelation time in pure SU(3) lattice gauge theory

Results of our autocorrelation measurement performed on Fujitsu AP1000 are reported. We analyze (i) typical autocorrelation time, (ii) optimal mixing ratio between overrelaxation and pseudo-heatbath and (iii) critical behavior of autocorrelation time around cross-over region with high statistic in wide range of $\beta$ for pure SU(3) lattice gauge theory on $8^4$, $16^4$ and $32^4$ lattices. For the mixing ratio K, small value (3-7) looks optimal in the confined region, and reduces the integrated autocorrelation time by a factor 2-4 compared to the pseudo-heatbath. On the other hand in the deconfined phase, correlation times are short, and overrelaxation does not seem to matter For a fixed value of K(=9 in this paper), the dynamical exponent of overrelaxation is consistent with 2 Autocorrelation measurement of the topological charge on $32^3 \times 64$ lattice at $\beta$ = 6.0 is also briefly mentioned.Comment: 3 pages of A4 format including 7-figure

Representations of SU(1,1) in Non-commutative Space Generated by the Heisenberg Algebra

SU(1,1) is considered as the automorphism group of the Heisenberg algebra H. The basis in the Hilbert space K of functions on H on which the irreducible representations of the group are realized is explicitly constructed. The addition theorems are derived.Comment: Latex, 8 page

Noncompact Gauge-Invariant Simulations of U(1), SU(2), and SU(3)

We have applied a new gauge-invariant, noncompact, Monte Carlo method to simulate the $U(1)$, $SU(2)$, and $SU(3)$ gauge theories on $8^4$ and $12^4$ lattices. The Creutz ratios of the Wilson loops agree with the exact results for $U(1)$ for $\beta \ge 0.5$ apart from a renormalization of the charge. The $SU(2)$ and $SU(3)$ Creutz ratios robustly display quark confinement at $\beta = 0.5$ and $\beta = 2$, respectively. At much weaker coupling, the $SU(2)$ and $SU(3)$ Creutz ratios agree with perturbation theory after a renormalization of the coupling constant. For $SU(3)$ the scaling window is near $\beta = 2$, and the relation between the string tension $\sigma$ and our lattice QCD parameter $\Lambda_L$ is $\sqrt{\sigma} \approx 5 \Lambda_L$.Comment: For U(1), we switched from beta = 2 / g^2 to beta = 1 / g^2; 3 pages; latex and espcrc2.sty; one figure generated by PiCTeX; our contribution to Lattice '9

Series of Abelian and Non-Abelian States in C>1 Fractional Chern Insulators

We report the observation of a new series of Abelian and non-Abelian topological states in fractional Chern insulators (FCI). The states appear at bosonic filling nu= k/(C+1) (k, C integers) in several lattice models, in fractionally filled bands of Chern numbers C>=1 subject to on-site Hubbard interactions. We show strong evidence that the k=1 series is Abelian while the k>1 series is non-Abelian. The energy spectrum at both groundstate filling and upon the addition of quasiholes shows a low-lying manifold of states whose total degeneracy and counting matches, at the appropriate size, that of the Fractional Quantum Hall (FQH) SU(C) (color) singlet k-clustered states (including Halperin, non-Abelian spin singlet states and their generalizations). The groundstate momenta are correctly predicted by the FQH to FCI lattice folding. However, the counting of FCI states also matches that of a spinless FQH series, preventing a clear identification just from the energy spectrum. The entanglement spectrum lends support to the identification of our states as SU(C) color-singlets but offers new anomalies in the counting for C>1, possibly related to dislocations that call for the development of new counting rules of these topological states.Comment: 12 pages with supplemental material, 20 figures, published versio