Small Instantons in CP1CP^1 and CP2CP^2 Sigma Models

The anomalous scaling behavior of the topological susceptibility $\chi_t$ in two-dimensional $CP^{N-1}$ sigma models for $N\leq 3$ is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of $\chi_t$ in these models is traced to the presence of small instantons with a radius of order $a$ (= lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of L\"{u}scher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in $CP^1$ and $CP^2$, leading to a divergent topological susceptibility in the continuum limit. For the \CP models with $N>3$ the topological susceptibility is observed to scale properly with the mass gap. These larger $N$ models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or ``Wilson bag'' excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.Comment: 33 pages, 12 figure

A Note on ODEs from Mirror Symmetry

We give close formulas for the counting functions of rational curves on complete intersection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the instanton corrected Yukawa coupling.Comment: 24 pages using harvma

Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points

In this paper, we derive the virtual structure constants used in mirror computation of degree k hypersurface in CP^{N-1}, by using localization computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1} with two marked points. We also apply this technique to non-nef local geometry O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff factorization.Comment: 10 pages, latex, a minor change in Section 4, English is refined, Some typing errors in Section 3 are correcte

Decreasing erucic acid level by RNAi-mediated silencing of fatty acid elongase 1 (BnFAE1.1) in rapeseeds (Brassica napus L.)

The β-ketoacyl CoA synthase encoded by fatty acid elongase 1 gene (BnFAE1.1) is a rate-limiting enzyme regulating biosynthesis of erucic acid in rapeseeds (Brassica napus). To develop low level of erucic acid in rapeseeds by intron-spliced hairpin RNA, an inverted repeat unit of a partial BnFAE1.1 gene interrupted by a spliceable intron was cloned into pCAMBIA3301, and a seed-specific (Napin) promoter was used to control the transcription of the transgene. Four transgenic plants harboring a single copy of transgene were generated. Expression of endogenous BnFAE1.1 gene in developing T3 seeds was significantly reduced. In mature T3 seeds, erucic acid was decreased by 60.8 to 99.1% compared with wild type seeds, and accounted for 0.36 to 15.56% of total fatty acids. The level of eicosenoic acid was also greatly decreased. Furthermore, it resulted in a significant increase in the level of oleic acid, but total fatty acid content in T3 seeds was the same with that in wild type seeds. In conclusion, the expression of endogenous BnFAE1.1 was efficiently silenced by the designed RNAi silencer, causing a significant down-regulation in the level of erucic acid. Therefore, the RNAi-mediated post-transcriptional silencing of FAE1 gene to reduce oleic acid in rapeseeds was an efficient method to breed some new B. napus lines.Key words: Brassica napus L., fatty acid elongase, intron-spliced hairpin RNA, down-regulation, erucic acid

The structure of the Kac-Wang-Yan algebra

The Lie algebra $\mathcal{D}$ of regular differential operators on the circle has a universal central extension $\hat{\mathcal{D}}$. The invariant subalgebra $\hat{\mathcal{D}}^+$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $\hat{\mathcal{D}}^+$-module with central charge $c\in\mathbb{C}$, and its irreducible quotient $\mathcal{V}_c$, possess vertex algebra structures, and $\mathcal{V}_c$ has a nontrivial structure if and only if $c\in \frac{1}{2}\mathbb{Z}$. We show that for each integer $n>0$, $\mathcal{V}_{n/2}$ and $\mathcal{V}_{-n}$ are $\mathcal{W}$-algebras of types $\mathcal{W}(2,4,\dots,2n)$ and $\mathcal{W}(2,4,\dots, 2n^2+4n)$, respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group $\text{O}(n)$ and the symplectic group $\text{Sp}(2n)$, respectively. Based on Sergeev's theorems on the invariant theory of $\text{Osp}(1,2n)$ we conjecture that $\mathcal{V}_{-n + 1/2}$ is of type $\mathcal{W}(2,4,\dots, 4n^2+8n+2)$, and we prove this for $n=1$. As an application, we show that invariant subalgebras of $\beta\gamma$-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.Comment: Final versio

Heat flux operator, current conservation and the formal Fourier's law

By revisiting previous definitions of the heat current operator, we show that one can define a heat current operator that satisfies the continuity equation for a general Hamiltonian in one dimension. This expression is useful for studying electronic, phononic and photonic energy flow in linear systems and in hybrid structures. The definition allows us to deduce the necessary conditions that result in current conservation for general-statistics systems. The discrete form of the Fourier's Law of heat conduction naturally emerges in the present definition