Duality and Symmetry in Chiral Potts Model

We discover an Ising-type duality in the general $N$-state chiral Potts model, which is the Kramers-Wannier duality of planar Ising model when N=2. This duality relates the spectrum and eigenvectors of one chiral Potts model at a low temperature (of small $k'$) to those of another chiral Potts model at a high temperature (of $k'^{-1}$). The $\tau^{(2)}$-model and chiral Potts model on the dual lattice are established alongside the dual chiral Potts models. With the aid of this duality relation, we exact a precise relationship between the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts model and the $sl_2$-loop-algebra symmetry of its associated spin-$\frac{N-1}{2}$ XXZ chain through the identification of their eigenstates.Comment: Latex 34 pages, 2 figures; Typos and misprints in Journal version are corrected with minor changes in expression of some formula

On τ(2)\tau^{(2)}-model in Chiral Potts Model and Cyclic Representation of Quantum Group Uq(sl2)U_q(sl_2)

We identify the precise relationship between the five-parameter $\tau^{(2)}$-family in the $N$-state chiral Potts model and XXZ chains with $U_q (sl_2)$-cyclic representation. By studying the Yang-Baxter relation of the six-vertex model, we discover an one-parameter family of $L$-operators in terms of the quantum group $U_q (sl_2)$. When $N$ is odd, the $N$-state $\tau^{(2)}$-model can be regarded as the XXZ chain of $U_{\sf q} (sl_2)$ cyclic representations with ${\sf q}^N=1$. The symmetry algebra of the $\tau^{(2)}$-model is described by the quantum affine algebra $U_{\sf q} (\hat{sl}_2)$ via the canonical representation. In general for an arbitrary $N$, we show that the XXZ chain with a $U_q (sl_2)$-cyclic representation for $q^{2N}=1$ is equivalent to two copies of the same $N$-state $\tau^{(2)}$-model.Comment: Latex 11 pages; Typos corrected, Minor changes for clearer presentation, References added and updated-Journal versio

Fusion Operators in the Generalized τ(2)\tau^{(2)}-model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin

We construct the fusion operators in the generalized $\tau^{(2)}$-model using the fused $L$-operators, and verify the fusion relations with the truncation identity. The algebraic Bethe ansatz discussion is conducted on two special classes of $\tau^{(2)}$ which include the superintegrable chiral Potts model. We then perform the parallel discussion on the XXZ spin chain at roots of unity, and demonstrate that the $sl_2$-loop-algebra symmetry exists for the root-of-unity XXZ spin chain with a higher spin, where the evaluation parameters for the symmetry algebra are identified by the explicit Fabricius-McCoy current for the Bethe states. Parallels are also drawn to the comparison with the superintegrable chiral Potts model.Comment: Latex 33 Pages; Typos and errors corrected, New improved version by adding explanations for better presentation. Terminology in the content and the title refined. References added and updated-Journal versio

Spin operator matrix elements in the quantum Ising chain: fermion approach

Using some modification of the standard fermion technique we derive factorized formula for spin operator matrix elements (form-factors) between general eigenstates of the Hamiltonian of quantum Ising chain in a transverse field of finite length. The derivation is based on the approach recently used to derive factorized formula for Z_N-spin operator matrix elements between ground eigenstates of the Hamiltonian of the Z_N-symmetric superintegrable chiral Potts quantum chain. The obtained factorized formulas for the matrix elements of Ising chain coincide with the corresponding expressions obtained by the Separation of Variables Method.Comment: 19 page

Eigenvectors of Baxter-Bazhanov-Stroganov \tau^{(2)}(t_q) model with fixed-spin boundary conditions

The aim of this contribution is to give the explicit formulas for the eigenvectors of the transfer-matrix of Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. These formulas are obtained by a limiting procedure from the formulas for the eigenvectors of periodic BBS model. The latter formulas were derived in the framework of the Sklyanin's method of separation of variables. In the case of fixed-spin boundaries the corresponding T-Q Baxter equations for the functions of separated variables are solved explicitly. As a particular case we obtain the eigenvectors of the Hamiltonian of Ising-like Z_N quantum chain model.Comment: 14 pages, paper submitted to Proceedings of the International Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007

Factorized finite-size Ising model spin matrix elements from Separation of Variables

Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter--Bazhanov--Stroganov or $\tau^{(2)}$-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy

Comment on the Generation Number in Orbifold Compactifications

There has been some confusion concerning the number of $(1,1)$-forms in orbifold compactifications of the heterotic string in numerous publications. In this note we point out the relevance of the underlying torus lattice on this number. We answer the question when different lattices mimic the same physics and when this is not the case. As a byproduct we classify all symmetric $Z_N$-orbifolds with $(2,2)$ world sheet supersymmetry obtaining also some new ones.Comment: 28 pages, 9 figures not included, available in postscript at reques

Eigenvectors in the Superintegrable Model I: sl_2 Generators

In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic tau_2(t)model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of tau_2(t) in the Q=0 sector is 2^r, with r=(N-1)L/N when the size of the transfer matrix L is a multiple of N. We introduce chiral Potts model operators, different from the more commonly used generators of quantum group U-tilde_q(sl-hat(2)). From these we can form the generators of a loop algebra L(sl(2)). For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra sl(2).Comment: LaTeX 2E document, 11 pages, 1 eps figure, using iopart.cls with graphicx and iopams packages. v2: Appended text to title, added acknowledgments and made several minor corrections v3: Added reference, eliminated ambiguity, corrected a few misprint

Multigene interactions and the prediction of depression in the Wisconsin Longitudinal Study

Objectives: Single genetic loci offer little predictive power for the identification of depression. This study examined whether an analysis of gene-gene (G x G) interactions of 78 single nucleotide polymorphisms (SNPs) in genes associated with depression and agerelated diseases would identify significant interactions with increased predictive power for depression. Design: A retrospective cohort study. Setting: A survey of participants in the Wisconsin Longitudinal Study. Participants: A total of 4811 persons (2464 women and 2347 men) who provided saliva for genotyping; the group comes from a randomly selected sample of Wisconsin high school graduates from the class of 1957 as well as a randomly selected sibling, almost all of whom are non-Hispanic white. Primary outcome measure: Depression as determine by the Composite International Diagnostic Interview-Short-Form. Results: Using a classification tree approach (recursive partitioning (RP)), the authors identified a number of candidate G 3 G interactions associated with depression. The primary SNP splits revealed by RP (ANKK1 rs1800497 (also known as DRD2 Taq1A) in men and DRD2 rs224592 in women) were found to be significant as single factors by logistic regression (LR) after controlling for multiple testing (p=0.001 for both). Without considering interaction effects, only one of the five subsequent RP splits reached nominal significance in LR (FTO rs1421085 in women, p=0.008). However, after controlling for G x G interactions by running LR on RP-specific subsets, every split became significant and grew larger in magnitude (OR (before) → (after): men: GNRH1 novel SNP: (1.43 → 1.57); women: APOC3 rs2854116: (1.28 → 1.55), ACVR2B rs3749386: (1.11 → 2.17), FTO rs1421085: (1.32 → 1.65), IL6 rs1800795: (1.12 → 1.85)). Conclusions: The results suggest that examining G x G interactions improves the identification of genetic associations predictive of depression. 4 of the SNPs identified in these interactions were located in two pathways well known to impact depression: neurotransmitter (ANKK1 and DRD2) and neuroendocrine (GNRH1 and ACVR2B) signalling. This study demonstrates the utility of RP analysis as an efficient and powerful exploratory analysis technique for uncovering genetic and molecular pathway interactions associated with disease aetiology

Interferon priming is essential for human CD34+ cell-derived plasmacytoid dendritic cell maturation and function

Plasmacytoid dendritic cells (pDC) are essential for immune competence. Here we show that pDC precursor differentiated from human CD34+ hematopoietic stem and progenitor cells (HSPC) has low surface expression of pDC markers, and has limited induction of type I interferon (IFN) and IL-6 upon TLR7 and TLR9 agonists treatment; by contrast, cGAS or RIG-I agonists-mediated activation is not altered. Importantly, after priming with type I and II IFN, these precursor pDCs attain a phenotype and functional activity similar to that of peripheral blood-derived pDCs. Data from CRISPR/Cas9-mediated genome editing of HSPCs further show that HSPC-pDCs with genetic modifications can be obtained, and that expression of the IFN-α receptor is essential for the optimal function, but dispensable for the differentiation, of HSPC-pDC percursor. Our results thus demonstrate the biological effects of IFNs for regulating pDC function, and provide the means of generating of gene-modified human pDCs