Notes on Integral Identities for 3d Supersymmetric Dualities

Four dimensional $\mathcal{N}=2$ Argyres-Douglas theories have been recently conjectured to be described by $\mathcal{N}=1$ Lagrangian theories. Such models, once reduced to 3d, should be mirror dual to Lagrangian $\mathcal{N}=4$ theories. This has been numerically checked through the matching of the partition functions on the three sphere. In this article, we provide an analytic derivation for this result in the $A_n$ case via hyperbolic hypergeometric integrals. We study the $D_4$ case as well, commenting on some open questions and possible resolutions. In the second part of the paper we discuss other integral identities leading to the matching of the partition functions in 3d dual pairs involving higher monopole superpotentials.Comment: 16 pages, 1 figure, typos corrected, references added and improved discussio

Quantum phase transition as an interplay of Kitaev and Ising interactions

We study the interplay between the Kitaev and Ising interactions on both ladder and two dimensional lattices. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. It is confirmed by the degeneracy of the entanglement spectrum and non-trivial phase factors (inequivalent projective representations of the symmetries), which are obtained within infinite matrix-product representation of numerical density matrix renormalization group. We derive the effective theory to describe the topological phase transition on both ladder and two-dimensional lattices, which is given by the transverse field Ising model with/without next-nearest neighbor coupling. The ladder has three phases, namely, the Kitaev SPT, symmetry broken ferro/antiferromagnetic order and classical spin-liquid. The non-zero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point, change of the entanglement spectrum degeneracy and a drop in the ground-state fidelity. The central charge of the critical points are either c=1 or c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. In the absence of frustration, the 2D lattice shows a topological phase transition from the $\mathbb{Z}_2$ spin-liquid state to the long-range ordered Ising phase at finite ratio of couplings, while in the presence of frustration, an order-by-disorder transition is induced by the Kitaev term. The 2D classical spin-liquid phase is unstable against the addition of Kitaev term toward an ordered phase before the transition to the $\mathbb{Z}_2$ spin-liquid state.Comment: 16 pages, 18 figure

Relationship between serum level of interleukin-2 in patients with systemic lupus erythematosus and disease activity in comparison with control group

Background: Despite the large number of surveys, there are not any validated biomarkers for SLE disease activity till now. This study aimed to evaluate the relationship between serum level of IL-2 in patients with SLE and disease activity in comparison with control group. Materials and Methods: In this case-control study, 73 patients with lupus and 73 healthy subjects referred to the rheumatology clinic of 5 Azar Hospital in Gorgan (North of Iran).They were studied via convenience sampling during 2011-2012. Blood samples were taken from both groups and serum levels of interleukin -2 measured by Avi Bion Human IL-2 ELISA kit. Serum Level of IL-2 greater than 15 pg/ml defined positive and lesser than this amount defined negative. Disease activity evaluated with SLE disease activity index. Score greater than or equal to three or four defined as active disease. Data analysis conducted by SPSS software (version 16) and by using descriptive statistics and statistical tests. Results: Serum level of IL-2 was positive in 45.2% of sample studied and negative in 54.8% in case group, while in control group, serum level of IL-2 only in 11% of sample studied was positive and in 89% was negative. Statistical analysis indicated a significant relationship between serum level of IL-2 and the SLE disease activity index (p=0.025). Conclusion: This study showed the relationship between serum levels of IL-2 and disease activity, so this biomarker can be used as a clinical indicator for assessing disease activity in patients with SLE