Superconformal indices of three-dimensional theories related by mirror symmetry

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.Comment: 17 pages; minor change

Loss of spiritual and moral values

В работе была рассмотрена проблема духовно-нравственного воспитания молодежи Российского общества. В процессе изучения были выявлены проблемы, которые были связаны с утратой духовных и нравственных ценностей, перед нами стояла задача выявить пути и методы решения этой проблемы.In this study we consider the problem of spiritual and moral education of the youth of the Russian society. In the course of studying the problems were identified that were associated with the loss of spiritual and moral values, our task was to identify ways and means of solving this problem

Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission

S-duality and 2d Topological QFT

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.Comment: 25 pages, 11 figure

Relation between the 4d superconformal index and the S^3 partition function

A relation between the 4d superconformal index and the S^3 partition function is studied with focus on the 4d and 3d actions used in localization. In the case of vanishing Chern-Simons levels and round S^3 we explicitly show that the 3d action is obtained from the 4d action by dimensional reduction up to terms which do not affect the exact results. By combining this fact and a recent proposal concerning a squashing of S^3 and SU(2) Wilson line, we obtain a formula which gives the partition function depending on the Weyl weight of chiral multiplets, real mass parameters, FI parameters, and a squashing parameter as a limit of the index of a parent 4d theory.Comment: 20 pages, LaTeX; v2: comments added; v3: minor corrections, version published in JHE

q-Ultraspherical polynomials for q a root of unity

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of $q$-beta integrals of Ramanujan.Comment: 7 pages, LATE

Distinct Patterns of Expression and Evolution of Intronless and Intron-Containing Mammalian Genes

Comparison of expression levels and breadth and evolutionary rates of intronless and intron-containing mammalian genes shows that intronless genes are expressed at lower levels, tend to be tissue specific, and evolve significantly faster than spliced genes. By contrast, monomorphic spliced genes that are not subject to detectable alternative splicing and polymorphic alternatively spliced genes show similar statistically indistinguishable patterns of expression and evolution. Alternative splicing is most common in ancient genes, whereas intronless genes appear to have relatively recent origins. These results imply tight coupling between different stages of gene expression, in particular, transcription, splicing, and nucleocytosolic transport of transcripts, and suggest that formation of intronless genes is an important route of evolution of novel tissue-specific functions in animals

An E7 Surprise

We explore some curious implications of Seiberg duality for an SU(2) four-dimensional gauge theory with eight chiral doublets. We argue that two copies of the theory can be deformed by an exactly marginal quartic superpotential so that they acquire an enhanced E7 flavor symmetry. We argue that a single copy of the theory can be used to define an E7-invariant superconformal boundary condition for a theory of 28 five-dimensional free hypermultiplets. Finally, we derive similar statements for three-dimensional gauge theories such as an SU(2) gauge theory with six chiral doublets or Nf=4 SQED.Comment: 27 page