Modularity of supersymmetric partition functions

We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional $$ \mathcal{N} $$ N = 1 supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.

Arithmetic of generalized Dedekind sums and their modularity

Dedekind sums were introduced by Dedekind to study the transformation properties of Dedekind η function under the action of SL2(ℤ). In this paper, we study properties of generalized Dedekind sums si,j(p, q). We prove an asymptotic expansion of a function on ℚ defined in terms of generalized Dedekind sums by using its modular property. We also prove an equidistribution property of generalized Dedekind sums.

MerCat: a versatile k-mer counter and diversity estimator for database-independent property analysis obtained from metagenomic and/or metatranscriptomic sequencing data

MerCat (“ Mer - Cat enate”) is a parallel, highly scalable and modular property software package for robust analysis of features in next-generation sequencing data. Using assembled contigs and raw sequence reads from any platform as input, MerCat performs k-mer counting of any length k, resulting in feature abundance counts tables. MerCat allows for direct analysis of data properties without reference sequence database dependency commonly used by search tools such as BLAST for compositional analysis of whole community shotgun sequencing (e.g., metagenomes and metatranscriptomes).

MerCat: a versatile k-mer counter and diversity estimator for database-independent property analysis obtained from metagenomic and/or metatranscriptomic sequencing data

MerCat (“ Mer - Cat enate”) is a parallel, highly scalable and modular property software package for robust analysis of features in next-generation sequencing data. Using assembled contigs and raw sequence reads from any platform as input, MerCat performs k-mer counting of any length k, resulting in feature abundance counts tables. MerCat allows for direct analysis of data properties without reference sequence database dependency commonly used by search tools such as BLAST for compositional analysis of whole community shotgun sequencing (e.g., metagenomes and metatranscriptomes).

ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.

Modularity of strong normalization in the algebraic-λ-cube

In this paper we present the algebraic-λ-cube, an extension of Barendregt's λ-cube with first- and higher-order algebraic rewriting. We show that strong normalization is a modular property of all the systems in the algebraic-λ-cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. We also prove that local confluence is a modular property of all the systems in the algebraic-λ-cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.