Related papers: The theory of vector-modular forms for the modular…
For an irreducible affine variety $X$ over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on $X$ - gauge modules and Rudakov modules, which admit a compatible…
This is an expository article invited for the ``Commentary'' section of PNAS in connection with Y.-Z. Huang's article, ``Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,'' appearing in the same issue of…
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with…
In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by…
We extend the previously introduced constructive modular method to nonperturbative QFT. In particular the relevance of the concept of ``quantum localization'' (via intersection of algebras) versus classical locality (via support properties…
In this expository paper, we present a construction of tree modules and combine it with (infinite dimensional) tilting theory and relative Mittag-Leffler conditions in order to explore limits of the approximation theory of modules. We also…
In this paper we follow the constructions of Turaev's book [Tu] closely, but with small modifications, to construct of a modular functor, in the sense of Kevin Walker, from any modular tensor category. We further show that this modular…
Cl\'ery and van der Geer determined generators for some modules of vector valued Picard modular forms on the two dimensional ball. In this paper we consider the case of a three dimensional ball with the action of the Picard modular group…
In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a…
The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in…
The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We…
For a nice-enough category $\mathcal{C}$, we construct both the morphism category ${\rm H}(\mathcal{C})$ of $\mathcal{C}$ and the category ${\rm mod}\mbox{-}\mathcal{C}$ of all finitely presented contravariant additive functors over…
This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the…
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion…
We introduce the new concept of cartesian module over a pseudofunctor $R$ from a small category to the category of small preadditive categories. Already the case when $R$ is a (strict) functor taking values in the category of commutative…
The theory of quaternionic modular forms has been studied for decades as an example of the modular forms of many variables. The purpose of this study is to provide some congruence relations satisfied by such quaternionic modular forms.
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the effective…
A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for…
Motivated by the study of meromorphic vector fields, a model theory of "compact complex manifolds equipped with a generic derivation" is here proposed. This is made precise by the notion of a differential CCM-structure. A first-order…
In this article, we establish an $L^2$ extension theorem for Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles, and the strong openness and stability properties of the multiplier submodule sheaves associated to…