Related papers: Modular differential equations for characters of R…
In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its…
We present a low entry-level introduction to the Conformal Bootstrap. We review and obtain several basic bounds using Linear Programming in machine precision in Mathematica, making the results accessible even to the most uneducated computer…
Recently, the modular linear differential equation (MLDE) for level-two congruence subgroups $\Gamma_\theta, \Gamma^{0}(2)$ and $\Gamma_0(2)$ of $\text{SL}_2(\mathbb{Z})$ was developed and used to classify the fermionic rational conformal…
$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
We describe the construction of the genus-zero parts of conformal field theories in the sense of G. Segal from representations of vertex operator algebras satisfying certain conditions. The construction is divided into four steps and each…
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…
In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…
We prove existence and uniqueness of complex Hodge structures on modular functors. The proof is based on the non-Abelian Hodge correspondence and Ocneanu rigidity. Given a modular functor, we explain how its Hodge numbers fit into a…
It is shown that every weak Jacobi form of weight zero and index one on a congruence subgroup of the full Jacobi group can be decomposed into $N=4$ superconformal characters. Additionally, a simple expression for the mock modular form…
We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in…
We study non-trivial deformations of the natural action of the Lie algebra $\mathrm{Vect}({\mathbb R}^n)$ on the space of differential forms on ${\mathbb R}^n$. We calculate abstractions for integrability of infinitesimal multi-parameter…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and…
In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their H\"older…
We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formul{\ae} for…
We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal…
The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra…
Fermionic-type character formulae are presented for charged irreduciblemodules of the graded parafermionic conformal field theory associated to the coset $osp(1,2)_k/u(1)$. This is obtained by counting the weakly ordered `partitions'…
We find modular transformations of normalized characters for the following $W$-algebras: (a) $W^{min}_k(\frak{g})$, where $\frak{g}=D_n \, (n \geq 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\geq -2$, or $\geq…