Related papers: Congruence Property in Orbifold Theory

This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras…

Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures the failure of…

We study a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type whose simple current modules are graded by $\mathbb{Z}_{2k}$. Based on those simple current modules, a vertex operator algebra associated with a…

We apply the factorization and vector bundle propositionerty of the sheaves of conformal blocks on $\overline{\mathscr{M}}_{g,n}$. defined by vertex operator algebras (VOAs) and give geometric proofs of essential results in the…

We show in this work that homology in degree d of a congruence group, in a very general framework, defines a weakly polynomial functor of degree at most 2d and we describe this functor modulo polynomial functors of smaller degree. Our main…

Let $G$ be a simple algebraic group over an algebraically closed field of prime characteristic. If $M$ is a finite dimensional $G$-module that is projective over the Frobenius kernel of $G$, then its character is divisible by the character…

Let G be a finite group acting tamely on a proper reduced curve C over an algebraically closed field. We study the G-module structure on the cohomology groups of a G-equivariant locally free sheaf F on C, and give formulas of…

Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed…

A super-modular category is a unitary pre-modular category with M\"uger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories as any unitary…

Monstrous moonshine relates the representation of the Monster finite sporadic simple group to the distinguished modular functions, called Hauptmoduln. Chen-Yui~\cite{Chen-Yui} showed that the CM values of Hauptmoduln which appeare in…

Let $G$ be a connected reductive group. In a previous paper, arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a $G$-variety $X$ admits a natural homomorphism with finite kernel to the Langlands dual group $G^\vee$…

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that…

Let $R$ be a commutative ring. Roughly speaking, we prove that an $R$-module $M$ is flat iff it is a direct limit of $R$-module affine algebraic varieties, and $M$ is a flat Mittag-Leffler module iff it is the union of its $R$-submodule…

In a recent article (2022) we proved with L. Zaj\'i\v{c}ek that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave…

Modular symbols for the congruence subgroup $\Gamma_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to…

The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to…

In this paper, for a vertex operator algebra $V$ with an automorphism $g$ of order $T,$ an admissible $V$-module $M$ and a fixed nonnegative rational number $n\in\frac{1}{T}\Bbb{Z}_{+},$ we construct an $A_{g,n}(V)$-bimodule $\AA_{g,n}(M)$…

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$ at level $-\frac{5}{4}$ are investigated. After classifying the…

In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra…

We prove an $\text{SL}_2 (\mathbb{Z})$-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another…