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Geoffrey Mason

Publications and source records attributed to Geoffrey Mason.

At least 37 records · Page 2Linked to original sources
arXiv2017

Generalized Twisted Quantum Doubles of a Finite Group and Rational Orbifolds

In previous work the authors introduced a new class of modular quasi-Hopf algebras $D^ω(G, A)$ associated to a finite group $G$, a central subgroup $A$, and a $3$-cocycle $ω\in Z^3(G, C^x)$. In the present paper we propose a description of the class of orbifold models of rational vertex operator algebras whose module category is tensor equivalent to $D^ω(G, A)$-mod. The paper includes background on quasi-Hopf algebras and a discussion of some relevant orbifolds.

math.QA↗
arXiv2016

The 290 fixed-point sublattices of the Leech lattice

We determine the orbits of fixed-point sublattices of the Leech lattice with respect to the action of the Conway group Co_0. There are 290 such orbits. Detailed information about these lattices, the corresponding coinvariant lattices, and the stabilizing subgroups, is tabulated in several tables.

math.GR↗
arXiv2015

On the structure of modules of vector valued modular forms

If $ρ$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(ρ)$ of vector valued modular forms for $ρ$ is free and of finite rank over the ring $M$ of scalar modular forms of level one. This paper initiates a general study of the structure of $M(ρ)$. Among our results are absolute upper and lower bounds, depending only on the dimension of $ρ$, on the weights of generators for $M(ρ)$, as well as upper bounds on the multiplicities of weights of generators of $M(ρ)$. We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free-basis for $M(ρ)$ in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series $E_6$ of weight six.

math.NT↗
arXiv2015

Jacobi trace functions in the theory of vertex operator algebras

We describe a type of n-point function associated to strongly regular vertex operator algebras V and their irreducible modules. Transformation laws with respect to the Jacobi group are developed for 1-point functions. For certain elements in V, the finite-dimensional space spanned by the corresponding 1-point functions for the inequivalent irreducible modules is shown to be a vector-valued weak Jacobi form. A decomposition of 1-point functions for general elements is proved, and shows that such functions are typically quasi-Jacobi forms. Zhu-type recursion formulas are proved; they show how an n-point function can be written as a linear combination of (n-1)-point functions with coefficients that are quasi-Jacobi forms.

math.QA↗
arXiv2015

Three-dimensional imprimitive representations of the modular group and their associated modular forms

This paper uses previous results of the authors on vector-valued modular forms to study certain non-congruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of Atkin--Swinnerton-Dyer type satisfied by the Fourier coefficients of these forms. Our results rest on group-theoretic facts about the modular group, a detailed study of its imprimitive three-dimensional representations, and the theory of their associated vector-valued modular forms. For the proof of the congruences we also make essential use of a result of Katz.

math.NT↗
arXiv2014

Cleft Extensions and Quotients of Twisted Quantum Doubles

Given a pair of finite groups $F, G$ and a normalized 3-cocycle $ω$ of $G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension $\Bbbk^G_ω\#_c\,\Bbbk F$ where $c$ denotes some suitable cohomological data. When $F\rightarrow \overline{F}:=F/A$ is a quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type $\Bbbk^G_ω\#_c\, \Bbbk F\rightarrow \Bbbk^G_ω\#_{\overline{c}} \, \Bbbk \overline{F}$. Our construction is particularly natural when $F=G$ acts on $G$ by conjugation, and $\Bbbk^G_ω\#_c \Bbbk G$ is a twisted quantum double $D^ω(G)$. In this case, we give necessary and sufficient conditions that Rep($\Bbbk^G_ω\#_{\overline{c}} \, \Bbbk \overline{G}$) is a modular tensor category.

math.QA↗
arXiv2013

Leibniz Algebras and Lie Algebras

This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.

math.RA↗
arXiv2013

On the structure of $\mathbb{N}$-graded Vertex Operator Algebras

We consider the algebraic structure of $\mathbb{N}$-graded vertex operator algebras with conformal grading $V=\oplus_{n\geq 0} V_n$ and $\dim V_0\geq 1$. We prove several results along the lines that the vertex operators $Y(a, z)$ for $a$ in a Levi factor of the Leibniz algebra $V_1$ generate an affine Kac-Moody subVOA. If $V$ arises as a shift of a self-dual VOA of CFT-type, we show that $V_0$ has a `de Rham structure' with many of the properties of the de Rham cohomology of a complex connected manifold equipped with Poincaré duality.

math.QA↗
arXiv2013

C-Graded Vertex Algebras and Conformal Flow

We consider C-graded vertex algebras, which are vertex algebras V with a C-grading such that V is an admissible V-module generated by 'lowest weight vectors'. We show that such vertex algebras have a 'good' representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras, which are C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (eg a lattice theory, or WZW model) is a path in a space whose points are PVOAs.

math.QA↗
arXiv2013

Fourier coefficients of vector-valued modular forms of dimension 2

We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

math.NT↗
arXiv2012

$FSZ$-groups and Frobenius-Schur Indicators of Quantum Doubles

We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, PSL_2(q), M_{11}, M_{12} and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5^6 with non-integer indicators.

math.RA↗
arXiv2011

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces II

We continue our program to define and study $n$-point correlation functions for a vertex operator algebra $V$ on a higher genus compact Riemann surface obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas for the genus two partition function for free bosonic theories and lattice vertex operator algebras $V_L$. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. We also compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity. We compare our results with those obtained in the companion paper, when a pair of tori are sewn together, and show that the partition functions are not compatible in the neighborhood of a two-tori degeneration point. The \emph{normalized} partition functions of a lattice theory $V_L$ \emph{are} compatible, each being identified with the genus two theta function of $L$.

math.QA↗
arXiv2011

Lattice subalgebras of strongly regular vertex operator algebras

We prove a sharpened version of a conjecture of Dong-Mason about lattice subalgebras of a strongly regular vertex operator algebra $V$, and give some applications. These include the existence of a canonical conformal subVOA $W\otimes G\otimes Z \subseteq V$, and a generalization of the theory of minimal models.

math.QA↗
arXiv2011

Vertex operator algebras and weak Jacobi forms

Let $V$ be a strongly regular vertex operator algebra. For a state $h \in V_1$ satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr$_Mq^{L(0)-c/24}ζ^{h(0)} ($M$ a $V$-module) is a vector-valued weak Jacobi form of weight 0 and a certain index $ /2$. We discuss refinements and applications of this result when $V$ is holomorphic, in particular we prove that if $g = e^{h(0)}$ is a finite order automorphism then Tr$_V q^{L(0)-c/24}g$ is a modular function of weight 0 on a congruence subgroup of $SL_2(Z)$.

math.QA↗