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Geoffrey Mason

Publications and source records attributed to Geoffrey Mason.

At least 55 records · Page 3Linked to original sources
arXiv2010

On the Fourier coefficients of 2-dimensional vector-valued modular forms

Let $ρ: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $ρ(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(τ)$ of integral weight associated to $ρ$ which have \emph{rational} Fourier coefficients. (These span the complex space of all integral weight vector-valued modular forms associated to $ρ$.) As a special case of the main Theorem, we prove that if $N$ does \emph{not} divide 120 then every nonzero $F(τ)$ has Fourier coefficients with \emph{unbounded denominators}.

math.NT↗
arXiv2010

On the canonical decomposition of generalized modular functions

The authors have conjectured (\cite{KoM}) that if a normalized generalized modular function (GMF) $f$, defined on a congruence subgroup $Γ$, has integral Fourier coefficients, then $f$ is classical in the sense that some power $f^m$ is a modular function on $Γ$. A strengthened form of this conjecture was proved (loc cit) in case the divisor of $f$ is \emph{empty}. In the present paper we study the canonical decomposition of a normalized parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1, f_0$ such that $f_1$ has \emph{unitary character} and $f_0$ has \emph{empty divisor}. We show that the strengthened form of the conjecture holds if the first "few" Fourier coefficients of $f_1$ are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either $f_0=1$ or if the divisor of $f$ is concentrated at the cusps of $Γ$.

math.NT↗
arXiv2009

Generalized Twisted Quantum Doubles and the McKay Correspondence

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if $G$ is a subgroup of $SU_2(\mb{C})$ then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of $G$ stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when $\bar{G} = 1$.

math.RA↗
arXiv2009

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.

math.QA↗
arXiv2009

Logarithmic vector-valued modular forms

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $ρ: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $ρ(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to $ρ$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(ρ)=\oplus_k \mathcal{H}(k, ρ)$ of all holomorphic logarithmic vector-valued modular forms associated to $ρ$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

math.NT↗
arXiv2009

Vertex Operators and Modular Forms

The leitmotif of these Notes is the idea of a vertex operator algebra (VOA) and the relationship between VOAs and elliptic functions and modular forms. This is to some extent analogous to the relationship between a finite group and its irreducible characters; the algebraic structure determines a set of numerical invariants, and arithmetic properties of the invariants provides feedback in the form of restrictions on the algebraic structure. One of the main points of these Notes is to explain how this works, and to give some reasonably interesting examples.

math.QA↗
arXiv2009

Structure of the module of vector-valued modular forms

Let $V$ be a representation of the modular group $Γ$ of dimension $p$. We show that the $\mathbb{Z}$-graded space $\mathcal{H}(V)$ of holomorphic vector-valued modular forms associated to $V$ is a free module of rank $p$ over the algebra $\mathcal{M}$ of classical holomorphic modular forms. We study the nature of $\mathcal{H}$ considered as a functor from $Γ$-modules to graded $\mathcal{M}$-lattices and give some applications, including the calculation of the Hilbert-Poincaré of $\mathcal{H}(V)$ in some cases.

math.NT↗
arXiv2008

Remarks on the McKay Conjecture

The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to $p$ ($p$ a prime) of a finite group $G$ and those of the subgroup $N$, the normalizer of Sylow $p$-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is \emph{equivalent} to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is \emph{equivalent} to parallel statements about holomorphic orbifolds $V^G, V^N$. There is a uniform formulation of MC covering these different situations which involves quantum dimensions of objects in pairs of ribbon fusion categories.

math.RT↗
arXiv2007

The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras

We define the $n$-point function for a vertex operator algebra on a genus two Riemann surface in two separate sewing schemes where either two tori are sewn together or a handle is sewn to one torus. We explicitly obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and lattice vertex operator algebras in both sewing schemes. We prove that the partition functions are holomorphic in the sewing parameters on given suitable domains and describe their modular properties. Finally, we show that the partition functions cannot be equal in the neighborhood of a two-tori degeneration point where they can be explicitly compared.

math.QA↗
arXiv2007

Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.

math.QA↗
arXiv2006

On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups

We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group H by an abelian group, with 3-cocycle inflated from a 3-cocycle on H. We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle w is gauge equivalent to a twisted quantum double of an elementary abelian 2-group if, and only if, w^2 is trivial; second, we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8, and classify the braided tensor equivalence classes of these quasi-triangular quasi-bialgebras. It turns out that there are exactly 20 such equivalence classes.

math.QA↗
arXiv2006

On Genus Two Riemann Surfaces Formed from Sewn Tori

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane $\mathbb{H}_{2}$. Equivariance of these maps under certain subgroups of $Sp(4,\mathbb{Z)}$ is shown. The invertibility of both maps in a particular domain of $\mathbb{H}_{2}$ is also shown.

math.QA↗
arXiv2006

Integrability of C_2-cofinite vertex operator algebras

The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one subspace V_1 is isomorphic to the irreducible highest weight \hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable \hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.

math.QA↗
arXiv2004

Shifted Vertex Operator Algebras

We study the properties of shifted vertex operator algebras, which are vertex algebras derived from a given theory by shifting the conformal vector. In this way, we are able to exhibit large numbers of vertex operator algebras which are regular(rational and C_2-cofinite) and yet are pathological in one way or another.

math.QA↗
arXiv2004

Local and Semilocal Vertex Operator Algebras

We investigate a general structure theory for a vertex operator algebra. We discuss the center and blocks, the Jacobson radical and solvable radical and local vertex operator algebras. The main consequence of our structure theory is that if the vertex operator algebra satisfies some mild conditions then it is necessarily semilocal, i.e. a direct sum of local vertex operator algebras.

math.QA↗
arXiv2003

Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras

In this paper, we obtain a canonical central element $ν_H$ for each semi-simple quasi-Hopf algebra $H$ over any field $k$ and prove that $ν_H$ is invariant under gauge transformations. We show that if $k$ is algebraically closed of characteristic zero then for any irreducible representation of $H$ which affords the character $χ$, $χ(ν_H)$ takes only the values 0, 1 or -1, moreover if $H$ is a Hopf algebra or a twisted quantum double of a finite group then $χ(ν_H)$ is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-Hopf algebra over any field $k$. Using this result, we establish the relationship between the antipode $S$, the values of $χ(ν_H)$, and certain associated bilinear forms when the underlying field $k$ is algebraically closed of characteristic zero.

math.QA↗
arXiv2001

Computing the Frobenius-Schur indicator for abelian extensions of Hopf algebras

In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.

math.QA↗