[Paper Review] Moonshine for Rudvalis's sporadic group I

John F. R. Duncan|ArXiv.org|Sep 15, 2006

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces enhanced vertex operator superalgebras (VOSAs) with refined conformal structures, constructing a self-dual VOSA of rank 28 whose full symmetry group is the direct product of a cyclic group of order 7 and the sporadic Rudvalis group. It establishes two-variable McKay–Thompson series for the group's action, proving their modular invariance and providing explicit expressions, thus realizing a non-Monstrous Moonshine phenomenon for a sporadic group not contained in the Monster group.

ABSTRACT

We introduce the notion of vertex operator superalgebra with enhanced conformal structure, which is a refinement of the notion of vertex operator superalgebra. We exhibit several examples, including a particular one which is self-dual, and whose full symmetry group is a direct product of a cyclic group of order seven with the sporadic simple group of Rudvalis. We thus obtain an analogue of Monstrous Moonshine for a sporadic group not involved in the Monster. Two variable analogues of the usual McKay--Thompson series are naturally associated to the action of the Rudvalis group on this object, and we provide explicit expressions for all the series arising.

Motivation & Objective

• To develop a refined framework for vertex operator superalgebras with enhanced conformal structures, extending the theory of VOAs beyond the Monster group.
• To construct a self-dual VOSA of rank 28 whose full automorphism group is the direct product of C7 and the Rudvalis sporadic group.
• To define and compute two-variable McKay–Thompson series for the Rudvalis group, analogous to Monstrous Moonshine.
• To establish modular invariance of these series under the action of SL2(Z), generalizing Zhu’s modular theory to Jacobi forms.
• To provide evidence for a characterization of the VOSA as the unique self-dual enhanced VOSA of rank 28 with no small elements, analogous to V♮ and VB♮.

Proposed method

• Introduces the notion of vertex operator superalgebra with enhanced conformal structure, refining standard VOAs by incorporating additional superconformal symmetries.
• Constructs a specific VOSA, denoted A_Ru, using untwisted and twisted constructions from linear groups and Clifford algebra modules.
• Utilizes the monomial description of the Rudvalis group via the Cayley algebra to define group actions on the VOSA.
• Defines two-variable characters as Jacobi forms in the class E_m, with m = 28, and proves their transformation properties under SL2(Z) via Poisson summation and theta function identities.
• Applies Zhu’s modular theory to VOAs and extends it to U(1)-VOAs, using Jacobi forms in place of modular forms.
• Derives explicit expressions for the two-variable McKay–Thompson series by analyzing the graded character of A_Ru and its decomposition into irreducible representations of the Rudvalis group.

Experimental results

Research questions

• RQ1 Can a sporadic group not contained in the Monster group be realized as the full symmetry group of a self-dual vertex operator superalgebra with enhanced conformal structure?
• RQ2 What are the explicit expressions for the two-variable McKay–Thompson series associated with the action of the Rudvalis group on such a VOSA?
• RQ3 How do these series transform under the action of SL2(Z), and do they satisfy modular invariance properties analogous to those in Monstrous Moonshine?
• RQ4 Can the VOSA A_Ru be characterized as the unique self-dual enhanced VOSA of rank 28 with no non-trivial small elements?
• RQ5 To what extent can Zhu’s modular theory for VOAs be generalized to U(1)-VOAs using Jacobi forms?

Key findings

• The two-variable character of the VOSA A_Ru lies in the class E_28 and spans a one-dimensional representation of the subgroup of SL2(Z) generated by S and T².
• The character of A_Ru is a Jacobi form of weight 0 and index 28, with coefficients matching irreducible representations of the Rudvalis group, such as 784 = 1 + 783 and 92512 = (2)378 + 406 + 91350.
• The series exhibit modular invariance under the action of SL2(Z), with transformation laws derived from the Poisson summation formula applied to the theta function.
• The character of A_Ru has vanishing odd-charge subspaces and symmetric coefficients for p^m and p^{-m}, with the first non-zero terms appearing at degree 0 and 1/2.
• The decomposition of homogeneous subspaces into irreducible representations of the Rudvalis group mirrors the structure seen in Monstrous Moonshine, such as 196884 = 1 + 196883.
• The VOSA A_Ru is self-dual and has no non-trivial small elements (of degree 1/2 or 1), supporting the conjecture that it is the unique such object of rank 28 with this symmetry group.