arXiv 2203.030522022-03-06math.RThep-thmath.NT

Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

Miranda C. N. Cheng, Gabriele Sgroi

英文摘要：

We describe a family of indefinite theta functions of signature $(1,1)$ that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell-Lerch sums. We use these relations to write the McKay-Thompson series of umbral moonshine at lambency $\ell=8,12,16$ in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Generalised Umbral Moonshine

Umbral moonshine describes an unexpected relation between 23 finite groups arising from lattice symmetries and special mock modular forms. It includes the Mathieu moonshine as a special case and can itself be viewed as an example of the more general moonshine phenomenon which connects finite groups and distinguished modular objects. In this paper we introduce the notion of generalised umbral moonshine, which includes the generalised Mathieu moonshine [Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number Theory Phys. 7 (2013), 145-223] as a special case, and provide support…

2019-03-02hep-thmath.NTmath.RT

The Umbral Moonshine Module for the Unique Unimodular Niemeier Root System

We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.

2014-12-28hep-thmath.NTmath.RT

Classical and Umbral Moonshine: Connections and $p$-adic Properties

The classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group $M$, the largest of the simple sporadic groups, and certain modular functions, called Hauptmodln. In particular, the $n$-th Fourier coefficient of Klein's $j$-function is the dimension of the grade $n$ part of a special infinite dimensional representation $V$ of the monster group. More generally the coefficients of Hauptmoduln are graded traces $T_g$ of $g \in M$ acting on $V$. Similar phenomena have been shown to hold for the Matthieu group $M_{24}$, but inste…

2015-08-18hep-thmath.NTmath.RT

Proof of the Umbral Moonshine Conjecture

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases.

2015-11-26math.NTmath.RT

类数、同余数与伞影月光

Class Numbers, Congruent Numbers and Umbral Moonshine

本文继续研究有限群与模曲线算术几何不变量的关系，通过伞影月光与虚二次类数的联系，分类最小散在单马蒂厄群的特殊模，并由此导出古代同余数问题的新结论。

2023-06-28hep-thmath.NTmath.RT

Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine

The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We …

2019-04-04hep-thmath.RT

更多相关论文收起更多论文

K3 Elliptic Genus and an Umbral Moonshine Module

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of $K3$ string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the $K3$ elliptic genus. Inspired by the above two relations between moonshine and $K3$ string theory, we construct a chiral CFT by orbifolding the free theory of 24 chiral fermions and two pairs of fermionic and bosonic ghosts. In this paper we mainly focus on the c…

2017-09-06hep-thmath.RT

Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules

In this work we consider an association of meromorphic Jacobi forms of half-integral index to the pure D-type cases of umbral moonshine, and solve the module problem for four of these cases by constructing vertex operator superalgebras that realise the corresponding meromorphic Jacobi forms as graded traces. We also present a general discussion of meromorphic Jacobi forms with half-integral index and their relationship to mock modular forms.

2017-07-05hep-thmath.RT

Umbral Moonshine and the Niemeier Lattices

In this paper we relate umbral moonshine to the Niemeier lattices: the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattic…

2014-07-22hep-thmath.NTmath.RT

Umbral Moonshine and K3 Surfaces

Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. In this paper we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val s…

2017-07-18hep-thmath.AGmath.RT

ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine

We consider double scaled little string theory on $K3$. These theories are labelled by a positive integer $k \ge 2$ and an $ADE$ root lattice with Coxeter number $k$. We count BPS fundamental string states in the holographic dual of this theory using the superconformal field theory $K3 \times \left( \frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/ \mathbb{Z}_k$. We show that the BPS fundamental string states that are counted by the second helicity supertrace of this theory give rise to weight two mixed mock modular forms. We compute the helicity supertraces using two s…

2014-10-22hep-thmath.NTmath.RT

Umbral Moonshine

We describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterised naturally by the divisors of twelve. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterises the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight two. These extremal Jacobi forms are closely related to certain …

2013-10-17hep-thmath.RT

Landau-Ginzburg Orbifolds and Symmetries of K3 CFTs

Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3 elliptic genera twined by discrete symmetries that are manifest in the UV description, though often inaccessible in the IR. We obtain various twining functions coinciding with mo…

2017-07-18hep-thmath.AGmath.RT

An Overview of Penumbral Moonshine

As Mathieu moonshine is a special case of umbral moonshine, Thompson moonshine (in half-integral weight) is a special case of a family of similar relationships between finite groups and vector-valued modular forms of a certain kind. We call this penumbral moonshine. We introduce and explain some features of this phenomenon in this work.

2022-02-18hep-thmath.RT

Traces of Singular Moduli and Moonshine for the Thompson Group

We describe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in work of Borcherds and Zagier on Borcherds products and traces of singular moduli. We conjecture the existence of an infinite dimensional graded module for the Thompson group and provide evidence for our conjecture by constructing McKay--Thompson series for each conjugacy class of the Thompson group that coincide with weight one-half modular forms of higher level. We also observe a discriminant property in this moonshine for the Tho…

2015-04-30hep-thmath.NTmath.RT

Mock Modular Mathieu Moonshine Modules

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4 superconformal algebra. Similarly, any subgroup of Co_0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2 superconformal algebra. Through the decomposition of the corre…

2015-08-10hep-thmath.RT

Moonshine

Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of the modular function j-744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and …

2015-04-05hep-thmath.NTmath.RT

Module constructions for certain subgroups of the largest Mathieu group

For certain subgroups of $M_{24}$, we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. The construction is related to the Conway moonshine module and employs a technique introduced by Anagiannis--Cheng--Harrison. With this construction we are able to give concrete vertex algebraic realizations of certain cuspidal Hecke eigenforms of weight two. In particular, we give explicit realizations of trace functions whose integraliti…

2019-12-09math.NTmath.RT

Derived Equivalences of K3 Surfaces and Twined Elliptic Genera

We use the unique canonically-twisted module over a certain distinguished super vertex operator algebra---the moonshine module for Conway's group---to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective complex K3 surface that fixes a stability condition in the distinguished space identified by Bridgeland. According to work of Huybrechts, following Gaberdiel--Hohenegger--Volpato, any such derived equivalence determines a conjugacy class in Conway's group, the automorphism group of the Leech lattice. Conway's group acts naturally on the m…

2015-12-30hep-thmath.AGmath.NT

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did no…

2022-08-09hep-thmath.NTmath.RT