Jae-Hyun Yang
In this article, we discuss the relation between Kac-Moody algebras, the monstrous moonshine, Jacobi forms and infinite products. We also review Borcherds' solution of the Moonshine Conjecture and his work of constructing automorphic forms on the orthogonal group which can be written as infinite products.
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
Monstrous Moonshine was extended in two complementary directions during the 1980s and 1990s, giving rise to Norton's Generalized Moonshine conjecture and Ryba's Modular Moonshine conjecture. Both conjectures have been unconditionally resolved in the last few years, so we describe some speculative conjectures that may extend and unify them.
This paper shows how to use Computational Algebra techniques, namely the decomposition of rational functions in one variable, to explore a certain set of modular functions, called replicable functions, that arise in Monstrous Moonshine. In particular, we have computed all the rational relations with coefficients in Z between pairs of replicable functions. ----- En este articulo mostramos como usar tecnicas de Algebra Computacional, concretamente la descomposcion de funciones racionales univariadas, para estudiar un cierto conjunto de funciones modulares, llamadas funciones replicables, que apa…
For each element of the Fischer-Griess Monster sporadic simple group, we construct an infinite dimensional Lie algebra equipped with a projective action of the centralizer of that element. Our construction is given by a string-theoretic "add a spacetime torus and quantize" functor applied to an abelian intertwining algebra that is formed from a family of twisted modules of the Monster vertex operator algebra. We prove that for all Fricke elements in the Monster, the characters of centralizers acting on the corresponding irreducible twisted modules are Hauptmoduln. From these results, we resolv…
This article is a short and elementary introduction to the monstrous moonshine aiming to be as accessible as possible. I first review the classification of finite simple groups out of which the monster naturally arises, and features of the latter that are needed in order to state the moonshine conjecture of Conway and Norton. Then I motivate modular functions and modular forms from the classification of complex tori, with the definitions of the J-invariant and its q-expansion as a goal. I eventually provide evidence for the monstrous moonshine correspondence, state the conjecture, and then pre…
We provide a physics derivation of Monstrous moonshine. We show that the McKay-Thompson series $T_g$, $g\in \mathbb{M}$, can be interpreted as supersymmetric indices counting spacetime BPS-states in certain heterotic string models. The invariance groups of these series arise naturally as spacetime T-duality groups and their genus zero property descends from the behaviour of these heterotic models in suitable decompactification limits. We also show that the space of BPS-states forms a module for the Monstrous Lie algebras $\mathfrak{m}_g$, constructed by Borcherds and Carnahan. We argue that $\…
Topological modularity of Monstrous Moonshine
本研究探究魔群月光、轨形、Kitaev链与拓扑模形式的关联，通过分析Monster CFT的各类对称轨形的可分性性质，拓展了魔群月光与拓扑模形式领域的研究边界。
Monstrous Moonshine for integral group rings
本文将怪兽月光与模月光中的亏格为零断言推广到整群环情形，利用 Borcherds 整形式无鬼定理将猜想归结为拟可复制函数的亏格为零问题，并证明了 4A 类循环子群的情形。
Generalized modular equations and the CM values of Hauptmoduln
本文研究了Hauptmoduln的CM值性质，证明其在q系数为循环整数时为代数整数，并探讨了完全复制性与模不变性的关系。
Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly …
We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in Generalized Monstrous Moonshine, using a Borcherds-Kac-Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group.
In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera to obtain number theoretic generalizations of important results in Monstrous moonshine. Furthermore, we show the utility of the extended notions in yielding uniform proofs for numerous arithmetic properties of Fourier coe…
Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in weight zero. This construction serves as a foundation for a companion paper in which we explore the connection between penumbral Thompson moonshine and a special case of generalized monstrous moons…
The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the $j$-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the $j$-function satisfy congruences modulo $p^n$ for $p \in \{2, 3, 5, 7, 11\}$, which led to the theory of $p$-adic modular forms. We combine these two aspects of the $j$-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in…
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…
Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Math…
The commencement of monstrous moonshine is a connection between the largest sporadic simple group---the monster---and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.
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 …
In this note, we expand on some technical issues raised in \cite{PPV} by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0+1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras $\mathfrak{m}_g$---som…
Two New Avatars of Moonshine for the Thompson Group
本文证明了Thompson群月光现象的两种新化身的存在性，通过Borcherds乘积和奇异模迹揭示新旧模的关系，并扩展了广义魔群月光与半影月光的对应。
In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of $M_{12}$ to Jacobi forms of weight one and index zero. We construct Jacobi forms for all conjugacy classes of $M_{12}$ that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy…
We show that there is a dual description of conformal blocks of $d=2$ rational CFT in terms of Hecke eigenfields and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related $d=2$ spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier latti…
We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients …
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。