A Short Introduction to Monstrous Moonshine
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Arxiv:Math/0612474V3 [Math.NT] 23 May 2017 20D08
KAC-MOODY ALGEBRAS, THE MONSTROUS MOONSHINE, JACOBI FORMS AND INFINITE PRODUCTS Jae-Hyun Yang Table of Contents 1. Introduction Notations 2. Kac-Moody Lie Algebras Appendix : Generalized Kac-Moody Algebras 3. The Moonshine Conjectures and the Monster Lie Algebra Appendix : The No-Ghost Theorem 4. Jacobi Forms 5. Infinite Products and Modular Forms 6. Final Remarks 6.1. The Fake Monster Lie Algebras 6.2. Generalized Kac-Moody Algebras of the Arithmetic Type 6.3. Open Problems Appendix A : Classical Modular Forms Appendix B : Kohnen Plus Space and Maass Space arXiv:math/0612474v3 [math.NT] 23 May 2017 Appendix C : The Orthogonal Group Os+2,2(R) Appendix D : The Leech Lattice Λ References This work was in part supported by TGRC-KOSEF. Mathematics Subject Classification (1991) : Primary 11F30, 11F55, 17B65, 17B67, 20C34, 20D08. Typeset by AMS-TEX 1 2 JAE-HYUN YANG 1. Introduction Recently R. E. Borcherds obtained some quite interesting results in [Bo6-7]. First he solved the Moonshine Conjectures made by Conway and Norton([C-N]). Secondly he constructed automorphic forms on the orthogonal group Os+2,2(R) which are modular products and then wrote some of the well-known meromorphic modular forms as infinite products. Modular products roughly mean infinite prod- ucts whose exponents are the coefficients of certain nearly holomorphic modular forms. The theory of Jacobi forms plays an important role in his second work in [Bo7]. More than 10 years ago Feingold and Frenkel([F-F]) realized the connec- tion between the theory of a special hyperbolic Kac-Moody Lie algebra of the type (1) HA1 and that of Jacobi forms of degree one and then generalized the results of H. -
From String Theory and Moonshine to Vertex Algebras
Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential). -
Arxiv:1911.10534V3 [Math.AT] 17 Apr 2020 Statement
THE ANDO-HOPKINS-REZK ORIENTATION IS SURJECTIVE SANATH DEVALAPURKAR Abstract. We show that the map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an E1-ring B and a map B ! MString such that the composite B ! MString ! tmf is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed. 1. Introduction The goal of this paper is to show the following result. Theorem 1.1. The map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This integral result was originally stated as [Hop02, Theorem 6.25], but, to the best of our knowledge, no proof has appeared in the literature. In [HM02], Hopkins and Mahowald give a proof sketch of Theorem 1.1 for elements of π∗tmf of Adams-Novikov filtration 0. The analogue of Theorem 1.1 for bo (namely, the statement that the map π∗MSpin ! π∗bo induced by the Atiyah-Bott-Shapiro orientation is surjective) is classical [Mil63]. In Section2, we present (as a warmup) a proof of this surjectivity result for bo via a technique which generalizes to prove Theorem 1.1. We construct an E1-ring A with an E1-map A ! MSpin. The E1-ring A is a particular E1-Thom spectrum whose mod 2 homology is given by the polynomial subalgebra 4 F2[ζ1 ] of the mod 2 dual Steenrod algebra. The Atiyah-Bott-Shapiro orientation MSpin ! bo is an E1-map, and so the composite A ! MSpin ! bo is an E1-map. -
Our Mathematical Universe: I. How the Monster Group Dictates All of Physics
October, 2011 PROGRESS IN PHYSICS Volume 4 Our Mathematical Universe: I. How the Monster Group Dictates All of Physics Franklin Potter Sciencegems.com, 8642 Marvale Drive, Huntington Beach, CA 92646. E-mail: [email protected] A 4th family b’ quark would confirm that our physical Universe is mathematical and is discrete at the Planck scale. I explain how the Fischer-Greiss Monster Group dic- tates the Standard Model of leptons and quarks in discrete 4-D internal symmetry space and, combined with discrete 4-D spacetime, uniquely produces the finite group Weyl E8 x Weyl E8 = “Weyl” SO(9,1). The Monster’s j-invariant function determines mass ratios of the particles in finite binary rotational subgroups of the Standard Model gauge group, dictates Mobius¨ transformations that lead to the conservation laws, and connects interactions to triality, the Leech lattice, and Golay-24 information coding. 1 Introduction 5. Both 4-D spacetime and 4-D internal symmetry space are discrete at the Planck scale, and both spaces can The ultimate idea that our physical Universe is mathematical be telescoped upwards mathematically by icosians to at the fundamental scale has been conjectured for many cen- 8-D spaces that uniquely combine into 10-D discrete turies. In the past, our marginal understanding of the origin spacetime with discrete Weyl E x Weyl E symmetry of the physical rules of the Universe has been peppered with 8 8 (not the E x E Lie group of superstrings/M-theory). huge gaps, but today our increased understanding of funda- 8 8 mental particles promises to eliminate most of those gaps to 6. -
§2. Elliptic Curves: J-Invariant (Jan 31, Feb 4,7,9,11,14) After
24 JENIA TEVELEV §2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let M = isom. classes of elliptic curves . 1 { } We are going to assign to each elliptic curve a number, called its j-invariant and prove that 1 M1 = Aj . 1 1 So as a space M1 A is not very interesting. However, understanding A ! as a moduli space of elliptic curves leads to some breath-taking mathemat- ics. More generally, we introduce M = isom. classes of smooth projective curves of genus g g { } and M = isom. classes of curves C of genus g with points p , . , p C . g,n { 1 n ∈ } We will return to these moduli spaces later in the course. But first let us recall some basic facts about algebraic curves = compact Riemann surfaces. We refer to [G] and [Mi] for a rigorous and detailed exposition. §2.1. Algebraic functions, algebraic curves, and Riemann surfaces. The theory of algebraic curves has roots in analysis of Abelian integrals. An easiest example is the elliptic integral: in 1655 Wallis began to study the arc length of an ellipse (X/a)2 + (Y/b)2 = 1. The equation for the ellipse can be solved for Y : Y = (b/a) (a2 X2), − and this can easily be differentiated !to find bX Y ! = − . a√a2 X2 − 2 This is squared and put into the integral 1 + (Y !) dX for the arc length. Now the substitution x = X/a results in " ! 1 e2x2 s = a − dx, 1 x2 # $ − between the limits 0 and X/a, where e = 1 (b/a)2 is the eccentricity. -
K3 Surfaces, N= 4 Dyons, and the Mathieu Group
K3 Surfaces, =4 Dyons, N and the Mathieu Group M24 Miranda C. N. Cheng Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural inter- pretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new “moonshine” for the largest Mathieu group is con- nected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3 T 2-compactified type II string theory. This insight on the symmetry× of the theory sheds new light on the gener- alised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spec- arXiv:1005.5415v2 [hep-th] 3 Jun 2010 trum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d = 4, = 4 string theory, twisted by elements of the group of stringy K3N isometries. 1 1 Introduction and Summary Recently there have been two new observations relating K3 surfaces and the largest Mathieu group M24. They seem to suggest that the sporadic group M24 naturally acts on the spectrum of K3-compactified string theory. -
Around Symmetries of Vertex Operator Algebras
Around Symmetries of Vertex Operator Algebras Atsushi Matsuo The University of Tokyo Corrected and revised version of the slides presented for Graphs and Groups, Representations and Relations August 9, 2018 Novosibirsk State University A. Matsuo Around Symmetries of Vertex Operator Algebras 1 x1 Vertex algebras and vertex operator algebras | R.E. Borcherds: Vertex algebras, Kac-Moody algebras and the mon- ster, Proc. Nat. Acad. Sci. U.S.A. 83, (1986), 3068{3071. | I. Frenkel, J. Lepowsky, and A. Meurman: Vertex operator algebras and the Monster, Academic Press, New York, 1988. A. Matsuo Around Symmetries of Vertex Operator Algebras 2 Vertex algebras. A vertex algebra is a vector space V , say over C, equipped with X / −n−1 • A bilinear map V × V V ((z)), (a; b) 7! a(n)bz . n2Z • A nonzero vector 1 2 V , called the vacuum vector. satisfying the following axioms: 2 (1) Borcherds( ) identities. The following holds for all p; q; r Z, X1 p (a b) − c i (r+i) (p+q i) i=0 X1 ( )( ) i r r = (−1) a − (b c) − (−1) b − (a c) : i (p+r i) (q+i) (q+r i) (p+i) i=0 (2) Creation property. { 0; (n ≥ 0), a(n)1 = a; (n = −1). A. Matsuo Around Symmetries of Vertex Operator Algebras 3 Vertex operators. A vertex algebra is specified by the triple (V; Y; 1), where X / −1 −n−1 Y : V (End V )[[z; z ]]; a 7! Y (a; z) = a(n)z : n The generating function Y (a; z) is often called the vertex operator, al- though this term is also used in broader context. -
No. 2, 435-459. Richard E. Borcherds,∗ Mathematics Department, University of California at Berkeley, CA 94720-3840 U
Modular Moonshine II. 24 July 1994, corrected 19 Sept 1995 Duke Math. J. 83 (1996) no. 2, 435-459. Richard E. Borcherds,∗ Mathematics department, University of California at Berkeley, CA 94720-3840 U. S. A. e-mail: [email protected] Alex J. E. Ryba, Dept of Mathematics, Marquette University, Milwaukee, WI 53233, U. S. A. e-mail: [email protected] The monster simple group acts on the monster vertex algebra, and the moonshine conjectures state that the traces of elements of the monster on the vertex algebra are Hauptmoduls. Ryba [R94] conjectured the existence of similar vertex algebras over fields of characteristics p acted on by the centralizers of certain elements of prime order p in the monster, and conjectured that the Brauer traces of p-regular elements of the centralizers were certain Hauptmoduls. We will prove these conjectures when the centralizer involves a sporadic group (p 11, corresponding to the sporadic groups B, F i0 , T h, HN, He, and M12). ≤ 24 Contents. 1. Introduction. Notation. 2. Cohomology and modular representations. 3. Vertex superalgebras mod p. 4. A vanishing theorem for cohomology. 5. The case p = 2. 6. Example: the Held group. 7. Open problems and conjectures. 1. Introduction. The original “moonshine conjectures” of Conway, Norton, McKay, and Thompson said that the monster simple group M has an infinite dimensional graded representation V = n ZVn such that the dimension of Vn n 1 ⊕ ∈ is the coefficient of q of the elliptic modular function j(τ) 744 = q− +196884q+ , and more generally the n − ··· McKay-Thompson series Tg(τ) = n Z Tr(g Vn)q is a Hauptmodul for some genus 0 congruence subgroup P ∈ | of SL2(R). -
Moonshines for L2(11) and M12
Moonshines for L2(11) and M12 Suresh Govindarajan Department of Physics Indian Institute of Technology Madras (Work done with my student Sutapa Samanta) Talk at the Workshop on Modular Forms and Black Holes @NISER, Bhubaneswar on January 13. 2017 Plan Introduction Some finite group theory Moonshine BKM Lie superalgebras Introduction Classification of Finite Simple Groups Every finite simple group is isomorphic to one of the following groups: (Source: Wikipedia) I A cyclic group with prime order; I An alternating group of degree at least 5; I A simple group of Lie type, including both I the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; I the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type). I The 26 sporadic simple groups. The classification was completed in 2004 when Aschbacher and Smith filled the last gap (`the quasi-thin case') in the proof. Fun Reading: Symmetry and the Monster by Mark Ronan The sporadic simple groups I the Mathieu groups: M11, M12, M22, M23, M24; (found in 1861) I the Janko groups: J1, J2, J3, J4; (others 1965-1980) I the Conway groups; Co1, Co2, Co3; 0 I the Fischer groups; Fi22. Fi23, Fi24; I the Higman-Sims group; HS I the McLaughlin group: McL I the Held group: He; I the Rudvalis group Ru; I the Suzuki sporadic group: Suz; 0 I the O'Nan group: O N; I Harada-Norton group: HN; I the Lyons group: Ly; I the Thompson group: Th; I the baby Monster group: B and Sources: Wikipedia and Mark Ronan I the Fischer-Griess Monster group: M Monstrous Moonshine Conjectures I The j-function has the followed q-series: (q = exp(2πiτ)) j(τ)−744 = q−1+[196883+1] q+[21296876+196883+1] q2+··· I McKay observed that 196883 and 21296876 are the dimensions of the two smallest irreps of the Monster group. -
Arxiv:1307.5522V5 [Math.AG] 25 Oct 2013 Rnho Mathematics
JORDAN GROUPS AND AUTOMORPHISM GROUPS OF ALGEBRAIC VARIETIES ∗ VLADIMIR L. POPOV Steklov Mathematical Institute, Russian Academy of Sciences Gubkina 8, Moscow 119991, Russia and National Research University Higher School of Economics 20, Myasnitskaya Ulitsa, Moscow 101000, Russia [email protected] Abstract. The first section of this paper is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The ap- pendix contains formulations of some open problems and the relevant comments. MSC 2010: 20E07, 14E07 Key words: Jordan, Cremona, automorphism, birational map This is the expanded version of my talk, based on [Po10, Sect. 2], at the workshop Groups of Automorphisms in Birational and Affine Geometry, October 29–November 3, 2012, Levico Terme, Italy. The appendix is the expanded version of my notes on open problems posted on the site of this workshop [Po122]. Below k is an algebraically closed field of characteristic zero. Variety means algebraic variety over k in the sense of Serre (so algebraic group means algebraic group over k). We use without explanation standard nota- arXiv:1307.5522v5 [math.AG] 25 Oct 2013 tion and conventions of [Bo91] and [Sp98]. In particular, k(X) denotes the field of rational functions of an irreducible variety X. Bir(X) denotes the group of birational self-maps of an irreducible variety X. Recall that if X is the affine n-dimensional space An, then Bir(X) is called the Cremona group over k of rank n; we denote it by Crn (cf. -
Vertex Operator Algebras, the Latter Being a Slightly More Restricted Notion Than the Former (See the Main Text)
Introduction to Vertex Algebras Christophe NOZARADAN† Mathematics department, Universit´eCatholique de Louvain, Louvain-la-Neuve, Belgium e-mail: [email protected] Abstract These lecture notes are intended to give a modest impulse to anyone willing to start or pursue a journey into the theory of vertex algebras by reading one of the books [K1] or [LL]. Therefore, the primary goal is to provide required tools and help being acquainted with the machinery which the whole theory is based on. The exposition follows Kac’s approach [K1]. Fundamental examples relevant in Theoretical Physics are also discussed. No particular prerequisites are assumed. arXiv:0809.1380v3 [math.QA] 11 Nov 2008 Last edited on November 11, 2008. † Research Fellow F.R.S. - FNRS (Aspirant) ii Prologue The notion of vertex algebra1 was first axiomatized by Richard Borcherds in 1986. In retrospect, this algebraic structure has been proved to be equivalent to Conformal Theory of chiral fields in two dimensions [BPZ], for which it provides a rigorous mathematical formulation. Thereby, a new area was born, stressing beautifully the interplay between physics and mathematics. Originally, vertex operators arose in String Theory. They are used to describe certain types of interactions, between different particles or strings, localized at vertices – hence the name“vertex” – of the corresponding Feynman diagrams. From this point of view, the notion of vertex operator thus appeared before the underlying concept of vertex algebra. The concept of vertex algebra happens to be a powerful tool – and initially introduced by Borcherds to this purpose – in the proof [Bo] of the so-called “Moonshine Monstrous” con- jectures, formulated by Conway and Norton. -
Monstrous Moonshine and Monstrous Lie Superalgebras
Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405-444 (1992). Richard E. Borcherds, Department of pure mathematics and mathematical statistics, 16 Mill Lane, Cam- bridge CB2 1SB, England. We prove Conway and Norton’s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meur- man. To do this we use the no-ghost theorem from string theory to construct a family of generalized Kac-Moody superalgebras of rank 2, which are closely related to the monster and several of the other sporadic simple groups. The denominator formulas of these su- peralgebras imply relations between the Thompson functions of elements of the monster (i.e. the traces of elements of the monster on Frenkel, Lepowsky, and Meurman’s repre- sentation), which are the replication formulas conjectured by Conway and Norton. These replication formulas are strong enough to verify that the Thompson functions have most of the “moonshine” properties conjectured by Conway and Norton, and in particular they are modular functions of genus 0. We also construct a second family of Kac-Moody super- algebras related to elements of Conway’s sporadic simple group Co1. These superalgebras have even rank between 2 and 26; for example two of the Lie algebras we get have ranks 26 and 18, and one of the superalgebras has rank 10. The denominator formulas of these algebras give some new infinite product identities, in the same way that the denominator formulas of the affine Kac-Moody algebras give the Macdonald identities. 1 Introduction. 2 Introduction (continued). 3 Vertex algebras.