## Monstrous moonshine and monstrous Lie superalgebras (1992)

Venue: | INVENT. MATH |

Citations: | 127 - 0 self |

### BibTeX

@ARTICLE{Borcherds92monstrousmoonshine,

author = {Richard E. Borcherds},

title = {Monstrous moonshine and monstrous Lie superalgebras },

journal = {INVENT. MATH},

year = {1992},

volume = {109}

}

### OpenURL

### Abstract

We prove Conway and Norton’s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meurman. 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 superalgebras 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 representation), 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 superalgebras 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

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Citation Context ...E) ⊕ Λ 2 (E) . . . is a virtual vector space which is the alternating sum of the exterior powers of E, and similarly H(E) is the alternating sum of the homology groups Hi(E) of the Lie algebra E (see =-=[10]-=-). This identity is true for any finite dimensional Lie algebra E because the Hi(E)’s are the homology groups of a complex whose terms are the Λ i (E)’s. The left hand side corresponds to a product ov... |

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Citation Context ... (1−e r ) pg(1−r 2 /2) ∏ r∈2L ′+ (1−e r ) pg(1−r 2 /4) = ∑ w∈W det(w)w(e ∏ ρ (1−e iρ ) 8 (1−e 2iρ ) 8 ) where L is the Lorentzian lattice which is the sum of the 16-dimensional Barnes-Wall lattice Λ2 =-=[12]-=- and the two dimensional even Lorentzian lattice II1,1, W is its reflection group which has Weyl vector ρ, L ′ is the dual of L and pg is defined by ∑ n pg(n)qn ∏ = n>0 (1 − qn ) −8 (1 − q2n ) −8 . Th... |

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Citation Context ...n of the monster. 4 Generalized Kac-Moody algebras We summarize the results about generalized Kac-Moody algebras that we use, which can be found in [4], [5], [20], and the third edition of Kac’s book =-=[23]-=-. We modify the original definition of generalized Kac-Moody algebras in [4] slightly so that these algebras are closed under taking universal central extensions, as in [5]. (This is not necessary for... |

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Citation Context ... algebras Hi(E) turns out to have dimension equal to the number of elements in the Weyl group of length i; for finite dimensional Lie algebras this was first observed by Bott, and was used by Kostant =-=[26]-=- to give a homological proof of the Weyl character formula. The sum over the homology groups can therefore be identified with a sum over the Weyl group. 5For Kac-Moody algebras the same is true and w... |

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Citation Context ...ical states). See section 6 for more details. The space of physical states is a subquotient of a vertex algebra constructed from the vertex algebra V ; vertex algebras are described in more detail in =-=[3,16,18]-=-, and the properties we use are summarized in section 3. This subquotient can be identified using the no-ghost theorem from string theory ([21] or section 5), and is as described above. We need to kno... |

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Citation Context ...d on naturally by the monster sporadic simple group. (The referee has asked me to explain why I have never published my (long and messy) proof of the assertion in [3] that the module V constructed in =-=[17]-=- has the structure of a vertex algebra. The reason is that my proof used many results from the announcement [17], and the only published proof of this announcement, given in the book [16], incorporate... |

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Citation Context ...e Weyl character formula. The sum over the homology groups can therefore be identified with a sum over the Weyl group. 5For Kac-Moody algebras the same is true and was proved by Garland and Lepowsky =-=[20]-=-. For generalized Kac-Moody algebras things are a bit more complicated. The sum over the homology groups can still be identified with a sum over the Weyl group, but the things we sum are more complica... |

37 |
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Citation Context ...24 copies of each positive multiple of ρ. This Lie algebra was first constructed in [3], and the properties stated above were proved in [8]; this construction depended heavily on the ideas in Frenkel =-=[15]-=-. The fake monster Lie algebra is acted on by the group 2 24 .2.Co1 in the same way that the monster Lie algebra is acted on by the monster group, and we construct a superalgebra for many elements of ... |

30 | The Monster Lie algebra
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(Show Context)
Citation Context ... Lie algebra in [8] and is now called the fake monster Lie algebra. (The Kac-Moody algebra whose Dynkin diagram is that of the reflection group of II25,1 has also been called the monster Lie algebra (=-=[7]-=-); it is a large subalgebra of the fake monster Lie algebra, and does not seem to be interesting, except as an approximation to the fake monster Lie algebra.) The root lattice of the fake monster Lie ... |

30 |
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Citation Context ...re are some calculations for the case when VL is the vertex algebra of L = E8 in Kac, Moody and Wakimoto [24]. The monster Lie algebra can also be constructed as a semi-infinite cohomology group; see =-=[19]-=-. 22The construction of the monster Lie algebra above looks bizarre at first sight, so we briefly explain some of the motivation behind it. The fake monster vertex algebra MΛ [8] is the Lie algebra P... |

23 |
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Citation Context ... for j is j(τ) = (1 + 240 ∑ n>0 σ3(n)q n ) 3 q ∏ n>0 (1 − qn ) 24 where σ3(n) = ∑ d|n d3 is the sum of the cubes of the divisors of n; see any book on modular forms or elliptic functions, for example =-=[30]-=-. Another way of thinking about j is that it is an isomorphism from the quotient space H/SL2(Z) to the complex plane, which can be thought of as the Riemann sphere minus the point at infinity. We can ... |

22 |
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Citation Context ... conditions characterize V as a graded representation of the monster. 4 Generalized Kac-Moody algebras We summarize the results about generalized Kac-Moody algebras that we use, which can be found in =-=[4]-=-, [5], [20], and the third edition of Kac’s book [23]. We modify the original definition of generalized Kac-Moody algebras in [4] slightly so that these algebras are closed under taking universal cent... |

12 |
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(Show Context)
Citation Context ...few coefficients of both functions are the same. Unfortunately this final step of the proof (in section 9) is a case by case check that the first few coefficients are the same. Norton has conjectured =-=[28]-=- that Hauptmoduls with integer coefficients are essentially the same as functions satisfying relations similar to the ones above, and a conceptual proof or explanation of this would be a big improveme... |

9 |
On a class of non-linear functional equations connected with modular functions
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(Show Context)
Citation Context ... determined by its 12 coefficients ai, i = 1, 2, 3, 4, 5, 7, 8, 9, 11, 17, 19, 23.) For the case of the elliptic modular function j(q), where g = 1, these recursion formulas were discovered by Mahler =-=[27]-=-. We do not make any use of the formulas (9.1), apart from the fact that the coefficients for n = 1, 2, 3 and 5 determine all the coefficients. (This 27is easy to check without calculating the formul... |

7 | The automorphism group of the 26 dimensional even Lorentzian lattice - Conway - 1983 |

6 | Borcherds, Central Extension of Generalized Kac-Moody algebras - E - 1991 |

4 |
Borcherds, Lattices like the Leech lattice
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Citation Context ...tion 11 we see that the numbers mult(r) are given by mult(r) = ∑ µ(s)Tr(g d |Er/ds)/ds (13.2) ds|((r,L),N) where (r, L) is the highest common factor of the numbers (r, a) for a ∈ L. By theorem 2.2 of =-=[6]-=- there is a reflection group W g acting on the lattice L, which may be taken as any of the following groups. 1 The subgroup of W of elements that commute with g. 2 The subgroup of W of elements that m... |

4 |
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Citation Context ...dic simple groups. A second method of constructing some of them is to replace the monster vertex algebra V by the vertex algebra of the Leech lattice VΛ. From this we get the fake monster Lie algebra =-=[8]-=- (where it is called the monster Lie algebra) and several variations of it. The Lie superalgebras we construct form two families as follows: (1) A Lie algebra or superalgebra of rank 2 for many conjug... |

4 |
Vertex operator algebras and the Monster, Academinc
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(Show Context)
Citation Context ...t half of this paper is the following. Theorem 1.1. Suppose that V = ⊕n∈ZVn is the infinite dimensional graded representation of the monster simple group constructed by Frenkel, Lepowsky, and Meurman =-=[16,17]-=-. Then for any element g of the monster the Thompson series Tg(q) = ∑ n∈Z Tr(g|Vn)q n is 1a Hauptmodul for a genus 0 subgroup of SL2(R), i.e., V satisfies the main conjecture in Conway and Norton’s p... |

3 |
Lectures on modular forms Annals of Mathematics Studies 48
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(Show Context)
Citation Context ...dular forms in this section are usually forms of level 2, i.e. modular forms for the group Γ(2) = { ( ) a b c d ∈ SL2(Z)| ( ) ( ) a b 1 0 c d ≡ 0 1 mod 2}. We recall a few facts about this group from =-=[22]-=-. The group Γ(2) has no elliptic elements, 3 cusps (represented by 0, 1, and ∞), genus 0, and has index 6 in SL2(Z). The dimension of the space of forms of even nonnegative weight 2k is equal to k+1 a... |

3 |
On E10, preprint
- Kac, Moody, et al.
(Show Context)
Citation Context ...or −4 have multiplicity 0 or 1 so there may be something interesting going on in this case. There are some calculations for the case when VL is the vertex algebra of L = E8 in Kac, Moody and Wakimoto =-=[24]-=-. The monster Lie algebra can also be constructed as a semi-infinite cohomology group; see [19]. 22The construction of the monster Lie algebra above looks bizarre at first sight, so we briefly explai... |

3 |
On replication formulas and Hecke operators, Nagoya University preprint
- Koike
(Show Context)
Citation Context ...ely replicable function in [28] is that the function should satisfy some identities equivalent to 8.3. Norton’s conjectures that these modular functions are completely replicable were proved by Koike =-=[25]-=-. In the next section we use this to prove that the functions ∑ m Tr(g|Vm)qm are these modular functions. 269 The moonshine conjectures. In this section we complete the proof of theorem 1.1, i.e., we... |

2 | Borcherds. Vertex algebras - E |

1 |
Completely replicable functions, preprint
- Alexander, Cummins, et al.
(Show Context)
Citation Context ...ssible to prove Norton’s conjecture by a very long and tedious case by case check, because all functions which are either Hauptmoduls or which satisfy the relations above can be listed explicitly. In =-=[1]-=- the authors use a computer to find all “completely replicable” functions with integer coefficients, and they all appear to be Hauptmoduls. Roughly half of them correspond to conjugacy classes of the ... |

1 |
On axiomatic formulations of vertex operator algebras and modules
- Frenkel, Huang, et al.
(Show Context)
Citation Context ...ical states). See section 6 for more details. The space of physical states is a subquotient of a vertex algebra constructed from the vertex algebra V ; vertex algebras are described in more detail in =-=[3,16,18]-=-, and the properties we use are summarized in section 3. This subquotient can be identified using the no-ghost theorem from string theory ([21] or section 5), and is as described above. We need to kno... |

1 |
A finiteness theorem for subgroups of P SL(2, R) which are commensurable with
- Thompson
- 1979
(Show Context)
Citation Context ... correspond to elements of the monster. (However the genus of Γ0(N) tends to infinity as N increases, so there are only a finite number of integers N for which it has genus 0; more generally Thompson =-=[31]-=- has shown that there are only a finite number of conjugacy classes of genus 0 subgroups of SL2(R) which are commensurable with SL2(Z).) So we want to calculate the Thompson series Tg(τ) and show that... |