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Vertex Operator Algebras, Modular Forms and Moonshine

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Vertex Operator Algebras, Modular Forms and Moonshine Vertex Operator Algebras, Modular Forms and Moonshine Geoffrey Mason University of California, Santa Cruz Contents Lecture 1. 1. The Monster simple group 2. J and V \ 3. Monstrous Moonshine 4. Vertex algebras 5. Locality and quantum fields 6. CFT axioms Lecture 2. 7. Lie algebras and local fields 8. Vertex operator algebras 9. Super vertex operator algebras Lecture 3. 10. Modules over a VOA 11. Lattice theories 12. Partition functions and modular-invariance Lecture 4. 13. The Lie algebra on V1 14. Automorphisms 15. Twisted sectors 16. The Moonshine module 17. AutV \ References. 1 Lecture 1 §1 The Monster simple group Group theorists conceived the Monster sporadic simple group M in the early 1970s, although it was not officially born until 1982. Many features of M were understood well before that time, however. In particular the complete character table was already known. Here is a small part of it. 1 2A 2B χ1 1 1 1 χ2 196883 4371 275 χ3 21296876 91884 − 2324 Let Vi be the M-module that affords the character χi. From the character table one can compute branching rules Vi ⊗ Vj = ⊕kcijkVk. In particular, ⊗2 the tensor square V2 decomposes into the sum of symmetric and exterior 2 2 squares S (V2) ⊕ Λ (V2), and the branching rules show that c222 = 1 with 2 V2 ⊆ S (V2). So there is a canonical M-invariant surjection V2 ⊗ V2 → V2, and it gives rise to a commutative, nonassociatve algebra structure on V1 whose automorphism group contains M. We can formally add an identity element 1 to obtain a unital, commutative algebra B = V1 ⊕ V2 (1) (V1 = C1) with M ⊆ Aut(B). §2 J and V \ Up to an undetermined constant, there is a unique modular function of weight 0 on the full modular group Γ := SL2(Z) with a simple pole of residue 1 at ∞. Such functions can be represented as quotients of holomorphic modular forms of equal weight. For example we have θ (τ)3 J + 744 = E8 = q−1 + 744 + 196884q + 21493760q2 + ... (2) ∆(τ) θ (τ) J + 24 = Λ = q−1 + 24 + 196884q + 21493760q2 + .... (3) ∆(τ) 2 Here, J = q−1 + 196884q + 21493760q2 + ... (4) is the modular function with constant 0, ∆(τ) is the discriminant ∞ Y ∆(τ) = q (1 − qn)24, (5) n=1 X (α,α)/2 θL(τ) = q (6) α∈L is the theta function of an even lattice L, and E8, Λ denote the E8 root lattice and Leech lattice respectively. John McKay noticed that the first few Fourier coefficients in (4) are sim- ple linear combinations of dimensions of the irreducible M-modules Vi with nonnegative coefficients. This suggests that we replace the coefficients by the putative M-modules that correspond to them - a process that sometimes goes by the abysmal name of ‘categorification’. From (3)-(6), the coefficients of J are all nonnegative, so at least they correspond to linear spaces. Shifting the grading by 1 for later convenience, we obtain a Z-graded linear space \ \ \ \ V := V0 ⊕ V2 ⊕ V3 ⊕ ... (7) with \ V0 = V1 \ V2 = V1 ⊕ V2 (8) \ V3 = V1 ⊕ V2 ⊕ V3 ... \ n−1 and with dim Vn = coefficient of q in (4). McKay’s observation was pro- \ moted to the conjecture that each Vn carries a ‘natural’ action of M. Note \ that V2 is identified with the algebra B. 3 §3 Monstrous Moonshine With the conjectured Z-graded M-module V \ in hand, for each g ∈ M we can take the graded trace of g and obtain another q-series ∞ −1 X n Zg = Zg(q) := q Tr \ (g)q . (9) Vn n=0 It was John Thompson who first asked what one can say about these additional q-expansions. (There are 174 of them, one for each conjugacy class of M.) We have Z1A(1, q) = J by construction, and from the character table and (8) we see that −1 2 Z2A(q) = q + 4372q + 96256q + ... −1 2 Z2B(q) = q + 276q − 2048q + ... In a celebrated paper, John Conway and Simon Norton resoundingly an- swered Thompson’s question. They gave overwhelming evidence for the con- jecture that each of the trace functions (9) was a hauptmodul for a subgroup of SL2(Q) commensurable with SL2(Z). This means that for each g we have a subgroup Γg ⊆ SL2(Q) with |Γg :Γg ∩ Γ|, |Γ:Γg ∩ Γ| < ∞ such that the following hold: (i) Each Zg is the q-expansion of a modular function of weight zero on Γg. (10) (ii) If H is the complex upper half-plane, the compact Riemann surface ∗ Γg \ H is a Riemann sphere whose function field is C(Zg). If g = 1A then of course Z1 = J and Γ1 = Γ. Conway-Norton proposed formulae for each Zg. For example, η(τ)24 Z (q) = + 24, (11) 2B η(2τ)24 where η(τ) is the Dedekind eta-function ∞ Y η(τ) = q1/24 (1 − qn). (12) n=1 Z2B(q) is a hauptmodul for the index 3 subgroup Γ0(2) ⊆ Γ. 4 §4 Vertex algebras The problem is now to define a natural action of the Monster M on V \ (59) so that the graded traces Zg satisfy the Conway-Norton moonshine conjec- tures (10). Borcherds’ radical proposed solution involved the idea of a vertex algebra, which may be defined as follows. It is a pair (V, 1) consisting of a nonzero C-linear space V and a distinguished vector 1 6= 0. Moreover, V is equipped with bilinear products µn : V ⊗ V → V (n ∈ Z), u ⊗ v 7→ u(n)v (u, v ∈ V ), satisfying the following axioms for all u, v, w ∈ V : 1) There is n0 = n0(u, v) ∈ Z such that u(n)v = 0 for n ≥ n0, (13) 2) v(n)1 = 0 (n ≥ 0) and v(−1)1 = v, (14) 3) For all p, q, r ∈ Z we have ∞ X p {u(r + i)v}(p + q − i)w = (15) i i=0 ∞ X r (−1)i {u(p + r − i)v(q + i)w − (−1)rv(q + r − i)u(p + i)w}. i i=0 Thanks to 1), both sums in 3) are finite so that the identity in question is sensible. At this point the reader may well be asking, where did these identities come from, what are they good for, and what do they have to do with Mon- strous Moonshine? The point of these lectures is to address these questions. We begin by specializing (15) in various ways. It is convenient to consider u(n) ∈ End(V ) to be the linear operator such that v 7→ u(n)v (v ∈ V ). r Taking r = 0, the binomial i vanishes unless i = 0 and (15) reduces to the operator identity ∞ X p [u(p), v(q)] = {u(i)v}(p + q − i), (16) i i=0 called the commutator formula. Similarly, taking p = 0 yields the associativ- ity formula ∞ X r {u(r)v}(q) = (−1)i {u(r − i)v(q + i) − (−1)rv(q + r − i)u(i)}. (17) i i=0 5 With n0(u, v) as in (13), we obtain ∞ X r (−1)i {u(p + r − i)v(q + i) − (−1)rv(q + r − i)u(p + i)} = 0 (18) i i=0 whenever r ≥ n0, which is sometimes referred to as commutivity. Assuming (13), it is not too hard to show that (15) is a consequence of the commutator and associativity formulas, and thus is equivalent to them. There are other equivalent ways to reformulate (15) that are useful. We explain one of them (cf. (24)) in the next Section. §5 Locality and Quantum fields In the succeeding two Sections we will explain how the idea of a vertex algebra corresponds to the physicist’s conformal field theory. The important idea of a vertex operator, or quantum field, or simply field, defined on an arbitrary linear space V is as follows. It is a formal series X −n−1 −1 a(z) := anz ∈ End(V )[[z, z ]] n∈Z of operators an on V such that if v ∈ V then anv = 0 for all large enough n. Set F(V ) = {a(z) ∈ End(V )[[z, z−1]] | a(z) is a field}. F(V ) is a linear subspace of End(V )[[z, z−1]]. If (V, 1) is a vertex algebra, we set X Y (u, z) := u(n)z−n−1 (u ∈ V ), n∈Z where we are using notation introduced in the previous Section. By con- struction, {Y (u, z) | u ∈ V } ⊆ F(V ), (19) and we can think of Y as a linear map Y : V → F(V ), u 7→ Y (u, z). (20) 6 We use obvious notation when manipulating fields, eg., X Y (u, z)v := {u(n)v}z−n−1 ∈ V [[z]][z, z−1]. n In this language, (14) reads X Y (u, z)1 = u + {u(n)1}z−n−1. n≤−2 In particular, it follows that the Y map (20) is injective. A pair of fields a(z), b(z) ∈ F(V ) are called mutually local if k (z1 − z2) [a(z1), b(z2)] = 0 (some integer k ≥ 0). (21) p q This means that the (operator) coefficients of each monomial z1 z2 in the following identity coincide: k k (z1 − z2) a(z1)b(z2) − (z1 − z2) b(z2)a(z1) = 0. (22) Indeed, k (z1 − z2) a(z1)b(z2) k X k X X = (−1)i zk−izi a z−m−1 b z−n−1 i 1 2 m 1 n 2 i=0 m n ( ) X X X X k = (−1)i a b z−p−1z−q−1 i m n 1 2 p q k−i−m=−p i−n=−q ( k ) X X X k = (−1)i a b z−p−1z−q−1.
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