By M. Aizenman (Chief Editor)
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Additional info for Communications in Mathematical Physics - Volume 214
1, that again σ (g) = 1. In this case, the precise description of Z(g, h, τ ) for gh = hg and h of odd order is given in [DLM1]. As discussed in [DM1] and [DM2], the uniqueness of V (g) leads to a projective representation of the centralizer CM (g) on V (g). ). The conjectures of Conway–Norton–Queen state that there are (projective) representations of CM (g) on suitably graded spaces such that all graded traces are either Hauptmoduln or zero. There is no doubt that the twisted modules V (g) are the desired spaces.
We first need to define n-point correlation functions. These are multi-linear functions T (v1 , . . , vn ), vi ∈ V , defined for vi homogeneous via T (v1 , . . , vn ) = T ((v1 , z1 ), . . , (vn , zn ), (g, h), q) = z1wtv1 · · · znwtvn trYM (v1 , z1 ) · · · YM (vn , zn )φ(h)q L(0)−c/24 . 9) We only need the case n = 2. 4. Let v, v1 ∈ V be homogeneous with gv = µ−1 v, hv = λ−1 v and (µ, λ) = (1, 1). 31). 5. Let k ∈ 1 T Z. 12) (1 − λq k )trYM (v1 , z1 )v(wtv − 1 + k)φ(h)q L(0) = λq k ∞ i=0 wtv − 1 + k wtv−1+k−i z1 trYM (v(i)v1 , z1 )φ(h)q L(0) .
As ω˜ = L[−2]1, the lemma follows. If we write V = ⊕n Vn , then of course we have Z(1, g, τ ) = q −c/24 n T (ω, ˜ 1, g, τ ) = q −c/24 n (tr|Vn g)q n , (n − c/24)(tr|Vn g)q n , and we may think of T (ω, ˜ 1, g, τ ) as arising from a sequence of virtual characters of G. ” This is relevant because of the work of Devoto [De] in which such things are interpreted as being elements of degree 2 in the elliptic cohomology of BG. Similarly suppose that v ∈ V satisfies wt[v] = k with gv = v for all g ∈ G.
Communications in Mathematical Physics - Volume 214 by M. Aizenman (Chief Editor)