Integrality properties of the Kontsevich-Witten genus Following work of Di Francesco, Itzykson, and Zuber, we construct a ring homomorphism kw: MU•(pt) ~~. ® (Q where ~. = 7l[qili ~l]/(Li(-1)iqr -iq"r > 0) is a Hopf algebra with coproduct qi ~Ei+k=i qj ® qk• The generating function q(x) = L qk x " satisfies q(x)q( -x) = 1. So log(q(x» is an odd power series, and the genus is defined by the formal group law kw-1 (kw(X) + kw(Y» = X +kw Y with kw(T) = T 1 / 2 Iogq(T 1 /2 a This defines a candidate

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... ~for a quantum cohomology spectrum; we can begin to see. few things about the diagram Mg monodromy Brg BUq