Communications in Number Theory and Physics

Volume 6 (2012)

Number 4

Moonshine for $M_{24}$ and Donaldson invariants of $\mathbb{C} \mathrm{P}^2$

Pages: 759 – 770



Andreas Malmendier (Department of Mathematics, Colby College, Waterville, Maine, U.S.A.)

Ken Ono (Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia, U.S.A.)


Eguchi et al. recently conjectured a new moonshine phenomenon. They conjecture that the coefficients of a certain mock modular form $H(\tau)$, which arises from the $K3$ surface elliptic genus, are sums of dimensions of irreducible representations of the Mathieu group $M_{24}$. We prove that $H(\tau)$ surprisingly also plays a significant role in the theory of Donaldson invariants. We prove that the Moore–Witten $u$-plane integrals for $H(\tau)$ are the $\mathrm{SO}(3)$-Donaldson invariants of $\mathbb{C} \mathrm{P}^2$. This result then implies a moonshine phenomenon where these invariants conjecturally are expressions in the dimensions of the irreducible representations of $M_{24}$. Indeed, we obtain an explicit expression for the Donaldson invariant generating function $\mathrm{Z}(p,S)$ in terms of the derivatives of $H(\tau)$.

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