Elliptic genera and the moonshine module (Q5961417)

The Fischer-Griess monster is a finite simple group with about \(8\times 10^{53}\) elements, which acts on a graded vector space \(V^\sharp= \bigoplus_{n\geq-1} V_n\) constructed by \textit{I. B. Frenkel, J. Lepowsky} and \textit{A. Meurman} [Vertex operator algebras and the monster. Boston: Academic Press (1988; Zbl 0674.17001)]. \(V^\sharp\) has the structure of a vertex operator algebra (or \(\mathbb{Z}_2\)-orbifold conformal field theory) and is the direct sum of an untwisted space \(V^+\) and a twisted space \(V^-\), both built canonically from the Leech lattice \(\Lambda\). \(\Lambda\) is the nonorthogonal direct sum of three rescaled copies of the \(E_8\)-root lattice. Hence the vector spaces \(V^+\) and \(V^-\) are closely related to the Lie algebra \(E_8\). Using the Dynkin diagram of \(E_8\), \textit{J. W. Milnor} and \textit{M. A. Kervaire} constructed an 8-dimensional almost-parallelizable spin manifold \(M_0^8\) [Proc. Int. Congr. Math. 1958, 454-458 (1960; Zbl 0119.38503)]. Here the author now shows that \(\dim V_n^+\) and \(\dim V_n^-\) are twisted \(\widehat{A}\)-genera of this manifold. In the physics language this means that the numbers of states at the \(n\)th level in the spaces \(V^+\) and \(V^-\) can be described by twisted \(\widehat{A}\)-genera. Theorem. Let \(T_{\mathbb{C}}\) be the complexified tangent bundle of the spin manifold \(M_0^8\). Then we have for \(n\geq 1\) (a) \(\dim V_n^+= 2\widehat{A} (M_0^8, R_{4n+1} (T_{\mathbb{C}}))+ (-1)^{n+1} \widehat{A} (M_0^8, R_{n+1} (T_{\mathbb{C}}))\), (b) \(\dim V_n^-= 2\widehat{A} (M_0^8, R_{4n+1} (T_{\mathbb{C}}))\). In the proof he uses normalized elliptic genera which are (for \(8k\)-dimensional oriented manifolds) modular functions on \(\Gamma_0(2)\). The employment of \(\Gamma_0(2)\) is necessary for the modular consistency of the orbifold construction of \(V^\sharp\). Finally, he proves the following close connection between \(M_0^8\) and \(E_8\): \[ q^{-1} \widehat{A} (M_0^8, \theta_q (T_{\mathbb{C}}))= \vartheta_{E_8} (q^2) q^{-1} \dim_* \theta_q (T_{\mathbb{C}})- 240. \] Here \(\vartheta_{E_8}\) denotes the theta function of the \(E_8\)-root lattice and \[ \dim_* \theta_q (T_{\mathbb{C}})= \sum_{n\geq 0} (-1)^n \dim_* R_n (T_{\mathbb{C}}) q^n. \]