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Book Chapter

Note on the Landweber-Stong elliptic genus


Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1988). Note on the Landweber-Stong elliptic genus. In Elliptic curves and modular forms in algebraic topology (pp. 216-224). Berlin: Springer.

Cite as: http://hdl.handle.net/21.11116/0000-0004-394C-8
[For the entire collection see Zbl 0642.00007.] \\par Recently Landweber, Stong, Ochanine, Witten and others studied elliptic genera φ: Ω\\sb*\\spSO(n)→ R from the oriented bordism ring to a commutative \\bbfQ-algebra with 1. An elliptic genus φ is characterized by the property that φ(M) vanishes if M is the total space of a complex projective bundle of an even-dimensional complex vector bundle over an oriented compact manifold. There are three power series associated to a genus φ: the logarithm of the formal group law of φ, which is given by g(x)=\\sum\\sp∞\\sbn=0φ (\\bbfCP\\sp2n)x\\sp2n+1/2n+1, the Hirzebruch characteristic power series P(u) of φ, and the KO-theory characteristic power series F(y) of φ. These three power series are related to one another by the formulas u/g\\sp-1(u)=P(u)=(u/2)/\\sin h(u/2)\\quad F(e\\sp u+e\\sp-u- 2). \\it P. S. Landweber and \\it R. E. Stong [Topology 27, 145--161 (1988; Zbl 0647.57013)] introduced a particular elliptic genus with values in the power series ring R=\\Bbb Q[[q]]. The author gives elementary proofs of a variety of formulas for the power series g, F, P which are associated to the Landweber-Stong elliptic genus, and he gives in particular proofs of some properties of these power series. one of these is the property that P(u) has coefficients in \\Bbb Z[[q]]. Moreover the author studies other genera introduced by Witten. The proofs use ideas from the theory of elliptic functions and modular forms.