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Elliptic spectra, the Witten genus and the theorem of the cube. (English) Zbl 1031.55005

There have been several attempts to generalize the elliptic genus of E. Witten [Lect. Notes Math. 1326, 161-181 (1988; Zbl 0679.58045)] to families of spin manifolds with trivialized Pontryagin class \(p_{1}/2\). From a topological point of view, this amounts to constructing a canonical map of ring spectra of the associated bordism theory \(MO \langle 8\rangle \) to each elliptic spectrum \(E\) (compare for instance [M. J. Hopkins, Proceedings of the International Congress of Mathematicians, Zürich 1994, Basel, Birkhäuser, 554-565 (1995; Zbl 0848.55002)]).
This paper gives a convenient description of all multiplicative maps from \(MU \langle 6\rangle \) to \(E\). Here, \(MU \langle 6\rangle \) is the bordism theory associated to \(SU\) manifolds with trivialized second Chern class. The final step to the {\` real world\'} is postponed to a different paper. In fact, an alternative approach to the Witten genus of a family is suggested in [M. J. Hopkins, Proceedings of the international congress of mathematicians, Vol. I (Beijing, 2002), Higher Education Press, Beijing, 291-317 (2002; Zbl 1031.55007)]. However, the results of the paper are still interesting and important for several reasons: they reveal how natural the Witten genus really is and how close bordism theory and homotopy theory are related to elliptic curves and modular forms.
In the sequel I will describe the results of the paper in more detail. For \(E\) an even periodic ring spectrum, it is shown that \(E_0 BU \langle 6\rangle \) carries the universal cubical structure from the formal group \(E^0 CP^0\) associated to \(E\) to the multiplicative formal group. The proof takes the largest part of the paper and involves an interpretation of the singular homology of \(K (Z,3)\) as Weil pairings on the additive formal group law.
The Thom isomorphism shows that \(E_0 MU \langle 6\rangle \) is a torsor for \(E_0 BU \langle 6\rangle \). The Thom space of the tautological bundle over \( CP^\infty\) gives a sheaf of sections of the line bundle \(I(0)\) over \(E^0 CP^\infty\). The authors identify the cubical structures on \(I(0)\) over the formal group of \(E\) with maps of ring spectra from \(MU \langle 6\rangle \) to \(E\). The classical theorem of the cube says that there is a unique way to assign to each generalized elliptic curve a natural cubical structure. Hence, when applied to elliptic spectra, it furnishes the desired orientation.

MSC:

55N34 Elliptic cohomology
14H52 Elliptic curves
58J26 Elliptic genera
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