Conformal field theory and elliptic cohomology. (English) Zbl 1071.55004
In [G. Segal, Sémin. Bourbaki, 40éme Année, Vol. 1987/88, Exp. No. 695, Astérisque 161–162, 187–201 (1988; Zbl 0686.55003)], the author proposed a geometric construction of a cohomology theory using conformal field theories. The theory should be related to the homotopy theoretic constructions of elliptic cohomology theories and their real version, the spectrum of topological modular forms [M. J. Hopkins, Li, Ta Tsien (ed.), Proceedings of the international congress of mathematicians, ICM 2002, Beijing: Higher Education Press. 291–317 (2000; Zbl 1031.55007)].
The paper at hand constructs an \(A_\infty\)-spectrum \(E=\omega^{-1} \Sigma^\infty {\mathcal E}_+\) and a map from \(E\) to \(K [\! [ q]\! ] [ q^{-1}]\) which gives the \(q\)-expansion of objects. The space \({\mathcal E}\) is the homotopy fibre of a map \(p_1: \tilde{{\mathcal E}} \to K ({\mathbb Z},4)\) with \( \tilde{{\mathcal E}}= \Omega B (B_{ell} \phi)\). The space \(B_{ell} \phi\) is the union of all spaces of so called elliptic bundles on a complex elliptic curve \(E_\tau\). More precisely, \(B_{ell} (\phi )\) is some type of classifying space associated with conformal field theories. In this context, a very rigorous and detailed definition of a conformal field theory is formulated. Specifically, the coherence conditions of modular functions are carefully worked out. There are several candidates for the class \(\omega\) in terms of theta functions of suitable lattices. One of them is the discriminant \(\Delta\) but there is no geometric reasoning for picking \(\Delta\).
The construction is quite in the spirit of [V. Snaith, Math. Proc. Camb. Philos. Soc. 89, 325–330 (1981; Zbl 0464.55007)] where \(K\)-theory was obtained by the same method from \( \mathbb CP^\infty\). The coefficients of \(E\) are not calculated but the relations to the Moonshine module and the Monster are discussed and many conjectures are formulated.
The paper at hand constructs an \(A_\infty\)-spectrum \(E=\omega^{-1} \Sigma^\infty {\mathcal E}_+\) and a map from \(E\) to \(K [\! [ q]\! ] [ q^{-1}]\) which gives the \(q\)-expansion of objects. The space \({\mathcal E}\) is the homotopy fibre of a map \(p_1: \tilde{{\mathcal E}} \to K ({\mathbb Z},4)\) with \( \tilde{{\mathcal E}}= \Omega B (B_{ell} \phi)\). The space \(B_{ell} \phi\) is the union of all spaces of so called elliptic bundles on a complex elliptic curve \(E_\tau\). More precisely, \(B_{ell} (\phi )\) is some type of classifying space associated with conformal field theories. In this context, a very rigorous and detailed definition of a conformal field theory is formulated. Specifically, the coherence conditions of modular functions are carefully worked out. There are several candidates for the class \(\omega\) in terms of theta functions of suitable lattices. One of them is the discriminant \(\Delta\) but there is no geometric reasoning for picking \(\Delta\).
The construction is quite in the spirit of [V. Snaith, Math. Proc. Camb. Philos. Soc. 89, 325–330 (1981; Zbl 0464.55007)] where \(K\)-theory was obtained by the same method from \( \mathbb CP^\infty\). The coefficients of \(E\) are not calculated but the relations to the Moonshine module and the Monster are discussed and many conjectures are formulated.
Reviewer: Gerd Laures (Bochum)
MSC:
| 55N34 | Elliptic cohomology |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 20C34 | Representations of sporadic groups |
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