*(English)*Zbl 0625.57008

It is a great delight when a new development in mathematics can be clarified and further elaborated with the help of recently developed methods in physics, as is the case here. Work by *S. Ochanine* [Topology 26, 143-151 (1987; Zbl 0626.57014)] and the reviewer and *R. Stong* [Topology 27, No.2, 145-161 (1988; Zbl 0647.57013)], aided by *D. Chudnovsky* and *G. Chudnovsky* [Topology 27, No.2, 163-170 (1988; Zbl 0653.57015)], had led to the notion of elliptic genus, in which one assigns a level 2 modular form of weight $2n$ to a closed oriented smooth manifold of dimension $4n$. What business does one have assigning modular forms to manifolds?

The startling answer given here is that the supercharge of the supersymmetric nonlinear sigma model, $F\left(q\right)$, is (up to an explicit factor formed from Dedekind’s eta function $\eta \left(q\right)$) precisely the elliptic genus of the manifold. The function $F\left(q\right)$ has an explicit and illuminating expression obtainedby use of the ordinary Atiyah-Singer index theorem:

where ${{\Lambda}}_{t}T$ and ${S}_{t}T$ denote $1+tT+{t}^{2}{{\Lambda}}^{2}T+\xb7\xb7$. and $1+tT+{t}^{2}{S}^{2}T+\xb7\xb7\xb7$, respectively. Here M has dimension d, and T is the complexification of its tangent bundle.

Several further possibilities are suggested. There is an alternative nonlinear sigma model leading to

in place of F(q), for which ${\Phi}\left(q\right)=\eta {\left(q\right)}^{d}G\left(q\right)$ is a modular form of weight d/2 for SL(2, $\mathbb{Z})$ provided that M is a spin manifold with vanishing first rational Pontryagin class. There are further variants, in which one makes use of a vector bundle addition to the tangent bundle, leading to modular forms of levels 1 and 2.

Moreover, there is an illuminating discussion of the question which motivated the development of elliptic genera, namely the problem of the constancy of equivariant elliptic genera for ${S}^{1}$ actions on spin manifolds. The argument offered here has since been made rigorous in work by *C. Taubes* $[``{S}^{1}$ actions and elliptic genera”, preprint (Harvard Univ. 1987)] and later by R. Bott and C. Taubes. Earlier work on the same problem was done by *S. Ochanine* [“Genres elliptiques équivariants”, in Elliptic curves and modular forms in algebraic topology, Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326, 107-122 (1988; Zbl 0649.57023)].

This paper is written largely in “physical” terms. The author has since written an account of these topics in mathematical terms [“The index of the Dirac operator in loop space”, in Elliptic curves and modular forms in algebraic topology, Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326, 161-181 (1988; Zbl 0679.58045)].

##### MSC:

57R20 | Characteristic classes and numbers (differential topology) |

58J22 | Exotic index theories (PDE on manifolds) |

57S15 | Compact Lie groups of differentiable transformations |

81T99 | Quantum field theory |

11F11 | Holomorphic modular forms of integral weight |

##### Keywords:

elliptic cohomology; Circle actions on Spin manifolds; modular function; elliptic genera; elliptic genus; supercharge of the supersymmetric nonlinear sigma model; Dedekind’s eta function; Atiyah-Singer index theorem; modular form of weight d/2; rational Pontryagin class; equivariant elliptic genera; index of the Dirac operator in loop space##### References:

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