##
**Elliptic genera and quantum field theory.**
*(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, \(F(q)\), is (up to an explicit factor formed from Dedekind’s eta function \(\eta(q)\)) precisely the elliptic genus of the manifold. The function \(F(q)\) has an explicit and illuminating expression obtainedby use of the ordinary Atiyah-Singer index theorem: \[ F(q)=q^{-d/16} \hat A(M,\prod^{\infty}_{k=1}\Lambda_{q^{k- }}T\cdot S_{q^ k}T), \] where \(\Lambda_ tT\) and \(S_ tT\) denote \(1+tT+t^ 2\Lambda^ 2T+..\). and \(1+tT+t^ 2S^ 2T+...\), 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 \[ G(q)=q^{-d/24} \hat A(M,\prod^{\infty}_{k=1}S_{q^ k}T) \] in place of F(q), for which \(\Phi (q)=\eta (q)^ d G(q)\) 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)].

The startling answer given here is that the supercharge of the supersymmetric nonlinear sigma model, \(F(q)\), is (up to an explicit factor formed from Dedekind’s eta function \(\eta(q)\)) precisely the elliptic genus of the manifold. The function \(F(q)\) has an explicit and illuminating expression obtainedby use of the ordinary Atiyah-Singer index theorem: \[ F(q)=q^{-d/16} \hat A(M,\prod^{\infty}_{k=1}\Lambda_{q^{k- }}T\cdot S_{q^ k}T), \] where \(\Lambda_ tT\) and \(S_ tT\) denote \(1+tT+t^ 2\Lambda^ 2T+..\). and \(1+tT+t^ 2S^ 2T+...\), 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 \[ G(q)=q^{-d/24} \hat A(M,\prod^{\infty}_{k=1}S_{q^ k}T) \] in place of F(q), for which \(\Phi (q)=\eta (q)^ d G(q)\) 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)].

Reviewer: P.Landweber

### MSC:

57R20 | Characteristic classes and numbers in differential topology |

58J22 | Exotic index theories on manifolds |

57S15 | Compact Lie groups of differentiable transformations |

81T99 | Quantum field theory; related classical field theories |

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
Full Text:
DOI

### Digital Library of Mathematical Functions:

1st item ‣ §23.21(iv) Modular Functions ‣ §23.21 Physical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions### References:

[1] | Landweber, P.S., Stong, R.: Circle actions on spin manifolds and characteristic numbers, Rutgers University preprint 1985 |

[2] | Atiyah, M.F., Hirzebruch, F.: In: Essays on topology and related subjects, pp. 18-28. Berlin, Heidelberg, New York: Springer 1970 |

[3] | Atiyah, M.F., Singer, I.M.: Ann. Math.87, 484, 586 (1968) · Zbl 0164.24001 · doi:10.2307/1970715 |

[4] | Atiyah, M.F., Segal, G.B.: Ann. Math.87, 531 (1968) · Zbl 0164.24201 · doi:10.2307/1970716 |

[5] | Atiyah, M.F., Bott, R.: Ann. Math.87, 456 (1968) |

[6] | Witten, E.: Fermion quantum numbers in Kaluza-Klein theory. In: Shelter Island II: Proceedings of the 1983 Shelter Island conference on quantum field theory and the fundamental problems of physics. Khuri, N. et al. (eds.). Cambridge, MA: MIT Press 1985 |

[7] | Borsari, L.: Bordism of semi-free circle actions on spin manifolds. Trans. Am. Math. Soc. (to appear) · Zbl 0622.57024 |

[8] | Ochanine, S.: Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology (to appear) · Zbl 0626.57014 |

[9] | Chudnovsky, D.V., Chudnovsky, G.V.: Elliptic modular functions and elliptic genera. Columbia University preprint (1985) · Zbl 0561.10016 |

[10] | Ochanine, S.: Elliptic genera forS 1 manifolds. Lecture at conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986) |

[11] | Landweber, P.S., Ravenel, D., Stong, R.: Periodic cohomology theories defined by elliptic curves. Preprint (to appear) · Zbl 0920.55005 |

[12] | Landweber, P.S.: Elliptic cohomology and modular forms. To appear in the proceedings of the conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986) |

[13] | Hopkins, M., Kuhn, N., Ravenel, D.: Preprint (to appear) |

[14] | Hopkins, M.: Lecture at the conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986) |

[15] | Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B261, 678 (1985) · doi:10.1016/0550-3213(85)90593-0 |

[16] | Schellekens, A., Warner, N.: Anomalies and modular invariance in string theory, Anomaly cancellation and self-dual lattices (MIT preprints 1986). Anomalies, characters and strings (CERN preprint TH 4529/86) |

[17] | Pilch, K., Schellekens, A., Warner, N.: Preprint, 1986 |

[18] | Witten, E.: J. Differ. Geom.17, 661 (1982), Sect. IV. In: Anomalies, geometry, and topology. Bardeen, W., White, A. (eds.). New York: World Scientific, 1985, pp. 61-99, especially pp. 91-95 · Zbl 0499.53056 |

[19] | Atiyah, M.F., Singer, I.M.: Ann. Math.93, 119 (1971) · Zbl 0212.28603 · doi:10.2307/1970756 |

[20] | Zagier, D.: A note on the Landweber-Stong elliptic genus (October 1986) · Zbl 0653.57016 |

[21] | Asorey, M., Mitter, P.K.: Regularized, continuum Yang-Mills process and Feynman-Kac functional integral. Commun. Math. Phys.80, 43 (1981) · Zbl 0476.58008 · doi:10.1007/BF01213595 |

[22] | Bern, Z., Halpern, M.B., Sadun, L., Taubes, C.: Continuum regularization of QCD. Phys. Lett.165 B, 151 (1985) |

[23] | Eichler, M., Zagier, D.: The theory of Jacobi forms. Boston: Birkhäuser 1985 · Zbl 0554.10018 |

[24] | Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys.92, 455 (1984) · Zbl 0536.58012 · doi:10.1007/BF01215276 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.