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**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
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### 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:

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