## 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)].
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
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### 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 [4] Atiyah, M.F., Segal, G.B.: Ann. Math.87, 531 (1968) · Zbl 0164.24201 [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) [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 [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 [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
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