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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 {\it S. Ochanine} [Topology 26, 143-151 (1987; Zbl 0626.57014)] and the reviewer and {\it R. Stong} [Topology 27, No.2, 145-161 (1988; Zbl 0647.57013)], aided by {\it D. Chudnovsky} and {\it 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: $$ F(q)=q\sp{-d/16} \hat A(M,\prod\sp{\infty}\sb{k=1}\Lambda\sb{q\sp{k- }}T\cdot S\sb{q\sp k}T), $$ where $\Lambda\sb tT$ and $S\sb tT$ denote $1+tT+t\sp 2\Lambda\sp 2T+..$. and $1+tT+t\sp 2S\sp 2T+...$, 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 $$ G(q)=q\sp{-d/24} \hat A(M,\prod\sp{\infty}\sb{k=1}S\sb{q\sp k}T) $$ in place of F(q), for which $\Phi (q)=\eta (q)\sp d G(q)$ is a modular form of weight d/2 for SL(2, ${\bbfZ})$ 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 $S\sp 1$ actions on spin manifolds. The argument offered here has since been made rigorous in work by {\it C. Taubes} $[``S\sp 1$ 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 {\it 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

57R20Characteristic classes and numbers (differential topology)
58J22Exotic index theories (PDE on manifolds)
57S15Compact Lie groups of differentiable transformations
81T99Quantum field theory
11F11Holomorphic modular forms of integral weight
Full Text: DOI
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