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The \(\widehat{\Gamma} \)-genus and a regularization of an \(S^{1}\)-equivariant Euler class. (English) Zbl 1209.57018
A multiplicative characteristic class for complex vector bundles is uniquely determined by its values for line bundles \(L\), which can be written as a formal power series in the first Chern class \(c_1(L)\). In the present article, a multiplicative class \(\hat\Gamma\) for complex vector bundles is considered that is associated to the power series \(\hat\Gamma(z)^{-1}\) with \(\hat\Gamma(z)=e^{\gamma z}\Gamma(1+z)\), where \(\Gamma\) is the Gamma function, and \(\gamma=-\Gamma'(1)\).
Heuristically, the class \(\hat\Gamma(E)\) for a complex vector bundle \(E\to M\) equals the \(S^1\)-equivariant Euler class of the formal normal bundle \(i^*LE/E\to M\), where \(i: M\to LM\) is the inclusion of \(M\) into its free loop space, and \(LE\to LM\) is the loop bundle of \(E\). The same heuristic argument shows that if \(\eta\to M\) is a real vector bundle, then the \(\hat\Gamma\)-class of its complexification equals \(\hat A(\eta)\).
The associated genus of almost complex manifolds proves to vanish on Riemann surfaces. Computations show that it is a smooth invariant on almost complex manifolds of real dimensions 4 and 8, whereas it depends on the almost complex structure in real dimension 6.

MSC:
57R20 Characteristic classes and numbers in differential topology
40A20 Convergence and divergence of infinite products
55P35 Loop spaces
58J26 Elliptic genera
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