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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.

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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