On mod 2 and higher elliptic genera
type: Item
instance of: scholarly article
title: On mod 2 and higher elliptic genera
zbMATH Open document ID: 0763.58026
DOI: 10.1007/BF02096624
published in: Communications in Mathematical Physics
publication date: 1993-01-16T00:00:00Z
review text: Elliptic genera have usually come in two ways: either by topological methods or by invoking properties of some differential operator, frequently an index formula. Although cobordism plays a crucial role, this paper essentially belongs to the latter category in that it uses the Atiyah-Singer mod 2 index formula for Dirac operators to construct mod 2 elliptic genera on compact smooth \(8k+1\)- and \(8k+2\)-dimensional spin manifolds. The basic idea is to lift the classical Dirac operator mod 2 index formula to a homomorphism from \(KO(X)\) to \(\mathbb{Z}_ 2\), where it is readily checked that it equals the standard cobordism \(KO\)-invariant. Cobordism is then invoked to arrive at the relationship with mod 2 elliptic genera as geometric objects. Higher elliptic genera are considered in the second part of the paper. It is interesting to note that in the course of the work an integral formula for the mod 2 index is obtained in the case of (compact) \(8k+2\)- dimensional spin manifolds.
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 58J26
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 95695
zbMATH Keywords: Atiyah-Singer mod 2 index
zbMATH Keywords: mod 2 elliptic genera
zbMATH Keywords: Dirac operator
zbMATH Keywords: cobordism
author: Kefeng Liu
MaRDI profile type: Publication
cites work: Q4831225
cites work: Clifford modules
cites work: Q5589103
cites work: Q5647673
cites work: Spectral asymmetry and Riemannian geometry. II
cites work: Cobordism span of a manifold and elliptic genera
cites work: On the Rigidity Theorems of Witten
cites work: G-actions and the fundamental group
cites work: Remark on Witten's Modular Forms
cites work: THE CHERN-DOLD CHARACTER IN COBORDISMS. I
cites work: Q3695591
cites work: Elliptic genera, involutions, and homogeneous spin manifolds
cites work: Spin cobordism determines real K-theory
cites work: Spin bordism and elliptic homology
cites work: Generalized elliptic genera and Baker-Akhiezer functions
cites work: Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres
cites work: THE INDEX OF ELLIPTIC OPERATORS OVERC*-ALGEBRAS
cites work: Sur les genres multiplicatifs définis par des intégrales elliptiques. (On multiplicative genera defined by elliptic integrals)
cites work: Elliptic genera, modular forms over \(KO_ *\), and the Brown-Kervaire invariant
cites work: C*-algebras, positive scalar curvature, and the Novikov conjecture
cites work: Q5571606
cites work: Simply connected manifolds of positive scalar curvature
cites work: Elliptic genera and quantum field theory
cites work: Periods of modular forms and Jacobi theta functions
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description: scientific article; zbMATH DE number 95695