scholarly journals The Atiyah–Patodi–Singer index theorem for Dirac operators over $C*$-algebras

2013 ◽  
Vol 17 (2) ◽  
pp. 265-320 ◽  
Author(s):  
Charlotte Wahl
Keyword(s):  
Dirac Operators ◽  
Index Theorem ◽  
C Algebras

scholarly journals An L2-Index Theorem for Dirac Operators on S1×R3

2000 ◽  
Vol 177 (1) ◽  
pp. 203-218 ◽  
Author(s):  
Tom M.W. Nye ◽  
Michael A. Singer
Keyword(s):  
Dirac Operators ◽  
Index Theorem ◽  
L2 Index

Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds

2018 ◽  
Vol 12 (04) ◽  
pp. 897-939 ◽  
Author(s):  
Simone Cecchini
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Complete Riemannian Manifold ◽  
Positive Scalar Curvature ◽  
Type Operator ◽  
Spin Manifold ◽  
Compact Set ◽  
Positive Scalar ◽  
C Algebras

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert [Formula: see text]-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when [Formula: see text] is a closed spin manifold, we show that if the cylinder [Formula: see text] carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on [Formula: see text] must vanish.

scholarly journals GENERALIZED GINSPARG–WILSON ALGEBRA AND INDEX THEOREM ON THE LATTICE

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1931-1941
Author(s):  
KAZUO FUJIKAWA
Keyword(s):  
General Class ◽  
Dirac Operators ◽  
Index Theorem ◽  
Explicit Construction ◽  
Negative Integer ◽  
Chiral Anomaly ◽  
Gauge Fields ◽  
Topological Properties

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form γ5 (γ5D) + (γ5D)γ5 = 2a2k+1(γ5D)2k+2 where k stands for a non-negative integer. The choice k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.

scholarly journals A K-theoretic relative index theorem and Callias-type Dirac operators

1995 ◽  
Vol 303 (1) ◽  
pp. 241-279 ◽  
Author(s):  
Ulrich Bunke
Keyword(s):  
Dirac Operators ◽  
Index Theorem ◽  
Relative Index

scholarly journals The noncommutative geometry of k-graph C*-algebras

2007 ◽  
Vol 1 (2) ◽  
pp. 259-304 ◽  
Author(s):  
David Pask ◽  
Adam Rennie ◽  
Aidan Sims
Keyword(s):  
Fixed Point ◽  
Isomorphism Class ◽  
Lower Semicontinuous ◽  
Index Theorem ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Local Index ◽  
C Algebras ◽  
Local Index Theorem ◽  
Fixed Point Algebra

AbstractThis paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C* (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the Tk action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

scholarly journals An index theorem for non-standard Dirac operators

1999 ◽  
Vol 16 (7) ◽  
pp. 2537-2544 ◽  
Author(s):  
Jan-Willem van Holten ◽  
Andrew Waldron ◽  
Kasper Peeters
Keyword(s):  
Dirac Operators ◽  
Index Theorem
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