scholarly journals Untwisting twisted spectral triples

2019 ◽  
Vol 30 (14) ◽  
pp. 1950076 ◽  
Author(s):  
Magnus Goffeng ◽  
Bram Mesland ◽  
Adam Rennie
Keyword(s):  
Functional Calculus ◽  
Higher Order ◽  
Spectral Triple ◽  
Index Formula ◽  
Spectral Triples ◽  
Local Index ◽  
Index Data ◽  
Twisted Spectral Triples

We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be “logarithmically dampened” through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici’s ansatz for a twisted local index formula is identically zero.

scholarly journals Spectral Flow for Nonunital Spectral Triples

2015 ◽  
Vol 67 (4) ◽  
pp. 759-794 ◽  
Author(s):  
A. L. Carey ◽  
V. Gayral ◽  
J. Phillips ◽  
A. Rennie ◽  
F. A. Sukochev
Keyword(s):  
Index Theory ◽  
Spectral Triple ◽  
Index Formula ◽  
Spectral Flow ◽  
Spectral Triples ◽  
Local Index ◽  
C Algebra ◽  
Odd Index ◽  
Definition Of ◽  
Special Case

AbstractWe prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisûes the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow

scholarly journals Real Part of Twisted-by-Grading Spectral Triples

2020 ◽  
Author(s):  
Manuele Filaci ◽  
◽  
Pierre Martinetti ◽  
◽  
◽  
...  
Keyword(s):  
Standard Model ◽  
Spectral Triple ◽  
Spectral Triples ◽  
The Real ◽  
Initial Algebra ◽  
The Standard Model ◽  
Twisted Spectral Triples

After a brief review on the applications of twisted spectral triples to physics, we adapt to the twisted case the notion of real part of a spectral triple. In particular, when one twists a usual spectral triple by its grading, we show that - depending on the KO dimension - the real part is either twisted as well, or is the intersection of the initial algebra with its opposite. We illustrate this result with the spectral triple of the standard model.

scholarly journals Gauge transformations for twisted spectral triples

2018 ◽  
Vol 108 (12) ◽  
pp. 2589-2626 ◽  
Author(s):  
Giovanni Landi ◽  
Pierre Martinetti
Keyword(s):  
Spectral Triples ◽  
Gauge Transformations ◽  
Twisted Spectral Triples

NONUNITAL SPECTRAL TRIPLES ASSOCIATED TO DEGENERATE METRICS

2004 ◽  
Vol 16 (01) ◽  
pp. 125-146
Author(s):  
A. RENNIE
Keyword(s):  
Finite Dimension ◽  
Index Theorem ◽  
Smooth Manifolds ◽  
Spectral Triples ◽  
Local Index ◽  
Dimension Spectrum ◽  
Spectrum Hypothesis ◽  
Local Index Theorem ◽  
K Theory

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

scholarly journals The local index formula in semifinite Von Neumann algebras I: Spectral flow

2006 ◽  
Vol 202 (2) ◽  
pp. 451-516 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Spectral Flow ◽  
Local Index ◽  
Von Neumann

scholarly journals The local index formula in semifinite von Neumann algebras II: The even case

2006 ◽  
Vol 202 (2) ◽  
pp. 517-554 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Local Index ◽  
Von Neumann

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

The local index formula for a Hermitian manifold

1997 ◽  
Vol 180 (1) ◽  
pp. 51-56
Author(s):  
Peter B. Gilkey ◽  
S. Nikčević ◽  
J. Pohjanpelto
Keyword(s):  
Hermitian Manifold ◽  
Index Formula ◽  
Local Index

An application to local index theory

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Compact Manifold ◽  
Index Theory ◽  
Heat Kernels ◽  
Index Formula ◽  
Torsion Free Group ◽  
Local Index ◽  
Orbital Integrals ◽  
Index Formulas ◽  
Torsion Free

This chapter verifies the compatibility of the formula for the orbital integrals of heat kernels introduced in the previous chapter to the index formula of Atiyah-Singer, to the fixed point formulas of Atiyah-Bott, and to the index formula for orbifolds of Kawasaki. Given that the McKean-Singer formula expresses the index of a Dirac operator over a compact manifold Z as the supertrace of a heat kernel, if Z is the quotient of X by a cocompact torsion free group, this supertrace can be evaluated explicitly by the formulas provided in the previous chapter. This chapter directly checks these formulas to be compatible with the index formulas.

scholarly journals The Local Index Formula in Noncommutative Geometry Revisited

2013 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fedor A. Sukochev
Keyword(s):  
Noncommutative Geometry ◽  
Index Formula ◽  
Local Index
Sign in / Sign up

Export Citation Format

Share Document