Chiral Asymmetry and the Spectral Action

Abstract

We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.

A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary

“…where definitions ofR(x), A iin , see [20]. In addition, we proved the Kastler-Kalau-Walze type theorems for foliations with or without boundary associated with sub-Dirac operators in [16] Wres…”

“…Meanwhile, Pfäffle and Stephan considered compact Riemannian spin manifolds without boundary equipped with orthogonal connections, and investigated the induced Dirac operators in [19]. In [20], Pfäffle and Stephan considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds, and for the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type they computed the spectral action. For the associated Dirac operators with torsion D * T , D T [21], we got the Kastler-Kalau-Walze theorem associated to Dirac operators with torsion on 4-dimensional compact manifolds with boundary…”

A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary

“…Denote by σ l (A) the l-order symbol of an operator A. An application of (2.1.4) in [11] shows that 20) and the sum is taken over r − k + |α| + ℓ − j − 1 = −n, r ≤ −p 1 , ℓ ≤ −p 2 .…”

Dirac operators with torsion and the noncommutative residue for manifolds with boundary

“…Recently, Pfäffle and Stephan considered compact Riemannian spin manifolds without boundary equipped with orthogonal connections, and investigated the induced Dirac operators in [14]. In [15], Pfäffle and Stephan considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds, and for the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type they computed the spectral action.…”

“…[15] Let M be a 4-dimensional compact manifold without boundary and∇ be an orthogonal connection with torsion. Then we get the volumes associated to D * T D T on compact manifolds without boundary…”