Conformal Perturbations of Twisted Dirac Operators and Noncommutative Residue

Recently Dabrowski et al. [Adv. Math. 427, 109128 (2023)] obtained the metric and Einstein functionals by two vector fields and Laplace-type operators over vector bundles, giving an interesting example of the spinor connection and square of the Dirac operator. Pfäffle and Stephan [Commun. Math. Phys. 321, 283–310 (2013)] considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds and computed the spectral action. Motivated by the spectral functionals and Dirac operators with torsion, we give some new spectral functionals which is the extension of spectral functionals to the noncommutative realm with torsion, and we relate them to the noncommutative residue for manifolds with boundary. Our method of producing these spectral functionals is the noncommutative residue and Dirac operators with torsion.

Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for existence of associated heat trace expansions with power and power-log terms. The first log coefficient is a noncommutative residue, vanishing when the smearing function is 1. For Dirac operators with general well-posed spectral boundary conditions, it follows that the zeta function is regular at 0. In the selfadjoint case, the eta function has a simple pole at zero, and the value of zeta as well as the residue of eta at zero are stable under perturbations of the boundary projection of order at most minus the dimension.

Under the announcement by Alain Connes that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein–Hilbert action of general relativity, we derive the Lichnerowicz-type formula for the Witten deformation of the non-minimal de Rham–Hodge operator and the gravitational action in the case of n-dimensional compact manifolds without boundary. Finally, we present the proof of the Kastler–Kalau–Walze-type theorem for the Witten deformation of the non-minimal de Rham–Hodge operator on four- and six-dimensional oriented compact manifolds with boundary.

A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors w_1(t), w_2(t), w)3(t), and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients a_0 ,a_2 ,a_4 of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki’s noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to a phase of the Dirac determinant - that agrees with very recent calculations appeared in the mathematical literature - and to a multiplicative anomaly - also in agreement with other calculations, in the coinciding situations. After explicit, nontrivial resummation of the mass series expansions involving zeta and eta functions, the results are expressed in terms of quite simple formulas.

In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boundary.

We develop the deformation theory of instantons on asymptotically conical Spin(7)-manifolds where the instanton is asymptotic to a fixed nearly G2-instanton at infinity. By relating the deformation complex with spinors, we identify the space of infinitesimal deformations with the kernel of the twisted negative Dirac operator on the asymptotically conical Spin(7)-manifold.Finally we apply this theory to describe the deformations of Fairlie–Nuyts–Fubini–Nicolai Spin(7)-instantons on R8, where R8 is considered to be an asymptotically conical Spin(7)-manifold asymptotic to the cone over S7. We calculate the virtual dimension of the moduli space using Atiyah–Patodi–Singer index theorem and the spectrum of the twisted Dirac operator.

We give a brute-force proof of the fact, announced by Alain Connes, that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein-Hilbert action of general relativity. We show that this also holds for twisted (e. g. by electrodynamics) Dirac operators, and more generally, for Dirac operators pertaining to Clifford connections of general Clifford bundles.

A short proof of the local Atiyah-Singer index theorem

We establish a mod 2 index theorem for real vector bundles over 8k + 2 dimensional compact pin− manifolds. The analytic index is the reduced η invariant of (twisted) Dirac operators and the topological index is defined through KO-theory. Our main result extends the mod 2 index theorem of Atiyah and Singer (1971) to non-orientable manifolds.

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the Spinc Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results.

On Riemannian or spin manifolds, there are geometric first order differential operators called generalized gradients. In this article, we prove that the Dirac operator twisted with an associated bundle is a linear combination of some generalized gradients. This observation allows us to find all the homomorphism type Weitzenbock formulas. We also give some applications.

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

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.