Signature Operator Research Articles

The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integral involving the fermionic signature operator as well as a double integral which takes into account the flux of Dirac currents across the event horizon. The spectrum of the fermionic signature operator is computed. The corresponding generalized fermionic projector states are analyzed.

Transverse noncommutative geometry of foliations

We define an L2-signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals–Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes–Chern character. The Connes–Moscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our “L2-signature”. For Galois coverings of bounded geometry foliations, we also define an Atiyah–Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L2-index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [7], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids.

Two-cocycle twists and Atiyah–Patodi–Singer index theory

AbstractWe construct η- and ρ-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

The refined analytic torsion and a well-posed boundary condition for the odd signature operator

Abstract In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator B in the sense of Seeley. We then show that the zeta-determinants of B 2 and eta-invariants of B subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

The fermionic signature operator and space-time symmetries

We show that and specify how space-time symmetries give rise to corresponding symmetries of the fermionic signature operator and generalized fermionic projector states.

Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For G=Z/mZ and for a regular G-cover S→S¯ (possibly branched), a finite-index subgroup Γ<Mod(S¯) acts on H1(S;Z) commuting with the deck group action, thus inducing a homomorphism Γ→Sp2gG(Z) to an arithmetic group. The induced map H∗(Sp2gG(Z);Q)→H∗(Γ;Q) can be understood using index theory. To this end, we describe a families version of the G-index theorem for the signature operator and apply this to (i) compute H2(Sp2gG(Z);Q)→H2(Γ;Q), (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to SU(p,q) arising from Atiyah–Kodaira constructions, and (iv) describe how classes in H∗(Sp2gG(Z);Q) give equivariant cobordism invariants for surface bundles with a fiberwise G action, following Church–Farb–Thibault.

The fermionic signature operator and quantum states in Rindler space-time

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling–Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.

The Fermionic Signature Operator and Hadamard States in the Presence of a Plane Electromagnetic Wave

We give a nonperturbative construction of a distinguished state for the quantized Dirac field in Minkowski space in the presence of a time-dependent external field of the form of a plane electromagnetic wave. By explicit computation of the fermionic signature operator, it is shown that the Dirac operator has the strong mass oscillation property. We prove that the resulting fermionic projector state is a Hadamard state.

Mapping Surgery to Analysis II: Geometric Signatures

We give geometric constructions leading to analytically controlled Poincarcom- plexes in the sense of the previous paper. In the case of a complete Riemannian manifold we identify the signature of the associated complex with the coarse index of the signature operator. Mathematics Subject Classifications (2000): 19J25, 19K99.

The chiral index of the fermionic signature operator

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators thereon for which our index is non-trivial.

The surgery exact sequence, K-theory and the signature operator

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.

Lorentzian spectral geometry for globally hyperbolic surfaces

The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.

Twisted Dirac operators and the noncommutative residue for manifolds with boundary

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.

Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

In Huang and Lee (in The Refined Analytic Torsion and a Well-posed Boundary Condition for the Odd Signature Operator, 2010) introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of global boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map which is a local isometry on the boundary and has only simple fixed points. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

Large time limit and local $L^2$-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut–Lott type superconnections in the L^2 -setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2 -index formulas. As applications, we prove a local L^2 -index theorem for families of signature operators and an L^2 -Bismut–Lott theorem, expressing the Becker–Gottlieb transfer of flat bundles in terms of Kamber–Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2 -eta forms and L^2 -torsion forms as transgression forms.

Algebraic Structures, Physics and Geometry from a Unified Field Theoretical Framework

Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P(G, M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein’s theorem is incomplete for that description. The importance of the Clifford algebras in the description of all symmetries, mainly the interaction of gravity with the other fields, is briefly discussed.

Geometric construction of Hopf cyclic characteristic classes

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra Hn, playing the role of a ‘quantum structure group’ for the ‘space of leaves’ of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand–Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cyclic classes. This paper provides an effective method for constructing the Hopf cyclic cohomology classes of Hn and of Hn relative to On, in the spirit of the Chern–Weil theory, which completely elucidates their relationship with the characteristic classes of foliations.

The Higson–Roe exact sequence and [formula omitted] eta invariants

The goal of this paper is to solve the problem of existence of an ℓ2 relative eta morphism on the Higson–Roe structure group. Using the Cheeger–Gromov ℓ2 eta invariant, we construct a group morphism from the Higson–Roe maximal structure group constructed in [35] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin ℓ2 rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the ℓ2 rho invariants of the corresponding signature operators. We thus get new proofs for the classical ℓ2 rigidity theorems of Keswani obtained in [41].

The index of geometric operators on Lie groupoids

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the “topological side” of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.