The ζ-Determinant and the Additivity of the η-Invariant on the Smooth, Self-Adjoint Grassmannian

Wojciechowski, Krzysztof

doi:10.1007/s002200050561

Cited by 42 publications

(43 citation statements)

References 21 publications

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“…The methods of [17] show that the assumption P ∈ Gr ∞ (A) can be somewhat relaxed [16]. Finally, that ζ(D P ; 0) is independent of P ∈ Gr ∞ (A) is [35,Prop. 0.5].…”

Section: The η-Invariant and Spectral Flowmentioning

confidence: 99%

“…That the ζ-and η-functions extend meromorphically has been proved in increasing generality in [15], [18], [19], [7], [35], and [17]. The definitive treatment of all wellposed boundary value problems is given in [17].…”

Section: The η-Invariant and Spectral Flowmentioning

confidence: 99%

“…The definitive treatment of all wellposed boundary value problems is given in [17]. The proof of the statement about regularity at s = 0 can be found in [35]. The methods of [17] show that the assumption P ∈ Gr ∞ (A) can be somewhat relaxed [16].…”

Section: The η-Invariant and Spectral Flowmentioning

confidence: 99%

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The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Kirk¹,

Lesch²

2004

Forum Mathematicum

230

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Abstract. Several proofs have been published of the mod Z gluing formula for the η-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the η-invariant is left obscure in the literature. In this article we present a gluing formula for the η-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón projectors of the two parts of the decomposition. The main ingredients of our presentation are the ScottWojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Brüning-Lesch to the mod Z gluing formula.Our presentation includes careful constructions of the Maslov index and triple index in a symplectic Hilbert space. As a byproduct we give intuitively appealing proofs of two theorems of Nicolaescu on the spectral flow of Dirac operators.As an application of our methods, we carry out a detailed analysis of the η-invariant of the odd signature operator coupled to a flat connection using adiabatic methods. This is used to extend the definition of the Atiyah-Patodi-Singer ρ-invariant to manifolds with boundary. We derive a "non-additivity" formula for the AtiyahPatodi-Singer ρ-invariant and relate it to Wall's non-additivity formula for the signature of even-dimensional manifolds.

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“…The methods of [17] show that the assumption P ∈ Gr ∞ (A) can be somewhat relaxed [16]. Finally, that ζ(D P ; 0) is independent of P ∈ Gr ∞ (A) is [35,Prop. 0.5].…”

Section: The η-Invariant and Spectral Flowmentioning

confidence: 99%

Section: The η-Invariant and Spectral Flowmentioning

confidence: 99%

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The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Kirk¹,

Lesch²

2004

Forum Mathematicum

230

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“…Pi are well defined by the results of Grubb [11] and Wojciechowski [35]. The orthogonal projections P 1 and P 2 determine maps κ 1 and κ 2 as in (10), and we define…”

Section: Tr(dementioning

confidence: 99%

Gluing Formulae of Spectral Invariants and Cauchy Data Spaces

Park

2006

Analysis, Geometry and Topology of Elliptic Operators

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“…In particular both functions are holomorphic in a neighborhood of s = 0. Therefore det ζ D P is a well-defined, smooth function on Gr * ∞ (D) (see [ 52]). We will discuss the regularity of η-function of the operator D P , with P ∈ Gr * ∞ (D) in Section 9.…”

Section: Determinants Of Dirac Operators On a Manifold With Boundarymentioning

confidence: 99%

Analytic surgery of the ζ-determinant of the Dirac operator

Park¹,

Wojciechowski

2002

Nuclear Physics B - Proceedings Supplements

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The ζ-Determinant and the Additivity of the η-Invariant on the Smooth, Self-Adjoint Grassmannian

Cited by 42 publications

References 21 publications

The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Gluing Formulae of Spectral Invariants and Cauchy Data Spaces

Analytic surgery of the ζ-determinant of the Dirac operator

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