[Paper Review] The Atiyah-Patodi-Singer index on manifolds with non-compact boundary

Maxim Braverman, Pengshuai Shi|arXiv (Cornell University)|Jun 21, 2017

Advanced Operator Algebra Research42 references7 citations

TL;DR

This paper establishes the Atiyah-Patodi-Singer index theorem for strongly Callias-type operators on complete odd-dimensional manifolds with non-compact boundary. It introduces a relative eta-invariant that behaves like the difference of individual eta-invariants despite their individual values being undefined, and proves that this relative invariant equals twice the spectral flow of a family of operators, extending index theory to non-compact settings without bounded geometry assumptions.

ABSTRACT

We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold $M$. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds $N_0$ and $N_1$. If the dimension of $M$ is odd we show that the latter index depends only on the restrictions $A_0$ and $A_1$ of $D$ to $N_0$ and $N_1$ and thus is an invariant of the boundary. We use this invariant to define the relative eta-invariant $η(A_1,A_0)$. We show that even though in our situation the eta-invariants of $A_1$ and $A_0$ are not defined, the relative eta-invariant behaves as if it was the difference $η(A_1)-η(A_0)$.

Motivation & Objective

• To generalize the Atiyah-Patodi-Singer index theorem to complete odd-dimensional manifolds with non-compact boundary.
• To define an invariant index for strongly Callias-type operators under elliptic boundary conditions in the non-compact setting.
• To introduce a relative eta-invariant that behaves as if it were the difference of individual eta-invariants, even when those are not defined.
• To establish a spectral flow formula relating the relative eta-invariant to the index of a cobordism between boundary operators.
• To prove a splitting theorem for the index on manifolds decomposed along a non-compact hypersurface.

Proposed method

• Introduces strongly Callias-type operators on complete manifolds with non-compact boundary, defined by stronger growth conditions on the potential ensuring discrete spectrum.
• Constructs a scale of Sobolev spaces on the boundary using eigensections of the boundary operator, enabling the definition of elliptic boundary conditions dependent on the boundary operator.
• Defines an essentially cylindrical manifold as a replacement for compact manifolds in the non-compact setting, where the manifold is cylindrical outside a compact set.
• Uses a cobordism construction by extending the manifold with a cylinder to relate the index of a boundary value problem to the spectral flow of a family of operators.
• Applies the APS boundary condition on the boundary components and uses spectral sections to define a continuous family of boundary conditions.
• Proves the index is constant along a continuous family of boundary conditions via isomorphism of domains, leading to the spectral flow formula.

Experimental results

Research questions

• RQ1 Can the Atiyah-Patodi-Singer index theorem be extended to manifolds with non-compact boundary without assuming bounded geometry or compactness?
• RQ2 Is there a well-defined invariant that captures the difference between eta-invariants of two operators on non-compact manifolds when the individual invariants are not defined?
• RQ3 How does the index of a boundary value problem on a non-compact manifold relate to the spectral flow of a family of operators on the boundary?
• RQ4 Can the index on a non-compact manifold be decomposed via a splitting theorem along a non-compact hypersurface?
• RQ5 What is the precise relationship between the relative eta-invariant and the spectral flow of a family of self-adjoint operators on the boundary?

Key findings

• The index of a strongly Callias-type operator with elliptic boundary condition on a complete odd-dimensional manifold with non-compact boundary is Fredholm.
• The index of the boundary value problem on a manifold with non-compact boundary equals the sum of the indices on two parts when the manifold is split along a non-compact hypersurface.
• The relative eta-invariant η(𝒜₁,𝒜₀) is well-defined even when η(𝒜₁) and η(𝒜₀) are not, and behaves as if it were their difference.
• The relative eta-invariant satisfies η(𝒜¹,𝒜⁰) = 2·sf(𝒜), where sf(𝒜) is the spectral flow of the family 𝒜.
• The index of the extended operator on the cobordism manifold is constant along the family, leading to the spectral flow formula.
• The spectral flow of a family of operators on the boundary equals half the difference of the relative eta-invariants, establishing a deep link between index theory and spectral invariants.