Boundary operators and the Atiyah–Patodi–Singer index

Let M be a compact oriented even-dimensional Riemannian spin manifold with boundary Y, with metric and twisting connection of product type on a collar. Orient Y by the outward-normal-first rule and choose an inward distance coordinate u≥0. Write D⁺=J(∂_u+B), where J is inward Clifford multiplication from S⁺ to S⁻ and B is the self-adjoint tangential operator on S⁺ restricted to Y. This factorization fixes B's sign; using an outward coordinate without changing the formulas would reverse it.

Use the domain of H¹ positive spinors whose H^(1/2) trace is annihilated by the spectral projection Π_(≥0)(B). Thus negative boundary modes are allowed, while zero and positive modes are forbidden. With this domain D⁺ to L² negative spinors is Fredholm, and its adjoint has the corresponding complementary boundary condition, not an independently imposed copy of the same one. The spectral projection is global along Y, which makes this a nonlocal elliptic boundary condition.

For the closed odd-dimensional boundary Dirac operator, define eta initially by summing sign(λ)|λ|^(−s) over nonzero eigenvalues for sufficiently large real part of s. The sum has meromorphic continuation and is regular at zero in this Dirac setting. Its value there is η(B), while h(B) is the ordinary finite dimension of the zero eigenspace. The reduced invariant is the real number (η+h)/2 for this index formula; taking it only modulo integers would lose the exact integer index.

With the declared APS convention, the twisted spin formula is the bulk integral of  times ch(E) minus reduced eta. Product geometry removes additional boundary transgression terms; a nonproduct metric or connection requires appropriate corrections or a deformation argument that accounts for them. Eta is not the difference of two separately finite counts of positive and negative modes. Most boundary spectra have infinitely many of each sign, and analytic continuation is essential even in a simple circle example.

If the boundary operator stays invertible along a smooth variation, its spectral projection varies continuously in the appropriate calculus. At a zero crossing, the strict negative-mode domain changes by a finite-dimensional space. The bulk and eta terms encode this change through spectral flow. Allowing zero modes by replacing Π_(≥0) with Π_(>0) enlarges the domain by h(B) dimensions and increases the index by that amount. The replacement formula therefore has η−h rather than η+h in the boundary numerator.

Reversing B alone sends η to its negative while keeping h unchanged, so reduced eta does not simply change sign when a kernel is present. Reversing the geometric orientation also changes chirality, bulk orientation and boundary conventions, which must be handled together. These observations matter when gluing manifolds or comparing formulas from different sources. They also explain why a formally attractive expression containing only the bulk integral is generally not an integer on a manifold with boundary.