03.09.24 · modern-geometry / spin-geometry

Eta invariant and Atiyah-Patodi-Singer index theorem

shipped3 tiersLean: none

Anchor (Master): Atiyah-Patodi-Singer 1975 *Spectral asymmetry and Riemannian geometry I* (Math. Proc. Camb. Phil. Soc. 77); APS 1976 II and III; Bismut-Cheeger 1989 *η-invariants and their adiabatic limits* (J. AMS 2); Bismut-Freed 1986 *The analysis of elliptic families I/II* (Comm. Math. Phys. 106 + 107); Melrose 1993 *The Atiyah-Patodi-Singer Index Theorem*; Berline-Getzler-Vergne Ch. 9

Intuition [Beginner]

A self-adjoint elliptic operator on a compact closed manifold has a discrete spectrum of real eigenvalues, with finitely many of any given size. Some are positive, some are negative, and the question of whether the positives outnumber the negatives is the question of spectral asymmetry. The eta invariant is the regularised numerical answer.

If you imagine the spectrum drawn on the real line, the positive and negative eigenvalues might look balanced at first glance, but a careful weighted count can detect a tilt. The weight is $1/∣ λ ∣^{s}$ for large real $s$ — heavy on small eigenvalues, light on large ones — and the tilt is the sum of $sign (λ) /∣ λ ∣^{s}$ . This sum converges for $s$ large and extends analytically to $s = 0$ , where it reads off the regularised count.

The eta invariant matters because it measures a piece of geometry that no ordinary integral over the manifold can see. On a closed manifold without boundary, every characteristic-class integral is symmetric in a strong sense and cannot detect spectral tilt. The eta invariant is the boundary-tied correction that appears when the manifold has a boundary, and it carries the parity of the operator's spectrum as a real number.

Visual [Beginner]

Picture the real line with the eigenvalues of a self-adjoint operator marked as dots: a few on the right (positive), a few on the left (negative), accumulating to infinity in both directions. The eta invariant is the regularised value of "positive count minus negative count", read off after weighting each dot by an inverse power of its absolute value.

When the spectrum is symmetric about zero, the eta invariant vanishes. When the spectrum tilts to one side — even by a small amount, even after only a careful regularised count — the eta invariant records the tilt.

Worked example [Beginner]

Take the simplest case: the unit circle $S^{1}$ of circumference $2 π$ , and the operator $D = - i d / d θ$ acting on smooth complex-valued functions. Its eigenvalues are the integers: the function $e^{in θ}$ has eigenvalue $n$ for every $n \in Z$ .

Step 1. List the spectrum: $\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots$ . The spectrum is symmetric about the origin, so every positive eigenvalue $n$ is paired with a negative eigenvalue $- n$ .

Step 2. The eta series, for large real $s$ , asks you to add $1/ n^{s}$ once for each positive integer $n$ (these contribute $+ 1$ to the sign) and to subtract $1/ n^{s}$ once for each positive integer $n$ (the absolute values of the negative eigenvalues, contributing $- 1$ to the sign). The two halves cancel term by term, so the eta series is zero for every large real $s$ . By analytic continuation, the eta invariant is $η_{D} (0) = 0$ .

Step 3. Now twist by a complex line bundle with a flat $U (1)$ -connection of holonomy $e^{2 π i α}$ , with $α$ in the open interval $(0, 1)$ . The new operator is $D_{α} = - i d / d θ + α$ , with eigenvalues $n + α$ for $n \in Z$ . The spectrum is shifted: now every eigenvalue is positive when $n \geq 0$ (since $n + α > 0$ ) and negative when $n \leq - 1$ (since $- 1 + α < 0$ ).

Step 4. The eta series adds $1/ (n + α)^{s}$ for each $n \geq 0$ (the positive eigenvalues) and subtracts $1/ (n - α)^{s}$ for each $n \geq 1$ (the absolute values of the negative eigenvalues). Each of the two collected expressions is a Hurwitz zeta function. Using the special value of the Hurwitz zeta function at $s = 0$ , the answer at $s = 0$ is $η_{D_{α}} (0) = 1 - 2 α$ .

What this tells us: a symmetric spectrum gives eta zero. A spectrum shifted by an amount $α$ gives an eta that depends linearly on $α$ , taking values in the open interval $(- 1, 1)$ as $α$ sweeps $(0, 1)$ . The eta invariant records the asymmetry of the spectrum after regularisation, and it is sensitive to the flat-bundle holonomy that produced the shift.

Check your understanding [Beginner]

Exercise (easy, multiple choice).

In the Atiyah-Patodi-Singer index formula for a Dirac operator on a manifold with boundary, what role does the eta invariant of the boundary operator play?

A. It is the bulk integral of the index density
B. It is the boundary correction term added to the bulk integral
C. It is the dimension of the kernel of the bulk operator
D. It is unrelated to the boundary

Hint

The APS theorem replaces the closed-manifold index formula by a bulk integral plus a boundary correction.

Answer

B. The APS formula equals the bulk characteristic-class integral over $X$ minus a boundary correction equal to one-half of $η (D ∣_{Y}) + h$ , where $Y$ is the boundary, $D ∣_{Y}$ is the boundary Dirac operator, and $h = dim ker (D ∣_{Y})$ . The eta invariant of the boundary Dirac operator is exactly the boundary correction.

Formal definition [Intermediate+]

Eta series and eta invariant. Let $X$ be a compact closed Riemannian manifold and let $D$ be a self-adjoint elliptic differential operator of positive order acting on sections of a Hermitian vector bundle $E \to X$ . The spectrum $Spec (D) \subset R$ is discrete, real, and bi-infinite (accumulating to $\pm \infty$ in general). For complex $s$ with $ℜ s$ sufficiently large, define the eta series $$ \eta_D(s) := \sum_{\lambda \in \mathrm{Spec}(D), \lambda \neq 0} \mathrm{sign}(\lambda) , |\lambda|^{-s}. $$ The sum is taken with multiplicities. Convergence for $ℜ s > dim X / ord (D)$ follows from Weyl's eigenvalue counting law, which gives $∣ λ_{k} ∣ \sim c k^{ord (D) / d i m X}$ for the $k$ -th eigenvalue ^{[Atiyah-Patodi-Singer 1975 I]}.

Theorem (meromorphic continuation, APS 1975 I §2). The function $η_{D} (s)$ extends meromorphically to all of $C$ , with only simple poles, and $s = 0$ is not a pole. The eta invariant of $D$ is $$ \eta(D) := \eta_D(0). $$

Some authors include the kernel contribution and define $η (D) + h$ where $h = dim ker D$ ; we follow the original APS convention where the kernel is tracked separately. The combination $ξ (D) := \frac{1}{2} (η (D) + h)$ is the reduced eta invariant, the natural object in the APS boundary correction.

Definition (Dirac-type operator in product form on a collar). Let $X$ be a compact Riemannian manifold with boundary $\partial X = Y$ and assume a collar neighbourhood $[0, 1) \times Y ↪ X$ of $\partial X$ in which the metric is the product $d u^{2} + g_{Y}$ for the coordinate $u \in [0, 1)$ and a fixed metric $g_{Y}$ on $Y$ . A Dirac-type operator $D : Γ (E) \to Γ (E)$ is in product form on the collar if there exist a bundle isomorphism $E ∣_{[0, 1) \times Y} ≅ [0, 1) \times (E ∣_{Y})$ and a self-adjoint Dirac-type operator $A : Γ (E ∣_{Y}) \to Γ (E ∣_{Y})$ on $Y$ such that on the collar $$ D = \sigma \left( \frac{\partial}{\partial u} + A \right), $$ with $σ$ a bundle isomorphism intertwining Clifford multiplication by $d u$ . The operator $A$ is the boundary operator associated to $D$ ^{[Atiyah-Patodi-Singer 1975 I §2]}.

APS boundary condition. Let $Π_{\geq 0}$ be the orthogonal projection of $L^{2} (\partial X, E ∣_{\partial X})$ onto the non-negative-spectrum closed subspace $$ L^2_{\geq 0}(\partial X) := \overline{\mathrm{span}{\varphi_\lambda : \lambda \in \mathrm{Spec}(A), \lambda \geq 0}}, $$ where $φ_{λ}$ is an eigensection. The APS domain of $D$ is $$ \mathrm{Dom}{\mathrm{APS}}(D) := { u \in H^1(X, E) : \Pi{\geq 0}(u|{\partial X}) = 0 }. $$ The condition $\Pi{\geq 0}(u|_{\partial X}) = 0 $m e an s t h e b o u n d a r y v a l u eo f$ u$ has no non-negative-spectrum component — it is a global, non-local boundary condition.

Counterexamples to common slips.

The APS boundary condition is global in the sense that it projects onto a spectral subspace of the boundary operator. Standard local boundary conditions (Dirichlet, Neumann, Robin) do not give a Fredholm Dirac operator in even dimensions; APS conditions do.
The eta invariant $η (D)$ on a closed manifold is in general a real number — not an integer. This is in stark contrast to the closed-manifold Atiyah-Singer index, which is integer-valued. Combined with the bulk integral and the kernel correction $h$ , the APS formula does produce an integer.
The eta invariant is not a local invariant of the metric. Two metrics that differ in the interior of $X$ give the same bulk integral $\int_{X} A ch (E)$ but different eta invariants, because eta encodes the boundary spectrum globally. By contrast, the modulo- $Z$ reduction of $η$ is a topological invariant.
The product-form assumption on the collar is essential; without it the proof of the APS theorem requires an additional gluing argument. The cohomological content of the theorem is unaffected, but the integration-by-parts manipulations break down without product geometry near the boundary.

Key theorem with proof [Intermediate+]

Theorem (Atiyah-Patodi-Singer 1975 I). Let $X$ be a compact even-dimensional Riemannian spin manifold with boundary $Y = \partial X$ , and let $E \to X$ be a Hermitian vector bundle with metric connection. Assume the Riemannian metric on $X$ and the connection on $E$ are products on a collar $[0, 1) \times Y$ of $Y$ , and let $D = D^{+} : Γ (S^{+} \otimes E) \to Γ (S^{-} \otimes E)$ be the twisted Dirac operator on $X$ . The operator $D$ equipped with APS boundary conditions is Fredholm, and $$ \mathrm{Ind}(D, \mathrm{APS}) = \int_X \widehat A(X) \wedge \mathrm{ch}(E) - \frac{\eta(A) + h}{2}, $$ where $A$ is the boundary Dirac operator, $h = dim ker A$ , and $\widehat A(X) \in \Omega^(X) $,$ \mathrm{ch}(E) \in \Omega^(X) $a r e t h e$ \widehat A$-genus and twisting Chern character of the Levi-Civita and bundle connections respectively ^{[Atiyah-Patodi-Singer 1975 I §3]}.

Proof. The argument follows the heat-kernel route of Atiyah-Bott-Patodi adapted to manifolds-with-boundary. Five steps:

Step 1: Fredholmness with APS conditions. The APS boundary condition pairs with the elliptic interior to produce a Fredholm operator on $H^{1} (X, S^{+} \otimes E)$ . The key analytic input is that on the collar $[0, 1) \times Y$ the operator $D = σ (\partial_{u} + A)$ admits an explicit solution operator using Fourier decomposition along $Y$ in the eigenbasis of $A$ . A boundary value $u_{0} \in L^{2} (Y, S \otimes E)$ extends to a $u \in H^{1}$ on the collar if and only if its non-negative-spectrum components are absent, because the positive-eigenvalue Fourier modes grow exponentially into the interior. The APS condition $Π_{\geq 0} (u ∣_{Y}) = 0$ exactly excludes those modes; the resulting domain is Fredholm by an Atkinson-type argument using the parametrix from 03.09.22.

Step 2: McKean-Singer for manifolds with boundary. On $X$ with APS conditions, the heat operators $e^{- t D^{*} D}$ and $e^{- t D D^{*}}$ are trace-class for $t > 0$ , and the supertrace $$ \mathrm{Str}(e^{-t \mathcal{D}^2}) := \mathrm{Tr}(e^{-tD^D}|_{S^+ \otimes E}) - \mathrm{Tr}(e^{-tDD^}|_{S^- \otimes E}) $$ is constant in $t$ in the closed-manifold case. On the manifold with boundary, the McKean-Singer formula picks up a boundary correction. The limit $t \to + \infty$ still computes the index of $D$ with APS conditions. The limit $t \to 0^{+}$ is no longer the bulk integral alone — a boundary heat-flux term survives.

Step 3: small- $t$ asymptotic expansion with boundary. The heat kernel on $X$ has an asymptotic expansion $$ \mathrm{Str}(e^{-t \mathcal{D}^2}) \sim \int_X a_n(x) , \mathrm{vol}_X + \int_Y b_n(y) , \mathrm{vol}_Y , t^{-1/2} + \mathrm{(lower)}, \qquad t \to 0^+, $$ where the interior density $a_{n} (x)$ is the local index density $A \land ch (E)$ by Getzler rescaling (the same calculation as the closed-manifold case from 03.09.20), and the boundary density $b_{n} (y)$ is a local invariant of the boundary geometry computable from the boundary operator $A$ alone. The leading boundary contribution comes from the heat kernel of the half-line problem $\partial_{u}^{2} + A^{2}$ on $[0, \infty) \times Y$ with APS conditions.

Step 4: boundary contribution as eta plus kernel. The boundary heat-flux term, integrated from $t = 0$ to $t = \infty$ using the Mellin transform, identifies as $$ \int_0^\infty \frac{1}{\sqrt{4\pi t}}, \mathrm{Tr}(A , e^{-tA^2}) , dt = \frac{1}{\sqrt{\pi}} \int_0^\infty \mathrm{Tr}(A , e^{-tA^2}) , \frac{dt}{\sqrt{t}}. $$ The Mellin transform of $Tr (A e^{- t A^{2}})$ at the right exponent recovers the eta series of $A$ : $$ \eta_A(s) = \frac{1}{\Gamma((s+1)/2)} \int_0^\infty t^{(s-1)/2} , \mathrm{Tr}(A , e^{-tA^2}) , dt. $$ Evaluating at $s = 0$ gives $η (A)$ , and adding the kernel contribution $h = dim ker A$ produces the reduced combination $η (A) + h$ . The factor of $\frac{1}{2}$ comes from the half-line geometry: only half of the full-line spectral contribution feeds the boundary correction.

Step 5: assembling the formula. The McKean-Singer constancy of the supertrace, applied between the $t \to 0^{+}$ small-time expansion and the $t \to + \infty$ index limit, gives $$ \mathrm{Ind}(D, \mathrm{APS}) = \lim_{t \to +\infty} \mathrm{Str}(e^{-t \mathcal{D}^2}) = \int_X \widehat A(X) \wedge \mathrm{ch}(E) - \frac{\eta(A) + h}{2}. $$ The sign of the boundary correction is fixed by the orientation convention; the factor $\frac{1}{2}$ is forced by the half-line geometry of the collar. $□$

Bridge. The Atiyah-Patodi-Singer index theorem builds toward the entire boundary-corrected index theory and identifies the eta invariant as the canonical boundary correction. The foundational reason it works is exactly that the heat-kernel proof of 03.09.10 (Atiyah-Singer on closed manifolds) localises to the diagonal, and on a manifold-with-boundary the diagonal localisation picks up a boundary heat-flux term whose Mellin transform is precisely the eta series of the boundary operator. This is exactly the spectral-asymmetry input the closed-manifold story cannot see: the eta invariant generalises the closed-manifold integer index to a real number that captures the parity of the boundary spectrum. The central insight is that putting these together — bulk integral plus boundary eta — restores the integer index of the bulk Dirac operator under APS conditions; the bulk integral alone need not be an integer, the eta correction restores integrality. The bridge is the recognition that eta is not an auxiliary regulariser but the exact boundary contribution to the heat-kernel proof, and this same eta-form pattern appears again in 03.09.23 (Bismut superconnection) as the family eta-form of Bismut-Cheeger, the higher analogue identifying the family APS correction with a transgression of the superconnection Chern character.

Exercises [Intermediate+]

Exercise 4 (medium, symbolic).

Express the eta invariant as a Mellin transform of the heat-kernel quantity $Tr (A e^{- t A^{2}})$ and identify the value at $s = 0$ .

Hint

The Mellin transform $\int_{0}^{\infty} t^{s - 1} f (t) d t$ converts a heat-kernel decay into a Dirichlet-type series in $s$ . Use the integral representation $∣ λ ∣^{- s} = \frac{1}{Γ ( s )} \int_{0}^{\infty} t^{s - 1} e^{- t ∣ λ ∣} d t$ .

Answer

Using $∣ λ ∣^{- s} = \frac{1}{Γ ( s /2 )} \int_{0}^{\infty} t^{s /2 - 1} e^{- t λ^{2}} d t$ for $ℜ s > 0$ , $$ \mathrm{sign}(\lambda) |\lambda|^{-s} = \frac{1}{\Gamma((s+1)/2)} \int_0^\infty t^{(s-1)/2} \lambda , e^{-t \lambda^2} , dt. $$ Summing over eigenvalues (justified by exponential decay for $t > 0$ ): $$ \eta_A(s) = \frac{1}{\Gamma((s+1)/2)} \int_0^\infty t^{(s-1)/2} , \mathrm{Tr}(A , e^{-tA^2}) , dt. $$ The value at $s = 0$ is $$ \eta(A) = \frac{1}{\sqrt{\pi}} \int_0^\infty t^{-1/2} , \mathrm{Tr}(A , e^{-tA^2}) , dt, $$ provided the integral converges, which holds when the small- $t$ asymptotics of $Tr (A e^{- t A^{2}})$ has no $t^{- 1/2}$ pole (an APS-specific local invariant-theory cancellation, established in 1975 I §2). Rubric: full credit for the Mellin formula and identification of $η (0)$ as the regularised integral.

Exercise 5 (medium, short-answer).

Apply the APS index theorem to the Dirac operator on the disk $D^{2} = {∣ z ∣ \leq 1}$ with the round metric, twisted by the rank-one identity bundle. The boundary is the unit circle $S^{1}$ .

Hint

In two dimensions $A = 1$ and $ch (E) = 1$ for a flat identity twisting bundle, so the bulk integral reduces to the integral of $1$ over the disk, which is its volume.

Answer

Two dimensions are odd-dimensional from the spin point of view in the sense that the boundary Dirac operator is the one we computed in the worked example. The bulk integral $\int_{D^{2}} A \land ch (E)$ for a flat product metric with identity twisting reduces — after correctly accounting for the spin structure on the disk — to a vanishing form-degree contribution at top degree two only if $A$ has a degree-two piece. In this dimension $A = 1 + O (curvature^{2})$ , so the bulk integral is zero for the flat metric. The boundary operator is the Dirac operator on $S^{1}$ with the spin structure inherited from the bounding disk (the bounding spin structure on $S^{1}$ , distinct from the periodic one). Its eta invariant is zero by the worked example. The kernel of the boundary operator on the bounding spin structure is zero (the spectrum is $Z + 1/2$ ). So $Ind (D, APS) = 0 - 0 = 0$ . Rubric: full credit for the bulk + eta + kernel accounting; partial credit for any one piece.

Exercise 6 (medium, short-answer).

Explain why a local boundary condition (e.g., Dirichlet $u ∣_{Y} = 0$ ) does not produce a Fredholm Dirac operator in even dimensions, but APS conditions do.

Hint

Compute the index of the half-plane Dirac operator with Dirichlet conditions versus APS conditions. The Dirichlet case has infinite-dimensional kernel by Fourier decomposition.

Answer

On the half-line $[0, \infty) \times Y$ with product geometry, Dirac in product form is $D = σ (\partial_{u} + A)$ . Fourier-decompose along $Y$ in the eigenbasis ${φ_{λ}}$ of $A$ . A section $u (u, y) = \sum_{λ} c_{λ} (u) φ_{λ} (y)$ satisfies $D u = 0$ iff each $c_{λ}$ solves $c_{λ}^{'} (u) + λ c_{λ} (u) = 0$ , i.e., $c_{λ} (u) = c_{λ} (0) e^{- λ u}$ . The $L^{2}$ condition on the half-line requires $λ > 0$ , so the kernel of the half-line Dirac operator is the span of ${e^{- λ u} φ_{λ} : λ > 0}$ . Imposing Dirichlet $u (0, y) = 0$ forces $c_{λ} (0) = 0$ for all $λ$ , killing the kernel — but the cokernel (computed by the dual problem) becomes infinite-dimensional, and the operator is not Fredholm. APS conditions $Π_{\geq 0} (u (0, y)) = 0$ project out exactly the positive-spectrum modes, matching the half-line $L^{2}$ growth condition, and produce a Fredholm domain by Atkinson's theorem. Rubric: full credit for the Fourier decomposition and the contrast.

Exercise 7 (hard, short-answer).

Show that for the operator $D_{α} = - i d / d θ + α$ on $S^{1}$ with $α \in (0, 1)$ , the eta invariant $η (D_{α}) = 1 - 2 α$ via Hurwitz zeta values.

Hint

The Hurwitz zeta function $ζ (s, a) = \sum_{n \geq 0} (n + a)^{- s}$ extends meromorphically to $C$ , with $ζ (0, a) = 1/2 - a$ .

Answer

The spectrum of $D_{α}$ is ${n + α : n \in Z}$ . For $α \in (0, 1)$ , eigenvalues $n + α$ are positive for $n \geq 0$ and negative for $n \leq - 1$ . For $ℜ s$ large, $$ \eta_{D_\alpha}(s) = \sum_{n \geq 0} (n+\alpha)^{-s} - \sum_{n \geq 1} (n - \alpha)^{-s} = \zeta(s, \alpha) - \zeta(s, 1 - \alpha). $$ Using the special value $ζ (0, a) = 1/2 - a$ (Hurwitz): $$ \eta(D_\alpha) = \zeta(0, \alpha) - \zeta(0, 1 - \alpha) = (1/2 - \alpha) - (1/2 - (1 - \alpha)) = 1 - 2\alpha. $$ Rubric: full credit for the Hurwitz zeta identification and the evaluation. Partial credit for the symmetric setup with sign-tracking.

Exercise 8 (hard, short-answer).

The signature theorem of Hirzebruch states that for a closed oriented $4 k$ -manifold $M$ , the signature is $sig (M) = \int_{M} L (M)$ where $L$ is the Hirzebruch $L$ -polynomial. State the corresponding APS signature theorem for a manifold with boundary, identifying the boundary correction.

Hint

The signature operator is the chirality-graded $d + d^{*}$ on even forms minus odd forms. Its boundary operator at the boundary $Y$ is a self-adjoint operator on $Y$ , and the APS theorem applies.

Answer

For a compact oriented Riemannian $(4 k)$ -manifold $X$ with boundary $Y$ , product metric on a collar of $Y$ , the signature operator $D_{sig}$ on $X$ with APS boundary conditions has index equal to the signature of $X$ , and $$ \mathrm{sig}(X) = \int_X L(X) - \frac{\eta(A_{\mathrm{sig}}) + h}{2}, $$ where $A_{sig}$ is the boundary signature operator on $Y$ . The eta invariant of $A_{sig}$ on $Y$ records the spectral asymmetry of the boundary signature, and the formula identifies the deviation $sig (X) - \int_{X} L (X)$ — which is in general non-zero on a manifold with boundary — as exactly the eta correction. This is the case treated in detail in APS 1975 I §4 and 1976 II for flat-bundle twists. Rubric: full credit for the precise formula with $η$ and $h$ correctly placed; partial credit for the bulk-plus-eta structure.

Lean formalization [Intermediate+]

Mathlib does not yet ship the analytic infrastructure required to state the APS theorem. A schematic of the eventual formalisation reads:

import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.Zeta
import Mathlib.Geometry.Manifold.VectorBundle.Basic

/-- The eta series of a self-adjoint elliptic operator. -/
noncomputable def etaSeries
    {X : Type*} [Manifold X] (D : SelfAdjointElliptic X) (s : ℂ) : ℂ :=
  ∑' (lambda : D.spectrum.Nonzero),
    (Real.sign lambda.val) * (Complex.abs lambda.val) ^ (-s)
  -- convergence for Re s large via Weyl's law; meromorphic continuation
  -- via Mellin transform of the heat kernel

/-- The eta invariant: value of the meromorphic continuation at s = 0. -/
noncomputable def etaInvariant
    {X : Type*} [Manifold X] (D : SelfAdjointElliptic X) : ℝ :=
  Complex.re (etaSeries D 0)

/-- The Atiyah-Patodi-Singer index theorem for a Dirac-type operator
    in product form on a collar. -/
theorem aps_index
    {X : ManifoldWithBoundary} (D : DiracTypeProductForm X)
    (E : HermitianBundle X) :
    fredholmIndexAPS D =
      (∫ x in X, ahatGenusForm X x ∧ chernCharacter E x) -
      (etaInvariant D.boundaryOperator + Real.ofNat
        (Nat.card (kernel D.boundaryOperator))) / 2 :=
  sorry  -- heat-kernel proof: collar product form, boundary heat-flux,
         -- Mellin transform identifying eta as boundary correction

The required upstream contributions to Mathlib include: spectral theory of self-adjoint elliptic operators on a closed manifold (eigenvalue counting law and meromorphic continuation of zeta-type series); manifold-with-boundary infrastructure with product collars; the APS spectral projection $Π_{\geq 0}$ as a Fredholm boundary condition; heat-kernel infrastructure on manifolds with boundary; and the Mellin transform identification of eta with a heat-kernel integral. Each piece is formalisable in isolation but the integrated APS statement is several layers of dependency away from current Mathlib.

Advanced results [Master]

Theorem (Atiyah-Patodi-Singer 1975 I, full form). Let $X$ be a compact even-dimensional Riemannian spin manifold with boundary $Y$ , equipped with a product metric on a collar of $Y$ , and let $E \to X$ be a Hermitian vector bundle with a connection that is a product on the collar. Let $D : Γ (S^{+} \otimes E) \to Γ (S^{-} \otimes E)$ be the twisted Dirac operator on $X$ , and let $A$ be its boundary Dirac operator on $Y$ . The operator $D$ equipped with APS boundary conditions $Π_{\geq 0} (u ∣_{Y}) = 0$ is Fredholm, and $$ \mathrm{Ind}(D, \mathrm{APS}) = \int_X \widehat A(X) \wedge \mathrm{ch}(E) - \frac{\eta(A) + h(A)}{2}, $$ where $h (A) = dim ker (A)$ .

The bulk integral is the same characteristic-class integrand as in the closed-manifold Atiyah-Singer theorem (03.09.10). The boundary correction $\frac{1}{2} (η (A) + h (A))$ is the contribution forced by the non-empty boundary. The total expression is an integer (the index of a Fredholm operator), but neither summand is integer-valued in general: the bulk integral can be any real number, and so can the eta correction, but their difference is forced to be integer-valued.

Theorem (regularity of eta at $s = 0$ ). For a self-adjoint Dirac-type operator $A$ on a compact closed odd-dimensional manifold $Y$ , the function $η_{A} (s)$ is holomorphic at $s = 0$ . The value $η (A) = η_{A} (0)$ is a well-defined real number.

The proof rests on the small-time heat-kernel asymptotic $$ \mathrm{Tr}(A , e^{-tA^2}) \sim \sum_{j \geq 0} a_j(A) , t^{(j - \dim Y)/2}, \qquad t \to 0^+. $$ A potential pole of $η_{A} (s)$ at $s = 0$ would arise from a non-vanishing $t^{- 1/2}$ coefficient $a_{d i m Y - 1} (A)$ . The APS local-invariant-theory cancellation (1975 I §2 and the Bismut-Freed refinement of 1986) shows that the coefficient $a_{d i m Y - 1} (A)$ , which is locally computable from the geometry near a point, vanishes pointwise. The vanishing uses a clever symmetry argument: under reversal of orientation, the local invariant changes sign, but it must be invariant up to sign under reflection — hence zero.

Theorem (APS 1975 I §2, meromorphic continuation). For any self-adjoint elliptic operator $D$ of positive order on a compact closed manifold, $η_{D} (s)$ extends meromorphically to $C$ , with poles only at the points $s = (dim X - j) / ord (D)$ for $j \in Z_{\geq 0}$ , all simple. The residues are local geometric invariants computable from the symbol of $D$ and finitely many of its covariant derivatives. The point $s = 0$ is never a pole for Dirac-type operators on an odd-dimensional manifold.

Theorem (Bismut-Cheeger 1989 — eta-form and adiabatic limit). Let $π : M \to B$ be a smooth fibration with closed odd-dimensional fibres equipped with a family of self-adjoint Dirac-type operators $A = {A_{b}}$ . The family eta-form $η \in Ω^{odd} (B)$ is $$ \widehat \eta := \frac{1}{\sqrt{\pi}} \int_0^\infty \mathrm{Str}_{\mathrm{vert}} !\left( \frac{d \mathbb{A}t}{dt} , e^{-\mathbb{A}t^2} \right) dt, $$ where $A_{t}$ is the Bismut superconnection of the family. The form is closed in $\Omega^(B) $, an d i t s d e g r ee - z er oco m p o n e n t r eco v er s t h e f ib r e w i see t ain v a r ian t :$ \widehat \eta_{[0]}(b) = \eta(A_b) $. I n t h e a d iaba t i c l imi t$ g{M/B} \to \varepsilon , g{M/B} $a s$ \varepsilon \to 0$, the bulk Bismut superconnection separates into a horizontal piece (computing the index density of a fictitious bulk operator) and a vertical piece (whose eta-form is the APS-style correction to the family index).*

The eta-form is the family analogue of the eta invariant, and its construction unifies the APS boundary correction (single operator) with the Bismut superconnection framework (03.09.23) for families. Bismut-Cheeger 1989 used this construction to prove the family-APS theorem and to identify the adiabatic-limit behaviour of family-index forms.

Theorem (APS 1975 I §3, additivity under gluing). Let $X_{1}$ and $X_{2}$ be two compact even-dimensional manifolds with isomorphic boundaries $\partial X_{1} ≅ \partial X_{2} = Y$ , and let $X = X_{1} \cup_{Y} X_{2}$ be the closed manifold obtained by gluing along the common boundary. Then $$ \mathrm{Ind}(D|X) = \mathrm{Ind}(D|{X_1}, \mathrm{APS}) + \mathrm{Ind}(D|_{X_2}, \mathrm{APS}) + h(A), $$ where $A = D ∣_{Y}$ and $h (A) = dim ker A$ .

The additivity formula expresses the closed-manifold index as the sum of APS indices on the two pieces plus a kernel correction. The eta invariants $η (A)$ on each side cancel by orientation reversal: $η (A ∣_{X_{1}}) + η (- A ∣_{X_{2}}) = 0$ . This is the fundamental gluing principle for APS.

Theorem (Chern-Simons interpretation, APS 1975 I §6 + 1976 II). Let $Y$ be a closed odd-dimensional Riemannian manifold and let $A_{0}, A_{1}$ be two flat $U (1)$ -connections on $Y$ . The difference of eta invariants of the twisted Dirac operators $$ \eta(D_{A_1}) - \eta(D_{A_0}) \pmod{\mathbb{Z}} $$ is the Chern-Simons functional $CS (A_{1}) - CS (A_{0})$ modulo integers, where $CS$ is the Chern-Simons three-form integrated over $Y$ (in dimension $3$ , with analogous higher-degree forms in higher odd dimensions).

This is the foundational link between the eta invariant and Chern-Simons gauge theory. On a closed odd-dimensional manifold, $η (mod Z)$ is a topological invariant of the underlying spin structure and the gauge equivalence class of the connection. The Chern-Simons functional and the eta invariant agree modulo integers and define the same anomaly in gauge theory; this is the origin of the term "Chern-Simons phase" for $e^{iπ η}$ in topological field theory.

Synthesis. The eta invariant is the foundational reason that index theory on manifolds with boundary requires a real-valued correction term, and the central insight is exactly that the closed-manifold Atiyah-Singer integrand $A \land ch (E)$ , while perfectly defined locally, fails to integrate to an integer on a manifold with boundary; the eta invariant $\frac{1}{2} (η (A) + h (A))$ is precisely the correction needed to restore integrality, identifying analysis with topology at the boundary. Putting these together, the APS theorem decomposes the boundary index into three pieces — bulk characteristic-class density, eta spectral asymmetry, and boundary kernel dimension — each computable from local data on $X$ or spectral data on $\partial X$ , with the sum forced to be an integer by the Fredholm index. This is exactly the structural pattern that appears again in 03.09.23 (Bismut superconnection) as the family eta-form, the higher analogue identifying the family APS correction with a transgression of the superconnection Chern character, and the same pattern recurs in the Chern-Simons interpretation of $η (mod Z)$ : putting these together, the eta invariant is the spectral shadow of the Chern-Simons functional, and the APS theorem identifies index theory on manifolds-with-boundary with bulk topology plus a boundary Chern-Simons phase. The bridge is the recognition that eta interpolates between two domains — analysis (spectral asymmetry of a self-adjoint operator) and topology (Chern-Simons / secondary characteristic class) — and the modulo-integer reduction is exactly where the two pictures meet.

Full proof set [Master]

Proposition (meromorphic continuation of $η_{D} (s)$ , APS 1975 I §2). Let $D$ be a self-adjoint elliptic operator of positive order $m$ on a compact closed manifold $X$ of dimension $n$ , acting on a Hermitian vector bundle $E$ . Then $η_{D} (s)$ has a meromorphic continuation to $C$ with simple poles only at $s = (n - j) / m$ for $j \in Z_{\geq 0}$ , and is holomorphic at $s = 0$ when $D$ is Dirac-type on an odd-dimensional manifold.

Proof. Express $η_{D} (s)$ as a Mellin transform of a heat-kernel quantity. For $ℜ s > 0$ and $λ > 0$ , $$ \lambda^{-s} = \frac{1}{\Gamma(s/2)} \int_0^\infty t^{s/2 - 1} e^{-t \lambda^2} , dt. $$ Hence $$ \mathrm{sign}(\lambda) |\lambda|^{-s} = \frac{1}{\Gamma((s+1)/2)} \int_0^\infty t^{(s-1)/2} \lambda , e^{-t \lambda^2} , dt $$ (the factor of $λ$ in the integrand absorbs the sign). Summing over the non-zero spectrum and using uniform decay of the heat trace, $$ \eta_D(s) = \frac{1}{\Gamma((s+1)/2)} \int_0^\infty t^{(s-1)/2} , \mathrm{Tr}(D , e^{-t D^2}) , dt, $$ where the integral is taken with the kernel contribution removed (or equivalently $D$ replaced by the restriction to the orthogonal complement of $ker D$ ).

The heat-kernel small-time asymptotic $$ \mathrm{Tr}(D , e^{-t D^2}) \sim \sum_{j \geq 0} a_j(D) , t^{(j - n)/m - 1/2}, \qquad t \to 0^+, $$ provides the local input. Split the integral at $t = 1$ : $$ \eta_D(s) = \frac{1}{\Gamma((s+1)/2)} \left( \int_0^1 t^{(s-1)/2} \mathrm{Tr}(D , e^{-t D^2}) , dt + \int_1^\infty t^{(s-1)/2} \mathrm{Tr}(D , e^{-t D^2}) , dt \right). $$ The second integral is entire in $s$ because the heat trace decays exponentially as $t \to + \infty$ . The first integral is analysed by substituting the asymptotic expansion: for each term $a_{j} (D) t^{(j - n) / m - 1/2}$ , the integral $\int_{0}^{1} t^{(s - 1) /2 + (j - n) / m - 1/2} d t$ converges for $ℜ s > (n - j) / m$ and equals $1/ (s /2 + (j - n) / m)$ , contributing a simple pole at $s = (n - j) / m$ . The full eta function is the sum of these contributions (after multiplication by the entire $1/Γ ((s + 1) /2)$ ), giving the asserted meromorphic structure.

For Dirac-type $D$ on odd-dimensional $X$ , the pole at $s = 0$ would arise from $j = n$ , i.e., the coefficient $a_{n} (D)$ . The APS local-invariant-theory cancellation shows $a_{n} (D) = 0$ pointwise: by Getzler-style symbol-rescaling adapted to the Dirac case in odd dimensions, the local index density of an odd-dimensional Dirac operator vanishes (the Atiyah-Singer integrand $A \land ch (E)$ is even-degree and integrates to zero on an odd-dimensional manifold). Hence $η_{D}$ is holomorphic at $s = 0$ and $η (D) := η_{D} (0)$ is well-defined. $□$

Proposition (Fredholmness under APS conditions). Let $X$ be a compact manifold with boundary $Y$ , product collar metric, and $D$ a Dirac-type operator in product form $σ (\partial_{u} + A)$ on the collar. The operator $D$ with domain $Dom_{APS} (D)$ is Fredholm $H^{1} \to L^{2}$ .

Proof sketch. Set up a parametrix for $D$ on $X$ in three pieces: an interior parametrix (from the elliptic interior, using the closed-manifold parametrix construction of 03.09.22), a boundary parametrix on the collar using the explicit Fourier decomposition along $Y$ in the spectral basis of $A$ , and a partition-of-unity assembly. The interior parametrix gives a Fredholm operator on the interior at the cost of compactly supported correction terms. The boundary parametrix solves the half-line problem $\partial_{u} + A$ exactly: on the collar, $D u = f$ has a unique solution in $L^{2} (R_{+} \times Y)$ with APS conditions for any $f$ , given by Fourier integration in the eigenbasis of $A$ . The two parametrices glue via cut-off functions; the gluing error is compact (smoothing). Atkinson's theorem then identifies $D$ as Fredholm.

The Fredholm index is finite because both kernel and cokernel are finite-dimensional: the kernel consists of $H^{1}$ sections of $S^{+} \otimes E$ that solve $D u = 0$ with $Π_{\geq 0} (u ∣_{Y}) = 0$ , and on a compact manifold the solution space is finite-dimensional by elliptic regularity; the cokernel is identified via the dual problem with a corresponding finite-dimensional space. $□$

Proposition (heat-kernel identification of the boundary contribution). On a manifold with boundary in product form on the collar, the McKean-Singer supertrace satisfies the small- $t$ asymptotic $$ \mathrm{Str}(e^{-t \mathcal{D}^2}) \sim \int_X \widehat A(X) \wedge \mathrm{ch}(E) - \frac{\eta(A) + h(A)}{2} + O(t^{1/2}), \qquad t \to 0^+, $$ and the McKean-Singer index identity gives $Ind (D, APS) = lim_{t \to + \infty} Str (e^{- t D^{2}}) = lim_{t \to 0^{+}} Str (e^{- t D^{2}})$ .

Proof. Decompose the heat kernel $e^{- t D^{2}}$ near the diagonal into an interior contribution and a boundary contribution. The interior contribution, computed by Getzler rescaling exactly as in the closed-manifold case (03.09.20), gives the bulk integral $\int_{X} A \land ch (E)$ . The boundary contribution is computed by the half-line model $\partial_{u}^{2} + A^{2}$ on $[0, \infty) \times Y$ with APS conditions. The model heat kernel is $$ K_t(u, u'; y, y') = \frac{1}{\sqrt{4\pi t}} \left( e^{-(u-u')^2 / 4t} - \sum_{\lambda \geq 0} \cdots \right) $$ with the reflection-and-spectral-projection term explicitly computable in the eigenbasis of $A$ .

The boundary supertrace at small $t$ , after Mellin transformation, identifies with $$ -\frac{1}{2\sqrt{\pi}} \int_0^\infty t^{-1/2} \mathrm{Tr}(A , e^{-tA^2}) , dt = -\frac{\eta(A) + h(A)}{2}, $$ where the factor of $\frac{1}{2}$ is the half-line measure factor, the sign is fixed by the chirality grading, and the $h (A)$ correction comes from the contribution of zero modes of $A$ to the boundary heat kernel (these zero modes are not detected by the $Tr (A e^{- t A^{2}})$ integrand because $A$ kills them, but they contribute to the boundary projection).

Combining the interior and boundary contributions gives the small- $t$ asymptotic; the McKean-Singer constancy of the supertrace under APS conditions (Step 1 of the Key Theorem proof) identifies the small- $t$ limit with the index. $□$

Proposition (eta computation on $S^{1}$ with flat $U (1)$ twist). Let $Y = S^{1}$ with circumference $2 π$ and let $D_{α} = - i d / d θ + α$ for $α \in (0, 1)$ . Then $η (D_{α}) = 1 - 2 α$ .

Proof. The spectrum of $D_{α}$ is ${n + α : n \in Z}$ . Every eigenvalue is non-zero for $α \in (0, 1)$ . Positive eigenvalues are ${n + α : n \geq 0}$ and negative eigenvalues are ${n + α : n \leq - 1}$ . The eta series for $ℜ s$ large is $$ \eta_{D_\alpha}(s) = \sum_{n \geq 0} (n + \alpha)^{-s} - \sum_{n \geq 1} (n - \alpha)^{-s} = \zeta(s, \alpha) - \zeta(s, 1 - \alpha), $$ where $ζ (s, a) = \sum_{n \geq 0} (n + a)^{- s}$ is the Hurwitz zeta function. The Hurwitz zeta extends meromorphically to $C$ with a single simple pole at $s = 1$ and special value $ζ (0, a) = 1/2 - a$ (a classical computation via the Bernoulli polynomial expansion of $ζ (s, a)$ at $s = 0$ ). Hence $$ \eta(D_\alpha) = \zeta(0, \alpha) - \zeta(0, 1 - \alpha) = (1/2 - \alpha) - (1/2 - (1 - \alpha)) = 1 - 2\alpha. $$ The value sweeps the open interval $(- 1, 1)$ as $α$ sweeps $(0, 1)$ . $□$

Proposition (Chern-Simons interpretation modulo integers, APS 1976 II). Let $Y$ be a closed Riemannian spin 3-manifold and let $A_{0}, A_{1}$ be flat $U (1)$ -connections. Then $η (D_{A_{1}}) - η (D_{A_{0}}) = 2 (CS (A_{1}) - CS (A_{0})) (mod Z)$ , where $CS (A) = \frac{1}{4 π ^{2}} \int_{Y} A \land d A + \frac{2}{3} A \land A \land A$ is the Chern-Simons three-form.

Proof sketch. Consider the 4-manifold $X = [0, 1] \times Y$ with the cylindrical metric and a connection on $X$ interpolating between $A_{0}$ (at $u = 0$ ) and $A_{1}$ (at $u = 1$ ). Apply the APS index theorem on $X$ with both boundary components. The bulk integral $\int_{X} A \land ch (E)$ , for a $U (1)$ -twist, reduces to a Chern-Simons-type integral by Stokes; modulo integers it equals $CS (A_{1}) - CS (A_{0})$ . The boundary correction is $\frac{1}{2} (η (A_{1}) + h_{1}) - \frac{1}{2} (η (A_{0}) + h_{0})$ (the two ends of the cylinder contribute with opposite orientation). The APS index on $X$ is an integer. Therefore $η (D_{A_{1}}) - η (D_{A_{0}})$ equals $2 (CS (A_{1}) - CS (A_{0}))$ modulo integers, after accounting for the kernel and orientation factors. Details are in APS 1975 I §6 and 1976 II §2-§3. $□$

Connections [Master]

Atiyah-Singer index theorem 03.09.10. The APS theorem is the manifold-with-boundary extension of Atiyah-Singer. On a closed manifold the boundary is empty, the eta and kernel corrections drop out, and the formula reduces to the closed-manifold integer index $Ind (D) = \int_{X} A \land ch (E)$ . The bulk integral in the APS formula is exactly the closed-manifold Atiyah-Singer integrand, computed locally from the same characteristic-class data. The boundary correction is the genuinely new content of APS, and it is non-zero precisely when the manifold has a boundary.
Bismut superconnection 03.09.23. The Bismut superconnection framework extends the heat-kernel proof of Atiyah-Singer to families, producing a closed differential form on the parameter space whose cohomology class is the Chern character of the family index. The Bismut-Cheeger eta-form, constructed from the transgression of the superconnection Chern character, is the family analogue of the eta invariant. The single-operator APS theorem corresponds to the degree-zero piece of the family eta-form; higher-degree pieces give the family APS theorem on a fibration with boundary.
Heat-kernel proof of Atiyah-Singer 03.09.20. The heat-kernel proof of the closed-manifold index theorem provides the analytic machinery the APS theorem extends. The Getzler rescaling, McKean-Singer constancy of the supertrace, and small-time asymptotic expansion of the heat trace all carry over to the manifold-with-boundary case, with one new ingredient: the boundary heat-flux term whose Mellin transform identifies as the eta invariant. The closed-manifold proof from 03.09.20 is the bulk-integral piece of the APS formula; eta is the boundary-only modification.
Dirac operator 03.09.08. The fundamental operator in the APS theorem is the twisted Dirac operator $D = D^{+}$ , with its boundary operator $A$ the restriction of $D$ to the collar product structure. The Lichnerowicz-Weitzenböck identity for $D$ on the closed manifold underlies the heat-kernel asymptotic expansion; the same identity restricted to the boundary collar produces the half-line model that computes the boundary correction. The eta invariant is intrinsically a Dirac-type construction in the original APS framework, although it generalises to any self-adjoint elliptic operator.
Elliptic operators 03.09.09. The eta invariant is defined for any self-adjoint elliptic operator of positive order, not just Dirac-type. The meromorphic continuation of $η_{D} (s)$ is a general statement of Seeley-Gilkey type elliptic theory. The regularity at $s = 0$ (and hence the existence of $η (D)$ as a well-defined real number) does require a Dirac-type assumption for the cleanest version of the local-invariant-theory cancellation, but the meromorphic continuation itself is a general elliptic-theory fact built from the heat-kernel small-time expansion.
Pseudodifferential operators 03.09.22. The parametrix construction for the Fredholm property of $D$ with APS boundary conditions uses the pseudodifferential / Sobolev-space framework, supplemented by the boundary spectral projection $Π_{\geq 0}$ . Melrose's b-calculus refinement (1993) extends pseudodifferential calculus to a smooth class adapted to manifolds with boundary, recovering the APS theorem as a special case and clarifying the role of the eta invariant as a contribution from the indicial family at the boundary.
Chern-Simons gauge theory. The eta invariant modulo integers equals (up to a factor of $2$ ) the Chern-Simons functional on an odd-dimensional closed manifold. The original APS 1975 I §6 and 1976 II §2 results identify $e^{iπ η}$ as the Chern-Simons phase in three-dimensional gauge theory. In topological field theory, the eta invariant computes the partition function of a free Dirac fermion coupled to a flat connection on a closed odd-dimensional manifold; this is the foundational link between spectral asymmetry and quantum field theory anomalies. Witten's interpretation of Chern-Simons theory as a topological field theory rests on this identification.
Kirillov character formula via the equivariant index 03.09.25. Sibling BGV-track unit. Both this unit and 03.09.25 are equivariant-index theorems built on the heat-kernel / Bismut-superconnection machinery, and both produce closed-form geometric expressions for spectral data of Dirac-type operators. Where APS computes the integer / half-integer index of a manifold with boundary as a bulk + boundary spectral sum, the Kirillov formula computes the equivariant index of a Dirac operator on a homogeneous space $G / T$ as an orbit-integral Fourier transform. The two results share the equivariant superconnection apparatus and the small-time heat-kernel localisation argument, and the Bismut-Cheeger family eta-form of APS is the family version of the equivariant Chern character whose evaluation at $g = e^{X}$ in the Kirillov formula reduces to the orbit integral on $O_{λ}$ . Read together, the two units exhibit the equivariant-index theorem operating at two ends of its applicability: APS on the spectral side of boundary problems, Kirillov on the representation-theoretic side of homogeneous spaces.

Historical & philosophical context [Master]

The eta invariant and the Atiyah-Patodi-Singer index theorem first appeared in a three-paper sequence published in Mathematical Proceedings of the Cambridge Philosophical Society in 1975-1976 ^{[Atiyah-Patodi-Singer 1975 I; ref: TODO_REF Atiyah-Patodi-Singer 1976 II; ref: TODO_REF Atiyah-Patodi-Singer 1976 III]}. The first paper introduced the eta series, established its meromorphic continuation, defined the eta invariant as the value at $s = 0$ , and proved the APS index theorem for the twisted Dirac operator on a compact manifold with boundary under product-collar conditions. The second paper extended the analysis to flat-bundle twists and developed the secondary-invariant theory: differences of eta invariants for distinct flat connections produce real-valued invariants of representations of the fundamental group, refining the integer-valued Atiyah-Singer index. The third paper applied the framework to signature theorems and Hirzebruch's number-theoretic identities, including the eta invariant's role in computing the signature defect of a manifold with corners.

The conceptual breakthrough was the recognition that the boundary contribution to the heat-kernel proof of the index theorem is not a local geometric integral but a global spectral invariant of the boundary operator — the eta invariant. Prior approaches to manifold-with-boundary index theory had attempted to find a local boundary integrand analogous to the bulk $A \land ch (E)$ and had run into the obstruction that no local formula could produce a half-integer correction. Atiyah, Patodi, and Singer resolved this by abandoning the search for a local boundary term and instead introducing the spectral asymmetry of the boundary Dirac operator as the boundary correction. The non-local APS boundary condition was the analytic compromise that made the resulting bulk operator Fredholm.

Jean-Michel Bismut and Jeff Cheeger extended the framework to family-index theory in their 1989 paper η-invariants and their adiabatic limits ^{[Bismut-Cheeger 1989]}, constructing the family eta-form as a transgression of the Bismut superconnection Chern character. The Bismut-Cheeger eta-form refines the single-operator eta invariant into a differential form on the parameter space and identifies the family-APS index theorem with a cohomological identity in $Ω^{*} (B)$ . Earlier work by Bismut and Daniel Freed on the determinant line bundle of a family of Dirac operators ^{[Bismut-Freed 1986]} paved the way: they showed that the eta-form arises as the curvature of the determinant line bundle, and that the holonomy of the determinant line bundle around a loop is computed by the eta invariant of the family.

Richard Melrose's 1993 monograph The Atiyah-Patodi-Singer Index Theorem ^{[Melrose 1993]} reorganised the APS theory using the b-calculus, a pseudodifferential framework adapted to manifolds with cylindrical ends. The b-calculus replaces the non-local APS boundary condition with a microlocal smoothness condition at the boundary's cotangent fibre, and the eta invariant emerges as a contribution from the indicial family of the b-operator. This reformulation makes the APS theorem look more like the closed-manifold theorem analytically, at the cost of working with a more elaborate pseudodifferential calculus. Modern treatments in spin geometry, including Berline-Getzler-Vergne Ch. 9 ^{[Berline-Getzler-Vergne 1992]}, use a hybrid of the original APS heat-kernel approach and Melrose's b-calculus picture.

Bibliography [Master]

@article{AtiyahPatodiSinger1975I,
  author  = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title   = {Spectral asymmetry and {R}iemannian geometry {I}},
  journal = {Math. Proc. Cambridge Philos. Soc.},
  volume  = {77},
  year    = {1975},
  pages   = {43--69}
}

@article{AtiyahPatodiSinger1976II,
  author  = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title   = {Spectral asymmetry and {R}iemannian geometry {II}},
  journal = {Math. Proc. Cambridge Philos. Soc.},
  volume  = {78},
  year    = {1976},
  pages   = {405--432}
}

@article{AtiyahPatodiSinger1976III,
  author  = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title   = {Spectral asymmetry and {R}iemannian geometry {III}},
  journal = {Math. Proc. Cambridge Philos. Soc.},
  volume  = {79},
  year    = {1976},
  pages   = {71--99}
}

@article{BismutCheeger1989,
  author  = {Bismut, Jean-Michel and Cheeger, Jeff},
  title   = {$\eta$-invariants and their adiabatic limits},
  journal = {J. Amer. Math. Soc.},
  volume  = {2},
  year    = {1989},
  pages   = {33--70}
}

@article{BismutFreed1986I,
  author  = {Bismut, Jean-Michel and Freed, Daniel S.},
  title   = {The analysis of elliptic families {I}: Metrics and connections on determinant bundles},
  journal = {Comm. Math. Phys.},
  volume  = {106},
  year    = {1986},
  pages   = {159--176}
}

@article{BismutFreed1986II,
  author  = {Bismut, Jean-Michel and Freed, Daniel S.},
  title   = {The analysis of elliptic families {II}: Dirac operators, eta invariants, and the holonomy theorem},
  journal = {Comm. Math. Phys.},
  volume  = {107},
  year    = {1986},
  pages   = {103--163}
}

@book{Melrose1993,
  author    = {Melrose, Richard B.},
  title     = {The {A}tiyah-{P}atodi-{S}inger Index Theorem},
  series    = {Research Notes in Mathematics},
  volume    = {4},
  publisher = {A K Peters},
  year      = {1993}
}

@book{BerlineGetzlerVergne1992,
  author    = {Berline, Nicole and Getzler, Ezra and Vergne, Mich\`ele},
  title     = {Heat Kernels and Dirac Operators},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {298},
  publisher = {Springer-Verlag},
  year      = {1992}
}

@book{LawsonMichelsohn1989,
  author    = {Lawson, H. Blaine and Michelsohn, Marie-Louise},
  title     = {Spin Geometry},
  series    = {Princeton Mathematical Series},
  volume    = {38},
  publisher = {Princeton University Press},
  year      = {1989}
}

@book{Gilkey1995,
  author    = {Gilkey, Peter B.},
  title     = {Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem},
  edition   = {2nd},
  publisher = {CRC Press},
  year      = {1995}
}

@article{APSBriefAnnouncement,
  author  = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title   = {Spectral asymmetry and {R}iemannian geometry},
  journal = {Bull. London Math. Soc.},
  volume  = {5},
  year    = {1973},
  pages   = {229--234}
}

Prerequisites

03.09.10
03.09.20
03.09.23
03.09.08
03.09.09

Tier anchors

beginner: spectral asymmetry of a self-adjoint operator; signed count of positive minus negative eigenvalues, regularised
intermediate: Atiyah-Patodi-Singer 1975 I §1-§3; Berline-Getzler-Vergne Ch. 9 §9.5; Melrose *The Atiyah-Patodi-Singer Index Theorem* Ch. 1-2
master: Atiyah-Patodi-Singer 1975 *Spectral asymmetry and Riemannian geometry I* (Math. Proc. Camb. Phil. Soc. 77); APS 1976 II and III; Bismut-Cheeger 1989 *η-invariants and their adiabatic limits* (J. AMS 2); Bismut-Freed 1986 *The analysis of elliptic families I/II* (Comm. Math. Phys. 106 + 107); Melrose 1993 *The Atiyah-Patodi-Singer Index Theorem*; Berline-Getzler-Vergne Ch. 9

References

TODO_REF
Atiyah-Patodi-Singer — Spectral asymmetry and Riemannian geometry I · Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69 — originator paper, eta invariant and APS index theorem
TODO_REF
Atiyah-Patodi-Singer — Spectral asymmetry and Riemannian geometry II · Math. Proc. Camb. Phil. Soc. 78 (1976), 405-432 — flat bundles, secondary invariants
TODO_REF
Atiyah-Patodi-Singer — Spectral asymmetry and Riemannian geometry III · Math. Proc. Camb. Phil. Soc. 79 (1976), 71-99 — number-theoretic and signature applications
TODO_REF
Bismut-Cheeger — η-invariants and their adiabatic limits · J. Amer. Math. Soc. 2 (1989), 33-70 — eta-form, family APS
TODO_REF
Bismut-Freed — The analysis of elliptic families I · Comm. Math. Phys. 106 (1986), 159-176 — determinant line and superconnection background
TODO_REF
Bismut-Freed — The analysis of elliptic families II · Comm. Math. Phys. 107 (1986), 103-163 — Dirac case, eta-form precursor
TODO_REF
Melrose — The Atiyah-Patodi-Singer Index Theorem · Research Notes in Mathematics 4, A K Peters, 1993 — b-calculus / cusp-calculus proof
TODO_REF
Berline-Getzler-Vergne — Heat Kernels and Dirac Operators · Grundlehren 298, Springer 1992, Ch. 9 §9.5 (eta invariant), Ch. 9 §9.6 (APS theorem)
TODO_REF
Lawson-Michelsohn — Spin Geometry · Princeton 1989, §III.14 (APS index theorem outline)
TODO_REF
Gilkey — Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem · 2nd ed., CRC Press 1995, Ch. 3 §3.7 + Ch. 4 — heat-kernel proof of APS

Reviewer

TBD

Estimated time

beginner: 22m
intermediate: 55m
master: 100m