Spectral flow and the Floer grading mod 8

Intuition Beginner

At a resting point of the Chern-Simons landscape, the Hessian tells which nearby directions go uphill and which go downhill. If the resting point changes along a path, the Hessian changes too. Its eigenvalues move on the number line.

Spectral flow counts how many eigenvalues cross zero during the path, with signs. An eigenvalue crossing from negative to positive contributes one sign; crossing the other way contributes the opposite sign. This count measures how the number of downhill directions changes.

Floer grading uses that count. It assigns relative degrees to flat connections so that a flow line between two critical points has the expected dimension. For instanton Floer homology, the grading is naturally taken modulo eight.

Visual Beginner

Each curve is one eigenvalue of a changing self-adjoint operator. Spectral flow is the signed crossing count through zero, not the total number of eigenvalue curves.

Worked example Beginner

Suppose three eigenvalue curves cross zero as a parameter moves from left to right. The first crosses upward, from negative to positive. The second also crosses upward. The third crosses downward, from positive to negative.

The signed count is two upward crossings minus one downward crossing, so the spectral flow is one. If this path connects two flat connections, the relative Floer grading changes by one, up to the convention and the later modulo-eight reduction.

What this tells us: the grading is not assigned by looking only at one endpoint. It records how the linearized Chern-Simons problem changes along a path between endpoints.

Check your understanding Beginner

Formal definition Intermediate+

Let $Y$ be a closed oriented Riemannian three-manifold and let $\alpha ,\beta$ be irreducible flat $\mathrm{SU}(2)$ connections, regarded as critical points of the Chern-Simons functional on ${\mathcal{B}}^{\ast }(Y)$ 03.07.17, 03.07.18. Choose a smooth path $A_t$ of connections from $\alpha$ to $\beta$, transverse to gauge directions by a slice condition.

At a flat connection $A$, the Hessian of Chern-Simons in Coulomb gauge is represented, up to a conventional constant and gauge-fixing stabilization, by the self-adjoint elliptic operator $$ B_A = *_Y d_A $$ on adjoint-valued one-forms satisfying $d_A^{\ast }a=0$. A common stabilized form acts on pairs $(\varphi ,a)\in {\Omega }^0(Y;\mathrm{ad}P)\oplus {\Omega }^1(Y;\mathrm{ad}P)$ by $$ K_A(\phi,a)=\bigl(d_A^a,\ d_A\phi+_Yd_Aa\bigr). $$ This operator is self-adjoint and elliptic; it removes gauge degeneracy and is the standard operator used for spectral flow.

For a continuous path $\{K_{A_t}{\}}_{t\in [0,1]}$ of self-adjoint Fredholm operators with invertible endpoints, the spectral flow $$ \operatorname{sf}{K_{A_t}}\in\mathbb Z $$ is the signed count of eigenvalues crossing zero from negative to positive as $t$ increases, counted with multiplicity. Downward crossings contribute negatively.

The relative Floer grading is $$ \mu(\alpha,\beta)\equiv \operatorname{sf}{K_{A_t}}\pmod 8, $$ with the sign convention chosen consistently with the expected dimension formula for ASD trajectories from $\alpha$ to $\beta$.

Counterexamples to common slips

• Spectral flow is not the difference between two individual eigenvalues. It is a signed count of all crossings through zero along a path.
• The path matters at the integer level. In instanton Floer theory the ambiguity changes by multiples of eight, giving the intrinsic mod-eight grading.
• The gauge-fixed Hessian is used because Chern-Simons is constant along gauge orbits. Without removing gauge directions, zero modes from the group action obscure the Morse index.

Key theorem with proof Intermediate+

Theorem (spectral flow as an APS index). Let $\{K_t{\}}_{t\in [0,1]}$ be a smooth path of self-adjoint elliptic operators on a closed manifold, with invertible endpoints. Then the spectral flow of $K_t$ equals the Fredholm index of the cylinder operator $$ \mathcal D=\frac{d}{dt}+K_t $$ on $[0,1]\times Y$ with Atiyah-Patodi-Singer boundary conditions: $$ \operatorname{ind}(\mathcal D_{\mathrm{APS}})=\operatorname{sf}{K_t}. $$

Proof. First treat a path with only regular crossings, meaning the kernel of $K_t$ is finite-dimensional at each crossing and the crossing form on the kernel is nonsingular. Between crossings, no eigenvalue crosses zero, so the dimension of the negative spectral subspace is locally constant.

Near one regular crossing $t_0$, perturbation theory diagonalizes the crossing eigenvalues to first order. An eigenvalue with positive derivative crosses from negative to positive and contributes $+1$ to spectral flow; one with negative derivative contributes $-1$. The operator $\frac{d}{dt}+K_t$ on a small cylinder segment has local kernel and cokernel controlled by these crossing modes, so its local index contribution is the same signed crossing number.

Adding the cylinder segments gives additivity of both sides: the Fredholm index is additive under gluing intervals, and spectral flow is additive under concatenating paths. The APS boundary conditions remove endpoint spectral subspaces so that no endpoint correction remains when $K_0$ and $K_1$ are invertible. Summing the local contributions gives the equality. The general case follows by a small homotopy to regular crossings; both spectral flow and Fredholm index are homotopy invariant with fixed invertible endpoints ^{[Atiyah-Patodi-Singer spectral flow 1976]}. $\square$

Bridge. Spectral flow builds toward 03.07.23 because it supplies the relative degree of Floer generators, and it appears again in 03.07.22 through determinant-line orientations of the same Fredholm operators. The foundational reason is that the Chern-Simons Hessian identifies local Morse index change with eigenvalue crossings; this is exactly what the APS theorem identifies with a cylinder index. Putting these together, the mod-eight grading generalises finite-dimensional Morse index differences, and the bridge is dual to the slice-theorem separation of gauge directions in 03.07.18.

Exercises Intermediate+

Advanced results Master

The stabilized Hessian at a flat connection $\alpha$ is the odd signature-type operator $$ K_\alpha= \begin{pmatrix} 0 & d_\alpha^*\ d_\alpha & *Yd\alpha \end{pmatrix} $$ on ${\Omega }^0(Y;\mathrm{ad}P)\oplus {\Omega }^1(Y;\mathrm{ad}P)$. It is self-adjoint elliptic. Nondegeneracy of the Chern-Simons critical point is equivalent, after stabilizer directions are removed, to invertibility of this operator. Holonomy perturbations are introduced to achieve this condition for the critical set used in Floer theory.

For flat connections $\alpha ,\beta$, a path $A_t$ gives a cylinder operator $$ \mathcal D_{A_t}=\frac{d}{dt}+K_{A_t}. $$ This is the linearization of the ASD equation on $[0,1]\times Y$ in temporal gauge, after adding gauge-fixing. The APS theorem identifies its index with the spectral flow of the Hessians. On a long cylinder from $\alpha$ to $\beta$, the same index gives the expected dimension of the trajectory moduli space before quotienting by translation on an unparametrized cylinder.

The mod-eight ambiguity is a topological index calculation. Changing the path from $\alpha$ to $\beta$ changes spectral flow by the spectral flow around a loop in ${\mathcal{B}}^{\ast }(Y)$. For an integral homology sphere and $\mathrm{SU}(2)$, this loop determines a bundle over $S^1\times Y$, and the relevant index is divisible by eight. Hence $\mu (\alpha ,\beta )\in \mathbb{Z}/8$ is well-defined.

In examples such as the Poincare homology sphere $\Sigma (2,3,5)$, the irreducible flat connections form a finite set after the standard perturbative choices. The Chern-Simons values and rho-invariants determine relative gradings through the APS formula. The commonly quoted instanton Floer groups for the oriented Poincare sphere have two basic generators separated by four degrees under one standard convention; orientation reversal shifts the convention and dualizes the resulting group.

Synthesis. The foundational reason spectral flow appears is that Floer theory needs a replacement for the finite-dimensional Morse index. This is exactly what the Chern-Simons Hessian provides after the slice theorem removes gauge directions. The central insight identifies changes of Morse index with eigenvalue crossings, while APS identifies those crossings with the index of the cylinder linearization. Putting these together, the mod-eight grading supplies the bridge between analytic trajectory dimensions and algebraic degrees in the Floer complex.

Full proof set Master

Proposition 1 (additivity of spectral flow). If $K_t$ is a path on $[0,1]$ with invertible endpoints and $c\in (0,1)$ is not a crossing time, then $$ \operatorname{sf}(K_t;[0,1])=\operatorname{sf}(K_t;[0,c])+\operatorname{sf}(K_t;[c,1]). $$

Proof. By definition, spectral flow is the sum of signed zero-crossing multiplicities. Since $c$ is not a crossing time, no crossing is split at $c$. The crossings in $[0,1]$ are exactly the disjoint union of crossings in $[0,c]$ and crossings in $[c,1]$. Summing signed multiplicities over the union gives the formula. $\square$

Proposition 2 (regular crossing formula). If $t_0$ is an isolated regular crossing and the crossing form on $\ker K_{t_0}$ has signature $\sigma$, then the local contribution of $t_0$ to spectral flow is $\sigma$.

Proof. Restrict the path to the finite-dimensional crossing space given by the spectral projection for a small interval around zero. Analytic perturbation theory gives eigenvalue branches ${\lambda }_j(t)$ with ${\lambda }_j(t_0)=0$. Regularity means the derivatives ${\lambda }_j^{\prime }(t_0)$ are nonzero after diagonalizing the crossing form. A branch with positive derivative crosses upward and contributes $+1$; a branch with negative derivative crosses downward and contributes $-1$. The sum of these signs is the signature of the crossing form. $\square$

Proposition 3 (path reversal). For the reversed path ${\tilde{K}}_t=K_{1-t}$, $$ \operatorname{sf}(\widetilde K_t)=-\operatorname{sf}(K_t). $$

Proof. After a small homotopy, assume all crossings are regular and endpoints remain invertible. A crossing at $t_0$ for $K_t$ becomes a crossing at $1-t_0$ for ${\tilde{K}}_t$. The crossing form changes sign because differentiating $K_{1-t}$ gives $-{\dot{K}}_{1-t}$. Hence every local signature changes sign. Summing the local signatures gives the result. Homotopy invariance removes the regular-crossing assumption. $\square$

Connections Master

• Chern-Simons functional 03.07.17. Spectral flow is computed from the Hessian of Chern-Simons at flat critical points, so it is the grading refinement of the functional's Morse theory.

• Configuration space and slice theorem 03.07.18. The gauge-fixed Hessian and its Fredholm properties rely on working transverse to gauge orbits in ${\mathcal{B}}^{\ast }(Y)$.

• APS index theorem 03.09.24. The equality between spectral flow and the index of $\frac{d}{dt}+K_t$ is the cylinder form of the APS index theorem.

• Instanton Floer homology 03.07.23. The mod-eight class of spectral flow assigns degrees to generators and predicts dimensions of trajectory moduli spaces counted by the differential.

Historical & philosophical context Master

Atiyah, Patodi, and Singer introduced spectral flow as the index-theoretic count of eigenvalues crossing zero in their work on spectral asymmetry and boundary problems ^{[Atiyah-Patodi-Singer spectral flow 1976]}. The same mechanism appears in the APS theorem for cylinder operators, where a one-parameter family of boundary operators becomes a Fredholm operator in one higher dimension.

Floer used spectral flow to grade the instanton chain complex in 1988 ^{[Floer 1988]}. Donaldson's account systematized the construction for flat $\mathrm{SU}(2)$ connections on homology three-spheres, tying the mod-eight grading to the ASD deformation complex and the APS index formula ^{[Donaldson 2002]}.

Bibliography Master

@article{AtiyahPatodiSinger1976SpectralFlow,
  author = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title = {Spectral flow and the index of elliptic operators},
  journal = {Bulletin of the London Mathematical Society},
  volume = {8},
  pages = {1--12},
  year = {1976}
}

@article{AtiyahPatodiSinger1976III,
  author = {Atiyah, Michael F. and Patodi, Vijay K. and Singer, Isadore M.},
  title = {Spectral asymmetry and Riemannian geometry. III},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
  volume = {79},
  pages = {71--99},
  year = {1976}
}

@article{Floer1988InstantonGrading,
  author = {Floer, Andreas},
  title = {An instanton-invariant for 3-manifolds},
  journal = {Communications in Mathematical Physics},
  volume = {118},
  pages = {215--240},
  year = {1988}
}

@book{Donaldson2002FloerGrading,
  author = {Donaldson, Simon K.},
  title = {Floer Homology Groups in Yang-Mills Theory},
  series = {Cambridge Tracts in Mathematics},
  volume = {147},
  publisher = {Cambridge University Press},
  year = {2002}
}