FiniteEnergyPlane.czIndex_eq_czIndexOfPath

Data. Let $\psi : [0,1] \to M$ be a continuous path, where $M$ is the space of real $2\times 2$ matrices whose rows and columns are indexed by the two-element set $\{\mathrm{inl}\,0, \mathrm{inr}\,0\}$ (the "$n=1$" case of the $2n\times 2n$ setting). Write $\psi(t) = \begin{pmatrix} a(t) & b(t) \\ c(t) & d(t)\end{pmatrix}$ with first row/column $\mathrm{inl}\,0$ and second $\mathrm{inr}\,0$.

Hypothesis. $\psi \in \mathcal{SP}(1)$, meaning all three of:

• $\psi(t)$ is symplectic for every $t \in [0,1]$, i.e. $\psi(t)\, J_0\, \psi(t)^{\mathsf T} = J_0$ with $J_0 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ (for $2\times 2$ matrices this is $\det \psi(t) = 1$);
• $\psi(0) = I$;
• $\det(I - \psi(1)) \neq 0$.

Conclusion. The integer $\mu_{CZ}(\psi)$ equals the integer $\mathrm{cz}(\varphi)$, where both sides are defined below.

Left side $\mu_{CZ}(\psi)$. For a $2\times 2$ matrix $A = \begin{pmatrix} a & b\\ c & d\end{pmatrix}$ define its complex-linear part $\tfrac12(A - J_0 A J_0)$ and the complex number

$$\det\nolimits_{\mathbb C}(A) = \tfrac{a+d}{2} + i\,\tfrac{c-b}{2},$$

and $\hat\rho(A) = \det_{\mathbb C}(A)/|\det_{\mathbb C}(A)|$ (with the convention that this is $0$ when $\det_{\mathbb C}(A) = 0$, since division by zero returns $0$). An integer $k$ is a CZ value of $\psi$ if there exist

• a continuous path $\chi : [0,1] \to M$ with $\chi(0) = \psi(1)$, with $\chi(t)$ symplectic and $\det(I - \chi(t)) \neq 0$ for every $t$, and with $\chi(1) = -I$ or $\chi(1) = \mathrm{diag}(2, \tfrac12)$;
• continuous functions $\theta_1, \theta_2 : [0,1] \to \mathbb R$ with $\hat\rho(\psi(t)) = e^{i\theta_1(t)}$ and $\hat\rho(\chi(t)) = e^{i\theta_2(t)}$ for all $t$;

such that

$$2\big((\theta_1(1) - \theta_1(0)) + (\theta_2(1) - \theta_2(0))\big) = 2\pi k .$$

Then $\mu_{CZ}(\psi)$ is defined as: if some CZ value exists, an arbitrarily chosen one (via the axiom of choice; the definition itself does not assert that the CZ value is unique); otherwise $0$.

Right side $\mathrm{cz}(\varphi)$. The path $\varphi : \mathbb R \to \mathbb R^{2\times 2}$ (now indexed by $\{0,1\}$, with $\mathrm{inl}\,0 \mapsto 0$, $\mathrm{inr}\,0 \mapsto 1$) is

$$\varphi(\tau) = \psi(\pi_{[0,1]}(\tau)),$$

where $\pi_{[0,1]}$ clamps $\tau$ to $[0,1]$ (so $\varphi(\tau) = \psi(0)$ for $\tau \le 0$ and $\psi(1)$ for $\tau \ge 1$). For $s \in \mathbb R$ let $e(s) = (\cos s, \sin s)$. A function $\theta : \mathbb R \to \mathbb R$ is an angle lift from $s$ if $\theta$ is continuous on $[0,1]$, $\theta(0) = s$, and for every $t \in [0,1]$ there is $r > 0$ with $\varphi(t)\, e(s) = r\, e(\theta(t))$. The winding set is

$$W = \Big\{ \tfrac{\theta(1) - s}{2\pi} \;:\; s \in [0, 2\pi],\ \theta \text{ an angle lift from } s \Big\} \subseteq \mathbb R .$$

Let $\alpha = \inf W$ and $\beta = \sup W$, taken as real-number infimum/supremum (which by convention return $0$ when $W$ is empty or unbounded in the relevant direction). Then

$$\mathrm{cz}(\varphi) = \begin{cases} 2(\lceil \beta\rceil - 1) & \text{if } \alpha \le \lceil\beta\rceil - 1,\\ 2\lceil\beta\rceil - 1 & \text{otherwise.}\end{cases}$$

So the theorem asserts: for every continuous path $\psi$ of $2\times 2$ symplectic matrices starting at $I$ with $1$ not an eigenvalue of $\psi(1)$, the (choice-defined, default $0$) integer $\mu_{CZ}(\psi)$ coincides with the integer $\mathrm{cz}(\varphi)$ computed from the inf/sup of the winding set of the clamped, reindexed path $\varphi$.