Axiomatic characterization of the Conley–Zehnder index

Throughout, for a natural number $n$ (including $n=0$), $\mathrm{Mat}_n$ denotes the real $2n\times 2n$ matrices whose rows and columns are indexed by two copies of $\{0,\dots,n-1\}$: a first ("$q$") block and a second ("$p$") block. $\mathrm{Id}$ is the identity matrix. $J_0\in\mathrm{Mat}_n$ is the block matrix

$$J_0=\begin{pmatrix}0&-\mathrm{Id}\\ \mathrm{Id}&0\end{pmatrix}.$$

A matrix $A$ is called symplectic if $A\,J_0\,A^{\mathsf T}=J_0$. A path is a continuous map $[0,1]\to\mathrm{Mat}_n$ (with the usual Euclidean topology on matrix entries); the set of paths carries the compact-open topology (equivalently the topology of uniform convergence). Products, inverses and transposes of paths below are always taken pointwise in $t$. For $n=0$ all matrices are the empty $0\times 0$ matrix, whose determinant is $1$.

$\mathrm{SP}(n)$ is the set of paths $\psi$ such that $\psi(t)$ is symplectic for every $t\in[0,1]$, $\psi(0)=\mathrm{Id}$, and $\det(\mathrm{Id}-\psi(1))\neq 0$.

$\mathrm{Sp}^*(n)$ is the set of symplectic $A\in\mathrm{Mat}_n$ with $\det(\mathrm{Id}-A)\neq 0$.

A symplectic loop is a path $\varphi$ with $\varphi(t)$ symplectic for every $t$ and $\varphi(0)=\varphi(1)=\mathrm{Id}$.

$W^+=-\mathrm{Id}$, and $W^-$ is the diagonal matrix whose $q$-block diagonal is $(2,-1,\dots,-1)$ and whose $p$-block diagonal is $(\tfrac12,-1,\dots,-1)$ (the entry $2$ resp. $\tfrac12$ sits at index $0$; all other diagonal entries are $-1$). For $n=0$, $W^+=W^-$ is the empty matrix.

For $A\in\mathrm{Mat}_n$, let $C_A=\tfrac12\,(A-J_0AJ_0)$. Let $X$ be the upper-left ($q$-row, $q$-column) $n\times n$ block of $C_A$ and $Y$ its lower-left ($p$-row, $q$-column) block, and set

$$d(A)=\det{}_{\mathbb C}\,(X+iY)\in\mathbb C,$$

the determinant of the complex $n\times n$ matrix $X+iY$ (equal to $1$ when $n=0$). Then

$$\hat\rho(A)=\frac{d(A)}{|d(A)|}\in\mathbb C,$$

with the convention that division by zero gives $0$, so $\hat\rho(A)=0$ whenever $d(A)=0$.

For a function $f:[0,1]\to\mathbb C$, a function $\theta:[0,1]\to\mathbb R$ is called an argument lift of $f$ if $\theta$ is continuous and $f(t)=e^{i\theta(t)}$ for every $t\in[0,1]$. (No lift exists if $|f(t)|\neq 1$ for some $t$, e.g. if $\hat\rho$ takes the junk value $0$; continuity of $f$ is not separately assumed.)

For a path $\psi$ and an integer $k$, "$k$ is a CZ value of $\psi$" means: there exist a path $\chi$ with $\chi(0)=\psi(1)$, $\chi(t)\in\mathrm{Sp}^*(n)$ for all $t\in[0,1]$, and $\chi(1)=W^+$ or $\chi(1)=W^-$; and argument lifts $\theta_1$ of $t\mapsto\hat\rho(\psi(t))$ and $\theta_2$ of $t\mapsto\hat\rho(\chi(t))$ with

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

(The lifts are of $\hat\rho$ itself, not of $\hat\rho^2$; the factor $2$ appears only in this equation. No condition $\psi\in\mathrm{SP}(n)$ is part of this notion.)

The index $\mu_{CZ}(\psi)\in\mathbb Z$ of a path $\psi$ is defined as: if some integer $k$ is a CZ value of $\psi$, then $\mu_{CZ}(\psi)$ is one such $k$ chosen by the axiom of choice (an unspecified one if several exist); if no CZ value exists, $\mu_{CZ}(\psi)=0$. It is defined for every path, not only those in $\mathrm{SP}(n)$.

For a path $\varphi$ and an integer $k$, "$k$ is a Maslov value of $\varphi$" means: there is an argument lift $\theta$ of $t\mapsto\hat\rho(\varphi(t))$ with $\theta(1)-\theta(0)=2\pi k$. The Maslov index $\mu(\varphi)$ is a choice-selected such $k$ if one exists (an unspecified one if several exist), and $0$ otherwise; it is defined for every path, not only loops.

For a real symmetric matrix $S\in\mathrm{Mat}_n$ (symmetric = Hermitian over $\mathbb R$), let $\lambda_1,\dots,\lambda_{2n}$ be its real eigenvalues listed with multiplicity. $\mathrm{Sign}(S)$ is the number of indices $i$ with $\lambda_i>0$ minus the number with $\lambda_i<0$.

A function $\mu$ from paths in $\mathrm{Mat}_n$ to $\mathbb Z$ satisfies the homotopy property if for every $\psi\in\mathrm{SP}(n)$ and every $\psi'$ in the connected component of $\psi$ in $\mathrm{SP}(n)$ (with the subspace topology inherited from the compact-open topology on paths), $\mu(\psi')=\mu(\psi)$.

$\mu$ satisfies the loop property if for every symplectic loop $\varphi$ and every $\psi\in\mathrm{SP}(n)$,

$$\mu(\varphi\psi)=\mu(\psi)+2\,\mu_{\mathrm{Maslov}}(\varphi),$$

where $(\varphi\psi)(t)=\varphi(t)\psi(t)$ and $\mu_{\mathrm{Maslov}}$ is the specific Maslov index defined above (not something derived from $\mu$).

$\mu$ satisfies the signature property if for every real symmetric $S\in\mathrm{Mat}_n$ with $\det S\neq 0$ and all eigenvalues satisfying $|\lambda_i|<2\pi$, and for every path $\psi$ with $\psi(t)=\exp(t\,J_0S)$ for all $t\in[0,1]$ (matrix exponential, $\exp(M)=\sum_{m\ge0}M^m/m!$), one has

$$2\,\mu(\psi)=\mathrm{Sign}(S).$$

Membership $\psi\in\mathrm{SP}(n)$ is not a hypothesis here. For $n=0$ the only such $S$ is the empty matrix ($\det=1$, $\mathrm{Sign}=0$), so the property demands $\mu(\psi)=0$ for the unique path.

Claim. For every $n\in\mathbb N$, both of the following hold:

1. $\mu_{CZ}$ (in dimension $n$) has the homotopy property, the loop property and the signature property.
2. Every function $\mu$ from paths in $\mathrm{Mat}_n$ to $\mathbb Z$ having the homotopy, loop and signature properties satisfies $\mu(\psi)=\mu_{CZ}(\psi)$ for every $\psi\in\mathrm{SP}(n)$ (nothing is asserted outside $\mathrm{SP}(n)$).