# Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices

Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $D_{\mathrm{APS}}^W$ once equipping $D^W$ with the APS-boundary condition. The index formula from Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry. I." states that $$\mathrm{ind}(D_{\mathrm{APS}}^W)=\int_W \alpha_D-\overline{\eta}(D^{\partial W}),$$ where $\alpha_D$ is a local term determined by the principal symbol of $D^W$, $D^{\partial W}$ is the boundary operator and $\overline{\eta}(D^{\partial W})$ its reduced eta invariant. See the paper of Atiyah-Patodi-Singer for details. In analogy with the Atiyah-Singer index theorem on closed manifolds, this index theorem can be viewed as an “even-dimensional” index theorem. The “odd-dimensional” version was considered by Dai-Zhang in “An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary“ (see http://arxiv.org/abs/math/0103230).

The paper by Dai-Zhang considers $W$ and $D^W$ as above, but $D^W$ is not necessarily graded, and a smooth function $g:W\to U(N)$. The product structure near the boundary guarantees that $D^{\partial W}$ is graded and after a choice of Lagrangian subspace $L\subseteq \ker(D^{\partial W})$, one can define an APS-type projection on the boundary $P(L)$. The projection $P(L)$ gives rise to self-adjoint boundary conditions on $D^W$. Denote the associated self-adjoint Fredholm operator $D^W_{P(L)}$. The details of this construction can be found in the paper of Dai-Zhang. It is shown by Dai-Zhang that for $P_{P(L)}:=\chi_{[0,\infty)}(D^W_{P(L)})$ and $P_{gP(L)g^{-1}}:=\chi_{[0,\infty)}(D^W_{gP(L)g^{-1}})$, the operator $$T_g(L):= P_{gP(L)g^{-1}}g P_{P(L)}: P_{P(L)}L^2(W,E) \to P_{gP(L)g^{-1}}L^2(W,E)$$ is Fredholm. They compute the index by an APS-type formula: $$\mathrm{ind}(T_g(L))=-\int_W \alpha_D\wedge \mathrm{ch}(g)-\overline{\eta}(D^{\partial W},g)+\tau_\mu(gP(L)g^{-1}, P(L),P_W).$$ Here $\overline{\eta}(D^{\partial W},g)$ is an $\eta$-type invariant defined from $D^{\partial W}$ and $g$. The operator $P_W$ is the Calderon projection, a pseudo-differential operator of order $0$ on $\partial W$ differing from $gP(L)g^{-1}$ and $P(L)$ by a lower order pseudo-differential operator. The term $\tau_\mu(gP(L)g^{-1}, P(L),P_W)$ is the Maslov triple index, considered in greater detail in the paper “The $\eta$-invariant, Maslov index, and spectral flow for Dirac–type operators on manifolds with boundary” by Kirk-Lesch (https://arxiv.org/abs/math/0012123).

Structurally, the difference between the two index theorems is the appearance of the Maslov triple index in the Toeplitz case.

After this lengthy pre-amble, we arrive at a question. Is there an explicit example where the Maslov triple index $\tau_\mu(gP(L)g^{-1}, P(L),P_W)$ has been computed to be non-zero?