Holonomy of the Dirac operator

Question

Let $(M,g)$ be a spin manifold and $X=\dot{\gamma}$ be a vector field of $\gamma$, $1$-periodic. The flow of spinors $\psi_{\gamma}$ is defined by:

$$\psi_{\gamma}(0,\psi)=\psi$$

$$\psi_{\gamma}(t,\psi_{\gamma}(t',\psi))=\psi_{\gamma}(t+t',\psi)$$

$$\nabla_{\dot{\gamma}} \psi_{\gamma}=\dot{\gamma}.{\cal D}\psi_{\gamma}$$

where $\cal D$ is the Dirac operator and $\nabla$ is the spinorial connection. The holonomy of $\gamma$ is:

$$g_{\gamma}(\psi)= \psi_{\gamma}(1,\psi)$$

Can we define invariants of the manifold $M$ by the holonomy of the Dirac operator?