Maslov Anomaly and the Morse Index Theorem

$$ {e^{{\rm{i}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \Gamma } \left[ {\tilde M} \right]}} = \int {{D_{{X^a}}}{e^{iSn\left[ {x,\tilde M} \right]}}} $$

$$ {S_{{\rm{fl}}}}\left[ {x,\tilde M} \right] = {1 \over 2}\int_0^T {dt{{\bar x}_a}\left( t \right){{\left[ {{\partial \over {\partial t}} - \tilde M\left( t \right)} \right]}^a}_b{x^b}\left( t \right)} . $$