The $\hat{A}$ genus as a projective volume form on the derived loop space

In the present work, we extend our previous work with Gwilliam by realizing $\hat{A}(X)$ as the projective volume form associated to the BV operator in our quanitization of a one-dimensional sigma model. We also discuss the associated integration/expectation map. We work in the formalism of $L_\infty$ spaces, objects of which are computationally convenient presentations for derived stacks. Both smooth and complex geometry embed into $L_\infty$ spaces and we specialize our results in both of these cases.