Geometric responses of the Pfaffian state

We define and study the Pfaffian state on Riemann surfaces with arbitrary metrics and an inhomogeneous magnetic field and derive its universal transport coefficients. Following a path integral approach, we compute the generating functional which encodes the linear response of the system to a variation of the background metric and the magnetic field and use it to compute the leading and subleading corrections to the charge density in a large-$N$ expansion. We also present a derivation of gravitational anomaly contribution at $O\left(k^6\right)$ to the static structure factor for the Pfaffian state in the long wavelength limit.