# Geometric responses of the Pfaffian state

## TL;DR

This paper investigates the geometric and transport properties of the Pfaffian quantum Hall state on arbitrary Riemann surfaces, deriving universal coefficients and analyzing responses to metric and magnetic field variations.

## Contribution

It introduces a path integral framework to compute the generating functional and response coefficients for the Pfaffian state on complex geometries, including gravitational anomaly effects.

## Key findings

- Derived universal transport coefficients for the Pfaffian state.
- Computed leading and sub-leading charge density corrections.
- First derivation of gravitational anomaly contribution to the structure factor.

## Abstract

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 sub-leading corrections to the charge density in a large-$N$ expansion. We also present the first derivation of gravitational anomaly contribution at O$(k^6)$ to the static structure factor for the Pfaffian state in the long wavelength limit.