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Title:
Symplectic geometry of the fractional quantum Hall fluid
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Abstract:
Fractional quantum Hall (FQH) states can be described by an effective hydrodynamics featuring a ``FQH constraint'' which relates the vorticity of the fluid to its density. In addition, fundamental properties of the FQH state, including the filling fraction $\nu$ and shift $\mathcal{S}$, appear as coefficients of the different terms in this constraint. In this talk we address the mathematical interpretation of this constraint. We show that the FQH constraint is equivalent to a certain initial condition on a conserved moment map associated with the action of the group SDiff of area-preserving diffeomorphisms of 2D space on the phase space of a charged fluid in a magnetic field. In this way we obtain the original FQH constraint derived by Stone. We then explain how to modify the original fluid action to obtain Abanov's version of the FQH constraint, which incorporates the effect of ``Hall viscosity''. For a FQH fluid on a closed 2D manifold $\Sigma$, we find that the conserved moment map discussed above only exists when the FQH relation $N=\nu(N_{\phi}+\frac{\mathcal{S}}{2}\chi)$ is satisfied, where $N$ is the number of particles, $N_{\phi}$ is the number of flux quanta of the magnetic field piercing $\Sigma$, and $\chi$ is the Euler characteristic of $\Sigma$. If time allows we will discuss the extension of these results to manifolds with boundary. Our results show that the structure of the FQH constraint is far from arbitrary. This talk is based on joint work with Gustavo Monteiro.
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