Framing Anomaly in Lattice Chern-Simons-Maxwell Theory

TL;DR

This paper demonstrates the presence of framing anomaly in a lattice $U(1)$ Chern-Simons-Maxwell model without boundary, confirming its topological properties and aiding in defining lattice chiral topological orders.

Contribution

It provides the first explicit lattice computation of framing anomaly in a $U(1)$ Chern-Simons theory, validating the lattice model as a proper topological quantum field theory.

Findings

The expectation value of the modular T operator has a universal phase of -2π/12.

The phase decomposes into contributions from the Gauss-Milgram sum and framing anomaly.

The spectrum of T confirms the framing anomaly phase of 2π/24.

Abstract

Framing anomaly is a key property of $(2+1)d$ chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the $U(1)$ Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular $T$ operator in the ground subspace on a spatial torus, from which we extract that $\langle T \rangle$ has a universal phase of $-2\pi/12$ as expected: $-2\pi/8$ from the Gauss-Milgram sum of the topological spins of the ground states, and $2\pi/24$ from the framing anomaly; we can also extract the $2\pi/24$ framing anomaly phase alone from the full spectrum of $T$ in the ground subspace by computing $\langle T^m \rangle$. This pins down the last and most crucial property required for a valid lattice definition of $U(1)$ Chern-Simons theory.