Abstract
The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.
Highlights
Tensor factors must be invisible to the bulk state
In the present paper we extend the notion of a topological boundary conditions (TBCs) to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases
The short-range correlations that erase or “gap out” these imaginary edge modes is counted by the area law of the bulk entanglement entropy; it is the global constraints on the state that lead to the subleading correction and provides the signal of topological order
Summary
We begin with the action of Chern-Simons theory with a simple gauge group G on a compact manifold M : SCS. This is not what we are interested in: we are looking for interfaces upon which such degrees of freedom pair up and become massive as a result of interactions between them Such a situation is sketched in figure 1; associated with the Chern-Simons on the left is the Lie algebra gL and level-Killing form κL and that on the right with gR and κR. Lagrangian subspaces of this type are known as Lagrangian Lie subalgebras (with respect to κ) [37,38,39,40,41] As restrictive as this condition is, there can still exist non-trivial interfaces even when gL gR (as we find in example 1 of section 2.1). After imposing topological boundary conditions, we must restrict fields at the interface to lie in the isotropic subalgebra g, with its own irreducible highest weight representations (irreps). The fact that these are constant for conformal embeddings indicates that this would-be coset is trivial: this is consistent with the statement that the TBCs are gapping out the degrees of freedom at the interface
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