JHEP08(2018)020
the naive factorization even on the lattice. One way to resolve this problem is to relax
Gauss’s law so that it can be violated at the boundary of A and B. Formally this requires
embedding H into an extended Hilbert Space [6]
H ⊂ H
A
⊗ H
B
, (1.4)
where H
A
and H
B
are allowed to contain surface charges at their respective boundaries
that transform non-trivially under the boundary gauge group. These additional degrees of
freedom, which we refer to as “entanglement edge modes”, represent the minimal extension
needed to construct a factorizable Hilbert space from H. The physical Hilbert space is
contained inside a gauge invariant subspace of H
A
⊗ H
B
: within the enlarged Hilbert
space, one can trace over B and define the entanglement entropy as
S
A
= − tr ρ
A
log ρ
A
(1.5)
In the physical, gauge invariant state, the Gauss law requires a matching of the electric field
across the boundaries which induces entanglement between the edge modes on ∂A and ∂B.
For example, in a lattice gauge theory where each link supports a Hilbert space spanned
by the representation matrix elements of the gauge group, the entanglement entropy (EE)
takes the general form [7].
S
A
= −
X
R
∂
P (R
∂
) log P (R
∂
) +
X
R
∂
P (R
∂
)
X
e∈∂
log dim R
e
+ bulk EE (1.6)
where R
∂
denotes the set of representations labeling the links on the boundary of region
A, e denotes the individual edges on the boundary and dim R
e
is the dimension of the
representation R
e
associated with an edge e. The first two term come from the edge
modes:the first is the Shannon entropy associated with the probability distribution P (R
∂A
)
of representations over the boundary, and the second is a term extensive in the size of the
boundary that appears in non-abelian gauge theories. It may seem strange to include
the entropy due to edge modes, which are gauge variant degrees of freedom added for
the sake of factorizing the Hilbert space. However, recent work have shown that these
terms have a real significance. For example in U(1) Maxwell theory, the Shannon-entropy
of the edge modes is necessary to reconcile the replica trick and thermal calculation of
entanglement entropy [8, 9], and in topological gauge theories like Chern Simons theory,
the log dim R term is responsible for the topological entanglement entropy, which is an
important diagnostic for topological phases.
In [1],the authors developed a classical phase space analog of extended Hilbert space for
Yang- Mills and Einstein gravity and showed how the edge degrees of freedom are needed
to define the theories in a subregion in a gauge invariant way. So far, few examples of
the extended Hilbert space have been worked out in the continuum with the exception of
two-dimensional Yang-Mills and U(1) Maxwell theory. In this work, we show how to derive
the extended Hilbert space for Chern Simons theory in the frame work of [1] and explain
how this naturally arises from a proper regularization of the Euclidean path integral which
prepares the quantum state in question. Many of the results in this work have appeared in
– 2 –
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