soft open fences

Entanglement in BF theory II:
edge-modes

Jackson R. Fliss1,fish and Stathis Vitouladitis2,monkey face

1 Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge CB3 0WA, United Kingdom

2 Institute for Theoretical Physics, University of Amsterdam,

1090 GL Amsterdam, The Netherlands

fish<EMAIL_ADDRESS>monkey face<EMAIL_ADDRESS>
## Abstract

We consider the entanglement entropy arising from edge-modes in Abelian
$p$-form topological field theories in $d$ dimensions on arbitrary spatial
topology and across arbitrary entangling surfaces. We find a series of
descending area laws plus universal corrections proportional to the Betti
numbers of the entangling surface, which can be taken as a higher-dimensional
version of the “topological entanglement entropy.” Our calculation comes in
two flavors: firstly, through an induced edge-mode theory appearing on the
regulated entangling surface in a replica path integral and secondly through a
more rigorous definition of the entanglement entropy through an extended
Hilbert space. Along the way we establish several key results that are of
their own merit. We explain how the edge-mode theory is a novel combination of
$(p-1)$-form and $(d-p-2)$-form Maxwell theories linked by a chirality
condition, in what we coin a _chiral mixed Maxwell theory_. We explicitly
evaluate the thermal partition function of this theory. Additionally we show
that the extended Hilbert space is completely organized into representations
of an infinite-dimensional, centrally extended current algebra which naturally
generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We
construct the Verma modules and the representation characters of this algebra.
Lastly, we connect the two approaches, showing that the thermal partition
function of the chiral mixed Maxwell theory is precisely an extended
representation character of our current algebra, establishing an exact
correspondence of the edge-mode theory and the entanglement spectrum.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
  2. 2 The BF path integral
    1. 2.1 On a closed manifold
    2. 2.2 On a manifold with boundary
  3. 3 Entanglement from the replica path integral
    1. 3.1 The edge-mode theory
    2. 3.2 Extracting the entanglement entropy
  4. 4 The extended Hilbert space and a current algebra
    1. 4.1 Gauge-(in)variance, entanglement, and the extended Hilbert space
    2. 4.2 A current algebra
    3. 4.3 Mode expansion of the current algebra
    4. 4.4 Verma modules, characters, and extended characters
    5. 4.5 The entanglement entropy
  5. 5 Discussion
  6. A Partition functions of higher-form gauge theories
    1. A.1 The partition function of $(p-1)$-form Maxwell theory
    2. A.2 The partition function of chiral mixed Maxwell theory
  7. B Laplacians, zeta functions, and heat kernels

##  1 Introduction

Long-range entanglement is a notion with potent conceptual and practical
utility in characterizing quantum phases of matter. In gapped ground states,
short-range correlations manifest in an ‘area law’ entanglement coming from
degrees of freedom localized and straddling the boundary of the region of the
interest (within a correlation length), the so-called “entangling surface.”
Important, however, are potential long-range corrections to this area law.
This is exemplified in (2+1)-dimensional gapped systems where a constant
negative correction to the area law can arise from non-local features of the
ground state, which constrain the short-ranged correlations at the entangling
surface. This is the celebrated “topological entanglement entropy” and is a
smoking gun of (2+1)d topological order [1, 2].

This story is mirrored beautifully in topological quantum field theory (TQFT),
which provides IR effective field theories of topological order: when
restricted to a region of spacetime, TQFTs are host to a robust spectrum of
“edge-modes” localized to the boundary of the region [3, 4, 5, 6]. These edge-
modes are the inheritance of bulk gauge transformations, which are broken by
the existence of a boundary. When “gluing” a region to its complement to form
a complete state, the edge-modes on either side of the common boundary are
maximally entangled up to global constraints [7, 8], giving a divergent area-
law entanglement entropy with universal constant corrections. TQFT also
provides powerful avenues to corroborate these results, such as through the
replica trick and surgery [9, 8].

By this point, the role of edge-modes in topological entanglement entropy is
extremely well understood in (2+1) dimensions; while generally believed to
extend, their role is still relatively unexplored in general dimensions. In
this work we make progress in this direction, focussing on the Abelian
topological phases described by $p$-form BF theories in $d$ dimensions:

$\displaystyle
S_{\text{BF}}\coloneqq\frac{\mathbb{K}^{\mathsf{I}\mathsf{J}}}{2\pi}\int
B_{\mathsf{I}}\mathbin{\scalebox{0.8}{$\wedge$}}\differential A_{\mathsf{J}}.$
(1.1)

Above $A_{\mathsf{J}}$ is a vector of $p$-forms and $B_{\mathsf{I}}$ is a
vector of $(d-p-1)$-forms and $\mathbb{K}$ is a rank $\kappa$ symmetric
matrix111In principle, this action doesn’t require $\mathbb{K}$ to be
symmetric, however only the symmetric part will participate in the edge mode
entanglement and this saves us introducing extra notation. of integer entries.
These are theories whose ground states are $p$-form membrane condensates. In a
companion paper, we explained how such theories display an extreme long-range
form of entanglement, what we name “essential topological entanglement” (ETE),
which must be present in the strict IR limit of a TQFT [10]. The ETE is
entirely finite: absent from it are all contributions from UV degrees of
freedom and their subsequent area law. However, it is still important to know
the UV contributions to entanglement entropy which provide benchmarks for
simulating topological order with a given UV model222A real-world material
being a pertinent example! (such as through tensor networks or matrix product
states). In this direction, the potential contributions to topological
entanglement entropy in $p$-form condensates were deduced by Grover, Turner,
and Vishwanath (GTV) [11] by focussing on lattice gauge theories as a
particular UV realization. Despite various example calculations333See e.g.
[12, 13, 14, 15, 16, 17]. See [18] for a very general lattice gauge theory
calculation., to date there has not been a comprehensive computation performed
in the corresponding continuum TQFT for generic entangling surfaces and any
dimension. In this paper, we fill this gap.

We will explain how the edge-mode spectrum leads to area (plus subleading
area) law entanglement entropy with constant pieces that are sensitive to
topological features of the entangling surface. To be concrete, we find

$\displaystyle\mathcal{S}_{\text{EE}}$
$\displaystyle=\sum_{k=1}^{\left\lfloor\frac{d-1}{2}\right\rfloor}\mathrm{C}^{(p-1)}_{k}\quantity(\frac{\ell}{\varepsilon})^{d-2k}+\frac{\kappa}{2}\quantity(\mathcal{I}_{\frac{d-2}{2}}^{(p-1)}+\mathcal{I}_{\frac{d-2}{2}}^{(d-p-2)})\delta_{d,{\text{even}}}\log(\frac{\ell}{\varepsilon})$
$\displaystyle\phantom{=~{}}-\frac{1}{2}\quantity(\mathrm{b}_{p}+\mathrm{b}_{d-p-1})\log\absolutevalue{\det\mathbb{K}}.$
(1.2)

where $\varepsilon$ is a short-distance regulator, $\mathrm{C}^{(p-1)}_{k}$
are non-universal dimensionless numbers, that we compute,
$\mathcal{I}_{\frac{d-2}{2}}^{(k)}$ is the $\quantity(\frac{d-2}{2})$-th heat
kernel coefficient for a particular spectral zeta function, $\ell$ is a
characteristic length scale, and $\mathrm{b}_{n}$ denotes the $n$-th Betti
number of the entangling surface. This is the first main result of this paper.
Our result differs from the GTV result; in the discussion, Section 5, we
discuss to what extent the two results can be reconciled through local terms
to the entangling surface. We arrive at this result through two independent,
yet conceptually complementary, avenues.

  1. (I)

Firstly we utilize the replica trick and perform path integral on a replica
manifold, regulated by excising a small region (of circumference
$\varepsilon$) about the entangling surface. The resulting edge-mode theory on
this regulated entangling surface is a novel mix of $(p-1)$-form and
$(d-p-2)$-form Maxwell theories, tied together by a “chirality” condition. We
coin this theory “chiral mixed Maxwell theory.” The existence of such theories
as well as their thermal partition function, which we explicitly compute, is a
second main result of this paper. This edge-mode theory contributes an entropy
given by Section 1.

  2. (II)

Secondly, and more rigorously, we address the subtle issues of gauge-
invariance in defining entanglement entropy by moving to an “extended Hilbert
space” (EHS) [19, 20, 21, 22, 23]. We show that the EHS is organized by an
infinite-dimensional current algebra, akin to the Kac–Moody algebras that
arise at the edge of (2+1) dimensional phases. This algebra has also appeared
in the context of 4d Abelian Maxwell theory [24], where fixes the spectrum of
the theory and can, for instance, be used to establish a state-operator
correspondence for non-local operators [25]. However, to our knowledge, the
existence of these algebras for general $p$-forms and in general dimensions
has not been explored. Here we will elucidate them in detail, construct their
Verma modules, and compute their representation characters. We regard this as
a third main result of this paper. Unlike in (2+1)d, these algebras are not
necessarily conformal. Regardless, they completely fix the computation of
Section 1 which arises from the high-temperature limit of the representation
character of this algebra (i.e., a regulated count of the representation
dimension).

Along the way, we explain the connection between these two approaches: the
chiral mixed Maxwell theory appearing in (I) has a spectrum that is completely
fixed by the infinite-dimensional algebras of approach (II). Correspondingly,
their partition function is given exactly by a representation character, which
is the ultimate source of the match in Section 1. This is a precise analogue
of “edge spectrum = bulk entanglement spectrum” promoted to higher dimensions.
We regard this as a final major result of this paper.

An organizing summary of the paper is as follows. In Section 2 we introduce
the BF theory and perform its path integral on manifolds with and without
boundary. While there are many results for the BF path integral on closed
manifolds, we will be very careful keeping track of factors of $\mathbb{K}$,
which, to our knowledge, had not been fully nailed down in the previous
literature. We will use these results in Section 3 to evaluate the replica
integral and describe how the computation ultimately results from the
partition function of the chiral mixed Maxwell theory. In Section 4 we shift
gears and describe the more systematic definition of the entanglement entropy
through the extended Hilbert space and describe features of the resulting
current algebra that organizes it. We then use the representation characters
of this algebra to compute the entanglement entropy, finding a match with
Section 3. In the discussion, Section 5, we will put our results in context
with the known results of GTV, as well as the ETE of this theory, and possible
future extensions of our computation.

###  1.1 Notation

For the reader’s ease, we introduce here some basic notation that will be used
in what follows.

We will analyze theories on torsion-free manifolds of spacetime dimension $d$,
which we shall collectively denote as $X$. These manifolds may have a
boundary, which we will embed into $X$ using the map
$i_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}:{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X\hookrightarrow X$. Theories will be quantized on
manifolds of dimension $D\equiv d-1$, which we will refer to as $\Sigma$,
often calling it the “Cauchy slice” without reference to any causal structure
of the TQFT. We will consider a subregion $R$, which is the closure of a
$D$-dimensional embedded open submanifold of $\Sigma$. The interior of $R$ is
$\overline{R}\coloneqq R\setminus{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$, and its complement is ${R}^{\mathsf{c}}$, the
closure of $\Sigma\setminus R$. Note that
$R\cap{R}^{\mathsf{c}}={\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$. We will denote the space of forms of degree $p$
as $\Omega^{p}(\;\cdot\;)$. Unless stated otherwise, these forms will be real
valued. Cohomology groups will be denoted with their degree placed upstairs,
$\mathrm{H}^{p}(\;\cdot\;)$, while homology groups will be denoted with their
degree placed downstairs, $\mathrm{H}_{p}(\;\cdot\;)$, and these groups are
always defined with integer coefficients, unless stated otherwise. For
compact, boundary-less manifolds, we notate the dimensions of the groups by
the Betti number, i.e.:

$\mathrm{b}_{p}(\;\cdot\;)\coloneqq\dim\mathrm{H}^{p}(\;\cdot\;)=\dim\mathrm{H}_{p}(\;\cdot\;).$
(1.3)

For (co)homology groups on manifolds with boundary or for relative homology
groups, we will always explicitly write the dimension.

##  2 The BF path integral

In this section, we describe the path integral quantization of multi-
component, Abelian, $p$-form BF theory, on a $d$-dimensional, torsion-free
manifold, $X$, with a potentially nonempty boundary,
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}X$. We will start by
reminding the reader of the procedure in the case where $X$ is a closed
manifold, presenting general results with the added benefit of a careful
accounting of the level matrix. We then move on to modify the situation in the
presence of boundaries.

###  2.1 On a closed manifold

Let us set the stage by starting with a closed manifold. We consider BF theory
defined by the action

$\displaystyle
S_{\text{BF}}[A,B]\coloneqq\frac{\mathbb{K}^{\mathsf{I}\mathsf{J}}}{2\pi}\int_{X}B_{\mathsf{I}}\mathbin{\scalebox{0.8}{$\wedge$}}\differential
A_{\mathsf{J}}.$ (2.1)

In the above, $A_{\mathsf{I}}\in\Omega^{p}(X)$ and
$B_{\mathsf{J}}\in\Omega^{d-p-1}(X)$ are vectors of $p$\- and $(d-p-1)$-form
gauge fields respectively. The level matrix, $\mathbb{K}$ is a symmetric,
integer, and non-degenerate matrix of rank $\kappa$. In what follows, we will
drop the indices, wherever not necessary, to simplify the notation. Note that,
as emphasized above, this definition makes sense whenever $X$ is torsion-free.
We will comment on the case of torsion manifolds at the end of this
subsection. For a more general and precise definition of BF theory, we refer
the reader to [10, Appendix A].

The equations of motion arising from varying the action 2.1 are flatness
conditions:

$\displaystyle{\text{{eom}}}[A]=\frac{\mathbb{K}}{2\pi}\differential{A}=0\qquad\qquad{\text{{eom}}}[B]=(-1)^{(d-p)(p+1)}\frac{\mathbb{K}}{2\pi}\differential{B}=0.$
(2.2)

Moreover, the action 2.1 possesses a gauge redundancy of the form

$\updelta A=\alpha\mbox{\quad and\quad}\updelta B=\beta,$ (2.3)

where $\alpha\in\Omega_{\text{cl}}^{p}(X)$ and
$\beta\in\Omega_{\text{cl}}^{d-p-1}(X)$, are closed $p$\- and $(d-p-1)$-forms
respectively. Note that there are two types of gauge shifts. Shifts by
harmonic forms correspond to large gauge transformations, in the sense that
they are not continuously connected to the identity, and shifts by exact forms
correspond to the usual infinitesimal gauge transformations. To properly
quantize the theory in the path integral formalism, we must therefore divide
by the volume of the gauge groups. However, note that there is a tower of
reducibility of the gauge parameters. For example, splitting $\alpha$ off as
$\alpha=\alpha_{0}+\differential\alpha_{1}$, with
$\alpha_{0}\in\operatorname{Harm}^{p}(X)$ and $\alpha_{1}\in\Omega^{p-1}(X)$,
the parameter $\alpha_{1}$ generates the same gauge transformation as
$\alpha_{1}+\tilde{\alpha}_{1}$, with
$\tilde{\alpha}_{1}\in\Omega^{p-1}_{\text{cl}}(X)$. This tower continues until
one reaches a zero-form gauge redundancy. This redundancy is encoded in the BF
path integral in the form of the volume of the gauge group:

$\displaystyle\mathcal{Z}_{\text{BF}}[X]=\int\frac{\mathinner{\mathrm{D}A}\,\mathinner{\mathrm{D}B}missing}{\opbraces{\operatorname{vol}}(\mathcal{G})}\
\mathrm{e}^{\mathrm{i}S_{\text{BF}}[A,B]}$

where, $\mathcal{G}$ is the total gauge group,
$\mathcal{G}=\mathcal{G}_{p}\times\mathcal{G}_{d-p-1}$, with

$\displaystyle\mathcal{G}_{k}\coloneqq\Omega^{k}_{\text{cl}}(X)\big{/}\mathcal{G}_{k-1},$
(2.4)

and $\mathcal{G}_{0}=\operatorname{Harm}^{0}(X)$. In what follows we will be
careless with overall numerical coefficients, such as factors of $2\pi$ in the
partition function, as they can be absorbed in an overall normalization of the
path integral measure. However, we will pay extra attention to factors of
$\mathbb{K}$, as they play a crucial role in the universal terms of the
entanglement entropy.

One commonly employed method to properly quantize such higher-gauge theories
is to include ghosts and ghosts-for-ghosts, and so on, until the gauge
transformations are completely resolved [26]. An alternative approach, due to
[27], which was shown to be equivalent, at the level of determinants, to that
of Blau and Thompson [26] (which is in turn also equivalent to Schwarz’s
method of resolvents [28]), is to Hodge-decompose the fields as

$\begin{split}A&=A_{0}+\differential{A_{\perp}}+\differential^{\dagger}{A_{\parallel}},\\\
B&=B_{0}+\differential{B_{\perp}}+\differential^{\dagger}{B_{\parallel}}.\end{split}$
(2.5)

with $A_{0}\in\operatorname{Harm}^{p}(X)$, $A_{\perp}\in\Omega^{p-1}(X)$,
$A_{\parallel}\in\Omega^{p+1}(X)$, $B_{0}\in\operatorname{Harm}^{d-p-1}(X)$,
$B_{\perp}\in\Omega^{d-p-2}(X)$, and $B_{\parallel}\in\Omega^{d-p}(X)$ and
perform the path integral directly. In what follows, we will show that, with a
bit of care, this method also correctly reproduces the harmonic correction to
the partition function444This corresponds to the refinement of the Ray–Singer
torsion as an element of the determinant line bundle,
$\det\mathrm{H}^{\bullet}$, of $\mathrm{H}^{\bullet}(X)$. and we will obtain
the level dependence of the partition function. With the decomposition 2.5,
taking into account the Jacobians of the transformation, the path integral
measure decomposes as

$\displaystyle\mathinner{\mathrm{D}A}=\mathinner{\mathrm{D}A_{0}}\
\mathinner{\mathrm{D}A_{\perp}}\mathinner{\mathrm{D}A_{\parallel}}\
\quantity(\opbraces{\det{}^{\prime}}_{\Omega^{p-1}(X)}(\differential^{\dagger}\differential))^{1/2}\quantity(\opbraces{\det{}^{\prime}}_{\Omega^{p+1}(X)}(\differential\differential^{\dagger}))^{1/2},$
(2.6)

and similarly for $\mathinner{\mathrm{D}B}$, while the action takes the form

$\displaystyle
S_{\text{BF}}[A,B]=\frac{\mathbb{K}}{2\pi}\left\langle\differential^{\dagger}B_{\parallel},\star\differential\differential^{\dagger}A_{\parallel}\right\rangle,$
(2.7)

where
$\left\langle{\color[rgb]{0.65625,0.2890625,0.2890625}\definecolor[named]{pgfstrokecolor}{rgb}{0.65625,0.2890625,0.2890625}\bullet},{\color[rgb]{0.23046875,0.53515625,0.765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.53515625,0.765625}\bullet}\right\rangle$
is the Hodge inner product,
$\int_{X}{\color[rgb]{0.65625,0.2890625,0.2890625}\definecolor[named]{pgfstrokecolor}{rgb}{0.65625,0.2890625,0.2890625}\bullet}\mathbin{\scalebox{0.8}{$\wedge$}}\star{\color[rgb]{0.23046875,0.53515625,0.765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.53515625,0.765625}\bullet}$.
Following [29] we will normalize the path integral measure as

$\displaystyle\mathinner{\mathrm{D}A}=\prod_{x\in
X}\mathbb{E}\differential{A_{x}}\qquad\qquad\mathinner{\mathrm{D}B}=\prod_{x\in
X}\mathbb{E}\differential{B_{x}},$ (2.8)

where $\mathbb{E}$ is a frame matrix, satisfying $\mathbb{E}^{2}=\mathbb{K}$.
This has the effect of removing the $\mathbb{K}$ dependence from the
functional determinants, arising upon integrating over $A_{\parallel}$ and
$B_{\parallel}$. Note that this is not the only choice. Normalizing the modes
as $\sim\mathbb{K}^{\alpha}\differential{A_{x}}$ and
$\sim\mathbb{K}^{1-\alpha}\differential{B_{x}}$, for some real $\alpha$, still
takes care of the $\mathbb{K}$-dependence of the functional determinants, but
as will become evident below, it rescales the partition function as

$\mathcal{Z}_{\text{BF}}[X]\longmapsto\absolutevalue{\det\mathbb{K}}^{\alpha\,\upchi(X)}\mathcal{Z}_{\text{BF}}[X],$
(2.9)

where $\upchi(X)$ is the Euler characteristic of $X$. This ambiguity is on the
one hand, well documented and understood [30] and on the other hand harmless
if one is interested in topological entanglement entropy. The reason is,
ultimately, that it can be seen to arise from (or equivalently can be absorbed
into) a locally integrated contribution [11]. We will return to this point in
the discussion, Section 5.

Ignoring this ambiguity, for the reasons mentioned above, the integral over
$A_{\parallel}$ and $B_{\parallel}$ with the measure 2.8 produces

$\mathcal{Z}_{\text{BF}}^{\parallel}[X]=\quantity(\opbraces{\det{}^{\prime}}_{\Omega^{p}(X)}(\differential^{\dagger}\differential))^{-\frac{3}{2}}.$
(2.10)

Integrating over $A_{\perp}$ and $B_{\perp}$, modulo small gauge
transformations, i.e. the part of 2.4 generated by
$\differential\Omega^{k-1}(X)$, gives an alternating product of determinants,
which can be combined with 2.10 to give the analytic torsion555Actually we get
this expression for odd-dimensional manifolds. For even-dimensional manifolds
we get a different power of the (determinant expression of the) analytic
torsion. However, on even-dimensional manifolds the analytic torsion is unity,
so we can write 2.11 for all dimensions. [27]:

$\displaystyle\mathcal{Z}_{\text{BF}}^{\perp}[X]\mathcal{Z}_{\text{BF}}^{\parallel}[X]=\prod_{k=0}^{d}\quantity(\opbraces{\det{}^{\prime}}_{\Omega^{k}(X)}\triangle_{n})^{\frac{k}{2}(-1)^{p+k}}\eqqcolon\mathrm{T}_{\text{A}}[X]^{(-1)^{p-1}}.$
(2.11)

As it stands, $\mathrm{T}_{\text{A}}[X]$ depends on the metric, whenever not
all cohomology groups are trivial. So this cannot be the final expression of
the partition function; it is the zero-modes that will provide the fix, as is
usually the case. Let us see what their contribution is. The integral over
$A_{0}$ and $B_{0}$, modulo large gauge transformations, with the measure 2.8
gives [29, 31]:

$\displaystyle\mathcal{Z}_{\text{BF}}^{0}[X]$
$\displaystyle=\quantity(\prod_{k=0}^{p}\absolutevalue{\det\quantity(\mathbb{K}\otimes\mathbb{G}_{k})}^{\frac{1}{2}(-1)^{p-k}})\quantity(\prod_{\ell=0}^{d-p-1}\absolutevalue{\det\quantity(\mathbb{K}\otimes\mathbb{G}_{\ell})}^{\frac{1}{2}(-1)^{d-p-1-\ell}}),$
(2.12)

where $\mathbb{G}_{k}$ is the metric on the moduli space of harmonic
$k$-forms, i.e. the Gram matrix of the topological basis of harmonic
$k$-forms. More explicitly, the topological basis,
$\quantity{\tau^{(k)}_{\mathsf{i}}}_{\mathsf{i}=1}^{\mathrm{b}_{k}(X)}$, is
defined by the relation

$\displaystyle\int_{C^{\mathsf{i}}_{(k)}}\tau^{(k)}_{\mathsf{j}}=\delta^{\mathsf{i}}_{\mathsf{j}},$
(2.13)

for a fixed basis,
$\quantity{C^{\mathsf{j}}_{(k)}}_{\mathsf{i}=1}^{\mathrm{b}_{k}(X)}$, of
$k$-cycles. Then $\mathbb{G}_{k}$ is defined as

$\displaystyle\matrixquantity[\mathbb{G}_{k}]_{\mathsf{i}\mathsf{j}}\coloneqq\left\langle\tau^{(k)}_{\mathsf{i}},\tau^{(k)}_{\mathsf{j}}\right\rangle.$
(2.14)

Poincaré duality implies that (up to a unimodular matrix, which can be set to
one by a suitable choice of basis of the $k$-cycles)
$\mathbb{G}_{d-k}={{\mathbb{G}_{k}}^{-1}}$. Utilizing this, we get:

$\displaystyle\mathcal{Z}_{\text{BF}}^{0}[X]=\absolutevalue{\det\mathbb{K}}^{\mathrm{h}_{p}(X)}\mathrm{T}_{\mathrm{H}^{\bullet}}[X]^{(-1)^{p-1}},$
(2.15)

where

$\displaystyle\mathrm{h}_{p}(X)$
$\displaystyle\coloneqq\frac{1}{2}(-1)^{p}\sum_{k=0}^{p}(-1)^{k}\mathrm{b}_{k}(X)+\frac{1}{2}(-1)^{d-p-1}\sum_{k=0}^{d-p-1}(-1)^{k}\mathrm{b}_{k}(X)$
$\displaystyle\overset{\upchi}{=}(-1)^{p}\sum_{k=0}^{p}(-1)^{k}\mathrm{b}_{k}(X),$
(2.16)

where the last equality is modulo removing factors of the Euler
characteristic, which can be done by adding a local counterterm, as we alluded
to above. Moreover, we have defined

$\displaystyle\mathrm{T}_{\mathrm{H}^{\bullet}}[X]\coloneqq\prod_{k=0}^{d}\quantity(\det\mathbb{G}_{k})^{\frac{1}{2}(-1)^{k+1}}.$
(2.17)

$\mathrm{T}_{\mathrm{H}^{\bullet}}[X]$ is precisely the cohomological
correction to the analytic torsion, necessary to form the metric-independent
Ray–Singer torsion:

$\displaystyle\mathrm{T}_{\text{RS}}[X]=\mathrm{T}_{\mathrm{H}^{\bullet}}[X]\mathrm{T}_{\text{A}}[X].$
(2.18)

Putting everything together, the partition function for multi-component,
$p$-form BF theory on a compact, closed manifold $X$, reads:

$\mathcal{Z}_{\text{BF}}[X]=\absolutevalue{\det\mathbb{K}}^{\mathrm{h}_{p}(X)}\;\mathrm{T}_{\text{RS}}[X]^{(-1)^{p-1}}.$
(2.19)

This form of the partition function has a number of desirable features which
we list below.

  * •

It is topological. Of course, this was anticipated since the very first
appearance of the theory, in [28]. However, to obtain it in the path integral
formalism requires some care, regardless of the method one chooses. The method
of resolutions [28] was unable to reproduce it, in cases with non-trivial
cohomology groups. In the BRST formulation, one needs to append the action by
a BRST-closed, but not necessarily BRST-exact, quantum action [26, 31]. In the
Batalin–Vilkovisky (BV) [32] formulation of the theory [33] one needs to pay
special attention to the residual superfields, furnishing the fiber of the BV
integral. Finally, in the direct method that we employed one must be carfeul
about the zero-modes appearing in the tower of gauge-for-gauge-for-…-gauge
volumes; note, for example, that in [27] the zero-mode piece is not correctly
reproduced and hence the formulas there depend implicitly on a choice of
metric on $X$.

  * •

On a three-dimensional manifold, with $p=1$, 2.19 is the square of the
partition function of Abelian Chern–Simons theory. More explicitly, on
$\mathbb{S}^{3}$ we obtain:

$\displaystyle\mathcal{Z}_{\text{BF}}\quantity[\mathbb{S}^{3}]=\absolutevalue{\det\mathbb{K}}^{-1},$
(2.20)

which is to be compared with
$\mathcal{Z}_{\text{CS}}[\mathbb{S}^{3}]=\absolutevalue{\det\mathbb{K}}^{-1/2}$
in [34]. For a more general case, in $d=3$ with $p=1$, allowing also for
torsion, see also [35].

  * •

On $X=\mathbb{S}^{1}\times\Sigma$,
$\mathcal{Z}_{\text{BF}}[\mathbb{S}^{1}\times\Sigma]$ simply counts the number
of ground states on $\Sigma$. Equivalently, it measures the dimension of the
Hilbert space on $\Sigma$. This is given by
$\dim\mathcal{H}_{\Sigma}=\absolutevalue{\det\mathbb{K}}^{\mathrm{b}_{p}(\Sigma)}$
[10]. From 2.19, making use of the Künneth formula:
$\mathrm{b}_{k}(X)=\mathrm{b}_{k}(\Sigma)+\mathrm{b}_{k-1}(\Sigma)$, if
$k\geqslant 1$, while $\mathrm{b}_{0}(X)=\mathrm{b}_{0}(\Sigma)$, we correctly
find

$\mathcal{Z}_{\text{BF}}\quantity[\mathbb{S}^{1}\times\Sigma]=\absolutevalue{\det\mathbb{K}}^{\mathrm{b}_{p}(\Sigma)}.$
(2.21)

Lastly, let us mention that if we allow $X$ to have torsion, in which case we
should take into account the differential cohomology definition of BF theory
[10, Appendix A], the expression for the partition function is modified as
follows. Let
$\mathrm{H}_{p}(X;\mathbb{Z})=\mathbb{Z}^{\mathrm{b}_{p}(X)}\oplus\mathrm{T}_{p}(X)$,
where
$\mathrm{T}_{p}(X)=\mathbb{Z}_{p_{1}(X)}\oplus\cdots\oplus\mathbb{Z}_{p_{N}(X)}$
is the torsion part. Then, we can obtain from [36], upon a slight modification
to account for the multi-component case and to scale away numerical
coefficients, that

$\mathcal{Z}_{\text{BF}}[X]=\absolutevalue{\det\mathbb{K}}^{\mathrm{h}_{p}(X)}\;\absolutevalue{\opbraces{\operatorname{Hom}}(\mathrm{T}_{p}(X),\mathbb{Z}^{\kappa}\big{/}\opbraces{\operatorname{im}}\mathbb{K})}\;\mathrm{T}_{\text{RS}}[X]^{(-1)^{p-1}}.$
(2.22)

Note that Poincaré duality and the universal coefficient theorem give
$\mathrm{T}_{p}(X)\cong\mathrm{T}_{d-p-1}(X)$, making this expression
invariant upon exchanging $A$ and $B$. At this stage, the inclusion of torsion
seems like a mathematical curiosity. Indeed, we will not discuss further
manifolds with torsion in this paper. However, this result may be of relevance
for a future utilization of “surgery,”666 In this method one needs to compute
the partition function of BF theory on a branched cyclic cover of $X$,
ramified over the entangling surface. Although the topology of the replica
manifold is completely encoded in the topology of the entangling surface and
the original manifold, the replica manifold can have non-trivial torsion, even
if the original manifold does not [37]. see the discussion, Section 5.

###  2.2 On a manifold with boundary

Consider now the case where $X$ has a boundary, ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X\neq\varnothing$. As explained in [10], the action
2.1, gives rise to a boundary symplectic form, which as it stands is
consistent with fixing $A$ as a boundary condition. In general, the action is
supplemented by a boundary term, $S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$. This leads to a modified variational problem,
specified by symplectic potential on the boundary,
$\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}$. Explicitly, the variation of the total action
$S_{\text{full}}=S_{\text{BF}}+S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$ reads:

$\displaystyle\mathinner{\updelta
S_{\text{full}}}=\left\langle{\text{{eom}}}[A],\mathinner{\updelta
B}\right\rangle_{X}+\left\langle{\text{{eom}}}[B],\mathinner{\updelta
A}\right\rangle_{X}+\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X},$ (2.23)

where $\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}$ can be put in Darboux form:

$\displaystyle\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}=\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}\mathbf{p}\mathbin{\scalebox{0.8}{$\wedge$}}\star_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathinner{\updelta\mathbf{q}},$ (2.24)

with $\mathbf{q}$ and $\mathbf{p}$ being functions of the boundary values of
the fields, $A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}\coloneqq
i^{*}_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}A$,
$B_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}\coloneqq
i^{*}_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}B$. Having the
symplectic potential in the form of 2.24 we have two options to proceed, in
order to have a well-defined variational problem: fix
$\mathbf{p}\overset{!}{=}0$ or $\updelta\mathbf{q}\overset{!}{=}0$. Without
loss of generality, we will proceed with the Dirichlet approach,
$\mathbf{p}\overset{!}{=}0$. The other boundary condition can be attained by a
boundary symplectomorphism and can be easily shown to give equivalent results.
Note that the symplectic potential is, as it stands, degenerate due to
boundary gauge transformations. We will take care of that below when we
discuss the boundary path integral measure.

The object one would like to study is the partition function of the combined
system:

$\mathcal{Z}_{{\text{BF}}+{\text{edge}}}[X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X]\coloneqq\int_{\mathcal{C}_{\mathbf{p}}}\frac{\mathinner{\mathrm{D}A}\mathinner{\mathrm{D}B}missing}{\opbraces{\operatorname{vol}}(\mathcal{G})}\
\mathrm{e}^{-S_{\text{full}}^{\text{E}}[A,B]},$ (2.25)

where $\mathcal{C}_{\mathbf{p}}\coloneqq{\quantity{A\in\Omega^{p}(X),\
B\in\Omega^{d-p-1}(X)\;\middle|\;\mathbf{p}=0}}$ and
$S_{\text{full}}^{\text{E}}[A,B]$ is the Euclideanized partition
function777note that the bulk piece is not affected by the Wick rotation,
since it is a one-derivative action. However, the boundary terms can change in
the usual way.. Note also that here the gauge group $\mathcal{G}$ consists of
gauge transformations that vanish on the boundary. The next step is to
decompose the fields $A$ and $B$ as follows:

$\displaystyle A$ $\displaystyle=\widetilde{A}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}+\hat{A},$ (2.26) $\displaystyle B$
$\displaystyle=\widetilde{B}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}+\hat{B},$ (2.27)

where $\hat{A}$ and $\hat{B}$ are off-shell gauge fields with Dirichlet
boundary conditions, $i^{*}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\,\hat{A}=0=i^{*}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\,\hat{B}$. The boundary conditions are absorbed
completely by $\widetilde{A}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$ and $\widetilde{B}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$. The latter are, in the spirit of [33], potentially
discontinuous extensions of the boundary fields,
$A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$ and
$B_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$ and are chosen to
be on-shell, i.e. flat. This has the effect of disentangling the bulk, from
the boundary contribution:

$S_{\text{full}}[A,B]=S_{\text{BF}}\quantity[\hat{A},\hat{B}]+S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\quantity[A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}},B_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}].$ (2.28)

Similarly, the measure has a piece coming from the hatted fields and giving
only bulk contributions, and one coming from the tilded fields, giving all the
edge contributions.

The hatted fields are very easy to deal with, they only contribute in the
bulk, and they give the full bulk contribution. The computation is completely
analogous to that of Subsection 2.1. The only difference is that we make use
of the Hodge decomposition theorem for Dirichlet forms on a manifold with
boundary [38]:

$\Omega_{\text{D}}^{\bullet}(X)=\mathrm{H}^{\bullet}(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)\oplus\differential{\Omega_{\text{D}}^{\bullet-1}(X)}\oplus\quantity(\differential^{\dagger}{\Omega_{\text{N}}^{\bullet+1}}\cap\Omega_{\text{D}}^{\bullet}(X)),$
(2.29)

where

$\displaystyle\Omega_{\text{D}}^{\bullet}(X)$
$\displaystyle\coloneqq\quantity{\omega\in\Omega^{\bullet}(X)\;\middle|\;i_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}^{*}\omega=0}$ (2.30)
$\displaystyle\Omega_{\text{N}}^{\bullet}(X)$
$\displaystyle\coloneqq\quantity{\omega\in\Omega^{\bullet}(X)\;\middle|\;i_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}^{*}\star\omega=0}$ (2.31)

denote the spaces of Dirichlet and Neumann forms respectively. Altogether we
have that

$\displaystyle\mathcal{Z}_{\text{bulk}}[X]=\int\frac{\mathinner{\mathrm{D}\hat{A}}\mathinner{\mathrm{D}\hat{B}}missing}{\opbraces{\operatorname{vol}}(\mathcal{G})}\
\mathrm{e}^{\mathrm{i}S_{\text{BF}}\quantity[\hat{A},\hat{B}]}=\absolutevalue{\det\mathbb{K}}^{\mathrm{h}_{p}\quantity(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)}\mathrm{T}_{\text{RS}}\quantity[X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X]^{(-1)^{p-1}},$ (2.32)

where $\mathrm{h}_{p}\quantity(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)$ is the relative version of $\mathrm{h}_{p}(X)$,
appearing in 2.19, i.e.:

$\displaystyle\mathrm{h}_{p}(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)$
$\displaystyle\coloneqq\frac{1}{2}(-1)^{p}\sum_{k=0}^{p}(-1)^{k}\dim\mathrm{H}^{k}(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)+\frac{1}{2}(-1)^{d-p-1}\sum_{k=0}^{d-p-1}(-1)^{k}\dim\mathrm{H}^{k}(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X)$
$\displaystyle\overset{\upchi}{=}(-1)^{p}\sum_{k=0}^{p}(-1)^{k}\dim\mathrm{H}^{k}(X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X).$ (2.33)

The contribution of the tilded fields is a little more subtle. We will
calculate their contribution, with a specific choice of
$S_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$, in detail in
Section 3, but let us already give a rough sketch of the computation for a
general boundary action. First of all, it is clear from the above discussion
that they only contribute on the boundary. Secondly, the flatness constraint
becomes a Bianchi identity for the boundary gauge fields. In particular,
$A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$ and
$B_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$ are curvatures of
$(p-1)-$ and $(d-p-1)$-form gauge fields, $a$ and $b$, respectively. All in
all, the contribution of $\widetilde{A}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$ and $\widetilde{B}_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}$, including the boundary condition $\mathbf{p}=0$,
in the measure is

$\sum_{{\text{instantons}}}\int\frac{\mathinner{\mathrm{D}a}\mathinner{\mathrm{D}b}missing}{\opbraces{\operatorname{vol}}(\mathcal{G}^{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}})}\ \delta[\mathbf{p}],$ (2.34)

where the sum over instantons is a sum over all topologically non-trivial
configurations, i.e. over cohomology $p$\- and $(d-p-1)$-classes,
respectively, of ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}X$,
and $\opbraces{\operatorname{vol}}(\mathcal{G}^{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}})$ indicates the volume of the boundary gauge
transformations. Since $a$ and $b$ appear only through their curvatures, there
is a redundancy upon shifting them by flat gauge fields. Relatedly, dividing
by this volume takes care of the redundancy in the symplectic potential,
$\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}$.

Putting everything together, we find that the partition function, 2.25, takes
the form

$\displaystyle\mathcal{Z}_{{\text{BF}}+{\text{edge}}}[X,{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X]=\mathcal{Z}_{\text{bulk}}[X]\mathcal{Z}_{{\text{edge}}}[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X],$ (2.35)

with $\mathcal{Z}_{\text{bulk}}[X]$ as in 2.32 and
$\mathcal{Z}_{{\text{edge}}}[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X]$ is the edge-mode partition function, computed
with the measure 2.34.

##  3 Entanglement from the replica path integral

In this section, we will calculate the entanglement entropy using path
integral techniques via the replica trick. We begin with a brief primer on
replica path integrals and how they are calculated. The basic ingredient is
the path integral representation of a state. We begin with a Cauchy slice,
$\Sigma$, and a Hilbert space of states associated to it,
$\mathcal{H}_{\Sigma}$. We can imagine generating a state in
$\mathcal{H}_{\Sigma}$ by finding a $d$-dimensional manifold, $X_{-}$, whose
boundary is $\Sigma$. According to the standard rules of topological field
theory, the path integral with specified boundary conditions produces the
wavefunction of a state in $\mathcal{H}_{\Sigma}$. That is,

$\Psi_{X_{-}}[\varphi]\coloneqq\int_{\mathcal{C}[X_{-};\varphi]}\mathrm{D}\Phi\
\mathrm{e}^{\mathrm{i}S[\Phi]},$ (3.1)

is the wavefunction associated with the state

$\displaystyle\ket{\Psi_{X_{-}}}=\int_{\mathcal{C}[\Sigma]}\mathinner{\mathrm{D}\varphi}\Psi_{X_{-}}[\varphi]\ket{\varphi}\in\mathcal{H}_{\Sigma},$
(3.2)

in the Hilbert space $\mathcal{H}_{\Sigma}$ assigned to $\Sigma$. In the
above, we have schematically indicated all fields by $\Phi$ and
$\mathcal{C}[X_{-};\varphi]$ is an appropriate functional space (including
quotienting out gauge redundancies) over $X_{-}$, with boundary conditions
$i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X_{-}}^{*}\Phi=\varphi$. $\mathcal{C}[\Sigma]$ is,
similarly, an appropriate functional space over $\Sigma$. The dual Hilbert
space,
$\mathcal{H}_{\Sigma}^{\smash{\scalebox{0.9}[1.4]{\rotatebox{90.0}{\guilsinglleft}}}}$
is canonically isomorphic with the Hilbert space associated to the orientation
reversal, $\overline{\Sigma}$, of $\Sigma$. Hence, the norm of the state
$\ket{\Psi_{X_{-}}}$, is given by conjugating $\Psi_{X_{-}}$ and integrating
over the boundary conditions. At the level of the path integral, this has the
action of gluing $X_{-}$ onto $X_{+}$, its time-reversal about $\Sigma$,

$\norm{\Psi_{X_{-}}}^{2}=\int_{\mathcal{C}[X]}\mathrm{D}\Phi\
\mathrm{e}^{\mathrm{i}S[\Phi]}=\mathcal{Z}[X],$ (3.3)

yielding the partition function on $X\coloneqq X_{-}\cup_{\Sigma}X_{+}$.
Similarly, the pure density matrix
$\rho\coloneqq\outerproduct{\Psi_{X_{-}}}{\Psi_{X_{-}}}\in\mathcal{H}_{\Sigma}\otimes\mathcal{H}_{\Sigma}^{\smash{\scalebox{0.9}[1.4]{\rotatebox{90.0}{\guilsinglleft}}}}$
has the path integral expression

$\displaystyle\rho[\varphi_{-},\varphi_{+}]$
$\displaystyle\coloneqq\matrixelement{\varphi_{-}}{\rho}{\varphi_{+}}=\Psi_{X_{-}}[\varphi_{-}]\Psi_{X_{-}}^{*}[\varphi_{+}]=$
$\displaystyle=\int_{\mathcal{C}[X_{\Sigma{\text{-cut}}};\varphi_{-},\varphi_{+}]}\mathinner{\mathrm{D}\Phi}\
\mathrm{e}^{\mathrm{i}S[\Phi]},$ (3.4)

where $X_{\Sigma{\text{-cut}}}\coloneqq X_{-}\sqcup X_{+}\cong
X\setminus\quantity(\Sigma\times[0,1])$, which has as its boundary two copies
of $\Sigma$ with opposite orientation, $\Sigma_{\pm}$, on which we impose
boundary conditions $\varphi_{\pm}$, respectively. This density matrix could
possibly be unnormalized.

We now choose a subregion $R\subset\Sigma$ and imagine demarcating

$\varphi_{\pm}=\Theta_{R}\,\varphi_{\pm}+\Theta_{{R}^{\mathsf{c}}}\,\varphi_{\pm}\eqqcolon\varphi_{R,\pm}+\varphi_{{R}^{\mathsf{c}},\pm},$

where $\Theta_{R}$ is the characteristic function of $R$ (taking values 1
inside $R$ and 0 everywhere else) and $\Theta_{{R}^{\mathsf{c}}}=1-\Theta_{R}$
is the characteristic function for the complement region. The reduced density
matrix, $\rho_{R}$, is described, at least at a formal level, by identifying
$\varphi_{{R}^{\mathsf{c}},+}=\varphi_{{R}^{\mathsf{c}},-}$ and integrating
over their values:

$\displaystyle\rho_{R}[\varphi_{R,-},\varphi_{R,+}]$
$\displaystyle=\int_{\mathcal{C}[{R}^{\mathsf{c}}]}\mathrm{D}\varphi_{{R}^{\mathsf{c}}}\rho[\varphi_{R,-}+\varphi_{{R}^{\mathsf{c}}};\overline{\varphi}_{R,+}+\varphi_{{R}^{\mathsf{c}}}]=\int_{\mathcal{C}[X_{R{\text{-cut}}};\varphi_{R,-},\varphi_{R,+}]}\mathrm{D}\Phi\
\mathrm{e}^{\mathrm{i}S[\phi]},$ (3.5)

resulting in a path integral on $X_{R{\text{-cut}}}\coloneqq
X_{-}\cup_{{R}^{\mathsf{c}}}X_{+}$, possessing a cut along $R$, and boundary
conditions $\varphi_{\pm}$, imposed on either side of the cut. This density
matrix is unnormalized: its trace is simply
$\norm{\Psi_{X_{-}}}^{2}=\mathcal{Z}[X]$, which arises from identifying
$\varphi_{R,+}$ with $\varphi_{R,-}$ and then integrating, effectively
“gluing” the cut closed.

We now aim to calculate the von Neumann entropy of $\rho_{R}$. We will make
use of the replica trick. First we compute the $n$-th Rényi entropy as

$\mathcal{S}_{n}\coloneqq\frac{1}{1-n}\log\frac{\Tr\left(\rho_{R}^{n}\right)}{\left(\Tr\rho_{R}\right)^{n}}~{},$
(3.6)

where the denominator arises to normalize $\rho_{R}$, if it was not already
normalized. The von Neumann entropy is then given as the limit:

$\mathcal{S}_{\text{EE}}=\lim_{n\rightarrow 1}\mathcal{S}_{n}~{}.$ (3.7)

The denominator of 3.6 is simply given by $\mathcal{Z}[X]^{n}$, so we now make
sense of the numerator. It arises from copying $X_{R{\text{-cut}}}$ $n$ times;
the trace identifies the boundary conditions,
$\varphi^{(i)}_{R,+}=\varphi^{(i+1)}_{R,-}$ (where $i$ indexes the replicas
mod $n$) and their subsequent integration glues the replicated manifold into
an $n$-fold branched cover over ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$, which we denote $X_{n}$. This gives a path
integral expression of the entanglement entropy as

$\mathcal{S}_{\text{EE}}=\lim_{n\rightarrow
1}\frac{1}{1-n}\log\frac{\mathcal{Z}[X_{n}]}{\mathcal{Z}[X]^{n}}=-\evaluated{\partialderivative{n}(\log\frac{\mathcal{Z}[X_{n}]}{\mathcal{Z}[X]^{n}})}_{n=1}.$
(3.8)

We are now tasked with evaluating the path integrals in question taking care
with the codimension-2 surface fixed by the $\mathbb{Z}_{n}$ replica symmetry
— the entangling surface. We will address this by regulating the entangling
surface and looking at the edge theory that arises there. To be precise, we
will literally excise a tubular neighborhood around the entangling surface: in
terms of the state on $\Sigma$, this has the interpretation of putting a small
buffer region between $R$ and ${R}^{\mathsf{c}}$. This will serve as a UV
regulator. Specifically, we replace the bulk replica manifold $X_{n}$ with
$\mathcal{X}_{n,\varepsilon}\coloneqq
X_{n}\setminus\left(\mathbb{D}^{2}_{n\varepsilon}\times{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R\right)$ and keep in mind the limit as
$\varepsilon\rightarrow 0$. The interpretation of $\varepsilon$ as a regulator
on the unreplicated state mandates that the circumference of the disk scales
as $n\varepsilon$. We will perform a similar excision of the original
manifold: $X\leadsto\mathcal{X}_{1,\varepsilon}\coloneqq
X\setminus\left(\mathbb{D}^{2}_{\varepsilon}\times{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R\right)$. This yields

$\mathcal{S}_{\text{EE}}=\lim_{\varepsilon\rightarrow 0}\lim_{n\rightarrow
1}\frac{1}{1-n}\log\frac{\mathcal{Z}_{{\text{BF}}+{\text{edge}}}\quantity[\mathcal{X}_{n,\varepsilon}]}{\mathcal{Z}_{{\text{BF}}+{\text{edge}}}\quantity[\mathcal{X}_{1,\varepsilon}]^{n}}~{}.$
(3.9)

Importantly, these are path integrals on manifolds with boundary,
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}=\mathbb{S}^{1}_{n\varepsilon}\times{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$. As we have discussed in Section 2, we need to
supplement these path integrals with boundary conditions and boundary terms on
$\mathcal{X}_{n,\varepsilon}$ enforcing those boundary conditions through the
variational principle. As we have shown in Subsection 2.2, the BF path
integral on this regulated, replica geometry naturally splits into a product
of “bulk” and “edge” terms

$\mathcal{Z}_{{\text{BF}}+{\text{edge}}}\quantity[\mathcal{X}_{n,\varepsilon},{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}]=\mathcal{Z}_{\text{bulk}}\quantity[\mathcal{X}_{n,\varepsilon}]\;\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}].$ (3.10)

We now insert the relevant
$\mathcal{Z}_{{\text{BF}}+{\text{edge}}}\quantity[\mathcal{X}_{n,\varepsilon},{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}]$ into the ratio 3.9. In
this paper, we will focus on the contribution of the edge-modes to the
entanglement entropy. It is precisely these terms that we expect to contribute
an area law. In the discussion, Section 5, we will comment on possible bulk
contributions. Since the above appears as a product, we can neatly isolate the
contribution of the edge-modes as:

$\displaystyle\mathcal{S}_{\text{EE}}=\lim_{\varepsilon\to 0}\lim_{n\to
1}\frac{1}{1-n}\log\frac{\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}]}{\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]^{n}}.$ (3.11)

###  3.1 The edge-mode theory

We now consider the boundary action. The bare BF action, 2.1, is consistent
with either fixing $A$ or annihilating $B$ as a boundary condition. However,
the subsequent integration over the remaining fluctuating boundary degrees of
freedom is now unbounded: we must supplement this theory with a boundary
Hamiltonian888The need to supplement an edge Hamiltonian as a regulator is a
theme that will be echoed in Section 4.. We will focus on quadratic boundary
actions. In general dimensions these quadratic actions will be dimensionful
and general lessons of effective field theory guide us to write the most
relevant one. Without loss of generality we can assume that this comes from
the quadratic action for the $p$-forms and write the following boundary
action:

$S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}[A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}},B_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}]=\frac{\mathbb{K}}{4\pi}\mu^{\Delta}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}}A_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}}\mathbin{\scalebox{0.8}{$\wedge$}}\star
A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}=\frac{\mu^{\Delta}}{4\pi}\left\langle
A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}},\mathbb{K}\,A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\right\rangle,$ (3.12)

where $\mu$ is an arbitrary energy scale and $\Delta=d-1-2p$, as required by
dimensional analysis. With this action, the boundary symplectic form (taking
also into account the boundary terms from the variation of the bulk action)
becomes:

$\boldsymbol{\Theta}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}}=\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}}\quantity(B_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}+\mu^{\Delta}(-1)^{(d-p-1)(p+1)}\star
A_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}})\mathbin{\scalebox{0.8}{$\wedge$}}\mathinner{\updelta
A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}}.$ (3.13)

As such, the variational principle, with the above boundary action, 3.12, is
compatible with demanding

$B_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}+\mu^{\Delta}(-1)^{(d-p-1)(p+1)}\star
A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}\overset{!}{=}0$
(3.14)

as boundary condition. We will call 3.14 a “generalized chiral boundary
condition.”

This boundary action, 3.12, and the corresponding boundary condition, 3.14,
have been known for a long time to appear on the boundary of BF theories, [39,
40]. Ignoring global issues, the flatness condition that the bulk path
integral imposes on $A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$
makes this is a simple theory of a free $(p-1)$-form gauge field, also known
as a singleton mode in the string theory literature. Naturally (Hodge-dually),
it is also a theory of a free $(d-p-2)$-form. However, on a non-trivial
topology, the boundary condition 3.14 imposes a constraint on the fluxes. On a
manifold of the form $\mathbb{S}^{1}_{\varepsilon}\times\Sigma$ — as is our
case here — both fields have fluxes obeying Dirac quantization around purely
spatial cycles, but fractional charges, when wrapping the thermal circle. This
is entirely reminiscent of the story of the two-dimensional chiral boson on a
torus, which can be seen to arise at the edge of a Chern–Simons theory, which
is _not_ modular invariant: upon demanding periodicity around the spatial
circle, one cannot retain periodicity around the thermal circle. It also
reflects the story of self-dual gauge fields in higher dimensions [41,
42]999Of course our approach is much less rigorous than the original story. A
rigorous analysis would require a revisit from the lens of differential
cohomology. We take this as an opportunity to stress its importance for a
solid understanding of gauge theories.. In our case, the theory becomes a
“chiral half” of a $(p-1)$-form and a $(d-p-2)$-form Maxwell theory, which we
call, relatively unimaginatively, _chiral mixed Maxwell theory_. The role of
chiral fields as edge-modes of BF theories was recently reemphasized in [43,
44]. Note, however, that there, as well as in all other instances where chiral
gauge fields make an appearance (see e.g. [45, 42, 41, 48, 49, 50, 51, 52, 53,
46, 47] for an incomplete list) it concerns $k$-form fields in $2k+2$
dimensions, where one can construct a genuinely chiral combination of the
gauge fields. This is, to our knowledge, the first time a chiral pure gauge
theory is constructed in generic dimensions101010and as a bonus, arbitrary
topology.

Let us be more concrete. First, we Hodge-decompose
$A_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$ as
$A_{\text{h}}+\differential{a}$, where $A_{\text{h}}$ is a harmonic $p$-form
and $a$ is the (globally-well defined) $(p-1)$-form gauge field. We do the
same for $B_{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}$. Now, if
we denote by $\differential{\tau}$ the volume element of the thermal circle,
we can split the harmonic parts of the two fields as

$\displaystyle A_{\text{h}}$
$\displaystyle=\mathcal{A}+\differential{\tau}\mathbin{\scalebox{0.8}{$\wedge$}}\widetilde{\mathcal{A}}$
(3.15) $\displaystyle B_{\text{h}}$
$\displaystyle=\mathcal{B}+\differential{\tau}\mathbin{\scalebox{0.8}{$\wedge$}}\widetilde{\mathcal{B}},$
(3.16)

where $\mathcal{A}\in\operatorname{Harm}^{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$,
$\widetilde{\mathcal{A}}\in\operatorname{Harm}^{p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$,
$\mathcal{B}\in\operatorname{Harm}^{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$, and
$\widetilde{\mathcal{B}}\in\operatorname{Harm}^{d-p-2}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$. Then the boundary condition 3.14 relates
$\widetilde{\mathcal{A}}$ to $\mathcal{B}$ (and similarly
$\widetilde{\mathcal{B}}$ to $\mathcal{A}$) as follows:

$\mathcal{B}=\mu^{\Delta}(-1)^{(d-p)p}\star_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\widetilde{\mathcal{A}}.$ (3.17)

And the boundary action becomes

$\displaystyle S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\quantity[\mathcal{A},\mathcal{B}]=\frac{\mu^{\Delta}}{4\pi}\left\langle\mathcal{A},\mathbb{K}\,\mathcal{A}\right\rangle+\frac{\mu^{-\Delta}}{4\pi}\left\langle\mathcal{B},\mathbb{K}\,\mathcal{B}\right\rangle+\frac{\mu^{\Delta}}{4\pi}\left\langle\differential{a},\mathbb{K}\,\differential{a}\right\rangle.$
(3.18)

We take this as the definition of the action of chiral mixed Maxwell theory.

Let us proceed with quantizing this theory, by performing the path integral on
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}$. For notational
simplicity we will perform the path integral on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}$ and we will rescale the
radius in the final formulas to obtain the partition function on
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}$. First, observe that
writing the action in the form 3.18 is equivalent to performing the path
integral over $\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{B}}$ over the
delta function that enforces the boundary conditions. This leaves us with

$\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]=\mathcal{Z}_{\text{inst}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]\,\mathcal{Z}_{\text{osc}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}],$ (3.19)

where

$\displaystyle\mathcal{Z}_{\text{inst}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]$
$\displaystyle\coloneqq\int\mathinner{\mathrm{D}\mathcal{A}}\mathinner{\mathrm{D}\mathcal{B}}\exp(-\frac{\mu^{\Delta}}{4\pi}\left\langle\mathcal{A},\mathbb{K}\,\mathcal{A}\right\rangle+\frac{\mu^{-\Delta}}{4\pi}\left\langle\mathcal{B},\mathbb{K}\,\mathcal{B}\right\rangle)$
(3.20)
$\displaystyle\mathcal{Z}_{\text{osc}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]$
$\displaystyle\coloneqq\int\frac{\mathinner{\mathrm{D}a}missing}{\opbraces{\operatorname{vol}}(\mathcal{G}^{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}_{p-1})}\exp(-\frac{\mu^{\Delta}}{4\pi}\left\langle\differential{a},\mathbb{K}\,\differential{a}\right\rangle).$
(3.21)

The instanton integral is over the space of harmonic $p$\- and $(d-p-1)$-forms
on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ and we will
evaluate it shortly. The oscillator piece is an integral over a $(p-1)$-form
gauge fields modulo their gauge transformations. This is a regular set of
$(p-1)$-form Maxwell oscillators.

We first deal with the oscillators. We can expand in modes of the Hodge
Laplacian and integrate over those, separating the zero-modes. We must do so
for the tower of ghosts that follow from the reducible gauge invariances of
the $(p-1)$-form gauge fields. We show in Appendix A, that the oscillators
contribute as

$\displaystyle\mathcal{Z}_{\text{osc}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]=\quantity(\upeta^{(p-1)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\quantity[q])^{-\kappa}\;\quantity(\upeta^{(d-p-2)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\quantity[q])^{-\kappa}.$ (3.22)

In the above, $q$ denotes the nome, $q\coloneqq\mathrm{e}^{-\varepsilon\mu}$,
and we have defined an analogue of the Dedekind eta function,

$\displaystyle\upeta^{(k)}_{Y}[q]\coloneqq
q^{-\tfrac{1}{2}E_{0}}\quantity(\prod_{\mathsf{n}\in\mathscr{N}^{\perp}_{k}}\sum_{N_{\mathsf{n}}=0}^{\infty}q^{N_{\mathsf{n}}\sqrt{\lambda_{\mathsf{n}}}})^{-1/2},$
(3.23)

associated to the $k$-form Hodge Laplacian on a closed, compact manifold $Y$.
In the above, $\mathscr{N}_{k}^{\perp}$ denotes the index-set of the non-zero
spectrum of the $k$-form Laplacian acting on coclosed forms,
$\lambda_{\mathsf{n}}$ are the corresponding eigenvalues, measured in units of
$\mu$, and $E_{0}$ is the (potentially divergent) zero-point energy (also in
units of $\mu$),
$E_{0}\coloneqq\sum_{\mathsf{n}\in\mathscr{N}^{\perp}_{k}}\lambda_{\mathsf{n}}$.
Moreover, the above is, strictly speaking, defined only up to a phase. This
will not concern us; the existence or choice of a complex structure of $Y$ is
insignificant for us, since for our purposes it suffices to only consider real
nomes. The definition 3.23 is also equivalent to

$\displaystyle\upeta_{Y}^{(k)}[q]=\quantity(\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}^{\perp}}\sinh(\frac{1}{2}\varepsilon\mu\sqrt{\lambda_{\mathsf{n}_{k}}}))^{1/2}=\quantity(\prod_{\mathsf{n}_{k}\in\mathscr{N}_{k}^{\perp}}q^{-\frac{1}{2}\sqrt{\lambda_{\mathsf{n}_{k}}}}\quantity(1-q^{\sqrt{\lambda_{\mathsf{n}_{k}}}}))^{1/2},$
(3.24)

Finally it is easy to see that if we take $Y=\mathbb{S}^{1}$, our generalized
eta function reduces to the usual Dedekind eta function:
$\upeta_{\mathbb{S}^{1}}^{(0)}[q]=\upeta[q]$. Moreover, since the non-zero
spectra of the Laplacians acting on transversal $(p-1)$\- and $(d-p-2)$-forms
coincide, the eta functions associated with those are equal, we write them
separately, however for reasons that will become apparent shortly.

Moving on to the instantons, we can further expand $\mathcal{A}$ and
$\mathcal{B}$ in terms of the topological basis of harmonic forms on
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$:

$\mathcal{A}_{\text{h}}=\mathcal{A}_{\mathsf{i}}\tau_{\mathsf{i}}^{(p)}\mbox{\quad
and\quad}\mathcal{B}_{\text{h}}=\mathcal{B}_{\mathsf{i}}\tau_{\mathsf{i}}^{(d-p-1)}.$
(3.25)

Enter Dirac quantization. Magnetic fluxes along $p$\- and $(d-p-1)$-cycles on
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ are quantized as

$\displaystyle\int_{\eta}\mathcal{A}\in
2\pi\mathbb{Z}^{\kappa},\quad{{}^{\forall}\\!}\eta\in\mathrm{H}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)\mbox{\quad and\quad}\int_{\gamma}\mathcal{B}\in
2\pi\mathbb{Z}^{\kappa},\quad{{}^{\forall}\\!}\gamma\in\mathrm{H}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R).$ (3.26)

This means that $\mathcal{A}_{\mathsf{i}}=2\pi n_{\mathsf{i}}$ and
$\mathcal{B}_{\mathsf{i}}=2\pi m_{\mathsf{i}}$, where
$n_{\mathsf{i}},m_{\mathsf{i}}\in\mathbb{Z}^{\kappa}$. As such, the action
3.18 becomes:

$\displaystyle S_{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\quantity[\mathcal{A},\mathcal{B}]$
$\displaystyle=\pi\varepsilon\mu\quantity(\boldsymbol{n}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{n}+\boldsymbol{m}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{m}).$ (3.27)

In the above, we have combined $n_{\mathsf{i}}$ and $m_{\mathsf{i}}$ into
vectors
$\boldsymbol{n}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}$ and
$\boldsymbol{m}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}$. Moreover,
$\widetilde{\mathbb{G}}_{k}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$ is the dimensionless version of
$\mathbb{G}_{k}^{{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R}$,
the Gram matrix of harmonic $k$-forms on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$. This comes from the fact that $\mathbb{G}_{k}$
has dimension $\quantity[\mathbb{G}_{k}]=\mu^{1+(2k-1-d)}$, so
$\widetilde{G}_{p}\coloneqq\mu^{\Delta-1}$ and
$\widetilde{\mathbb{G}}_{d-p-1}\coloneqq\mu^{1+\Delta}\mathbb{G}_{d-p-1}$ are
dimensionless. Therefore, the instanton contribution reads

$\mathcal{Z}_{\text{inst}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]=\sum_{\begin{subarray}{c}\boldsymbol{n}\in\mathbb{Z}^{\kappa\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}\\\
\boldsymbol{m}\in\mathbb{Z}^{\kappa\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}\end{subarray}}\exp[-\pi\varepsilon\mu\quantity(\boldsymbol{n}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{n}+\boldsymbol{m}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{m})].$ (3.28)

To arrive at the final expression for the edge-mode partition function, let us
also introduce a Siegel-type Theta function, denoted by
$\Theta[q;\mathbb{O}_{N}]$:

$\displaystyle\Theta[q;\mathbb{O}_{N}]\coloneqq\sum_{\boldsymbol{r}\in\mathbb{Z}^{N}}q^{\pi\boldsymbol{r}\cdot\mathbb{O}_{N}\cdot\boldsymbol{r}}.$
(3.29)

If $\mathbb{O}_{N}$ is a real, $N\times N$ matrix, when
$q\in\mathbb{R}_{\geqslant 0}$, it truly becomes a Siegel Theta function and
inherits all the modular properties of those. Expressing the instanton
contribution, 3.20, in terms of the Theta functions, 3.29, and combining with
the oscillator contribution, 3.22, gives the following simple form for the
edge-mode partition function:

$\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}]=\frac{\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}]}{\quantity(\upeta^{(p-1)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}[q])^{\kappa}}\
\frac{\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}]}{\quantity(\upeta^{(d-p-2)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}[q])^{\kappa}}.$ (3.30)

We briefly re-emphasize the chiral nature of this result and comment on its
importance. To get a feel for it, note first that if
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R=\mathbb{S}^{1}$, with
$p=1$, this becomes exactly the partition function of two (multi-component)
chiral bosons. This resonates entirely with the usual story. In $d=3$ BF
theory can be written as a sum of two Chern–Simons theories, each of which
supports a chiral scalar on its boundary. The partition function of this
chiral boson is not modular invariant111111This fact,
$\mathcal{Z}_{\text{chiral boson}}(\beta)\neq\mathcal{Z}_{\text{chiral
boson}}(1/\beta)$, is actually crucial for extracting the correct entanglement
entropy of bulk Chern-Simons theory [7]!. In $d=5$ dimensions, a smoking gun
signal that the edge theory is chiral is the fact that it is not
$\mathsf{S}$-duality invariant, as can be seen by comparing to the regular
Maxwell partition function (cf. Appendix A). In [40], $\mathsf{S}$-duality of
the edge-mode theory was crucial in proving an obstruction to symmetry-
preserving regulators of edge-states of BF theory121212This was, in turn
interpreted as a gauge theory version of the Nielsen–Ninomiya theorem [54,
55].. Our analysis provides evidence that this may not be the case; i.e. the
fact that the UV lattice models regularizing Maxwell theory break
$\mathsf{S}$-duality, while correct, does not imply that these models are
symmetry breaking from the point of view of edge states of BF theory. Moving
away from $d=5$, for generic dimensions, one can take a half of our partition
function as the definition of an Abelian chiral gauge theory, but note that
only this combination is well-defined, at least from a path integral
perspective. A systematic understanding of this chiral theory deserves further
research.

In the next section, we will move on to extract the entanglement entropy. To
do that, we will need to compare the edge-mode partition function on
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}$ to that on
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{1,\varepsilon}$. The
${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}$ partition function can
be easily obtained by rescaling the radius of the $\mathbb{S}^{1}$ to
$n\varepsilon$. This results in

$\mathcal{Z}_{\text{edge}}\quantity[{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}\mathcal{X}_{n,\varepsilon}]=\frac{\Theta\quantity[q^{n};\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}]}{\quantity(\upeta^{(p-1)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\quantity[q^{n}])^{\kappa}}\
\frac{\Theta\quantity[q^{n};\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}]}{\quantity(\upeta^{(d-p-2)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\quantity[q^{n}])^{\kappa}}.$ (3.31)

###  3.2 Extracting the entanglement entropy

We can now simply insert 3.31 and 3.30 into 3.11 to extract the entanglement
entropy. We have two kinds of terms to deal with; the Theta functions and the
eta functions. Let us deal with them separately.

We treat first the instantons, i.e. the Theta functions, for it will be easier
to extract their high-temperature behavior. Fortunately, Siegel Theta
functions are very well behaved and obey a sort of modularity equation, which
can be easily proved using the Poisson summation formula. Explicitly, they
obey the following transformation equation [56]:

$\displaystyle\Theta[\mathrm{e}^{-\varepsilon\mu};\mathbb{O}_{N}]=\quantity(\varepsilon\mu)^{-\frac{N}{2}}\absolutevalue{\det\mathbb{O}_{N}}^{-\frac{1}{2}}\Theta\quantity[\mathrm{e}^{-\frac{1}{\varepsilon\mu}};\mathbb{O}_{N}^{-1}].$
(3.32)

The high-temperature behavior is then straightforward to obtain:

$\displaystyle\Theta[\mathrm{e}^{-\varepsilon\mu};\mathbb{O}_{N}]\overset{\varepsilon\to
0}{\sim}\quantity(\varepsilon\mu)^{-\frac{N}{2}}\absolutevalue{\det\mathbb{O}_{N}}^{-\frac{1}{2}}.$
(3.33)

With that, the total contribution of the Theta functions to the entanglement
entropy is

$\displaystyle\evaluated{\mathcal{S}_{\text{EE}}}_{\Theta}$
$\displaystyle=\lim_{\varepsilon\to 0}\lim_{n\to
1}\frac{1}{1-n}\log\frac{\Theta\quantity[q^{n};\mathbb{K}\otimes\widetilde{\mathbb{G}}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}_{p}]\
\Theta\quantity[q^{n};\mathbb{K}\otimes\widetilde{\mathbb{G}}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}_{d-p-1}]}{\quantity(\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}_{p}]\
\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}_{d-p-1}])^{n}}=$
$\displaystyle=\frac{1}{2}\quantity(\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))\quantity\big[\kappa\quantity(1-\log(\varepsilon\mu))-\log\absolutevalue{\det\mathbb{K}}].$
(3.34)

As we shall see shortly, this term will be partially cancelled by the
contribution of the eta functions, ultimately resulting in the topological
piece of the entanglement entropy.

Let us get to it then. To evaluate the contribution of the eta functions, we
will use the following trick [57, 58]. In words, we will Taylor expand
$\log\quantity(\upeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})^{-1}$ and on each term perform an inverse Mellin
tranform131313note that the piece containing the zero-point-energy, $E_{0}$,
in 3.23 or 3.24 obviously does not contribute to the entropy so we will focus
on the rest. Written as a Mellin integral, the sum over the eigenvalues,
$\quantity{\lambda_{\mathsf{n}}}$ is easily performed to yield the spectral
zeta function,

$\displaystyle\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}(s)\coloneqq\sum_{\mathsf{n}\in\mathscr{N}^{\perp}_{k}}\lambda_{\mathsf{n}}^{-s}.$
(3.35)

In equations:

$\displaystyle-\sum_{\mathsf{n}\in\mathscr{N}_{k}^{\perp}}\log(1-\mathrm{e}^{-\varepsilon\mu\sqrt{\lambda_{\mathsf{n}}}})^{-1}$
$\displaystyle\overset{{\text{Taylor}}}{=\joinrel=\joinrel=}\sum_{\mathsf{n}\in\mathscr{N}_{k}^{\perp}}\sum_{m=1}^{\infty}\frac{1}{m}\mathrm{e}^{-\varepsilon\mu\;m\sqrt{\lambda_{\mathsf{n}}}}$
$\displaystyle\overset{{{{\text{Mellin}}}^{-1}}}{=\joinrel=\joinrel=}\sum_{\mathsf{n}\in\mathscr{N}_{k}^{\perp}}\sum_{m=1}^{\infty}\frac{1}{m}\int_{c+\mathrm{i}\mathbb{R}}\frac{\differential{u}}{2\pi\mathrm{i}}\Gamma(u)\quantity(\varepsilon\mu\;m\sqrt{\lambda_{\mathsf{n}}})^{-u}$
$\displaystyle\overset{{\text{resum}}}{=\joinrel=\joinrel=}\int_{c+\mathrm{i}\mathbb{R}}\frac{\differential{u}}{2\pi\mathrm{i}}\;\upzeta(u+1)\;\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(p-1)}\quantity(\frac{u}{2})\;\Gamma(u)\;\quantity(\varepsilon\mu)^{-u},$
(3.36)

where $\upzeta(u)$ is the ordinary Riemann zeta function. In the above
integral $c>0$ is a positive number lying to the right of the rightmost pole
of the integrand. To extract the leading terms as $\varepsilon\rightarrow 0$,
we then imagine pushing the integration contour to $-1<c<0$, so that the
integrand vanishes as $\varepsilon\to 0$. The price we pay is that, as we
sweep to the left, we pick up the residues from all of the poles of the
integrand.

$\displaystyle\lim_{\varepsilon\rightarrow
0}\log\quantity(\upeta^{(k)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}(q))^{-1}=\frac{1}{2}\sum_{u_{*}\geqslant
0}\underset{u=u_{*}}{\Res}\upzeta(u+1)\;\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}\quantity(\frac{u}{2})\;\Gamma(u)\;\quantity(\varepsilon\mu)^{-u}.$
(3.37)

In short, evaluating the descendant contribution to $\mathcal{S}_{\text{EE}}$
reduces to analyzing the pole structure of the above expression. In Appendix B
we carefully analyze the poles of $\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}$ which all lie at $u>0$. Inside 3.37 there
is a double pole at $u=0$ from $\Gamma(u)\upzeta(u+1)$, with residue

$\displaystyle\Res_{0}^{(k)}$
$\displaystyle=\frac{1}{2}{\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}}^{\prime}(0)-\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}(0)\log\quantity(\varepsilon\mu)$
$\displaystyle=\quantity(-\mathcal{I}_{\frac{d-2}{2}}^{(k)}\delta_{d,{\text{even}}}+\mathrm{b}_{k}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))\log\quantity(\varepsilon\mu).$ (3.38)

In the above, $\mathcal{I}_{\frac{d-2}{2}}^{(k)}$ is the
$\quantity(\frac{d-2}{2})$-th heat kernel coefficient for the spectral zeta
function arising from the value of $\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(k)}(0)$ and
$\mathrm{b}_{k}({\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R)$, the
$k$-th Betti number enumerates the number of zero-modes of the $k$-form
Laplacian, again reviewed in Appendix B. We see then that this term, evaluated
for the eta functions associated to the $(p-1)$\- and the $(d-p-2)$-form
Laplacians, contributes to the entropy as:

$\displaystyle\evaluated{\mathcal{S}_{\text{EE}}}_{\Res_{0}}=\frac{\kappa}{2}\quantity[\quantity(\mathcal{I}_{\frac{d-2}{2}}^{(p-1)}+\mathcal{I}_{\frac{d-2}{2}}^{(d-p-2)})\delta_{d,{\text{even}}}-\quantity(\mathrm{b}_{p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\mathrm{b}_{d-p-2}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))]\quantity(1-\log\quantity(\varepsilon\mu)).$
(3.39)

We see a happy cancellation between (part of) this term and (part of) the
contribution of the Thetas, Subsection 3.2, owing to Poincaré duality
$\mathrm{b}_{p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)=\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$ and
$\mathrm{b}_{d-p-2}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)=\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$.

The residues away from zero come from the poles of
$\upzeta_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{(p-1)}(u/2)$141414Recall that apart from the
zero-modes, the eta functions for $(p-1)$\- and for $(d-p-2)$-forms are
identical.. These lie at $d-2k$, for integer $k\geqslant 1$151515Note that
there is a shift $k\to k-1$ compared with Appendix B. (with $k\leqslant d/2-1$
when $d$ is even) and give in total

$\displaystyle\Res_{2d-k}=(4\pi)^{1-\frac{d}{2}}\upzeta(d-2k+1)\frac{\Gamma(d-2k)}{\Gamma\quantity(\frac{d}{2}-k)}\mathcal{I}_{k-1}^{(p-1)}\quantity(\varepsilon\mu)^{2k-d},$
(3.40)

where, again, $\mathcal{I}_{k-1}^{(p-1)}$ are heat kernel coefficients for the
spectral zeta function. They can be written as integrals of local geometrical
(but not topological) data of the entangling surface. These residues
contribute area law (and subleading) terms to the entropy:

$\displaystyle\evaluated{\mathcal{S}_{\text{EE}}}_{\Res_{d-2k}}$
$\displaystyle=\mathrm{I}^{(p-1)}_{k}\quantity(\varepsilon\mu)^{2k-d},$ (3.41)

where

$\displaystyle\mathrm{I}^{(p-1)}_{k}$
$\displaystyle\coloneqq\kappa(4\pi)^{1-\frac{d}{2}}\upzeta(d-2k+1)\frac{\Gamma(d-2k)}{\Gamma\quantity(\frac{d}{2}-k)}(d-2k+1)\
\mathcal{I}_{k-1}^{(p-1)}.$ (3.42)

As expected, the area-law terms are non-universal, depend on the geometry of
the entangling surface (through the heat-kernel coefficients,
$\mathcal{I}_{k-1}^{(p-1)}$) and the regulator, $\varepsilon$.

Assembling the oscillator contributions and adding the contribution of the
instantons, we arrive at the main result for the entanglement entropy:

$\displaystyle\mathcal{S}_{\text{EE}}$
$\displaystyle=\sum_{k=1}^{\left\lfloor\frac{d-1}{2}\right\rfloor}\mathrm{C}^{(p-1)}_{k}\quantity(\frac{\ell}{\varepsilon})^{d-2k}+\frac{\kappa}{2}\quantity(\mathcal{I}_{\frac{d-2}{2}}^{(p-1)}+\mathcal{I}_{\frac{d-2}{2}}^{(d-p-2)})\delta_{d,{\text{even}}}\log(\frac{\ell}{\varepsilon})$
(3.43)
$\displaystyle\phantom{=~{}}-\frac{1}{2}\quantity(\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))\log\absolutevalue{\det\mathbb{K}},$

where $\mathrm{C}_{k}\coloneqq\mathrm{I}_{k}\mathrm{e}^{2k-d}$ and we traded
the energy scale for a length scale $\ell\equiv\mu^{-1}\mathrm{e}$ as is more
common in the presentation of entanglement entropies. In the above formula, we
have ignored terms with $k>\left\lfloor\frac{d-1}{2}\right\rfloor$ (which are
present only when $d$ is odd), since they give vanishing contributions in the
limit $\varepsilon\to 0$.

#### On extracting universal features:

Some comments are in order regarding our main result, 3.43. In odd dimensions
we have a universal subleading correction given by

$\displaystyle\mathcal{S}_{\text{TEE}}=-\frac{1}{2}\quantity(\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))\log\absolutevalue{\det\mathbb{K}}.$ (3.44)

In even dimensions, however, the log term

$\displaystyle\frac{\kappa}{2}\quantity(\mathcal{I}_{\frac{d-2}{2}}^{(p-1)}+\mathcal{I}_{\frac{d-2}{2}}^{(d-p-2)})\log(\frac{\ell}{\varepsilon})$
(3.45)

spoils the universality of the $\log\absolutevalue{\det\mathbb{K}}$ as
rescalings of the cutoff result in constant shifts of
$\mathcal{S}_{\text{EE}}$. Under general arguments, there is nothing that
prohibits $\log$ contributions to the entanglement entropy in even dimensions
and, indeed, here we find such a contribution. The coefficient of this $\log$
is a potentially universal piece of data, but in fact it depends on the
geometry of the entangling surface through its heat kernel coefficient. On top
of this, throughout the paper we had been neglectful of terms proportional to
the Euler characteristic, $\upchi({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$. The main reason is that, while topological, they
ultimately arise (or can be absorbed into) ambiguities in the edge theory path
integral measure and so are non-universal. This non-universality is reflected
in the entropy: the Euler characteristic can be recast as the integral of a
local quantity, through the generalized Gauss–Bonnet theorem. Nevertheless, it
was explained in [11] that a generalization of the Kitaev–Preskill / Levin–Wen
protocol [1, 2] is possible in higher dimensions with an explicit construction
in $d=4$. By this, it is meant that one can decompose the entangling surface,
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$, into pieces,
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R_{i}$ of specific
geometry. Then one can consider a linear combination of entanglement entropies
across ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R_{i}$, such
that, when surgered together, the locally integrated contributions to the
entropy cancel out. The upshot is that we expect, through such a construction,
one can extract in all dimensions the topological entanglement entropy:

$\mathcal{S}_{\text{TEE}}=-\frac{1}{2}\quantity(\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)+\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R))\log\absolutevalue{\det\mathbb{K}}~{},$ (3.46)

although we do not do so in this paper.

As an example, consider four-dimensional BF theory and take $p=1$ or $p=2$.
The two cases are interchangeable, as they are mapped to each other by
interchanging the roles of the $A$ and $B$ fields in the original action 2.1.
Generally, it holds that
$\mathcal{I}^{(0)}_{1}=\frac{1}{6}\upchi({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$ [59]. Taking the entangling surface to be a
torus, ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R=\mathbb{T}^{2}$, for $p=1$ we have
$\mathcal{I}^{(0)}_{1}=0$. One can explicitly show that the same holds for
$p=2$, as expected, by analyzing the spectrum of the transverse Laplacian
acting on one-forms. Therefore, the topological entanglement entropy for the
four-dimensional case with an entangling surface with the topology of a torus
is given by

$\displaystyle\mathcal{S}_{\text{TEE}}=-2\log\absolutevalue{\det\mathbb{K}}.$
(3.47)

Note that in four dimensions the subleading part of the entropy is always
topological (although possibly not universal), since
$\mathcal{I}^{(p-1)}_{\frac{d-2}{2}}$ is given by the Euler characteristic. In
contrast, in higher dimensions, this is not the case. In six dimensions, for
example,

$\displaystyle\mathcal{I}^{(0)}_{2}=\frac{1}{180}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\differential\opbraces{\operatorname{vol}}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\
\quantity(10\mathcal{R}^{2}-\mathcal{R}_{\mu\nu\rho\sigma}\mathcal{R}^{\mu\nu\rho\sigma}+2\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}),$

which depends sensitively on the geometry of the entangling surface, and
therefore one truly needs a suitable Kitaev–Preskill / Levin–Wen-like
subtraction scheme to extract the universal pieces of data.

##  4 The extended Hilbert space and a current algebra

Above we have presented a replica path integral calculation of the
entanglement entropy. We have focussed on the entropy that comes entirely from
an “edge-mode theory,” which we have coined as “chiral mixed Maxwell,” in the
high-temperature limit; this essentially counts a regulated Hilbert space
dimension of this edge theory. In this section, we provide a more honest
accounting of the entanglement entropy, accounting for the subtleties of gauge
invariance. This will provide an alternative view of the role of these edge-
modes as providing an arena, the “extended Hilbert space” (EHS), by which the
entanglement entropy can be precisely defined. We will see that a physical
state is embedded as a maximally entangled state in this EHS and thus its
entanglement entropy is naturally interpreted as a dimension of the EHS, which
we will regulate. The upshot of this section is that we will reproduce the
topological entanglement entropy from a more rigorous starting point, while
also providing a, perhaps, intuitive view on the need for the edge theory and
why it contributes to the entanglement entropy.

Along the way we will show that the EHS is completely organized by an
infinite-dimensional spectrum-generating current algebra, entirely analogous
to the Kac-Moody algebras appearing at the edge of three-dimensional
topological phases. In general dimensions this algebra is non-conformal.
Regardless, we will show that it is powerful enough to fix the entire
computation of the entanglement entropy. We will also show that this algebra,
remarkably, generates the entire spectrum of the edge-mode theory from Section
3. The fact that the spectrum of the chiral Maxwell theory is entirely fixed
by a (potentially non-conformal) current algebra echoes the results of [24,
25] for 1-form Maxwell theories in four-dimensions. Here we present the
generic story which we regard as a major result of this paper.

###  4.1 Gauge-(in)variance, entanglement, and the extended Hilbert space

The typical starting point of the entanglement entropy is the supposition of a
Hilbert space factorization between a region, $R$, and its complement,
${R}^{\mathsf{c}}$:

$\mathcal{H}=\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}~{}.$ (4.1)

This supposition typically fails in quantum field theory due to an infinite
number of correlated short-distance modes161616More modernly, and more
precisely, it fails due to the type-III nature of the von Neumann algebra
associated to a subregion; see e.g. [60] and references therein., however, one
can imagine regulating this computation by a short-distance regulator,
$\varepsilon$, (say by putting the system on a lattice, or utilizing a mutual
information regulator [61]). However, even in a regulated scenario, 4.1, fails
for quantum gauge theories due to the global nature of gauge constraints that
physical states must satisfy [19, 20, 21, 22, 23, 61, 62]. It is, by now, well
understood how to properly define entanglement entropy in gauge theories. One
method is to utilize an algebraic definition of the von Neumann entropy
applied to the reduced density matrix realized directly as a gauge-invariant
operator [61]. We have explored that aspect of entanglement entropy in this
family of theories in a companion paper, [10]. Here we focus on the
alternative formulation, which goes by the name of the “extended Hilbert
space” [19, 20, 21, 22, 23].

In short, while the physical Hilbert space, $\mathcal{H}$, cannot be realized
as a tensor product, we can embed $\mathcal{H}$ (let us call the embedding
$\mathcal{I}$) into an EHS, $\mathcal{H}_{\text{ext}}$, which admits a tensor
product.

$\mathcal{H}\overset{\mathcal{I}}{\longhookrightarrow}\mathcal{H}_{\text{ext}}\coloneqq\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}~{}.$
(4.2)

By definition, $\mathcal{H}_{\text{ext}}$ is furnished with unphysical, gauge-
variant states. These are states in either $\mathcal{H}_{R}$ or
$\mathcal{H}_{{R}^{\mathsf{c}}}$ which carry the action of gauge
transformations acting on the entangling surface,
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$. That is to say, it
is furnished with edge-mode states living on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$. Given a physical state $\ket{\psi}\in\mathcal{H}$
it embeds to a state $\ket*{\widetilde{\psi}}\in\mathcal{H}_{\text{ext}}$
which we can then tensor-factorize and reduce upon $R$:

$\widetilde{\rho}_{R}\coloneqq\Tr_{\mathcal{H}_{{R}^{\mathsf{c}}}}\outerproduct*{\widetilde{\psi}}{\widetilde{\psi}}$
(4.3)

and the entanglement entropy can defined in the typical way:

$\mathcal{S}_{\text{EE}}\coloneqq-\Tr_{\mathcal{H}_{R}}\left(\widetilde{\rho}_{R}\log\widetilde{\rho}_{R}\right)~{}.$
(4.4)

In practice, this can be computed utilizing the replica trick we reviewed
above. The important step in this process, however, is identifying the
embedding map, $\mathcal{I}$, and thus the appropriate embedded state
$\ket*{\widetilde{\psi}}\in\mathcal{H}_{\text{ext}}$. Since
$\mathcal{H}_{\text{ext}}$ is furnished (by definition) by gauge-variant
states, this embedding is determined by demanding gauge-invariance by hand:
i.e. if a gauge transformation labeled by $\alpha$ with support on
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ acts on
$\mathcal{H}_{R}$ and $\mathcal{H}_{{R}^{\mathsf{c}}}$ with charge operators
$Q_{R}[\alpha]$ and $Q_{{R}^{\mathsf{c}}}[\alpha]$, respectively, then we will
demand

$\left(Q_{R}[\alpha]\otimes 1+1\otimes
Q_{{R}^{\mathsf{c}}}[\alpha]\right)\ket*{\widetilde{\psi}}\overset{!}{=}0~{}.$
(4.5)

We will refer to 4.5 as the “quantum gluing condition.” Because $\alpha$ can
be arbitrarily local to ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$, this induces correlations to
$\ket*{\widetilde{\psi}}\vphantom{\displaystyle\sum}$ that are local to
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ and maximally
entangled. As a result $\widetilde{\rho}_{R}$ will be maximally mixed amongst
an infinite number of edge-modes that are local to
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$. Regulating this
maximal mixture will result in a (divergent) area law (with possibly
subleading area laws) to the entanglement entropy, but there may also exist
universal corrections. We will make these ideas concrete in this theory below.
To begin, we will first identify a suitable $\mathcal{H}_{R}$ with which to
build the extended Hilbert space.

###  4.2 A current algebra

We will construct $\mathcal{H}_{R}$ by quantizing the theory on
$X=\mathbb{R}\times R$. Many of the details here follow the opening sections
of [10] for quantizing BF theory, however, now allowing for the existence of a
boundary ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R\neq\varnothing$. We refer the reader there for a
more detailed accounting of the procedure and notation and point out only the
necessary features special to the current situation here. We will often use
the embedding map of the boundary $i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}:{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\hookrightarrow R$, and
$i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}:{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X\hookrightarrow X$, as well as the embedding map of
$R$ into $X$: $i_{R}:R\hookrightarrow X$.

We split $A=A_{0}+a$ and $B=B_{0}+b$ with $A_{0}$ and $B_{0}$ with “a leg”
along $\mathbb{R}$ and $a$ and $b$ along $R$. Correspondingly the action 2.1
splits,

$\displaystyle
S_{{\text{BF}}}[A_{0}+a,B_{0}+b]=\frac{\mathbb{K}}{2\pi}\int_{X}\left((-1)^{d-p}\mathbf{d}b\wedge
A_{0}+B_{0}\wedge\mathbf{d}a+b\wedge da\right),$ (4.6)

once we impose boundary conditions $i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}^{\ast}\,A_{0}=i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}^{\ast}B_{0}=0$. Above $\mathbf{d}$ should be
regarded as the exterior derivative along $R$. $A_{0}$ and $B_{0}$ impose
flatness of $a$ and $b$ on $R$:

$\mathbf{d}a=\mathbf{d}b=0$ (4.7)

which we will refer to as the Gauss law constraints. The residual gauge
transformations, preserving the boundary conditions act on $a$ and $b$ as

$\displaystyle\updelta b=\mathinner{\mathbf{d}\beta},$ $\displaystyle\qquad
i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{*}\differential_{\mathbb{R}}\beta=0$ (4.8)
$\displaystyle\updelta a=\mathinner{\mathbf{d}\alpha},$ $\displaystyle\qquad
i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{*}\differential_{\mathbb{R}}\alpha=0.$ (4.9)

Recall that higher-form gauge theories come with a tower of lower (secondary,
tertiary, quaternary, etc.) gauge-invariances. These lower invariances
indicate some redundancy in what arises as a global symmetries when $R$ has a
boundary. Later on, we will impose restrictions on the gauge parameters that
survive all the way to the boundary, to take care of this.

The charges associated to 4.8 can be built from the symplectic form

$\displaystyle\Omega_{R}=(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{R}\updelta
b\wedge\updelta a.$ (4.10)

Above, we view $\updelta b$ and $\updelta a$ as variational one-forms. We
define variational vectors, $\boldsymbol{v}_{\alpha}$ and
$\boldsymbol{v}_{\beta}$ such that their interior product gives 4.8:

$\displaystyle\boldsymbol{v}_{\alpha}\mathbin{\lrcorner}\updelta
a=\mathbf{d}\alpha\qquad\mbox{\quad
and\quad}\qquad\boldsymbol{v}_{\beta}\mathbin{\lrcorner}\updelta
b=\mathbf{d}\beta.$ (4.11)

The charges, given by

$\displaystyle\updelta
Q[\alpha]\coloneqq\boldsymbol{v}_{\alpha}\mathbin{\lrcorner}\Omega_{R}\qquad\mbox{\quad
and\quad}\qquad\updelta
Q[\beta]\coloneqq\boldsymbol{v}_{\beta}\mathbin{\lrcorner}\Omega_{R},$ (4.12)

can then be found as

$\displaystyle\begin{split}Q[\alpha]&\coloneqq(-1)^{d-p}\frac{\mathbb{K}}{2\pi}\int_{R}b\mathbin{\scalebox{0.8}{$\wedge$}}\mathbf{d}\alpha=-\frac{\mathbb{K}}{2\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}b\mathbin{\scalebox{0.8}{$\wedge$}}\alpha\\\
Q[\beta]&\coloneqq(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{R}\mathbf{d}\beta\wedge
a=(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\beta\wedge a\end{split}$ (4.13)

up to a total variation171717This total variation ambiguity doesn’t affect the
algebra of the charges, of course.. The second equality in each line
emphasizes that the charges localize to ${{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$ upon imposing the Gauss law constraint, 4.7. When
these charges do not vanish identically, then they are genuine, global,
symmetries of the system, i.e. they act on and transform states. As alluded to
above, there is still a need to fix an additional gauge redundancy: if
$i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{\ast}\alpha=\mathbf{d}\gamma$ for some
$\gamma\in\Omega^{p-2}({{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})$ then $Q[\alpha]$ is identically zero by pulling
the Gauss constraint on $b$ back to ${{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$ (a similar argument follows for $\beta$). We fix
this by imposing

$\displaystyle\differential^{\dagger}i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{\ast}\alpha=\differential^{\dagger}i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}^{\ast}\beta=0$ (4.14)

This constraint fixes completely all the lower-invariances. The algebra of the
charges is also given by the symplectic form as

$\displaystyle\anticommutator{Q[\alpha]}{Q[\beta]}_{R}$
$\displaystyle=\Omega_{R}(\boldsymbol{v}_{\beta},\boldsymbol{v}_{\alpha})=(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{R}\mathbf{d}\beta\wedge\mathbf{d}\alpha$
(4.15)

The canonical quantization of the charges,
$\anticommutator{{\color[rgb]{0.65625,0.2890625,0.2890625}\definecolor[named]{pgfstrokecolor}{rgb}{0.65625,0.2890625,0.2890625}\bullet}}{{\color[rgb]{0.23046875,0.53515625,0.765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.53515625,0.765625}\bullet}}\to-\mathrm{i}\commutator{{\color[rgb]{0.65625,0.2890625,0.2890625}\definecolor[named]{pgfstrokecolor}{rgb}{0.65625,0.2890625,0.2890625}\bullet}}{{\color[rgb]{0.23046875,0.53515625,0.765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.53515625,0.765625}\bullet}}$,
then yields a centrally extended algebra

$\displaystyle\commutator{\hat{Q}[\alpha]}{\hat{Q}[\beta]}_{R}$
$\displaystyle=\mathrm{i}(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{R}\mathinner{\mathbf{d}\beta}\mathbin{\scalebox{0.8}{$\wedge$}}\mathinner{\mathbf{d}\alpha}=\mathrm{i}\frac{\mathbb{K}}{2\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\mathinner{\mathbf{d}\beta}\mathbin{\scalebox{0.8}{$\wedge$}}\alpha.$
(4.16)

###  4.3 Mode expansion of the current algebra

We want to turn the algebra 4.16 into a countable current algebra. For that we
want to decompose our various forms into modes of a common basis and find
their mode algebra.

In the following, we restrict ourselves within ${{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$ and drop the bold differential. It is understood
that everything below has been pulled back to ${{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$. We will keep in mind our restriction that the
forms $\alpha$ and $\beta$ are coclosed (though not necessarily co-exact) in
${{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R}$. The set of
eigen-$k$-forms of the transversal Laplacian,
$\evaluated{\differential^{\dagger}\differential}_{\Omega^{k}}$, provides a
basis for the coclosed $k$-forms. Poincaré duality and the Hodge decomposition
tell us that the non-zero spectrum of $\differential^{\dagger}\differential$
acting on $k$-forms on a $D$-dimensional compact manifold is the same as its
spectrum acting on $(D-k-1)$-forms, since they are both related to the
spectrum of $\differential\differential^{\dagger}$ on $(D-k)$-forms.

So, as far as the non-zero-modes are of concern, in our case, we need the
bases provided by the $(p-1)$-forms and $(d-p-2)$-forms. Since the non-zero
spectra of $\differential^{\dagger}\differential$ on the $(d-2)$-dimensional
manifold ${{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R}$ on these
two spaces coincide, these forms will be labeled by the same index set
$\quantity{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}$. Namely, we have two
orthonormal bases

$\displaystyle\quantity{\varphi_{\mathsf{n}}\in\Omega^{p-1}({{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})}_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}$
$\displaystyle\mbox{\quad
and\quad}\quantity{\chi_{\mathsf{n}}\in\Omega^{d-p-2}({{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})}_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}},$ (4.17)
$\displaystyle\left\langle\varphi_{\mathsf{n}},\varphi_{\mathsf{m}}\right\rangle$
$\displaystyle=\delta_{\mathsf{n}\mathsf{m}}=\left\langle\chi_{\mathsf{n}},\chi_{\mathsf{m}}\right\rangle,$

with

$\displaystyle\matrixquantity{\differential^{\dagger}\differential\varphi_{\mathsf{n}}=\lambda_{\mathsf{n}}\varphi_{\mathsf{n}}\\\\[10.00002pt]
\differential^{\dagger}\differential\chi_{\mathsf{n}}=\lambda_{\mathsf{n}}\chi_{\mathsf{n}}}\
,\qquad\lambda_{\mathsf{n}}\neq 0\
{{}^{\forall}\\!}\mathsf{n}\in\mathscr{N}_{\perp}^{*}.$ (4.18)

Now, coming to the zero-modes, these are simply the harmonic $(p-1)$\- and
$(d-p-2)$-forms on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$.
The natural bases to expand those are the topological bases

$\displaystyle\quantity{\tau^{(p-1)}_{\mathsf{i}}}_{\mathsf{i}=1}^{\mathrm{b}_{p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}$ $\displaystyle\mbox{\quad
and\quad}\quantity{\tau^{(d-p-2)}_{\mathsf{i}}}_{\mathsf{i}=1}^{\mathrm{b}_{d-p-2}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)},$ (4.19) satisfying
$\displaystyle\int_{\eta^{\mathsf{j}}}\tau^{(p-1)}_{\mathsf{i}}$
$\displaystyle=\delta^{\mathsf{j}}_{\mathsf{i}}=\int_{\sigma^{\mathsf{j}}}\tau^{(d-p-2)}_{\mathsf{i}},$
(4.20)

for all $(p-1)$\- and $(d-p-2)$-cycles $\eta^{\mathsf{j}}$ and
$\sigma^{\mathsf{j}}$ respectively.

We can then expand:

$\displaystyle\alpha$
$\displaystyle=\sum_{\mathsf{n}\in\mathscr{N}^{*}_{\perp}}\alpha^{\mathsf{n}}\varphi_{\mathsf{n}}+\sum_{\mathsf{i}=1}^{\mathrm{b}_{p-1}}\alpha^{\mathsf{i}}\tau^{(p-1)}_{\mathsf{i}}$
(4.21) $\displaystyle\beta$
$\displaystyle=\sum_{\mathsf{n}\in\mathscr{N}^{*}_{\perp}}\beta^{\mathsf{n}}\chi_{\mathsf{n}}+\sum_{\mathsf{i}=1}^{\mathrm{b}_{d-p-2}}\beta^{\mathsf{i}}\tau^{(d-p-2)}_{\mathsf{j}}.$
(4.22)

Similarly, we can expand $\hat{a}$ and $\hat{b}$ as

$\displaystyle\begin{split}\hat{a}&=(-1)^{d-p-1}2\pi\quantity[{{\mathbb{K}}^{-1}}]\quantity(\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\hat{a}_{\mathsf{n}}\star\chi_{\mathsf{n}}+\sum_{\mathsf{i}=1}^{\mathrm{b}_{p}}\hat{a}_{0\mathsf{i}}\star\tau^{(d-p-2)}_{\mathsf{i}})\\\
\hat{b}&=-2\pi\quantity[{{\mathbb{K}}^{-1}}]\quantity(\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\hat{b}_{\mathsf{n}}\star\varphi_{\mathsf{n}}+\sum_{\mathsf{i}=1}^{\mathrm{b}_{d-p-1}}\hat{b}_{0\mathsf{i}}\star\tau^{(p-1)}_{\mathsf{i}}).\end{split}$
(4.23)

so that the charges become

$\displaystyle\hat{Q}[\alpha]$
$\displaystyle=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\hat{b}_{\mathsf{n}}\alpha^{\mathsf{n}}+\sum_{\mathsf{i},\mathsf{j}=1}^{\mathrm{b}_{p-1}}\hat{b}_{0\mathsf{i}}\quantity[\mathbb{G}_{p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\alpha^{\mathsf{i}}$ (4.24)
$\displaystyle\hat{Q}[\beta]$
$\displaystyle=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\hat{a}_{\mathsf{n}}\beta^{\mathsf{n}}+\sum_{\mathsf{i},\mathsf{j}=1}^{\mathrm{b}_{d-p-2}}\hat{a}_{0\mathsf{i}}\quantity[\mathbb{G}_{d-p-2}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\beta^{\mathsf{i}},$ (4.25)

where we note again that by Poincaré duality it holds that
$\mathbb{G}_{k}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}=\quantity[\mathbb{G}_{d-2-k}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]^{-1}$, up to an invertible matrix which can be
set to unity by a choice of basis.

Equivalently, we can invert 4.23 to write the individual modes as

$\displaystyle\hat{a}_{\mathsf{n}}$
$\displaystyle=(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\chi_{\mathsf{n}}\mathbin{\scalebox{0.8}{$\wedge$}}\hat{a}$
(4.26) $\displaystyle\hat{a}_{0\mathsf{i}}$
$\displaystyle=(-1)^{d-p-1}\frac{\mathbb{K}}{2\pi}\sum_{\mathsf{j}=1}^{\mathrm{b}_{p}}\quantity[\mathbb{G}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\tau^{(d-p-2)}_{\mathsf{j}}\mathbin{\scalebox{0.8}{$\wedge$}}\hat{a}$
(4.27) $\displaystyle\hat{b}_{\mathsf{n}}$
$\displaystyle=-\frac{\mathbb{K}}{2\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\varphi_{\mathsf{n}}\hat{b}$ (4.28)
$\displaystyle\hat{b}_{0\mathsf{i}}$
$\displaystyle=-\frac{\mathbb{K}}{2\pi}\sum_{\mathsf{j}=1}^{\mathrm{b}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}}\quantity[\mathbb{G}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\tau^{(p-1)}_{\mathsf{j}}\mathbin{\scalebox{0.8}{$\wedge$}}\hat{b}.$
(4.29)

With this mode expansion, the current algebra 4.16 is written in modes as

$\displaystyle\commutator{\hat{b}_{\mathsf{n}}}{\hat{a}_{\mathsf{m}}}$
$\displaystyle=\frac{\mathrm{i}\,\mathbb{K}}{2\pi}\;C_{\mathsf{n}\mathsf{m}},\qquad\mathsf{n},\mathsf{m}\in\mathscr{N}_{\perp}^{*},\mbox{\quad
where\quad}$ (4.30) $\displaystyle C_{\mathsf{n}\mathsf{m}}$
$\displaystyle\coloneqq\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\differential{\chi_{\mathsf{m}}}\mathbin{\scalebox{0.8}{$\wedge$}}\varphi_{\mathsf{n}}.$
(4.31)

Of course, the zero-modes have trivial commutators with all the other modes as
well as amongst themselves. The final step to pin down the algebra completely
is to analyze the matrix $C_{\mathsf{n}\mathsf{m}}$. Observe that the two
bases of the non-zero sectors are related by

$\displaystyle\differential{\chi_{\mathsf{n}}}=\sum_{\mathsf{m}\in\mathscr{N}_{\perp}^{*}}C_{\mathsf{m}\mathsf{n}}\star\varphi_{\mathsf{m}}.$
(4.32)

Let us momentarily assume that there is no degeneracy of eigenvalues, i.e.
$\lambda_{\mathsf{n}}=\lambda_{\mathsf{m}}\Rightarrow\mathsf{n}=\mathsf{m}$
and act on 4.32 with $\differential\differential^{\dagger}$. We get that for
all $\mathsf{n}\in\mathscr{N}_{\perp}^{*}$ we must have

$\displaystyle\sum_{\mathsf{m}\in\mathscr{N}_{\perp}^{*}}C_{\mathsf{m}\mathsf{n}}(\lambda_{\mathsf{n}}-\lambda_{\mathsf{m}})=0.$
(4.33)

This can only be achieved if $C_{\mathsf{m}\mathsf{n}}$ is diagonal:
$C_{\mathsf{m}\mathsf{n}}=C_{\mathsf{m}}\delta_{\mathsf{m}\mathsf{n}}$. Now
consider the inner-product
$\left\langle\differential{\chi_{\mathsf{n}}},\differential{\chi_{\mathsf{m}}}\right\rangle$.
We have

$\displaystyle\left\langle\differential{\chi_{\mathsf{n}}},\differential{\chi_{\mathsf{m}}}\right\rangle=\left\langle\chi_{\mathsf{n}},{\differential^{\dagger}}\differential\chi_{\mathsf{m}}\right\rangle=\lambda_{\mathsf{m}}\delta_{\mathsf{n}\mathsf{m}}$
$=$ (4.34) $\displaystyle
C_{\mathsf{n}}C_{\mathsf{m}}\left\langle\star\varphi_{\mathsf{n}},\star\varphi_{\mathsf{m}}\right\rangle=C_{\mathsf{n}}^{2}\delta_{\mathsf{n}\mathsf{m}}.$

This tells us that
$\absolutevalue{C_{\mathsf{n}}}=\sqrt{\lambda_{\mathsf{n}}}$. We can
arbitrarily choose the sign of the square root. Different signs will
correspond to different choices of raising and lowering operators. We choose
the following convention, which will make the choice of raising and lowering
operators consistent across all $d$ and $p$:
$C_{\mathsf{n}}=\sqrt{\lambda_{\mathsf{n}}}$.

Returning to the general case and allowing for degeneracy of the eigenvalues,
we note that the two bases only mix elements of the same eigenvalue, giving
thus $C_{\mathsf{m}\mathsf{n}}$ in a block-diagonal form. Then we can
reshuffle the basis within a subspace of a fixed eigenvalue to fully
diagonalize $C_{\mathsf{m}\mathsf{n}}$ and hence the result is unchanged.
Altogether we can write the final expression for the current algebra:

$\displaystyle\commutator{\hat{b}_{\mathsf{n}}}{\hat{a}_{\mathsf{m}}}$
$\displaystyle=\frac{\mathrm{i}\,\mathbb{K}}{2\pi}\
\sqrt{\lambda_{\mathsf{n}}}\delta_{\mathsf{n}\mathsf{m}}.$ (4.35)

We note the similarity to the Kac–Moody algebras appearing in 2d, however it
is important to emphasize that the modes here are not labeled, necessarily, by
integers, but instead by the countable set of eigenfuctions
$\mathscr{N}_{\perp}^{\star}$.

###  4.4 Verma modules, characters, and extended characters

The current algebra, 4.35, allows us to build $\mathcal{H}_{R}$ as a direct
sum of Verma modules corresponding to integrable representations of this
algebra. It will be useful to supplement this algebra with a positive-definite
Hamiltonian. Recall that since the bulk theory is first order in derivatives,
its Hamiltonian is identically zero and so this is an ingredient we will add
in by hand. The Hamiltonian we will supplement will be the natural
generalization of the “Sugawara Hamiltonian” in 2d Kac–Moody algebras.

Let us define ladder operators of the non-zero-modes as

$\hat{K}_{\mathsf{n}}\coloneqq\frac{1}{\sqrt{2}}\quantity(\mu^{\Delta/2}\hat{a}_{\mathsf{n}}+\mathrm{i}\mu^{-\Delta/2}\hat{b}_{\mathsf{n}})\mbox{\quad
and\quad}\hat{J}_{\mathsf{n}}\coloneqq\frac{1}{\sqrt{2}}\quantity(\mu^{\Delta/2}\hat{a}_{\mathsf{n}}-\mathrm{i}\mu^{-\Delta/2}\hat{b}_{\mathsf{n}})~{}.$
(4.36)

where $\mu$ is an energy scale and $\Delta=d-1-2p$. We then write

$\hat{H}=\hat{H}_{\text{zero}}+\hat{H}_{\text{osc}}$ (4.37)

with

$\displaystyle\hat{H}_{\text{zero}}$
$\displaystyle=\mu^{\Delta}\pi{{\mathbb{K}}^{-1}}\sum_{\mathsf{i},\mathsf{j}=0}^{\mathrm{b}_{p}}\hat{a}_{0\mathsf{i}}\quantity[\mathbb{G}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]^{-1}_{\mathsf{i}\mathsf{j}}\hat{a}_{0\mathsf{j}}+\mu^{-\Delta}\pi{{\mathbb{K}}^{-1}}\sum_{\mathsf{i},\mathsf{j}=0}^{\mathrm{b}_{d-p-1}}\hat{b}_{0\mathsf{i}}\quantity[\mathbb{G}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]^{-1}_{\mathsf{i}\mathsf{j}}\hat{b}_{0\mathsf{j}}$
$\displaystyle\hat{H}_{\text{osc}}$
$\displaystyle=\pi{{\mathbb{K}}^{-1}}\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\quantity(\mu^{\Delta}\hat{a}_{\mathsf{n}}\hat{a}_{\mathsf{n}}+\mu^{-\Delta}\hat{b}_{\mathsf{n}}\hat{b}_{\mathsf{n}})=2\pi{{\mathbb{K}}^{-1}}\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\hat{J}_{\mathsf{n}}\hat{K}_{\mathsf{n}}+\mu
E_{0},$ (4.38)

with $E_{0}$ a potentially divergent zero-point energy that will play no role
in the entanglement entropy. This Hamiltonian is motivated by three very
important points: firstly, it is positive-definite, and secondly, when
expressed in terms of the ladder-operators it plays a natural algebraic role,
extending 4.35 to

$\displaystyle\commutator{\hat{J}_{\mathsf{n}}}{\hat{K}_{\mathsf{m}}}$
$\displaystyle=\frac{\mathbb{K}}{2\pi}\sqrt{\lambda_{\mathsf{n}}}\delta_{\mathsf{n}\mathsf{m}}$
(4.39)
$\displaystyle\commutator{\hat{H}}{\hat{K}_{\mathsf{n}}}=\sqrt{\lambda_{\mathsf{n}}}\hat{K}_{\mathsf{n}}$
$\displaystyle\mbox{\quad
and\quad}\commutator{\hat{H}}{\hat{J}_{\mathsf{n}}}=-\sqrt{\lambda_{\mathsf{n}}}\hat{J}_{\mathsf{n}}.$
(4.40)

It is evident in this writing, that the operator $\hat{J}_{\mathsf{n}}$, with
$\mathsf{n}\in\mathscr{N}_{\perp}^{*}$, lowers the energy by
$\sqrt{\lambda_{\mathsf{n}}}$ units and $\hat{K}_{\mathsf{n}}$ raises the
energy by the same amount.

Thirdly, and perhaps most importantly, when expressed in terms of the original
field variables on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$,
$\hat{H}$ is _local_. Namely writing

$\displaystyle i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}^{*}A$
$\displaystyle=a+\differential{t}\mathbin{\scalebox{0.8}{$\wedge$}}\widetilde{a},$
(4.41) $\displaystyle i_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}X}^{*}B$
$\displaystyle=b+\differential{t}\mathbin{\scalebox{0.8}{$\wedge$}}\widetilde{b},$
(4.42)

with $a\in\Omega^{0}(\mathbb{R})\otimes\Omega^{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$,
$\widetilde{a}\in\Omega^{0}(\mathbb{R})\otimes\Omega^{p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$,
$b\in\Omega^{0}(\mathbb{R})\otimes\Omega^{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$, and
$\widetilde{b}\in\Omega^{0}(\mathbb{R})\otimes\Omega^{d-p-2}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)$, $\hat{H}$ takes the form

$\displaystyle\hat{H}=\frac{\mu^{\Delta}}{4\pi}\left\langle
a,\mathbb{K}\,a\right\rangle_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}+\frac{\mu^{-\Delta}}{4\pi}\left\langle
b,\mathbb{K}\,b\right\rangle_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}.$ (4.43)

One can already recognize $\hat{H}$ as the Hamiltonian of the chiral mixed
Maxwell theory appearing in Section 3181818Indeed, this matching is made
explicit by “adding in” the time component, $\widetilde{a}$, of
$i_{{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}X}^{*}A$ to
$\hat{H}$ by replacing $b$ through the “chiral” boundary condition
$b=\mu^{\Delta}(-1)^{(d-p)p}\star_{\scriptscriptstyle{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\widetilde{a}$..

The structure of the $\mathcal{H}_{R}$ factor of the extended Hilbert space is
now clear. The eigenstates of the zero-modes $\hat{a}_{0\mathsf{i}}$ and
$\hat{b}_{0\mathsf{j}}$ define primary, lowest-weight states, $\ket{\omega}$,
that are annihilated by all $\hat{J}_{\mathsf{n}}$. We can act on each such
state with $\hat{K}_{\mathsf{n}}$, repeatedly to construct a whole Verma
module,

$\displaystyle\mathcal{V}_{\omega}$
$\displaystyle\coloneqq\operatorname{span}\quantity{\ket{\omega,\quantity{N_{\mathsf{n}}}}}\coloneqq\operatorname{span}\quantity{\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\prod_{\mathsf{I}=1}^{\kappa}\quantity(\hat{K}_{\mathsf{I}\mathsf{n}})^{N_{\mathsf{I}\mathsf{n}}}\ket{\omega}}.$
(4.44)

The full Hilbert space is then a direct sum of all Verma modules

$\displaystyle\mathcal{H}_{R}=\bigoplus_{\omega}\mathcal{V}_{\omega}.$ (4.45)

The primary states are labeled by the eigenvalues of the current algebra zero-
modes. These we can easily find by using Dirac quantization restricting to
${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$:

$\displaystyle\int_{\sigma_{\mathsf{i}}}a=2\pi
n^{\mathsf{i}},\qquad{{}^{\forall}\\!}\sigma_{\mathsf{i}}\in\mathrm{H}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)\qquad n^{\mathsf{i}}\in\mathbb{Z}~{},$ (4.46)

and

$\displaystyle\int_{\eta_{\mathsf{i}}}b=2\pi
m^{\mathsf{i}},\qquad{{}^{\forall}\\!}\eta_{\mathsf{i}}\in\mathrm{H}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)\qquad m^{\mathsf{i}}\in\mathbb{Z}.$ (4.47)

Now, invoking the mode expansion, 4.23, this amounts to

$\displaystyle\hat{a}_{0\mathsf{i}}\ket{\omega}$
$\displaystyle=\mathbb{K}\sum_{\mathsf{j}=1}^{\mathrm{b}_{p}}n^{\mathsf{j}}\quantity[\mathbb{G}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\ket{\omega}$ (4.48)
$\displaystyle\hat{b}_{0\mathsf{i}}\ket{\omega}$
$\displaystyle=\mathbb{K}\sum_{\mathsf{j}=1}^{\mathrm{b}_{d-p-1}}m^{\mathsf{j}}\quantity[\mathbb{G}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]_{\mathsf{i}\mathsf{j}}\ket{\omega}.$ (4.49)

Therefore, the primary states, are labeled by two vectors of integers,
$\boldsymbol{n}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}$ and
$\boldsymbol{m}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}$,
$\ket{\omega}\equiv\ket{\boldsymbol{n},\boldsymbol{m}}$, and have energy

$\displaystyle\Delta_{\boldsymbol{n}\boldsymbol{m}}=\pi\mu\quantity(\boldsymbol{n}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{n}+\boldsymbol{m}\cdot\quantity(\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R})\cdot\boldsymbol{m}),$ (4.50)

where we have, again absorbed the powers of $\mu$ into the dimensionless Gram
matrices $\widetilde{\mathbb{G}}_{k}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}$. From here on we will also measure our energies
in units of $\mu$, i.e. $\sqrt{\lambda_{\sf
n}}\rightarrow\mu\sqrt{\lambda_{\sf n}}$ with the new $\lambda_{\mathsf{n}}$’s
dimensionless.

We can now define extended characters of our algebra. Having decomposed the
Hilbert space as

$\displaystyle\mathcal{H}_{R}=\bigoplus_{\begin{subarray}{c}\boldsymbol{n}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{p}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}\\\
\boldsymbol{m}\in\mathbb{Z}^{\kappa\,\mathrm{b}_{d-p-1}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)}\end{subarray}}\mathcal{V}_{\boldsymbol{n},\boldsymbol{m}},$
(4.51)

we can first compute the characters of each Verma module,
$\opbraces{\operatorname{ch}}_{\mathcal{V}_{\boldsymbol{n},\boldsymbol{m}}}[q]\coloneqq\tr_{\mathcal{V}_{\boldsymbol{n},\boldsymbol{m}}}q^{\hat{H}/\mu}$.
This is straightforward. Remembering that $\hat{K}_{\mathsf{n}}$ raises the
energy by $\mu\sqrt{\lambda_{\mathsf{n}}}$ we obtain

$\displaystyle\opbraces{\operatorname{ch}}_{\mathcal{V}_{\boldsymbol{n},\boldsymbol{m}}}[q]\coloneqq{\color[rgb]{0.23046875,0.53515625,0.765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.53515625,0.765625}\underset{\smallmatrixquantity{{\text{vacuum}}\\\
{\text{contribution}}}}{\underbrace{\phantom{-}\vphantom{\quantity(\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}})}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q^{\kappa
E_{0}}}\phantom{-}}}}\
{\color[rgb]{0.65625,0.2890625,0.2890625}\definecolor[named]{pgfstrokecolor}{rgb}{0.65625,0.2890625,0.2890625}\underset{\text{descendants}}{\underbrace{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\quantity(\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\sum_{N_{\mathsf{n}}=0}^{\infty}q^{N_{\mathsf{n}}\sqrt{\lambda_{\mathsf{n}}}})^{\kappa}}}}}\
{\color[rgb]{0.23046875,0.48828125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.23046875,0.48828125,0.14453125}\underset{\text{primaries}}{\underbrace{\phantom{-}\vphantom{\quantity(\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}})}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}q^{\Delta_{\boldsymbol{n},\boldsymbol{m}}}}\phantom{-}}}},$
(4.52)

Finally, summing over all the Verma modules gives us an extended character:

$\displaystyle\boldsymbol{\opbraces{\operatorname{ch}}_{0}}[q]$
$\displaystyle\coloneqq\sum_{\begin{subarray}{c}\boldsymbol{n}\in\mathbb{Z}^{\mathrm{b}_{p}}\\\
\boldsymbol{m}\in\mathbb{Z}^{\mathrm{b}_{d-p-1}}\end{subarray}}\opbraces{\operatorname{ch}}_{\mathcal{V}_{\boldsymbol{n},\boldsymbol{m}}}[q]$
$\displaystyle=q^{\kappa
E_{0}}\quantity(\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\sum_{N_{\mathsf{n}}=0}^{\infty}q^{N_{\mathsf{n}}\sqrt{\lambda_{\mathsf{n}}}})^{\kappa}\sum_{\begin{subarray}{c}\boldsymbol{n}\in\mathbb{Z}^{\mathrm{b}_{p}}\\\
\boldsymbol{m}\in\mathbb{Z}^{\mathrm{b}_{d-p-1}}\end{subarray}}q^{\Delta_{\boldsymbol{n},\boldsymbol{m}}}.$
(4.53)

Here, we once again encounter the generalized Dedekind eta function from
Section 3;

$\displaystyle\upeta^{(p-1)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}[q]\coloneqq
q^{-\tfrac{1}{2}E_{0}}\quantity(\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\sum_{N_{\mathsf{n}}=0}^{\infty}q^{N_{\mathsf{n}}\sqrt{\lambda_{\mathsf{n}}}})^{-1/2}~{}.$
(4.54)

Additionally, recalling the form of the primary state energies, 4.50, the sum
over the primaries of $q^{\Delta_{\boldsymbol{n},\boldsymbol{m}}}$ organizes
into the Siegel Theta functions appearing in Section 3. Consequently, we find
a curious and powerful correspondence: the partition function, 3.30, of the
chiral mixed Maxwell theory, from Section 3 takes the form of an extended
character of our current algebra!

$\displaystyle\boldsymbol{\opbraces{\operatorname{ch}}_{0}}[q]=\frac{\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}_{p}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]}{\quantity(\upeta^{(p-1)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}[q])^{\kappa}}\
\frac{\Theta\quantity[q;\mathbb{K}\otimes\widetilde{\mathbb{G}}_{d-p-1}^{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}]}{\quantity(\upeta^{(d-p-2)}_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}[q])^{\kappa}}=\mathcal{Z}_{{\text{edge}}}\quantity[\mathbb{S}^{1}_{\varepsilon}\times{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}].$ (4.55)

###  4.5 The entanglement entropy

We have now finally set the stage to calculate the entanglement entropy.
Consider partitioning the Cauchy slice, $\Sigma$, into two parts
$\Sigma=R\cup_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}{R}^{\mathsf{c}}$, where
$\cup_{{\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R}$ denotes
gluing along their common boundary. We would like to compute the entanglement
entropy of a state in $\mathcal{H}_{\Sigma}$ upon tracing out the degrees of
freedom on ${R}^{\mathsf{c}}$. However, we have seen that
$\dim\mathcal{H}_{\Sigma}=\absolutevalue{\det\mathbb{K}}^{\mathrm{b}_{p}(\Sigma)}$,
which is obviously finite, but each of $\mathcal{H}_{R}$ and
$\mathcal{H}_{{R}^{\mathsf{c}}}$ is a sum of Verma modules, 4.45, and thus
infinite-dimensional. Thus we starkly see that
$\mathcal{H}_{\Sigma}\neq\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$.
Instead, we will embed $\mathcal{H}_{\Sigma}$ inside the tensor product and
impose the quantum gluing condition, 4.5, to ensure that gauge transformations
acting on ${\text{\scalebox{1.14}{\char 64\relax}}\mspace{1.0mu}}R$ annihilate
the physical states living in
$\mathcal{H}_{\Sigma}\subset\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$.
Let us denote

$\displaystyle\ \hat{Q}_{R}[\alpha]$
$\displaystyle\coloneqq(-1)^{d(p+1)}\frac{\mathbb{K}}{4\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\alpha\mathbin{\scalebox{0.8}{$\wedge$}}\hat{b}=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\alpha^{\mathsf{n}}\hat{b}_{\mathsf{n}}\mbox{\quad
and\quad}$ (4.56) $\displaystyle\hat{Q}_{R}[\beta]$
$\displaystyle\coloneqq\frac{\mathbb{K}}{4\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R}\beta\mathbin{\scalebox{0.8}{$\wedge$}}\hat{a}=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\beta^{\mathsf{n}}\hat{a}_{\mathsf{n}},$
(4.57)

the operators with support on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R$. Similarly for ${R}^{\mathsf{c}}$ there are charge
generators

$\displaystyle\hat{Q}_{{R}^{\mathsf{c}}}[\overline{\alpha}]$
$\displaystyle\coloneqq(-1)^{d(p+1)}\frac{\mathbb{K}}{4\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}{R}^{\mathsf{c}}}\overline{\alpha}\mathbin{\scalebox{0.8}{$\wedge$}}\hat{b}=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\overline{\alpha}^{\mathsf{n}}\hat{b}_{\mathsf{n}}\mbox{\quad
and\quad}$ (4.58)
$\displaystyle\hat{Q}_{{R}^{\mathsf{c}}}\quantity[\overline{\beta}]$
$\displaystyle\coloneqq\frac{\mathbb{K}}{4\pi}\int_{{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}{R}^{\mathsf{c}}}\overline{\beta}\mathbin{\scalebox{0.8}{$\wedge$}}\hat{a}=\sum_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\overline{\beta}^{\mathsf{n}}\hat{a}_{\mathsf{n}},$
(4.59)

with support on ${\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}{R}^{\mathsf{c}}$. Upon gluing $R$ to
${R}^{\mathsf{c}}$ along their common boundary, we will identify $\alpha$ and
${\beta}$ with their parity opposites, $\overline{\alpha}$ and
$\overline{\beta}$, respectively. We can then define the generators of gauge
transformations on the entire $\Sigma$ as

$\displaystyle\hat{\mathcal{Q}}[\alpha]$
$\displaystyle\coloneqq\hat{Q}_{R}[\alpha]\otimes\hat{1}_{{R}^{\mathsf{c}}}+\hat{1}_{R}\otimes\hat{Q}_{{R}^{\mathsf{c}}}[\overline{\alpha}],$
(4.60) $\displaystyle\hat{\mathcal{Q}}[\beta]$
$\displaystyle\coloneqq\hat{Q}_{R}[\beta]\otimes\hat{1}_{{R}^{\mathsf{c}}}+\hat{1}_{R}\otimes\hat{Q}_{{R}^{\mathsf{c}}}\quantity[\overline{\beta}],$
(4.61)

where $\hat{1}_{R}$ $\quantity(\hat{1}_{{R}^{\mathsf{c}}})$ is the identity
operator on $\mathcal{H}_{R}$ $\quantity(\mathcal{H}_{{R}^{\mathsf{c}}})$. The
condition that the physical states should be uncharged under the charge
generators, identifies $\mathcal{H}_{\Sigma}$ as a specific subspace of
$\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$, namely

$\displaystyle\mathcal{H}_{\Sigma}=\opbraces{\operatorname{ker}}\hat{\mathcal{Q}}[\alpha]\cap\opbraces{\operatorname{ker}}\hat{\mathcal{Q}}[\beta]\subset\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}.$
(4.62)

At the level of the ladder operators, 4.36, the quantum gluing condition can
be expressed as

$\displaystyle\quantity(\hat{J}_{\mathsf{n}}\otimes\hat{1}_{{R}^{\mathsf{c}}}+\hat{1}_{R}\otimes\hat{K}_{\mathsf{n}})\ket{\psi}$
$\displaystyle\overset{!}{=}0$ (4.63)
$\displaystyle\quantity(\hat{K}_{\mathsf{n}}\otimes\hat{1}_{{R}^{\mathsf{c}}}+\hat{1}_{R}\otimes\hat{J}_{\mathsf{n}})\ket{\psi}$
$\displaystyle\overset{!}{=}0.$ (4.64)

which roughly states that raising operators on $R$ are lowering operators on
${R}^{\mathsf{c}}$ and vice versa. Due to this, it is clear that
$\ket*{\widetilde{\psi}}$ will be maximally entangled in terms of the
oscillator occupation numbers, $\quantity{N_{\mathsf{n}}}$. The quantum gluing
condition, 4.5, applied at the level of zero-modes enforces a matching of the
integer charges. The result of this is that physical states of this
$\mathcal{H}_{\Sigma}$ embed into generalized Ishibashi states of
$\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$:

$\displaystyle\ket*{\widetilde{\psi}}=\left.\ket{\boldsymbol{0}}\\!\right\rangle\coloneqq\sum_{\boldsymbol{n},\boldsymbol{m}}\sum_{\quantity{N_{\mathsf{n}}}}\ket{\boldsymbol{n},\boldsymbol{m},\quantity{N_{\mathsf{n}}}}_{R}\otimes\overline{\ket{-\boldsymbol{n},-\boldsymbol{m},\quantity{N_{\mathsf{n}}}}_{{R}^{\mathsf{c}}}},$
(4.65)

where the overline denotes an anti-linear conjugate of the state191919We have
written this for $\left.\ket{\boldsymbol{0}}\\!\right\rangle$, corresponding
to the ground state without any Wilson surfaces piercing $\Sigma$. The
computation with a generic $\left.\ket{\boldsymbol{r}}\\!\right\rangle$ can be
easily obtained, by shifting the argument of the Siegel Theta functions by a
constant fractional charge. This will not affect the end entanglement
entropy.. The reduced density matrix is given by tracing over the
$\mathcal{H}_{{R}^{\mathsf{c}}}$ tensor factor:

$\displaystyle\widetilde{\rho}_{R}\coloneqq\tr_{\mathcal{H}_{{R}^{\mathsf{c}}}}\left.\ket{\boldsymbol{0}}\\!\right\rangle\\!\left\langle\\!\bra{\boldsymbol{0}}\right..$
(4.66)

However the above object is not well defined, per se. We need to first address
the subtle issue of normalizability. Given that
$\left.\ket{\boldsymbol{0}}\\!\right\rangle$ is maximally entangled over an
infinite number of modes (labeled by occupation numbers, $N_{\mathsf{n}}$),
its norm is divergent. To obtain a normalizable vector in
$\mathcal{H}_{R}\otimes\mathcal{H}_{{R}^{\mathsf{c}}}$ we need to first
regularize $\left.\ket{\boldsymbol{0}}\\!\right\rangle$ and we will do so with
the Sugawara-type Hamiltonian we discussed earlier, 4.37:

$\displaystyle\left.\ket{\varepsilon}\\!\right\rangle\coloneqq\exp(-\frac{1}{4}\varepsilon\left(\hat{H}_{R}+\hat{H}_{{R}^{\mathsf{c}}}\right))\left.\ket{\boldsymbol{0}}\\!\right\rangle,$
(4.67)

where we have explicitly denoted which factor of $\mathcal{H}_{\text{ext}}$
the Hamiltonians act.

From here it is easy to see that the regulated Ishibashi state, 4.67, takes
the form of a canonical purification of a thermal density matrix with inverse
temperature $\varepsilon$. This thermal density matrix is exactly the
(unnormalized) reduced density matrix

$\displaystyle\widetilde{\rho}_{R}(\varepsilon)\coloneqq\tr_{\mathcal{H}_{{R}^{\mathsf{c}}}}\mathrm{e}^{-\frac{\varepsilon}{4}\left(\hat{H}_{R}+\hat{H}_{{R}^{\mathsf{c}}}\right)}\left.\ket{\varepsilon}\\!\right\rangle\\!\left\langle\\!\bra{\varepsilon}\right.\mathrm{e}^{-\frac{\varepsilon}{4}\left(\hat{H}_{R}+\hat{H}_{{R}^{\mathsf{c}}}\right)}$
(4.68)

and whose norm is given by the extended character, Subsection 4.4

$\displaystyle\tr_{\mathcal{H}_{R}}\widetilde{\rho}_{R}(\varepsilon)=\boldsymbol{\opbraces{\operatorname{ch}}_{0}}[q_{\varepsilon}],\qquad\qquad
q_{\varepsilon}\coloneqq\mathrm{e}^{-\varepsilon\mu}.$ (4.69)

As $\varepsilon\rightarrow 0$ this is simply the regulated dimension of
$\mathcal{H}_{R}$. Thus, the von Neumann entropy of the reduced density matrix
is the regulated $\log\dim\mathcal{H}_{R}$. More suggestively, however, given
the correspondence 4.55, this is also the high-temperature limit of the
thermodynamic entropy of the edge-mode theory, i.e.

$\displaystyle\mathcal{S}_{\text{EE}}=\lim_{\varepsilon\to
0}\quantity(1-\varepsilon{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}_{\varepsilon})\mathcal{Z}_{{\text{edge}}}\quantity[\mathbb{S}^{1}_{\varepsilon}\times{\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R].$ (4.70)

At this point the technical computations follow that of Subsection 3.2. We
arrive again at the main result for the entanglement entropy:

$\displaystyle\mathcal{S}_{\text{EE}}$
$\displaystyle=\sum_{k=1}^{\left\lfloor\frac{d-1}{2}\right\rfloor}\mathrm{C}^{(p-1)}_{k}\quantity(\frac{\ell}{\varepsilon})^{d-2k}+\frac{\kappa}{2}\quantity(\mathcal{I}_{\frac{d-2}{2}}^{(p-1)}+\mathcal{I}_{\frac{d-2}{2}}^{(d-p-2)})\delta_{d,{\text{even}}}\log(\frac{\ell}{\varepsilon})$
(4.71)
$\displaystyle\phantom{=~{}}-\frac{1}{2}\quantity(\mathrm{b}_{p}+\mathrm{b}_{d-p-1})\log\absolutevalue{\det\mathbb{K}}.$

where $\mathrm{C}^{(p-1)}_{k}$ are non-universal, dimensionless, numbers,
$\mathcal{I}_{\frac{d-2}{2}}^{(k)}$ is the $\quantity(\frac{d-2}{2})$-th heat
kernel coefficient for the spectral zeta function, and we have exchanged
$\ell=\mu^{-1}\mathrm{e}$ as a characteristic length scale (see Section 3 for
details).

##  5 Discussion

In this paper, we considered the edge contributions to the entanglement
entropy in higher-dimensional Abelian topological phases described by $p$-form
BF theories. These are phases whose ground states are condensates of $p$-form
surface operators. We found that the entanglement entropy coming from
localized edge-modes at the entangling surface takes the form of a non-
universal, divergent, area law decreasing in powers of two with a possible log
divergence in even dimensions. The constant corrections to this area law are
given in terms of topological features of the entangling surface, namely its
$(p-1)$-st and $(d-p-2)$-st Betti numbers. Our result is upheld through two
separate, but complementary, fronts: we have performed a replica path integral
calculation where the entropy arises as a high-temperature thermal entropy of
an edge-mode partition function living on a regulated entangling surface (with
the regulator playing the role of the inverse temperature). This is edge-mode
theory is a chiral combination of $(p-1)$\- and $(d-p-2)$-form Maxwell
theories, which we call “chiral mixed Maxwell theory.” We followed this
calculation with a more rigorous definition of the entanglement entropy
through an extended Hilbert space and showed that this extended Hilbert space
is organized by a novel infinite-dimensional current algebra, which has not
appeared in the literature (to this degree of generality) before. We
elucidated features of this current algebra and extracted the entanglement
entropy through its representation characters. Along the way we have shown
that these characters account for the spectrum of the Maxwell edge-theory and
match its thermal partition function.

There are several features of our main result that require elaboration and
that we would like to highlight at this point.

#### Comparing to the GTV result

Let us comment on the discrepancy of our result with the result found by
Grover, Turner, and Vishwanath [11] in states of discrete gauge theories
described by condensates of $p$-form membranes:

$\mathcal{S}_{\text{GTV}}=\mathcal{S}_{\text{local}}-\sum_{n=0}^{p-1}(-1)^{p-1+n}\mathrm{b}_{n}({\text{\scalebox{1.14}{\char
64\relax}}\mspace{1.0mu}}R)\;\log{\absolutevalue{G}}$ (5.1)

where $\mathcal{S}_{\text{local}}$ is built out of integrating local
quantities (and so includes possible logs and Euler characteristics in even
dimensions), and $\absolutevalue{G}$ is the order of a discrete gauge group
(this is the analogue of $\absolutevalue{\det\mathbb{K}}$ in our computation).
This result was arrived at by counting the constraints implied by the
intersection of $p$-form membranes with the entangling surface.202020We are
grateful the authors of [11] for correspondence and explaining their result to
us. In order to compare Section 1 and 5.1 and better highlight the
discrepancy, it is useful to write our result in terms of the analogous
alternating sum. We can do this either in the language of differential forms
or in the language of chains. The calculations from Sections 3 and 4 are
natural in the language of differential forms, so let us start there. We note