there is, is a Wilson loop:

$\mathrm{W}_{n}(\gamma)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\,n\,\int_{\gamma}a),\qquad
n\in\mathbb{Z}.$ (5.3)

These operators are electrically charged under the one-form symmetry of Maxell
theory, which served as the starting point of photonisation (cf. section 2).
In particular, if

$U^{\text{elec}}_{C}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\int_{C}\star
f),$ (5.4)

is the generator of the electric one-form symmetry, where $C$ is a closed two-
dimensional surface, $\mathrm{W}_{n}(\gamma)$ carries charge $n$:

$U^{\text{elec}}_{C}\cdot\mathrm{W}_{n}(\gamma)=\exp(\frac{2\pi\,\mathrm{i}\,n}{\imaginary\mathrm{t}}\;{\text{Link}}\quantity(C,\gamma))\mathrm{W}_{n}(\gamma),$
(5.5)

where ${\text{Link}}\quantity(C,\gamma)$ denotes the linking number between
the closed surface $C$ and the loop $\gamma$.

Furthermore, there exist magnetically charged line operators. As we explained
in section 2, and as is well-understood in Maxwell theory [17], there is also
a magnetic one-form symmetry, with generators (in the absense of a theta term)

$U^{\text{mag}}_{C}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\int_{C}f).$
(5.6)

The operators charged under this symmetry are ’t Hooft loops. In the electric
presentation of the theory these are described by excising a loop from
spacetime, providing boundary conditions for the gauge-fields that fix the
value of $\int f$ on a thin tube surrounding the loop. They can be written
more compactly, however, in the magnetic presentation, where we exchange the
gauge-field for a magnetic gauge-field, $\check{a}$, whose curvature is
$\check{f}=\real\mathrm{t}\,f+\mathrm{i}\imaginary\mathrm{t}\star f$. The ’t
Hooft loops become, in this presentation, Wilson loops for the magnetic
photon:

${\text{'tH}}_{m}(\gamma)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\,m\int_{\gamma}\check{a}),\qquad
m\in\mathbb{Z}.$ (5.7)

and carry “electric” charge in the magnetic presentation:

$\exp(\mathrm{i}\int_{C}\star\check{f})\cdot{\text{'tH}}_{m}(\gamma)=\exp(\frac{2\pi\,\mathrm{i}\,m\,\absolutevalue{\mathrm{t}}^{2}}{\imaginary\mathrm{t}}{\text{Link}}\quantity(C,\gamma))\
{\text{'tH}}_{m}(\gamma).$ (5.8)

In the original, electric, presentation, they carry magnetic charge:

$U^{\text{mag}}_{C}\cdot{\text{'tH}}_{m}(\gamma)=\exp(2\pi\,\mathrm{i}\,m\;{\text{Link}}\quantity(C,\gamma))\,{\text{'tH}}_{m}(\gamma).$
(5.9)

In the presence of a theta term, they also carry electric charge, due to the
Witten effect. This is given by:

$U^{\text{elec}}_{C}\cdot{\text{'tH}}_{m}(\gamma)=\exp(2\pi\,\mathrm{i}\,m\,\frac{\real\mathrm{t}}{\imaginary\mathrm{t}}\;{\text{Link}}\quantity(C,\gamma))\,{\text{'tH}}_{m}(\gamma).$
(5.10)

More generally, we can consider dyonic, or Wilson–’t Hooft, line operators
labelled both by an electric and a magnetic charge:

$\mathrm{W}_{n,m}(\gamma)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\,n\int_{\gamma}a+\mathrm{i}\,m\int_{\gamma}\check{a}),\qquad
n,m\in\mathbb{Z}.$ (5.11)

Simultaneously, we will combine the one-form symmetries into their
irreducible, self- and anti-self-dual, incarnations, as in section 2:

$U^{\pm}_{C}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\exp(\mathrm{i}\int_{C}J^{\pm}).$
(5.12)

The Wilson–’t Hooft lines carry charge under $U^{\pm}_{C}$ as:

$U^{\pm}_{C}\cdot\mathrm{W}_{n,m}=\exp(\pm
2\pi\frac{n+\mathrm{t}^{\pm}m}{\imaginary\mathrm{t}}{\text{Link}}(C,\gamma))\mathrm{W}_{n,m}.$
(5.13)

These are the basic, or primary, in a sense to be made precise below, line
operators. But this is not the complete story. We can obtain composite, or
descendant (in the same to-be-made-precise sense) line operators by smearing
functions of gauge-invariant local operators on $\gamma$. The only gauge-
invariant local operators in our disposal are polynomials of (derivatives of)
$J^{\pm}$. An unambiguous way to smear the local operators over the basic line
operators, is to excise an infinitesimal tube around $\gamma$, i.e. an
$\mathbb{S}^{2}_{\varepsilon}\times\gamma$, on which we spread the desired
local operator, and shrink it onto $\gamma$ by taking the limit
$\varepsilon\to 0$. This is completely analogous to how disorder operators,
including the aforementioned basic ’t Hooft loops (in the electric
presentation), are defined in quantum field theory. Using the mode expansion
of $J^{\pm}$ on a three-dimensional surface 4.58, and the OPE of the currents
we can convert the complicated local operator into sums of powers of the
modes, $J^{\pm}_{\mathsf{n}}$, of $J^{\pm}$. A sketch of this construction,
for a specific operator, is shown in figure 3. In the next few sections we
will make this picture precise.

$\displaystyle\int_{\mathbb{S}^{2}_{\varepsilon}\times\gamma}\alpha\wedge
J^{\pm}$
---

${\text{W}}_{n,m}(\gamma)$
---
$J^{\pm}_{\mathsf{n}}{\text{W}}_{n,m}(\gamma)$
---
$\displaystyle\ =\ \sum_{\mathsf{n}}\alpha_{\mathsf{n}}$
---
${\text{W}}_{n,m}(\gamma)$
---
$J^{\pm}_{\mathsf{n}}{\text{W}}_{n,m}(\gamma)$
---
$\displaystyle\int_{\mathbb{S}^{2}_{\varepsilon}\times\gamma}\alpha\wedge
J^{\pm}$
---
Figure 3: An example of descendant line operators. On the left-hand-side of
the equation, a small tube (depicted in blue, modulated to indicate different
“density” along the circle) of the operator $J^{\pm}$ smeared over a function,
$\alpha$, surrounds the curve $\gamma$ which supports a Wilson–’t Hooft line.
On the right-hand-side is the result of shrinking the tube on the line, namely
a series of new, descendant, line operators.

###  5.2 Radial evolution

A crucial feature of the Hilbert space on
$\mathbb{S}^{2}\times\mathbb{S}^{1}$, is its Verma module structure under the
Kac–Moody algebra. In order to reproduce this structure, via a path integral
on $Y=\mathbb{B}^{3}\times\mathbb{S}^{1}$, we need to find an expression of
the current modes, $J^{\pm}_{\mathsf{n}}$, that can be inserted in the path
integral. Key to doing so is solving the self-duality and conservation
equations, for $J^{\pm}$ on $Y$. We take the metric on $Y$ to be,

$\differential{s_{Y}^{2}}=\differential{r^{2}}+\underset{{\color[rgb]{0.23046875,0.53515625,0.765625}\smallmatrixquantity{\text{\rotatebox{90.0}{$\mathrel{\hbox
to0.0pt{\raisebox{0.90417pt}{$\cdot$}\hss}\raisebox{-0.90417pt}{$\cdot$}}=$}}\\\
\differential{s_{\Sigma_{r}}^{2}}}}}{{\color[rgb]{0.23046875,0.53515625,0.765625}\underbrace{{\color[rgb]{0,0,0}r^{2}\differential{s_{\mathbb{S}^{2}}}+\differential{s_{\mathbb{S}^{1}}^{2}}}}}},$
(5.14)

i.e. we fix the $\mathbb{S}^{1}$ radius to be one, throughout, and label the
spatial slice where the $\mathbb{S}^{2}$ is of radius
$r\in\interval{0}{r_{0}}$ by $\Sigma_{r}$, where $\Sigma_{r_{0}}$ is the slice
on which the states, in subsection 4.4, are prepared. With these conventions,
it is useful to decompose the currents as,

$J^{\pm}=\differential{r}\mathbin{\scalebox{0.8}{$\wedge$}}J_{r}^{\pm}+J^{\pm}_{\Sigma_{r}},$
(5.15)

where $J_{r}^{\pm}\in\Omega^{1}\quantity(\Sigma_{r})$ and
$J_{\Sigma_{r}}^{\pm}\in\Omega^{2}\quantity(\Sigma_{r})$. In this
decomposition, the self-duality equations, $\star_{Y}J^{\pm}=\pm J^{\pm}$, are
expressed as a relation between $J^{\pm}_{r}$ and $J^{\pm}_{\Sigma_{r}}$:

$J^{\pm}_{r}=\pm\star_{r}J^{\pm}_{\Sigma_{r}}.$ (5.16)

Here, and in the following, we denote by $\star_{r}$, the Hodge star on
$\Sigma_{r}$, induced by the metric on the slices. Moreover, from the
conservation equations, $\differential{J^{\pm}}=0$, we read a Gauss law and a
dynamical equation controlling the radial evolution of the current. The Gauss
law, written equivalently in terms of $J_{\Sigma_{r}}^{\pm}$ or $J^{\pm}_{r}$,
using 5.16, reads:

$\mathinner{\mathbf{d}J^{\pm}_{\Sigma_{r}}}=0\qquad\Leftrightarrow\qquad{\mathbf{d}^{\dagger}}J^{\pm}_{r}=0,$
(5.17)

The bold differential, $\mathbf{d}$, denotes the differential on $\Sigma_{r}$
and the codifferential is defined with respect to the Hodge-star on
$\Sigma_{r}$. Similarly, the radial evolution equations both in terms of
$J_{\Sigma_{r}}^{\pm}$ and $J^{\pm}_{r}$ read:

$\partial_{r}J^{\pm}_{\Sigma_{r}}\mp\mathbf{d}\star_{r}J^{\pm}_{\Sigma_{r}}=0\qquad\Leftrightarrow\qquad\star_{r}\partial_{r}\star_{r}J^{\pm}_{r}\mp\star_{r}\mathbf{d}J^{\pm}_{r}=0.$
(5.18)

As explained in subsections 4.3 and A, $J^{\pm}_{\Sigma_{r}}$, as a closed
two-form, can be expanded in the (orthonormalised) basis,
$\mathscr{V}^{2}_{\parallel}\quantity(\Sigma_{r})$, given by A.25, of Hodge-
duals of eigen-one-forms of the Beltrami operator on $\Sigma_{r}$:

$J^{\pm}_{\Sigma_{r}}=J^{\pm}_{0}(r)\star_{r}\phi_{0}(r)+\sum_{\begin{subarray}{c}\mathsf{n}\in\mathscr{N}_{\perp}^{*}\\\
\sigma=\pm\end{subarray}}J^{\pm}_{\mathsf{n}\sigma}(r)\star_{r}\phi_{\mathsf{n}\sigma}(r),$
(5.19)

where we now explicitly denote the dependence of the coefficients,
$J^{\pm}_{\mathsf{n}}(r)$, and the basis forms, $\phi_{\mathsf{n}\sigma}(r)$,
on $r$. Plugging this expansion into 5.18, yields a system of first order
ordinary differential equations for the coefficients:

$\displaystyle\partial_{r}J_{0}^{\pm}(r)+\frac{1}{r}J_{0}^{\pm}(r)$
$\displaystyle=0$ (5.20)
$\displaystyle\partial_{r}J_{\mathsf{n}\sigma}^{\pm}(r)+\quantity[\mathbb{A}^{\pm}_{\mathsf{n}}(r)]_{\sigma\tau}J_{\mathsf{n}\tau}^{\pm}(r)$
$\displaystyle=0,$

together with boundary conditions fixing $J_{0}^{\pm}(r_{0})=Q_{0}^{\pm}$, and
$J^{\pm}_{\mathsf{n}\sigma}(r_{0})=Q^{\pm}_{\mathsf{n}\sigma}$. These are
precisely the modes that were used to build creation and annihilation
operators on $\Sigma_{r_{0}}$. Note that this differential equation factorises
modes of different energy, i.e. different $\mathsf{n}$, but mixes modes on the
$\sigma\tau$-plane which we shall call the Beltrami plane.

The differential equation for the zero-mode, can be immediately integrated,
yielding

$J_{0}^{\pm}(r)=\frac{r_{0}}{r}\,Q_{0}^{\pm}.$ (5.21)

For the non-zero-modes, the matrix $\mathbb{A}^{\pm}_{\mathsf{n}}(r)$ is given
as

$\quantity[\mathbb{A}^{\pm}_{\mathsf{n}}(r)]_{\sigma\tau}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\left\langle\phi_{\mathsf{n}\sigma},\star_{r}\partial_{r}\star_{r}\phi_{\mathsf{n}\tau}\right\rangle\mp\left\langle\phi_{\mathsf{n}\sigma},\star_{r}\mathbf{d}\phi_{\mathsf{n}\tau}\right\rangle.$
(5.22)

We provide details about the explicit construction of the basis
$\mathscr{V}\quantity(\Sigma_{r})$, as well as all the entries of the matrix
$\mathbb{A}^{\pm}_{\mathsf{n}}$ in appendices A and C. The solution of the
above equation takes the form of a radially ordered integral:222222That is,
simply, a time-ordered integral, with the radius playing the role of time.

$\displaystyle J^{\pm}_{\mathsf{n}\sigma}(r)$
$\displaystyle=\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}Q^{\pm}_{\mathsf{n}\tau},\mbox{\quad
with\quad}$ (5.23) $\displaystyle\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\opbraces{\operatorname{Rexp}}(\int_{r}^{r_{0}}\differential{r^{\prime}}\mathbb{A}^{\pm}_{\mathsf{n}}(r)).$
(5.24)

Altogether $J^{\pm}_{\Sigma_{r}}$ reads, then

$J^{\pm}_{\Sigma_{r}}=Q_{0}^{\pm}\frac{r_{0}}{r}\star_{r}\phi_{0}(r)+\sum_{\mathsf{n},\sigma,\tau}Q^{\pm}_{\mathsf{n}\sigma}\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}\star_{r}\phi_{\mathsf{n}\tau}(r).$
(5.25)

For illustration purposes, let us concentrate temporarily on a subfamily of
the basis $\mathscr{V}\quantity(\Sigma_{r})$, returning subsequently to the
general case, deferring the details to the appendix. This subfamily concerns
the eigenforms with no momentum along the $\mathbb{S}^{1}$ of $\Sigma_{r}$,
i.e. the one-forms

$\phi_{\ell m\sigma}(r)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{2}}\quantity(\Psi_{\ell
m}\differential{\eta}+\sigma\quantity(\frac{\ell(\ell+1)}{r^{2}})^{-1/2}\star_{r}\mathinner{\mathbf{d}\quantity(\Psi_{\ell
m}\differential{\eta})}missing).$ (5.26)

In the above, $\Psi_{\ell m}$, are the real spherical harmonics, normalised on
$\mathbb{S}^{2}_{r}$, with $\ell$ and $m$ the usual angular momentum and
magnetic quantum numbers, with $\ell\neq 0$, and $\eta$ is the coordinate on
$\mathbb{S}^{1}$. One can check that these are orthonormalised eigenforms of
the Beltrami operator on $\Sigma_{r}$, with eigenvalue
$\sigma\quantity(\frac{\ell(\ell+1)}{r^{2}})^{1/2}$. For this family the
matrix $\mathbb{A}^{\pm}_{\mathsf{n}}(r)$ becomes

$\mathbb{A}^{\pm}_{\ell m}(r)=\frac{1}{2r}\matrixquantity(1\mp
2\sqrt{\ell(\ell+1)}&1\\\ 1&1\pm 2\sqrt{\ell(\ell+1)}).$ (5.27)

In this case $\mathbb{A}^{\pm}_{\ell m}(r)$ commutes with itself at different
radii so the radially ordered exponential reduces to a regular exponential,
yielding, finally as solution

$\displaystyle\mathbb{U}_{\ell
m}^{\pm}(r,r_{0})=\frac{1}{2+4\ell}\Bigg{[}\quantity(\frac{r}{r_{0}})^{-\ell-1}\matrixquantity(1+2\ell\mp
2\sqrt{\ell(\ell+1)}&1\\\ 1&1+2\ell\pm 2\sqrt{\ell(\ell+1)})$ (5.28)
$\displaystyle+\quantity(\frac{r}{r_{0}})^{\ell}\matrixquantity(1+2\ell\pm
2\sqrt{\ell(\ell+1)}&-1\\\ -1&1+2\ell\mp 2\sqrt{\ell(\ell+1)})$
$\displaystyle\Bigg{]}.$

Let us return to the general case. With the solutions 5.21 and 5.23, we can
equivalently express the operators acting on $\Sigma_{r_{0}}$ in terms of the
fields $J^{\pm}$, as

$\displaystyle Q_{0}^{\pm}$
$\displaystyle=\int_{\Sigma_{r}}\frac{r}{r_{0}}\phi_{0}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}$
(5.29) $\displaystyle Q_{\mathsf{n}\sigma}^{\pm}$
$\displaystyle=\int_{\Sigma_{r}}\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}^{-1}\phi_{\mathsf{n}\tau}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}.$
(5.30)

Note how, as they should, the modes $Q_{i}^{\pm}$ (where $i$ stands either for
$0$ or $\mathsf{n}\sigma$), are actually independent of $r$. In other words,
connecting to the discussion of subsection 4.3,
${{\mathbb{U}^{\pm}_{i}(r,r_{0})}^{-1}}\cdot\phi_{i}(r)$ provide a basis of
(anti-)chiral one-forms on $Y$:

$\mathbb{P}_{\pm}\differential\quantity({{\mathbb{U}^{\pm}_{i}(r,r_{0})}^{-1}}\cdot\phi_{i}(r))=0,$
(5.31)

subject to a gauge condition. In this case, the natural gauge condition is
given by the Coulomb gauge, that they lie in the kernel of
$\mathbf{d}\star_{r}i^{*}_{\Sigma_{r}}$.

Knowing the expression of the charges in terms of the field content, we can
now obtain how such charges act on states, $\ket{\mathcal{L}}$, of the form
5.1. We simply perform the path integral inserting $Q^{\pm}_{i}$ on a small
$\Sigma_{r}$, and perform the OPE with the path integral insertion sitting at
the centre, sending subsequently $r\to 0$:

$Q_{i}^{\pm}\ket{\mathcal{L}}=\lim_{r\to
0}\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\int_{\Sigma_{r}}{{\mathbb{U}^{\pm}_{i}(r,r_{0})}^{-1}}\cdot\phi_{i}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}\times\mathcal{L}\quantity(\mathbb{S}^{1}\times\quantity{0}),$
(5.32)

where, in 5.32, only smooth configurations of $J^{\pm}_{\Sigma_{r}}$
contribute, for the same reasons as in section 3. This is a direct four-
dimensional analogue of 3.39. The behaviour of this insertion as $r\to 0$,
requires some clarification. Let us illustrate the situation for the empty
state:

$\ket{\mathbf{1}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}.$
(5.33)

The radial evolution matrix, $\mathbb{U}_{\mathsf{n}}^{\pm}(r,r_{0})$, gives
rise to a smooth mode, behaving as

$\displaystyle\sim\quantity(\frac{r}{r_{0}})^{\ell}$ (5.34) at the origin,
$r\to 0$, and a divergent one, behaving as
$\displaystyle\sim\quantity(\frac{r}{r_{0}})^{-\ell-1}.$ (5.35)

In order to single out the configurations of the currents that contribute to
the path integral, we must project out the divergent modes. This is
implemented by the projector to the kernel of
$\mathbb{U}_{\mathsf{n}}^{\pm}(0,r_{0})$, which we will denote as
$\boldsymbol{\Pi}_{\mathsf{n}}^{\pm}$.232323Note that
$\mathbb{U}_{\mathsf{n}}^{\pm}(0,r_{0})$ from 5.28 (cf. also appendix C) has
rank 1, reflecting precisely the fact that there is a smooth and a divergent
mode. Therefore, the part of $J^{\pm}_{\Sigma_{r}}$ that contributes in the
path integral has an expansion:

$\evaluated{J^{\pm}_{\Sigma_{r}}}_{{\text{missing}}}{smooth}=Q_{0}^{\pm}{\color[rgb]{0.23046875,0.53515625,0.765625}\underset{\sim
r\sin\theta\,\differential{\varphi}\mathbin{\scalebox{0.8}{$\wedge$}}\differential{\theta}}{\underbrace{{\color[rgb]{0,0,0}\frac{r_{0}}{r}\star_{r}\phi_{0}(r)}}}}+\sum_{\mathsf{n},\sigma,\rho,\tau}Q^{\pm}_{\mathsf{n}\sigma}\,{\color[rgb]{0.23046875,0.48828125,0.14453125}\underset{{\text{basis
of smooth two-
forms}}}{\underbrace{{\color[rgb]{0,0,0}\boldsymbol{\Pi}_{\mathsf{n}\sigma\tau}^{\pm}\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\tau\rho}\star_{r}\phi_{\mathsf{n}\rho}(r)}}}},$
(5.36)

From this it immediately follows that the empty state is uncharged:

$Q_{0}^{\pm}\ket{\mathbf{1}}=\lim_{r\to
0}\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\int_{\Sigma_{r}}\frac{r}{r_{0}}\phi_{0}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}=0,$
(5.37)

as the smooth part of $J^{\pm}_{\Sigma_{r}}$ goes to zero at $r\to 0$. The
action of the non-zero modes is also straighforward:

$Q_{\mathsf{n}\sigma}^{\pm}\ket{\mathbf{1}}=\lim_{r\to
0}\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\int_{\Sigma_{r}}\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}^{-1}\phi_{\mathsf{n}\tau}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\boldsymbol{\Pi}^{\pm}_{\mathsf{n}\sigma\tau}Q^{\pm}_{\mathsf{n}\tau},$ (5.38)

where here, and henceforth, repeated indices $\quantity{\sigma,\tau,\ldots}$
on the Beltrami plane are contracted. From this it follows that, if
$v^{\pm}_{\mathsf{n}}$ is any vector in the image of
$\mathbb{U}_{\mathsf{n}}^{\pm}(0,r_{0})$, i.e. orthogonal to
$\boldsymbol{\Pi}_{\mathsf{n}}^{\pm}$, the combination
$v_{\mathsf{n}\sigma}^{\pm}Q_{\mathsf{n}\sigma}^{\pm}$ annihilates the empty
state:

$v_{\mathsf{n}\sigma}^{\pm}Q_{\mathsf{n}\sigma}^{\pm}\ket{\mathbf{1}}=0.$
(5.39)

This behaviour is in fact general. This linear combination of charges will
take the role of annihilation operators, for path integral states, with its
Hermitian conjugate becoming the new creation operators.

###  5.3 Squeeze, squeeze, squeeze

Let us make the above intuition precise. As before, we take
$v^{\pm}_{\mathsf{n}}$ to be a vector orthogonal to
$\boldsymbol{\Pi}_{\mathsf{n}}^{\pm}$,

$\boldsymbol{\Pi}_{\mathsf{n}}^{\pm}\cdot v^{\pm}_{\mathsf{n}}=0.$ (5.40)

Of course, since $\boldsymbol{\Pi}_{\mathsf{n}}^{\pm}$ is a rank-1 projector
on a two-dimensional plane, $v_{\mathsf{n}}^{\pm}$ is unique, up to rescaling
and a global phase. We will take $v_{\mathsf{n}}^{\pm}$ to be such that

$\quantity(v_{\mathsf{n}}^{\pm})^{\dagger}\cdot\boldsymbol{\sigma}_{z}\cdot
v_{\mathsf{n}}^{\pm}=\pm 1,$ (5.41)

where $\boldsymbol{\sigma}_{z}$ is the Pauli $z$-matrix. With this choice of
$v^{\pm}_{\mathsf{m}}$, it is straightforward to check that the operators

$\begin{array}[]{ccccc}\mathcal{B}_{\mathsf{n}\pm}&\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=&v_{\mathsf{n}\sigma}^{\pm}Q_{\mathsf{n}\sigma}^{\pm}&=&v_{\mathsf{n}\pm}^{\pm}\mathcal{A}_{\mathsf{n}\pm}+v_{\mathsf{n}\mp}^{\pm}\mathcal{A}_{\mathsf{n}\mp}^{\dagger},\\\\[3.00003pt]
\mathcal{B}^{\dagger}_{\mathsf{n}\pm}&\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=&{\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}\sigma}^{\pm}Q_{\mathsf{n}\sigma}^{\mp}&=&{\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}\pm}^{\pm}\mathcal{A}^{\dagger}_{\mathsf{n}\pm}+{\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}\mp}^{\pm}\mathcal{A}_{\mathsf{n}\mp},\\\ \end{array}$
(5.42)

where the overline denotes complex conjugation, satisfy the correct Kac–Moody
algebra of ladder operators:

$\commutator{\mathcal{B}_{\mathsf{n}\sigma}}{\mathcal{B}_{\mathsf{m}\tau}^{\dagger}}=\mathrm{k}\sqrt{\lambda_{\mathsf{n}}(r_{0})}\delta_{\mathsf{n}\mathsf{m}}\delta_{\sigma\tau},$
(5.43)

with all other commutators vanishing.

Notice, however, that 5.42 is a Bogoliubov transformation. The new set of
oscillators can be obtained by a unitary transformation of the old
oscillators:

$\displaystyle\mathfrak{S}_{\mathsf{n}}\quantity(v_{\mathsf{n}})\mathcal{A}_{\mathsf{n}\pm}\mathfrak{S}^{\dagger}_{\mathsf{n}}\quantity(v_{\mathsf{n}})$
$\displaystyle=\mathcal{B}_{\mathsf{n}\pm}\mbox{\quad and\quad}$ (5.44)
$\displaystyle\mathfrak{S}_{\mathsf{n}}\quantity({\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}})\mathcal{A}_{\mathsf{n}\pm}^{\dagger}\mathfrak{S}^{\dagger}_{\mathsf{n}}\quantity({\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern 1.3mu}_{\mathsf{n}})$
$\displaystyle=\mathcal{B}^{\dagger}_{\mathsf{n}\pm}.$

The operator $\mathfrak{S}_{\mathsf{n}}(v_{\mathsf{n}})$ is known as a
squeezing operator, and in particular as a as _non-separable two-mode_
squeezing operator [75, 76, 77, 78]. Its explicit form, in terms of the
original set of oscillators, follows by an application of the
Baker–Campbell–Hausdorff formula, and is given as:

$\mathfrak{S}_{\mathsf{n}}\quantity(v_{\mathsf{n}})=\exp(-\rho_{\mathsf{n}}\quantity(\mathrm{e}^{\mathrm{i}\xi_{\mathsf{n}}}\mathcal{A}^{\dagger}_{\mathsf{n}+}\mathcal{A}^{\dagger}_{\mathsf{n}-}-\mathrm{e}^{-\mathrm{i}\xi_{\mathsf{n}}}\mathcal{A}_{\mathsf{n}+}\mathcal{A}_{\mathsf{n}-}))\,\exp(\mathrm{i}\psi_{\mathsf{n}+}\mathcal{A}^{\dagger}_{\mathsf{n}+}\mathcal{A}_{\mathsf{n}+}+\mathrm{i}\psi_{\mathsf{n}-}\mathcal{A}^{\dagger}_{\mathsf{n}-}\mathcal{A}_{\mathsf{n}-}),$
(5.45)

where $\rho_{\mathsf{n}},\xi_{\mathsf{n}},\psi_{\mathsf{n}\pm}\in\mathbb{R}$
are related to $v^{\pm}_{\mathsf{n}}$ as

$\displaystyle v^{\pm}_{\mathsf{n}\pm}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\psi_{\mathsf{n}\pm}}\cosh(\rho_{\mathsf{n}})$
(5.46) $\displaystyle v^{\pm}_{\mathsf{n}\mp}$
$\displaystyle=\mathrm{e}^{\mathrm{i}\quantity(\psi_{\mathsf{n}\pm}+\xi_{\mathsf{n}})}\sinh(\rho_{\mathsf{n}}).$

Apart from the phases $\psi_{\mathsf{n}\pm}$ — which are nevertheless
necessary to capture the correct vectors $v_{\mathsf{n}}^{\pm}$ — this
corresponds to an $\mathfrak{su}(1,1)$ squeezing transformation.

A salient feature of squeezing transformations is that applying the squeezing
operator to the a ground state of the original set of operators,

$\mathcal{A}_{\mathsf{n}\pm}\ket{\boldsymbol{0}}_{\mathcal{A}_{\mathsf{n}}}=0,$
(5.47)

yields a ground state of the new set of operators. Namely, the state

$\ket{\boldsymbol{0}}_{\mathcal{B}_{\mathsf{n}}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathfrak{S}_{\mathsf{n}}(v_{\mathsf{n}})\ket{\boldsymbol{0}}_{\mathcal{A}_{\mathsf{n}}},$
(5.48)

satisfies

$\mathcal{B}_{\mathsf{n}\pm}\ket{\boldsymbol{0}}_{\mathcal{B}_{\mathsf{n}}}=0.$
(5.49)

These states are known as _squeezed vacua_. Excited states can be built in the
standard way, by applying creation operators of the new set of modes on the
squeezed vacuum. They are related to the excited states of the original set of
modes by a squeezing transformation:

$\mathcal{B}_{\mathsf{n}\pm}^{\dagger}\ket{\boldsymbol{0}}_{\mathcal{B}_{\mathsf{n}}}=\mathfrak{S}_{\mathsf{n}}(v_{\mathsf{n}})\mathcal{A}_{\mathsf{n}\pm}^{\dagger}\ket{\boldsymbol{0}}_{\mathcal{A}_{\mathsf{n}}}$
(5.50)

In other words, they remain squeezed.

Finally, recall that our setup involves infinitely many modes. Demanding that
the new ground states are annihillated by $\mathcal{B}_{\mathsf{n}\pm}$ for
all $\mathsf{n}\in\mathscr{N}_{\perp}^{*}$, identifies it with an all-mode
squeezed state that is only pairwise non-separable. In other words there
exists a squeezing operator acting on all modes defined as:

$\mathfrak{S}\underset{{\text{\tiny for}}}{\overset{{\text{\tiny
shorthand}}}{\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\joinrel=\joinrel=\joinrel=\joinrel=}}\mathfrak{S}\quantity(\quantity{v_{\mathsf{n}}}_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\mathfrak{S}_{\mathsf{n}}\quantity(v_{\mathsf{n}}),$
(5.51)

Note here, that there is no ordering amgiguity in defining $\mathfrak{S}$, as
modes at different levels commute. It is for the same reason that it also acts
on each energy level separately, i.e.

$\mathfrak{S}\mathcal{A}_{\mathsf{n}\pm}{\mathfrak{S}^{\dagger}}=\mathcal{B}_{\mathsf{n}\pm},\qquad{{}^{\forall}\\!}\mathsf{n}\in\mathscr{N}_{\perp}^{*},$
(5.52)

without mixing ladder operators at different levels. As a consequence, there
are all-mode squuezed vacua

$\ket{\boldsymbol{0}}_{\mathcal{B}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\mathfrak{S}\ket{\boldsymbol{0}}_{\mathcal{A}},$
(5.53)

annihillated by all $\mathcal{B}$ modes,

$\mathcal{B}_{\mathsf{n}\pm}\ket{\boldsymbol{0}}_{\mathcal{B}}=0,\qquad{{}^{\forall}\\!}\mathsf{n}\in\mathscr{N}_{\perp}^{*},$
(5.54)

where $\ket{\boldsymbol{0}}_{\mathcal{A}}$ is the ground state annihilated by
all $\mathcal{A}$ modes,

$\mathcal{A}_{\mathsf{n}\pm}\ket{\boldsymbol{0}}_{\mathcal{A}}=0,\qquad{{}^{\forall}\\!}\mathsf{n}\in\mathscr{N}_{\perp}^{*}.$
(5.55)

As an explicit illustration of this construction, let us focus again on the
family without momentum on the $\mathbb{S}^{1}$, corresponding to
$\mathsf{n}=(\ell,m,0)$, and governed by the evolution matrix 5.28. The
magnetic quantum number, $m$, does not enter any of the formulas, so we will
suppress it, denoting the modes just by the angular momentum number, $\ell$.
The Bogoliubov coefficients are given by

$\displaystyle v^{+}_{\ell}$
$\displaystyle=\frac{-1}{2\quantity(\ell(\ell+1))^{1/4}}\matrixquantity(\sqrt{\ell}+\sqrt{\ell+1}\\\\[5.0pt]
\sqrt{\ell}-\sqrt{\ell+1}),$ (5.56) $\displaystyle v^{-}_{\ell}$
$\displaystyle=\frac{-\mathrm{i}}{2\quantity(\ell(\ell+1))^{1/4}}\matrixquantity(\sqrt{\ell}-\sqrt{\ell+1}\\\\[5.0pt]
\sqrt{\ell}+\sqrt{\ell+1}),$ (5.57)

corresponding to the squeezing parameters

$\rho_{\ell}=\frac{1}{4}\log(\frac{\ell+1}{\ell}),\quad\xi_{\ell}=\pi,\quad\psi_{\ell+}=\pi,\mbox{\quad
and\quad}\psi_{\ell-}=\frac{3\pi}{2}.$ (5.58)

This yields a two-mode squeezing operator as:

$\mathfrak{S}_{\ell}(v_{\ell})=\exp(\frac{1}{4}\log(\frac{\ell+1}{\ell})\quantity(\mathcal{A}_{\ell+}^{\dagger}\mathcal{A}_{\ell-}^{\dagger}-\mathcal{A}_{\ell-}^{\dagger}\mathcal{A}_{\ell-}))\exp(\mathrm{i}\pi\quantity(\mathcal{A}_{\ell+}^{\dagger}\mathcal{A}_{\ell+}+\frac{3}{2}\mathcal{A}_{\ell-}^{\dagger}\mathcal{A}_{\ell-})).$
(5.59)

###  5.4 The correspondence

After this interlude on squeezed states, we return to line operators, and the
states they prepare, via the path integral. First we consider the states of
the Wilson–’t Hooft operators, 5.11,

$\ket{\mathrm{W}_{n,m}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}).$ (5.60)

Let us see the action of the charges on these states.

Starting wit the zero-mode, we have:

$Q_{0}^{\pm}\ket{\mathrm{W}_{n,m}}=\lim_{r\to
0}\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\int_{\Sigma_{r}}\frac{r}{r_{0}}\phi_{0}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}}\times\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}).$
(5.61)

We will need the OPE between $J^{\pm}_{\Sigma_{r}}$ and the Wilson–’t Hooft
loop. In fact, here we will only need the Wick contraction, as all the regular
terms are bound to vanish, by regularity of the rest of the zero-mode
integrand. The desired OPE is:

$J^{\pm}_{\Sigma_{r}}\times\mathrm{W}_{n,m}(\mathbb{S}^{1}\times\quantity{0})\sim\quantity(\mathrm{i}\,n\expectationvalue{J^{\pm}_{\Sigma_{r}}\int_{\mathbb{S}^{1}}a}+\mathrm{i}\,m\expectationvalue{J^{\pm}_{\Sigma_{r}}\int_{\mathbb{S}^{1}}\check{a}})\mathrm{W}_{n,m}(\mathbb{S}^{1}\times\quantity{0}),$
(5.62)

where the twiddle indicates that it is considered up to regular terms. To
continue on we will need the two-point function of the gauge-fields on
$\mathbb{R}^{3}\times\mathbb{S}^{1}$.242424Remember, we are taking this OPE in
the limit $r\to 0$, so the three-ball, $\mathbb{B}^{3}_{r_{0}}$ is identical
to the whole $\mathbb{R}^{3}$. Parametrising $\mathbb{R}^{3}$ by coordinates
$x$ and the circle by an angle, $\eta$, the two-point function reads:

$\expectationvalue{a_{\mu}(x,\eta)a_{\nu}(0,0)}=\frac{4}{\imaginary\mathrm{t}}\;\frac{g_{\mu\nu}\;\sinh\norm{x}}{\norm{x}\quantity(\cosh\norm{x}-\cos\eta)}+{\text{gauge-
dependent terms}}.$ (5.63)

Integrating one of the gauge fields on the circle, to get the holonomy, and
differentiating the other one to get the current yields immediately:

$\expectationvalue{\quantity(\int_{\Sigma_{r}}\frac{r}{r_{0}}\phi_{0}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}})\quantity(\int_{\mathbb{S}^{1}}a)}=\pm\frac{\pi}{\sqrt{2}\,r_{0}}\frac{1}{\imaginary\mathrm{t}}.$
(5.64)

From here, electric–magnetic duality implies

$\expectationvalue{\quantity(\int_{\Sigma_{r}}\frac{r}{r_{0}}\phi_{0}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}})\quantity(\int_{\mathbb{S}^{1}}\check{a})}=\pm\frac{\pi}{\sqrt{2}\,r_{0}}\frac{\mathrm{t}^{\mp}}{\imaginary\mathrm{t}},$
(5.65)

and therefore, finally:

$Q_{0}^{\pm}\ket{\mathrm{W}_{n,m}}=\pm\pi\mathrm{i}\;\frac{1}{\sqrt{2}r_{0}}\frac{n+\mathrm{t}^{\mp}m}{\imaginary\mathrm{t}}\ket{\mathrm{W}_{n,m}}.$
(5.66)

Compare this to 4.86. The state $\ket{\mathrm{W}_{n,m}}$ has exactly the same
zero-mode charges, as the Kac–Moody primaries $\ket{n,m}$. It is tempting,
therefore, to identify the Wilson–’t Hooft state with $\ket{n,m}$.

This is almost the right answer. To iron out that almost we need the action of
the non-zero modes. First, note that the OPE 5.62 cannot give any singular
contribution upon integrated against any of the higher-modes. The reason is,
simply, that $\expectationvalue{J^{\pm}_{\Sigma_{r}}\int_{\mathbb{S}^{1}}a}$
and $\expectationvalue{J^{\pm}_{\Sigma_{r}}\int_{\mathbb{S}^{1}}a}$ are
proportional to the volume form on the sphere. But the one-forms
$\phi_{\mathsf{n}}(r)$, that it will be integrated against, are periodic on
the circle. Hence the net answer is zero. Therefore, as far as non-zero modes
are concerned, the OPE produces only regular contributions. We are therefore
in the same territory as for the empty state that we discussed previously.
Hence again we have that

$\mathcal{B}_{\mathsf{n}\pm}\ket{\mathrm{W}_{n,m}}=0,\qquad{{}^{\forall}\\!}\mathsf{n}\in\mathscr{N}_{\perp}^{*}.$
(5.67)

This is the precise sense, in which Wilson–’t Hooft lines are _primary
operators_. They have definite one-form charge, and the states they generate
are primary states of the Kac–Moody algebra 5.43, sitting at the bottom of the
Verma module labelled by $n$ and $m$.

Reconciling the above fact with our digression on squeezed states, the
Wilson–’t Hooft lines correspond to _squeezed primary states_ :

$\ket{\mathrm{W}_{n,m}}=\mathfrak{S}\ket{n,m}.$ (5.68)

Inverting this relation, we have, equivalently that the vacua $\ket{n,m}$
correspond to:

$\ket{n,m}={\mathfrak{S}^{\dagger}}\ket{\mathrm{W}_{n,m}}=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
{\mathfrak{S}^{\dagger}}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}),$
(5.69)

where ${\mathfrak{S}^{\dagger}}\mathrm{W}_{n,m}$ denotes the line operator
obtained by shrinking ${\mathfrak{S}^{\dagger}}$ — expressed in terms of the
currents — onto the Wilson–’t Hooft loop, à la figure 3, or equivalently via
the OPE:

${\mathfrak{S}^{\dagger}}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\lim_{r\to
0}{\mathfrak{S}^{\dagger}}[J^{\pm}_{\Sigma_{r}}]\times\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}).$
(5.70)

Let us pause here to comment on the ground state, i.e. the state $\ket{0,0}$,
of zero energy. By the above discussion, it corresponds to the operator

$\ket{0,0}\quad\leftrightsquigarrow\quad{\mathfrak{S}^{\dagger}}\quantity(\mathbb{S}^{1}\times\quantity{0}).$
(5.71)

This is clearly not the identity operator. In one sense, it is almost the
identity operator, as this is the only primary line operator it sees. It is
completely transparent to the one-form charges. However, it is also as far as
one can get from the identity operator as it contains photon excitations of
arbitrary frequency — indeed, of all frequencies. This was, to some extent,
anticipated in [26], where it was shown that, in a generic CFT, the identity
operator, cannot prepare the vacuum state on the torus (or generally, on any
spatial slice other than the sphere). Our result is consistent with that
statement, while still tractable in this example.

Continuing on, and moving up in the Verma module, all the other states can be
reached by acting with $\mathcal{B}_{\mathsf{n}\pm}^{\dagger}$ on the squeezed
primaries, i.e.

$\displaystyle\ket{\mathrm{W}_{n,m};\quantity{N_{\mathsf{n}\sigma}}_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}^{\sigma=\pm}}$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\prod_{\begin{subarray}{c}\mathsf{n}\in\mathscr{N}_{\perp}^{*}\\\
\sigma=\pm\end{subarray}}\quantity(\mathcal{B}^{\dagger}_{\mathsf{n}\sigma})^{N_{\mathsf{n}\sigma}}\ket{\mathrm{W}_{n,m}}$
$\displaystyle=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
\prod_{\begin{subarray}{c}\mathsf{n}\in\mathscr{N}_{\perp}^{*}\\\
\sigma=\pm\end{subarray}}\quantity(\mathcal{B}^{\dagger}_{\mathsf{n}\sigma})^{N_{\mathsf{n}\sigma}}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}).$
(5.72)

The modes $\mathcal{B}_{\mathsf{n}\pm}$ and
$\mathcal{B}_{\mathsf{n}\pm}^{\dagger}$ can be written, in terms of the local
operators $J^{\pm}$ as

$\displaystyle\mathcal{B}_{\mathsf{n}\pm}\quantity(\mathbb{S}^{1}\times\quantity{0})$
$\displaystyle=\lim_{r\to
0}\int_{\Sigma_{r}}v_{\mathsf{n}\sigma}^{\pm}\quantity[\mathbb{U}^{\pm}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}^{-1}\phi_{\mathsf{n}\tau}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}},$
(5.73)
$\displaystyle\mathcal{B}_{\mathsf{n}\pm}^{\dagger}\quantity(\mathbb{S}^{1}\times\quantity{0})$
$\displaystyle=\lim_{r\to 0}\int_{\Sigma_{r}}{\mkern
1.3mu\overline{\mkern-1.3muv\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}\sigma}^{\pm}\quantity[\mathbb{U}^{\mp}_{\mathsf{n}}(r,r_{0})]_{\sigma\tau}^{-1}\phi_{\mathsf{n}\tau}(r)\mathbin{\scalebox{0.8}{$\wedge$}}J^{\pm}_{\Sigma_{r}},$
(5.74)

respectively. Acting them on Wilson–’t Hooft operators gives precisely the
_descendant operators_ that we discussed in subsection 5.1. They are related
to the descendant states 4.75, by a squeezing transformation. For example, a
single excited mode is

$\mathcal{A}^{\dagger}_{\mathsf{n}\pm}\ket{n,m}={\mathfrak{S}^{\dagger}}\mathcal{B}^{\dagger}_{\mathsf{n}\pm}\ket{\mathrm{W}_{n,m}}=\int_{\mathcal{C}[\,\cdot\,]}\mathinner{\mathrm{D}a}\mathrm{e}^{-S[a]}\
{\mathfrak{S}^{\dagger}}\mathcal{B}_{\mathsf{n}\pm}^{\dagger}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0}),$
(5.75)

and similarly for the higher-excited states.

To recap, we have just constructed a one-to-one map between line operators on
$\mathbb{R}^{3}\times\mathbb{S}^{1}$ and states on
$\mathbb{S}^{2}\times\mathbb{S}^{1}$. The line operators that we have in our
disposal are the Wilson–’t Hooft lines and modulated versions thereof, i.e.
smeared with modes of the basic gauge-invariant operator, $f$, or equivalently
$J^{\pm}$. Each different allowed smearing, that is, each different smooth
configuration of $J^{\pm}$ gives a different state. The Wilson–’t Hooft lines
are the primary operators of the Kac–Moody algebra. Modulated operators are
their descendants. They prepare states on
$\mathbb{S}^{2}\times\mathbb{S}^{1}$, that are orthonormal and span the entire
Hilbert space. These states are related to the standard energy eigenstates on
the Hilbert space by a squeezing transformation. This map is displayed in
figure 4.

$\displaystyle\ket{0,0}\quad$
$\displaystyle\leftrightsquigarrow\quad\vbox{\hbox{{
\begin{picture}(1.0,0.2802)\put(0.0,0.0){\includegraphics[width=173.44534pt,page=1]{vacuum-
operator.pdf}}\put(4.29798,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=2]{vacuum-
operator.pdf}}\put(0.42312,0.12682){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=3]{vacuum-
operator.pdf}}\put(8.35326,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\mathcal{B}_{\mathsf{n}\pm}^{\dagger}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=4]{vacuum-
operator.pdf}}\end{picture}}}}$ $\displaystyle\ket{n,m}\quad$
$\displaystyle\leftrightsquigarrow\quad\vbox{\hbox{{
\begin{picture}(1.0,0.2802)\put(0.0,0.0){\includegraphics[width=173.44534pt,page=1]{nm-
primary.pdf}}\put(0.42312,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=2]{nm-
primary.pdf}}\put(-3.45175,0.12682){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=3]{nm-
primary.pdf}}\put(4.4784,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\mathcal{B}_{\mathsf{n}\pm}^{\dagger}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=4]{nm-
primary.pdf}}\end{picture}}}}$
$\displaystyle\mathcal{A}_{\mathsf{n}\pm}^{\dagger}\ket{n,m}\quad$
$\displaystyle\leftrightsquigarrow\quad\vbox{\hbox{{
\begin{picture}(1.0,0.2802)\put(0.0,0.0){\includegraphics[width=173.44534pt,page=1]{nm-
descendant.pdf}}\put(-3.63216,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=2]{nm-
descendant.pdf}}\put(-7.50703,0.12682){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=3]{nm-
descendant.pdf}}\put(0.42312,0.12749){\color[rgb]{0,0,0}\makebox(0.0,0.0)[lt]{\smash{\begin{tabular}[t]{l}${\mathfrak{S}^{\dagger}}\mathcal{B}_{\mathsf{n}\pm}^{\dagger}{\text{W}}_{n,m}\quantity(\mathbb{S}^{1}\times\quantity{0})$\end{tabular}}}}\put(0.0,0.0){\includegraphics[width=173.44534pt,page=4]{nm-
descendant.pdf}}\end{picture}}}}$ $\displaystyle\hskip 10.00002pt\vdots$
Figure 4: The states and their corresponding operators. (Top) The squeezing
operator,
${\mathfrak{S}^{\dagger}}\quantity(\mathbb{S}^{1}\times\quantity{0})$,
containing photons of all frequencies, represented by a rainbow-coloured line
prepares the vacuum state. (Middle) The squeezing operator, surrounding a
Wilson–’t Hooft line of charges $n,m$ prepares the primary state $\ket{n,m}$.
(Bottom) The squeezing operator, on top of the mode
$\mathcal{B}^{\dagger}_{\mathsf{n}\pm}$ on top of a Wilson–’t Hooft line
prepares the descendant $\mathcal{A}^{\dagger}_{\mathsf{n}\pm}\ket{n,m}$.

#### Energies and overlaps

In this last paragraph, we will, briefly, compare the squeezed states we have
arrived at to the energy eigenstates. We begin by calculating the average
energy of the squeezed primary states $\ket{\mathrm{W}_{n,m}}$. A quick
computation shows that their average energy is

$\matrixelement{\mathrm{W}_{n,m}}{H_{\Sigma_{r_{0}}}}{\mathrm{W}_{n,m}}=\Delta_{n,m}+E_{0,\mathfrak{S}}(r_{0}),$
(5.76)

where

$E_{0,\mathfrak{S}}(r_{0})=\sum_{\mathsf{n},\sigma}\sinh[2](\rho_{\mathsf{n}})\sqrt{\lambda_{\mathsf{n}}(r_{0})},$
(5.77)

is essentially measuring a zero-point energy, which has to do with the fact
that the Hamiltonian $H_{\Sigma_{r_{0}}}$ is not normal-ordered with respect
to the new set of ladder operators. More importantly,

$\Delta_{n,m}=\frac{\pi}{2r_{0}^{2}}\frac{\absolutevalue{n+\mathrm{t}\,m}^{2}}{\imaginary\mathrm{t}}$
(5.78)

is the same as the energy of the (unsqueezed) primaries 4.87, as well as the
instanton weights in the path integral 4.38. This property reflects and
refines the observation of Verlinde [34] and the argument of Kapustin [33],
that the Wilson–’t Hooft lines have a quantum number $\Delta_{n,m}$ akin to a
scaling weight.252525Do notice, however, that these are not the eigenvalues of
the dilation operator of the theory under consideration.

For the first excited states, an elementary computation reveals their average
energy

$\matrixelement{\mathrm{W}_{n,m}}{\mathcal{B}_{\mathsf{n}\sigma}H_{\Sigma_{r_{0}}}\mathcal{B}^{\dagger}_{\mathsf{n}\sigma}}{\mathrm{W}_{n,m}}=\Delta_{n,m}+\quantity(\cosh[2](\rho_{\mathsf{n}})+\sinh[2](\rho_{\mathsf{n}}))\sqrt{\lambda_{\mathsf{n}}(r_{0})}+E_{0,\mathfrak{S}}(r_{0}).$
(5.79)

In the no-squeezing limit, $\rho_{\mathsf{n}}\to 0$, this becomes precisely
the energy of the first excited energy eigenstates.

Finally we can also compute overlaps between the squeezed and the unsqueezed
primaries. This is very much facilitated by the disentangling formula of
$\mathfrak{s}\mathfrak{u}(1,1)$ squeezing operators [79, 80]:

$\displaystyle\exp(\alpha_{\mathsf{n}}\mathcal{X}_{\mathsf{n}}^{+}-{\mkern
1.3mu\overline{\mkern-1.3mu\alpha\mkern-1.3mu}\mkern
1.3mu}_{\mathsf{n}}\mathcal{X}_{\mathsf{n}}^{-})$
$\displaystyle=\exp(\mathrm{e}^{\mathrm{i}\arg(\alpha_{\mathsf{n}})}\tanh\absolutevalue{\alpha_{\mathsf{n}}}\mathcal{X}_{\mathsf{n}}^{+})$
$\displaystyle\phantom{=~{}}{\color[rgb]{0.6,0.6,0.6}\times}\exp(-2\log\cosh\absolutevalue{\alpha_{\mathsf{n}}}\mathcal{X}_{\mathsf{n}}^{0})\exp(-\mathrm{e}^{-\mathrm{i}\arg(\alpha_{\mathsf{n}})}\tanh\absolutevalue{\alpha_{\mathsf{n}}}\mathcal{X}_{\mathsf{n}}^{-}),$
(5.80)

where

$\displaystyle\mathcal{X}_{\mathsf{n}}^{+}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{2}}\frac{\mathcal{A}^{\dagger}_{\mathsf{n}+}\mathcal{A}^{\dagger}_{\mathsf{n}-}}{\mathrm{k}\sqrt{\lambda_{\mathsf{n}}}},$
(5.81) $\displaystyle\mathcal{X}_{\mathsf{n}}^{-}$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{2}}\frac{\mathcal{A}_{\mathsf{n}+}\mathcal{A}_{\mathsf{n}-}}{\mathrm{k}\sqrt{\lambda_{\mathsf{n}}}},$
(5.82) $\displaystyle\mathcal{X}_{\mathsf{n}}^{0}$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{2}\quantity(1+\frac{\mathcal{A}^{\dagger}_{\mathsf{n}+}\mathcal{A}_{\mathsf{n}+}+\mathcal{A}^{\dagger}_{\mathsf{n}-}\mathcal{A}_{\mathsf{n}-}}{\mathrm{k}\sqrt{\lambda_{\mathsf{n}}}})$
(5.83)

are the generators of $\mathfrak{s}\mathfrak{u}(1,1)$:

$\commutator{\mathcal{X}_{\mathsf{n}}^{+}}{\mathcal{X}_{\mathsf{n}}^{-}}=-2\mathcal{X}_{\mathsf{n}}^{0},\qquad\commutator{\mathcal{X}_{\mathsf{n}}^{0}}{\mathcal{X}_{\mathsf{n}}^{\pm}}=\pm\mathcal{X}_{\mathsf{n}}^{\pm}.$
(5.84)

In this case, the phases, $\psi_{\mathsf{n}\pm}$, of the squeezing operator,
5.45, do not matter as the phase-shift operator they furnish is normal
ordered, and thus, acts as the identity on $\ket{n,m}$. In total we find:

$\innerproduct{n,m}{\mathrm{W}_{n,m}}=\prod_{\mathsf{n}\in\mathscr{N}_{\perp}^{*}}\quantity(\cosh(\rho_{\mathsf{n}}))^{-2}.$
(5.85)

##  6 Discussion

In this paper, we studied CFTs with continuous generalised global symmetries.
We showed that an invertible continuous $(p+1)$-form symmetry in a
$(2p+2)$-dimensional unitary CFT automatically enhances to an infinite
collection of codimension-one, i.e. zero-form, conserved charges, labelled by
(anti-)chiral $p$-forms. The algebra of these charges is spectrum-generating
(up to decoupled neutral factors) and characterises completely the CFT. The
dynamics of the CFT are those of free $p$-forms. Along a similar vein, we
showed that a non-invertible continuous $(p+1)$-form symmetry, leads to a non-
invertible current algebra, which we describe in terms of the fusion rules of
the symmetry generators. As before, it characterises completely the dynamics,
leading, in this case, to an $\mathrm{O}(2)$ $p$-form gauge theory. In the
invertible case, and focussing on $p=1$, hence in four-dimensional CFTs, we
constructed the representation theory of this algebra (for the invertible
case), which we showed reproduces the path integral calculation, as a non-
trivial check. This allowed us to describe explicitly the Hilbert space of
Maxwell theory on arbitrary closed spatial manifolds. On non-trivial
topologies, the full Hilbert space consists of the usual photon Hilbert space,
as well as of Verma modules, built on top of states with one-form symmetry
charge. Additionally, we showed that the full spectrum of states on
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ can be obtained by a path integral on
$\mathbb{B}^{3}\times\mathbb{S}^{1}$, with various operator insertions. The
radial evolution on the ball acts as a squeezing transformation between the
path integral states and the energy eigenstates, requiring that line operators
are subsequently dressed with a squeezing operator, containing all photon
frequencies, to reach the energy eigenstates. Notably, the vacuum state of
Maxwell theory on $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is not produced by the
path integral with no insertions, but by the path integral with a squeezing
operator inserted. Nevertheless, this construction leads to a one-to-one
correspondence between line operators on $\mathbb{R}^{3}\times\mathbb{S}^{1}$
and states on $\mathbb{S}^{2}\times\mathbb{S}^{1}$ and a classification of
line operators in terms of the current algebra. In short, Wilson–’t Hooft
lines,

$\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})=\exp(\mathrm{i}n\int_{\mathbb{S}^{1}}a+\mathrm{i}m\int_{\mathbb{S}^{1}}\check{a}),$
(6.1)

are charged under the one-form symmetries, $Q_{0}^{\pm}$, 5.29:

$Q_{0}^{\pm}\times\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})\sim\pm\;\frac{\pi\mathrm{i}}{\sqrt{2}r_{0}}\frac{n+\mathrm{t}^{\mp}m}{\imaginary\mathrm{t}}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1}),$
(6.2)

and annihilated by the lowering operators $\mathcal{B}_{\mathsf{n}\pm}$, 5.73,
of the Kac–Moody algebra:

$\mathcal{B}_{\mathsf{n}\pm}\times\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})\sim
0.$ (6.3)

This defines them as primary operators. They have definite scaling weight, as
defined in the body of the paper, given by

$\displaystyle\frac{1}{\mathrm{k}}Q_{0}^{+}Q_{0}^{-}\times\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})$
$\displaystyle\sim\Delta_{n,m}\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})$ (6.4)
$\displaystyle\Delta_{n,m}$
$\displaystyle=\frac{\pi}{2r_{0}^{2}}\frac{\absolutevalue{n+\mathrm{t}\,m}^{2}}{\imaginary\mathrm{t}}.$
(6.5)

The path integral on $\mathbb{B}^{3}\times\mathbb{S}^{1}$ with an insertion of
$\mathrm{W}_{n,m}\quantity(\mathbb{S}^{1})$, dressed by a squeezing operator,
5.70, prepares the primary energy eigenstates, $\ket{n,m}$ 4.67. Descentant
line operators are given by acting with raising operators,
$\mathcal{B}_{\mathsf{n}\pm}^{\dagger}$, 5.74, on the primaries. Again, the
path-inegral on $\mathbb{B}^{3}\times\mathbb{S}^{1}$, with a descendant,
dressed with a squeezing operator, produces the descendant states 4.75.

We comment on several questions we have left unanswered and possible
generalisations.

##### Non-invertible symmetries and orbifold branches

In subsection 4.3 we described a non-invertible current algebra. This was
described in terms of the fusion rules of non-invertible topological
operators. The free theory that realises the algebra is, in that case, that of
an $\mathrm{O}(2)$ $p$-form gauge theory. One very interesting question is to
construct the representation theory of these non-invertible current algebras.
At $p=0$, or equivalently in $d=2$, an important role is played by tube
algebras [81, 82] and lasso actions [83] associated to the fusion of the
topological operators. A careful study of this, should land exactly on the
orbifold branch of a compact scalar, which has a much simpler algebraic
description [53]. Similarly, exploiting and generalising representation theory
of higher-dimensional non-invertible symmeties [54, 55, 56, 57] should give an
algebraic point of view to orbifold branches of gauge-theories, operning a
window to new BKT-like phase transitions at points of enhanced symmetry.

##### Non-abelian current algebras

Despite the absence of non-abelian higher-form symmeries, one could still
consider the physics of a four-dimensional CFT with a non-abelian version of
our higher-dimensional current algebra, schematically of the form:

$\commutator{Q_{\mathsf{n}}^{\mathfrak{a}}}{Q_{\mathsf{m}}^{\mathfrak{b}}}=\mathsf{f}^{\mathfrak{a}\mathfrak{b}}_{~{}~{}\mathfrak{c}}\mathsf{T}_{\mathsf{n}\mathsf{m}}^{\mathsf{l}}Q_{\mathsf{l}}^{\mathfrak{c}}\;+\;\mathrm{k}\sqrt{\lambda_{\mathsf{n}}}\delta^{\mathfrak{a}\mathfrak{b}}\delta_{\mathsf{n}\mathsf{m}},$
(6.6)

where the $\mathfrak{u}(1)\times\mathfrak{u}(1)$ indices, $\pm$, got replaced
by some Lie algebra, $\mathfrak{g}$, indices,
$\quantity{\mathfrak{a},\mathfrak{b},\mathfrak{c},\cdots}$. There are a few
indicators that such an algebra might be hiding behind four-dimensional
supercofnormal field theories. For instance, the form of the Vafa–Witten
partition functions [84], for $\mathcal{N}=4$ super-Yang–Mills (SYM) theory
and their relation to two-dimensional RCFTs is reminiscent of our exact
formulas for the Maxwell partition fuction and their two-dimensional
analogues. A more concrete indicator is Kapustin’s definition of scaling
weight for $\tfrac{1}{2}$BPS operators in $\mathcal{N}=4$ SYM [33], which is
arrived at in a similar way as the one for Wilson–’t Hooft operators, which as
we saw, is intimately linked to the current algebra.

##### Modularity and factorisation

The form, 4.46 of the partition function begs for an investigation of the
modularity properties of the partition function, as well as a possible
“holomorphic factorisation” of the partition function. Both of these
directions deserve separate attention. On the modularity side, if higher-
dimensional CFTs behave like two-dimensional, then swapping the thermal circle
for a spatial one-cycles should leave the partition function unaffected.
Harnessing that statement leads to a generalised Cardy formula [22] and a
universal form of the Casimir energy [27]. Our exact form of the partition
function of Maxwell theory, can serve as a testing ground for modularity in
higher-dimensional CFTs. Relatedly, but with a different goal in mind, one
could imagine making $q$ complex and study whether the Maxwell partition
function factorises in a holomorphic and anti-holomorphic piece:

$\mathcal{Z}_{\text{Maxwell}}\quantity(q,\bar{q})\overset{?}{=}\mathcal{Z}(q)\,\mathcal{Z}\quantity(\bar{q}).$
(6.7)

This would open a new window towards the physics of chiral one-forms (or more
generally chiral $p$-forms), whose importance in six-dimensional
superconformal field theories and in M-theory makes them subject of constant
study [85, 86, 87, 88, 89, 90, 91, 92, 32].

##### Other topologies

The choice of $\mathbb{S}^{2}\times\mathbb{S}^{1}$ for our state-operator
correspondence is a natural one for that the unique one-cycle allows line
operators wrapping it and the unique two-cycle allows a single flux. It is not
clear that this is the only choice, though. For example, states on a
$\mathbb{T}^{3}$ topology are prepared via a four-dimensional path integral
with some two-cycle filled in. It is likely that to get the complete set of
states we have to consider a mixture of states prepared by filling in every
two-cycle. Therefore, the state-operator map, would, in that case, include a
sum over topologies, with a given boundary, in the path integral. This is not
the case here, because of the requirement that the $\mathbb{S}^{1}$ remains
non-trivial.

##### Surgery and overlaps

Given our state-operator correspondence, we now have a resolution of the
identity on $\mathbb{S}^{2}\times\mathbb{S}^{1}$, in terms of line operators:

$\text{1}=\sum_{n,m\in\mathbb{Z}}\sum_{\quantity{N_{\mathsf{n}\sigma}}}\outerproduct{\mathrm{W}_{n,m};\quantity{N_{\mathsf{n}\sigma}}}{\mathrm{W}_{n,m};\quantity{N_{\mathsf{n}\sigma}}}.$
(6.8)

We can use this to simplify correlation functions on manifolds that can be cut
along an $\mathbb{S}^{2}\times\mathbb{S}^{1}$. Relatedly, states can be glued
to produce partition functions or correlation functions on manifolds surgered
along $\mathbb{S}^{2}\times\mathbb{S}^{1}$. This fact opens up a new
perspective on partition functions. Take, for example, partition functions in
$d$ Euclidean dimensions with the topology of the sphere. These are known to
be related to counting problems through F-theorem arguments [93], at least in
$d=3$. On the other hand, by the logic above, this partition function should
be related to the overlap between states prepared by computing path integrals
on some half spaces which do not necessarily produce the vacuum of the theory.
An example would amount to considering the Hopf fibration of the sphere and
consider states defined on the $\mathbb{T}^{2}$ gluing surface. Now we know
those states, while having a simple path integral construction, are described
by complicated superpositions of high-energy eigenstates. Therefore, these
partition functions could have a more natural counting interpretation shedding
light on the exact nature of the F-theorem. At this point this is just a
plausibility argument, as we have only focused on theories with higher-form
symmetries in even dimensions. It would be interesting to study this in more
depth.

##### Acknowledgements

We thank Chen Yang for discussions and an initial collaboration. It is a
pleasure to acknowledge helpful discussions with Jeremias Aguilera Damia,
Dionysios Anninos, Jackson Fliss, Keivan Namjou, Ryan Thorngren, Luigi
Tizzano, David Tong, and Yifan Wang. SV is supported by the NWO Spinoza prize
awarded to Erik Verlinde. SV thanks ULB Brussels and International Solvay
Institutes for hospitality during the workshop “Symmetries, Anomalies and
Dynamics of Quantum Field theory,” as well as CERN, for hospitality during the
“GenHET meeting in String Theory,” where some of the results in this paper
were communicated. Both authors would like to credit Physics Sessions
Initiative, and acknowledge the hospitality of KITP at UCSB. Research at KITP
was supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958.

##  Appendix A The transversal Laplacian

In this appendix we will give some outline some spectral properties of the
transversal Laplacian on three dimensional manifolds, which we denote
generically by $\Sigma$, and, subsequently, explicitly construct its spectrum
on $\Sigma_{r}=\mathbb{S}^{2}_{r}\times\mathbb{S}^{1}$.

###  A.1 Spectral properties

First, by transversal Laplacian, on $p$-forms, on a $d$-dimensional, closed
manifold $M$, we mean:

$\Box_{p}^{M}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\evaluated{\triangle}_{\Omega^{p}(M)\cap\opbraces{\operatorname{ker}}\differential^{\dagger}},$
(A.1)

where $\triangle$ denotes the Hodge Laplacian,
$\triangle=\quantity(\differential+\differential^{\dagger})^{2}$. On non-zero-
modes, the transversal Laplacian is equivalent to
$\differential^{\dagger}\differential$ on $p$-forms. However these operators
have, potentially, different number of zero-modes. We care about
$\Box_{p}^{M}$ which has

$\dim\opbraces{\operatorname{ker}}\Box_{p}^{M}=\dim\opbraces{\operatorname{ker}}\triangle_{p}=\mathrm{b}_{p}(M),$
(A.2)

where $\mathrm{b}_{p}(M)$ is the $p$-th Betti number of $M$. The zero-modes
are given, simply, by the harmonic $p$-forms on $M$. Finally, the transversal
Laplacian is a self-adjoint operator, and hence, its eigenforms provide a
basis for coclosed (also known as transversal) $p$-forms. By Poincaré duality,
the spectrum of the Hodge Laplacian for $0\leqslant p\leqslant d$ is
determined by the spectrum of the transversal Laplacian for $0\leqslant
p\leqslant\left\lfloor\frac{d-1}{2}\right\rfloor$.

Let us now focus on three dimensional manifolds, $\Sigma$, one-forms (i.e.
$p=1$), and non-zero-modes.262626The story can be adapted, more generally, to
$p$-forms on $(2p+1)$-dimensional manifolds, mutatis mutandis [94]. There is
another seld-adjoint operator acting on one-forms, the Beltrami operator:

$\star\differential:\Omega^{1}(\Sigma)\to\Omega^{1}(\Sigma).$ (A.3)

The transversal Laplacian is simply the square of the Beltrami operator,
$\Box_{1}^{\Sigma}=\quantity(\star\differential)^{2}$. As such, they commute,
so they can be diagonalised simultaneously, and, moreover, the non-zero
spectrum of the transversal Lalpacian consists of the squares of the
eigenvalues of the Beltrami operator. Since $\star\differential$ is self-
adjoint its eigen-one-forms provide an (orthonormalisable) basis of coclosed
one-forms.

On a generic closed, oriented three-dimensional manifold, $\Sigma$, the
Beltrami operator has simple spectrum [68]. The same holds for the Hodge
Laplacian [68] with an appropriate clarification on the word
generic.272727More precisely, it is simple on the complement of a
codimension-1 set of metrics, but not on the complement of a codimension-2 set
of metrics [95]. For most of our applications, and importantly, for the state-
operator correspondence, we are interested in very non-generic manifolds, with
high-degree of symmetry, and therefore degeneracy, such as products of
spheres. There, the spectrum of the Hodge Laplacian is actually (at least)
twofold degenerate. On $\mathbb{S}^{3}$ it is a classic result, see e.g. [96,
97, 98] for an account of it. On $\mathbb{S}^{2}\times\mathbb{S}^{1}$, we
construct explicitly the spectrum below, in subsection A.2. Finally, on
$\mathbb{T}^{3}$, it is straightforward to construct the spectrum and see that
it is spanned by

$\exp(\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{\theta})\,\omega(\boldsymbol{k}).$

In the above, $\boldsymbol{\theta}=(\theta_{1},\theta_{2},\theta_{3})$ are the
three angles of the torus, $\boldsymbol{k}=(k_{1},k_{2},k_{3})$, with
$k_{i}=\frac{n_{i}}{L_{i}}$, where $n_{i}\in\mathbb{Z}$ and $L_{i}$ are the
radii of each circle, and
$\omega(\boldsymbol{k})=\omega(\boldsymbol{k})_{i}\differential{x^{i}}$ is an
eigenform — or equivalently $\omega(\boldsymbol{k})_{i}$ is an eigenvector —
of the matrix $\mathrm{i}\,\varepsilon_{ij\ell}k^{\ell}$. It is then easy to
verify that there are two such eigenvectors, both corresponding to the
eigenvalues $\norm{\boldsymbol{k}}^{2}$ for the transversal Laplacian,
verifying its twofold degeneracy.

###  A.2 Eigenforms and eigenvalues on $\mathbb{S}^{2}\times\mathbb{S}^{1}$

WNow we move on to constructing the spectrum on
$\Sigma_{r}=\mathbb{S}^{2}_{r}\times\mathbb{S}^{1}$. We use the following
metric and coordinates:

$\differential{s}^{2}_{\Sigma_{r}}=r^{2}\quantity(\differential{\theta^{2}}+\sin\theta\differential{\varphi^{2}})+\differential{\eta^{2}},$
(A.4)

where $\varphi\in\interval[openright]{0}{2\pi}$ and
$\theta\in\interval[openright]{0}{\pi}$ are the angles on the sphere and
$\eta\in\interval[openright]{0}{2\pi}$ is the angle on the circle. We will
denote the eigen-one-forms of $\Box_{1}^{\Sigma_{r}}$ as $\Phi$, subscripted
by their various quanutm numbers.

#### Zero-mode

The easiest to construct is the zero-mode. This is given by the unique
harmonic form on $\Sigma_{r}$, $\differential{\eta}$. Therefore, the
normalised zero-mode is given by

$\Phi_{0}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{\differential{\eta}}{\sqrt{\opbraces{\operatorname{vol}}(\Sigma_{r})}}=\frac{\differential{\eta}}{2\sqrt{2}\pi\,r}.$
(A.5)

#### Non-zero modes

Let us introduce some convenient notation. First, we write

$T_{k}(\eta)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{cases}\frac{1}{\sqrt{\pi}}\cos(k\eta),&k>0\\\\[10.00002pt]
\frac{1}{\sqrt{2\pi}},&k=0\\\\[10.00002pt]
\frac{1}{\sqrt{\pi}}\sin(\absolutevalue{k}\eta),&k<0,\end{cases}$ (A.6)

with $\eta\in\interval[openright]{0}{2\pi}$. These are the real, orthonormal
eigenfunctions of the Laplacian on $\mathbb{S}^{1}$, with eigenvalue $k^{2}$:

$\Box_{0}^{\mathbb{S}^{1}}T_{k}(\eta)=-\nabla^{2}T_{k}(\eta)=k^{2}\;T_{k}(\eta).$
(A.7)

Similarly, we denote by $\Upsilon_{\ell m}(\theta,\varphi)$ the real spherical
harmonics on $\mathbb{S}^{2}_{r}$:

$\Upsilon_{\ell m}(\theta,\varphi)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{\mathcal{N}_{\ell
m}}{r}\mathrm{P}_{\ell}^{\absolutevalue{m}}(\cos\theta)\begin{cases}\cos(m\varphi),&m>0\\\\[10.00002pt]
1,&m=0\\\\[10.00002pt] \sin(\absolutevalue{m}\varphi),&m<0,\end{cases}$ (A.8)

where,

$\mathcal{N}_{\ell
m}=(-1)^{m}\sqrt{2}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-\absolutevalue{m})!}{(\ell+\absolutevalue{m})!}}\,,$
(A.9)

and $\mathrm{P}_{\ell}^{m}(x)$ are the associated Legendre polynomials. In the
above, $\ell\\!\in\\!\mathbb{Z}_{\geqslant 0}$ and $m$ are integers, with
$-\ell\leqslant m\leqslant\ell$. These are the real, orthonormalised
eigenfunctions of the Laplacian on $\mathbb{S}^{2}_{r}$:

$\displaystyle\Box_{0}^{\mathbb{S}^{2}_{r}}\Upsilon_{\ell
m}(\theta,\varphi)=-\nabla^{2}\Upsilon_{\ell
m}(\theta,\varphi)=\underset{\matrixquantity{\text{\rotatebox{90.0}{$\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=$}}\\\
\rho_{\ell m}(r)}}{{\frac{\ell(\ell+1)}{r^{2}}}}\Upsilon_{\ell
m}(\theta,\varphi).$ (A.10)

The degeneracy of the eigenvalue $\rho_{\ell m}(r)$ is

$D_{\ell m}=2\ell+1.$ (A.11)

Using these two building blocks, we can build all the non-zero eigen-one-
forms.

##### Momentumless.

A first family is given by eigenforms with no momentum along the
$\mathbb{S}^{1}$, i.e. $k=0$. These take the form

$\Phi^{(1)}_{\ell m}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Upsilon_{\ell
m}(\theta,\varphi)\frac{\differential{\eta}}{\sqrt{2\pi}},$ (A.12)

with $\ell>0$ and $-\ell\leqslant m\leqslant\ell$, and have eigenvalue
$\rho_{\ell m}(r)$:

$\Box_{1}^{\Sigma_{r}}\Phi^{(i)}_{\ell
m}=\frac{\ell(\ell+1)}{r^{2}}\Phi^{(i)}_{\ell m}.$ (A.13)

One can explicitly check that

$\displaystyle\Phi^{(2)}_{\ell m}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{\rho_{\ell
m}(r)}}\star_{r}\mathbf{d}\Phi^{(1)}_{\ell m}$ (A.14)
$\displaystyle=\frac{1}{\sqrt{2\pi}}\frac{r}{\sqrt{\ell(\ell+1)}}\quantity(\frac{1}{\sin\theta}\,\partial_{\varphi}\Upsilon_{\ell
m}(\theta,\varphi)\differential{\theta}-\sin\theta\,\partial_{\theta}\Upsilon_{\ell
m}(\theta,\varphi)\differential{\varphi}),$

is also an eigen-one-form of $\Box_{1}^{\Sigma_{r}}$, with the same
eigenvalue, $\rho_{\ell m}(r)$. All of these modes are orthonormalised:

$\left\langle\Phi_{\ell
m}^{(i)},\Phi_{\ell^{\prime}m^{\prime}}^{(i^{\prime})}\right\rangle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\delta_{ii^{\prime}},\qquad
i,i^{\prime}\in\quantity{1,2}.$ (A.15)

##### Momentumful.

The rest of the modes, are modes with momentum along the $\mathbb{S}^{1}$,
i.e. $k\neq 0$. They are given by

$\displaystyle\Phi^{(1)}_{\ell mk}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{\rho_{\ell
m}(r)+k^{2}}}\quantity(\sqrt{\frac{\rho_{\ell m}(r)}{k^{2}}}\Upsilon_{\ell
m}(\theta,\varphi)\differential T_{k}(\eta)-\sqrt{\frac{k^{2}}{\rho_{\ell
m}(r)}}\differential\Upsilon_{\ell m}(\theta,\varphi)\,T_{k}(\eta))$ (A.16)
$\displaystyle=\quantity(\frac{\ell(\ell+1)}{r^{2}}+k^{2})^{-\frac{1}{2}}\left[\frac{\sqrt{\ell(\ell+1)}}{r}\frac{\partial_{\eta}T_{k}(\eta)}{k}\Upsilon_{\ell
m}(\theta,\varphi)\differential{\eta}\right.$ $\displaystyle\hskip
50.00008pt\left.-kT_{k}(\eta)\,\frac{r}{\sqrt{\ell(\ell+1)}}\quantity(\partial_{\varphi}\Upsilon_{\ell
m}(\theta,\varphi)\,\differential{\varphi}+\partial_{\theta}\Upsilon_{\ell
m}(\theta,\varphi)\differential{\theta})\right],$

and

$\displaystyle\Phi^{(2)}_{\ell mk}$ $\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{\rho_{\ell
m}(r)+k^{2}}}\star_{r}\mathbf{d}\Phi^{(2)}_{\ell mk}$ (A.17)
$\displaystyle=\frac{1}{\sqrt{\rho_{\ell
m}(r)\,k^{2}}}\star_{r}\quantity[\differential\Upsilon_{\ell
m}(\theta,\varphi)\mathbin{\scalebox{0.8}{$\wedge$}}\differential
T_{k}(\eta)]$
$\displaystyle=\frac{r}{\sqrt{\ell(\ell+1)}}\frac{\partial_{\eta}T_{k}(\eta)}{k}\quantity(\frac{1}{\sin\theta}\,\partial_{\varphi}\Upsilon_{\ell
m}(\theta,\varphi)\,\differential{\theta}-\sin\theta\,\partial_{\theta}\Upsilon_{\ell
m}(\theta,\varphi)\differential{\varphi}).$

Their eigenvalue is $\lambda_{\ell mk}(r)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\rho_{\ell
m}(r)+k^{2}$:

$\Box_{1}^{\Sigma_{r}}\Phi^{(i)}_{\ell
mk}=\quantity(\frac{\ell(\ell+1)}{r^{2}}+k^{2})\Phi^{(i)}_{\ell
mk}\mathrel{=\hbox to0.0pt{\hss\raisebox{1.29167pt}{$\cdot$}}\hbox
to0.0pt{\hss\raisebox{-1.29167pt}{$\cdot$}}}.\lambda_{\ell
mk}(r)\Phi^{(i)}_{\ell mk}$ (A.18)

and are again orthonormalised:

$\left\langle\Phi_{\ell
mk}^{(i)},\Phi_{\ell^{\prime}m^{\prime}k}^{(i^{\prime})}\right\rangle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\delta_{kk^{\prime}}\delta_{ii^{\prime}}.$
(A.19)

We write collectively, both for the case with momentum and without momentum,
the modes as $\Phi^{(i)}_{\ell mk}$, by defining $\Phi^{(i)}_{\ell
m0}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Phi^{(i)}_{\ell
m}$. Therefore, a complete orthonormal basis of the space of coclosed one-
forms on $\Sigma_{r}$ is given by

$\mathscr{B}^{1}_{\perp}(\Sigma_{r})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Bigg{\\{}\Phi_{0},\Big{\\{}\Phi_{\ell
mk}^{(i)}\Big{\\}}_{\ell\in\mathbb{Z}_{>0},\;m\in\interval{-\ell}{\ell},\;k\in\mathbb{Z}}^{i=1,2}\Bigg{\\}}.$
(A.20)

The degeneracy of the eigenvalue $\lambda_{\ell mk}(r)$ on one-forms is

$D_{\ell mk}^{(1)}=2(2\ell+1).$ (A.21)

Finally, the Hodge duals of the above forms provide the spectrum of the
longitudinal Laplacian on two-forms,
$\evaluated{\triangle}_{\Omega^{2}(\Sigma_{r})\cap\opbraces{\operatorname{ker}}\mathbf{d}missing}$.
Relatedly, an orthonormal basis for those is:

$\mathscr{B}^{2}_{\parallel}(\Sigma_{r})\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Bigg{\\{}\star_{r}\Phi_{0},\Big{\\{}\star_{r}\Phi_{\ell
mk}^{(i)}\Big{\\}}_{\ell\in\mathbb{Z}_{>0},\;m\in\interval{-\ell}{\ell},\;k\in\mathbb{Z}}^{i=1,2}\Bigg{\\}}.$
(A.22)

###  A.3 A convenient basis

As we discussed above, the transversal Laplacian commutes with the operator
$\star_{r}\mathbf{d}$. Therefore they can be diagonalised simultaneously. The
one-forms that diagonalise the operator $\star_{r}\mathbf{d}$ are related to
the above basis as follows:

$\phi_{\ell mk\sigma}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\frac{1}{\sqrt{2}}\quantity(\Phi^{(1)}_{\ell
mk}+\sigma\,\Phi^{(2)}_{\ell mk}),\qquad\sigma=\pm.$ (A.23)

They are again orthonormal and statisfy

$\star_{r}\mathbf{d}\phi_{\ell mk\sigma}=\sigma\sqrt{\lambda_{\ell mk}(r)}\
\phi_{\ell mk\sigma}.$ (A.24)

Therefore, we can also write the following two bases for coclosed one-forms
and closed two-forms respectively:

$\displaystyle\mathscr{V}^{1}_{\perp}(\Sigma_{r})$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Bigg{\\{}\phi_{0}(r),\Big{\\{}\phi_{\mathsf{n}\sigma}(r)\Big{\\}}_{\mathsf{n}=(\ell,m,k)}^{\sigma=\pm}\Bigg{\\}},$
(A.25) $\displaystyle\mathscr{V}^{2}_{\parallel}(\Sigma_{r})$
$\displaystyle\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\Bigg{\\{}\star_{r}\phi_{0}(r),\Big{\\{}\star_{r}\phi_{\mathsf{n}\sigma}(r)\Big{\\}}_{\mathsf{n}=(\ell,m,k)}^{\sigma=\pm}\Bigg{\\}},$

where we also renamed the zero-mode to $\phi_{0}(r)\equiv\Phi_{0}(r)$. When we
need to expand both one- and two- forms in these bases, we will refer to them
collectively as $\mathscr{V}(\Sigma_{r})$. This basis has the advantage that
it makes the Kac–Moody structure clear.

##  Appendix B Current algebra in the generic case

In this appendix we explain how the Kac–Moody algebra of subsection 4.3 gets
modified in the generic — in the sense of [68, 95] — case of a closed three-
dimensional manifold, where the spectrum of the transversal Laplacian is
simple, i.e. the eigenvalues are non-degenerate (cf. appendix A). We denote
the three-dimensional Riemannian manifold we are treating, by a slight abuse
of notation, as $\Sigma$, where it should be understood that we are actually
considering the pair $(\Sigma,g)$ where $g$ is a Riemannian metric on $\Sigma$
used to define the Hodge-star. We are not interested in varying the metric to
obtain genericity results, like in [68, 95], so we suppress it.

The Beltrami operator, $\star\differential$, will still be used to diagonalise
the current algebra 2.27/4.48. However, in the generic case, if
$\sqrt{\lambda_{\mathsf{n}}}$ appears in the spectrum of $\star\differential$,
$-\sqrt{\lambda_{\mathsf{n}}}$ doesn’t, and vice versa. So we will label the
orthonormalised eigen-one-forms of $\star\differential$ as

$\quantity{\phi_{\mathsf{n}_{\pm}}}_{\mathsf{n}_{\pm}\in\mathscr{N}_{\pm}},$
(B.1)

(where $\mathscr{N}_{\pm}$ is the index-set containing the labels
$\mathsf{n}_{\pm}$) satisfying

$\star\differential{\phi_{\mathsf{n}_{\pm}}}=\pm\sqrt{\lambda_{\mathsf{n}_{\pm}}}\phi_{\mathsf{n}_{\pm}},$
(B.2)

where $\lambda_{\mathsf{n}_{\pm}}>0$ is the corresponding (non-zero)
eigenvalue of the transversal Laplacian on $\phi_{\mathsf{n}_{\pm}}$. The
labelling above, is such that eigenforms with label $\mathsf{n}_{-}$ contain
the negative spectrum of the Beltrami operator, and those with
$\mathsf{n}_{+}$, the positive spectrum. The genericity result mentioned
above, implies that typically,
$\lambda_{\mathsf{n}_{+}}\neq\lambda_{\mathsf{n}_{-}}$ for all
$\mathsf{n}_{+},\mathsf{n}_{-}$. This is unlike the case we have treated in
the main text — valid in highly symmetric, and fine-tuned, manifolds, such as
products of spheres. Moreover, on a three-dimensional manifold, the spectrum
of $\star\differential$ accumulates at $+\infty$ and at $-\infty$ [99, 100,
101], which guarantees that both signs appear in B.2. As before, we exclude
the kernel of $\star\differential$, as this is treated, unambiguously, via the
harmonic forms,
$\quantity{\phi_{0\mathsf{i}}}_{\mathsf{i}=1}^{\mathrm{b}_{1}(\Sigma)}$, of
$\Sigma$. The one-forms $\phi_{\mathsf{n}_{\pm}}$ are such that

$\left\langle\phi_{\mathsf{n}_{\pm}},\phi_{\mathsf{m}_{\pm}}\right\rangle_{\Sigma}=\delta_{\mathsf{n}_{\pm}\mathsf{m}_{\pm}},\mbox{\quad
and\quad}\left\langle\phi_{\mathsf{n}_{+}},\phi_{\mathsf{n}_{-}}\right\rangle_{\Sigma}=0.$
(B.3)

As explained in appendix A,
$\quantity{\phi_{0\mathsf{i}},\phi_{\mathsf{n}_{\pm}}}$ provide a complete
basis of coclosed one-forms on $\Sigma$. We use this basis to expand
$\Lambda^{\pm}$ in 2.27/4.48, and the dual, two-form basis, given by the Hodge
stars of the above, to expand $J^{\pm}$:

$\displaystyle i^{*}_{\Sigma}\Lambda^{\mp}$
$\displaystyle=\sum_{\mathsf{i}=1}^{\mathrm{b}_{1}(\Sigma)}\Lambda^{\mp}_{0\mathsf{i}}\phi_{0\mathsf{i}}+\sum_{\mathsf{s}=\pm}\;\sum_{\mathsf{n}_{\mathsf{s}}\in\mathscr{N}_{\mathsf{s}}}\Lambda^{\mp}_{\mathsf{n}_{\mathsf{s}}}\star_{\Sigma}\phi_{\mathsf{n}_{\mathsf{s}}},$
(B.4) $\displaystyle i^{*}_{\Sigma}J^{\pm}$
$\displaystyle=\sum_{\mathsf{i}=1}^{\mathrm{b}_{2}(\Sigma)}Q^{\pm}_{0\mathsf{i}}\star_{\Sigma}\phi_{0\mathsf{i}}+\sum_{\mathsf{s}=\pm}\;\sum_{\mathsf{n}_{\mathsf{s}}\in\mathscr{N}_{\mathsf{s}}}Q^{\pm}_{\mathsf{n}_{\mathsf{s}}}\star_{\Sigma}\phi_{\mathsf{n}_{\mathsf{s}}}.$
(B.5)

The difference with 4.58 is subtle, but important. This expansion leads to the
mode algebra:

$\commutator{Q^{\pm}_{\mathsf{n}_{\mathsf{s}}}}{Q^{\mp}_{\mathsf{m}_{\mathsf{t}}}}=\pm\mathrm{k}\mathsf{s}\sqrt{\lambda_{\mathsf{n}_{\mathsf{s}}}}\delta_{\mathsf{n}_{\mathsf{s}}\mathsf{m}_{\mathsf{t}}}.$
(B.6)

Defining ladder operators

$\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{cases}Q^{+}_{\mathsf{n}_{\mathsf{s}}}&\mathsf{s}=+\\\
Q^{-}_{\mathsf{n}_{\mathsf{s}}}&\mathsf{s}=-\\\ \end{cases},\mbox{\quad
and\quad}\mathcal{A}^{\dagger}_{\mathsf{n}_{\mathsf{s}}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{cases}Q^{-}_{\mathsf{n}_{\mathsf{s}}}&\mathsf{s}=+\\\
Q^{+}_{\mathsf{n}_{\mathsf{s}}}&\mathsf{s}=-,\\\ \end{cases},$ (B.7)

leads to the algebra

$\commutator{\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}}{\mathcal{A}^{\dagger}_{\mathsf{m}_{\mathsf{t}}}}=\mathrm{k}\sqrt{\lambda_{\mathsf{n}_{\mathsf{s}}}}\delta_{\mathsf{n}_{\mathsf{s}}\mathsf{m}_{\mathsf{t}}}.$
(B.8)

Note that we still have two decoupled algebras, one for each sign of
$\mathsf{s}$, i.e. one for each side of the spectrum of $\star\differential$.
In terms of these modes, the Hamiltonian takes the form

$H_{\Sigma}=\frac{1}{\mathrm{k}}\sum_{\mathsf{i}=1}^{\mathrm{b}_{2}(\Sigma)}Q_{0\mathsf{i}}^{+}Q_{0\mathsf{i}}^{-}+\frac{1}{\mathrm{k}}\sum_{\mathsf{s}=\pm}\;\sum_{\mathsf{n}_{\mathsf{s}}\in\mathscr{N}_{\mathsf{s}}}\mathcal{A}^{\dagger}_{\mathsf{n}_{\mathsf{s}}}\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}+E_{0},$
(B.9)

with

$E_{0}=\frac{1}{2}\sum_{\mathsf{s}}\sum_{\mathsf{n}_{\mathsf{s}}}\sqrt{\lambda_{\mathsf{n}_{\mathsf{s}}}}.$
(B.10)

As can be seen from B.8, the operators
$\mathcal{A}^{\dagger}_{\mathsf{n}_{\mathsf{s}}}$ raise the energy by
$\sqrt{\lambda_{\mathsf{n}_{\mathsf{s}}}}$, and
$\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}$ lower it by the same amount.

From here on, it is an easy exercise to repeat the steps explained in the main
text and arrive at the conclusion that the Hilbert space is spanned by

$\ket{\boldsymbol{n},\boldsymbol{m};\quantity{N_{\mathsf{n}_{\mathsf{s}}}}_{\mathsf{n}_{\mathsf{s}}\in\mathscr{N}_{\mathsf{s}}}^{\mathsf{s}=\pm}}\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\prod_{\mathsf{s}=\pm}\prod_{\mathsf{n}\in\mathscr{N}_{\mathsf{s}}}\quantity(\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}^{\dagger})^{N_{\mathsf{n}_{\mathsf{s}}}}\ket{\boldsymbol{n},\boldsymbol{m}},$
(B.11)

with $\ket{\boldsymbol{n},\boldsymbol{m}}$ the primary states, given by 4.67
and 4.68, with the appropriate tweak that they are annihilated by all
$\mathcal{A}_{\mathsf{n}_{\mathsf{s}}}$. The rest of the discussion, including
4.80 and its implications follow immediately.

##  Appendix C Details on the radial evolution

Here we collect some details about the radial evolution on
$\mathbb{B}^{3}\times\mathbb{S}^{1}$. For the reader’s convenience we repeat
the radial evolution equation, 5.20 here:

$\partial_{r}J_{\mathsf{n}\sigma}^{\pm}(r)+\quantity[\mathbb{A}^{\pm}_{\mathsf{n}}(r)]_{\sigma\tau}J_{\mathsf{n}\tau}^{\pm}(r)=0.$
(C.1)

The matrix $\mathbb{A}_{\mathsf{n}}^{\pm}(r)$ is given, in the basis
$\mathscr{V}(\Sigma_{r})$ A.25, by

where $\mathsf{n}=(\ell,m,k)$. Writing

$J^{\pm}_{\mathsf{n}\sigma}(r)=\quantity[\mathbb{U}_{\mathsf{n}}^{\pm}(r,r_{0})]_{\sigma\tau}J^{\pm}_{\mathsf{n}\sigma}(r_{0}),$
(C.2)

the initial value problem consisting of C.1 and its boundary conditions at
$r=r_{0}$ gets mapped to the matrix ordinary differential equation:

$\dot{\mathbb{U}}_{\mathsf{n}}^{\pm}(r,r_{0})+\mathbb{A}_{\mathsf{n}}^{\pm}(r)\mathbb{U}_{\mathsf{n}}^{\pm}(r,r0)=0,$
(C.3)

where the dot indicates derivative with respect to $r$, together with boundary
conditions

$\mathbb{U}_{\mathsf{n}}^{\pm}(r_{0},r_{0})=\text{1},$ (C.4)

This has the unique solution

$\mathbb{U}_{\mathsf{n}}^{\pm}(r,r_{0})=\opbraces{\operatorname{Rexp}}\quantity(\int_{r}^{r_{0}}\differential{r^{\prime}}\mathbb{A}_{\mathsf{n}}^{\pm}\quantity(r^{\prime})),$
(C.5)

where $\opbraces{\operatorname{Rexp}}$ denotes the radially ordered
exponential:

$\opbraces{\operatorname{Rexp}}\quantity(\int_{r}^{r_{0}}\differential{r^{\prime}}\mathbb{O}\quantity(r^{\prime}))\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\sum_{N=0}^{\infty}\frac{1}{N!}\int_{r}^{r_{0}}\\!\differential{r_{1}}\int_{r}^{r_{0}}\\!\differential{r_{2}}\cdots\int_{r}^{r_{0}}\\!\differential{r_{N}}\mathrm{R}\quantity\big(\mathbb{O}(r_{1})\mathbb{O}(r_{2})\cdots\mathbb{O}(r_{N})),$
(C.6)

with

$\mathrm{R}\quantity(\mathbb{O}_{1}(r_{1})\mathbb{O}_{2}(r_{2}))\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=\begin{cases}\mathbb{O}_{1}(r_{1})\mathbb{O}_{2}(r_{2})&{\text{if}}\
r_{1}<r_{2},\\\ \mathbb{O}_{2}(r_{2})\mathbb{O}_{1}(r_{1})&{\text{if}}\
r_{2}<r_{1}.\end{cases}$ (C.7)

When $k=0$, it holds that
$\commutator{\mathbb{A}_{\mathsf{n}}^{\pm}(r_{1})}{\mathbb{A}_{\mathsf{n}}^{\pm}(r_{2})}=0$
for all radii, and hence the ordered exponential reduces to a regular one. The
solution is given in this case by 5.28:

$\displaystyle\mathbb{U}_{\ell
m}^{\pm}(r,r_{0})=\frac{1}{2+4\ell}\Bigg{[}\quantity(\frac{r}{r_{0}})^{-\ell-1}\matrixquantity(1+2\ell\mp
2\sqrt{\ell(\ell+1)}&1\\\ 1&1+2\ell\pm 2\sqrt{\ell(\ell+1)})$ (C.8)
$\displaystyle+\quantity(\frac{r}{r_{0}})^{\ell}\matrixquantity(1+2\ell\pm
2\sqrt{\ell(\ell+1)}&-1\\\ -1&1+2\ell\mp 2\sqrt{\ell(\ell+1)})$
$\displaystyle\Bigg{]}.$

Here and onwards we suppress the magnetic quantum number $m$ in the labelling
of the radial evolution data, as they don’t depend on it. The eigenvalues of
$\mathbb{U}_{\ell}^{\pm}(r,r_{0})$ are

$\delta_{(1)\ell}^{\pm}=\quantity(\frac{r}{r_{0}})^{-\ell-1}\mbox{\quad
and\quad}\delta_{(2)\ell}^{\pm}=\quantity(\frac{r}{r_{0}})^{\ell},$ (C.9)

with eigenvectors

$u_{(1)\ell}^{\pm}=\matrixquantity(1+2\ell\mp 2\sqrt{\ell(\ell+1)}\\\\[5.0pt]
1)\mbox{\quad and\quad}u_{(2)\ell}^{\pm}=\matrixquantity(-1-2\ell\mp
2\sqrt{\ell(\ell+1)}\\\\[5.0pt] 1),$ (C.10)

respectively. It is evident that at $r\to 0$ one of the eigenvalues vanishes
and, thus, the evolution matrix becomes rank one. Its kernel is spanned by
$v_{(2)\ell}^{\pm}$. The projector $\mathbf{\Pi}_{\ell}^{\pm}$, appearing from
5.36 onwards, is, in this case:

$\mathbf{\Pi}_{\ell}^{\pm}=\frac{u_{(2)\ell}^{\pm}\otimes
u_{(2)\ell}^{\pm}}{\norm{u_{(2)\ell}^{\pm}}^{2}}=\frac{1}{2(1+2\ell)}\matrixquantity(1+2\ell\pm
2\sqrt{\ell(\ell+1)}&-1\\\\[5.0pt] -1&1+2\ell\mp 2\sqrt{\ell(\ell+1)}).$
(C.11)

For $k\neq 0$, $\mathbb{A}_{\mathsf{n}}^{\pm}(r)$ does not commute with itself
at different radii and thus we have to use the radially ordered exponential.
While this we cannot solve exactly for arbitrary radius, the crucial feature
our the state-operator correspondence is its behaviour as $r\to 0$. In that
limit, the differential equation C.3 reduces to the differential equation for
$k=0$, and reveals that $\mathbb{U}_{\ell mk}^{\pm}(r,r_{0})$, behaves
identically to its $k=0$ eigenpart. Namely it has a singular and a regular
part, scaling as

$\sim\quantity(\frac{r}{r_{0}})^{-\ell-1}\mbox{\quad
and\quad}\sim\quantity(\frac{r}{r_{0}})^{\ell},$ (C.12)

respectively. Correspondingly, $\mathbb{U}_{\ell mk}^{\pm}(0,r_{0})$ becomes
rank-1.

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