# Radiative heat exchange driven by acoustic vibration
modes between two solids at the atomic scale

M. Gómez Viloria  Laboratoire Charles Fabry, UMR 8501, Institut d’Optique,
CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127 Palaiseau
Cedex, France.  Y. Guo  Institut Lumière Matière, Université Claude Bernard
Lyon 1, CNRS, Université de Lyon, 69622 Villeurbanne, France.  S. Merabia
Institut Lumière Matière, Université Claude Bernard Lyon 1, CNRS, Université
de Lyon, 69622 Villeurbanne, France.  R. Messina  Laboratoire Charles Fabry,
UMR 8501, Institut d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin
Fresnel, 91127 Palaiseau Cedex, France.  P. Ben-Abdallah
<EMAIL_ADDRESS>Laboratoire Charles Fabry, UMR 8501, Institut
d’Optique, CNRS, Université Paris-Saclay, 2 Avenue Augustin Fresnel, 91127
Palaiseau Cedex, France.

(September 3, 2024)

###### Abstract

When two solids are separated by a vacuum gap of thickness smaller than the
wavelength of acoustic phonons, the latter can tunnel across the gap thanks to
van der Waals forces or electrostatic interactions. Here we show that these
mechanical vibration modes can also contribute significantly, at the atomic
scale, to the nonlocal radiative response of polar materials. By combining
molecular-dynamics simulations with fluctuational-electrodynamics theory we
investigate the near-field radiative heat transfer between two slabs due to
this opto-mechanical coupling and we highlight its dominant role at cryogenic
temperatures. These results pave the way to exciting avenues for the control
of heat flux and the development of cooling strategies at the atomic scale.

The physics of heat transfer between two solids separated by a vacuum gap in
the transition regime between conduction and radiation remains today largely
unknown. When solids are separated by gaps having thicknesses of tens of
nanometres or more heat transfer is exclusively driven by photons exchange. In
the far-field regime (distances larger than the thermal wavelength, around
10$\,\mu$m at ambient temperature) this transfer is limited by Stefan-
Boltzmann’s law defining the blackbody limit Planck . At subwavelength scale
and down to distances of about ten nanometers, the heat flux exchanged between
the solids can overcome this limit by several orders of magnitude Rytov ;
Polder ; Joulain ; Volokitin ; RMP ; Hargreaves ; Narayanaswamy ; Shen ;
Rousseau ; Ottens ; Kralik  thanks to the tunneling of evanescent photons
which superimposes to the flux driven by propagative photons. Below this
separation distance, heat transfer can be mediated by multiple carriers arxiv
; Francoeur1 ; Francoeur2 ; Guo ; MGV . More specifically, at sub-nanometer
scale, acoustic vibration modes of solids participate in the transfer. In
2015, Chiloyan et al. highlighted Chiloyan , by means of atomistic
simulations, the dominant role played by these modes on the transfer between
polar materials. In that work, it was claimed that this transfer results from
the tunneling of vibration modes thanks to surface forces. Such a transfer has
been described theoretically in the continuum limit for isotropic media by
Pendry et al. Pendry  (see also Refs. Volokitin2 ; Volokitin3 ) using the
classical elasticity theory. More recently this description has been extended
to anisotropic piezoelectric materials Maasilta . In the present Letter we
show that acoustic vibration modes, which are traditionally purely mechanical
modes in the long-wavelength (LW) limit, are not only able to tunnel through
the separation gap thanks to the surface forces existing between the two
solids, but also contribute, at atomic-scale separation distances (i.e. short
wavelengths limit), to the nonlocal radiative response of materials. In the LW
limit, it is well-known that the optical phonons are the only excitations that
give rise to local electric dipole moments inside the material owing to the
motion of neighboring atoms with opposite partial charges in opposite
directions. In this limit, the optical phonons are the only link between the
atomic vibrations within the solid and the surrounding electromagnetic field.
These optical vibration modes entirely drive the radiative response of
material. Although a mesoscopic theory describing the nonlocal response of
polar materials has been recently introduced De Liberato  in an analogous way
as the hydrodynamic description of electron gas in nonlocal plasmonics Fuchs ;
Cirac , this theory suffers from a fundamental limitation to properly describe
light-matter interactions at the atomic scale. Indeed, it ignores the crucial
role played by the acoustic vibration modes in the radiative response of
material. However, as we will see, at atomic scale, acoustic vibration modes
play a major role in this response. Here, we make a detailed description of
this opto-mechanical coupling and we highlight its importance in the radiative
heat exchanges between two solids close to the physical contact. We
demonstrate that, contrary to wide belief, these acoustic vibration modes can
significantly contribute to the radiative heat exchanges and can even be
dominant, in front of the contribution coming from optical phonons, in the
cryogenic regime.

Figure 1: (a) Dispersion relation of eigenmodes in a diatomic chain with a
reduced mass $\mu=\frac{M_{1}M_{2}}{M_{1}+M_{2}}$, chain stiffness $C$ and
lattice constant $a$. (b) Imaginary part of the dielectric permittivity of a
MgO cubic crystal (bulk) in the (001) direction of reciprocal space,
calculated by molecular dynamics. (c) Dispersion relation of eigenmodes in the
MgO crystal obtained by solving its secular equation. The four eigenmodes are
known as the longitudinal optical (LO; solid red), transverse optical (TO;
solid black), longitudinal acoustic (LA; dashed red) and transverse acoustic
(TA; dashed black) branches.

To start, let us consider a polar material of arbitrary crystalline structure.
To describe light interaction with this crystal in the infrared frequency
range and at atomic-scale separation distances, a nonlocal model of the
radiative response of the material must be introduced. To go beyond the Gubbin
and De Liberato theory De Liberato  a fully atomistic calculation of the
dielectric response has been performed (see Supplemental Material suppmat  and
references thompson ; matsui ; chalopin ; Callen  therein), based on the
analysis of the fluctuations of the polarization density within the crystal.
In spatially invariant crystals this density is related to local electric
field $\mathbb{E}(\mathbb{r},t)$ through the nonlocal relation (assuming the
system stationary)

$\mathbb{P}(\mathbb{r},t)=\epsilon_{0}\int\int\mathrm{d}\mathbb{r^{\prime}}\mathrm{d}t^{\prime}\;\stackrel{{\scriptstyle\leftrightarrow}}{{\chi}}\\!(\mathbb{r}-\mathbb{r^{\prime}},t-t^{\prime})\cdot\mathbb{E}(\mathbb{r^{\prime}},t^{\prime}),$
(1)

where $\epsilon_{0}$ is the vacuum permittivity, whereas
$\stackrel{{\scriptstyle\leftrightarrow}}{{\chi}}\\!(\mathbb{r}-\mathbb{r}^{\prime},t-t^{\prime})$
denotes the electric susceptibility tensor of the crystal,
$\mathbb{r}-\mathbb{r}^{\prime}$ and $t-t^{\prime}$ being the spatial and
temporal distance, respectively, between polarization and electric field.
According to the fluctuation-dissipation theorem Chandler , the spatial
Fourier components of the susceptibility at temperature $T$ read

$\chi_{mn}(\mathbb{k},t)=-H(t)\frac{1}{\epsilon_{0}k_{\rm
B}T}\frac{\mathrm{d}}{\mathrm{d}t}\langle\delta P_{m}(\mathbb{k},0)\delta
P^{*}_{n}(\mathbb{k},t)\rangle,$ (2)

where $\delta P_{m}=P_{m}-\langle P_{m}\rangle$, $\langle.\rangle$ denotes an
ensemble average, $*$ is the conjugation operation and $H$ is the Heaviside
function. For a statistically stationary process (i.e. $\mathrm{d}\langle
P_{m}\rangle/\mathrm{d}t=0$) we have equivalently

$\chi_{mn}(\mathbb{k},t)=-H(t)\frac{1}{\epsilon_{0}k_{\rm
B}T}\frac{\mathrm{d}}{\mathrm{d}t}\langle
P_{m}(\mathbb{k},0)P^{*}_{n}(\mathbb{k},t)\rangle.$ (3)

It turns out by time Fourier transformation of this expression that the
electric susceptibility and the relative dielectric permittivity of the
crystal read

$\begin{split}\chi_{mn}(\mathbb{k},\omega)&=\frac{1}{\epsilon_{0}k_{\rm
B}T}\bigg{[}\langle P_{m}(\mathbb{k},0)P^{*}_{n}(\mathbb{k},0)\rangle\\\
&\,\left.+\,\mathrm{i}\omega\int_{0}^{\infty}\mathrm{d}t\;e^{\mathrm{i}\omega
t}\langle P_{m}(\mathbb{k},0)P^{*}_{n}(\mathbb{k},t)\rangle\right]\end{split}$
(4)

and
$\epsilon_{mn}(\mathbf{k},\omega)=\chi_{mn}(\mathbf{k},\omega)+\delta_{mn}$,
respectively, $\delta_{mn}$ being the Kronecker delta. These expressions
relate the nonlocal radiative response of material to the correlations
functions of local dipolar moments. The latter are in turn related to the
motion of partial charges which are associated to each atom. In contrast, we
define the local response as the limit
$\chi_{mn}(\mathbf{k},\omega)\to\chi_{mn}(\mathbf{k}=0,\omega)$ valid for
large separation distances. Relation (4) can be generalized to the quantum
regime by relating the correlation function of fluctuating polarization
density operator to the electrical susceptibility through the general Kubo
formula Zwanzig .

To give insight into the link between the vibration modes and the radiative
response of material we detail below the case of the diatomic chain Kittel
which is the simplest polar crystal. In this particular case, it is well known
that the dispersion relation of resonant modes reads

$\begin{split}\omega^{2}&=C\left(\frac{1}{M_{1}}+\frac{1}{M_{2}}\right)\\\
&\,\pm
C\left[\left(\frac{1}{M_{1}}+\frac{1}{M_{2}}\right)^{2}-\frac{4\sin^{2}(ka/2)}{M_{1}M_{2}}\right]^{1/2},\end{split}$
(5)

where $C$ denotes the chain stiffness between the atoms of mass $M_{1}$ and
$M_{2}$ while $a$ is the lattice period and $k$ is the mode wavenumber. As for
the amplitudes $u_{l}=u_{k}\exp(\mathrm{i}[kla-\omega t])$ and
$v_{l}=v_{k}\exp(\mathrm{i}[kla-\omega t])$ of the normal modes associated
with the masses $M_{1}$ and $M_{2}$, respectively, in the unit cell $l$ they
satisfy the relation

$\frac{u_{k}}{v_{k}}=\frac{2Ce^{-\mathrm{i}ka/2}\cos(ka/2)}{2C-M_{1}\omega^{2}}.$
(6)

In the LW limit ($k\to 0$), the dispersion relation of optical (high
frequency) and acoustic (low frequency) branches read

$\omega^{2}=2C\left(\frac{1}{M_{1}}+\frac{1}{M_{2}}\right),\qquad\omega^{2}=\frac{C}{2(M_{1}+M_{2})}(ka)^{2},$
(7)

and the amplitudes of optical and acoustic normal modes satisfy respectively
the relations $u_{k}/v_{k}\approx-M_{2}/M_{1}$ (i.e. out-of-phase atomic
vibration) and $u_{k}/v_{k}\approx 1$ (i.e. in-phase atomic vibration),
showing that only the optical modes give rise to dipole moments. On the other
hand, close to the upper bound of the Brillouin zone (i.e. $k\approx\pi/a$),
that is in the extreme near-field regime, the situation radically changes. As
shown in Fig. 1(a), an anticrossing of acoustic and optical branches appears
in this zone showing a strong coupling between these modes with a frequency
splitting (assuming here $M_{2}<M_{1}$) $\Gamma=\omega_{\rm o}-\omega_{\rm
a}$, with $\omega_{\rm o}=\sqrt{2C/M_{2}}$ and $\omega_{\rm
a}=\sqrt{2C/M_{1}}$. As far as the amplitudes of normal modes are concerned,
we see from the general expression (6) that

$\frac{u_{k}}{v_{k}}\approx\frac{\mathrm{i}C(ka-\pi)}{2C-M_{1}\omega^{2}_{\rm
o,a}},$ (8)

so that $u_{k}/v_{k}\to 0$ for the optical modes and $u_{k}/v_{k}\to\infty$
for the acoustic modes. These relations demonstrate that both types of modes
give rise to dipole moments and therefore both contribute to the radiative
response of the chain. Moreover these relations also demonstrate that for both
acoustic and optical modes one atom is motionless in the unit cell while the
second is free to oscillate making these modes identical in nature.

Figure 2: Thermal conductance spectra between two MgO samples for different
separation distances at (a) $T=300$ K and (b) for different temperatures at
$d=6$ Å. The conductance is calculated using both the local (red) and nonlocal
(red) dielectric response of material. The dashed curves represent Planck’s
law in arbitrary units.

The radiative contribution of acoustic modes to the nonlocal radiative
response can be directly observed in a concrete scenario. In Fig. 1(b) we show
the dielectric response of a magnesium oxide (MgO) crystal in the $(001)$ axis
of reciprocal space obtained by molecular-dynamics simulation suppmat . This
material has been chosen to get well-separated optical and acoustic branches
making the analysis and interpretation of results easier. The comparison of
this mapping with the dispersion relations of resonant vibration modes [Fig.
1(c)] calculated by solving the secular equation of the crystal clearly shows
the contribution of optical branches at high frequencies but also the one of
acoustic branches at low frequencies. We also observe, at the edge of the
Brillouin zone, the anticrossing, previously mentioned for the diatomic chain,
between the longitudinal optical (LO) and the longitudinal acoustic (LA)
branches, true signature of strong coupling between these vibration modes in
this region. It is worthwhile to note that the contribution of acoustic modes
is not limited to the edge of the Brillouin zone. In Fig. 1(b) we see that the
acoustic mode can couple to the electromagnetic field relatively far away from
this region. For MgO, the contribution of acoustic modes to the nonlocal
response of crystal can be observed down to wavectors $k\approx 1/2a$, $a=4.2$
Å being the lattice constant of crystal.

The role played by these modes on the radiative heat transfer can then be
investigated thanks to fluctational-electrodynamics theory. According to this
framework, the conductance of radiative heat exchanged at temperature $T$
between two solids separated by a vacuum gap of thickness $d$ can be written
in the Landauer-like form Polder ; pba ; Biehs

$G(T,d)=\int_{0}^{\infty}\\!\frac{\mathrm{d}\omega}{2\pi}\frac{\mathrm{d}\Theta}{\mathrm{d}T}(\omega,T)\int\\!\frac{\mathrm{d}\boldsymbol{\kappa}}{(2\pi)^{2}}\;\sum_{\alpha=\mathrm{s},\mathrm{p}}\mathcal{T}_{\alpha}(\mathbb{\kappa},\omega,d),$
(9)

where $\Theta(\omega,T)={\hbar\omega}/[\exp(\hbar\omega/k_{\rm B}T)-1]$ is the
mean energy of Planck oscillator at temperature $T$ and
$\mathcal{T}_{\alpha}(\mathbb{\kappa},\omega,d)$ is the transmission
coefficient in polarization $\alpha\in\\{\mathrm{s,p}\\}$ of mode
($\mathbb{\kappa},\omega$), $\mathbb{\kappa}$ being the parallel component of
the wavector. Assuming a system with azimuthal symmetry this coefficient reads

$\begin{split}&\mathcal{T}_{\alpha}(\kappa,\omega,d)\\\
&\,=\begin{cases}\displaystyle\frac{(1-|r_{\alpha,1}|^{2})(1-|r_{\alpha,2}|^{2})}{|1-r_{\alpha,1}r_{\alpha,2}\exp[2\mathrm{i}k_{z}d]|^{2}},&\kappa<\omega/c,\\\
\vspace{-0.3cm}\\\
\displaystyle\frac{4\,\mathrm{Im}\,r_{\alpha,1}\mathrm{Im}\,r_{\alpha,2}\exp[-2\,\mathrm{Im}\,(k_{z})d]}{|1-r_{\alpha,1}r_{\alpha,2}\exp[-2\,\mathrm{Im}\,(k_{z})d]|^{2}},&\kappa\geq\omega/c.\\\
\end{cases}\end{split}$ (10)

Here, $r_{\alpha,i}$ denotes the reflection coefficient of medium $i=1,2$ from
vacuum and $k_{z}=\sqrt{(\omega/c)^{2}-\kappa^{2}}$ is the normal component of
wavector in vacuum while $\kappa=|\boldsymbol{\kappa}|$. The reflection
coefficients can be written in terms of surface impedances $Z_{\alpha,i}$ as
follows Ford

$r_{\mathrm{s},i}(\kappa,\omega)=\frac{\displaystyle
Z_{\mathrm{s},i}(\kappa,\omega)-\frac{\omega}{c^{2}k_{z}}}{\displaystyle
Z_{\mathrm{s},i}(\kappa,\omega)+\frac{\omega}{c^{2}k_{z}}},$ (11a)
$r_{\mathrm{p},i}(\kappa,\omega)=\frac{\displaystyle\frac{k_{z}}{\omega}-Z_{\mathrm{p},i}(\kappa,\omega)}{\displaystyle\frac{k_{z}}{\omega}+Z_{\mathrm{p},i}(\kappa,\omega)},$
(11b)

with Esquivel

$\displaystyle
Z_{\mathrm{s},i}(\kappa,\omega)=\frac{2\mathrm{i}}{\pi\omega}\int_{0}^{\infty}\frac{\mathrm{d}q_{z}}{\epsilon_{\mathrm{t},i}(k,\omega)-(ck/\omega)^{2}},$
(12a) $\displaystyle Z_{\mathrm{p},i}(\kappa,\omega)$
$\displaystyle\,=\frac{2\mathrm{i}}{\pi\omega}\int_{0}^{\infty}\frac{\mathrm{d}q_{z}}{k^{2}}\left[\frac{q_{z}^{2}}{\epsilon_{\mathrm{t},i}(k,\omega)-(ck/\omega)^{2}}+\frac{\kappa^{2}}{\epsilon_{\mathrm{l},i}(k,\omega)}\right],$
(12b)

where $k^{2}=q_{z}^{2}+\kappa^{2}$. Here $\epsilon_{\mathrm{l},i}(k,\omega)$
and $\epsilon_{\mathrm{t},i}(k,\omega)$, denote the longitudinal and
transverse dielectric functions which are calculated by molecular-dynamics
simulations suppmat . The computed spectra of heat conductances calculated
from the integrand over the $\omega$-integral of expression (9) with the
nonlocal response of material (dependent on wavevector $k$) for different
separation distances and different temperatures are presented in (Figs. 2 a,b)
and compared with conductances calculated with the local dielectric
permittivity $\epsilon(\omega)$ (independent of $k$).

At gaps $d>1$ nm we see that the heat transfer mainly stems from modes at high
frequencies. The comparison of spectra with the dielectric permittivity
plotted in Fig. 1 shows that these modes are in the spectral range of optical
phonons. However, below this critical distance we observe that the lower-
frequency modes also participate in the transfer. We also note in Fig. 2(b)
that the relative weight of these modes in comparison with the high-frequency
modes increases at low temperature. These modes even become dominant in the
cryogenic regime ($T<100$ K). The inspection of the transmission coefficients
plotted in Fig. 3 clearly shows that these low-frequency modes correspond to
acoustic vibration modes.

Figure 3: Local (left) and nonlocal (right) transmission coefficient of
p-polarized radiative heat exchange between two MgO samples separated by a
vacuum gap of $d=6$ Å

.

This result unquestionably demonstrates that the acoustic modes contribute
radiatively to the transfer at small separation distances. Unlike the
conductive-like heat transfer due to the mechanical tunneling of acoustic
modes mediated by van der Waals forces between the two solids, this transfer
is purely radiative and is related to dipole-dipole interactions induced by
the acoustic vibration modes. Also, it must be noted (see Fig.1(b)) that the
acoustic modes with very small wavevector do not play any role in the
radiative response of the material. These modes can in principle participate
to the heat transfer by tunneling but not to the radiative one. However, as
shown in Chiloyan  this tunneling is negligible because of the weakness of
surface forces.

Finally, we analyze the ratio of the nonlocal radiative conductance to the
local one. The results plotted in Fig. 4 with respect to the separation
distance for different temperatures show, at large distance, that the local
and non local conductances become, as expected, identical. Moreover we note
that nonlocal effects dominate at low temperature, in the cryogenic regime
owing to the dipoles generated by the acoustic vibration modes. This result
would be trivial if these modes were known to give rise to electric dipoles.
But this is not generally the case for small wavevectors. Hence, at relatively
large distances [see Fig. 2(a)] these modes do not contribute to the transfer.
On the other hand, at close separation distance (large wavectors), this is not
true anymore and these mechanical vibration modes contribute significantly to
the heat transfer, this contribution being purely radiative in nature. In the
inset of Fig. 4 we see at ambient temperature that the conductance follows the
usual power law scaling in $1/d^{2}$, $d$ being the separation gap between the
two plates (see suppmat ). On the the other hand, in the cryogenic regime the
scaling changes since the transfer is not mediated anymore by the surface
phonon-polaritons.

Figure 4: Nonlocal vs local radiative conductance with respect to the
separation distance for different temperatures. Inset: Full radiative
conductance with respect to the separation distance.

In this work we shed light on the radiative heat transfer between polar
materials close to the physical contact. We have shown that the acoustic
vibration modes play a major role in the nonlocal radiative response of
material and even become the dominant channel for radiative heat exchanges at
the atomic scale in the cryogenic regime. Since the acoustic vibration modes
can be excited with the help of piezoelectric transducers or using Raman or
Brillouin light scattering, the radiative heat exchanges could, in principle,
be actively tuned at the atomic scale. Future developments in this direction
will certainly benefit from recent progress in instrumental opto-mechanics
Clivia .

###### Acknowledgements.

This research was supported by the French Agence Nationale de la Recherche
(ANR), under grant ANR-20-CE05-0021-01 (NearHeat).

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# Langlands duality for skein modules of 3-manifolds

David Jordan

(Date: June 2022)

###### Abstract.

I introduce new Langlands duality conjectures concerning skein modules of
3-manifolds, which we have made recently with David Ben-Zvi, Sam Gunningham,
and Pavel Safronov. I recount some historical motivation and some recent
special cases where the conjecture is confirmed. The proofs in these cases
combine the representation theory of double affine Hecke algebras and a new
1-form symmetry structure on skein modules related to electric-magnetic
duality. This note is an expansion of my talk given at String Math 2022 in
Warsaw, and is submitted to the String Math 2022 Proceedings publication.

##  1\. Introduction

In his 1967 letter to André Weil, Robert Langlands proposed a mysterious
conjectural correspondence – now known as Langlands reciprocity – between what
are now called automorphic representations of a simple algebraic group $G$ and
Galois representations to its _Langlands dual_ group ${}^{L}G$. Let us fix a
number field $\mathbb{F}$ with ring of adeles $\mathbb{A}_{\mathbb{F}}$.
Automorphic representations of $G$ are, loosely speaking, certain
$G(\mathbb{A}_{\mathbb{F}})$-representations which may be realised inside
$L^{2}(G(\mathbb{A}_{\mathbb{F}})/G(\mathbb{F}))$, and hence are closely
related to automorphic forms. The Langlands dual group ${}^{L}G$ is another
simple algebraic group obtained from $G$ by interchanging root data. Galois
representations to ${}^{L}G$ are group homomorphisms from the absolute Galois
group $\Gamma(\overline{\mathbb{F}}/\mathbb{F})$ to ${}^{L}G(\mathbb{C})$.
Among the many remarkable features of Langlands reciprocity already evident is
that the objects it relates are of a priori very different nature. Langlands
reciprocity and its many consequences and relatives – including variants for
function fields and local fields – became collectively known as arithmetic
Langlands duality.

Decades after Langlands circulated his conjectures, Beilinson and Drinfeld
discovered a new kind of Langlands duality – also conjectural – taking place
between certain moduli spaces of bundles over a smooth projective curve $X$.
Their proposal, now known as the de Rham geometric Langlands duality, asserts
an equivalence between different algebro-geometric categories living over two
such moduli spaces. On the automorphic side lies a certain category of
$\mathcal{D}$-modules – i.e. systems of polynomial differential equations – on
a moduli space of holomorphic $G$-bundles on $X$. On the Galois side lies a
category of coherent sheaves – i.e., finite quotients of maps between vector
bundles – on a moduli space of flat ${}^{L}G$-bundles (i.e. ${}^{L}G$-bundles
equipped with a flat connection 1-form) on $X$. As in the arithmetic setting,
the geometric Langlands duality relates objects of a priori very different
nature, and bearing no elementary relationship.

The thread connecting the arithmetic and geometric conjectures passes through
a deep series of analogies – known as Weil’s Rosetta stone – which relate the
arithmetic of function fields to the geometry of complex curves. One imagines
the ring of adeles $\mathbb{A}_{\mathbb{F}}$ of a function field $\mathbb{F}$
to be a complex curve, with the primes as points, and hence regards
$G(\mathbb{A}_{\mathbb{F}})$ as giving “local coordinates” of a $G$-bundle in
the formal neighborhood of each prime; on the geometric side this corresponds
to the specification of a $G$-bundle by its Taylor expansion at each point of
a curve. One thereby situates the category of $\mathcal{D}$-modules on
$\mathrm{Bun}_{G}$ on the geometric side opposite to a category of $\ell$-adic
sheaves on the arithmetic side; these in turn are a kind of categorification
of automorphic forms, via the “sheaf-to-function” correspondence for function
fields. On the other side of Langlands duality, the absolute Galois group is
well-understood to be an arithmetic analog of the fundamental group of the
curve.

Beilinson and Drinfeld’s conjectures were later deformed in the language of
twisted $\mathcal{D}$-modules by Feigin, Frenkel and Gaitsgory, following a
proposal of Stoyanovsky. In their quantum de Rham geometric Langlands duality
a new symmetry emerges which is not present in the arithmetic or geometric
settings: the distinction between automorphic and Galois sides evaporates, and
both moduli spaces which appear are those of holomorphic $G$-bundles
(respectively, ${}^{L}G$-bundles), while the categories appearing are both
categories of ($\Psi$\- or ${{}^{L}\Psi}$-twisted, respectively)
$\mathcal{D}$-modules. Here the twisting depends on a possibly infinite
complex parameter $\Psi\in\mathbf{CP}^{1}$, and ${{}^{L}\Psi}=-1/(n_{G}\Psi)$,
where $n_{G}\in\\{1,2,3\\}$ is the lacing number of $G$. The classical
geometric Langlands conjecture is the special case
$\Psi=\infty,{{}^{L}\Psi}=0$.

A physical manifestion of quantum geometric Langlands duality was discovered
by Kapustin and Witten, as an instance of $S$-duality, also known as
Montonen–Olive duality. This duality relates certain low-energy approximations
called “twists” of $\mathcal{N}=4$ super Yang-Mills quantum field theory in
four dimensions. $S$-duality is a sort of non-abelian Fourier transform, whose
existence is predicted by $\mathcal{M}$-theory, and which can be understood as
a generalisation of electric-magnetic duality in Maxwell theory. In this
telling $G$ and ${{}^{L}G}$ each appear as gauge groups of distinct theories,
and the parameters $\Psi,{{}^{L}\Psi}$ each give $\mathbb{CP}^{1}$ charts on
the space of twists. $S$-duality asserts an equivalence of physical theories,

$\mathcal{Z}_{G,\Psi}\simeq\mathcal{Z}_{{{}^{L}G},{{}^{L}\Psi}},$

and Kapustin and Witten explained how to derive the categories appearing in
both the classical and quantum geometric Langlands conjectures as categories
of surface operators
$\mathcal{Z}_{G,\Psi}(X),\mathcal{Z}_{{{}^{L}G},{{}^{L}\Psi}}(X)$. This
reformulation has been very influential in both mathematics and physics
communities, and motivates much of the discussion which follows.

Ben-Zvi and Nadler proposed a further combinatorial incaranation of the
geometric Langlands duality, now known as the Betti geometric Langlands
duality. The key idea is to apply the Riemann–Hilbert correspondence on both
sides of de Rham conjectures. On the Galois side, to a de Rham local system on
$X$ is associated its monodromy homomorphism $\pi_{1}(X)\to{{}^{L}G}$. On the
automorphic side, to a $\mathcal{D}$-module on $\mathrm{Bun}_{G}(X)$ is
associated its sheaf of flat sections over $\mathrm{Bun}_{G}(X)$. The Betti
geometric Langlands duality conjectures assert an equivalence between a
suitable category of coherent sheaves on the character variety – the moduli
space of homomorphisms $\pi_{1}(X)\to{{}^{L}G}$ – and a category of sheaves of
vector spaces on $\mathrm{Bun}_{G}(X)$. In particular the latter is
conjectured, like the former, to depend only on the topological surface
$\Sigma$ underlying the algebraic curve $X$.

The Galois side of the Betti correspondence was subsequently $q$-deformed in
the framework of quantum groups in our works with Brochier, Ben-Zvi, Snyder
and Safronov, where it was called the quantum character theory TFT, and given
the structure of a fully extended four-dimensional TFT (with divergent
partition function, what is sometimes called a once-categorified 3-dimensional
TQFT, or a (3+1)-TQFT). The essential idea in these constructions was that the
Betti Galois moduli spaces may be regarded as mapping stacks
$\operatorname{Maps}(\Sigma,BG)$ into the classifying stack $BG$, with
$\mathrm{Coh}(BG)=Rep(G)$. The stack $BG$ has a canonical 2-shifted symplectic
structure, whose deformation quantization $BG_{q}$ is given algebraically by
the braided tensor category $\mathrm{Rep}_{q}(G)$. Using the language of
higher Morita theory and factorization homology, it was possible to make
formal sense of a functor “$\operatorname{Maps}(\Sigma,BG_{q})$”, and to
establish that this extends to a 4-dimensional TQFT with divergent partition
function.

The purpose of this note is to share a recent conjecture we have made jointly
with David Ben–Zvi, Sam Gunningham and Pavel Safronov, which envisions
Langlands duality between the skein modules of closed oriented 3-manifolds.
The $G$-skein module $\mathrm{Sk}_{G,q}(M)$ of a closed, oriented 3-manifold
is the formal linear span of $G$-labelled ribbon graphs, modulo local
relations modelled on the ribbon braided tensor category $\mathrm{Rep}_{q}(G)$
of representations of the quantum group $U_{q}\mathfrak{g}$; skein categories
are defined in a similar spirit (see Section 2 for precise definitions and
some examples). In its simplest form, our conjecture states:

######  Conjecture 1.1.

Let $G$ be a semisimple algebraic group, and let ${{}^{L}G}$ denote its
Langlands dual group. Let $M$ be a closed, oriented 3-manifold, suppose that
$\Psi\in\mathbb{C}^{\times}$ is transcendental, and let $q=e^{\mathrm{i}\Psi}$
and ${{}^{L}q}=e^{\mathrm{i}{{}^{L}\Psi}}$. Then we have a linear isomorphism,

$\mathrm{Sk}_{G,q}(M)\cong\mathrm{Sk}_{{{}^{L}G},{{}^{L}q}}(M).$

See Section 4 for the statement of various refinements and strengthenings of
the conjecture, and Section 5 for proofs in a few special families of
examples.

######  Remark 1.2.

It was conjectured by Edward Witten and proved in our work with Gunningham and
Safronov that the skein modules are in fact finite-dimensional under the above
assumptions, so that our conjecture can be rephrased as an equality of natural
numbers:

$\dim\mathrm{Sk}_{G,q}(M)=\dim\mathrm{Sk}_{{{}^{L}G},{{}^{L}q}}(M).$

Moreover our method of proof established that the dimension is independent of
$q$ so long as it is taken to be transcendental (it is expected to suffice to
require only that $q$ is not root of unity). Hence the precise specification
of parameters $q$ and ${{}^{L}q}$ in the conjecture is somewhat superfluous,
and is included only to give the reader a sense for the parallel with de Rham
quantum geometric Langlands conjectures.

In particular, to confirm the conjecture as stated it is enough to compute the
generic dimension of the skein module for $G$ and for ${{}^{L}G}$. In fact,
all the cases where we can confirm the conjecture are by independently
computing both dimensions. Because of the transcendental relationship between
$q$ and ${{}^{L}q}$, a canonical isomorphism of vector spaces (e.g. one
intertwining the action of the diffeomorphism group on each side) is likely to
depend analytically, rather than algebraically, on the parameter $q$.

The following table summarises the many instances of Langlands duality
recounted so far. Note the symmetry shared between the quantum de Rham and
quantum skein module formulations.

Shorthand | Object of study | Automorphic/A side |  | Galois/B side
---|---|---|---|---
Arithmetic |  Number field $\mathrm{F}$ | $\\{V\subseteq L^{2}(G(\mathbb{A}_{\mathrm{F}})/G(\mathbb{F}))\\}$ | $\leftrightarrow$ | $\\{\rho:\Gamma(\overline{\mathrm{F}}/\mathrm{F})\to{{}^{L}G}\\}$
Classical |  |  |  |
de Rham |  Complex curve $X$ | $\mathcal{D}(\mathrm{Bun}_{G}(X))$ | $\leftrightarrow$ | $\mathrm{Coh}(\mathrm{Loc}_{{{}^{L}G}}(X))$
Betti |  Real surface $\Sigma$ | $\mathrm{Shv}(\mathrm{Bun}_{G}(\Sigma))$ | $\leftrightarrow$ | $\mathrm{Coh}(\mathrm{Ch}_{{{}^{L}G}}(\Sigma))$
Quantum |  |  |  |
de Rham |  Complex curve $X$ | $\mathcal{D}_{\Psi}(\mathrm{Bun}_{G}(X))$ | $\leftrightarrow$ | $\mathcal{D}_{{{}^{L}\Psi}}(\mathrm{Bun}_{{{}^{L}G}}(X))$
Betti |  Real surface $\Sigma$ | $\mathrm{Shv}_{q}(\mathrm{Bun}_{G}(\Sigma))$ | $\leftrightarrow$ | $\mathrm{Coh}_{q}(\mathrm{Ch}_{G}(\Sigma))$
Skein | 3-manifold | $\mathrm{Sk}_{G,q}(M)$ | $\cong$ | $\mathrm{Sk}_{{{}^{L}G},{{}^{L}q}}(M)$

The most basic motivation for the conjecture is that the ${{}^{L}G}$-skein
module of a 3-manifold $M$ specialises at $q=1$ to the algebra of functions on
the ${{}^{L}G}$-character variety of $M$, and the skein category of a surface
$\Sigma$, specialises at $q=1$ to the ${{}^{L}G}$-character stack of a surface
$\Sigma$. Indeed, a ${{}^{L}G}$-labelled ribbon graph defines in a
straightforward way a holomorphic function on the ${{}^{L}G}$-character
variety, essentially by sending a ${{}^{L}G}$-local system $E$ to the trace of
an associated bundle living along the support of the graph: these functions,
and their deformations, are called Wilson loop observables. It is then natural
to hope – though far from automatic – that the one-parameter family of quantum
deformations given by the skein module indeed coincides with the one-parameter
family of twists in the Kapustin–Witten theories. The rich structure of
extended topological field theory enjoyed by skein modules – they are expected
to coincide with the value of the quantum character theory TFT on 3-manifolds,
and hence to describe the Kapustin–Witten state space – as well as their
natural interpretation as $q$-deformed Wilson loop operators lends support for
this hope.

Another source of motivation for our conjecture comes from the deep analogies
of Mazur, Kapranov, and Reznikov, relating 3-manifold topology to algebraic
number theory, and from the subsequent development of _arithmetic quantum
field theory_. In this “MKR dictionary” – which builds on Weil’s Rosetta stone
– one imagines a number field (more precisely its ring $\mathbb{O}$ of
integers) to be a closed 3-manifold; one treats ideals in $\mathbb{O}$ as
links, and prime ideals as knots. Units in $\mathbb{O}$ are treated as
embedded surfaces, field extensions are treated as branched covers, homology
groups as ideal class groups, etc. Building on this dictionary, the impetus of
arithmetic field theory is to compute arithmetic invariants of number fields
as if they were partition functions for a quantum field theory. This idea was
explored first by Kim, who formulated an arithmetic analog of 3D Chern-Simons
TQFT as a mechanism for constructing $L$-functions, and taken up more recently
by Ben-Zvi, Sakellaridis, and Venkatesh who have proposed to study various
aspects of arithmetic Langlands duality using ideas from Kapustin and Witten’s
4D TQFT.

It is perhaps remarkable in hindsight given the MKR dictionary that our
conjecture so significantly post-dates the geometric Langlands conjectures.
Just as skein theory attaches a vector space to each 3-manifold, arithmetic
Langlands duality attaches vector spaces – the space of automorphic forms, and
the space of algebraic functions on the arithmetic character variety, on each
side of the duality – to each number field. The geometric Langlands dualities
differ in two key respects: they involve complex curves ($\leftrightarrow$
function fields) as opposed to 3-manifolds ($\leftrightarrow$ number fields),
and they replace vector space-valued invariants with categorical ones. This of
course makes Langlands duality for skein modules more elementary to formulate,
and to falsify, than geometric Langlands duality, since it reduces to a
statement about equality of integer dimensions rather than about equivalences
of $\infty$-categories. From a more physical perspective, it is interesting to
note that Kapustin and Witten’s expressed quantum geometric Langlands duality
for _2-dimensional manifolds_ using a _four-dimensional_ QFT, but largely left
aside the intervening question of mathematically describing the Hilbert spaces
attached to _3-manifolds_.

This sense of anachronism can be partly explained by posing the following two
natural and (to our knowledge) unanswered questions concerning Langlands
duality in the arithmetic and physical settings. The answers to each question
would fill in some more of the dots between our conjectures and the canon of
Langlands duality which I have surveyed above. On the side of arithmetic:

######  Question 1.3.

What sense, if any, can be made of a quantum arithmetic Langlands duality?

In particular, what role does the symplectic structure on character varieties,
and its deformation quantization, play in number theory, and what is the
arithmetic meaning of the deformation parameter $q$? On the physical side:

######  Question 1.4.

What is the Hilbert space attached to a closed oriented 3-manifold by the
Kapustin–Witten $A$-side twist at $\Psi=0$?

The answer to this question would sit across Langlands duality from the
${{}^{L}G}$-character variety of $M$, and should have an $A$-side flavour
involving symplectic/contact geometry, Fukaya categories, Floer theory, etc.
One may contemplate possible answers either by degenerating our understanding
for generic $\Psi$ (as a skein module), or by extrapolating up in dimension
from the case of surfaces, where the $A$-side twist at $\Psi=0$ is a category
of $\mathcal{D}$-modules on $\mathrm{Bun}_{G}(X)$.

One among many complications which arise when contemplating this $A$-side
twist at $\Psi=0$ for 3-manifolds is that while $S$-duality predicts an
equivalence of theories
$\mathcal{Z}_{G,\Psi}\simeq\mathcal{Z}_{{{}^{L}G},{{}^{L}\Psi}}$, it
nevertheless _interchanges_ the Dirchlet and Nahm pole boundary conditions.
The skein module is, in some sense by definition, the orbit of the Dirichlet
boundary condition by the Wilson loop operators of the theory; and for
irrational non-zero $\Psi$ (equivalently, $q$ not a root of unity) we
anticipate that these orbits will coincide. We however do not expect such a
coincidence at $\Psi=0,\infty$, and therefore one must formulate the Nahm pole
boundary condition on 3-manifolds in mathematical terms.

### References

Many of the conjectures recalled in this introduction were circulated first as
private correspondence, were never published, or were published only much
later as folklore. For this reason, I have avoided in-line citations
throughout the introduction, even in those cases which appeared promptly in
print, such as Kapustin and Witten’s paper [KW07] on electric-magnetic duality
and Ben-Zvi and Nadler’s paper [BN18] introducing Betti geometric Langlands
conjectures. For a more thorough recounting of Langlands duality in its
arithmetic, geometric, and physical manifestions, and for an introduction to
arithmetic topology, I recommend:

  * •

_The genesis of the Langlands program_ [MS21], an LMS Lecture series which
covers Robert Langlands’s biography, his mathematics, and his letter to Andre
Weil,

  * •

_Between electric-magnetic duality and the Langlands program_ [BV], an
unpublished series of lecture notes, on the relations between geometric
Langlands, arithmetic Langlands, and electric-magnetic duality.

  * •

_Knots and primes_ [Mor11], a Springer textbook introducing the foundations of
arithemetic topology.

### Acknowledgements

I thank my collaborators David Ben-Zvi, Sam Gunningham, Pavel Safronov, Monica
Vazirani, and Haiping Yang for our many discussions surveyed here, and for
their encouragement to share our conjectures and work in progress in this
forum. I also thank Dan Freed, Du Pei and Edward Witten for several
enlightening discussions and correspondences regarding electric-magnetic
duality, and Minhyong Kim for inspiring discussions about arithmetic quantum
field theory. Finally, I thank the organisers of String Math 2022 for a great
conference, and for the opportunity to prepare these notes.

##  2\. Quantum groups and their skein theory

The construction of skein modules and skein categories depends on the
specification of a cp-rigid ribbon braided tensor category111For experts:
equivalently, we may consider instead the subcategory of compact projective
objects, which will be a (not-necessarily-unital!) ribbon braided monoidal
tensor category in which all objects are dualizable. We note that unless
$\mathcal{A}$ is semisimple, the compact-projective objects do not form an
abelian category, but instead an additive idempotent complete category.
$\mathcal{A}$. This means that $\mathcal{A}$ is a locally presentable abelian
category with enough compact projective objects, equipped with a bilinear
monoidal functor $\otimes:\mathcal{A}\times\mathcal{A}\to\mathcal{A}$, a
braiding natural isomorphism $\sigma$, and a ribbon element $\nu$, in which
all compact projective objects are dualizable.

The idea will be that we can encode all the algebraic structure of a cp-rigid
ribbon braided tensor category in a diagrammatic form, using the _graphical
calculus_ of Reshetikhin and Turaev. These diagrams and all relations between
them take place in a decorated cylindrical ball $D^{2}\times I$, but we can
then allow our diagrams to permeate an entire oriented 3-manifold, imposing
only locally the relations coming from the graphical calculus in some ball.
The resulting vector space is called the skein module. We may additionally
allow these diagrams to end at boundary surface components of our 3-manifold;
this leads to the notion of a skein category, and of the induced functor
between skein categories.

###  2.1. Typical examples of ribbon braided tensor categories

Our main examples of interest will be the categories $\mathrm{Rep}_{q}(G)$
attached222More precisely the specification of ribbon structure depends on
choosing a $d$th root of $q$, where $d$ is the determinant of the Cartan
matrix, but we will suppress this point from the notation. to an algebraic
group $G$ at transcendental values of the parameter $q$. This means we
consider the category of locally finite-dimensional representations of the
quantum group $U_{q}(\mathfrak{g})$ attached to the Lie algebra $\mathfrak{g}$
of $G$, and within it we consider a tensor subcategory consisting of those
$U_{q}(\mathfrak{g})$-representations, all of whose weights lie in the weight
lattice of $G$. We recall that the weight lattice of $G$ lies between those of
the adjoint and the simply connected group of $\mathfrak{g}$, and hence is
determined by specifying $Z(G)$, the quotient of the weight lattice of $G$ by
the root lattice; or equivalently by $\pi_{1}(G)$, the quotient of the
coweight lattice by the coroot lattice of $G$.

######  Example 2.1.

We will treat the examples $\mathrm{Rep}_{q}(\mathrm{SL}_{2})$ and
$\mathrm{Rep}_{q}(\mathrm{PGL}_{2})$ in detail throughout these notes. We
recall that $\mathrm{Rep}_{q}(\mathrm{SL}_{2})$ is a semisimple category with
simples $\mathrm{V}(0),\mathrm{V}(1),\mathrm{V}(2),\ldots$ with each
$\mathrm{V}(k)$ having dimension $k+1$ and highest weight $k$. The category
$\mathrm{Rep}_{q}(\mathrm{PGL}_{2})$ is the full subcategory of
$\mathrm{Rep}_{q}(\mathrm{SL}_{2})$ consisting of direct sums of the
irreducible representations
$\mathrm{V}(0),\mathrm{V}(2),\mathrm{V}(4),\ldots$.

Let us recall some further standard algebraic facts about these two
categories. First, in the case of $\mathrm{Rep_{q}}(\mathrm{SL}_{2})$, we
recall that the defining two-dimensional representation $\mathrm{V}(1)$ is a
_tensor generator_ : every representation $\mathrm{V}(k)$ appears as a direct
summand in the kth tensor power $\mathrm{V}(1)^{\otimes k}$. Secondly, we have
that the $\mathrm{Hom}$ spaces $\mathrm{Hom}(\mathrm{V}(1)^{\otimes
k},\mathrm{V}(1)^{\otimes\ell})$ are freely generated by the evaluation and
coevalution maps depicted in the first term of the RHS below. In particular,
we can express the braiding in terms of the evaluation, coevaluation and
identity maps:

We have omitted the source and target $\mathrm{V}(1)$ from the generators. Let
us note that the evaluation and coevaluation ought usually to involve
$\mathrm{V}(1)^{*}$, but that we have identified
$\mathrm{V}(1)\cong\mathrm{V}(1)^{*}$. There are two natural candidates for
such isomorphisms, and it is customary to favor the skew-symmetric pairing,
hence the negative sign in the formula for the value of the circle, i.e. the
quantum dimension of $\mathrm{V}(1)$.

######  Example 2.2.

For $\mathrm{Rep_{q}}(\mathrm{PGL}_{2})$, the situation is similar, but
somewhat more complicated. It is again easy to show that
$\mathrm{Rep_{q}}(\mathrm{PGL}_{2})$ is tensor generated by the generator
$\mathrm{V}(2)$. Recalling the isomorphism
$\mathrm{\mathrm{V}}(2)\otimes\mathrm{V}(2)\cong\mathrm{V}(0)\oplus\mathrm{V}(2)\oplus\mathrm{V}(4)$,
we have now a three-dimensional space
$\operatorname{End}(\mathrm{V}(2)\otimes\mathrm{V}(2))$, spanned by the three
diagrams on the RHS below. In particular, we may express the braiding:

for suitable constants $a,b,c\in\mathbf{C}(q)$ which can be determined by
direct computation in the given basis. Here again we have omitted the label
$\mathrm{V}(2)$ from the generators.

######  Example 2.3.

For the family of groups $G=\mathrm{SL}_{N}$, we also have that
$\mathrm{Rep_{q}}(\mathrm{SL}_{N})$ is tensor generated by the defining
representation $\mathrm{V}(\omega_{1})$ associated to the first fundamental
weight of $\mathrm{SL}_{N}$. It is possible to give a similar explicit
diagrammatic description for $\mathrm{Hom}$ spaces of tensor powers of
$\mathrm{V}(\omega_{1})$, using either MOY calculus [MOY98] or the
spiders/webs descriptions of [Kup96, CKM14], to which we refer for details.

###  2.2. Graphical calculus, skein modules and skein categories

The key fact we will use about ribbon braided tensor categories is that they
admit a _graphical calculus_. We have already seen some examples of graphical
calculus above; let us recall what it means more precisely.

######  Definition 2.4 (Sketch. See Figure 1 and [Coo19, Section 4.2]).

Let $\Sigma$ be an oriented surface.

  * •

An $\mathcal{A}$-labeling of $\Sigma$ is the data, $X$, of an oriented
embedding of finitely many disjoint disks
$x_{1},\dots,x_{n}\colon\mathbb{D}^{2}\rightarrow\Sigma$ labeled by objects
$V_{1},\dots,V_{n}$ of $\mathcal{A}$. We denote by $\vec{x_{i}}$ the $x$ axis
sitting inside each disk $x_{i}$, and denote $\vec{X}=\cup_{i}\vec{x_{i}}$.

  * •

A ribbon graph has “ribbons” connecting “coupons”. A coupon is simply a zero-
simplex in $\Sigma\times I$, equipped with the choice of a normal vector. A
ribbon is an embedded 1-simplex in $\Sigma\times I$, whose endpoints each lie
either in the set $X$, or at some coupon, and which is equipped with a normal
vector along it. We furthermore require that the tangent vector of each
embedded simplex points directly up when ending at $X\times\\{0,1\\}$ or else
in the direction of the coupon’s normal vector when ending at a coupon.
Finally, we require a cyclic ordering of the incoming and outgoing ribbons at
each coupon.

  * •

An $\mathcal{A}$-coloring of a ribbon graph is a labelling of each ribbon by
an object of $\mathcal{A}$, and of each coupon by a morphism from the
(ordered) tensor product of incoming edges to the (ordered) tensor product of
outgoing edges.

  * •

We say that an $\mathcal{A}$-colored ribbon graph $\Gamma$ is compatible with
an $\mathcal{A}$-labeling if $\partial\Gamma=\vec{X}$, and denote by
$\mathrm{Rib}_{\mathcal{A}}(M,X)$ the vector space with basis the
$\mathcal{A}$-colored ribbon graphs on $M$ compatible with $X$.

######  Remark 2.5.

The graphical calculus for ribbon tensor categories has appeared in the
literature with many differing but essentially equivalent formulations. The
formulation we present here is adapted from our presentation in [GJS19],
however the customary “ribbons” and “coupons” formulation has been replaced
with a topologically equivalent formulation which will be more compatible with
simplicial homology arising later in the paper.

Let us now describe the defining relations of skein modules. Consider the
3-ball written as the cylinder $\mathbb{D}^{2}\times I$ over the 2-disk
$\mathbb{D}^{2}$, and fix a labeling $X\cup Y$ with disks
$X=(x_{1},V_{1}),\ldots,(x_{n},V_{n})$ embedded in
$\mathbb{D}^{2}\times\\{0\\}$ and
$Y=\\{(y_{1},W_{1}),\ldots(y_{m},W_{m})\\}\times\\{1\\}$. Then we have a well-
defined surjection,

$\mathrm{Rib}_{\mathcal{A}}(\mathbb{D}^{2}\times I,X\cup
Y)\to\mathrm{Hom}_{\mathcal{A}}(V_{1}\otimes\cdots\otimes
V_{n},W_{1}\otimes\cdots\otimes W_{m}),$

called the Reshetikhin–Turaev evaluation map, see [Tur16]. In essence this map
takes a ribbon graph in a cylinder, and a generic projection, and interprets
each crossing as a braiding, and each coupon as a morphism between each tensor
product, and computes the corresponding composition: that this is well-defined
independent of crossings is perhaps the most fundamental algebraic property of
ribbon categories. We will call the kernel of this map the skein relations
between $X$ and $Y$.

######  Definition 2.6.

Let $M$ be an oriented 3-manifold equipped with a decomposition of its
boundary $\partial M\cong\operatorname{rev}{\Sigma}_{in}\coprod\Sigma_{out}$,
and $\mathcal{A}$-labelings $X_{in}$ of $\Sigma_{in}$ and $X_{out}$ of
$\Sigma_{out}$.

  * •

The relative $\mathcal{A}$-skein module
$\mathrm{Sk}_{\mathcal{A}}(M,X_{in},X_{out})$ is the vector space spanned by
isotopy classes of $\mathcal{A}$-colored ribbon graphs in $M$ compatible with
$X_{in}\cup X_{out}$, taken modulo isotopy and the skein relations between
$X_{in}$ and $X_{out}$ determined by any oriented ball $\mathbb{D}^{2}\times
I\subset M$333Here we assume without loss of generality that
$\mathbb{D}^{2}\times\\{0\\}\subset\Sigma\times\\{0\\}$ and
$\mathbb{D}^{2}\times\\{1\\}\subset\Sigma\times\\{1\\}$..

  * •

When $\partial M=\emptyset$ (hence $\partial\Gamma=\emptyset$), we call this
the $\mathcal{A}$-skein module, and denote it by
$\mathrm{Sk}_{\mathcal{A}}(M)$.

######  Definition 2.7.

Let $\Sigma$ be an oriented surface. The skein category
$\operatorname{SkCat}_{\mathcal{A}}(\Sigma)$ of $\Sigma$ has:

  * •

As its objects, $\mathcal{A}$-labelings of $\Sigma$.

  * •

As the 1-morphisms from $X$ to $Y$ the relative $\mathcal{A}$-skein module of
$(\Sigma\times[0,1],X,Y)$.

The composition of 1-morphisms is by concatenation of relative skeins in the
evident sense.

Figure 1. At left: the skein module is spanned by $\mathcal{A}$-labelled
ribbon diagrams; relations; the skein relations state that the diagram within
the cylinder can be replaced by any other diagram whose RT-evaluation gives
the same element of $\mathrm{Hom}(U\otimes V\otimes W,V\otimes X)$. At right:
a defect is introduced along the dotted line, and a braided module category is
inserted along it (see Section 3). The skein module now includes labels and
relations from the braided tensor category away from the defect, and from its
braided module category along the defect.

######  Example 2.8.

In the cases $\mathcal{A}=\mathrm{Rep_{q}}(\mathrm{SL}_{2})$ and
$\mathcal{A}=\mathrm{Rep_{q}}(\mathrm{PGL}_{2})$ the rather intricate
definition of the skein module involving labelled ribbon graphs simplifies
considerably, owing to the simplifications highlighted in Example 2.1. In both
cases, one can dispense with labels and orientations on strands: for labels,
one takes $\mathrm{V}(1)$ or $\mathrm{V}(2)$, respectively as the label always
(and thus omits it), and one further applies the self-duality of
$\mathrm{V}(1)$, respectively $\mathrm{V}(2)$ to eliminate orientation data.

For coupons, in the $\mathrm{Rep_{q}}(\mathrm{SL}_{2})$ case, Schur Weyl
duality implies that all morphisms can be expressed in terms of the braidings,
so no coupons are necessary at all. For $\mathrm{Rep_{q}}(\mathrm{PGL}_{2})$,
we require only the trivalent coupon from Example (2.2).

###  2.3. Relations to deformation quantization

The starting point for studying skein modules using geometric representation
theory comes from the following description of skein modules and skein
categories at $q=1$.

######  Definition 2.9.

The $G$-character stack $\underline{\mathrm{Ch}}_{G}(X)$ of a topological
space $X$ is the moduli stack of flat $G$-bundles on $X$, equivalently, it is
the quotient stack obtained by quotienting the $G$-representation variety
$\mathrm{Hom}(\pi_{1}(X),G)$, by the conjugation action of $G$. The
$G$-character variety $\mathrm{Ch}_{G}(X)$ is the GIT quotient, i.e. the set
of closed $G$-orbits on the representation variety.

######  Theorem 2.10.

The idempotent completion (respectively, free cocompletion) of the skein
category $\operatorname{SkCat}_{G,q=1}(\Sigma)$ is naturally equivalent to the
category of vector bundles (respectively quasi-coherent sheaves) on the
$G$-character stack $\underline{\mathrm{Ch}}_{G}(\Sigma))$. Moreover, the
deformation to the skein category is flat in the parameter $q$.

Here, flatness in the parameter $q$ means that $\mathrm{Hom}$ spaces defined
in $\operatorname{SkCat}_{G}(\Sigma)$ are actually free
$\mathbf{C}[q,q^{-1}]$-modules; informally, one says then that the vector
space supporting the skein module is “constant” in $q$ and that only the
algebra structure changes as a function of $q$: in such cases, we obtain a
Poisson bracket on the $q=1$ limit, and we say that the skein algebra/skein
category is a deformation quantization of the character variety/character
stack. The resulting Poisson bracket quantizes the Atiyah–Bott/Goldman/Fock-
Rosly Poisson bracket (a single Poisson bracket on the character variety
independently constructed by each grouping of authors in three different
ways). One has a similar interpretation for the $q=1$ limit of
$\mathrm{Sk}_{G}(M)$ for any 3-manifold. The following is proved for
$G=\mathrm{SL}_{2}$ in [Bul97, PS00], but holds for any group $G$.

######  Theorem 2.11.

The skein module $\mathrm{Sk}_{G,q=1}(M)$ of an oriented 3-manifold $M$ is
naturally isomorphic to the algebra $\mathcal{O}(\mathrm{Ch}_{G}(M))$ of
algebraic functions on the $G$-character variety of $M$.

However, when we move the quantum parameter $q$ away from one, skein modules
behave very differently than skein categories in elementary terms. While skein
categories of surfaces (more precisely the relative skein algebras which
appear as their $\mathrm{Hom}$ spaces) depend _flatly_ on the quantum
parameter $q$, quite the opposite is true for skein modules of closed
3-manifolds. We have noted that at $q=1$ the skein module of a closed
3-manifold may be identified with the algebra of algebraic functions on the
$G$-character variety of $M$ – this is in particular typically an infinite-
dimensional vector space. By contrast, we have

######  Theorem 2.12 ([GJS19]).

Let $q\in\mathbb{C}^{\times}$ be generic, and let $M$ be a closed, oriented
3-manifold. Then the $G$-skein module of $M$ is finite-dimensional.

This statement first appeared as a conjecture of Edward Witten in 2014, and
came largely as a surprise to the skein theory community: because of the many
known results asserting that skein algebras and skein categories of surfaces
deform flatly, it was generally expected that skein modules of 3-manifolds
should as well. With the benefit of hindsight and modern approaches to
deformation quantization – specifically the framework of shifted symplectic
structures [Pan+13] – the finiteness asserted in the theorem is not at all
surprising.

The character stacks of surfaces, which are deformed by skein categories, are
symplectic (what is now called “0-shifted symplectic”), where as the character
stacks of 3-manifolds have a very different structure: they are what is now
called $(-1)$-shifted symplectic. One way to understand $(-1)$-shifted
symplectic stacks is by a shifted analog of Darboux coordinates: whereas
Darboux coordinates equip a 0-shifted symplectic stack with a local
symplectomorphism to the unique symplectic vector space in that dimension,
Darboux coordinates for $(-1)$-shifted stack identify the neighborhood of any
point with the intersection of two Lagrangian subspaces of the symplectic
vector space.

The finite-dimensionality of skein modules can be understood in this context
from the fact that the notion of deformation quantization for 0-shifted and
$(-1)$-shifted symplectic structures is very different in nature. Whereas the
familiar notion for $0$-shifted structures involves a non-commutative algebra,
with a flatness criterion on the underlying vector space of the quantization,
the quantization of a $(-1)$-shifted symplectic structure defines a perverse
sheaf on the classical variety, whose global sections describes the
deformation. There are (at least) two general avenues to construct this
perverse sheaf: an algebraic construction of deformation quantizations and
their tensor products, following Kashiwara and Schapira [KS12], or a geometric
construction following [Ben+15].

In the algebraic construction, we suppose that the $(-1)$-shifted structure
really does arise as the intersection of two Lagrangian subvarieties
$L_{1},L_{2}$, and that the symplectic variety $X$ has a deformation
quantization into some non-commutative algebra $A$ (or sheaf of algebras), and
that the Lagrangians each quantize to holonomic (left, and right,
respectively) modules $M_{1}$, $M_{2}$ over $A$. In this case, Kashiwara and
Schapira show that the non-commutative intersection – i.e. the relative tensor
product $M_{1}\otimes_{A}M_{2}$ – defines a perverse sheaf over the classical
intersection $L_{1}\cap L_{2}$. A careful analysis of the sections of this
perverse sheaf in the setting of skein modules yielded the finiteness result
of [GJS19].

A related geometric construction of Abouzaid and Manolescu [AM17] of
complexified instanton Floer homology also hinges on the construction of the
character stack of a 3-manifold, as an intersection of two standard
Lagrangians in the character stack of a surface of some genus $g$. The
Brav–Bussi–Ben-Bassat–Joyce quantization of such a $(-1)$-shifted symplectic
stack is controlled by a perverse sheaf of _vanishing cycles_ , and in good
situations this will have, indeed, a finite dimensional space of global
sections.

######  Remark 2.13.

Work in progress by Gunningham and Safronov gives an identification of the
Kashiwara–Schapira sheaf and the Brav–Bussi–Ben-Bassat–Joyce sheaf in great
generality, including in our setting of skein modules, thus unifying the two
approaches.

We might be tempted to understand the finiteness obtained in these two
constructions as follows: generically, two Lagrangian submanifolds should
intersect transversely, but of course in practice they may not. However, when
we quantize, even if the classical Lagrangians fail to intersect transversely
– giving a positive dimensional character variety, and hence an infinite-
dimensional algebra of functions – at the quantum level, and at generic
parameters, they exhibit the generic behavior.

Both the Kashiwara–Schapira and the Abouzaid–Manolescu quantizations are by
construction not only vector spaces but chain complexes; we have so far only
constructed on their cohomology in degree zero. This suggests that the skein
module also has a natural derived enhancement, with finite-dimensional
cohomologies in each degree, and satisfying a Langlands duality akin to 1.1.
We will not discuss this derived enhancement in the present note.

###  2.4. Skein topological field theory

Given a 3-manifold $M$ with oriented boundary $\partial
M=\Sigma_{+}\sqcup\Sigma_{-}$, we obtain a functor,

$\mathrm{Sk}_{\mathcal{A}}(M):\operatorname{SkCat}(\Sigma_{-})^{op}\otimes\operatorname{SkCat}(\Sigma_{+})\to\mathrm{Vect},$

called the skein bimodule of $M$, which assigns to a pair of skein objects the
relative skeins in $M$ compatible with the given marking at the boundary.

######  Theorem 2.14 (Walker, unpublished).

The assignments of surfaces and 3-manifolds to their skein category and skein
(bi)modules, respectively, defines a functor from the category
$\operatorname{Cob}^{3}$ of surfaces and 3-cobordisms, to the category of
categories and categorical bimodules.

It was shown in [Coo19] that skein categories satisfy a universal property
called excision, which implies that the skein category coincides with the
categorical factorization homology of Ayala–Francis [AF15], as was studied in
this context in [BBJ18, BBJ18a]. The excision property for skein modules
should be regarded as expressing the existence of an extension of skein theory
all the way down to the point: the $(3,2)$-TFT axioms allow us to cut
3-manifolds along surfaces, and excision allows us to cut surfaces up along
embedded 1-manifolds, and then those 1-manifolds up along 0-manifolds.

However, while it is known that skein modules define a $(3,2)$-TFT after
Walker, and that excision for skein categories implies a $(2,1,0)$-TFT via
factorization homology, I am not aware of a rigorous construction of the
desired $(3,2,1,0)$-TFT in the framework of fully extended topological field
theories. In [BJS21], we have constructed a $(3,2,1,0)$-TFT from the same
input, of a cp-rigid braided tensor category, however it remains to identify
the value of this theory on 3-manifolds with the skein module, as one expects.
We will nevertheless largely pass notions between the skein-theoretic and
fully extended formulations of the theory in Sections 3.

##  3\. Gradings and twists of skein modules

It is an elementary observation that the ribbon braided tensor category
$\mathrm{Rep}_{q}(G)$ is naturally graded (as an abelian monoidal category) by
the Pontryagin dual $Z(G)^{\vee}$ of the centre $Z(G)$ of the group $G$: the
central elements of $G$ define grouplike central elements of the quantum group
$U_{q}(\mathfrak{g})$, and by Schur’s lemma, any simple representation
uniquely determines a character of the centre. The group-like property implies
that the representations tensor compatibly with the group structure.

Because the skein relations all take place locally in a ball, the central
grading on $\mathrm{Rep}_{q}(G)$ extends naturally to a grading by
$H_{1}(M,Z(G)^{\vee})$ on the $G$-skein module: given a $G$-labelled ribbon
graph, we obtain a simplicial 1-cycle by replacing the label along each
1-simplex with its grading by $Z(G)^{\vee}$. We note that we always obtain a
cycle in this way, since at coupons, there can only be a nonzero morphism if
the sum of degrees is zero. If the manifold has boundary, and the skein is
defined relative to the boundary, then the 1-cycle will give a relative
$Z(G)^{\vee}$ 1-cycle, by similar reasoning. This assignment gives the grading
of the $G$-skein module by $H_{1}(M,Z(G)^{\vee})$, and this grading will play
an important role in what follows.

We will see that skein modules carry another structure: they can be twisted by
braided module categories inserted along codimension-two defects. The grading
and twisting constructions are dual in a precise sense, which we will soon
see. For now let us outline the construction of twisting of skein modules: the
complete details are presented in the forthcoming [GJS23], so we will be
somewhat brief.

######  Definition 3.1.

A braided module category $\mathcal{M}$ for a braided tensor category
$\mathcal{A}$ is a module category $\mathcal{M}$ for $\mathcal{A}$, together
with a ‘monodromy” natural automorphism,

$\alpha:(X,m)\mapsto(\alpha_{X,m}:X\otimes m\to X\otimes m),$

of the action functor of $\mathcal{A}$ on $\mathcal{M}$, which satisfies:

$\alpha_{X\otimes
Y,m}=(\sigma_{Y,X}\sigma_{X,Y}\otimes\mathrm{id}_{M})\circ\alpha_{X,Y\otimes
M}\circ(\mathrm{id}_{X}\otimes\alpha_{Y,M}).$

######  Remark 3.2.

To motivate this definition, we should imagine that the braided tensor
structure on $\mathcal{A}$ is encoded by the $E_{2}$-operad of little disks
(see Figure 2), and that $\mathcal{M}$ sits at a single point (a co-dimension
two defect) in the plane. The module action comes from bringing
$\mathcal{A}$-disks to $\mathcal{M}$, and the natural isomorphism $\alpha$ is
indeed the monodromy obtained by sliding those disks once counterclockwise
around $\mathcal{M}$ before colliding.

######  Definition 3.3.

The $E_{2}$-monoidal bicategory $\mathrm{BrMod}(\mathcal{A})$ has as objects
braided module categories, as its 1-morphisms braided module functors (i.e.
module functors $(F,J)$ satisfying an evident compatibility with $\alpha$),
and as 2-morphisms the braided module natural transformations. The
$E_{2}$-monoidal structure is given by sending the pair $\mathcal{M}_{1}$,
$\mathcal{M}_{2}$ to the relative tensor product
$\mathcal{M}_{1}\otimes_{\mathcal{A}}\mathcal{M}_{2}$, with the resulting
holonomy natural automorphism given as the composition of holonomies.

We will say that the braided module category $\mathcal{M}$ is dualizable if it
is fully dualizable as an object of $\mathrm{BrMod}(\mathcal{A})$.

Given a fully dualizable braided module category, and a knot $K$ in a
3-manifold $M$, we may define the skein module,
$\mathrm{Sk}_{\mathcal{M}}(M,K)$ as the formal span of all
$(\mathcal{A},\mathcal{M})$-labelled ribbon graphs, modulo
$(\mathcal{A},\mathcal{M})$-skein relations. Here, an
$(\mathcal{A},\mathcal{M})$-labelled ribbon graph is a ribbon graph in $M$,
equipped with a labelling by $\mathcal{A}$ in the open set $M\backslash K$,
and with a labelling by $\mathcal{M}$ along $K$. The
$(\mathcal{A},\mathcal{M})$-skein relations are given by imposing within any
open cylinder contained in $M\backslash K$ the Reshetikhin–Turaev evaluation
relations of $\mathcal{A}$, and within any open cylinder intersecting $K$
along $\\{0\\}\times I\subset D^{2}\times I$ the Reshetikhin–Turaev evaluation
relations of $\mathcal{M}$.

###  3.1. Higher form symmetry actions of abelian groups

The most important source of dualizable module categories for the present
discussion are the _invertible_ module categories, those for which the
evalution and coevaluation are equivalences of braided module categories,
identifying the dual braided module category as an inverse. These form an
abelian group under tensor product, and we can use this additional structure
to define twists not only along knots but along simplicial cycles in $M$. To
motivate the construction, we recall now the framework of higher form
symmetries in quantum field theory.

Recall that an ordinary action of an abelian group $A$ on a
$(d+1)$-dimensional QFT induces an action of $H_{d}(M,A)$ on the Hilbert space
$Z(M)$ attached to any $d$-manifold $M$ – in particular if $M$ is closed and
connected we have an action of the group $A\cong H_{d}(M,A)$ itself, on the
vector space $Z(M)$.

In purely local/field-theoretic terms, the $A$-action arises because we
redefine the field equations in the neighborhood of some $d$-cycle, requiring
that fields transform according by the $A$-action as they pass through the
cycle: in particular, this introduces a singularity to the field and so a
field in the twisted sense is not a field in the ordinary sense. The action of
$H_{d}(M,A)$ on $Z(M)$ which we highlighted arises by situating a $d$-cycle at
the halfway point of trivial mapping cylinders $M\times I$, thereby modifying
the notion of correlation functions.

In the literature on extended defects, such an action is called a 0-form
symmetry, the “0” expressing that it is not yet an extended symmetry. In this
context, we may alternatively contemplate the structure of $k$-form symmetry
action of a group $A$ on a $(d+1)$-dimensional QFT: in mathematical/functorial
terms this means we consider an action of the $(k+1)$-group $B^{k}A$ on the
theory (here $B^{k}$ denotes the $k$-fold delooping, a $k+1$-groupoid with
unique object, and unique $n$=morphism for $n<k$ and with $A$ as its group of
$k+1$-morphisms). From such an extended action we obtain (among other things)
an action of $H_{d-k}(M,A)$ on the Hilbert space $Z(M)$ attached to each
$d$-manifold $M$. Again in physical terms, the data of a $k$-form symmetry
action of the group $A$ allows for modifications of the field equations, now
along higher co-dimension submanifolds of $M$.

###  3.2. Twisting and grading together: the Picard groupoid

The homology groups $H_{d-k}(M,A)$ appearing above are _global_ invariants of
$M$ – that is, they depend on the global topology of $M$, and they allow us to
twist Hilbert spaces globally along some global cycle. It will be very useful
in practice to give a more local enhancement of a $k$-form symmetry action, by
introducing _twists_ $Z(M)[\gamma]$ of the Hilbert space $Z(M)$ indexed by
elements $\gamma\in H_{d-k-1}(M,A)$, and isomorphisms between homologous
twists (thereby refining the action of $H_{d-k}$). Very roughly, introduction
of twists allows us to consider cycles in $M^{d+1}$ relative to the boundary,
and therefore to “cut up” the $d-k$-cycles in $H_{d-k}(M,A)$ into more basic
pieces.

In order to simplify notation, we will fix $d=3$ and $k=1$ in what follows;
however it is straightforward to modify the general notions for arbitrary $d$
and $k$, and also to introduce higher groupoids facilitating twists in higher
codimension.

Recall that a simplicial 1-cycle $\gamma$ on $M$ with values in $A$ is by
definition an oriented graph embedded into $M$, with edges labelled by
elements of $A$, and with the property that at each vertex the sum of all
labels of incoming edges equals to the sum of all labels of outgoing edges.

######  Definition 3.4.

Fix an oriented $3$-manifold $M$. The Picard groupoid $H_{1,2}(M,A)$ has as
its objects the abelian group of simplicial $1$-cycles $C_{1}(M,A)$, with
symmetric monoidal product given by addition. Morphisms from $\gamma$ to
$\gamma^{\prime}$ are homology classes of $2$-chains $\eta$ satisfying
$d\eta=\gamma^{\prime}-\gamma$.

It follows immediately from the definition that the group of isomorphism
classes of objects of $H_{1,2}(M)$ is canonically isomorphic to $H_{1}(M,A)$,
and that the automorphism group of any object is canonically isomorphic to
$H_{2}(M,A)\cong H_{1}(M,A^{\vee})$. Hence, the data of a functor
$H_{1,2}(M,A)\to\mathrm{Vect}$ consists of a vector space $F([\gamma])$ for
each $[\gamma]\in H_{1}(M,A)$, each equipped with an action of he abelian
group $H_{1}(M,A^{\vee})$.

###  3.3. One-form actions on braided tensor categories.

Recall from [BJS21] that braided tensor categories form into a 4-category
$\mathrm{BrTens}$. This is the 4-category whose:

  * •

objects are cp-rigid braided tensor categories,

  * •

1-morphisms from $\mathcal{A}$ to $\mathcal{B}$ are
$\mathcal{A}$-$\mathcal{B}$-central algebras, i.e. cp-rigid tensor categories
$\mathcal{C}$ equipped with braided tensor functors
$\mathcal{A}\otimes\mathcal{B}^{op}\to\mathcal{Z}(\mathcal{C})$,

  * •

2-morphisms from $\mathcal{C}$ to $\mathcal{D}$ are
$\mathcal{C}$-$\mathcal{D}$ bimodule categories, compatibly with the central
action,

  * •

3-morphisms are bimodule functors, and

  * •

4-morphisms are natural transformations.

In particular, given a braided tensor category $\mathcal{A}$, its endomorphism
3-category consists of $\mathcal{A}$-$\mathcal{A}$-central algebras. The unit
endomorphism is $\mathcal{A}$ itself with its regular action. We may then
consider the endomorphisms 2-category of $\mathcal{A}$ thusly regarded, it is
precisely the 2-category $\mathrm{BrMod}(\mathcal{A})$ of braided module
categories we have met previously. See Figure 2 for a comparison with the
notion in Definition 3.1

$\mathcal{C}$$\mathcal{A}$$\mathcal{B}$$\mathbb{R}^{2}$$\mathbb{R}$(A)$\mathcal{C}$$\mathcal{D}$$\mathcal{M}$$\mathcal{A}$$\mathcal{B}$$\mathbb{R}^{2}$$\mathbb{R}$(B)$\mathcal{A}$$\mathcal{A}$$\mathcal{M}$$\mathcal{A}$$\mathcal{A}$$\mathbb{R}^{2}$$\mathbb{R}$(C)
Figure 2. (A) depicts a pair of braided tensor categories $\mathcal{A}$ and
$\mathcal{B}$, encoded by embeddings of disks $\mathbb{R}^{2}$, and a
1-morphism between them: an $\mathcal{A}$-$\mathcal{B}$-central algebra
$\mathcal{C}$ encoded by embeddings of intervals $\mathbb{R}$ sandwiched
between the $\mathcal{A}$\- and $\mathcal{B}$-regions. (B) depicts a
2-morphism: a $\mathcal{C}$-$\mathcal{D}$-bimodule structure on $\mathcal{M}$
is encoded by bringing disks into the central point. (C) depicts the special
case $\mathcal{A}=\mathcal{B}=\mathcal{C}=\mathcal{D}$, in which case we can
erase the $\mathbb{R}$, and $\mathcal{M}$ sits on an isolated point in
$\mathbb{R}^{2}$, and the braided module category structure comes from
approaching $\mathcal{M}$ in any direction with $\mathcal{A}$-disks.

######  Definition 3.5.

A 1-form symmetry action of the abelian group $A$ on the braided tensor
category $\mathcal{A}$ is the data of an $E_{2}$-monoidal functor
$A\to\mathrm{BrMod}(\mathcal{A})$, where $A$ is regarded as a discrete
symmetric monoidal 2-category.

More concretely, a 1-form symmetry action of $A$ on $\mathcal{A}$ is the data
of a braided module category $\mathcal{M}_{a}\in\mathrm{BrMod}(\mathcal{A})$
for each $a\in A$, and fixed equivalences
$\phi_{a,b}:M_{a}\otimes_{\mathcal{A}}M_{b}\to M_{a+b}$ of braided module
categories, satisfying an evident pentagon equation.

Here are the primary two sources of examples we will use:

######  Example 3.6.

Let $A$ be any subgroup of the group $Z_{gp}(\mathcal{A})$ of group-like
Bernstein-central elements of $A$,

$Z_{gp}(\mathcal{A})=\\{a\in\mathrm{Aut}(\mathrm{id}_{\mathcal{A}})\,\,|\,\,a_{V\otimes
W}=a_{V}\otimes a_{W}\\},$

and define $\mathcal{M}_{a}=\mathcal{M}$ as a category, with
$\alpha_{a,m}=(\mathrm{id}_{Y}\otimes a)\circ\sigma_{m,Y}\sigma_{Y,m}$. A
particular case of this example arises when we take $\mathcal{A}$ to be
$\mathrm{Rep_{q}}(G_{sc})$, and $A=Z(G)$, the universal grading group of
$\mathrm{Rep_{q}}(G_{sc})$. In physical literature (motivated by the Maxwell
theory), the resulting action is called the electric 1-form symmetry.

######  Example 3.7.

Let $\widetilde{A}$ be any $A$-graded braided tensor extension of
$\mathcal{A}$, and define $\mathcal{M}_{a}$ to be the degree $a$ component
$\mathrm{Rep_{q}}(\widetilde{G})_{a}$, with $\alpha$ and $\phi$ the
restrictions of the double braiding and the multiplication functor on
$\widetilde{A}$. A particular case of this example arises when we take
$\mathcal{A}$ to be $\mathrm{Rep_{q}}(G_{ad})$, $A=\pi_{1}(G)$, and
$\widetilde{A}=\mathrm{Rep_{q}}(G_{sc})$. Again in physical literature, the
resulting action is called the magnetic 1-form symmetry.

###  3.4. The twisted skein module

Let us now translate the abstract notions above into the formalism of skein
modules, and ultimately linear algebra.

######  Definition 3.8.

Fix a 1-form action $\pi:\mathcal{A}\to\mathrm{BrMod}(\mathcal{A})$. For each
simplicial cycle $\gamma\in C_{1}(M,A)$, the twisted skein module
$\operatorname{TwSk}(M)[\gamma]$ is the span of all
$(\mathcal{A},\pi,\gamma)$-labelled ribbon graphs, modulo the
$(\mathcal{A},\pi,\gamma)$-skein relations.

Here, a $(\mathcal{A},\pi,\gamma)$-labelled ribbon graph is a ribbon graph in
$M$, with strands and coupons labelled by objects and morphisms of
$\mathcal{A}$ on the complement of the support of $\gamma$, and with strands
and coupons labelled by objects and morphisms of $\mathcal{M}_{a}$ along any
1-simplex of $\gamma$ weighted by $a$. The skein relations are defined in the
evident way, by treating separately cylinders which do not intersect the
support of $\gamma$, cylinders which intersect a 1-simplex of $\gamma$, and
cylinders which intersect a 0-simplex of $\gamma$. We omit complete
definitions.

######  Proposition 3.9 ([GJS23]).

We have a well-defined functor,

$\operatorname{TwSk}(M):H_{1,2}(M,Z(G)^{\vee})\to\mathrm{Vect},$

mapping each $\gamma\in Z_{1}(M,Z(G)^{\vee})$ to the $\gamma$-twisted skein
module, and mapping each $\eta:\gamma_{1}\to\gamma_{2}$ to an isomorphism
induced by Reshetikhin-Turaev evaluation on a bounding disk.

This means simply that for each choice of $\gamma\in C_{1}(M,A)$ we will
define a twisted skein module $\operatorname{TwSk}(M)[\gamma]$, and for each
$\eta\in C_{2}(M,A)/dC_{3}(M,A)$ with $d\eta=\gamma^{\prime}-\gamma$ we define
an isomorphism,

$\operatorname{TwSk}(M)[\eta]:\operatorname{TwSk}(M)[\gamma]\to\operatorname{TwSk}(M)[\gamma^{\prime}],$

and that these isomorphisms will compose compatibly.

Finally, we can consider the two flavours of twisted skein modules coming from
Example 2.1.

######  Example 3.10.

Let $A=Z(G)$. We have seen in Example 3.6 that we have an action of $Z(G)$ on
$\mathrm{Rep_{q}}(G)$ by 1-form symmetries; the resulting action by
$H_{2}(M,Z(G))$ on $\mathrm{Sk}(M)=\operatorname{TwSk}(M)[0]$ allows us to
decompose $\mathrm{Sk}(M)$ according to the characters
$H_{2}(M,Z(G))^{\vee}\cong H_{1}(M,Z(G)^{\vee})$. The skein-theoretic
interpretation is depicted in Figure 3. The resulting graded decomposition is
the elementary one observed at the start of Section 3. This grading, in its
elementary form, has featured prominently in the computations of [Car17],
[Gil18], [GM18] and [DW21].

To our knowledge, the twisted skein modules $\operatorname{TwSk}(M)[\gamma]$
for non-trivial $\gamma\in H_{1}(M,Z(G))$ are new, and so do not have an a
priori elementary description. However, it is easy to describe the twists in
this case: the condition on defects says that when a 1-simplex of a ribbon
graph labelled with a simple representation $V$ passes through a 1-simplex of
the defect with weight $z\in Z(G)$, it picks up a scalar factor equal to the
evaluation of $z$ on $V$.

Figure 3. Top left: each character $\chi$ determines a braided module category
$\mathcal{M}_{\chi}$, whose underlying category is $\mathcal{A}$, and where
the double braiding is given by the central action of $\chi$ on each object
$V\in\mathcal{A}$ marking a skein. Top right: this braided module category is
invertible; closed loops contract trivially. At bottom: the pairing between
some 2-cycle $\Sigma$ and some skein is obtained by first condensing within
$\Sigma$ some network of triangles, and contracting each triangle through the
skein. The result is multiplication by the Poincare pairing
$\langle\Sigma,\gamma\rangle$ between the cycle $\Sigma$ and the support
1-cycle $\gamma$ of the skein. We note that invertibility of $\chi$ implies
already that adjacent edges of the triangulation cancel pairwise, hence the
resulting skein module is not twisted despite the introduction of defects, so
that we indeed obtain an endomorphism.

######  Example 3.11.

Let $A=\pi_{1}(G)$. We have seen in Example 3.7 that we obtain an action of
$\pi_{1}(G)$ on $\mathrm{Rep_{q}}(G)$ by 1-form symmetries. To our knowledge
both the resulting action of $H_{2}(M,Z(G))$ and hence the resulting
decomposition of $H_{1}(M,\pi_{1}(G)^{\vee})$ on $\mathrm{Sk}(M)$, and the
twisted skein modules are new, and so do not have an a priori elementary
description. However, it is again easy to define twists in this case. Given a
$G^{sc}$-labelled ribbon graph with simple labels, we can produce a 1-simplex
in $C_{1}(M,\pi_{1}(G))$, by replacing each simple label $V$ by its grading
with respect to $\pi_{1}(G)$, regarding $\mathrm{Rep_{q}}(G^{sc})$ as a
$\pi_{1}(G)$-graded extension of $\mathrm{Rep_{q}}(G)$. The space
$\operatorname{TwSk}(M)[\gamma]$ is the span of all $G^{sc}$-labelled ribbon
graphs, whose underlying $1$-cocycle equals $\gamma$, and modulo only those
skein relations also with underlying $1$-cocycle $\gamma$.

For any group $G$, and for $A=Z$ or $\pi_{1}$, and for $a\in
H_{1}(M,A^{\vee})$ and $b\in H_{1}(M,A)$ we denote by
$\mathrm{Sk}_{G,A}^{a,b}(M)$ the $a$-graded component of the $b$-twisted skein
module. It follows from the definition that we have a natural linear map of
vector spaces,

$\Phi:\mathrm{Sk}_{G^{ad},Z}(M)\to\mathrm{Sk}_{G^{sc}}(M),$

which regards a $\mathrm{Rep_{q}}(G^{ad})$-labelled ribbon graph as a
$\mathrm{Rep_{q}}(G^{sc})$-labelled ribbon graph. An important observation is
that while $\mathrm{Rep_{q}}(G^{ad})\to\mathrm{Rep_{q}}(G^{sc})$ is fully
faithful (so that in a ball this map is simply an isomorphism), it is _not_ an
isomorphism globally, and has both kernel and cokernel. The cokernel is clear:
by construction the image of $\Phi$ is contained in the degree zero part of
$\mathrm{Sk}_{G^{sc}}(M).$ The kernel is more subtle, but it stems from the
fact that on the source, by definition, only skein relations which have
trivial underlying 1-cycle are imposed, whereas in the target they are not. In
the skein module $\mathrm{Sk}_{G^{sc}}(M)$, two ribbon graphs of degree zero
can be related in a way that nevertheless involves elements of nontrivial
degree. This is captured cleanly in the following

######  Theorem 3.12 ([GJS23]).

Let $M$ be an oriented $3$-manifold (not necessarily closed), and let
$q\in\mathbb{C}^{\times}$ be an arbitrary complex number. Then we have a
canonical isomorphism,

$\mathrm{Sk}_{G/Z(G),\pi_{1}}^{a,b}(M)\cong\mathrm{Sk}_{G,Z}^{a,b}(M).$

Unpacking, it says that the map $\Phi$ above actually extends firstly to a map

$\mathrm{Sk}_{G^{ad},\pi_{1}}(M)\to\bigoplus_{b\in
H_{1}(M,Z(G))}\mathrm{Sk}^{0,b}_{G^{ad},Z}(M),$

which is injective, and then again to a map,

$\bigoplus_{a,b}\mathrm{Sk}^{a,b}_{G^{sc},\pi_{1}}(M)\to\bigoplus_{a,b}\mathrm{Sk}^{a,b}_{G^{sc},Z}(M),$

which is an isomorphism. We note that the indices $a$ and $b$ appear on botht
he LHS and RHS above. This is justified by the natural isomorphisms,
$\pi_{1}(G^{ad})\cong Z(G^{sc})^{\vee}$.

The reader familiar with the language of gauging of higher form symmetries may
not find this surprising: Indeed, we have $G^{ad}=G/Z(G)$, hence
$\mathrm{Rep_{q}}(G^{ad})$ is a gauging of the electric $1$-form symmetry by
$Z(G)$; gauging an electric 1-form symmetry induces on the new theory a dual
magnetic 1-form symmetry, with an isomorphism like that above.

##  4\. The bigraded refinement of the skein conjecture

In total, we have defined bi-graded skein module,

$\mathrm{Sk}_{G^{sc},Z}^{a,b}(M),\qquad\textrm{for $a\in
H_{1}(M,Z(G^{sc})^{\vee}),b\in H_{1}(M,Z(G^{sc}))$},$

using the electric 1-form symmetry, and

$\mathrm{Sk}_{G^{ad},\pi_{1}}^{a,b}(M),\qquad\textrm{for $a\in
H_{1}(M,\pi_{1}),b\in H_{1}(M,\pi_{1}(G^{ad})^{\vee})$},$

using the magnetic 1-form symmetry, where in each case $a$ denotes the grading
degree, and $b$ denotes the twist. Now recall that under Langlands/S-duality,
the electric and magnetic 1-form symmetries are interchanged. Thus the same
electric-magnetic duality principle which informs Conjecture 1.1 also
predicts:

######  Conjecture 4.1 (Bigraded refinement of skeins conjecture).

Suppose that $q\in\mathbb{C}^{\times}$ is transcendental and that $M$ is a
closed oriented $3$-manifold. Then we have an equality of dimensions,

$\dim\mathrm{Sk}_{G,Z}^{a,b}(M)=\dim\mathrm{Sk}_{{{}^{L}G},\pi_{1}}^{b,a}(M).$

Now suppose that $G$ is simply connected of ADE type, so that
${{}^{L}\mathfrak{g}}=\mathfrak{g}$. In this case the Langlands dual group
${{}^{L}G}=G^{ad}$ has a second manifestation as a quotient $G/Z(G)$ of $G$ by
the electric 1-form symmetry we exploited to define the bi-grading. Hence we
may apply Theorem 3.12, to obtain the following equivalent

######  Conjecture 4.2 (Reformulation of refined skein conjecture).

Suppose that $G$ is simply connected of ADE type, that
$q\in\mathbb{C}^{\times}$ is transcendental and that $M$ is a closed oriented
$3$-manifold. Then we have an equality of dimensions,

$\dim\mathrm{Sk}_{G,Z}^{a,b}(M)=\dim\mathrm{Sk}_{G,Z}^{b,a}(M),$

for all $a\in H_{1}(M,Z(G)^{\vee}),b\in H_{1}(M,Z(G))$.

######  Remark 4.3.

The reformulation of Conjecture 4.1 is very convenient: applying it to the
case $G=\mathrm{SL}_{N}$, it means we can check the refined conjecture
completely internally to either the $Z$-twists on $\mathrm{SL}_{N}$ or
alternatively the $\pi_{1}$-twists on $\mathrm{PGL}_{N}$. Because more is
known about $\mathrm{SL}_{N}$ skein theory, we work with the $Z$-twists in the
next section.

######  Remark 4.4.

The isomorphism in Theorem 3.12 is completely canonical, holds for arbitrary
$q$ and also for 3-manifolds with boundary: it is a very general principle,
and in particular, there was an a priori natural alignment of the indices $a$
and $b$ appearing on either side of the asserted isomorphism. By contrast, the
indices $a$ and $b$ do not themselves live in the same group naturally, but
rather live in Pontryagin dual groups. Hence Conjecture 4.2 seems not to parse
semantically.

Indeed, to make sense of Conjecture 4.2, one must choose an isomorphism
$Z(G)\cong Z(G)^{\vee}$, though clearly the statement itself does not depend
on the choice. This should be regarded as a further symptom of the fact that
Langlands duality should be swapping $Z(G)$ with $\pi_{1}({{}^{L}G})^{\vee}$,
but that we are identifying $\pi_{1}({{}^{L}G})^{\vee}$ with $Z(G)$ in the ADE
case.

Recall that the untwisted skein modules in Conjecture 1.1 are obtained as sums
over the untwisted graded components. Hence, Conjecture 1.1 follows from the
following weakening of Conjecture 4.2.

######  Conjecture 4.5.

Suppose that $G$ is simply connected of ADE type, that
$q\in\mathbb{C}^{\times}$ is transcendental and that $M$ is a closed oriented
$3$-manifold. Then we have an equality of dimensions,

$\dim\mathrm{Sk}_{G,Z}^{a,0}(M)=\dim\mathrm{Sk}_{G,Z}^{0,a}(M),$

for all $a\in H_{1}(M,Z(G)^{\vee}$.

##  5\. Some evidence for the conjecture

Dimensions of skein modules are very difficult to compute in elementary terms.
Indeed, this is underscored by the twenty five years which passed between the
two fundamental examples $S^{2}\times S^{1}$ [HP95] and $T^{3}$ [Car17,
Gil18]. Since then a dam has broken and these dimensions are being computed
using a combination of elementary and categorical techniques. This allows us
now to prove our conjecture in a number of special cases. Recall that the Lie
algebra $\mathfrak{sl}_{N}$ is fixed under Langlands duality; however on the
group level, Langlands duality interchanges the simply connected form with the
adjoint form on the dual group, so it is meaningful to contemplate Langlands
duality for the pair ($\mathrm{SL}_{N}$, $\mathrm{PGL}_{N}$).

We focus on two families of skein modules, where complete computations can be
done. In the first family, we fix the Langlands dual pair to be
($\mathrm{SL}_{2}$, $\mathrm{PGL}_{2}$), and consider mapping cylinders
$\Sigma_{g}\times S^{1}$ of arbitrary genus $g$, while in the second family we
instead fix the 3-manifold to be $T^{3}$, and vary the Langlands dual pair to
be ($\mathrm{SL}_{N}$, $\mathrm{PGL}_{N}$). For the first family, we recall
the following:

######  Theorem 5.1 ([GM18], [DW21]).

Suppose that the quantum parameter is generic. Then we have:

$\dim\mathrm{Sk}_{SL_{2}}(\Sigma_{g}\times S^{1})=2^{2g+1}+2g-1.$

With respect to the electric 1-form symmetry action, we have:

$\dim\mathrm{Sk}_{\mathrm{SL}_{2}}^{a,0}(\Sigma_{g}\times
S^{1})=\left\\{\begin{array}[]{ll}g+1,&a=(0,0)\\\ g,&a=(1,0)\\\
1,&\textrm{otherwise}\end{array}\right.,$

where the cases refer to the Künneth decomposition,

$\displaystyle H_{1}(\Sigma_{g}\times S^{1},\mathbb{Z}/2)$
$\displaystyle=H_{0}(\Sigma,\mathbb{Z}/2)\otimes H_{1}(S^{1})\oplus
H_{1}(\Sigma,\mathbb{Z}/2)\otimes H_{0}(S^{1})$
$\displaystyle=H_{0}(\Sigma,\mathbb{Z}/2)\oplus H_{1}(\Sigma,\mathbb{Z}/2).$

The first non-trivial confirmation of Langlands duality for skein module is
then given by the following

######  Theorem 5.2 ([GJS23], to appear).

Suppose that the quantum parameter is generic. Then we have:

$\dim\mathrm{Sk}_{\mathrm{PGL}_{2}}(\Sigma_{g}\times S^{1})=2^{2g+1}+2g-1$

In fact the proof is not by computing with $\mathrm{Sk}_{\mathrm{PGL}_{2}}$,
but instead by applying Corollary 4.5. More precisely we prove:

$\dim\mathrm{Sk}_{\mathrm{SL}_{2}}^{0,a}(\Sigma_{g}\times
S^{1})=\left\\{\begin{array}[]{ll}g+1,&a=(0,0)\\\ g,&a=(1,0)\\\
1,&\textrm{otherwise}\end{array}\right..$

The first case is, of course, already proved by Carrega and Gilmer, we repeat
it only for emphasis. For the second case, we use the fact that the twisted
skein category at $q=1$ becomes the category of quasi-coherent sheaves on the
_twisted character variety_. This is in fact a _smooth affine variety_ , hence
the Hochschild homology of its quantization to the skein category is
identified with the middle homology group of the twisted character variety,
which was shown to be $g$-dimensional by Hausel–Rodriguez-Villegas [HR07],
following Hitchin.

Turning now to the second family of skein modules, we recall the following
notation: $\mathcal{P}(d)$ denotes the partition number of $d$, i.e. the
number of ways to write $d$ as a sum of non-negative integers,
$J_{3}=C\ast\mu$ is the third Jordan totient function, obtained as the Möbius
inversion of the cube function $C(n)=n^{3}$.

######  Theorem 5.3 ([Gun+23], to appear).

Suppose that the quantum parameter is generic. Then we have:

$\dim\mathrm{Sk}_{SL_{N}}(T^{3})=(\mathcal{P}\ast J_{3})(N)=\sum_{N=d\cdot
e\cdot f}\mathcal{P}(d)\cdot e^{3}\cdot\mu(f).$

With respect to the electric 1-form symmetry action we have

$\mathrm{Sk}_{SL_{N}}(T^{3})^{(a,b,c),0}=\operatorname{gcd(a,b,c,N)},$

where we write $(a,b,c)$ for an element of
$(\mathbb{Z}/N)^{3}=H_{1}(T^{3},\mathbb{Z}/N)$.

This theorem is proved by first obtaining an equivalence between the skein
category of the torus and the category of modules for a double affine Hecke
algebra, subsequently giving an isomorphism between this double affine Hecke
algebra and a smash product of a quantum torus and the symmetric group, and
finally computing the Hochschild homology directly by combinatorial means.

Our second non-trivial confirmation of Langlands duality is given by the
following

######  Theorem 5.4 ([GJS23]).

Suppose that the quantum parameter is generic, and that $N$ is prime. Then we
have:

$\dim\mathrm{Sk}_{PGL_{N}}(T^{3})=(\mathcal{P}\ast J_{3})(N)=\sum_{N=d\cdot
e\cdot f}\mathcal{P}(d)\cdot e^{3}\cdot\mu(f).$

With respect to the magnetic 1-form symmetry action we have

$\mathrm{Sk}_{\mathrm{PGL}_{N}}(T^{3})^{(a,b,c),0}=\operatorname{gcd(a,b,c,N)}=1.$

Our proof is similar to that of Theorem 5.2. We apply Conjecture 4.5, and
instead prove:

$\dim\mathrm{Sk}_{\mathrm{SL}_{N}}^{0,(a,b,c)}(T^{3})=1,\quad\textrm{if
$(a,b,c)\neq 0$}.$

For this, we rely again on the fact that the twisted character variety is
smooth, and that its middle homology is known to be one-dimensional. Of
course, we expect the theorem to hold when $N$ is not prime, but in this case
the twisted character variety is not smooth, so we can’t easily apply the same
techniques for the proof.

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