where their commutation relations coincide. To realize this idea we start by
considering a Hamiltonian quadratic in the $\tilde{R}_{\mathbf{n}}$ and
satisfying translation invariance, which we can write most generally as

$H=\frac{1}{2\rho(2\pi)^{2}r^{4}}\,{\sum_{{\mathbf{n}}}}^{\prime}\,\frac{F({\mathbf{n}})}{{\mathbf{n}}^{2}}\,\tilde{R}_{\mathbf{n}}\tilde{R}_{-{\mathbf{n}}}\;.$
(120)

$F({\mathbf{n}})$ is a regulator function which can be viewed as defining the
moment of inertia of the $SU(N)$ rigid body: more precisely we have the
inverse proportionality
$I_{{\mathbf{n}},{\mathbf{n}}}\propto{\mathbf{n}}^{2}/F({\mathbf{n}})$. The
equation of motion then takes the form
(${\tilde{\Omega}}_{\mathbf{n}}\equiv-\tilde{R}_{\mathbf{n}}/(\rho r^{2})$)

$\dot{\tilde{\Omega}}_{\mathbf{n}}=\frac{1}{2(2\pi
r)^{2}}{\sum_{{\mathbf{m}}}}^{\prime}\,\frac{F({\mathbf{m}})}{{\mathbf{m}}^{2}}\frac{N}{\pi}\sin{\left(\frac{\pi}{N}({\mathbf{n}}\wedge{\mathbf{m}})\right)}\,\big{(}\tilde{\Omega}_{-{\mathbf{m}}}\tilde{\Omega}_{{\mathbf{n}}+{\mathbf{m}}}+\tilde{\Omega}_{{\mathbf{n}}+{\mathbf{m}}}\tilde{\Omega}_{-{\mathbf{m}}}\big{)}\;.$
(121)

This matches the long distance part of the Euler equation, eq. (70) at
$\bar{v}_{I}=0$, provided $F({\mathbf{m}})=1$ for
$|{\mathbf{m}}|\lesssim\sqrt{N}$ and $F({\mathbf{m}})\rightarrow 0$ for
$|{\mathbf{m}}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\sqrt{N}$. The second request
guarantees that the charges $\tilde{R}_{{\mathbf{n}}}$, with
$|{\mathbf{m}}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\sqrt{N}$ decouple from the
macroscopic dynamics. In the rigid body analogy, this choice corresponds to
assigning an infinite moment of inertia to the UV modes. Moreover, even though
at finite $N$ rotational symmetry is broken both microscopically, by the fine
lattice, and globally, by the torus geometry, we would like to preserve it in
the limit $N\rightarrow\infty$ with $a_{1}$ fixed. This can be simply achieved
by choosing $F({\mathbf{n}})\equiv G(|{\mathbf{n}}|)$. We will thus consider
systems defined by a regulator function $F({\mathbf{n}})$ satisfying

$F({\mathbf{n}})\equiv G(|{\mathbf{n}}|)\qquad\qquad
F({\mathbf{n}})=1\,\,{\mathrm{for}}\,\,|{\mathbf{n}}|\ll\sqrt{N}\qquad\qquad
F({\mathbf{n}})\rightarrow 0\,\,{\mathrm{for}}\,\,|{\mathbf{n}}|\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}\sqrt{N}$ (122)

A possible choice is

$F({\mathbf{n}})=\left(\frac{N}{{\mathbf{n}}^{2}+N}\right)^{p}\qquad\qquad
p>0\,.$ (123)

These requests define a family of UV completions of quantum hydrodynamics. We
shall also illustrate what changes by relaxing some of these constraints and
show that the crucial aspects of IR physics are shared by a broader class of
Hamiltonians.

In order to make the scaling at large $N$ explicit, it is convenient to
rewrite the Hamiltonian in terms of the canonically normalized $SU(N)$
generators $J_{\mathbf{n}}$. By eqs. (78) and (80) we can then write

$H=\frac{\Lambda^{4}}{32\pi^{4}\rho}{\sum_{{\mathbf{n}}}}^{\prime}\,\frac{F({\mathbf{n}})}{{\mathbf{n}}^{2}}\,J_{\mathbf{n}}J_{-{\mathbf{n}}}\;.$
(124)

The prefactor $\frac{\Lambda^{4}}{\rho}$ coincides, as indeed dictated by
dimensional analysis, with Landau’s guess [29, 48] for the gap of the
transverse modes. Indicating by $m=\rho a_{1}^{2}$ the mass of a fluid
element, i.e. the mass in a coarse lattice cell, we can associate the energy
$\frac{\Lambda^{4}}{4\pi^{2}\rho}=\Lambda^{2}/m$ to the motion of a fluid
element with cut-off scale momentum.

Let us now study the spectrum of the Hamiltonian of eq. (124) focusing on the
simplest $SU(N)$ representations. The very simplest one, the singlet,
obviously has zero energy and provides the ground state of the system.
Consider next the fundamental. The generators, as discussed in subsection
5.1.1, are the ‘t Hooft matrices of eq. (75). Since
$J_{-{\mathbf{n}}}J_{\mathbf{n}}={\mathbb{1}}$, the Hamiltonian is here
proportional to the identity operator, all the $N$ states are degenerate:
$H\left|\alpha\right\rangle=E_{\Box}\left|\alpha\right\rangle$, with

$E_{\Box}=\frac{\Lambda^{4}}{32\pi^{4}\rho}{\sum_{{\mathbf{n}}}}^{\prime}\,\frac{F({\mathbf{n}})}{{\mathbf{n}}^{2}}\simeq\frac{\Lambda^{4}}{16\pi^{3}\rho}\,\ln{(\sqrt{N})}\;.$
(125)

Here, in the last step, we used that $F({\mathbf{n}})$ cuts off the
logarithmic divergent sum, roughly at $|{\mathbf{n}}|\sim\sqrt{N}$. Of course
while the log term is robust for large $N$, the finite $O(1)$ contributions
depend on the detailed choice of $F$. In terms of the physical length scales
we can equivalently write

$E_{\Box}=\frac{\Lambda^{4}}{16\pi^{3}\rho}\,\ln{(\Lambda r)}\;.$ (126)

We see that the overall scale of the energy matches the gap predicted by
Landau. However, the result is enhanced by a logarithmic factor that depends
on the size of the system. This result does not come unexpected, as we have
already seen that the fundamental representation describes localized vortices:
a logarithmic dependence of the energy on the total area is typical of vortex
flows where the velocity falls off with the inverse of the distance, see eq.
(102). Indeed inserting eq. (102) in the classical energy density $\rho
v^{2}/2$ we find

$E_{vortex}\simeq\frac{1}{2}\rho\int_{a_{1}}^{2\pi
r}d^{2}x\,{\mathbf{v}}^{2}\simeq\frac{\Lambda^{4}}{16\pi^{3}\rho}\,\ln{(2\pi
r/a_{1})}\;.$ (127)

in perfect agreement with eq. (126).

As the sum in eq. (125) depends logarithmically on the number of modes over
which $F({\mathbf{n}})$ is approximately constant, the result does not change
qualitatively when a different choice for $F({\mathbf{n}})$ is made. We are of
course excluding extreme choices like, for instance, $F({\mathbf{n}})$ growing
in the UV. For example, if one extends $F({\mathbf{n}})$ to be constant for
all ${\mathbf{n}}$, the final answer will change by a factor of two. Note
that, as the states in the fundamental are exactly degenerate, the dynamics of
this sector is trivial: any superposition is also an energy eigenstate and all
observables have time-independent expectation values.

The above discussion carries out unchanged to the antifundamental, given its
generators satisfy $\bar{J}_{\mathbf{n}}=-J^{T}_{\mathbf{n}}$. Thus we have
again $\bar{J}_{-{\mathbf{n}}}\bar{J}_{\mathbf{n}}=1$ $\forall{\mathbf{n}}$ so
that $E_{\bar{\Box}}=E_{\Box}$. Moreover, since all other unitary irreducible
representations can be obtained by considering tensor products of the defining
representation, the same argument shows that the energy spectra of any pair of
conjugate representations coincide.

The fact that the fundamental representation is gapped at the energy cut off
$\Lambda^{4}/\rho$ matches our conclusion in section 5.1.2 that the
fundamental of $SU(N)$ should be discarded when focusing on
${S\mathrm{Diff}}(T^{2})$. The validity of the simple semiclassical estimate
of its energy in eq. (127) suggests that in fact all states with resolvable
vortices will be gapped. We recall that this class of states fully includes
the representations that cannot be written as the tensor product of the same
number of $\Box$ and $\bar{\Box}$. We do not have a mathematical proof of
that, but, as illustrated in the appendix, all the explicit cases we studied
numerically confirm this expectation. Again this matches our conclusion that
the only states that should be kept consist of non-resolvable vortex-
antivortex pairs. The smallest representation containing these is the adjoint.
Somewhat expectedly, one finds that its spectrum is indeed not gapped, as we
now show.

####  5.2.1 The spectrum of the adjoint: vorton states

As we already discussed, the adjoints states $\left|{\mathbf{n}}\right\rangle$
are labelled by the fine lattice wave vectors while, as shown in eq. (108),
the action of the charges is given by the structure constant themselves. It is
then straightforward to see that the Hamiltonian of eq. (124) is diagonal in
the $\\{\left|{\mathbf{n}}\right\rangle\\}$ basis

$\left\langle{\mathbf{m}}\right|H\left|{\mathbf{n}}\right\rangle=E_{\mathbf{n}}\delta_{{\mathbf{m}},{\mathbf{n}}}$
(128)

with eigenvalues given by

$E_{\mathbf{m}}=\frac{\Lambda^{4}}{8\pi^{4}\rho}\sum_{{\mathbf{n}}\neq\mathbf{0}}F({\mathbf{n}})\frac{\sin^{2}{\left(\frac{\pi}{N}({\mathbf{m}}\wedge{\mathbf{n}})\right)}}{{\mathbf{n}}^{2}}\;.$
(129)

In contrast to the case of the fundamental representation, the spectrum now
substantially depends on the choice of $F({\mathbf{n}})$.

Let us consider first a sharp cut-off function satisfying eq. (122):
$F({\mathbf{n}})=1$ for $|n^{I}|<\frac{1}{2}(\sqrt{N}-1)$ and
$F({\mathbf{n}})=0$ for $|n^{I}|>\frac{1}{2}(\sqrt{N}-1)$. We drop for ease of
computation the request $F({\mathbf{n}})=F(|{\mathbf{n}}|)$. The behaviour of
the sum in eq. (129) depends on whether $|{\mathbf{m}}|$ is below or above
$\sqrt{N}$ (i.e. whether $|{\mathbf{p}}|$ is below or above the cut-off scale
$\Lambda$). For $|{\mathbf{m}}|\ll\sqrt{N}$ the sine has an argument smaller
than $O(1)$ and can be expanded in powers of
$\frac{m^{I}}{\sqrt{N}}=\frac{p^{I}}{\Lambda}$. In this regime the dispersion
relation then reads

$E_{\mathbf{m}}\simeq\frac{\Lambda^{4}}{8\pi^{4}\rho}\sum^{|n^{I}|\leq\frac{1}{2}(\sqrt{N}-1)}_{{\mathbf{n}}\neq\mathbf{0}}\frac{\pi^{2}}{N^{2}}\frac{({\mathbf{m}}\wedge{\mathbf{n}})^{2}}{{\mathbf{n}}^{2}}=\frac{\Lambda^{2}}{16\pi^{2}\rho}\left(\frac{{\mathbf{m}}^{2}}{r^{2}}\right)=\frac{p^{2}}{4\rho
a_{1}^{2}}\;,$ (130)

corresponding to the non-relativistic kinetic energy of a particle with mass
$M=2\rho a_{1}^{2}$. That is twice the mass of a fluid element. Notice however
that, given the sum in eq. (130) diverges quadratically, the precise result
strongly depends on the choice of regulator.

To evaluate instead the energy for $|{\mathbf{m}}|\gg\sqrt{N}$, we can choose
$m^{I}=(0,m)$ with $m\gg\sqrt{N}$. At leading order in $1/N$ and $1/m$ we can
then write

$E_{(0,m)}=\frac{\Lambda^{4}}{8\pi^{4}\rho}\sum^{|n^{I}|\leq\frac{1}{2}(\sqrt{N}-1)}_{{\mathbf{n}}\neq\mathbf{0}}\frac{\sin^{2}{\left(\frac{\pi}{N}\,m\,n^{1}\right)}}{{\mathbf{n}}^{2}}\simeq\frac{\Lambda^{4}}{2\pi^{4}\rho}\sum^{\frac{1}{2}\sqrt{N}}_{n^{1},n^{2}=1}\frac{\sin^{2}{\left(\frac{\pi}{N}\,m\,n^{1}\right)}}{{\mathbf{n}}^{2}}\\\
\simeq\frac{\Lambda^{4}}{4\pi^{3}\rho}\sum^{\frac{1}{2}\sqrt{N}}_{n^{1}=1}\frac{\sin^{2}{\left(\frac{\pi}{N}\,m\,n^{1}\right)}}{n^{1}}\simeq\frac{\Lambda^{4}}{8\pi^{3}\rho}\sum^{\frac{1}{2}\sqrt{N}}_{n^{1}\sim\frac{N}{m}}\frac{1}{n^{1}}\simeq\frac{\Lambda^{4}}{8\pi^{3}\rho}\ln{\left(\frac{m}{\sqrt{N}}\right)}\;.$
(131)

Choosing an arbitrary orientation of the wave vector ${\mathbf{m}}$ one can
also check that the result is rotationally invariant up to corrections
suppressed by inverse powers of $N$. In the continuum/infinite volume limit,
$N\rightarrow\infty$ with $a_{1}=2\pi/\Lambda$ fixed, the energy of the states
in the adjoint representation is then given by

$E_{\text{adj}}(p)\simeq\begin{cases}\frac{p^{2}}{4\rho a_{1}^{2}}&\text{for
}p^{2}\ll\Lambda^{2}\\\ \frac{\pi}{\rho
a_{1}^{4}}\ln{\left(\frac{p^{2}}{\Lambda^{2}}\right)}&\text{for
}p^{2}\gg\Lambda^{2}\end{cases}\;.$ (132)

A few comments are here in order. Notice, first of all, that the
$N\rightarrow\infty$ limit with $a_{1}$ fixed is well defined. The dependence
on both the fundamental lattice length $a$ and torus radius $r$ drop out.
However, in addition to $\rho$, which fully defined the classical theory, a
new parameter, $a_{1}=2\pi/\Lambda$, appears quantum mechanically. Secondly
the adjoint states are gapless, as intuitively befits the universal long
distance quanta of hydrodynamic vorticose motion. Indeed, the states
encompassing the gapless region, that is $p<\Lambda$ and $E\propto p^{2}$, are
precisely the vortons we previously identified. Somewhat expectedly the states
that approximate a representation of ${S\mathrm{Diff}}(T^{2})$ within $SU(N)$
are the ungapped ones. The existence of vortons and their energy spectrum is
the main result of our study.

The long wavelength behaviour $E\propto p^{2}$ gives way to a logarithmic
behaviour $E\propto\log p$ at momenta of the order of the cut-off $\Lambda$.
The velocity of the quanta at this change of regime is of order

$|\frac{\partial E}{\partial p}|\sim\frac{\Lambda}{\rho
a_{1}^{2}}=\frac{2\pi}{\rho a_{1}^{3}}\equiv c_{v}\,.$ (133)

It is instructive to compare this result to the dispersion relation of
longitudinal phonons. By zooming on incompressible flows, we have excluded the
phonons from the very beginning. However, in any fluid they do exist and have
linear dispersion relation: $E_{\text{ph}}(p)=c_{s}p$. For small enough
momenta, we then always have $E_{\text{ph}}(p)\gg E_{\text{adj}}(p)$, which
justifies the decoupling of the longitudinal modes. The energies, however,
become comparable at momenta of order $\Lambda_{\text{ph}}={\rho
a_{1}^{2}c_{s}}=\Lambda\,\frac{c_{s}}{c_{v}}$, which provides another limit of
applicability of our model. The simplest option, $c_{s}\sim c_{v}$, then
identifies $\Lambda$ with the very scale at which fluid compression cannot be
neglected.

Consider now different choices for $F$. The quadratic and logarithmic
divergence of the relevant sums in respectively eqs. (130) and (131) are
easily seen to determine what happens for different choices satisfying eq.
(122). For $p\ll\Lambda$ the coefficient controlling $E\propto p^{2}$ changes
by $O(1)$, while for $p\gg\Lambda$ only the finite part of the leading
logarithmic contribution is modified.

More dramatic changes occur for regulators not respecting eq. (122), even
though these choices are not physically sensible. For instance, for
$F({\mathbf{n}})\equiv 1\,\,,\forall{\mathbf{n}}$, the energy behaves
logarithmically at all momenta:
$E_{\text{adj}}(p)\simeq\frac{\Lambda^{4}}{8\pi^{3}\rho}\ln{(p\,r)}$. Moreover
the gap is finite: the energy of the state with the softest possible momentum
is $E_{\text{adj}}(p=\frac{1}{r})\approx 7\frac{\Lambda^{4}}{(2\pi)^{4}\rho}$.
One can also check that any function $F({\mathbf{n}})$ that shuts off between
$|{\mathbf{n}}|\sim\sqrt{N}$ and $|{\mathbf{n}}|\sim N$ leads to a spectrum
similar to eq. (132), but where the transition from quadratic to logarithmic
behaviour happens at a scale lower than $\Lambda$.

The main message is that, for the physically sensible regulators eq. (122),
the vorton spectrum, while UV physics dependent, is robustly controlled by a
quadratic dispersion relation, as shown in eq. (132). For definiteness in what
follows we will stick to the choice $F({\mathbf{n}})=1$ for
$|n^{I}|<\frac{1}{2}(\sqrt{N}-1)$ and $F({\mathbf{n}})=0$ for
$|n^{I}|>\frac{1}{2}(\sqrt{N}-1)$.

####  5.2.2 Multi-vorton states

With the results obtained so far, we can now show that tensor products of
adjoints are also ungapped. Again this matches what we argued in subsection
5.1.2, that only the low momentum portion of these representations properly
represents ${S\mathrm{Diff}}$ in the $N\rightarrow\infty$ limit.

To study the spectrum let us consider a tensor product of a number $M$ of
adjoints
$\left|{\mathbf{n}}_{1}\right\rangle\otimes\left|{\mathbf{n}}_{2}\right\rangle\otimes\dots\otimes\left|{\mathbf{n}}_{M}\right\rangle\equiv\bigotimes_{j}^{M}\left|{\mathbf{n}}_{j}\right\rangle$.
The vorticity operator of the tensor product space is
$\Omega_{tot}=\oplus_{j}^{M}\Omega_{j}$ where

$\Omega_{1}({\boldsymbol{x}})=\Omega({\boldsymbol{x}})\otimes{\mathbb{1}}\otimes\dots\otimes{\mathbb{1}}\,,\quad\Omega_{2}({\boldsymbol{x}})={\mathbb{1}}\otimes\Omega({\boldsymbol{x}})\otimes\dots\otimes{\mathbb{1}}\,,\quad{\mathrm{etc.}}\,.$
(134)

To study the energy on these states we can consider a wave packet

$\left|\\{\psi\\}\right\rangle=\sum_{{\mathbf{n}}_{1}\dots\mathbf{n}_{M}}\left(\bigotimes_{J}^{M}\hat{\psi}_{j}({\mathbf{n}}_{j})\left|{\mathbf{n}}_{j}\right\rangle\right)$
(135)

and estimate $\left\langle\\{\psi\\}\right|H\left|\\{\psi\\}\right\rangle$. As
we are aiming at the low energy limit, we will focus on wave packets supported
in the vorton region $|{\mathbf{p}}|=|{\mathbf{n}}/r|\ll\Lambda$. The
Hamiltonian consists of a sum over pairs $jk$ involving the bilinears products
$\Omega_{j}\Omega_{k}$. One can easily see that the diagonal terms
$\Omega_{1}\Omega_{1}$, …$\Omega_{M}\Omega_{M}$ give each the energy
expectation $E_{j}$ on the corresponding adjoint state

$E_{j}=\sum_{n}|\hat{\psi}_{j}({\mathbf{n}})|^{2}E_{\mathbf{n}}$ (136)

with $E_{\mathbf{n}}$ given by eq. (130). The sum of these contributions
$E_{1}+\dots+E_{M}$ thus corresponds to the energy of $M$ non-interacting
vortons: as vortons are ungapped, this energy can be made arbitrarily small
(in the infinite volume limit) by picking individual vorton wave functions
$\psi_{j}$ supported at softer and softer momenta.

Consider now instead the cross terms involving $\Omega_{i}\Omega_{j}$, with
$i\not=j$, which can be interpreted as giving the interaction energy between
vortons. One finds that the contribution of these terms is controlled by the
products of vorticity expectation values in eq. (117) of each separate vorton
state. In terms of the position space wave packets and already taking the
continuum limit we have

$\left\langle\psi_{j}\right|\Omega({\boldsymbol{x}})\left|\psi_{j}\right\rangle=\frac{i\,\boldsymbol{\partial}\psi_{j}\wedge\boldsymbol{\partial}\psi_{j}^{*}}{\rho}\equiv\partial_{I}J_{j}^{I}$
(137)

with

$J_{j}^{I}=\frac{1}{2\rho}\left(-i\psi^{*}_{j}\tilde{\partial}^{I}\psi_{j}+i\tilde{\partial}^{I}\psi^{*}_{j}\psi_{j}\right)\,.$
(138)

Unlike for the diagonal terms, one finds that the $i\not=j$ contributions are
not quadratically UV divergent. For wave packets supported at momenta
$\ll\Lambda$, we can then remove the regulator and use the long distance
Hamiltonian of eq. (69). To estimate the energy we can also directly work at
infinite volume. The interaction energy is then given by

$\frac{\rho}{2}\sum_{i\not=j}\int\Omega_{i}\frac{1}{-\nabla^{2}}\Omega_{j}\,=\,\frac{\rho}{2}\sum_{i\not=j}\int\,d^{2}{\boldsymbol{x}}\,d^{2}\mathbf{y}\,J_{i}^{K}({\boldsymbol{x}})G_{KL}({\boldsymbol{x}}-\mathbf{y})J_{j}^{L}(\mathbf{y})$
(139)
$G_{KL}(x-y)=\frac{2}{\pi}\left[\delta^{KL}-\frac{2(x-y)^{K}(x-y)^{L}}{|{\boldsymbol{x}}-\mathbf{y}|^{2}}\right]\frac{1}{|{\boldsymbol{x}}-\mathbf{y}|^{2}}$
(140)

where in the last step we used the explicit expression of the 2D propagator
$-1/\nabla^{2}$, then eq. (137) and finally we integrated by parts. The final
expression corresponds to the electrostatic interaction energy among a set of
localized charge dipoles with polarization densities $-J^{I}_{i}$. As $G_{KL}$
decreases like $|{\boldsymbol{x}}-\mathbf{y}|^{-2}$, the interaction energy
can be made arbitrarily small by making the wave packets arbitrarily separated
from one another. In the limit of infinite reciprocal separation the only
remaining contribution is the diagonal one $E_{1}+\dots+E_{M}$, corresponding
to $M$ non-interacting vortons.

The obvious corollary of this discussion is that multi-vorton states are also
ungapped.

##  6 Vorton Interactions

Working directly at infinite volume, we have just shown that vortons behave
like particles with interactions that vanish at infinite mutual separation. We
can then think of a scattering experiment, where, for instance, two originally
infinitely separated vorton waved packets are time evolved to within a finite
distance where they interact. In this section we will compute the scattering
amplitude of such two-vorton process. We shall arrive at that in three steps.
The first is a detailed study of the two vorton state at finite $N$. This step
is in fact not strictly necessary, given we are in any case interested in
scattering at infinite volume. However, we believe it naturally completes the
matrix model description of the fluid. The second step is the derivation of a
vorton field theory, which describes both the spectrum and the interactions of
vortons in the $N\rightarrow\infty$ limit. This result summarizes our
understanding of the universal long distance dynamics of a perfect quantum
fluid. As an application of the vorton field theory we shall then compute the
scattering amplitude.

###  6.1 Two vorton states

We want to consider representations that belong in the tensor product of two
adjoints. Using Young Tableaux such tensor product is seen to decompose into
the direct sum of seven irreducible representations:

$\ytableau~{}&~{}\\\ ~{}\none\\\ ~{}\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\\\ ~{}\none\\\
~{}\none\otimes~{}\ytableau~{}&~{}\\\ ~{}\none\\\ ~{}\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\\\ ~{}\none\\\
~{}\none=\left(~{}\ytableau~{}&~{}\\\ ~{}\none\none\\\ ~{}\none\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\none\\\
~{}\none\none\\\ ~{}\none\none\oplus\raisebox{3.99994pt}{ \ytableau ~{} & ~{}
\\\ ~{} ~{} \\\ ~{} \none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}] \none\\\ ~{}
\none}~{}\oplus~{}\ytableau~{}&~{}\\\ ~{}\none\\\ ~{}\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\\\ ~{}\none\\\
~{}\none\oplus~{}{\boldsymbol{\bullet}}\right)_{S}\oplus\left(~{}\ytableau~{}&~{}\\\
~{}~{}\\\ ~{}~{}\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\\\
~{}\none\\\ ~{}\none~{}\oplus\raisebox{3.99994pt}{ \ytableau ~{} & ~{} ~{} \\\
~{} \none \none\\\ ~{} \none \none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}] \none \none\\\ ~{} \none
\none}\oplus~{}\ytableau~{}&~{}\\\ ~{}\none\\\ ~{}\none\\\
\none[\raisebox{-0.5pt}{\scalebox{0.8}{$\vdots$}}]\none\\\ ~{}\none\\\
~{}\none~{}\right)_{A}\;,$ (141)

where the longest columns have $N-1$ boxes and the bullet stands for the
trivial representation. The first four representations are symmetric under
exchange of the two adjoints and the latter three are antisymmetric. This
decomposition is easy to understand by representing adjoints states as
traceless matrices $f^{\alpha}_{~{}\beta}$, corresponding to the traceless
product of a vortex–anti-vortex pair, see eq. (110). The first two terms are
respectively symmetric and antisymmetric under exchange of any pair of
constituent vortices or constituent anti-vortices. They have dimensions of
order $\frac{1}{4}N^{4}$ each. They are self-conjugated and thus invariant
under interchanging the roles of vortices and anti-vortices. Analogously, the
first two terms among antisymmetric representations have opposite symmetry
properties under exchange of vortices and exchange of anti-vortices. These two
representations have also each dimensions of order $\frac{1}{4}N^{4}$ and are
conjugated to one another. Apart from these four large representations, one
finds also two adjoint representations and the singlet.

What we have just discussed is purely group theory. We must now view this
result from the standpoint of physics. In particular eq. (141) is at odds with
the interpretation of the adjoint states in the left hand side as particles in
a Fock space: in the tensor product of two particles we would find both single
particle states, the adjoints, and the vacuum, the singlet. Notice that, given
the Hamiltonian only depends on the $SU(N)$ (or ${S\mathrm{Diff}}(T^{2})$)
generators, the adjoints and the singlet on the right hand side are
dynamically indistinguishable from the same representations we already studied
242424It is interesting to contrast this situation with, for instance, the
case of isospin in QCD: unlike $SU(N)$ here, isospin merely constrains the
dynamics without fully determining it.. Indeed in the particular case of
hypothesis A of section 3.2, the Peter-Weil theorem implies that each
representation $(r,r)$ of $SU(N)_{L}\times SU(N)_{R}$ features only once, in
such a way that the adjoints and singlets on the right hand side of eq. (141)
correspond to states we already considered.

The above obstruction to a particle interpretation has however a rather simple
solution. If we fold the left hand side with vorton wave packets, i.e.
localized at momenta $p\ll\Lambda$, and then take the $N\rightarrow\infty$
limit, the overlap with the adjoints and the singlet on the right hand side
goes to zero. Furthermore, up to $O(p^{2}/\Lambda^{2})$ effects, the overlap
with the remaining four large representations, reduces in the same limit to
two combinations, the symmetric and antisymmetric one:
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{S/A}=(\left|{\mathbf{m}}\right\rangle\otimes\left|{\mathbf{n}}\right\rangle\pm\left|{\mathbf{n}}\right\rangle\otimes\left|{\mathbf{m}}\right\rangle)/\sqrt{2}$.
These corresponds to the two-particle subsector of a Fock space of either
bosonic or fermionic identical quanta. Let us see how this works in detail.

The adjoints in the right hand side of eq. (141) arise from tracing one
constituent vortex and one anti-vortex from different adjoints, while the
vacuum corresponds to tracing both. The states belonging to the two adjoints
are then explicitly written as

$\left|{\boldsymbol{\ell}}\right\rangle_{S/A}\propto{\sum_{{\mathbf{m}},{\mathbf{n}}}}^{\prime}\left(J_{{\boldsymbol{\ell}}}\right)^{\alpha}_{~{}\beta}\left(J_{\mathbf{m}}^{*}\right)_{\alpha}^{~{}\gamma}\left(J_{\mathbf{n}}^{*}\right)_{\gamma}^{~{}\beta}\,\big{(}\left|{\mathbf{m}}\right\rangle\otimes\left|{\mathbf{n}}\right\rangle\pm\left|{\mathbf{n}}\right\rangle\otimes\left|{\mathbf{m}}\right\rangle\big{)}\;.$
(142)

Tracing over the three $J$-matrices and normalizing the states we can write
252525In the normalization factor of
$\left|{\boldsymbol{\ell}}\right\rangle_{S}$ we have approximated
$\sqrt{N^{2}-2}\rightarrow N$.

$\displaystyle\left|{\boldsymbol{\ell}}\right\rangle_{S}$
$\displaystyle=\frac{\sqrt{2}}{N}{\sum_{{\mathbf{m}}}}^{\prime}\,\cos{\left(\frac{\pi}{2N}\,{\mathbf{m}}\wedge{\boldsymbol{\ell}}\right)}\,\left|\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{m}})\right\rangle\otimes\left|\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{m}})\right\rangle\;,$
(143) $\displaystyle\left|{\boldsymbol{\ell}}\right\rangle_{A}$
$\displaystyle=\frac{\sqrt{2}}{N}{\sum_{{\mathbf{m}}}}^{\prime}\,i\sin{\left(\frac{\pi}{2N}\,{\mathbf{m}}\wedge{\boldsymbol{\ell}}\right)}\,\left|\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{m}})\right\rangle\otimes\left|\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{m}})\right\rangle\;.$
(144)

In the above equations the sums over ${\mathbf{m}}$ actually extend over the
double fundamental domain, that is $-N+1\leq m_{I}\leq N-1$, and only the
${\mathbf{m}}$’s for which the arguments of the kets are integer vectors
should be retained. Moreover, the twisted periodicity of eq. (77) should be
applied to the kets, whenever their argument is outside the fundamental
domain.

The vacuum state corresponds to the symmetric zero-momentum state

$\left|\bullet\right\rangle=\left|{\boldsymbol{\ell}}=\mathbf{0}\right\rangle_{S}/\sqrt{2}=\frac{1}{N}{\sum_{{\mathbf{m}}}}^{\prime}\left|{\mathbf{m}}\right\rangle\otimes\left|-{\mathbf{m}}\right\rangle=\frac{a^{2}}{N}\sum_{{\boldsymbol{x}}}\left|{\boldsymbol{x}}\right\rangle\otimes\left|{\boldsymbol{x}}\right\rangle\;,$
(145)

where in the last step we used the position eigenstates defined in eq. (111).
The above three equations all define largely entangled states of two vortons,
the majority of which have momenta above the fluid cut-off $\Lambda$. We thus
expect their overlap with vorton wave packets to be suppressed. Consider
indeed a two vorton wave-packet
$\left|f_{1}({\boldsymbol{x}})\right\rangle\otimes\left|f_{2}({\boldsymbol{x}})\right\rangle$
where $f_{1}$ and $f_{2}$ are localised on a distance $L$ larger than the
fluid cut-off length $a_{1}$. The overlap with the “small” representations
subspace for such a state is approximately given by

$\sum_{{\boldsymbol{\ell}},S,A}\left|(\left\langle
f_{1}\right|\otimes\left\langle
f_{2}\right|)\,\left|{\boldsymbol{\ell}}\right\rangle_{S/A}\right|^{2}\simeq\sum_{\boldsymbol{x}}|f_{1}({\boldsymbol{x}})|^{2}|f_{2}({\boldsymbol{x}})|^{2}\sim\frac{a^{2}}{L^{2}}<\frac{a^{2}}{a_{1}^{2}}=\frac{1}{N}\;,$
(146)

and vanishes for $N\rightarrow\infty$. A similar conclusion holds for the
overlap with $\left|\bullet\right\rangle$. In the continuum limit, we can thus
neglect the overlap of $\left|{\boldsymbol{\ell}}\right\rangle_{S,A}$ and
$\left|\bullet\right\rangle$ with vorton wave packets.

Consider now the four “big” representation in eq. (141). We can also
explicitly write them using projectors. For example, the projector on the
symmetric-symmetric states is

$P_{SS}\propto{\sum_{\begin{subarray}{c}{\mathbf{m}}_{1},{\mathbf{m}}_{2},\\\
{\mathbf{n}}_{1},{\mathbf{n}}_{2}\end{subarray}}}^{\prime}\left|{\mathbf{m}}_{1},{\mathbf{m}}_{2}\right\rangle\left\langle{\mathbf{n}}_{1},{\mathbf{n}}_{2}\right|\left\\{\left(\left(J_{{\mathbf{m}}_{1}}\right)^{\alpha_{1}}_{~{}\beta_{1}}\left(J_{{\mathbf{m}}_{2}}\right)^{\alpha_{2}}_{~{}\beta_{2}}\left(J_{{\mathbf{n}}_{1}}^{*}\right)_{\alpha_{1}}^{~{}\beta_{1}}\left(J_{{\mathbf{n}}_{2}}^{*}\right)_{\alpha_{2}}^{~{}\beta_{2}}\right.\right.\\\
+(\alpha_{1}\leftrightarrow\alpha_{2})\Big{)}+(\beta_{1}\leftrightarrow\beta_{2})\Big{\\}}\;,$
(147)

where we neglected the subtraction of trace terms that correspond to the
singlet and the adjoints, which we already established to be negligible. The
other three projectors are obtained by exchanging symmetrisations with anti-
symmetrisations accordingly. Two terms in eq. (147) contain a double trace of
products of $J$-matrices. They give two momentum conservation Kronecker
symbols and correspond to the identity operator on the symmetric or anti-
symmetric subspaces, respectively

$I_{S/A}=\frac{1}{4}{\sum_{{\mathbf{m}}_{1},{\mathbf{m}}_{2}}}^{\prime}\big{(}\left|{\mathbf{m}}_{1},{\mathbf{m}}_{2}\right\rangle\left\langle{\mathbf{m}}_{1},{\mathbf{m}}_{2}\right|\pm\left|{\mathbf{m}}_{1},{\mathbf{m}}_{2}\right\rangle\left\langle{\mathbf{m}}_{2},{\mathbf{m}}_{1}\right|\big{)}\;.$
(148)

The other two terms involve the single trace of a product of four
$J$-matrices, which produces only conservation of the total momentum. These
terms are thus off-diagonal in the individual vorton momentum labels. In
respectively the symmetric and anti-symmetric subspaces they correspond to the
following operators (the same comments we made below eq. (144) apply to these
summations)

$\displaystyle\Delta_{S}$
$\displaystyle=\frac{1}{2N}{\sum_{{\boldsymbol{\ell}},{\mathbf{m}},{\mathbf{n}}}}^{\prime}\cos{\left(\frac{\pi}{2N}\,{\mathbf{m}}\wedge{\mathbf{n}}\right)}\,\left|\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{n}}),\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{n}})\right\rangle\left\langle\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{m}}),\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{m}})\right|\;,$
(149) $\displaystyle\Delta_{A}$
$\displaystyle=\frac{1}{2N}{\sum_{{\boldsymbol{\ell}},{\mathbf{m}},{\mathbf{n}}}}^{\prime}\,\sin{\left(\frac{\pi}{2N}\,{\mathbf{m}}\wedge{\mathbf{n}}\right)}\,\left|\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{n}}),\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{n}})\right\rangle\left\langle\frac{1}{2}({\boldsymbol{\ell}}+{\mathbf{m}}),\frac{1}{2}({\boldsymbol{\ell}}-{\mathbf{m}})\right|\;,$
(150)

Using these elements, the full projectors can be written as

$\displaystyle P_{SS}$ $\displaystyle=I_{S}+\Delta_{S}\;,$ $\displaystyle
P_{AA}$ $\displaystyle=I_{S}-\Delta_{S}\;,$ (151) $\displaystyle P_{AS}$
$\displaystyle=I_{A}+\Delta_{A}\;,$ $\displaystyle P_{SA}$
$\displaystyle=I_{A}-\Delta_{A}\;.$ (152)

Very much as we did previously with the “small” representations, one can study
the matrix elements of $\Delta_{S}$ and $\Delta_{A}$ on vorton wave packets
localised on a distance $L$ larger than the fluid cut-off length $a_{1}$. In
the same notation used previously we find

$|\left\langle f_{1}\right|\otimes\left\langle
f_{2}\right|\Delta_{S,A}\left|g_{1}\right\rangle\otimes\left|g_{2}\right\rangle|\,\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\,\frac{a_{1}^{2}}{L^{2}}$ (153)

The overlap is not as suppressed as in eq. (146), but it still becomes
negligible for wave packets that are deeply in the vorton region $L\gg a_{1}$.
When projected on vorton wave packets states the four “large” representations
reduce in reality to just two representations, corresponding to the symmetric
and antisymmetric product of two vortons
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{S/A}=(\left|{\mathbf{m}}\right\rangle\otimes\left|{\mathbf{n}}\right\rangle\pm\left|{\mathbf{n}}\right\rangle\otimes\left|{\mathbf{m}}\right\rangle)/\sqrt{2}$.
As already anticipated, when considering vorton wave packets, the content of
the seven irreps in eq. (141) reduces to
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{S}$ and
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{A}$, corresponding to the two
particle Fock-subspace for respectively a boson and a fermion field.

One can go on and investigate the same questions for representations that can
be written as tensor products of more than two adjoints. We have not done the
exercise, but for a product of $M$ adjoints
$\left|{\mathbf{n}}_{1}\right\rangle\otimes\dots\otimes\left|{\mathbf{n}}_{M}\right\rangle$
we expect the irreducible subspaces to correspond to Young tableaux with $M$
boxes, modulo the subtraction of traces. The simplest options, consisting of a
row or a column of boxes, correspond respectively to identical bosons or
identical fermions. However, the more complex options correspond to mixed
statistics.

These results motivate the investigation of a field theoretic realization of
vortons.

###  6.2 Vorton Field Theory

The results of the last two subsections indicate that it should be possible to
bypass the truncation of ${S\mathrm{Diff}}(T^{2})$ to $SU(N)$ and focus on a
theory where the vortons are the sole degrees of freedom. In order to achieve
that, we need to go back to section 3 and seek a different route to
quantization. All we need is to find an explicit realization of the algebra of
$R$-charges, use it to write the Hamiltonian of eq. (69) and then study the
dynamics. The first step is easily taken, as any Lie algebra can be explicitly
realized by either bosonic or fermionic ladder operators through the Jordan-
Schwinger map [49, 50]. In practice this works as follows. Given any explicit
representation $(T^{A})^{\beta}_{~{}\alpha}$ of the generators $Q^{A}$, one
considers either bosonic or fermionic ladder operators transforming in the
same representation, that is either $\Phi^{\alpha}$ or $\Psi^{\alpha}$ with
commutation relations

$[\Phi^{\alpha},\Phi^{\dagger}_{\beta}]=\delta^{\alpha}_{\beta}\,,\qquad\qquad\qquad\\{\Psi^{\alpha},\Psi^{\dagger}_{\beta}\\}=\delta^{\alpha}_{\beta}\,.$
(154)

The operators
$Q^{A}_{b}\equiv\Phi^{\dagger}_{\beta}\Phi^{\alpha}(T^{A})^{\beta}_{~{}\alpha}=\Phi^{\dagger}T^{A}\Phi$
and
$Q^{A}_{f}\equiv\Psi^{\dagger}_{\beta}\Psi^{\alpha}(T^{A})^{\beta}_{~{}\alpha}=\Psi^{\dagger}T^{A}\Psi$
are then both easily seen to represent the algebra.

In our case, a basis of the Lie algebra is given by the local charges
$R({\boldsymbol{x}})$, whose commutator is the Fourier transform of eq. (66)

$[R({\boldsymbol{x}}),R(\mathbf{y})]=-i\int
d^{2}z\,R(\mathbf{z})\boldsymbol{\partial}_{z}\delta^{2}(\mathbf{z}-{\boldsymbol{x}})\,\wedge\,\boldsymbol{\partial}_{z}\delta^{2}(\mathbf{z}-\mathbf{y})\,.$
(155)

We can then introduce ladder operators, either bosonic or fermionic,
transforming in the adjoint of ${S\mathrm{Diff}}$, that is position space
scalar fields $\Phi({\boldsymbol{x}})$ or $\Psi({\boldsymbol{x}})$ satisfying

$[\Phi({\boldsymbol{x}}),\Phi^{\dagger}(\mathbf{y})]=\delta^{2}({\boldsymbol{x}}-\mathbf{y})\,,\qquad\qquad\qquad\\{\Psi({\boldsymbol{x}}),\Psi^{\dagger}(\mathbf{y})\\}=\delta^{2}({\boldsymbol{x}}-\mathbf{y})\,.$
(156)

Applying the above construction, one then finds that the $R$-algebra is
equally well represented as

$R_{b}({\boldsymbol{x}})=i\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi\,\qquad{\mathrm{or}}\qquad
R_{f}({\boldsymbol{x}})=i\boldsymbol{\partial}\Psi^{\dagger}\wedge\boldsymbol{\partial}\Psi$
(157)

Of course also the sum $R_{b}+R_{f}$ offers a representation. In the boson
(fermion) case, the Hilbert space is a Fock space constructed from a vacuum
state $\left|0\right\rangle$, satisfying $\Phi\left|0\right\rangle=0$
($\Psi\left|0\right\rangle=0$) by repeated action with $\Phi^{\dagger}$
($\Psi^{\dagger}$). As we will now discuss, the particle excitations of either
Fock space offer specific incarnations of the vortons.

Focusing for definiteness on the bosonic case, the Hamiltonian is obtained by
replacing $\Omega({\boldsymbol{x}})=-R_{b}({\boldsymbol{x}})/\rho$ in the
Hamiltonian of eq. (69). Of course we should also take into account the need
for a UV regulation of the intermediate $1/\nabla^{2}$, as evidenced by the
study of the $SU(N)$ truncation. Working directly in the infinity volume limit
(thus dropping the velocity zero mode $\bar{v}\rightarrow 0$) we can write

$\displaystyle H$ $\displaystyle=$
$\displaystyle{\frac{1}{2\rho}}\int\,d^{2}{\boldsymbol{x}}\,R_{b}\frac{F(-\nabla^{2}/\Lambda^{2})}{-\nabla^{2}}R_{b}=$
$\displaystyle=$ $\displaystyle{\frac{1}{2\rho}}\int
d^{2}{\boldsymbol{x}}d^{2}\mathbf{y}[i\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi]({\boldsymbol{x}})G^{\Lambda}({\boldsymbol{x}}-\mathbf{y})[i\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi](\mathbf{y})$

where, as previously, $F(\tau)$ is a UV regulator function satisfying
$F\rightarrow 0$ for $\tau\rightarrow\infty$ and $F\rightarrow 1$ for
$\tau\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}1$. The UV regulated Green’s function
$G^{\Lambda}({\boldsymbol{x}}-\mathbf{y})$ is still logarithmically IR
divergent. However, as already discussed in section 5.2.2, $R_{b}$ is a total
derivative, and the divergent piece drops out. More precisely we can write

$R_{b}({\boldsymbol{x}})=\partial_{I}J^{I}\qquad\qquad
J^{I}=\frac{i}{2}\left(\Phi^{\dagger}\tilde{\partial}^{I}\Phi-\tilde{\partial}^{I}\Phi^{\dagger}\Phi\right)$
(159)

so that we can express the Hamiltonian as

$H=\frac{1}{2\rho}\int
d^{2}{\boldsymbol{x}}d^{2}\mathbf{y}J^{K}({\boldsymbol{x}})G_{KL}^{\Lambda}({\boldsymbol{x}}-\mathbf{y})J^{L}(\mathbf{y})$
(160)

where the long distance behaviour of $G_{KL}^{\Lambda}$ is given by eq. (140),
while at coinciding points its quadratic divergence is regulated to

$G_{KL}^{\Lambda}(0)=\left[\frac{\Lambda^{2}}{8\pi}\int
dxF(x)\right]\delta_{KL}\equiv c\frac{\Lambda^{2}}{8\pi}\,\delta_{KL}$ (161)

To study the dynamics it is further convenient to write the Hamiltonian in
normal ordered form. A straightforward computation then gives

$H=\frac{1}{2\rho}\left\\{c\frac{\Lambda^{2}}{8\pi}\int
d^{2}{\boldsymbol{x}}\,\boldsymbol{\partial}\Phi^{\dagger}\boldsymbol{\partial}\Phi+\int
d^{2}{\boldsymbol{x}}d^{2}\mathbf{y}\,:J^{K}({\boldsymbol{x}})G_{KL}^{\Lambda}({\boldsymbol{x}}-\mathbf{y})J^{L}(\mathbf{y}):\,\right\\}\,.$
(162)

This results makes the dynamics manifest. The degrees of freedom are bosonic
vortons with a dispersion relation set by the term quadratic in the field:
$E_{\mathbf{p}}=(c\Lambda^{2}/16\pi\rho){\mathbf{p}}^{2}$, which nicely
matches the result in section 5.2.1. The second term in the Hamiltonian,
vanishes on single vorton states, and represents a dipolar interaction among
vorton pairs. Also for this term one can check that the matrix elements
perfectly match the $SU(N)$ model matrix elements among symmetric bi-adjoints
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{S}=(\left|{\mathbf{m}}\right\rangle\otimes\left|{\mathbf{n}}\right\rangle+\left|{\mathbf{n}}\right\rangle\otimes\left|{\mathbf{m}}\right\rangle)/\sqrt{2}$
at low momentum ($p\ll\Lambda$). This is not surprising given the computation
is fully controlled by the algebra of the $R$-charges. Furthermore, as one can
immediately appreciate, eq. (162) makes a specific prediction for the
structure of the vorton scattering amplitude. In fact, at Born level the
regulator can be dropped, and the amplitude is fully determined by one
parameter, $\rho$, with a specific dependence on the external momenta. We will
present this computation in the next subsection.

The above discussion goes through basically unchanged for the case of
fermionic vortons: the result amounts to replacing $\Phi\rightarrow\Psi$ in
eqs. (159) and (162), while the matrix elements now match those for the anti-
symmetric bi-adjoint
$\left|{\mathbf{m}},{\mathbf{n}}\right\rangle_{A}=(\left|{\mathbf{m}}\right\rangle\otimes\left|{\mathbf{n}}\right\rangle-\left|{\mathbf{n}}\right\rangle\otimes\left|{\mathbf{m}}\right\rangle)/\sqrt{2}$
in the $SU(N)$ model. Similarly in a realization with both a boson and a
fermion, the Hamiltonian would consist, at the quadratic level, of the sum of
the bosonic and fermionic kinetic term, while the current appearing in the
interaction would be the sum of the bosonic and fermionic currents. That would
make specific predictions for the scattering amplitudes $bb\rightarrow bb$,
$ff\rightarrow ff$ and $bf\rightarrow bf$. Notice that individual vorton
number is preserved, so that processes like $bb\rightarrow ff$ do not occur.

###  6.3 Vorton scattering

We are now ready to apply our results to compute the simplest instance of
vorton $S$-matrix: $2\rightarrow 2$ scattering. As stressed in the previous
section, the matrix elements in the vorton field theory match those of the
$SU(N)$ model in the low energy limit. To perform computations we can thus
work directly in the field theory. Writing the $S$-matrix as
$S={\mathbb{1}}+i{\mathbb{M}}$, the reduced $2\rightarrow 2$ scattering
amplitude $\cal M$ is defined by

$\left\langle\mathbf{k}_{1},\mathbf{k}_{2}\right|{\mathbb{M}}\left|\mathbf{q}_{1},\mathbf{q}_{2}\right\rangle=(2\pi)^{3}\delta(E_{{\mathbf{k}}_{1}}+E_{{\mathbf{k}}_{2}}-E_{{\mathbf{q}}_{1}}-E_{{\mathbf{q}}_{2}})\delta^{2}({\mathbf{k}}_{1}+{\mathbf{k}}_{2}-{\mathbf{q}}_{1}-{\mathbf{q}}_{2})\,{\cal
M}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})$ (163)

In the Born approximation, $\cal M$ is directly given by the matrix elements
of the interaction Hamiltonian. The corresponding Feynman diagrams for the
three cases of interest $bb$, $ff$ and $bf$ are shown in Fig. 6. As also
evidenced by the figure, the basic building block is the $bf$ amplitude, which
has the structure of a $t$-channel amplitude.

Figure 6: Born level Feynman diagrams relevant for the vorton scattering.

Defining

${\cal
M}_{t}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})=\frac{1}{\rho}\,\frac{(\mathbf{k}_{1}\wedge\mathbf{q}_{2})(\mathbf{k}_{2}\wedge\mathbf{q}_{1})}{(\mathbf{q}_{1}-\mathbf{k}_{1})^{2}}\,,$
(164)

we then have

$\displaystyle{\cal
M}_{bf}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})$
$\displaystyle=$ $\displaystyle 2{\cal
M}_{t}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})$
(165) $\displaystyle{\cal
M}_{bb}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})$
$\displaystyle=$ $\displaystyle{\cal
M}_{t}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})+{\cal
M}_{t}({\mathbf{k}}_{2},{\mathbf{k}}_{1},{\mathbf{q}}_{1},{\mathbf{q}}_{2})\equiv{\cal
M}_{t}+{\cal M}_{u}$ $\displaystyle{\cal
M}_{ff}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})$
$\displaystyle=$ $\displaystyle{\cal
M}_{t}({\mathbf{k}}_{1},{\mathbf{k}}_{2},{\mathbf{q}}_{1},{\mathbf{q}}_{2})-{\cal
M}_{t}({\mathbf{k}}_{2},{\mathbf{k}}_{1},{\mathbf{q}}_{1},{\mathbf{q}}_{2})\equiv{\cal
M}_{t}-{\cal M}_{u}$

where, in an obvious notation, we defined a $u$-channel amplitude ${\cal
M}_{u}$ obtained by exchanging
${\mathbf{k}}_{1}\leftrightarrow{\mathbf{k}}_{2}$ in ${\cal M}_{t}$. Notice
that all these amplitudes vanish when any of the external momenta vanish. Like
for the vanishing of the vorton energy at zero momentum, this can be
intuitively understood by the fact that, at zero momentum, adjoint states
becomes locally indistinguishable from the singlet. While eq. (164) makes this
property manifest, it is also useful to implement energy and momentum
conservation and write it in terms of unconstrained kinematic variables. In
particular, momentum conservation allows to write
$\mathbf{q}_{1/2}=\frac{1}{2}\mathbf{(P\pm Q)}$ and
$\mathbf{k}_{1/2}=\frac{1}{2}(\mathbf{P}\pm\mathbf{K})$, while energy
conservation corresponds to $\mathbf{K}^{2}=\mathbf{Q}^{2}$. Substituting in
eq. (165) and taking into account that crossing now corresponds to
$\mathbf{K}\rightarrow-\mathbf{K}$, we can write (no need to show ${\cal
M}_{bf}=({\cal M}_{bb}+{\cal M}_{ff})/2$, whose expression is too long)

$\displaystyle{\cal M}_{bb}$ $\displaystyle=$
$\displaystyle\frac{1}{4\rho}\left\\{{\mathbf{Q}}^{2}-{\mathbf{P}}^{2}\right\\}\,,$
(166) $\displaystyle{\cal M}_{ff}$ $\displaystyle=$
$\displaystyle\frac{1}{4\rho}\left\\{\frac{({\mathbf{Q}}^{2}-{\mathbf{P}}^{2})({\mathbf{P}}\cdot{\mathbf{Q}})({\mathbf{P}}\cdot{\mathbf{K}})+({\mathbf{Q}}^{2}+{\mathbf{P}}^{2})({\mathbf{P}}\wedge{\mathbf{Q}})({\mathbf{P}}\wedge{\mathbf{K}})}{{\mathbf{P}}^{2}{\mathbf{Q}}^{2}}\right\\}\,$
(167) $\displaystyle=$
$\displaystyle\frac{1}{4\rho}\left\\{({\mathbf{Q}}^{2}-{\mathbf{P}}^{2})\,\cos\alpha\,\cos\beta+({\mathbf{Q}}^{2}+{\mathbf{P}}^{2})\,\sin\alpha\,\sin\beta\right\\}.$
(168)

Where $\alpha$ $(\beta)$ is the angle between $\mathbf{P}$ and $\mathbf{Q}$
$(\mathbf{K})$. The dependence of the amplitudes on the total momentum
$\mathbf{P}$ makes evident, as it was to be expected, the absence of Galilean
invariance. The vorton dispersion relation
$E_{\mathbf{k}}\sim{\mathbf{k}}^{2}$ only accidentally complies with Galilean
symmetry. Notice also that, even though the interaction Hamiltonian is non-
local, the bosonic amplitude ${\cal M}_{bb}$ is polynomial in the momenta.
That is not the case for the fermionic one ${\cal M}_{ff}$, whose singularity
at zero momentum perfectly reflects the non-locality of the Hamiltonian. There
must of course exist a change of variables that makes the locality of the Born
level bosonic $2\rightarrow 2$ amplitude manifest. We have not investigated
that, but it would be interesting to do so and see if all the bosonic tree
level amplitudes are made local that way.

Another property of ${\cal M}_{bb}$ is its independence of the angular
variables: in terms of partial waves for $\mathbf{Q}$ (and $\mathbf{K}$) it is
a pure $s$-wave. This result follows from the quadratic dependence on the
momenta and from Bose symmetry, which dictates invariance of ${\cal M}_{bb}$
under the independent sign flip of $\mathbf{Q}$ and $\mathbf{K}$. These
properties constrain ${\cal M}_{bb}$ to be a linear function of
${\mathbf{P}}^{2}$ and ${\mathbf{Q}}^{2}$. In fact ${\cal M}_{bb}$ has the
additional property of vanishing for $|{\mathbf{P}}|=|{\mathbf{Q}}|$, that is
for incoming, and outgoing, orthogonal momenta:
${\mathbf{q}}_{1}\cdot{\mathbf{q}}_{2}={\mathbf{k}}_{1}\cdot{\mathbf{k}}_{2}=0$.
A pair of bosonic vortons thus has the peculiarity of not diffusing when
crossing at $90$ degrees. The fermionic amplitude ${\cal M}_{ff}$ does not
share this property, though it has other zeroes. In particular it vanishes for
$(\alpha,\beta)$ equal to either $(\pi/2,0)$ or $(0,\pi/2)$. In terms of
momenta, this means that in the scattering of two vortons with the same
energy, i.e. $|{\mathbf{k}}_{1}|=|{\mathbf{k}}_{2}|$, the final state momenta
${\mathbf{q}}_{1}$ and ${\mathbf{q}}_{2}$ cannot be parallel. Similarly for
parallel initial state momenta, the two final state vortons cannot have the
same energy. Of course these properties follow from the dipolar nature of the
vorton-vorton interaction, see eq. (162), and from the relation between dipole
and momentum, see eq. (159).

The Born level is a good approximation as long as the coupling is weak. A
natural measure of coupling strength is provided by the scattering phases.
These are readily read off the $S$-matrix on states labelled by energy,
momentum and angular momentum, where the latter is just the the quantum number
canonically conjugated to the angles $\alpha$, or $\beta$. One finds that,
given the vorton mass $\mu=c(8\pi\rho/\Lambda^{2})$, the scattering phases
scale like ${\cal M}\mu\sim k^{2}/\Lambda^{2}$. This result also simply
follows from dimensional analysis, given that $\cal M$ has dimension
$[k^{2}/\rho]=1/[mass]$ and given that the vorton mass is the only mass
parameter we can use to normalize the amplitude so as to make it
dimensionless. A quick study of the 1-loop corrections also shows that the
loop expansion is controlled by $k^{2}/\Lambda^{2}$. We conclude that the
coupling strength is $\sim k^{2}/\Lambda^{2}$ so that the system is strongly
coupled at the cut-off, i.e. for $k\sim\Lambda$, but weakly coupled at all
lower momenta. Notice that, even though the interaction Hamiltonian, when
written in terms of the canonical field $\Phi$, purely depends on the
classical parameter $\rho$, it should not come as a surprise that the
interaction strength is instead controlled by the cut-off $\Lambda$. That is
because the existence of asymptotic states and of an $S$-matrix is guaranteed
precisely by the existence of the UV cut-off $\Lambda$. That makes it more
intuitively clear that the strength of the interaction is in the end
controlled by $k^{2}/\Lambda^{2}$.

###  6.4 Effective Field Theory approach

The results of the last two sections compel us to try and investigate vorton
QFT from the modern effective field theory (EFT) standpoint. In this section
we will make a brief foray into that, focusing on the bosonic vorton QFT. The
need for a deeper study will emerge. We leave that to future work [47].

The idea is to formulate vorton QFT in the path integral approach on the basis
of symmetry principles and by working order by order in the derivative
expansion. Notice however that the use of spacial locality as a constitutive
principle will be limited, given vortons are endowed with long range dipolar
interactions.

In the path integral approach, the first step is to write the classical
Lagrangian corresponding to the Hamiltonian of eq. (6.2). Adding an auxiliary
field $A({\boldsymbol{x}},t)$ such Lagrangian can be written as

${\cal
L}=i\Phi^{\dagger}\dot{\Phi}+\frac{1}{\rho}\left[i(\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi)A+\frac{1}{2}\boldsymbol{\partial}A\frac{1}{F(-\boldsymbol{\partial}^{2}/\Lambda^{2})}\boldsymbol{\partial}A\right]$
(169)

where $F(\tau)$ is the same regulator function defined in eq. (6.2). By the
above Lagrangian, $\Phi$ and $\Phi^{\dagger}$ are canonically conjugated
variables, and moreover one can easily see that by eliminating the auxiliary
field $A$ one gets back the Hamiltonian pf eq. (6.2). Notice that the
Lagrangian is non-local only at distance scales $\sim 1/\Lambda$, and because
of the non-trivial regulator function $F$. On the other hand, the exchange of
$A$ produces a specific long-distance non-locality in $H$. As, by hypothesis,
$F(\tau)$ is regular around $\tau=0$, it’s low momentum expansion defines a
subset of the higher derivative corrections to the leading long-distance
dynamics.

The above Lagrangian is invariant under two sets of continuous
transformations. The first is given by

$A({\boldsymbol{x}},t)\rightarrow A({\boldsymbol{x}},t)+\alpha(t)$ (170)

for any $\alpha(t)$. While the invariance of the quadratic term is obvious,
that of the term linear in $A$ is proven by noticing that
$\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi$ is a
total derivative and by integrating by parts. Notice that this symmetry
forbids ${\dot{A}}^{2}$, thus making $A$ an auxiliary field. The second set of
transformations consists of volume preserving mappings in
$(\Phi^{\dagger},\Phi)$ space, that is transformations that preserve the two
form $d\Phi^{\dagger}\wedge d\Phi$. As $\Phi({\boldsymbol{x}})$ and
$\Phi^{\dagger}({\boldsymbol{x}})$ are canonically conjugated, these
correspond to canonical transformations that do not mix variables at different
space points. At the infinitesimal level, such transformations can be written
in terms of the Poisson bracket with a generating function $\cal F$

$\delta\Phi=\partial_{\Phi^{\dagger}}{\cal
F}(\Phi^{\dagger},\Phi)\qquad\qquad\delta\Phi^{\dagger}=-\partial_{\Phi}{\cal
F}(\Phi^{\dagger},\Phi)$ (171)

with ${\cal F}^{\dagger}=-{\cal F}$ in order to satisfy the consistency
condition $\delta\Phi^{\dagger}=(\delta\Phi)^{\dagger}$. The vorticity
$\boldsymbol{\partial}\Phi^{\dagger}\wedge\boldsymbol{\partial}\Phi$ is easily
seen to be exactly invariant, while for the kinetic term one finds exact
invariance up to a time derivative262626This result is the field theoretic
version of the invariance of $\int p\dot{q}$ under canonical transformations
in analytical mechanics. It can also be quickly understood by noticing that
the kinetic term involves the integral along the time line of the 1-form
$K=\Phi^{\dagger}d\Phi$, whose external derivative is just the volume 2-form:
$dK=d\Phi^{\dagger}\wedge d\Phi$. As the latter is invariant under volume
preserving transformations, the variation $\delta K$ must be closed, $d\delta
K=0$. Since the topology of $\Phi$-space is trivial, the variation is actually
exact $\delta K=dG$, for some $G$, and therefore the action is invariant up to
boundary terms.

$\delta(i\Phi^{\dagger}\dot{\Phi})=-i\partial_{\Phi}{\cal
F}\dot{\Phi}+i\Phi^{\dagger}\frac{d}{dt}(\partial_{\Phi^{\dagger}}{\cal
F})=i\frac{d}{dt}\left(-{\cal F}+\Phi^{\dagger}\partial_{\Phi^{\dagger}}{\cal
F}\right)\,.$ (172)

The vast majority of choices for ${\cal F}$ leads to non-linear
transformations, but the subset ${\cal
F}=i\alpha\Phi^{\dagger}\Phi+\beta{\Phi^{\dagger}}^{2}-\beta^{*}\Phi^{2}$
(with real $\alpha$) leads to a linearly realized $SL(2,R)$ subgroup

$\delta\Phi=i\alpha\Phi+2\beta\Phi^{\dagger}\qquad\qquad\delta\Phi^{\dagger}=-i\alpha\Phi^{\dagger}+2\beta^{*}\Phi\,.$
(173)

The constant shift $\Phi\rightarrow\Phi+c$,
$\Phi^{\dagger}\rightarrow\Phi+c^{*}$, is obviously also a symmetry, generated
by ${\cal F}=c\Phi^{\dagger}-c^{*}\Phi$.

While we arrived at eq. (169) following the thread of our $SU(N)$ rigid body
construction, the symmetries we have just identified offer an alternative
route. One can indeed convince oneself that eq. (169) with
$F(-\boldsymbol{\partial}^{2}/\Lambda^{2})\rightarrow 1$ is the lowest order
local action, counting powers of fields and derivatives, that is compatible
with eqs. (170) and (172). This result places vorton field theory in the
standard framework of effective field theory, according to which the dynamics
of systems is constructed on the basis of symmetries and of an expansion in
powers of fields and derivatives.

Notice that, in particular, the symmetries forbid a pure gradient term
$\boldsymbol{\partial}\Phi^{\dagger}\boldsymbol{\partial}\Phi$ in the
quadratic action. Because of that, the solutions of the linearized classical
equations of motion272727That is solutions for which $\Phi$ is small enough
that the quartic term can be neglected. satisfy a trivial dispersion relation
$\omega_{\mathbf{k}}=0$ for any ${\mathbf{k}}$. Thus they do not propagate.

We just outlined the situation when treating $\Phi$ and $\Phi^{\dagger}$ as
classical variables. As symmetries underlie the structure of eq. (169), it
matters to investigate their fate in the quantum mechanical treatment. In
particular, for non-linear symmetries the concern is the path integral
measure, or equivalently the UV regulator 282828In the canonical approach that
role is played by operator ordering and by the associated possible UV
divergences.. In many standard examples of effective field theories based on
symmetries and derivative expansion, like the pion Lagrangian of QCD,
dimensional regularization offers the most convenient approach, as it
generally respects all the symmetries. An exception is represented by the
symmetries that rely on the dimensionality of space-time, like chiral
symmetry. Unfortunately vorton field theory appears to belong in this class,
given the symmetries and the Lagrangian are based on the 2-dimensional Levi-
Civita tensor $\epsilon^{IJ}$. Moreover dimensional regularization eliminates
all power divergences (or, better, consistently hides them in the definition
of composite operators) but in the vorton system a quadratic divergence
saturated at the UV cut-off scale $\Lambda$ crucially generates a gradient
kinetic term, see eq. (162), which makes quanta propagate, bringing asymptotic
states into existence. How should we then proceed in our case? Indeed we
already know of one option, which is precisely where we started from: UV-
regulate by truncating ${S\mathrm{Diff}}(T^{2})$ to $SU(N)$. Space is
consequently latticized and the vorton field $\Phi$ reduced to the discrete
set of the $N^{2}-1$ components of the $SU(N)$ adjoint. The procedure would
then amount to repeating the steps we followed in section 6.2 but working with
$SU(N)$ rather than directly with ${S\mathrm{Diff}}(T^{2})$. We could then
proceed either by path integral or by canonical quantization and study the
quantum mechanical fate of the analogue of eq. (171) in the $SU(N)$ model. In
the canonical approach, the main effect of quantization is that the ordering
of $\Phi$ and $\Phi^{\dagger}$ matters when considering a generic generating
function ${\cal F}$. One can however easily see that for the case of harmonic
generator ${\cal F}=f(\Phi)-f(\Phi)^{\dagger}$ there is no ordering problem
given there is no operator product involving both $\Phi$ and $\Phi^{\dagger}$.
This subclass of transformations can then straightforwardly be adapted to the
$SU(N)$ regulated theory. We leave the study of the general case to future
work.

Another fact of the quantum theory is that symmetries can be spontaneously
broken. More precisely, symmetries that act linearly on the classical fields
may end up being non-linearly realized in the quantum theory. In our case the
linear symmetry subgroup $SL(2,R)$ is indeed spontaneously broken to $U(1)$ in
the Fock vacuum. Specifically, the transformations associated with $\beta$ in
eq. (173) are broken, while phase rotations $\Phi\rightarrow e^{i\alpha}\Phi$
stay unbroken. An order parameter of the breaking is for instance given by the
local operator $\Phi\Phi^{\dagger}$, given one has

$\left\langle
0\right|\delta_{\beta}(\Phi^{2})({\boldsymbol{x}})\left|0\right\rangle=\left\langle
0\right|2\Phi\Phi^{\dagger}({\boldsymbol{x}})\left|0\right\rangle>0\,.$ (174)

Indeed the expectation value in the last expression is UV divergent and
formally equal to $2\delta^{2}({\mathbf{0}})$. When considering UV regulated
quantities, like for instance the Hamiltonian, the relevant equation would
entail a UV smeared composite operator of the form

$\displaystyle\left\langle 0\right|\delta_{\beta}\left[\int
d^{2}\mathbf{y}\,f(\mathbf{y})\Phi({\boldsymbol{x}}+\mathbf{y}/2)\Phi({\boldsymbol{x}}-\mathbf{y}/2)\right]\left|0\right\rangle=$
(175) $\displaystyle=2\left\langle 0\right|\int
d^{2}\mathbf{y}\,f(\mathbf{y})\Phi({\boldsymbol{x}}+\mathbf{y}/2)\Phi^{\dagger}({\boldsymbol{x}}-\mathbf{y}/2)\left|0\right\rangle=2f({\mathbf{0}})\,.$
(176)

For a regulator function $f(\mathbf{y})$ that integrates to unity and is
localized over the fluid cut-off length, we have
$f({\mathbf{0}})\sim\Lambda^{2}/4\pi$. Notice that $SL(2,R)$ forbids the
addition of an arbitrary gradient term
$\boldsymbol{\partial}\Phi^{\dagger}\boldsymbol{\partial}\Phi$ in eq. (162).
Therefore the term that appears after normal ordering and its effects on the
spectrum must be controlled by the spontaneous breaking of $SL(2,R)$. A
remarkable aspect here is that spontaneous symmetry breaking leads to the
appearance of effects that depend on UV physics. It would definitely be
interesting to study in detail the regulated Ward identities (for instance
using the $SU(N)$ truncation) and explore the role of single and double vorton
states in the light of Goldstone theorem. We leave this study to future work.

##  7 Summary

As this is a long and winding paper, we shall here offer a detailed summary.

The initial object of our study has been a dynamical system whose
configuration space is ${S\mathrm{Diff}}(T^{2})$, the group of area preserving
mappings of the 2-torus $T^{2}$ onto itself. In practice, given two sets of
coordinates $\phi^{a}$, $a=1,2$, and $x^{I}$, $I=1,2$ both parametrizing
$T^{2}$, the dynamical variables are given by the time-dependent mappings
$\phi\rightarrow x(\phi,t)$ subject to the constraint
${\mathrm{det}}(\partial\phi/\partial x)=1$. With the velocity
$v^{I}=\dot{x}^{I}(\phi,t)$ and the Lagrangian of eq. (47), the system
classically defines the incompressible perfect 2D fluid. The $\phi$’s and the
$x$’s, correspond to respectively the Lagrangian and the Eulerian descriptions
of the flow. The goal has been to study the system quantum mechanically.

Our system belongs to the general class of mechanical systems whose
configuration space is a Lie group $G$ [30, 31]. This class generalizes the
notion of rigid body, given the case $G=SO(3)$ coincides with the ordinary
rigid body of mechanics. At least when $G$ is finite dimensional, once the
irreducible representations of $G$ are known, the steps to derive the quantum
description are clear. The first step is dictated by the Peter-Weyl theorem,
which establishes that the collection of entries
$D^{r}_{\alpha_{r}\beta_{r}}(g(\pi))$ of all irreducible representations
offers a complete orthonormal basis of functions on $G$ (see the discussion
around eq. (34)). Each irrep $r$ of dimension $d_{r}$ then corresponds to a
Hilbert subspace of dimension $d_{r}^{2}$. The action of $G$ on each index of
$D^{r}_{\alpha_{r}\beta_{r}}(g(\pi))$ realizes a doubling $G_{L}\times G_{R}$
of $G$, with respect to which the subspace transforms like $(r,r)$. This
doubling already exists classically. Moreover even though only $G_{L}$ is
exact in general, $G_{R}$ is crucial to describe the dynamics. Indeed the
Hamiltonian is a quadratic form in the generators of $G_{R}$ (see eq. (69)).
The second step in the solution of the quantum theory, is the diagonalization
of the Hamiltonian in each separate $(r,r)$ block. Each block then gives
$d_{r}$ generally distinguished eigenvalues, with each eigenvalue
$d_{r}$-degenerate because of $G_{L}$.

The extension of the above construction to the fluid entails a technical
difficulty and a conceptual conundrum. The technical difficulty is that
${S\mathrm{Diff}}(T^{2})$ is an infinite Lie group for which we do not possess
ready made explicit irreducible representations. We dealt with this issue by
using that ${S\mathrm{Diff}}(T^{2})=\lim_{N\rightarrow\infty}SU(N)$. Working
at finite $N$, we then identified a physically sensible way to take the
$N\rightarrow\infty$ limit.

The conceptual conundrum is tied to the very existence of dual coordinates
$\boldsymbol{\phi}$ (Lagrangian) and ${\boldsymbol{x}}$ (Eulerian), with
${S\mathrm{Diff}}(T^{2})_{L}$ and ${S\mathrm{Diff}}_{R}(T^{2})$ acting
respectively on the first and on the second. The position space velocity
$\boldsymbol{v}({\boldsymbol{x}},t)$, and more precisely its vorticity
$\boldsymbol{\nabla}\wedge\boldsymbol{v}$, coincides with the
${S\mathrm{Diff}}(T^{2})_{R}$ generators and is consequently
${S\mathrm{Diff}}(T^{2})_{L}$ invariant. As discussed in section 3, this
implies that there exist twice as many local degrees of freedom than we can
measure with $\boldsymbol{v}({\boldsymbol{x}},t)$ and associated with the
internal variables $\boldsymbol{\phi}$. Classically this seems irrelevant, as
each $\boldsymbol{\phi}$ configuration lives a life of its own, but not so
when considering quantum superpositions: measurements of
$\boldsymbol{v}({\boldsymbol{x}},t)$ are then generally described by a density
matrix. We considered two alternative attitudes towards this fact (labelled A
and B2). The first is just to accept it. The second is to do away with
${S\mathrm{Diff}}(T^{2})_{L}$ and to only represent the
${S\mathrm{Diff}}(T^{2})_{R}$ generators
$\boldsymbol{\nabla}\wedge\boldsymbol{v}({\boldsymbol{x}},t)$. The latter
choice is equivalent to giving up the group manifold configuration space we
started with. Our impression is that this second option is more likely to be
concretely realized in a laboratory.

As the energy spectrum and the position space dynamics are determined by the
${S\mathrm{Diff}}_{R}(T^{2})$ algebra, the above two options only differ in
the eigenvalue degeneracy. The study of their dynamics thus technically
coincide. Using the truncation ${S\mathrm{Diff}}_{R}(T^{2})\rightarrow SU(N)$,
we pursued that study starting in section 4. A crucial consequence of the
truncation is the reduction of $T^{2}$ to an $N\times N$ lattice. The $N^{2}$
values of the now discretized vorticity on each lattice site provide the
$N^{2}-1$ generators of $SU(N)$ (the apparent mismatch of $N^{2}$ and
$N^{2}-1$ is accounted for by the vanishing of the vorticity zero mode on
$T^{2}$). However, the discretized vorticity algebra ($\equiv SU(N)$)
approximates ${S\mathrm{Diff}}(T^{2})$ only at wavelengths $\gg\sqrt{N}a$,
with $a$ the fundamental lattice length. This led us to identify
$\Lambda=1/\sqrt{N}a$ with the fundamental cut-off of the fluid effective
theory and to construct a suitably regulated Hamiltonian (see eq. (120)).

Armed with our construction, the study of the spectrum reduced to computing,
irrep by irrep, the eigenvalues of a quadratic form in the $SU(N)$ generators.
A detailed study of the fundamental and the adjoint representations allowed us
to draw a general picture. Our results are neatly stated in the limit where
$N\rightarrow\infty$ with $\Lambda$ fixed, in which the lattice becomes a
continuum and the volume of $T^{2}$ goes to infinity like $N/\Lambda^{2}$. We
found that fundamental $\Box$ and antifundamental $\bar{\Box}$ are fully
degenerate and gapped at $\omega_{\Box}\sim\Lambda^{4}/\rho$. This result,
which is fully dictated by dimensional analysis when taking $\Lambda$ and the
mass density $\rho$ as the sole inputs, coincides with Landau’s estimate for
the gap of the transverse hydrodynamic modes. Landau then concluded that the
only ungapped mode of quantum hydrodynamics is the longitudinal one, in
compliance with the observed properties of superfluids at zero temperature.
Our finding however differs from Landau’s when considering the states in the
adjoint representation. These are not degenerate and satisfy an ungapped
dispersion relation that at low momentum takes the quadratic form
$\omega(k)=c\Lambda^{2}k^{2}/\rho$. That is the main result of our paper.

The absence of a gap in the adjoint can be qualitatively understood on the
basis of group theory and locality. As we show, the $N$ basis states of $\Box$
can be chosen to correspond to single vortices localized on each of the $N$
sites of the coarse grained $\sqrt{N}\times\sqrt{N}$ lattice, see Fig. 3.
Analogously, the antifundamental $\bar{\Box}$ has a basis of localized anti-
vortices. Now, $\Box\otimes\bar{\Box}={\mathbf{Adj}}\oplus{\mathbf{1}}$ and
locality imply that the adjoint states are locally indistinguishable from the
trivial representation ${\mathbf{1}}$ when their momentum is sent to zero. By
continuity, in this limit their energy must then also vanish like it vanishes
for ${\mathbf{1}}$. This argument can easily be extended to prove the
gaplessness of any representation that can be written as a tensor product of
an equal number of $\Box$ and $\bar{\Box}$. That is precisely the subset that
trivially realizes the $Z_{N}$ center of $SU(N)$ and which can equivalently be
written as tensor products of adjoints. Indeed, as we prove in section 5.2.2,
the low end of the energy spectrum for the tensor product of $q$ adjoints has
the form $E=\omega(k_{1})+\dots+\omega(k_{q})$ compatible with $q$ quanta that
are weakly coupled at long distance and have the same dispersion relation as
the adjoint. We named these quanta vortons.

As it is implicit from ${\mathbf{Adj}}\in\Box\otimes\bar{\Box}$, the vortons
are a vortex-antivortex pair. More precisely, for momentum ${\mathbf{p}}$ they
correspond to a pair separated by a distance
$\pi\tilde{\mathbf{p}}/\Lambda^{2}$ (see eq. (114)). In the low momentum
regime $p\ll\Lambda$, they can be viewed as point-like objects, given they
carry a vorticity dipole smaller than the cut-off length $1/\Lambda$. Instead,
for $p\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}\Lambda$, the dipole length exceeds $1/\Lambda$ and the
components of the pair can be resolved within the effective theory. In this
regime, their dispersion relation also shifts to
$\omega\sim(\Lambda^{4}/\rho)\log p/\Lambda$ which suits expectations for a
macroscopic vortex antivortex configuration. Indeed the high momentum
($p\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}\Lambda$) adjoint states correspond to $SU(N)$ generators that
do not properly approximate the ${S\mathrm{Diff}}(T^{2})$ algebra. In view of
that, we concluded that only vortons with momentum $p\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{\hskip 0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\Lambda$ should be considered a universal consequence of
${S\mathrm{Diff}}(T^{2})$, while the $p\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\Lambda$ modes
should be discarded as a spurious feature of the truncation. Moreover, as the
latter modes feature well separated vortices, we also inferred that $\Box$,
$\bar{\Box}$ and all the states involving well separated vortices should be
discarded. In particular this implies that irreps that do not feature vortons
should be fully discarded.

That conclusion also fits well our general expectation that states that do not
purely involve vortons are gapped. While we did not provide a general
mathematical proof, that expectation is based on the analytical and numerical
study of various non trivial irreps (see the Appendix) and on the absence of
any obvious reason for these states to be ungapped. A deeper perspective would
undoubtedly be gained by exploring in which way the gapless vortons are
mandated by Goldstone theorem [47] 292929We thank Alberto Nicolis for rightly
pressuring us on this.. The relevant symmetries would either be
${S\mathrm{Diff}}_{L}$ in the original 2D fluid or the transformation of eq.
(171) in vorton QFT. We left that study to future work and focused on the
study of multi-vorton states.

Our construction, besides implying vortons behave as free particles when
infinitely separated, also completely fixes the structure of their interaction
up to a proportionality constant. In section 6 we investigated the most basic
implications. For that purpose we made specific choices for the structure
($\equiv$ irrep content) of the Hilbert space. Notice the $(r,r)$ blocks of
the original perfect fluid system imply (for $N\rightarrow\infty$) an infinite
number of vorton flavors, given every flux configuration is in turn degenerate
under the action of ${S\mathrm{Diff}}_{L}$. That is another perspective on the
bizarre nature of this system. We thus considered it more plausible to realize
${S\mathrm{Diff}}_{R}$ only. Furthermore, motivated by the particle nature of
vortons, we realized ${S\mathrm{Diff}}_{R}$ through ladder field operators
that create and destroy vortons. We provided two examples based on
respectively bosonic and fermionic vortons. Within this set up, see section
6.3, we studied $2\rightarrow 2$ scattering at Born level. The amplitude
dependence on the kinematical variables is completely fixed by the algebra,
with distinctive features that would unmistakably identify these systems
experimentally. For instance the scattering amplitude for identical bosons
vanishes when the incoming momenta are mutually orthogonal. In the scattering
of identical fermions, instead, one finds that two collinear vortons with
different energies won’t scatter into two vortons with identical energies. All
these amplitudes are quadratic in the overall momentum and vanish in the soft
limit as it suits hydrodynamic modes. We also found that the Born
approximation breaks down for momenta of order $\Lambda$, at which the vortons
are therefore strongly coupled. Remarkably, in spite of this fact, our
construction is technically consistent also at momenta above $\Lambda$, only
its interpretation as a fluid becomes untenable.

##  8 Outlook

We can force ourselves to a synthesis by considering three questions:
Did we solve our original problem? What did we learn? Where do we go from
here?

The answer to the first question is: yes, at least in 2D, there exists a
consistent quantum mechanical perfect fluid. This affirmative answer however
comes with a qualification that brings us to the second question. In our
original incompressible fluid, each velocity/vorticity configuration has an
internal infinite degeneracy associated with the action of
${S\mathrm{Diff}}_{L}$ on the Lagrangian coordinates of the fluid. We regard
this property as suspect. Interestingly, however, we could construct a variety
of systems that do not possess the internal infinite degeneracy and yet are
endowed with a local vorticity satisfying the quantum mechanical Euler’s
equation (eq. (71)) of ordinary fluids. We aided ourselves in the construction
of these systems with a regulation ${S\mathrm{Diff}}(T^{2})\rightarrow SU(N)$,
but in the end our results do not depend structurally on the details of the
regulation. Our main result is that all these systems feature ungapped quanta,
the vortons, whose flow configuration is a vorticity dipole and whose
interactions are fully fixed. We thus labelled vorton QFT the simplest and
perhaps most plausible incarnations. The vortons sharply distinguish our
systems from superfluids, where all modes carrying vorticity are gapped.

The obvious goal at this point is to try and find out if vorton QFT can be
realized experimentally. A first issue to investigate towards that goal
concerns the robustness of the symmetries at the basis of vorton QFT. A
preliminary study was undertaken in section 6.4, though a more thorough study
is warranted. In particular one should understand the occurrence on the
ungapped vortons according to the finite density analogue of Goldstone’s
theorem. In parallel, it also seems mandatory to explore the relation between
vorton QFT and the quantum Hall system, where occupied Landau levels do behave
like a sort of incompressible fluid. An even simpler but related system to
consider is 2D electrodynamics in a constant magnetic field, which is dually
equivalent to a 2D superfluid. As charged particles are dual to vortices,
particle-antiparticle bound states seem the natural candidates to reproduce
the dynamics of vortons. Indeed a preliminary study [47] indicates that is the
case, but only up to contact interaction terms and up to the obvious constant
gap in energy associated with the rest mass of particle and antiparticle.
These facts further motivate the study of the robustness of the symmetries
that constrain vorton QFT.

If the above program ends up consolidating vorton QFT, it will then be
interesting to consider further properties of its dynamics. One particularly
interesting issue is the semiclassical limit of the flow, which should arise
for sufficiently large vorton density. One is easily convinced that the
critical request is that the flow velocity collectively generated by the
vortons is larger than their individual velocity. A simple estimate shows this
requires vorton densities $\gg\Lambda^{2}$. It would be interesting to study
quantum effects by expanding around such semiclassical flows. In fact that
brings to mind the possible use of our insights in the effective description
of fluids, like those occurring in finite temperature QFT, that do not
correspond to QFTs around a vacuum (see e.g. [51]). At this stage we have no
concrete idea to offer, but a fresh look at those systems with our results in
mind may teach us something. The extension of our analysis to 3D fluids is
also in our mind, even though it appears a long shot at this moment.

In the end, we have as many questions as when we started. Luckily they are
different.

### Acknowledgements

We would like to thank Alexander Abanov, Andrea Cappelli, Vadim Cheianov,
Gabriel Cuomo, Sergey Dubovsky, Angelo Esposito, Fanny Eustachon, Eren Firat,
Ben Gripaios, Brian Henning, Alexander Monin, Alberto Nicolis, Riccardo Penco,
Federico Piazza, Tudor Ratiu, Slava Rychkov and George Savvidy for insightful
dicussions and for honest criticism. RR offers one very special thank to
Solomon Endlich, Walter Goldberger and Ira Rothstein for the hard work and
endless debates that animated the abandoned project upon which much of the
present paper was based. R.R. is partially supported by the Swiss National
Science Foundation under contract 200020-188671 and through the National
Center of Competence in Research SwissMAP.

##  Appendix A Energy spectrum beyond fundamental states

Obtaining an analytic formula for the spectrum of representations beyond the
fundamental and the adjoint is a challenging task. However, there are still
interesting features of the spectrum that one can derive for an arbitrary
representation. In what follows we discuss two such results: the $N$-fold
degeneracy associated with most of the energy eigenstates and the statistics
of the energy eigenvalues. The latter will allow us to put a lower bound on
the number of vortices in a gapless state. At the end of the section we
describe the results of the numerical diagonalisation of the Hamiltonian on
two-vortex states.

###  A.1 $N$-fold degeneracy of the states

The first interesting feature of the spectrum is that for most $SU(N)$
representations all energy eigenstates are $N$-fold degenerate, as we have
seen for the fundamental. To see this, let us consider the products of $n$
fundamental states, the $n$-vortex states. The basis states
$\left|\alpha_{1},\dots,\alpha_{n}\right\rangle\equiv\left|\alpha_{1}\right\rangle\otimes\dots\otimes\left|\alpha_{n}\right\rangle$
are eigenstates of the elementary translations $T_{2}$ of eq. (87) with
eigenvalues $\omega^{\sum_{i}\alpha_{i}}\equiv e^{-ip_{2}a}$. The second
component of the momentum $p_{2}=-\frac{1}{r}(\sum_{i}\alpha_{i}\mod N)$ is
periodic and takes $N$ different integer values in units of $\frac{1}{r}$,
which we chose to be in the range $-\frac{1}{2}(N-1)\leq
p_{2}\,r\leq\frac{1}{2}(N-1)$. The space of all $n$-vortex states can be split
into $N$ subspaces of fixed $p_{2}$ value. Since the Hamiltonian of eq. (124)
commutes with $T_{2}$ it does not mix the subspaces with different values of
$p_{2}$ and takes a block-diagonal form:

$\left\langle
p_{2}^{\prime},\mu^{\prime}\right|H\left|p_{2},\mu\right\rangle=\delta^{p^{\prime}_{2}}_{p_{2}}\,H^{(p_{2})}_{\mu^{\prime},\mu}\;.$
(177)

Here the $\mu$ label distinguishes among the states with fixed $p_{2}$. The
other translation acts on the basis states as
$T_{1}\left|\alpha_{1},\dots,\alpha_{n}\right\rangle=\left|\alpha_{1}+1\mod
N,\dots,\alpha_{n}+1\mod N\right\rangle$ and maps each fixed $p_{2}$ subspace
to another one with $p_{2}$ reduced by $n\mod N$ in units of $\frac{1}{r}$. If
$n$ and $N$ do not have any non-trivial common divisors then shifting by
$n\mod N$ runs all $N$ different values of $p_{2}$. The Hamiltonian also
commutes with $T_{1}$, which means that all $N$ blocks in the Hamiltonian are
identical and do not depend on the value of $p_{2}$. This introduces an
$N$-fold degeneracy of energy eigenstates that is labeled by $p_{2}$. If the
greatest common divisor of $n$ and $N$ is larger than one, then the $N$
$p_{2}$ blocks in the Hamiltonian are split into $\gcd(n,N)$ equal subsets
connected by the action of $T_{1}$. The degeneracy of such states is smaller
and is given by $\frac{N}{\gcd(n,N)}$.303030In the magnetic field analogy the
$n$-vortex state behaves like a particle of charge $n$ that lives in a
magnetic field. The flux per plaquette $\Phi=2\pi n/Ne$ is such that
$T_{1}T_{2}=e^{-ie\Phi}T_{2}T_{1}$, which holds for the translations acting on
the $n$-vortex states. Making the parallel with the quantum Hall literature
[52] we have $\Phi=\frac{n}{N}\Phi_{0}$, with $\Phi_{0}$ the fundamental flux.
If $n$ and $N$ share a common divisor we have to consider $N/\gcd(n,N)$
instead of $N$, such that the spectrum admits an $N/\gcd(n,N)$ degeneracy. The
basis states are eigenstates of $T_{2}$ with $-\frac{1}{2}(N-1)\leq
p_{2}\,r\leq\frac{1}{2}(N-1)$ and are eigenstates of $T_{1}^{N/\gcd(n,N)}$
such that $p_{1}$ is confined to take one of $\gcd(n,N)$ values. This defines
the magnetic Brillouin zone.

We have seen before that for the fundamental representation, one-vortex, this
degeneracy fixes the spectrum completely. For two-vortex states, the
degeneracy is also $N$, since we take $N$ to be odd. Then the space of $N^{2}$
two-vortex states splits in $N$ subspaces of $N$ states each, which can be
further reduced to $(N-1)/2$ antisymmetric and $(N+1)/2$ symmetric states. We
shall discuss the results of numerical diagonalisation of the Hamiltonian on
the two-vortex states at the end of this section. Since the adjoint states are
made of $N$ fundamental vortices, we expect no large $N$ degeneracy in the
energy eigenvalues given $gcd(N,N)=N$. This matches the absence of large
degeneracies observed in the analytic expression of the spectrum in eq. (129).
The same is true for any other representation in the tensor product of any
multiple of $N$ fundamentals, for which no degeneracy is required by
translational invariance.

###  A.2 Statistics of the energy eigenvalues

More information on the spectrum can be obtained from the statistics of energy
eigenvalues. Indeed, it is easy to obtain a mean energy over all $n$-vortex
states. Starting from the eq. (124), and using the same regulator as in eq.
(130), the Hamiltonian on these states takes the form

$\left\langle\alpha_{1},\dots,\alpha_{n}\right|H\left|\beta_{1},\dots,\beta_{n}\right\rangle\\\
=\frac{\Lambda^{4}}{32\pi^{4}\rho}\sum^{|n^{I}|\leq\frac{1}{2}(\sqrt{N}-1)}_{{\mathbf{n}}\neq\mathbf{0}}\frac{1}{{\mathbf{m}}^{2}}\left(n\,\delta^{\alpha_{1}}_{\beta_{1}}\cdots\delta^{\alpha_{n}}_{\beta_{n}}+\sum_{i\neq
j}^{n}(J_{\mathbf{m}}^{\dagger})^{\alpha_{i}}_{\beta_{i}}(J_{\mathbf{m}})^{\alpha_{j}}_{\beta_{j}}\,\overbrace{\delta^{\alpha_{1}}_{\beta_{1}}\cdots\delta^{\alpha_{n}}_{\beta_{n}}}^{\text{no
}i,j}\right)\;.$ (178)

Since the $J_{\mathbf{m}}$ matrices are traceless only the first term
contributes to the trace of the Hamiltonian over all $n$-vortices states:

$\sum_{\alpha_{1},\dots,\alpha_{n}}\left\langle\alpha_{1},\dots,\alpha_{n}\right|H\left|\alpha_{1},\dots,\alpha_{n}\right\rangle=N^{n}\,n\,\frac{\Lambda^{4}}{32\pi^{4}\rho}\sum^{|n^{I}|\leq\frac{1}{2}(\sqrt{N}-1)}_{{\mathbf{n}}\neq\mathbf{0}}\frac{1}{{\mathbf{m}}^{2}}=N^{n}\,n\,E_{\Box}\;.$
(179)

Since there are $N^{n}$ $n$-vortex states the mean energy is $n$ times the
energy of a single vortex and is thus growing logarithmically with $N$ in the
continuum limit. In order to place bounds on the energy eigenvalues, we have
to further look into the maximal deviation from the mean, hoping this way to
identify the gapped representations. In order to implement such bounds let us
consider traces of higher powers of the Hamiltonian or, rather, of higher
powers of the deviation of the Hamiltonian from its mean $\delta
H=H-nE_{\Box}\mathbbm{1}$. The latter is simply given by the second term in
eq. (178). The quadratic dispersion of the energy eigenvalues is given by

$\overline{\delta H^{2}}\equiv
N^{-n}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{n}\\\
\beta_{1},\dots,\beta_{n}\end{subarray}}\left\langle\alpha_{1},\dots,\alpha_{n}\right|\delta
H\left|\beta_{1},\dots,\beta_{n}\right\rangle\left\langle\beta_{1},\dots,\beta_{n}\right|\delta
H\left|\alpha_{1},\dots,\alpha_{n}\right\rangle\\\
=N^{-n}\left(\frac{\Lambda^{4}}{32\pi^{4}\rho}\right)^{2}\sum_{\begin{subarray}{c}\alpha_{1},\dots,\alpha_{n}\\\
\beta_{1},\dots,\beta_{n}\end{subarray}}\sum^{\sqrt{N}}_{{\mathbf{n}},{\mathbf{m}}\neq\mathbf{0}}\frac{1}{{\mathbf{m}}^{2}\,{\mathbf{n}}^{2}}\sum_{\begin{subarray}{c}i\neq
j\\\ i^{\prime}\neq
j^{\prime}\end{subarray}}^{n}(J_{\mathbf{m}}^{\dagger})^{\alpha_{i}}_{\beta_{i}}(J_{\mathbf{m}})^{\alpha_{j}}_{\beta_{j}}\,(J_{\mathbf{n}}^{\dagger})_{\alpha_{i^{\prime}}}^{\beta_{i^{\prime}}}(J_{\mathbf{n}})_{\alpha_{j^{\prime}}}^{\beta_{j^{\prime}}}\,\overbrace{\delta^{\alpha_{1}}_{\beta_{1}}\cdots\delta^{\alpha_{n}}_{\beta_{n}}}^{\text{no
}i,j}\,\overbrace{\delta_{\alpha_{1}}^{\beta_{1}}\cdots\delta_{\alpha_{n}}^{\beta_{n}}}^{\text{no
}i^{\prime},j^{\prime}}\\\
=2n(n-1)\,\left(\frac{\Lambda^{4}}{32\pi^{4}\rho}\right)^{2}\,\sum^{\sqrt{N}}_{{\mathbf{m}}\neq\mathbf{0}}\frac{1}{{\mathbf{m}}^{4}}\;,$
(180)

where we have used a shorthand to indicate the endpoints of the sum over
${\mathbf{n}}$. After taking the traces only the terms with
$i=i^{\prime},j=j^{\prime}$ or $i=j^{\prime},j=i^{\prime}$ survive and all
give equal contributions proportional to a momentum conservation
$\delta({\mathbf{m}}+{\mathbf{n}})$ or $\delta({\mathbf{m}}-{\mathbf{n}})$
respectively. The remaining momentum sum is converging and does not depend on
$N$ in the continuum limit. The fact that energy dispersion remains finite
while the mean is growing with $N$ is not enough to prove the absence of the
gapless modes since the total number of eigenvalues is growing as $N^{n}$,
which is much faster than the mean. Eq. (180) only implies a conservative
bound on the maximal deviation of the energy eigenvalues from the mean,
$\delta H_{\text{max}}^{2}\leq N^{n}\,\overline{\delta H^{2}}$, or

$\frac{\delta H_{\text{max}}}{\overline{H}}\leq
const\cdot\sqrt{\frac{n-1}{n}}\,\frac{N^{n/2}}{\ln{N}}\;.$ (181)

Since the ratio on the right hand side is growing with $N$ we cannot constrain
the smallest energy eigenvalue at large $N$. The inequality is saturated if
just one of the $N^{n}$ eigenvalues deviates maximally from the mean while the
rest are exactly equal to the mean, which is not expected to represent the
actual spectrum. The $N$-fold degeneracy discussed in the previous section
would soften the bound slightly, replacing $N^{n/2}\rightarrow N^{(n-1)/2}$ if
the spectrum were precisely $N$ times degenerate as it is for $n=2$.

In order to obtain a useful constraint one should consider the trace of higher
powers of $\overline{\delta H}$. The maximal deviation of the eigenvalues from
the mean can be constrained as $\delta H_{\text{max}}^{k}\leq
N^{n}\,\overline{\delta H^{k}}$, and choosing $k\gtrsim n\ln N$ should be
enough to counter the growth in the dimension of the $n$-vortex subspace. For
a general $k$ we propose a very crude estimate of $\overline{\delta H^{k}}$.
Starting from eq. (178), we see that every term in the momentum sum in $\delta
H$ is in turn a sum of $n(n-1)$ terms corresponding to the choice of two
different indices out of $n$. After taking the trace over $\delta H^{k}$, we
end up with a product of at least one and at most $k$ momentum conserving
$\delta$-symbols. For instance at $k=3$ we expect momentum conservation
contributions
$\delta({\mathbf{m}}_{1}+{\mathbf{m}}_{2})\delta({\mathbf{m}}_{1}-{\mathbf{m}}_{3})\delta({\mathbf{m}}_{2}-{\mathbf{m}}_{3})$
or $\delta({\mathbf{m}}_{1}+{\mathbf{m}}_{2}+{\mathbf{m}}_{3})$. Since every
momentum label is included in at least one momentum conservation $\delta$, the
momentum sums will converge at large momenta with a result independent of $N$
in the large $N$ limit. The momentum sums can thus grow at most as $const^{k}$
with the constant independent of both $N$ and $n$. We can therefore place a
conservative upper bound on the trace of $\delta H^{k}$ as

$\overline{\delta
H^{k}}\leq\left(\frac{\Lambda^{4}}{32\pi^{4}\rho}\right)^{k}\,const^{k}\,n^{2k}\;.$
(182)

It leads to the following constraint on the maximal relative deviation of the
energy eigenvalues from the mean:

$\frac{\delta H_{\text{max}}}{\overline{H}}\leq const\cdot
n\,\frac{N^{n/k}}{\ln{N}}\xrightarrow[k\rightarrow\infty]{}const\,\frac{n}{\ln{N}}\;.$
(183)

The right hand side monotonically decreases as $k$ grows. Since the bound is
valid for any $k$ we can optimise it by taking the limit of large $k$, or
$\frac{k}{n\ln N}\rightarrow\infty$ to be precise. The latter bound implies
that the maximal relative deviation from the mean energy for the states with a
fixed number of vortices $n$ is vanishing in the continuum limit and there
could not be any gapless states unless the number of vortices grows as well,
at least as $n\gtrsim\ln N$. In particular, no states with a finite number of
vortices can be gapless in the continuum limit. Unfortunately, the bound is
not strong enough to imply that the number of elementary vortices in a gapless
state should be at least $N$.

###  A.3 Numerical diagonalisation: two-vortex and vortex-anti-vortex states

To confirm our theoretical prediction on the gapped nature of the 2-vortex
spectrum, we here numerically diagonalise the Hamiltonian at fixed $N$. For
the 2-vortex state, the $N$ fold degeneracy actually reduces the problem to a
diagonalisation of a simple $N\times N$ matrix, as opposed to $N^{2}\times
N^{2}$. In particular we can focus on states at a fixed momentum value $p_{2}$
and diagonalise the Hamiltonian in this subspace: given all subspaces with
fixed $p_{2}$ have identical spectrum, this will suffice. In practice we
considered the states $\left|\alpha,-\alpha\right\rangle$ that correspond to
$p_{2}=0$. The numerical results are displayed in Fig. 7 where we plot the
distribution of the eigenvalues for $N=91^{2}$. The eigenvalues are
concentrated around $\bar{H}=2E_{\Box}$, that is twice the energy of a single
fundamental vortex. This agrees with the result in eq. (179) for the mean
energy of a 2-vortex state. There is a sharp fall off of the eigenvalue
density at some finite value below $2E_{\Box}$, which hints at the fact that
there are no gapless states in the large $N$ limit. This matches our
expectation.

The numerical results can be well understood if we model the 2-vortex state
using a Coulomb interaction between vortices localized on a single fluid cell.
The energy of a state with one vortex located at the origin and another at the
position ${\boldsymbol{x}}$ is written as
$E=2E_{\Box}+E_{int}({\boldsymbol{x}})$. The interaction energy is associated
with the second term of eq. (178) and depends only on the relative separation
between the vortices. This translation invariance is identified with the
aforementioned $N$-fold degeneracy in the 2-vortex spectrum. The
configurations with nearby vortices will correspond to the highest energy. The
maximum energy, $E=4E_{\Box}$, is reached for superposed vortices which act as
a single vortex with twice the vorticity. At larger separation, the energy the
vortex pair decreases, while the number of allowed configurations goes up.
That explains the fall-off towards large energies of the energy eigenvalue
density observed in Fig. 7. For very large separations the picture changes as
the number of available states decreases again. Namely the points on the torus
with $\pi r\leq|{\boldsymbol{x}}|\leq\sqrt{2}\pi r$ are found only in the
corners of the square $|x^{I}|\leq\pi r$. The minimal energy corresponds to
the maximal possible vortex separation $|{\boldsymbol{x}}|=\sqrt{2}\pi r$. The
corresponding interaction energy, even if negative, does not feature a
logarithmic enhancement and thus cannot compensate for the free part
$2E_{\Box}$. In the continuum limit the gap of the 2-vortex states is
logarithmically large. More precisely the majority of the states has energy
just below $2E_{\Box}$ and corresponds to vortices separated by large
distances $|{\boldsymbol{x}}|\sim r$.

Figure 7: Energy eigenvalue distribution functions of two-vortex states for
$N=91^{2}$ obtained by exact numerical diagonalisation. Energy is measured in
the units of single vortex energy. The $N$ fold degeneracy of the spectrum is
not represented.

The same logic can be used to explain the spectrum of adjoint states of eq.
(129) represented here in Fig. 8. Indeed, an adjoint state is a product of a
fundamental and anti-fundamental state, i.e., a vortex–anti-vortex pair. Given
that the vorticity field of an anti-vortex is negative, the interaction energy
is now precisely the opposite of the 2-vortex case. The interpretation is now
different however, as the naive Coulomb model predicts an $N$ fold degeneracy
that is not present in the adjoint spectrum. In particular the adjoint state
energy of eq. (129) depends on ${\mathbf{m}}$, defined on the finer lattice,
whereas the Coulomb model for the two vortex state assumed localization over
the coarser lattice. The naive model, however, can explain the main features
of the adjoint states spectrum. The minimum energy is zero and it is attained
when the vortex and anti-vortex are on top of each other. The corresponding
configuration has vanishing vorticity and is just the vacuum state. The rest
of the $N^{2}-1$ states belong to the adjoint representation and their energy
spectrum has a gap that vanishes in the continuum limit. The low lying modes
with the quadratic dispersion relation of eq. (132) correspond to the tail of
the interaction energy distribution function. The peak is centered slightly
above $2E_{\Box}$ and corresponds to adjoint modes associated with large
momenta and an energy with a logarithmic momentum dependence.

Figure 8: Energy eigenvalue distribution function of adjoint states for
$N=25^{2}$ from eq. (129). Energy is measured in the units of single vortex
energy. The absence of $N$ fold degeneracy in the spectrum implies that it is
more expensive to compute than for the two vortex state.

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