KCL-2024-039

Confinement in Five Dimensions

B.S. Acharya
Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11,
34151, Trieste, Italy

and

Department of Physics, Kings College London, London, WC2R 2LS, UK

Abstract

Five dimensional super conformal field theories can be studied using their
geometric realisation as a limit of $M$-theory on a metrically conical Calabi-
Yau threefold. We utilise this framework to investigate the phases of such
theories that arise by varying the couplings away from the conformal point. We
demonstrate that many 5d SCFTs, including strongly coupled gauge theories,
have couplings giving rise to massive, confining vacua with confining strings
and corresponding unbroken 1-form symmetries. The simplest examples arise by
considering the parameter space of complete Ricci flat metrics on discrete
quotients of the standard conifold singularity. Varying other couplings
produces coupled 5d SCFTs interacting via massive BPS instanton particle
states.

##  1 Introduction.

A remarkable fact about string/$M$-theory concerns the physics of spacetime
singularities. If space develops certain, rather special kinds of singularity,
light, interacting degrees of freedom can arise, typically described by an
interacting conformal field theory. One might therefore like to understand
more generally which singularities in spacetime make sense physically and have
this kind of interpretation, perhaps eventually providing an interpretation of
cosmological singularities, though we are rather far from such an
understanding today.

The existence of interacting conformal field theories in five and six
dimensions is also a remarkable fact which was first discovered in the
string/$M$-theory framework. In particular, five dimensional superconformal
theories (SCFTs) were discovered by Seiberg [26]. These were then interpreted
in [23, 21, 18] as arising from $M$-theory on Calabi-Yau threefolds with
certain special kinds of singularity. Since then, this geometric framework has
been one of the main tools for studying such theories.

In this paper we will study the geometric $M$-theory description of the
conformal theories which arise at strong coupling in five dimensional super
Yang-Mills theory with ADE gauge group (and zero Chern-Simons term). Said
differently, we will study geometrically the conformal field theories which
admit deformations whose infrared limit are five dimensional super Yang-Mills
theories. The singular Calabi-Yau spaces corresponding to the conformal fixed
points turn out to be discrete quotients of the standard conifold singularity
with its standard Ricci flat cone metric [9]. This enables a simple analysis
of the physics as we vary the various coupling constants. As we will show, in
addition to the Yang-Mills coupling constant, there are two additional
couplings whose roles are privileged in the geometric description. In
particular, there is a one parameter family of complete, asymptotically
conical Calabi-Yau metrics on a smooth 6-manifold which desingularises the
space and hence this parameter deforms the conformal field theory. Along this
branch, an analysis of the spacetime metric reveals that there are no
$L^{2}$-normalisable zero modes and hence the resulting theory is massive.
However, the 6-manifold has a non-trivial fundamental group and
correspondingly the five dimensional theory has stable strings whose charges
are given by the centre of the original gauge group, which shows that the
theory has an unbroken 1-form symmetry and exhibits confinement at low
energies. The strings are $M$2-branes which wrap incontractible loops in the
extra dimensions.

The results of this paper are the five dimensional analogues of the four
dimensional theories which arise by studying $M$-theory on certain
asymptotically conical spaces of $G_{2}$-holonomy. There one was able to study
super Yang-Mills theory geometrically and prove that the theory has a mass gap
and exhibits confinement [1, 7, 8, 2, 4].

Having studied these examples, for completeness we also study the theories
obtained from general quotients of the conifold. These also deform to produce
confining vacua in five dimensions as well as new five dimensional theories
which consist of pairs of interacting SCFTs which are coupled together non-
conformally and contain massive BPS instanton particles whose mass goes to
zero at the SCFT point. The results suggest that the existence of massive,
confining deformations of five dimensional conformal theories might be a
fairly common phenomenon.

The main results of this paper are summarised in the figure. In the geometric
approach to 5d SCFTs, strong coupling fixed points are characterised by
special codimension six conical singularities in spacetime. For the
singularities corresponding to the conformal theories studied in this paper,
we will see that they can be reached in three different ways via degenerations
of three topologically distinct, smooth, complete spacetimes. Physically this
implies that there are three different low energy theories that arise by
deforming the same conformal theory, indicated by the three lines in the
figure. Since the geometric analysis is only really valid suitably far out
along the three lines, we can’t really say what happens in the centre of the
figure, apart from saying that the CFT point exists there.

In the figure, the conformal field theory is indicated by the cyan disk at the
centre. There are three distinguished couplings, $a_{1},a_{2}$ and $\mu$ with
$(a_{1},a_{2},\mu)=(0,0,0)$ being the conformal point. Switching on these
parameters individually we obtain three different theories at low energies.
The red line is the weakly coupled gauge theory. Far along the blue line we
obtain a massive, confining theory. Finally, along the green line the system
consists of a pair of conformal theories which are non-conformally coupled via
massive BPS instanton particles.

Figure 1: Three geometric couplings of the 5d CFT and their weak coupling
interpretations.

The rest of the paper begins with a review of the geometry of Calabi-Yau
conifolds, their symmetries and their parameters. We then describe in detail
the complete asymptotically conical Calabi-Yau spaces which correspond to
weakly coupled super Yang-Mills theory and describe the geometry of the strong
coupling limit. Next, we analyse the parameter space emanating from the strong
coupling limit and show that there are two additional parameters beyond the
Yang-Mills coupling that deform the theory in non-trivial ways whose
interpretation is given. In section four one of these deformations is shown to
produce massive confining theories whilst in section five the other
deformation is shown to produce the coupled conformal theories. We close the
paper with a preliminary analysis of the more general theories that arise in
this framework.

##  2 5d Superconformal Theories and Calabi-Yau Cones

We will consider $M$-theory spacetimes of the form $M=Z\times\mathbb{R}^{5}$
with, $Z$ a 6-manifold with a background metric $g(M)=g(Z)+\eta$, where $g(Z)$
is a Calabi-Yau metric on $Z$ and $\eta$ is the flat Minkowski metric. When
$Z$ is compact, the low energy effective theory iss a five-dimensional
supergravity theory on $\mathbb{R}^{5}$ with eight supercharges. This theory
has a moduli space of vacua, closely related to the moduli space of Calabi-Yau
metrics on $Z$. At special points in the moduli space, $Z$ may develop very
particular kinds of singularities and additional light degrees of freedom
arise there. We are interested in the conical singularities that $Z$ may
develop since these are expected to correspond to 5d superconformal field
theories. Near such a singularity, $Z$ looks like
$M\cong\mathbb{R}^{+}\times\Sigma$ , with $\Sigma$ a compact 5-manifold. The
metric on $M$ looks conical around this point:
$ds_{0}^{2}=dr^{2}+r^{2}g(\Sigma)$ and it has holonomy group $SU(3)$.

In order to completely decouple gravity from the degrees of freedom localised
at the conical singularity, one apparently requires a complete asymptotically
conical Calabi-Yau metric on a non-compact 6-manifold, $M$, which looks like
$ds_{0}^{2}=dr^{2}+r^{2}ds(\Sigma)^{2}$ as $r$ becomes very large.
Additionally, physically, $M$ (and its five-dimensional cross-section
$\Sigma$) are allowed to have orbifold singularities and the metric is
required to be complete and Calabi-Yau in the orbifold sense. We are thus
interested complete asymptotically conical Calabi-Yau orbifolds. This is the
precise definition of the geometric framework we will study. Note that
asymptotically conical Calabi-Yau manifolds have recently been classified
[10]. Much of the existing physics literature on the nature of the
singularities of Calabi-Yau threefolds corresponding to five dimensional
theories is rather topological and algebraic in flavour with very little
reference to the Calabi-Yau metric. Instead we will insist on the completeness
and asymptotically conical criteria which correspond to decoupling gravity and
conformality respectively. These properties of the metric will actually lead
to some of the most physically significant results in what follows.

##  3 5d Super Yang-Mills and Hyperconifolds

In Yang-Mills theories in five dimensions, $1/g^{2}_{YM}$ has mass dimension
one. Consequently, any perturbative process is suppressed by the mass scale
$1/g^{2}_{YM}$ and becomes less (or more) probable as the momenta/energies of
external particles are decreased (or increased). The theory is thus weakly
coupled at low energies. One remarkable fact is that supersymmetric Yang-Mills
theory at weak coupling arises by deforming a strongly coupled interacting
super conformal field theory. We now describe the geometric realisation of
these theories in the $M$-theory framework. In order to do that we will need
to review known facts about conifolds, their parameters and their symmetries.

###  3.1 The Conifold, its Desingularisations and Calabi-Yau metrics

We will require several facts about the conifold and its resolutions and
smoothings. Most of the material in this subsection is in the original paper
of Candelas-de la Ossa [9]. The conifold, ${\cal C}_{0}$, is a six-dimensional
space with an isolated singular point which admits a conical Calabi-Yau
metric. One can describe ${\cal C}_{0}$ topologically as $\mathbb{R}^{+}\times
S^{2}\times S^{3}$ as well as algebraically as a quadric in $\mathbb{C}^{4}$:

$P_{0}:z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=0$ (1)

The conical Calabi-Yau metric on ${\cal C}_{0}$ is given by

$g({\cal C}_{0})=dr^{2}+r^{2}\;g_{H}(S^{2}\times S^{3})$ (2)

where $g_{H}(S^{2}\times S^{3})$ is the homogeneous Einstein metric on
$(SU(2)\times SU(2))/U(1)$, with the $U(1)$ embedded symmetrically in each
$SU(2)$, both regarded as degree one line bundles over $SU(2)/U(1)$. $r$ is a
coordinate on $\mathbb{R}^{+}$ and the space has a conical singularity at
$r=0$. Since a certain $\mathbb{Z}_{2}$ subgroup acts trivially, the metric
has an $SO(4)=(SU(2)\times SU(2))/\mathbb{Z}_{2}$ group of symmetries.

${\cal C}_{0}$ has three distinct desingularisations to smooth 6-manifolds,
${\cal C}_{a_{1}}$, ${\cal C}_{a_{2}}$ and ${\cal{C}}_{\mu}$, labelled by
parameters $(a_{1},a_{2},\mu)\in(\mathbb{R}^{+},\mathbb{R}^{+},\mathbb{C})$.
These are each obtained by removing a neighbourhood of the singular point in
${\cal C}_{0}$ and replacing it with a suitable compact set whose boundary
matches that of ${\cal C}_{0}\setminus\\{0\\}$. Hence the boundaries of these
three manifolds are all topologically $S^{2}\times S^{3}$. The parameters
$(a_{1},a_{2},\mu)\in(\mathbb{R}^{+},\mathbb{R}^{+},\mathbb{C})$ have
magnitudes which fix the size of the compact region and are free parameters.
We will need to describe these three manifolds in more detail, since these
three parameters will eventually be the coupling constants of the 5d theories
we discuss.

  * •

$\mu\neq 0$: ${\cal{C}}_{\mu}$ is simply obtained by deforming the algebraic
equation to

$P_{\mu}:z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=\mu$ (3)

This explicitly shows that ${\cal{C}}_{\mu}=T^{*}(S^{3})$, the cotangent
bundle of the sphere and that the manifold contains a compact $S^{3}$ of
radius $|\mu|^{1/2}$ at the centre.

  * •

$a_{1}\neq 0$: To describe ${\cal C}_{a_{1}}$, we rewrite the quadric as
$\det(W)=0$, with $W$ the matrix

$W=\begin{pmatrix}X&U\\\
V&Y\end{pmatrix}={1\over\sqrt{2}}\begin{pmatrix}z_{1}+iz_{2}&-z_{3}+iz_{4}\\\
z_{3}+iz_{4}&z_{1}-iz_{2}\end{pmatrix}$ (4)

and one replaces it with the simultaneous solutions of $\det(W)=0$ and

$W=\begin{pmatrix}X&U\\\ V&Y\end{pmatrix}\begin{pmatrix}\lambda_{1}\\\
\lambda_{2}\end{pmatrix}$ (5)

where $(\lambda_{1},\lambda_{2})$ are coordinates on $\mathbb{CP}^{1}$. This
determines a point in $\mathbb{CP}^{1}$ for every non-zero point on ${\cal
C}_{0}$, whilst the origin is replaced by the entire $\mathbb{CP}^{1}$. ${\cal
C}_{a_{1}}$ is in fact a rank two complex vector bundle over $\mathbb{CP}^{1}$
which is the sum of two degree minus one line bundles, ${\cal
C}_{a_{1}}=({\cal O}(-1)\oplus{\cal O}(-1))|_{\mathbb{CP}^{1}}$. The real
parameter $a_{1}$ gives the size of the $\mathbb{CP}^{1}$ at the origin and
will be important when we discuss the physics.

Since it is a vector bundle over $\mathbb{CP}^{1}$, ${\cal C}_{a_{1}}$ can be
covered by two coordinate patches, $\lambda_{1}\neq 0$, $\lambda_{2}\neq 0$.
Explicitly, these are

${\cal C}_{a_{1}}=U_{\lambda_{1}\neq 0}\cup U_{\lambda_{2}\neq
0}=\\{X,V,U/X=Y/V\\}\cup\\{Y,U,X/U=V/Y\\}$ (6)

  * •

$a_{2}\neq 0$: Similarly, to describe ${\cal C}_{a_{2}}$,

we again consider $\det(W)=0$ but blow up a different $\mathbb{CP}^{1}$:

$W=\begin{pmatrix}\lambda_{1}&\lambda_{2}\end{pmatrix}\begin{pmatrix}X&U\\\
V&Y\end{pmatrix}$ (7)

where $(\lambda_{1},\lambda_{2})$ are coordinates on another
$\mathbb{CP}^{1}$. This is a distinct resolution but with the same topology
${\cal C}_{a_{2}}=({\cal O}(-1)\oplus{\cal O}(-1))|_{\mathbb{CP}^{1}}$.
Coordinate charts can be given by

${\cal C}_{a_{2}}=U_{\lambda_{1}\neq 0}\cup U_{\lambda_{2}\neq
0}=\\{X,U,V/X=Y/U\\}\cup\\{Y,V,X/V=U/Y\\}$ (8)

###  3.2 Asymptotically Conical Calabi-Yau metrics:

Candelas and de la Ossa [9] showed that all three manifolds admit smooth,
complete Calabi-Yau metrics, $g_{a_{1}},g_{a_{2}}$ and $g_{\mu}$ all of which
asymptote to the conical metric at infinity. These metrics also have the full
$SO(4)$ symmetry of the conical model. Furthermore, the parameters
$a_{1},a_{2},\mu$ control the sizes of the respective spheres at the origin,
which turn out to be volume minimising supersymmetric (i.e. calibrated)
cycles.

Finally we can come to the physical theories of interest in this paper. They
all arise by considering discrete quotients of these three spaces and the
conifold by subgroups of the $SO(4)$ isometry group. Such quotients have
previously been considered by Gubser et al. [19] and Davies [15, 16] and the
results of this paper provide a physical interpretation of some of the results
and ideas of those papers.

###  3.3 Super Yang-Mills from $({{\cal O}(-1)\oplus{\cal
O}(-1)\over{\Gamma_{\mathbb{ADE}}}})|_{{\mathbb{C}P}^{1}}$

As a 6-manifold, ${\cal C}_{a_{1}}=\mathbb{R}^{4}\times S^{2}$ and the
quotients we will initially consider replace each $\mathbb{R}^{4}$ fibre by
the orbifold ${\mathbb{R}^{4}}/{\Gamma_{\mathbb{ADE}}}$, where
$\small{\Gamma}_{\mathbb{ADE}}$ is a finite subgroup of $SU(2)$ which fixes
the $0\times S^{2}$ at the origin in $\mathbb{R}^{4}\times S^{2}$. Since the
Candelas-de la Ossa confold metric $g_{a_{1}}$ is
$\small{\Gamma}_{\mathbb{ADE}}$-invariant, ${\cal
C}_{a_{1}}/\small{\Gamma}_{\mathbb{ADE}}=({{\cal O}(-1)\oplus{\cal
O}(-1)\over{\Gamma_{\mathbb{ADE}}}})|_{{\mathbb{C}P}^{1}}={\mathbb{R}^{4}}/\small{{\Gamma}}_{\mathbb{ADE}}\times
S^{2}$ inherits a complete asymptotically conical Calabi-Yau orbifold metric,
asymptotic to a metric cone over $(S^{2}\times S^{3})/\Gamma_{\mathbb{ADE}}$.

We can see this more explicitly by calculating the action of
$\small{\Gamma}_{\mathbb{ADE}}$ on ${\cal C}_{a_{1}}$ using the previous
subsection. The original coordinates $z_{i}$ transform in the fundamental
representation of $SO(4)$. The $\small{\Gamma}_{\mathbb{ADE}}$ discrete
subgroups of interest are defined as the finite subgroups of $SU(2)_{L}\subset
SO(4)\equiv(SU(2)_{L}\times SU(2)_{R})/\mathbb{Z}_{2})$. $SU(2)_{L}$ acts on
the left on the pair $(X,V)$ in the fundamental representation. Similarly, it
acts on $(Y,-U)$ in the conjugate representation. Because of this, $SU(2)_{L}$
acts trivially on the $\mathbb{CP}^{1}$ coordinates $X/U$ and $Y/V$. To show
this we parametrise $SU(2)_{L}$ matrices $g_{L}$ as

$g_{L}=\begin{pmatrix}a&b\\\ -b^{*}&a^{*}\end{pmatrix}$ (9)

with $aa^{*}+bb^{*}=1$. Then

$X/U\longrightarrow(aX+bV)/(bY+aU)=(aX+bXY/U)/(bY+aU)=X/U$ (10)

This proves that ${\cal C}_{a_{1}}/\small{\Gamma}_{\mathbb{ADE}}=({{\cal
O}(-1)\oplus{\cal
O}(-1)\over\small{\Gamma}_{\mathbb{ADE}}})|_{{\mathbb{C}P}^{1}}={\mathbb{R}}^{4}/\small{\Gamma}_{\mathbb{ADE}}\times
S^{2}$. For simplicity, consider the examples
$\Gamma_{\mathbb{A}}=\mathbb{Z}_{N}$. The action on the coordinates is then
given by

$\mathbb{Z}_{N}:(X,Y,U,V)\longrightarrow(\bar{\alpha}X,\alpha
Y,\bar{\alpha}U,{\alpha}V)$ (11)

where $\alpha^{N}=1$ is a primitive $N$-th root of unity.

Hence, under the $\mathbb{Z}_{N}$ action the local coordinates in the two
patches transform with weights which are $(-1,1,0)$ and $(1,-1,0)$
respectively, which shows that the $\mathbb{CP}^{1}$ at the origin i.e. the
zero-section is fixed by $\mathbb{Z}_{N}$. The discussion is similar for all
$\Gamma_{\mathbb{ADE}}\subset SU(2)_{L}$.

At low energies, $M$-theory on ${\cal C}_{a_{1}}/\Gamma_{\mathbb{ADE}}=({{\cal
O}(-1)\oplus{\cal O}(-1)\over{\Gamma_{\mathbb{ADE}}}})|_{{\mathbb{C}P}^{1}}$
is described by five dimensional super Yang-Mills theory with ADE gauge group:
the ADE singularity is along a copy of $\mathbb{CP}^{1}\times\mathbb{R}^{5}$
and supports a 7d ADE super Yang-Mills theory. At low energies, integrating
over the $\mathbb{CP}^{1}$ we obtain the fields of the pure 5d super Yang-
Mills theory. The gauge coupling of the 5d theory is given, in 11d units, by
${1\over g^{2}_{YM}}\propto 4\pi a_{1}^{2}$, the volume of the minimal
$\mathbb{CP}^{1}$ in the Candelas-de la Ossa metric. Thus, the theory is
weakly coupled at large $a_{1}$.

Massive BPS particles and monopole strings.

The super Yang-Mills theory has a Coulomb branch of vacua along which the
adjoint scalars take non-zero vacuum expectation values and break the gauge
symmetry to the maximal torus $U(1)^{rk(ADE)}$. This produces massive vector
multiplets containing $W$-bosons. These are in fact BPS states. In $M$-theory
this can be seen by realising that along the Coulomb branch the
$\mathbb{CP}^{1}$ of ADE singularities gets resolved by a configuration of
rk(ADE) copies of $S^{2}\times\mathbb{CP}^{1}$, which intersect according to
the ADE Dynkin diagram. The rk(ADE) independent Coulomb vevs, $v_{i}$ are the
differences of the volumes of these 2-spheres111For small $v_{i}$ there exists
a smooth, complete asymptotically conical Calabi-Yau metric and the 2-sphere
volumes are minimal wrt to this metric. In particular, each of these 2-spheres
minimises volume within its homology class. The $W$-boson particles are
identified as $M2$-branes which wrap these various spheres and are BPS
precisely because the spheres are volume minimising within homology - they
give the smallest mass particles for a given charge. $m_{W}=q|v|$ is the
schematic formula for the mass-to-charge ratio.

Similarly, $M2$-branes which wrap the original $\mathbb{CP}^{1}$ at the origin
are also BPS particles. These are the so-called instantonic particles of the
5d theory since they are charged under the $U(1)$ global symmetry associated
to the conserved current, $j_{I}=\star Tr(F\wedge F)$ [26] and give rise to 4d
instantons upon compactification to four dimensions. The instanton particle
masses at generic points on the Coulomb branch is $m_{I}=1/g^{2}_{YM}$ ,
though they can fractionate at the origin. See [22, 27] for other aspects of
these instantons. In $M$-theory, along the Coulomb branch where the Calabi-Yau
threefold is smooth, the current $j_{I}$ arises from the $C$-field equation of
motion integrated over the Calabi-Yau.

Finally, there are BPS monopole strings in the spectrum. These arise in
$M$-theory as $M$5-branes wrapping the 4-manifolds of the form
$S^{2}\times\mathbb{CP}^{1}$. The monopole tensions are given by an exact
formula which is schematically $T_{m}=|v|/g^{2}_{YM}$.

The conformal theory of interest arises when all of these BPS particles and
strings become massless, namely when all the $v_{i}$ and $1/g^{2}_{YM}$ go to
zero. The fact that electric, magnetic and instanton particle states are
becoming massless simultaneous seems to be a ubiquitous signal for the
appearance of a conformal fixed point at strong coupling. In this limit,
${\cal C}_{a_{1}}/\small{\Gamma}_{\mathbb{ADE}}$ formally degenerates to the
hyperconifold ${\cal C}_{0}/\small{\Gamma}_{\mathbb{ADE}}$ with the conifold
metric $g_{0}$. The key statement is that $M$-theory on the hyperconifold
${\cal C}_{0}/\small{\Gamma}_{\mathbb{ADE}}$ with the conifold metric $g_{0}$
gives rise to a 5d SCFT which is the 5d ADE super Yang-Mills theory at strong
coupling.

From this point of view, the weakly coupled super Yang-Mills theory is
obtained from the 5d SCFT by switching on the coupling $1/g^{2}_{YM}$ .
Geometrically it corresponds to desingularising ${\cal
C}_{0}/\small{\Gamma}_{\mathbb{ADE}}$ to ${\cal
C}_{a_{1}}/\small{\Gamma}_{\mathbb{ADE}}$ with the Candelas-de la Ossa metric,
$g_{a_{1}}$. This makes sense because $g_{a_{1}}$ is both complete and
asymptotic to $g_{0}$ at infinity. However, as we have reviewed above, there
are two other complete metrics $g_{a_{2}}$ and $g_{\mu}$ asymptotic to $g_{0}$
at infinity and these also admit the $\Gamma_{\mathbb{ADE}}$ symmetry. Hence
the parameters $a_{2}$ and $\mu$ also correspond to couplings in the 5d theory
and one can ask about the physics of these deformations of the 5d conformal
theories.

##  4 Confinement in Five Dimensions

We now consider the theory for $\mu\neq 0$, which leads to the main results of
this paper.

In this case we obtain $M$-theory on a smooth 6-manifold which is
${{\cal{C}}_{\mu}}/\small{\Gamma}_{\mathbb{ADE}}=T^{*}({S^{3}}/\small{\Gamma}_{\mathbb{ADE}})$,
again with a complete, asymptotically conical Calabi-Yau metric, $g_{\mu}$.
This is clear from the description of ${\cal C}_{\mu}$ as a quadric 3:
$\Gamma_{\mathbb{ADE}}$ acts freely on the sphere when $\mu\neq 0$. What is
the physics of $M$-theory on ${{\cal{C}}_{\mu}}/\small{\Gamma}_{\mathbb{ADE}}$
with this metric? The background spacetime is completely smooth so is
amenable, at suitably large $\mu$, to a field theory analysis.

In analysing the low energy spectrum around a fixed, smooth Calabi-Yau
background, massless states arise from harmonic 2-forms and 3-forms. However,
since ${{\cal{C}}_{\mu}}/\small{\Gamma}_{\mathbb{ADE}}$ is non-compact, we are
actually interested in the $L^{2}$-normalisable harmonic forms with respect to
the Candelas-de la Ossa metric $g_{\mu}$ , since it is these which give rise
to massless fields in five dimensions with finite kinetic terms. In actual
fact, there are none of these, as can be seen from the analysis of [25] and
[20] which analyses the $L^{2}$-cohomology of asymptotically conical spaces in
general.

Theorem ([25], [20]): If $(M,g)$ is an asymptotically conical Riemannian
n-manifold, the $L^{2}$-cohomology of $(M,g)$ is given by:

$\displaystyle L^{2}H^{k}(M)\;\;\;\;\;$ $\displaystyle\cong$
$\displaystyle\;\;\;\;\;H^{k}(M,\partial M),\;\;k<n/2$ (12) $\displaystyle
L^{2}H^{n/2}(M)\;\;\;\;$ $\displaystyle\cong$
$\displaystyle\;\;\;\;\;Im(H^{n/2}(M,\partial M)\rightarrow H^{n/2}(M))$ (13)
$\displaystyle L^{2}H^{k}(M)\;\;\;\;$ $\displaystyle\cong$
$\displaystyle\;\;\;\;\;\;H^{k}(M),\;\;k>n/2$ (14)

In our case, $M$ is a quotient of $T^{*}(S^{3})$ and $\partial M$ a quotient
of $S^{2}\times S^{3}$ so its real cohomology comes from that of
$T^{*}(S^{3})$. To compute the cohomology groups above one can use the long
exact cohomology sequence for the pair $(M,\partial M)$:

$\dots.H^{p}(M,\partial M)\rightarrow H^{p}(M)\rightarrow H^{p}(\partial
M)\rightarrow H^{p+1}(M,\partial M)\rightarrow\dots$ (15)

In our example, $(M,\partial M)=(T^{*}(S^{3}),S^{2}\times S^{3})$ and
explicitly working out the exact sequences we learn that the entire
$L^{2}$-cohomology is empty. For instance, to compute the 3rd
$L^{2}$-cohomology we consider the piece of the above sequence with $p=3$.
Since, in the case at hand, the map from $H^{3}(T^{*}(S^{3}))$ to
$H^{3}(S^{2}\times S^{3})$ is actually an isomorphism and the kernel is
trivial, which shows that the image of $H^{3}(M,\partial M)$ in $H^{3}(M)$ is
trivial in this case and hence there are no $L^{2}$-normalisable harmonic
3-forms on $(C_{\mu},g_{\mu})$.

Intuitively, the Poincare dual of the compact 3-cycle, $S^{3}$ in $M$ is a
compactly supported harmonic 3-form, $\alpha$. The image of $\alpha$ in the
$L^{2}$-cohomology is actually proportional to the self-intersection number of
the zero-section $S^{3}$ in ${\cal C}_{\mu}$. However, this is just the Euler
number of $S^{3}$, which is zero. In fact, explicit calculation shows that
$\alpha$ has a norm which diverges logarithmically [13].

Physically this means that the theory for non-zero $\mu$ has a mass gap.
However, the theory is still interesting as it has an unbroken 1-form symmetry
(in all cases except $\Gamma_{E_{8}}$). This is due to the fact that
$M$2-branes wrapped on non-trivial 1-cycles appear as stable strings in five
dimensions and these have charges classified by the first homology group. This
group, as was first shown in [2], is
$H_{1}({{\cal{C}}_{\mu}}/\small{\Gamma}_{\mathbb{ADE}})\cong Z(ADE)$ the
centre of the compact, simply connected ADE group. Therefore these five
dimensional theories are confining at low energies. The $E_{8}$ theory does
not confine as $E_{8}$ has trivial centre.

Notice that all of the BPS particles and monopole strings from the Yang-Mills
theory have disappeared when $\mu\neq 0$: there are no compact, incontractible
even-dimensional cycles in $C_{\mu}/\Gamma_{\mathbb{ADE}}$ . This is
intuitively consistent with the confinement of the original electric degrees
of freedom. Furthermore, the instanton symmetry has been explicitly broken
suggesting that the deformation to the confining phase involves the instanton
operators explicitly and is reminiscent of confinement in three dimensional
QED where monopole operators induce the confining phase [24]. In [11], the
authors have analysed all supersymmetric operator deformations of these 5d
SCFTs and it would be very interesting to connect the results here with those
as well as with the results on the Higgs branch of the SCFTs in [5, 12].

There are however two classes of BPS objects in this theory: instantons and
membranes. The instantons arise from Euclidean $M$2-branes wrapping the
supersymmetric $S^{3}/\small{\Gamma_{\mathbb{ADE}}}$ at the centre. These will
certainly give rise to effects of order $\exp(-c|\mu|^{3})$ and hence might
also be consistent with a small dynamically generated scale. For large $\mu$,
which is where our analysis is valid, we can neglect the effects of these
instantons. Their presence, however, suggests that the formal $\mu\rightarrow
0$ limit could have an interesting structure since in principle the geometry
could be deformed from the classical one by the instanton effects.

BPS membranes arise from $M$5-branes which wrap the minimal volume
$S^{3}/{{\Gamma}_{\mathbb{ADE}}}$ in the centre of the manifold. The world-
volume theory of these branes is a three dimensional Yang-Mills Chern-Simons
theory with four supercharges. In fact it is the twice supersymmetric version
of the theory first discussed in [4] as governing the dynamics of domain walls
in four dimensional pure supersymmetric Yang-Mills theories. These theories
have a non-zero Witten index and, hence, there are non-trivial bound states
formed by these membranes. This suggests that there is a non-trivial 2-form
generalised symmetry in these theories and aspects of this could be analysed
using the techniques of [6, 14, 17].

##  5 $M$-theory on ${\cal C}_{a_{2}}/\small{\Gamma}_{\mathbb{ADE}}$: Coupled
5d SCFTs

Next we consider the theory obtained by deforming the SCFT by $a_{2}$. ${\cal
C}_{a_{2}}$ can also be covered by two coordinate charts:

${\cal C}_{a_{2}}=U_{\lambda_{1}\neq 0}\cup U_{\lambda_{2}\neq
0}=\\{X,U,V/X=Y/U\\}\cup\\{Y,V,X/V=U/Y\\}$ (16)

Hence, under the $\mathbb{Z}_{N}$ symmetry the two coordinate charts have
weights $(-1,-1,2)$ and $(1,1,-2)$. This is very different to the action on
${\cal C}_{a_{1}}$. Unlike in section three, here the symmetry acts non-
trivially on the $\mathbb{CP}^{1}$ zero section. Therefore, if $N$ is prime,
the singularities of ${\cal C}_{a_{2}}/\mathbb{Z}_{N}$ consist of two isolated
fixed points, located at the north and south poles of the $\mathbb{CP}^{1}$
and each modelled on the singularity at the origin in
$\mathbb{C}^{3}/\mathbb{Z}_{N}$ given in coordinates as

$(z_{1},z_{2},z_{3})\rightarrow(\alpha z_{1},\alpha z_{2},\alpha^{-2}z_{3})$
(17)

Now, $M$-theory on $\mathbb{C}^{3}/\mathbb{Z}_{N}$ itself produces a 5d SCFT
located at the origin e.g. $N=3$ is the mysterious Seiberg $E_{0}$ theory
[26]. Therefore, in $M$-theory on ${\cal C}_{a_{2}}/\mathbb{Z}_{N}$ one has a
pair of $\mathbb{C}^{3}/\mathbb{Z}_{N}$ SCFTs which are coupled via the non-
zero size $\mathbb{CP}^{1}$ of radius $a_{2}$. Notice also that if $N$ is
even, then the order two elements of $\mathbb{Z}_{N}$ fix the
$\mathbb{CP}^{1}$ and the low energy theory also includes $SU(2)$ gauge fields
with $a_{2}\sim 1/g_{YM}$. This demonstrates that $a_{2}$ is a coupling in
these models. The theory also contains instanton particles which wrap (the
$\mathbb{Z}_{N}$-quotient of) the $\mathbb{CP}^{1}$ and couple to the two 5d
theories. Since the two coupled SCFTs are isomorphic there is a one-to-one
correspondence between their operators and the geometry strongly suggests that
the coupling between them when $a_{2}\neq 0$ respects the symmetry between
them i.e. only diagonal couplings are present.

Notice that in the $N=2$ case, the $\mathbb{Z}_{2}$ fixes the
$\mathbb{CP}^{1}$ and the theory is isomorphic to the $E_{1}$ theory.

##  6 More general quotients and theories

Here we briefly consider the theories obtained by considering more general
quotients of the conifold, beyond $\Gamma_{\mathbb{ADE}}$. Therefore we wish
to consider generic finite subgroups $\Gamma\subset SO(4)$. A similar analysis
was given in [3] for four dimensional $N=1$ theories arising from
$G_{2}$-holonomy orbifolds.

We begin by noting that, essentially following the discussion of the previous
subsection, a generic element of $\Gamma$ will act on ${\cal C}_{a_{1}}$ and
${\cal C}_{a_{2}}$ with isolated fixed points because it will act on the base
$\mathbb{CP}^{1}$ fixing two points. Each of these isolated fixed points
itself supports a 5d SCFT. Hence, since $\Gamma$ will contain several such
elements, there will be several pairs of such isolated fixed points on the
$\mathbb{CP}^{1}$. Hence, the 5d theory will contain several pairs of 5d SCFTs
all coupled together via the BPS instanton particles corresponding to
$M$2-branes wrapping the $\mathbb{CP}^{1}/\Gamma$. Additionally there will
also be non-Abelian 5d gauge fields if $\Gamma$ contains elements which fix
the $\mathbb{CP}^{1}$. Finally, if $\Gamma$ also has elements which have
codimension four fixed points and act non-trivially on the base
$\mathbb{CP}^{1}$, then there will be pairs of non-Abelian flavour symmetries
present also.

How do these symmetries act on ${\cal C}_{\mu}=T^{*}(S^{3})$ ? By extending
slightly the analysis of section three and using the the fact that the $z_{i}$
coordinates transform in the fundamental representation of $SO(4)$ one can see
that whenever ${\cal C}_{a_{i}}/\Gamma$ has a pair of flavour symmetries,
${\cal C}_{\mu}/\Gamma$ has a codimension four singularity along a single copy
of $T^{*}(S^{1})\subset{\cal C}_{\mu}$. This indicates that the pair of global
symmetries in the theories obtained from ${\cal C}_{a_{i}}/\Gamma$ get broken
to a diagonal subgroup by appropriate vacuum expectation values to give the
theory on ${\cal C}_{\mu}/\Gamma$.

Furthermore, the elements of $\Gamma$ which have isolated fixed points on
${\cal C}_{a_{i}}$ tend to act freely on ${\cal C}_{\mu}=T^{*}(S^{3})$. Since
they act freely they will produce incontractible cycles which give rise to
confining strings. This gives a correspondence between the 5d CFT sectors
arising from $M$-theory on ${\cal C}_{a_{i}}/\Gamma$ and confining strings in
$M$-theory on ${\cal C}_{\mu}/\Gamma$. This also leads to an interesting
observation which we will finish with.

Consider the $\mathbb{Z}_{p}$ action on the conifold coordinates given by:

$(X,V,U,Y)\;\;\longrightarrow\;\;(\alpha
X,\alpha^{q}V,\bar{\alpha}^{q}U,\bar{\alpha}Y)$ (18)

where $\alpha^{p}=1$ and $q\neq 0\in\mathbb{Z}$. We will restrict to $p$ and
$q$ being relatively prime for simplicity.

Then, ${\cal C}_{\mu}/{\mathbb{Z}_{p}}=T^{*}(L(p,q))$, with $L(p,q)$ a general
three dimensional Lens space. In particular, since the first homology group of
$L(p,q)=\mathbb{Z}_{p}$ we have a massive confining theory with strings whose
charges are quantised mod $p$. When $|q|=1$ this is exactly the model we
obtained by deforming the 5d SCFT which arises from $SU(p)$ super Yang-Mills
at strong coupling. When $|q|\neq 1$ $M$-theory on ${\cal
C}_{a_{i}}/\mathbb{Z}_{p}$ gives a pair of 5d SCFTs coupled to each other via
a massive instanton operator. On the other hand $M$-theory on $T^{*}(L(p,q))$
looks like a massive confining theory for all non-zero $p$ and $q$. It would
be interesting to distinguish physically $M$-theory on $T^{*}(L(p,1))$ from
$M$-theory on $T^{*}(L(p,q))$ with $|q|\neq 1$.

Acknowledgements.

We would like to thank A. Antinucci, D. Baldwin, L. Foscolo, K. Intriligator,
J. Lotay and G. Rizi for discussions. In particular, we especially acknowledge
discussions with L. di Pietro, whose questions led to the main insight of this
paper. The work of BSA is supported by a grant from the Simons Foundation
(#488569, Bobby Acharya)

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the experimenter can not concentrate the power of the probe modes to one that
minimizes the error probability. Further, we have the following conditions:

$\displaystyle\sum_{j=0}^{m-1}\sqrt{\omega_{j}}\left(\sqrt{\eta_{j}^{(0)}}-\sqrt{\eta_{j}^{(1)}}\right)=\mathcal{O}(0),$
(149)

we will have a scaling advantage using the CtoD quantum strategy.

##  Appendix F Discussion on average phase sensing

Let’s reduce our discussion to single-parameter estimation. As a concrete
example, we consider the sensing of the average phase:

$\displaystyle\overline{\theta}=\sum_{j=0}^{m-1}\sqrt{\omega_{j}\eta_{j}}\,\theta_{j}.$
(150)

Next, there will be two distinct scenarios: (i) The differences between each
pair of the $m$ phases are known. (ii) There still exist $m-1$ independent and
unknown phases after estimation of the average phase. In the subsequent
subsections, we shall examine these two scenarios separately.

###  F.1 Degenerative case

If differences between each pair of phases are known, we may apply the
following transformation [58] of to the QFIM achievable by the QI network:

$\displaystyle F^{\rm qi}_{\overline{\theta}}$
$\displaystyle=\left(\begin{matrix}\frac{\partial\theta_{0}}{\partial\overline{\theta}},\cdots,\frac{\partial\theta_{m-1}}{\partial\overline{\theta}}\end{matrix}\right)\overline{F}\left(\begin{matrix}\frac{\partial\theta_{0}}{\partial\overline{\theta}}\\\
\vdots\\\
\frac{\partial\theta_{m-1}}{\partial\overline{\theta}}\end{matrix}\right)$
(151) $\displaystyle=\frac{4m\nu N_{\rm S}(N_{\rm S}+1)}{(N_{\rm
B}^{\prime}+1)(2N_{\rm S}+1)}+\mathcal{O}\left(\frac{m^{2}\nu N_{\rm
S}^{2}}{m_{\rm re}N_{\rm B}^{2}}\right).$ (152)

where $\overline{F}$ is the QFIM obtained by the CtoD, as shown in Eq. (D.1),
$m_{\rm re}^{-1}$ is defined in Eq. (2) in the main text and is given by the
premise $\omega_{j}\sim m_{\rm re}^{-1}$. Then, its RMSE is:

$\displaystyle\epsilon^{\rm qi}_{\overline{\theta}}=\sqrt{\frac{(N_{\rm
B}^{\prime}+1)(2N_{\rm S}+1)}{4m\nu N_{\rm S}(N_{\rm
S}+1)}}+\mathcal{O}\left(\sqrt{\frac{mN_{\rm S}}{m_{\rm re}^{2}\nu N_{\rm
B}}}\right).$ (153)

On the other hand, given by the convexity of Fisher information [31], the
output state from classical illumination network in Eq. (25) will achieve the
maximal value of quantum Fisher information (QFI) when the probe is pure. By
using the results of QFI for Gaussian states [53, 54, 55, 31], we will have
the maximal QFI for classical case:

$\displaystyle\mathcal{F}^{\rm ci}_{\overline{\theta}}$
$\displaystyle=\max_{\\{\beta_{j};|\beta_{j}|=N_{\rm S}\\}}\frac{4}{2N_{\rm
B}+1}\left|\frac{\partial\sum_{j=0}^{m-1}\sqrt{\omega_{j}\eta_{j}}e^{-i\theta_{j}}\beta_{j}}{\partial\overline{\theta}}\right|^{2},$
(154) $\displaystyle=\frac{4m^{2}N_{\rm S}}{2N_{\rm B}+1}.$ (155)

The corresponding root mean-square-error (RMSE) achievable by the classical
network is:

$\displaystyle\epsilon_{\overline{\theta}}^{\rm ci}$
$\displaystyle=\sqrt{\frac{2N_{\rm B}+1}{4m^{2}\nu N_{\rm S}}}.$ (156)

As shown in Eqs. (153) and (156), there is a disadvantage of QI network
average parameter sensing in the scaling of transmitter number $m$.

###  F.2 Non-degenerative case

Consider the case where there are $m$ unknown independent parameters
$\\{\overline{\theta},\theta_{1},\cdots,\theta_{m-1}\\}$, where
$\overline{\theta}$ is defined in Eq. (150). The partial Fisher information of
$\overline{\theta}$ can be obtain by computing the Schur’s complement [58] of
the initial FIM, i.e. that in Eqs. (D.1). The presence of $m-1$ nuisance
parameters in both quantum and classical scenarios leads to a reduction in the
value of partial Fisher information. Given the current focus on multi-
parameter sensing, we will leave this research question for future works.


We saw that $\mathcal{O}\langle n\rangle$ is a Zhang algebra in the category
${\rm Alg}^{\star}$ of involutive algebras. Let $\mathcal{D}$ be the category
with the same objects class as the category ${\rm Alg}^{\star}$ but with
morphisms class between two objects given by the set of all complex linear
involutive positive maps, if $A,B\in{\mathrm{Obj}}(\mathcal{C})$:

${\rm Hom}_{\mathcal{D}}(A,B)=\\{\tau\in{\rm
Hom}_{{\mathrm{Vect}}_{\mathbb{C}}}(A,B):\tau(aa^{\star})\geq
0,\leavevmode\nobreak\ \tau(a^{\star})=\tau(a)^{\star},a\in A\\}.$

The category $\mathcal{B}$ is the usual category of (complex involutive)
probability spaces, ${\rm Prob}$. Let $V$ be a unitary matrix in
$\mathcal{M}_{n}(\mathbb{C})\otimes\mathcal{A}$ and set, for all times $t\geq
0$, ${\textrm{{{U}}}^{\langle n\rangle}}^{V}(t)=V{\textrm{{{U}}}^{\langle
n\rangle}}(t)V^{-1}$. If $u$ is an element of the Voiculescu dual group
$\mathcal{O}\langle n\rangle$, ${\textrm{{{U}}}^{\langle n\rangle}}(t)(u)$ and
${\textrm{{{U}}}^{\langle n\rangle}}^{V}(t)(u)$ stand for the values on $u$ of
the morphisms on $\mathcal{O}\langle n\rangle$ induced by the two matrices
${\textrm{{{U}}}^{\langle n\rangle}}(t)$ and ${\textrm{{{U}}}^{\langle
n\rangle}}^{V}(t)$. We assume further that the involutive subalgebra of
$\mathcal{A}$ generated by the entries of $V$ is free from the algebra
generated by the entries of ${\textrm{{{U}}}^{\langle n\rangle}}(t)$ for all
times $t\geq 0$.

######  Lemma 27.

Let $t\geq 0$ a time, $u\in\mathcal{O}\langle n\rangle$, $V$ an unitary
element of $\mathcal{M}_{n}(\mathbb{C})\otimes\mathcal{A}$, with the notations
introduced so far,

(14) $\tau\circ{\textrm{{{U}}}^{\langle
n\rangle}}^{V}(t)(u)=\tau\circ{\textrm{{{U}}}^{\langle
n\rangle}}(t)(u),\leavevmode\nobreak\ t\geq,u\in\mathcal{O}\langle n\rangle.$

###### Proof.

The process ${\sf U}^{{V}}$ is solution of the following free stochastic
differential system, with obvious notation,

$\text{d}{\textrm{{{U}}}^{\langle n\rangle}}^{V}(t)=\frac{\sf
i}{\sqrt{n}}\text{d}\textrm{{{W}}}^{V}_{t}{\textrm{{{U}}}^{\langle
n\rangle}}^{V}(t)-\frac{1}{2}{\textrm{{{U}}}^{\langle
n\rangle}}^{V}(t),\leavevmode\nobreak\ t\geq 0.$

Hence, gauge invariance of the driving noise W implies (14), property that is
proved now. For an integer $m\geq 1$, we denote by $\textrm{{{NC}}}_{2m}$ the
set of matchings of the intervale $\llbracket 1,2m\rrbracket$. A matching
$\textrm{{{m}}}\in\textrm{{{NC}}}^{(2)}_{2m}$ is alternatively seen as a non-
crossing partition or as an involution of $\llbracket 1,2m\rrbracket$
verifying:

$\text{for all }k<l\in\llbracket 1,2m\rrbracket,\leavevmode\nobreak\
\textrm{{{m}}}(k)<\textrm{{{m}}}(l).$

We compute the cumulants of the family
$\\{\textrm{{{W}}}^{V}(\alpha,\beta),\leavevmode\nobreak\
1\leq\alpha,\beta\leq n\\}$ and prove that:

(15)
$\begin{split}&k_{2m+1}\left(\textrm{{{W}}}^{V}(\alpha_{1},\beta_{1}),\ldots,\textrm{{{W}}}^{V}(\alpha_{2m+1},\beta_{\beta_{2m+1}})\right)=0,\leavevmode\nobreak\
m\geq 1,\text{ for all }\alpha,\beta\in\\{1,\ldots,n\\}^{2m+1},\\\
&k_{2m}\left(\textrm{{{W}}}^{V}(\alpha_{1},\beta_{1}),\ldots,\textrm{{{W}}}^{V}(\alpha_{2m+1},\beta_{\beta_{2m}})\right)=k_{2m}\left(\textrm{{{W}}}(\alpha_{1},\beta_{1}),\ldots,\textrm{{{W}}}(\alpha_{2m},\beta_{\beta_{2m}})\right)\\\
&\phantom{k_{2m}\left(\textrm{{{W}}}^{V}(\alpha_{1},\beta_{1}),\ldots,\textrm{{{W}}}^{V}(\alpha_{2m+1},\beta_{\beta_{2m}})\right)}=\sum_{\textrm{{{m}}}\in\textrm{{{NC}}}_{2m}}\prod_{i=1}^{2m}\delta_{\alpha_{i},\beta_{\textrm{{{m}}}(i)}}\delta_{\alpha_{\textrm{{{m}}}(i)},\beta_{i}},\leavevmode\nobreak\
\alpha,\beta\in\\{1,\ldots,n\\}^{2m}.\end{split}$

To compute these cumulants, we use the following fondamental formula relating
cumulants with products of elements of $\mathcal{A}$ as entries to the
cumulants of these elements. Let $p\geq 1$ an integer and $k_{p}\geq 1$ an
other one. We define the interval partition
$\sigma=\\{\\{1,\ldots,k_{1}\\},\\{k_{1}+1,\ldots,k_{2}\\},\ldots,\\{k_{p-1}+1,\ldots,k_{p}\\}\\}$
and denote by $1_{k_{p}}$ the partition of $\llbracket 1,k_{p}\rrbracket$ with
only one block. Let $a_{1},\ldots,a_{k_{p}}\in\mathcal{A}$, then:

(16) $k_{k_{p}}(a_{1}\cdots a_{k_{1}},a_{k_{1}+1}\cdots
a_{k_{2}},\ldots,a_{k_{p-1}+1}\cdots
a_{k_{p}})=\sum_{\begin{subarray}{c}\pi\in\textrm{{{NC}}}_{k_{p}}\\\
\pi\vee\sigma=1_{k_{p}}\end{subarray}}k_{\pi}(a_{1},\ldots,a_{k_{p}}).$

By using formula $\eqref{formulaproductcumulant}$ and freeness of the
matricial entries of $V$ with $\textrm{{{W}}}_{t}$, it is easy to prove that
the odd cumulants in (15) are equal to zero. Let $p\geq 1$ and
$\alpha_{1},\ldots,\alpha_{2p}\in\\{1,\ldots,n\\}^{2p}$. Consider a word
$u=v_{1}w_{1}\tilde{v}_{1}\cdots v_{1}\tilde{w}\tilde{v}_{1})$ with the
$v^{{}^{\prime}s}$ and the $\tilde{v}^{s}$ in the algebra generated by the
matrix coefficients of $V$ and the $w^{{}^{\prime}s}$ in the algebra generated
by the matrix coefficients of $W_{t}$. We say that an integer $i\leq 3p$ is
white coloured if $u_{i}$ is equal to $v_{i}$ or $v^{\prime}_{i}$ and black
coloured if $u_{i}$ is equal to $u_{i}$.

Let $\sigma$ be the interval partition
$\sigma=\\{\\{1,2,3\\},\ldots,\\{6p-2,6p-1,6p\\}\\}$. Let
$\pi\in\textrm{{{NC}}}_{6p}$ such that $\sigma\vee\pi=1_{6p}$. By using
nullity of mixed cumulants having components of $V$ and $\textrm{{{W}}}_{t}$
as entries, we prove that

$k_{\pi}(V(\alpha_{1},k_{1}),\textrm{{{W}}}(k_{1},q_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V(\alpha_{2p},k_{2p}),\textrm{{{W}}}(k_{2p},q_{2p}),V^{\star}(q_{2p},\beta_{2p}))$

is equal to zero if a block of $\pi$ is white a black coloured. Also, the
trace of $\pi$ on the set of black coloured integers is a matching m, in that
case,

(17)
$\begin{split}&k_{\pi}(V(\alpha_{1},k_{1}),\textrm{{{W}}}(k_{1},q_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V(\alpha_{2p},k_{2p}),\textrm{{{W}}}(k_{2p},q_{2p}),V^{\star}(q_{2p},\beta_{2p}))=\\\
&\hskip
56.9055ptk_{\textrm{{{m}}}}(W(k_{1},q_{1}),\ldots,W(k_{2p},q_{2p}))k_{\textrm{{{K}}}(\textrm{{{m}}},\pi)}(V(\alpha_{1},k_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V^{\star}(q_{2p},\beta_{2p})).\end{split}$

The non crossing partition denoted $\textrm{{{K}}}(\textrm{{{m}}},\pi)$ is a
partition of the white coloured integers of $\llbracket 1,6p\rrbracket$ and
equal to the complement of m in the partition $\pi$:
$\textrm{{{m}}}\cup\textrm{{{K}}}(\textrm{{{m}}},\pi)=\pi$. We sum (17) over
non crossing partitions having the same trace
$\textrm{{{m}}}\in\textrm{{{NC}}}^{2}_{2p}$ over black coloured integers and
over integers $k_{1},\ldots,k_{2p}$,$q_{1},\ldots,q_{2p}$ in
$\\{1,\ldots,n\\}^{2}$. By using the moments-cumulants formula, we obtain:

$\begin{split}&\sum_{\begin{subarray}{c}1\leq k_{1},\ldots,k_{2p}\leq n,\\\
1\leq q_{1},\ldots,q_{2p}\leq
n\end{subarray}}\sum_{\begin{subarray}{c}\pi\in\textrm{{{NC}}}_{6p}\\\
\pi_{\bullet}=\textrm{{{m}}}\end{subarray}}k_{\pi}(V(\alpha_{1},k_{1}),\textrm{{{W}}}(k_{1},q_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V(\alpha_{2p},k_{2p}),\textrm{{{W}}}(k_{2p},q_{2p}),V^{\star}(q_{2p},\beta_{2p}))\\\
&=\sum_{\begin{subarray}{c}1\leq k_{1},\ldots,k_{2p}\leq n,\\\ 1\leq
q_{1},\ldots,q_{2p}\leq
n\end{subarray}}k_{\textrm{{{m}}}}(W(k_{1},q_{1}),\ldots,W(k_{2p},q_{2p}))\tau_{\textrm{{{K}}}(\textrm{{{m}}},1_{6p})}(V(\alpha_{1},k_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V^{\star}(q_{2p},\beta_{2p}))\\\
&=\sum_{\begin{subarray}{c}1\leq k_{1},\ldots,k_{2p}\leq n,\\\ 1\leq
q_{1},\ldots,q_{2p}\leq
n\end{subarray}}\prod_{i=1}^{2p}\delta_{k_{i},q_{\textrm{{{m}}}(i)}}\delta_{q_{i},k_{\textrm{{{m}}}(i)}}\tau_{\textrm{{{K}}}(\textrm{{{m}}},1_{6p})}(V(\alpha_{1},k_{1}),V^{\star}(q_{1},\beta_{1}),\ldots,V^{\star}(q_{2p},\beta_{2p}))\\\
\end{split}$

To compute right hand side of the last equation, pick a block $V$ of the
partition $\textrm{{{K}}}(\textrm{{{m}}},1_{6p})$, then by using cyclicity of
$\tau$, we have:

(18) $\sum_{\begin{subarray}{c}1\leq k_{1},\ldots,k_{2p}\leq n,\\\ 1\leq
q_{1},\ldots,q_{2p}\leq
n\end{subarray}}\tau_{V}(V^{\star}(q_{i_{1}},\beta_{i_{1}})V(\alpha_{i_{2}},k_{i_{2}})\cdots
V(\alpha_{i_{l}},k_{\textrm{{{m}}}(i_{l})})\prod_{l}\delta_{k_{i_{l}},q_{\textrm{{{m}}}(i_{l})}}\delta_{q_{i_{l}},k_{\textrm{{{m}}}(i_{l})}}=\prod_{l}\delta_{\alpha_{l},\beta_{\textrm{{{m}}}(l)}}\delta_{\beta_{l},\alpha_{\textrm{{{m}}}(l)}}.$

The proof of the formulas (15) is now complete. ∎

We saw that for a classical Levy process on the Zhang algebra of function on a
group, gauge invariance and independence of increments implies braid
invariance. It seems difficult to prove braid invariance of ${U^{\langle
n\rangle}}$ with the same arguments. However, we prove in [7] that
${U^{\langle n\rangle}}$ is a limit, in non-commutative distribution of a
braid invariant process: the process $U^{\mathbb{C}}_{n,d}$ (which is not a
Lévy process) that is defined in [7]. Braid invariance of the finite
dimensinal approximations of $U^{\langle n\rangle}$ implies braid invariance
of the latter.

We can apply Theorem 26: there exists an algebraic generalized master field
associated with the higher dimensional counterpart of the free unitary
Brownian motion ${U^{\langle n\rangle}}$.

######  Proposition 28.

Let $n\geq 1$ an integer. There exists a probabilistic generalized master
field (see Definition 20), denoted $\Phi^{\langle n\rangle}$, such that for
any one parameter family of simple loops $\gamma$ with $|\gamma_{t}|=t$ and
$\mathrm{Int}(\gamma_{s})\subset\mathrm{Int}(\gamma_{t})$ for all times $0\leq
s\leq t$, the process $\Phi^{\langle n\rangle}(\gamma_{t})$ is a free unitary
Brownian motion of dimension $n$.

###  5.3. Amalgamated algebraic master fields

Let $n\geq 1$ an integer. In Section 2.2, we defined the Zhang algebra
$\mathcal{R}\mathcal{O}\langle n\rangle$ as being the involutive algebra
generated by one unitary element $u$ and a complete set of mutually orthogonal
projectors $\\{p_{i},\leavevmode\nobreak\ 1\leq i\leq n\\}$. Set
$\mathcal{R}=\langle p_{i},1\leq i\leq n\rangle$. The algebra
$\mathcal{R}\mathcal{O}\langle n\rangle$ belongs to the category ${\rm
lCoMod}{\rm Alg}(\mathcal{R})$. Let $\mathcal{D}$ be the category having the
same objects class as ${\rm lCoMod}{\rm Alg}(\mathcal{R})$ and the class of
morphisms between two objets being given by the positive complex
$\mathcal{R}$-bimodule maps, for all $A,B\in{\mathrm{Obj}}(\mathcal{D})$:

(19) ${\rm
Hom}_{\mathcal{D}}(A,B)=\\{\phi\in{\mathrm{Vect}}_{\mathbb{C}}(A,B):\phi(a^{\star})=\phi(a)^{\star},\leavevmode\nobreak\
\phi(aa^{\star})\geq 0,\leavevmode\nobreak\
\phi(rar^{\prime})=r\phi(a)r^{\prime}\\}$

With the notations of Definition 20, the category $\mathcal{B}$ is the usual
category of $\mathcal{R}$-amalgamated probability spaces, also known as
rectangular probability spaces. Let $r=(r_{1},\ldots,r_{n})$ be positive real
numbers such that $r_{1}+\cdots+r_{n}=1$. In [7], we define the semi-group
$\textrm{{{E}}}_{r}$ on the Zhang algebra $\mathcal{R}\mathcal{O}\langle
n\rangle$ and prove tha $\textrm{{{E}}}_{r}$ is a free, with amalgamation,
semi-group. We explain briefly how this semi-group is obtained.

We fix a integer $n\geq 1$. For each integer $N\geq 1$, we pick
$(d_{N}^{1},\ldots,d_{N}^{n})$ a partition of $N$ into $n$ parts and we assume
that the ratio $\frac{d_{N}^{i}}{N}$ converges to $r_{i}$ for each integer
$1\leq i\leq n$ as $N\to+\infty$. Let $\mathbb{K}$ be one of the three
division algebras $\mathbb{R},\leavevmode\nobreak\ \mathbb{C}$ and
$\mathbb{H}$. In [7], we define for a partition d of $N$ with length $n$ the
quantum process $U^{\mathbb{K}}_{\textrm{{{d}}}}$ on the Zhang algebra
$\mathcal{R}\mathcal{O}\langle n\rangle$ that extracts rectangular blocks from
the unitary Brownian motion of dimension $N$. This process is valued into a
rectangular probability space, which we denote by $\mathcal{M}_{d_{N}}$. As an
algebra, $\mathcal{M}_{d_{N}}$ is isomorphic to the space of matrices with
dimension $N\times N$ and entries in the algebra of bounded variables (or
random variables having moments of all order) with values in $\mathbb{K}$. The
algebra $\mathcal{M}_{d_{N}}$ is a bimodule algebra over the commutative
unital algebra generated by the projectors:

$p_{i}=\sum_{j=d_{1}+\ldots+d_{i-1}}^{d_{1}+\ldots+d_{i}}e^{j}\otimes
e_{j},\leavevmode\nobreak\ 1\leq i\leq n,$

where $(e_{1},\ldots,e_{n})$ denote the canonical basis of $\mathbb{K}^{N}$
(as a left $\mathbb{K}$-module and $(e^{1},\ldots,e^{n})$ is its dual basis.
If $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, the conditional expectation
$\mathbb{E}_{d_{N}}$ on $\mathcal{M}_{d_{N}}$ is the mean of the trace of the
diagonal blocks:

$\mathbb{E}_{d_{N}}(A)=\sum_{i=1}\frac{1}{d_{N}(i)}\mathbb{E}\left[\textrm{{{Tr}}}(p_{i}Ap_{i})\right]p_{i},\leavevmode\nobreak\
A\in\mathcal{M}_{d_{N}}$

In [7], we define the semi-group $\textrm{{{E}}}_{r}$ as the limiting
distribution of the non commutative distribution of the rectangular extraction
process:

$\textrm{{{E}}}_{d_{N}}=\mathbb{E}_{d_{N}}\circ
U^{\mathbb{K}}_{d_{N}}(t)\xrightarrow{N\rightarrow+\infty}\textrm{{{E}}}_{r_{1},\ldots,r_{n}}(t),\leavevmode\nobreak\
\text{ for all time }t\geq 0.$

We apply our main Theorem 26 to obtain a probabilistic generalized master
field $\Phi^{\langle r\rangle}$ associated with the free semi-group
$\textrm{{{E}}}_{r}$ that we name _amalgamated higher dimensional master
field_ with parameter $r$. The following proposition sums up this last
discussion.

######  Proposition 29.

Let $r_{1},\ldots,r_{n}\geq 1$ positive real numbers. There exists a
probabilistic generalized master field (see Definition 18), which we
$\Phi^{\langle r_{1},\ldots,r_{n}\rangle}$, such that for all one-parameter
family of simple loops $\gamma$ with $|\gamma_{t}|=t$ and
$\mathrm{Int}(\gamma_{s})\subset\mathrm{Int}(\gamma_{t})$ for all times $0\leq
s\leq t$, the process $\left(\Phi^{\langle
r_{1},\ldots,r_{n}\rangle}(\gamma_{t})\right)_{t\geq 1}$ is a free with
amalgamation quantum Lévy process which non-commutative distribution at time
$t\geq 0$ is $\textrm{{{E}}}_{r_{1},\ldots,r_{n}}(t)$.

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