UMN–TH–4102/21

Small Instantons in Weakly-Gauged
Holographic Models

Tony Gherghettaa and Alex Pomarolb,c

aSchool of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA
bIFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
cDepartament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona

##  1 Introduction

Instantons play a prominent role in Yang-Mills theories, with their
contribution to the path integral $\propto e^{-8\pi^{2}/g^{2}}$ becoming
especially important for large gauge couplings. For example, in QCD, the
Euclidean action of the BPST instanton [1], after including quantum
corrections, is given by $S_{E}=8\pi^{2}/g^{2}(1/\rho)$ with a running gauge
coupling $g(1/\rho)$ dependent on the instanton size $\rho$. Thus at large
$\rho$, corresponding to IR energy scales where the QCD gauge coupling is
large, nonperturbative effects due to “large” instantons can dominate the path
integral and lead to a variety of phenomena.

This instanton dependence on the gauge coupling suggests that if the gauge
coupling were to increase (or at least remain sizeable) at small distances (or
UV energy scales) then small-size instantons could also dominate the path
integral. This can be easily achieved by adding extra colored matter fields at
some high-energy scale $m$, which changes the sign of the $\beta$-function,
leading to a sizeable gauge coupling at small distances $\ll 1/m$. The matter
fields can be either fermions or scalars. However, when the extra states are
colored fermions, the axial anomaly causes the instanton contributions to
explicitly depend on the fermion masses, $\propto(\rho\,m_{F})^{N_{F}}$
(assuming a common mass $m_{F}$ and number of fermions $N_{F}$), which again
suppresses the small instanton contributions corresponding to $\rho\ll
1/m_{F}$! On the other hand, there is no extra suppression from the axial
anomaly if the matter fields are colored scalars, and therefore small
instanton contributions could be enhanced. However, in this case the scalar
masses, $m_{S}$ must be tuned to be small, $m_{S}\ll 1/\rho$, in order to
significantly affect the running of the gauge coupling. Thus, it is not so
straightforward to enhance the small instanton contributions of a Yang-Mills
theory in a natural way.111 Alternatives that enhance the action $S_{E}$ from
small instantons which is then used to increase the QCD axion mass are given
in [2, 3, 4, 5]. In particular, small-size instantons are enhanced by power-
law corrections if QCD propagates in a flat extra dimension at energies much
larger than the electroweak scale [5].

A third possibility to enhance the small instanton contributions is to
consider QCD propagating in five-dimensional (5D) Euclidean AdS. In this
theory, gluons on the UV boundary of the AdS space receive a well-known
(classical) logarithmic contribution [6, 7, 8] that is purely due to the AdS
geometry and does not rely on introducing extra colored states. Importantly,
this contribution enhances the gauge coupling of the gluons on the UV boundary
at small distances.

This logarithmic effect has a 4D interpretation based on the AdS/CFT
correspondence [9]. The 5D AdS QCD theory is dual to a 4D strongly-coupled CFT
with a global SU(3)c symmetry that contains a large number $N$ of constituents
charged under this SU(3)c. By weakly gauging SU(3)c (identified as QCD color),
this CFT charged matter would then be responsible for causing the QCD coupling
to grow at small distances. However, we should emphasize that this is only a
holographic interpretation of the 5D logarithmic running and there are no
charged scalar or fermion constituents in the 5D model. Thus, the inherent
drawbacks of extra colored matter fields, discussed previously, could in
principle be avoided.

Furthermore, by the AdS/CFT correspondence, we expect 5D holographic models to
have an instanton action similar to that found in 4D theories,
$S_{E}=8\pi^{2}/g^{2}(1/\rho)$, where $g(1/\rho)$ is identified with the
running coupling of the gluons on the UV boundary. This expected form of the
action can be directly interpreted in the 5D theory by using the fact that the
AdS/CFT dictionary relates the renormalization scale $\mu$ of the running
coupling $g(\mu)$ to the inverse of the 5th coordinate $z$, i.e. $\mu\sim
1/z$. Given that the running coupling in the 4D instanton action is
renormalized at $\mu=1/\rho$, therefore suggests that in the 5D AdS-Yang-Mills
theory there must be an instanton configuration localized on the UV boundary
that extends only a finite distance $z\sim\rho$ into the 5D bulk.

Figure 1: A 2D contour of the 5D Lagrangian density (26) (multiplied by the
volume factor $2\pi^{2}x^{3}$ and for $\rho=1$) evaluated at 10, depicting an
instanton and anti-instanton both located at $x_{\mu}=0$ (where two Euclidean
coordinates $x_{2,3}$ are not shown). As $z$ increases, the instanton size
remains fixed while the anti-instanton size grows until it annihilates the
instanton at $z\sim\rho$.

To look for such a solution we consider an SU(2) Yang-Mills theory in
Euclidean AdS5 whose limit on the AdS boundary is a 4D instanton. However, it
turns out that such a solution is, in fact, not possible because topological
charge conservation implies that the 4D instanton cannot “disappear” into the
AdS bulk (as can also be directly verified by examining the 5D AdS Yang-Mills
equations). Indeed the only instanton solution to the 5D AdS Yang-Mills
equations that we obtain is from uplifting the 4D instanton to the AdS5 bulk
as a “cylindrical” configuration with no dependence on the 5th coordinate. The
corresponding topological charge, defined on 4D slices at any given $z$, is
then always one for this solution. Since this uplifted instanton solution is
not localized, we obtain $S_{E}=8\pi^{2}/g^{2}(z_{\rm IR}^{-1})$, where
$z_{\rm IR}\gg\rho$ is an IR cutoff (identified with the confinement scale of
the CFT). This means that despite the fact that the gauge coupling increases
towards the UV boundary, small instantons are highly suppressed and
ineffective.

Alternatively, this suppression can be understood from a supersymmetric
argument. Interestingly, the “cylindrical” instanton configuration is also a
solution of the supersymmetrized 5D model, and therefore by the AdS/CFT
dictionary implies that the dual 4D theory necessarily contains colored
fermion constituents. The axial anomaly of these colored fermions then leads
to the usual chiral suppression (but saturated at $z_{\rm IR}^{-1}$). At large
$N$, where the effects from the colored CFT fermions dominate, this
suppression can be simply interpreted in the 5D theory as an extra running of
the gauge coupling from $1/\rho$ to $z_{\rm IR}^{-1}$ to give the
corresponding action $S_{E}=8\pi^{2}/g^{2}(z_{\rm IR}^{-1})$. Thus, the
absence of an expected localized instanton solution results from an accidental
supersymmetry of the bulk at the classical level.

These arguments therefore suggest that AdS5 Yang-Mills theory only admits UV
localized configurations with zero topological charge. Indeed, we discover new
instanton-anti-instanton configurations of the 5D AdS Yang-Mills theory which
extend only a finite distance into the 5D bulk. In particular, we find a
solution that for $z\ll\rho$, consists of an anti-instanton of size
$z^{2}/\rho$ located inside an instanton of size $\rho$. As $z$ increases, the
anti-instanton grows in size until it “pinches off” the instanton at a
distance in the bulk $z\sim\rho$. A two-dimensional contour of the Lagrangian
density of this solution is shown in Figure 1. This solution is analytically
obtained by taking advantage of the SO(5,1) isometry group of Euclidean AdS5,
corresponding to the conformal symmetry of the dual 4D CFT. In particular, the
instanton-anti-instanton solution is simply generated by performing a discrete
5D conformal transformation (or “inversion”) on the uplifted 4D instanton
solution.

This localized zero topological charge solution can also be understood using
the accidental supersymmetry in the bulk. However, contrary to the “cylinder”
instanton, the instanton-anti-instanton configuration does not preserve the 5D
supersymmetry and consequently there are no fermion zero modes in the dual 4D
theory (that would lead to extra chiral suppression from the axial anomaly).
Thus, the running gauge coupling only receives contributions from the fermion
nonzero modes, whose running ends at $1/\rho$ and not $z_{\rm IR}^{-1}$.

Our solution-generating technique can be extended by performing other 5D
conformal transformations on known 4D instanton solutions. For instance,
performing a 5D special conformal transformation on the uplifted 4D instanton
solution gives rise to other instanton-anti-instanton solutions. In fact by
performing multiple, independent, special conformal transformations on the 4D
instanton or multi-instanton solutions [10], an instanton-anti-instanton
solution “gas” can be generated. In all cases the topological charge is
preserved in the AdS bulk. Similarly, we find generalizations of known 4D
meron solutions [11, 12, 13] to 5D configurations. Curiously, we also discover
a new, simple analytical solution to the 5D AdS Yang-Mills equations that is
not obtained from any 5D conformal transformation of a known 4D solution. It
is a 5D meron configuration, that for any given $z$ has a topological charge
$\frac{1}{2}$ and size proportional to $z$. Unlike the known 4D meron which is
singular, the 5D meron is regular. Again, applying 5D special conformal
transformations on the new 5D meron solution further generates other meron-
antimeron configurations.

##  2 Yang-Mills theory interacting with a strongly-coupled CFT

We are interested in studying Euclidean topological configurations of Yang-
Mills theories interacting with a strongly-coupled CFT. For simplicity, we
consider an SU(2) theory with gauge bosons $A_{\mu}^{a}(a=1,2,3)$ whose 4D
Lagrangian is given by

${\cal L}_{4}=\frac{1}{4g_{4}^{2}}F_{\mu\nu}^{a}F^{a\mu\nu}+A_{\mu}^{a}J_{\rm
CFT}^{a\mu}+{\cal L}_{\rm CFT}~{},$ (1)

where $g_{4}$ is the Yang-Mills coupling, $J_{\rm CFT}^{a\mu}$ is the SU(2)
CFT current and ${\cal L}_{\rm CFT}$ is the CFT Lagrangian.

We will study the above Yang-Mills theory in the limit where it admits a 5D
holographic description. In this limit, the CFT is described by gravity in
AdS5 with spacetime coordinates $x_{M}=(x_{\mu},z)$ $(M=0,1,2,3,5)$ and metric

$ds^{2}=g_{MN}dx^{M}dx^{N}=\frac{L^{2}}{z^{2}}\left(\delta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right)\equiv
w^{2}(z)\left(\delta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right)~{},$ (2)

where $L$ is the AdS5 curvature length, $\delta_{\mu\nu}={\rm diag}(1,1,1,1)$
is the 4D Euclidean metric and the warp factor $w(z)=L/z$. Following the usual
holographic procedure, the AdS5 action is regularized by truncating the space
with a boundary at $z=z_{\rm UV}$ and taking the limit $z_{\rm UV}\to 0$ after
the proper counterterms have been introduced on this UV boundary to cancel UV
divergences. Also for convenience, we introduce an IR boundary at $z=z_{\rm
IR}\gg z_{\rm UV}$. This corresponds to an IR mass gap in the CFT at the scale
$\sim z_{\rm IR}^{-1}$ that also helps to regularize IR divergences.

In the 5D holographic model the SU(2) global symmetry of the CFT corresponds
to introducing an SU(2) local symmetry in the bulk with gauge bosons
$A_{M}^{a}$ $(a=1,2,3)$. The 5D Euclidean action is then given by

$S_{5}=\int d^{4}x\int_{z_{\rm UV}}^{z_{\rm
IR}}dz\sqrt{g}~{}\frac{1}{4g_{5}^{2}}F_{MN}^{a}F^{aMN}~{},$ (3)

where
$F_{MN}^{a}=\partial_{M}A_{N}^{a}-\partial_{N}A_{M}^{a}+\varepsilon^{abc}A_{M}^{b}A_{N}^{c}$
is the field-strength tensor with $\varepsilon^{abc}$ the Levi-Civita symbol,
and $g_{5}$ is the dimensionful 5D SU(2) gauge coupling. The 4D Yang-Mills
gauge bosons $A^{a}_{\mu}$, which are introduced in (1) as external fields
weakly coupled to the CFT, correspond in the 5D theory to the bulk gauge field
evaluated on the UV boundary $A^{a}_{\mu}\equiv A^{a}_{\mu}(x,z_{\rm UV})$. On
this boundary we can add the action

$S_{\rm UV}=\int
d^{4}x~{}\frac{1}{4g_{4}^{2}}F_{\mu\nu}^{a}F^{a\mu\nu}\Bigg{|}_{z_{\rm
UV}}~{},$ (4)

where the $\mu,\nu$ indices are contracted with the 4D Euclidean metric
$\delta_{\mu\nu}$ and the boundary gauge coupling has been identified with
$g_{4}$ in (1). As will be shown, the boundary kinetic term (4) acts as a
counterterm to regularize the divergent 5D contributions to $A^{a}_{\mu}$ in
the limit $z_{\rm UV}\to 0$.

The bulk equation of motion can be obtained by extremizing (3), $\delta
S_{5}/\delta A^{a}_{M}=0$ which gives

$\partial_{M}\left(w(z)F^{MNa}\right)+w(z)\varepsilon^{abc}A_{M}^{b}F^{MNc}+{\cal
B}_{\mu}^{a}\delta_{\mu}^{N}=0~{},$ (5)

where $\delta_{\mu}^{N}$ is the Kronecker delta and the boundary term is

${\cal B}_{\mu}^{a}=w(z)F_{\mu 5}^{a}\Big{|}_{z_{\rm UV}}^{z_{\rm IR}}~{}.$
(6)

The last term in (5), together with the contribution from (4), leads to the UV
boundary condition

$\frac{1}{g_{4}^{2}}\left(\partial_{\mu}F^{\mu\nu
a}+\varepsilon^{abc}A_{\mu}^{b}F^{\mu\nu
c}\right)-\frac{1}{g_{5}^{2}}w(z)F_{\nu 5}^{a}\Big{|}_{z_{\rm UV}}=0~{}.$ (7)

This corresponds to the equation of motion of a 4D Yang-Mills field coupled to
a current that can be identified with the CFT current in (1): $J_{\rm
CFT}^{a\mu}=-\frac{1}{g_{5}^{2}}w(z)F_{\mu 5}^{a}|_{z_{\rm UV}}$. The IR
boundary condition is obtained from (6) and given by $F_{\mu 5}^{a}|_{z_{\rm
IR}}=0$.

We will be working in the limit in which the 5D coupling $g_{5}\to 0$ and
gravitational interactions are neglected. This limit is well-defined and
corresponds in the dual 4D theory to a large $N$ expansion with the 5D
gravitational coupling scaling as $G_{\rm N}\sim 1/N^{2}$ while $g_{5}^{2}\sim
1/N$.

Using the above holographic model we can calculate the CFT contribution to the
4D Yang-Mills gauge coupling, as this will be useful later. This is done by
integrating out the 5D gauge boson as a function of the boundary field
$A_{\mu}^{a}$ at the quadratic level to give the UV-boundary term [6]

$\int d^{4}x\frac{1}{4}\left(\frac{L}{g_{5}^{2}}\log\left(\frac{z_{\rm
IR}}{z_{\rm UV}}\right)\right)F_{\mu\nu}^{a}F^{a\mu\nu}\Bigg{|}_{z_{\rm
UV}}~{}.$ (8)

This term diverges as $z_{\rm UV}\to 0$. However, this divergence can be
cancelled by the counterterm (4). Indeed, by defining the “running” gauge
coupling as

$\frac{1}{g_{4}^{2}(\mu)}=\frac{1}{g_{4}^{2}}-\frac{L}{g_{5}^{2}}\log(\mu
z_{\rm UV})~{},$ (9)

the bulk contribution (8) corresponds to the replacement $g_{4}^{2}\to
g_{4}^{2}(\mu=z_{\rm IR}^{-1})$ in $\eqref{eq:UVLag}$. This is the expected
result of an SU(2) Yang-Mills theory with $\beta$-function coefficient,
$b_{\rm CFT}\equiv 8\pi^{2}\frac{L}{g_{5}^{2}}\sim N\gg 1$ due to the CFT
222The $\beta$-function coefficient $b$ is defined from the renormalization
group equation:
$\frac{d}{d\log\mu}\left(\frac{8\pi^{2}}{g_{4}^{2}}\right)\equiv-b$. For SU(2)
QCD this definition corresponds to $b_{QCD}=-22/3+2/3n_{f}$, where $n_{f}$ is
the number of Weyl fermions while the CFT contribution is $b_{\rm
CFT}=8\pi^{2}L/g_{5}^{2}$., that changes the Yang-Mills gauge coupling down to
energies $\mu=z_{\rm IR}^{-1}$. Note that the running in (9) only includes the
CFT matter contributions, while the 4D gluon contributions, which are sub-
dominant in the large $N$ expansion, are neglected.

##  3 AdS5 Yang-Mills Euclidean configurations

###  3.1 Solution ansatz

To obtain solutions to (5) we assume the following ansatz:

$A_{\mu}^{a}(x,z)=2\eta_{\mu\nu}^{a}\frac{x_{\nu}}{x^{2}}f(x,z)~{},\quad
A_{5}^{a}(x,z)=0~{},$ (10)

where $\eta_{\mu\nu}^{a}$ is the ‘t Hooft symbol [14],
$x^{2}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ and $f(x,z)$ is an arbitrary
function. The matrix gauge potential corresponding to the ansatz (10) can be
written in terms of a gauge parameter $\Omega$ as

$A_{\mu}(x,z)=f(x,z)\left[-i(\partial_{\mu}\Omega)\Omega^{-1}\right]~{},$ (11)

where

$\Omega=\frac{1}{\sqrt{x^{2}}}\left({\vec{x}}\cdot{\vec{\sigma}}\mp
ix_{0}\mbox{1}\mbox{l}\right)~{},$ (12)

with $\Omega^{-1}=\Omega^{\dagger}$ and $\sigma_{i}$ ($i=1,2,3$) are the Pauli
matrices. Note that the matrix gauge potential can be gauge transformed as

$A_{\mu}^{\prime}=U^{\dagger}A_{\mu}U-i(\partial_{\mu}U^{\dagger})U~{},$ (13)

where $U=U(x)$ is a gauge transformation.

Substituting the ansatz (10) into the equation of motion (5) gives a second-
order nonlinear partial differential equation for $f(x,z)$:

$f^{(2,0)}+\frac{1}{x}f^{(1,0)}+f^{(0,2)}-\frac{1}{z}f^{(0,1)}-\frac{4}{x^{2}}f(f-1)(2f-1)=0~{},$
(14)

where the first (second) index in the superscript denotes differentiation with
respect to $x\ (z)$. Note that if $f$ is a solution, then $1-f$ is also a
solution. With the ansatz (10) we obtain the 5D Lagrangian density

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{6}{x^{2}}\left((f^{(0,1)})^{2}+(f^{(1,0)})^{2}+\frac{4}{x^{2}}f^{2}(f-1)^{2}\right)~{}.$
(15)

At a fixed, nonzero $z$, corresponding to a 4D slice of AdS5, we can also
define a topological charge density by

$D=\frac{1}{4}F_{\mu\nu}^{a}{\widetilde{F}}^{a\mu\nu}=\frac{24}{x^{3}}f^{(1,0)}f(1-f)=\frac{4}{x^{3}}\frac{\partial}{\partial
x}\left(3f^{2}-2f^{3}\right)~{},$ (16)

where
${\widetilde{F}}_{\mu\nu}^{a}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma
a}$ with $\varepsilon_{\mu\nu\rho\sigma}$ the Levi-Civita symbol. The
topological charge $q$ on the 4D slice is then given by

$q=\frac{1}{8\pi^{2}}\int d^{4}x\,D~{}.$ (17)

The solution $1-f$ has an opposite topological charge to $f$. For a 4D
spherically symmetric solution, the topological charge (17) can be written as

$q=(3f^{2}-2f^{3})\Big{|}_{0}^{\infty}~{}.$ (18)

In particular, configurations with $q=1$ have $f$ changing from $f(0,z)=0$ to
$f(\infty,z)=1$.

We next present solutions to the AdS5 Yang-Mills equation of motion (14) by
first considering the trivial case of $f(x,z)\equiv f(x)$, followed by
nontrivial $z$-dependent solutions.

Figure 2: Left: The uplifted 4D instanton solution (19) for $\rho=1$,
depicting the $f=0$ asymptotic region rising to $f=1$ at $x\sim\rho$,
independent of $z$. Right: The instanton-anti-instanton solution (24) for
$\rho=1$, depicting the merging of the asymptotic $f=1$ regions at $x\sim
z\sim\rho$ from the anti-instanton and instanton. The $f=1$ “plateaus” enclose
a “basin” corresponding to $f=0$.

###  3.2 Uplifted 4D Instanton

A trivial 5D solution of (14) can be obtained by assuming that $f(x,z)$ is $z$
independent. This leads to the solution

$f(x,z)=\frac{x^{2}}{x^{2}+\rho^{2}}~{},$ (19)

which is just the usual 4D instanton solution with size $\rho$ and topological
charge $q=1$, uplifted to the fifth dimension. The solution (19) is plotted in
Figure 2, while the topological charge density,
$D=48\rho^{4}/(x^{2}+\rho^{2})^{4}$ is shown in Figure 3. The Lagrangian
density is

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{48\rho^{4}}{(x^{2}+\rho^{2})^{4}}~{},$
(20)

which leads to the 5D action

$S_{5}=8\pi^{2}\frac{L}{g_{5}^{2}}\log\left(\frac{z_{\rm IR}}{z_{\rm
UV}}\right)~{}.$ (21)

Similarly, an anti-instanton solution with topological charge $q=-1$ arises
from the replacement $f\rightarrow 1-f$, and also leads to (21). It is
important to remark that for small but finite $g_{5}$, the cutoff of the 5D
model, $\Lambda_{5}$ (where loop corrections become of order one) scales as
$\Lambda_{5}\sim 1/g_{5}^{2}$ and then (21) is expected to be sensitive to the
cutoff for $z\gtrsim\rho/g_{5}^{2}L$. Here we are only considering the formal
limit $g_{5}\to 0$ such that $\Lambda_{5}\to\infty$.

The 5D action (21) diverges as $z_{\rm UV}\to 0$, but as outlined in Section
2, this divergence can be cancelled by the UV boundary term (4) evaluated on
the instanton solution (19) and using (9). Combining this with (21), we obtain
the finite action

$S_{5}+S_{\rm UV}=\frac{8\pi^{2}}{g^{2}_{4}(z_{\rm IR}^{-1})}~{},$ (22)

where the Yang-Mills gauge coupling is now evaluated at the scale $z_{\rm
IR}^{-1}$.333Recall from (9) that only the CFT matter contribution to the
gauge coupling running is considered, since this dominates at large $N$. This
result is unexpected. As explained in the Introduction, when matter fields are
added to a Yang-Mills theory, instantons receive loop corrections that can be
captured by the replacement $g_{4}\to g_{4}(1/\rho)$ [15]. However, the gauge
coupling in (22) does not exhibit this behaviour and instead the gauge
coupling dependence gives a much larger action for small instantons with
$\rho\ll z_{\rm IR}$, since $g_{4}(z_{\rm IR}^{-1})\ll g_{4}(1/\rho)$
according to (9). Therefore, we find that small instantons in weakly-gauged
holographic models are much more suppressed than expected.

The reason for the larger than expected action (22) boils down to the fact
that the 5D instanton solution extends from $z_{\rm UV}$ to $z_{\rm IR}$ (see
Figure 2) without any suppression. If the 5D instanton were localized towards
the UV boundary, and extended only a distance $\rho$ in the bulk, the
corresponding action would be of order $8\pi^{2}/g^{2}_{4}(1/\rho)$. However,
we find that a UV localized instanton solution is not possible. This is
because such a solution would require the topological charge (18) to change
from one at $z_{\rm UV}$, to zero at some $z>z_{\rm UV}$, which is not
consistent with topological charge conservation.444This can also be understood
from the equation of motion (14). A UV localized 5D solution would require
that the gauge field configuration becomes pure gauge in the bulk (either
$f\to 1$ or $0,~{}\forall x$) at some $z>z_{\rm UV}$. But starting on the UV
boundary with the 4D instanton solution (19), $f(x,z)$ interpolates between
$f=0$ at $x=0$ to $f=1$ at $x\rightarrow\infty$. Therefore, a UV localized
instanton would require $f$ to abruptly change at some finite $z>z_{\rm UV}$
from $f=0\to 1$ at $x=0$ (or from $f=1\to 0$ at $x\rightarrow\infty$).
Assuming $f$ is a smooth function, this behaviour is not possible as can be
seen by, for example, taking the limit $x\rightarrow 0$ in (14). In this
limit, the nonlinear and $x$-derivative terms dominate over the $z$ derivative
terms (assuming $f\simeq x^{2}/(x^{2}+\rho^{2})$ corresponding to a 4D
instanton), and therefore near $x=0$ the solution $f$ remains approximately
constant as $z$ increases. This implies that $f$ (near $x=0$) cannot change
from $f=0\to 1$ along the $z$ direction. A similar argument holds, if the
change $f=1\to 0$ were to occur in the limit $x\to\infty$. A loophole to this
argument could be to allow for singular configurations. In this case a
negative charged configuration emerging from the bulk could compensate the
positive charge of the UV boundary instanton.

Supersymmetry provides an alternative explanation of why the expected action
$S_{E}=8\pi^{2}/g^{2}_{4}(1/\rho)$ was not obtained in the 5D holographic
model. This follows by noting that the instanton configurations of our 5D
model (3) are also solutions of the supersymmetric version [16], since the
supersymmetric partners (gluinos and adjoint scalars) can be consistently put
to zero in the instanton configuration. This implies that these configurations
preserve supersymmetry at the classical level, and, by the AdS/CFT
correspondence, must be compatible with instanton solutions of the 4D dual
supersymmetric CFT at the quantum level. These theories necessarily contain
massless colored fermions, $\psi$ that require a ’t Hooft fermion vertex to be
included in the partition function [17]. Furthermore, these CFT fermions
contribute instanton zero modes to the partition function but importantly the
nonzero mode contributions cancel [18, 19]. This modifies the instanton
measure to give [14]

${\langle{\rm det}(\bar{\psi}_{L}\psi_{R})\rangle}{\rho^{b_{\rm
CFT}}}e^{-\frac{8\pi^{2}}{g^{2}_{4}(1/\rho)}}\sim\left(\frac{\rho}{z_{\rm
IR}}\right)^{b_{\rm
CFT}}e^{-\frac{8\pi^{2}}{g^{2}_{4}(1/\rho)}}=e^{-8\pi^{2}\left[\frac{1}{g^{2}_{4}(1/\rho)}-\frac{b_{\rm
CFT}}{8\pi^{2}}\log(\rho/z_{\rm IR})\right]}~{},$ (23)

where $b_{\rm CFT}\propto N$ and $1/z_{\rm IR}$ is identified with the
condensate scale of the CFT. The negative of the exponent of (23) then matches
the result (22) since $b_{\rm CFT}=8\pi^{2}L/g_{5}^{2}\sim N$ and gives the
running from $1/\rho$ to $z_{\rm IR}^{-1}$ (again this assumes that the CFT
matter dominates the running of the gauge coupling). In other words, a 5D
localized instanton with action $8\pi^{2}/g^{2}_{4}(1/\rho)$ cannot be a
solution in a 4D supersymmetric CFT at the quantum level.

In summary, we see that (nonsingular) UV localized solutions are only possible
if the topological charge is already zero on the UV boundary. This is possible
for instanton-anti-instanton solutions as we discuss in the next section.

###  3.3 5D Localized Instanton-Anti-Instanton solutions

New solutions to the 5D AdS Yang-Mills equations can be generated by using the
fact that the SU(2) gauge theory (3) and the AdS5 metric are invariant under
conformal transformations. This includes both discrete and continuous
transformations which are given in Appendix A. By transforming (19), new
solutions to the 5D AdS Yang-Mills equations will be generated.

####  3.3.1 An instanton-anti-instanton by 5D inversion

Consider the uplifted 4D anti-instanton solution,
$f(x,z)=\rho^{2}/(x^{2}+\rho^{2})$. Under a discrete 5D inversion (50) with
$\delta=\rho$, this solution transforms to

$f(x,z)=~{}\frac{(x^{2}+z^{2})^{2}}{x^{2}\rho^{2}+(x^{2}+z^{2})^{2}}~{},$ (24)

where the matrix gauge parameter part contained in the square brackets of (11)
remains invariant using (48). This is also a solution of (14), and when $z=0$
we recover a 4D instanton solution.555This is expected because a 4D coordinate
inversion transforms an anti-instanton into an instanton and vice-versa. The
solution (24) is shown in Figure 2. Note that performing a 5D inversion on the
uplifted 4D instanton solution would instead give rise to the solution $1-f$
(with $f$ given in (24)), which becomes an anti-instanton at $z=0$.

Figure 3: Contour plots of the topological charge density (multiplied by the
volume factor $2\pi^{2}x^{4}$ and for $\rho=1$) of the uplifted 4D instanton
(left) and the instanton-anti-instanton (25) (right). In the right plot the
diagonal line, $x=z$ corresponds to zero charge density with the upper (lower)
region depicting contours of negative (positive) charge density associated
with the anti-instanton (instanton).

The topological charge density corresponding to (24) is given by

$D=48\frac{(x^{2}-z^{2})(x^{2}+z^{2})^{3}\rho^{4}}{(x^{2}\rho^{2}+(x^{2}+z^{2})^{2})^{4}}~{}.$
(25)

For any nonzero $z$, the topological charge (17) is zero (since
$f(0,z)=f(\infty,z)=1$ as shown in Figure 2), indicating that the solution
consists of an instanton and anti-instanton. Indeed, by plotting (25) in
Figure 3 we see that, at fixed $z$, we have an instanton of size $\rho$ and an
anti-instanton of size $z^{2}/\rho$ both located at $x=0$. As $z$ grows the
anti-instanton size extends to $\sim\rho$ where it pinches off (or
“annihilates”) the instanton.666This exact instanton-anti-instanton solution
has similarities to the 4D solution obtained in Ref [20, 21] where a complex
scalar field transforming in the fundamental (doublet) representation was
added to an SU(2) gauge theory, albeit with a negative quartic coupling in the
scalar potential. In our case, the solution is supported by 5D Yang-Mills
fields and requires no negative couplings.

The 5D Lagrangian density is given by

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{48\rho^{4}(x^{2}+z^{2})^{4}}{(x^{2}\rho^{2}+(x^{2}+z^{2})^{2})^{4}}~{},$
(26)

and is shown in Figure 1. We can obtain the value of the action by performing
the integral over $x$ first, followed by the $z$ integral, giving

$S_{5}=16\pi^{2}\frac{L}{g^{2}_{5}}\left(\log\frac{\rho}{z_{\rm
UV}}-3\frac{z_{\rm UV}^{2}}{\rho^{2}}+{\cal O}\left(\frac{z_{\rm
UV}^{4}}{\rho^{4}}\right)\right)~{}.$ (27)

Again, adding the UV boundary contribution from (4) by substituting the
solution (24), we obtain a finite action in the limit $z_{\rm UV}\to 0$:

$S_{5}+S_{\rm UV}=\frac{16\pi^{2}}{g^{2}_{4}(1/\rho)}~{},$ (28)

which consists of an instanton and anti-instanton contribution (equivalent to
two instantons). However, notice that the gauge coupling is now evaluated at
the scale of the instanton size $\rho$ and not at the IR scale as in (22) for
the uplifted 4D instanton. This follows from the fact that the solution (24)
is localized on the UV boundary, and extends into the bulk a distance
$\sim\rho$ (see Figure 2). This provides the expected behaviour for a Yang-
Mills field interacting with matter where the running gauge coupling is
evaluated at the scale $1/\rho$ [15].777Instead, if we perform the $z$
integration first, followed by the $x$ integration, the 5D action becomes
$S_{5}=8\pi^{2}\left(\log\frac{\rho}{z_{\rm
UV}}+\frac{1}{2}\log\frac{\rho}{x_{\rm UV}}+{\cal
O}\left(\frac{z^{2}}{\rho^{2}}\right)+{\cal O}\left(\frac{x_{\rm
UV}}{\rho}\right)\right)~{}.$ (29) There is now a logarithmic singularity as
$x\rightarrow 0$. If $x_{\rm UV}=z_{\rm UV}^{2}/\rho$ is substituted into (29)
we recover the leading term in the action (27). This is because at the scale
$z_{\rm UV}^{2}/\rho$ we encounter the anti-instanton of size $\sim z_{\rm
UV}^{2}/\rho$ located at the origin.

The behaviour in (28) is also consistent with the supersymmetric argument
given in Section 3.2. The instanton-anti-instanton configuration (24) no
longer preserves the accidental supersymmetry in the bulk at the classical
level. This means there are no longer any fermion zero modes in the dual 4D
CFT at the quantum level and therefore there is no chiral suppression in the
instanton measure from the axial anomaly. Instead, above the scale $1/\rho$,
there are nonzero mode contributions at the quantum level which leads to the
(classical) localized instanton-anti-instanton configuration with the running
coupling behaviour in (28).

At this point we should note that (24) is only an approximate solution of the
UV-boundary equation of motion (7) that is valid in the limit $1/g_{4}^{2}\gg
L/g_{5}^{2}$, where the second term in (7) (CFT contribution) is neglected,
and the limit $z_{\rm UV}\to 0$, where (24) is approximately an instanton of
size $\rho$ and an anti-instanton whose size ($z_{\rm UV}^{2}/\rho$) tends to
zero:

$f(x,z_{\rm UV}\to 0)\simeq\frac{x^{2}}{x^{2}+\rho^{2}}+\frac{z_{\rm
UV}^{4}/\rho^{2}}{x^{2}+z_{\rm UV}^{4}/\rho^{2}}~{}.$ (30)

Similarly, the 4D $N$-instanton solutions [22, 23] can also be generalized to
the 5D bulk by transforming them with the 5D inversion (50). The analytic
expressions are complicated but one finds anti-instantons with $z$-dependent
sizes that appear at the location of the instantons, similar to the one
instanton case. Thus, each instanton extends into the 5D bulk a distance
$z\sim\rho_{i}$, where $\rho_{i}$ is the size of the $i$-th instanton.

Figure 4: A 2D contour of the Lagrangian density (33) (multiplied by the
volume factor $4\pi x^{3}$ and for $\rho=1,d=10$) evaluated at 1. An anti-
instanton appears at $x_{0}=-10$ and $x_{1}=x_{2}=x_{3}=0$ (Euclidean
coordinates $x_{2,3}$ are not shown) whose size grows and meets at $z\sim d$
the instanton located at $x_{0}=x_{1}=0$.

####  3.3.2 An instanton-anti-instanton by 5D special conformal transformation

More general solutions can be obtained by considering special conformal
transformations (53). Without loss of generality, we can choose the parameter
$d_{\mu}=(d,0,0,0)$ so that translations occur along the $x_{0}$ axis.
Assuming $\delta_{1}=\delta_{2}=d$, the uplifted 4D instanton transforms to a
solution dependent on $d$ and $z$ given by

$A_{M}^{a}=\frac{2d^{3}}{h(x,z)}\begin{pmatrix}-x_{1}(2x_{0}+d)&r_{5}^{2}-2x_{1}^{2}+x_{0}d&-2x_{1}x_{2}+x_{3}d&-2x_{1}x_{3}-x_{2}d&-2x_{1}z\\\
-x_{2}(2x_{0}+d)&-2x_{1}x_{2}-x_{3}d&r_{5}^{2}-2x_{2}^{2}+x_{0}d&-2x_{2}x_{3}+x_{1}d&-2x_{2}z\\\
-x_{3}(2x_{0}+d)&-2x_{1}x_{3}+x_{2}d&-2x_{2}x_{3}-x_{1}d&r_{5}^{2}-2x_{3}^{2}+x_{0}d&-2x_{3}z\end{pmatrix}~{},$
(31)

where $r_{5}^{2}=x^{2}+z^{2}$ and

$h(x,z)=-d^{4}z^{2}+d^{2}(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2})+\rho^{2}((\vec{x}+\vec{d})^{2}+z^{2})^{2}~{},$
(32)

with $(\vec{x}+\vec{d})^{2}=x^{2}+2x_{0}d+d^{2}$. Note that $A_{5}^{a}$ in
(31) (last column) is nonzero, but when $d\rightarrow\infty$, $A_{5}^{a}\to 0$
and we recover the uplifted 4D instanton solution. The 5D Lagrangian density
corresponding to (31) is

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{48d^{8}\rho^{4}((\vec{x}+\vec{d})^{2}+z^{2})^{4}}{(-d^{4}z^{2}+d^{2}(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2})+\rho^{2}((\vec{x}+\vec{d})^{2}+z^{2})^{2})^{4}}~{}.$
(33)

A two-dimensional contour of (33) is shown in Figure 4. The topological charge
density is

$D=\frac{48d^{8}\rho^{4}((\vec{x}+\vec{d})^{2}-z^{2})((\vec{x}+\vec{d})^{2}+z^{2})^{3}}{(-d^{4}z^{2}+d^{2}(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2})+\rho^{2}((\vec{x}+\vec{d})^{2}+z^{2})^{2})^{4}}~{},$
(34)

and is shown in Figure 5. For any $z\not=0$, the topological charge is zero,
indicating that this is an instanton-anti-instanton configuration. To
understand the behaviour of the topological charge density it is useful to
consider the two limits $d\gg\rho$ and $d\ll\rho$. In the first case, at any
fixed value of $z\ll d$, we have an instanton of size $\rho$ located at $x=0$
with an anti-instanton of size $z^{2}/\rho$ located at $x_{0}=-d$ (Figure 5,
left). In the $d\ll\rho$ limit, the instanton has a size $d^{2}/\rho$ and is
located at $x_{0}=-d$, with the anti-instanton of size $\rho z^{2}/d^{2}$
again located at $x_{0}=-d$ (Figure 5, right). In both cases, the anti-
instanton grows as $z$ increases and annihilates with the instanton at $z\sim
d$.

We can obtain different solutions by applying the special conformal
transformation (53) using other values of $\delta_{1},\delta_{2}$. However,
they give qualitatively similar solutions as the $\delta_{1}=\delta_{2}=d$
solution. Furthermore, by repeatedly applying the conformal transformation
(53) with different constant four-vectors $d^{(\alpha)}_{\mu}$
$(\alpha=1,2,\dots)$, one can obtain multi-instanton-anti-instanton solutions.

Figure 5: The topological charge density (34) (multiplied by the volume factor
$4\pi x^{4}$) of the instanton-anti-instanton (31) with $\rho=1$ and
$d=10\,(0.01)$ for the left (right) figure. The contours are drawn for
$x_{0}<0$ and $x_{1}=x_{2}=x_{3}=0$ and show an anti-instanton appearing at
$x_{0}=-10\,(-0.01)$ whose size grows with increasing $z$ and annihilates the
instanton located at $x_{0}=0$. Note the contours of the topological charge
density are much smaller in the right figure.

###  3.4 Meron solutions

####  3.4.1 Uplifted 4D meron

The 5D bulk also admits meron solutions. Uplifting the well-known 4D meron
[11, 12], we have the $z$-independent solution (for $x>0$)

$f(x,z)=\frac{1}{2}~{},$ (35)

which is singular at $x=0$ with $f(x,z)=0$. This has topological charge
$q=\frac{1}{2}$. The Lagrangian density is

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{3}{2x^{4}}~{},$ (36)

with 5D action

$S_{5}=3\pi^{2}\frac{L}{g_{5}^{2}}\log\left(\frac{x_{\rm IR}}{x_{\rm
UV}}\right)\log\left(\frac{z_{\rm IR}}{z_{\rm UV}}\right)~{},$ (37)

where $x_{\rm IR\,(\rm UV)}$ are the infrared (ultraviolet) cutoffs of $x$.
The 5D action is divergent in the $z_{\rm UV}\to 0$ limit but can be
renormalized by adding the UV contribution (4) to give

$S_{5}+S_{\rm UV}=\frac{3\pi^{2}}{g^{2}_{4}(z_{\rm
IR}^{-1})}\log\left(\frac{x_{\rm IR}}{x_{\rm UV}}\right)~{}.$ (38)

This reproduces the well-known 4D meron result which has a logarithmically
divergent action. The effect of the 5D bulk is again to have the Yang-Mills
coupling evaluated at the scale $z_{\rm IR}^{-1}$ (similarly to the instanton
result (22)).

Figure 6: A 2D contour of the Lagrangian density (41) for the 5D meron
(multiplied by the volume factor $2\pi^{2}x^{3}$) evaluated at 10. A point-
size meron appears at $x_{\mu}=0$ (where two Euclidean coordinates $x_{2,3}$
are not shown) which grows in size as $z$ increases.

####  3.4.2 5D meron

The solution (35) is a trivial extension of the 4D solution since it does not
depend on the 5th coordinate. It is therefore interesting to check whether
nontrivial $z$-dependent solutions exist which are not obtained from a 5D
conformal transformation of a known 4D solution. Remarkably, a simple and
nontrivial, analytic solution of the 5D AdS Yang-Mills equation of motion (14)
is found to be

$f(x,z)=\frac{x^{2}}{2x^{2}+\frac{3}{2}z^{2}}~{}.$ (39)

For $z=0$, this solution is just the 4D meron (35) with topological charge
$q=\frac{1}{2}$ that behaves as a (singular) point charge. However, for
nonzero $z$ the topological charge density becomes nonsingular:

$D=576\frac{z^{2}(2x^{2}+3z^{2})}{(4x^{2}+3z^{2})^{4}}~{}.$ (40)

This shows that the solution (39) represents a meron of finite size $\sim z$
and $q=\frac{1}{2}$, that extends into AdS5. Similarly, an antimeron
corresponds to the solution $1-f$.

Figure 7: The topological charge density (45) (multiplied by the volume factor
$4\pi x^{4}$) for $d=10$. The contours are drawn for $x_{0}<0$ and
$x_{1}=x_{2}=x_{3}=0$ and depict an antimeron appearing at $x_{0}=-10$ whose
size grows with increasing $z$ and annihilates the meron located at $x_{0}=0$
whose size also grows with increasing $z$. The topological charge density is
zero when $x_{0}=-d/2$.

The 5D Lagrangian density is given by

$\frac{1}{4}F_{MN}^{a}F^{aMN}=96\frac{4x^{4}+21x^{2}z^{2}+18z^{4}}{(4x^{2}+3z^{2})^{4}}~{}.$
(41)

For $z_{\rm UV}\to 0$ and $z_{\rm IR}\ll x_{\rm IR}$, the action becomes

$S_{5}\simeq\frac{3\pi^{2}}{2}\frac{L}{g_{5}^{2}}\log\left(\frac{x_{\rm
IR}^{2}}{z_{\rm IR}z_{\rm UV}}\right)\log\left(\frac{z_{\rm IR}}{z_{\rm
UV}}\right)~{}.$ (42)

Unlike the 4D meron, there is no singularity in the limit $x_{\rm
UV}\rightarrow 0$, consistent with the fact that the 5D meron is a nonsingular
solution for nonzero $z$.

####  3.4.3 5D localized meron-antimeron solutions

Conformal transformations can be applied to the meron solutions to generate
new 5D solutions. However, the uplifted 4D meron (35) and 5D meron (39) are
invariant under the 5D inversion and do not give new solutions. Instead, we
apply the special conformation transformation to obtain $z$-dependent
solutions. These solutions depend on the translation parameter $d$ that can be
understood as the separation between the two single pseudoparticles.

Consider first the 4D meron (35). Under a special conformal transformation
(53) with $\delta_{1}=\delta_{2}=d$, the 5D Lagrangian density is

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{3d^{4}}{2(d^{2}z^{2}-(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2}))^{2}}~{},$
(43)

while the topological charge density formally consists of a line in the bulk,
connecting the meron and antimeron, that is identically zero for any given
$z$.

Figure 8: A 2D contour of the Lagrangian density (46) with $b=10$ (multiplied
by the volume factor $4\pi x^{3}$ and assuming $x_{2}=x_{3}=0$) evaluated at
10. There are two merons at $x_{0}=\pm 10$, and two antimerons at the origin
$x_{0}=0$ which grow in size as $z$ increases and annihilate at $z\sim 10$.

Similarly the 5D meron (39) can be transformed to a new solution by a special
conformal transformation. Performing the inversions at
$\delta_{1}=\delta_{2}=d$, the 5D Lagrangian density is

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{96d^{4}(d^{4}z^{4}+13d^{2}z^{2}(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2})+4(x^{2}+z^{2})^{2}((\vec{x}+\vec{d})^{2}+z^{2})^{2})}{(d^{2}z^{2}-4(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2}))^{4}}~{}.$
(44)

The topological density is

$D=\frac{576d^{4}z^{2}((\vec{x}+\vec{d})^{2}-x^{2})(d^{2}z^{2}+2(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2}))}{(d^{2}z^{2}-4(x^{2}+z^{2})((\vec{x}+\vec{d})^{2}+z^{2}))^{4}}~{},$
(45)

and is shown in Figure 7. We see that the solution corresponds to an antimeron
appearing at $x_{0}=-d$ whose size grows with increasing $z$ and annihilates
the meron located at $x_{0}=0$ whose size also grows with increasing $z$. When
$d\rightarrow\infty$ we recover the 5D meron with $q=\frac{1}{2}$.

There are also 4D two-meron solutions that can be generalized to solutions in
AdS5 by transforming them under 5D conformal transformations. The 4D meron-
meron solution [11], which is just the sum of two 4D merons located at
$x_{0}=\pm\,b$ (without loss of generality), can be transformed using a 5D
inversion (50) (with $\delta=b$) to obtain a $z$-dependent solution.
Similarly, by replacing one of the merons by an anti-meron, we can obtain a 4D
meron-antimeron solution [13]. In both cases, after the 5D inversion, the
Lagrangian density is given by

$\frac{1}{4}F_{MN}^{a}F^{aMN}=\frac{24b^{4}(x^{2}+z^{2})^{4}}{\left(b^{4}x^{4}+(x^{2}+z^{2})^{4}-2b^{2}(2x_{0}^{2}-x^{2})(x^{2}+z^{2})^{2}\right)^{2}}~{}.$
(46)

A 2D contour of (46) is shown in Figure 8 for $b=10$, which depicts two merons
at $x_{0}=\pm 10$ and two antimerons at the origin $x_{0}=0$ which grow in
size as $z$ increases, and annihilate at $z\sim 10$.

A special conformal transformation can also be performed on both 4D meron-
meron and meron-antimeron solutions with independent translation parameters
$d_{\mu}^{(\alpha)}(\alpha=1,2,\dots)$, leading to multi-meron-anti-meron
solutions similar to the instanton case.

##  4 Conclusion

Weakly-gauged holographic models, which are dual to Yang-Mills theories
interacting with a strongly-coupled CFT, are an interesting class of theories
in which to study the effects of small instantons, since the gauge coupling of
the UV-localized gluon can grow at small distances without introducing extra
colored matter. However, contrary to expectations, we show that this feature
is nullified by topological charge conservation which forbids UV localized
instanton configurations and causes small instanton effects to be highly
suppressed. Surprisingly, given that the 5D theory has, in principle, no
knowledge of the underlying constituents in the dual 4D theory, we find that
this suppression is equivalent to that obtained from a 4D dual theory with
colored fermion constituents.

In fact this result can be understood by noting that the “cylinder” instanton
configuration remains a solution in a supersymmetric version of the 5D model,
leading to an accidental supersymmetry in the bulk. By the AdS/CFT dictionary,
the corresponding dual 4D theory is then a supersymmetric CFT that necessarily
contains colored fermion constituents. At the quantum level, the
supersymmetric CFT contributes fermion zero modes to the gauge coupling
running while the nonzero mode contributions cancel. Thus the chiral
suppression from the fermion zero modes in the instanton measure can be neatly
interpreted as a running in the gauge coupling from $1/\rho$ to $z_{\rm
IR}^{-1}$ to give the “cylinder” instanton action $S_{E}=8\pi^{2}/g^{2}(z_{\rm
IR}^{-1})$ and not the smaller action $S_{E}=8\pi^{2}/g^{2}(1/\rho)$ from a
UV-localized instanton.

Despite the absence of a UV localized instanton solution, we discover other
new solutions of the AdS5 Yang-Mills equations that are UV-localized but have
zero topological charge. These solutions are simply obtained by performing 5D
conformal transformations on known 4D solutions. This includes an instanton-
anti-instanton solution (obtained from a 5D inversion on the 4D instanton)
where an anti-instanton of size $z^{2}/\rho$ grows inside an instanton of size
$\rho$ until they annihilate at a distance $z\sim\rho$ in the 5D bulk. Other
instanton-anti-instanton solutions can be obtained by performing 5D special
conformal transformations on the 4D instanton and feature non-spherically
symmetric configurations. In fact the solution generating technique can be
extended to obtain an arbitrary number of instanton-anti-instanton
configurations.

These localized zero topological charge solutions can also be interpreted from
the perspective of accidental supersymmetry in the bulk. The instanton-anti-
instanton configurations do not preserve the 5D supersymmetry, and therefore
the corresponding dual 4D CFT does not contain any fermion zero modes at the
quantum level. This means that the contribution to the running gauge coupling
is only from nonzero modes above the scale $1/\rho$, which is consistent with
the localized 5D solutions that are truncated at a distance $z\sim\rho$ in the
bulk.

Moreover, we can apply 5D conformal transformations on known 4D meron and
multi-meron solutions. This again gives rise to UV localized meron-antimeron
configurations with zero topological charge. However, we also discovered a new
5D meron solution that is not obtained from any 5D conformal transformation on
a known 4D solution. At any fixed $z$ in the AdS bulk, it has topological
charge $1/2$ and is regular. Furthermore, performing 5D conformal
transformations on the 5D meron solution leads to new meron-antimeron
configurations.

Our analysis has made a number of simplifying assumptions on the 5D AdS Yang-
Mills theory. We have only studied smooth solutions in the bulk, neglected
gravitational interactions and worked in the limit $g_{5}\rightarrow 0$ such
that the 5D cutoff $\Lambda_{5}\rightarrow\infty$. Also at the UV boundary we
have neglected bulk effects. We expect that including these effects will
modify our solutions although the qualitative behaviour should remain
unchanged.

Despite these simplifications, our work has uncovered new nonperturbative
configurations of Yang-Mills theories interacting with strongly-coupled CFTs.
These solutions can be understood from a bulk topological charge conservation
or an accidental supersymmetry in the bulk that relates the presence of UV
localized (classical) 5D configurations to the quantum corrections in the dual
4D theory. While accidental supersymmetry necessarily implies colored fermion
constituents in the dual 4D theory, it remains unclear whether
nonsupersymmetric CFTs with colored scalar zero modes could also exist to
explain our results. In any case, the holographic approach has provided some
answers and opened a new way to study Euclidean Yang-Mills solutions.

## Acknowledgments

We thank Aleksey Cherman, Erich Poppitz, Misha Shifman and Arkady Vainshtein
for helpful discussions. The work of T.G. is supported in part by the DOE
Grant No. DE-SC0011842 at the University of Minnesota, and the Simons
Foundation. The work of A.P. is supported by the Catalan ICREA Academia
Program and grants FPA2017-88915-P, 2017-SGR-1069 and SEV-2016-0588. T.G.
acknowledges the Aspen Center for Physics which is supported by the National
Science Foundation grant PHY-1607611, where part of this work was done.

##  Appendix A 5D Conformal Transformations

Consider the transformation $x_{M}\rightarrow{\bar{x}}_{M}(x,z)$ where the
metric (2) is invariant i.e. $g_{MN}(x)=g_{MN}(\bar{x})$. Then the action
satisfies

$\int
d^{5}{\bar{x}}\sqrt{g({\bar{x}})}~{}\frac{1}{4g_{5}^{2}}F_{MN}^{a}({\bar{x}})F^{aMN}({\bar{x}})=\int
d^{5}x\sqrt{g(x)}~{}\frac{1}{4g_{5}^{2}}F_{MN}^{a}(x)F^{aMN}(x)~{},$ (47)

where

$\displaystyle A_{M}^{a}(x)$ $\displaystyle=$
$\displaystyle\frac{\partial{\bar{x}}^{N}}{\partial
x_{M}}A_{N}^{a}({\bar{x}})~{},$ (48) $\displaystyle F_{MN}^{a}(x)$
$\displaystyle=$ $\displaystyle\frac{\partial{\bar{x}}^{R}}{\partial
x_{M}}\frac{\partial{\bar{x}}^{S}}{\partial x_{N}}F_{RS}^{a}({\bar{x}})~{}.$
(49)

Thus if $A_{M}^{a}({\bar{x}})$ is a solution to the 5D Yang-Mills equations of
motion (5), then given (47), $A_{M}^{a}(x)$ will also be a solution.888Note
that the new solution gives a different value of the action because the
integration limits in (47) are not transformed. Also the new solution can have
a different topological charge, since the topological charge density is not
invariant under the 5D conformal transformation. Note that if
$\frac{\partial{\bar{x}}^{N}}{\partial x_{M}}=\delta_{MN}$ then no new
solution is generated.

We next summarize the possible isometries of AdS5. These consist of discrete
and continuous transformations which keeps the metric invariant.

###  A.1 Discrete Transformations

A discrete isometry of AdS5 space is the 5D coordinate inversion, given by

$x_{M}\rightarrow{\bar{x}}_{M}(x,z)=\frac{\delta^{2}}{x^{2}+z^{2}}x_{M}~{},$
(50)

where the inversion is performed at the distance scale (or radius) $\delta$. A
useful relation is

${\bar{r}}_{5}^{2}\equiv{\bar{x}}_{M}{\bar{x}}^{M}=\frac{\delta^{4}}{x^{2}+z^{2}}~{}.$
(51)

###  A.2 Continuous Transformations

The continuous isometry group of Euclidean AdS5 is $SO(5,1)$ which is a 15
parameter group consisting of the 10 parameter Poincare group (of 4D
translations, 3D rotations and Lorentz boosts) and an additional 5 parameters
that parameterize dilations and special conformal tranformations. These are
given by

####  A.2.1 Dilation

This consists of the transformation

$x_{M}\rightarrow{\bar{x}}_{M}(x,z)=\lambda\,x_{M}~{},$ (52)

where $\lambda$ is a real parameter. Note that two successive discrete
inversions (50), first with radius $\delta_{1}$, followed by a second with
radius $\delta_{2}$, is equivalent to a dilation with
$\lambda=\frac{\delta_{2}^{2}}{\delta_{1}^{2}}$.

####  A.2.2 Special conformal transformation

This transformation is given by

$x_{M}\rightarrow{\bar{x}}_{M}(x,z)=\frac{\delta_{2}^{2}}{\delta_{1}^{2}}\frac{x_{M}+d_{M}\frac{1}{\delta_{1}^{2}}(x^{2}+z^{2})}{\left(1+2\frac{d\cdot
x}{\delta_{1}^{2}}+\frac{d^{2}}{\delta_{1}^{4}}(x^{2}+z^{2})\right)}~{},$ (53)

where $d_{M}=(d_{\mu},0)$ is an arbitrary constant vector in the $x_{\mu}$
direction. The special conformal transformation corresponds to an inversion at
the distance scale $\delta_{1}$, followed by a translation ($x_{M}\to
x_{M}+d_{M}$), then a final inversion at the distance scale, $\delta_{2}$. The
radial Euclidean distance-squared transforms as

${\bar{r}}_{5}^{2}\equiv{\bar{x}}_{M}{\bar{x}}^{M}=\frac{\delta_{2}^{4}}{\delta_{1}^{4}}\frac{x^{2}+z^{2}}{\left(1+2\frac{d\cdot
x}{\delta_{1}^{2}}+\frac{d^{2}}{\delta_{1}^{4}}(x^{2}+z^{2})\right)}~{}.$ (54)

Note that the metric (2) remains invariant under (53) while the gauge fields
transform as (48).

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