Identifying topological structures with adjoint mode filtering

: We present new investigations on the Adjoint Filtering Method (AFM), a proposal for filtering gauge configurations by using adjoint zero modes. This method relies on the existence of the Supersymmetric Zero Mode (SZM), whose density follows the gauge action density for classical configurations. We review how a similar construction on the lattice is implemented based on the overlap operator and test the method against smooth configurations showing a remarkable agreement with the expected densities even when pairs of fractionals instanton/anti-instantons are present and noise is added. Then we explore the application of the method to Monte Carlo generated configurations based on SU(2) gauge group. The tuning of the parameters and quantitative results are explicitly shown for a T 3 × R lattice. We show explicit examples comparing the AFM to the density obtained from the Gradient Flow. The agreement is remarkable for some specific configurations containing fractional instantons with the advantage that the AFM does not modify the underlying structures.


Introduction and motivation
Lattice gauge theories have proven to be a very useful tool to investigate non-perturbative properties of gauge theories.This includes the phenomenologically interesting case of QCD, allowing a first-principles computation of many experimental quantities involving strong interactions. 1In addition, there are other field theories which are of great interest either for being candidate extensions of the Standard Model or for their theoretical interest.The latter includes the case of Yang-Mills theory and its supersymmetric extensions.The standard computational techniques used in Lattice Gauge Theory studies are dominated by the Monte Carlo method and related importance-sampling numerical techniques.There is however a challenge in trying to achieve a qualitative and semiquantitative understanding of the origin and magnitude of the non-perturbative phenomena involved.Our approach to describing strongly interacting theories is based on a semiclassical analysis, which has been demonstrated to work at least in certain regimes, as explained towards the end of this Section.Apart from searching for new formulations to describe strongly interacting phenomena and allow analytical computation, it is clear that analysing samples of lattice configurations can provide answers or clues regarding the underlying origin of these phenomena as well as new analytical methods.This is the idea which serves as a motivation for the present work.
Lattice configurations describe snapshots of the vacuum of the theory and this includes its structure at different length scales.It is to be expected that the most relevant length scale is given by the inverse mass gap or inverse Lambda parameter of the theory.Nonetheless, quantum fluctuations grow at short length scales as a result of the ultraviolet divergences of these theories.In practice, this implies that the action density, for example, is dominated by this short-wavelength fluctuations and it is hard to see any underlying long range structure appearing at the physical scale mentioned earlier.Extended semiclassical objects like instantons and fractional instantons are expected to be present but they are completely covered up by the noise.In this paper we are going to explore a filtering method that can eliminate the noise and allow to uncover these features.The ultimate goal would be to unveil secrets contained in these configurations.
The general idea has a longer history.A special role was initially played by different iterative smoothening techniques such as cooling or smearing [2][3][4][5][6].These methods are local and have a characteristic smoothening radius that has to be monitored carefully in order to erase the high frequency fluctuations without altering the contents at the relevant scales one wants to explore.Disregarding this careful analysis has sometimes led to unjustified general critiques of the method.The panorama has been greatly clarified with the introduction of gradient flow techniques in which the smearing radius becomes a continuous parameter [7][8][9].The effects are quite similar to other smoothening techniques, but the method has a well-defined continuum limit [8].More recently the effects of the gradient flow on instanton properties have been monitored and even an extrapolation in the flow time has been applied [10].Despite the fact that they have proven to be very useful in the analysis of configurations, smoothening methods have also well studied limitations.Smoothening provides a limited window for the underlying structures of the gauge configurations altering some structures outside of it as we explained in later sections of this work.
The criticism on cooling techniques has been an initial motivation to study alternative approaches.More generally, the limitations of the smoothening make an alternative point of view, which does not so much rely on the change of the gauge field, desirable.More importantly, the semiclassical analysis aims for a complete description of the relevant features of the QCD vacuum.Therefore one should include alternative observables and approaches to confirm that a consistent picture emerges.To demonstrate the presence of instanton structures in the configurations, several authors [11][12][13] made use of the solution of the Dirac equation in the gauge field background.In the continuum, solutions of the gauge field equations of motion like the BPST instanton [14] or the KvBLL calorons [15][16][17][18] have associated solutions of the Dirac equation in the fundamental [19][20][21][22][23] and adjoint representation [24][25][26], whose densities trace back the presence of these structures.The advantage is that under the presence of high frequency quantum noise, these densities are less affected than the corresponding gauge field observables.The situation becomes optimal if we consider the Dirac equation in the adjoint representation because in that case one particular solution of the Dirac equation (the supersymmetric zero-mode SZM) has a density which is identical to the action density of the gauge field.When adding noise to the gauge field, the corresponding SZM density implements a filtering of this noise with respect to the action density itself.This idea was the germ of a proposal made a few years ago by one of the authors and collaborators [27,28] to produce an effective filtering method applicable to lattice configurations.In those papers a special property of the SZM was found which distinguishes it from other solutions.In those papers tests were made to show that the filtering was working properly for some smooth configurations before and after applying noise to them.However, the ultimate purpose of applying this methodology to the relevant Monte Carlo generated configurations was not attempted and in this publication we will try to take the first steps in this direction.Note that the adjoint representation has been considered also in different context for a numerical analysis of gauge field configurations [29,30].
What features of the QCD or pure Yang-Mills vacuum are we aiming to identify?Our goal is to explore the validity of semiclassical ideas.This has a long history starting with the work of Callan, Dashen, and Gross [31,32] based on merons.Other ideas in the same spirit are the instanton liquid model of Shuryak [33], and the fractional instanton liquid model (FILM) proposed by one of the present authors [34,35].Our belief in semiclassical ideas is supported by theories in lower dimensions [36] and with additional matter content [37][38][39].In this program the analysis of lattice configurations is expected to play a crucial role.It is essential, in our opinion, to consider first regimes where the semiclassical analysis is under control.In such kind of regimes, the numerical method can be tested reliably.Moreover, such a controlled regimes can be even continuously connected to full vacuum and hence it is possible to conjecture more fundamental insights from such an analysis.Monitoring this process on the lattice one expects to find clues to validate these ideas.
An specific regime in which the semiclassical ideas and methodology is expected to work is that in which the theory is formulated on a compact manifold in which the size in some directions is small.The validity of semiclassical analysis in this regime is a consequence of asymptotic freedom.This has already been proven by the studies carried out many years ago by one of the authors and collaborators [40].The studies were performed for a lattice of size of size L 3 s × L 0 , in which L s ≪ L 0 .It corresponds to a Hamiltonian study of Yang-Mills theory in a small spatial box.This was inserted into a program initiated by Lüscher [41] of studying the transition from small to large volumes in asymptotically free theories.The results showed clearly how perturbative and semiclassical methods were successful in describing the lattice results in a range of physical spatial sizes.A leading role in this analysis was played by fractional instanton solutions which tunnel between perturbative vacua and lead to a recovery of isotropy and confinement [34,35,42].It is on the basis of these results that the authors proposed a picture of the Yang-Mills vacuum based upon a liquid of fractional instantons which could explain and relate a finite topological susceptibility and string tension.Recently there has been a renewed interest in fractional instantons and their dynamical role both in analytic work [43][44][45][46][47][48][49] and in numerical one [50,51].In most of the afore-mentioned studies at finite volume an important role is played by boundary conditions: the twisted boundary conditions (TBC) introduced by 't Hooft [52][53][54].Indeed, boundary conditions are crucial at small volumes but should become irrelevant at large ones.Semiclassical arguments suggest why TBC are more effective in revealing the transition from small to large volumes.Indeed, even in the absence of twist, solutions of the Yang-Mills classical equations of motion appear that are made out of assemblies of fractional instantons.Twist only imposes a constraint on the number of them.This will become clear with the configuration examples studied in this paper.
Apart from the previous Hamiltonian picture with three small directions, that we can describe as a T 3 × R topology, more general topologies can be considered.If two directions (L 1 and L 2 ) are much smaller that the other two (L 0 and L 3 ), a T 2 × R 2 topology is obtained.Again we consider twisted boundary conditions in this setup and fractional instanton structures also play a role here.These solutions were studied and described as vortex-like [55][56][57], since they can be looked upon as R 4 solutions periodic in two directions, which act as the world sheet directions in this setting.This two-dimensional picture is particularly attractive as the corresponding fractional instantons are well separated since their action-density profile is exponentially localized.This allows creating configurations containing both fractional instantons and anti-instantons.Although these configurations are not exact solutions of the classical equations of motion, when the separation of structures is large they give rise to quasi-zero modes of the Dirac operator that can be neatly identified.Furthermore, despite not being protected by a non-zero topological charge they can be extremely stable under gradient flow.The vortex-like fractional instantons do carry center flux as shown in the original papers [55][56][57].They can provide an underlying gaugeinvariant substrate [56] to the center-vortices obtained after gauge-fixing and studied by many lattice studies [58][59][60][61][62].The fact that there is a regime in which they can be studied by semiclassical methods is very attractive and connects with recent studies by other authors which also considered the T 2 × R 2 regime [46,63].
The third geometric situation R 3 ×S 1 is special in several ways.It is the only situation in which a Q = 1 analytic solution was found showing the decomposition of the action into well-separated fractional instanton structures: the KvBLL calorons [15][16][17][18].In pure gauge theory, this geometry corresponds to finite temperature field theory, where the temperature corresponds to the inverse of the S 1 radius.The semiclassical regime at small radius is therefore separated by the deconfinement transition from the physics of the confined vacuum.However, if the theories are deformed, the transition can be avoided and a continuous connection between the semiclassical regime and the confined vacuum is established; this is the basic approach behind different studies of adiabatic continuity [39,[64][65][66].The most well established deformation is N = 1 supersymmetric Yang-Mills theory on R 3 × S 1 with periodic boundary conditions for the fermions.From the lattice perspective there have been a lot of work studying correlations of the finite temperature behaviour of the theory with topological structures (See for example [67][68][69][70] and references therein).
In our opinion all of these investigations have shown in a very consistent way that the semiclassical approach can indeed provide an understanding of the QCD vacuum.They still need to be extended and improved in several ways and input from numerical simulations is essential.The final understanding of the underlying long-range structure of lattice configurations is the most difficult and complex task and remains a challenge.Whether the semiclassical regimes previously mentioned help in disentangling this problem in the spirit of the proposal made in Refs.[34,35] (see Ref. [78] for a recent summary) remains to be seen.Presumably, effort by many researchers would be needed to achieve this goal.Nevertheless, as this paper illustrates, the present study reveals rather beautiful structures appearing in Yang-Mills theory.
In the present work we improve the methods for the filtering required to study the longrange structures on Yang-Mills configurations.As explained, the main two categories are smoothening techniques and methods based on low modes of the Dirac-Wilson operator.We are comparing two methods which are, for the reasons discussed above, the most advanced techniques in each of the two categories: the Gradient Flow (GF) and the Adjoint Filtering Method (AFM).These two techniques are explained in Sec. 2. In Sec. 3 we introduce the semiclassical regime with twisted boundary conditions and how we prepared test configurations with a given semiclassical background.Smooth test configurations as well as configurations with a controlled noise level are investigated in Sec. 4. This confirms the correctness of our numerical methods and provides a first hint concerning the specific properties of the semiclassical contributions.Finally, in Sec. 5 Monte Carlo generated ensembles of configurations for pure Yang-Mills theory on T 3 × R are analyzed.We have chosen the simulation parameters such that a controlled semiclassical behaviour is expected, but still ensuring that the density of topological excitations is large enough.In this way we can show how the optimization of the parameters for the methods can be achieved, which is in general quite challenging for a complete ensemble of configurations.

Filtering techniques
As mentioned in the introduction, we will study different filtering techniques such that applied to Monte Carlo generated lattice configurations could resolve the underlying presence of topological structures that could explain the long-range features of the theory.All techniques have pros and cons and the use of them can show the robustness of the results and lead to an optimal combination.
The quantities that are of interest for our study are the action density S(x) and the topological charge density q(x) related to long range seminclassical contributions.These can be defined from the gauge fields as The simplest lattice implementation of these quantities comes from expressing the field strength components in terms of clover plaquettes.This definition we denote as q c (x), S c (x) if not combined with smoothing techniques.

Gradient flow
As mentioned in the introduction this method is an improvement of the cooling/smearing methods that were used in the early studies.In the continuum the gradient flow evolves a given gauge field configuration from an initial value following the gradient of the Yang-Mills action functional.Thus, the action decreases with the flow parameter τ .Intuitively, the effect is to smear out the field over an extension √ 8τ , the smearing radius.Thus, quantum fluctuations at scales smaller that this radius are strongly reduced without modifying substantially the long range structure present in the original configuration.The total topological charge of the configuration does not change under this continuous process.However, the flow process modifies the initial configuration, for example by erasing instanton/antiinstanton pairs separated by distance smaller that the smearing radius.This is not the only effect that the flow can have.If the base manifold is a torus, which is a particular relevant topology for lattice studies, there is a breaking of scale invariance leading to distortion of the size distribution of instantons.This is particularly dramatic for Q = 1 since the limiting instanton size tends to zero.Indeed, for this reason there are no (self-dual) instantons on the torus as proven in Ref. [74].
All this flow dynamics should be well understood before going over to the lattice version of the flow.On the lattice there are new challenges appearing.The choice of the lattice action used for the gradient is not irrelevant.Like the lattice size, also lattice artefacts destroy the scale-invariance of the theory.In practice, as studied in Ref. [79], this induces a modification of the size parameter of an instanton by the flow.Curiously, the growth or decrease can be tuned by choosing different lattice actions.In principle a naive tree-level improvement of the lattice action minimises these lattice artefacts.Nonetheless, one can even use this effect to compensate the flow of the continuum actions towards zero-size instantons with contributions from lattice artefacts [79].Once the size becomes of the order of the lattice spacing the instanton disappears altogether from the lattice and the topological charge changes along the flow.This phenomenon was observed in the early studies made with cooling [6,80] but it holds for gradient flow as well.
All these phenomena are present in our studies.Nonetheless, the flow produces very nice smooth configurations revealing the presence of structures larger than the smearing radius.
To distinguish the definition we denote as q c (τ ), S c (τ ) the quantities obtained at flow time τ .If not otherwise specified, the Wilson action is used in the flow and the observable is computed with the clover definition on the flowed observables.Further details of the implementation can be found in [8].

Adjoint filtering method (AFM)
Motivated by the previous considerations it would be interesting to find a filtering method which would produce nice smooth pictures without distorting the size distribution of in-stantons and eliminating instanton/anti-instanton pairs.This led one of the authors and collaborators to present a method based on zero modes and quasi-zero modes of the adjoint Dirac operator [27,28].The idea will be explained below.
Given a gauge field which is a solution of the classical equations of motion one can construct a solution of the Dirac equation in the adjoint representation as follows: for V an arbitrary 4-spinor.This can be called the supersymmetric zero-mode (SZM), since it corresponds to the gluino partner in the supersymmetric vector representation.
Using the projectors P ± = (I ± γ 5 )/2 one can obtain the positive and negative chirality counterparts which are solutions of the left and right-handed Weyl equation respectively.
Choosing an appropriate basis these bi-spinorial fields can be written as follows: 3) where σ µ is a quaternionic 2 × 2 matrix (σ µ = (I, −i⃗ τ )) given in terms of the Pauli matrices (τ i ), and σµ are their adjoints.Since, v ± are arbitrary bi-spinors, the solution runs over a two-dimensional space spanned by one mode and its charge conjugate.Choosing v ± of unit norm, the density of these modes satisfy (2.6) When ultraviolet fluctuations are added to the classical solution, the filtering property of these solutions becomes apparent.The AFM densities (S AF M (x) and q AF M (x)) are obtained from the lowest modes of the adjoint Dirac operator using (2.6).Despite the modification by the noise, these densities remain close to the original zero-mode values while the values computed from the gauge field directly show large deviations.This has been shown analytically in Ref. [28] and becomes also clear from the results presented in Sec.4.2.
In general a gauge field configuration can give rise to several different Dirac zeromodes.The index theorem relates the number of zero-modes in any representation to the total topological charge as follows: where n ± (R) is the number of left/right handed zero modes of the Dirac operator in representation R, and c R and d R are values of the quadratic Casimir and the dimension of the representation in question (For the fundamental representation R = F ). Thus, for nonzero topological charge there should be zero-modes even if the gauge field is not a solution of the classical equations of motion.One might wonder how this affects the method, given that the SZM is only a zero-mode for a classical solution, but it just implies that in general near zero-modes have to be considered.Furthermore, even if the gauge field is a classical solution there are 2N Q = n + (A) − n − (A) zero-modes in the adjoint representation.How can one distinguish the SZM within this set?Notice that the adjoint representation is real and zero-modes (and other eigenstates of the Dirac operator) come in CP pairs just as we saw for the SZM.Indeed, the two Weyl spinors of a CP pair can be combined using a quaternionic 2 × 2 matrix representation.Hence, the SZM is just one pair within these N Q zero-mode pairs.It is expected to select the most relevant information from the zero modes.
There is one particular feature of the SZM which allows one to distinguish it from other zero or quasi-zero modes of the adjoint Dirac operator.Taking for example v t + = (1, 0) the positive chirality SZM becomes Notice that the first component is real at all points of space.This is a very strong condition not shared by other zero-modes.A similar result applies for the negative chirality component ψ − (x).This property was the basic ingredient put forward in Ref. [27] to select the SZM from within other (quasi)zero modes.In Ref. [28] a more elegant way to define the method was proposed.The left/right SZM was identified with the eigenvector of lowest eigenvalue of the operators where P 0 is a projector onto the subspace of states satisfying the reality condition, while P ± projects onto the positive and negative chirality sectors.

Lattice implementation of the adjoint filtering method (AFM)
On the lattice the analysis of zero-modes and quasi-zero modes depends strongly upon the choice of lattice Dirac operator D L .It is clear that the construction relies strongly on chiral symmetry represented on the lattice by the Ginsparg-Wilson relation.This relation allows a lattice definition of an integer valued topological charge using the Dirac operator in the fundamental representation [81]: For smooth configurations this coincides with the topological charge obtained from the gauge field up to lattice artefacts.Similarly, one can also construct a corresponding lattice definition of the topological charge density as follows [82] q also consistent with gauge field definition up to lattice artefacts.This is a truly remarkable result.However, in practice, a cut on the number of modes that enters in the trace needs to be done.The cut can be understood as a certain filtering [12,13].In fact, a small number of modes might be enough to signal the presence of topological structures on the configurations.Nonetheless, the filtering not only reduces the ultraviolet noise but also distorts the structures even for smooth configurations.Zero-modes in the fundamental representation do not have densities that match that of the gauge field.This feature only appears for the SZM in the adjoint representation.
The cleanest definition of the AFM on the lattice is obtained via the massless overlap Dirac operator (D ov ) [83,84]. (2.12) with links V µ (x) in the adjoint representation.The Overlap operator satisfies the Ginsparg-Wilson relation This relation implies that D ov , in contrast to the Wilson-Dirac operator D w , is a normal operator and implements chiral fermions on the lattice, which leads to a cleaner identification of the zero modes.For this reason it has been used in many studies that tried to identify QCD vacuum structure such as [29,30,81,[85][86][87] and in particular earlier studies of the AFM [28,88].
We have tested different rational and polynomial approximations for the approximation ϵ(x) of the sign function.However, the optimizations of numerical methods for the AFM are not the focus or the current work and we decided to use the simple rational approximation [84,89] with coefficients The AFM is based on the lowest eigenmodes of the operator which is the lattice counterpart of (2.9).The chirality projections P ± can be consistently applied in case of the overlap operator and P 0 projects the real part of the first component to zero.
The lowest eigenvalues have been computed using the PRIMME library [90,91].The method provides the smallest N ev eigenvectors (|ψ λ i ⟩) and eigenvalues (λ i ) with a given accuracy δ defined by3 (2.17) We define the SZM on the lattice as the eigenvector of smallest eigenvalue (λ 1 ) of O ± assuming that it is sufficiently well separated from the rest of the eigenvalue spectrum.The vectors obtained from the eigenvalue algorithm are always normalized to one, which means a normalization factor has to be added.
There is not always a clear gap between a single SZM and the rest of the spectrum of O ± .In general several lowest eigemodes have to be considered in the construction of the topological charge densities.Therefore we define for a given N cut , which is ideally chosen according to a gap in the spectrum, with appropriate normalization factors α (±) i and q SZM (x) = q AF M (x) for N cut = 1.As explained later, especially for noisy configurations, allowing N cut > 1 turns out to be essential.In the analysis of Monte Carlo generated configurations, it turns out that instead of a fixed number of modes N cut a cut according to a fixed maximal size for the eigenvalues λ i < λ cut is more feasible.The normalization factors can be set to the same value for all modes and this overall normalization can be selected according to the total topological charge of the configuration.In case of the SZM density, we have applied this procedure.When several modes are included, we have found that a good choice for the normalization is α Nc , which we have used in our numerical studies.It is important to note that the overlap operator has an additional parameter, which is the mass (or Hopping parameter κ) in the Wilson-Dirac kernel.This parameter is not relevant in the continuum limit as long as it is in a certain range.In investigations of lowest eigenmodes of the overlap operator in the fundamental representation, it has been found that the κ dependence leads to an effective smearing radius [92].As we will see, in the AFM, κ will need to be tuned correctly, such that the correct number of zero modes are included in the overlap operator.

Semiclassical contributions and twisted boundary conditions
The main interest of our study is to be able to identify semiclassical structures within Monte Carlo generated configurations.These include normal instantons, but also objects having fractional topological charge known as fractional instantons.These objects appear naturally when considering a torus with twisted boundary conditions (TBC) [43,48,93,94].These are topological sectors that emerge when constructing gauge bundles with gauge group SU(N)/Z(N) over manifolds having non-trivial two-cycles.Physically these sectors correspond to quantized Z(N) fluxes through these two-cycles.On the torus one has one element of the center per plane z µν = e 2πinµν /N .It is convenient to decompose the antisymmetric matrix n µν into two 3-vectors in the standard way (n 0i = k i and n ij = ϵ ijl m l ).'t Hooft realized (see also [95]) that twist is related to the topological charge as follows: Thus, for non-orthogonal twists ( ⃗ k • ⃗ m ̸ = 0 mod N ) the topological charge is fractional.The self-dual (anti-self-dual) configurations in this bundle are the so-called fractional instantons (anti-instantons).Since they have minimum action they can be obtained by gradient flow.This process has been shown to converge in the continuum [96] and has been used on the lattice to obtain these configurations for different gauge groups, twist tensors and torus sizes [97][98][99].We apply the same methods here to generate fractional instantons having minimal action Q = 1/N .Notice that, as implied by the index theorem, they are unique up to translations, but their size is determined by the torus size.However, even if some of the torus periods go to infinity the solution has a stable limit.
Fractional instantons have been seen to play a major role in a semiclassical description of the physics of gauge fields in a T 3 ×R environment [40,100].They actually contribute to the restoration of rotational invariance, center-symmetry, and to recover the value of the string tension.We have chosen this regime to test the filtering methods on Monte Carlo generated configurations.We emphasize that, contrary to the fundamental representation case, the Dirac operator in the adjoint representation and its modes have no singularity or multivaluedness in the presence of twisted boundary conditions.Thus the AFM can be applied straightforwardly.
It is important to realize that fractional instantons are local structures which satisfy the classical equations of motion.This means that they appear also in configurations having no twist and vanishing topological charge.This will be very well exemplified in the explicit smooth configurations that we have used to test the AFM method, to be described below.

Constructing configurations
In this subsection we explain the basic procedure employed in generating test configurations for our analysis.They correspond to different situations organised with a varying degree of complexity.The simplest examples are smooth configurations corresponding to solutions of the classical equations of motion having non-zero values of the topological charge Q.These are interesting cases in which the number of zero-modes of the Dirac equation is dictated by the index theorem (3.1).
As we mentioned before, it is relatively easy to obtain the unit fractional instantons having Q = 1/N by cooling/gradient-flow techniques.Once they have been constructed, one can use them as a building block to construct more complex examples.One of the obvious ways is by replicating the torus in any direction.On the lattice the new torus size in µ direction becomes L ′ µ = p µ L µ , where L µ is the previous torus size and p µ an integer.The new configuration is also a solution of the classical equations of motion with topological charge Q ′ = p 0 p 1 p 2 p 3 /N and with a new twist tensor is n ′ µν = p µ p ν n µν .Since n ′ is defined modulo N , with appropriate choices the new configuration hase no twist n ′ = 0 and the topological charge Q ′ is an integer.All these lattice configurations are to a very good approximation self-dual and the action saturates the Bogomolny bound.
Another manipulation that can be done uses time reversal.This operation changes instantons into anti-instantons and viceversa.The interesting thing is that one can glue the instanton and anti-instanton together into a new configuration with larger temporal torus size in the time-direction.It has vanishing topological charge Q ′ = 0 and no twist in the temporal direction n ′ 0i = 0, but the action is twice the one of the original fractional instanton.At the time where the reflected configuration is connected, the chromoelectric field changes sign.Hence smoothness requires this field to be small at these points.Obviously one can combine these steps to get more general configurations.
Ultraviolet noise can be added to the configurations applying a number (N MC ) of standard heat bath Monte Carlo steps at a given gauge coupling β.The smaller β and the larger N MC the more noise is introduced in these heated configurations.In our test we choose the parameters such that the large distance content remains unchanged in this procedure.Combining several steps of heating and cooling/gradient flow steps one could also distort the symmetric structures.Furthermore, by choosing different lattice actions in the gradient flow one use lattice artefacts to our benefit in distorting the size distributions.The final goal of the paper is, of course the application of the method to thermalized Monte Carlo generated configurations.In Tab. 1 we summarize the configurations used in the present paper.1. Summary of the configurations used to test the method.Ξ AF M is the cross-correlation computed between q AF M (x) and a smooth reference density, q c (x).In case of the smooth configurations, this is directly computed from the gauge field.For 4.2 we used q c (x) from the configuration before the noise has been added.For MC 1 and MC 2, the clover definition at a late flow time q c (x, τ ′ = 4) is used as a reference.Furthermore, in MC 1 and MC 2 a small amount of gradient flow has been applied before the AFM (τ = 0.5).Ξ AF M in the table corresponds to the average of the complete ensembles in this case, see Sec. 5.

Comparison of numerical methods on tests configurations
In this section we present results of the AFM for different test configurations representing certain scenarios relevant for the final analysis ordered according to their complexity.These configurations have a known semiclassical background which can be used for cross-check and verification.This provides also some hints on how the results are affected by the parameters of the method and how to optimize them.The GF is used in these examples only as a tool to manipulate the configurations.We set the parameter for the convergence of the iterative solver (2.17) to δ = 10 −9 .
In order to estimate the quality of the filtering, the difference between the topological charge density obtained from the gauge field q c (x) and the one obtained using the AFM q AF M (x) has to be quantified.This can be done using the cross-correlation of the densities determined with two different methods A and B [12,92], To remove ambiguities related to the normalization, one can take the ratio of the latter with respect to its geometrical mean

.2)
A refers to some filtering technique and B always refers to the gauge definition q c (x) in the following, therefore we use the notation Ξ A .The quantity signals the correlation between the different densities: if the distributions obtained with both methods are the same then Ξ A = 1, while for completely different distributions Ξ A = 0.

Smooth configurations
On smooth configurations with a given semiclassical content, the topological charge density can be directly obtained with the field theoretical definition (2.1).This provides an unbiased cross-check, which does not rely on other methods or assumptions.The filtering technique should reproduce the semiclassical background chosen in the construction of the configurations.

Eight fractional anti-instantons, Q = −4
Using the methods explained in Sec.3.1, we have constructed a configuration on a V = 8 4 lattice with 8 fractional anti-instantons.We started with an V = 8 × 4 3 SU(2) fractional anti-instanton and replicated twice every spatial direction eliminating the twist and producing a configuration with Q I = −4 which is to a very good approximation anti-selfdual.We then measured the smallest eigenvalues of the chirally projected overlap operator P ± D ov P ± , Tab. 2, and found 16 zero modes in accordance with the index theorem.This configurations is challenging given the small lattice size occupied by each fractional anti-instanton implying large lattice artefacts.As a result, the topological charge defined on the lattice by clover plaquettes, Q c = −3.5, deviates significantly from the expected integer value.
Nevertheless the topological density defined in this way, still resembles the expected eight fractional anti-instantons.For better visualization, we show in Fig. 1a a two-dimensional slice of the lattice which captures two of them.In the AFM, we determined the four smallest eigenvalues of the O − operator Tab. 2. In this case there is a considerable gap between the smallest eigenvalue, corresponding to the SZM, and the rest of the spectrum.The topological charge density obtained with the AFM reproduces accurately the one of the gauge field definition as shown in Fig. 1b.To show the consistency at a more quantitative level, we have performed a parabolic fit around the maxima of each structure according to The results are summarized in Tab. 3. The fit parameters obtained in this way are indeed very similar which confirms the consistency of the method.Furthermore, the crosscorrelation between the two determination is close to one (Ξ AF M = 0.982).We also applied the GF for a long flow time τ = 100 to confirm the stability of the structures and their character as local minima of the action.
x 0 /a y 0 /a z 0 /a t 0 /a ρ 0 /a h 0 /a q c 1.50 2.78 0.887 3.50 59.47 0.0038 q AF M 1.50 2.78 0.88 3.51 51.38 0.0035 q c (τ = 100) 1.50 2.78 0.78 3.51 50.95 0.0038 Table 3. Parameters obtained by fitting to a parabola (Eq.(4.3)) using 5 points around a particular local maximum for the configuration discussed in Sec.4.1.1.Comparison of the results for the AFM (density of the SZM) and gauge field definition.The result for the large flow time shows the stability of the solutions.

Configuration with a fractional instanton/anti-instanton pair, Q = 0
The starting point for the construction of this configuration was a Q = 1/2 fractional instanton in a V = 8 4 lattice with non-orthogonal twisted boundary conditions.We glued this configuration to its time-reversed to produce a smooth configuration on a V = 16 × 8 3 lattice which contains a fractional instanton/anti-instanton pair.This is an interesting configuration to study, since the total topological charge vanishes and it is no longer selfdual or anti-self-dual.Furthermore, it is a situation that must be present in possible realistic descriptions of the Yang-Mills vacuum as predicted in the fractional instanton liquid model [78].
For any smoothing method this kind of configurations presents a major challenge, since the smoothening process tends to erase these structures.They do not correspond to minima of the classical action and smoothening methods relying on a minimization of the action, like GF or cooling, lead to important modifications and eventually to the annihilaton of the pair.As mentioned earlier, this is a process that occurs in the continuum and not due to the lattice discretization.Obviously, the rate of annihilation of a pair depends on the ratio of the separation with respect to the size of the objects.This process is illustrated in Fig. 2. showing the evolution of the ratio as a function of the GF time.At large flow times the total action and topological charge density drop rather steeply to very small values approaching the classical vacuum.It is important to stress that the annihilation is not a lattice artifact but rather a dynamical effect triggered by the GF itself.
Also for methods based on the eigenvalues, some additional difficulties are expected due to the absence of zero modes according to the index theorem.However, our results show that the AFM provides a more reliable representation of the structures than the GF.Instead of zero modes, which are excluded by the index theorem, the close to zero modes are reproducing the density of each one of the fractional (anti-)instantons, Fig. 3.We computed the spectrum of the projection operator O ± and the chirally projected overlap operator P ± D ov P ± for each chirality sector and found a gap of O(10 −2 ) between the smallest eigenvalues and the rest of the spectrum for both operators Tab. 4. This is considerably smaller than in the example presented in Sec.4.1.1,but nevertheless sufficient to obtain a reliable result with Ξ AF M = 0.989.One can apply the AFM method to the full sequence of configurations generated by GF.This allows us to study how well the AFM is able to discriminate the two structures along the annihilation process.For that purpose we have computed the lowest eigenvalues of O ± up to a flow time τ = 40, see Fig. 2. The SZM captures the instanton/anti-instanton pair until τ ≃ 22 where there is a separation of around d/ρ = 0.4 between them.At that point, the gap in the eigenvalue spectrum of O ± has dropped to zero and there is a crossing of levels.This is clear since now the lowest state does show a completely different density, but the first excited state displays a certain continuity in shape with the instanton structure.

A caloron configuration
Calorons are solutions of the classical equations of motion for R 3 × S 1 .This is an interesting case for which there are analytic formulas for the Q = 1 case which display the decomposition into separate fractional topological charge objects [15][16][17]101].As shown in Ref. [102] these can be approximated on the lattice by configurations with one direction much shorter than the others.In this particular case, we used twisted boundary conditions to produce a fractional instanton on a lattice of size V = 20 × 20 × 10 × 4 and replicated the z-direction to produce a Q = 1 configuration on a V = 20 3 × 4 lattice.
The lowest part of the spectrum of the P ± D ov P ± operator revealed four zero modes with positive chirality Tab. 5, which is consistent with the index theorem (3.1).To apply the AFM, we computed the four lowest modes of the O ± Tab. 5. We found a gap of order O(10 −3 ) on the O + operator between the smallest eigenvalue corresponding to the SZM and the rest of the spectrum.The AFM topological density obtained through (2.6) reproduced successfully the two separate fractional topological charge objects, as we can see from Fig. 4 and from the value of the cross-correlation to the clover definition with the topological charge density Ξ AF M = 0.995.

SU(3) configuration with several fractional instanton/anti-instantons
In order to test the method also for larger gauge groups, we present as the last smooth example an SU(3) configuration on a V = 6 × 6 × 40 × 40 lattice with Q = 0.This type of lattice geometry approximates the T 2 × R 2 case.The configuration contains two fractional instantons and two fractional anti-instantons.This is obtained by gluing along the temporal direction one configuration containing two fractional instantons and it's time-

Filtering of semiclassical background from noisy configurations
The first check on smooth configurations has confirmed that the method reproduces well the semiclassical features.As a next step, we test the filtering properties of the AFM method.The aim is to distinguish the semiclassical background from ultraviolet noise.We take the smooth configuration analyzed in Sec.4.1.4and add noise to it.This is done by applying 20 standard heat bath sweeps based on the plain Wilson action at β = 3.2.Afterwards, the noise has erased all signs of the semiclassical background if the topological density is computed directly from the gauge field of the "heated" configuration, see Fig. 6.
In comparison to the smooth configuration, the gap separating the two smallest eigenvalues O ± in each sector and the rest of the spectrum is drastically reduced, see Tab. 7.These smallest two eigenvalues remain nearly degenerate.Plotting the densities generated from each of the corresponding eigenvectors separately, as shown in Fig. 7, revealed that each mode peaks in only one of the four factional structures.This is different from the smooth configuration before the noise has been added, Sec.4.1.4.The property of the SZM that it provides an equally weighted average of the contributions from each structure  can hence be lost once the noise is added.We have made similar observations also for the configurations generated from Monte Carlo simulations.In several cases each individual lowest eigenmode contributing to q AF M (x) in Eq. (2.18) distinguishes between the different fractional topological objects which build up the complete topological charge density.This can, of course, only be valid in an approximate limit since, due to the non-linearity, q(x) can not be decomposed into a simple sum of fractional contributions.In Fig. 6 we show, how the complete AFM produces a clear filtered image re-obtaining the topological density of the smooth configuration and, consequently, the cross-correlation to the topological charge of the original smooth configuration before heating is high Ξ AF M = 0.897.This shows that the AFM is indeed a filtering method that can eliminate the UV fluctuations introduced by Monte Carlo simulation once the contribution from a sufficient number of modes is combined.
Before we discus the results on the Monte Carlo configurations we collect the main results we obtained so far on smooth configurations • The AFM reproduces correctly the topological density of self-dual smooth configurations.In particular, only the lowest mode of the O ± , the SZM, is needed.
• For smooth configurations which are not self-dual, i.e instanton/anti-instanton pairs, the AFM also reproduces the topological density without altering the configuration.
• In the presence of noise, a hierarchy on the spectrum of the O ± appears.The density of the (quasi)zero modes distinguish the different fractional instantons present on the configuration.Adding the densities of all (quasi)zero modes reproduces correctly the topological density.

Application in Monte Carlo simulations
In a final test, we apply the AFM to ensembles of Monte Carlo generated configurations.It is expected that these contain a combination of all phenomena discussed so far.The noise level also implies a less obvious dependence on the parameters of the method.Especially on a coarse lattice, an additional improvement of the Dirac operator might be required.The simplest improvement is a smearing of the link fields in the Dirac operator.We consider therefore a short GF before applying the AFM.The flow time should in this case be chosen such that the smearing radius remains below a few lattice spacings.As discussed below, this increases the gap in the spectrum of O ± and helps to distinguish the doublers from the physical branch in the spectrum of the Wilson-Dirac kernel.In that way, the parameters of the method can be tuned more easily and a much better performance can be achieved.
We study the combination of the AFM and the GF also at larger flow times.Such a study interpolates between a the noisy Monte Carlo generated and an ensemble of smooth configurations.The AFM performs very well on these smooth configurations, even though they contain less separated and symmetric structures than examples discussed in the last sections.

Yang-Mills theory in a twisted box
In these first studies we aim for a setting with a nontrivial semiclassical dynamics on rather small lattices in order to repeat the calculation with different parameters at a reasonable computational cost.In addition, the setup should allow to control the density and size of typical structures.For this reason we considered a SU(2) Yang-Mills theory in a small twisted box like previously studied in Ref. [40].It corresponds to a T 3 × R geometry with a spatial twist m = {1, 1, 1}, which is fully symmetric in all T 3 directions.On the lattice this is implemented by a large "time" direction representing R and small "spatial" T 3 directions.An additional advantage of this setting is that the relevant features of the densities can be shown as a function of the large time direction after integrating over the spatial coordinates.
In the following we discuss some features of this setting, see Ref. [40] for further details.The potential energy has two minima connected by a non-trivial singular gauge transformation.The order parameter distinguishing the two vacua is the Polyakov loop winding once in every spatial direction P (0,1,1,1) .The Polyakov line in one given direction µ is defined by the product of all links in that direction where x ⊥ are the coordinates orthogonal to direction µ.Different Polyakov loops are traces of products of P µ , in particular where x ⊥ is now orthogonal to all directions with a nontrivial winding of the loop, i. e. P (0,1,1,1) only depends on the time coordinate x 0 .Indeed, the same features hold if the straight line Polyakov loop (Eq.(5.1)) is replaced by other paths having the same winding.These are often more useful since they have smaller statistical fluctuations.Choosing one of the two vacua corresponds to the breaking of a Z(2) subgroup of center symmetry.The situation resembles the double-well potential in one dimension.It is easy to see that the minimum action configuration that tunnels between the two vacua is precisely a Q = 1/2 fractional instanton (or anti-instanton).This has a finite free-energy, thus restoring the center symmetry.This free energy grows with the physical size of the spatial torus.In a lattice Monte Carlo simulation of the system, the size depends on the spatial size of the lattice and on β.At larger values of β the probability to generate fractional instantons decreases and their average number per configuration is small.In this regime these topological structures are identified more easily and a semiclassical calculation of the observables is possible.
For our study we have chosen a regime with moderate values of β which provides already a nontrivial challenge for the detection and pairs of instanton-anti-instantons appear.We have also chosen in our first tests a rather small lattice to reduce the numerical costs of the method since we want to test larger numbers of eigenmodes and different parameters.The small lattice, however, leads to structures, which are less smooth compared to a setup closer to the continuum limit.More realistic cases would likely provide higher densities but also finer lattice spacings.In order to check the scaling of the results, we have considered two different lattice spacing.
Our concrete parameters are a coarse lattice of V = 4 3 × 32 and β = 2.44 and a finer lattice of V = 8 3 ×64 and β = 2.60.According to our rough scaling estimates they have the same physical lattice sizes but different lattice spacings.For each lattice we have selected a subset of 100 thermalized configurations well separated by a number of 1000 and 2000 heatbath sweeps for the coarse and finer lattice respectively.

Eigenvalue spectrum of Wilson-Dirac operator and O ± operator
To tune the κ parameter we computed the low-lying part of the spectrum of the Wilson-Dirac operator in the adjoint representation, see Figures 8 and 9.This already shows a highly nontrivial topological activity with real modes spread along the real axis.It is hence difficult to distinguish within the real modes those that are doublers, but the situation is significantly improved after GF with a small flow time is applied.Based on these findings  we have chosen κ = 0.24 as it seems reasonably aligned with a gap in the spectrum of the Wilson-Dirac operator.The effect of the nonzero flow time on the spectrum of the operator O ± can be seen in Figures 10 and 11.A cluster of low eigenvalues separated from a more dense region at higher values can be identified.The gap separating these two increases after some GF is applied.It seems that a suitable cut in the spectrum λ cut can be determined in order to single out the states that should enter for the computation q AF M (x) in Eq. (2.18).This is quantified in more detail in the following sections.

Results on the coarse lattice
At first we discuss in detail the ensembles on the coarse lattice with V = 4 3 ×32 at β = 2.44.
In the following we need to quantify the accuracy in order to optimize the parameters.The best way for a quantitative comparison that we have found is in terms of the correlation (4.2) with q c (x, τ ′ ), obtained at a certain reference flow time τ ′ .We have chosen τ ′ = 4 since the UV noise is sufficiently filtered out at that flow time and the configurations are rather smooth.At the end we will discuss some possible shortcomings of this approach.An important parameter of the AFM is the cut λ cut on the number of eigenmodes used in (2.18).A second relevant parameter is the flow time for the initial GF applied before the AFM.Hence we define q AF M (x, τ ) as the density obtained with the AFM after GF with flow time τ has been applied.Then, ⟨Ξ AF M (τ, τ ′ , λ cut )⟩ represents the average correlation of this density with q c (x, τ ′ ) at τ ′ ≥ τ .The dependence on these two parameters is shown in Fig. 12, where each line of points represents the dependence on λ cut at a fixed τ .In the smooth case at τ = 4.0 the information is contained in the near zero modes agrees well with the field theoretical definition.At lower τ the optimal λ cut remains small.In case of τ = 0 and τ = 0.25, the relevant information seems to be spread over a larger eigenvalue region.This observation can also be understood from Fig. 8a: without GF, the lowest eigenvalues of O ± are significantly larger and there is no clear separation from the rest of the spectrum.For each τ , we take λ cut with the maximum ⟨Ξ AF M (τ, τ ′ )⟩ max as the optimal value (λ opt ).At larger flow times it is given by the lowest considered value of λ cut = 0.01.The maximum ⟨Ξ AF M (τ, τ ′ )⟩ max starts at a rather small value of 0.4 at τ = 0, but increases significantly already at small flow times until it saturates at τ ∼ 3.This is shown in Fig. 13, where we also plotted ⟨Ξ GF (τ, τ ′ )⟩, the correlation between the gauge definition at different flow times q c (x, τ ) and at the reference flow time q c (x, τ ′ ).As expected the AFM and the GF converge to the same results at larger τ and the filtering property of the AFM leads to better results at smaller flow times.From these results we deduce that for the GF applied before the AFM a flow time of τ = 0.5 might be a good compromise: the smearing radius is still small (two lattice spacings), but the eigenvalue spectrum is sufficiently improved to find a reasonable λ cut for all configurations.
Note that in the rare cases of configurations with zero content of fractional instantons the correlation Ξ AF M (τ, τ ′ ) is not meaningful as (4.2) diverges due to q AF M being completely flat.We had to discard these empty configurations which were chosen according to |q c (t, τ ′ )| < 0.1 for all t.For this ensemble 27 were discarded.The AFM at τ = 0.5 has for 21 of these configurations no signal, meaning no eigenvalue below the optimal cut.Notice that for physically interesting regimes with high density of fractional instantons these cases are actually marginal.
The results obtained so far provide an indication of the viability of the AFM from the average correlation to the GF at large flow times.For a more detailed analysis it is necessary to take a closer look at individual configurations.Fig. 14 shows the distribution of Ξ AF M (τ, τ ′ , λ cut ) at τ = 0.5 and 2.0 for the whole ensemble with the optimal value λ cut = 0.03 and 0.1 respectively.It is remarkable that many configurations present a value very close to one Ξ AF M (τ, τ ′ ) ∼ 1, but there is also a considerable fraction of configurations with Ξ AF M (τ, τ ′ ) < 0.6.Our quantification of the AFM performance by comparison with the GF does not take into account possible failures of the GF method.We therefore have to inspect the topological charge distributions more closely.Fig. 15a presents some examples of the density profiles q AF M (x, τ = 0.5) and q c (x, τ ) for τ = 0.5, 2, 4 integrated over spacial coordinates.Several different scenarios can be identified:  These observations are also confirmed by the Polyakov loop winding in all spacial directions (P (0,1,1,1) ).It is expected to vanish at the center of each fractional instanton or antiinstanton and provides an independent signal, see Fig. 15b.We have seen that the AFM performs reasonably well in many cases, but the significant fraction of the configurations where some sctructures are missed by the AFM, scenario 3, needs further discussions.In some of these cases, we were able to identify the missing structures in the densities of higher modes or varying the value of κ.However, this does not provide a reasonable solution since "wrong" contributions are added at the same time by varying these parameters.The problem is hence related to a missing separation between signal and noise contributions.q − AF M (t, τ = 0.5), κ = 0.24 q + AF M (t, τ = 0.5), κ = 0.36 q c (t, τ = 0.5)) Figure 16.Spatial integrated topological charge density from q c (t, τ ) and q ± AF M (t, τ ) for τ = 0.5 to show the presence of doublers.
The wrong contributions can be in some cases related to the doublers in the Wilson-Dirac spectrum and effects of a tuning of the κ parameter.As an example, we computed the lowest eigenmodes of O ± at κ 1 = 0.24 and κ 2 = 0.36 on a configuration with a fractional instanton and a fractional anti-instanton, Fig. 16.This is reproduced both by the AFM at κ 1 and the GF density at large flow time.κ 2 introduces additional zero modes of the overlap operator.The AFM topological charge density in the positive chirality sector, q + AF M (t, τ = 0.5), contains an extra fractional instanton signal.Furthermore the extra instanton has the same position and shape as the fractional anti-instanton.The signal is hence cancelled by the additional wrong contribution.This shows that the densities related to the doublers seem to resemble with high accuracy the density of the physical modes.
We can push this investigations a bit further and check how much the AFM is affected by the doublers (now for κ = κ 1 and the compete ensemble).If the doublers lead to very similar signals in the opposite chirality sector, they can be identified by the crosscorrelations of these sectors, see Fig. 17.If we consider the complete number of eigenmodes (8 per configuration) there is a considerable correlation between the modes on both chiralities.Below our chosen value of λ cut , this correlations is, however, negligible.This indicates that our optimization of the cut leads to a removal of the doubling modes.At the same time it also removes relevant contributions from the signal.From these findings one can already conclude that the AFM results might improve towards the continuum limit.At finer lattices there is a better separation of typical fractional instanton size and lattice lowest eigenmodes is challenging.At a smaller lattice spacing the results are drastically improved.In most cases underlying semiclassical structures can be clearly identified.We were also able to show that the AFM tends to reproduce structures like instanton/antiinstanton pairs significantly better than the GF, which tends to induce annihilations.
As a next step we plan to explore how the method performs in other physical situations, for example at larger volumes.We are also going to investigate ways to reduce the costs of the numerical computation, for example reducing the precision of the representation of the overlap operator.
often referred as the kernel of the Overlap operator and ϵ(H) is the sign function defined through the spectral decomposition of H.For the kernel we took H = γ 5 D w , where D w is the standard Wilson-Dirac operator (DW) in the adjoint representation,

Figure 1 .
Figure 1.2D slice of the topological charge density for the Q = −4 configuration (Sec.4.1.1)obtained from the gauge field (a) and the supersymmetric zero mode (b).The slice was chosen to cut at the maxima of the two fractional instantons shown.For better visualization, the negative density (−q) is shown.

4 Figure 2 .
Figure 2. Detailed analysis of the Q = 0 configuration (Sec.4.1.2).Left: evolution d/ρ during gradient flow.Right: lowest modes of the O ± operator as a function of the gradient flow time.

Figure 3 .
Figure 3. 2D slice of the topological charge density obtained with AFM for an instanton/antiinstanton configuration with Q = 0 (Sec.4.1.2).The slice was chosen to show the maxima of the fractional instantons.Only the lowest mode (SZM) in each chirality is considered.

Figure 4 .
Figure 4. 2D slice of the topological charge density of the AFM applied to a Q = 1 caloron configuration (Sec.4.1.3).Only the lowest mode with positive chirality (SZM) is considered.

Figure 7 .
Figure 7. Density obtained from the two lowest modes of operators O − (left) and O + (right).

5 Figure 8 .
Figure 8. Lowest part of the spectrum of the DW operator for 100 Monte Carlo generated configurations with our coarse lattice V = 4 3 × 32 (β = 2.44, κ = 0.24) at two different flow times.

Figure 12 .
Figure12.Average correlation ⟨Ξ AF M (τ, τ ′ , λ cut )⟩ between AFM density at τ and q c obtained at a large flow time τ ′ = 4.The values are shown as a function of the cut on the sum of the eigenvalues (2.18).The lines end where for one of the configurations λ cut is larger than the largest computed eigenvalue.If there is no eigenvalue below λ cut for a configuration, Ξ AF M (τ, τ ′ , λ cut ) is set to zero.

1 . 2 .Figure 15 .
Figure 15.(a) Example configurations comparing the topological density integrated over spacial coordinates obtained from the AFM at τ = 0.5 with the topological density from the gauge field obtained at different flow times.From left to right, top to bottom, the values of Ξ AF M = 0.99, 0.61, 0.43, 0.24.(b) Polyakov loop P (0,1,1,1) measured on the same configurations.

3 . 4 .
Bottom-left of Fig. 15a: The AFM misses structures.Bottom-right of Fig. 15a: Both the AFM and the GF seem to have an inconsistent picture and it is difficult to identify the correct topological density.

Table 2 .
Lowest eigenvalues of the operators P ± D ov P ± and O ± for the configuration presented in Sec.4.1.1.The positive chirality sector, P + D ov P + contains no zero modes.Note that the eigenvalues of P ± D ov P ± are always doubly degenerate.

Table 4 .
Lowest eigenvalues of the O ± and P ± D ov P ± operators for the configuration discussed in Sec.4.1.2.

Table 5 .
Lowest eigenvalues of the O ± and P ± D ov P ± operators for the configuration discussed in Sec.4.1.3.

Table 6 .
Lowest eigenvalues of the O ± and P ± D ov P ± operators for the configuration discussed in (Sec.4.1.3).

Table 7 .
Lowest eigenvalues of the O ± and the P ± D ov P ± operator for the heated configuration discussed in Sec.4.2.