We study pure SU(3) gauge theory discretized on a four-dimensional Euclidean lattice of size V=Nσ3×Nτ with periodic boundary conditions in all directions. Focusing on qualitative, phenomenological aspects of observables based on TDA near the first-order deconfinement transition, we choose Nσ=32 and Nτ=12 throughout this work and postpone a detailed investigation of the dependence on the particular choice of lattice geometry to the future. For later use, the set of all lattice sites is denoted Λ and the set of all spatial sites Λσ.

Gluonic degrees of freedom are described by means of link variables Uμ(x) for x∈Λ, μ=1,…,4, that form SU(3) group elements. We employ the conventional Wilson action, S[U]=β3∑PRe Tr[1-UP],(1)where β is proportional to the inverse squared gauge coupling and the sum runs over all plaquette variables on the lattice, defined as UP≡Uμν(x)=Uμ(x)Uν(x+μ^)Uμ†(x+ν^)Uν†(x)(2)for an elementary square in the μ-ν plane at x∈Λ. We study the system for β ranging from 4.0 to 8.5, which corresponds to increasing the temperature of the system.

Configurations are generated using the well-known (pseudo-)heat-bath algorithm for SU(3) combined with overrelaxation [23–27]. In our setup, advancing the Markov chain by 1 step corresponds to 1 heat-bath sweep followed by 5 overrelaxation sweeps. We thermalize each process with 1000 steps using a warm start where all links are sampled uniformly from the Haar measure, and we discard 200 steps between recorded samples to minimize effects of autocorrelation. Expectation values of persistent homology observables are computed as ensemble averages over 80 samples from 8 independent Markov chains for each β. Furthermore, some additional reference results for certain observables are obtained from measurements performed every 10 steps, also using 8 parallel chains per value of β and a total of 600 steps, resulting in a total of 480 samples per run. In particular, we use this second ensemble to compute the volume average of the absolute Polyakov loop trace, as well as the standard volume-scaled two-point susceptibility for several quantities. For some observable O, the latter is defined as χO=V·(⟨O2⟩-⟨O⟩2).(3)In general, results are given in lattice units, and no scale setting is performed at this stage. Errors for all results are estimated using the statistical jackknife method; see Appendix B for details. For the determination of susceptibilities, we perform binning of measurements with a bin size of 10 before computing the associated errors.

As in our previous work [6], we also compare observables computed from the raw data to the same results obtained from smoothed gauge configurations, from now on referred to respectively as “(un-)cooled” results. This provides further insight into whether cooling is generally required in order to expose the infrared physics under investigation. To this end, we employ the Wilson flow [28] based on the Wilson action defined above and using a standard fourth-order Runge-Kutta discretization scheme with a step size of δt=0.01 and varying total flow times. Specifically, some persistent homology results obtained from the uncooled data are recomputed after nflow∈{1,2,5,10,20} steps.

An order parameter distinguishing between the confined and deconfined phases is given by the Polyakov loop, defined as the product of gauge links wrapping around the imaginary time direction, P(x)=𝒫∏τ=1NτU4(x,τ),(4)where 𝒫 denotes path ordering. Some examples of two-dimensional slices of Polyakov loop trace configurations deep in the confined and deconfined phases are shown in Fig. 1(a), before and after cooling. The cooling procedure reveals signals of extended structures for larger values of β, whereas the configurations appear to remain largely disordered at low β. The absolute volume average of the real part of the Polyakov loop trace, L=13Nσ3⟨|∑x∈ΛσRe TrP(x)|⟩,(5)acquires a nonzero expectation value above some critical coupling marking the location of the phase transition, corresponding to a change of the shape of its effective potential; see Fig. 1(b). For the particular setting chosen in the present work, the phase transition is located around βc≃6.2; see Fig. 1(c).¹

βc is provided here only as the approximate location where the Polyakov loop is observed to acquire a nonzero expectation value. A precise determination has been carried out in the literature [29] and is not the goal of the present work.

The schematic effective Polyakov loop potentials shown in Fig. 1(b) highlight the key difference among the SU(2) and SU(3) phase transitions: the former is second and the latter is first order. This is indicated by the different behavior of the location of the minimum around Tc: for SU(2) it transitions from L=0 continuously to L>0, while for SU(3) a jump occurs at Tc.

Geometrically, a substantial difference between first- and second-order phase transitions is a diverging correlation length for relevant excitations only in the second case. Our previous study [6] has shown that a wide variety of geometric structures as detected by persistent homology can qualitatively change near second-order phase transitions in non-Abelian lattice gauge theories. For first-order phase transitions, we therefore expect that fewer structures closely follow the phase transition dynamics with potentially less pronounced kinks at the critical temperature. Instead, as we will reveal, there can be finite-volume effects, which resemble the behavior near second-order phase transitions but vanish in the infinite-volume effect.

We introduce the concept of persistent homology for sublevel sets of lattice functions, focusing on an intuitive approach. We refer to the literature for comprehensive, mathematically more elaborate introductions [4,5].

The sublevel sets of a real-valued function f:Λ→R on the lattice Λ are given by Mf(ν)≔{x∈Λ|f(x)≤ν}(6)for any ν∈R. For ν below minx∈Λf(x) the sublevel set is empty and for ν above maxx∈Λf(x) the sublevel set is the entire lattice. Furthermore, for ν≤μ we have Mf(ν)⊆Mf(μ), so the family of sublevel sets {Mf(ν)}ν provides a filtration of the lattice Λ.

Yet, the sublevel sets Mf(ν) do not contain interesting topological information by themselves, since they merely are finite sets of lattice points. Instead, one constructs so-called cubical complexes from f, which resemble the sublevel sets Mf(ν) and are topologically less trivial. Again, we provide a rather intuitive approach to their construction and refer to the literature for more elaborate treatments, see, e.g., [31,32]. In general, a cubical complex is a set of cubes of different dimensions such as edges between points being dimension-1 cubes, squares being dimension-2 cubes and so forth, along with the requirement that the set be closed under taking boundaries. The full cubical complex CΛ of the lattice Λ consists of a 4-cube for each lattice point, where the lattice point is located at the cube’s center. It furthermore includes all 3-cubes, which appear as boundaries of the 4-cubes, all 2-cubes, which appear as boundaries of the 3-cubes, etc. Finally, CΛ includes all vertices of the 4-cubes.

We use certain subsets of the full complex CΛ to describe the sublevel sets of the lattice function f. Specifically, we construct a function f˜:CΛ→R, whose sublevel sets provide subcomplexes of CΛ, i.e., are again closed under taking boundaries. First, for all 4-cubes C∈CΛ we set f˜(C)≔f(x), where x∈Λ is the unique center point of C. Any 3-cube D∈CΛ is contained in the boundary of two 4-cubes. On the 3-cube D the function f˜ picks up the lower of the two corresponding 4-cube values: f˜(D)≔min{f˜(C)|D∈∂C},(7)which is repeated for all 3-cubes D∈CΛ. Analogously, all 2-cubes are contained in multiple 3-cubes, for which f˜ has been already defined. Equation (7) can thus be consistently applied to all 2-cubes D∈CΛ for corresponding 3-cubes C∈CΛ, similarly for the 1-cubes and the vertices. This defines the function f˜ on all CΛ. Through the inductive construction, its sublevel sets Cf(ν)≔{C∈CΛ|f˜(C)≤ν}(8)form subcomplexes of the full cubical complex CΛ. For ν<minx∈Λf(x): Cf(ν)=∅, while for ν≥maxx∈Λf(x): Cf(x)=CΛ. Furthermore, for ν≤μ we have Cf(ν)⊆Cf(μ), so the family {Cf(ν)}ν provides a filtration of the full cubical complex CΛ. This is called the sublevel set or the lower-star filtration of f, and ν is its filtration parameter.

The cubical complexes Cf(ν) can be viewed as a “pixelization” of the sublevel sets Mf(ν) and are generally topologically less trivial than mere point sets. Much of their topology is suitably described by means of homology [33], which is algorithmically efficiently computable and homotopy invariant. The Cf(ν) can give rise to nontrivial homology classes of different dimensions, as illustrated for three spatial dimensions in Fig. 2(a). Indeed, different connected components and looplike holes can appear, which form dimension-0 respectively dimension-1 homology classes. Cubes can enclose empty volumes, which are described as dimension-2 homology classes. Finally, in four dimensions enclosed empty 4-volumes can appear, which provide dimension-3 homology classes (not displayed).

Sweeping through the filtration {Cf(ν)}ν, the homology may generally change depending on ν. This is illustrated in Fig. 2(b) for the case of functions f defined on a two-dimensional lattice, where the vertical bar height indicates the function value. Considering the left-hand example, when ν=minxf(x) (green plane), the first connected component (dimension-0 homology class) is born with birth parameter b=minxf(x). Towards the larger filtration parameters indicated by the blue and purple planes, the single 2-cube evolves into a path-connected accumulation of 2-cubes. The homology class corresponding to the first connected component remains invariant. Yet, at the filtration parameter indicated by the purple plane, a second dimension-0 homology class is born. At the filtration parameter indicated by the red plane, a saddle point occurs and the first homology class merges with the second. The former dies with death parameter d. The second homology class lives up to infinite filtration parameter; its death parameter d can be formally set to ∞.

In addition to dimension-0 homology classes, dimension-1 homology classes may appear in the lower-star filtrations of functions on a two-dimensional lattice. An example is provided on the right-hand side of Fig 2(b), which resembles a vertically inverted “volcano.” While for low filtration parameters as indicated by the green plane, multiple connected components appear, these all merge towards the larger filtration parameter indicated by the blue plane to form a loop-like hole. A dimension-1 homology class has been born with birth parameter b. Increasing the filtration parameter, it gets thickened (purple plane) and becomes ultimately fully filled with squares (red plane); it dies with corresponding death parameter d.

The persistent homology of {Cf(ν)}ν is fully described by the collection of all birth-death parameter pairs (b,d) for the homological features of the different dimensions. The difference p=d-b provides a measure for their dominance and is called persistence. In this work we mostly focus on the dimension-ℓ Betti curves, which count dimension-ℓ homology classes depending on the filtration parameter ν: βℓ(ν)≔#{Dim.-ℓ homology classes of Cf(ν)}.(9)

Persistent homology comes with a number of advantageous properties. First, state-of-the-art algorithms allow for its efficient evaluation, computing the homology and related birth-death pairs for all filtration parameters at once. We utilize the versatile computational topology library GUDHI for python [35]. It facilitates cubical complexes with periodic boundary conditions, which we employ. Mathematically, persistent homology is provably stable with regard to perturbations of the input for a variety of persistent homology metrics, see, e.g., [36,37]. This theoretically underpins its suitability for applications to lattice gauge theories. Finally, persistent homology and the Betti curves can be well used in statistical analyses, giving rise to notions of ergodicity and large-volume asymptotics [38,39].

We turn to the Betti curves for the sublevel set filtrations of local electric and magnetic field energy densities, i.e., for f=TrE2 and f=TrB2. For the lattice gauge theory under consideration, the total energy density reads T00(x,τ)∼TrE2(x,τ)+14TrB2(x,τ).(10)Therefore, upon studying electric and magnetic energy density filtrations, we gain insights into the electromagnetic structures assembling the total energy density. On the lattice, we employ clover-leaf variants of SU(3)-valued electric and magnetic fields, which are provided by antisymmetric combinations of spatiotemporal and spatial-only plaquettes, respectively. Their construction has been outlined for the case of gauge group SU(2) in [6] and is not repeated here. The prefactor 1/4 for the magnetic contributions in (10) is due to the different normalization of the magnetic compared to the electric field.

Figure 3 shows the Betti curves of homology dimensions zero to three for the TrE2 filtration [panel (a)] and for the TrB2 filtration [panel (b)], evaluated for a range of inverse couplings β. All Betti curves provide sharply peaked distributions, which for increasing β values shift towards lower filtration parameters. The overall energy density decreases with increasing β, which explains this effect.

For increasing homology dimensions, the support of the distributions shifts towards larger filtration parameters and widens (from left to right). This is a geometric effect due to the formation mechanism of homology classes of different dimensions. For instance, multiple dimension-0 homology classes first need to be born in order to merge and form a dimension-1 feature, see also the right-hand example in Fig. 2(b). Similarly, many dimension-1 features first need to form a pierced surface, which then gets successively filled with cubes to form an enclosed volume, i.e., a dimension-2 homology class.

Comparing the Betti curves for the TrE2 and TrB2 filtrations, we notice that in the latter case the support of the curves is at approximately a factor of 4 larger filtration parameters than in the former case. This is due the different prefactor involved in the definition of B compared to E, which also appears in the total energy density (10).

For the TrE2 filtration, the Betti curve peak height increases in homology dimension zero (left) with increasing β, but decreases for homology dimensions two and most strongly for three (right). This is different for the TrB2 filtration, where for all homology dimensions the peak heights, after a brief decline for low β, increase with increasing β. We turn to a more detailed investigation of this phenomenon in the next subsection.

In Fig. 4(a) we show the peak values of the Betti curves of Fig. 3, plotted against inverse coupling β. The maximal value of a dimension-ℓ Betti curve is the maximal number of dimension-ℓ homology classes appearing in the filtration of cubical complexes. Figure 4(b) shows the corresponding variables evaluated for the cooled/smoothed lattice configurations, and Appendix A provides a more detailed discussion on the dependence of the Betti curve maxima on the number of flow steps. Figure 4(c) will be discussed later.

Naturally, the curves displayed in Fig. 4(a) match the descriptions of the previous Sec. III B. Moreover, throughout all homology dimensions and for both uncooled and cooled configurations, the maximal Betti numbers for the TrB2 filtration feature a local minimum near β≃5.6. The TrE2 filtration has this effect only for the cooled configurations, for which, generally, the maximal Betti numbers are much closer to those of the TrB2 filtration than for the uncooled case. Throughout homology dimensions, the range of maximal Betti numbers differs for the uncooled configurations by up to ∼10% within the displayed β interval. For the cooled configurations, the variation of the maximal Betti numbers depending on β is of the order ∼40% or more.

Most interestingly, at or near the critical βc≃6.2 we find a crossing among the maximal Betti numbers of the TrE2 filtration with those of the TrB2 filtration, in particular for homology dimensions one and larger. This effect is most clearly visible in the top homology dimension three (right column). Comparison among uncooled [panel (a)] and cooled [panel (b)] configurations shows that the crossing remains approximately stable against cooling, see also Fig. 7 in Appendix A.

We proceed with a discussion of possible interpretations of these findings. We focus first on the crossings of maximal Betti numbers for the TrE2 and TrB2 filtrations at or near βc≃6.2, subsequently providing an explanation for the minima near β≃5.6. The crossings being most clearly visible in homology dimension three indicates that it is local maxima in electric and magnetic energy densities, which are predominantly responsible for these. Indeed, unoccupied 4-cubes, which give rise to enclosed 4-volumes (dimension-3 homology classes), appear in the complexes CTrE2(ν) and CTrB2(ν) through local maxima in the corresponding lattice functions TrE2(x,τ) and TrB2(x,τ) for filtration parameters ν lower than these maximal values. Therefore, the crossings hint at the presence of local lumps of field energy density switching type from electric to magnetic across the (de)confinement phase transition at βc.

The crossings in the maximal Betti numbers for TrE2 and TrB2 filtrations are further reminiscent of electromagnetic dualities, such as semiclassically available for the Georgi-Glashow model [40,41]. Montonen-Olive dualities generally imply that the strong coupling behavior of the gauge theory can be determined by the dual theory at weak couplings, mapping the gauge bosons to magnetic monopoles and vice versa. With electromagnetic dualities in mind, the crossings suggest the following interpretation. Below βc and thus for bare couplings above the critical gc, there is a larger abundance of homology classes, which correspond to local lumps of electric energy density, than there are lumps of magnetic energy density. On top there are thermal fluctuations, which contribute a bit more to electric than to magnetic energy densities, cf. the TrE2 filtration data in Figs. 4(a) and 4(b). Near the phase transition, electric and magnetic excitations contribute approximately equally to the structures appearing in the energy densities, resulting in the crossing of their maximal Betti numbers near βc and an equipartition in the number of electric and magnetic structures. Above βc and therefore for bare couplings below the critical bare coupling gc, the picture is reversed, so that electric structures get more scarce and magnetic structures in energy densities get more abundant. Despite these considerations, we remind the reader that for the shown β interval the variation of the maximal Betti numbers for both the TrE2 and the TrB2 filtration is of order ∼10%, so there appear to be significantly more structures present than those providing duality signals.

Cooling suppresses the thermal fluctuations overlaying the electromagnetic duality signatures, enhanced for electric excitations, see Fig. 4(b). Approaching self-duality, cooled configurations are closer to satisfying the Bogomol’nyi bound, so the structures in the TrE2 and TrB2 filtrations become more similar. This is consistent with Fig. 4(b).

The minima in the maximal Betti numbers near β≃5.6 visible in Figs. 4(a) and 4(b) can be understood as follows, based on susceptibilities. The plaquette trace contributions to the action (1) are given by SP[U]=13∑PRe TrUP.(11)The β derivative of the expectation value of (11) is given by the plaquette susceptibility as defined in Eq. (3): d⟨SP⟩dβ=ddβ∫𝒟USP[U]exp(βSP[U])∫𝒟Uexp(βSP[U])=⟨SP2⟩-⟨SP⟩2≡1VχRe TrUP/3,(12)where 𝒟U is the lattice integral over all SU(3)-valued link variables [constructed from SU(3) Haar measures]. The plaquette susceptibility χRe TrUP/3 contains connected plaquette trace correlations which, in the vicinity of a second-order phase transition, are expected to converge to a nonzero constant at large separation. This is due to a peak in the correlation length at the transition, for which the peak height grows to infinity in the infinite-volume limit. The deconfinement phase transition of SU(3) being first order, no such diverging correlation length appears in our case. Yet, as a finite-volume effect χRe TrUP/3 peaks near β≃5.6. This can be inferred from the left panel in Fig. 4(c), where χRe TrUP/3 is shown as a function of β.²

For a lattice of size 124, corresponding data have been discussed in [42], see Fig. 4.2 therein.

Similar susceptibilities can be extracted from the local values of TrE2 and TrB2. These are as well displayed in Fig. 4(c). We notice a distinct peak around β≃5.6 in both susceptibilities, which is more pronounced for χTrB2 than for χTrE2. This indicates that the correlation lengths of TrE2 and TrB2 excitations and therefore the related homology classes are largest near β≃5.6. The maximal Betti numbers thus exhibit a minimum around β≃5.6, which we see in Figs. 4(a) and 4(b).

The somewhat large variations observed in the maximal Betti numbers shown in Fig. 4, in particular for homology dimensions zero and one, indicate that jackknife errors are likely underestimated, in particular when compared to the considerably larger uncertainties associated with the susceptibility results, despite the greater number of samples used there. This is potentially due to larger than expected autocorrelation times of the Betti curve observables. A detailed investigation of this issue is difficult due to the comparably high computational cost of the persistent homology analysis, and is beyond the scope of the present work.

Figure 5(a) shows the Betti curves for the Re TrP/3 filtration for a range of β values, computed for uncooled configurations. We find clearly peaked distributions of features across the filtration for all homology dimensions zero to two (left to right), whose support increases towards larger filtration parameters with increasing homology dimension. Again, this is a natural finding for the sublevel set filtration, since multiple connected components first need to exist and to merge, in order to form a dimension-1 feature, and similarly for dimension-2 features.

In terms of their dependence on β, we notice that the Betti curves stay roughly on top of each other up to β≃5.6, followed by a mild decline in peak heights for further increasing β values. This is more transparently visible in Fig. 5(b), where the maximal values of the Betti curves of Fig. 5(a) are displayed along with their dependence on the number of flow steps for cooling. Clearly, the maximal Betti numbers stay approximately constant up to β≃5.6, up to which value the peak heights remain also insensitive to cooling. Above the kink around β≃5.6, Betti curves begin to depend on the number of flow steps: increasing nflow can strongly enhance the decline in peak heights with increasing β.

We encountered kinklike behavior around β≃5.6 before: the maximal Betti numbers of TrE2 and TrB2, in particular for cooled configurations, gave rise to minima at this β value, see Fig. 3 and the discussion towards the end of Sec. III C. We attributed this behavior to a peak in plaquette trace correlations, which as a finite-volume effect occurs for our lattice near β≃5.6 and not near βc≃6.2. The plaquette trace correlations also correlating with the number of features in the Polyakov loop trace filtration, we expect the kinklike behavior in the related maximal Betti numbers near β≃5.6 to also be a finite-volume effect. The insensitivity of the maximal Betti numbers to cooling below β≃5.6 can be attributed to the Polyakov loop behavior itself: at low β cooling barely affects Polyakov loop traces and leaves the volume average zero, while at large β cooling enhances the appearance of nonzero volume averages, cf. Fig. 1. For β>βc, thermal fluctuations on top of larger domains in Re TrP(x) get increasingly suppressed through cooling.

For gauge group SU(2) the Polyakov loop trace filtration has been studied in [6]. It has been found that the Betti curves remain invariant to changes in β up to the (pseudo-)critical βc, above which Betti number distributions broaden and decrease in overall numbers for increasing β. Qualitatively, this is similar to the SU(3) case investigated in the present work, except for the kinklike behavior occurring around β≃5.6 and not near βc, which, again, we expect to be a finite-volume effect. For gauge group SU(3) this is possible, since the deconfinement phase transition is first order, while it is second order for gauge group SU(2). Yet, the dependence of the Betti curves on β above βc is much stronger in the SU(2) case than for SU(3) above β≃5.6. This can be explained at least partially via the Polyakov loop trace absolute volume average L(β) having for SU(3) maximally ∼5% of the value compared to SU(2), taking into account the studied β intervals. Therefore, it can be anticipated that for the given β interval the β dependence of the number of geometric structures associated with the Polyakov loop trace is weaker for SU(3) than for SU(2).

We turn to investigate in how far topological excitations coupling to Polyakov loops might play a role for the dynamics visible in the TrE2, TrB2 and Re TrP/3 filtrations. For this we consider filtering the lattice configurations by means of the Polyakov loop topological density: qP(x)=132π2ϵijkTr[(P-1(x)∂iP(x))×(P-1(x)∂jP(x))(P-1(x)∂kP(x))],with ϵijk the Levi-Civita symbol in three dimensions. Indeed, the nomenclature for qP is justified: for the pure gauge theory on a continuous space-time 4-torus, the topological charge can be computed as the integral of a topological density ∼Tr(E·B) over the entire 4-torus. The topological charge can be identically rewritten into an integral over the spatial 3-torus with integrand qP, for gauge group SU(2) [45] as well as a general special unitary gauge group SU(Nc) [44]. Similarly to our earlier work on persistent homology for gauge group SU(2) [6], it is expected that the persistent homology of the qP filtration is by construction less sensitive to lattice artifacts compared to a filtration using the usual topological density ∼Tr(E·B). This and the fruitful findings in [6] motivate the study of the qP filtration in the following.

In Fig. 6(a) we show the Betti curves of the qP filtration, again for a range of β values and mostly for uncooled configurations. We find clearly peaked distributions, which barely reveal any β dependence. For all homology dimensions, the peak height does not reveal kinklike behavior but instead only randomly scatters by up to ≲4% around mean values (maximal Betti numbers not displayed). In Fig. 6(a) the Betti curves for the uncooled configurations have been overlayed by the Betti curves for cooled configurations at β=8.5, which agree with the corresponding data for the uncooled configurations. This is consistent with the expected behavior of cooling: while it removes small-scale thermal fluctuations, it ideally leaves larger topological excitations untouched, therefore also the homological features appearing in the qP filtration.

The results for the Betti curves are complemented by the distributions of persistence values (p=d-b) for the qP filtration, shown in Fig. 6(b). Again, the curves with colors indicated by the color bar have been computed for uncooled configurations, overlayed by the persistence distributions for cooled configurations at β=8.5. We notice that all persistence distributions lay on top of each other, up to statistical fluctuations for the right-hand tail of the distributions. Low statistics in this regime is responsible for these scatterings, since individual persistent homology classes and their persistences become visible. Generally, the distributions for dimensions zero and two have similar shape, the latter coming with slightly larger support, while the dimension one persistence distributions look different. This can be indicative for local values of qP scattering evenly around zero. Indeed, if this is the case, then local minima in qP giving rise to signals in homology dimension zero and local maxima in qP dominating homology dimension two behave statistically alike, so do the related persistences.

Analogous to the Polyakov loop trace filtration discussed in Sec. IV A, we have studied the Polyakov loop topological density filtration for gauge group SU(2) in [6]. Interestingly, for SU(2) the persistence distributions of the qP filtration have revealed clear exponential behavior in homology dimensions zero and two. We loosely attributed this to the presence of dyons, with fitted exponents heuristically matching the predictions for the topological charge statistics of dyons. Furthermore, in the SU(2) case above βc, cooling had a substantial influence on the persistence distributions, where it enhanced the presence of features with large persistences. As we revealed, the situation for gauge group SU(3) is markedly different, since no exponential behavior occurs for the persistence distributions and cooling barely has any effect on the features occurring in the qP filtration.