Visualisation of Centre Vortex Structure

The centre vortex structure of the $SU(3)$ gauge field vacuum is explored through the use of novel visualisation techniques. The lattice is partitioned into 3D time slices, and vortices are identified by locating plaquettes with non-trivial centre phases. Vortices are illustrated by rendering vortex lines that pierce these non-trivial plaquettes. Non-trivial plaquettes with one dimension in the suppressed time direction are rendered by identifying the visible spatial link. These visualisations highlight the frequent presence of singular points and reveal an important role for branching points in $SU(3)$ gauge theory in creating high topological charge density regimes. Visualisations of the topological charge density are presented, and an investigation into the correlation between vortex structures and topological charge density is conducted. The results provide new insight into the mechanisms by which centre vortices generate non-trivial gauge field topology. This work demonstrates the utility of visualisations in conducting centre vortex studies, presenting new avenues with which to investigate this perspective of the QCD vacuum.


I. INTRODUCTION
In recent years the centre vortex perspective of the QCD vacuum [1,2] has emerged as the most fundamental aspect of QCD vacuum structure, simultaneously governing the properties of confinement and dynamical chiral symmetry breaking in quantum chromodynamics (QCD). Centre vortices have been shown to give rise to mass splitting in the low-lying hadron spectrum [3][4][5], a linear static quark potential [6][7][8][9][10], appropriate Casimir scaling [11], appropriate behaviour of the quark propagator [4,12] and infrared enhancement of the gluon propagator [13][14][15][16]. These results all support the theory that centre vortices capture the essence of QCD vacuum structure and contribute significantly to a full understanding of QCD.
Centre vortices naturally give rise to an area-law falloff in the Wilson loop expectation value [17], such that where A(C) is the minimal area spanned by the Wilson loop. This area-law behaviour is often taken to be an indicator of confinement [10,18]. The fact that one of the defining features of the vortex model is tied so intimately to the geometry of vortices in the vacuum indicates that visualising these structures may provide valuable insight [19,20]. To this end, we construct visualisations of centre vortices and topological charge density on the lattice. We use these visualisations to investigate the dynamics of the vortex model in an interactive and novel manner.
We begin this work in Sec. II with a description of how centre vortices are identified on Monte-Carlo generated lattice gauge fields. In Sec. III we then describe in detail our convention for plotting vortices in three dimensional space, and present the first interactive visualisations of centre vortices on the lattice.
In the digital version of this paper these visualisations are presented as interactive 3D models embedded in the document. To interact with these models, it is necessary to open the document in Adobe Reader or Adobe Acrobat (requires version 9 or newer). Linux users should install Adobe Acroread version 9.4.1, the last edition to have full 3D support. Note that 3D content must also be enabled for the interactive content to be available, and for proper rendering it is necessary to enable double-sided rendering in the preferences menu. To view the models, click on the figures marked as Interactive in the caption. To rotate the model, click and hold the left mouse button and move the mouse. Use the scroll wheel or shift-click to zoom. Some pre-set views of the model are also provided to highlight areas of interest. To reset the model back to its original orientation and zoom, press the 'home' icon in the toolbar or change the view to 'Default view'.
As projected centre vortices are inherently two dimensional objects embedded four dimensions, we describe the technique used to capture the behaviour of vortices in the fourth dimension in Sec. IV, allowing us to observe how vortices evolve over time. In Sec. V we present visualisations of topological charge density alongside vortex lines. In Secs. VI and VII we describe singular points and branching points; two of the unique vortex structures present in the vacuum. Finally, in Sec. VIII we investigate the correlation these structures have with topological charge density. This investigation lays the groundwork for the development of further visualisation techniques, and emphasises the importance of centre vortex geometry in a full understanding of the QCD vacuum.

II. VORTEX IDENTIFICATION
To visualise vortices, we first need to outline how they are identified on the lattice. Physical centre vortices are 'thick' objects, meaning that in four dimensions they form sheets of finite thickness [21]. In contrast, on the lattice we identify 'thin' or 'projected' vortices, known as P-vortices. These P-vortices have infinitesimal thickness. Whilst not physical, P-vortices have been shown to be highly correlated to the location of thick vortices, indicating that it is appropriate to concern ourselves with the behaviour of P-vortices in understanding vortex structure on the lattice [10].
We perform our calculations on 100 20 3 × 40 SU (3) gauge field configurations of lattice spacing a = 0.125 fm. To identify P-vortices, we first gauge-transform each configuration to maximal centre gauge. This is performed by creating a gauge transformation Ω(x) that maximises the functional [7] Maximising Eq. (2) serves to bring every link U µ (x) as close as possible to the centre of SU (3), which consists of the three elements Once the configuration is fixed to maximal centre gauge, we then project U µ (x) onto the nearest centre element to obtain our vortex-only configurations, Z µ (x). It is these vortex-only configurations that we shall be working with for constructing our visualisations.
Once we have obtained our vortex-only configurations, it is simple to identify centre vortices. As we are concerned with P-vortices, it is sufficient to calculate the smallest 1 × 1 Wilson loop on the lattice. In the µ, ν plane, this is known as the plaquette P µν (x). As the centre of SU (3) is closed and P µν is the product of four centre elements, P µν (x) is itself a centre element. If P µν (x) = exp ±2πi 3 I then we say that the plaquette is pierced by a centre vortex of charge m = ±1, otherwise if P µν (x) = I then we say it is not pierced by a vortex. Note however that because P µν = P † νµ , there is an ambiguity in the assignment of the vortex phase, as it is dependent on the ordering of the Lorentz indices. We will therefore employ a right-hand rule as discussed in the following section. The orientation of vortices is significant to the behaviour of vortex models as a whole, and is discussed in greater detail in Ref. [22]. Now that we have identified P-vortices on the lattice, we can begin to construct 3D visualisations. These visualisations aim to elucidate the properties of vortices, and serve as a guide to explaining how vortex structures give rise to the salient features of QCD.
1. An example of the plotting convention for vortices located within a 3D time slice. Left: A +1 vortex in the +ẑ direction. Right: A −1 vortex in the −ẑ direction.

III. SPATIALLY-ORIENTED VORTICES
As the lattice is a four-dimensional hypercube, we visualise the centre vortices on a set of 3D slices. The choice of dimension to take slices along is irrelevant at low temperature in Euclidean space where our lattice calculations take place, so to maximise the volume of each slice we choose to take time slices along the original x-axis, resulting in N t = 20 slices each with dimensions N x × N y × N z = 20 × 20 × 40. Within each slice we can visualise all vortices associated with an x − y, x − z or y−z plaquette by calculating P x y (x), P y z (x) and P z x (x) for all x in the slice. These vortices will be referred to as the 'spatially-oriented' vortices, as they are fixed in time.
As discussed in the previous section, the plaquettes are evaluated on a centre projected configuration, so we can identify the plaquettes with the group of integers modulo 3 according to the vortex centre charge m ∈ {−1, 0, +1}, such that P µν = exp m2πi 3 I. Hereafter we will refer to a plaquette simply by its centre charge.
For a charge m = +1 vortex, a blue jet is plotted piercing the centre of the plaquette, and for a charge m = −1 vortex, a red jet is plotted. The direction of the jet is set according to a right-hand rule, such that • P x y = ±1 =⇒ ±ẑ direction.
An example of this plotting convention is shown in Fig. 1.
Projected centre vortices are surfaces in four dimensional space-time, analogous to the centre line of a vortex in fluid dynamics that maps out a surface as it moves through time. As the surface cuts through the threedimensional spatial volume of our visualization, a Pvortex line is rendered mapping the flow of centre charge. The spatially-oriented vortices for the 3D slices with t = 1, 2 are illustrated in Figs. 2, 3. At first glance the vortex structure appears highly complex, and it is difficult to identify the significant features. As such, we make use of the 3D models to hone in and isolate the important features present in these slices. We present some of these features in Fig. 4.
We observe that the vortices do indeed form closed lines, as highlighted in the view 'Vortex path' in Fig. 2 and the middle panel of Fig. 4. This is essential, as centre vortices must be closed to conserve the centre flux and satisfy the Bianchi identity [23,24]. We also observe that the vortex loops tend to be large. This agrees with the determination made of SU (2) vortices in Refs. [25,26].
The presence of branching/monopole points is of particular interest, as previous studies have primarily focussed on SU (2) theory which is free from these structures. In SU (3) it is possible to conserve centre flux at the intersection of 3 or 5 vortex lines within a 3D slice. An example of a branching/monopole point in our visualisations is shown in the right panel of  A vortex branching point with centre charge +2 flowing into a vertex (left) is equivalent to a vortex monopole with charge +1 flowing out of the vertex (right). The arrows indicate the direction of flow for the labelled charge. Our illustrations adopt the right-hand notation with m ∈ {−1, 0, +1} and jets denoting the oriented flow of charge m = +1.
The ambiguity between monopoles and branching points [24] arises from the periodicity of the centre phase z = exp (2πi/3). By our conventions, each jet denotes the directed flow of +1 centre charge. However, because exp(2πi/3) = exp(−4πi/3), one unit of positive charge is equivalent to two units of negative charge (and vice-versa), and hence we could also interpret our illustrations as representing the directed flow of two units of negative charge. This results in a difficulty distinguishing between branching points and monopoles.
This ambiguity is highlighted in Fig. 5, where we see the equivalence between a branching point and a monopole. For the remainder of this work we will refer to intersections of 3 or 5 vortices as branching points rather than monopoles, as the terms are interchangeable without a strict vortex charge limit.
Branching points within a 3D lattice slice can be identified at sitesx on the dual lattice by counting the number of vortex lines piercing the 3D cube aroundx [24], denoted N cube (x). N cube (x) then takes values from 0 to 6. The interpretation of each value of N cube (x) is summarised in Table I. The ensemble average of the number of vortices piercing each 3D cube. As it is necessary to preserve continuity of the vortex flux, we see that there are no cubes with one vortex piercing them. The largest vortex contribution is from N cube = 2, arising from vortices propagating without branching or touching. We also see that N cube = 3 branching points dominate the N cube = 5 branching points. Simple three-way branching point. 4 Vortex self-intersection. 5 Complex five-way branching point. 6 Vortex self-intersection or double branching.
The distribution of N cube (x) over our ensemble is shown in Fig. 6. As required, we observe that N cube (x) = 1 points are not present.
Branching points correspond to N cube (x) = 3, 5, or 6. As some N cube (x) = 6 points cannot be unambiguously classified as branching points, they are excluded from our subsequent branching point analysis. This is an acceptable exclusion, as we can see from Fig. 6 N cube (x) = 6 points make up only 5.9 × 10 −5 % of the total number of 3D cubes in our ensemble, whereas N cube (x) = 3, 5 branching points are far more prevalent. Thus, we will only consider N cube (x) = 3, 5 branching points in the following sections.
It is clear from our visualisations and the data in Fig. 6 that branching points occur frequently in the confining phase, with an average of 110 (14) branching points per 3D slice. This corresponds to a physical density of ρ BP = 3.5(5) fm −3 . Further discussion of branching points and their relationship with topological charge is presented in Sec. VII.

IV. SPACE-TIME ORIENTED VORTICES
For each link in a given 3D slice there are two additional plaquettes that lie in the x i − t plane, pointing in the positive and negative time directions. Vortices associated with space-time oriented plaquettes contain information about the way the line-like vortices evolve with time, or equivalently, how the vortex surfaces appear in four dimensions.
In a given 3D slice we only have access to one link associated with a space-time oriented vortex, and as such we plot an arrow along this link to indicate its association with this vortex. We adopt the following plotting convention for these space-time oriented vortices: These conventions are shown diagrammatically in Fig. 7. The four dimensional Levi-Cevita permutation symbol is used to extend the right-hand rule to four dimensions. Utilising these conventions, we see that the first two time slices now appear as Figs. 8, 9. As we step through time, we expect to see the positively oriented space-time vortex indicator links retain their colour but swap direction as they transition from being forwards in time to backwards in time, as shown in Fig. 10 and in the views 'Forward/backward arrows' in Figs. 8 and 9.
The space-time oriented vortices act as predictors of vortex line motion between slices. To see this, consider Fig. 11. In Fig. 11a, we observe a line of four m = −1 (red) spatially-oriented vortices with no space-time oriented links associated with them, indicating that this line should remain fixed as we step through time. Alternatively, towards the top of the red line we observe a branching point with two associated −1 space-time indicator arrows. The forward-oriented arrow indicates that this branching point will move. That is, the sheet piercing the t = 1 slice is generating non-trivial space-time vortices as it proceeds to pierce the t = 2 slice. Observing the same region at t = 2 in Fig. 11b, we see that this is precisely what occurs. The vortex line has remained fixed, whereas the branching point has shifted. This vortex motion can also be examined in the views 'Vortex line behaviour' in Figs. 8 and 9.
Another example of space-time oriented vortices predicting the motion of spatially-oriented vortex lines is shown in Fig. 12. Here we see in Fig. 12a a line of three m = +1 spatially-oriented vortices each with an associated m = −1 space-time oriented vortex below them. As we step to t = 2 in Fig. 12b we observe the space-time oriented arrows change direction, and the spatially-oriented vortex line shifts one lattice spacing down such that the space-time oriented vortices are now above them.
The cases presented in Fig. 11 and Fig. 12 are ideal, where the spatially-oriented vortex lines shift only one lattice spacing between time slices. However, it is frequently the case where the spatially-oriented vortices shift multiple lattice spacings per time step, as demonstrated in Fig. 13. In Fig. 13a, we observe a large sheet of space-time oriented vortices with a line of spatially oriented vortices above them. As we transition to t = 2 in Fig. 13b, the line is carried along the sheet and now appears at the bottom.
To see how this occurs diagrammatically, consider appears to move three plaquettes in one time step. These multiple transitions make it difficult to track the motion of vortices between time slices. However, the space-time oriented vortices remain a useful tool for understanding how centre vortices evolve with time. Note that if a spatially-oriented vortex has no associated space-time oriented vortices then it is guaranteed to remain stationary. In this respect, the lack of space-time oriented vor-  tices is also a valuable indicator of vortex behaviour.

V. TOPOLOGICAL CHARGE
We now wish to explore the relationship between vortices and topological charge. The topological charge density is given by Topological charge is calculated on the lattice by evaluation of clover terms C µν . The simplest 1 × 1 clover term is given by [27] C µν (x) = 1 4 From these terms, we obtain To improve the topological charge density calculation by removing higher-order error terms, it is possible to make use of an improved lattice field-strength tensor by taking into account larger Wilson loops in the definition of the clover terms, as described in Ref. [27]. In this paper we make use of the simple 1 × 1 topological charge definition when analysing projected configurations, as the gauge link information is highly localised around the projected vortex locations. However, for the original and smoothed configurations we instead employ a 5-loop improved definition as it produces more accurate results and shows better convergence to integer values on smoothed configurations [27].
We calculate the topological charge density on an original lattice configuration after eight sweeps of three-loop O(a 4 )-improved cooling [27]. This smoothing is necessary to remove short-range fluctuations and associated large perturbative renormalisations, but is a sufficiently low number of sweeps so as to minimally perturb the configuration. Topological charge density obtained after minimal over-improved stout-link smearing is explored in Sec. VIII.
We plot regions of positive topological charge density in red, and regions of negative topological charge density in blue, with a colour gradient to indicate the magnitude. Only topological charge density of sufficient magnitude is plotted to better emphasise regions of significance.
Overlaying the topological charge density visualisation onto Figs. 8 and 9, we obtain Figs. 15 and 16.
Observing the percolation of non-trivial centre vortices in the context of topological charge density provides new insight into the instability of instanton-like objects to centre-vortex removal [8]. We can quantitatively evaluate the correlation between the locations of centre vortices and the regions of significant topological charge density by using the measure where V is the lattice volume, and Vortex associated with any plaquette touching x, 0 , Otherwise.
A value of C = 1 indicates no correlation. C < 1 and C > 1 indicate anti-correlation or correlation respectively. We can then calculateC by taking the ensemble average to obtain an indication of how well the vortices correlate to the locations of high topological charge density.
We can also compare the results of this calculation to the maximally correlated value for C, which can be obtained by postulating that the total number of vortices, N V , correlate to the N V highest values of |q(x)|, denoted |q i |. As we are assuming perfect correlation, L(x) = 1 for all i, and hence the numerator of Eq. 7 reduces to a sum over |q i |. Hence, We can then compareC toC Ideal to gauge how well correlated L(x) is to |q(x)|.
Evaluation of Eq. 7 and averaging over our ensemble of 100 configurations providesC = 1.46(3). Thus, there is a significant correlation between the positions of vortices and topological charge density. The small uncertainty indicates that this correlation is common across the ensemble. The ideal value ofC for optimal correlation isC Ideal = 3.96 (8). ThatC is smaller than this is in accord with the visualisations of Figs. 15 and 15, in that we do not observe perfect overlap between the locations of vortices and the regions of high topological charge density.
Finally, we visualise the vortex configurations after smoothing to investigate how the vortex structure changes. The results, presented in Fig. 17, follow eight sweeps of O(a 4 )-improved cooling. We note that an enormous amount of the vortex matter is removed. However, it is well established that, under smoothing, the vortexonly configurations retain many of the salient long-range features of QCD [3,8,14], suggesting that the removed vortices are in some way irrelevant to these long-range properties.

VI. SINGULAR POINTS
Given the presence of the antisymmetric tensor in the definition of topological charge density presented in Eq. (4), it is clear that for there to be non-trivial topological charge present on the projected vortex configurations, we require the vortex field strength to span all four dimensions. This condition is met at singular points, where the tangent vectors of the vortex surface span all four dimensions. The contribution to the topological charge from these singular points is discussed in detail in Refs. [22,[28][29][30].
In our visualisations, singular points appear as a spatially-oriented vortex running parallel to the link identifier of a space-time oriented vortex, as shown in Fig. 18. Points satisfying this condition, whilst being difficult to locate by eye in Figs. 8 and 9, actually occur frequently, as illustrated in Fig. 19. At these points we have vortices generating field strength in all four space-time dimensions. An example of a cluster of singular points from the visualisation in Fig. 9 (see view 'Collection of Singular Points') is shown in Fig. 20.
The vortex configuration in Fig. 18 spans all four dimensions because the jet indicates a vortex in the x − y plane generating non-zero field strength F xy (x) and the z-oriented indicator link denotes a vortex in the z − t plane, giving rise to non-zero F zt (x). Hence, at the point x the topological charge density can be non-trivial.
Around the lattice site x in Fig. 18 there are four x − y and four z − t plaquettes, allowing for a multiplicity of 16. As there are three unique combinations of orthogonal planes in 4D (x − y and z − t, x − z and y − t, y − z and x − t), this gives a total maximum multiplicity of 48 for each singular point. However, this maximum is highly unlikely, and the highest multiplicity in the configuration shown in Fig. 19  We can verify the relationship between singular points The signature of a singular point, in which the tangent vectors of the vortex surface span all four dimensions. In this case, the blue jet is associated with field strength in the x − y plane, and the orange space-time vortex indicator link is associated with a vortex generating field strength in the z − t plane. Hence, the vortex surface spans all four dimensions. and topological charge by identifying vortices satisfying the parallel condition shown in Fig. 18 and plotting these points against the results of the topological charge calculation performed on the projected vortex-only configurations. As seen in Fig. 21, when we apply the correct sign to the odd index permutations we observe that there is perfect agreement between the location of singular points and the identified topological charge.
To quantify the correlation between q(x) and singular points, we make use of a measure similar to that defined in Eq. (7), However, we redefine our identifier L(x) to be In the case of singular points and |q(x)| obtained from the projected configurations, we expect that the obtained correlation will be identical to the ideal value, calculated in the same manner as Eq. 9. This is indeed what we observe, withC =C Ideal = 17.9(7). In Sec. VIII we will make use of this measure again to examine the correlation between singular points and different topological charge density calculated prior to centre vortex projection where the expected values are less apparent.

VII. BRANCHING POINTS
As mentioned in Sec. III, the SU (3) gauge group permits branching points, identified as the intersection of three or five spatially-oriented vortices in an elementary 3D cube. The branching points are highlighted for t = 1 on our sample configuration in Fig. 22.
Branching points are of particular interest as they are important for generating regions of high topological charge density on the projected vortex configurations. To understand the reason why, consider a clover term C µν as defined in Eq. (5). On a projected configuration, each of the four imaginary parts of the plaquettes in Eq. (5) can take one of three possible values: ± √ 3/2 or 0. Topological charge density of the lowest magnitude will be given by each of the orthogonal clover terms in Eq. (6) contributing ± √ 3/2, either because the remaining plaquettes in each clover do not contribute, or because they contribute but cancel due to opposing signs. To obtain larger values of |q(x)|, it is therefore necessary for multiple plaquettes in at least one of the clover terms to contribute both non-trivially and with the same sign so that the magnitude of the topological charge density increases above the lowest non-trivial value. This is equivalent to requiring that multiple vortex jets pierce the clover parallel to each other, such that they form a pattern like that shown in Fig. 23. To conserve the vortex flux, the configuration in Fig. 23 is most simply achieved by placing a branching point immediately below the two parallel vortices. Hence, there is reason to suspect that branching points may The argument made above by no means claims that branching points must be associated with large values of |q(x)|, as there are most certainly alternative vortex arrangements that will lead to the same values. For example, a branching point could generate two parallel vortex lines that then continue parallel to one another for some distance, generating topological charge density away from the original branching point. Or alternatively, two separate vortex lines could come close to one another, running parallel without the need for any local branching point. Thus, the correlation between large values of |q(x)| and branching points is not expected to be perfect, however the presence of a correlation provides information on the role of branching points in generating large topological charge density. Inspection of the 3D model in Fig. 24 suggests a significant correlation, as is highlighted in the view 'Two branching points and their associated topological charge'.
To evaluate the correlation numerically we again make use of the measure defined in Eq. (7). As branching Branching points (dots) plotted alongside the topological charge density from the projected vortex configurations. It can be observed that the branching points are almost always neighbouring topological charge density. (Interactive) points are defined as the intersection of 3 or 5 vortices, they exist on the dual 3D lattice of each time-slice. The dual lattice sites are denoted byx. For the four unique combinations of three dimensions, xyz, xyt, xzt and yzt, we define our branching point indicator measure as Branching point associated withx in 3D slices of constant µ, The µ index in Eq. (12) indicates which dimension is playing the role of time; i.e. which dimension is not included in the 3D cubes. Similarly, we define q µ (x) to be the average of the topological charge over each 3D cube aroundx. We then have four correlation measures for each 3D combination that can be averaged over, giving a total correlation of By constructing the ideal correlation as defined in Eq. (9) for each choice of 3D coordinates and averaging as done in Eq. (13), we can also calculate the ideal correlation with which we can compare against.
With this measure now suitably defined, we find that we obtain an ensemble average ofC = 6.0(2), compared toC = 1 for no correlation and an ideal of C Ideal = 16.1 (3). This result indicates that there is a notable correlation between branching points and topological charge density and, as expected, they are not the only source of large topological charge. This result is interesting as it speaks to the tendency of vortex lines to either re-combine or diverge away from branching points, rather than remain in close proximity to one another, which provides an interesting consideration for the construction of SU (3) vortex models such as those presented in Refs. [23,29].

VIII. CORRELATION WITH TOPOLOGICAL CHARGE DENSITY
When considering correlations between vortex matter and topological charge density, it is natural to wonder whether the vortex structures identified on the projected vortex-only configurations correlate to the topological charge density identified on the original configurations. As is well established, to accurately identify topological charge density directly from the lattice gauge links it is necessary to first perform smoothing to filter short-range fluctuations [31,32].
To this end, we perform 5 sweeps of over-improved stout-link smearing, with smearing parameters = −0.25 and ρ = 0.06, to minimally smooth the configurations before extracting the topological charge density [31]. To ensure the smoothed configurations maintain information captured in the vortex projection, we also produce smeared configurations that are preconditioned in maximal centre gauge.
We also obtain vortices from these smoothed configurations by fixing them to maximal centre gauge and then centre projecting, giving us in total three vortex configurations and three topological charge configurations. The methods by which these ensembles are obtained are summarised in Fig. 25.
We now repeat our correlation calculations for the singular points, branching points and the vortices themselves for four new combinations of vortex and topological charge density configurations. These results, as well as the correlation results from the previous sections, are summarised in Fig. 26. We see that for all of the new correlations presented, there is a soft correlation between the vortex structures and the topological charge density, however compared to their respective ideal correlations we can see that the correlation is more subtle. Of all the correlations of q S (x) or q PS (x) with vortex information, the strongest correlation is with the original Z VO µ (x).
Of particular interest is the fact that the correlation does not improve when the vortex configuration is preconditioned by the same degree of smoothing as the topological charge, as shown in Fig. 26 (c) and (d). This suggests that the primary cause of the more subtle correlation is the vortex projection rather than the smoothing. In fact, we even observe that the correlation shifts closer to 1 and further from the ideal when the vortex configuration is obtained following 5 sweeps of smoothing. This arises because the number of vortex structures is reduced under smearing, as seen in Fig. 27, but the overlap with topological charge has clearly not improved substantially. As noted earlier in Fig. 17, under cooling this sparsity of vortices is further amplified, indicating that as the degree of smoothing FIG. 25. Summary of the processes used to obtain vortex and topological charge density configurations. 'MCG' denotes gauge fixing to maximal centre gauge and 'Smear' denotes application of 5 sweeps of over-improved stout-link smearing as described in the text. From these methods we obtain the vortex only (VO), smeared (S) and preconditioned smeared (PS) topological charge and vortex configurations. As the topological charge density is gauge invariant, it could equivalently be calculated following gauge fixing to maximal centre gauge.
increases, vortices are increasingly removed from the lattice.
An additional consideration for the observed correlation is the fact that projected vortices do not perfectly correlate with the location of the physical thick vortices. Rather, the projected vortices appear within the thick vortex core, but under different Gribov copies of maximal centre gauge they will be identified at different specific lattice sites [10]. This variability can contribute to the more subtle correlation observed in Fig. 26.
These findings reinforce the result that whilst centre vortices reproduce many of the salient features of QCD, vortex-only configurations are only subtly correlated with the topological charge density of the configurations from which they are obtained.

IX. CONCLUSIONS
In this work we have presented a new way to examine the four-dimensional structure of centre vortices on the lattice through the use of 3D visualisation techniques. These visualisations give new insight into the geometry and time-evolution of centre vortices, and reveal a prevalence of singular points and branching points in the vortex vacuum. It is especially remarkable how common branching points are in SU(3) gauge theory.
We have also explored the connection between these vortex structures and topological charge density. While demonstrating that the topological charge density obtained on projected vortex configurations is generated by singular points, we discovered an interesting correlation between branching points and topological charge; namely that branching points provide an important mechanism for generating large values of topological charge density.
We explored the connection with topological charge density obtained from the original configurations after varying degrees of smoothing. We deduced that the topological charge density of the gauge fields is significantly affected under centre projection, however the modification maintains a positive correlation with the original topological charge density identified on the lattice.
Future work exploring the nature of the Gribov copy problem in regard to SU (3) vortex locations is of interest, as is an exploration of the change in vortex structure as the temperature tends towards the deconfining phase. From this work, it is clear that visualisations of centre vortices provide valuable information about the structure of the QCD vacuum and provide an elegant complement to numerical results.