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the importance of the instanton content of @xmath0 gauge theory comes through both the intrinsic importance of understanding the ground state of qcd and the role instantons are conjectured to play in light hadron structure .
cooling is a technique for removing the high - frequency non - topological excitations of the gauge - field .
however , during cooling instantons are also removed ; either if they are very small ( lattice artifacts ) or through @xmath1 annihilation .
we use under - relaxed cooling to reduce the latter problem .
also , on the cooled configurations there is still the problem of extracting the instanton properties ; for this we have developed pattern - recognition algorithms .
we present results for 20 configurations at @xmath2 @xmath3 and @xmath4 @xmath5 lattices .
the gauge update for under - relaxed cooling @xcite is implemented in each cabibbo - marinari subgroup as @xmath6 where @xmath7 is the gauge link that minimises the action , @xmath8 the original link , @xmath9 is the under - relaxation parameter and @xmath10 a normalisation constant . under - relaxed
cooling increases the number of _ calibrated _ sweeps needed to annihilate an @xmath1 pair ; for a given value of @xmath9 a calibrated sweep is the number of sweeps needed to destroy a @xmath11 instanton . with no under - relaxation
one occasionally finds a very narrow instanton broadening out under cooling ( presumably because of its environment ) .
we have not observed this with ( significant ) under - relaxation .
we chose @xmath12 and our measurements were carried out between 23 and 46 cooling sweeps ( corresponding to between 10 and 20 cooling sweeps at @xmath13 ) .    on the cooled configurations we first find all the local extrema of the symmetrised topological charge density , @xmath14 , relative to the @xmath15 block surrounding each point .
( we do not consider the action , @xmath16 , as it clearly records less structure . )
each peak is treated as a linear superposition of the topological charge of the object at that point , calculated from a lattice - corrected formula , plus a contribution from every other object on the lattice , calculated from the continuum formula .
a self - consistent set of widths is then found by iteration .
these are our candidate instantons .
summing up @xmath14 over the lattice and comparing it to @xmath17 shows a discrepancy .
we define @xmath18 and impose filters on our candidate instantons .
the parameters of the filters are chosen to minimise @xmath19 .
we have a `` spatial '' filter to remove spurious peaks due to ripples on large objects and a `` width '' filter .
the latter compares the width calculated above with the width calculated from the charge within a radius 2 ( or 3 ) of the peak using a lattice - corrected formula ; a peak is only included if the various widths are in sufficiently good agreement .
full details will appear elsewhere  @xcite .    in figure 1
we show the instanton size distribution for @xmath4 at 23 sweeps .
[ figure1 ]    = 6.0 cm = 6.0 cm    the distribution is peaked around @xmath20 .
the best fit to the large-@xmath21 tail of the distribution is @xmath22 with @xmath23 .
as one would expect , the total number of instantons is found to vary rapidly with the amount of cooling .
however the average size and the form of the small / large @xmath21 tails varies much less .
we note that our results are consistent with those of @xcite but not with those of @xcite .
the fact that the impact of a cooling sweep does not scale complicates the scaling analysis .
figure  2 shows that we can tune the number of cooling sweeps so as to get scaling when comparing @xmath4 and @xmath2 .
[ figure2 ]    = 6.0 cm = 6.0 cm    examining the number of unlike charges a distance @xmath24 away from each peak ( normalised by the volume of the shell ) gives the distribution that is shown in fig 3 ( for @xmath4 after 23 sweeps ) .
[ figure3 ]    = 6.0 cm = 6.0 cm    it is uniform at long distances and amplified at short distances .
the corresponding distribution for like charges is uniform at long distances and suppressed at short distances .
this implies some screening of instantons by anti - instantons in the vacuum  as expected .    calculating @xmath25 where @xmath26 is
the number of objects of size @xmath21 and @xmath27 is the charge carried by objects of size @xmath21 shows that the charge carried by small ( large ) instantons is correlated ( anti - correlated ) with the sign of @xmath28 ( figure  4 ) .
indeed our results suggest over - screening of large instantons by small anti - instantons .
[ figure4 ]    = 6.0 cm = 6.0 cm
the wilson - dirac operator , or equivalently @xmath29 , may have vanishing or almost - zero eigenvalues for certain values of @xmath30 . moreover , it has been shown that the lowest eigenvalues of @xmath31 are strongly localised in space - time @xcite .
`` exceptional '' configurations are assumed to be related to the appearance of ( almost-)zero eigenvalues at some @xmath32 .
they seem to be more frequent at smaller @xmath33 and with sw improvement . on the other hand ,
it is unclear if and how these zero modes are related to the topology of the underlying gauge field , in particular because the chirality @xmath34 of the corresponding eigenvectors @xmath35 of @xmath36 is typically much smaller than one . to clarify this relation we investigate the low - lying eigenmodes of @xmath37 using a modified conjugate gradient method  @xcite .
we verified that the index theorem is realized on single instanton configurations ( generated as in ref .
@xcite ) , for both improved and unimproved wilson fermions , with one right - handed ( 4 right - handed plus 3 left - handed ) zero modes in the range @xmath38 for anti - periodic ( periodic ) boundary conditions .
this is in complete analogy to the results for 2-dimensional wilson and for staggered fermions @xcite .
.position @xmath39 and chirality @xmath40 of the zero - mode for single instantons without ( with ) improvement .
[ cols="^,^,^,^",options="header " , ]     for anti - periodic boundary conditions , the eigenmodes are localised and centered on the instanton . moreover , we find considerable effects from discretization errors and sw improvement which are summarized in table  1 .
we also investigated the localisation of the lowest eigenmodes on four exceptional configurations encountered by ukqcd at @xmath2 on @xmath41 and @xmath42 lattices . in all cases
the ( almost-)zero mode is localized close to a small ( @xmath43 ) instanton ( less than @xmath44 away ) .
the chirality has the same sign as the topological charge of the topological object but is not correlated with the overall topological charge of the configuration .
this provides some evidence that exceptional configurations are related to small instantons in the underlying gauge field .
more detailed results will be presented elsewhere @xcite .
we acknowledge financial support by pparc grant gr / k41663 .
d.s . was funded by the carnegie trust for the universities of scotland .
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we present a study of the instanton size and spatial distributions in pure su(3 ) gauge theory using under - relaxed cooling . 
 we also investigate the low - lying eigenmodes of the ( improved ) wilson - dirac operator , in particular , the appearance of zero - modes and their space - time localisation with respect to instantons in the underlying gauge field .