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SU(N) caloron measure and its relation to instantons

Calorons of the SU(N) gauge group with non-trivial holonomy, i.e. periodic instantons with arbitrary eigenvalues of the Polyakov line at spatial infinity, can be viewed as composed of N Bogomolnyi--Prasad--Sommerfeld (BPS) monopoles or dyons. Using the metric of the caloron moduli space found previously we compute the integration measure over caloron collective coordinates in terms of the constituent monopole positions and their U(1) phases. In the limit of small separations between dyons and/or trivial holonomy, calorons reduce locally to the standard instantons whose traditional collective coordinates are the instanton center, size and orientation in the color space. We show that in this limit the instanton collective coordinates can be explicitly written through dyons positions and phases, and that the N-dyon measure coincides exactly with the standard instanton one.


I. INTRODUCTION
Belavin-Polyakov-Schwartz-Tyupkin (BPST) instantons [1,2] are known to play an important role in Quantum Chromodynamics, see Refs. [3,4] for reviews. The instanton liquid model [5] is especially helpful in providing a microscopic mechanism of the spontaneous chiral symmetry breaking, as due to the delocalization of the would-be zero fermion modes in the instanton ensemble [6].
At the same time, instantons do not lead to confinement, at least in the naive dilute limit. In the pure glue version of QCD, there are two well-known criteria of confinement: the area behavior of large Wilson loops, and the zero average of the Polyakov lines [7], with its subsequent restoration to the center-of-group values at temperatures above the deconfinement transition. To be more precise, one can formally obtain the area law for large Wilson loops from averaging over the instanton ensemble, provided instanton size distribution drops as dρ/ρ 3 for large-size instantons [8]. However, such distribution implies that large-size instantons overlap, which makes meaningless the description of the vacuum fluctuations in terms of the instanton collective coordinates: one has to use other degrees of freedom.
To study the temperature dependence of the average Polyakov line, one needs first to generalize the zero-temperature BPST instantons to the periodic Harrington-Shepard instantons [9]. The quantum weight of periodic instantons has been found by Gross, Pisarski and Yaffe [10], and the instanton ensemble at any temperatures has been built in Ref. [11], using the variational principle of Petrov and one of the authors [5]. Averaging the Polyakov line over this ensemble, it was found that it oscillates near zero at small temperatures and rapidly approaches the center-of-group value at T ≥ Λ [12]. However, it is neither exactly zero at small T , nor is there a sharp phase transition. This is the kind of behavior expected from the approximate treatment of large-size instantons. Again, one concludes that in order to observe mathematically the confinement-deconfinement phase transition, one needs to use the degrees of freedom appropriate for overlapping instantons. It was conjectured in Ref. [13] that the adequate description should be in terms of N monopoles constituting an SU (N ) instanton. In a more simple 2d CP N −1 model also possessing instantons, the appropriate degrees of freedom known as "instanton quarks" or "zindons" have long been availablesee Ref. [14] for references and for a detailed study of the CP N −1 instanton ensemble in terms of their constituents.
For the 4d Yang-Mills theory, a somewhat similar construction of instantons through their "constituents" became available more recently, owing to Kraan and van Baal [15] and Lee and Lu [16], first for the SU (2) gauge group and later for the general SU (N ) [17]. These authors have found explicitly an exact self-dual solution of the Yang-Mills equation of motion at any temperature with a unity topological charge and with arbitrary eigenvalues of the Polyakov line (or holonomy) at spatial infinity. We shall call this general solution the KvBLL caloron. The periodic Harrington-Shepard instanton is a limiting case of the KvBLL caloron at trivial holonomy corresponding to the Polyakov line assuming center-of-group values. A caloron with the double topological charge has been constructed in Ref. [18].
The fascinating feature of the SU (N ) calorons is that they can be viewed as composed of N Bogomolnyi-Prasad-Sommerfeld monopoles [19] or, more precisely, dyons since they carry both magnetic and electric charges; the composite calorons are electrically and magnetically neutral. Apart from N −1 eigenvalues of the Polyakov line at spatial infinity, the SU (N ) KvBLL caloron is characterized by 4N collective coordinates forming its moduli space. A natural choice of the collective coordinates is to use 3N positions of the dyons' centers in space, and N dyons' U (1) phases, 3N +N = 4N . If all N dyons are spatially far apart, the action density of the KvBLL caloron consists of N time-independent 3d lumps whose profile is the well-known energy density of individual BPS dyons. In the opposite limit when all dyons are within the spatial range ≤ 1/T from each other, the KvBLL caloron becomes a single 4d lump whose profile is close to the usual periodic instanton. As the temperature goes to zero with dyons separation fixed, the caloron action density tends to that of the standard BPST instanton. Contrary to the standard instanton, however, the holonomy (or the Polyakov line at infinity) remains non-trivial.
The average eigenvalues of the Polyakov line are determined by the dynamics of the ensemble of calorons with non-trivial holonomy. For example, in the N = 1 supersymmetric version of the Yang-Mills theory the dyon-induced superpotential can be computed exactly [20], and its minimum corresponds to the Polyakov line's eigenvalues such that Tr L = 0 as it should be in the confining phase. Moreover, the known exact v.e.v. of the gluino condensate corresponds to this particular holonomy, whereas the trivial-holonomy instantons lead to a wrong value [20,21]. This result is the more surprising that at T → 0 the local difference between gauge fields with trivial and non-trivial holonomy vanishes, implying that long-range fields are critical, at least in the supersymmetric gluinodynamics [21].
In the non-supersymmetric Yang-Mills theory, the question what average holonomy is dynamically preferred, is open. From lattice simulations we know that in the confining phase < Tr L >= 0 but we do not know what dynamics leads to it. Revealing it would be tantamount to understanding the mechanism of confinement. A step in that direction has been taken in Ref. [22] where the quantum weight of the KvBLL caloron has been computed exactly, as function of the holonomy, dyon separation and temperature for the SU (2) group. Based on this calculation, an argument has been presented that at T < T c = 1.125 Λ a trivial holonomy becomes dynamically unfavorable from the free energy minimization viewpoint. Below T c dyons repulse each other, and calorons presumably "ionize" into separate dyons. However, to find out their fate and whether the system prefers the "confining" holonomy (1), one has to study the dynamics of many dyons. To that end one has first of all to find the statistical or quantum weight of a dyon configuration as given by the combination of the collective coordinate measure and the small oscillation quantum determinant.
This paper is devoted to the study of the measure of a single SU (N ) KvBLL caloron, written in terms of the dyon coordinates and their U (1) phases. The metric tensor of the moduli space has been first conjectured by Lee, Weinberg and Yi [23,24] and then derived by Kraan [25] using the explicit Atiyah-Drinfeld-Hitchin-Manin-Nahm (ADHMN) construction [26,27] for the SU (N ) caloron [17]. We have independently reproduced the same result for the moduli space metric, however we do not present the derivation here as it is lengthy but is not qualitatively different from that by Kraan. Instead, we compute the determinant of the metric tensor, which defines the integration measure over the dyons' collective coordinates for the general SU (N ) caloron, and compare it with the long-known instanton measure [28] written in terms of the instanton position, size and group orientation. We demonstrate that the SU (N ) instanton measure written in these terms coincides exactly with the one written in terms of the coordinates and phases of the instanton constituents. This result is not altogether trivial, as in the first case the measure arises from the volume of the SU (N )/SU (N − 2) coset whereas in the second case it follows from the 3d geometry. We also find the direct relation between the instanton group orientation and the dyons positions and U (1) phases. We believe that it may be an important step in combining the success of the small-size instantons in physics related to the spontaneous chiral symmetry breaking, with the description of large-size instantons in terms of their dyon constituents, which is presumably necessary for the confinement physics. Since this paper concentrates mainly on the mathematical questions, we do not discuss here the very interesting recent studies of the KvBLL calorons on the lattice [29].

II. NOTATIONS
To help navigate and read the paper, we first introduce some notations used throughout. Basically we use the same notations as in Ref. [17]. In what follows we shall measure all quantities in the temperature units and put T = 1. The temperature factors can be restored in the final results from dimensions.
Let the Polyakov line at spatial infinity have the following eigenvalues We use anti-hermitian gauge fields The eigenvalues µ m are uniquely defined by the condition N m=1 µ m = 0. If all eigenvalues are equal up to the integer, implying µ m = k/N − 1, m ≤ k and µ m = k/N, m > k where k = 0, 1, ...(N − 1), the Polyakov line belongs to the SU (N ) group center, and the holonomy is then said to be "trivial". By making a global gauge rotation one can always order the Polyakov line eigenvalues such that which we shall assume done. The eigenvalues of A 4 in the adjoint representation, A ab 4 = if abc A c 4 , are ±(µ m − µ n ) and N − 1 zero eigenvalues. For the trivial holonomy all adjoint eigenvalues are integers. The difference of the neighbor eigenvalues in the fundamental representation ν m ≡ µ m+1 − µ m determines the spatial core size 1/ν m of the m th monopole whose 3-coordinates will be denoted as y m , and the spatial separation between neighbor monopoles in color space will be denoted by With each 3-vector ̺ m we shall associate a 2-component spinor ζ † α m built according to the Euler parametrization This spinor, together with its Hermitian conjugate ζ m α , forms a 2 × 2 matrix for any m = 1...N : These spinors are used in the construction of the caloron field. The Euler angle ψ m is fictitious in parameterizing the 3d vector ̺ m but enters explicitly the gauge field of the caloron and belongs to its moduli space, together with ̺ m . In fact ψ m has the meaning of the U (1) phase of the m th dyon. We shall also use the following notation for the variation For trivial holonomy, the KvBLL caloron reduces to the Harrington-Shepard periodic instanton at non-zero temperatures and to the ordinary Belavin-Polyakov-Schwartz-Tyupkin instanton at zero temperature. Instantons are usually characterized by the scale parameter (the "size" of the instanton) ρ. It is directly related to the dyons positions in space, actually to the perimeter of the polygon formed by dyons, In these notations the KvBLL caloron gauge field can be written as the following N × N matrix [17]: where the summation over k, l is understood and where The N × N matrix f mn is in fact the ADHMN Green function f (µ n , µ m ) found in [30]. In Appendix B we derive a simple expression for this quantity used to obtain certain limiting cases of the general eq.(9).

III. ZERO MODES IN THE YANG-MILLS THEORY
Here we remind what are zero modes and how the moduli space metrics arises from the path integral. In our notations the partition function for the pure Yang-Mills theory reads where A µ (x 4 , x) must obey the periodicity condition A µ (0, x) = A µ (1/T, x). The integration measure in eq.(11) is defined through the scalar product where A µ (x) = n α n u n µ (x) for the complete normalized set of functions u n µ (x). We want to compute the contribution to the partition function from some set of solutions of the classical Yang-Mills equation of motion A µ (Y, x) parameterized by the collective coordinates Y p . It means that we have to take into account only small fluctuations about the surface formed by this set of solutions in the configuration space. Usually S[A(Y )] = S cl is the same for the whole set, and is locally minimal. The integral over fluctuations is Gaussian only in the directions orthogonal to the surface. We have to separate Gaussian and non-Gaussian variables of integration.
The result in the quadratic order after fixing the background gauge D cl µ a µ = 0 is where a µ = it b a b µ (tr(t a t b ) = 1 2 δ ab ) are small fluctuations orthogonal to the zero modes of the operator W µν , χ andχ are the ghost fields from gauge fixing. The factor √ 2πg comes from the definition of the measure. The Jacobian J is in fact the determinant of the moduli space metric tensor, i.e.
where δ p A µ is a zero-mode of W µν , associated with the collective coordinate Y p through where Ω p is chosen such that the background gauge condition is satisfied, In the next section we present the result for the metric tensor g pq for the KvBLL caloron of the SU (N ) gauge group.

IV. CALORON MODULI SPACE METRIC
As mentioned in the Introduction, the metric of the moduli space of N different BPS monopoles of the SU (N ) gauge group has been first conjectured in Refs. [23,24] generalizing the previous work [31], and then confirmed by an explicit calculation in Ref. [25]. In these papers, the metric tensor was expressed in terms of the monopole -electric charge interaction potential w i (̺) satisfying the equation One introduces also a N × N matrix such that the separation between consecutive dyons is ρ m = y m − y m−1 = (S T y) m . In terms of the dyon interaction potential w( ̺) the metric found in Refs. [23,24,25] is (see e.g. Eq. (78) in [25] [32]) The metric (21) is implicit as it employs the notion of the monopole-electric charge interaction potential w which by itself is ambiguous as it does not exist without a Dirac string singularity. The combination ( w(̺) · d ̺) is however independent of the way one introduces the Dirac string singularity. Choosing it along the z axis and solving eq. (19) we find the monopole-electric charge interaction potential if one parameterizes ̺ = (sin θ cos φ, sin θ sin φ, cos θ). Hence, W · dy in eq. (21) can be rewritten as Being combined with eq.(24) it gives where dΣ m = cos θ m dφ m + dψ m according to eq.(7). Therefore, the second term in eq.(21) can be written as where, according to eq.(23), Since dξ 4 is, in this context, a N -vector (dξ 4 , ..., dξ 4 ), one has for all components (dξ 4 S) m = 0, see the definition of S in eq. (20). Hence, the term quadratic in dξ 4 in eq.(28) is simply where tr N = m ν m = 1 has been used. Because of (dξ 4 S) m = 0, the terms linear in dξ 4 in eq.(29) are zero. In the last term, quadratic in dΣ, we note that We introduce N × N matrices K, L, M : With the help of these matrices, the chain of eq.(31) can be continued as Thus, the last term in eq.(28) is Combining all terms from eq.(21) we obtain finally a simple and explicit expression for the moduli space metric: where dΣ m is given by eq. (7). As a matter of fact, we have independently derived the moduli space metric in precisely this form, using the ADHMN construction for the SU (N ) caloron. However, since the derivation is lengthy but not qualitatively different from that of Kraan [25] we do not present it here. Explicitly, the K, L, M matrices involved in eq.(37) are M mn ≡ 4π 2 ν n + π ̺n + π ̺n+1 δ mn − π ̺n δ m+1,n − π ̺m δ m,n+1 .
Notably K and M are symmetric and differ only by interchanging 4π 2 ν m and ρ m /π: this will be used in computing the determinants.
As an example, we give the matrix M for the SU (4) gauge group: The SU (2) gauge group is too "small" for the general formula (40). In this case the matrix M is simply where ̺ 1 = ̺ 2 = | y 1 − y 2 | and ν 1 + ν 2 = 1.

V. THE DETERMINANT OF THE METRIC TENSOR
In the previous section we have rewritten the moduli space metric in the explicit form (37). However only the determinant of the metric is needed in such calculations as the saddle point approximation, see. eq. (14). In this section we derive a compact expression for the volume of the general SU (N ) moduli space and then give examples for the specific cases of the SU (2) and SU (3) groups, as well as an asymptotic formula for the general SU (N ) group, valid at large separations between the dyons.
First of all, we need to check the dimension or the number of parameters of the moduli space. These are the 3N coordinates of dyon centers y i , one overall time position ξ 4 , and N − 1 relative color orientations ψ m entering the metric (37) from eq.(7). Therefore, the dimension of the caloron moduli space is 4N as it should be for a general self-dual solution with unity topological charge. We note that the transformation δψ 1 = δψ 2 = . . . = δψ N is a global U (1) gauge rotation leaving the gauge field unchanged. As a consequence, the matrix L has one zero eigenvalue L|1, . . . , 1 >= 0 which makes the size of the maximal non-degenerate minor of the metric tensor (37) equal to 4N . The determinant of the metric tensor is where det ′ L is the product of all non-zero eigenvalues of L. The corresponding volume form is is a set of variables that parameterize the relative U (1) orientations of the dyons. Note that α N corresponds to the trivial gauge transformation. The transformation matrix has the form The factor 2 N −1 √ N in eq.(45) comes from the equation where (QQ T ) −1 00 ≡ 1, 1, . . . , 1|(QQ T ) −1 |1, 1, . . . , 1 /N . It turns out that det ′ L can be expressed through det K. To show this let us introduce such that the matrix L from eq.(39) can be written as Using that < 1, 1, . . . , 1|RK −1 = π < 1, 1, . . . , 1| we have and from a simply calculable det ′ V = N N m=1 νm we obtain In its turn, det K has a simple relation to det M which follows from the symmetry between the two matrices mentioned at the end of Section IV.B. Thus, the final result for the element of the volume of the moduli space is where M is given by eq.(40). One can also rewrite it in terms of the "center of mass" position ξ = m ν m y m and the separations between dyons neighboring in color space, ρ m : We now integrate over α m in the fundamental domain specified by (57) and obtain Thus, the caloron measure integrated over the U (1) phases (denoted by G) is Below we find det M in the particular cases of the SU (2) and SU (3) gauge groups and in the limit of large dyon separations in a general SU (N ) case.

C. SU(N), large separations
In the general case, det M cannot be written in an easy form. However, for large separations between dyons in a caloron one can derive a simple asymptotic for det M , eq.(40). We expand it in inverse powers of ̺ m . Let us write where the matrix M 1 is composed of the inverse powers of ̺ m . We have Hence from eq.(59) we obtain the caloron measure We remind the reader that periodicity in the indices is assumed; Eq.(65) can be interpreted as Coulomb repulsion of dyons inside a caloron. However, not all dyons interact with each other but only those that are "neighbors" in the color space. In SU (3) all dyons are neighbors in this sense, while the SU (2) group is too small to see the effect.

VII. RELATION TO THE INSTANTON MEASURE IN THE TRIVIAL HOLONOMY LIMIT
In this limit, the KvBLL caloron becomes the Harrington-Shepard periodic instanton with the standard BPST instanton moduli space. It is basically an SU (2) configuration embedded into the SU (N ) group. The 4N -parameter moduli space is usually described as 4 'center-of-mass' coordinates ξ µ , one 'size' parameter ρ, and 4N − 5 'gauge orientation' collective coordinates determining the embedding. This has been the traditional parametrization of instantons for 25 years, starting from the work by Bernard [28] who computed the instanton measure and its volume for a general SU (N ) group.
At first glance, there is little in common between this moduli space and that of the non-trivial caloron, given in the previous sections in terms of the constituent dyons' 3d positions and U (1) phases. Our goal is to demonstrate that the measures of the two moduli spaces in fact coincide exactly, including the non-trivial normalization.
We shall do it in two steps. In this section we show that the volume of the dyon moduli space coincides with that found by Bernard in terms of the SU (2) embedding. In the next section we give an explicit construction of the instanton SU (N ) gauge orientation matrix (determining the SU (2) embedding) in terms of the 3d positions and U (1) phases of the constituent dyons.
The trivial holonomy limit corresponds to taking all Polyakov eigenvalues equal µ m = k/N − 1, m ≤ k and µ m = k/N, m > k, where k = 0, 1, ..., (N − 1), meaning all their differences ν m = µ m+1 − µ m are zero except one which is unity, see section II. First of all we note that in this limit one gets where s is the perimeter of the polygon formed by the dyons. It is directly related to the instanton size ρ by eq.(8): ρ = s/2π. To find the volume of the dyons moduli space and relate it to the standard instanton one, we have to integrate eq.(59) over dyons' 3d positions with the perimeter s fixed. More concretely, we have to evaluate Leaving unintegrated the center of mass 4-coordinate ξ µ and the instanton size ρ, the moduli space volume is, from eq.(59), The integral (67) is computed in Appendix A with the result Consequently coinciding exactly with Bernard's result [28]. It is interesting to note that it was obtained there in a completely different way -by computing the group volume for the embedding of SU (2) into SU (N ). There seems to be nothing near it in the present derivation.

VIII. LIMITING CASES OF THE CALORON GAUGE FIELD
In this section we give the trivial holonomy limit of the KvBLL gauge field. It is the Harrington-Shepard SU (2) instanton imbedded into SU (N ). The way it is embedded depends on the constituent dyons color orientations and their relative positions. As a byproduct, we give the gauge field of the KvBLL caloron with the exponential precision (i.e. dropping terms of the order of Ø(e −2πrmνm ), where r m is a distance to the m th dyon).

A. Far from the cores
With the exponential precision, the matrix F mn (see eq.(B5)) is diagonal and so is the matrix f mn : From eq.(6) one has and from eq.(10) The last term in eq.(9) is zero, and with the exponential precision we can write the gauge field This expression is similar to the one found in [22] for the SU(2) case. It is given in a non-periodical gauge. To pass to the periodical gauge one has to add 2πiµ m δ mn to A 4 (see the discussion at the end of the subsection D). A 4 has the Coulomb-like form. In the periodical gauge where e ϕm ≡ rm−1× rm | rm−1× rm| .
B. Reduction to the trivial holonomy case In the trivial holonomy limit (ν L = 1, ν m =L = 0) eq.(B1) simplifies. It becomes a Shrödinger equation on the unit circle with only one delta function in the left-hand-side. The solution is independent of N and can be found in Ref. [17]. It is given by where 2πρ 2 = ρ m and r ≡ r L . We now introduce a N × N unitary matrix U which plays the role of the 'color orientation' of the (periodic) instanton to which the KvBLL reduces in the trivial holonomy case. The first two columns of U are defined through the spinors The rest columns are constrained only by the unitarity condition U † U = 1 N ; they are not involved in the field construction. Correspondingly, the first two rows of U † are given by the hermitian conjugate spinors, This definition is non-contradictory if the two complex N -vectors U m 1 and U m 2 are orthogonal and normalized to unity. Indeed, using eqs. (6,8), we obtain To write down the gauge field of the trivial-holonomy instanton from the general expressions (9,10) we first replace there ζ → U according to eqs.(78,79): where (λ 0 , λ a ) are N × N matrices with (1 2 , τ a ) put into the left-upper corner. The last term in eq.(9) is again zero, and we arrive at the compact result for the trivial-holonomy caloron: This formula reproduces exactly the Harrington-Shepard instanton with arbitrary 'gauge orientation' U defined, as we see from eq.(78), by the dyon relative coordinates ̺ m and the relative U (1) orientation angles.

C. Small-size KvBLL caloron
Another important limit when the caloron field has a simple form is the case of small ρ = m ̺ m /(2πT ) ≪ 1/T implying that dyons' separations are small, ̺ m ≪ 1/T (in this subsection we restore the temperature factors). We are interested in the caloron field at the distances r from the center of the group of N dyons, larger than the separations between them, r ∼ ρ ≫ ̺ m . Therefore, we can put in the leading order r m = r for all m = 1, . . . , N . We also consider the range of x 4 ∼ ρ where the field is large. In this range, the ADHM Green function is simply Repeating the calculations from the previous subsection we arrive at a standard expression for the BPST instanton: where the instanton orientation matrix U is given by eq.(78). Corrections to eq.(86) die out as T in the range r ∼ x 4 ∼ ρ ≫ ̺ m where the field is large. Eq.(86) is the approximate gauge field for small-size calorons in the non-periodic gauge used in Ref. [17]. To obtain the approximate small-ρ field in the periodic gauge, one has to gauge-transform eq.(86): where g mn = δ mn e 2πiµmx4 . We note finally that when all dyons' separations ̺ m are small, the metric determinant is given by eq.(66) (even though the holonomy can be non-trivial!), and the caloron measure coincides with that of the standard instanton, as shown in Section VII.

IX. CONCLUSIONS
The metric of the 4N -dimensional moduli space of the general SU (N ) caloron with arbitrary eigenvalues of the Polyakov line at spatial infinity and at any temperature, is given in terms of the spatial coordinates of the N dyons that constitute the caloron, and their U (1) phases.
We have computed the determinant of the metric tensor, which defines the weight of the SU (N ) caloron contribution to the partition function. The metric determinant is a function of the 3d separations between dyons and of the Polyakov loop eigenvalues. When all those eigenvalues are equal, it is the "trivial holonomy" case, and the KvBLL caloron reduces to the usual periodic instanton whose moduli space is usually written in terms of the instanton position, size and orientation. We have shown that the SU (N ) instanton measure written in these variables coincides exactly with the one written in terms of the coordinates and phases of the instanton constituents, the dyons. This result is not altogether trivial, as in the first case the measure arises from the volume of the SU (N )/SU (N − 2) coset whereas in the second case it follows from the 3d geometry. We have also identified the instanton SU (N ) orientation matrix through the dyons positions and U (1) phases.
The following emerging physical picture may be plausible. The adequate degrees of freedom in the Yang-Mills vacuum are, at any temperatures, calorons with non-trivial holonomy, which are more general than the standard periodic instantons with trivial holonomy. The measure should be described in terms of dyons' positions and phases. The free energy of the ensemble of interacting dyons should be studied; hopefully at low temperatures it has a minimum at the "most non-trivial holonomy" corresponding to Tr L = 0, however at T > T c related to Λ there must be N degenerate minima corresponding to trivial holonomy. An indication that this may indeed be the case has been presented for SU (2) in Ref. [22]. If correct, it would serve as the microscopic mechanism of the confinementdeconfinement transition.
At low temperatures, although the correct description is still in terms of dyons with non-trivial holonomy supporting the confinement, statistical fluctuations will lead to a large portion of dyons that are not widely separated. If a group of N different-type dyons happen to be close to each other, the configuration is locally undistinguishable from the standard SU (N ) instanton. Small-size instantons can be described both in the "position-size-orientation" terms, and in terms of dyons. However, for large-size overlapping instantons the former language looses sense while the latter remains valid.
This physical picture (calling, of course, for a detailed mathematical study) may justify the adequacy of the small-size instantons in physics related to the spontaneous chiral symmetry breaking, while simultaneously explaining confinement as presumably due to dyons. APPENDIX A: VOLUME OF THE MODULI SPACE To get the moduli space volume, one has to integrate over the positions of N dyons with the perimeter of the N -polygon fixed. More precisely we have to evaluate the following integral (A1) To reduce the number of integrations in eq.(A1) we use the following trick. We introduce auxiliary integrals over Feynman parameters to reproduce the δ-functions: The infinitesimal iǫ is added to ensure convergence. Now we can integrate over ̺ m since the integrals are factorized: The iǫ shift makes each integral over ̺ m finite. One can easily calculate it: d̺ m d cos θ 2π̺ m e i̺m(β+iǫ) e i̺mα cos θ = 4π α 2 − (β + iǫ) 2 . (A4) Now the measure can be written as a 4d integral From dimensions, µ N (s) = α N s 2N −3 where α N is a constant to be computed; we find it by induction. We first consider the N = 2 case where µ 2 (s) can be found directly from eq.(A1): This implies α 2 = 2 3 π 4 . For general N we rotate the integration contour α → −iα in eq.(A5) since the poles are at ±(β + iǫ). We can then rewrite eq.(A5) in an SO(4) invariant form where α µ = ( α, β). The crucial step is the following recurrent relation: where we have used that the radial part of the 4d Laplace operator is ∂ 2 µ f (s) = 1 s 3 ∂ s s 3 ∂ s f (s) . The solution to this equation is .