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D0-Branes, Constrained Instantons and D=4 Super Yang-Mills Theories

We consider in more detail the role of D0-branes as instantons in the construction of SU(N) Super Yang-Mills and Super QCD theories in four space-time dimensions with D4, D6 and NS-branes. In particular, we show how the D0-branes describe both the exact and constrained instantons and reproduce the correct pattern of lifting of zero modes on the various branches of these models.

The basic construction of SYM instantons on a Dp-brane world-volume, is as bound states with D(p − 4)-branes [12,13,14,15]. In the case of the construction of ref. [2], the SU (n c ) SYM theory is described by n c D4-branes, so we must consider D0-branes bound to D4-branes. The five-dimensional world-volume of the D4-branes is then compactified to the four-dimensional physical space and, to make a bound D0-brane describe an instanton in the four dimensional physical space, the D0-brane euclidean-time world-line must lie along the compactified direction of the D4-brane. Although the details of such reduction depend on the amount of supersymmetry (N = 4, 2, 1, 0) that we want to preserve in spacetime, the basic properties of the instanton moduli space only depend on the geometry of D0-D4 branes bound states, which we review in this introduction.
The collective dynamics of the instantons is specified by a point action and an appropriate measure over instanton moduli space. In the present D0-branes construction, both ingredients appear as the dimensional reduction down to zero dimensions of the D0-brane one-dimensional collective dynamics. In particular, the instanton moduli space as a hyper-Kahler quotient appears as a Higgs branch of the world-line theory of the D0-branes. The basic terms in the world-line action are the static Born-Infeld action of the D0-brane and a source term coming from the Chern-Simons couplings on the D4-branes world-volume which relates the RR photon of the type-IIA string theory to the theta parameter of the four-dimensional instanton. 1 There are also non-trivial gauge interactions specifying the dynamics of 0-0 strings and 0-4 strings in the general case where we have k D0-branes bound to n c D4-branes. The 0-0 sector is just the dimensional reduction down to one dimension of ten-dimensional SYM with U (k) gauge group, corresponding to a system of k D0-branes, and has N = 4 supersymmetry in four-dimensional notation. The 0-4 sector breaks half of these unbroken 16 supercharges, leading to an action for the 0-4 strings with N = 2 supersymmetric couplings. To be more explicit, let us parametrize the relevant degrees of freedom in terms of a T-dual configuration of D3-branes inside D7-branes of type IIB string theory. We will take the D3-branes world-volume to lie in the (x 6 , x 7 , x 8 , x 9 ) plane, with x 6 euclidean time, and the D7-branes to extend further in the (x 0 , x 1 , x 2 , x 3 ), all euclidean, directions. (These conventions for the axes follow from the conventions of ref. [2] where the world-volume of the D4-branes is along the (x 0 , x 1 , x 2 , x 3 , x 6 ) directions, and the request that the euclidean-time world-line of the D0-branes is in the compactified Neumann direction of the D4-branes, that is along x 6 .) Using N = 1 superspace notation, the 0-0 sector contains a vector superfield built from the gauge field (A 6 , A 7 , A 8 , A 9 ) on the D3-brane world-volume, and the "gluino" superpartners, and also three chiral superfields in the adjoint of the U (k) gauge symmetry: W, Z, Z, whose bosonic components are the complex combinations X 4 + iX 5 , X 0 + iX 1 , X 2 + iX 3 respectively, and correspond to the transverse motion of the D3-brane. Upon dimensional reduction to the D0-brane worldline, we will gauge-fix A 6 to zero, and consider only configurations with A 7 = 0, so that the remaining components of the gauge field can be assembled into another chiral superfield A 8 + iA 9 = X 8 + iX 9 = V . The fermionic components of these superfields exhaust the 16k fermionic components of the ten dimensional U (k) SYM theory. If we denote the fields Φ 00 = (W, Z, Z) collectively by Φ i , i = 1, 2, 3, the 0-0 action is defined by the following N = 4 superpotential: The 0-4 sector contributes hypermultiplets in the fundamental and anti-fundamental representations of the U (k) symmetry, with the D4-brane gauge indices a, b, . . . = 1, . . . , n c playing here the role of "flavour". The 0-4 strings contribute a "D04-quark superfield" , and the orientation-reversed 4-0 sector gives the D40-anti-quark . By coupling these hypermultiplets to the previous 0-0 chiral superfields in the adjoint of U (k), we break N = 4 supersymmetry to N = 2 and the SO (6) global R-symmetry down to SU (2) R . Indeed the D04-quark and D40-anti-quark fields (H, H) couple in the superpotential to only one of the three 0-0 superfields. The 0-0 superfield which is singled out in this way is the W superfield according to its four-dimensional field theory interpretation. The corresponding superpotential reads where we have introduced a mass term inherited from the superpotential in the D4-branes, i.e. the 4-4 strings, coupling the SU (n c ) adjoint W ′ 44 = X ′ 4 + iX ′ 5 giving the position of D4branes. The N = 2 supersymmetry on the D0-brane world-line implies that [W ′ , W ′ † ] = 0, i.e. the W ′ field lies along the flat directions of the D4-brane theory, and can be taken as diagonal matrices by a SU (n c ) rotation 2 W ′b The instanton moduli space is then obtained as a maximal Higgs phase of this theory on the D0-brane world-line, where we consider the most general vacuum expectation values of the hypermultiplets H, H and the adjoint fields Z, Z. In this paper we will be mainly concerned with properties of the single instanton sector, or equivalently, of the dilute gas limit of instantons. In this case the overlap between the instantons can be neglected and the moduli space admits a much simpler description. Indeed, well separated D0-branes are characterized by the complete diagonalization of all the adjoint 0-0 fields corresponding to the space-time positions, i.e. we consider Z, Z diagonal generic matrices, which spontaneously break U (k) to the abelian subgroup U (1) k . It is intuitively clear that this produces just k copies of the single instanton moduli space. In this subspace, the 4k position degrees of freedom z i ,z i are completely decoupled from the rest of the variables.
The hypermultiplets D-equations collapse to 5) and the F -equations, for diagonal W and W ′ fields: The brane construction of the full gauge theory produces naively a U (n c ) group. Only the SU (n c ) subgroup is relevant to the instanton discussion, of course. In fact, on the Coulomb branch of the D4-branes world-volume, the extra U (1) is frozen, since fluctuations of the trace of W ′ have infinite norm [16].
where m = 1, . . . , k is an index labeling the D0-brane corresponding to each U (1) factor, and a, b, . . . are the flavour D4-indices. The interesting branch of solutions is the one with maximal Higgsing of the hypermultiplets, with w m = w ′ a = 0, all m, a (all D0 and D4branes on top of each other in the (x 4 , x 5 ) plane). Subtracting the 2k F -equations and the k D-equations from the total 4kn c real components in the scalars φ h , φ h , and further dividing by the U (1) k gauge symmetry, we find a total of 4kn c − 2k − k − k = 4k(n c − 1) real parameters. Adding the 4k degrees of freedom coming from the eigenvalues of Z and Z, representing the translation moduli in the physical dimensions, we end up with the correct 4kn c dimensional bosonic moduli space of YM instantons.
The previous parametrization is appropriate for the dilute instanton limit, in which the non-abelian structure of U (k) is not really probed. Indeed, when the instantons, or D0-branes "overlap", the off-diagonal entries in the Z, Z fields become important and eqs. (1.5) and (1.6) are modified by terms which mix the adjoint superfields with the hypermultiplets. In any case, in the Higgs branch relevant to our discussion, Z and Z contribute 4k degrees of freedom and the hypermultiplets 4k(n c − 1), describing as before the correct 4kn c dimensional bosonic moduli space of YM instantons.

D-brane construction of exact instantons in SYM
In this section we adapt the previous construction to the more detailed realizations of SYM theories in four dimensions with different amounts of supersymmetry, starting with the maximal one, N = 4 SYM.

N=4 SYM
Since the world-volume theory of the D4-branes already has N = 4 SU (n c ) SYM theory in five dimensions, we simply have to compactify the D4-D0 bound states to four dimensions, without any further breaking of supersymmetry. Consider wrapping both the D4 and D0-branes around a circle of length L 6 in the x 6 direction. 3 On the relevant bosonic moduli space of real dimension 2d = 4kn c , the D0-branes world-line has a set of d free superfields (ξ s , η s ), in four-dimensional notation, i.e. we have an action 0 dτ d s=1 |ξ s | 2 + i η sη s + massive .
(2.1) 3 Recall that this corresponds to the euclidean time of the D0-branes.
Each Weyl fermion η s has two complex or four real components off-shell so that, when the one-instanton contribution to the path-integral is dominated by the classical static trajectoryξ s =η s = 0, we get a factor where we defined M D0 L 6 ≡ 8π 2 /g 2 . In this expression, the bosonic measure dµ(ξ) for subgroup of the gauge group where the instanton sits.
The second check of (2.2) is given by the fit of the Yang-Mills coupling to the mass of the D0-brane, M D0 L 6 ≡ 8π 2 /g 2 , needed to reproduce the standard instanton factor.
Dimensional reduction from the D4-brane Born-Infeld action leads to . This is an important point, since it implies that the length of the D0-branes world-line to be considered is really L 6 , a statement which will become non-trivial in the next section, when considering the case of N = 2 and N = 1 SYM theories.

N=2 and N=1 SYM
More interesting is the D-brane construction of instantons with reduced supersymmetry in the four physical dimensions. Using the constructions of refs. [1] and [2], we compactify the five-dimensional theory to four dimensions on the D4-brane world-volume by means of a Kaluza-Klein reduction on a segment of length L 6 in the x 6 direction, bounded by NS five-branes with world-volume in the (x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ) directions. By a suitable complex rotation of the (x 4 , x 5 ) plane into the (x 8 , x 9 ) plane of one of the NS-branes [4], one breaks further the N = 2 supersymmetry to N = 1. In such constructions, the role of the NS-branes is simply to project out certain degrees of freedom, and the appropriate supersymmetries. They do not add any extra degrees of freedom, unless special classes of phase transitions are considered. In particular, the fermions propagating in the world-volume of the NS-branes are set to their vacuum values. Among them, we have the six-dimensional Goldstone fermions associated to the supercharges broken by the NS-brane. Therefore, in constructing the relevant Yang-Mills instantons using D0-branes, it is natural to project out those fermion zero modes which overlap with the above-mentioned six-dimensional Goldstone fermions. This makes sense because we consider constant fermion configurations,η = 0, on the D0-brane world-line, which extends between the NS branes, intersecting them at the end-points.
Recalling (2.3), we see that the D0-brane world-line has length L 6 in order to fit the bare Yang-Mills instanton action, and therefore it extends only once between the NS-branes. This world-line touching the NS-branes at the end-points must be considered as topologically stable, unlike the case of free D0-branes or the N = 4 setting without NS-branes, where a finite D0-action contribution requires wrapping the world-line around some non-contractible circle in the target space. This distinction between free D0-branes and bound D0-branes will reappear in our comments on the relation with the M-Theory lifting in the last section.
On the other hand, it is important that no extra constraints are imposed by the NSbranes on the bosonic collective coordinates ξ s , whose measure is completely determined in (2.2) in the N = 4 theory, up to Jacobian factors which are related to zero modes. In the absence of a detailed understanding of the microscopic couplings between D-branes and NS-branes, we have to rely on consistency checks with the field theory picture and geometric arguments. For example, the projection of certain bosonic degrees of freedom on the world-volume of the D4-branes follows from the global geometry once we assume the rigidity of the NS-branes. Alternatively, the bosonic projections follow from the fermionic ones by supersymmetry. In our case, the brane geometry projects out the scalars in the 0-0 sector, except those representing the motion of the D0-brane in the physical space- there is no hamiltonian representation of bosonic and fermionic degrees of freedom on the instanton "world-volume", and therefore no statement of equality between fermionic and bosonic "states", as it would be required by supersymmetry in non-vanishing dimensions.
Exactly the same mechanism is at work in the standard instanton superfield formalism [17,18,19].
In order to see how the fermion projections work, consider the case of two NS-branes and a set of n c D4-branes and k D0-brane world-lines suspended between them in the x 6 direction. Denote by NS' the second NS-brane, which is parallel to the first NS-brane but translated in the x 6 direction by L 6 in the N = 2 configuration, and rotated into the In general, in the present type-IIA context, the projector over the unbroken supersymmetries on a D-brane world-volume is best represented in M-Theory language as where Γ(Σ p+1 ) is the product of Dirac matrices in the world-volume directions, with the understanding that p-branes whose M-Theory description involves a wrapped or boosted M-brane in the eleventh direction include a factor of Γ 11 [20].
, the chirality operator reads , the fermion occupation numbers taking values 0 or 1. Acting on the 2 6/2 + 2 6/2 states in 0-4 and 4-0 sectors, built as polynomials On the same set, the projection imposed by the NS brane is +1 = iΓ NS = iΓ 0 Γ 1 · · · Γ 4 Γ 5 = −(1 − 2N w ). So, on this subspace the NS projector is P NS = N w , and the analogous projector for the rotated We see that, in all cases, the effect of each NS-brane is to divide by 2 all the bulk zero-modes democratically. This amounts to a total factor of 1/4 when the NS' brane is a rotation of the NS-brane into the (x 8 , x 9 ) plane. So, if N denotes the amount of space-time supersymmetry of the brane configuration, the resulting instanton leaves 2N supercharges unbroken, out of the total 4N which are linearly realized in the vacuum of the space-time theory. The total number of fermionic zero-modes is given by: the correct number, where we have already subtracted the massive degrees of freedom in the Higgs branch which defines the instanton moduli space, and we have split the zero modes between those corresponding to the 2N broken supercharges, and those of a fixed instanton, coming from the hypermultiplet degrees of freedom.
Notice that in the preceding sections, it was convenient to adopt the notation in terms of Weyl fermions with respect to the SO(1, 3) of the (x 6 , x 7 , x 8 , x 9 ) plane. However, the presence of the NS or NS'-branes breaks this group, and the fermion components left after projection need not fullfill representations of such group.

Constrained instantons and the Coulomb branch
Up to now we have considered the case of instantons in super Yang-Mills theory and showed that it can be described by a system of k D0-branes bound to n c D4-branes. In particular this means that the D0-branes and the D4-branes are all located at the same point in the (x 4 , x 5 , x 7 , x 8 , x 9 ) plane. In this section we will show that the geometrical operation of moving apart the D4-branes corresponds, as intuition would tell, to describing constrained instantons [21,22,23]. We postpone the introduction of space-time matter, i.e.
D6-branes, and its complications to the next section. Thus in this section we will study the Coulomb phase of N = 2 SYM models.
Let us start by considering what happens in moving apart the D0-branes. For example, separating one D0-brane from the others in the (x 4 , x 5 ) directions, just means that one is considering a bundle of (k − 1) instantons. In other words, only when the D0-branes are bound to the D4-branes (in the (x 4 , x 5 , x 7 , x 8 , x 9 ) plane) they can describe exact instantons.
It obviously follows that, in moving apart a D4-brane from the others, we will leave the D0branes bound to the (n c − 1) remaining D4-branes, and that we can only speak of exact instantons on the corresponding SU (n c − 1) subgroup. We can, however, still consider approximate ("constrained") instantons of the full SU (n c ) group, by moving a little apart, one by one, the D4-branes while keeping the D0-branes bound to at least two D4-branes. 4 In the following we shall concentrate on the single instanton sector, i.e. only one D0brane, and return at the end of the section to the more involved case of multi-instanton interactions. For a single D0-brane only the relative position with respect to the D4branes matters, and therefore we may fix the D0-brane at w = 0, and consider the effect of turning on the W ′ field, whose diagonal configurations correspond to well defined D4-brane positions.
The relevant coupling in the D0-brane world-line theory which is sensitive to the D4-branes motion is (see eq. (1.4)) This coupling is simply a mass term (the mass matrix given by φ 44 ∼ W ′ which describes the position of the D4-branes in the (x 4 , x 5 ) plane) for the instanton moduli (in the notation of (1.4) Φ 04 ∼ H and Φ 40 ∼ H). When considered in the Coulomb branch of the N = 2 configurations, it yields the familiar action of a constrained instanton: where we have rescaled the bare Yang-Mills coupling g 2 ∼ g s /L 6 into the collective coordinates ξ s and η s . Taking into account the projection at the boundaries of the world-line due to the NS and NS'-branes, we have a partition function of the form where Vol c (M) represents the bosonic "volume" of the moduli space, regulated with the gaussian term exp(−v 2 |ξ| 2 ), and the term v Nd c /2 comes from the fermionic integrals. We now briefly sketch the situation in the case of multi-instanton configurations. In the limit of dilute instantons, the lifting pattern of the zero-modes is just given by k copies of the single instanton case. Instead, beyond the dilute approximation, new terms appear in the effective lagrangian, due to the non-vanishing of the off-diagonal entries of the Z, Z fields. For example, in the parametrization given in ref. [24] of the Higgs branch of the effective theory of the D0-branes, one can exhibit particular configurations with diagonal Z, but non-trivial off-diagonal entries of the Z field, given by as a function of the eigenvalues z n of the Z fields, and the hypermultiplets' moduli. Now the term |[W, Z]| 2 of the effective potential vanishes unless at least one of the two fields has non-vanishing off-diagonal entries. Thus in presence of the non-trivial off-diagonal entries of the Z field given by eq. (3.5), and turning on a diagonal W field, representing definite separations of the D0-branes in the (x 4 , x 5 ) plane, we find positive energy contributions to the effective potential of the form which describe a runaway potential for the eigenvalues z n , and an analogous lifting term for the superpartners. Notice that this potential depends only on the relative separations z m −z n , so that the overall center of mass of all D0-branes is not lifted by these interactions, and furthermore (3.6) vanishes in the limit of instantons at very large distances, i.e. the dilute limit. This is in agreement with the known results in field theory (see for example ref. [19]) where it has been shown that on the Coulomb branch there remain only 4 bosonic and 4 fermionic zero modes, associated with the center of mass position of the instantons and with the broken supersymmetries, respectively.

Space-time flavour and the Higgs branch
In the D-brane formulation of super-QCD (SQCD), the matter (space-time) flavour is described by n f D6-branes with world-volume in the (x 0 , x 1 , x 2 , x 3 , x 7 , x 8 , x 9 ) directions.
We know from field theory that the main effect of the presence of matter on the instantons is the appearance of new fermionic zero modes, as well as new patterns of zero mode lifting when turning on the Higgs branch of the field theory. In this section, we will start by describing the flavour zero modes of the instanton in terms of the local dynamics of the D0-branes and the D6-branes. An important aspect of this dynamics, and a source of many subtleties, is the non-standard geometrical arrangement of the D0-D6 branes system. In our construction, the D0-branes are extended in euclidean time in the x 6 direction, while the D6-branes are completely localized on this axis. Therefore, D0 and D6-branes do not share any propagating dimension, and are purely instantonic relative to each other. This changes the standard statement that the D0-D6 branes system is not supersymmetric.
Here the relevant parameter ν = d ND + d DN , the sum of ND and DN directions in the boundary conditions of 0-6 and 6-0 strings, takes the value ν = 8 and we have some unbroken supersymmetry.
The local spectrum on the 0-6 and 6-0 strings is similar to the standard ν = 8 spectrum at the intersection between a D1-brane and a D9-brane, as explained for example in [25].
We can transform the D0- The two states at each D0-D6 branes intersection can be assembled into a complex field χ = χ 06 , with the orientation reversed component χ 60 representing the conjugated field χ * . In this way we may realize the U (1) gauge symmetry of the D6-brane as a global symmetry on the D0-D6 branes intersection. 6 Now we can consider the full system adding the D4-branes and the NS-branes. By adding the D4-branes the space-time theory becomes SQCD with n f flavours and we expect that in the world-line theory of the D0-branes there will appear induced couplings due to 6 Notice that, in this picture, the full U (n f ) global flavour symmetry is hard to visualize explicitly, since the D6-branes sit at different "instants" of time from the point of view of the D0-brane world-line. which can be written with no reference to the D6-branes. It simply corresponds to an expectation value of the Φ 44 fields in the appropriate directions. Now, in the notation adopted in section 1, in order to write the D0-brane world-line supersymmetry in fourdimensional superspace notation, we represented the D0-D4 fields as T-duals of a D3-D7 brane system whose intersection lied in the "space-time" (x 6 , x 7 , x 8 , x 9 ). In this notation, a motion of the D4-brane in the (x 8 , x 9 ) plane is represented as a background gauge field on the D7-brane world-volume. Going back to the D0-brane world-volume via dimensional reduction, we have to consider a non-zero expectation value of the "gauge field" V ′ 44 , coupling to the "flavour indices" of Φ 44 . So, from this point of view, the appropriate masses come now from D-terms rather than holomorphic couplings on the D0branes world-line. Taking φ v ′ ∼ φ 46 ∼ φ 64 ∼ v, we have induced masses of the form On the other hand, the 2n f flavour collective coordinates found above, χ 06 , χ 60 , are localized at each D0-D6 intersection, and must be lifted by means of a local coupling involving all three types of branes touching locally at that point, including the D4-branes.
The reason is that, as we know from the field theory analysis, the flavour zero modes are lifted by pairing them with the rest of the fermionic zero modes of the instanton. In the D-brane setting, we must pair up the flavour collective coordinates with the fermion zero modes from the 0-4 and 4-0 sectors. Now, the most general coupling respecting the orientation reversal symmetry of the string theory, hermiticity, and the U (1) global symmetry of the D6-branes, is given by where we have used the fact that the complex conjugate of a 0-4 field, ψ * 04 , transforms like a 4-0 field.
In order to exhibit in more detail the pattern of pairings, it is convenient to consider in turn the two cases with N = 2 and N = 1 space-time supersymmetry. In the first case, the NS projection leaves 4n c holomorphic fermionic collective coordinates of the 0-4 and 4-0 strings, prior to any Higgsing on the instanton moduli. We can denote them as ψ a , ψ * a , transforming as ψ 04 , and ψ a , ψ * a , transforming as ψ 40 . The part of the N = 2 Higgs branch accessible to the D-brane representation is the set of non-baryonic Higgs branches, which, according to ref. [26], can be parametrized, up to U (n c ) × U (n f ) transformations, by and we see clearly the geometric pattern of lifting characteristic of the N = 2 configurations: two D6-branes are necessary to lift one D4-brane, and also the zero mode lifting pattern is two to one, in the sense that two fermion components associated to the same D4-brane pair up with two flavour components associated to different D6-branes, those on which the D4-brane is sliding.
In the case of the N = 1 configuration, the NS-NS' projection leaves only 2n c fermionic components of the 0-4 and 4-0 strings, which we can denote by ψ a , ψ * a . Also, the Higgs constraints are different, and the Higgs phase is parametrized by φ 46 = φ 64 = diag(v). Therefore, the general coupling (4.3) reduces now to Now the pairing is one to one, as only one D6-brane is needed to lift each D4-brane.
The couplings (4.5) and (4.6) must be considered together with the induced masses in the bulk for the ψ 04 , ψ 40 fields, coming from the D-term couplings to the V ′ field. In to the appearence of the Affleck-Dine-Seiberg super-potential [27].
Thus in the case of N = 1 supersymmetry with n f = (n c − 1) D6-branes, the presence of a D0-brane bound to the D4-branes gives rise, on the Higgs branch, to an effective repulsive interaction between the D4-branes. This effect is a particular case of the "quantum rules" proposed in ref. [8] for even more general configurations with N = 1 supersymmetry and an arbitrary number of D4-branes and D6-branes. The D0-branes, which act as the field theory instantons in these models, give rise to this "quantum rule" only in the n f = n c − 1 case. In fact, a more complete motivation of such quantum rules in terms of instantons requires studying the system compactified on a circle, and exploiting T-duality [3,8]. Under a T-duality transformation, D0-branes are converted into D1-branes of type-IIB string theory, and the associated instanton configurations look like dimensionally reduced monopoles of the theory with an additional adjoint scalar. The superpotentials on the Coulomb branch of this scalar are saturated by three-dimensional instantons for any number of flavours. This "fractionalization" property of instantons upon compactification is a well known phenomenon [28,29,30].

Discussion
In this paper we have studied the microscopic model of instantons in SU (n c ) SYM theories as D0-branes bound to D4-branes in the Type-IIA D-brane description of the space-time theory. In particular, we have traced the phenomenon of zero-mode lifting of constrained (approximate) instantons, to certain interactions in the local quantum mechanics of the D0-branes world-line. In this way we provide a number of non-trivial checks on the D-brane construction of Yang-Mills instantons, as presented in [12,13].
Although our discussion applies strictly to the regime of weak type-IIA coupling, it is interesting to ask what happens of the D0-branes we have been discussing in this paper, in the lifting to M-theory. A free D0-brane of the type-IIA theory becomes a graviton Kaluza-Klein state carrying momentum along the circle in the eleventh direction, and electrically charged with respect to the U (1) gauge field corresponding to the rotations around this circle. As discussed in refs. [35,36], the physics of such free D0-branes seems to have nothing to do with the field theory SYM models we are interested in. Whereas in the M-theory picture it could be not so simple to decouple these modes, in the Type-IIA construction we can just declare that we do not consider free D0-branes. In this way, we are making an explicit distinction between the free D0-branes, and the D0-D4 branes bound states. As stated in section 2, this distinction is natural from the point of view of the type-IIA configuration, since there is no non-contractible circle where we could wrap a free D0-brane world-line to yield a semiclassical instanton contribution. On the other hand, the world-line of the bound D0-brane is just the euclidean world-line of the D4-brane with which it is solidary. In other words, once the D0-branes are bound to the D4-branes, they must be considered as part of the dynamical data on the D4-branes world-volume.
Thus, the only D0-branes of interest are those which are bound to at least two D4branes and are not allowed to become free. In the internal space, the D0-branes just completely adhere to the D4-branes since they differ only in the space-time directions where the D0-branes are at a point. In lifting to M-theory, only the internal space gets modified and from this point of view the bound D0-branes just completely merge in the D4-branes and they become part of the resulting fivebrane. Actually one could see this from the opposite point of view: in extending the D4-branes to become an M-theory fivebrane wrapped around the eleventh circle, one is effectively dressing them up with bound D0-branes.

Acknowledgements
This work is partially supported by the European Commission TMR programme ERBFMRX-CT96-0045 in which A.P. is associated to the Milano University.