Non-perturbative effective interactions from fluxes

# Non-perturbative effective interactions from fluxes

Marco Billò, Livia Ferro, Marialuisa Frau, Francesco Fucito, Alberto Lerda and Jose F. Morales

Dipartimento di Fisica Teorica, Università di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
Laboratoire de Physique Théorique
École Normale Supérieure
24, rue Lhomond, F- 75231 Paris Cedex 05, France
I.N.F.N. - sezione di Roma II
Via della Ricerca Scientifica, I-00133 Roma, Italy
Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale
and I.N.F.N. - Gruppo Collegato di Alessandria - sezione di Torino
Via V. Bellini 25/G, I-15100 Alessandria, Italy
###### Abstract:

Motivated by possible implications on the problem of moduli stabilization and other phenomenological aspects, we study D-brane instanton effects in flux compactifications. We focus on a local model and compute non-perturbative interactions generated by gauge and stringy instantons in a quiver theory with gauge group and matter in the bifundamentals. This model is engineered with fractional D3-branes at a singularity and its non-perturbative sectors are described by introducing fractional D-instantons. We find a rich variety of instanton-generated interactions, ranging from superpotentials and Beasley-Witten like multi-fermion terms to non-supersymmetric flux-induced instanton interactions.

Superstrings, D-branes, Gauge Theories, Instantons
preprint: DFTT/20/2008
ROM2F/2008/20
LPTENS 08/41

## 1 Introduction and motivations

Recently a lot of attention has been devoted to the study of four dimensional compactifications of Type II string theories with systems of intersecting or magnetized D-branes that preserve supersymmetry [1, 2, 3]. These compactifications provide, in fact, promising scenarios for phenomenological applications and realistic model building in which gauge interactions similar to those of the supersymmetric extensions of the Standard Model of particle physics are engineered using space-filling D-branes that partially or totally wrap the internal six-dimensional space. The effective actions of such brane-world models describe interactions of gauge degrees of freedom, associated to open strings, with gravitational fields, associated to closed strings, and have the generic structure of supergravity in four dimensions coupled to vector and chiral multiplets. As is well-known [4], four-dimensional supergravity theories are specified by the choice of a gauge group , with the corresponding adjoint fields and gauge kinetic functions, by a Kähler potential and a superpotential , which are, respectively, a real and a holomorphic function of some chiral superfields . The supergravity vacuum is parametrized by the expectation values of these chiral multiplets that minimize the scalar potential

where is the Kähler covariant derivative of the superpotential and the () are the D-terms. Supersymmetric vacua, in particular, correspond to those solutions of the equations satisfying the D- and F-flatness conditions .

When we consider Type IIB string theory on a Calabi-Yau three-fold in presence of D3-branes, which is the case discussed in this paper, the chiral superfields comprise the fields and that parameterize the deformations of the complex and Kähler structures of the three-fold, the axion-dilaton field

 τ=C0+ie−φ , (2)

where is the R-R scalar and the dilaton, and also some multiplets coming from the open strings attached to the D-branes. The resulting low-energy supergravity model has a highly degenerate vacuum. One way to lift (at least partially) this degeneracy is provided by the addition of internal 3-form fluxes of the bulk theory [5, 6, 7] via the generation of a superpotential [8, 9]

 Wflux=∫G3∧Ω . (3)

Here is the holomorphic -form of the Calabi-Yau three-fold and is the complex 3-form flux given in terms of the R-R and NS-NS fluxes and . The flux superpotential (3) depends explicitly on and implicitly on the complex structure parameters which specify . Insisting on unbroken supersymmetry requires the flux to be an imaginary anti-selfdual 3-form of type (2,1) [10], since the F-terms , and are proportional to the , and components of the -flux respectively. These F-terms can also be interpreted as the “auxiliary” -components of the kinetic functions for the gauge theory defined on the space-filling branes, and thus are soft supersymmetry breaking terms for the brane-world effective action. Such soft terms have been computed in various scenarios of flux compactifications and their effects, like for instance induced masses for the gauginos and the gravitino, have been analyzed relying on the structure of the bulk supergravity Lagrangian and on -symmetry considerations (see for instance the reviews [5, 6, 7] and references therein) and recently also by a direct world-sheet analysis in [11].

Beside fluxes, also non-perturbative contributions [12, 13] to the effective actions may play an important rôle in the moduli stabilization process [14, 15] and bear phenomenologically relevant implications for string theory compactifications. In the framework we are considering, non-perturbative sectors are described by configurations of D-instantons or, more generally, by wrapped Euclidean branes which may lead to the generation of a non-perturbative superpotential of the form

 Wn.p.=∑{kA}c{kA}(Φi)e2πi∑AkAτA . (4)

Here the index labels the cycles wrapped by the instantonic branes, while denotes the instanton number and the complexified gauge coupling of a D-brane wrapping the cycle . Finally, are some (holomorphic) functions of the chiral superfields whose particular form depends on the details of the model. In general, the ’s depend on the axion-dilaton modulus and the Kähler parameters that describe the volumes of the cycles around which the D-branes are wrapped111The explicit dependence of on and can be derived from the Dirac-Born-Infeld action as explained in Appendix A.1.. We remark that Eq. (4) holds both for gauge and stringy instantons corresponding, respectively, to the cases where the cycle is occupied or not by a gauge D-brane.

The interplay of fluxes and non-perturbative contributions, leading to a combined superpotential

 W=Wflux+Wn.p. , (5)

offers new possibilities for finding supersymmetric vacua. Indeed, the derivatives , and might now be compensated by , and [15] so that also the , and components of may become compatible with supersymmetry and help in removing the vacuum degeneracy [16].

Another option could be to arrange things in such a way to have a Minkowski vacuum with and broken supersymmetry. If the superpotential is divided into an observable and a hidden sector, with the flux-induced supersymmetry breaking happening in the latter, this could be a viable model for supersymmetry breaking mediation. If all moduli are present in , the number of equations necessary to satisfy the extremality condition for seems sufficient to obtain a complete moduli stabilization. To fully explore these, or other, possibilities, it is crucial however to develop reliable techniques to compute non-perturbative corrections to the effective action and determine the detailed structure of the non-perturbative superpotentials that can be generated, also in presence of background fluxes.

In the last few years there has been much progres in the analysis of non-perturbative effects in brane-world models and concrete computational tools have been developed using systems of branes with different boundary conditions [17, 18]. These methods not only allow to reproduce [18]-[22] the known instanton calculus of (supersymmetric) field theories [23], but can also be generalized to more exotic configurations where field theory methods are not yet available [24]- [46]. The study of these exotic instantons has led to interesting results in relation to moduli stabilization, (partial) supersymmetry breaking and even fermion masses and Yukawa couplings [24, 25, 33]. A careful analysis of the moduli of this kind of instantons is however required in order to be sure that unwanted neutral fermionic zero-modes are either absent, as in some orientifold models [28, 29, 30], or lifted [35, 39]. If really generated, such exotic interactions could also become part of a scheme in which the supersymmetry breaking is mediated by non-perturbative soft-terms arising in the hidden sector of the theory, as recently advocated also in [46]. Nonetheless, the stringent conditions required for the non-perturbative terms to be different from zero, severely limit the freedom to engineer models which are phenomenologically viable.

To make this program more realistic, in this paper we address the study of the generation of non-perturbative terms in presence of fluxes. Indeed fluxes not only lead to the perturbative superpotential (3) but also lift some zero-modes of the instanton background and allow for new types of non-perturbative couplings. In the following we will consider the interactions generated by gauge and stringy instantons in a specific setup consisting of fractional D3-branes at a singularity which engineer a quiver gauge theory with bi-fundamental matter fields. In order to simplify the treatment, still keeping the desired supergravity interpretation, this quiver theory can thought of as a local description of a Type IIB Calabi-Yau compactification on the toroidal orbifold . From this local standpoint, it is not necessary to consider global restrictions on the number and of D3-branes, which can therefore be arbitrary, nor add orientifold planes for tadpole cancelation. In such a setup we then introduce background fluxes of type and , and study the induced non-perturbative interactions in the presence of gauge and stringy instantons which we realize by means of fractional D-instantons. In this way we are able to obtain a very rich class of non-perturbative effects which range from ”exotic” superpotentials terms to non-supersymmetric multi-fermion couplings. We also show that, as anticipated in [11], stringy instantons in presence of -fluxes can generate non-perturbative interactions even for gauge theories. This has to be compared with the case without fluxes where an orientifold projection [28, 29, 30] is required in order to solve the problem of the neutral fermionic zero-modes. Notice also that since the and components of the are related to the gaugino and gravitino masses (see for instance [47, 48]), the non-perturbative flux-induced interactions can be regarded as the analog of the Affleck-Dine-Seiberg (ADS) superpotentials [49] for gauge/gravity theories with soft supersymmetry breaking terms. In particular the presence of a flux has no effect on the gauge theory at a perturbative level but it generates new instanton-mediated effective interactions [38].

For the sake of simplicity most of our computations will be carried out for instantons with winding number ; however we also briefly discuss some multi-instanton effects. In particular from a simple counting of zero-modes we find that in our quiver gauge theory an infinite tower of D-instanton corrections can contribute to the low-energy superpotential, even in the field theory limit with no fluxes, in constrast to what happens in theories with simple gauge groups where the ADS-like superpotentials are generated only by instantons with winding number . These multi-instanton effects in the quiver theories certainly deserve further analysis and investigations.

The plan of the paper is the following: in Section 2 we review the D-brane setup in the orbifold in which our computations are carried out. In Section 3 we discuss a quick method to infer the structure of the non-perturbative contributions to the effective action based on dimensional analysis and symmetry considerations. In Section 4 we analyze the ADHM instanton action and discuss in detail the one-instanton induced interactions in SQCD-like models without introducing -fluxes. Finally in Sections 5 and 6 we consider gauge and stringy instantons in presence of -fluxes and compute the non-perturbative interactions they produce. Some more technical details are contained in the Appendix.

## 2 D3/D(–1)-branes on C3/(Z2×Z2)

In this section we discuss the dynamics of the D3/D brane system on the orbifold where the elements of act on the three complex coordinates of as follows

 h1 :(z1,z2,z3)→(z1,−z2,−z3) , (6) h2 :(z1,z2,z3)→(−z1,z2,−z3) , h3 :(z1,z2,z3)→(−z1,−z2,z3) .

This material is well-known; nevertheless, we review it mainly with the purpose of setting our notations.

### 2.1 The gauge theory

A stack of D3-branes in flat space gives rise to a four-dimensional gauge theory with supersymmetry. Its field content, corresponding to the massless excitations of the open strings attached to the D3-branes, can be organized into a vector multiplet and three chiral multiplets (). These are matrices:

 {V,ΦI}u v (7)

with . In superspace notation, the action of the theory is

 (8)

where is the axion-dilaton field (2) and is the chiral superfield whose lowest component is the gaugino.

When the D3-branes are placed in the orbifold, the supersymmetry of the gauge theory is reduced to and only the -invariant components of and are retained. Since is a scalar under the internal group, while the chiral multiplets form a vector, it is immediate to find the transformation properties of these fields under the orbifold group elements . These are collected in Tab. 1, where in the first columns we have displayed the eigenvalues of and in the last column we have indicated the irreducible representation () under which each field transforms222Quantities carrying an index of the chiral or anti-chiral spinor representation of , like for example the gauginos or of the theory, transform in the representation of the orbifold group; thus there is a one-to-one correspondence between the spinor indices of and those labeling the irreducible representations of ; for this reason we can use the same letters in the two cases (see for example Ref. [11] for details)..

To each representation of one associates a fractional D3 brane type. Let be the number of D3-branes of type with

 3∑A=0NA=N (9)

so that the adjoint fields , of the parent theory break into blocks transforming in the representation . Explicitly, writing with , we have

 R0⊗RA=RAandRI⊗RJ=δIJR0+|ϵIJK|RK . (10)

The invariant components of and surviving the orbifold projection are given by those blocks where the non-trivial transformation properties of the fields are compensated by those of their Chan-Paton indices and are

 {V}uA vA ∪ {ΦI}uA vA⊗I . (11)

Here the symbol denotes the components of the block, and the subindex is a shorthand for the representation product , namely

 0⊗I=IandJ⊗I=|ϵJIK|K (12)

as follows from (10). Eq. (11) represents the field content of a gauge theory with gauge group and matter in the bifundamentals , which is encoded in the quiver diagram displayed in Fig. 1.

The projected theory is invariant under the global symmetries corresponding to the Cartan subgroup of the -symmetry invariance of the action (8):

 ΦI→eiζIΦI,V→V,Wα→ei2∑IζIWα (13) dθ→e−i2∑IζIdθ,d¯θ→ei2∑IζId¯θ ;

these transformations encode the charges of the various fields w.r.t. to the three ’s. The symmetry extends, as we will see, to the zero modes of the gauge fields in instantonic sectors, and can be exploited to constrain the form of the allowed non-perturbative interactions. To this aim, it will prove useful to take linear combinations of the symmetries (13) corresponding to introducing the charges

 q=q1+q2+q3 ,q′=q1−q2 ,q′′=q1−q3 . (14)

The values of these charges for the various gauge fields are displayed in Table 2.

The fractional D3-branes can also be thought of as D5-branes wrapping exceptional (i.e. vanishing) 2-cycles of the orbifold. Note that there are only three independent such cycles on which are associated to the three exceptional ’s corresponding to the non-trivial elements of . This implies that only three linear combinations of the ’s are really independent. Indeed, the linear combination is trivial in the homological sense since a D5 brane wrapping this cycle transforms in the regular representation and can move freely away from the singularity because it is made of a D5-brane plus its three images under the orbifold group. The gauge kinetic functions of the four factors can be expressed in terms of the three Kähler parameters describing the complexified string volumes of the three non-trivial independent 2-cycles and the axion-dilaton field . In the unresolved (singular) orbifold limit, which from the string point of view corresponds to switching off the fluctuations of all twisted closed string fields, we simply have [50, 11]

 τA=θA2π+i4π2g2A=14τ (15)

for all ’s. However, by turning on twisted closed string moduli, one can introduce differences among the ’s and thus distinguish the gauge couplings of the various group factors.

### 2.2 The instanton moduli space

A very similar analysis applies to instantonic sectors of the gauge theory. In this framework instantons are realized with D-branes and their moduli space is described by the lowest modes of open strings with at least one end-point on the D-branes. The gauge theory dynamics in the sector with instanton number can be efficiently described by the limit of the action of a system of D-branes and D3-branes on . This system is described in terms of a matrix theory whose action is [23]

 SD3/D(−1)=Trk[12g20SG+SK+SD+Sϕ] (16)

with

 SG= DcDc−12[χm,χn]2−iλ˙αA[χAB,λ˙α B] , (17) SK= χm¯w˙αw˙αχm−[χm,aμ]2−iMαA[¯¯¯¯χAB,M Bα]+i2¯¯¯¯χAB¯μAμB , SD= iDc(¯w˙α(τc)˙α ˙βw˙β−i¯ηcμν[aμ,aν])+iλ˙αA(¯μAw˙α+¯w˙αμA−[aα˙α,M Aα]) , Sϕ= 18ϵABCD¯w˙α¯¯¯ϕAB¯¯¯ϕCDw˙α+12¯w˙αϕABw˙α¯¯¯¯χAB+i2¯μA¯¯¯ϕABμB .

The fields entering in (17) represent the lowest modes of open strings with D/D and D/D3 boundary conditions and are

 M={aμ,χm,Dc,MαA,λ˙αA}i j ∪ {w˙α,μA}u i ∪ {¯w˙α,¯μA}i u (18)

with and labeling the D and the D3 boundaries respectively. The other indices run over the following domains: ; ; ; , labeling, respectively, the vector and spinor representations of the Lorentz group and of the internal-symmetry group333See footnote 2., while . We have also defined

 χAB =χm(Σm)AB,¯¯¯¯χAB=χm(¯¯¯¯Σm)AB=12ϵABCDχCD , (19) ϕAB =ϕm(Σm)AB,¯¯¯ϕAB=ϕm(¯¯¯¯Σm)AB=12ϵABCDϕCD

where and are the chiral and anti-chiral blocks of the Dirac matrices in the six-dimensional internal space, and are the six vacuum expectation values of the scalar fields in the real basis. Finally,

 1g20=πgs(2πα′)2 (20)

is the coupling constant of the gauge theory on the D branes. The scaling dimensions of the various moduli appearing in (17) are listed in Tab. 3.

The action (16) follows from dimensional reduction of the six-dimensional action of the D5/D9 brane system down to zero dimensions and, as discussed in detail in Ref. [18], it can be explicitly derived from scattering amplitudes of open strings with D/D or D/D3 boundary conditions on (mixed) disks. In the field theory limit (i.e. ), the term in (16) can be discarded and the fields and become Lagrange multipliers for the super ADHM constraints that realize the D- and F-flatness conditions in the matrix theory.

Now let us consider the orbifold projection. The group breaks down to with and being the numbers of fractional D3 and D branes of type such that

 N=3∑A=0NAandk=3∑A=0kA . (21)

Consequently, the indices and break into and , while the spinor index splits into with denoting the unbroken symmetry. For the sake of simplicity from now on we always omit the index 0 and write

 Mα0≡Mα,λ˙α0≡λ˙α,μ0≡μ,¯μ0≡¯μ . (22)

Furthermore we set

 ¯¯¯¯χ0I ≡¯¯¯¯χI, χ0I=12ϵIJK¯¯¯¯χJK≡χI , (23) ¯¯¯ϕ0I ≡¯¯¯ϕI=⟨¯ΦI⟩, ϕ0I=12ϵIJK¯¯¯ϕJK≡ϕI=⟨ΦI⟩ .

This notation makes more manifest which zero-modes couple to the holomorphic superfields and which others couple to the anti-holomorphic ones. Indeed, the action in (17) becomes

 Sϕ= 12¯w˙α(ϕI¯ϕI+¯ϕIϕI)w˙α+¯w˙αϕIw˙α¯¯¯¯χI+χI¯w˙α¯ϕIw˙α (24) −i2¯μ¯ϕIμI+i2¯μI¯ϕIμ−i2ϵIJK¯μIϕJμK .

Taking into account the transformation properties of the various fields and of their Chan-Paton labels, one finds that the moduli that survive the orbifold projection are

 M = {aμ,Dc,Mα,λ˙α}iA jA  ∪  {w˙α,μ}uA jA  ∪  {¯w˙α,¯μ}iA uA (25) ∪  {χI,¯¯¯¯χI,MαI,λ˙αI}iA jA⊗I  ∪  {μI}uA iA⊗I  ∪  {¯μI}iA uA⊗I .

Like for the chiral multiplets in (11), the non-trivial transformation of the instanton moduli carrying an index is compensated by a similar transformation of the Chan-Paton labels making the whole expression invariant under the orbifold group, as indicated in the second line of (25).

The projected moduli action is invariant under the symmetry whose properties on the D3/D3 sector we already discussed in Sec. 2.1, see Table 2. The charges of the moduli with respect to the same choice of made in Eq. (14) are given in Table 4.

Among the moduli in , the bosonic combinations

 xμ≡1k3∑A=0kA∑iA=1{aμ}iA iA (26)

represent the center of mass coordinates of the D-branes and can be interpreted as the Goldstone modes associated to the translational symmetry of the D3-branes that is broken by the D-instantons. Thus they can be identified with the space-time coordinates, and indeed have dimensions of a length. Similarly, the fermionic combinations

 (27)

are the Goldstinos for the two supersymmetries of the D3-branes that are broken by the D-branes, and thus they can be identified with the chiral fermionic superspace coordinates. Indeed they have dimensions of (length). Notice that neither nor appear in the moduli action obtained by projecting (16).

The moduli (25) account for both gauge and stringy instantons. Gauge instantons correspond to D-branes that sit on non-empty nodes of the quiver diagram, so that their number can be interpreted as the second Chern class of the Yang-Mills bundle of the component of the gauge group. Stringy instantons correspond instead to D-branes occupying empty nodes of the quiver, so that in this case we can set for all ’s. From (25) one sees that in the stringy instanton case the modes , , and are missing since there are no invariant blocks and thus the only moduli are the charged fermionic fields and . Recalling that the bosonic -moduli describe the instanton sizes and gauge orientations, one can say that exotic instantons are entirely specified by their spacetime positions. The presence (absence) of bosonic modes in the D3/D sector for gauge (stringy) instantons can be understood in geometric terms by blowing up 2-cycles at the singularity and reinterpreting the fractional D3/D system in terms of a D5/E1 bound state wrapping an exceptional 2-cycle of . Gauge (stringy) instantons correspond to the cases when the D5 brane and the E1 are (are not) parallel in the internal space. In the first case the number of Neumann-Dirichlet directions is 4 and therefore the NS ground states is massless. In the stringy case, the number of Neumann-Dirichlet directions exceeds 4 and therefore the NS ground state is massive and all charged moduli come only from the fermionic R sector.

## 3 D-instanton partition function

In this section we study the non-perturbative effects generated by fractional D-instantons on the gauge theory realized with space-filling fractional D3 branes. For simplicity from now on we take (see Fig. 2)

 N2=N3=0withN0 and N1 % arbitrary , (28)

which is the simplest configuration that allows us to discuss both gauge and stringy instanton effects.

This brane system describes a theory with gauge group and a single bifundamental multiplet

 Φ1(x,θ)≡Φ(x,θ)=ϕ(x)+√2θψ(x)+θ2F(x) , (29)

which in block form is

 Φ=⎛⎝0Qu f˜Qf u0⎞⎠ (30)

with and . The two off-diagonal blocks and represent the quark and anti-quark superfields which transform respectively in the fundamental and anti-fundamental of , and in the anti-fundamental and fundamental of . Both quarks and anti-quarks are neutral under the diagonal factor of the gauge group, which decouples. On the other hand [51] the relative group, under which both and are charged, is IR free and thus at low energies the resulting effective gauge group is . Therefore, from the point of view of, say, the factor this theory is just SQCD with colors and flavors. In the following we will study the non-perturbative properties of this theory in the Higgs phase where the gauge invariance is completely broken by giving (large) vacuum expectation values to the lowest components of the matter superfields. This requires . The moduli space of this SQCD is obtained by imposing the D-flatness conditions. As remarked in [52], even if the effective gauge group is , we have to impose the D-term equations also for the (massive) 444For a discussion of this point in the same orbifold model we are considering, see for example the beginning of Section 3 of the published version of Ref. [54]. factors to obtain the correct moduli space of the quiver theory; in our case these D-term conditions lead to the constraint

 Q¯Q−¯˜Q˜Q=ξ1lNc×Nc (31)

where is a Fayet-Iliopoulos parameter related to twisted closed string fields which vanish in the singular orbifold limit. For the D-term constraints allow for flat directions parameterized by meson fields

 Mf1 f2≡˜Qf1 uQu f2 (32)

and baryon fields

 Bf1…fNc=ϵu1…uNcQu1 f1…QuNc fNc,˜Bf1…fNc=ϵu1…uNc˜Qf1 u1…˜QfNc uNc (33)

which are subject to constraints whose specific form depends on the difference (see for instance Ref. [53]). These are the good observables of the low-energy theory in the Higgs phase. For , instead, the baryons cannot be formed and only the meson fields are present.

To have a quick understanding of the non-perturbative effects that can be obtained in our stringy set-up, it is convenient to use dimensional analysis and exploit the symmetries of the D3/D brane system; we will see that besides the well-known one-instanton effects like the ADS superpotential at [49], in the quiver theory an infinite tower of multi-instanton corrections to the superpotential are in principle allowed. This is what we are going to show in the remainder of this section. In Section 4 we specialize our discussion to the one-instanton sector and, using again dimensional analysis and symmetry considerations, we analyze various types of non-perturbative effects in the low-energy theory, as a preparation for the study of the flux-induced terms presented in Sections 5 and 6.

### 3.1 The moduli space integral

The non-perturbative effects produced by a configuration of fractional D-instantons with numbers can be analyzed by studying the centered partition function

 Wn.p.=∫dˆM3∏A=0(MkAβAse2πikAτAe−TrkA[SK+SD+Sϕ]) , (34)

where the integration is over all moduli listed in (25) except for the center of mass supercoordinates and defined in (26) and (27). These centered moduli are collectively denoted by . The action is obtained by taking the field theory limit in (16) and restricting the moduli to their invariant blocks for each , while the term represents the classical action of fractional D-instantons of type (see Eqs. (4) and (15)). Finally the power of the string scale compensates for the scaling dimensions of the measure over the centered moduli space so that the centered partition function has mass dimension 3, as expected. Indeed, using Tab. 3 one can easily show that the mass dimension of the instanton measure is

 D[dˆM]=3−3∑A=0kAβA (35)

where is the one-loop -function coefficient of the gauge theory with fundamentals and anti-fundamentals, namely

where denotes the index555The index is defined by . For gauge groups, the indices in the adjoint, fundamental, symmetric and antisymmetric representations are, respectively, given by , , and . of the representation . It is interesting to remark that the explicit expression of is well-defined even in the case where it cannot be interpreted as the -function coefficient of any gauge theory. Keeping this in mind, all formulas in this section can be applied to both gauge and stringy instanton configurations.

Coming back to the centered partition function (34) one can ask which dependence on the scalar vacuum expectation values is generated by the integral over the instanton moduli. A quick answer to this question follows by requiring that the form of be consistent with the symmetries of the D3/D system. In particular we can exploit the symmetries left unbroken by the orbifold projection that we have discussed in Section 2. These are symmetries of the D3/D action but not of the instanton measure. Indeed, since there are unpaired moduli, like and , which transform in the same way under , the charges of the centered instanton measure, and hence of , are non-trivial. In particular, the charge is 666As one can see from Tab. 4, the moduli and have opposite charges, like and . For this reason they do not contribute to the total charges of the centered measure which thus depend only on the charges of unpaired moduli.

 q[dˆM]=−2nμq(μ)−2nμIq(μI)−2q(λ) (37)

where and are the numbers of and moduli, and the factors of account for identical contributions from and . Finally, the term comes from the two components of the anti-chiral fermion

 λ˙α≡1k3∑A=0kA∑iA=1{λ˙α}iA iA (38)

which are unpaired since their partners, namely the fermionic superspace coordinates defined in (27), have been taken out from the centered measure . The minus signs in (37) come from the fact that a fermionic differential transforms oppositely to the field itself. Using the charges listed in Tab. 4, it is easy to rewrite (37) as

 q[dˆM]=3−3∑A=0kA(3NA−3∑I=1NA⊗I)=3−3∑A=0kAβA . (39)

In a similar way one finds

 q′[dˆM] (40) q′′[dˆM] =−2nμq′′(μ)−2nμIq′′(μI)−2q′′(λ)=−23∑A=0kA(NA⊗1−NA⊗3) .

One can check that the charges of the ADHM measure coincide with the ones of the moduli space of instanton zero-modes. In fact, since in an instanton background the bosonic zero-modes always come together with their complex conjugates, the charges , and of the instanton measure depend only on the number of fermionic zero-modes, namely on the number of gaugino zero-modes, and on the number of zero-modes of the fundamental matter fields777Remember that in an instanton background . These numbers are given by the index of the Dirac operator evaluated, respectively, in the adjoint and fundamental representations under which the fields transform, i.e.

 nΛ =2kAℓ(AdjA)=2kANA , (41) nψI =2kA(2NA⊗I)ℓ(NA)=2kAN