Superpotentials From Stringy Instantons Without Orientifolds
aa r X i v : . [ h e p - t h ] M a r Preprint typeset in JHEP style - HYPER VERSION
Superpotentials From Stringy InstantonsWithout Orientifolds
Christoffer Petersson
Fundamental PhysicsChalmers University of TechnologySE 412 96 G¨oteborg, Sweden [email protected]
Abstract:
In this paper we show that it is possible to derive non-perturbative superpoten-tial terms from a stringy instanton without introducing orientifold planes. The instantonis realized by a Euclidean D brane wrapping a non-trivial cycle upon which we also wrapa single space-filling D brane. The standard problem of unwanted neutral fermionic zeromodes is evaded by the appearance of couplings to charged bosonic zero modes in theinstanton moduli action. Since the Euclidean D brane wraps a cycle which is not associ-ated to any low energy gauge dynamics, it can not be interpreted as an ordinary gaugeinstanton, but rather as a stringy one. By considering such a brane configuration at anorbifold singularity, we can explicitly evaluate the instanton moduli space integral and finda holomorphic superpotential term with the structure of a baryonic mass term. ontents
1. Introduction 12. The Orbifold Projection of the k = 1 Instanton Sector of N =4 SYM 3
3. Determining the Prefactor 7
4. Evaluating the Moduli Space Integral 10
5. Implications for the Gauge Dynamics 12
1. Introduction
There has been interesting recent developments in the context of string theory realizationsof instanton effects in gauge theories [1, 2, 3, 4, 5, 6]. Non-perturbative superpotential termswhich are known to be generated by a single instanton, such as the ADS superpotentialin N =1 SQCD for the case N f = N c −
1, have been explicitly derived using boundaryconformal field theory [7, 8, 9]. Such a gauge instanton can be realized by a Euclidean Dbrane (ED brane) wrapped on a non-trivial cycle upon which the space-filling D branesthat make up the gauge group, in an engineered SQCD theory, have also been wrapped.It has further been shown that certain stringy realizations lead to non-perturbativesuperpotential terms generated by instantons which do not admit an obvious interpretationfrom an ordinary gauge theory point of view [10, 11, 12, 13, 14, 15, 17, 18, 19, 20]. Suchcases are realized by an ED brane wrapping a cycle which has no gauge dynamics associatedto it and is called a stringy instanton. It has been shown that it is possible to generatephenomenologically interesting superpotential terms, such as Majorana mass terms forright handed neutrinos [10, 11, 15, 21].In the stringy case of an ED brane on a cycle which has no space-filling D brane alreadywrapped on it, there generically arises a problem due to an excess of neutral fermionic zeromodes. The reason is that for an open string, with both endpoints on an ED brane– 1 –hat wraps an otherwise “unoccupied” cycle, does not feel the presence of the space-fillingD branes and therefore, gives rise to 4 fermionic massless modes, corresponding to thesupertranslations broken by the ED brane in an N =2 background. This implies thatinstead of the 2 (Goldstino) zero modes required for the generation of a superpotentialterm we get additional fermionic moduli fields that do not appear in the moduli actionand hence make the instanton moduli space integral vanish. The standard way to get ridof these unwanted fermionic zero modes is to introduce an orientifold plane and therebyproject these extra modes out [16, 17, 18, 22, 23]. There have also been investigationsconcerning the possibility of lifting these fermionic moduli fields by including backgroundfluxes together with gauge flux on the world volume of the ED brane [24, 25, 26, 27].In this paper, we will show that it is in fact possible to generate non-perturbativesuperpotential terms from stringy instantons without introducing orientifolds or takingclosed string modes into account. As our main focus will be to show how the problemof unwanted neutral fermionic zero modes can be evaded we will throughout the paperonly be considering local configurations and not be concerned with global issues such ascancellation of D brane induced tadpoles which in general require the presence of orientifoldplanes. We will work in a type IIB Z × Z orbifold background, where we can use a simpleCFT description when studying the interactions between the massless modes of the openstrings stretching between the various branes. Although the N =1 non-chiral world volumegauge theory this orbifold background gives rise to is not of particular phenomenologicalimportance, we believe that the results we obtain are quite general and applicable to manyother D brane systems in various Calabi-Yau backgrounds [28, 29, 30].The key point in generating the non-perturbative superpotential term will be to con-sider branes wrapping 3 different 2-cycles. We wrap N D5 branes on the first cycle, N on the second and a single D5 brane on the third, N = 1. By also wrapping an ED1 onthe third cycle we are in the situation where we have an instanton which is not associatedto any low energy gauge dynamics, since there is only an IR free U(1) factor here. How-ever, due to the presence of bosonic zero modes between the ED1 brane and the single D5brane, there will appear couplings in the effective instanton moduli action which involvethe unwanted fermionic zero modes. The integration over these extra fermionic zero modesimposes constraints on the remaining moduli fields, analogous to the fermionic ADHM[31] constraints one imposes on the moduli fields of an conventional gauge instanton, andwe are left with an integral which has the correct number (two) of neutral fermionic zeromodes to make up the integration measure over chiral superspace. When performing theremaining integrations, we find that a holomorphic superpotential term with the structureof a baryonic mass term is obtained for the case when N = N , without introducing anyorientifolds. We regard the computation done in this paper as an explicit confirmation ofthe expectations raised in [32, 33] for related configurations. We will see that, by includingthe baryonic mass term as a non-perturbative part of the superpotential, the R-chargeassignment of the chiral superfields is uniquely fixed in the non-chiral quiver gauge theoryunder consideration and moreover, an axial U(1) symmetry is broken. Note however that,in other configurations where orientifolds were used, it has been shown that the inclusion ofsuch a baryonic mass term, in some instances, leads to dynamical supersymmetry breaking– 2 –16, 34, 35, 36].The plan of this paper is as follows. In section 2 we first review the field content ofthe k = 1 instanton sector of the N =4 SYM theory, realized by a D(-1) instanton in theworld volume of N D3 branes, and then perform a Z × Z orbifold projection to obtain our N =1 SQCD-like gauge theory with one instanton. We review the open string spectrum forsuch a configuration and write down the corresponding effective instanton moduli actionin the ADHM limit. In section 3 we discuss the prefactor of the possibly generated non-perturbative superpotential term and also the power to which the chiral superfields shouldappear in such a term. In section 4 we explicitly evaluate the moduli space integral for theconfiguration when one of the cycles is wrapped by a single space-filling D brane togetherwith an instanton ED brane, and we find a non-vanishing holomorphic result. In section5 we give a brief discussion of the implications on the gauge theory dynamics we shouldexpect from the non-perturbative superpotential term found in section 4.
2. The Orbifold Projection of the k = 1 Instanton Sector of N =4 SYM In this section we will review the open string spectrum for a system with N D3 branes andone D(-1) brane ( k = 1) in a type IIB background [8, 5, 6, 37]. Since we are interested ininstanton calculus we Wick rotate our ten dimensional Minkowski spacetime, according to[8]. In the gauge sector the massless modes of the open strings, with its endpoints attachedto two of the N D3 branes, form an N =4 SYM multiplet [38]. In the NS sector, we obtainthe gauge field A µ from the oscillators with spacetime indices pointing along the D3 brane.The oscillators pointing in the 3 complex directions transverse to the D3 brane give, in N =1 language, the three chiral superfields Φ , , . All fields in the gauge sector are in theadjoint representation of U( N ).The fields in the neutral sector correspond to the zero modes of the strings withboth ends on the D(-1) brane. These fields do not transform under the gauge groupof the D3 branes but instead in the adjoint representation of the instanton gauge groupwhich, for a single ( k = 1) instanton, is simply U(1). In the same way as the N =4SYM theory in 4 dimensions can be obtained from a dimensional reduction of the N =1SYM theory in ten dimensions [39], the neutral sector can be obtained by continuing thereduction down to zero dimensions. We denote the four bosonic moduli fields, longitudinalto the D3 branes world volume, by a µ . Also, from the oscillators with spacetime indicesin the directions transverse to the D3 branes we get another six bosonic moduli fieldswhich we will however not be concerned with since they will be projected out by theorbifold projection in the configurations we will consider later on. In the R sector, thefermionic zero-modes are denoted M αA and λ ˙ αA , where α / ˙ α denote SO(4) Weyl spinorindices of positive/negative chirality transforming in the fundamental representation underthe respective factor of SU(2) × SU(2) ∼ =SO(4) and A upstairs/downstairs denote SO(6)Weyl spinor indices of negative/positive chirality which transform in the fundamental/anti-fundamental representation of SU(4) ∼ =SO(6). Hence, the presence of the D3 branes havebroken the Euclidean Lorentz group SO(10) to SO(4) × SO(6) and the ten dimensional– 3 – z z h z − z − z h − z z − z Table 1:
The action of the orbifold generators. chirality of both fermionic fields have been chosen to be negative. We will also introducea triplet of auxiliary fields D c that can be used to decouple quartic moduli interactionsand linearize supersymmetry transformations but most importantly, it is crucial in orderto recover the standard ADHM results in the field theory ( α ′ →
0) limit [8, 6]. Since wewill only be dealing with a single instanton, the neutral sector fields are not matrix valued.The charged sector fields come from the zero-modes of the strings stretching betweenthe D(-1) brane and one of the N D3 branes. For each such open string we have twoconjugate sectors distinguished by the orientation of the string. In the NS sector, wherethe world-sheet fermions have opposite moding compared to the bosons, we obtain a bosonicSO(4) Weyl spinor ω ˙ α in the first four directions where the GSO projection picks out thenegative chirality. In the conjugate sector, we will get an independent bosonic SO(4) Weylspinor ¯ ω ˙ α of the same chirality. In the R sector, we obtain two independent SO(6) Weylspinors µ A and ¯ µ A , one for each conjugate sector, with chirality fixed by the GSO projectionsuch that both spinors transform in the fundamental representation of SU(4) ∼ =SO(6). Notethat the moduli fields in the charged sector with(out) a “bar” are in the anti-fundamental(fundamental) representation of U( N ), the world volume gauge group of the D3 branes.Let us now perform an orbifold projection [40, 41] on the configuration described above.We will choose the orbifold group to be Z × Z , which will give us a non-chiral N =1 quivergauge theory [42]. Since the orbifold projection was done in detail in [17] we will here onlystate the action of the two generators h and h of the two Z (see Table 1) and its regularrepresentations γ ( h ) acting on the Chan-Paton factors, γ ( h ) = − − , γ ( h ) = − − (2.1)where the 1’s denote N ℓ × N ℓ unit matrices, ℓ = 1 , ..., P ℓ =1 N ℓ = N . For a reviewon fractional branes, see [43]. In the gauge sector, the orbifold projection implies that the vector superfields are blockdiagonal matrices of different size ( N , N , N , N ), one for each node of the quiver. Sincewe will throughout the paper never occupy node 4 with fractional D3 branes, we set N = 0from now on. Thus, our gauge group is U( N ) × U( N ) × U( N ). The three chiral superfields– 4 – λ α α ααµ µ µµµ µµ ωω Φ ΦΦΦ ΦΦ
13 .. . M Figure 1:
The Z × Z orbifold quiver gauge theory where the fractional D3 branes (green circles)have been given rank assignment ( N , N , N ,0). We have also included all neutral and chargedzero modes of the fractional instanton (red square) which is located at node 3, together with N fractional D3 branes. Φ i will have the following formΦ = , Φ =
00 0 0 Φ , Φ =
00 Φ (2.2)where the non-zero entries Φ ℓm denote chiral superfields transforming in the fundamentalrepresentation of gauge group U( N ℓ ) and in the anti-fundamental of gauge group U( N m ).The associated quiver diagram is displayed in Figure 1. The Chan-Paton structure for the fractional D(-1) instantons will be the same as for thegauge sector. However, since we will only be considering a single instanton at node 3, alloff diagonal neutral modes are absent, as they connect instantons at two distinct nodes.Thus, we keep only the third diagonal component in the 4 × k = 1 and k = k = k = 0. The 4 bosoniczero modes in the NS sector that remains, corresponding to the location of this fractionalinstanton in the world volume of the fractional D3 branes, will (also here) be denoted by a µ . – 5 –or the fermionic moduli fields M αA and λ ˙ αA we can choose a representation of theDirac matrices such that M αA and λ ˙ αA for A = 1 , , A = 4 they are block diagonal [17]. Thus, since we are only interestedin a diagonal component of the Chan-Paton matrix, the only neutral fields that survivethe projection are those with SU(4) index 4. We denote the third component of theseremaining fermionic moduli fields by M α and λ ˙ α . The charged sector is now described by the open strings stretching from the fractionalinstanton at node 3 to the fractional D3 branes, and vice versa. Since the charged bosonicmoduli fields do not carry indices which point in any of the directions the orbifold actson, the Chan-Paton factor will have a block diagonal structure and thus we will only findsurviving fields among the zero modes of the open strings between the fractional D(-1) and the D3 branes at node 3. Hence, we obtain 4 N bosonic zero modes ω ˙ α , ¯ ω ˙ α .The charged fermionic zero modes µ A and ¯ µ A will display the same structure as in(2.2) for A = 1 , ,
3, but they will be block diagonal for A = 4. This means that betweenthe fractional instanton and the N D3 branes at node 3 we have 2 N fermionic zero-modes µ and µ . As the SU(4) indices will not be written explicitly, we simply note that thesecharged fermions correspond to the SU(4) index 4. Between the instanton and the N D3 branes at node 1 there are 2 N fermionic zero-modes µ and ¯ µ , corresponding toSU(4) index 2. Finally, between the instanton and the N D3 branes at node 2, we have2 N fermionic zero-modes µ and ¯ µ , with SU(4) index 3. Note that, in order to easethe notation, we do not write out the fundamental indices of the charged moduli fields(without a “bar”), corresponding to the gauge group of the fractional D3 node the stringstretches from, and similarly for the anti-fundamental indices of the charged fields (with a“bar”) stretching to the D3 branes. We can now calculate tree level open string scattering amplitudes by inserting the vertexoperators for the moduli fields at the boundary of a disk, corresponding to the world volumeof the open string. In order to recover the standard ADHM result for an ordinary gaugeinstanton we take the “ADHM limit”, implying that we, in addition to taking the fieldtheory limit α ′ →
0, perform a particular rescaling of the moduli fields, see [6], and thensend g → ∞ while keeping the 4-dimensional D3 brane world volume gauge coupling fixed.By summing over all amplitudes that survive this limit we recover the following effectiveinstanton moduli action for a single fractional instanton, S = i ( µ ω ˙ α + ω ˙ α µ ) λ ˙ α − iD c (cid:0) ω ˙ α ( τ c ) ˙ β ˙ α ω ˙ β (cid:1) (2.3) The block diagonal structure of the Chan-Paton factors can also be understood from the fact that thecharged open strings stretching from the fractional D(-1) instanton to one of the fractional D3 branes,which is not at node 3, would behave as 8 Dirichlet-Neumann strings with massive NS ground state sincethe ED1 and D5 then wrap different 2-cycles. – 6 –nd the interaction terms between the charged sector and the chiral superfields are givenby S = 12 ω ˙ α (cid:0) Φ Φ + Φ Φ + Φ Φ + Φ Φ (cid:1) ω ˙ α + i µ Φ µ − i µ Φ µ + i µ Φ µ − i µ Φ µ − i µ Φ µ + i µ Φ µ (2.4)where we have both holomorphic and anti-holomorphic couplings. All terms in (2.3) and(2.4) can be obtained by performing the Z × Z orbifold projection of the parent k = 1instanton sector of the N = 4 theory [17].We will throughout the paper assume that it makes sense to take the ADHM limit of theinstanton moduli action. For a conventional gauge instanton, this is the limit that yields,first of all, the standard ADHM measure on the instanton moduli space of the N = 4 D3world volume gauge theory before the orbifold projection [8, 6], but also the one instantongenerated ADS superpotential of the N = 1 fractional D3 world volume gauge theoryafter the Z × Z orbifold projection [17]. Even though we will later on be concerned withinstantons that do not admit an obvious interpretation in terms of ordinary commutativegauge theory they will however have similarities with ordinary gauge instantons since they,for example, have charged bosonic moduli associated to them.As suggested by [10, 8], if a non-perturbative superpotential is generated in the config-uration described above, its form can be obtained by evaluating the moduli space integral S W = C Z d { a, M, λ, D, ω, ω, µ, µ } e − S − S . (2.5)We will in the following two sections, first discuss the prefactor C , here inserted in order tocompensate for the dimension of the moduli space measure, and then explicitly evaluatethe integral (2.5).
3. Determining the Prefactor
In this section we will start by considering the case when there are N > N = 1, and discuss the structure we expect the generated superpotential term tohave, using dimensional analysis. In order to check that the action term in (2.5) is dimensionless we need to know the scalingdimension of all the moduli fields that appear in the instanton measure. This can beobtained by demanding a dimensionless moduli action in (2.3) [6, 18],[ a µ ] = [ ω ˙ α ] = [ ω ˙ α ] = M − s , [ D c ] = M s (cid:2) M α (cid:3) = [ µ ] = [ µ ] = M − / s , [ λ ˙ α ] = M / s . (3.1)– 7 –here M s = 1 / √ α ′ . For the configuration under consideration, with fractional D3 branerank assignment ( N , N , N ,0) and fractional D(-1) rank assignment (0,0,1,0), the dimen-sion of the instanton measure is given by h d { a, M, λ, D, ω, ω, µ, µ } i = M − ( n a − n M + n λ − n D + n ω,ω − n µ,µ ) s = M − ( n ω,ω − n µ,µ ) s = M − (3 N − N − N ) s = M − β s (3.2)since we have n ω,ω = 4 N charged bosons ( ω ˙ α and ω ˙ α ) and n µ,µ = 2 N + 2 N + 2 N charged fermions ( µ , µ , µ , µ , µ and µ ) in addition to the instanton gaugefield a µ ( n a = 4), its superpartners M α and λ ˙ α ( n M = 2 and n λ = 2) and the n D = 3auxiliary fields D c . In (3.2) we have denoted the dimension of the instanton measure by β since we recognize it as the one loop β -function coefficient for the gauge coupling constant g of the N =1 U( N ) vector multiplet, together with the contribution from N + N generations of bi-fundamental chiral superfields, β = 3 N − N − N [44, 45]. The oneloop β -function coefficient β can also be obtained by calculating the annulus vacuumamplitude for the open strings between the D( − instanton and the fractional D3 branes[10, 7, 18, 46, 47, 48]. This one loop running of the gauge coupling constant g is dueto the massless states circulating the loop and because we take the strictly local pointof view, there are no threshold corrections due to higher string states [49, 50, 51]. Theabsence of higher string state contribution together with the fact that we are performingthe integration over the instanton zero modes explicitly in (2.5) implies that, in order tonot overcount, there is no contribution from the annulus diagrams to the prefactor C in(2.5).The only missing piece of the prefactor is obtained by taking into account the vacuumdisk diagrams which have their boundaries completely on the fractional instanton at node3 and contribute by an exponential of the topological normalization of a D( − disk , − π g , where again g is the U( N ) gauge coupling constant, at the string scale M s . Thus,by combining the dimensionful factor M β s , compensating the dimension of the instantonmoduli measure (3.2), together with the vacuum disk exponential we can now identify theprefactor in (2.5) with the one loop renormalization group invariant scale Λ of the U( N )gauge theory on the world volume of the fractional D3 branes at node 3, C = M β s e − π g = Λ β (3.3)where we have suppressed the θ -angle dependence.Since we can identify the neutral zero modes a µ and M α as coming from the super-translations broken by the fractional instanton we will henceforth denote them by x µ = a µ and θ α = M α . Thus, we can pull out these modes from the moduli integral in (2.5) andobtain the measure over chiral superspace d xd θ . This allows us to determine to whichpower the chiral superfields will appear in the instanton generated superpotential term, S W = Z d xd θ W np (3.4) Note that such a vacuum disk amplitude is also given by minus the classical instanton action [52]. – 8 –here the non-perturbative superpotential is given by W np = Λ β Z d { λ, D, ω, ω, µ, µ } e − S − S ∼ Λ β Φ − β +3 . (3.5)This is the usual form of the ( N f = N c −
1) ADS superpotential [44, 45], which is generatedby an instanton when N + N = N −
1, where N f = N + N is the number of flavorsand N c = N is the number of colors. Using this constraint, we note that the power towhich the chiral superfields in (3.5) appear is negative, implying that the majority of fieldswill appear in the denominator, as expected, since such a term is generated by a gaugeinstanton. In the remainder of this section, we will consider the N = 1 case, where thenon-perturbative superpotential term is generated by an instanton which does no longerhave an obvious gauge theory interpretation. Let us now turn to the main focus of our study, which is the case when we only have asingle fractional D3 brane at node 3, N = 1, where the instanton is located. There is nolonger any low energy gauge dynamics associated with the third node since the U(1) factoris IR free.From the dimensional counting of the moduli measure in (3.2) we see that, for N = 1,the coefficient to which dynamical scale Λ in (3.3) appears is (3 − N − N ). Although thiscoefficient can no longer be interpreted as an ordinary one loop β -function coefficient, wecan conclude that if it was possible to generate a holomorphic superpotential term for thisconfiguration, it would have the following structure, W snp ∼ Λ − N − N string Φ N + N . (3.6)We have here labeled the scale Λ with the subscript “string” in order to indicate the factthat it no longer has an ordinary gauge theory interpretation, but is of stringy origin.Since the power to which the chiral superfields in (3.6) appear is positive we conclude thatmajority of fields will appear in the numerator and hence such a superpotential term cannot be generated by an ordinary gauge instanton. Note that the only way to satisfy theADS constraint N + N = N − N = 1 is to set the number of flavors N = N = 0.Note also that the instanton disk vacuum amplitude can no longer be interpreted as theinstanton classical action for an ordinary gauge instanton, but rather as the normalizationof a D( − disk, since there exists no instanton solutions for ordinary commutative U(1)gauge theory.It is interesting to note that the coefficient (3 − N − N ) and the instanton disk vacuumamplitude obtained in this case have the same appearance as one would expect the one-loop β -function coefficient and the instanton action to have for a noncommutative U(1) gaugetheory with N + N flavors [53, 54, 55]. Moreover, it is known that noncommutativity inU( N ) gauge theories have a particular dramatic effect for the case N = 1 since it is only ona noncommutative background that abelian gauge theories become non-trivial and allowfor instanton solutions [56]. Therefore, one might expect that the case under considerationis related to such configurations. We leave these issues for future work. The author would like to thank Jose Francisco Morales for pointing this out. – 9 –f we did not have any fractional D3 branes at all at node 3, only the chiral superfieldsΦ and Φ would exist [17]. In that case, there are no charged bosonic zero modes, theinstanton moduli action in (2.3) vanishes and we only have the couplings in the last lineof (2.4) left. Hence, if it was not for the two neutral fermionic λ ˙ α -fields, the moduli spaceintegral would yield a contribution of the form det[Φ ] det[Φ ] for N = N , since thecharged fermionic zero modes appear symmetrically. We note that this term has the samedimension as the chiral superfields should have according to (3.6). The problem with thecase when there are no fractional D3 branes at node 3 is of course that the λ ˙ α -modes makethe integral vanish since they do not appear in the effective moduli action.As will be shown in the following section, a non-perturbative superpotential term likedet[Φ ] det[Φ ] is exactly what we find when we evaluate the moduli space integral forthe case when we do have a single fractional D3 brane at node 3. In this case, there is nolonger any problems with the unwanted λ ˙ α -modes, since we now have charged bosonic zeromodes which make these neutral fermions appear as Lagrange multipliers, implementingconstraints completely analogous to the fermionic ADHM constraints one obtains for or-dinary gauge instantons in the ADHM limit. The difference in this configuration is thatthe instanton here can not be interpreted as an ordinary gauge instanton, but rather as astringy one.
4. Evaluating the Moduli Space Integral
In this section, we will explicitly evaluate the instanton moduli space integral for the stringy N = 1 configuration described above, W snp = Λ − N − N string Z d D c d ω ˙ α d ω ˙ α dµ dµ d N µ d N µ d N µ d N µ × δ (cid:0) µ ω ˙ α + ω ˙ α µ (cid:1) e − S − S (4.1)where we have performed the integrals over the λ ˙ α variables in (2.3) and thereby imple-mented the fermionic ADHM constraints in terms of two δ -functions. Let us also express S , from (2.4), in the following way S = 12 ω ˙ α { ΦΦ } ω ˙ α + i µ Φ µ − i µ Φ µ + i µ Φ µ − i µ Φ µ − i µ Φ µ + i µ Φ µ (4.2)where { ΦΦ } = Φ Φ + Φ Φ + Φ Φ + Φ Φ . Due to the fermionic nature of the two δ -functions brought down by the λ -integration, wecan simply drop the “ δ F ” and obtain the following two terms, (cid:0) µ ω ˙1 + ω ˙1 µ (cid:1)(cid:0) µ ω ˙2 + ω ˙2 µ (cid:1) = µ (cid:16) ω ˙1 ω ˙2 − ω ˙1 ω ˙2 (cid:17) µ = µ (cid:16) ω ˙1 ω ˙1 + ω ˙2 ω ˙2 (cid:17) µ (4.3)– 10 –here we have raised indices using ω ˙ α = ǫ ˙ α ˙ β ω ˙ β , with ǫ ˙1˙2 = − ǫ ˙2˙1 = −
1. In (4.3) we havealso used the fact that the terms in which either µ or µ appear twice vanish since theseGrassmann variables anticommute to zero. Since the terms in (4.3) appear in front of theexponential in (4.1), these terms soak up both µ and µ , implying that we are alreadydone with the integration over these two variables. This means that, in order to get anon-vanishing result, we can only expand the terms in the exponent of (4.1) containing µ and µ to zeroth order. Thus, we forget about the last four couplings in the firstline of (4.2) and instead study the last two couplings which include all the remainingcharged fermionic moduli fields, µ , µ , µ and µ . Since these remaining fields appearsymmetrically we must expand both these terms to N th1 = N th2 order to be able to soakup the remaining charged fermionic moduli fields. Hence, from the fermionic integrationwe get the constraint N = N . (4.4)The integration over the remaining fermions brings down determinants of Φ and Φ andwe arrive at the following result, W snp = Λ − N string det[Φ ] det[Φ ] × I for N = N = N (4.5)where the remaining task is to evaluate the following bosonic integral, I = Z d D c d ω ˙ α d ω ˙ α (cid:0) ω ˙1 ω ˙1 + ω ˙2 ω ˙2 (cid:1) e iD c ( ω ˙ α ( τ c ) ˙ β ˙ α ω ˙ β ) − ω ˙ α { ΦΦ } ω ˙ α . (4.6)Note that dimensional analysis of (4.5) tells us that the bosonic integral I must be dimen-sionless. Hence, since I can only depend on the dimensionful quantity { ΦΦ } , we concludethat I must be a simple number, independent of { ΦΦ } . In the following section we willshow that this number is non-zero. Since the charged bosonic moduli fields appear quadratically in the exponent of (4.6) wecan simply insert the the components of the three Pauli sigma matrices τ c and arrive atthe following expression for the bosonic integral, I = Z d Dd ωd ω (cid:16) − ∂∂M − ∂∂M (cid:17) exp − (cid:2) ω ˙1 ω ˙2 (cid:3) " M M M M ω ˙1 ω ˙2 (4.7)where we have denoted M = − iD + { ΦΦ } , M = − iD − D , M = − iD + D and M = iD + { ΦΦ } . Performing the Gaussian integrals over the charged bosonic modulifields and taking the derivatives with respect to M and M , we obtain Z d D (cid:16) − ∂∂M − ∂∂M (cid:17) M M − M M = Z d D { ΦΦ } (cid:2) D + { ΦΦ } (cid:3) (4.8)where we have inserted back the expressions for the M ’s and denoted D = P c =1 ( D c ) . Ifwe now change to spherical coordinates ( R d D = 4 π R dD D ), rescale ˜ D = D { ΦΦ } and use– 11 –he fact that R ∞ d ˜ D ˜ D [ ˜ D +1] = π , we can conclude that the bosonic integral I from (4.6)only results in an irrelevant numerical factor which can be absorbed in the prefactor Λ string in (4.5).Thus, we have now shown that the final result from the complete moduli space integralwas given in (4.5) and reads W snp = Λ − N string B ˜ B (4.9)for N = N = N . In (4.9), we interpret the determinants from (4.5) as baryons, B = det[Φ ] and ˜ B = det[Φ ], and the superpotential term (4.9) as a stringy instantongenerated baryonic mass term . Note that we have generated a holomorphic superpotentialterm without using the D-term constraints for the matter fields, although we, of course,have to implement them in order to ensure supersymmetry.Let us summarize our findings in slightly more general terms: We assumed that wehad a background geometry with (at least) three non-trivial cycles, with two of themwrapped by space-filling D branes such that they, on their world volume, realized an N = 1U( N ) × U( N ) gauge theory with bi-fundamental matter. In that case, a non-perturbativesuperpotential term like (4.9) was generated by wrapping a single space-filling D branetogether with an instanton ED brane on the third cycle. We believe that this result isquite general and should be applicable to many D brane systems in various backgrounds.
5. Implications for the Gauge Dynamics
We have seen in the previous section that it is possible to generate a non-perturbativesuperpotential term for a U( N ) × U( N ) gauge theory with an additional U(1) factor whichhas an instanton associated to it. In this section we will discuss how we should expect thedynamics to change when we include the stringy instanton generated superpotential term(4.9).In an attempt to make contact with the standard analysis of SQCD, let us for themoment consider the same configuration but in a background where we can take the limitwhere the volume of the second cycle, upon which N D3 branes are wrapped, is large. Wecan then, using this limit, think of the case when the U( N ) group associated to the largecycle acts, together with the IR free U(1) factor from the third cycle, as a flavor group forthe N D3 branes wrapped on the first cycle that make up the U( N ) gauge group. Thissystem is reminiscent of the N f = N c + 1 (where N c = N ) SQCD case where we expectconfinement but unbroken chiral symmetry at the origin of the moduli space [57]. From anon-perturbative analysis of this specific SQCD theory we know that the classical modulispace, which in this case is the same as the quantum moduli space, is described by mesons M ij and baryons B i and ˜ B j where the flavor indices i, j = 1 , ..., N + 1, supplemented bycertain constraints. These constraints can be implemented as equations of motion by the Note that such a mass term can also be written as the determinant of the meson field, det[ M ] =det[Φ Φ ], and we will in fact show in the next section that the relation det[ M ] = B ˜ B can be obtainedas an equation of motion. – 12 –ollowing non-perturbative superpotential term [57], W np = Λ − N +1 (cid:0) M ij B i ˜ B j − det[ M ij ] (cid:1) . (5.1)In order to make contact with our D brane configuration, which is a Z × Z orbifoldprojection of the N = 4 theory we must also remember to include the cubic tree levelsuperpotential that explicitly breaks the flavor SU( N + 1) × SU( N + 1) chiral symmetry ofthe SQCD theory, W tree = Φ f Φ c Φ c f − Φ f c Φ c Φ f (5.2)where the gauge indices c = 1 , ..., N and the flavor indices f = 1 , ..., N . We will fromnow on only be interested in the part of the flavor group which has been broken down toSU( N ) × U(1) by (5.2). Since our configuration is in the SQCD regime where N f = N c + 1we expect the presence of the non-perturbative superpotential in (5.1) although it will heredescribe fields with flavor indices decomposed into representations of SU( N ) × U(1). Themesons M ij , which are in the ad + of SU( N + 1), are decomposed into the followingrepresentations of SU( N ) × U(1), M ff ′ = Φ f c Φ c f ′ : ad + M f = Φ f c Φ c : N − M f = Φ c Φ c f : N + M = Φ c Φ c : ′ . (5.3)The baryons B i and ˜ B j , in the N + and N + of SU( N + 1), decompose according to B f = ǫ ff f ··· f N − ǫ c ··· c N Φ c Φ f c · · · Φ f N − c N : N + ˜ B f = ǫ ff f ··· f N − ǫ c ··· c N Φ c Φ c f · · · Φ c N f N − : N − B = ǫ f ··· f N ǫ c ··· c N Φ f c · · · Φ f N c N : ˜ B = ǫ f ··· f N ǫ c ··· c N Φ c f · · · Φ c N f N : . (5.4)In order to see what effect the stringy instanton generated superpotential term from(4.9) might have on the SQCD theory described above, let us write out the superpotentialin terms of the decomposed fields in (5.3) and (5.4), and simply add the non-perturbativesuperpotential term from (4.9), W tree + W np + W snp = (cid:16) Λ − N +1 M + Λ − N +3string (cid:17) B ˜ B − Λ − N +1 M det[ M ff ′ ] + · · · . (5.5)The dots in (5.5) refer to terms which are not important for our discussion, for example,those that include the fields M f and M f which are both set to zero by the equations ofmotion for the fields Φ f and Φ f , see (5.2). Note that the stringy term affects only theterm from (5.1) which is of the same form and the effect can be seen as a shift in the flavorsinglet field M , without removing the moduli space singularities. Further note that if weinterpret the stringy superpotential from (4.9) as det[ M ff ′ ], instead of B ˜ B , we get a similarshifting effect for M , W tree + W np + W snp = (cid:16) − Λ − N +1 M + Λ − N +3string (cid:17) det[ M ff ′ ] + Λ − N +1 M B ˜ B · · · . (5.6)– 13 –egardless of how we interpret the stringy superpotential in (4.9), the equation of motionfor M gives the constraint, det[ M ff ′ ] = B ˜ B .From (5.5) we see that the stringy instanton would break R-symmetry unless we canassign R charge zero to the field M . And as expected in this non-chiral gauge theory, thereis a non-anomalous R-charge assignment such that the fields Φ c and Φ c have R-chargezero, see Table 2. Thus, by including the stringy instanton, we fix the the non-anomalousU(1) R current uniquely. Moreover, there is a non-anomalous axial U(1) A symmetry, underwhich the charge of B ˜ B is compensated by the opposite charge of M , but which is herenon-perturbatively broken by the stringy instanton. U (1) R Φ
12 1 N Φ
21 1 N Φ N Φ N Table 2:
The non-anomalous R-charge assignment.
In conclusion, we have in this paper shown that it is possible to generate an interestingnon-perturbative superpotential term (4.9) by a stringy instanton in a simple Z × Z orbifold background without introducing orientifold planes. In more generic Calabi-Yaubackgrounds, it has been shown that similar terms as the one we have found here willhave dramatic effects on the gauge dynamics and, in some cases, give rise to dynamicalsupersymmetry breaking [16, 34, 35, 36]. We regard this computation as an example ofhow such an effect could arise in more realistic theories. Acknowledgements
It is a pleasure to thank Riccardo Argurio, Gabriele Ferretti, Alberto Lerda and DanielPersson for fruitful discussions and for reading the manuscript. The author would also liketo thank Matteo Bertolini, Jose F. Morales, Bengt E.W. Nilsson and Niclas Wyllard forinteresting conversations.
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