Jhep03(2009)040

JHEP03(2009)040 March 5, 2009 : January 20, 2009 February 10, 2009 : : Published Received Accepted an isolated cycle invariant ch instanton configurations ear condensate are generated 10.1088/1126-6708/2009/03/040 auge theory brane. Moreover, etric mass terms. including non-perturbative con- Technology, doi: al interactions are not completely Published by IOP Publishing for SISSA a,b = 1 abelian gauge theory on the world volume of a N [email protected] , and Christoffer Petersson 0901.1182 a D-branes, Nonperturbative Effects In this note we study the [email protected] SISSA 2009 412 96 Sweden Göteborg, PH-TH Division, CERN, CH-1211 Geneva, Switzerland E-mail: Department of Fundamental Physics, Chalmers University of b a c

ArXiv ePrint: single fractional D3-brane. In thedecoupled limit we where find gravitation that aby superpotential a and D-brane a fermionic instantonunder bilin effect. an orientifold projection, A evenin related in presence situation the of absence arises supersymmetry ofinduce for any breaking new g couplings background in fluxes, thetributions 4-dimensional su effective to action, the cosmological constant and non-supersymm Keywords: Abstract: Gabriele Ferretti Non-perturbative effects on a fractional D3-brane JHEP03(2009)040 3 7 9 1 2 11 12 ctive ] has helped elu- 6 – 1 ave several non-trivial sly the picture of the ow for various D-brane s interpretation in terms to influence the dynamics heory [ pen and closed strings, that ly free running of the gauge empts of constructing semi- field theoretical counterpart. ] and allowed for an explicit l theory within string theory, moduli space integral, we find 7 ary gauge instanton effects and clidean branes, give rise to ad- supersymmetric theories. Since ton profile. Recently, instanton ry, such as Majorana masses for framework, we should not expect re U(1) gauge theory corresponds lso in the gauge theories to which = 1 world volume theory of a single space- N – 1 – = 1 world volume theories Yukawa coupling in GUT SU(5). 5 N × ] since Euclidean branes can give rise to couplings in the effe 15 10 – 8 × 10 In this note we analyze simple brane configurations which all In many cases of interest these effects arise by taking seriou 2.1 Fractional D3-branes2.2 at an orbifold Non-perturbative singularity effects2.3 in pure U(1) Computation gauge of2.4 theory the superpotential and The the pure condensates Sp(0) case to the limiting case between gaugethose theories that that admit admit ordin instantonof effects ordinary which field do theory. notwe By are have embedding an provided this with obviou seeminglyfeatures. a trivia First UV of complete all versionthis we which pure note, from turns U(1) the gauge out pointcoupling theory to of constant seems h at view to high of exhibit energies. both an o Then, asymptotical by evaluating the filling fractional D3-brane probing a singularity. Such a pu instanton effects. In particular we focus on the 3 Instanton effects in flux backgrounds 1 Introduction The construction of the instanton action by means of string t Contents 1 Introduction 2 D-instanton effects in instanton as an independentof wrapped theories brane that and would ordinarily by notcalculus allowing support it in a gauge string instan theoryrealistic string has vacua found [ many applications in att treatment of a large class ofstring non-perturbative theory phenomena is in a consistent enlargement ofthe the effects field theory of theseIn instantons fact, to instantons be in limited stringditional to theory, effects, those realized not of as only their in wrappedthey the Eu couple. gravitational sector, but a action that are forbidden toneutrinos all and orders the in perturbation theo cidating the physical meaning of the ADHM construction [ JHEP03(2009)040 = 1 N utions to higher . By “local” we 6 K en put forward in the interesting that these × cted when we turn on till have a well-defined 1 , 3 vanishing 2-cycle which is sults one obtains in stan- ing mass terms in the 4- onding fermionic bilinear persymmetric mass terms R background. ory limit in order to obtain se residual instanton effects These papers argued that a ly find non-vanishing results e field theory/ADHM inter- instanton effect. Moreover, tion. 0) gauge theory we find that d Euclidean D1-brane (ED1- dependence on several of the ings in the effective instanton ty. 1 trix models and D-brane instantons. where the branes are present, erest in the context of moduli ]. e 4-dimensional effective action sence of fluxes. We comment on ntibrane pairs have been added, at low energies in UV complete the superpotential in the ce-filling D-branes are wrapping o the cosmological constant and 6 effective action was also used, for 18 K , 17 rivations are different. d superpotentials are related and in agreement ]. 16 – 2 – = 1 world volume theories N = 1 IIB brane configuration on = 4 abelian gauge theory on a D3-brane in flat space, we will N N ]. However, while that paper discussed D-instanton contrib 5 ] for a recent discussion concerning the relation between ma 19 These results are of course in contrast with the vanishing re Finally, we discuss how these low rank gauge theories are affe The results we find agree with previous arguments that have be Analogous arguments can be applied to the case of an isolated See [ 1 we set upcondensate and and perform find the it to calculation be concerning non-vanishing in the adard corresp one-instanton commutative pure abelian gauge theory,in and indeed the we limit on wherepretation gravity is is not abandoned. completely Thenon-vanishing decoupled idea instanton and corrections of th to working the inexample, 4-dimensional the in string [ the that a non-perturbative superpotential is generated by a D- Veneziano-Yankielowicz superpotential should be present versions of pure U(1) andarise along Sp(0) the theories. Higgs branch Theand of reason these the is theories gauge becau after brane-a groupeffects has can also been be embedded explained into by a a supergroup. direct D-instanton computa It is The arguments in that paper that involve instanton generate with the corresponding results found in this note, but the de context of geometric transitions and matrix models [ some background fluxes thatdimensional theory. induce Since soft these supersymmetry fluxesaction also break we induce obtain new non-vanishing coupl instantonfrom corrections configurations that to give th a vanishingconfigurations contribution in that ab give a non-perturbativealso contribution on t configurations whereare generated both by supersymmetric a and D-instanton effect. non-su 2 D-instanton effects in derivative terms in the Consider a generic, local, be concerned with fractional D-instanton contributions to mean that we are considering only a small region of invariant under an orientifold projection.the Although cycle no spa and thereinstanton is action no and notion modulia of space non-perturbative any integral. gauge superpotential Also dynamics, is inbrane). we generated this can Since by Sp( s a thisresolved procedure wrappe 2-cycle in volumes general in inducesstabilization the an in explicit superpotential, type IIB it flux is compactifications of [ int abelian gauge theory on a fractional D3-brane at a singulari JHEP03(2009)040 is he can n (2.2) (2.1) − n 3 ]. ]. Such − le space- 34 or any of ne space- 34 , orbifold as – space-filling 2 33 26 i Z seen as a strong , N 9 × , 2 8 orresponds to the Z ted as an ordinary xt section, Λ n be interpreted as s not the dynamical / dimension 3 , such configurations 3 ntifold projection. In y from the integration ly as C then out the instanton p amplitudes. denotes a generic gauge ws it [ 3 der the gauge group where it. Even though we choose − b ase can straightforwardly be case, since there are no (non- ring a rane instanton, relative to the . criptions given in [ 0 = 0 case arises as an interesting , connected to the instanton node, al possibilities: 0 b ≥ ed if we keep the string scale finite, − ] superpotentials which schematically ) a configuration with , n 22 = 3 4 , b> – n n 3 Φ , N b 20 − n 3 b Λ − Φ 3 , N 2 – 3 – = = Λ , N np 1 np N W W -function. The expression Φ β However, such configurations may still give rise to a term in t 2 ] for a discussion on how some of these instanton effects can be no longer corresponds to the coefficient of the beta function f 25 – n ]. Let us denote by ( 23 : The instantonic D-brane wraps a cycle upon which more than o : The ED-brane wraps a cycle which is either occupied by a sing − 35 the nodes to which Φscale couples for and these the dimensionful nodes.well-defined constant in Λ terms However, i of as the will D-brane be instanton action discussed and in the the ne Here, 3 terms are polynomial inover the the matter fermionic fields zero-modes since and they we arise write them onl schematical have the structure: Case A filling D-brane are alsoan wrapped. ordinary gauge In instanton.can this generate case Affleck-Dine-Seiberg If (ADS) the [ the ED-brane matter ca content allows it superpotential if the matter content of the other nodes allo invariant combination of the chiral matter fieldsthe charged instanton un resides. Case B filling D-brane, or isboth unoccupied cases but the invariantgauge wrapped under instanton. ED-brane an can orie not be directly interpre be understood in terms of non-vanishing open string one-loo where Λ iscoefficient the of dynamically the generated one-loop scale and its exponent c We will also see in the next section that the See however [ • • 2 space-filling branes, we distinguish between two non-trivi ignoring global issues. Depending on the position of the D-b an example [ 2.1 Fractional D3-branes atWe will an now orbifold illustrate singularity the physics outlined above by conside coupling effect by using Seiberg dualities. limiting case of bothvanishing these vacuum types expectation values of of) configurations. chiralwe superfields For need this another waymoduli to space singularities. introduce This a scaleand finite is thus naturally scale refrain provid in from order takingto the work to strict in smoo field ainferred theory/ADHM simple from lim orbifold this setting, construction, the using more for general example toric the c pres JHEP03(2009)040 . - i . ). is 1 1 1 τ vac- , we τ 2.3 from D3 (2.4) (2.3) (2.5) 1 ) and 3 i i 1 1 bosonic N πiτ N µ 1 2 , ¯ , although ) are well- U( 1 1 N − i along with 3 . Q µ 2.5 annulus = ) µ nd the instan- -instanton am- 4 1 x 1 N -branes at node − ED i 1 3 ) S N ) and ( µ D3 − µ, the open string begin- 2 i n 2.4 1 2 N ( . N is the string frame metric − − 1 1 ω upon which the ED1-brane ]) we obtain the dimension s wrapped on Σ ensional action we need to g ups, there are 4 N . -instanton and the 6 πiτ ω, 6 xified vacuum instanton disk 1 -instanton in such a system 2 (3 n al D(-1)-instanton at node 1, tes for the dimension in ( i nt to which we can relate 1 e o this instanton configuration, K fermionic modes − + g ture s with gauge group 4 D i om the orbifold limit, where in M n N N 2 det 1 − . In the charged sector, consisting − 3 = ˙ α nt vacuum Möbius amplitudes between the λ p N ) n λ φ µ − 3 2 2 − µ, + n N θ ie 1 2 − n and 1 − 1 2 + -instanton and the N ω α -instanton and the other end on one of the 1 − 2 3 is the dilaton and s 1 θ x ω, C n n φ ( ( – 4 – M h 4 in the quiver. On the world volume of these , − − 1 s s 3 = Σ , M M Z 4 2 , transforming in the bifundamental representations. ′ , N j = = α − 2 3 1 i 6= π = 1 N µ i 4 − i 2 -instanton, there are 4 bosonic zero modes N = 1 − 1 ω,µ, 1 The dimension of the prefactor of the D(-1) with τ N 3 ij Λ at nodes i and 4 fermionic modes c . 1 D x,θ,λ,D,ω, from the strings stretching between the D(-1) ]. A non-vanishing dimension corresponds to non-vanishing { ˙ α d ], given by (minus) the node 1 instanton classical action 37 h ω ) can also be obtained by studying one-loop fluctuations arou 1 , , ¯ ˙ α 2.4 36 is the RR 2-form gauge potential, ω ] for details). In the neutral sector, consisting of modes of , 2 -instanton and the orientifold. C 28 22 1 From the scaling dimension of the moduli fields (see e.g. [ Let us first recall the zero mode structure of a D(-1) In the presence of orientifolds one must also take into accou 3 the open strings stretching between the D(-1) One-loop corrections. plitude ( of massless modes charged under the 4-dimensional gauge gro moduli (see [ is wrapped, branes. Furthermore, in the charged sector there are 2 the complexified volume of the node 1 resolved 2-cycle Σ In this expression we refer to the general situation, away fr in that case there is no 4-dimensional gauge coupling consta conclude that the prefactor should have the following struc auxiliary modes Throughout this note we will only consider a single fraction denoted by D(-1) In order to obtainhave a a prefactor dimensionless for term the in moduli space the integral effective that 4-dim compensa pulled back onto the world volume of the ED1-brane. Note that D(-1) with chiral superfields, Φ fractional branes D3 By also taking into accountamplitude the [ contribution from the comple ton [ ning and ending on the D(-1) of the measure for the moduli space integral corresponding t defind even in the case when there are no spacefilling D5-brane uum amplitudes with one end on the D(-1) fractional D3-branes we obtain a (non-chiral) gauge theory Here JHEP03(2009)040 , ) e 2 1 π g 4 ion 2.1 i g side + for the -branes orbifold 1 1 π 1 θ 2 2 b Z = ). × ry. Since each 1 = 1 abelian gauge e induced by a 2.4 2 τ Z N of the D3 / and we interpret it 3 1 the same 2-cycle as his implies that the 5 -brane world volume C g ogarithmic correction 1 discussed around ( ) in the world volume theory. give us an expression for onstant, 4 e gauge kinetic term, and B isted sectors, they act as N dentified with logarithmic ractional D3-branes: t agree with the vanishing mension of the D-instanton osed string tree-level channel due inserted along the boundary = one-loop amplitudes between es at a profile for them. By inserting 3 -function coefficient and of the D3 N inetic term we recover the same = 3, indicating that the abelian β ess open string modes in the annulus 1 A = 1 g b 2 N , in agreement with ( assive modes to the prefactor of the gauge 4 for the massless closed (supergravity) modes = N 1 g one-loop channel as an ultraviolet logarithmic N − ]. 3 42 The cases N , 41 − – 5 – 2 -brane exhibits an asymptotically free behavior at 1. For the case we will be mostly interested in later N = 1 pure abelian gauge theory 1 ]. = 0, we get that − > N 4 37 1 1 N , N N disk amplitude, and the sum of the coefficients of these 36 = 1 , 4 incorporates the twisted scalars that correspond to the flux 3 1 N 22 g ]. By expanding the square root in the Dirac-Born-Infeld act = 39 2 , = 3 does however agree with the result one finds for a pure N of the gauge group at node 1. Furthermore, the dimension of th 1 38 b -function coefficient can also be obtained on the closed strin 1 = 1 as well as β g 1 -field through the vanishing 2-cycles of the orbifold geomet N 2 -function coefficient for the gauge coupling constant B β = 1 and ]. Therefore, the infrared logarithmic divergence in the cl 1 40 N ) are known to arise in this particular orbifold and can both b -instanton for the following surrounding of space-filling f 2.2 1 The key point for our purposes is that all of these procedures In the case when there are space-filling D5-branes wrapped on The one-loop -branes. The massless modes circling the loop give rise to a l The reason why the result we obtained from circling the massl Note that the result i 4 5 from the dual supergravity solution for fractional D3-bran that is valid for high energy. Thisresult non-vanishing one coefficient obtains of from an course ordinary does no world volume theory on the single D3 singularity, given in [ on, when is because of thekinetic term absence [ of threshold corrections from the m calculation can be precisely mapped to the tree level result to the tree level vacuum D(-1) D3 D(-1) divergence due to the lack of conformal invariance (unless and ( to the twisted tadpoles is exactly reflected in the open strin the instanton we can relate this effect to the gauge coupling c to quadratic order inthereby the an gauge expression field for we the get gauge the coupling prefactor constant of th the one-loop as being due to the stringy UV completion. Instanton generated superpotentials. for the world volumelogarithmic gauge corrections theory to ofcorrections the to the instanton the D5-branes action gauge atamplitude can coupling prefactor node now constant, can 1. and be be hence i identified T the with di the one-loop corrections is precisely given by 3 of the NS-NS theory defined on a noncommutative background [ theory. This expression for prefactor can now alternativelyspacefilling be D5-branes obtained with by two considering of gauge the field D5-branes vertex at operators node 1 [ type of fractionalsources D3-brane for is all charged thethis twisted under supergravity scalars all solution and into the induce thelogarithmic a three prefactor behavior of logarithmic tw as the above. gauge k coupling constant JHEP03(2009)040 - - 1 B × = × 1 1 k 3 1 2 ) (2.9) (2.7) (2.8) (2.6) N anton N D(-1) − + 1) does not 1 U( 2 k B N N × and of type − 1 A = 3 -function coefficient B 1 rated by the D(-1) t the superpotentials , b rmionic modes, from β 1 condensates we should , as discussed above, Λ Nk ) factors but Λ 0) in case s required by the following an arbitrary number of uperpotential is generated , ed to one of the N 0 + 4 . ] 1 , k , N, 23 i , ] Φ np = 2 + 1 12 32 transforming in the bifundamental W µ Φ N h N . µ, and the one-loop 21 21 n 0). The gauge group is U( Λ 1 N 3+2 A ∂ , det[Φ ∂ Λ 0 = Λ N λ 2 + 1) 1 0). The gauge group is U(1) det[Φ 1 and Φ b n − , N, 3 B N -instanton amplitude and = ’s since they act as Lagrange multipliers enforcing the – 6 – − =3+2 12 1 λ 6 From the general relation between gaugino con- µ 2 , -instanton background. Let us perform a simple i≈ + 1 np ] A = Λ 1 µ, N B α ,N,N, n W N Λ np B − α + since there is an equal number of fractional D3-branes 1 W and θ are in one-to-one correspondence with the formation of N n B A ) = ( tr [Λ ) = (1 B 4 h ) is a D(-1) 4 = 3 ]= and , N A 1 , N 2.7 F 3 b 3 and A M are in the bifundamental of the two U( A , N , N 2 2 . This gives us the dimension of the fermionic part of the inst 32 , N dim [ , N B 1 1 ]: -instanton effect: N N ] for both case 32 1 is the dynamical scale of U( and Φ : ( 29 : ( A and we have chiral fields Φ and the following non-perturbative superpotential is gene . 23 2 3 N ) ) is the gaugino of the vector multiplet at node 1, we expect tha 2 0) in case N N α − -instantons at node 1. by a D(-1) representation. In this configuration a non-perturbative s U( correspond to the dynamical scaledenote of the either fact of that them. ( Instead Here Φ where Λ 3 Case A instanton [ U( Case B is correctly given by 1 We first of all note that there is an equal number of massless fe Remember that we subtract the number of • • 6 ,N,N, moduli space [ (1 which we will now compare totwo the types number of of fermionic condensates: zero mode fermionic ADHM-constraints. in both configurations, although they are of type ( Instanton-generated condensates. densates and low energy effective superpotentials, instantons, for both case the various types of open strings with at least one end attach D(-1) where Λ generated in both case a vacuum expectation value in a D(-1) counting of fermionic zeroexpect. modes Moreover, let in us order discuss the to case see when which we type have placed of JHEP03(2009)040 . 1 ). 2 1) > 2.9 − (2.12) (2.10) (2.11) 1 k 0) from N plicit D- , 1 U( , = 3 for the 3 = 0, on a 1 × 1 , b 1 − fermionic zero ) b N he more general N = ) each chiral super- n ]. Since each gaugino ) to be generated. 12 node. Similarly, we can Φ , , 2.10 21 rpolate between these two i N = 0 case, a non-perturbative fold we used, such a theory ] ranes of type (0 ) where 23 N 3+2 A ) refers to the abelian fermions ) would require the presence of Φ 2.2 the theory to U( ) would require the presence of = 0, corresponds to a pure U(1) 32 = Λ 2.11 2.11 ], N i 2.10 ] 21 ] in and out from infinity. For instance, in ( 12 ) and ( we get that the following condensate is det [Φ , ) with det [Φ . gauge theory h α Φ 43 3 A 20 Λ 2.1 21 2.8 N α = 1. If we instead were to place , when 2 = Λ − B U(1) 1 3 B ) indicates that the following condensate should k – 7 – np ] det [Φ 2.8 and = Λ W α ) for case i Λ A ) for α α = 2 branes” [ ) fermionic zero modes. Hence, since this dimension it is possible to move a fractional D3-brane of type Λ 2.8 = 1 it is only in a one-instanton background we expect B 1 2.9 α b N 1 A 3) fermionic zero modes, which does not agree with ( Λ 1 tr [Λ ) to be generated. k h h k − A 1 − 2.11 b -instanton background, 1 ) for 1 , the relation ( k -instantons, the condensate ( -instanton background [ 1 B 2.9 1 1 we do not expect a condensate of type ( D(-1) we will show that these expectations are fulfilled by doing ex > 1 1 ] = 2 + 2(3 ] = 2 + 2( k k . By using renormalization group matching and the fact that 2.3 F F : From the relation ( : For case B -instantons at node 1, the condensate ( M M [ [ 1 B A modes, which agrees with ( where we have multiplied both sides of ( formed in a D(-1) soaks up one fermionicfield zero mode soaks and up (the two, scalar we component need of an instanton background with 2 + 4 a condensate of type ( only agrees with ( Thus, for case with dim be formed in a D(-1) in the U(1) vector multiplet at node 1. If we were to consider t dim Case B where we have removed the trace since Λ Case A D(-1) In section • • In a more generalcorresponds situation, to quite the independent intermediate from case the between orbi ( superpotential with the following structure is generated, configuration pure U(1) case, we are led to believe that also in the limiting The limiting situation for both case starting from the configuration instanton computations. 2.2 Non-perturbative effects in pure Further Higgsing leaves usget with to a the pure U(1) U(1) theory theory by on successively the removing first fractional b (1,1,0,0) away from the singularity, implying a Higgsing of gauge theory. Forconfiguration by our moving various specific “ orbifold it is possible to inte JHEP03(2009)040 , ] for gauge (2.13) (2.14) (2.15) 50 ), we also r the U(1) 2.8 ]). One might ) is generated 30 , , 2 2.12 19 − (Sp(0)) = 1. Thus, – h of the 2-cycles in the N 17 ], stanton effect on a 2-cycle 18 )) = e reflected in the low energy Sp(0) case) is also wrapped. , d the relation ( N ode, and if so, we must be able e added in the U(1) case, and . In a UV complete framework z (VY) superpotential [ eory which does have non-singular ing vacuum expectation value, 17 , [ there are no instanton effects in G back into the VY superpotential resulting in a finite sized 3-cycle -instanton background, 8 (SO( 3 ], (see also [ 1 orbifold singularity we are using here are S Λ 49 2 ) can be generated directly by a one- – Z . , h log (SO(2)) = 0 and × 3 47 h 2.13 2 − + 1 Z 1 / 6= = Λ 3 N i C α S – 8 – ) ) nor ( Λ = 1 SYM. In this case however, when the number = 0 that the superpotential ( α )) = G ) are expected from the point of view of geometric ) has the structure of a one-instanton amplitude it ]. ( Λ N N h 2.12 h N 39 in the next section we show that in this string theory 2.13 -function coefficient does not agree with the dimension = 7 2.12 β (Sp( (U(1)) = 1 h VY ) have the same structure as in the case of the usual non- ) because of the incomplete decoupling of gravity. Moreover ], corresponding to the size of the 3-cycle: W α ) and ( Λ N, h 2.13 2.12 α 2.12 ), in agreement with the fact that the 3-cycle in the IR regime )) = tr [Λ N ) and ( 0 and not just for those values corresponding to non-abelian ≈ ]. 2.13 ] and the Dijkgraaf-Vafa theory [ ). In the following section we will show that these results fo S 44 (U( ≥ 2.12 h 46 , N 2.12 45 ) is the dual Coxeter number of the gauge group all G ( h Note that ( Also note that both ( This is in contrast to a noncommutative pure abelian gauge th Note that even the fractional D3-branes at the 7 8 this prescription tells ushence that that a the VY fermion superpotential bilinearcorresponding should should b to acquire a ( non-vanish groups. For instance, one gets valid for instanton solutions [ of colors is greater than one, neither ( cycle without chiral matter. Since ( where one obtains ( case (and the Sp(0)upon case) which can a be single precisely D5-brane explained (or an by orientifold a plane D-in for the acquires a finite size. Moreover, when inserting this result should be generated by a one-instantonto effect calculate on an it isolated using n thestandard D-instanton pure techniques. abelian Although gauge theory, realization we do generate ( in accordance with the discussion in thewhenever previous the section following an condensate is formed in a D(-1) instanton effect since the one-loop for example argueresolved that geometry, even should trigger a a single geometric D5-brane, transition wrapped on one the glueball field abelian gaugino condensation for pure expect for the particular case when transitions [ with a single unit ofeffective 3-form theory flux through by it. the This presence should of then a b Veneziano-Yankielowic of the superpotential or the condensate. expected to deform the geometry in the IR [ it is expected to have the following generalized definitions JHEP03(2009)040 ) 2.13 Thus, (2.16) (2.18) (2.17) sionless and the , ents are 1 we would , 11 2 µ > . Note that ) ) and ( d s 3 1 inary ADHM − -terms would k 0 D moduli , 2 ( /g . Thus, when we ˙ S ) α nstraints. From 2 2 0 2.12 c or ′ 11 1 − g λ 2 µ pear more than once e ) α D out we bring down 3 − 11 3 ” π 11 ˙ µ β space integral for the D and ¯ µ ˙ ) ω α is rendered massive at d 2 ˙ 4 ˙ β α ω = 4 11 y ) lowing bosonic integral: 9 11 c µ − t stringy UV completion, 2 0 ], τ + ¯ 2 3 ly gets a mass upon com- ( 6 the integral must have the y dµ However, since this U(1) is ˙ ˙ 2 nding instanton action. α α utions like ( /g ) ˙ , − α ¯ ω 1 ω 5 2 2 “ ω y c D s 1 2 11 ( + d 2 0 µ iD 2 1 1 ˙ g (¯ α y e is no contribution from configurations − 2 ( i 2 i ω ) − 2 − c 1 + d e D D c ( ) ˙ 4 2 0 2 β D charged fermions 1 y g ) 3 2 2 ω 2 -brane, 1 ˙ y d ˙ 1 β α − D k ˙ α ) + ( e c 3 2 λ 0 y 1 g τ fixed, then the quadratic ( 2 1 2 ( ˙ d α y ˙ – 9 – s α ( − i ω 2 g ¯ 2 ω ˙ Z α D − . If we were to take the field theory/ADHM limit, ) 3 ω c e 3 4 but only 2 Λ y ˙ 2 α iy ˙ α iD y ~y λ θ ω · − + 2 0 with + 2 4 ~y y 3 d 2 1 ˙ xd y ) α y → 4 c ( ω y i d = ′ 2 2 4 D -instanton and a D3 ( d d ˙ α − 2 c c Z 1 2 0 e -functions we get a cross-term that contain both the ω g 1 δ D D ) already saturates the dimension of a superpotential term. = 2 × 3 3 -instanton at node 1. This can be explicitly seen here since f d d 1 np = and 2.16 Z Z ], which can be much smaller than the string scale. Our statem 2 3 3 -fields would act as Lagrange multipliers, enforcing the ord -modes we always get that each charged fermionic zero mode ap ), up to a dimensionless constant. Let us show that this dimen θW d iy c 53 ˙ in ( α 2 − – λ D 0 3 moduli + -functions in the measure, enforcing the fermionic ADHM-co = Λ = Λ xd δ 51 S 2.12 1 4 y d -variables only appears linearly and when we integrate them np ˙ α = Z W λ , or equivalently, ˙ fermionic Lagrange multipliers 1 = ω 2 1 k d The The superpotential can be computed by evaluating the moduli Finally, one might be worried that the U(1) vector multiplet variable which we integrate out. We are then left with the fol − As argued before from the fermionic zero mode counting, ther → ∞ 4 9 np 11 S 0 the prefactor Λ where the instanton action for the moduli fields is given by [ with more that one D(-1) where and the dimensionful 0-dimensional coupling constant read µ applicable within this range of masses. 2.3 Computation ofHaving the argued superpotential from and many different theare points condensates expected of view when that the contrib we U(1) now node proceed is to embedded an into explicit computation a using constisten the correspo from dimensional analysis westructure can of conclude ( thatconstant the is result non-zero. of configuration with a D(-1) the string scale bynon-anomalous its it coupling is to massless thepactification background in [ RR-fields. the non-compact limit and on two fermionic have 2 g integrate over the vanish and the the product of these two in the measure and hence anti-commutes to zero. JHEP03(2009)040 her (2.19) (2.20) ons from ). This term is ]. This shifts 5 ], 2.17 54 ) for the U(1) case. -branes at node 1, 1 2.13 D3 auginos. The profile for 2 . From this analysis we 1 ) ition of Λ. What is im- 1-point function from the , to zero. In this case we α -instanton [ ~y ]. N 4 ~y · θ ves rise to a non-vanishing 1 , 6 amical moduli fields merely ρ s and thereby introduces an ~y g the mixed disk amplitude · [ ( 2 0 ree level amplitudes on mixed on which introduces a mini- 2 ution from when we act with g 2 g structure [ 0 ~y 2 g dynamical 4-dimensional fields nd smoothen out the moduli space · · · − eld and the remaining insertions and also ( − ionic bilinears. We will do this in here on we are considering a true e = e + ˙ 5 B α 4 ckground and it implements a deformation ω 10 ˙ 2 α ~y ρ · dρ ρ and ω ~y α + A Z = θ 2 0 and thereby give up the ordinary ADHM ) . This implies that the procedure of not insertions. In addition to the profile con- α ) 2 3 0 y θ x ρ 4 g 4 µ S → to the effective instanton action ( d ρ ′ − c α ξ – 10 – Z (vol c X and 0 and obtain the following simple expression, 3 3 2 2 ( ) for case ˙ iD α 2 2 0 0 ω a> 2.11 We can perform a similar computation to show the 4 πg πg ]= -brane world volume theory. We can now use the fact π 2 2 α 1 2 for Λ 3 3 3 ]. α /a 44 ) and ( 2 b = Λ = Λ = Λ tr [Λ 2.10 np -instanton. The gaugino has tadpoles on mixed disks with eit 1 W π/a e ]. Although such a mixed disk amplitude has multiple inserti p 6 , = 5 dx 2 ax − moduli insertions or with ¯ bx µ 2 e -instanton effect in the D3 1 R For this we need the instanton profile for the 4-dimensional g We begin by considering the case where we have placed Thus, we refrain from taking the limit and ¯ One way to prevent the instanton from shrinking to zero size a 10 ˙ α obtain an expression for a pair of gauginos with the followin singularity is to add a Fayet-Iliopoulos term constraints and hence set the instanton size, any of the 4-dimensional fields candisks be with obtained one by vertex computing operator t insertion forfor a moduli gauge fields theory fi [ added when the gauge theoryof is defined the on bosonic a ADHM non-commutative ba constraints [ the zero modes that correspondextra to term the in broken supersymmetrie the gaugino profile that depends explicitly on the point of view ofpoint the of worldsheet view it of shoulddescribe the be the 4-dimensional non-trivial thought gauge instanton of theory. background asdepend. The on a which non-dyn The the instanton profilewith a is massless then propagator obtained and by taking multiplyin the Fourier transform The trivial numericalportant constant is can that be the absorbed result into is the independent defin of would not get any contribution to the superpotential. instanton moduli space interpretation, implying thatD(-1) from that together with the D(-1) mal scale, smoothens outcontribution the to the moduli superpotential. space singularity andComputing gi the condensates. formation of a corresponding condensate involving theall ferm generality, recovering ( decoupling gravity completely can be seen as a regularizati ω tribution these amplitudes give rise to we also get a contrib the supersymmetry generators that were broken by the D(-1) JHEP03(2009)040 i - e ]. ), 1 28 2.6 (2.21) (2.23) (2.22) the size of -instanton ]. For one 1 ρ ) is precisely 28 . 2.23 -instanton and a d 0) is given in [ 1 − , , 0 moduli 0 2 S . 6 ]. π − e 32 , N, = ] d α − 4 that will not be important ). 0 moduli Λ + 1 S 2 α o groups of symplectic type α ding − θ ρ . They do however appear in al correction generated by the N calculated in ( e 2.10 modes of the neutral sector are r a configuration with a D(-1) + ) when performing the integrals } c ns to the case when orientifolds d tr [Λ 2 µ 4 − ) D ρ 0 } ) and when we multiply both sides moduli x 2.20 µ S ), corresponding to the superpoten- − 2.6 0) is given in [ was treated in detail in [ ω,µ, ), and the result we found implies that, X 2 2.11 ( ω,µ, still denotes the position and Z and the 2.16 µ x ˙ = 3) for the pure U(1) theory the relevant × α 4 x ,N,N, λ d 2 1 λ, D, ω, b { Z -instanton effect. of the remaining integral – 11 – Z d / 1 ), corresponding to the ADS superpotential ( 1 3 b Z C ]= 1 = 0 ( b 2.10 α x,θ,λ,D,ω, cancel off and we simply get a dimensionless constant { Λ d N α ρ = Λ Z i ] 1 b tr [Λ α Λ θ get identified. Moreover, the two conjugate sectors among th and the condensate ( 2 α = Λ ) for the pair of gauginos in terms of the unconstrained modul ji B case xd i ] 4 α tr [Λ d correspond to the supertranslations broken by the D(-1) 2.20 h Λ α α Z and Φ θ Sp(0) ij ) is generated by a D(-1) and the condensate ( tr [Λ h and A 2.13 µ is the space-time coordinate while x µ ), the moduli space integral for a configuration with a D(-1) ) with the product of chiral superfields we recover ( X 6= 0, ( 2.7 ′ Similarly, for case Finally, in the limiting case For case The expression ( α 2.23 which we can absorb in the prefactor Λ the expression for the gaugino pair and we can use ( As usual, and do not appear explicitly in the instanton action 2.4 The pure Now, the crucial point is that the integral that remains to be It is straightforward to generalize the above consideratio are present. The specific example of where we see that the factors of where the zero mode structure and the effective instanton action fo The result of theof moduli ( space integral is given in ( the integral one evaluates wheninstanton computing configuration. the superpotenti tial ( for particular choice of O3-plane,and the all fields the Φ gaugecharged zero groups modes turn also get int identified while fields can now be inserted into the moduli space integral yiel over these two variables, fractional D3-brane with rank assignment (1 instanton and a fractional D3-brane with rank assignment ( integrals were performed above, starting from ( projected out in the one-instanton case by the O3-plane. the instanton. The ellipses denote terms with less powers of for our purposes. JHEP03(2009)040 - ˙ α = 1 λ 1 b ˜ (2.25) (2.24) alization , again in 3 moduli fields, = Λ should be = 0 and the ] and instanton ˙ α λ 1 W 61 b ˜ N ], , 0) brane and move ]. Furthermore, this ] and use the results 28 , we must have that a 60 0 63 64 ˜ B -flux of type (0,3) which , s well-defined and non- verse complex direction nce, there are no ADHM eutral , 3 great relevance to string 61 factor Λ G 63 h one end on the D(-1) coefficient of a logarithmic , ation since in this way we N,N, n tential without the need for empty for verify that e obeys the above conditions. e D-brane world volume the- 60 thus important to understand 2. In the case when there are / ], ) and case 4 0) for which [ . 55 ˜ N A = 0. 23 . − 3 N = 3. This non-vanishing dimension, due 12 N N ,N,N, 1 b ˜ detΦ ˜ 3+ A = 0) and hence no gauge dynamics we still − N Λ 4 2 detΦ ]. − ˜ N 3 B N 29 – 12 – = ] about the Sp(0) case. In a more general setup = − flavors present [ ˜ np 1 3 = Λ A 18 ) we can start from a ( N N , N W ˜ np B 0) configuration since we expect an ADS superpotential = 17 , W 2.24 0 2 N -instanton at node 1 we get by dimensional analysis of the 1 = N,N, 1 ) = (6 + 3 -instanton action and µ ) from ( N 1 n ]. + 65 2.25 – θ (0,1,1,0) branes into the fixed point. In order for the renorm n 62 is given by the configuration (0 N 2)( ˜ B / (1 ) theory when there are ], such as flux-induced supersymmetry breaking terms [ now requires a ( − ˜ N 59 A ) – ω fields have already been projected out by the orientifold. He n c 56 It is obvious that the corresponding moduli space integral i We will in this section follow the world sheet approach of [ Case By again placing a D(-1) D + (1,1,0,0) branes away from the orbifold fixed point in a trans x n group matching to continuously take us between case vanishing for the pure Sp(0) case since the charged sector is gives a soft supersymmetry breaking mass to the gravitinos [ N superpotential is generated also for the case when no fractional D3-branes ( ( moduli space measure that the dimension of the instanton pre have a well-defined D(-1) and then move In order to recover ( type of flux induces a coupling in the instanton action to the n zero mode lifting [ and notation from those papers. In the first example, we turn o we expect contributions of thisThis type phenomena to arise is whenever ofinduce a an interest cycl explicit when moduli Kähler studyingany dependence space-filling moduli in D-branes. stabiliz the superpo 3 Instanton effects inCompactifications flux backgrounds in the presencephenomenology in of the background context of fluxesthe moduli interplay are stabilization. between of fluxes Itories is and [ effective interactions in th to the neutral zerocorrection mode structure, from can a be non-vanishing identified vacuum Möbius with diagram the wit for an Sp( Similarly, case instanton and the other on the O3-plane [ and constraints, no integrals to perform and we can immediately agreement with the discussion in [ JHEP03(2009)040 ne do and ˙ α (3.3) (3.1) (3.2) µ λ -brane mionic x and be 1 ), we are 12 . background 3) , 2.17 ′ (0 ) does not play α G s 3.2 g -component of the ne includes 2 1 θ . πiτ dimensionless 2 11 ary e µ ≈ 11 y additional interactions for oft supersymmetry breaking ftly broken. o Grassmann variables µ ˙ α -flux. The way we implement discuss backgrounds where we λ 0) 3 to the effective instanton action ˙ , e 4-dimensional effective action, α utes to the superpotential even λ (3 G . ere are no annulus diagrams and hence 2 ′ G s α g s 0) s 3 tral fermionic zero modes. ]. The effect of turning on (3,0) flux -branes or orientifold planes. In this , ifolds and fluxes a D-instanton breaks 4 of g √ g π 1 2 (3 √ 63 √ -variables as well. Hence, by using our i 3) G ] and the moduli space integral becomes, , , α ) in the instanton action ( 3 (0 + θ 64 61 Λ iG 3.2 , x + α . Thus, the last term in ( 2 4 60 θ ) d 11 c α – 13 – disk amplitude. µ θ D ( Z 1 . ). We will ignore any kind of backreaction of the 2 0) , ′ s ≈ α 11 s g (3 However, if we turn on some supersymmetry breaking g 3 and 2.5 µ d π √ G 2 − 2 11 11 4 np − or µ e S ) as a non-perturbative contribution to the cosmological ˙ 11 α πi µ λ 3.3 2 Let us now consider the configuration with a single D3 d Let us begin by considering an instanton configuration with o = 2 c d 0) D , ) they still act as Lagrange multipliers and pull down the fer − 3 0 (3 d S 2.17 . Z 1 1 ) we are able to evaluate the moduli space integral in this flux πiτ ], πiτ 2 2 e 2.12 e 64 chiral superfield from ( -instanton, but in a background with (3,0) , 1 1 = τ 63 ) we see that this moduli space measure is dimensionless (if o 3) , ) [ np variables in ( 2.3 (0 -instanton as usual, but with no fractional D3 ˙ 1 α W However, by including the first term of ( λ 2.17 Recall that in the absence of space-fillingThis D-branes, agrees orient with the fact that the charged sector is empty, th in the counting), implying that the prefactor should also be 11 12 α -functions which soak up both case we expect no superpotential to be generated since the tw no logarithmic corrections to the vacuum D(-1) the 8 background supercharges and therefore has too many neu this background flux is by addingin the ( following interactions We can view the term in ( From ( θ implying that a single fractional D(-1)-instanton contrib without any fractional D3-branes orhave turned orientifolds. on Then, flux we ofmass type terms for (3,0), the which 4-dimensional generically gauginos induces [ s Note that since this particularthe type of flux does not induce an and a D(-1) Turning on (3,0)-flux. “spurion” derivation of ( constant in the effective theory in which supersymmetry is so can be seen as giving a vacuum expectation value to the auxili (0,3)-flux, a coupling to these variables appears [ any role in the integration of given the opportunity to explicitly soak up the and obtain the following one-instanton generated term in th background geometry due to the presence ofTurning fluxes. on (0,3)-flux. given only by D(-1) not appear in the instanton action. δ JHEP03(2009)040 (3.5) (3.4) , (1996) 343 , (1994) 6041 . This suggests 0) , B 474 -brane (3 D 50 3 (2003) 045 terms and moreover, G D s ) 02 g µ ), only the lowest com- ( √ (1996) 541 B modes. If we do not make Council (Vetenskapsr˚adet) aluable conversations. The ]. alistic configuations where 3.2 und (3,0)-flux in this setting ≈ α a for ongoing discussions and Nucl. Phys. JHEP θ f C.P. is supported by an EU g Phys. Rev. . Uranga for many helpful dis- . 0), where we have also placed , , and , , Construction of instantons m B 460 1 32 µ , SPIRES φ 1 [ , -instanton gives rise to the following 23 . 1 φ 32 0) s , ]. g Φ (3 ], √ 23 Nucl. Phys. G 32 , ) survive and we are left with the following Λ x 3.4 SPIRES – 14 – 4 = ΛΦ d ] [ ]. hep-th/9512077 np , Z W ), then the D(-1) ) are reminescent of ≈ d 3.2 D-instanton induced interactions on a 3.5 − SPIRES [ 4 np ]. ]. ]. ]. S ) and ( hep-th/0002011 [ SPIRES SPIRES SPIRES SPIRES 3.4 (1978) 185 ] [ ] [ ] [ ] [ Classical gauge instantons from open strings Branes within branes Combinatorics of boundaries in string theory A 65 Small instantons in string theory Non-perturbative superpotentials in string theory (2000) 014 of the chiral superfields Φ in ( 02 φ hep-th/0211250 hep-th/9511030 hep-th/9604030 hep-th/9407031 Phys. Lett. JHEP [ [ [ [ Let us finally consider a related configuration, (1 [5] M.B. Green and M. Gutperle, [6] M. Billóet al., [7] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, [1] J. Polchinski, [2] E. Witten, [3] M.R. Douglas, [4] E. Witten, that the flux-induced mass for the gauginos is given by, non-perturbative supersymmetric mass term [ non-supersymmetric mass term, fractional D3-branes at nodes 2 andwe 3. are By given turning two on different a opportunities backgro to soak up the two ponents Note that the terms ( that these three observablesboth might fluxes also and D-instantons be are related taken into in account. more re Acknowledgments It is a pleasure to thankcollaborations R. on Argurio, related M. issues. Bertolini, andcussions C.P. A. and would Lerd also like J. to F. thankresearch Morales, A of D. G.F. Persson is andcontracts E. supported Witten 622-2003-1124 in for and part v 621-2006-3337. byMarie the The Curie Swedish research EST Research o fellowship. References On the other hand, if we do make use of the first term in ( use of the flux induced terms in ( JHEP03(2009)040 ]. ]. ] , Adv. string , D , 4 , SPIRES SPIRES (2007) 060 [ ] [ 09 (2004) 45 UT models (2007) 076 ]. (2007) 113 04 JHEP arXiv:0707.1871 On low rank classical [ B 682 , Non-perturbative Yukawa B 771 SPIRES [ JHEP , (2007) 011 , ]. , arXiv:0810.1482 arXiv:0711.0209 ]. ]. [ ory ]. , 06 eigand, The glueball superpotential d M. Schmidt-Sommerfeld, tons Nucl. Phys. , ]. and C. Vafa, , (2008) 061602 SPIRES SPIRES Nucl. Phys. (1983) 493 JHEP SPIRES ] [ Stringy instantons and quiver gauge SPIRES , , ] [ ] [ De Sitter vacua in string theory ]. 100 ] [ Instanton induced neutrino Majorana SPIRES Supersymmetric QCD and its massless limit: ]. ] [ ]. Spacetime instanton corrections in (2008) 251601 B 218 Stringy instantons as strong dynamics SPIRES 100 – 15 – ] [ Computation of D-brane instanton induced SPIRES SPIRES [ [ Dynamical supersymmetry breaking in supersymmetric hep-th/0304271 hep-th/0301240 Phys. Rev. Lett. Nucl. Phys. [ arXiv:0803.2829 hep-th/0703028 [ [ , , Neutrino Majorana masses from string theory instanton hep-th/0609213 [ ]. ]. Stringy instantons and Yukawa couplings in MSSM-like Stringy instantons and cascading quivers Hierarchies from D-brane instantons in globally defined ]. ]. ]. ]. (1984) 493 Phys. Rev. Lett. , (2004) 1045 [ arXiv:0809.3432 D-brane instantons and matrix models [ SPIRES SPIRES 7 SPIRES SPIRES SPIRES SPIRES arXiv:0811.1583 B 241 [ ] [ ] [ (2003) 046005 (2007) 052 (2008) 066004 (2007) 086002 , ] [ ] [ ] [ 03 On the (non) perturbative origin of quark masses in D-brane G D 68 D 78 D 76 (2008) 041 A gauge theory analog of some ’stringy’ D-instantons ]. JHEP 11 Nucl. Phys. , , Phys. Rev. hep-th/0612132 arXiv:0707.3126 hep-th/0311181 SPIRES arXiv:0704.1079 hep-th/0609191 groups in string theory, gauge theory and matrix models Theor. Math. Phys. Phys. Rev. JHEP [ [ an effective Lagrangian analysis [ superpotential couplings — Majorana masses from string the [ arXiv:0812.4804 Phys. Rev. masses in CFT orientifolds[ with MSSM-like spectra QCD Non-perturbative SQCD superpotentials from string instan [ Calabi-Yau orientifolds effects orientifold models vacua — the seesaw mechanism for D-brane models couplings from string instantons [8] R. Blumenhagen, M. Cvetiˇcand T. Weigand, [9] and L.E. A.M. Uranga, Ibáñez [24] D. Krefl, [26] B. Florea, S. Kachru, J. McGreevy and N. Saulina, [17] M. Aganagic, K.A. Intriligator, C. Vafa and N.P. Warner [23] O. Aharony and S. Kachru, [25] A. Amariti, L. Girardello and A. Mariotti, [19] I. Garcia-Etxebarria, [21] T.R. Taylor, G. Veneziano and S. Yankielowicz, [10] M. R. Cvetiˇc, Richter and T. Weigand, [12] R. Blumenhagen, M. D. R. Cvetiˇc, Lüst, Richter and T. W [16] S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, [18] K.A. Intriligator, P. Kraus, A.V. Ryzhov, M. Shigemori [20] I. Affleck, M. Dine and N. Seiberg, [22] N. Akerblom, R. Blumenhagen, D. E. Lüst, Plauschinn an [13] M. Cvetiˇcand T. Weigand, [15] C. Kokorelis, [11] A.N. L.E. Schellekens and Ibáñez, A.M. Uranga, [14] and L.E. R. Richter, Ibáñez JHEP03(2009)040 , ]. ]. , gauge , s on the , SPIRES , SPIRES =1 ]. y ] [ Yang-Mills ] [ N superconformal Adv. Theor. (2005) 4699 U(1) orientifold (2007) 091 , 0) ]. , SPIRES 3 , ]. (2 Z 10 ] [ / A 20 ]. , 6 olds ]. T ]. and arXiv:0803.2514 SPIRES Stringy instantons at 4 , JHEP ] [ SPIRES hep-th/9802068 hep-th/9903077 R , [ [ ] [ SPIRES SPIRES ] [ SPIRES ] [ ]. ]. ]. ]. tersson, ] [ Fractional branes and On the gauge / gravity ]. ]. hep-th/0011218 duality via a geometric transition [ (1998) 689 (1999) 476 N Int. J. Mod. Phys. SPIRES ]. ]. ]. Fractional branes and dynamical SPIRES SPIRES , SPIRES ]. ] [ 83 SPIRES ] [ ] [ ] [ hep-th/0505040 198 SPIRES [ ] [ orbifold of type-I strings ] [ arXiv:0704.0262 [ A large- hep-th/0702015 (2001) 026 brane models and the metric Kähler of twisted [ SPIRES SPIRES SPIRES Instantons and toric quiver gauge theories =1 SPIRES arXiv:0711.1430 hep-th/0112187 – 16 – D-brane Instantons on the [ [ ] [ ] [ ] [ Non-perturbative superpotentials across lines of 01 Geometric transitions and dynamical SUSY breaking ] [ N =1 magnetized D-brane worlds N (2006) 011 The one-loop UV divergent structure of =4 JHEP =2 (2007) 26 01 (2007) 067 Instantons on noncommutative hep-th/9605049 D Phys. Rev. Lett. Wilsonian effective actions and the IR/UV mixing in hep-th/0610003 , [ [ arXiv:0709.0245 , [ N hep-th/0103067 arXiv:0709.4277 A 4 On the gauge theory/geometry correspondence (2008) 033 (2002) 222 ]. [ [ 06 hep-th/9811131 Non-perturbative and flux superpotentials for type I string ]. R Non-perturbative gauge superpotentials from supergravit 01 JHEP Stringy instantons in IIB brane systems B 782 , Commun. Math. Phys. , hep-th/0310157 arXiv:0803.2041 arXiv:0704.0784 arXiv:0711.1837 JHEP [ [ [ [ B 630 SPIRES , (2001) 3 (1997) 187 (2008) 1 SPIRES (2007) 024 JHEP ] [ (2007) 051 ] [ , 05 Instanton effects in Instantons in (1999) 1415 [ 12 Superpotentials from stringy instantons without orientif 3 Nucl. Phys. B 603 B 483 B 796 , (2003) 051 (2008) 078 (2008) 123 (2007) 038 Nucl. Phys. JHEP ]. JHEP , , 12 05 07 07 , orbifold 3 hep-th/0503156 SPIRES arXiv:0708.3806 Math. Phys. theories six dimensional theory matter theory on noncommutative Nucl. Phys. JHEP correspondence and the open/closed string[ duality supersymmetry breaking [ Nucl. Phys. [ noncommutative gauge theories JHEP JHEP JHEP Nucl. Phys. theories Z marginal stability orbifold singularities [46] F. Cachazo, K.A. Intriligator and C. Vafa, [39] E. Imeroni and A. Lerda, [40] P. Di Vecchia, A. Liccardo, R. Marotta and F. Pezzella, [44] N. Nekrasov and A.S. Schwarz, [45] R. Gopakumar and C. Vafa, [37] M. Billóet al., [41] C.P. Martin and D. Sánchez-Ruiz, [28] R. Argurio, M. Bertolini, G. Ferretti, A. Lerda[29] and C. M. Pe Bianchi, F. Fucito and J.F. Morales, [35] M. Berkooz and R.G. Leigh, [36] M. Billóet al., [38] M. Bertolini, P. Di Vecchia, G. Ferretti and R. Marotta, [42] V.V. Khoze and G. Travaglini, [43] S. Franco, A. Hanany, F. Saad and A.M. Uranga, [30] M. Aganagic, C. Beem and S. Kachru, [33] R. Argurio, G. Ferretti and C. Petersson, [34] S. Kachru and D. Simic, [27] M. Bianchi and E. Kiritsis, [31] I. Garcia-Etxebarria and A.M. Uranga, [32] C. Petersson, JHEP03(2009)040 ]. , ] ]. , ] , , , SPIRES ] [ SPIRES An index for , ] [ (2008) 102 ctifications , arXiv:0810.5660 =1 ]. 12 , (2008) 112 ]. N D-branes at hep-th/9505006 [ hep-th/0610007 (2004) 21 10 [ (2005) 102 JHEP Dirac actions for D-branes SPIRES ]. , [ hep-th/0504041 SPIRES [ 10 JHEP ] [ B 690 , eyen, . Wijnholt, arXiv:0708.0403 (1995) 471 ]. [ ]. (2007) 107 Lifting D-instanton zero modes by SPIRES in type-I superstring vacua The calculus of many instantons JHEP in and A. Tomasiello, Non-perturbative F-terms across ]. , ] [ s 01 (1982) 231 B 353 (2005) 2745 SPIRES U(1) Soft supersymmetry breaking in ]. SPIRES Nucl. Phys. ] [ ]. , 22 (2007) 098 ] [ JHEP SPIRES B 113 , theories with an antisymmetric tensor Flux-induced SUSY-breaking soft terms ] [ 10 arXiv:0805.0713 Towards realistic string vacua ) [ SPIRES N SPIRES – 17 – Phys. Lett. ] [ Anomalous , hep-th/0503072 ] [ Sp( [ JHEP , brane action and fermion zero modes in presence of Phys. Lett. An effective Lagrangian for the pure 3 , Exact superpotentials, quantum vacua and duality in (2008) 028 hep-th/0311241 D [ hep-th/0204153 [ 07 hep-th/0206063 ]. ]. On the matter of the Dijkgraaf-Vafa conjecture [ Class. Quant. Grav. ]. ]. (2005) 066 A perturbative window into non-perturbative physics , ]. ]. branes with background fluxes gauge theories 3 06 ) JHEP c D arXiv:0808.2918 hep-th/0303104 [ [ , SPIRES SPIRES N (2004) 195 (2002) 92 ( SPIRES SPIRES ] [ ] [ SPIRES SPIRES [ [ ] [ ] [ (2002) 231 JHEP SP Flux interactions on D-branes and instantons Non-perturbative effective interactions from fluxes , D-brane instantons and the effective field theory of flux compa B 689 B 637 Notes on supersymmetric 371 (2009) 048 (2003) 052 ]. ]. ]. 01 04 Nucl. Phys. Nucl. Phys. arXiv:0807.4098 arXiv:0807.1666 hep-th/0312232 hep-th/0507069 SPIRES SPIRES SPIRES supersymmetric Yang-Mills theory Calabi-Yau orientifolds with D-branes and fluxes [ [ JHEP [ recombination and background fluxes [ Phys. Rept. [ on backgrounds with fluxes [ lines of BPS stability hep-th/0208048 JHEP hep-th/0307063 singularities, compactification and hypercharge [ supersymmetric background flux the Dirac operator on [64] M. et Billó’ al., [65] A.M. Uranga, [57] L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Pro [61] M. T.W. Graña, Grimm, H. Jockers and J. Louis, [63] M. et Billó’ al., [59] R. Blumenhagen, M. R. Cvetiˇc, Richter and T. Weigand, [62] I. Garcia-Etxebarria, F. Marchesano and A.M. Uranga, [49] F. Cachazo, [51] I. Antoniadis, E. Kiritsis and J. Rizos, [54] N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, [56] P.K. Tripathy and S.P. Trivedi, [58] E. Bergshoeff, R. Kallosh, A.-K. Kashani-Poor, D. Sorok [60] P.G. and Camara, L.E. A.M. Ibáñez Uranga, [50] G. Veneziano and S. Yankielowicz, [53] J.P. Conlon, A. Maharana and F. Quevedo, [48] P. Kraus and M. Shigemori, [52] M. Buican, D. Malyshev, D.R. Morrison, H. Verlinde and M [55] K.A. Intriligator and P. Pouliot, [47] R. Dijkgraaf and C. Vafa,