Jhep07(2009)017

Home , Supergroup (physics)

JHEP07(2009)017 July 6, 2009 May 11, 2009 : June 22, 2009 : : ix model tech- s, Gauge-gravity Published Received Accepted me interesting consequences sults in hep-th/0311181. We 10.1088/1126-6708/2009/07/017 tive way of computing stringy ne instanton eﬀects. This pro- vania, etation of D-brane instantons as doi: Published by IOP Publishing for SISSA 0810.1482 Matrix Models, Solitons Monopoles and Instantons, D-brane We point out that in some situations it is possible to use matr [email protected] SISSA 2009 Philadelphia, PA 19104-6396, U.S.A. E-mail: Department of Physics and Astronomy, University of Pennsyl c niques a la Dijkgraaf-Vafa tovides perturbatively an compute explanation D-bra incheck terms this of proposal stringy in instantons someof of simple this the scenarios. observation, re such We point asmulti-instanton out the eﬀects. so fact It that also itresidual gives provides instantons a a of perturba further higgsed supergroups. interpr Keywords:

Abstract: ãiGarc´ıa-Etxebarria Iñaki correspondence ArXiv ePrint: D-brane instantons and matrix models JHEP07(2009)017 3 8 1 3 10 13 15 is small. ding in a eed differ N nic) matrix ntons in the original theory gree. The situation was N that in which e gauge theory an infinite ed in a simple way by the gy point of view, where it erms this definition of the d from the planar limit of ry: the matrix model gives e computed by taking the graaf and Vafa. t has the advantage that it te to the particular geometric s of the original theory, and a nk, but the disagreement could icular cases the results obtained y. For completeness, we include . 2.1 – 1 – ], Dijkgraaf and Vafa have argued that the exact 3 – 1 orbifold of the conifold n Z ] of the original supersymmetric theory. It has a set of (boso ], where it was argued that the matrix model results might ind 4 6 , 5 ] the UV completion of the gauge theory was defined by its embed 6 , not necessarily large). A particularly interesting case is 5 N Remarkably, even if the effective superpotential is obtaine In [ 2.1 The Dijkgraaf-Vafa2.2 correspondence D-brane instantons 1 Introduction In a series of remarkable papers [ 4 A quiver example: 5 Conclusions and further directions 3 The glueball superpotential for low rank and D-brane insta (with Contents 1 Introduction 2 Background material the matrix model, it is the low energy superpotential for fini from the field theory expectations forbe gauge attributed factors to of the low ra choiceresults of for a UV the definition definitionembedding of of the of the gauge the gauge theo gauge theory in theory string corresponding theory chosen by Dijk low energy superpotential for a class of gauge theories can b planar limit of adeconstruction matrix [ model. Thisvalued matrix fields model in is one to determin potential one correspondence given with by the the superfield superpotentiala short of review the of original the theor Dijkgraaf-Vafa results in section (higgsed) supergroup of highgauge enough theory rank. coincides with Atis the the well usual level defined gauge of in theorycan F-t the one, be UV. but understood It i as is addingset also to of very the brane-antibrane original natural pairs DV from with embedding trivial the of K-theory strin th charge. This case led originally to somefrom puzzles, the since matrix for model some part clarified and in the [ pure gauge theory seemed to disa JHEP07(2009)017 l ], the F-terms ] also provides 4 4 ble set of examples, tributions of D-brane tion in terms of matrix another embedding of D- n the duality cascade, so e we have exotic D-brane the superpotential. This ]. Our claim is that the ons to some of the ones we rything can be formulated ane instantons to the low 9 . There are by now different ing the system such that it nd the matrix model results – e comments on how could it elatively easy and systematic lanar limit of the associated 7 configuration and the matrix nstantons. The matrix model brane pairs to the system. In y remarks about the interpre- nstanton effects are typically een gauge theory and matrix so the perturbative approach -perturbative effects. Another lated, and they can be shown ond the Dijkgraaf-Vafa setup. ffects, at least when a matrix he context of matrix models. ith residual instanton effects of s mention that [ s of the supergroup construction he gauge theory. o extract the relevant information. rly, as described in [ – 2 – ]. In that paper, the geometric transition is used 14 ], Seiberg duality in the case of ADE quiver theories 3 ], in which Seiberg duality (as in the duality cascade) is 12 – matrix model description, which turns out to be rather usefu 10 super ], where Seiberg duality is combined with higgsing. 13 ], which essentially consists of adding infinite brane-anti An interesting consequence of this result is that it gives a r More precisely, we claim that it is possible to obtain the con We can find some other interesting implications of the result Some of the constructions above also admit a nice interpreta The purpose of this note is to point out, in the simplest possi From this perspective, the matrix model viewpoint offers yet We would also like to point out that very similar considerati 5 instantons to the effectivematrix superpotential model. by In the studying conclusionstation the we of include p the some non-planar preliminar contributions,be and possible we to also generalize make our som results to some geometries bey can be understood as Weylare reflections essentially of insensitive the to quiver this diagram, Weyl reflection. a Simila and the geometric transitionto approaches give are the very closely same re result in the planar limit, which determines to compute non-perturbative effects due to stringy D-brane i that it is alsomodels possible as to coming understandmatrix this from discrepancy model stringy betw computation D-braneenergy includes instanton superpotential effects the of the [ contribution string of theory realization D-br of t way of computing perturbativelymodel stringy description multi-instanton of e therather system involved is to available. studybased using on These the stringy multi-i matrix instanton model calculus, will be a welcomeways tool. of computing exoticadmits D-brane a instanton gauge effects theory interpretation by whileLet modify still allowing us t mention as examplesused [ to turn D-brane instantonsexample into conventional is gauge [ non this setup stringy D-brane instantonthe effects higgsed are supergroup. identified w models. For example, as explained in [ computed by the matrixwe model can are do insensitive our toin calculations where terms we either of are “up gauge i theinstanton theory, effects cascade”, or with at where no the gauge eve a theory bottom interpretation. nice of Let intermediate the u viewpoint cascade,model between wher in the the form brane-antibrane of the brane instanton effects into gauge theory.of It is [ given in term for understanding the physics of these duality cascades in t discuss here have already appeared in [ JHEP07(2009)017 . W ]. W The 17 2 – 15 ddings of ) to be a z ( W by summarizing 2 ee for the simplest ]. Since we will be 3 – he explicit connection 1 to them in future work. here refers to the rank of the contains our conclusions lueball superfields. ed connection only in the ing the superpotential fa [ 5 ) SYM with no flavors and n section N ase, a subtle and interesting brane instanton calculus. In N ion a la Dijkgraaf and Vafa. in pure field theory [ = 1. We take the matrix model provides a tion gy derivation and interpretation t low energy degrees of freedom, are . The ve superpotential. If we restrict N ]. we study a more interesting and nton superpotential contributions 4 14 = 2 U( nks of the physical gauge group. We review st assume them to hold. N (Φ) (2.1) W Tr – 3 – θ 2 d , let us indicate this by writing the derivative of Z i a and we expect that we have to use the matrix model 1 + 1. We will denote the dynamical scale of this theory n effect, extrema = 1 theory obtained from n N N . The configurations we will be concerned with are stringy embe 2.1 we see how the D-brane instanton and matrix model results agr 3 The classical vacua of these theory are obtained by extremiz The remainder of this note is organized as follows. We start i Let us mention that in this note we will be checking the propos We feel that the previous reasons make it worth pointing out t We hope that the terminology we use does not lead to confusion The existence of the mass gap, and glueballs being the correc = 1 gauge theories with a mass gap, the lightest states being g 1 2 by Λ. polynomial function of degree N this point in section matrices in the matrix model, and has nothing to do with the ra statements which are very hard to show rigorously. Here we ju is, nevertheless, a large between D-brane instantons and matrixconnection models. between It two is, rich in fields any of c study. simplest possible scenario: we will befor computing quiver one-insta gauge theoriesThis admitting leaves a many issues matrix to model be descript studied, and we hope to come back dealing with stringy instantons we concentrateof on the the results, strin although the same results canGauge also theory. be derived canonical example is a for computing higher derivativeourselves interactions in to the the effecti superpotential,different but and equivalent thus viewpoint the of the planar discussion limit, in [ the main points ofsection the Dijkgraaf-Vafa correspondence and D- one-instanton sector in conifoldinvolved geometries. example, In the section abelianand orbifold some of discussion the of conifold. interesting Sec open problems. 2 Background material 2.1 The Dijkgraaf-Vafa correspondence Let us start by reviewing the main results of Dijkgraaf and Va a potential for the adjoint superfield: This superpotential breaks down the supersymmetry to This superpotential has JHEP07(2009)017 i → N ) (2.6) (2.5) (2.4) (2.3) (2.2) N ). The i N erivative of the U( ], for pure SYM ng relations lets ) and Sp( i 18 =1 N n i , # Q erpotential will be of U( ) i ic action, having lowest grees of freedom of this S i rpotential involving the ( Q re given by the glueball 0 gument below, but let us ∂S F ut of the low energy theory, × ∂ ing theory. ) is the dynamical scale of the j i 0 i N N 2 N NS α − i +           (Φ) = 0 is obtained by considering + i N W n ′ SO( . It is conventional and convenient ) S a α i i i a W a , and Λ . N ). W i . N → i ) . i S ) theory. − can be computed perturbatively from a /S ) N + N j 4 extrema: : N 0 N a i N a 3 ! U( ( F n i SU( i N 2 Λ i 6= a N ]. N =1 i j Y i 3 n i Tr 6 1 i – 4 – S Λ 2 a Q N log π 1 g

1 S a 32 N → 2 ) log           − i )= N S = S = Λ ( " i i Φ= S eff N n =1 i 3 i X W Λ = 1 SYM with the Higgsed gauge group ). A classical solution of Tr )= . As explained by Veneziano and Yankielowicz [ i N α S a ( λ − eigenvalues of Φ into the eff ), and the residual ranks z z ( ) cases can be found in [ W ( N i ′ =1 N n i W in terms of the dynamical scale Λ of the original theory, the d , one for each of the gauge factors: Q U( ) after integrating out the massive matter. The usual matchi i i i i g S ]: N is a perturbative series on the glueballs 1 Q the spinorial chiral superfield appearing in the gauge kinet 0 )= × α z F ) ( 0 ′ W What Dijkgraaf and Vafa showed is that In the case of the theory we are dealing with, the effective sup This theory confines, and it is believed that the low energy de N W Sp( factor SU( Similar matching relations for the related SO( us express Λ a partition of the superpotential where as where Λ denotes the dynamical scale of the SU( first describe how we will embed these gauge theories into str related matrix model. We describe the main points of their ar the low energy dynamicsglueball can superfield: be obtained from an effective supe with Around any of these vacua Φwhich is is massive so then it can given be by integrated o here count how many eigenvalues of Φ are equal to the form [ theory after confinement,superfields similarly to the case of pure SYM, a to denote this Higgsing process U( component the gluino JHEP07(2009)017 . ) ) ], z 2 to N 21 N ings (2.9) (2.8) , in such e ofΦ 1 U( ated 2- 2 × S ) dying was 1 N u defined by l homology class ), but otherwise n N to be well separated i U( a ) more clearly as: g to the same homology × 2.7 rpolating between the two . . . . ics of such a system is U( rthermore the adjoint chiral ) . n tunnel from U( um in its homology class. gineering, more on this below. × ld to the geometry above. The ns of the brane, gets a positive ons of the matrix model [ iB ) = 0 2 + D5 branes, which we take to wrap 2 different 2-spheres of the resolved N = 0 (2.7) ) wrapped by the brane: (we take the i J 2 N ( n i a U( S ity of the conifold by a finite size a )) ] for a more detailed description. S ) its first derivative with respect to − z ) theories with a superpotential for the Z × z ( 20 ( ′ z ) s ′ N ( 1 W 1 n of the Yang-Mills theory living on the brane W =1 N πig ]. i Y 2 τ 6 2 + ( g , so we can rewrite ( Sp( – 5 – 2 i is close to = a + w z → ), and 2 = πi + ) 2 YM w ) case [ 4 2 z 2.1 g N . The size of the resolved 2-cycles is determined by the y + N interpolation describes the 3-chain that makes the two i − 2 a + 2 y ) reduces to the one for the conifold. We take the conifold a π 2 θ 2 x + 2.8 → 2 = ) for the adjoints. Classical vacua of the system correspond x 1 τ a 2.1 branes get separated into the different cycles, some open str N + 1) in the gauge theory just by smoothly taking one eigenvalu ) are located at 2 D-branes in different ways in the z ( N so the geometry is no longer singular, but instead has an isol ( N W ]. It is given by Minkowski spacetime times a local Calabi-Ya U 3 A type IIB configuration realizing the gauge theory we are stu 19 × . For such a tunneling to be possible in string theory the tota 2 a 1) ) the same function as in ( z − to ( 1 1 N a W Let us briefly mention that all the resolved two spheres belon We will also be interested in engineering Sp( The bare holomorphic coupling constant Let us consider this background in the presence of It is easy to see from here that when Resolution means, roughly, that we substitute the singular 3 the resolution of: with adjoint. They can be obtainedbreaking by pattern adding a in suitable this orientifo case is Sp( get massive, and the gaugemultiplet group living on on the the brane, branesmass and factorizes. since which parametrizes Fu the motio resolution two-sphere is a local volumeclass. minim One simple way to see this in physics terms is that we ca everything is very similar to the U( Geometry. for simplicity), the equation ( is given by the complexified volume of the two cycle conifolds. When the which are related tovacua, and domain coming walls from in branes the wrapping the effective 3-chain. theory inte cycle at each of the degenerations distributing the to be resolved, a way that the resulting space is smooth. See for example [ complexified gauge coupling constant for the theory we are en The extrema of described in [ with the superpotential ( to U( should stay the same, the from Minkowski and two internal directions. The low energy dynam cycles homologous. Such a tunneling is described by instant JHEP07(2009)017 is 2 S (2.10) below), 2.2 stanton can 0: ≥ N ct one when we are amics of string theory there is a single brane We see that up to now o consider dynamical at a glueballs for describing h is a successful check of eory are very similar, and n fact, one way of reading er to reproduce the known ). We will refer to this last rees of freedom in order to groups would give rise to a scribed what plays the role ves, in agreement with the e gauge group on the cycle, no-Yankielowicz part of the ffects. Other candidates one g. One way out is to include ts, which are not present in N ntial (see section ted processes. The reason is -dynamical. Nevertheless, as e effects in the matrix model four neutral zero modes, and ese are commonly called U(1) g theory side than those that ur point of view is that these πiτS ld for any m in the string side are those This is a subtle problem, and with our prescription that the 2 cribing its dynamics. These are − the single instanton case (as we are NS + N /S N – 6 – 3 0 ] for when to include the extra stringy glueballs, 6 Λ log S = ) N U( W branes wrapped on it, would have been Sp( ], multi-instanton processes involving a Sp(0) and a U(0) in N , giving rise to a U(1) gauge group, and in the case on which the 2 11 S These extra glueballs must be included for the cases in which It turns out that in order to account for all the low energy dyn Note that the prescription given in [ We should point out that the previous discussion is the corre wrapped by no branes, buthad there there been is an orientifold such that th wrapping a one needs to includewould more be “glueball” necessary superfields just inthe from the results the strin of gauge theory thisnecessary point note in of is view. order that these to I gauge extra account theory. degrees for of the freedo stringy instanton effec the tree level dynamicscan of be the matched stringy quite systemof easily. and the Nevertheless, the all-important we gauge have glueballcentral th not superfield to yet our in de discussion. the string side. The low energy degrees of freedom of the string theory. so we need to makestringy the instanton corresponding effects. glueballs dynamic in ord matrix model shouldour compute proposal. D-brane instanton effects, whic restricting ourselves to superpotential contributionsdoing in in this note),least but the U(0) we glueballs expectthe in that following. the Consider it instantons description might ininstantons of be a in more U(0) necessary the gauge complica t D-brane groupthus (th instanton they literature). do They notfact have contribute that to we the arediscussed superpotential considering in by the [ themsel correspondingcontribute glueball to non the superpotential.side We if cannot we just reproduce setthe thes the U(0) U(0) glueball glueball in to thesuperpotential. zero game, Namely, from and we the just take beginnin set the to following zero formula the to Venezia ho which was obtained via the geometric transition, coincides case as the Sp(0) case.glueball In and gauge a theory, Veneziano-Yankielowicz none superpotentialalso of des the this only two gauge cases onmatch with which one the has superpotential tocould due include to think these one of extra stringy such deg the instanton as e stringy SO(0), dynamics. SO(2) or Theare U(0) justification the do of D-brane not this instantons require result that extr from contribute o to the superpote JHEP07(2009)017 ] D5 25 , N (2.12) (2.11) 24 tly), and ory: fixed. This has is not related to This means that i morphic coupling ). 4 M M s (Φ) g . The prefactor takes 2.4 (Φ), and distributing ne-instanton sector in n W = ing the multi-instanton W M i sen vacuum. S ition to a flux background. + x dimensional holomorphic the main focus of this note, ]. Since we are interested in ome back to this interesting Φ Tr matrix ( ical glueballs are associated th action: = 0. For the one-instanton in from further discussion of 3 scribed above, we have hysical theory. As above, the zing ply the arguments of [ d – . . . otential ( the F-term physics is captured 1 M N Z re the ones without momentum inser- + ve the matrix model in the saddle s × 2 = 0 due to the effect of an extra 1 g limit of the matrix model. As usual observation in the conclusions. M S − M M + Let us now come back to the connection ). The topological theory living on the 1 lve momentum insertions, are computed by the z ( ], but for multi-instanton processes there πiτ/N M exp 6 2 W − = )) e n 3 0 ) vanish when M – 7 – = 0 [ M S U( = Λ 2.10 3 Λ 1 ×···× ) 1 ] that only the planar diagrams contribute. M 23 ] to the two dimensional worldvolume of the brane. Partially extrema. We take , its relevance to the physical theory will be explained shor 22 n limits, we will keep the ’t Hooft coupling N Vol(U( M = Z in order to give the usual holomorphic scale Λ of the gauge the πi 2 YM 4 g is the cutoff for the theory, which combines with the bare holo − vacua into the 0 π θ 2 For the particular case of the superpotential, which will be Note that the first two terms in ( This prescription for the U(0) glueball has a chance of match M The argument boils down to the fact that the planar diagrams a is a overall scale unrelated to the coupling constant of the p 4 = s the into account the volume of the unbroken gauge groupit in can the be cho further argued [ This formula requires some explanation. Here Φ is a tions. It also impliesnon-planar that the part higher of F-terms, the which matrix do model. invo We will come back to this classical vacua of this theory will be determined by extremi g two desirable consequences. First it tellspoint us approximation, that we which can is sol easier; and second, one can ap Here Λ reproducing in this way the Veneziano-Yankielowicz superp τ the number of branes to argue that the result can be computed via a geometric trans results, and also fitswith well instanton effects with in the thequestion heuristic corresponding node. idea in that We the hope dynam future.this to c note, so In this anythis subtlety case, point. will we not be will important not and goMatrix we models beyond refra as the the theory o between on the the string branes. theory andcomputing the F-terms, matrix we model, can following go [ to the topological string. As de solving the equations of motionby for the this partition theory function tells of us the that holomorphic matrix model wi the required information can be obtained in the large could be nontrivial contributions. when taking large worldvolume of the D5 branesChern-Simons is theory the [ reduction of three comple branes in a geometry defined by the function superpotential, such a prescription dynamically sets light hypermultiplet appearing close to JHEP07(2009)017 r n- on in (2.14) (2.13) (2.15) he free : i S 0 F + of the matrix model i 2 2 S Φ rap D-brane instantons, the matrix model ampli- ng just the one instanton is the bare holomorphic o be cycles not wrapped calculus most relevant to residual gauge group into glueballs. The conjecture rmined by the volume of i ers τ ersymmetric gauge theories. i is a polynomial in m the free (conifold) theory. in the superpotential (notice Φ Tr S les and localized at a point in given by: 0 d expansion for future work. abi-Yau has cycles over which i F 0 Z . We can then expand the planar spacetime filling D-brane wrapping the n πiτ F i n s 2 ) and 1 S g i − N − · · · ) l makes sense, and it is the one we will always 1 S i i 1 ( S 0 exp ∂S n Z ∂ ...i )) denotes the planar part of the matrix model i 1 n i 0 c – 8 – N M Z n ...i U( 5 n =1 X 1 i X i ). of the physical string. In this way, the perturbative = i 2.9 )= 0 S 1 ) match nicely once we take into account the following. S F ( ×···× as a gauge coupling constant does not make sense. The definiti ) 2.5 determines the number of fractional instantons in the facto 1 τ eff can be understood as a systematic fractional instanton expa i M i W S S ). ) and ( ] from the mirror Chern-Simons perspective, the answer for t Vol(U( 2.12 2.13 26 = 0 Z denotes the rank of the gauge factor U( i N ) we are taking into account. In this note we will be consideri i Expressions ( Let us now describe how the prescription goes for connecting When computing instanton effects, sometimes there will be no 5 N terms of the (complexified)be volume implicitly of using. the resolved cycle stil cycle, and thus the definition of The first term gives risealso to that the it Veneziano-Yankielowicz is term the partition function for the conifold), and where As explained in [ tudes to F-terms in the physical string. The ’t Hooft paramet planar part of the matrixaccount, model, gives once the we Veneziano Yankielowicz take superpotential partition the volume function of into the a free part and a correction: partition function ( get identified with the glueballs gauge coupling on thethe branes wrapped wrapping two the cycle cycle as in which eq. is ( dete expansion of the matrixis model that is the an low expansion energy in superpotential powers of of the the string theory is This series expansion in 2.2 D-brane instantons Let us now proceedour to discussion. review As the reviewed aspectswe in of can the wrap D-brane previous space-filling instanton section, D-branesDepending the in Cal on order to the engineer particularby 4d sup vacuum any we spacetime choose fillingwhich there D-brane. in might this Over als context these are branes cycles wrapping we these might internal w cyc sion, where the power of sector, which is determinedWe by leave the the universal study term of coming multi-instanton fro effects due to the U( JHEP07(2009)017 rane ), with N ). The D- (1). Such a N nton point of O es. If the D-brane erspace. The kind which the couplings tons do not exist in the superpotential in have no simple gauge nsparent, here we have fold actions and their effect spacetime filling branes f the instanton and the no wrapped space-filling hus has 4 fermionic zero dly rather empty) Sp(0) ft the two extra goldstinos N of using the more common en there is a single D-brane nd the ADHM couplings lift rojection acts oppositely on 6 e instanton is es implementing the ADHM space-filling D-branes wrap- types. We describe them in ge instanton for Sp( gauge group Sp( on. Furthermore, the sign of ce of branes and orientifolds, ro modes, which will play the hould include some mechanism N ]. ), it works for U(1) gauge theories too 33 , N 14 , = 1, and thus it has the two desired Goldstinos ) contributions to the 4d effective action living s 11 – 9 – g N Goldstinos, as we want. ˙ α ¯ θ ) gauge theory. Intuitively, since it is on top of the D- N is bigger than 1 for U( N Goldstinos. ]. The notation Sp(0) is thus intended to suggest that the D-b ] for a nice discussion of the details of the possible orienti ˙ α ¯ θ 27 32 (1)”. The justification for this denomination from the insta – O 28 corresponding to the 4 spontaneously broken supersymmetri variables in the integration of the superpotential over sup ˙ α θ ¯ θ , α modes. θ ˙ α One interesting observation one could make is that the way in The basic condition for a D-brane instanton to contribute to Let us now describe instantons of the U(1) type. These instan In order to make the connection with the matrix model more tra In order to explain why this happens, let us consider first The simplest configurations in which this happens, and which An instanton of Sp(0) type occupies a cycle in which there are ¯ θ See for example [ only. More precisely, the couplings between the zero modes o 6 α between the zero modes implement thedoes ADHM not constraints and require li that fields living in the branesthe implement the ADHM constraints, a θ this setting is thatrole it of has the exactly two unlifted fermionic ze on the spacetime wrapping branes. spacetime. They give nonperturbative (in of backgrounds where are8 considering supercharges. preserve, in A the D-branemodes absen instanton is a 1/2 BPS object, and t branes, the D-brane instanton “feels” just pure field theory either,wrapping but the they same do cycle exist as in the string instanton theory [ wh instanton can be thoughtgauge of group. as a gauge instanton of a (admitte on the instanton zero modes chosen to refer todenomination this “ kind of instanton as “Sp(0)”, instead wrapping the cycle, giving a U( branes, and which isthe mapped orientifold to projection itself is by such the that orientifold acti the gauge group on th theory interpretation, are instantonsthe of following. the U(1) and Sp(0) instanton is going to contributethat to lifts the the superpotential, two we s brane instanton in this context can be interpreted as the gau view is the following.ping the Consider same the cycle orientifoldspace-filling as action and the instanton on instanton. branes, these Since branes the would orientifold have p the couplings for theconstruction string modes [ between the different bran projection does indeed remove the JHEP07(2009)017 , ) M 2.7 (3.1) (2.16) matrix , with p M ntons Ω indices, and these ⊗ j n j p M el and the D-brane Φ Ω eement between the , configurations with old, so let us take a ) into equation ( D x · ( ding into string theory. 2.1 atrix model result does n i W ]. One important example Φ le and interesting problem superpotential due to these is very much related to the wo zero modes, the general sting. What happens here is tion D 35 ortant remarks. Although we l, but can generate couplings , es which we have not discussed 2 btained from the matrix model of more than two fermions (or 2 34 Φ · · · t that matrix models can be used is a section of Tr 1 j θ = 0. As explained above, we will be ω Φ 2 d 2 D . These instantons do not contribute to · w c Z indices, antisymmetric in the 1 i + N i Φ 2 = z D ]. Modifying the matrix model side of the story – 10 – f (Φ) = + 40 N ] and also of this work: such a completion is not 2 – W 6 y Φ) 36 , Tr + θ (Φ 2 2 n x d j n i Z ... 1 j 1 i ]. U(1) theories do not have instanton effects in gauge theory 33 θ w 2 is for later convenience. Plugging this 2 / We include some preliminary remarks on the relation between xd 4 7 d Z Another interesting possibility for lifting extra zero mod The geometry we will be focusing on will be the resolved conif Before going on to check the agreement between the matrix mod In other words, it is a form antisymmetric in the 7 the superpotentials, but induceequivalently, derivatives). operators They with are insertions of the form: where Φ represents the moduli of the system, and as studied in detail in [ the moduli space. quadratic superpotential for the gauge theory: where the factor ofwe 1 obtain the conifold equation indices live in the cotangent bundle over the moduli space. instanton calculus in these cases,restrict let here us make to a the couplecase case of in imp which in the which instantonthat the has the more instanton zero instanton has modes does exactlyinvolving is not t more also contribute than intere to two fermions, theis called superpotentia the higher gauge F-terms [ instanton for SQCD with models and higher F-terms in the conclusions. here is by using fluxes,in as order studied in to [ incorporatewhich the we effect will not of attempt these to fluxes solve is here. a worthwhi 3 The glueball superpotentialIn for this low section rank we andnot will match D-brane the study insta gauge the theorygauge two result, group basic namely, as U(1) cases reviewed inresult and in of which sec Sp(0). the the D-brane m instanton Weprescription. calculation will This and gives the see evidence result thatto for o the compute there D-brane advertised is resul instanton effects. a nice agr as innocuous as it looks, since it also changes the low energy and the fact thatfact D-brane that instantons we contribute are inTo completing restate this this one U(1) case of theory in the the main UV points by ofexotic embed U(1) [ instantons. JHEP07(2009)017 . ) ¯ θ ): , in 2.1 2.5 (3.5) (3.4) (3.3) (3.2) 2.9 2.1 , where A α θ = 1 in eq. ( N . This reduces the etime filling branes, 2 orientifold action, so it mz rcharges on the bulk and 2 1 eviewed in section of the SU(2) R-symmetry as discussed in section zero modes. he matrix model free energy ur neutral zero modes, and ce to see how the saturation ulk. The Sp(0) instanton is viving Goldstinos us on the Sp(0) case, so the in how this deformation acts tanton can now contribute to nable to guess that two of the ) = eter determined by the theory omorphic one-matrix model, so πiτS t we must set z ]. 2 order to avoid the complications ( 33 − W 3 S iB = 0. In this case we will have 8 back- + + 3 0 = Λ Λ J 3 0 W 3 0 2 Λ S πiτ S Λ 2 Let us now describe in some detail how the Z − πiτ s 2 e 1 g – 11 – − log e = = Let us start by the matrix model side of the story, S S = out. Its equation of motion gives: πiτ W )= 2 S ) gives: S,τ the bare coupling of the configuration, defined in eq. ( 3.2 ( τ eff W the cutoff scale and modes goes for U(1) instantons is for example [ 0 ¯ θ We will start by considering the case with Let us start by studying the case were we have background spac We now switch on the quadratic superpotential is an index taking values in the fundamental representation where we have introduced the cutoff independentThe scale D-brane Λ. instantonstringy computation. instanton result reproducesof this having contribution. to impose In the (super)ADHM constraints, we will foc considering the resolved conifold, with a resolution param we are engineering. The matrix model computation. which is the simplest. Wein have to this compute background. the The planar conifold limitwe is of just described t obtain by the the Veneziano-Yankielowicz result. free hol Note tha which upon substitution into ( ground supercharges close to the orientifold, and 16 in the b a 1/2 BPS objectwill wrapping have a 4 Goldstinos 2-cycle in mapped its to worldvolume. itself Let under us the call the sur in order to captureWe Sp(0) get: and U(1) D-brane instanton effects, We can now proceed to integrate supersymmetry of the background by4 half, close so to now we the havefour orientifold. 8 zero In supe Goldstinos such a ofthe configuration the superpotential. it instanton is get This reaso lifted, isfrom indeed and the the the point case, of ins let view of us the now effective expla actionso on we the are instanton dealing with ordinary gauge theory instantons. As r modes are simply projected outof by the the orientifold. A good pla A with Λ group preserved by thethus orientifold. cannot contribute Such to an the instanton superpotential. has fo JHEP07(2009)017 (3.7) (3.8) ], and (3.10) = 2, so , let us in front 33 A α , θ strings as N m 32 ). This goes ], this can be , get taken care 41 28 3.9 e the low energy s [ YM g ve adjoint, is by now construction in such a and ds in string theory, but es associated to the field ]). π s a breaking of the super- ng in order to saturate the t of as a gaugino from the 41 ector multiplet of le wrapped by the instanton. modes as the addition of the ting out the adjoint is given in ) (3.9) ) holds seems to be reasonable. ctors of is one of the Goldstinos 3.7 πiτ 2 α α 2 α θ λ − λ α α 2 πiτ λ θ 2 Tr Φ (3.6) 2 2 − ) also exists for our Sp(0) instanton. We e exp ( m . We have chosen this notation in order to YM 2 YM S term in the instanton measure, allowing the ˙ 1 2 2 α mπ g s mπ g 3.7 ¯ λ M λ − − 2 – 12 – d = mM = = and (Φ) = 2 adj α inst ] for a careful derivation for the more involved U(1) )= inst Λ W λ τ ( 33 δS δS modes. This integration will give us a factor of eff 2 α = 1 by the addition of the superpotential for the adjoint W = 1 algebra. As reviewed in section VI.4 of [ θ ) with the microscopic scales appearing in ( N N 3.5 ], with a natural identification of the instanton-instanton = 2 to 43 N , . The coupling induced by the deformation would then be: 2 α 42 θ factors come from the measure of integration of the zero mode s M The rest of the calculation, once we have dealt with the massi From the point of view of the instanton brane In order to match with the matrix model result we need to relat We now claim that a term similar to ( chiral multiplet: symmetry from the gauge theory the deformation of the background appears a Such a term can be used to saturate the take it to be well understood (see for example [ as follows. The physicalterms scale of for the the string theory scale before quantities integra as: nicely encoded in thefollowing effective term: action for the instanton zero gauge theory zero modes, assuming that an analog of ( instanton). The end result is simply given by: the exponential suppression comes from the volume of the cyc physical Λ appearing in ( cannot compute the existence of such a coupling via CFT metho given the fact thatnatural string way theory [ gives rise to the whole ADHM instanton to contribute to the superpotential. One consequence of such aΦ coupling get is lifted, that let some us of call the such zero modes mod Just as in the gaugefermionic theory integration over case, the we can bring down this coupli The of the resulting contribution to the superpotential (the fa remind the reader that the adjoint superfield comes from the v of once we integrate over the rest of the zero modes properly [ the highest component fermionpoint of of the view multiplet of can the broken be though JHEP07(2009)017 = 2 (4.1) (3.11) (3.12) vanish, N 10 X and t the orientifolded 01 X e using matrix model fore and after integrating orbifold quivers by giving theory also has a quartic in the game, and treating y generalizable to een describing. n agram denote gauge factors, ne instanton intersects some rically give rise to couplings scuss in the main text. We have ing. This means that we can W A e theory of the brane, which ingularity of an abelian orbifold he brane. Integration over them ], there are fermionic zero modes n the original paper of Dijkgraaf 9 and node to zero, – 10 7 , where we are omitting the part of , the ranks that go into the matrix 10 X 1 Sp X 21 2.1 , X m 3 01 12 ), which will not be relevant for our discussion. 2 adj X 2 X N = Λ 01 = Λ – 13 – X W 3 Λ = Tr W orbifold of the conifold n Z ) beautifully. ]. We will choose an assignment of ranks of the quiver such tha 3.5 2 . Quiver for the orientifolded orbifold of the conifold we di = 0. Nevertheless, as we saw in section prefactor gives the matching relation between the theory be W m In our case, since we are setting the rank of the In this section we will focus on a model which is easy to analyz The relevant quiver diagram is shown in figure quiver theories with superpotentialsand for Vafa the [ adjoints, as i The resulting superpotential can thus be written as: 4 A quiver example: The out the adjoint: large masses to the adjoints. The analysis we do here is easil matching ( Figure 1 omitted the part of the quiver to the right of U( arising from strings stretching betweentypically the instanton induces and t interesting couplingsmight be in potentially the relevantwith world for are volum model forbidden building. in perturbation They theory. gene model computation are unrelatedcompute to those the in effective the superpotential physical keeping str techniques. It is the theoryof corresponding to the branes conifold. at the These s theories can be easily obtained from node is empty, giving rise to one of thethe Sp(0) quiver factors that we will have notand b be relevant the for arrows us.superpotential, denote The nodes given bifundamental of for chiral the generic di ranks multiplets. in the The nodes by: Let us proceed to the more interesting case in which the D-bra space-time filling branes. In this case, as discussed in [ and JHEP07(2009)017 ) n n: A K (4.4) (4.5) (4.2) (4.3) (4.6) ] for 2 ] that gives 48 0, extract the ) can be taken 2 N +1 1 K > K U( putation. The tech- i ) confine while taking × )) arising from strings superpotential is then ] for calculations along 12 (these are anticommut- the matrix model point K 1 ) ) (4.7) 1 X 47 β N e prescription in [ – , N 21 prefactor in the Veneziano Sp( n the Sp(0) node. One way inst α i 44 X finity, keeping the dynamical f U( h ]. An instanton sitting on the on: ) factor before and after inte- , which we can now integrate δS 12 9 ut just the leading term due to le [ – 12 X K det i 7 i X 21 12 exp ( 12 , X +1) X h β X 01 K ( 21 21 21 X / det 1 X X dαdβ X h 1 N h 12 N − Z 12 3 high − det X 1 αX πiτ +1) N 01 2 – 14 – = − K − , and two zero modes X 3 + 1)Λ e 3( high α 1 θ inst 0. The only place where the actual rank of Sp( ) brane. Essentially the same physics [ K N 1 = Λ − = Λ are background vevs. From this point of view, we can = Tr δS 2 s N ) with no flavors. It confines, and the low energy super- = ( eff 21 ). W K K > , and apply it to the Sp(0) case. We obtain the result: +1) W X 3 low mM K ) gives rise to the following coupling in the instanton actio 2.5 , low 3( = /∂S 12 is the mass matrix for Λ 0 4.1 X eff i Z + 1)Λ ) with ∂ 12 W K K are the dynamical scales of the Sp( X 21 = ( low X h W and Λ ) to mean: high 4.1 ), we can take instead the following shortcut. We can now use this result computed in the gauge theory with Let us now compare this result with the D-brane instanton com The leading term we are interested in corresponds to letting Let us proceed to compute the low energy superpotential from The remaining theory is Sp( The contribution of the instanton to the low energy effective K take ( to be a flavor group, and going from the instanton to the U( matrix model result for enters in the matrixYankielowicz model term, computation as is in in eq. ( determining the quivers and taking the limitscales where of the the adjoint masses resulting go conifold to theory in finite. See for examp nology for dealing with this problem is by now standard [ In this equation out. It is important to take into account the matching relati them as if we had Sp( potential is given by: ing scalars living in the fundamental and antifundamental o the rest of the quiver theory to be an spectator. In this way U( Sp(0) node has two neutral zero modes rise to the superpotential ( these lines. Since we areSp( not interested in the full answer, b of view. We want to obtainwe could the go effect about induced by computing one this instanton effect i would be by applying th grating out the massive matter. where Λ given by: JHEP07(2009)017 ns lti- = 1 (4.8) N , and we can 3 jkgraaf and Vafa are is note, let us list here h the higher orders in d (D-brane) instanton d the D-brane instanton ious calculation are those r as the (in general more deal with (when it is ap- ] for progress in the de containing the instanton od sign, and indicates that tive effects in string theory e matrix model. We checked namely the one (fractional) her orders in the instanton etimes involved even in the ects from the point of view of Since the time when the DV ndicate that, in the same way on effects in the string side, sly, reducing the problem to 49 21 y a perturbative calculation in 8 tudied in section or the instanton node and ex- iver gauge theories admitting a . X 12 dy D-brane instanton contributions also X ) = det inst – 15 – δS exp ( , where we studied the case of the isolated instanton. 3 dαdβ Z ] for some recent progress in more stringy systems. It would ) obtained from the matrix model. 53 ] for a review of some powerful techniques for dealing with mu – 4.5 above. The second term can be used to saturate the integratio 41 50 1 , N − 37 3 , 36 , , and gives rise to: β 11 and α The geometries on which one can apply directly the ideas of Di We have restricted ourselves to matching the matrix model an In this same line of argument, one technical difficulty to matc We see that the essential ingredients that went into the prev There are many ways in which one could extend the results in th When we are interested just in the superpotential, we can stu 8 over case, and [ The overall prefactor works exactly as in the conifold case s instantons in the case of gauge theory withbe extended susy, interesting [ toexpansion check using the some proposed of these correspondence techniques. for hig that appeared already in section a few of them. results just up toinstanton leading sector. order in Thiswriting the simplified down instanton the the expansion, Veneziano-Yankielowicz calculationstremizing superpotential enormou the f complete superpotential.the matrix This model is formalismplicable) already might than a be the go significantly D-braneone-instanton simpler sector. instanton to calculus, which is som the matrix model perturbationis expansion that with we non-perturba and will have these to are dealcalculus. somewhat with See cumbersome however multi-(fractional) [ to instant deal with using standar rather restricted, and not very suitable for model building Namely, once we considered the glueball superfield for the no identify it with Λ dynamical, the matrix modelinvolved) instanton correctly calculation. gave us the same answe 5 Conclusions and furtherIn directions this work we have startedmatrix the models. study of We D-brane found instanton that eff atthat least ordinary the gauge simplest non-perturbative examples i effectsthe are matrix captured model, more b stringythis effects are result also in computed by some th matrix detail model for description. isolated conifolds, and for qu reproducing the result ( JHEP07(2009)017 ge ], where a 54 ential terms in udy of higher F- of type IIB string . There have been some ]. 2.1 14 e computed by the non- ons. It would be rather rted by the High Energy ]. o thank Nao Hasegawa for effect of the C-deformation nton effects across moduli sley-Witten higher F-terms cular, this includes mirrors iagrams and Beasley-Witten turbative amplitudes of the ents in the study of matrix ons theory in the appropriate roduced these higher F-terms ward reason is that we need to ing to see if and how D-brane ons with Marta Gomez-Reino, us diagrams are associated with g to see if and how these higher effects across the whole moduli ves, or more than two fermions, bviously too sketchy, so it would nt for our story is [ SPIRES ] [ ion and section ently, starting with [ ], so if non-planar diagrams are also related to – 16 – 55 hep-th/0206255 [ (2002) 3 ). Recently it has become clear that these higher F-terms Matrix models, topological strings and supersymmetric gau 2.2 B 644 ]. They have a fascinating interplay with ordinary superpot 40 Nucl. Phys. , , 37 , 11 theories We would also like to make some preliminary comments on the st One seemingly reasonable possibility is that these terms ar As a last remark, all of our discussion has been in the context [1] R. Dijkgraaf and C. Vafa, via the geometric transition, as discussed in the introduct models and topological string theory. Particularly releva results appeared there have been very interesting developm terms in the context oftowards matrix the models (we end have of succinctly int section interesting model building applications of these ideas rec higher F-terms then we would have a nice relation between Bea matrix-model inspired formalism isB-model on given a for very computingof rich toric per set Calabi-Yau of spaces). geometricinstanton backgrounds It effects (in would appear parti be in extremely this interest setup. space [ order to give aspace consistent of description the of compactification. non-perturbative ItF-terms would are be encoded rather in interestin the matrix model description. planar diagrams in the matrix model.compute terms A naive in and the straightfor effectiveand superpotential it with is well derivati known thatoperators in the of topological this string kind higher in gen be the good physical to superstring. have a This more ishigher direct o F-terms. connection between We non-planar know d thatof non-planar diagrams the encode gluino the anticommutation algebra [ and the C-deformation. theory. By mirror symmetry, one couldbackground expect that might Chern-Sim also computeinteresting to the see effect how this of works. exotic E2 instant Acknowledgments I am happy toDaniel acknowledge interesting Krefl, and Sergio and fruitful Montañez Angel discussi kind Uranga. support I also and want constantPhysics t encouragement. Research Grant This DE-FG05-95ER40893-A020. work is suppo References form an integral part of the global picture for D-brane insta JHEP07(2009)017 , , Adv. string , D ]. 4 , (2007) 060 (2002) 21 ]. (1990) 246 , (2004) 45 09 ]. (2003) 161 =1 ]. SPIRES N B 644 ] [ SPIRES (2007) 113 B 342 JHEP B 682 On low rank classical ] [ ]. , ]. B 648 SPIRES SPIRES ] [ [ B 771 , Perturbative computation of ]. ]. SPIRES SPIRES n, Nucl. Phys. [ ] [ Nucl. Phys. The glueball superpotential , ]. ]. Nucl. Phys. nd C. Vafa, , ]. Open string instantons and Nucl. Phys. , hep-th/0211017 Chiral rings and anomalies in , (1982) 231 [ SPIRES SPIRES hep-th/0211170 ] [ Stringy instantons and quiver gauge [ Nucl. Phys. ] [ ]. SPIRES SPIRES , hep-th/9912151 SPIRES [ ] [ ] [ B 113 ] [ Spacetime instanton corrections in (2003) 138 Stringy instantons as strong dynamics SPIRES arXiv:0711.1954 arXiv:0711.1430 – 17 – (2002) 071 [ ] [ Non-perturbative superpotentials across lines of Nonperturbative effects and the large-order behavior , Geometric transitions and dynamical SUSY breaking 12 B 573 Phys. Lett. hep-th/0304271 An effective Lagrangian for the pure Comments on conifolds , (2000) 026001 supersymmetry, deconstruction and bosonic gauge theories arXiv:0803.2829 [ hep-th/0610003 Neutrino Majorana masses from string theory instanton [ JHEP arXiv:0709.4277 hep-th/0609213 [ =1 [ (2008) 033 ]. , Stringy instantons and cascading quivers ]. ]. ]. ]. D 62 On geometry and matrix models N A perturbative window into non-perturbative physics ]. ]. 01 Phys. Lett. , (2004) 1045 [ arXiv:0809.3432 [ SPIRES (2008) 1 7 (2007) 024 SPIRES SPIRES SPIRES SPIRES JHEP ] [ SPIRES SPIRES (2007) 052 (2008) 066004 [ [ ] [ ] [ ] [ ] [ , Phys. Rev. 05 , 03 B 796 D 78 On exact superpotentials in confining vacua (2008) 041 A gauge theory analog of some ’stringy’ D-instantons JHEP ]. , JHEP 11 , Nucl. Phys. SPIRES hep-th/0210135 arXiv:0707.3126 hep-th/0609191 hep-th/0311181 hep-th/0207106 supersymmetric Yang-Mills theory supersymmetric gauge theory of matrix models and topological strings [ [ marginal stability superpotentials JHEP Phys. Rev. [ [ [ effects [ hep-th/0208048 hep-th/0302011 theories groups in string theory, gauge theory and matrix models glueball superpotentials vacua - the seesaw mechanism for D-brane models Theor. Math. Phys. [2] R. Dijkgraaf and C. Vafa, [5] M. Aganagic, K.A. Intriligator, C. Vafa and N.P. Warner, [4] R. Dijkgraaf and C. Vafa, [6] K.A. Intriligator, P. Kraus, A.V. Ryzhov, M. Shigemori a [7] R. Blumenhagen, M. Cvetiˇcand T. Weigand, [8] and L.E. A.M. Uranga, Ibáñez [9] B. Florea, S. Kachru, J. McGreevy and N. Saulina, [3] R. Dijkgraaf and C. Vafa, [13] D. Krefl, [14] M. Aganagic, C. Beem and S. Kachru, [15] F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, [17] F. Ferrari, [19] S. Kachru, S.H. Katz, A.E. Lawrence and J. McGreevy, [21] M. R. Mariño, Schiappa and M. Weiss, [11] I. Garcia-Etxebarria and A.M. Uranga, [18] G. Veneziano and S. Yankielowicz, [10] O. Aharony and S. Kachru, [12] A. Amariti, L. Girardello and A. Mariotti, [16] R. Dijkgraaf, M.T. Grisaru, C.S. Lam, C. Vafa and D. Zano [20] P. Candelas and X.C. de la Ossa, JHEP07(2009)017 , ]. , , gs on , SPIRES ] [ (2008) 102 (1995) 637 ctifications (2006) 060 (2001) 2798 orientifold 12 (2007) 076 , ]. 02 , ]. 3 (1994) 311 133 (2008) 112 ]. 42 (2003) 045 04 /Z olds 6 10 02 165 T JHEP , JHEP Stringy instantons at , SPIRES SPIRES , SPIRES JHEP ] [ (2007) 011 ] [ , JHEP ] [ JHEP , arXiv:0708.0403 Prog. Math. 06 duality , [ d M. Schmidt-Sommerfeld, tons , Lifting D-instanton zero modes by tersson, The calculus of many instantons N J. Math. Phys. ]. ]. Non-perturbative F-terms across , duality via a geometric transition N JHEP Kodaira-Spencer theory of gravity and , N ]. ]. ]. ]. Instanton induced neutrino Majorana (2007) 098 SPIRES SPIRES hep-th/0702015 ] [ ] [ [ 10 arXiv:0805.0713 Commun. Math. Phys. [ arXiv:0704.0262 [ , A large- SPIRES SPIRES SPIRES SPIRES – 18 – D-brane instantons on the ] [ ] [ ] [ ] [ JHEP , (2007) 26 (2008) 028 (2007) 067 07 hep-th/0103067 hep-th/0205297 ]. ]. ]. 06 [ [ B 782 Non-perturbative and Flux superpotentials for Type I strin ]. ]. ]. ]. ]. ]. New instanton effects in supersymmetric QCD New instanton effects in string theory Worldsheet derivation of a large- JHEP arXiv:0808.2918 arXiv:0704.0784 arXiv:0711.1837 hep-th/0409149 JHEP [ [ [ [ , SPIRES SPIRES SPIRES , (2001) 3 (2002) 3 SPIRES SPIRES SPIRES SPIRES SPIRES SPIRES ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ Nucl. Phys. Non-perturbative effective interactions from fluxes Classical gauge instantons from open strings Flux interactions on D-branes and instantons , D-brane instantons and the effective field theory of flux compa Superpotentials from stringy instantons without orientif B 603 B 641 Chern-Simons gauge theory as a string theory (2009) 048 (2007) 038 (2008) 078 (2005) 056 Superstrings and topological strings at large- orbifold 01 07 05 01 3 Z arXiv:0807.4098 arXiv:0807.1666 hep-th/0512039 hep-th/0612132 arXiv:0704.1079 hep-th/0211250 hep-th/9207094 hep-th/9309140 hep-th/0008142 [ [ JHEP [ JHEP [ recombination and background fluxes lines of BPS stability JHEP JHEP [ [ [ Nucl. Phys. Nucl. Phys. masses in CFT orientifolds with MSSM-like spectra Non-perturbative SQCD superpotentials from string instan [ [ exact results for quantum string amplitudes orbifold singularities the [41] N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, [39] M. et Billó’ al., [40] A.M. Uranga, [31] R. Argurio, M. Bertolini, G. Ferretti, A. Lerda and C. Pe [33] C. Petersson, [34] C. Beasley and E. Witten, [37] I. Garcia-Etxebarria, F. Marchesano and A.M. Uranga, [35] C. Beasley and E. Witten, [36] R. Blumenhagen, M. R. Cvetiˇc, Richter and T. Weigand, [26] H. Ooguri and C. Vafa, [27] A.N. L.E. Schellekens and Ibáñez, A.M. Uranga, [28] M. Billóet al., [29] N. Akerblom, R. Blumenhagen, D. E. Lüst, Plauschinn an [30] M. Bianchi and E. Kiritsis, [32] M. Bianchi, F. Fucito and J.F. Morales, [38] M. Billóet al., [24] C. Vafa, [25] F. Cachazo, K.A. Intriligator and C. Vafa, [23] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, [22] E. Witten, JHEP07(2009)017 ] , =1 bifold N (2003) quiver r Adv. Theor. D , , B 661 and ]. hep-th/9604198 [ r A quiver gauge theory and (2004) 014 ]. 2 arXiv:0803.2514 SPIRES A (1996) 541 Nucl. Phys. 01 , ]. ]. ] [ , superpotentials from (1998) 255 =1 SPIRES 28 N ]. =1 ]. ] [ JHEP B 460 ]. , ]. SPIRES SPIRES N Remodeling the B-model ] [ ] [ ]. ]. SPIRES SPIRES [ SPIRES SPIRES ] [ hep-th/0510173 Power towers of string instantons for ] [ [ ] [ Nucl. Phys. SPIRES , J. Geom. Phys. ] [ SPIRES Four-dimensional wall-crossing via , – 19 – ] [ arXiv:0709.1453 Higher order loop equations for (Non-)BPS bound states and D-brane instantons [ hep-th/0307054 hep-th/0311258 [ [ (2006) 031 01 ]. arXiv:0807.4723 arXiv:0806.3102 Phases and geometry of the [ hep-th/0206063 arXiv:0803.1562 , [ Multi-instanton and string loop corrections in toroidal or [ (2009) 117 ]. ]. (2003) 063 (2004) 033 hep-th/0302109 JHEP Stringy instantons in IIB brane systems The C-deformation of gluino and non-planar diagrams , SPIRES 287 09 03 arXiv:0803.2513 [ ] [ gauge theory with flavor from fluxes SPIRES SPIRES (2008) 069 (2002) 231 (2008) 027 ] [ ] [ =1 JHEP JHEP Gauge fields and D-branes 08 (2003) 53 [ 07 , , N 371 7 Small instantons in string theory (2008) 012 ]. ]. Comments on quiver gauge theories and matrix models JHEP JHEP 07 , , hep-th/0212079 Phys. Rept. hep-th/0211287 SPIRES hep-th/9511030 SPIRES matrix models multi-instanton calculus models Math. Phys. Commun. Math. Phys. [ JHEP [ [ three-dimensional field theory [ matrix models 257 [ vacua [52] P.G. Camara and E. Dudas, [42] E. Witten, [49] F. Fucito, J.F. Morales, R. Poghossian and A. Tanzini, [50] R. Blumenhagen and M. Schmidt-Sommerfeld, [53] D. Gaiotto, G.W. Moore and A. Neitzke, [54] V. Bouchard, A. Klemm, M. and Mariño S. Pasquetti, [55] H. Ooguri and C. Vafa, [44] S. Seki, [45] R. Casero and E. Trincherini, [48] S. Kachru and D. Simic, [43] M.R. Douglas, [46] S. Chiantese, A. Klemm and I. Runkel, [47] Y. Ookouchi, [51] M. R. Cvetiˇc, Richter and T. Weigand,