JHEP10(2013)007 0 , 4 Y and Springer all exhibit 0 F July 16, 2013 1 : October 1, 2013 dP : ) self-duality of September 26, 2013 Z : , and 3 Received Z / 3 10.1007/JHEP10(2013)007 Published C c Accepted doi: . We present extensive checks of the 3 Published for SISSA by = 1 gauge theory dualities which relate Z / [email protected] 3 N , C and Timm Wrase b singularity, as well as dualities for the 1 dP [email protected] Ben Heidenreich , a dualities from orientifold transitions 1210.7799 = 1 Duality in Gauge Field Theories, String Duality, D-branes, Supersymmetric We report on a broad new class of [email protected] N Department of Physics, CornellIthaca, University, NY 14853, U.S.A. Stanford Institute for Theoretical Physics,Stanford, Stanford CA University, 94305, U.S.A. E-mail: Theory Group, Physics Department, CERN, CH-1211, Geneva 23, Switzerland b c a Open Access ArXiv ePrint: such dualities, and arguetype that IIB their string theory. ten-dimensional origin is the SL(2 Keywords: gauge theory duality, including anomaly matching, partialsymmetries, moduli and space matching matching, of matching thethen of present superconformal discrete a indices related between duality for thesingularities, the proposed illustrating duals. the breadth We ofwe show this that new certain class infinite of classes dualities. of geometries In which a include companion paper, Abstract: the worldvolume gauge theoriesCalabi-Yau of singularity. D3 In branesdualities, this probing arising paper, from different the orientifolds we orbifold of focus singularity the on same the simplest example of these new I˜nakiGarc´ıa-Etxebarria, New Part I: field theory JHEP10(2013)007 7 37 47 50 61 22 25 13 41 71 15 SU(3) duality × SU(2) duality 56 20 × SU(7) 15 4 40 – i – 36 68 9 ←→ 34 10 USp(6) 30 3 70 Z N > 57 17 / 3 ←→ 0 C 1 SU(2) theory , 0 4 65 × F dP 63 Y SU(6) theory 29 37 SU(4) theory 3 t × × Z / = 4 orientifolds 3 57 1 53 C 55 N N A.2.1 A.2.2 Orientifolds of 6.1.1 Moduli6.1.2 space and Higgsing Case study: the SU(5) 3.3.2 The SU(5) theory 3.3.1 The USp(6) C.1 On nonperturbative effects A.2 Examples A.3 General quiverfolds 6.3 The real cone over A.1 Orientifolding a quiver gauge theory 6.1 Complex cone over 6.2 Complex cone over 5.1 The USp(8) 5.2 The USp(10) 5.3 The infrared behavior for 4.1 Expansion in 4.2 Large 3.1 Classic checks3.2 of the duality Limitations from3.3 the perturbativity of Case the study: string the coupling SU(5) B Negative rank duality C Exactly dimensionless couplings D Coulomb branch computation of the string coupling 7 Conclusions A Quiverfolds 6 Further examples 5 Infrared behavior 4 Matching of superconformal indices 3 Duality for Contents 1 Introduction 2 Review of Montonen-Olive duality JHEP10(2013)007 ], 26 – ], gen- 24 4 , 3 , 1 75 76 80 SU(4) = 4 super-Yang Mills ].) String theory also × 28 N ) self-duality of type IIB ]. As such, the two fields ) self-duality. In this paper we Z , Z 26 , ], the duality cascade [ ], models with antisymmetric ) duality of type IIB string theory. 7 ) Montonen-Olive duality in the 23 Z – , 74 – Z 5 , 20 ]. (Many of these are related manifes- = 7 30 74 N , ], which relates ], to yet more complicated examples (see 29 33 13 – , 31 12 – 1 – element of the SL(2 efficiently /τ 21 1  i (0) → − B τ  ), which nonetheless acts nontrivially on the worldvolume gauge field ], “self-dual” theories [ Z ], and geometric transitions [ , 11 – 28 , In particular, the appearance of an SL(2 8 27 1 ]), in addition to the classifications of various types of confining gauge theo- ] where the confined phase has a weakly coupled dual description without a dual 19 15 , – 14 16 Montonen-Olive duality is different from Seiberg duality in some important ways. Un- Another gauge theory duality of a different nature also enjoys a close relationship Seiberg duality often admits a very natural and enlightening embedding in string the- The term “S-duality” is sometimes used to refer to the entire SL(2 E.1 A note onE.2 computing Check of the superconformal index calculation for USp(8) 1 ory, with each of the formulations mostconstant. suitable for There certain values is of the nofixed Yang-Mills flow coupling point. wherein Indeed, distinct due to gauge maximal theories supersymmetry, there converge is on no the flow whatsoever,will same and use infrared when it to refer specifically to the (as an electromagnetic duality) and gaugethe coupling action (as a of strong/weak Montonen-Olive duality), duality reproducing on the gauge theory. like Seiberg duality, Montonen-Olive dualityvarious is superficially an distinct exact but quantum duality, in equivalent formulations the of sense a that single it physical the- gives to string theory. Montonen-Oliveto duality itself [ at differentstring couplings, theory. is directlyworldvolume related gauge to theory of the D3 SL(2 the branes D3 in under a SL(2 flat background follows from the invariance of toric duality [ tations of the same phenomenon,NS5 where branes Seiberg through duality each issupplies realized other some as in contexts the where a effecthave Seiberg enjoyed particular of a passing duality T-dual largely can picture symbiotic be relationship. [ exact [ e.g. [ ries [ gauge group. ory, where it appears in the context of brane systems [ The success of Seibergtype, duality has ranging motivated from a thorougheralizations natural study with generalizations of adjoint to further matter dualities SOtensor of matter and and this [ a USp superpotential gauge [ groups [ 1 Introduction F On the decomposition of certainG generalized A Specht conjectured modules identity for elliptic hypergeometric integrals E Details of the superconformal index for JHEP10(2013)007 ) 3 ]. Z D3 = 4 , 46 by a k ) can N N Z Z , / + 1), or ) ) is again = 1 gauge k N k ]. By contrast, 3), a natural N 36 O 2 ), SO(2 k ) dual SU( Z , ) self-duality as well. ) duality [ Z Z , , ) are exchanged under the k = 4 SO(2 ]. While this is not obviously related N 44 – 1 theories. Recently, there has been a lot ) acting on the worldvolume gauge 37 > = 4 gauge theories, they can exhibit Z D3 branes in smooth background is actually , N N N ) differs from its SL(2 N + 1) and USp(2 k 5-branes [ ] that S-duality nonetheless acts simply on the M – 2 – 47 ]). These global subtleties do not affect the class of checks we = 2 theories with a similar SL(2 = 4 case discussed above, we argue that the collapsed 45 = 1 analog, we will consider the worldvolume gauge = 1 analogs of Montonen-Olive duality. N N N N ], and thus is not self-dual. However, this is fully consistent with the 31 ]) and of certain 35 , 34 = 1 variants of Montonen-Olive duality provide an interesting counterpoint N = 1 dualities coming from wrapped N ) gauge theory (depending on the type of O3 plane). Whereas SO(2 k ) self-duality of D3 branes, as the gauge group on = 1 examples of Montonen-Olive duality known in the literature include mass deformations of Z While such a construction generally involves collapsed O7 planes, rather than O3 Fortunately, other types of Montonen-Olive duality are possible. By placing Since Montonen-Olive duality arises from SL(2 In this paper, we construct , ), which is self-dual (see for example [ Since they often decouple and/or acquire a mass, St¨uckelberg it is common to ignore the U(1) factors N factor coming from its center [ 3 2 N N Z SL(2 U( will perform, so wetheories. will freely Nevertheless, remove these the factorscase U(1) can we factors in expect when principle convenient, thatplay be while the an still detected important proper talking by role. inclusion about a of self-dual more detailed the analysis, U(1) and factors, in as that dictated by the brane construction, will our examples are chiral andof are work not on obvious deformationsto of our work, it would be very interestingin to search D-brane for gauge connections. groupscontext of when Montonen-Olive discussing duality, the since low the energy group SU( effective theory. This can be confusing in the planes, and the appearancesituation, of we argue fractional in a branes companionentire at paper system. small [ Analogously volume to the further complicates the theories (see e.g. [ duality due to theThus, S-duality in transformation order properties totheory of construct of the D3 an respective branesact probing nontrivially O3 on an planes the orientifold orientifolded [ plane, Calabi-Yau leading to singularity, dual where theories SL(2 with distinct gauge groups. are trivial and hence meaningless in the case ofbranes a atop self-duality. an O3USp(2 plane in flatself-dual under space, Montonen-Olive duality, one SO(2 obtains an place to look for analogoustheory dualities of with less D3 supersymmetryinvariant, branes is these in probing gauge the a worldvolume theoriesUnfortunately, gauge Calabi-Yau are there singularity. expected are to Sincechecks virtually exhibit we the no an perform geometry available in SL(2 is checks this SL(2 paper, of such this as conjecture. anomaly matching and The moduli class space of matching, to known examples of Seibergtheories duality, while via illuminating the the dynamics ofaforementioned duality. interesting features gauge of Moreover, Montonen-Oliveand our duality which examples are persist also with due less to serve supersymmetry. extended to supersymmetry, illustratetheory which of of D3 branes the in a flat background (with the possible addition of an all energy scales). theories are interesting forchirality, many confinement, reasons: and dynamical unlike supersymmetrynew breaking, class of among other things. Our one description is weakly coupled S-dual descriptions are necessarily strongly coupled (at JHEP10(2013)007 and − , we com- can lift D3 4 N we review the singularity. The A 1 gauge theories, and ], whereas very few dP 3 48 Z / 3 C ]). Furthermore, the gauge 49 singularity. We present several 3 , we review some basic facts about Z 2 / 3 C we review a useful mathematical tool for B – 3 – . . We also briefly discuss dualities which arise in the 7 ], though pitfalls abound due to the necessity of low N , we show that holomorphic combinations of couplings 52 C , we discuss further examples of the duality coming from ]. Some recent results hint that an AdS/CFT description , we show how the string coupling can be related to the 6 , we discuss the infrared physics of these gauge theories us- D 51 , 5 50 ]. ). We leave a detailed treatment of these dualities to a future work. Z 47 , geometries, some of which appear to have a different origin in string theory, 0 , 4 Y , we present the simplest example of a new duality, relating two possible gauge planes while emitting/absorbing a number of fractional branes during the process. We provide several appendices for the reader’s benefit. In appendix The outline of our paper is as follows. In section These gauge theories are also interesting from the point of view of moduli stabilization, There are numerous additional motivations for studying duality in this context. While 3 and + in this context to obtain gauge theories which are not approximately superconformal. 0 language of quiverfolds, athe generalization presence of of quiver orientifold gauge planes.anomaly theories In matching. which appendix arise In naturally appendix which in are invariant underRGE all invariant. possible spurious In and/or appendix anomalous global symmetries are corresponding gauge theories can beexhibit blown interesting down behavior to at recover the F low unrelated to SL(2 We present our conclusions in section pute the superconformal indicesmatch for the (up proposed to dual theproposed gauge limits duality. theories of In and section show ouring that computational Seiberg they resources), duality. a Indifferent section very ten-dimensional geometries, nontrivial with check particular of attention the to the Montonen-Olive duality which illustratetion how it istheories distinct for from D3 Seiberg branesnontrivial duality. consistency probing checks In and the sec- discuss orientifolded an example of the duality. In section Yau singularities correspond to rigid shrinkinghave divisors, played whereas an blown-up important versions role ofof in these type stabilizing IIB K¨ahlermoduli in string geometricof theory compactifications the [ dynamics mayN be possible [ models we analyze, webreaking, find a a free runaway magnetic superpotential, phase, confinement or with a chiral nontrivial symmetry superconformalas fixed the point. nonperturbative dynamics ofbrane moduli these and gauge potentially as K¨ahlermoduli theories well. for Indeed, sufficiently a low number of interesting Calabi- tively studied, orientifolded singularities havetematic received tools comparatively little for attention. theexamples Sys- construction have of been many studied examplestheories in we are any study available are detail highly [ (see nontrivialtensor for chiral matter, gauge example and theories [ with a tree-level nontrivialbranes, superpotentials, flow. a Depending range on of the interesting singularity IR and the behavior number arises. of D3 In particular in the limited sample of O7 Understanding such a transitionfurther is details one to important [ motivation for our work,worldvolume but gauge we theories defer on D3 branes probing (toric) singularities have been exhaus- O7 planes undergo an “orientifold transition” at strong coupling, exchanging O7 JHEP10(2013)007 (2.2) (2.1) -invariant j (enforcing the n at weak coupling, + τ πiτ 2 − → e τ , we relate the matching of G ): Z , , = 4 theory has not just one but an theories to a group theoretic conjecture πi 2 YM . N 4 3 g b d Z + / D3 branes probing a smooth background is + + 3 C N π cτ aτ YM 2 – 4 – θ = 0 = τ , we relate the matching of certain baryonic directions YM τ F ], we discuss the construction of these orientifold gauge we discuss some technical details of the computation of the 53 E , 47 , though the extended supersymmetry may be broken by irrelevant ]). φ − ) self-duality of type IIB string theory. 55 = 4 gauge theory to strong coupling, we encounter an infinite number , ie Z , + N 54 0 of strongly-coupled dual descriptions. Alternately phrased, by deforming C ) gauge theory with holomorphic gauge coupling equal to the type IIB axio- = 4 gauge theories are characterized by their gauge group and by their ) gauge theory. Such gauge theories are easily realized in string theory; for N = ] for definitions and basic properties) is approximately N N 56 10d τ = 4 U( To illustrate this intricate and fascinating behavior, we observe that Klein’s It is important to bear in mind that Montonen-Olive duality is not literally a “duality” Montonen-Olive duality relates such a gauge theory to a dual theory at a different Rigid In companion papers [ ) (see [ N τ ( infinite number a weakly-coupled of phases with a weakly-coupled dual description. j periodicity of the thetadescriptions angle), cannot both it be iscase weakly straightforward where coupled. to the The check gauge stringbranes, that realization is theory the the of arises original SL(2 this as and duality, in the dual the world-volume gauge(a theory word on whose a root stack is of “two”): D3 a weakly coupled coupling under the action of the modular group, SL(2 In particular, unless the modular transformation is of the form an dilaton, operators in a smoothly curved(see background for or example by relevant [ operators in the presence of flux holomorphic gauge coupling, which takes the form for an SU( instance, the world-volume gauge theory on 2 Review of Montonen-OliveIn duality this section, we review certainfor aspects of our Montonen-Olive duality paper. which will be important theories using exceptional collectionsstring as theoretic well arguments as that details theS-duality. of dual We their gauge also gravity theories discuss duals, arethe the focusing connected duality, and nature on by construct of ten-dimensional infinite the families orientifold of transition geometries which which seems exhibit to similar dualities. govern Coulomb branch. In appendix superconformal index. In appendix in the moduli space of theand prospectively provide dual evidence for thisthe conjecture. superconformal Finally, indices in appendix to a conjectural identity for elliptic hypergeometric integrals. gauge and superpotential couplings of a D-brane gauge theory by moving out along the JHEP10(2013)007 . ) + on Z = 0 , | ) τ τ ). We ( ) gauge j ) S-dual Z is confor- k | , and are , /τ π H 1 2 — is an O3 θ/ . The infinite ). 4 plane is SL(2 → − 1 Z , , where − τ , serve to divide the H 3 12 < | ) τ ( j | (weak coupling) lies at the top edge = +1 respectively, and the Im = 0 (strong coupling) at the bottom, τ ∞ τ i 0 for rational values of = + 1 and τ → − τ = 2, a fundamental domain under the identification / τ 1 – 5 – plus a single (pinned) D3 brane < , so that − | is equivalent to inversion through the center of the disk, τ iτ τ plotted across the upper half plane + i 1+ | /τ ) 1 Re ), colored blue/purple. The superimposed curved grey lines | τ = ( j | w axis. → − τ + 1) gauge group, however, the dual descriptions have gauge τ | → ∞ ) k τ + 1) gauge group. ( j k | is a modular-invariant definition of weak coupling. A plot of ) gauge group, all the dual descriptions have the same perturbative gauge (intermediate coupling) in the center and ) leads to a fractal structure, as seen in the figure. Thus, the behavior of N i Z ). In string theory, this corresponds to the fact that the ( , k = | → ∞ ]. For an SO(2 — another name for an O3 ) τ τ 46 − ( . The modular invariant j | f O3 + 1. The transformation τ For an SU( The single additional D3 brane is “pinned” to the orientifold plane because it is its own orientifold 4 → group USp(2 of the image. A pinnedplane, brane giving arises rise whenever to an an odd SO(2 number of (upstairs) branes is placed atop an orientifold group. Thisnow is consider a Montonen-Olive duality consequence forgroup, SO of the and duals USp the likewise have gauge the invariance groups. sameinvariant gauge [ of For group; an equivalently, the the SO(2 O3 D3 brane under SL(2 order of SL(2 these theories at strong couplingin is the very strong rich, coupling with limit,coupled many depending dual dual on weakly descriptions the coupled become exacttherefore descriptions value free dense of along as the the Im theta Re angle. The weakly τ and the shaded triangular region is the canonical fundamental domain forso SL(2 that the upper half plane (conformally mapped to a disk) is shown in figure of the disk, whereas the left andaxis right edges spans correspond the to perimeter.plane into an The infinite black numbercoupled of regions, dual disjoint corresponding colored description regions, to ( illustrate each containing the a boundary subregion of with the a region weakly Figure 1 mally mapped to the unit disk via JHEP10(2013)007 − is = 1 and − f O3 (2.3) + N f O3 wherein 5 , where the + . The f /τ O3 is a perturbative 1 , leaving the + as T + f = 1, it cannot be true O3 → − exchanges the O3 3 τ are related by S-duality, ) . : S and − ]. We do not mean to imply that ST S + f 57 O3 ) gauge groups to each other. D3’s k k and image of O3 + 2 + + 1 and + ST τ O3 ]. We summarize the resulting structure → 46 τ ) gauge group, and is perturbatively equiv- −→ )[ k : . However, since ( exchanges the O3 Z + 1) and USp(2 + – 6 – , T T k ], and play an important role in the new dualities to O3 47 + 1) D3’s ) on the triplet is as follows: − . We denote the Z k , f − O3 f ) multiplet. However, the multiplet is as yet incomplete. To ) generators O3 + (2 Z Z − ]. It is possible to rephrase this by saying that the two different , , maps 58 O3 invariant, whereas cyclically permutes the three O3 planes. Since ST back to + + ST f also gives rise to an USp(2 O3 , the two configurations being distinguished non-perturbatively by their + + f O3 = 4 gauge theories with SO(2 . maps O3 . Summary of the four gauge theories that arise from placing D3 branes on top of the N 2 ST , leaving the The complete action of SL(2 The upshot of the previous paragraph is that Montonen-Olive duality relates strongly The term “orientifold transition” was used in a different context in [ − 5 -invariant; therefore f duality, the alent to the O3 spectrum of BPS states [ our physical mechanism is the same, just that we also have a change in the orientifold type. three O3 planes formin a figure triplet under SL(2 O3 invariant, so that This is not thethey whole story, must however. form Because some the SL(2 see O3 this, consider theT SL(2 that pled orientifold plane. Examplessupersymmetry of this are phenomenon considered with collapseddiscussed in O7 in [ planes this and paper. coupled Thus, S-duality maps This is a wellstrongly coupled known orientifold example planes of recombine with what branes we to will form call a different, an weakly “orientifold cou- transition”, Figure 2 four different O3 planes. JHEP10(2013)007 or (2), 6 = 4 0 R ) gauge can be N k are odd, + d , where red, + 1) is the ). f τ O3 1 k and and a + . As can be seen in + 1) and USp(2 3 k = 1 dualities. In the is even (hence = 4 SO(2 c N N + 1 exchanges the two O3 plane types. τ , respectively, and the latter two possibilities D3 branes atop an O3 + → for which . The different colors indicate the type of the O3 k f 2 in the dual theory. Thus, the red regions have a O3 τ
1 / – 7 – 1 or c d a b ≤ + τ ) it is the conjugate subgroup consisting of elements k , O3 Re − + 1) theory. Note that the region where each dual theory is < k f O3 2 / 1 − ) of elements + 1) description, whereas the blue/purple striped regions have a dual 3 Z k , patterned after figure , Z τ ) description. The thin grey lines outline a fundamental region for Γ / k 3 C SL(2 transverse space, leading to gauge theories with maximal supersymmetry, ⊂ 3 = 1), whereas for USp(2 C (2) is even. 0 bc b . A schematic illustration of the phase structure of − + 2 leaves the O3 plane type invariant, and defines the periodicity of the theta angle ad planes give rise to the same gauge theory at different theta angles. In particular, τ To illustrate the nature of these dualities, we show how the weakly coupled description + → 3 Duality for In this section,examples we discussed examine above, the the sixequivalently simplest directions transverse of to our the D3 new branes form a flat relate the different theories.subgroup Γ For example,since the self-duality groupfor for which SO(2 Thus, the gauge theoriesidentified with corresponding each to other 2 upon shifting the theta anglechanges by as a a half period. functionthe of figure, the each holomorphic gauge gauge theory coupling has in additional figure self-dualities as well as the dualities which perturbative is most likely smaller than the colored region indicated hereO3 (see figure τ in the corresponding gauge theory, whereas plane (and hence the gauge group)blue in and the purple dual correspond weaklyare coupled to distinguished phase an by for requiring each valuedual of weakly coupled SO(2 weakly coupled USp(2 the self-duality group for the SO(2 Figure 3 theories as a function of JHEP10(2013)007 . 4 = 1 (3.1) . To 6 N R . As the orbifold, 2 3 P . Counting i Z z / we study the 3 C → − i A.2.2 U(1), with the U(1) z × ]. . 3 47 plane with and without a Z / − 3 = 4 orientifolds considered in C N 2 ) gauge theories discussed above. k SU(N) exceptional divisor. Placing D3 branes . i 2 quiver gauge theory shown in figure z P 3 3 1 ) πi/ N 2 ) and SO(2 plane or to an O7 e k – 8 – orbifold of the SU(N) + → 3 = 1 gauge theory. i Z z 3 N = 1 SU( N SU(N) . The quiver gauge theory for + 1), USp(2 k ], where we will discuss many more examples. orbifold theory for the orientifold involution 3 Figure 4 53 , Z / SO(6) R-symmetry is just the rotational isometry group of + 1) as two separate cases, we find that there are three possible gauge which corresponds to an O7 plane wrapping the shrunken = 4 SO(2 ) dual descriptions of gauge theories arising from D3 branes at sin- 47 3 i we discuss in general how to “orientifold” a quiver gauge theory and Z C N z ∼ = , N A that is wrapped by an O7 = 1 theory at low energies, we must either switch on flux or introduce 2 → − i P N z ) and SO(2 N We will argue that two of these gauge theories are dual, whereas the third is self-dual, In appendix We consider an orientifold of this configuration, since, as we argued in the previous A simple and well-known example of such a transverse space is the analogous to the The dualities studied heredualities between are orientifold merely gauge thepaper theories, as simplest some well of examples as which of in we [ a will study large in class detail of in this apply this procedure to twoorientifolds explicit of examples. the InSO(2 particular in section theories arising on D3a branes shrunken probing thispinned orientifolded D3 singularity. brane respectively. They correspond to resulting configuration is essentiallythe a previous section, we willIn argue this that paper the we strongdetailed focus coupling discussion behavior of on is the the closely analogy characteristics analogous. between of the gravity the duals resulting to gauge [ theories, deferring a appearing as an R-symmetry in the section, the SL(2 gularities all have theinvolution same gauge group and matter content. We choose the orientifold The singularity can be resolvedat by blowing the up singularity a The leads orbifold to reduces the the isometry of the transverse space to SU(3) obtain an singularities. We choose to do the latter. where the orbifold action on the transverse complex coordinates is where the SU(4) JHEP10(2013)007 ˜ N − Z N (3.4) (3.3) (3.2) or planes, N . + Z symmetry. ) family of B 3 N ) factor in the Z k ] and applied to is the number of and O7 SU( k − × and related abelian 49 , ) decouples from the 2 ˜ ˜ 3 N N 2 3 4) , k N N 3 − 3 3 48 Z − 3 3 Z k Z / ω ω − ω ω ˜ 3 B j C N ) where ˜ A ˜ ˜ 2 4 R N N k i 2 4 R N N ˜ A − + U( + − 2 3 2 3 Tr 2 3 2 3 × ) = 4 gauge theory corresponding to ijk k discrete symmetry times a constant gauge  2 even though the cube of the generator N ˜ 2 λ k 3 Z − Z = SU(3) U(1) U(1) isometry of the transverse space. A SU(3) U(1) symmetry upon composing the generator N ˜ W 3 × ]. Orientifolds of ) 7 ) Z SU( ˜ N , N 63 – 9 – , × k ) 52 B k j 2 ) gauge theories for collapsed O7 A ˜ i N − ) whose cube is the inverse of the cube of the generator. A 4) SU( 4 ˜ ]. For the involution discussed above, one finds SO( N + 4) SU( − SU( Tr − ) reproduces the original gauge theory at a different rank 68 ˜ 1 1 N k × – N ijk N ) gauge theories have a classical moduli space which includes 2  are superpotential couplings. 64 2 λ N ), so we obtain a ˜ − λ th power of the generator lies within the gauge group. SO( ˜ k ) or SU( USp( + 4) N = N ˜ N SU( i is even, and the tree-level superpotentials are i N i and i W ˜ A B × ˜ ˜ A B 6 SU( λ N , 4) × orientifold we discuss here was, to the best of our knowledge, first studied ) or SU( ) − symmetry is equivalent to the discrete symmetry indicated in the charge k 3 πi/n N . Thus, we see that the moduli space of the SO( 3 2 2 N Z e k Z / 2 ) and USp( − 3 ≡ ] and recently revisited from the dimer point of view in [ 4 C − N n 62 − ω N – SU( The SO( Note that we label the discrete symmetry as a The N = These two gauge theoriesIn are related general, by a a discrete negative symmetry rank duality can as be explained rewritten in as appendix a 59 6 7 × D3 branes probing a smooth region of the Calabi-Yau cone. Meanwhile, the remaining 0 SO( N transformation whenever the directions corresponding to moving D3is branes then away from Higgsed the down singularity.(downstairs) to The D3 SO( gauge branes group removed from theHiggsed O-plane, gauge corresponding group. to the Afterrest U( integrating of out the massive gauge matter, groupk the in U( the IR, giving a separate center of SU( with an element of SU( This latter table up to a constant gauge transformation. respectively, where is not the identity. This is because the cube of the generator lies within the where respectively. Both theoriesSU(3) have symmetry, corresponding a to non-anomalouscareful the R-symmetry SU(3) analysis in reveals addition thatThe two both to models models a are also global have a discrete “baryonic” in [ the problem of moduliorbifolds stabilization have in also proven [ to bedynamics an interesting in testing string ground theory for studying4) [ non-perturbative JHEP10(2013)007 ) ). ]) is is ˜ n N 3 3 69 , we , we Z Z + 1), 4 SU( k 3.2 theories. × , whereas ) N , we discuss ,N respectively, 5 + 4) ), SO(2 N ˜ gcf(6 k N Z plane. Regardless ) theories for even , leading to a more × − N R = 0 mod 3 the 3.1 N SU( ) gauge group. example of the proposed U(1) ˜ N × N × 4) 8 − ) or SU( N N . Since the global symmetry groups ) ˜ not divisible by three the baryonic N , + 4) to exist. ˜ N ˜ N gcf(3 “color conjugation” symmetry (see e.g. [ or ) monodromy. Z 2 Z Z , N × – 10 – R singularity, all corresponding to the same geometric plane rather than a compact O7 3 + Z U(1) / ) theories are interconnected by an analogous motion in 3 we discuss a specific, finite × ˜ C N , which comes from the outer automorphic group of SO(2 = 4 case, and we therefore hypothesize that one of the SO 3.3 N SU( N ) self-duality, whereas the other SO family and the USp family × ) duality. In the remainder of this section and in section Z Z must be even for USp( , , ˜ N + 4) center of SU(3) composed with a constant gauge transformation, and ˜ 3 N theories are connected by the above process, as are all odd Z ] that the geometric duals are distinguished from each other by different (S-dual) choices N 47 , are distinguished from each other by the presence or absence of a pinned D3 Moreover, there is an additional N 9 ) appearing in the k , we provide further strong evidence for the proposed duality by comparing the su- We continue our discussion of these gauge theories in the following sections. In sec- We begin by discussing ’t Hooft anomaly matching in section The situation is closely analogous to the three gauge theories SO(2 In all, we have obtained three distinct families of gauge theories corresponding to D3 4 We argue in [ In this case the center of SU(3) lies within center of the SU( 8 9 for the USp theoriesmust it match between is dual SU(3) descriptions, this suggests that the ranksof of discrete the torsion. dual pair must equivalent to the therefore lies within the continuousdistinct. symmetry group, whereas for for the SO theory withIn even net, the global symmetry group for the SO theories is SU(3) their infrared physics using Seiberg duality. 3.1 Classic checks ofAs the a duality precursor to anomalyglobal matching, symmetry we groups. note In that particular, the for dual theories should have the same of the duality, andduality. in section tion perconformal indices between the prospectively dual theories, and in section various computable infrared observables, and explore some of itsprecise properties. statement of the proposedby duality. We a then partial provide further matching evidencehighlight between for an the the important duality limitation moduli of spaces our of methods which the is two nonetheless theories. linked to In the nature section USp(2 families enjoys an SL(2 are related by anpresent SL(2 strong field theoretic evidence for the latter duality, based on the matching of orientifold involution. Twoand of odd these theories, thebrane and SO( its corresponding half-integraltheory D3 corresponds brane to charge, a while compact theof O7 USp( the O-plane type, thethe seven-brane two configurations tadpole have is the cancelled same locally SL(2 by (anti-)D7 branes, and where all even Similarly, all USp( moduli space, where branes probing the orientifolded gauge theories falls into two disconnected components for even and odd JHEP10(2013)007 ] = N 47 k (3.6) (3.7) (3.5) Z 2 G 6 33 − nonabelian, as ]. In general, − G 71 + 3) 9 + 3) We see that the 1 1 ˜ + 3) N − ˜ ) theory: is necessarily even, ( ) theory for odd N ˜ ], and do not appear as ( ˜ 10 ˜ ) Levi-Civita symbol N N th Weyl fermion under . ˜ N N ˜ ( i N 69 N 3 2 ], it is shown that this N ˜ 1 1 2 N 47 SU( 4 3 SU( − × × , the Jacobian for the symmetry R η 4) 3 + 4) 3 Z − ) theory takes the form 3 R R 2 ˜ U(1) 3 N discrete anomalies for N N , 2 Z k B +3, in agreement with the restriction /N Z U(1) U(1) SU(3) ) N 2 ˜ SU( N SU(3) B B k , × is self-dual. SU(3) N A = 2 is the instanton number for the background in N 4) N + 3 N A − n − ˜ N A = = 1. and (grav) – 11 – N N η = B k is the multiplicative charge of ( 6 orientifold gauge theories. In our notation, Z ]. i ,N 33 2 N η ≡ − A 3 , where G 70 n − N N Z , A η / N O 3 3) 69 Q N 3) 9 3) theory. It will also follow from the arguments of [ C 1 1 − 3) − − , where − ) theory: USp( 2 i − N η ) theory for even ( N N N i ( 3 2 N N 1 2 ( Q 4 3 SU( − SU( = to be determined. SU( × k × k discrete symmetry. For a discrete anomaly Z × R k 2 4) 3 Z 4) 3 Z − 3 R R + 1) 2 U(1) − 3 N 2 N Z N U(1) U(1) SU(3) SO( SU(3) and (grav) SU(3) ) . The anomalies for the i r ( T We now consider the moduli spaces of the prospectively dual theories, which is clas- The global anomalies for the two models are shown in table 2 i Here and in future we only write the η i 10 the remaining discrete “anomalies”anomalies need in not the match path between integral two measure dual [ theories [ for some particularclassified choice as of “mesons” contraction or “baryons”, ofis depending irreducibly the involved on in the whether gauge contraction the of indices.charge gauge SU( indices or not, Such i.e. operators on whether the may baryonic be sically equivalent to theoperators affine identified variety under parameterized the by F-terma the conditions holomorphic holomorphic and gauge gauge classical invariant of constraints invariant the [ SO( from matching the global symmetryrank groups relation discussed agrees with above. D3this In charge is [ conservation, evidence as for itand a must. the possible Since USp( duality betweenthat the the SO( SO( for some odd integer anomalies match between the two theories for the generator of the transformation in the pathquestion; integral therefore, is the anomaly vanishes iff be related as follows: Table 1 Q JHEP10(2013)007 , , . ) 3 as 6) 3) Z N kN − − 3 3 11 (3.8) (3.9) ˜ B N N (3.12) (3.10) (3.11) SU( 4( 4( = ) ⊂ B ( k N and and O Z 3) 6) − − 3 3 ˜ . . N N and 5. Thus, baryonic ˜ N 2( 2( N 2 2 ˜ kN B A k , k , A N > 6) ) theory can be similarly ≡ and and ˜ + 6) − ) N 3 3 A ˜ N ˜ N ( k N N ˜ A B O SU( 2( 2( 0, which can be “factored” -charges are -charges are = 1 with × R R > , A , ) = ) = 2 A 2 N ˜ N n n + 4) -charges Q k kN ) ) ˜ ˜ R B B N , ( ( is anomalous, with an anomaly-free A R R AAB A AAB Q 3 ( ( ω 1 1 n n ) ) = – 12 – N N k , Q k , Q 3 ) which appears. A B Q 3) , only mesonic operators are possible. , only mesonic operators are possible. ) theory: 0 0 3 3 ˜ − + 3) N = ( = ( ω ω 3 3 AAB ˜ N N O . “Reducible” baryons are similar, but with an additional = = O , the minimum possible , the minimum possible k 3 SU( 3 3 2( 2( 1 1 ω − − 3 3 Q Q × . ω ω 2 ) = ) = -charge exists. In particular, this occurs in the following cases ) theory: n ˜ = = N R + 4) k kN N 3 3 charge ˜ A A ˜ Q Q 3) and 6) and N 3 ( ( and , as before. 0, which can be “factored” as R R Z k SU( 3 1 − − ω < n Q Q ˜ × and and N N A 3 3 4) gauge invariants exist for the case 4) 2( 2( Q ω ω − charge of a gauge invariant operator is necessary integral, since − < < -charges = = charge A R N R R 3 3 N 3 Q Q Q Q Q Z respectively, corresponding to the irreducible baryons respectively, corresponding to the irreducible baryons For For For For In general, mesons and reducible and irreducible baryons can all be intermixed in the The holomorphic gauge invariants of the USp( We will focus on the “irreducible” baryons, of the form No SO( We do not mean to imply that the gauge index contractions factorize in the indicated manner. 1. 2. 1. 2. 11 -charge of +2 for every factor of ( Similarly, for the USp( duality relations between theoperator with two the theories. correct for However, the in SO( certain cases only one type of classified, where now the irreducible baryons have with the and in both cases the R for integral powers These have operators can be further subdivided into those with and those with subgroup that was identified above: where the lies within the gauge group. is vanishing or not. The corresponding U(1) JHEP10(2013)007 ˜ ˜ ˜ N N N < / ˜ ˜ A A 4 also ’s in R − ˜ = 21. 3 (3.16) (3.15) (3.13) (3.14) A Q N / 4 B ) ) theory, N ˜ N , the SU(3) , ˜ N ˜ SU( A -charge in both × R , n + 4) k ˜ A directly. Using the more 3, whereas checking that N n j , − B } -charge, i.e. transforms as ( LiE , is totally symmetric in its N T F-term condition implies that n R 2 ) , ˜ n S = ˜ A ˜ i N B ⊗ 2 ˜ A takes the form of a “Pfaffian” of = 2 N ˜ , ( A 2 ... R 3) 2 , and while we do not have a general ... ˜ S N/ Q . The 1 ) {z − ˜ N ←→ 2 k ⊗ N/ 2 3 ˜ ˜ A ˜ A N N N/ 1 2 ) j 2( 2 S we have verified agreement up to B ˜ , with = ( + 3 that we saw from the anomaly matching A ⊗ n + 4) invariant T = ) ˜ ˜ N ,B – 13 – O 1 ˜ N 11 and 13 and i E.1 ˜ N | 6) ˜ , A N operators of minimum possible A 9 = ˜ ≡ − , A 1 3 7 ) − 3 N , Tr ( N ω ( 2( ←→ ≡ 2 = 5 = n . Thus, N ˜ k N/ R 3 N n B j Q n Sym ...i U(1) ) for 1 ], one can show that the gauge invariant component of k and × 1 is too computationally expensive using j 72 ( 1 [ 3 we show how agreement between the SU(3) representations of 3) between the two theories. Such operators can be factored into i n 2 ω N O F ˜ − N/ = LiE SU(3) ˜ N 3 × Q ) , the F-term conditions impose no additional constraints. Using the computer ˜ N denotes the symmetric tensor product. N follows from a certain conjectural mathematical identity involving representations S B 6) = 2( ⊗ N − B As a further check, we should be able to match the mesonic operators with For The SU(3) representations of these operators should also match. For SU(3) indices. A similar argument goes through for the USp( , which is symmetric in its factors. The nonvanishing gauge-invariant component of N 2 n ˜ resulting in the same spectrum of single-trace3.2 operators. Limitations fromBefore the turning perturbativity to of specific the examplestions string of to coupling the having duality, we a briefly perturbative review some description general of obstruc- the D-brane gauge theories obtained from products of single-trace operators of the form: where the F-term conditions3 imply that proof, in appendix and of the symmetric group, and we provide additional evidence2( for this identity. this holds for larger efficient approach explained inIt appendix would be desirable to have an argument for all algebra package transforms in Sym where representation can be determined aspairs, follows: and the the operator symplectic therefore invariantthe contracts factors nonvanishing the as component ( of the USp( which reproduces the rankconditions. relation between the theories. In particular, these operators must have the same This suggests the matching: therefore transforms in the SU(3) representation under SU( A JHEP10(2013)007 or or , i SU SO (3.19) (3.17) (3.18) (3.20) ) Λ ˜ Λ N ]: / )) for the 9  73 − SU( SU Λ SO 3.17 1) somewhere +4) ˜ N  is taken to be the ˜ λ g 9 USp( , ) = 1 for SO gauge groups. Λ ! Adj denotes the adjoint ( 1 or ) i T +2) r ˜ 13 N ( ,  r 3( T λ ˜ λ h i X ln , . − 1 9 πi − = 9 . The result is also intuitive: a pertur- = = D (Adj) SU 10d T SU b 3 ˜ , b

, τ , 2 – 14 – 18 i 3 π ) − g N 16 = 9 = − 18 SU( for SU and USp gauge groups, and USp SO Λ = b ˜ b 1 2 4) µ − ) = N ln dg ( d 18 T − SO( ≡ Λ ) 2) g − ( β N The nature of these obstructions will also serve to illustrate how our 6( λ h 12 ln , along with a small superpotential coupling ( πi 1 ) denotes the Dynkin index for the representation 2 , this can be very difficult in practice, and we will not attempt to do so. USp r ˜ ( Λ SU = T ˜ Λ /  Conversely, the duality we propose partly addresses the question of what happens to The one-loop beta function coefficients (the term within parentheses in ( The one-loop beta function for a supersymmetric gauge theory is given by [ 10d As we shall see, all of the D-brane gaugeWe theories employ considered the above conventions are strongly coupled in the infrared, τ 12 13 USp SU ˜ ˜ a Coulomb branch computation ofbative it gauge in theory appendix necessarily corresponds to a weak string coupling. so the latter requirement is too strong. see why, note that the string coupling is given by for the prospective dual theories,brackets. up This to result a can multiplicative numerical be factor established within in the a square variety of ways; for completeness, we present between these two scales.to incorporate We corrections willΛ work which in are this subleading limit. in an While expansion itthe in gauge is theory small possible in in Λ the principle limit where the dynamical scales have an inverted hierarchy. To Since the beta functions fortheory the is two either gauge IR group free factors orat have asymptotically most opposite free, in and signs, a the neither finite perturbative gauge scale description range is will of be only energy valid scales. possible ifΛ More there precisely, a is perturbative a description separation at between any the dynamical scales, Λ whereas for those of the USp theory they are result for the physicalsuperfields. gauge coupling depends on the anomalous dimensions ofgauge the group chiral factors of the SO theory are: where representation, and the sumholomorphic is gauge taken coupling, over then all this chiral result superfields. is exact, If whereas the corresponding exact some energy scale at whichin the the gauge theory infrared. isproposed weakly coupled, duality rather differs than from weak Seiberg coupling duality. quantization of open strings. For a “perturbative description”, we require that there exists JHEP10(2013)007 , SU ˜ Λ / (3.21) (3.22) USp ˜ Λ SU(2) gauge , R 3 or 3 / × k / ; 1 8 SO providing further − mn Λ , mapping any per- ˜ ) with the algebraic / 15 B n ; 10d SU j b τ 3.20 ˜ A SU(3) U(1) m . ; i a ˜ A = 0 ijk  mn 14 ab . ]; k Ω duality SU ˜ ˜ λ 1 B ˜ Λ 1 2 n ; . j USp(6) SU(2) [ b  = ˜ 5 A SU(2) i i limit. ˜ ˜ 0 to have gauge groups of non-negative rank, W A B × USp , N ˜ Λ ≥ – 15 – via the AdS/CFT correspondence, although this definition ˜ = 0 N or N n ] ; j b USp(6) SO ←→ ˜ A Λ 4 and m ; , i  ←→ [ a ≥ R ˜ theory k 15 A 15 ; SU / / N ab 2 mn Ω 16 − SU(5) B SU(2) j n A × i m SU(3) U(1) A ijk USp(6) λ  1 2 SU(5) = i i A B W We discuss higher rank examples in section With these considerations in mind, we turn to a specific example of the proposed By contrast, Seiberg duality is generally used to understand the infrared behavior The duality we propose acts as a modular transformation on While the Lagrangian definition of the gauge theory may be insufficient in thisRepeated case, application in of principle Seiberg string duality to these gauge theories requires a seemingly never-ending chain 15 14 theory provides a complete definitionis for impractical any for computations except in the large of deconfinements, leading tofactors more eventually and reconfining moreexplorations after gauge of group several the factors. steps, matter. we While have one not can imagine found some this of to these be the case in our limited 3.3.1 The The F-term conditions are where Ω denotes theboth symplectic theories, invariant. and show We that characterize both the generate classical a moduli runaway superpotential. space of 3.3 Case study:Since the we are constrained to the lowest rank duality wetheories: expect to find is between the SU(5) and USp(6) if we were able to do so,relations we which would have result to from somehow reconcile applyingrelationships the modular complicated between gauge transformations dynamical coupling to scales ( predicted by Seiberg duality. duality. of a gaugenatures theory of which these is(together two perturbative with types at deconfinement) of some tosuch duality. scale, an the exercise While individual an never gauge wecircumstantial illustration reproduces evidence group can of the that the factors, prospective repeatedly the duality dual is in apply different not gauge our a Seiberg theory, Seiberg experience duality duality in the usual sense. Indeed, turbative string coupling to acoupling nonperturbative and one. applying Conversely, deforming the topling. duality, strong we string Thus, obtain since athe the duality dual string provides description at coupling least with iswith partial a linked information an weak about inverted to string the hierarchy the behavior Λ cou- hierarchy of these Λ gauge theories JHEP10(2013)007 , ij ij B . A ij (3.28) (3.27) (3.25) (3.26) (3.24) (3.23) B ), which , a larger SU(2) for det ij , 1 2 × rank two or U(1) for A SU(2) SU(2) breaks , ns = ; Λ ij l . In particular, → × rs 2 B ˜ ; ij B k  B B ˜ B mq ; k pq ]. ; ˜ j and B 65 ). For USp(6) ˜ n B ; . USp(6) ij B j b j ˜ A v mn A i ; SU(3)). i m ; ˜ θ v B × i a 2, where this branch intersects ˜ rank one and SU(2) A sm , π/ sin  qs 6= 0. For generic (full-rank) ij n  6= 0 (and , qr ; iφ = j b  A . The remaining holomorphic gauge ab ij − n , ˜ (SU(2) k ij θ e A R np A . ˜ m  B m A × j ; = 9 Sp ˆ i a B ijk = 2Ω Λ ˜ U(1) ij = 0  A det l 1 6 ijk B rs ] ; ×  = 0 or k l mn = 1 2 SU(2) for  = ˜ B θ B – 16 – j [ ab and B × i = pq ; ADS j A k v , ˜ i W i = Ω mn B ˜ n B pq ij ; θ v ; j b 16 , respectively, and obey the constraint j A ˜ 8 A ˜ B under SU(3) 1 cos m ; 3 iφ mn i a / ; e ˜ 6 matrix over USp(6) 2 i A and = 0, parameterized by − ˜ = 0, parameterized by B × qs = 3  / ij ij nq ij  pr 16 B A  A mp  U(1), except for when mn  = × ab SU(2) breaks to a diagonal SU(2), whereas for rank-deficient ij × in the form: B = Ω ˜ B rank two. kl is viewed as a 6 B ij ˜ A ij to USp(4) branches 1 and 2, respectively. three, the SU(2) gaugerank factor one. is completely broken, whereas SU(2) USp(6) gauge symmetry remains: USp(4) A A A branch with A branch with A branch with We now consider the effect of the ADS superpotential on the moduli space. It is helpful We now discuss quantum corrections to this picture. The USp(6) gauge factor is The second F-term condition implies a constraint relating The same superpotential was obtained via a direct string computation in [ 2. 3. 1. 16 to rewrite generates an ADS superpotential: where asymptotically free, whereas thedynamics SU(2) are primary gauge governed factor by is USp(6) infrared (to free. leading order Thus, in the Λ infrared Thus, the classical moduli space has three distinct branches: where we apply theF-term condition first then F-term implies condition the constraint to simplify the right-hand side. The second which transform as invariants are easily cataloged: built from which transforms as a The first condition implies that all nonvanishing USp(6) holomorphic gauge invariants are JHEP10(2013)007 (3.34) (3.35) (3.36) (3.30) (3.33) (3.29) (3.31) (3.32) 17 . While other ij A F-term condition ˜ A , ˜ A 9 Sp Λ det . + , SU(3) flavor symmetry, which ij MN × a M . . M  , ). Therefore, supersymmetry is  . Finally, the 1 n 1 1 j ij ij − − − ij = 0 m  ˜ i A 3.29 M A ˜ SU(2) M ˆ ab A , B  . ]; Ω mn ⊃ k ˜  mn F-term condition is then mn T A A   B ˜ = 0 9 Sp 9 Sp − A j ˆ 1 2 ˜ [ a B Λ Λ , the classical superpotential dominates, and n A ˜ λ | det det = k 9 Sp  + m n Sp Λ  j ∼ j b = det – 17 – Λ ˆ ˜ ,A B A ˜ 9 − A MN eff / m 9 Sp 1 ˆ i a MN mn B = W ˜ − = 0  Λ ˆ A B det N b ˜ ] λ n ˜ j b j ab N b A − m ˜ A Ω i A i M a [ a = ˆ ˜ λ ˜ B ab |  | A A ˜ Ω A ab | ˜ λ MN Ω = 0. We obtain a semiclassical “moduli-space” parameterized ˆ B ˜ λ ˜ B 1 2 under a (fictitious) SU(6) = theory W ), we obtain SU(5) index the (fictitious) SU(6). The 3.31 : transforms as is related to the holomorphic gauge invariant ij ˆ M,N , subject to a runaway scalar potential generated by the ADS superpotential: B ij ij M In particular, for generic In particular, branch 1 of the classical moduli space is approximately flat for large det A M 17 approximately flat regions correspondingsemiclassical, in to that the the otherfrom classical branches the superpotential ADS of must superpotential. moduli be made space to may cancel exist, the they large are vacuum energy not arising 3.3.2 The The F-term conditions are: broken. the classical F-terms set by Applying ( However, this is incompatible with the constraint ( or so reads: where is broken by the constraint: as well as the (weak) gaugingfield of SU(2). We impose the constraint via a Lagrange multiplier where JHEP10(2013)007 = λ (3.37) (3.38) (3.39) (3.41) (3.40) (3.42) = 0. We = 0. This i 6 i i a A . A h , h n ; = 0 exists with all m 2 ; e i 2 ef d Φ h , B B q ;  m ; p cd , we find that this also ; 1 R 3 n 15 e i k 15 3 / B 1 / / / 2 p d B U 2 ; h n − ; B m ab − . j r ; B , 2 U c i a 3 16 5 2 b ]: ˆ and 2 1 5 m a p b ; ;ˆ A − − i i bcdef i b  17 B , U A q ; h ef = 1 ; bcdef 16 c l ijk  1 b ˆ 1 ,  b a Adj 0 2 B ;ˆ a 1 3 i B npq cd u  p ; i ; − k = 0, another solution with v 2 1 a 12 i 6 B 1 1 pq 1 ,B = a Φ i ab h V ≡ SU(3) SU(3) U(1) U(1) i ; – 18 – v B n i a j q ; ˆ a 2 i n m b e A ; B i 18 2 h U i a d = 2 m A i c i SU(5) 2 T 1). Thus, the second F-term constraint implies 5 b = , a 2 3 = 0 once the D-term conditions are imposed, which can ;ˆ 5 a i 0 B i  , ,U 2 mnp B 1 B 0 ijkl A AB e  h = 0 theory (without a superpotential) where , A h m = 1 without loss of generality. Suppose that 1 i i ; O =  0 d λ = 1 , A a B ab = 9 c ∗ 1 1 m B Λ mn u b n = 0 in all supersymmetric vacua for some field Φ, it is sufficient to show that ; m i i k b 1 a i V a T = (0 u A = U Φ  is the following: h j a a ], but such a gauge transformation will never regenerate a vev for Φ. u A pqr A W 71  4. ijk  1 160 1 2 ... ≡ ≡ = 1 m i mn b ˆ T V a, Since the F-term conditions are then identically satisfied, the classical moduli space Due to the first F-term constraint, the only possible nonvanishing holomorphic gauge In fact, to show that 18 is the subset oftheory that is of s-confining, the with the confined description [ However, applying the abovevanishes. gauge-fixed This forms suggests that for be verified by an explicit computation. where ˆ invariant involving where we may choose gauge fix such that We now characterize the classical moduli space. The first F-term constraint implies that Thus, the above spectrum of gauge invariants describes the classical moduli space offor the every solution toholomorphic gauge the invariants taking F-term the conditions samethe vevs. with unique This D-flat is solution because with thegauge the latter transformation same solution [ must holomorphic-gauge-invariant vevs be under equivalent to an extended complexified One feature of s-confining theories is that their classical and quantum moduli spaces match. with the dynamical superpotential: up to an overall multiplicative factor, where JHEP10(2013)007 R 4) − 6= 0 . N (3.44) (3.46) (3.43) (3.45) U(1) λ V 18 × Λ = 0, with 3 i . λ 4 det U  h can always be + ii n 1 ˆ = ij U . . One can show − i ij V ij R km V T . δ h z δ  1 − = 0 is the only solution − kl k kl = ˆ is full rank. We are then U V ˆ U im ij which can now be written i under the SU(3) ij l ij ˆ ijk U V ˆ 3 = U  19 V / , 2 kn 9 ˆ T − δ − Λ lj k − λ , we obtain ˆ = 0, we find U AAB 1 2 m i ik l nm k − , but could play a role for very large vevs. A ˆ ˆ U U h = 0 (3.47) ↔ ij ,  k SU , but preserves U(1) ) , i + ˆ n i Λ U 2 λ ii m ˆ ∼ diag U λT = mk + n = 0 . Note that in this case, n ( ˆ U ij = l V k  lj k  δ ˆ U kn m – 19 – z SU(3) λ − ˆ 2 U class ijk ) ,U . Thus, the classical moduli space of the  i j W → + jm 1 and vevs V l δ nm ˆ (  k U  V ˆ il = 0 is the only solution for 18 U n s jm l ˆ Λ g U ˆ U(1) U ˆ 2 U 1 3 , which transforms as a after a complexified SU(3) transformation. As the F-term λ il n 4 det ij = × ˆ + ij U V ˆ δ + T F-term conditions are: i ijk j mk  ˆ ∝ T ˆ V U F-term conditions reduce to SU(3) . + ij lj = m ˆ z ˆ V i T × U j i i k l 3.1 T ˆ ˆ T T . The  ik j inm ˆ SU(2) theory described above, and both are parameterized by the baryon  U ijk   F-term conditions for full-rank × = 9 9 1 ]. ˆ 1 U Λ ij Λ k ˆ 74 U = and 0 = In this case, the For simplicity, we restrict our attention to the case where To obtain a quantum description of this theory, we perturb the s-confining theory W There may also be instanton corrections to the superpotential, due to the completely broken SO( ˆ T 19 Extracting the component which is symmetric in gauge group. These are subleading for so that brought to the form conditions are appropriately covariantit under is this sufficient complexified to symmetry show transformation, that To show that there are noto F-flat solutions, we first show that The resulting superpotential is which breaks SU(3) that the resulting F-termbroken [ conditions cannot be solved, and thereforeentitled to supersymmetry make is the field redefinitions: discussed in section (without superpotential) by the classicalin superpotential the form: arising from the dynamicalno superpotential. remaining Setting F-term constraintstheory is on parameterized by preserved by the superpotential.the This USp(6) matches branch 1 of the classical moduli space of 0 theory, subject to the classical constraints, which are equivalent to the F-term conditions JHEP10(2013)007 . . ] R 3 86 – × (4.2) (4.1) (3.48) (3.49) 3 77 S . Define R ], while being 76 , = 0, so the above 75 ii j . = 0. The remaining [ = 1 theories [ ˆ U 5 ˆ R U N × 3 , we may “integrate out” S ˆ U we find ]. It also proves to be a very . = 0. By the above argument, (associated to rotations on the -symmetry generator and km 91 ˆ T , 3 δ R ˆ ≡ H T J , special superconformal generators 87 , µ R ± . P , 3 2 J , M, V 3 − 18 = 0 H 3 β Λ ,J ] for very readable expositions of the topic), J 3 − ± . 2 nm k λ e 4 det J ˆ 83 V U F − , and the U(1) = – 20 – 1) + ]. In this section we will present evidence for the = 4 theories [ H 78 − , H eff 76 , translations = N mn k ˙ α W ˆ 76 } U Tr ( , . The superconformal algebra then gives: Q 1 , = 1 theories compactified on 75 ,Q α S F-term conditions for † ] and Q N = Q ˆ U 90 { † – 2 Q 87 ] and 76 ˆ T F-term condition then imply that = 2 [ F-term condition. Contracting with F-term condition, this is sufficient to show that ˆ U ˆ N T T SU(2) theory. × , superconformal dilatations ˙ α S , which implies [ , which has no F-flat solutions and generates a runaway scalar potential, much like , 1 α ij Q It is not our intention to give a detailed discussion of the superconformal index here Superconformal indices for Having solved the V ,S ), supersymmetry generators = µ 3 The superconformal index is then defined by: in order to settleThe notation. superconformal algebra Consider has aS generators four dimensional theoryK compactified on Q As we will see momentarily, theidentities, agreement providing relies very on strong extremely non-trivial support group-theoretical for our conjectured duality. (we refer the interested reader tobut [ we will briefly review in this section the basic elements that enter into its computation a relatively recent development, have alreadydualities. provided In important particular, insights equality into offor the the a topic superconformal of number index of providesand known very S-dualities and strong conjectured in support Seiberguseful dualities tool between in the studyagreement of of the holography superconformal [ indices of the dual pair of theories presented in section In this section we discuss anotherbetween very the nontrivial superconformal test of indices the of proposedsomewhat the duality: technical two in the gauge matching nature, theories. andsequences The readers discussion of primarily is the interested inherently proposed in duality the may gauge wish theoretic con- to skip ahead to section for the USp(6) 4 Matching of superconformal indices these results apply for arbitrary (full-rank) these fields, leaving the effective superpotential: condition reduces to Together with the components of the after applying the JHEP10(2013)007 . , . † F M M Q 4.2 × (4.4) (4.5) (4.3) , one 3 G S and of it is inde- Q i F 3 , which plays R S . β ⊗  in the represen- ], which gives a ) i G g f 77 R ( ], for instance — in ] that, exactly as in i F of the . R 96 76 r χ # ) ) k g ( , f i G fg , k 3 ) R J 1 χ , g 2 i − k , neither necessarily simple, and r x − tx F R , x 2 t t k − t H is obtained by taking the plethystic ( ) β , and gauge and flavor group elements − g 3 20 T − T ) . ( )(1 i e J I in the representation f ) receives contributions only from states 1 ) k G ( , F tx i 1 3 Adj i F r any symmetry commuting with − 1) J 4.3 χ − =1 R ∞ k ) denotes the character of X − χ tx )) g (1 M 1 " ) +2 ( − g − – 21 – R ( Tr ( R x χ i G exp -charge and flavor group )(1 ) we follow the prescription in [ R + dg R ). χ tx x dg G i Z R ≡ 4.3 ( r t t 4.3 − Z  ], the index ( − ) = i (1 2 Λ coupling, and thus it is invariant under the RG flow and 92 t P ) = r ] for nice references to the Lie group representation theory that we will (2 + t, x, f 94 ( , , and thus the index does not actually depend on I t, x, f † ) = 93 ( Q -charge — determined using a-maximization [ T , the superconformal index I R T with gauge group and to be generated by with superconformal t, x, g, f . T ( Q i ) for are the same as in ( M T ] that since the index is independent of the radius X i M 95 4.4 , and we denote with bars the complex conjugate representations. Once we have denotes the Haar measure on the group the fermion number operator, and R ) constant along the flow, and in particular it should agree with the value of t, x, g, f dg F We will thus compute the index in a weakly coupled, non-conformal description of the In order to actually compute ( 4.2 . Introducing appropriate chemical potentials, the refined index is thus given by: We refer the reader to [ 20 at the IR superconformalthe fixed superconformal point. Inconstructing particular, we need to choose the right valueneed. of We will give explicit expressions for the Haar measure of the groups of interest to us in section superconformal fixed point incan the argue IR. [ Inpendent the of case the dimensionless ofchanges the in theory the compactified coupling on in constant. ( In order for this to be the case, we need to choose Here theory, and will assume that this gives the right index for the theory at its (presumed) Here tation the letter ( exponential, and integrating over the gauge group: systematic way of computing thecoupled superconformal Lagrangian index description in of terms of thecoupled the theory, theory when fields in one the ismatter weakly available. fields For a generalnot weakly necessarily irreducible, one constructs the letter have more to saythe about case this of integration the below).annihilated Witten by It index was [ arguedthe in role [ of a regulator only. Let us choose g, f where we have integrated over the gauge group in order to count singlets only (we will with JHEP10(2013)007 we (4.8) (4.9) (4.6) (4.7) ) one (4.10) (4.11) 4.2 4.5 ), and then . g ( ). The act of 2 n G ) is elementary, R g momentarily): ( ⊗ 1 R 4.5 g R χ and section χ 4.1 ) in the two dual phases, ) = g ( 21 4.5 2 R ). Similarly, for the symmetric , . As an example, consider the χ 2 i ) N R t . g . . ( . Unfortunately we have not been ) ij 1 ) N =1 to g δ ) just counts the number of singlets i = 0. Expanding ( N g R X ( ( n t χ , n + . 4.7 χ A i R ) = j j t t t g χ − i i ( t t ) j N =1 i g X ··· R N N ( ) χ ≤ ≤ ) j j g ( g ) = χ ≤ ≤ – 22 – ) with the second term being the character of the X X 1 ( i i g i R ( ≤ ≤ ), which has character: R 1 1 4.6 ) = χ N 2 dg χ g ) = ) = , or in terms of characters: dg χ ( g g . When expanding the plethystic exponential ( Z ( ( χ Z G − χ for SU( χ -th Adams operator on the maximal torus of SU( n g ) we will also encounter terms of the form ) = representations we have: ]. 4.5 ( 93 t 2 A are irreps of . j n R R ⊗ and are the elements of i ... i R t ⊗ 1 When expanding ( Ideally, one would compute a closed form expression for ( Characters for representations of Lie groups can be worked out systematically using the Weyl character and antisymmetric R is known as applying the 21 formula, see for example [ It is thus clear that where in decomposing such terms into charactersg of irreducible representations withfundamental group representation element By using the wellplugging know the resulting property expression of into thetrivial ( characters representation (i.e. just 1), we obtain that ( where encounters expressions of the form (we will deal with higher powers of 4.1 Expansion in The first limit correspondsbut to an the expansion integration around overparticular, one the needs gauge to group use orthogonality requires of some the characters more under advanced integration: technology. In and then show thatable the two to expressions prove agree thewill for equality provide all of very non-trivial the evidencetractable resulting for limits. indices, the matching but The of in exactelliptic the matching section hypergeometric functions will in integrals, two which thus particularly we remain formulate a precisely well in motivated appendix conjecture about JHEP10(2013)007 . ↔ (4.13) (4.12) ) SU(7)  f ) ( 3 f × , ( 6 0 , χ 8 ... χ ) + f ) + (  f 3 ) ) + , ( ] for the polynomial 4 3 0 f , f ( χ 6 97 2) Dynkin labels can be ( 2 [ , , χ ... 4 χ χ + 1 4 ) + 3 http://cern.ch/inaki/SCI.tar.gz  f ) + 4 ) Sage ( f f 2 ) + ) + 2 , ( ( 2 2 2 f 0  , , f ( ) χ 5 ( 1 12 , f χ 2 3 ( χ χ 0 χ , 3 8 4 ) + 5 ) + SU(4) theory the relevant contribution χ f ) + f  . On the other hand, the corresponding ( ) + . We have denoted the SU(3) representa- ( ) t f 4 t × ) + 4 , ( f f , 2 1 ( 2 ) + ( 4 , 0 f 0 χ 8 , f , χ ( 2 ( 4 χ 2 χ , χ χ 0 – 23 – ) + ) χ f ] for doing the relevant group decompositions and ) + 3 f  ( ( ) + 2 ) + f ) 6 2 ) + 2 72 , ( f 1 f [ 0 1 ( f ( χ , − ( 1 2 χ 4 1 4 , , x 4 7 0 , χ 0 LiE χ χ + χ  ) + 2 2 3 x + + 2 4 + 3 f  ( t  ( 4 3 5 3 2 4 t t t χ term. In the USp(8) + + + 1 8 2 3  t 2 3 ) = 1 + t in terms of ordinary Young tableaux. Notice that, as we were indicating ) = t, x, f ( 22 x, t, f up to the degrees that we checked. In particular, for SO(3) USp ( / SU(4), we obtain the index: 3 SO USp × I I With this technology in place, the actual computation of the indices is straightforward, When the problem is formulated in this way the rest of the computation is concep- The actual code we used for the calculation can be downloaded from 22 where we have ignored terms of other orders in obtained after some ratherexample, involved consider group theory the cancellations.after doing As the a gauge particularly integration simple is of the form: before, even at this relativelyrather low order complicated the character matching polynomials of the appearing. indices is Furthermore, very the non-trivial, agreement with is only where we have omitted termstion of by higher its Dynkin order labels, in sodescribed for as example the representation with (2 manipulations. if lengthy. Wein obtain section perfect agreementUSp(8) of the indices between the two dual theories tually straightforward, but doingcomputer this systems is by required handof for quickly the doing computer turns any algebra non-trivial impossible, package computations.Adams and We operations, the took and aid advantage the of mathematics software system any order, into sums of productsover of the characters gauge of irreps, group. whichthe can The final then flavor be results characters easily can in integrated the be terms decomposition dealt into of with irreducible irreps. similarly,between representations and terms, since In we we there will will the are give give important flavor an cancellations case example it below. is particularly important to do Proceeding systematically in this way, one can decompose the plethystic exponential, up to JHEP10(2013)007 × In and B (4.14) 23 . ), where 3 N reducing J , USp(10) ± SU( and therefore J 3.3 ↔ × ) + 1) + 1) H 2 r l l 4) ( ( f + 2) + 2) ( ) − l l 3 ⊕ ⊕ + 1 + 1 + 1 + 1 5 ( ( SU(9) χ l l l l f N ) ( SU(2) ⊕ ⊕ × f l l ( + 1) + 1) χ and found in both cases ... ) l l χ 4 ( ( f t + ( 1 8 2  ) χ − l l l l 7 ) 1 f + + + + 10 2 l l l l ( f 2 4 2 4 N N N N ( χ ) + 2 + 1 + 2 + 1 + 1 7 χ + − − + 4 -charge, which would violate unitarity if there ) exponent f 2 3 2 3 4 3 4 3 f R t ( ( ) + 5 5 χ χ ) f 2 ( . We clearly see that both expressions look 1 40 f t ), which makes our method of computation ill-defined. 1 1 1 1 ( χ ). They are the scalar components of the chiral SU(8) up to order ) – 24 – 4.4 SU(3) . 2 χ × ) + 4.3 ) f 2 ( f f ( 5 leads to negative R-charges for the chiral multiplets ) ( 7 ) with negative χ 3 ≤ N χ χ − 1 1 4 10 -maximization gives rise to charge assignments allowing for gauge N N 1 a ˜ A SU( 140 − + USp(12)  ) ) )  ) ) in the letter ( ∼ × 2 3 1 1 t t ↔ , , , , , , N Adj) Adj) 4) SU(7) theory is given by: , , 1 1 B ( ( ( ( ( 1 1  − × ( ( ) = , in which c N SU(11) < N ) are shown in table x, t, f SO( ( f × N SO ) ) ) ) < N I l ) ) ) ) l l l B l l SU( ( l l i ( i ( A ( SU SO SU ( SO ( ( ( ¯ ¯ ψ ψ A λ B λ F F × Field column denotes the representation under the SU(2) group generated by r 4) . The fields which contribute to the superconformal index for SO( − It is interesting to construct explicitly the states that are annihilated by One can proceed similarly for other ranks. As we have seen in section N In terms of the index itself, setting 23 , and thus to negative powers of ˜ A The physical origin of this divergenceas is SQCD the with same as 0 ininvariant operators simpler (in examples with our runaway case superpotentials,were such a superconformal vacuum withoutsymmetries chiral in symmetry breaking the (assuming IR). the absence of accidental U(1) contribute to the superconformal indexmultiplets, ( the right-handed chiralwell as fermions the in gauginos and theSO( field complex strengths conjugate of the representation gauge as groups. The fields that contribute for the rank leadsparticular to we a have runaway calculated theory,SU(6) the so and superconformal we SO(7) index for willperfect SO(5) restrict agreement. ourselves to larger ranks. again ignoring terms ofrather different different, degree and in onlyinvolving the agree Adams after operator. using some non-trivial group theoretical identities Table 2 the SU(2) expression for the SO(3) JHEP10(2013)007 -th m SU(7) ) SU(7) ) f × × f ( ( 0 , . 0 ) , following , ) 8 12 f f E ( χ χ ( 2 SU(4) theory. , 2 , 2 we find perfect 5 × → ∞ χ ) + ) + χ F f f N ), the explicit form ( ( 1) 2 2 , , ) + N ) + 5 4 − 8 f denotes taking the f χ χ ( ) ) ( 4 ). For the SO(3) m , 4 f f , 4 ( ( 1 ) ) ) 0 0 . χ , , χ f f f ) + 2 as desired, limited only by ) + 2 4 4 4.12 ( ( ( f f χ χ t 1 1 1 ( , , , ( E.1 1 1 1 1 1 ) + 1 1 , , ) + 3 3 χ χ χ 7 f f ) + ) + ( χ χ ( and 1 f f , 1 ( ( , 3) theories when 1 1 + 1 + 1 + 4 2 2 = 0, where () , , N χ ) + ) + χ 0 0 t − SU(3) character f f χ χ ( ( N 0 3 ) + , , 2 6 ) (such as a function of group characters) f ) + 2 ( g χ χ f SU( 6 ( -th antisymmetrized tensor product. , ( 0 f 3 × , m ) as is shown in detail in appendix χ 3 ) + ) + 2 – 25 – χ f ( f 0 ( + 1) 4.12 , +3 4 6 3 + 3 , limit of the Haar measures for group integration over 0 N χ Taking into account the factor ( χ N 2 +4 taking the 24 . m 3 3 1 0 0 0 0 0 0 0 0 J ± 2 ]. ) and USp( a direct computation of the representation under the flavor group takes very 95 N , 21
3 5 3 2 3 4 2 2 2 2 2 2 exp. ). 78 i (0) , SU( t B × 76 4.12 4) i (1) (0) 7 21 14 − N B 2 2 B A (0) (0)  ] ] B   6 2 ψ ψ N )  i (0) SU (0) SO (0) i i (0) (0) Lie groups appearing in our construction. Starting with SU( (0) (0) λ λ B (0) i = 0 are shown in table B B [ [ (0) . The gauge-invariants contributing to the superconformal index of the SO(3)  A B operator   t A B (  Let us start by taking the large Likewise we can check the gauge invariant contributions for the USp(8) The superconformal index counts gauge invariant combinations of these fields and we ABC For the operator 24 the of the integral of a gauge invariant function long so that we devised a refined method that is explained in appendix Using the tools givencomputing resources above, and one patience. can Inindex go this from section as a we high will complementary approachpair in perspective, the of computation SO( namely of we the the will discussion compute in the [ index in the dual We again find perfect agreement with ( 4.2 Large can explicitly construct these combinations to checktheory, our the result gauge ( invariants whichabout contribute at the lowest ordersagreement in with the ( Taylor expansion Table 3 theory at the lowest orderssymmetrized in tensor the Taylor product expansion and about [] JHEP10(2013)007 (4.21) (4.16) (4.17) (4.19) (4.20) (4.15) (4.18) . ) θ ( . f , ) ) θ   θ N. ( ) ( , , j ) for simplicity, we will f f θ ) − ) ) ρ − ρ x iθ i ( with a continuous integral !  ( θ ) − ( f i f ). It is convenient to change ] e ) ) inθ in ρ α 1 N : ρ inθ P e 2 − ( ( e − d θ j ( N )∆( − f [ x 6= iθ i X   Z e Adj dθρe 2 j , | 1 n )∆( χ n 2 ≡ x | ∆( Z ρ inθ 1 n | ), where we have denoted the Jacobian ) n =1 ∞ − j 1 n θ ρ ρ X n ∆( e ( ( =1 − ∞ i ) can be equivalently rewritten as:   dθ X f n − e i ] j 1 → | n n X dθρ −   inθ − =1 πix dx ρ ) ρ Y in the adjoint. This structure also applies to 4.15 j j N 2 2

2 R πn θ d 2 n  1 [ − exp i i d Z x i − – 26 – → ), this can be conveniently rewritten as: dθρ e =1 . With these changes, we have that: θ k exp Y j =1 1  ( x N ∞ inθ Y n k Z − e dθ in inθ dα − =1 e 1 I ∞ N  dθ Y 1 Z n ) i R ! j − 2 =1 1 π X Y 1 k 6= N − ≡ i N Z =1 X N  dθ ρ e Y (2 k N log(1 ! Z R → = → 1 , the integral ( − 1 itself: Z ) = N ) ) ]: N iθ − g j g θ becomes a continuous function θ e ( = ( ∝ N 78 = − 1 θ ) ) i ) = = θ n π g /n g ( ), and the integration can be taken to be on the unit circle around dgf ( dgf ρ ( x n j (2 in t x ], ! e Z Z 1 − dgf j dgf N 78 =1 i 6= limit, we replace the sum over eigenvalues t ∞ n i X Z ( Z N P ) = ) denotes the character of i

) = ) = ) = dγ i USp(8) SU(4) = 9 in section 5.1 = , we saw the both the SO and USp and i i N ˜ ˜ 1 A B W g 3.3 , 2 2 1 g = 7 and − ←→ – 30 – 2 1 N 1 g , R = 7 are: k ; 2 21 20 21 N mn B n ; j b ) bifundamental “flavors” and no superpotential. If theory A , SU(3) U(1) m ; ( i a . A ⊕ SU(4) 4 ) ijk × , and the two fixed points should be the same, as in Seiberg duality.  , ab 10d ( τ 1 λ δ F 1 2 SO(3) SU(7) N USp(8) , we speculate about the infrared behavior for = i i and section A B W 5.3 3 0 the difference between the gauge couplings . It turns out that the USp theories are in general somewhat more tractable than the We now consider specific examples. In section In some cases, the infrared behavior may be different. In particular, the string cou- Thus, in general the two fixed points reached by the dual theories in their respective → N 1 5.1 The The prospective dual theories for of SO theories, so welatter. focus We on begin the by former,respectively, discussing extracting where the predictions cases the for infraredsection the IR behavior behavior can of be the determined using known dualities. In then the two gauge theories are infrared free, whereas cases, since the string couplingnot is depend irrelevant on at the fixed point, the infraredtheories physics have should a dynamicallynow generated attempt runaway to superpotential determine for the infrared behavior of these gauge theories for larger values is an exactly dimensionlessg coupling. However, whiledeep ( infrared the theory is free, independent of the initial values of the couplings. In these can happen when the string couplingconstituent becomes couplings ill-defined at approach that some point,group limit. for instance with As when its a toy example, consider an SU( should match between thesition” two does fixed not points, occursection provided in that between; a we discontinuous have “phase argued tran- thatpling, these despite data being exactly do dimensionlessdeformation indeed of along the match the fixed flow, in point. does Instead, not the always flows correspond may to converge a to a single fixed point; this corresponding deformation interpolating between the dual descriptions in theperturbative infrared. regimes will occur atized different locations by along the a string line coupling.the of Since fixed global points the parameter- theories anomalies, are the connected by superconformal a index, continuous deformation, and the topology of the moduli space of self-dualities is closely tied to that of exactly marginal operators [ JHEP10(2013)007 3 / into +2 (5.6) (5.9) (5.3) (5.4) (5.5) (5.7) (5.8) 1 − Sp acquire a ⊕ 3 ˜ decomposes / B +2 M SU(3), respectively, × SU(3). × ˜ B, Ψ SU(4) Sp , R Λ transforms as
⊂ in the original theory. In terms of under SU(4) ˜ λ , 2 3 2 3 2 4 2  , + ˜ A Q J b
, ˜ , A A M, 6 I a 2 ... + det Φ Φ ˜  λ A 3 Pf ⊕ ) 3 Sp ab 1 + 2 Ψ + 9 Sp Λ 1 3 SU(3) U(1) 5 (Φ Λ – 31 – A ) and ∼ = Ω 1 = , λ + Φ 3 Sp IJ 1 W 6 Λ ∼ W 1 Φ M SO(6) gauge theory in the confined description which ∼ SO(4) W ∼ = 3 Sp 1 Q A W Λ ∼ SU(4) theory, showing that it has an infrared-free dual description W × parameterize a fictitious SU(12) , corresponding to the baryon R I,J U(1) × It is now instructive to rewrite the gauge group SU(4) as SO(6), under which the Φ Applying Seiberg duality, we obtain the SO(4) gauge theory: The IR dynamics of this theory are particularly easy to describe, as the USp(8)The factor dynamics of the s-confined USp(8) can be described in terms of the meson where the remaining terms are exactly those generated bytransform the as Pfaffian a for Ψ vector. = 0. The gauge-invariant meson Φ where we suppress thethe index definition structure for of simplicity, Φmass and and and we Ψ can absorb to be a make integrated factor out, them of leaving dimension-one Λ the fields. superpotential Thus, Ψ and with the superpotential where the indices into irreps Ψ and Φwhere transforming the as superpotential ( now takes the form where we suppress thedenotes index the structure lone and SO(6) numerical baryon, prefactors which is for automatically simplicity, and SU(3) invariant. det Φ with the superpotential under SU(3) this meson, the superpotential takes the form: with a quantum moduli space. is s-confining, leaving ancan SU(4) be Seiberg dualized to obtain an IR free description. We focus on the USp(8) JHEP10(2013)007 1 ), is λ λ ) = k r 5.12 ( (5.10) (5.11) (5.13) (5.14) (5.15) (5.16) (5.12) C , ) = 3, and Q , where γ i , n G 2 2 | i + 2 2 π ) maps to a glueball λ P A | γ ]) 2 336 5.7 ( k 2 99 λ + with is not an irrep, and therefore 1 2 2 i ). Thus, 2 A | n i , which take the form 2 2 1 π φ g ) into the beta function ( λ A ) = , | i , . 2 1 ) + ) ) 112 λ 2 Q k i 2 , 2 ), is nonperturbatively suppressed 2 1 ( τ r 5.15 g ) λ and g ( , + 2 2 = 1 2 + g C τ / Q of the gauge group 5.10 = ( A 2 28( 2 1 , β 2 + i g πi π 10d g ( r 2 A ∼ 1 4 − W 3 i g γ e 2 g πiτ , γ ( 1 | 6 − e ) 2 2 2 λ − 2 2 | 2 ) 2 λ π 3 2 i | λ 1 | 2 ∼ g r 2 1 1 2 – 32 – λ | π 24 2 | k λ 25 2 + τ λ (8 π ) = 2 1 k = 1 i − g ( 16 ∼ − λ ' n 1 ( 2 τ ) 10d Q SO(4) gauge group. Performing scale matching at each i 0 and π ' 3 γ g i πiτ ( ∼ = 16 ∼ ], which causes a splitting between the two gauge couplings, e γ , β β 1 2 | 2 − 1 π 2 λ 2 | 4 | in the representation 2 / π 1 SU(2) Q 2 i λ k i | γ g × 2 φ 192 k . The baryon det Φ in the superpotential ( − 2 1 = Φ 2 3 i are the anomalous dimensions of Q Sp g π γ Λ 3 8 A 1 3 γ / − 1 1 λ is the quadratic Casimir operator, in the dual theory [ is the ten-dimensional axio-dilaton, which is related to the other couplings by | and ) = kl = r i , of the SU(2) | 2 g Q W 10d / ( τ A ) γ τ correspond to more than one physical coupling. However, it is straightforward to account for the ij β r 2 ( A weakly-coupled flow can be approximated as follows. The gauge coupling does not We now consider the infrared behavior of this theory. Since the beta function coefficient W λ The argument given here is somewhat of an oversimplification since T and | 25 1 ijkl G giving and additional complications which arise in a more careful treatment. at the end of theloop one-loop level, flow. and The can remainder be of approximated the by flow occurs substituting more ( slowly, at the two- a positive real constantsuperpotential, which which we depends will on not the need to index compute. structurerun and at normalization one loop, of so the run we to initially the treat “fixed it point” as a constant, whereas the superpotential couplings at the one-loop level, where we use the one-loop result (see e.g. [ for a chiral superfield | where of SO(4) vanishes, theattractive. theory The exact has beta a functions free are: fixed point. We argue that this fixed point is where Thus, the splitting betweenat the weak gauge string coupling. couplings, (  τ step in this chain of dualities, we find that where JHEP10(2013)007 2 a I Z Q (5.18) (5.19) (5.17) SU(7) × in terms IJKLMN , I 2 b kl Q Q , a ij , and , corresponding Q A JI ab δ A ) = 0 , ) run to zero in the  2 ). In particular, the = λ MN mn = 0 IJ : A 5.17 A b J A up to a sign, and therefore A Q KL jln 2 ) is an irrelevant deformation  IJ A Q has rank at most four. A ikm 5.11  IJ baryon is nonvanishing, and the moduli ) , 2 + 4 ), ( Q Q IJKLMN = 0 ijkl I 5.10 = 0 gives a mass to too many flavors, b J A i  which are related by the spontaneously broken Q of SU(3), so that Q a I 4 cof ( h + Q Q ) are necessary and sufficient for a choice of ij ab , – 33 – δ A 5.18 + = 0 det and applying the second condition, and the second obeys a classical constraint of the schematic form . 2 4 vev with c MN NP JP Q 3.1 A + A A A KL ijkl LM A ), and therefore characterize the classical moduli space of the A A kl ) with JK vevs can be extended to solutions to ( A 5.17 A ij has (maximal) rank four, then the A 5.17 A IJKLMN 1 MN c I A + KL IKLMNP index the six components of a , its vev is fixed in terms of that of the meson A ijkl I 4 J ) A 2 However, we have not yet demonstrated that any nontrivial solutions to these Q and 26 IJKLMN I = ( . Since the SO(4) baryon = det I 2 2 To address these issues, it is more convenient to write the superpotential as One can show that the constraints ( However, not all We now consider the moduli space of this theory. The F-term conditions take the form: While the chain of dualities we have employed to arrive at this infrared-free description This is one example where the string coupling ( ) If A 4 W 26 Q space has two branches correspondingouter to automorphism the of sign SO(4). of theory. equations exist. Moreover, the quantumspace moduli if, space may for differ instance,generating from a the an dynamical classical F-flat moduli superpotential. where cof denotes thefirst matrix condition of cofactors, from the ( constraint first follows constraint from arises the upon classical contracting constraint the that ( to exist which satisfies ( to the baryon discussed in section complete F-term conditions imply the following constraints on where is an appropriate SU(3)of invariant. The first equation( fixes the SO(4)the meson classical moduli space is locally parameterized by the gauge invariant gauge theory. It would be interesting to pursue this point further. the infrared fixed point,family and of therefore flows, the all exactly of dimensionless which coupling converge parameterizes to a theis same valid free at fixed weak point. stringis coupling, perturbatively the independent above of discussion the string suggeststhe coupling. that same SO(4) the If gauge this infrared theory persists fixed should nonperturbatively, point also then describe the infrared behavior of the SO(3) spect to one-loop running.infrared, and Thus, the the theory gauge becomes couplings free. (and hence at the infrared fixedcoupling point, corresponds as to a discussed ratio previously. of couplings This which is remains constant consistent as because the the flow approaches string in the weak-coupling limit, where two-loop running can be treated adiabatically with re- JHEP10(2013)007 and (5.21) (5.24) (5.22) (5.23) (5.20) R . k 1 3 4 3 ; mn ˜ B , to the Pfaffian n d ; ) j b 3 p ˜ which generalizes (2 A A SU(3) U(1) m ...i ; M d i a 1 i ˜ A ijk  ...M ab )1 . Ω p R 1 denote the irreducible ˜ i λ (2 , 1 2 2 3 2 3 ij SU ...i -index tensor USp(10) SU(6) ij p = mn A 11 i πiτ i i ψ 2 ˜ ˜ n e W A M B ; d b j and Sp ) p φ ), if any, are lifted by quantum effects. SU(3) U(1) (2 πiτ m ijkl 4 ; a i ...i e A 5.18 φ )1 24 SU(4) theory has a dual description with a free p ←→ copies of a 2 ˆ λ ab (2 – 34 – i × h d Ω are numerical prefactors corresponding to exactly ln ˆ λ 2 , R . . .  c 1 2 1 = 9 are: k πi 1 d ; 9 8 9 2 1 = N = USp(4) SU(6) ...i mn and W 11 B i 1 i ij c n 10d  φ ; theory ψ ! τ j b 1 d SU(6) theory, showing that it has a line of infrared fixed points A SU(3) U(1) m × ≡ ; i a p SU(7) theory has these features as well, and it would be interesting 2 SU(6) A × ...i × , respectively, and . 1 ijk i  3 2 A ab M ) invariant formed from = d 2 1 λ δ c det 1 2 SO(5) SU(9) USp(10) = i i and = 0 with all other vevs vanishing satisfies the F-term conditions. As this gives A B W 3 4 i 6 − We Seiberg-dualize the USp(10) gauge group to obtain the theory In summary, we find that the USp(8) It is now straightforward to find directions in the classical moduli space. For instance components of 1111 = 1 We focus on the USp(10) including a free fixed point. to check this in more detail to5.2 gain a better understanding The of theThe proposed prospective duality. dual theories for after integrating out massive matter.vanish, and The beta the function (exactly coefficients marginal) for string both coupling gauge takes groups the form a mass to onlymoduli one space, SO(4) which flavor, is thiswhich therefore suggests parts nonempty. that of It the this moduli would direction space be is defined interesting part by to of ( infrared better the fixed understand quantum point andimply a that the quantum SO(3) moduli space. Our proposed duality would seem to with the superpotential marginal couplings, whose explicit valuessuperpotential can be generated determined by by the relating c s-confinement of USp(8). An explicit computation gives hA denotes an SU( the determinant of a matrix. in a non-canonically-normalized basis, where JHEP10(2013)007 (5.30) (5.25) (5.26) (5.27) (5.28) (5.29) , with 2 SU g ). 240 , , where weak 5.30 , 2 ' 10d π
τ 4 ψ 2 | , 2 / Sp γ , ˆ λ g | , 2 2 SU
k 2 SU ) − + 2 g π g ψ 3 6 2 SU φ 7 γ g 2 γ SU − ) in the adiabatic approx- 7 and g − + 1 7 − 2 φ SU(9) theory from the above 2 2 2 SU γ π 5.27 | 2 3 SU g 2 Sp π ) × ˆ λ g (2 7 g 2 | 8 3 SU 3 . ˆ λ k π g ) 1440 1 2 ' − 2 15 π 2 , Sp 5 (16 ' 2 g Sp 48 g ) = ) = ψ 1 2 ' SU ˆ λ g + 5 ) ( ' SU , we separate the flow into one-loop and g β + ψ ( SU , γ 2 SU g g 2 ( 2 Sp 2 take the form 5.1 2 Sp π g is constant along the two-loop flow under the g , γ , β ψ – 35 – 5 16 γ
2 ∼ , β 1 2 SU 30 (21 π 2 − s Sp g
8 g / ∼ 1 2 and πg 2 + SU . Thus, the two-loop flow lines lie along contours of 2 SU 2 − φ 8 2 π Sp g 2 | φ g γ g 7 ˆ λ 2 2 γ Sp | 48 3 2 /g g SU 35 k − g − 7 − 2 Sp 1 2 g ). As in section 2 2 2 | π 3 π Sp π 5 ˆ λ 2 g | SU(6) description. ) 96 9 16 k 5.14 ) along this flow, 2 576 3 Sp × π − g ' ' φ 5.24 15 φ , and converge on the fixed line γ ) = 2(16 γ 1 2 SU Sp g ' g ( ) + β Sp 2 2 Sp g g ( β Since the superpotential coupling and theta angles define one physical phase among By inspection, we see that the previous example, theinfrared string fixed coupling point, corresponds andcomplete to affects infrared a behavior the of marginal physics thetreatment, deformation there. prospectively at as dual SO(5) As the this such,coupled corresponds we in to cannot the a USp(4) readily portion infer the of the infrared fixed line which is strongly constant the final position along the fixed line dictated bythem, the there string is coupling, a asstring complex in coupling line ( corresponds of to a infrared weakly fixed coupled points gauge theory parameterized and by vice versa. Thus, unlike since the logarithm of theimation, superpotential is coupling, fixed small by compared ( to 1 under the same assumption of adiabaticity as before. stated assumptions. Indeed,marginal this coupling combination ( corresponds approximately to the exactly Putting these into the beta functions for the gauge couplings, we obtain coupling runs to the “fixed point” Thus, after the one-loop running, we have at one loop, applyinghigher-loop portions. ( At one loop, the gauge couplings do not run, and the superpotential where the anomalous dimensions We find the exact beta functions JHEP10(2013)007 ) 3. 3.9 / 0 and (5.31) (5.32) (5.33) (5.34) n > for are distinct in n . 2 3 5.2
= R 4. This is suggestive , R ≥ . 81 Q U(1) . χ ˜ 3 interacting via a gauge N ). One possibility, which − / 2 N ) − 9 1 3 3) 3 R − 2 5.33 and section N − = +2 ( | − there are many possible product N R G U(1) ( 5.1 | Q N N ], whereas the assumption of a free = anomalies further constrain its form. 8 27 9 2 SO(4) 3. If we assume that no accidental U(1) R R 96 / × = = 7 χ χ − 2 U(1) N , ,N ,N and U(1) χ – 36 – 9
36 3 R N R SU(6)] 1 − 27 × N > 3) U(1) | − − − 7, is chiral superfields with G | is semisimple, for large 3 R N χ ≥ ( = N G N N 3 R 3 2 = [USp(4) , there remain many possible spectra for this gauge group with 9U(1) = 9 examples treated in section must have vanishing beta function coefficient, whereas any U(1) G N = U(1) | 1 8 N G G | = 9. | 6. G , | N > = 7 and = 4 N ˜ N have the following anomalies ), it is easy to check that all such operators have R-charge G 7, whereas a similar argument applies to the USp theory for Should such an infrared description be found, it would be interesting to understand if Even if we assume that Indeed, since an arbitrary gauge invariant of the SO theory takes the form ( If such a fixed point exists, the U(1) At any free fixed point, all the fundamental chiral superfields will have dimension one 3.10 ≥ one can apply toMoreover, any the specific possibilities candidate are description, yet broader no if obvious we candidate allowit presents for has itself. accidental a U(1) direct symmetries. stringof theory the interpretation, infrared e.g. behavior in of terms these of theories branes. to We a leave further future study work. explains the pattern for all However, for any fixed vanishing beta function coefficients. While there are many further consistency checks for the case at hand.simple component Conservation of of thefactors superconformal must R-charge decouple. implies that the semi- gauge groups which can reproduce the dimension formula ( Thus, In particular, a collection of group Therefore, or ( N and nontrivial evidence forthe a cases free fixed point, which we have already shown to occur for and the corresponding superconformal R-charge +2 symmetries appear along theoperators flow, can then be the determinedfixed superconformal point via R-charge requires a-maximization that of [ the gauge R-charge invariant of such an operator be an integer multiple of 2 While the a number of ways, theyaccessible both in share some the dualat feature description, least that for i.e. the certain that infraredgenerally, values physics there for of is is the perturbatively string a coupling. weakly coupled We now dual ask description, whether this can hold more 5.3 The infrared behavior for JHEP10(2013)007 , ≡ 5 0 F which , up to 27 A . ), 1 , p,q 2 Y , which can be Y 1 dP ) duality which arises singularity. While this Z ) theories for the two sign , 3 Z A / 3 , with the involution of interest C 1 (a real cone over dP 1 = 1 SL(2 ) dualities we have focused on. The dP N Z , theories by Higgsing, and exhibit interest- = 4 examples, making the parallels easier 3 Z / N 3 – 37 – C 1 dP ]. at a point. We are interested in orientifolds of this configuration 2 is the same as the Calabi-Yau cone over the zeroth Hirzebruch surface 53 P , 0 , 2 47 both of which exhibit different, more complicated patterns of dualities. Y 28 , 0 , 4 ] for more on the infinite class of Sasaki-Einstein manifolds known as the Y 101 , and , before treating a specific example where the quantum moduli spaces can be . The left side shows the quiver gauge theory for 0 100 , N 2 . 1 Y P We then briefly discuss two other non-orbifold examples given by the Calabi-Yau cones We begin by discussing the Calabi-Yau cone over See [ The real cone over × 27 28 1 only one involution ofthe the choice parent of quiver fixed exists element signs.which which can satisfies Moreover, only be rule two anomaly I choices free for of without these appendix the signs lead addition to of theories noncompact “flavor” D7 branes.P As we 6.1 Complex coneWe over begin by considering the complexobtained cone by over the blowing first up del Pezzocorresponding surface to a compact O7 plane wrapping the del Pezzo base. As shown in figure ing infrared physics. We discussfor anomaly all matching, moduli spaceshown matching, to and match Higgsing exactly. over which appear to befurther of examples to a [ different origin. We focusprovides on a a simple, few non-orbifold example simpleresulting of gauge examples, theories the are and SL(2 related defer to the on the worldvolume ofexample D3 branes is probing closely the analogousto orientifolded to grasp, the it known isIn but this one section, example ofout we other a aim new previously to dualities unexplored which briefly class arise illustrate of from the D3 dualities branes breadth of probing this of orientifolded type. this singularities but class, and also to point indicated by the dashedchoices line. are shown The on resulting the quiverfold right. (see appendix 6 Further examples So far we have focused on a single example of a new Figure 5 JHEP10(2013)007 the (6.3) (6.4) (6.5) (6.1) (6.2) N ]. 103 4) 30 4) R R 8 8 +4) − 4 4 4 +4) − 8 8 ˜ − − N N ˜ ˜ ˜ +4 +4 +4 − − − ˜ 2 2 N N N N N N ( ( N N ˜ ˜ ˜ ( ( ˜ is odd. For even N N N N N N ˜ N N N N ˜ U(1) U(1) . N N − − N B . We see that the anomalies 4) 4 1) , , 4) regime our results agree with those B B +4) − 2) 4 +1) − ] ] geometric flavor symmetries of +4) − ˜ U(1) j j ˜ N N ˜ ˜ +4 − +2) − 3 1 1 1 1 3 N N ˜ N N N N N ( ( N N ˜ B ˜ ˜ R A ( ( N N N N ˜ 2( 2( a N N ( ( ˜ ˜ U(1) U(1) N N Z ZA i i − − + ˜ A U(1) , has no dual in the A-theory. 2 4 4 X X X ˜ 2 4 ˜ ˜ +4 +4 +2 − − − X XA N 1 1 N N × +2 +4 − − ˜ ˜ ˜ 0 0 N N N N N N ˜ ˜ − − + + N N N N X 2 provided that − − − − U(1) ˜ Y Y X j j − -maximizing superconformal R-charge, which -maximization over the symmetries preserved by the a ˜ a A A this theory is the negative rank dual of the first theory U(1) a i i 1 1 1 1 1 1 N – 38 – + B ˜ × B B SU(2) U(1) SU(2) U(1) R = ) ) ˜ Tr [ Tr [ ˜ N N N ij ij 1 1 , defined for even   2 / = U(1) = = +4) ˜ N 4) SU( W W (sc) R ( ˜ Z + 4) SU( − 1 1 1 1 ˜ N N U(1) 31 ], it is important to do 29 SU( SU( i i i i 102 ˜ ˜ ˜ ˜ ˜ Z Z Y Y X X A A B B for details on negative rank duality). B ] using brane tiling methods, these involutions also satisfy rule II and lead to 53 Witten anomalies do not match, and the theories are not dual. 3 ], as they should since the orientifold corrections are subleading in this limit [ For completeness, we also present the The global anomalies for both theories are shown in table The two possible sign choices lead to the orientifold gauge theory Up to charge conjugation of the global U(1) One can also show that holomorphicAs gauge remarked invariants do in not [ match between the two theories. For 102 30 31 29 (see appendix instance, the B-theory operator superpotential, as we doin here. [ It is easy to verify that in the large SU(2) is a linear combination: of a better labelFor ease we of refer presentation to we thesea-maximization, have chosen as theories a the as basis superconformal “Theory for R-charges the A” are R-symmetry and irrational. which “Theory does not B”,match satisfy between respectively. the two theories for with superpotential where in either case the gauge indices are cyclically contracted. Henceforward, for want with superpotential as well as the theory superpotentials which inherit thethe SU(2) parent theory, as expected for a compact O7 plane. show in [ JHEP10(2013)007 . 3.1 (6.6) (6.7) (6.8) , . We will also asymptotically 2 N 4) 216 X . a 4 − 6 , − X N − 2070 a +3) . ˜ N + 2) + 1) ( + 1) + 2) + 1) 8 + 1) + 2) ˜ 34 10 + 2) N 2 ˜ ˜ ˜ ˜ 1940 N N ˜ − 3 2 N + 1)( ˜ ˜ N . − − N ( ( ˜ N N ( ( N ( 3 2 4( ˜ 1) X ( ˜ N 4( − ˜ N a , ˜ − N − 2 − N ( 2 1780 . 1) 1. For large − − − − 4 6 = 1550 N . − 3) − ( B 2) 1) − 1) 2) 1) 1) 2) a 2) N + 4] + 16(2 − − ( − − + 4 − 1220 8 − − 4) . 34 10 − N 2 X N N − 3 2 N X N − − a N − ( ( N N in the next section. a ( ( N ( 3 2 4( 1) ( 8 – 39 – N 16 X 4( − Theory A Theory B 0740 N N − N − 2 . − a N = 0 and − ( 2 − 4( − − 2 X X 2 X a a a anomalies vanish in both theories. , whereas exactly one solution lies in this range for any R
113 0 0 = X . R B X 4)[3 R 2 R 0 B 2 B R 2 R B 13) U(1) a − − 3 √ B 3 R R U(1) U(1) 2 X U(1) U(1) U(1) U(1) U(1) U(1) U(1) − 263. The same considerations apply in theory B upon replacing 270 a . 2 2 2 . ( 2 B B 2 X X 2 X X 0 0 (4 2 U(1) U(1) U(1) , and U(1) SU(2) 2 3 − ≈ 1) B , X 1 2 U(1) U(1) U(1) U(1) U(1) U(1) SU(2) SU(2) SU(2) − 431 Witten anomaly analogous to that used for discrete symmetries in section 2. 3 4 5 6 7 8 9 10 11 12 − 13) 0 3 N U(1) √ , U(1) = 5 is exact, giving − ∈ 3 X − X N N X a (4 0 = ( a + 1. 2 3 is a solution to the quartic equation ˜ , U(1) N 3 B X . The nonvanishing anomalies for theory A and theory B, where we use a multiplicative no- → a 1. For example, we obtain approximately 1 In the following sections, we explore the prospective duality for odd − The result for approaches N have more to say about the case of even N > where in the range A-maximization in theory A gives Table 4 tation for the SU(2) The U(1) JHEP10(2013)007 : N (6.9) (6.10) (6.11) k . 3 3 3 0 R k B , and we employ a − − − 1 + 4 5 5 5 y 0 R ˜ ˜ ˜ N N N − U(1) − − − 3 + 4 U(1) ˜ N and N N N gauge-group center, and − U(1) − ] x R ˜ p N 2 ] Z ] ] in − p 2 1 1 2 × k n +1 − − 2 2 ] ] − 4 ˜ ˜ ˜ 2 3 3 1 N N N 0 X 3] 8 2] 4 = U(1) +4 k N − − − , , , , , 2 2 − ˜ 4 1 1 3 2 p N 0 R N N N 0 X 2] 4 3] 8 1 2 − − 2 2 Z , , , , , − − − − − 2 U(1) ˜ ˜ 3 3 2 3 p N N 2 ˜ 3 2 N − − 2 2 or +3 − − U(1) − − − − 2 U(1) ˜ N N 2 N N N +1 − − , − Z [ N B p × − B [ 2 2 [ 1 [ 4 – 40 – gives 013 [ [ p − B − − − U(1) 2 ˜ 12 [ [ N N U(1) 1 1 013 [ [ Z − − − p − 2 2 U(1) 1 4 − ) 2 p 1 2 2 N 3 ) k 2 ˜ +1 ) ) − 1 +2 X 2 3 k − − | ˜ ˜ ) ˜ X − + 2 N X X N − | N ˜ ) | | N ) ˜ ) B N ) X ) N ( X ˜ ˜ ˜ B ˜ ) Z B B 2 Z ) X denotes a monomial of degree 2 ( ( ( ˜ X ˜ X 2 ˜ ) Z 4 | A | A 2 2 Y n X | ) | ˜ ˜ ˜ ) ) ( | Z Y A ) B B | ˜ k ( ˜ ˜ ˜ Y ( y Y Z ( B A 2 | k ( ( ˜ | Y ˜ ( AZ 2 − B 2 A AZ 4) x ˜ = U(1) ( | Z | | ˜ 4 4 Y − 2 − − X ( Y ˜ − − ,( 2 Y Y ˜ 0 X +1 Z N ( +3 ( ( 1 ( N N ˜ +6 B +2) 2 2 ˜ ˜ N p N Y N − 2 ˜ ˜ ˜ Z A ˜ N N A A ˜ A A A N ( Z ˜ U(1) p A ˜ A ≤ p ≤ Using these methods, we obtain the following minimal operators in theory A for odd To find minimal operators, we begin by classifying irreducible “gauge-invariant” mono- To obtain this map, we consider certain “minimal” operators, i.e. operators whose U(1) A similar analysis in theory B for odd where 2 slightly different basis for the U(1) charges: indices) which are neutralwhich under cannot the be factored intogenerates two all or gauge-invariant monomials, more a gauge-invariant subset pieces.invariant of operators. which The will Using resulting correspond finite this tooperators list classification, actual are gauge- it minimal. is possible to show that certain candidate operators. Operators ofduality, this whereas generically type no can two minimal onlyto operators a mix share the unique with same matching other U(1) between charges, minimal the leading minimal operators operators under ofmials the the in dual theories. the fundamental fields, i.e. formal products of the fields (disregarding gauge- We begin by considering the mappingis between equivalently the moduli expressed spaces as ofthe the a F-term two map theories, conditions. which between the holomorphic gauge invariantscharges subject cannot to be obtained as the sum of the U(1) charges of two or more nonvanishing 6.1.1 Moduli space and Higgsing JHEP10(2013)007 , k b ), ), + 2 n k ) ˜ Y odd N ˜ 2 N a X m SU(3) 2 − Y SU( Y theories × N ( mn × k 2 N X − SU( must be 4 +4) the symmetry × − p ˜ factors appear, ) N N p is also self-dual. k ) 2 Z 2 ˜ N X − 2 XY 4 is symmetric, a gauge ˜ Y ( duality, we next discuss − p ab baryonic symmetry precisely ) to USp( − Z N N 3 ˜ N +4 Z 32 ˜ = 3. N ˜ B Z SU( Q × ) to SO( once again corresponds to Higgsing duality N k +4) 2 ) Consistent with the proposed operator ˜ N B theory. Here, the simplest Higgsing, SU( X factors must appear, i.e. 33 2 ˜ × N 4) indices. Since SU(3) ˜ Y Z 1). ( 4) − × k − 2 − N corresponds to Higgsing the B theory SU( − k ), where now the resulting theory is conjectured – 41 – 4 2 N . In particular, the operator k − 2 +1 − factor must appear in the combination 3 N , breaks SU( k SU(7) theories enjoy an unbroken 2 ˜ 2 B theory is inconsistent in string theory, potentially − Z N / ) 0 a singlet under S-duality, inconsistent with self-duality X ˜ ˜ N N X dP +4) 2 ←→ SU( ˜ not N ˜ Y ( × ( SU( ˜ ) k Z 2 . Each × k p − 2 ) SU(5) ) k − +3 2 X ˜ 2 N − ˜ Z Y +3 ( 4 ˜ is even. By contrast, in the operator p N − − p 4 − N N Z theories studied in section ) to USp( 2, a highly nontrivial check of the proposed duality. 0 is odd). ˜ . This is another nontrivial consistency check. N − dP ˜ 3 N N SU( Having discussed some generic features of the proposed odd- We hypothesize that the even- At this point, it is also instructive to consider the behavior of the even- These particular operators are also interesting in that they correspond to Higgsing Several comments are in order. Firstly, while this may not comprise a complete list It would be instructiveNote to also that compute from the this SU(2) viewpoint, representations of the these operators. = 5. This example turns out to be particularly tractable, and we now show that the = × 32 33 ˜ dual theories have biholomorphictheory quantum-deformed turns moduli out spaces. to The be SU(7) somewhat more intuitive, so webecause begin they are by obtained discussing by turning this on theory, a vev for an operator with a particularly tractable example with a deformed quantum6.1.2 moduli space. Case study:The the lowest rank exampleN of the proposed duality between the A and B theories is for for the parent theory,theory. while on the other hand there isdue no to candidate an dual uncanceled for K-theorycomputation (discrete) the of tadpole. the parent K-theory We hope tadpoles to in verify future this work. through explicit theory A to SO( to be self-dual underHowever, S-duality, things suggesting are that quite thecorresponding different A to in theory the operator the for even- even where now the resulting theory is 4) mapping, we observe thatsection the resulting theories are related byunder Higgsing. the Turning duality on a proposed vev for in (since to the corresponds to Higgsing the Awhereas theory SU( the operator of the gauge-index contractionthe as operator well aswhich the is therefore F-term antisymmetric in conditions. theinvariant SU( index For contraction example, exists consider i.e. if if and and only only ifproperties if an are even reversed, number and of an even number N of minimal operators, oneno can show other that minimal allSecondly, of operators to the share listed obtain operators the this are same matching, minimal, it U(1) and is that charges, necessary so to the carefully matching account is for the reliable. structure Thus, the spectrum of minimal operators appears to match between the two theories for JHEP10(2013)007 T ⊃ ] 1 − (6.15) (6.16) (6.12) (6.13) (6.18) (6.17) (6.14) M )[ M , fg = (det ˜ Z , de M n j ˜ . Z n p i ; . , 0 R c ] ˜ j index a fictitious SU(6) , Y 2 2 0 0 0 fg 14 SU(7) , ˜ n A ˜ ; Z b , ij ...M ˜ U(1) p )Φ Z de i ˜ 2 n Y Z j Z j ˜ ˜ 0 X Z A 2 n m i 1 3 1 ; i , so that on the moduli space we 21 13 21 ] ˜ 2 3 3 7 bc a X )Φ = Λ I, J, . . . mnp − Z ˜ M − − Z ˜  Y , much like cof Z n U(1) 104 = (Pf j m T Tr [ ; ] = n a ...M p B 1 ij and (Pf j 8 1  ˜ ...i − 21 Y Q n j 1 3 1 3 1 3 7 SU(3): 2 21 i 1 µ − j Y abcdefg 1 i − 2 M 2˜ m Jn  × p )[ Y M Z iji Q  n ] + j mnp abcdefg M Im Jn  n  1 1 1 ˜ , – 42 – Y 1)! Y Y i j p 1 1 ...i and “Pcf” stands for “Pfaffian cofactor”, since for 48 48 SU(2) U(1) ˜ 1 − Y A IJ j i Im ij ) 1 = (Pf 1 i n ≡ ≡ index SU(3), and ˜ Y B . (  Z M mnp ! 1 1 Φ m  IJ M n − M ) 1 Tr [ Q n n = (Pcf 2 2 Z ij , n , i ab ˜ λ  ≡ ≡ 1 1 mnp m, n, . . . m ˜ m (Pcf Z  ; = ij Y SU(7) SU(3) a J b 1 2 ) M ˜ i ˜ i Y A ˜ ˜ mnp ˜ ˜ ˜ W Z I a I a Y M X Pf A B  ˜ ˜ A A 1 2 ≡ ≡ (Pcf IJ Im index SU(7), Z We consider the B-theory SU(7) Y antisymmetric matrix n 2 SU(2). There is a classical constraint: × a, b, . . . × n We now account for the finiteness of these couplings. In particular, the superpotential The classical constraint is quantum modified to [ invertible it takes the form Pcf take the SU(3) gauge coupling and superpotential couplings tocouplings zero. give a mass tomust have certain components of for an arbitrary invertible matrix This equation describes the quantum moduli space of the SU(7) gauge theory when we for a 2 M SU(3) where we define where with the superpotential: All possible SU(7) gauge invariants are products of the following: space. Theory B. after which we briefly explain how to show that the SU(5) theory has the same moduli JHEP10(2013)007 ∼ = . (6.21) (6.22) (6.19) (6.23) (6.20) = +1 12  Therefore, the = ) contains only 12 with the SU(2) everywhere in the 34 6.17 ij 0 R m ,  Z 0 R kj α ij  0 0 0 σ 1 2 α k U(1) i σ ≡ 0 X 2 3 1 0 = 0 R α m 1 3 2 3 − − . α Z ij U(1) U(1) B , σ , 1 3 1 3 0 X 4 3 1 } 1 0 1 0 α j k − − 1 = 10 α m σ − − − Z ik U(1) ,  8 SO(3), and i B m 2 3 1 1 = − 1 11 0 1 0 0 × 11 11 0 0 0 , there is an unbroken U(1) SU(2) U(1) ij − − − α – 43 – 0 R ≡ {Y = 19 , σ 2 1 gauge anomaly, but this is fine because SU(3) is completely A m SU(2) ¯ 3 3 1 3 2 1 Q ⊕ M SU(3) o ⊃ 4 ) SU(2) U(1) 1 j ) ) ) 2 − ˜ i dim 3 ˜ A Z 3 Y ˜ i A 2 3 1 0 0 ˜ Φ Y ˜ Z YQ A ZQ ˜ ˜ 2 Y YZ Z Y  ˜ ˜ , Z Z  = ( = ( i = ( 0 i m m − ij m Φ = ( 1) indexes the fictitious SU(6). Since the l.h.s. of ( Y Q i 0 Z −  i ,  +3( 0 1 1 0 m = n indexes a fictitious SU(5) i = A m A complete list of SU(3) gauge invariants formed from these four fields is: Thus, the moduli space is parameterized by the operators: α j i Note that this spectrum has an SU(3) σ 34 broken on the modulianomaly free space. spectrum Adding forthe SU(3). an s-confined description, SU(7) Upon whereupon flavor adding thethe to a additional moduli mass fields the given are for original here. set the theory to additional zero and flavor, by one s-confining the F-term obtains leads conditions, a to leaving tadpole an in where SO(3) conventions: Since there are a total ofthat 16 there invariants, are still five subject further to “classical”these one constraints constraints modified relating explicit, constraint, these we we SU(3) conclude define: composites. To make moduli space. subject to the gaugingdimension of of the SU(3) moduli and space the is: quantum-modified constraint. Since all operators are neutral under U(1) SU(3) baryons built fromthe SU(3) moduli fundamentals, space, SU(3) leading is toremove completely a 8 broken confined Higgsed everywhere description degrees in where of the freedom effect and of their gauging superpartners. SU(3) is to where JHEP10(2013)007 (6.28) (6.30) (6.31) (6.29) (6.24) (6.25) (6.26) (6.27) , , α p , which is attained Z j 0 R n Y . 14 SU(7) (1) i , m Λ U Y i = 0 . × 2Ψ = 0 . mnp X γ CD  + ] = 0 Φ anomalies match those of the ij 3 0 B β Φ =  γ j ¯ Q 0 R 14 SU(7) 1 2 [ Ψ 0 R Ψ (1) β i , ≡ U AB j γ Ψ αβγ ] U(1) α  3 × ij U(1) Ψ  ¯ i β Ψ Q × + [ X ΦΨ = Λ Ψ j i + X αβγ ij , Φ 2  1 0 0 B +  02 0 0 i α 2 1 m α 0 B − − ] Ψ . αβγ α − Z ABCDE ij  – 44 – α m Ψ α U(1) , and Φ: α 1 Z 2 ,  α [cof 2Ψ ,  Φ × m i ΨΨ m 1 3 ⊕ = − ,Ψ 4 = 0 , i SU(2) U(1) α = 0 α ij ≡ Q ≡ Y α i α α σ Ψ B Ψ Φ α i α ] j α Φ = 0 Ψ j α ¯ Φ Ψ Q Ψ α , i ) completely describe the deformed moduli space of the quan- Ψ = 0 satisfying these constraints gives only three independent Q α Ψ Φ [ , i − ¯ i α , Φ Q 6 j 6.29 i α α Ψ i m AB m ] Ψ Z Y 3 ), ( ΨΦ ¯ 1 m Ψ Q [ det = 6.28 i ≡ ≡ Q Φ i ), ( Ψ Φ 6.27 The maximal unbroken flavor symmetry is SU(2) We define: One can check thatoriginal the description, global as expected. SU(2) whereupon the remaining constraints areare trivially satisfied. therefore: The light modes along this line when we take Φsolve and the Ψ constraints to to eliminate be Φ nonvanishing with all other fields vanishing. We can then the form: Together, ( tum theory. and After a somewhat longer computation, we find that the quantum modified constraint takes In terms of these gauge invariants, the classical constraints becomes: constraints from the secondconstraints, equation, as and expected. a further two from the first, for a total of five Although both equations appearabout a to background have with five components, examining small fluctuations The classical constraints then take the form: JHEP10(2013)007 a (6.37) (6.32) (6.33) (6.34) (6.35) (6.36) . b (1) where U(1) , U 3 b 3 + a (1) , ZT U 1 µ (1)  ) × U N s + ; M ( R a 3 3 3 i i K . = / / 0 / 2 U (1) λT 0 X mn U(1) U M L 0 R ; + ) X J (1) s × 0 2 0 0 j n ( B  U 3 5 1 4 A 1 5 3 2 1 0 N L U(1) N ; ; − − M i m I ,U K U(1) 0 X U a 3 5 1 5 1 5 SU(2) U b ZA 2 5 − − − M ij (1) L ; → IJK  U(1) 1 J  U b Adj 0 2 µ 1 3 1 3 U B 35 2 SU(3) 8 5 L N − ; 1 5 7 5 1 5 3 1 2 − ) a I + s − 0, the SU(5) gauge theory has an s-confined ( B U SU(3) 1 1 1 1 – 45 – MK (1) jmn × → 3) elements of each SU(3), and the unbroken U(1) V IJK SU(3) a U B  / 1 1 1 J i n M ; 2 W 1 3 = L SU(2) U(1) A − 1 − , B U m 3 I L Y SU(5) / T 1 ij (1) 1 3 MK , 5 SU(5) B is charged. 3 λ  V , taking 2 IJK B i i / AB J  Z M ; Z A ,U Y X I A I B = = L  a Z = B A = 9 U J 1 IJ W 3.3.2 I ; K I L Λ (1) J I 3, and we omit an unimportant U(1) global symmetry under which V T T U , U = 2 2 , 2. The superpotential partially breaks the flavor symmetries of the pure − IJK , W )  s ] discusses a similar theory in the context of dynamical supersymmetry breaking. ( R  = 1 We now consider the dual theory: = 1 9 105 1 (1) Λ denote the diag(1 U b = = I, J, . . . i, j, . . . W 0 R Thus, deforming the resulting theory by the tree-level superpotential, we obtain: Reference [ (1) 35 U and U(1) linear combinations are: As reviewed in section description: where s-confining theory. In particular, SU(3) with the superpotential: where only the additional singlet with the dynamical superpotential: Theory A. JHEP10(2013)007 ] for , α (6.42) (6.44) (6.38) (6.39) (6.41) (6.43) (6.40) 107 Φ [ 2 1 m → . , . . α SINGULAR , = 0 , , = 0 = 0 i = 0 , φ φ 3 N 3 J J ; i i ;3 M ; i k = 0 = 0 L = Φ that solutions to these U j γ γ 3 U 1 i m , I 36 ψ ψ N L M ; i 0 R β β i ,U T αj k ], which uses → ψ φ ), and the fields transform as U σ i ij i ,V α ik 106  IJK = 0 U(1) αβγ ψ α ) 6.23 , φ φ j i i ; 0 X − αβγ i ij Φ ,  α ( 1 0 1 0 + α i k σ − − j ψ − package [ φ = U(1) = 0 ij α → − i α α ,U σ ,T,  ij B ψ 1 3 1 11 1 0 0 0 Mk . ij 11 11 0 0 0 9 − − − V = 0 − = 0 = 0 , φ Λ ) ,V i α M L j 3 j 3 αi k N λ ; ; ,  , ψψ ; i δ α J i Ψ σ 2 ( 3 I − – 46 – K 4 ψ j U α Stringvacua 1 3 3 2 1 1 ⊕ U ψ αi j m jk = 0 = 0 = Zδ 1 3 4 σ SU(2) U(1) M 1 N µ ; α α α ,U I → 2 φ φ = = ψ . Thus, the moduli spaces are biholomorphic. 2 ,  3 + X i U α α 3 i i α α X j 2 k j ; ;3 X L i 4 = 0 ψ ψ i X i ψ 3 δ 2 2 I B i = 0 I AB J ILK + 9 SU(5) ABX B AB B 3; , ψ  l i − Λ = δ λδ = i α 5 = = = ,U 9 = − ψφ function of the α i Ψ + ψ i α α Λ ij φ φ ψφ φ 3 + ,U λ ψ ψ i iλm j MK ψ φ , U m + , we recover the exact constraint equations for the moduli space of MK φ − V i = 0 α = = I V m L → = ψ j J T 3 ; M ; MK α 33 α ij i , our remaining conventions are given in ( L σ V V U jk U ,Z IJK  M 14 SU(7) J  ; , ψ j α Elimination[] l ψ IJK U Ψ = 0  5 I ≡ 9 J 2 1 m T iJK Λ α k  ψ → The F-term conditions involving nonvanishing fields are: We use the ψ 36 for some mass scale theory B for Λ computations. Upon replacing: Applying the above decomposition and simplifying, we eventually obtain: under the global symmetries. where We decompose the non-vanishing fields as follows: It is straightforward to show,equations using must the satisfy: Gr¨obnerbasis algorithm, The F-term conditions now read: JHEP10(2013)007 The SU(2) (c) × + 2) ) ) N ( N N ( ( SU SU SU 2) ] that these dualities − . The red dashed lines 47 N 0 , there are two different 4) ( F − + 4) SU (b) N N ( – ( orbifold of the conifold which SU SU 2 (c) The resulting quiverfolds ) Z a 1) 2) 6(a) SU(2) isometry of the base. b. b. , a × 1 38 P . We argue in [ × B 1 P = 4 dualities between SO and USp theories, = 0 N F 39 – 47 – . . As shown in figure 6(c) 0 , (b) Phase II 2 Y . This is equivalent to the requirement that the orientifold 0 0 F F orientifolds we now study, the negative rank duals are either examples studied previously, there was only one toric phase, 0 1 F dP SU(2) isometry of the base. Only the involutions pictured in fig- ) self-duality of type IIB string theory. Interestingly, negative rank × Z and , The two Seiberg-dual quiver gauge theories for 3 (b) Z – / 3 do so, and of the fixed-element sign choices compatible with SU(2) C = 1 analogues. (a) (a) Phase I ] for a more detailed, brane tiling-based derivation of these orientifolds. (b) . – SU(2)-preserving anomaly-free quiverfolds that result from orientifolding these theories. N 53 quiver gauge theories (“phases”) which describe D3 branes probing this singularity. × 37 As in our previous examples, we wish to consider orientifolds corresponding to compact By contrast, for the In the See [ In this context, a toric quiver gauge theory isWe cannot one for eliminate which the the possibility number that of arrows a entering chain and of exiting deconfinements, dualizations, and reconfinements 6(a) 39 37 38 invariance, only one choice forgiving phase the I theories and pictured two in for figure phase II leadeach to node anomaly-free is theories, the same. might relate the two theories via known dualities, but we have not been able to find such a chain. this way, giving yet another new field theory duality. O7 planes wrapping the base preserves the SU(2) ure and its trivially equivalent or relatedbetween orientifolds by of Seiberg the duality.Seiberg two duality different in Instead, phases. the we parent Although theory, find the the a two resulting phases orientifolds nontrivial are are duality not related obviously by related in by exchanging SO and“negative rank USp duality” groups as and explainedrelate symmetric in to appendix and the antisymmetric SL(2 duality tensor also partly matter, “explains” the a patternsuggesting of that it may have some physical interpretation relating to Montonen-Olive duality toric These theories, which we denoteone by of phase the I and nodes. phase II, are related byand Seiberg duality we on found a duality relating two different orientifolds of that phase which differed SU(2) 6.2 Complex coneWe over now consider the Calabi-Yau coneis over the same as the real cone over Figure 6 indicate orientifold involutions compatible with the SU(2) JHEP10(2013)007 , 1 (6.45) (6.46) ). The , where we 3.1 0 F . While it led . This invari- B B . 2) 2) 3 center of SU(2) − − 34 N 2 + 8) N 1 N N 2 2 Z 1) 2 1 − 2 N 10 − 2( 2( 1) 1 Z 2 − − − ω ω N ± − ( − ( ( N 4 1 2 − 2 3 2 − . Note that the anomalies +2 − 3 3 6 R − 2 N N N to the R B R − + − 4(a) U(1) 4 4 40 2 . 1 2 1 2 1 2 / Z Z 3 R R 3 1 U(1) U(1) (b) Phase II  2 1 2 2 2 l 4 U(1) 2 2 B 4 / / . +2 − 1 1 Z C − 2 1 2 B 2 1 ) 2 N j N N U(1) U(1) ,N N A SU(2) − − U(1) SU(2) SU(2) k U(1) 2 B SU(2) SU(2) i gcf(2 Z A 1  × SU(2) – 48 – R Tr 1 kl  34 1 1 N ij + 8) U(1) N N  1) 2 − 2 N 10 SU(2) 1) 1) 1 × 2 − − = N ± − − ( − B ( ( ( N 1 2 − 3 2 W combined with a charge conjugation of U(1) anomalies all vanish. + 2) − 1 2 R U(1) N N R B R × 2 2 2 2 U(1) / Z Z → − (a) Phase I 3 R R 3 1 U(1) U(1) B 2 1 2 2 2 U(1) 2 2 2) SU( it is a distinct global symmetry. Thus, the global symmetry group is discrete symmetry is gauge equivalent / / N Z 2 1 2 B 2 1 SU(2) 2 − U(1) U(1) N Z SU(2) 1 SU(2) SU(2) × N U(1) 1 SU(2) SU(2) , and U(1) SU( We obtain the orientifold theory: 3 B , the i i i N A C B . The nonvanishing anomalies for the orientifolds of the different phases of , U(1) The global anomalies for this theory are shown in table Notice that the sole orientifold of phase I is its own negative rank dual, whereas the two B Here and in future by “gauge equivalent” we mean that the two generators are related by composition 40 ance corresponds to taking theto negative two rank different dual gauge as theories explainedmaps in in the appendix the above previous theory examples, to one itself. can check that inwith this a case (constant) it gauge transformation, see the discussion at the beginning of section For odd whereas for even actually SU(2) are invariant under with superpotential given by Phase I. orientifolds of phase II aretheories related are by Seiberg negative rank dual duality.each upon phase As dualizing in we the turn, shall providing left-node. see, evidencephases. the for We latter a now two duality discuss between the the orientifolds orientifolds of of the different Table 5 use a multiplicative notationU(1) for discrete and Witten anomalies as before (see section JHEP10(2013)007 . , 1 = N (6.49) (6.50) (6.47) (6.48) phase I center of N 2 Z 4) 4) +4) +4) − − N N N N 1 1 2 center of SU(2) N N 4 4 − − 4( 4( 4( 4( − 2 4 4 Z Z 2 ω ω ω ω ω ω Z 1 1 N N N N − − 4 4 4 4 ω ω ω ω ] that the embeddings in 4) gauge group factor and 2 2 2 2 R R 4 4 N N N N N N 53 ± − − + + − 1 1 1 2 2 1 2 2 N U(1) U(1) ], we also argue that the action . , 4 B B   53 4 l l +4 − 1 1 . +4 − 1 1 0 1 + 0 1 ˜ N N ) B B N N k k ; ; − − U(1) U(1) ,N + 2, by removing a single D3 brane we reduce j j k ˜ 2 2 C C i i gcf(4 ˜ = 2 A A 1 1 The global anomalies for these theories are Z   . N SU(2) SU(2) × +2 2 Tr Tr 1 1 – 49 – R N kj kj subgroup is gauge-equivalent to the C   1 1 4 ij ij U(1)   Z SU(2) SU(2) × = = ) ) ⊂ B N N ˜ is a distinct global symmetry. Thus, the global symmetry 2 W W Z 4 Z U(1) × 4) SU( the 2 + 4) SU( − k the global symmetry groups do not match, and the theories are not + 2 the 1 1 discrete symmetry is gauge-equivalent to the N N , where for simplicity we do not display the anomalies of the Seiberg k N = 4 4 SU(2) Z SU( SU( N × = 4 j j 4(b) 1 i i i i ; ; i i ˜ ˜ A A B B We obtain the orientifold theory: ˜ N C C , the N 2, after which the global symmetry groups no longer match, so there is no duality for even . For even − , and for 1 N 41 It is interesting to understand better the nature of this prospective duality between It is straightforward to check that these two theories, which are related by negative Although the global symmetry groups match for → 41 phase II duality closely analogous toderivable it. from However, known we examples emphasize of that Seiberg this duality. duality In is [ N not obviously One can also showother this theory, for by example constructing the holomorphic phase gauge I operator invariants in one theory with no dual in the dual. orientifolds of the twostring phases. theory for We the willso two we present phases can are evidence expect related in the as nature [ of in the realizations duality of relating ordinary the Seiberg two phases duality, phases to be an infrared Relationship between the twoshown phases. in table dual theories separately; one can verifysee that that they the match as phase expected.N I More and importantly, phase we II orientifolds have matching anomalies for odd group is SU(2) rank duality, are also relatedintegrating by out Seiberg massive matter. dualizing the SU( with superpotential For odd whereas for SU(2) with superpotential as well as the theory Phase II. JHEP10(2013)007 + 4) ) N N ( ( example SU SU 0 F II considered above, singularity. ) 0 N , p,q . The red dashed lines ( + 4) 2 0 , Y Y N 4 SU ( Y SU 43 ). The phase II quiverfold has 7 2) − + 2) N N ( (b) Phase II ( SU SU b . I 2) – 50 – which, like the cone over b + 2) − 0 , N N 4 ( ( a Y SU SU examples discussed previously. all of which are Seiberg dual. We focus on the two phases 1 0 , 42 4 4) dP Y − N ( (a) Phase I and 3 SU Z / 3 and on the involutions also pictured there. ) ) C 7 N N ( ( a . ] for a classification of the toric phases of D3 branes probing a I SU SU 109 , . Two of the five Seiberg-dual toric quiver gauge theories for . Quiverfolds for the anomaly-free orientifold gauge theories we will consider, arranged by . One can show that these orientifolds correspond to compact O7 planes, and 108 8 + 4) For simplicity, we restrict our attention to the anomaly-free orientifolds pictured in We consider the real cone over See [ Two of the remaining three phases also admit involutions, and several of the resulting orientifold theories N ( 42 43 SU pictured in figure figure are manifestly Seiberg dual to those considered here. studied above, this exampledistinct from exhibits the interesting new patterns of dualitiesis which an appear orbifold of thebranes conifold. probing this There singularity, are five toric quiver gauge theories which describe D3 it exchanges the two theories in phase II. 6.3 The realBefore cone concluding, over we present onegauge final example theories of of new D3 dualities branes relating the probing world-volume a Calabi-Yau singularity. Much like the a negative rank dual whichis is shown. not The pictured, phase as I it quiverfolds is are manifestly “self-dual” Seiberg under dual negativeof to rank the IIB duality. quiverfold S-duality which ontheory the dualities D-brane inside configuration each describing phase each that phase we just reproduces studied: the field it is a self-duality in phase I and Figure 8 the parent quiver and involution used to generate them (see figure Figure 7 indicate the orientifold involutions we will consider. JHEP10(2013)007 4 4 4 4 2 2 (6.54) (6.53) (6.51) (6.52) − − − − − +2 +2 − 7 7 7 7 R 2 2 2 2 14 14 14 14 N N N N N N N N + − − + − + − + 4 4 4 U(1) 1 2 1 2 1 2 1 2 − +4 − +4 +4 − 1 2 1 2 1 2 1 2 3 3 3 3 6 6 R N N N N N N 1 2 4 4 4 2 4 4 X . 2 4 2 − − − − +4 +2 N − + − + − + 2 2 2 − − U(1) N +4 +2 − − 2 N . 2 2 41 N N N N 1 2 1 2 1 2 1 2 1 2 1 2 N 2 N N N N N N N N P N − − U(1) 4 − − X j , which we omit from 34 4 +4 − N N 4 4 − +4 P 2 2 0 0 0 N N N B 4 4 N N − − − +2 +2 − N N 1 1 1 1 . We now briefly discuss 23 2 2 N N − − 0 − − U(1) 2 2 N N , N N N N , P N N 4 N N i − − − − 34 12 4) 4) − − Y 2 2 B P P − +4) − +4) 4 − +2 − +2 j ij N N N N 23 +4 − 1 1 1 N ( ( ( (  N N N N P 1 1 1 1 1 1 N N N N N N + 24 SU(2) U(1) − − − − 4 P i T 2) S j symmetry for even 34 ij 1 1 1 1 − P )  i 1 1 SU(2) U(1) 34 ,N symmetry for even N + P ) 3 4) 13 T ,N gcf(4 P isometry group of − – 51 – ij Z j  23 1 1 R 2 1 N P gcf(4 + 2) SU( Z 12 + 1 1 1 1 P 2 U(1) i N T ) SU( A × j 12 ij N  P 1 1 1 X i 12 = 2) SU( P We obtain the orientifold theory: We obtain the orientifold theory: 1 ) SU( − U(1) W T 1 1 1 1 1 N × ij N 1 1 1 1  1 2 = W + 4) SU( + 2) SU( 1 1 1 1 1 1 1 1 1 1 11 11 1 1 N N SU( SU( i i i 23 24 34 13 12 3 2 4 1 i i 23 34 41 12 S A P P P P P T T T T P P P P with superpotential As before, there is an additional discrete Phase I, Involution b. with superpotential where there is anthe additional charge discrete table for simplicity, as it will not play a large role in our analysis. each of the threethem theories using anomaly in matching. turn, after which we illustratePhase a I, Involution potential a. duality between preserve the full SU(2) JHEP10(2013)007 for ) (6.55) (6.56) ,N gcf(4 Z +8) +8) +4) +4) +4) +4) +4) +4) +8) +8) +8) +8) 6= R 8 8 N N N N N N N N N N N N N N ) ( ( ( ( ( ( . 2( 2( 2( 2( N N N N N N 2(3 2(3 ,N N U(1) . + − − + − − 1 2 1 2 1 2 1 2 1 2 1 2 gcf(8 j 41 Z T X , whereas we argue that no 34 +4 +6 +4 +2 +4 2 1 1 + theories. We omit discrete not divisible by eight, the +2 +4 +6 +4 +4 2 N P N N N N N − 0 N N N N N , i 23 − − − N U(1) 4 T , since Y B 12 N One can show that the anoma- +4 +4 +4 +4 1 1 1 1 +4 +4 P 1 1 00 1 1 + N N N N ij N N 24  − − − − 164 + 32) N − symmetry for even 4 20 + N − 2 4 2 ) − 2 2 − j − N 41 1 1 1 1 N ,N T N 2( SU(2) U(1) − 3 41 − ) gcf(8 P i 4 Z R N B R B R 1 1 . Thus, consistency along the Coulomb branch – 52 – X k ij 2 3 R R  U(1) U(1) U(1) U(1) U(1) − 2 2 + 2 B 2 X 2 X j U(1) U(1) 23 N + 4) SU( T 1 1 1 1 U(1) U(1) N U(1) 23 → SU(2) SU(2) P . i 2 N N X ) SU( ij  N 1 1 1 1 = W there are no discrete symmetries and therefore all three theories have + 4) SU( N 1 11 1 1 1 1 1 We obtain the orientifold theory: N SU( . The nonvanishing anomalies of the three different . . i i 2 4 p i i 34 41 12 23 41 23 6 X X T T P P P P For odd = 8 D3 branes we can reduce matching global symmetry groupsglobal and symmetry anomalies. groups For again even k match between allrules three out theories. a duality However, betweenN phase by I removing and phase II for even Relationship between thelies different involving orientifolds. the continuousered global above, symmetries where the match nonvanishing betweentable anomalies all (excluding discrete three anomalies) theories are consid- shown in with superpotential In this case, there is an additional discrete dualities are possible for even Phase II. Table 6 anomalies for simplicity, as there are no discrete symmetries for odd JHEP10(2013)007 2 ) 23 have T (6.57) 2 23 ) P 2 23 2 12 P is possible, P 2 12 ( B P N Q 41 1 T P ( N 3 T , which transform as 2 ) self-duality of type IIB A and Z . Consistent with the pro- 4 2 13 , ) R P 2 4 41 +2 2 . Together, the success of these − P +4 2 N N 3 2 U(1) 34 N 34 G 3 T P 2 P ) then acts in the usual way on a − × / 3 Z − 1 or T X , and or ( 4 − N and N F 1 +2 2 +4 2 T N 2 U(1) 1 N A T × 4 12 B P 2 4 2 +4 2 − – 53 – − / 0 N I.a I.b N 1 U(1) 24 singularity, providing extensive checks for the pro- 4 = 1 gauge theories arising on D3 branes probing N A P 3 × Z N / 3 we show that SL(2 C R B X is possible in all three theories. As an additional check Q Q Q 3.2 B Baryon . Moreover, in the phase II theory only integral . Q N N under SU(2)
4 − 2 , where the F-term conditions play a nontrivial role in the latter two 2 N ) , 41 4) , these issues do not arise, as the above operators are no longer well- T N 41 − P N . However, one can show that the operator spectra do not match in this case. 2 34 ( ], we argue that this duality originates from the SL(2 N P ± ( 53 , N 1 23 = 4 theories. In section P − In [ For odd The duality between the two orientifolds of phase I is an intriguing new feature of this This leaves open the possibility of a duality between the two orientifolds of phase I , N checks presents a compelling argument for the existence ofstring a theory, duality. and is thereforeof a close cousin ofparticular combination the of more holomorphic familiar couplings Montonen-Olive which duality is constant along the RG flow and viously explored in thecorresponding literature. to the We well-known focused onposed duality, a including particular anomaly matching, example matchingmatching, of of and these discrete matching symmetries, dualities, of moduli the space various superconformal quantities between indices. the In two some theoriesmathematical identities, instances follows see from, the for or matching example would of appendices imply, some remarkable that a duality can occurFor for theory this case, I.a, we we again consider find the the baryons of baryons minimal R-charge. 7 Conclusions In this paper, weorientifolded Calabi-Yau showed singularities that exhibit a the rich class of gauge theory dualities not pre- these two theories for even so these operatorssymmetries have explained no above. dualdifferent Thus, there theories we either, for conclude even consistent that with there the are no mismatchdefined, dualities in between and discrete the only integral and cases. geometry which does notleave further appear discussion in of the it simpler to a examples future we work. considered previously. We Clearly the operators do not match each other, which is inconsistent with a duality between posed duality, we find that thethe theory same I.b charges baryons under the global symmetries, as do the theory II baryons for even Specifically, we can compare the baryons of minimum R-charge in both theories: JHEP10(2013)007 ) ]. Z , 53 singu- 0 F ) dualities Z , = 4 theories and theories by Higgsing ] we provide infinite N 3 Z 53 , / 3 ) duality group therefore 47 Z C = 2 or , theories already understood ∗ N . In [ singularity is a closely related ge- N = 1 1 base). These theories exhibit inter- 1 , all of which exhibit similar dualities. dP N 1 dP dP = 1 duality is of a different type than the N . It would be interesting to better understand and 0 , 3 4 – 54 – Z Y / 3 C ) duality so long as the O7 planes are compact, though in Z , singularity is but one example among many geometries that 3 ) dualities, more complicated geometries (such as the Z / Z 3 ], which are mass deformations of , C 36 ) duality directly from that of the parent theory. In certain special , Z , 34 We anticipate that further study of these dualities will lead to new insights concerning In addition to SL(2 The orientifolded As the axio-dilaton corresponds to a holomorphic combination of couplings which is helps to substantially expand thegroups universe of play known a dualities pivotal to role, casesare and seen where to product radically arise. gauge broadens The thefurther infinite contexts study, variety of which in Calabi-Yau could which singularities reveal SL(2 provides furtherwe plenty types of have room of considered for duality here. or further illuminate the dualities not readily derivable fromdications of the further Seiberg dualities dualsrelating whose the known two string orientifolds in theoretic of phase the originthe I nature is literature. of and unclear, origin such We of as also these the dualities. find duality in- both string theory and gauge theories. In particular, on the gauge theory side, our work classes of geometries which generalize larity) also exhibit otherrelated interesting to dualities. Seiberg duality, The at least simplestHowever, from from of the the these perspective of field appear string theory to theory, as perspective, be we closely they argue in are [ new (presumably infrared) dualities ometry giving rise to a dual(corresponding pair to of blowing gauge down theories a relatedesting to two-cycle in the dynamics, the such aswhich a we were quantum-deformed able moduli to space completelyto match for understand between the the the dynamics dual lowest theories. rank of these example It would theories be for interesting larger string coupling as in ordinary Seiberg duality. exhibit these dualities.singularity We will expect exhibit an that SL(2 D3some branes cases it probing is any only orientifolded a self-duality. Calabi-Yau For example, the in the literature [ inherit their SL(2 cases, however, the flows correspondingto to a different single values fixed ofduality the point. gives string rise In coupling to an these converge infrared cases, duality one relating of the two which dual we theories, discuss both in taken the at weak text, the SL(2 richer dynamics. RGE invariant, in generalby these the theories axio-dilaton. will (We flowpart have to of demonstrated a that the this complex fixedacts occurs fixed line nontrivially in line is on a parameterized perturbatively specific the accessible.) example fixed where line, The SL(2 much as in the descriptions we find are differentbranes weakly at coupled the limits orientifolded ofvevs. singularity a These single — features theory valid make — formore it the complementary usually theory clear ranges considered of that of this Seibergto axio-dilaton Montonen-Olive (infrared) duality, differing duality. only by Rather, the it reduced is supersymmetry and more consequently closely analogous which corresponds to the axio-dilaton of type IIB string theory. We conclude that the dual JHEP10(2013)007 ) 44 Z ]. , 110 ]. 112 = 0 dualities. Indeed, this has N ) matter, quiverfold gauge theories also , – 55 – ) or ( , ], based on ideas reviewed in [ 111 While D-brane gauge theories are quiver gauge theories, the introduction of O-planes Finally, given a clear understanding of when dualities are expected to occur in string Other interesting work somewhat related in spirit, although focusing on non-supersymmetric analogues 44 theories. Quiverfold gauge theoriesthan admit quiver gauge more theories. general While gaugeas quiver groups well gauge and as theories adjoint matter allow only and content SU bifundamental gauge ( group factors As the main focus ofumes our of paper D3 is branes dualitiessome probing relating general gauge orientifolded facts theories Calabi-Yau about arising singularities, these on it the gauge is worldvol- theories. useful toleads establish to a slightly more general class of theories which we refer to as “quiverfold” gauge Phenomenology for likewise providingthanks a N. stimulating Hasegawa and for productive kind environment. encouragement I.G.-E. and constant support. A Quiverfolds cent Travel Award. We wouldStrings like to 2011, thank and the String organizerswhich Phenomenology of part String 2012 of Phenomenology for this 2011, the providing work stimulating Nordita was environments string carried in phenomenology out.Theory workshop, and B.H. the would 2012 String also Carg`esesummer Theory, school like and on to the Gauge thank Simons the Center organizers Graduate of Summer School on String under grant PHY-0757868. TheBoochever work of Prize B.H. Fellowship was in supportedsupported Fundamental in by Theoretical part a Physics. Research by Fellowship a The (GrantFoundation John number work (DFG). and WR of B.H. 166/1-1) David T.W. of gratefullyFoundation the was for acknowledges German International Research support Cooperation for in Research this and work Higher Education by and the by a Swedish Lu- of Seiberg duality, can be found in [ We would like to thankholkar, P. Aspinwall, N. M. Berg, Halmagyi, P. Berglund,line, S. M. E. Kachru, Buican, Silverstein, C. F. G. Csaki,L. Marchesano, Torroba A. McAllister, and Dab- L. G. A. Torroba McAllister, and Uranga A.thank N. Uranga for L. for illuminating McAllister Mekareeya, comments and discussions on B. G. the and Torrobawork manuscript. Pio- for C. of We initial particularly Csaki, B.H. collaboration and and T.W. extensive discussions. was supported The by the Alfred P. Sloan Foundation and by the NSF dualities, this seems extraordinarytopic from for the further field research. theory perspective, making it a natural Acknowledgments theory, it might berecently possible been attempted to for construct theOur case examples where work of supersymmetry suggests is broken alarities, by related antibranes or [ program from branes oforbifolds. probing dualities SUSY-breaking from singularities, While anti-branes such string at as theory non-supersymmetric Calabi-Yau seems singu- to suggest that both cases should lead to SL(2 JHEP10(2013)007 0 45 to B A and 0 ]. While A 48 , where 0 A connecting node , which we call a “quiv- to ]. B 0 53 A.3 B → . Note that not every quiver A 9 : X connects 0 A automorphism of the quiver which , respectively. 2 → , we first motivate their introduction by “de- Z B 0 B : A.3 – 56 – and 0 symmetry, which is the inspiration for the term and applying these rules to a few simple examples A 2 X Z A.1 ], these rules are equivalent to well-established results in the 53 ]) is somewhat more intuitive, and we focus on it here for that 113 ) representations. open strings. Thus, the involution also defines an order-two permutation , ) gauge group. Arrows in the quiver, bifundamental matter in the quiver N . As shown in [ the O-plane involution defines a ) or ( A.2 , , the arrow’s orientifold image Rule I: reverses the directions of arrows. B D-brane gauge theories generically come with a classical (tree-level) superpotential, The observations of the last paragraph may be summarized as follows: To this picture, we now add orientifold planes (O-planes). The associated involution, Before discussing quiverfolds in section Such gauge theories cannot be described by standard quiver diagrams (directed graphs), We restrict our attention to supersymmetric brane configurations and orientifolds. 45 , must map the background and the collection of branes onto itself (up to certain signs corresponds to the factbranes must that then not come all in image brane pairs configurations under can thewhich be involution. is orientifolded, determined since by the the geometry and brane configuration. Since these objects are An example of thehas resulting an involution involution, is inbe shown the in isomorphic sense figure to defined its above. charge conjugate A (the necessary same condition quiver is with that the the arrows reversed). quiver This oppositely oriented on the arrows ofnode the quiver, suchare that the for orientifold any images arrow of the nodes intersections. σ and orientations), and squarespermutation to on the the identity. nodes Thus, of the the involution defines quiver. an Moreover, order-two the involution maps open strings to A.1 Orientifolding aWe quiver consider gauge a theory quiver gaugeground. theory describing Each a node collectionassociated of in U( D-branes the probing some quivergauge back- corresponds theory, to correspond a to stack open of strings identical stretched D-branes, between the with stacks an of branes at their literature on orientifolding toricthe Calabi-Yau brane tiling singularities method using hasour some brane approach computational tilings (following advantages relating [ [ to thereason, superpotential, deferring further discussion of brane tiling methods to [ ing” a quiver in“quiverfold”. half along a line of riving” a set of rulesfree) for quiver obtaining gauge orientifold theory gauge in theoriesin section from section their parent (orientifold- and we develop a more generalerfold diagrammatic diagram”. notation in One section can showquiver that can any be connected thought quiverfold of diagram (in which a is precise not way a which strict we later make clear) as the result of “fold- in the ( allow SO and USp groups, as well as two-index tensor matter and bifundamental matter JHEP10(2013)007 The 46 up to some W ]). 69 symmetry. ]) for a set of rules relating is equivalent to 0 113 W = 1 description with three adjoint color-conjugation N parallel D3 branes in flat space, which is , where 0 N W → – 57 – W ]. 114 orientifolds the orientifold gauge theory is derived from the parent theory by the superpotential of the parent theory is invariant under the involu- ) super-Yang-Mills. This theory has an (but potentially insufficient) conditions which must be satisfied by consistent = 4 N singularity [ 2 . An example of an involution of a quiver. The quiver theory pictured here describes the N Rule III: gauging the involution. Rule II: tion. P dP We have presented a heuristic argument (following [ Since in general there are multiple gauge groups, the theory can still be chiral (cf. [ = 4 SU( necessary 46 A.2.1 We consider the worldvolume gaugeN theory of the worldvolume theories of stacksorientifolds. of D-branes To to the the extentas worldvolume that gauge these theories of arguments their orientifold hold, involutions. the We above now illustrate rules these should arguments be withA.2 viewed a pair of Examples examples. Note that the abovethe rules gauge should group only ranks compatible beparent with and interpreted anomaly orientifold at cancellation theories, the are correspondingO-planes. classical generally to level. different the in For tadpoles the instance, (RR charge) carried by the orientifold theory results frominvolution. identifying This the can chiral be and restated vector as: superfields related by the Notice that if we impose the samethis requirement implies on that the (generally the unknown) corresponding K¨ahlerpotential, gauge theory has a appropriately covariant under thealso involution, be appropriately we covariant: conclude thatsymmetry is, that transformation. the In superpotential the examplesrestriction must which is follow, in we fact: shall see that the appropriate Figure 9 toric JHEP10(2013)007 ∈ is 1). i 4 , M (A.4) (A.6) (A.1) (A.2) (A.3) i ± , ψ ) gauge 1 ). Con- 3 N N ± , 1 2 ± . The remaining , 1 1 ) or USp( 1 = ± N M . In the case where = diag( M symmetry is manifest, where i j ± to be unitary, which can be di- , R i j =         . ). We then choose the involution T , . . k , ··· N M , ∗ Φ † j I Φ T 1 0 Ω (A.5) i M ) T j T − T A ψ , the corresponding Weyl fermion projects ) gauge groups are possible in perturbative A ( i j Tr Φ Ω N I 1 0 0 0 MA 1 so that . . . 0 1 0 0 0 00 0 0 1 i j ijk → − − – 58 – → ±  Λ         A 1 3 A = → ± → ∗ subgroup of the SU(4) i = A ), and we choose = Ω, respectively, up to an overall factor which can be ψ R W I N = Ω = , and the superpotential: ) and USp( MM } or N 3 U(1) M , 1 2 × , transformation, taking the form Λ = 1 R I ), SO( ∈ { . N 3 i / , i +2 . Invariance of the kinetic term requires Λ i j is antisymmetric, it can be put into the form M must be unitary to leave the gauge kinetic term invariant. Since the involution acts on the gauge indices. Invariance under the remaining SO( ). M I N 4: We now consider the action of the involution on the (adjoint) Weyl fermions We consider orientifolds of this theory, imposing the rules from the previous section. transforms as i ... where symmetry requires that absorbed into Λ agonalized after an SU(4) For each positive (negative) eigenvalue of Λ USp( 1 and the remaining unbroken gauge symmetry is USp( once again to ensure that the invariant gauge bosons correspond to the generators of to ensure that theversely, invariant if gauge bosons correspond to the generators of SO( where squares to the identity, wesymmetric, it find can be diagonalized byunbroken a gauge gauge transformation, symmetry giving is SO( We first consider theonly action (products of the of) involution U( string on theory. the In gauge particular, bosons. the involution It must is act well on known the that gauge bosons as follows: up to a superpotential couplingthis which language can only be an SU(3) removedΦ by rescaling the fields. However, in chiral superfields Φ JHEP10(2013)007 +) = 1 , = 2 (A.7) (A.8) (A.9) + (A.10) , N N ). This ) gauge + )) world- , N ) and Λ = A.9 N is invariant. − , W i − ∗ , I = 1 language, the − . Thus, invariance ) (USp( T , 1 ) gauge theory and ) = 4 USp( N 0 − N k Sp N N ± (Φ ) gives rise to an I = ∗ + ∗ I = 4 SO( T ) II (O7 = 4 SO( 0 j . N . − n ) WLOG. In N i 1 (Φ 4 ). We form gauge invariant single i Φ I − , ± ∗ for instance, we would have obtained Φ ∗ j 0 I A.7 ... I T Φ ) = 2 T ) W k i 3 0 ) i brane can be computed directly from the j Φ ± 1 i p (Φ 1 (+1) to form a vector multiplet with the )( (Φ Tr Φ → − ]: a restriction on the form of the involution, 2 I I − h ± i j ijk 48 Tr Φ – 59 – W  ˆ Λ )( W, ) gives rise to an Tr 3 1 0 ≡ ) + k k ± → ˆ Λ) n Sp ˆ Λ i )( 0 j j ...i ± (O3 Φ 2 Sp D3 branes atop an O7 ˆ i Λ 0 1 − i i ± ) det( i ˆ Λ)( ( ˆ N Λ Z Sp ) gauge theory with a hypermultiplet in the antisymmetric ± ijk det( N (  ). The superpotential transforms as: 3 1 3 1 3 − ± , +), corresponding to the spectrum of an = = = 2 , 0 = 1 supersymmetry, which are not realized in string theory as the 1) for an SO (USp) projection, so that ± + , , W − 1 N = 2 USp( − ) gauge theory with a hypermultiplet in the symmetric representation, ± , → )) worldvolume gauge theory, in agreement with the sign rule ( )) gauge bosons, which we take to be ( N − ), corresponding to the spectrum of an N N N W − ). For an SO projection, the possibilities are Λ = diag( , ) is +1 ( D3 branes atop an O3 − Sp = 4 language, the above sign rule amounts to the requirement det Λ = 1, since = diag( , Sp ± i j ( + N ± ) (SO( N , ˆ Λ = 2 SO( − ) (USp( N In fact, the geometric involution of the O By comparison, D3 branes are mutually supersymmetric with coincident O3 and O7 In N = N under the involution. Had we imposed 4 action of the involutiontrace on mesons: the open string fields, ( agreement relies on our choiceW of rule II asspectra the correct with restriction only on theworldvolume transformation gauge of theory ofbackground. a stack of D3 branes coincident with an O-plane in a flat theory and an representation, respectively. planes: volume gauge theory, whereas USp( ± diag(+ an respectively. Similarly, for theand USp Λ projection, = the diag( possibilities are Λ = diag(+ This is our firstand example thus of the a spectrum of “sign the rule” orientifold [ theory, due to the requirement that where ( of the superpotential requires that where supersymmetry, at least oneSO( sign must be remaining signs specify the action of the involution on the adjoint chiral superfields: down to its invariant symmetric (antisymmetric) component. To preserve at least JHEP10(2013)007 , i i z z (A.14) (A.15) (A.16) (A.17) (A.11) (A.12) (A.13) → − +) with i is totally , z is positive + n , where the , i 3 ...i − ± φ 2 i 2 1 i , φ 1 1 which appear in Z 0 n φ − ˆ ...i Λ = ( ), the superpotential 0 2 i = 0 1 )Tr i 3 2 A.9 Z ± i )( 0 n n i 2 i , ˆ ± Λ ) 1 )( , corresponding to an O7 plane ) is exactly that required by the φ 1 3 ] = 0, so that Sp ), we obtain: 2 j z corresponds to the coordinates ± ± φ , i Φ ( 3 )( , A.7 h k → A.17 φ Z i . , φ Sp , 3 j j 3 ... z Tr z ± , T φ φ i j i i ) 1 2 , 2 ∗ 0 2 j ˆ 2 φ 2 Λ φ i I i ) z φ i = ( 1 − ( ˆ Ω. Thus, imposing ( Λ φ φ Tr 3 Sp ) 1 i j − come from factors of Ω → φ φ ˆ ± Λ Sp ijk 2 → 2 2 by convention, the overall normalization being ( – 60 – 2  = φ / Sp i φ ± z +) with an USp projection, we obtain = 1 6 1 = Tr ( 3 , T , Φ ± → i is symmetric (antisymmetric) when φ φ 1 i φ + = i h z i W , z 0 1 1 Tr i φ i 1 2 W ˆ Λ ), ) → − 3 = Sp 1 and following the same procedure, we would obtain a van- ± z ˆ , Λ = (+ ± W 2 ( h W ± , − 1 → ) comes from Ω ± implies that Tr = n Sp 0 ...i T ± 2 i W 1 M i = ( = 2 cases considered above. For example, choosing Z 3 ) N = Tr ˆ Sp Λ = diag( ± in which the D3 branes are embedded. Thus, the geometric involution is simply: M 3 C = 0, whereas choosing To obtain the superpotential of the orientifold theory, we replace the fields with their 1 = 4 and z whereas imposing ishing superpotential. Moreover, theextended superpotential supersymmetry ( of the corresponding brane configurations. while Tr first sign ( reduces to where for USp projectionsfields, the and trace we include implicitly an includesarbitrary extra factors up factor to of of 1 field Ω redefinitions. between each Written out pair explicitly, of we obtain: so that for (negative), as previously noted. Applying this replacement to the superpotential, we obtain projections: where invariance under the involution requires that N an SO projection, weat obtain corresponding to an O3 plane at the origin. of the It is straightforward to check that this reproduces the O3 and O7 involutions for the symmetric in its indices. Acting with the involution ( modulo F-terms, where the extrathe signs trace for symplectic projections. Geometrically, Upon imposing the F-term conditions, we obtain [Φ JHEP10(2013)007 ), × N (A.22) (A.19) (A.20) (A.21) (A.18) SU( , so that , 1 ∗ 1 (which we → ∗ 1 I = 4 I 3 T ) T 2 ) ) N 0 j 12 k 12 X X , with the orbifold ( SU( ( 3 . 32 3 Z 32 × δ / / δ † 2 3 ) +2 ) ∗ 23 j C i δ N T (Λ ) symmetry, we take the fixed 0 . j 23 3 0 . → k Z 0 0 X i j 31 ( i 0 2 31 . i , X 23  0 δ ) k 31 j 23 B,C breaks SU( . ∗ 32 Sp X X ,X SU(N) transform as ) δ 0 ∝ (leaving the superpotential invariant). 1 ± j 23 k 12 T ∗ 23 ( Sp ) A δ X X i i = 0 AB 31 − ± i 1 31 T i ( 12 ) X X where the phase factor can be removed by T Tr = 32 X 2 − X j 23 ( δ † † 1 , then ) ) 47 0 0 iθ/ X 0 I SU(N) – 61 – Tr = k k ( k k 0 − k Σ, j e 0 23 Tr i ijk 23 iθ (Λ δ  (Λ  0 † δ 0 e = 1 quiver gauge theory, shown in figure , we search for orientifolds of this configuration. In- 3 j j ) i j j j → det Σ = 0 ∝ = k Σ Σ 0 k Σ N 0 k k 0 i i i i 31 i A.1 C W (Λ → 0 Λ , with the involution of interest indicated by the dashed line. 0 SU(N) Λ j j X 3 j 3 j B i 23 Z Z 0 Σ ijk ijk i i / 0 /   i 3 i 3 A ) and C Λ C ijk Sp ,X  i 12 ijk ± ∗ 32 6=  X D3 branes probing the orbifold singularity δ for convenience), is well known. The corresponding superpotential is: 2 T = = ( . The resulting ) i N iθ/ 10 0 z j e 31 3 W X ( πi/ → 1 2 → I e or Ω, depending on whether we choose an SO or USp projection for the fixed i j . The quiver for 12 i 1 W → Λ X i = symmetry is manifest under which the z → 1 R I We compute the orientifold image of the superpotential: Applying the rules of section i 12 In general whenever 0 X 47 Figure 10 rotating Thus, we take Λ = Σ, where the invariance of the superpotential requires Therefore, invariance of the superpotential requires: In fact, this is only possible if Λ = node, respectively. Moreover, sinceΣ the is involution both squares to unitary and the Hermitian. identity, Σ node to be node 1 WLOG. The action of the involution on thewhere chiral Λ superfields is and then: Σand are unitary matrices, up to a superpotential couplingU(1) which can be removed by rescaling the fields.specting An the SU(3) quiver, oneone can node easily and exchange check the that other rule two. I As implies the that quiver has the a involution must fix Next, we consider action reproduce in figure A.2.2 Orientifolds of JHEP10(2013)007 , i ]. z . 49 3 z ) and +) for (A.29) (A.26) (A.27) (A.28) (A.23) (A.24) (A.25) +) are , → − − , → , i + − z , 3 − is positive , z , i − , , − 2 ± 0 z k 0 j 0 i → ) and (+ Z 2 − ] z 0 , , k k 1 − ; thus, we read off the z , 3 )Σ − Z Sp / ), and the requirement that , 3 so we will focus on the first 3 ± → − T +) and Σ = diag( C ± 1 ) , ) for SO and USp respectively, 48 ] [( l , z 0 2 . + − j of j A , , is totally symmetric in its indices. ( 1 ± . j k , − )Σ , ]. We verify this by computing the − z 1 , B k 31 ijk ∗ j 1 . i Sp , ± , 49 z I j Z X i A ± T z T ) = z ∗ 23 j ) i j 23 ) δ j , j Tr ] [( Sp X j 0 j l i B k A i )Σ ± ( i ( 12 B A Σ j )( Sp i i j i k )Σ X 32 3 +) and (+ ± ijk Σ Σ δ 1 which appears in the trace for USp projections. , Sp – 62 – ( ±  Tr − = Σ + ± )( → → → , is symmetric (antisymmetric) when 2 → i ≡ = i − i ± B i i i 12 23 31 z 2 = [( ijk B )( 0 X X X 1 (det Σ) k Z 0 1 2 ), ± j 0 2 ( i = ± ), we obtain: Z , 0 j 2 j W ± Σ 0 A.19 , j j 1 Σ ± 0 i i comes from the Ω correspond to the coordinates )Σ Sp Sp +) up to an SU(3) transformation. The spectrum of the latter theory turns ijk ± ± , ( Z + , → − ijk Z To derive the superpotential of the orientifold theory, we replace: The anomalous orientifolds correspond to noncompact O7 planes, whereas the remain- The anomaly can be cancelled by introducing noncompact “flavor” D7 branes into the geometry [ 48 so that for Σ(negative). = Applying these diag( replacements to the superpotential, we obtain: where invariance under the involution requires that: whereas the anomalous orientifolds, ( correspond to noncompact O7 planes, with the involution geometric involution From this, it is easy toSO check and that USp the respectively, anomaly-free correspond orientifolds, to ( compact O7 planes, with the involution Applying the involution ( where the sign The mesons ing possibilities correspond togeometric compact involution. O7 Consider planes mesons of [ the form: Upon imposing the F-term conditions, we find that out to be anomalouspossibility. for Similarly, for any an choice USp ofpossible, projection, gauge again Σ group = up diag(+ ranks, tochoice of an ranks. SU(3) transformation, with the latter being anomalous for any the superpotential be invariant takes the form of a sign rule: Thus, for anΣ SO = projection, diag( there are two possible involutions Σ = diag( After an SU(3) transformation, we obtain Σ = diag( JHEP10(2013)007 49 . 9 , this (A.30) (A.31) T M is invariant ]. = Tr W in a straightfor- 53 M W. charge conjugation A.1 2 Z )(det Σ) Sp ± ( +) for SO and USp projections, , − ] and reviewed in [ + , = , k 48 T ) B i j A A ( i m A B l ijk  A – 63 – 1 2 Tr = j m ) is necessary to ensure that the superpotential of the ) and Σ = diag(+ Σ W ) matter in the usual way, while each arrow intersecting l k − ¯ , Σ , − A.22 , ijk  − ) node along the boundary (“half” node) corresponds to an − . (det Σ) 3 flavor symmetries as the parent quiver theory, and are discussed more 1 2 ) R Sp ± U(1) × = ( W such that this symmetry is manifest as a reflection through a fixed line, as in figure To obtain the gauge group and spectrum of the orientifold theory we cut the plane in Rule I implies that the quiver of the parent theory possesses a For the purposes of this discussion, we mainly ignore the superpotential, though we For the cases Σ = diag( 2 While this is always possible to do, in general there are many possible embeddings. For a fixed involu- R 49 group, whereas eachSO + (USp) ( gauge group.corresponds Each to arrow bifundamental away ( from the boundary (“uncrossed” (whole) edge) tion, all embeddings will give the same quiverfold, as discussed below. two along the fixed line,(fixed) discarding edge one along half the of boundary it withwe a and call sign. labeling a The each “quiverfold”, resulting node diagram specifies andfollows: on the perpendicular the each half-plane, gauge which node group and away from spectrum the of the boundary orientifold (“whole” theory node) as corresponds to an SU gauge (arrow reversing) symmetry representing thein involution in question. WeIn embed the the resulting quiver figure, fixed nodesit must lie perpendicularly, along the whereas fixed line, anyanother and unfixed edge fixed edges (its edge will image) intersect which at crosses the the point of fixed crossing. line must intersect emphasize that rule II isleave generally the very restrictive, superpotential and invariant. notunder all An involutions the of explicit involution the computation can quiver to will to be check brane tedious, that tiling and methods, for as toric originally singularities formulated the in problem [ is well suited In the simple examples discussed above,ward we (if applied tedious) the fashion rules to ofof section rederive this known program results. applied Wederive now to the discuss gauge arbitrary some group quiver general and gaugediagram, features spectrum and theories. of define the Specifically, a orientifold suitable we theory generalization graphically show using of how the the to quiver quiver to represent these data. where we leave the contractionssame of SU(3) gauge indices implicit.thoroughly in The section resulting theories have the A.3 General quiverfolds respectively, the superpotential simplifies: can also be written as: Thus, as before, theorientifold sign theory rule does ( not vanish. where for USp projections the use of Ω in the trace is implicit. Since Tr JHEP10(2013)007 ), ¯ , ¯ . (b) The ) or ( 9 embeddings , (b – c), the 2 R 11 SU SO USp ( , ) ( , ) ( , ) (c) ) edge ending perpendicularly − (a). As shown in figure 11 symmetric embedding of the quiver. In a 2 (b) Z – 64 – it is straightforward to show that different 50 – – + + (a) . (a) An example of a quiverfold. The parent quiver is shown in figure Note that some apparently different quiverfolds are isomorphic. In particular, any An example quiverfold is shown in figure There is moreover an isomorphism between a crossed edge connecting a whole node to itself (or a whole 50 quiverfold. Moreover, given a quiverfold, it is possible toedge uniquely connecting reconstruct a the half-nodethe to parent itself) node in and two question.related half While by edges the a of involutions nonabelian opposite which flavor sign give symmetry and rise of like these orientation the configurations connected parent to appear theory, and different, the they resulting are spectra are the same. conjugation can only beof applied a to quiverfold is theequivalent. directed quiver Thus, at as edges connecting both a a half ends,cannot whole. node since be to a Furthermore, arrows crossed), whole not entering whereas nodeaccount and have every edges a these exiting connecting edge single isomorphisms, direction two half (they half-nodes nodesof are are the undirected. same Taking into involution (with the same choice of fixed-element signs) lead to the same whole node of the quiverfold canwith be charge different conjugated, yielding crossed a new,of and equivalent quiverfold uncrossed a edges; nodestrict this and quiver, corresponds its there is to image no swapping in analogous the the operation: positions original since crossed edges are not allowed, charge “enhanced” quiver, with aas few additional the representations worldvolume and gaugea gauge theory quiver groups on gauge allowed. intersecting theory, Just D-branes orientifoldsquiverfold of can (to these always the configurations be extent canstating that represented the always rule by be gauge I group represented holds), and by which spectrum. a is then a very useful tool for concisely depending on the orientation of the arrows.on the Finally, each boundary + ( (“half” edge) corresponds to symmetric (antisymmetric)quiverfold matter. can be drawnnote without the the fixed boundary elements line and by crossed using edges. appropriate From symbols this to perspective, de- a quiverfold is just an Figure 11 quiverfold can be redrawn without the fixed line using appropriate symbols,the defined in boundary (c). obliquely isedge) with joined opposite orientations to associated its to each image end, and arrow corresponding to to ( form an edge (“crossed” (whole) JHEP10(2013)007 j i ). N ^ subject ) gauge SO( ) groups N 51 ∼ = N − ) N ], this approach − ) antisymmetric 53 N ) and USp( ), USp( N N ) theories we can likewise N ^ USp( ] for more details and further by the USp( ), SO( ∼ = N ab is antisymmetric). 115 ) δ N M − ) and USp( N where : SO( dual in the physical sense. In particular the N M . This leads to a new gauge theory we denote − – 65 – N not by − ], for SO( N 116 with N ) symmetric bilinear invariant ] for further details and references. As shown in [ ) and (in the latter case) the alignment of the flavor rotations Λ N and replacing 48 ab ) gauge theory and is often referred to as negative rank duality although A.23 ) gauge theory with matter in certain representation we can exchange N N ), ( ^ SU( A.9 , the invariance of the superpotential imposes important constraints, such as the . j i ). As was first noticed by [ For an SU( It is possible to reformulate rule II graphically by describing the parent gauge theory While quiverfolds are useful for computing and representing the gauge group and It should be emphasized that just as a quiver is a direct pictorial representation of a A.2 N If the loop includes a half-edge, it doubles back on itself at this point, reentering the same node it just 51 ^ obtain a negative rank dualand theory replacing by exchanging the symmetrization SO( andbilinear antisymmetrization invariant Ω exited. two related gauge theories have generically different anomalies. symmetrization and antisymmetrization (i.e. reflect theand Young at tableau across the the same diagonal) SU( time replace to negative rank thatthe turns main out text. to bereferences. We very As refer useful we the in explain the readertheory below, to anomaly to this an matching continuation chapter discussion relates 13the for in in two example related [ an theories SU( are generically theories. B Negative rank duality In this appendix we review a fact about continuing SU( and the involution ininterested terms reader of to a [ braneis tiling, equivalent rather to than the a onequiverfold outlined quiver diagrams here. diagram. provide an Regardless We intuitive of refer andspectrum the the precise method of representation used of the to the orientifold apply gauge these group gauge and rules, theory, much like quiver diagrams for D-brane gauge a superset oftion those involutions consistentsign with rules both ( rulesand I Σ and II: as we saw in sec- quiver diagram, gauge invariant (mesonic)to operators the are requirement loops that inedges. the the However, quiverfold, loop in some enter casesdue and the to exit mesonic symmetry operator each (e.g. corresponding whole it to takes node such the a on form loop oppositely Tr vanishes spectrum directed of a given orientifold, the set of involutions consistent with rule I is usually on the boundary, as above. certain class of gauge theoriesrepresentation (quiver of gauge a theories), certain a (somewhat quiverfolderfold broader) is gauge class also theories. of a Just gauge direct as theories, pictorial gauge which invariant we (mesonic) call operators quiv- are directed loops in the quiver and involution by embedding the quiverfold on the half-plane with fixed elements JHEP10(2013)007 N (B.2) (B.1) denotes r . However, 1 and similarly are the gen- N scaling for the − a r ), we us the fol- . The leading which completes T N . N 1 2 , which is true for N 1 1 ) = − boxes in the Young boxes and ˜ r − p (˜ p p p p ) = . In particular we should N N p N ( − ∝ ∝ T A 1 ) ) r − r ( p ( ). Together with the fact that , where the A 1 T 1) r ab = 1 and not ( − δ p ) independent A . ) r fundamental representations. The ) ( term 2 N r r p ) = ( (˜ T ( scaling so that the overall sign under changing r N d N ( scaling by calculating the Dynkin index p − = N 1-times the tensor product of the above N d N )+ ) and taking the tensor product of a fun- N p 2 r − boxes. Similarly one can explicitly work out n m ) as having r ( ) 1) 1)), which leads to ( p 52 p b N r T − A − T ) – 66 – ( ,A : any representation with 1 in p ) m n N r ) under the negative rank duality. These are again ) ( r ( ) = ( boxes is the same as the leading r 1) (˜ a N d r ) with ( − r − N ( T , p which has a Young tableau with A N 1 − N r ). For the anomaly coefficient of SU( = 2. ) = T d 2 1 p N . Taking the tensor product with another fundamental r − r ] that we mentioned above, the proof is simple since we only scaling. Contrary to the dimension the Dynkin index and p ,..., ⊗ 1 1 1) , N r 115 − ( A scaling that is given by ]: ) = ( diag(1 N ) and USp( fundamentals. Taking r 1) fundamental representation leads to the above result. Explicitly, 1) ( 118 N we obtain a generator for the representation that is given by the tensor ) is defined by ( p − , which gives the stated result. − N r p N T ( N p + 1 1) T 2 − 1 1) fundamentals and we find the leading − ] it is proven that under these dualities any scalar quantity becomes the dual p independent this leads to the leading is ( N ) and anomaly coefficient ) we can choose one of the generators in the fundamental representation to be r ] for related results). For that we need the transformation properties of the Dynkin 2(( 115 N ( N N T √ 117 We now show that for any gauge theory the negative rank dual is free of gauge anoma- Below we study the anomalies of negative rank dual theories of a generic gauge theory In [ ) is This statement only holds for representations with fixed, = so that one finds → − 52 ( 1 product of all irreducible representation of SU( the scaling for SO( lowing result from [ A lies if all chiral matter representations have dimensionsthink that of are the even anti-fundamentalfor under representation the the of adjoint SU( negative representation we take for SU( T scaling for any representationtensor with product of generator with the proof. and anomaly coefficientDynkin for index the tensorerators product for of therepresentation representation introduces a factordamental of with ( similarly to the dimensionN any extra box in the Young tableau leads to an extra factor of To prove this, we can again derive the leading (see [ index determined by the leading anomaly coefficient of the fundamental representation are independent of Thanks to the theorems ofneed [ to determine thetableau overall has sign a leading ( N scalar quantity up totransforming potentially in an the representation overall sign.the transposed In tableau particular, obtained ifcorresponding by we representations a have are flip a related across by matter the field diagonal, then the dimensions of the JHEP10(2013)007 × ) = r ... (B.6) (B.7) (B.8) (B.3) (B.4) (B.5) ( . The . The a a × /d G G ) 1 r ( T = U(1) G , ) under the group , change sign. This can lead to an χ . , χ and the matter dimension by k χ gauge q χ i ∀ )) χ j q 54 , G , a q ( χ χ i i do not change sign under the negative )) χ χ i i q d G q a , q q ( χ i ) ) 53 G q χ 2 χ + ) r χ χ the r 3 ( χ ( ( ( 1) i ( d d . T A 1) d i − ) ) ) a a χ χ − χ . Whenever all chiral matter fields satisfy R X X G G χ χ χ q χ χ ( R r r ( i ( ( q d charges by = = q d d i )( ) i k – 67 – χ χ ) to find that both of the anomalies above pick χ χ ( ( r X X d d (˜ U(1) U(1) = = are SU, SO or USp groups. We denote the chiral χ χ /d j ) X X i 3 a a r (˜ G = = G A U(1) denotes the representation of 2 R 3 R U(1) i − a 2 a G χ G symmetry a negative rank dual theory is also anomaly free if all matter r U(1) U(1) U(1) ) = R i r , where ( n /d G undergoes a negative rank duality we use the fact that ) U(1) r ) where × a ( , the corresponding U(1) a χ G A G χ anomalies are r ... 3 ( d × G 1 ), then the above anomalies are unchanged after dualizing any of the =1 ) and n a χ G r (˜ (˜ Q d × and U(1) anomalies are given by /d does not undergo a negative rank transition then the above anomalies are unchanged. ) m 3 r Next we calculate the anomalies that involve the R-symmetry We take the combined gauge and global symmetry group to be a ) = ) = In the absence of an U(1) We assume for simplicity in the discussion below that the (˜ U(1) and G χ χ 53 54 2 T ( ( representation have dimensions that are oddpick under the up negative an rank dual. extra overall In minus that case sign. all globalrank symmetries duality. This conditionextra can overall be minus relaxed sign so in that global for anomalies fixed involving U(1) that do not involve theexamples R-symmetry the still global vanish non-abelian after gaugeso the groups that negative will none rank not of transition. undergoby the In replacing a global the our negative anomalies ranks rank pick of transition their up all negative the an ranks. gauge extra groups minus that sign. undergo the They negative are rank simply duality given with If In the case that − up an extra minus sign. In particular this means that all the gauge and mixed anomalies G where the sums ared over all chiral superfields U(1) matters fields by d U(1) of all gauge theoriestwo we theories are are related dualizing bywith is replacing its even. the negative rank and Furthermore, ofa adding the each non-abelian an gauge global gauge overall group anomalies group minus factor of that sign we is are the whenever also dualizing the being global dualized. anomaly involves rank dual i.e. whenever for every chiral field the number of all boxes in the Young tableaux JHEP10(2013)007 ) is N a − (B.9) G (B.10) (B.11) . Since , where SU( , if /b N ) a × µ ( G undergoes the πiτ 2 . a + 4)) ) µe a G N G ( ≡ − (Adj , T ) is SO( 1) + ) is odd, if N gauge a − G G + 3) we see that the two negative , ( χ R ) = 0. For the SU, SO and USp d q a N 1) ( G )( (Adj a − N T G χ 1) + χ R r ]), holomorphic couplings are not pertur- q (Adj ( − ( 73 j T T χ R q ) becomes the symmetric representation of ) i q ) a q N ) )( G χ χ ( r – 68 – − χ χ ( d ( ( d d d ). We also have to flip the Young tableaux so that undergoes the negative rank transition. We thus χ χ χ N a X X X G = = = SU( 3) which becomes is part of the global symmetry group, then there are no ) extrapolated to negative = 2. This means that only the last of the anomalies above ) denotes the Dynkin index of the adjoint of R R R − × a a N p G G N ( dual in the physical sense since they have different anomalies. SU( U(1) U(1) U(1) + 4) ) with the holomorphic dynamical scale Λ N j 2 a (Adj × µ ( N G not T τ 4) U(1) − i N ) denotes the dimension of the entire gauge group (excluding the global U(1) ) is even and as mentioned above for both theories, since they are related by changing the sign of gauge gauge 3.1 G . Both theories are related by taking the negative rank dual of both gauge group ( G ( d 3 d ). The usefulness of this duality is that we did not have to calculate the anomalies We focus first on the case where there are no nonabelian flavor symmetries. Due to A simple example of two negative rank dual theories has already appeared above in N various nonrenormalization theorems (see e.g.batively [ renormalized apart fromThus, the in the one-loop absence runningcouplings of of nonperturbative are holomorphic renormalization independent gauge ofphic of couplings. these gauge couplings, scale, couplings holomorphic provided we replace the scale-dependent holomor- C Exactly dimensionless couplings Under certain assumptions, a sufficient conditionalong for the a RG holomorphic coupling flow to isthose be which that constant are it spurious be and/or neutral anomalous. under all possible flavor symmetries, in particular the anomalies depend on rank dual theories are In this particular caseshifting the the negative ranks rank of dual the is gauge however groups. dual to the original theory after factors. The SO( which dualizes to USp( the antisymmetric representationSU( of SU( in section can furthermore conclude that allby anomalies replacing of the the negative ranksnegative. rank of dual theory all are gauge obtained group factors that undergosection the transition with their Thus negative rank transition since picks up an overallconclude that minus all sign gauge and if none mixed anomalies of vanish the after the global transition. non-abelian In symmetry the case groups where undergo a negative rank transition we Above symmetry group) and part of the gaugegauginos group. that contribute If andgroups we the have group to dimension set has always even parity under the negative rank transition. JHEP10(2013)007 R 55 (C.1) chiral χ (simple) N G N symmetries R + 1) matrix of U(1) χ N = 0 then Λ is ill-defined symmetries (not broken ( b R × . ) W N + 1) χ + N G ( N − 3. / W N + anomaly to be a “good” U(1). G a straightforward counting argument gives the – 69 – + 1 linearly independent spurious and/or anoma- N 2 global χ 56 + N U(1) U(1) N gauge = 0 N is a linear combination of an arbitrary U(1) with the “canonical” R linearly independent “good” U(1) or U(1) holomorphic couplings are neutral, and can be represented by vec- symmetries in general, whereas the “good” U(1)’s are those under superpotential terms (each with a corresponding coupling), W R U(1) + 1 which are annihilated by the ( N W N (mat) is the one-loop beta function coefficient (if χ N + T N G − of exactly dimensionless couplings of this type for a model with N 0 ]. However, if a certain holomorphic combination of couplings is neutral under ]. N 73 considered above), assuming that none of the constituent couplings are nonpertur- (Adj) itself is independent of scale). 119 T R If the gauge group is semi-simple, Thus, a holomorphic coupling corresponds to an exactly dimensionless physical (canon- However, non-holomorphic couplings are not likewise protected against renormaliza- τ This is closely related to theIf criteria the for gauge an group exactly contains marginal a U(1) operator factor, at then the this superconformal argument fixed still applies so long as we consider a 55 56 = 3 charges of the holomorphic couplings acting on the left. The rank of thispoint matrix [ is therefore global U(1) with a nonvanishing U(1) The argument is as follows:lous there U(1) are orwhich U(1) the tors of length number gauge groups, superfields, and by gauge anomalies or by the superpotential): marginal holomorphic coupling which violates theseon conditions imposes the a anomalous nontrivial relation dimensionsnot along be the computed flow. exactly awayextended from Since supersymmetry an anomalous must infrared dimensions satisfy fixed typically these point, can- computable conditions. examples without ically normalized) coupling if it(since is an neutral arbitrary under U(1) allU(1) possible U(1) and U(1) batively renormalized. While the converse need not be true, the existence of an exactly all of these U(1)’s,physical then coupling it is is scale unaffectedpling independent by (has is the vanishing exactly rescaling, anomalous dimensionless andshown dimension). if therefore to and Such the be a only corresponding cou- equivalent ifunder to it which the is all requirement classically chiral superfields that dimensionless. carry the This charge coupling is +2 is readily neutral under the U(1) phic couplings, leading tocounterparts. a nontrivial In running particular, for thesymmetry their rescaling under physical which may (canonically the be chiral normalized) superfield realizeding in as U(1) question a is may charged, complexification be whereasspurions of the spurious (superpotential a correspond- couplings) and/or U(1) and/or anomalous, the holomorphic leadingtheory dynamical scale(s) to [ of a the gauge rescaling of the corresponding but tion, and inthrough particular corrections chiral to superfields thecanonical normalization K¨ahlerpotential. are leads to subject rescaling Rescaling anomalies to the which alter wave-function chiral the superfields renormalization values of to the holomor- restore b JHEP10(2013)007 . W as F N F G G , we need only multiplets, F F G , with poles in the G ) covariance requires that Z , ) (C.3) meromorphic τ ) = 0 due to the lack of perturba- ( f ∞ 2 i ) (C.2) − ) symmetries which commute with (+ d 10d f τ R + ]. ( , where SL(2 f 2. However, no such holomorphic modular cτ N 120 = − symmetries as a sufficient condition for exact = ( – 70 – R 10d  τ b d d dµ counts the number of irreducible + + µ ) = 0. χ of weight cτ aτ ∞ invariant combinations of superpotential couplings, and N i  ), reproducing the above formula. 57 (+ F f ) — which is not perturbatively renormalized by the above f G U(1) N 3.20 ( − = 1 super-Yang-Mills [ 10d cannot depend on any other holomorphic couplings due to constraints τ N + 1 where the beta function blows up at finite coupling. Such poles signal a f which is annihilated by the same matrix acting on the right. Since row χ H N . By contrast, an exactly dimensionless coupling is a product of holomor- W ( N − ) is a holomorphic function satisfying -invariant combinations of superpotential couplings. We can then treat denote the semisimple component of the spurious flavor symmetries. Since only U(1) + F ) is a modular form W F N 10d τ G G τ N ( G ( − N f f counts the number of “good” U(1) or U(1) + We first consider the SO theory for even In particular, for the gauge theories studied in this paper, we are interested in whether Let These arguments must be modified to include any (potentially spurious) non-abelian G + 1 In fact it is a cusp form, since -singlet combinations of couplings can appear in our candidate exactly dimensionless N 57 U(1) χ F Hence form exists. Instead,upper such half a plane modular form is necessarily tive running, and imposed by the spurious and/or anomalous U(1) symmetries. criteria — can run nonperturbatively. Abeta spurion function analysis reveals must that take the the exact (Wilsonian) form: where turbative effects. While itconstrained is by nonrenormalization not theorems, possiblecases, and to such are exclude as known this pure to in be general, absent such in effects somethe are simple string also coupling the number of independent N C.1 On nonperturbativeSo effects far we have ignored the possibility that the holomorphic couplings run due to nonper- G couplings and the holomorphicconsider gauge couplings areif all it neutral were under gaugedargument (without still the holds, corresponding where gauge now coupling). Thus, the above counting flavor symmetries, since chiral multipletsject with to the same kinetic gaugecandidate mixing quantum numbers combination (unless are of forbidden sub- couplingsmetries by in must the addition also global to bemarginality. the symmetries). neutral U(1) under and In U(1) these particular, non-abelian the sym- phic couplings which is neutrallength under all the U(1)’s,rank and and can be column represented rank byis are a equal, vector of the number of linearly independent vectors of this type N JHEP10(2013)007 ) ) 2 k k N ) is k (D.5) (D.1) (D.4) (D.2) (D.3) + 5 SU( k × 3) into irreps 4) − i is a level-two ) theory and , B − 2 k ) N f , k k N  + + 2 0 and U( only, which implies , i vev will then break 3 N 3) ×  A Adj ) 2 B (2) and A − − 0 ⊕ N SU( 2 ) N 2 1 ( and × − denotes the reducible U( ⊕ − SU( 3 ) N 2 0 is not renormalized in either 3 k A − × ⊕ ⊕ = 10d 4) +2 τ 2 +1 We have broken 3(2 4 + 2 ⊕ k − ) becomes Γ , 5 , 0 1 Z − 58 ), whereas an , N ), where the last factor corresponds to , , − , k 1 1 k N Adj k − (1  ), since Higgsing a bifundamental breaks SO( ) for completeness. A similar computation U( 3) k ⊕ ⊕ ⊕ 0 ⊕ × ). Thus, decomposing ) ) − U( ) = 0, and USp(2 1 1 k ) 3.20 −→ τ − − +1 N ( × N ) × Adj U( – 71 – f k ) 4) ⊕ 3(2 ⊕ ⊕ → N → SU( D3 branes from the orientifold plane and breaking , containing both singlet and trace-free irreps. − + 2 ) 2 − +2 +1 +1 × k k k N N SU( 3) D3 branes. The holomorphic gauge coupling of U( , , 4) ⊕ 1 k − , , − → SU( SO( +2 k ( (1 ), we find ) N × k k → ⊕ ⊕ ) + 2 , ) k ). For a suitable normalization of the U(1) component, we have U( 1 k 1) 1) + 2 k , N 2 , , ) and USp(2 × 1 )( k )  N U( , , k 4 + 2 N ( ⊕ (1 U( ). → USp(2 − + 2 ) → → → × N k SU( (and for the USp theory) where SL(2 N 3.20 ) ) A B k k × 3( N SO( 4) USp(2 − 4 + 2 × = 4 gauge theory on the − N ) k We choose to turn on a vev for the singlet components of We now sketch the details of this argument. For simplicity, we routinely drop numerical To establish this result, we consider the SO( N N One can show by explicit computation that a vev of this type satisfies the D-term conditions. vev will break SU( 58 that only components of thesegenerators, fields therefore can be Higgsed. for each of the threeadjoint representation SU(3) of “flavors” dimension of each field, where Adj B SO( SO(2 the decomposition factors throughout the computation, only keepingWe track aim of the to dependence turn on on the couplings. a vev which breaks corresponding to removing D3 branes from the orientifold plane. In particular, a suitable the therefore equal to theeach ten-dimensional step axio-dilaton, of and the by computation,theory, performing we giving can scale ( relate matching it at to the couplings of the SO( In this appendix, we provide acan derivation be of ( done for gauge theories arising from moreswitch complicated on geometries. a mesonicthe vev, gauge removing group down to SO( for both SO(2 of SO( hold for odd modular form. Hence,theory. we conclude that D Coulomb branch computation of the string coupling breakdown of the Wilsonian description, and are likely inconsistent. Analogous statements JHEP10(2013)007 . λv and  (D.6) (D.8) (D.7) (D.9) SU(2) (D.10) N ,... 0 ) k + 2 k 4 N ( 1 1 1 1 # N − 0 R 2 4 , , but decouples from N N ] 4 N 0 R 2 1 1 1 1 11 1 1 11 2 3 k N − − Φ 1 + 1 j 2 4 R ,..., N N U(1) Φ k 2 3 i + − × , 1 1 1 0 1 2 3 2 3 2 + 2 ) 0 SU(2) U(1) Tr[Φ N 2 ijk , with 1 1 1 2  − Adj SU(2) ) , , as do two of the three Adj 0 R − − − − SU(3) U(1) k 1 λ ε ) 0 0 , invariance, and the superpotential now ) 0 k +2 → 1 1 1 1 + k ⊕ ⊕ ⊕ ⊕ ⊕ N (1 1 1 R U(1) R Φ, setting the superpotential coupling to 1 ⊕ λAAB SU( +1 +1 +1 +2 +2 mn 3 × ; / ) U( k 1 1) U(1) U(1) , N − B ) U( – 72 – n × λ ; , × j b ) charged fields all receive masses at the scale N + diag is precisely: k 1 (1 → A v  m ⊕ ; U( 2 3 4) SU( i a − 1) − A ⊂ , 4) SU( , 1 1 1 Adj ab 1 3 N δ , − , 1 3 1 1 1 1 11 1 1 11 1 1 Adj Adj ( ijk N SO(  breaks SU(3) i i i ) gauge group factor decouples from the rest of the theory and λ ε 1 k SO( Φ A B i ∼ v SU(3) ,B W , and a linear combination of the two, respectively. Thus, the matter = 2 3 A A A A A B B B i × A, B A/B i B origin A + diag Φ , = 4 superconformal fixed point in the infrared. To make the enhanced h 3 R A N = U(1) R 0 One can check that the U(1) is if the unHiggsed fields are U(1) k coming from content just below the Higgsing scale chiral superfields remain unHiggsed. The only way to get the correct scaling with from the superpotential whichdoublets, descends leaving from Note that, due to thethe unbroken line global cannot symmetries, couple the to chiral each superfields other above at and the below renormalizable level. where the unbroken flavor symmetry is SU(2) where the vev the other fields. Theflows U( to an supersymmetry manifest, we rescale Φ (up to a numerical factor) in the holomorphic basis. where we can now formallytakes restore the SU(3) form: JHEP10(2013)007 ⊂ ) and k (D.20) (D.16) (D.17) (D.18) (D.19) (D.11) (D.12) (D.13) (D.14) (D.15) λv = 4 super- , we obtain and v λv N , . 18 . i  ) ) k ) , k N i gives: ) 4+2 +2 to make k , 18 SU( − v N v i , i Λ +2 N ) b 4) N 18 SU( N  − − SO( Λ 00 N 18 SU( Λ k . 18 SU(  , λv Λ k 4 18 Λ Λ 4 ) ) − SO( 18  k λ k 4) − , ) is 1 whereas it is 2 for SU( − ) Λ − k k = =  2) 4+2 N 0 . ) b − = 9 = 9 − k 4) b 18 N  N − − SO( Λ 2)+10 0 SO(2 0 2) + 10 6( SU 0 SU N 18 4+2 b Λ − Λ λv λ k − SO( − − 18 − N ⊂ v h 2 b We have  Λ − SO( N − λ ) k , b (6( ln N Λ k 2) 59 − k , b = SO( , − . However, rescaling Φ – 73 – 18 4 πi 1  (6( 0 b Λ , N 2 ) b ) − 2)+10 ) λv − − k 00  6(  k − above, between, and below the scales 0 = = λ = N v λv 18 (Λ = h SU( +2 Λ k ) ) factor. 6( 2 k − Λ N ] for SU( k SO mat  ln k λ λ b T 10  h = = − 9 SU( SU( ln U( πi 1 121 − b ln ln b 2 2) Λ 0 SO ⊂ k −  πi 1 b due to a rescaling anomaly. We find: 4 , whereas scale matching at the scale πi πi 1 1 = ) 2 Adj N v Λ λ 2 2 k τ T ) henceforward for simplicity. λv 6( +  k = − = = k,N ) τ  U( = 3 ) N k,N and b k ⊂ τ k,N v τ v 9 SU( = ). Evaluating the holomorphic gauge coupling at the scale SU( Λ k ˆ τ Λ  ). Due to the vanishing of the beta function coefficient, the holomorphic gauge SU(2 ⊂ ) D.15 k Now consider the SU( To determine the gauge couplings of the low energy theory, we compute the beta We ignore the U(1) 59 coupling does not runsymmetry below manifest the alters scale which can be rewritten as using ( USp(2 between the scales since the index of embedding [ so that Thus, in net In either case, we have the scale matching relations function coefficients apply the scale matching relations. Above or below both scales, we have: whereas between the two scales we find: JHEP10(2013)007 , ). of N F 4.1 (E.1) direct (D.21) (D.22) for the ⊗ 21 21 E   i (0) i (0) B B   we present some = 21) representation E.1 m = 4 examples discussed , i N ) + 1 (D.23) i ˜ ) is also independent of ) N i ˜ ) N SU(4) theory (cf. section 18 F. ˜ ) as the ten-dimensional axio- N 9 λ − SU( × = 7 9 D.20 − SU( ˜ L Λ − SU( ˜ Λ ⊗ D.20 ˜ Λ N +4) ˜ +4) E N ) theory is closely analogous, and we ˜ +4) λ N ˜ ˜ N L N planes in the 18 USp( m + 9 USp( ˜ , as expected from the constant axio-dilaton Λ = | SU( 9 USp( ˜ v Λ X λ f | ˜ O3 Λ ]: +2) × we present the rather lengthy results related to ˜ +2) N – 74 – SU(4) dual pair. In section efficiently ˜ +2) N 6( ) = ˜ ]. However, the computation becomes more and ). 123 and N × E.2 ˜ 3( , λ . F 21 + 4) h ˜ 3( λ lives in a tensor product representation 72 +  [ N h ˜ λ ⊗ ˜ ln 122 N h D.15 i (0) B ln E ln ( B πi 1 disappears. Moreover, ( LiE 1 2  πi USp(8) m 1 k πi = = ↔ ), depending on which sign we take for the square root. The = Sym 10d 10d τ τ 10d D.21 τ -th symmetric tensor product (in our case SU(7) m × ). However, at this point an important subtlety arises, since the factor D.20 . 2 evaluated in that background, we interpret ( SU(3). The SU(7) theory whereas in section × × 10d τ The first observation is that The computation for the USp( Since the holomorphic gauge coupling on D3 branes probing a smooth background is SU(7) of a tensor product decomposes as [ tributing to the superconformalputer index algebra can program such bemore as computed expensive straightforwardly as using onecomputation a studies becomes com- larger intractable. and One larger canwhich then baryons, we use now and a explain. already different for and more efficient method, SO(3) the calculation of the superconformal index for the USp(8) E.1 A noteIn on the computing simplest cases, the representation under the flavor group of the gauge singlets con- E Details of theIn superconformal this appendix index we discuss for someindex technical details for of the the SO(3) computation of thetechnical superconformal details related to the calculation of the SU(3) representation of is also consistent withresolution ( to this puzzle ismuch that like the the two distinction answers betweenin correspond O3 section to different types of O-planes, but there is an ambiguity, since obtain the result in place of ( inside the log is a perfect square. This can be rewritten as which can be verified by applying ( just dilaton. Note that the resultprofile is independent of of the dual geometry at large Note that the dependence on JHEP10(2013)007 | , ) ) | F ). λ m | λ ˜ n | N 1) S x S E.2 ( (E.3) (E.4) (E.2) − SU(4) is just µ SU( the fun- . It also in terms × F λ × λ ), and the f ) = λ c L s x ◦ ( b , µ + 4) )( . This formula SU(4) ◦ + c ˜ κ n ]: ) N ), with ◦ × p µ are the symmetric f a ( i 2 122 µ κ s V ). = ( A ( ) c λ κ i USp(8) ◦ ( L 4.1 has more than 3 rows this ` and O ) . λ = * λ ) ab in the symmetric group s we have [ µ E   ( κ i . If λ κ F κ λ . L λ A κ ], in terms of generalized Kostka numbers. L ) p κ ], in which the left hand side of ( λ κ =1 ( λ κ i 122 ` O χ χ evaluated in the representation of respectively. Decomposing λ κ ) . Taking into account the factor ( 122 m κ µ κ χ = (i.e. all standard Young tableaux with | 9 ( ) X λ | for a more analytic approach to the problem based , the fact that ( κ C m i ( F κ and   m and C p of = ) λ | ) – 75 – 8 X m κ κ . This expression already provides an important κ λ | ( ( is the fundamental of SU(3), and thus =1 = λ ` i | ! SU(4) theory (cf. section X κ C 1 | F Q m × ! m ). 60 1 = = m = | ]: X κ κ | indexed by the partition λ 4.12 p = s ! κ 124 1 µ p m the 1 + 1 partition of 2 corresponding to the antisymmetric L . The gauge invariant contributions for the USp(8) λ ” is the plethysm operator, and µ ◦ L 7 ) = ]. ) is the number of parts (rows) of the partition F of elements of cycle type κ ( are shown in tables ⊗ 127 λ ` – 2 χ t E ( , with “ 125 µ m s is the Schur functor for , and ] for similar formulas, and appendix ◦ λ λ ) now follows using λ Sym L s 126 , E.2 ) denotes the order of the elements of cycle class κ 125 ( C The fields that contribute to the superconformal index for the USp( The second simplification in the calculation now comes from observing that the tensor We are left with computing the number of singlets in One could alternatively use the formula in example I.8.9 of [ is the character , we can apply the formula [ 2 60 λ κ on the discussion in [ theory are shown intheory table up to order we find perfect agreement with ( See also [ where the brackets indicate taking the singletE.2 part only. Check ofIn the this superconformal section index calculation wesuperconformal for index present for some the rather USp(8) lengthy results related to the calculation of the happens to be the mostonce. expensive part Making of this the manifest, computation, so the it final just formula needs we to computed be is calculated effectively: Formula ( definition of plethysm with a fundamental power symmetric polynomial: product of Adams operators appearing in this formula is actually independent of is given by Schur functions indexed by theof partitions power symmetric functions Here χ associated to follows from well knownrepresentation facts, in let terms us of symmetric give polynomials a [ quick proof. It is convenient to switch to the damental representation of SU(7). ThisIn can particular, beV denoting done by from general properties of plethysms. boxes), and simplification of the calculation, since the SU(3) representation described byterms the Young vanishes, tableau and we can ignore it in the sum. Here we are summing over all partitions JHEP10(2013)007 - R . ), where 0 , 3 ˜ 5 N J 0 χ , , 8 ± + SU( χ J SU(4) of order 2 , + 1) + 1) × + 4 × r l l 2 ( ( χ , + 2) + 2) 4 0 l l , + 4) 0 ⊕ ⊕ + + 1 + 1 + 1 + 1 , χ ( ( 5 4 l l l l 1 ˜ , N χ SU(2) χ ⊕ ⊕ 3 + 3 l l χ the antisymmetric tensor + 3) theory transforms in + 1) + 1) + 1 , 1 l l ∗ , 1 1 3 ( ( , , 0 1 1 − 3 , , , + 3 2 2 χ 4 2 2 χ 0 χ χ , N χ χ χ 1 l l l l 2 , + + 3 0 χ + + + 2 + 2 + 2 + + + + , 0 l l l l χ , 2 0 0 SU( 1 , , , 0 0 2 ˜ ˜ ˜ ˜ , , 4 2 2 4 0 1 1 N N N N χ + 3 1 1 χ × χ χ χ 2 χ χ 1 + 2 + 1 + 2 + , − + − + 1 exponent + 2 SU(3) character + 5 + 4 3 4 3 2 3 2 3 χ t 1 , 2 2 + 1) , , 0 1 0 χ + 4 χ χ N for the agreement between the two dual 2 4 , 0 1 1 1 1 3 + 2 χ SU(3) 4 – 76 – , 0 + 3 χ ) 1 5 , ˜ 0 N χ in the USp( SU( 3 3 1 1 1 1 − ¯ ) ) J 0 0 0 0 0 × ) ) 1) 1) ± ± ± ± N 2 , , ˜ , , Adj) Adj) A , , , , ( ( + 4) ( ( (1 (1 (1 (1 2 3 4 3 4 3 4 3 4 3 5 3 5 3 5 3 5 3 ex. ˜ N t denotes the symmetric tensor product and [] ∗ 2 3 2 ) ] ] 4 USp( 4 ) ˜ ) (1) B ˜ ˜ (0) (0) B B the baryon (0) (0) 4 8 ˜ ˜ ) ) (0) ψ A A (0) ψ ψ 6 ˜ 3 ) ( ) [ [ ) ) ) ˜ ) l A l l l ) ˜ ˜ ) N ) l ) 2 4 l A i (0) (0) ( i ( B A ( ( SU l ( l SU ( ( ˜ ) ) USp ( USp ( ( B ˜ ˜ ˜ (0) ¯ ¯ ˜ ψ ψ A A A λ B , where () (0) F (0) column denotes the representation under the SU(2) group generated by λ Field F ( ( ψ 2 ˜ (0) (0) SU (0) ˜ operator t USp (0) A A r ˜ ˜ λ ( ( A A λ . Gauge invariant contributions to the superconformal index for USp(8) SU (0) . The fields which contribute to the superconformal index for USp( ( ( λ Table 7 the SU(2) One of thetheories arguments relied presented on in thecharge section matching between the of two the descriptionsall flavor of values representation the of of theory. baryons In with particular, minimal we could argue that for Table 8 less than product. F On the decomposition of certain generalized Specht modules JHEP10(2013)007 . ] , 2 t on 127 124 N , B 0 , 61 3 0 6 , 122 2 , , 2 5 0 6 , , , 5 χ 2 , 12 2 2 9 4 χ 2 , 4 χ , χ χ χ χ 4 χ ) denotes the 4 + 4 + χ χ + + R 2 0 , , ( + 5 SU(4) of order 2 2 2 + 68 + 2 5 6 , , , k + 42 2 + 6 8 8 × 2 , 4 2 χ χ 0 + 8 , , , 4 2 χ , 1 χ χ 1 2 6 , 1 1 , χ 0 4 , χ χ χ 4 χ + + 2 3 χ 0 , χ + 3 + 13 + 9 1 2 ) (F.1) χ + 9 , , 1 4 is odd. 4 7 5 , , , 1 χ + 46 , 7 + 27 + 22 + 4 1 χ χ + 13 4 4 ( + 41 N χ , 1 4 2 χ χ + 20 + 2 , , , 1 χ 1 , 3 1 1 1 / 1 1 2 , , , , 3 1 1 , , χ + 6 + + 3 + 3 1 1 1 χ χ χ 3) 3 7 0 + 4 χ , 3 1 1 1 χ χ χ χ + 2 − , , , , χ χ 3 1 1 3 6 4 4 1 , 3 N χ , 6 + 7 + 2 ( χ χ χ χ + 21 3 + 54 χ 1 1 1 + + 3 1 + 1 + + 22 , , 1 χ -charge baryon of the form + 32 , 6 + + 19 1 1 , 3 6 1 , , 0 + 9 R 0 0 , 0 χ χ + 6 0 , Sym , + 2 χ + 35 + 14 3 3 χ 6 3 SU(3) character , 0 χ χ 0 , 3 3 χ 0 , , , χ ∼ = χ 3 6 + 3 3 χ χ + 5 + 26 χ χ χ 1 + 2 + 8 + 52 3 3 E . We also argued, and checked in a number , , 3 3 + 37 , , + 14 +3 + 12 0 0 (3) flavor group, where Sym (3)), where the angle brackets denote taking 0 0 R 5 + 7 2 , 5 χ χ 4 + 3 , , + 15 χ χ + 22 2 0 5 2 , – 77 – SU SU(3) 2 0 χ 3 SU , , χ χ 5 3 χ + χ χ +4 ⊗ 6 + 18 ) +17 16 + 8 6 + 35 16 + 28 +8 +21 N +43 SU( 3 ¯ J 0 0 0 0 0 0 0 0 0 0 0 0  2 ). In this appendix we would like to demystify this expression N N ( 2 2 2 2 2 2 2 2 2 2 2 2 ex. Sym ) side, transforming in the same representation of the flavor group. SU t D N 2 5 3 ) representation of the 4 ] ] ] ] SU( ˜ B (0) (0) ˜ ˜ ˜ ˜ B B B 2 (0) (0) (0) ( B (0) 6 2 12 ψ × ] B A (0) (0) ] ] B 2 ) ψ ψ ψ ψ 2 [ [ [ / ψ ψ 10 [ ˜ ) 4) 8 6 2 ) B (0) 4 SU (0) 3) (0) ) ) ) USp (0) ) (0) (0) ˜ λ ˜ ψ − ˜ − λ (0) A [ [ (0) [ ( (0) (0) (0) A B N operator (0) ˜ A ( A ˜ ˜ ˜ N ˜ ( A A A . Gauge invariant contributions to the superconformal index for USp(8) ( A ( ( ( ( In this appendix, as in the rest of the paper, we will be assuming that ]. We will also make use of the generalized Specht modules introduced by Doran in [ , and give some additional evidence for its validity based on this new viewpoint. The 61 N -th symmetric power of the representation somewhat by reformulating it as an statementS about representations of the symmetric group interested reader can find128 nice reviews of the required introductory material in [ to hold (as representationsthe of singlet the part under flavor The duality conjectured in this paper then requires the group theoretical identity k of examples, that therethe is SO( a corresponding minimal the Sym Table 9 JHEP10(2013)007 ] , λ and . In 122 , we (F.5) (F.6) (F.7) (F.3) (F.4) (F.2) 2 2 N e N ) of the ρ ( rather than C ) singlets. As 2 h is just giving rise to a ) singlet part: is the character N ( N N N ,N ( , 0 exchanging κ ρ SU λ s in the representation χ SU ω (in standard notation i . ρ 2 κ N ] acts on e ], which we will describe 2 ◦ for its decomposition into , ρ ≡ µ contains 127 ) follows from a conjectured 126 λ p F. [ , h s S F, λ µ λ i ... χ λ L F.1 c ,N 2 λ ρ 125 L 0 L e i µ , κ χ . Taking the ) ⊗ ◦ κ E S L ρ s λ . ) as: λ ( E s L : ,N λ h ∼ = C h 0 ,N ρ SU(3) F.3 2 κ ρ L p N N N , and the sum is over partitions of χ N ρ ,N = N = 2 + 2 + = = λ | | 2 | = for which χ X X N X = ρ λ λ | | | | N is a (in general reducible) representation of X ]: λ ) we have introduced the order λ N F = | =2 S N ]: | X ∼ = = i κ = | ,N 2 | of elements of cycle type – 78 – denotes the transpose of

E.2

2 X λ e 129 ) ) 0 ) = | , and λ ρ = N 123 ◦ κ .) There is a single partition of 2 ! F , F F ) χ 2 S κ 1 ρ e N N ⊗ p ⊗ ⊗ 126 h 122 ◦ , ) by acting with the involution ∼ = E E E SU( ρ , and ( ( ( p

125 ) N N N is the Schur symmetric function indexed by the partition N F.6 = F λ s , and as in ( E ⊗ Sym Sym Sym , so let us write

N

E N ( S ), it is the partition 2 N N ( in the generalized Specht module , and the character Sym ]). We will show that in this context ( | ρ

ρ | SU S into power symmetric polynomials 126 , in order to do this it is convenient to work with symmetric polynomials [ , . We can now use [ in λ E λ s ρ is the Schur functor for the partition 125 is a partition of ” denotes plethysm, runs over partitions of 2 λ ◦ ], which gives the transpose of is the elementary symmetric function or order 2 associated with the antisymmetric. ρ L κ 2 e We start by using the decomposition of the symmetric power of a tensor product into 122 [ , but we can easily obtain ( 2 2 finally get: Now, the generalized Spechtthe module permutation group further momentarily. (Notice thate the formula given inh [ gauge singlet of for partitions). Taking into account that the transpose partition of 2 where of cycles of type where cycle class indexed by where “ and Expanding We thus see that we arein left appendix to enumerate the instead of directly representations, so we rewrite ( where our particular case we have decomposition of certain generalized Specht module into ordinary Spechta modules. sum of ordinary tensor products [ (see also [ JHEP10(2013)007 1, of has and − µ i . We (F.8) (F.9) 0, we µ 2 κ (F.11) (F.10) S λµ ≥ are odd N δ = i i i κ κ ˜ κ Furthermore → i 62 . κ λ s + 1 and ,N 2 ]: Z   N 2 µ ρ S χ ∈ 123 such that all , λ ρ i χ κ N ) : 122 ρ ( of N , C 3 ]) shows that the identity holds λ , indexed by the partitions κ N µ S 72 = | [ + S 3 X . ρ | 2 − µ ρ ! , κ N χ 1 κ LiE N = M µ + | . So in order for the dimensions of the S c λ 1 |   λ κ k ) as SU(3) representations, as we wanted. µ N ∼ = X M , = ) | X µ λ = ∼ = | s F.11 ), we have found various pieces of evidence for – 79 – i µ µ ( 2 ,N c c 2 e 2 / N N ◦ N 3) F.10 = = | | ρ − S 3 with all parts even. We have thus just obtained a X X µ µ ) and ( p | | N we can associate in the usual way the SU(3) representation with h ( − λ are all even numbers. Using the fact that we are interested , the Specht modules ∼ = = S n F.10 λ N

S ) Sym F is a sum of ordinary Specht modules we associate to it the corresponding ). The right hand side is given by [ ), we obtain ⊗ ,N F.1 E 2 F.5 ( N has a basis indexed by the semistandard tableaux of shape N S ,N 2 Sym N

S between 3 and 21. More conceptually, it is possible to show (by using a . Since is a partition of λ N κ 1. Removing a column of three boxes on the leftmost column of a Young tableau . On the other hand it is well known that any ordinary Specht module runs over the three element partitions N ≥ κ i κ Coming back to our conjecture ( To each ordinary Specht module . Using linearity of characters, we find that 62 a basis indexed bycorresponding standard modules tableaux to match of it shape should hold that the numberYoung of tableau semistandard tableaux sum of SU(3) representations. its validity. First of all,for direct all computation odd (using straightforward modification of the straighteningexample) procedure that based on Garnirweight elements, 2 for rest vanish as SU(3) representations).have We also notice that since gives rise to SU(3) isomorphic representations,where so we now may just ˜ as wellnatural send isomorphism between ( where the parts of the partition in representations of SU(3), we can restrict the sum to partitions with 3 parts at most (the where numbers. Before presentingshow the that evidence this implies that ( we have found for this conjecture, let us where we have used orthogonalitythus find of the characters remarkably simple to result set thatSU(3) the the representation flavor representation term associated of in our to baryon parenthesiswe is the to just conjecture the generalized that Specht the module following decomposition holds for all N Plugging this back in ( irreducible representations of JHEP10(2013)007 ) , . 4.37 ] for a (G.1) (G.2) (G.3) (G.4) ) ) 1 1 )). The 132 − − c c , N z z 1 1 =1 − − b b a ) (so one has z SU( , z , z . L c c Q ) ×

z z ) 1 1 c , f 4) SU( − − b b z 2 , in terms of elliptic 1 , z , z − × − b 1 1 SO ) − − c c I N , z 1 1 z z t, x, z 1 M b b ], and we reproduce them ( = − c z , z , z 83 b c c I 1 z z z ]), one may expect that ( ) USp b b I z z Γ( 47 , f 2 ), N Γ( Γ( ≤ ] (we will not attempt to prove it in , z N N 1 ≤ ≤ Q 98 4.37 b

SU side, which we will parametrize as USp(2 t, x, f N ! ( 1 × I I N ≡ ! ! USp 1 1 L N N I I and weight 2 ≡ ≡ ≡ 2 ) ] ] ] ]), and by direct computation it is easy to see that the dimensions match up to ) factorizes into: ) + 1, +1) N N ], we have absorbed the contribution to the index coming from vector bosons into N N dz dz dz 97 [ [ [ 4.5 N 83 SU( USp(2 ≡ It is by now an standard exercise to reformulate the superconformal index in terms Z SO(2 We refer the reader to [ Z = 45. Z 63 M nice review of the field. here for the convenience of the reader, adapted to our notation: As in [ the integration measure. Explicit expressions can be found in [ of elliptic hypergeometric functionsstart (following on [ the USp 2 index ( our conjectured duality seems(this to is particularly be clear qualitatively when formulated differentis in a from string new theory ordinary fundamental [ identity Seiberg betweenthe duality elliptic one hypergeometric proven functions, by independent Rains from [ G A conjectured identity for ellipticIn this hypergeometric appendix integrals we willhypergeometric reformulate integrals, the conjecture ( tions that could perhaps bethis proven along paper). the lines One of point [ of mathematical interest is that, since the physical process behind shapes as in formula ( SAGE [ N of shape JHEP10(2013)007 . χ (G.5) (G.6) (G.7) (G.8) (G.9) (G.10) (G.11) (G.12) . . i ) 1 , and express the integrals χ − q 1 64 ) ), and its conjugate η r − ! f ) c − r ( tx , f − (2 ) t = χ (2 ) , 3 =1 t q 2 +1 c − X b , a z − ( − y tx b a χ , 1 q . r x χ − ! η ) = ) − t ) a i r u ,b p h 1 t x u +2 1 2 ( z b b ) t, x z b ( − 1 ; + + Γ( − a a − u a . χ t an unified index. The subindices denote L =1 c k 1 x =1 )(1 ) b X ut tx ). i Γ( Y f b q a a ,b = − from the bifundamental, and the contribu- 1 2 + ,a − u − . This fields transforms in the bifundamental ≡ ≡ − z a 1 uy uy variables we have been using so far. t ) G.8 I 1 χ ! z ) ) 1 − I ) f x k u − − 1 )(1 ( q 1 ,a – 81 – − 1 X 0 Γ( X tx a,b,c (1 (1 χ z 0 and 0 0 ≥ ) ≥ 2 − t + ≥ ≥ Y + Y Y z ), we have: a,b a a X , . . . , u c ( a,b ,a 1 (1 f ,b 1 u 2 χ . ,a z ) = ) = ) = ) z 1 ( q G.10 ) = Γ( 1 y y z η ) = 2 z ; ; = 1, which we will not indicate explicitly in what follows. =1 2 M ( t, x a u u X , z q ; ( ( , z 1 X χ a,b,c X i,a u θ 1 z = = = ≡ Γ( Q χ ) we will always mean ( t, x, f, z ), and accordingly its one-letter index is given by: t, x, f, z ( u ( L i i SU( × ) is a monomial in the expansion, and M q from the symmetric. Let us start by η I We are left to evaluate the contribution It is common in the literature to introduce the new variables 64 Expanding the denominator in ( in terms of these, but we will keep using the Where projection of the groupconstraints elements of into the the maximal form torus, and for SU characters there are Here we have introduced theIt total is character convenient to expand these characters into elementary monomials: tion of USp(2 The ordinary Γ functiondiscussion, (i.e. so the by Γ( generalization of the factorial) will play no role in our Finally, we have also introduced the short-hand notation where we have introduced the following standard special functions: JHEP10(2013)007 ) N ), we (G.16) (G.18) (G.15) (G.17) (G.13) (G.14) SU( × G.11 +1) . ) . M , f ! 2
) c k . ! , z f − q 1 1 ,i  η 2 2 − q ) q z ,b r η η r 2 b b − t c z + t, x, z − f (2 . a ,a ( a , k Γ 1 x t x z 2 z,i r I r ! L z , =1 ) − − i + Y 1 b c b k q f − q +

, f ,b N =1 q + back from its definition ( η i a 2 η X 2 · ,a a η b t z q 1 b kr 2+ , z z η + + t   − t 1 a ( − a
 ) ,j x c b x 1 2 − r f r Γ z − , − 1 ,j a + ,i ) ) b ( 2 b 3 2 t, x, z q =1 k k

+ z ( Y c z + η a x 0 ,i ) a r ) , f 2 t N I t ≥ b L =1 k z 2+ 2 Y ≤ E b Y t – 82 – + r ] − a,b Γ( t a 2 , z ( 1 − k =1 X 1 i

≤ = = G.5 Z X ) = ) then becomes: ) , x ) = exp( ] 0 G.1 ) = 1 k Y i