JHEP09(2019)111 Uality in Springer August 17, 2019 : September 3, 2019 September 16, 2019 : : 3 Orientifold

Home , Orientifold, Seiberg duality

JHEP09(2019)111 uality in Springer August 17, 2019 : September 3, 2019 September 16, 2019 : : 3 orientifold. We use ′ , Received f a Hanany-Witten brane Accepted m and argue that ﬂavour e identify various phases of Published ion of the dual squark ﬁeld. otti’s O B string dynamics from QCD with a Chern-Simons term in 3 . In particular the symmetry breaking 3 Published for SISSA by https://doi.org/10.1007/JHEP09(2019)111 ) QCD c N from type 0 strings and [email protected] , 3 QCD ) c . 3 N 1908.04324 U( The Authors. Brane Dynamics in Gauge Theories, Chern-Simons Theories, D c

We propose an embedding of U( , [email protected] [email protected] Department of Physics, SwanseaSingleton University, Park, Swansea, SA2 8PP,E-mail: U.K. Gauge Field Theories ArXiv ePrint: the brane configuration to proposethe magnetic a theory magnetic with Seiberg conjectured phases dual. of QCD W We also discuss thesymmetry is abelian not broken. theory Finally, withoutdynamics. we also Chern-Simons predict ter novel type 0 Keywords: and bosonization phases are both associated with condensat string theory. The UVconfiguration gauge in theory type 0B lives string on theory the in worldvolume the o presence of Sagn Seiberg duality Phases of Open Access Article funded by SCOAP Abstract: Mohammad Akhond, Adi Armoni and Stefano Speziali JHEP09(2019)111 5 7 8 1 3 8 16 16 10 11 12 13 15 15 16 17 13 c QFTs, and their rela- ries in various dimensions, ying dualities. Most of the ring theory and field theory with a Chern-Simons term. tions in strong coupling dy- eld theories often follow from 3 ogress made on the field theory ng theory. Having independent sions. There has been significant – 1 – 3 ′ with vanishing CS-term 3 4.2.1 Region4.2.2 II Region II 3.2.1 Electric3.2.2 theory Magnetic theory One may naturally ponder the ubiquity of dualities in generi 5.1 QED 4.1 Region I:4.2 bosonization Symmetry breaking 3.2 Including flavours 2.1 A pseudo-moduli2.2 space Hanany-Witten setup 3.1 Level-rank duality tionship to string theory. Indeed,front for recent non-supersymmetric years gauge have theories seen in threeprogress pr dimen in the understanding of the phase diagram of QCD are better understood in that setting. String theory has longnamics been of quantum a field sourceproperties theory. of of In insight the particular, for corresponding dualitiesevidence investiga brane in from configuration fi field in theory stri effort and so string far theory hasowing is been to a the largely step fact focused in that on verif non-perturbative phenomena supersymmetric in theo both st 1 Introduction 5 Comments about QED 6 Conclusions 4 Phase diagram 3 3d dualities from non-supersymmetric brane configurations Contents 1 Introduction 2 Overview of type 0B JHEP09(2019)111 , 3 ′ (1.1) (1.2) ]. The . This 8 3 . ], while the the theory ion of type 11 n particular , t the infrared Chern-Simons ]. 10 scalars . In the case of eformations. It > K 13 K f 2 , In order to realise / N 12 f > K hat realises the mag- ⊕ 2 2 ) gauge theories. For y. / 2) > N / are taken to infinity the N f . f the theory admits a dual ⋆ ich will be our main focus, flavour branes and an O ) N ecks: as in the symplectic N ioned squark condensation , k N + f K K f d, namely the dual “squark”, ollows start with a UV theory, which ory and the magnetic theory c and magnetic theories form K ing theory phenomena teaches N ( ≤ ,N ∓ c 2 + c theory are given in [ D3 branes suspended between an symmetric brane configuration of 2 / N N / ) and Sp( c f − f , N N c N N N U( 1 − ] that when ualities were explored in [ × 7 f ]: when ) 2 N K 16 are kept fixed, the theory exhibits rich vacua [ , + − f – 2 – ]) that for 15 2 K 6 / ] also for SO( f K,N , massless Dirac fermions and a level 7 5 N U f ]. and U( N 14 theory. The various IR phases of the electric theory (and ←→ → can be understood in terms of a non-SUSY Seiberg duality. . The electric theory also admits a Seiberg dual description ) 3 → ∞ 3 f c N N ] (see also [ theory, whose Lagrangian is more complicated than QCD 4 U( magnetic – ) fivebrane. In addition, there exits ) QCD fermions 1 c , k f ) theory with N c N N electric ⊕ ] which concerned the symplectic gauge group, we propose tha c 9 we embed the gauge theory in a Hanany-Witten brane configurat N ± 3 ) can then be identified with the phases of the magnetic dual. I 3 the theory is expected to flow to a CFT. K,K ⋆ ) c ] it satisfies global anomaly matching and RG flows after mass d ) gauge symmetry, it was conjectured in [ N 9 c ) QCD N c Our proposal of Seiberg duality is motivated by string theor However, one may wonder whether something changes for By swapping the fivebranes we obtain the brane configuration t Following [ Consider a U( Our Seiberg duality proposal passes several non-trivial ch N ≥ Other approaches to obtain 3d duality with relation to string In the ’t Hooft limit, when U( 2 1 N f can be understood in termsin of the the magnetic condensation of theory. a scalar fiel both the bosonized phase and the symmetry breaking phase, wh theory flows in the IR to QCD which we refer to as the so of QCD NS5 branes and a (1 electric theory becomes equivalentbecomes to equivalent a to supersymmetric a supersymmetric the theory. The electri we refer to as the Our proposal involves a modification of the UV theory, i.e. we orientifold plane. It isGiveon similar and to the Kutasov in corresponding type super IIB [ 0B string theory. The brane configuration consists of term. It was argued [ SU( discussion of this limit is beyond the scopepossibility of of this relating paper. these dualities to supersymmetric d is also supported by planar equivalence [ U( phase diagram of U( description in terms of a gauge theory coupled to scalars as f admits a flavour symmetry breaking phase where A similar picture was developed in [ N case [ us about non-supersymmetric branetranslate dynamics. into a The reconnection aforement of colour and flavour branes. netic Seiberg dual. The relation between field theory and str JHEP09(2019)111 , . - 3 B 22 pro- (1.3) R sector i action R tation, f 2 Z 1) − string theory, d by replacing 0. In addition, ≥ type 0 string does k f the Ω( ere exists a limit in ssential properties of l set of R-R fields. The this with reflection in 6 y is by using non-critical string worldsheet action, , thus requiring an equal , with al Wilson line exp ilities associated with the rsymmetric dual pair. ′ k articular there are now two how how the phase diagram ifold plane. Note that unlike section 5 we focus on QED f pproaches. branes was worked out in [ 2 n 3 we consider a certain brane ′ N action. The doubled set of R-R is the operator that counts the pe IIB theory. The twisted sector R − D3 f κ ]. The field theory that lives on the respectively. 3 planes in type 0 string theory. For 1) m ′ ′ 20 ≡ − orbifold of type IIB, with the , ]. 2 19 Z 21 D3 and ,K n – 3 – + 2 c orientifold, the (3+1) dimensional fixed hyperplane N ± 3 − ′ k action. The existence of two types of orientifold planes fol ≡ 6 I κ R f 1) ) gauge theory with 3 complex scalars in the adjoint represen − we get an O m , the mod 2 spacetime fermion number operator. The untwisted s 6 U( I F × 1) ]. Instead of swapping the fivebranes, the duality is obtaine ) − ]. Here, Ω is worldsheet parity and ( 18 n 25 , ]. The method relies on the embedding of SQCD in non-critical . The advantage of using this method is that the non-critical 24 17 µ = 2 supersymmetric Giveon-Kutasov dual pair. Therefore, th The worldvolume theory on a stack of Type 0B string theory can be obtained by a The paper is organised as follows: in section 2 we review the e In the following we will always denote the bare CS level by Another method of obtaining Seiberg duality in string theor In order to project out the closed string tachyon we make use o N ]. It is a U( → − and the signs are chosenthe to reflect O3-planes the of R-R type chargeR-R of IIB the discrete we orient torsion. do not Under have the the action additional of possib Ω, D3 turns into D3 with respect to the Ω( lows from the fact that the NS-NS two form can have a non-trivi In this section we review aspects of D3 branes and O of the electric theorySection manifest 6 itself is in devoted the to magnetic conclusions. and in 2 Overview of type 0B type 0B string theory andconfiguration its brane and configurations. propose In a sectio Seiberg duality. In section 4 we s branes is the same in both the critical and the non-critical a pioneered in [ is composed of atachyon tachyon will in eventually the be NS-NS projectedfields sector out lead as by in well the effect as orientifold totypes a a of new doubling threebranes ful of the which D-brane we spectrum. denote by In D3 p and D3 is therefore identical to the bosonic sector of the parent ty number of right movingspatial worldsheet directions fermions mod 2. Combining the relevant background we refer the reader to [ 23 which our non-supersymmetric dual pair becomes a known supe an we define the frequently occurring combination generated by ( jection [ and a pair of bifundamental Weyl fermions. not contain a closed string tachyon in the bulk [ µ the sign of the coefficient in front of the Liouville term in the strings [ JHEP09(2019)111 | ± ions X ]. In | (2.2) (2.1) (2.3) plane. -plane 23 ± ± 3 s) on top 3 ′ ′ ′ c D3 ut in [ 4 ) respectively, v N · ⊕ ( 6 s − SO(6) 4 ]. Y ting the component , 26 ), ! ) )[ ) ) gauge group one lands 2 doubled set of R-R fields ± ( N ′ ⊕ N = 4 SYM ) in the configuration with ( ( Adj Adj + ermions which transform in tX U( as the representation of the N πα 2 + Y + F N = 4 SYM. separated by a distance 2 ( poses). We summarise this in is the symplectic form − Y Y ′ ) 1) N

J ) − N D3 branes on top of an O parallel to that of the O ( ′ J + + µ N + ( . Sp( exp ξ B X ) ) + ϕ , ! iq iq Y F ( ( N 0 8 8 2 1 1 1) , to the constraints f f theory with SO(2 − ϕ 3 N t 0 1 dt 2 = ( – 4 – − ) theory leads to ∞ T

N 0 parent Z = 4 theory descends to the choice of fermion repre- = c D3 branes (together with their image 2 JϕJ ) 4 J ′ N v N · α 4 ⊕ 6 2 V s π SO(6) 4 ) (8 = 4 SYM . The “orientifold” daughters of ( ± ) ) gauge field and 6 adjoint scalars parameterising the direct − N N F = ⊕ ) Y N Adj Adj 1) U( M N ] − ( A 23 Figure 1 ) J ); starting from the 1 SO(2 − − ( µ is [ − respectively. We will denote these theories by ξ B X − ± is the mod 2 fermion number operator and − Y 3 ′ 3 ′ ). The worldvolume theory of such a configuration was worked o F = 4 SYM, collectively denoted by 2 1) ), and the supersymmetric Sp( N . The field content of the world volume theory of , with the worldvolume directions of D3 and D3 − . The Lagrangian for these theories can be obtained by subjec ( ± 1 and O 3 − ′ We are interested in stacks of The amplitude Möbius for a single D3 and its image D3 + Y 3 ′ Table 1 sentation (figure The choice of gauge group for the (see table of O in the closed string sector. number of each type of brane. In fact Ω projects out half of the where ( on fields of highlighting the orientifold typeworldvolume fermions on (the which two they featurestable live relevant for as our well pur both cases one has a U( the 2-index symmetric or antisymmetric representation of U transverse to the worldvolume. There are also a pair of Weyl f O across the O JHEP09(2019)111 (2.6) (2.7) (2.4) (2.5) nsion ]. From nces from 28 0 is given by plane carries he exponential −→ ± p t negative) for the p, the scalar vevs, or to see that the orientifold ert some lessons back to relative to the D3 will be sis tells us nothing about 2 0 nslates in the worldvolume . κ ± ) theory is unstable, namely ]. We will state the relevant , 3 eld theory, this is encoded in ′ contributions from the bosons D3 D3 2 . es where an O 28 T lsive) force between the branes T ( on. ions in the antisymmetric (sym- 0 ± . This is obvious from the second 4 ± ntial. We note that the integrand orward to detect in string theory. 3 κ ± + π ′ e). Note that instabilities of non- ± , 2 X O Y √ = X ) √ T ) is attractive (repulsive). For small 2 ) ± ± + O 2 3 ± = ) of the O 3 X ′ ′ ]. We would like to extract the charge of ( + O X 3 6 ( 2.7 D 2 , the leading order term as 27 ± (O 6 G ± 4 G − 4 X 3 ′ πV V MX – 5 – ,T − D3 2 π, T ∼ ± Λ Q √ M ± ) with 4 ± ). = A Γ(2) is the 6d scalar propagator. We see that the long = 3 = 2.4 4 D ± − 4.2.2 3 | M Q ′ are both positive, with the explicit form given in [ ± O A ) are defined as in [ X q Q | ( M i 1 f − ) ], the threebranes in type 0 carry the following charge and te 3 π 29 = 0 brane charge. The charges ( ± p ) plane. The nature of the interaction at short and long dista X and the + ) = (4 3 ′ πt 2 ± ), where the sign of the mass term for the scalars is positive ( − e (O X ( 2.5 ) is no longer a valid approximation, instead one can expand t 6 = − units of D 3 G q ) is, up to a sign, identical to the case analysed in [ ) around ′ 2.4 5 As observed in [ Notice that the (in)stability of the brane configuration tra − 2.3 2.3 ,( p ± 2 ± term in ( charge and tension are perturbative nature may still arise,Instead, but we are may less rely straightf the on brane the setup field (as in theory section analysis and try to rev the orientifold is similar.metric) Therefore, representation the is theory perturbatively with stable ferm (unstabl the D3s are repelledwhere away the from stable the vacuum orientifold. of But the the theory analy lies. In a non-SUSY setu This is clearly different from the situation in type II theori It is then a matter of comparing ( theory with anti-symmetric (symmetric)the fermions. 1-loop In Coleman-Weinberg potential, the which Fi and gets fermions unequal in each theory. field theory to statements about the vev of the scalars this, it follows thatand there O is a short range attractive (repu in ( the orientifold plane as wellin as ( the brane-orientifold pote results in the following. For large separation 2.1 A pseudo-moduliThe space discussion in the previous section shows that the crucial in constructing seiberg dual pairs in the next secti X range potential between the branes and O where where where the coefficients Λ, JHEP09(2019)111 . i ) ) n a n + X (2.8) (2.9) U( ( (2.10) V maximal ) factor of gative vevs N denotes the m v . U( U(1) eigenvalues take . As in the cases × ) ) e are also adjoint takes the diagonal n ) m m n + + + 3’s are given positive n the adjoint of the n n rmions thanks to the ( ( X he eigenvalues coincide − − . n U( t of the unbroken gauge N N a configuration where all ase (iii) corresponds to R values of the eigenvalues , a s of equal magnitude. These ve a completely kinematical ∈ cts a scalar potential . U(1) eory and field theory. i tion where rection is physical. When we iewed as moduli but are rather t is, none of the D3s coincide. m field theory discussed in (iii). ··· n a , × eigenvalues, i.e. − 1 ) N m m { v, a U( − N ); U(1) , N × D3s from the orientifold in the negative × ) m }| 0, then in the brane picture , a ) ··· n m n v, in the bulk. The worldvolume theory of this ··· v > U( − z , v – 6 – 2 { → , v, of the brane system before imposing the dynamical , a ) 1 n a }| N ··· z v, U( = diag ( i D3 branes coinciding in the bulk (away from the orientifold) + = diag n pseudo-moduli X i eigenvalues become exactly degenerate there is an enhanced h + N X h ) matrices, the most generic vev for the scalars N D3 branes in the transverse space. On the other hand giving ne of the n are all distinct. In this case the gauge group is broken to its n D3’s coinciding at coordinate i a m D3s to a negative point in the transverse space, their image D of the scalars corresponds to separating the opposite sign of an exactly degenerate set of torus and the(charge worldvolume 0) fermions scalars for all each become U(1) factor massive. in U(1) Ther symmetry. The breaking pattern in this case takes the form All worldvolume fermionsunbroken are gauge massive group. A but specialand there case the of are this entire type scalars gauge is symmetry i when is all unbroken. t symmetry. Unlike thosecancellation cases, between there the positive are and negative nowfermions eigenvalue also transform massless in the fe bi-fundamental of the non-abelia (i), (ii) above there are scalars transforming in the adjoin The unbroken gauge symmetry is now U( the unbroken gauge group. m Using the U( From the string theory perspective, case (i) corresponds to m (i) The (ii) When (iii) There is a more exotic possibility. Consider the situa form we encounter 3 possibilities: From a field theoretic point of view, depending on the specific configuration beautifully matches what one would expect fro constraints. We will examine the situation both in string th branes are at distinctCase points (ii) away from corresponds the to orientifold, tha coordinates of to be induced viadiscussion loop of the corrections. possible It is however useful to ha Case (iii) is more interesting. Suppose that direction. But onlysend the quotient space,coordinates i.e. and the appear positive in di D3s the and physical space. So we see that c to dictated by the dynamics of the theory. Generically one expe correspondingly the coordinates of the branes are not to be v JHEP09(2019)111 ]. In in two 33 = 2 setup figurations N NS5 ]. (a) to (b) a pair of D3s to separate them in a (b) 32 ther in the presence of ossible. The NS5 branes ′ ] which follows from a re- n setup. In constructions ogue of the 3d . All objects extend along the her [ es, which requires the intro- 31 9 and vice versa on either side , ny-Witten transition [ NS5 1 R − angement without the need for 3 8 9 ′ + 7 8 9 3 ′ to O O . In the presence of an orientifold, the | ′ + 6 6 3 | ′ ). and label only the asymptotic charge of the 2 + – 7 – 3 ′ , we distinguish them by referring to one as an 2 θ i 33 4 5 8 9 3 7 h ′ ) 3 ′ , k ′ D5 D3 O NS5 NS5 (1 (a) NS5 ]). In particular, we have NS5 branes which are non-parallel 30 which intersects the orientifold. We will only consider con ′ NS5 directions as well as those indicated below. 2 , 1 . The Hanany-Witten effect. In passing from the configuration , 0 . The various extended objects and their orientation in + x 3 ′ . The orientifold charge is switched from O The result of moving non-parallel fivebranes through one ano Seiberg duality has a standard string theory derivation [ ′ O Figure 2 in type IIB (see e.g. [ duction of NS5 branes. Our construction is the non-SUSY anal are created between the non-parallel NS5s. 2.2 Hanany-Witten setup We are interested in Hanany-Witten setups to study 3d theori NS5s are bound towill the inevitably orientifold intersect as plane we and try this to is move them no past longer one p anot where the orientifold is asymptotically O of an NS5 or NS5 orientifold plane in our diagrams (see figure direction mutually transverse to the NS5 and NS5 an orientifold is well understood. This is the so called Hana arrangement of non-parallel NS5without an branes orientifold, in it thethe is Hanany-Witte NS5 possible branes to to achieve intersect. this rearr This is done by using the freedom of their spatial coordinatesNS5 as in table shared Table 2 JHEP09(2019)111 . CS are limit Adj ′ (3.1) k N . The → theory. 4 l be more ⊕ 2 ) it is easy to s D e IIB analogue. electric + . 2.7 ˜ λ ˜ λ D3 branes stretched ¯ ˜ λσ ¯ ˜ λ ) subject to suitable c i eads to the non-SUSY N cussion of 3d dualities − ( − brane charges ending on − of D5 branes the linking ) and two antisymmetric ¯ λλ c Y our setup. The discussion p. ncludes quarks and squarks with ¯ λσλ N − expect from the fact that the he creation/annihilation of a o three-dimensional CS theo- i 3 ]. The difference here, besides ture as level-rank dualities. ssion around ( = 2 counterpart. In the large − 30 Dσ with a YM term and level ˜ λ N / t µ D ing rule is expected to hold: + 2 . The low-energy theory of such a ¯ ˜ λ A ), respectively. The Lagrangian takes i = 2 CS theories without flavours of 3 flavour D5-branes, see figure c ρ + N f N A N ν / the linking number of the NS5 and NS5 Dλ A ¯ λ µ of U( 2 i A 3 orientifold discussed previously. – 8 – i + ′ 3 2 in the adjoint of U( 2 ) ) gauge field − c σ σ µ ρ N A D ) 5-brane. We will refer to this as the and the ν ∂ , k + ( µ depending on the orientifold type, a fact that follows from 2 A ′ ) µν F of D3s are created in between them as we go from (a) to (b). This ( µνρ ǫ 1 2 . The construction is analogous to [ − 4 pair Tr we consider the setup of figure we consider the addition of 3 π ], and we provide a more refined account. In particular, we wil Tr k and 4 2 e 34 1 3.1 3 3.2 g relative to the type 0 D3 is a factor of two greater than the typ + ± = 3 ′ ) =0 f E ( N L In section In the next section we discuss the Hanany-Witten setup that l The starting point is the brane configuration (a) of figure In section Such a Lagrangian is understood as descending from its paren 3 3 3d dualities fromIn non-supersymmetric this brane section we configuration considerries. Hanany-Witten See setups figure that lead t gauge theories of interest with and without flavours. the following form (complex) Dirac fermions in the interactions, as well as a real scalar boundary conditions. There is a U( follows that of [ careful about the CS level of the U(1) factor of the gauge grou low-energy theory emerging fromin such the a fundamental brane representation configuration of i the gauge group. 3.1 Level-rank duality We begin by discussing how level-rank duality is realised in configuration is that(s)quarks. of non-SUSY Such analogue awithout of matter. setup These turns dualities out are also to known be in the meaningful litera for the dis being in type 0B, is the presence of the O see that for the type 0 configuration of figure imposing the conservation ofnumber of linking an number. NS5 isit proportional In from to the the the left absence difference and of right the respectively. net Following D3 the discu D3 between the NS5 and NS5 type IIB constructions with an orientifold this amounts to t between an NS5 brane and a (1 we expect to recover a supersymmetric YM-CS theory. The follow conserved provided a is twice the corresponding situationcharge in of O type IIB as one would The worldvolume theory is the dimensional reduction of the JHEP09(2019)111 he . 3 (3.3) (3.2) ] with 32 , 30 2 D NS5 . + . After reshuffling l ˜ CS interactions as D3 2 , where the number l ˜ ls ¯ ˜ (b) . The Lagrangian is l ¯ ˜ 3 k i l ˜ κ absence of supersym- theory. The worldvol- − − ) − nstructions in figure ll ¯ k , k and which give rise to level-rank lsl ¯ − l -loop mass of the order of − i (1 ) κ ry following e.g. [ ds in the representations of − Adj Adj Ds l ˜ resentation of the gauge group U( magnetic / ssed in figure D + is the covariant derivative. The l ¯ ˜ 3 . + 2 i ′ µ µ l l ˜ s Λ O a + iA ρ 2 m a g / ν − Dl l ¯ a i ∼ µ µ ∂ 2 s a + i 3 2 ≡ 2 ) s µ with YM term and level − µ – 9 – , m D ρ µ Λ D a a 2 e ν g ∂ + ( µ ∼ 2 a ) k 2 σ ) c µν m f +2. We refer to this as the N ( Adj Adj µνρ c 1 2 ǫ and antisymmetric Dirac fermions U( N s − is the auxiliary field of the vector multiplet borrowed from t − ) Tr µ k ˜ λ λ σ , k D Tr π A k ≡ 4 (1 2 m 1 κ g + D3 = ) fivebrane we arrive at the configuration (b) in figure c (a) N ) , k =0 f ] M ( N 34 NS5 L is the gauge field strength and . The field content of the worldvolume theories of the brane co . The brane setup for the (a) electric and (b) magnetic theory µν F ) they belong to. c + It is straightforward to obtain the Seiberg dual of this theo We are interested in the IR dynamics of these theories. In the 3 ′ N Table 3 O just like the gauge field and scalar gaugino. supersymmetric parent theory. It belongs to the adjoint rep well as a real adjoint scalar ume theory is now that of a gauge field covariant derivative is understoodU( to act on the various fiel a slight modification that takes into accountthe the NS5 effect and discu (1 of colour D3s is now Here Figure 3 duality. metry, the scalars onthe the cutoff two [ sides are expected to acquire a 1 JHEP09(2019)111 , k κ 2 + g c (3.4) (3.6) (3.5) ± N ). Due − , = c 3.2 N − CS ) NS5 κ M D3 ) and ( κ . c . ˜ 3.1 N + c ) and U( of our proposal. Here c N c , k N (b) CS TQFT where N − (1 e subleading due to the − − 2 CS TQFT with κ lay a role. Note that the re nicely encoded in terms 2 ,K κ,κ 1 evels in ( ≡ y separately semi-classically ) ,L o the setup, the worldvolume 2 = . c symmetric fermions shift the K 1 evel CS masses ) metric fermions) get a mass at . The IR phases of the electric L κ N c − ) 2 D5 the IR, we recover the following + + 2 κ c N κ f c N N N − , f the configuration. At energies well + 2 2 c k N − + − − 3 k ′ ) κ = O = 2 U( 2 – 10 – )) factors of the gauge group by disproportionate ←→ κ c ,L N c κ, K − N ≡ ). After integrating out the scalars we are left with gauge − κ,κ ) ) this translates to an attractive force between the branes ) c , k 3.3 2.5 N + 2 2 = (1 c ) (resp. SU( U( − c N κ N − + k D5 k = . (a) f − c 1 N N = K D3 1 − c L N + 2 k NS5 . The brane setup for the (a) electric and (b) magnetic theory + f N + 3 = ′ c O ˜ N As in the discussion following ( theory turn out to beof the richer dual than magnetic the theory. casesbefore We studied begin mapping above by out and analysing the a each phase theor diagram. one-loop and can belevels integrated of out. the U(1) Integrating and out SU( the anti scalars also have treestringy level nature of CS the masses masses, infields but ( and we antisymmetric expect fermions, them both to of b which haveto tree-l the lack of supersymmetry, also the gauginos (the antisym and the orientifolds, signalling perturbativebelow stability the o cutoff scales, the scalars are decoupled and do not p where the sign of the mass follows from the sign of the bare CS l In fact, these theories arelevel-rank dual duality to each other. Therefore, in We can include flavours in the discussiondirections by spanned adding D5 by branes the t flavour D5 branes are as in table 3.2 Including flavours Figure 4 While the IR of the magnetic theory is described by a U( Putting everything together we end up with the TQFTs U( amounts. As a result the IR of the electric theory is a U( JHEP09(2019)111 . e 4 (3.8) (3.7) dom, Dirac . The 4 f ) group. f N N ) f b i . N Φ · · · · a b Adj ) 0 and decouple from 2 SU( D > ( ouples to the gauge field i a + h.c.) 2 Φ k ¯ ranes is an SU( Φ − bi M ) ration (a) of figure ative contributions from its t to the gauge and flavour c Ψ + · · · ˜ i a are flavour indices. The inter- b i N pagator get positive contribu- uark fields to be as in table Adj Adj ¯ Φ Φ f ] , a b U( complex scalars Φ and ab ) [ Magnetic Theory ,N 2 f ˜ λ σ i µ ( l l ˜ N ˜ s φ χ χ ψ a M i a ··· + matter , ¯ Φ bi L ¯ − Ψ = 1 + a i ai Φ ] ) =0 i, j f Ψ) E ab [ ( N / – 11 – D ( ). The additional flavour terms are described by iλ L ( ai ). = 3.1 ¯ − Ψ i ) ) . Since there are more bosonic than fermionic degrees bi f 3.8 E λ ( + ¯ Ψ N · · · · 2 a b does not play a role in the IR dynamics of the theory and | L σ a i σ SU( ai Φ of the gauge multiplet of the electric theory is similar to th . These are essentially determined by their coupling to the µ Ψ 4 σ D k | − . Since there are more bosonic than fermionic degrees of free ) c λ . The field content of the electric and magnetic theory. = N are colour indices and Adj Adj c U( Electric Theory ,N matter and the gaugino = 0 is stable; µ Table 4 L ˜ λ λ σ σ Φ Ψ i A is, as before, given by ( ··· σ , h ) =0 f E = 1 ( N L a, b The fate of the scalar The tree level Lagrangian is given by A similar story pans out for the squark Φ. Indeed, the squark c , the scalar µ The representations ofgroups the are scalars listed and in fermions table with respec fermions Ψ. The relevant flavour symmetry emerging from the b flavourless case. The one-looptions corrections from its to coupling thecoupling to scalar to itself pro the and gaugino to the gauge field and neg Here where worldvolume theory on the D3 branes now includes antisymmetric gauginos, see later ( the vacuum actions with the gauginos fix the representations of the (s)q can be integrated out. 3.2.1 Electric theory The flavoured electric theory is realised on the brane configu the IR physics. A of freedom, one expects the squark to acquire a positive mass JHEP09(2019)111 . ). 4 f N (3.9) 1. We (3.12) (3.11) (3.10) The IR ological ≫ 4 ] modified k of SU( 32 to the appro- , 1 30 . It is obtained + h.c. 4 aj } ¯ and a Dirac fermion ψ ij a i { bi directions. These give φ ¯ φ ¯ , ψ } 9 / quark flavours. , ∂χ b a ij 8 ) fivebrane is } ) f figure f { x s alogue on the electric side, ij groups are given in table ( χ , k { N ), as well as the fundamental tasov move [ ¯ ai χ ard to say anything concrete ical to the magnetic theory of + ˜ i ψ transforming as κ . 3.4 . One can easily verify that the ) adjoint and its fermionic part- 3 branes in table the semiclassical regime aj + ( fermions , 2 f − χ with + 2 ψ f | N 3 i a f bi i j N φ φ N } b a M ⊕ matter ij µ ) and ˜ ≡ D { c ∂ f a i L | χ N ¯ φ . This can be obtained in a similar fashion N − + + 4 + 2 y ). + ) we have c ) i b =0 − K,K N f φ M 1.1 1.3 ) a k ( b a c N – 12 – of SU( ¯ ) φ − 2 N k L j which is an SU( k + h.c. s ( M = bi a i + j . M i ¯ ) φ ψ f ¯ a i , unless something drastic happens. M M ¯ k − N ( φ i a ] = 0, the IR theory is that of YM theory coupled to the φ ai = L ab 2 k [ ) y c il + h.c. ˜ N / − 6= 0, the gauge field and the gaugino acquire a Chern-Simons Dψ + aj ( j b ). The matter Lagrangian is ¯ fundamental quarks, i.e. QCD ¯ k bi ψ φ ψ j ¯ i i b j ψ f ¯ 3.2 i a φ electricIR: U( transforming as N + M i a φ ] 2 φ ai χ | a i ab i a ψ [ ¯ φ l ˜ φ i . We therefore expect the IR physics to be dominated by the top 2 µ y y k 2 D is as in ( | g − − − ) = =0 = f M ( N CS = 0 case. The matter multiplet consists of a complex scalar L f M matter The magnetic field content is given in table On the other hand, when For a non-zero level The tree level Lagrangian for this theory is N Integrating out the gauge sector is somewhat more natural in L 4 . Their representations with respect to the gauge and flavour CS theory coupled to where mass ψ priate boundary conditions. We havethe a gauge multiplet ident There are in addition newcorresponding degrees to of the freedom, motion which of have the no flavour an D3 branes along the resulting number of colour branes between the NS5 and the (1 to the electric theory, i.e. by subjecting the theory on the D from the flavoured electric theoryso by as the to standard account Giveon-Ku for the brane creation described in figure levels of the electric theory arequarks. shifted by In the summary, gaugino using as the in dictionary ( 3.2.2 Magnetic theory The flavoured magnetic theory lives on the configuration (b) o gaugino and the fundamentalabout quarks. the IR It dynamics is of less this straightforw theory. ners, the “mesinos” rise to two gauge singlets; the meson expect this to remain true also at finite which is nothing but the left hand side of ( JHEP09(2019)111 . In This . 3 k (3.14) (3.15) (3.13) . This f fixed. In he details N uples, just . There are f ≥ N ely QCD ] c 9 N becomes massive. eson multiplet. − φ can be understood in he assumption that in chyonic. Since at large 3 + 2 k CD ind. This will be crucial in efore, the squark aquires a m the dynamics. The main ultiplet, and more fermionic while keeping e as we move to finite are similar to the symplectic ≡ f the colour branes near the in all phases [ k the squark κ , we now have another coupling . In this region the rank of the k i 5 m δ g 2 . u . )Λ = . a k 2 m ¯ 0 E φ g k k j j < + M M 2 j j 2 i i M y ¯ ¯ − M M – 13 – M ( i a 2 , and flavour symmetry is unbroken, the mesons φ D ) comes from the meson multiplet, which is indeed i a φ φ ∼ 2 M is automatically positive. Following the discussion . vδ y 3 2 φ κ = M + ) for the magnetic theory includes a coupling between the

i a f φ N

3.12 = c ˜ N squarks are assumed to be tachyonic throughout this region. f N couples to the gauge multiplet as well as the meson multiplet ]. For this reason we will be somewhat brief and focus only on t of the magnetic gauge multiplet gets a positive mass and deco ), the IR theory on the electric brane configuration is precis 9 φ s 3.9 ), the which controls interactions between the (s)quarks and the m y 3.13 The squark The matter Lagrangian ( The scalar the gauge field becomes heavy and decouples we operate under t If the squark acquires a vev corresponds to region I in the phase diagram of figure If the meson acquires a vev of the form around ( become massive. Therefore, the most likely scenario is that In the following we willobtaining always the work phase with diagram this of assumption QCD in m As we saw in ( 4 Phase diagram as it did inorientifold. the flavourless case. This signals the stability o which are new to the unitary theory. 4.1 Region I:We bosonization start with the region ofmagnetic the gauge parameter group space where meson field and the scalar quarks negative. Thus, our main assumption is that this remains tru this regime the gaugecontribution sector to becomes the heavy mass of and the decouples squark fro ( constant Note that in addition to the magnetic gauge coupling is reasonable as one can go to arbitrarily large values of this limit the squark is tachyonic. terms of the dual magneticcase description. analysed in Many [ of the features more bosonic than fermionic degreesthan of freedom bosonic in degrees the of gauge1-loop m freedom mass in of the the meson form multiplet. Ther this section we argue that the conjectured phase diagram of Q k The two effects compete and the squark may become massive or ta JHEP09(2019)111 f N ion, (4.1) (4.2) he CS ) and can fundamental | f 3.14 κ | N . κ flavour D3 branes. = paragraph in terms f brane configuration, f N N ns like ( colour D3 branes which ∗ ) global symmetry. We contains, in addition to scalars f N κ f e world-volume of the alars and gauge field, also om open strings stretched N scalars in the fundamental y end on D5s from one side N without breaking the global f d, the Higgsing corresponds . ⊕ ng out the gaugino according I N 3 f 2 N + , K ) + c κ N II − U( III , c N → − ) f f – 14 – f N ), the IR of the magnetic theory in this region of N 2 N + 1.3 + κ . Phase diagram of QCD of the Higgsed gauge group as well as ′ ′ Higgsed D3 branes. K U( II f III colour D3 branes via reconnection to N U f , and we can integrate out the gauge field and gauginos at and k N Figure 5 magnetic quarks become massive due to Yukawa terms. In addit gauge theory with massive gauge field and massive gauginos. T . The reconnection preserves the original U( k k f 2 magnetic IR: − g N No seiberg duality ) κ ) so that, using the dictionary ( ). The gauge symmetry breaking pattern is given by f Let us try to understand the phenomenon described in the last Let us then assume that the magnetic squarks condense. In the 3.5 N leaving the gauginos in the to giving a colour-flavour lockingU( vev to the magnetic squark of the field theory description of the magnetic theory. Indee the meson and the mesino all become massive due to interactio This is the Higgs mechanism in the string theory language. Th energies below be integrated out.to The ( IR levels getthe parameter shifted space after is integrati described by Such a bosonic duala is CS described term in with the appropriate IR levels by and a coupling Lagrangian between that the sc mass is still proportional to Higgsed D3 branes no longerand supports end a on the gauge NS5 multiplet brane as from the the other. However, we still have after the Higgsing.between In the the colour brane branes set-up and these can only come fr squarks. The support a U( this corresponds to Higgsing will shortly argue, from the field theory side, that there are JHEP09(2019)111 . t 2 0. ) i a (4.3) (4.4) (4.5) φ a i as well ¯ κ < φ ) and in i a φ onetheless, a i 3.9 ¯ φ K ). − c f N 2 icipate breaking of N eory in ( i . K − G flow consistently with in the phase diagram of K . In this region ions is still captured by a f 5 ′ 2 connected D3 branes. The + K N nes than flavour D3 branes: nclude CS terms in the effective ). Such a Grassmannian will ]. As a consequence, the IR ge group. Once again this is an ] is for SU( ge symmetry is fully broken. ) and (double-trace) ( κ f 7 − 7 2 k b ) ction, but the implications and N 4.3 U + f φ s terms of the form f n I. Note that the electric theory 2 b j f N ¯ × N φ ]. N 7 )( flavour D3 branes stretched between j a U SU( K | φ ) factor corresponding to the symmetry ) = 2 ) factor from the remaining flavour D3 | a i κ l in [ κ | × κ + ¯ κ φ + f 2 f colour and flavour D3 branes (we stress that N − | K N κ f + + N U ( – 15 – f h | f 2 κ N S N | = = 2 U f 1 ) we have that in this region the global symmetry breaking S − ,N ), we recover a well-established duality. This is nothing bu 5 1.3 f → 2 2 κ ) 4.2 N ) fivebrane, as well as the f + , k + N 2 ) , which corresponds to region II and II K κ SU( ) gauge theory, while the result in ref. [ + c ′ > κ f . We will assume that the squarks condense also in this case. N N M f N f ). ( reconnected branes as well as a U( κ 1.1 − > N + κ < N 1 ⋆ f + − f , we expect rather different dynamics for the system and we ant N N 2 f 5 N N 0 here). After the Higgsing, we are left with The global symmetry now consists of a U( As a final step, tuning the mass terms both in the electric IR th In order to be consistent with the UV symmetries one must also i = 5 c ˜ branes. Using the dictionary ( on the latter no longer support a gauge multiplet and therefore gau κ < the D5 brane and the (1 figure squark condensation leads inrealised this in case string theory to by a reconnecting fully Higgsed gau corresponding to the symmetry breaking pattern given in ( This symmetry breaking pattern is the one anticipated in [ N be essentially parametrised by physics of this phase is described in terms of the Grassmanni pattern is Therefore, on the magnetic side, there are less colour D3 bra tachyonic squark, colour-flavour locking andthe brane resulting reconne physics will bewe different discuss with is respect to a regio U( the duality ( the flavour symmetry. As we shall see, the physics in these reg the magnetic IR theory in ( Let us begin with region II’ in the phase diagram of figure 4.2.1 Region II self-interactions for the squarks. These correspond to mas 4.2 Symmetry breaking When as quartic interaction of the form (single-trace) ( These terms can beglobal generated, symmetries. if not already present, by the R description. The required modification is discussed in detai JHEP09(2019)111 for 5 (4.6) is the ]. 9 5 are slightly 5 . According to ). This picture = 0 there is no IR is maximised rk condensation φ , κ k f N ( picture of figure ) correspond to scalar ry of massless Nambu- M e seeking is nothing but agram of figure 4.6 esino, due to the presence ( try breaking can occur for tor. Because of this, some ldstone bosons as the mass- an 2 that modify parts of the ugino. Therefore, the IR of ram in figure re our proposal is that these f flavour branes. ction we have a U(1) gauge condensation. Regardless, in , φ . s holds for a general number already discussed in [ f g tachyon condensation. We do N = 1 ⊕ ), after reconnection the theory in c ] upon condensation of the squarks ⋆ 7 f N 2 N + < N K fermions. The magnetic dual has a gauge f + f c N N − , < N c – 16 – N f 2 − N + f 2 N massless Dirac fermions. The reason that in this specific + 3 < K ) for large negative masses of the squarks f = 2 the electric gaugino is a singlet of the SU(2) factor of K N c 4.6 theory coupled to N (or 0 U 0 2. ⋆ ≥ c < N N . In particular, as we shall see momentarily, when f 3 with vanishing CS-term . However, “accidents” happen when + 1) with vanishing CS level at tree-level. Previously, squa c 3 ]. f since the window for which a Grassmannian phase exists in the N 7 N k < κ < N = 0 [ Indeed, after reconnection, the scalars in the bosonic dual On the other hand, we start to see deviations from the general = 1 i.e. QED k c symmetry breaking phase. Thisnon-zero in turn suggests that no symme is consistent with the mass deformations of the brane setup, for modified, the end resultcorrect picture is for however unaffected and the phaseN di The discussion of theof phase diagram colours in the preceding section the field theory limit one eventually lands on the Grassmanni 5 Comments about QED we land on the symmetry breaking phase. modes of the open stringsscalars in are tachyonic the and brane are configuration.not to know Therefo be whether stabilised a via nice open geometric strin picture emerges after this Naively, we seem toGoldstone bosons have we a obtain puzzle:the bosonization. effective description instead The of NG of ( theory obtaining we a ar theo When the electric gaugethe group electric is theory U(1), is U(1) there is no electric ga 5.1 QED the gauge group, butintermediate it steps carries taken charge to 2 arrive under at the abelian the fac general phase diag lead to masses being generatedof for Yukawa the interactions. quarks, mesontheory However, and with in the no this m CS case term after and reconne group U( the IR is discussion. In the case of less modes of open strings stretched between4.2.2 the two stacks o Region II When 0 massless Nambu-Goldstone bosons. We identify the Nambu-Go the field theory analysis of Komargodski and Seiberg [ JHEP09(2019)111 ). f N ommons ]. 35 [ 6= 2 the naive 3 dual quarks. f flavours. Flavour f N f N = 0 the theory looks ng of the symmetry N nce flavour symmetry d without fine-tuning that admits the same hout a Chern-Simons k ], for tion as a magnetic dual pported by STFC grant ] we anticipate that it is 7 s of fields that acquire a mpson for pointing out a and f having such a UV theory no when the gauge group 9 [ k redited. ntaneous breaking of U( ory with l leads to new insights about s. For k the magnetic gauge theory nd Zohar Komargodski for a − haviour of QED a mass due to Yukawa coupling = scalar quarks acquire a mass and flavours admits a magnetic dual ′ k f N . This regime is hard to analyse both ⋆ > N f – 17 – N based on a unitary group and its embedding in string 3 ), which permits any use, distribution and reproduction in 0) U(1) theory with ≥ k = 2 the self duality is well understood [ f CC-BY 4.0 N (with This article is distributed under the terms of the Creative C k . In particular the bosonized theory admits a simple realisa 3 In addition, we learned about the abelian theory, with or wit The Seiberg dual also enables us to gain a better understandi We haven’t discussed the regime of The brane setup is such that the flavour branes coincide and he breaking phase. Triggered byis condensation completely of Higgsed the and dual flavour squar symmetry gets broken. that upon Higgsing flows to another U(1) theory with This is consistent with existing conjectures about the IR be in field theory and in string theory. As in the symplectic case decouple, in the magnetic sideand the fermionic decouple. quarks acquire of the electric fermionic theory. While in the electric side self-dual. While for remains unbroken. Thus, our magnetic theory predicts no spo theory. The UVmass field and theory decouple as on the the theoryis flows brane that to it the configuration IR. admits consist The aQCD advantage Seiberg o duality. The magnetic Seiberg dua symmetry is not broken, as expected from field theory analysi self-duality deserves further investigation. ST/P00055X/1. Open Access. any medium, provided the original author(s) and source are c Acknowledgments We thank Vasilis Niarchos forcareful numerous reading useful of discussions thetypographical a manuscript. error We in are the grateful draft. to Dan The Tho work of A.A. has been su described by meson condensation. 6 Conclusions In this manuscript we discussed QCD term. The level Attribution License ( is U(1) and nothe Yukawa squarks term. acquire In amatter the mass. content absence as So of the we supersymmetry electric end theory, an up namely with a a dual U(1) magnetic the theory case the fermions do not acquire a mass is that there is no glui JHEP09(2019)111 ]. N ] ] N SPIRE gauge ]. IN string ][ ]. = 1 , , , N SPIRE ]. IN SPIRE Chern-Simons , ][ heories ]. IN 3 SPIRE ][ ]. IN ]. arXiv:1110.4386 [ , , ][ SPIRE Metastable vacua in large- IN QCD domain walls, SPIRE CERN-PH-TH-2004-022 ][ SPIRE arXiv:1806.08292 IN [ ]. ][ ski, IN ][ ]. ][ dia and X. Yin, ]. . Niro, ]. hep-th/0302163 [ (2012) 2112 bosonization SPIRE d IN SPIRE arXiv:1110.4382 3 [ ]. ]. ]. ]. Nonsupersymmetric dualities from mirror SPIRE IN ][ SPIRE (2018) 090 IN C 72 ][ IN arXiv:1609.02149 ][ [ bosonic vector models coupled to 09 ][ from non-SUSY Seiberg duality and brane SPIRE SPIRE SPIRE SPIRE (2003) 170 3 hep-th/0403071 arXiv:1711.04832 Exact results in non-supersymmetric large From super Yang-Mills theory to QCD: planar IN IN IN IN D-branes and SQCD in non-critical superstring [ hep-th/0606203 = 3 [ ][ ][ ][ ][ d (2012) 037 – 18 – JHEP arXiv:1606.01912 , M. Shifman, A. Vainshtein and J. Wheater eds., [ , 03 B 667 ]. From fields to strings: circumnavigating theoretical Eur. Phys. J. , A symmetry breaking scenario for QCD (2017) 011602 arXiv:1606.01893 , in [ (2006) 019 JHEP SPIRE hep-th/0504010 arXiv:1709.07872 arXiv:0808.0360 [ [ Particle-vortex duality in topological insulators and Phases of QCD , [ (2018) 106001 IN Seiberg duality in Chern-Simons theory Three dimensional bosonization from supersymmetry 118 (2017) 159 10 [ Level/rank duality and Chern-Simons-matter theories D-branes in non-critical superstrings and duality in Embedding three-dimensional bosonization dualities into Nucl. Phys. 05 Particle-vortex duality from , arXiv:1507.04378 arXiv:1706.08755 arXiv:1512.00161 arXiv:1607.07457 D 97 [ [ [ [ JHEP (2009) 1 , (2005) 081 (2017) 031 JHEP (2016) 031043 , Singapore (2004), pg. 353 [ , 10 12 Baryons, monopoles and dualities in Chern-Simons-matter t B 812 X 6 Phys. Rev. Lett. Phys. Rev. (2015) 013 (2018) 109 (2016) 093 (2016) 095 , , ]. ]. arXiv:1905.01797 JHEP JHEP 11 01 02 09 , , , 3 SPIRE SPIRE IN IN theories with flavor physics. Ian Kogan memorial collection [ theory orientifold field theories equivalence and its implications World Scientific JHEP Nucl. Phys. theory Chern-Simons theories and holography symmetry Phys. Rev. superconductors JHEP JHEP JHEP [ QCD Chern-Simons gauge theories theory with vector fermion matter dynamics [3] O. Aharony, [8] A. Armoni, T.T. Dumitrescu, G. Festuccia and Z. Komargod [9] A. Armoni and V. Niarchos, [6] J. Murugan and H. Nastase, [7] Z. Komargodski and N. Seiberg, [4] P.-S. Hsin and N. Seiberg, [5] A. Karch and D. Tong, [2] S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wa [1] O. Aharony, G. Gur-Ari and R. Yacoby, [17] S. Murthy and J. Troost, [18] A. Fotopoulos, V. Niarchos and N. Prezas, [15] A. Armoni, M. Shifman and G. Veneziano, [16] A. Armoni, M. Shifman and G. Veneziano, [13] G. Gur-Ari and R. Yacoby, [14] A. Giveon and D. Kutasov, [11] R. Argurio, M. Bertolini, F. Bigazzi, A.L. Cotrone[12] and P S. Kachru, M. Mulligan, G. Torroba and H. Wang, [10] K. Jensen and A. Karch, References JHEP09(2019)111 ] ] 376 l gauge , amples in D , 4 ]. , ]. strings ]. 0 ling arXiv:0801.0762 B strings in [ Erratum ibid. [ 0 SPIRE arXiv:1408.4633 SPIRE [ , 95, Palaiseau, France, IN ]. IN SPIRE ][ ][ ]. IN hep-th/9602052 ]. , ][ ]. (2002) 1 SPIRE IN , SPIRE SPIRE (2014) 387 ][ ]. IN ]. 371 (2008) 105009 IN ]. duality in string theory SPIRE ]. ][ ][ , rkshop, SUSY IN Supersymmetry and unification of ][ = 1 SPIRE ]. SPIRE B 739 D 77 SPIRE hep-th/9904069 Non-supersymmetric Seiberg duality, IN , in SPIRE [ IN hep-th/9906101 N [ IN IN ][ ]. ][ orientifold equivalence hep-th/9611230 ][ [ ][ Notes on D-branes Phys. Rept. SPIRE N Orientifolds, branes and duality of , IN hep-th/9702093 ][ SPIRE [ Phys. Rev. Phys. Lett. IN hep-th/9703210 – 19 – (1999) 66 (1999) 159 , [ , Nonsupersymmetric gauge theories from D-branes in Tachyon free orientifolds of type Time-reversal symmetry, anomalies and dualities in ]. ][ Branes and arXiv:1712.08639 [ (1997) 152 arXiv:0705.2140 [ D-branes and dual gauge theories in type Open strings SPIRE B 560 hep-th/0608180 B 558 hep-th/9802067 [ [ hep-th/9811035 (1997) 332 IN [ hep-th/9702014 [ ][ B 492 (1997) 251 Brane dynamics and gauge theory Type IIB superstrings, BPS monopoles and three-dimensiona Tree-level stability without spacetime fermions: novel ex Level-rank duality in Chern-Simons theory from a hep-th/9509080 (In)validity of large (2018) 006 B 56 (2007) 065 5 hep-th/9912145 [ B 505 07 Nucl. Phys. (1999) 983 Nucl. Phys. (1999) 155 , , (1997) 269 (2006) 105019 71 Nucl. Phys. , Comments on nonsupersymmetric orientifolds at strong coup JHEP Surprises in open string perturbation theory Some properties of open string theories , hep-th/0204089 B 546 B 400 D 74 (2000) 041 SciPost Phys. Nucl. Phys. ]. ]. ]. , , 02 d string theory 0 ¨ Unsal and L.G. Yaffe, SPIRE SPIRE SPIRE IN IN IN (2 + 1) theories gauge dynamics non-supersymmetric brane configuration [ Phys. Lett. Nucl. Phys. Rev. Mod. Phys. Phys. Rev. [ JHEP Nucl. Phys. Proc. Suppl. type string theory orientifold QCD and non-critical[ strings (2003) 407] [ various dimensions fundamental interactions. Proceedings, International Wo 15–19 May 1995, pg. 473 [ [35] C. P.-S. Córdova, Hsin and N. Seiberg, [28] A.M. Uranga, [21] C. Angelantonj and A. Sagnotti, [34] A. Armoni and E. Ireson, [31] S. Elitzur, A. Giveon and D. Kutasov, [32] N.J. Evans, C.V. Johnson and A.D. Shapere, [33] A. Hanany and E. Witten, [29] I.R. Klebanov and A.A. Tseytlin, [30] A. Giveon and D. Kutasov, [26] M. [27] J. Polchinski, S. Chaudhuri and C.V. Johnson, [24] A. Sagnotti, [25] A. Sagnotti, [22] R. Blumenhagen, A. Font and D. Lüst, [23] R. Blumenhagen, A. Font and D. Lüst, [20] A. Armoni, D. Israel, G. Moraitis and V. Niarchos, [19] D. Israel and V. Niarchos,