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JID:PLB AID:32005 /SCO Doctopic: Theory [m5Gv1.3; v1.178; Prn:26/05/2016; 15:27] P.1(1-5) Letters B ••• (••••) •••–•••

1 Contents lists available at ScienceDirect 66 2 67 3 68 4 Physics Letters B 69 5 70 6 71 7 www.elsevier.com/locate/physletb 72 8 73 9 74 10 75 11 76

12 Studying critical emerging from non-Abelian vortex in four 77 13 78 14 79 15 P. Koroteev a,b, M. Shifman c, A. Yung c,d,e 80 16 81 17 a Perimeter Institute for Theoretical Physics, Waterloo, ON N2L2Y5, Canada 82 b 18 Department of , University of California, Davis, CA 95616, USA 83 c 19 William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA 84 d National Research Center “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300, Russia 20 85 e St. Petersburg State University, Universitetskaya nab., St. Petersburg 199034, Russia 21 86 22 87 23 a r t i c l e i n f o a b s t r a c t 88 24 89 25 Article history: Recently a special vortex string was found [5] in a class of soliton vortices supported in four-dimensional 90 26 Received 13 May 2016 Yang–Mills theories that under certain conditions can become infinitely thin and can be interpreted as 91 27 Accepted 23 May 2016 a critical ten-dimensional string. The appropriate bulk Yang–Mills theory has the U (2) gauge group and 92 Available online xxxx 28 the Fayet–Iliopoulos term. It supports semilocal non-Abelian vortices with the world-sheet theory for 93 Editor: M. Cveticˇ R4 × 29 orientational and size moduli described by the weighted CP(2, 2) model. The full target space is Y6 94 30 where Y6 is a non-compact Calabi–Yau space. 95 We study the above vortex string from the standpoint of , focusing on the massless states 31 96 in four dimensions. In the generic case all massless modes are non-normalizable, hence, no massless 32 97 or vector fields are predicted in the physical spectrum. However, at the selfdual point (at strong 33 98 coupling) weighted CP(2, 2) admits deformation of the complex structure, resulting in a single massless 34 hypermultiplet in the bulk. We interpret it as a composite “baryon.” 99 35 © 2016 Published by Elsevier B.V. This is an open access article under the CC BY license 100 36 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 101 37 102 38 103 39 104 40 1. Introduction In this paper we explore the spectrum of the massless modes 105 41 of the above critical closed string theory. In [5] a hypothesis was 106 42 Studies of non-Abelian vortex solitons1 [1–4] supported by formulated regarding parameters of the bulk theory and the cor- 107 43 some four-dimensional Yang–Mills theories with N = 2 supersym- responding world sheet model necessary to make the thickness of 108 44 metry, resulted in identification of a bulk theory which under sev- the vortex string vanish at strong coupling, which is required in 109 45 eral conditions gives rise to a critical ten-dimensional string [5]. order to neglect higher derivative corrections in the world sheet 110 46 The appropriate four-dimensional Yang–Mills theory has the U (2) theory on the vortex. Here, using duality we derive an exact for- 111 47 gauge group, the Fayet–Iliopoulos term ξ and four matter hyper- mula for the relation between the two-dimensional (2D) coupling 112 48 multiplets. Its non-Abelian sector would be conformal if it were β and four-dimensional (4D) gauge coupling of N = 2SQCD. 113 49 114 not for the parameter ξ . It supports semilocal non-Abelian vortices Moreover, we identify the critical point β∗ = 0at which the vortex 50 115 with the world-sheet theory for orientational and size moduli de- string can become infinitely thin. At this point the resolved coni- 51 2 116 scribed by the weighted CP(2, 2) model. The target space is a fold mentioned above becomes a singular conifold. As the only 4D 52 117 six-dimensional non-compact Calabi–Yau Y6, namely, the massless mode of the string which emerges at β∗ = 0we identify 53 R4 118 resolved conifold. Including the translational moduli with the a single matter hypermultiplet associated with the deformation of 54 119 target space one obtains a bona fide critical string, a seemingly the complex structure of the conifold. Other states arising from the 55 promising discovery of [5]. 120 56 ten-dimensional are not dynamical in four dimensions. In 121 57 particular 4D graviton and unwanted vector multiplets are absent. 122 58 This is due to non-compactness of the Calabi–Yau manifold we 123 E-mail address: [email protected] (M. Shifman). deal with and non-normalizability of the corresponding modes. 59 1 The non-Abelian vortex strings are understood as strings carrying non-Abelian 124 60 moduli on their world sheet, in addition to translational moduli. Then we discuss the question of how the states seen in the 125 2 61 A more accurate mathematical term is the two-dimensional sigma model with bulk theory at weak coupling are related to what we obtain from 126 O − ⊕(2) 62 the target space ( 1)CP1 . For brevity we will use WCP(2, 2). the string theory at strong coupling. In particular we interpret the 127 63 128 http://dx.doi.org/10.1016/j.physletb.2016.05.075 64 129 0370-2693/© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 65 130 JID:PLB AID:32005 /SCO Doctopic: Theory [m5Gv1.3; v1.178; Prn:26/05/2016; 15:27] P.2(1-5) 2 P. Koroteev et al. / Physics Letters B ••• (••••) •••–•••

1 hypermultiplet associated with the deformation of the complex 3. Bulk duality vs world sheet duality 66 2 structure of the conifold as a monopole–monopole baryon. 67 3 Since our vortex string is BPS saturated, the tension T in Eq. (1) 68 4 2. World sheet model is given by the exact expression 69 5 70 6 T = 2πξ (6) 71 7 The basic bulk theory which supports the string under investi- 72 where ξ is the Fayet–Iliopoulos parameter of the bulk theory. 8 gation is described in detail in [6]. Let us briefly review the model 73 As we know [13,14] the bulk theory at hand possesses a 9 emerging on its world sheet. 74 strong–weak coupling duality4 10 The translational moduli fields (they decouple from other mod- 75 uli) in the Polyakov formulation [7] are given by the action 11 → =−1 = 4π + θ4D 76  τ τD , τ i . (7) 12 √ τ g2 2π 77 13 T 2 αβ μ 78 S0 = d σ hh ∂αx ∂β xμ + fermions , (1) 14 2 The bulk duality implies a similar 2D duality which manifests itself 79 15 in the world sheet theory as the interchange of the roles of the 80 where α ( = 1, 2) are the world-sheet coordinates, xμ ( = 16 σ α μ orientational and size moduli, 81 1, ..., 4) describe the R4 part of the string world sheet and h = 17 82 det h , where h is the world-sheet metric which is under- nP ↔ K , or, equivalently, β → β =−β, (8) 18 ( αβ ) αβ ρ D 83 stood as an independent variable. The parameter T stands for the 19 see Eq. (2). Equation (4) is valid only semiclassically and shows no 84 20 tension which will be discussed below. 85 = = sign of the strong–weak coupling duality (8). An obvious general- 21 In the bulk theory under consideration N f 2N 4, implying 86 P ization of (4) which possess duality (8) under (7) is 22 that in addition to orientational zero modes of the vortex string n 87 = K = 2 23 (P 1, 2), there are size moduli ρ (K 1, 2) [8,1,4,9–11]. 4π g 88 β = − . (9) 24 The gauged formulation of the non-Abelian part is as fol- g2 4π 89 25 lows [12]. One introduces the U (1) charges ±1, namely +1for 90 2 → 2 2 →− 26 n’s and −1for ρ’s, If g 16π /g then, obviously, β β as required by (8). The 91 4D selfdual point g2 = 4π is mapped onto β∗ = 0. The selfdual 27     92 √ point β = 0is a critical point at which the target space WCP(2, 2), 28 2 αβ ˜ ¯ P ˜ K 93 S1 = d σ h h ∇αnP ∇β n +∇αρ¯K ∇β ρ 29 which is the resolved conifold, becomes a singular conifold. 94 It was conjectured in [5] that the non-Abelian vortex string be- 30    95 e2 2 come infinitely thin at strong coupling and can be described by 31 + |nP |2 −|ρ K |2 − β + fermions , (2) 96 32 2 the string action (5). The condition necessary for the vortex string 97 2 33 in the bulk theory at hand to become infinitely thin is that m , 98 where 34 the square of the mass of the bulk Higgsed gauge bosons, is much 99 2 ∼ 2 35 larger than the string tension. At weak coupling m ξ g . It is 100 ∇ = ∂ − iA , ∇˜ = ∂ + iA (3) natural to assume that mass m goes to infinity at the selfdual point 36 α α α α α α = 101 37 β 0. This remains a hypothesis. An example of this behavior is 102 and A is an auxiliary gauge field. The limit e2 →∞ is implied. 38 α 4π 103 Equation (2) represents the WCP(2, 2) model.3 m2 = ξ. (10) 39 |β| 104 40 The total number of real bosonic degrees of freedom in (2) is 105 41 six, where we take into account constraint imposed by D-term. The expansion in derivatives of the action on the string world 106 2 42 Moreover, one U(1) phase is gauged away. These six internal de- sheet runs in powers of ξ/m , implying that higher derivatives are 107 2 43 grees of freedom are combined with four translational moduli from irrelevant if m →∞ [5]. Note that since the bulk theory has a 108 44 (1) to form a ten dimensional space needed for a superstring to be vacuum manifold, there are massless states in the bulk. Most of 109 45 critical. them are not localized on the string, and therefore are irrelevant 110 46 In the semiclassical approximation the coupling constant β in for the string study. The only localized zero modes are transla- 111 2 5 47 (2) is related to the bulk SU(2) gauge coupling g via tional, orientational and size moduli [16]. Other excitations of the 112 2 48 string have excitation energies ∼ m . 113 49 4π One can complexify the constant β in a standard way, by 114 β = . (4) θ 2 = + 2D 50 g adding the topological term in the action (5) βcompl β i 2π , 115 51 where θ2D is the two-dimensional theta angle which penetrates 116 52 Note that the first (and the only) coefficient of the beta functions is from the bulk theory [15]. In [20] we will present a complexified 117 53 the same for the bulk and world-sheet theories and equals to zero. version of the relation (9). 118 54 This ensures that our world sheet theory is conformal invariant. 119 55 The total world-sheet action is 4. Bulk 4D from the critical non-Abelian string 120 56 121 N = 57 The critical string discovered in [5] must lead to 2super- 122 S = S0 + S1 . (5) 58 symmetric spectrum in our bulk four-dimensional QCD. This is a 123 N = 59 priori clear because our starting basic theory is 2. The ques- 124 tion is how this symmetry emerges from the string world-sheet 60 3 Both the orientational and the size moduli have logarithmically divergent 125 61 norms, see e.g. [9]. After an appropriate infrared regularization, logarithmically di- model. 126 62 vergent norms can be absorbed into the definition of relevant two-dimensional 127 63 fields [9]. In fact, the world-sheet theory on the semilocal non-Abelian string is 128 not exactly the WCP(N, N˜ ) model [11], there are minor differences unimportant 4 Argyres et al. proved this duality for ξ = 0. It should allow one to study the 64 129 for our purposes. The actual theory is called the zn model. We can ignore the above bulk theory at strong coupling in terms of weakly coupled dual theory at ξ = 0 too. 65 differences. 5 More exactly they have logarithmically divergent norms, a marginal case. 130 JID:PLB AID:32005 /SCO Doctopic: Theory [m5Gv1.3; v1.178; Prn:26/05/2016; 15:27] P.3(1-5) P. Koroteev et al. / Physics Letters B ••• (••••) •••–••• 3

1 Non-Abelian vortex string is 1/2 BPS; therefore, out of eight Y6, respectively, and we use 4D metric with the diagonal entries 66 2 bulk supercharges it preserves four. These four supercharges form (−1, 1, 1, 1). 67 3 N = (2, 2) on the world sheet which is necessary to have N = 2 Next, we search for the solutions of (11) subject to the block 68 4 69 space-time supersymmetry [17,18]. form of δG MN, 5 70 6 71 5. Type IIA vs IIB δGμν = δgμν(x)φ6(y), δGμi = Bμ(x) V i(y), 7 72 δGij = φ4(x)δgij(y), (12) 8 Another question to ask is whether the string theory (5) be- 73 9 74 longs to Type IIA or IIB? We started with N = 2 supersymmetric 4 where xμ and yi are the coordinates in R and Y6, respectively. 10 QCD which is a vector-like theory and preserve 4D parity. There- 75 We see that δgμν(x), Bμ(x) and φ4(x) are the 4D graviton, vec- 11 fore we expect that the closed string spectrum in this theory 76 tor and scalar fields, while φ6(y), V i(y) and δgij(y) are the fields 12 should respect 4D parity. 77 on Y6. 13 On the other hand we know that Type IIB string is a chiral 78 Tor the fields δgμν(x), Bμ(x) and φ4(x) to be dynamical in 4D, 14 theory and breaks parity while Type IIA string theory is left-right 79 the fields φ6(y), V i(y) and δgij(y) should have finite norm over 15 symmetric and conserves parity [19]. Thus we expect that the 80 16 the six-dimensional internal space Y6. Otherwise 4D fields will 81 string theory of non-Abelian vortex is of Type IIA. In the subse- appear with infinite kinetic term coefficients, and, hence, will de- 17 quent paper [20] we confirm this expectations studying how parity 82 18 couple. 83 transformation acts on the 2D fermions. 19 Symbolically the Lichnerowicz equation (11) can be written as 84 20 85 6. Spectrum: 4D graviton and vector fields μ 21 (∂μ∂ + 6) g4(x)g6(y) = 0, (13) 86 22 87 As was mentioned above, our target space is R4 × WCP(2, 2) = 23 where 6 is the two-derivative operator from (11) reduced to Y6, 88 R4 × Y where Y is a non-compact Calabi–Yau conifold, which in 24 6 6 while g4(x)g6(y) symbolically denotes the factorized form in (12). 89 fact, at = 0isa resolved conifold, see [21] for a review. We will 25 β If we expand g6 in eigenfunctions 90 consider the string theory (5) at small . 26 β 91 Strictly speaking at small β the sigma model quantum correc- 27 −6 g6(y) = λ6 g6(y) (14) 92 28 tions blow up. In other words, we can say that at small β the 93 gravity approximation does not work. However, for the massless 29 the eigenvalues λ6 will play the role of the mass squared of the 94 states, we can do the computations at large β where supergrav- 30 4D states. Here we will be only interested in the λ6 = 0eigenfunc- 95 ity approximation is valid and then extrapolate to strong coupling. 31 tions. Solutions of the equation 6 g6(y) = 0for the Calabi–Yau 96 32 This is the reason why now we will limit ourselves only to mass- are given by elements of Dolbeault cohomology group 97 (p,q) 33 less states. The latter in the sigma model language corresponds to H (Y ), where (p, q) denotes the numbers of holomorphic and 98 N = 6 34 chiral primary operators. They are protected by (2, 2) world anti-holomorphic indices of the form. The numbers of these forms 99 sheet supersymmetry and their masses are not lifted by quantum (p,q) 35 h are called the Hodge numbers for a given Y6. Due to the fact 100 (0,0) 36 corrections. However, kinetic terms (the Kähler potentials) can be that h = 1, we can easily find (the only!) solution for the 4D 101 corrected. 37 graviton: it is a constant on Y6. This mode is non-normalizable. 102 38 We will argue that all massless string modes on the resolved Hence, no 4D graviton emerges from our string. 103 39 conifold (2) have infinite norm and thus the 4D-graviton and vec- This is a good news. We started from N = 2QCD in four di- 104 40 tor field states decouple. mensions without gravity and, therefore, do not expect 4D graviton 105 41 The massless 10D boson fields of type IIA string theory are to appear as a closed string state.6 106 graviton, and antisymmetric tensor B MN in the NS-NS sec- 42 The infinite norm of the graviton wave function on Y6 rules 107 43 tor. In the R-R sector type IIA string gives rise to one-form and out other 4D states of the N = 2 gravitational and tensor multi- 108 = 44 three-form. (Above M, N 1, ..., 10 are 10D indices). We start from plets: the vector field, the dilaton, the antisymmetric tensor and 109 45 massless 10D graviton and study what states it can produce in two scalars coming from the 10D three-form. 110 46 four dimensions. In fact, 4D states coming from other massless 10D Now, let us pass to the components of the 10D graviton δG 111 N = μi 47 fields listed above can be recovered from 2 supersymmetry in which give rise to a vector field in 4D. The very possibility of hav- 112 the bulk, see for example [22]. We will follow the standard string 48 ing vector fields is due to continuous symmetries of Y6, namely for 113 theory approach well developed for compact Calabi–Yau spaces. 49 our Y6 we expect seven Killing vectors,which obey the equations 114 50 The only novel aspect of the case at hand is that for each 4D state 115 51 we have to check whether its wave function is normalizable on Y6 m + m = = 116 Di V j D j V i 0, m 1, ..., 7 . (15) 52 keeping in mind that this space is non-compact. 117 Massless 10D graviton is represented by fluctuations of the 53 For Calabi–Yau this implies the equation  g (y) = 0, which takes 118 = − (0) (0) R4 × 6 6 54 metric δG MN G MN G MN, where G MN is the metric on Y6 j m = 119 the form D j D V i 0 and is equivalent to the statement that V i 55 which has a block form since R4 and Y are factorized. Moreover, 120 6 is a covariantly constant vector, D j V i = 0. This is impossible for 56 δG MN should satisfy the Lichnerowicz equation compact Calabi–Yau manifolds with the SU(3) holonomy [19], but 121 57 possible in the non-compact case. However it is easy to see that 122 A AB 58 D A D δG MN + 2R MANBδG = 0 . (11) m 123 the V i fields produced by rotations of the yi coordinates do not 59 2 124 A fall-off at large y (where the metric tends to flat). Thus, they are 60 Here D and R MANB are the covariant derivative and the Rie- i 125 (0) non-normalizable, and the associated 4D vector fields Bμ(x) are 61 mann tensor corresponding to the background metric G MN, where 126 (0) absent. 62 A − 1 A = 127 D A δG N 2 D N δG A 0. Given the block form of G MN, only the 63 six-dimensional part Rijkl of R MANB is nonzero while the op- 128 64 A A = μ + i 129 erator D A D is given by D A D ∂μ∂ Di D , where the in- 6 The alternative option that massless 4D spin-2 state has no interpretation in 65 dices μ, ν = 1, ..., 4 and i, j = 1, ..., 6belong to flat 4D space and terms of 4D gravity is ruled out by Weinberg–Witten theorem [23]. 130 JID:PLB AID:32005 /SCO Doctopic: Theory [m5Gv1.3; v1.178; Prn:26/05/2016; 15:27] P.4(1-5) 4 P. Koroteev et al. / Physics Letters B ••• (••••) •••–•••

1 7. Deformations of the conifold metric norm of the b modulus is marginal (i.e. logarithmically divergent), 66 2 in full accord with the analysis of [27]. Thus, this scalar mode is 67 3 It might seem that our 4D string does not produce massless 4D localized on the string in the same sense as the orientational and 68 4 states at all. This is a wrong impression. At the selfdual value of size zero modes are localized on the vortex-string solution. 69 5 β = 0it does! In type IIA superstring the complex scalar associated with de- 70 6 Consider the last option in (12), scalar 4D fields. In this case formations of the complex structure of the Calabi–Yau space enters 71 7 the appropriate Lichnerowicz equation on Y6 reduces to as a 4D hypermultiplet. Thus our 4D scalar b is a part of a hyper- 72 ˜ 8 multiplet. Another complex scalar b comes from 10D three-form, 73 k + kl = 9 Dk D δgij 2Rikjlδg 0 . (16) see [22] for a review. Together they form the bosonic content of 74 10 N = ˜ 75 The target space of the sigma model (2) is defined by the D term a 4D 2 hypermultiplet. The fields b and b being massless 11 76 condition can develop VEVs. Thus, we have a new Higgs branch in the bulk 12 which is developed only at the self-dual value of coupling constant 77 13 |nP |2 −|ρ K |2 = β. (17) g2 = 4π . The bosonic part of the full effective action for the b hy- 78 14 permultiplet to be presented in [20] takes the form 79 15 Also, a U(1) phase can be gauged away. We can construct the U(1)  80  4 8 16 gauge-invariant “mesonic” variables 4 2 2 T L 81 S(b) = T d x |∂μb| +|∂μb| log , (23) 17 |b|2 +| b|2 82 w PK = nP K . (18) 18 ρ 83 where L is the size of R4 introduced as an infrared regularization 19 In terms of these variables the condition (17) can be written as 84 of logarithmically divergent norm of b-field. 20 det w PK = 0, or 85 21 The logarithmic metric in (23) in principle can receive both per- 86 4 22 turbative and non-perturbative quantum corrections. However, for 87 w2 = 0, (19) N = 23 α 2theory the non-renormalization theorem of [14] forbids the 88 = 24 α 1 dependence of the Higgs branch metric on the 4D coupling con- 89 2 2 PK PK a stant g . Since the 2D coupling β is related to g we expect that 25 where w = σ wα, and σ -matrices are (1, −iτ ), a = 1, 2, 3. 90 α the logarithmic metric in (23) will stay intact. 26 Equation (19) define the conifold – a cone with the section S × S . 91 2 3 The presence of the “non-perturbatively emergent” Higgs branch 27 It has the Kähler Ricci-flat metric and represents a non-compact 92 at the self-dual point g2 = 4π at strong coupling is a success- 28 Calabi–Yau manifold [24,12,21]. 93 ful test of our picture. A hypermultiplet is a BPS state. If it were 29 At β = 0the conifold develops a conical singularity, so both S 94 2 present in a continuous region of τ at strong coupling it could 30 and S shrink to zero. The conifold singularity can be smoothed 95 3 be continued all the way to the weak coupling domain where 31 in two different ways: by a deformation of the Kähler form or by 96 its presence would contradict the quasiclassical analysis of bulk 32 a deformation of the complex structure. The first option is called 97 QCD [6,20]. 33 the resolved conifold and amounts to introducing the non-zero β 98 34 in (17). This resolution preserves the Kähler structure and Ricci- 99 K 8. String states interpretation in the bulk 35 flatness of the metric. If we put ρ = 0in (2) we√ get the CP(1) 100 36 model with the S target space (with the radius β). The explicit 101 2 To find the place for the massless scalar hypermultiplet we ob- 37 metric for the resolved conifold can be found in [24–26]. The re- 102 tained as the only massless string excitation at the critical point 38 solved conifold has no normalizable zero modes. In particular, we 103 β = 0let us first examine weak coupling domain g2 1. 39 will demonstrate in [20] that the 4D scalar β associated with de- 104 As well-known from the previous studies of the vortex-strings 40 formation of the Kähler form is not normalizable. 105 at weak coupling, non-Abelian vortices confine monopoles. The 41 If β = 0 another option exists, namely a deformation of the 106 elementary monopoles are junctions of two distinct elementary 42 complex structure [21]. It preserves the Kähler structure and Ricci- 107 non-Abelian strings [30,3,4]. As a result in the bulk theory we 43 flatness of the conifold and is usually referred to as the deformed 108 have monopole–antimonopole mesons in which monopole and an- 44 conifold. It is defined by deformation of Eq. (19), namely 109 45 timonopole are connected by two confining strings. For the U(2) 110 4 gauge group we have also “baryons” consisting of two monopoles, 46 2 = 111 47 wα b , (20) rather than of monopole–antimonopole pair. The monopoles ac- 112 = × 48 α 1 quire quantum numbers with respect to the global group SU(2) 113 SU(2) × U (1) of the bulk theory. Indeed, in the world sheet model 49 where b is a complex number. Now the S3 can not shrink to zero, 114 on the vortex-string confined monopole are seen as kinks inter- 50 its minimal size is determined by b. The explicit metric on the 115 polating between two different vacua [30,3,4]. These kinks are de- 51 deformed conifold is written down in [24,28,29]. The parameter 116 scribed at strong coupling by the nP and ρ K fields [31,32] (for 52 b becomes a 4D complex scalar field. The effective action for this 117 WCP(N, N˜ ) models see [33]) and therefore transform in the fun- 53 field is 118  × 7 54 damental representations of SU(2) SU(2) for WCP(2, 2). 119 4 2 As a result, the monopole–antimonopole mesons and baryons 55 S(b) = T d xhb|∂μb| , (21) 120 56 can be either singlets or triplets of both SU(2) global groups, 121 as well as in the bi-fundamental representations. With respect 57 where the metric hb(b) is given by the normalization integral over 122 to baryonic U (1)B symmetry which we define as a U(1) factor 58 the conifold Y6, 123    in the global SU(2) × SU(2) × U (1), the mesons have charges 59 √ ∂ ∂ = 124 = 6 li jk Q B (meson) 0, 1 while the baryons can have charges 60 hb d y gg gij g ¯ gkl , (22) 125 61 ∂b ∂b 126 Q B (baryon) = 0, 1, 2 . (24) 62 127 and gij(b) is the deformed conifold metric. 63 128 We calculated hb by two different methods, one of them [27] 64 129 was kindly pointed out to us by . Details of these cal- 7 Here we note that global group SU(2) × SU(2) × U (1) is the same for both bulk 65 culations will be presented in [20]. Here we only state that the and world sheet theories [6]. 130 JID:PLB AID:32005 /SCO Doctopic: Theory [m5Gv1.3; v1.178; Prn:26/05/2016; 15:27] P.5(1-5) P. Koroteev et al. / Physics Letters B ••• (••••) •••–••• 5

1 All the above non-perturbative√ stringy states are heavy, with mass [5] M. Shifman, A. Yung, Critical string from non-Abelian vortex in four dimen- 66 2 of the order of ξ , and can decay into screened quarks, which sions, Phys. Lett. B 750 (2015) 416, arXiv:1502.00683 [hep-th]. 67 3 are lighter, and, eventually, into massless bi-fundamental screened [6] For a review and extensive references see e.g. M. Shifman, A. Yung, Supersym- 68 metric Solitons, Cambridge University Press, 2009. 4 69 quarks. [7] A. Polyakov, Phys. Lett. B 103 (1981) 207. 5 Now we return from weak to strong coupling and go to the [8] For a review see e.g. A. Achucarro, T. Vachaspati, Phys. Rep. 327 (2000) 347. 70 6 self-dual point β = 0. We claim that at this point a new ‘exotic’ [9] M. Shifman, A. Yung, Phys. Rev. D 73 (2006) 125012. 71 7 Higgs branch opens up which is parameterized by the massless [10] M. Eto, J. Evslin, K. Konishi, G. Marmorini, et al., Phys. Rev. D 76 (2007) 105002. 72 [11] M. Shifman, W. 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Fine arXiv:hep-th/0410178. 94 30 Theoretical Physics Institute, University of Minnesota, by Russian [22] J. Louis, Generalized Calabi–Yau compactifications with D- and fluxes, 95 31 Foundation for Basic Research Grant No. 13-02-00042a and by Fortschr. Phys. 53 (2005) 770. 96 [23] S. Weinberg, E. Witten, Limits on massless particles, Phys. Lett. B 96 (1980) 59. 32 Russian State Grant for Scientific Schools RSGSS-657512010.2. The 97 [24] P. Candelas, X.C. de la Ossa, Comments on conifolds, Nucl. Phys. B 342 (1990) 33 work of A.Y. was supported by the Russian Scientific Foundation 246. 98 34 Grant No. 14-22-00281. P.K. would like to thank W. Fine Institute [25] L.A. Pando Zayas, A.A. Tseytlin, 3-branes on resolved conifold, J. High Energy 99 35 for Theoretical Physics at the University of Minnesota for kind hos- Phys. 0011 (2000) 028, arXiv:hep-th/0010088. 100 [26] I.R. Klebanov, A. 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