Hadrons of N=2 Supersymmetric QCD in Four Dimensions from Little String Theory
aa r X i v : . [ h e p - t h ] O c t FTPI-MINN-18/09, UMN-TH-3718/18October 4, 2018
Hadrons of N = 2 Supersymmetric QCD inFour Dimensions from Little String Theory M. Shifman a and A. Yung a,b,c a William I. Fine Theoretical Physics Institute, University of Minnesota,Minneapolis, MN 55455 b National Research Center “Kurchatov Institute”, Petersburg NuclearPhysics Institute, Gatchina, St. Petersburg 188300, Russia c St. Petersburg State University, Universitetskaya nab., St. Petersburg199034, Russia
Abstract
It was recently shown that non-Abelian vortex strings supported in aversion of four-dimensional N = 2 supersymmetric QCD (SQCD) becomecritical superstrings. In addition to four translational moduli, non-Abelianstrings under consideration have six orientational and size moduli. Togetherthey form a ten-dimensional target space required for a superstring to becritical, namely, the product of the flat four-dimensional space and conifold– a non-compact Calabi-Yau threefold. In this paper we report on furtherstudies of low-lying closed string states which emerge in four dimensionsand identify them as hadrons of our four-dimensional N = 2 SQCD. Weuse the approach based on “little string theory,” describing critical stringon the conifold as a non-critical c = 1 string with the Liouville field and acompact scalar at the self-dual radius. In addition to massless hypermultipletfound earlier we observe several massive vector multiplets and a massivespin-2 multiplet, all belonging to the long (non-BPS) representations of N =2 supersymmetry in four dimensions. All the above states are interpreted asbaryons formed by a closed string with confined monopoles attached. Ourconstruction presents an example of a “reverse holography.” Introduction
In this paper we continue studying the spectrum of four-dimensional “hadrons”formed by the closed critical string [1] which in turn can be obtained from asolitonic vortex string under an appropriate choice of the coupling constant[2]. One of our main tasks is to analyze the structure of the 4D supermulti-plets emerging from quantization of the closed string mentioned above. Wewill start though from a brief review of the setup.The problem of understanding confining gauge theories splits into twodifferent equally fundamental tasks. The first one is to understand the phys-ical nature of confinement and describe the formation of confining strings.There was a great progress in this direction in supersymmetric gauge theo-ries due to the breakthrough papers by Seiberg and Witten [3, 4] in whichthe monopole condensation was shown to occur in the monopole vacua of N = 2 supersymmetric QCD (SQCD). This leads to the formation of AbelianAbrikosov-Nielsen-Olesen (ANO) vortices [5] which confine color electric char-ges. Attempts to find a non-Abelian generalization of this mechanism led tothe discovery of the so called “instead-of-confinement” phase which occursin the quark vacua of N = 2 SQCD, see [6] for a review. In this phase the(s)quarks condense while the monopoles are confined.Once the nature of the confining string is understood the second task isto quantize this string in four-dimensional (4D) theory outside the criticaldimension to study the hadron spectrum. Most solitonic strings, such as theANO strings, have a finite thickness manifesting itself in the presence of aninfinite series of unknown higher-derivative corrections in the effective sigmamodel on the string world sheet. This makes the task of quantizing such astring virtually impossible.Recent advances in this direction [2] demonstrated that the non-Abeliansolitonic vortex in a particular version of 4D N = 2 SQCD becomes a criticalsuperstring. This particular 4D SQCD has the U(2) gauge group, four quarkflavors and the Fayet-Iliopoulos (FI) [7] parameter ξ .Non-Abelian vortices were first discovered in N = 2 SQCD with the U( N )gauge group and N f ≥ N flavors of quark hypermultiplets [8, 9, 10, 11]. Inaddition to four translational moduli characteristic of the ANO strings [5],the non-Abelian strings carry orientational moduli, as well as the size moduliif N f > N [8, 9, 10, 11] (see [12, 13, 14, 15] for reviews). If N f > N theirdynamics are described by effective two-dimensional sigma model on the1tring world sheet with the target space O ( − ⊕ ( N f − N ) CP , (1.1)to which we will refer to as the weighted CP model (WCP( N, N f − N )).For N f = 2 N the model becomes conformal. Moreover, for N = 2 thedimension of the orientational/size moduli space is six and they can be com-bined with four translational moduli to form a ten-dimensional space requiredfor superstring to become critical. In this case the target space of the world sheet 2D theory on the non-Abelian vortex string is R × Y , where Y is a non-compact six-dimensionalCalabi-Yau manifold, the so-called resolved conifold [16, 17].Since non-Abelian vortex string on the conifold is critical it has a perfectlygood UV behavior. This opens the possibility that it can become thin in acertain regime [2]. The string transverse size is given by 1 /m , where m isa typical mass scale of the four-dimensional fields forming the string. Thestring cannot be thin in a weakly coupled
4D theory because at weak coupling m ∼ g √ T and is always small in the units of √ T where T is the tension.Here g is the gauge coupling constant of the 4D N = 2 QCD and T is thestring tension.A conjecture was put forward in [2] that at strong coupling in the vicinityof a critical value of g c ∼ that m ( g ) → ∞ at g → g c .A version of the string-gauge duality for 4D SQCD was proposed [2]: atweak coupling this theory is in the Higgs phase and can be described in termsof (s)quarks and Higgsed gauge bosons, while at strong coupling hadrons ofthis theory can be understood as string states formed by the non-Abelianvortex string.The vortices in the U( N ) theories under consideration are topologicallystable and cannot be broken. Therefore the finite-length strings are closed. The non-Abelian vortex string is 1/2 BPS saturated and, therefore, has N =(2 ,
2) supersymmetry on its world sheet. Thus, we actually deal with a superstring inthe case at hand. At N f = 2 N the beta function of the 4D N = 2 QCD is zero, so the gauge coupling g does not run. Note, however, that conformal invariance in the 4D theory is broken bythe FI parameter ξ which does not run either. N = 2 SQCD.The first step of this program, namely, identifying massless string stateswas carried out in [18, 19] using supergravity formalism. In particular, asingle matter hypermultiplet associated with the deformation of the com-plex structure of the conifold was found as the only 4D massless mode ofthe string. Other states arising from the massless ten-dimensional gravitonare not dynamical in four dimensions. In particular, the 4D graviton andunwanted vector multiplet associated with deformations of the K¨ahler formof the conifold are absent. This is due to non-compactness of the Calabi-Yaumanifold we deal with and non-normalizability of the corresponding modesover six-dimensional space Y .The next step was done in [1] where a number of massive states of theclosed non-Abelian vortex string was found. This step required a change ofstrategy. The point is that the coupling constant 1 /β of the world sheetWCP(2,2) is not small. Moreover β tends to zero once the 4D coupling g approaches the critical value g c we are interested in. At β → To analyze the massive states the little string theories (LST) approach(see [23] for a review) was used in [1]. Namely, we used the equivalencebetween the critical string on the conifold and non-critical c = 1 string whichcontains the Liouville field and a compact scalar at the self-dual radius [24,25]. The latter theory (in the mirror Wess-Zumino-Novikov-Witten (WZNW)formulation) can be analyzed by virtue of algebraic methods. This leads toidentification of towers of massive states with spin zero and spin two [1].In this paper we focus on the 4D multiplet structure of the states foundearlier in [19, 1]. In addition to the massless BPS hypermultiplet associatedwith deformations of the complex structure of the conifold we identify severalmassive vector multiplets and a massive spin-2 multiplet, all belonging tolong non-BPS representations of N = 2 supersymmetry in four dimensions.We interpret all states we found as baryons formed by a closed string withconfined monopoles attached. This is in contradistinction to the massless states. For the latter, we can performcomputations at large β where the supergravity approximation is valid and then extrapo-late to strong coupling. In the sigma-model language massless states corresponds to chiralprimary operators. They are protected by N = (2 ,
2) world-sheet supersymmetry andtheir masses are not lifted by quantum corrections. c = 1 string and the spectrum of massive states. In Sec. 4 we introduce 4Dsupercharges and construct massless BPS hypermultiplet. In Sec. 5 we con-sider the lowest massive string excitations and show that they forms a longvector supermultiplet. Section 6 deals with the construction of N = 2 spin-2stringy supermultiplet. In Sec. 7 we discuss linear Regge trajectories, whileSection 8 summarizes our conclusions. In Appendix A we describe the BRSToperator and transitions between different pictures. In Appendix B we reviewlong N = 2 supermultiplets in 4D. N = 2 SQCD As was already mentioned non-Abelian vortex-strings were first found in4D N = 2 SQCD with the gauge group U( N ) and N f ≥ N flavors (i.e. thequark hypermultiplets) supplemented by the FI D term ξ [8, 9, 10, 11], see forexample [14] for a detailed review of this theory. Here we just mention that atweak coupling, g ≪
1, this theory is in the Higgs phase in which the scalarcomponents of the quark multiplets (squarks) develop vacuum expectationvalues (VEVs). These VEVs breaks the U( N ) gauge group Higgsing all gaugebosons. The Higgsed gauge bosons combine with the screened quarks to formlong N = 2 multiplets with mass m ∼ g √ ξ .The global flavor SU( N f ) is broken down to the so called color-flavorlocked group. The resulting global symmetry isSU( N ) C + F × SU( N f − N ) × U(1) B , (2.1)see [14] for more details.The unbroken global U(1) B factor above is identified with a baryonicsymmetry. Note that what is usually identified as the baryonic U(1) chargeis a part of our 4D theory gauge group. “Our” U(1) B is an unbroken bysquark VEVs combination of two U(1) symmetries: the first is a subgroup of4 a Figure 1:
Examples of the monopole “necklace” baryons: Open circles denotemonopoles. the flavor SU( N f ) and the second is the global U(1) subgroup of U( N ) gaugesymmetry.As was already noted, we consider N = 2 SQCD in the Higgs phase: N squarks condense. Therefore, non-Abelian vortex strings confine monopoles.In the N = 2 4D theory these strings are 1/2 BPS-saturated; hence, theirtension is determined exactly by the FI parameter, T = 2 πξ . (2.2)However, the monopoles cannot be attached to the string endpoints. Infact, in the U( N ) theories confined monopoles are junctions of two distinctelementary non-Abelian strings [28, 10, 11] (see [14] for a review). As a result,in four-dimensional N = 2 SQCD we have monopole-antimonopole mesonsin which the monopole and antimonopole are connected by two confiningstrings. In addition, in the U( N ) gauge theory we can have baryons appearingas a closed “necklace” configurations of N × (integer) monopoles [14]. Forthe U(2) gauge group the lightest baryon presented by such a “necklace”configuration consists of two monopoles, see Fig. 1.Both stringy monopole-antimonopole mesons and monopole baryons withspins J ∼ ∼ √ ξ and are heav-ier at weak coupling than perturbative states, which have mass m ∼ g √ ξ .However, according to our conjecture, at strong coupling near the criticalpoint g c m → ∞ , see [2] and Sec. 2.3 below. In this regime perturbativestates decouple and we are left with hadrons formed by the closed stringstates. All hadrons identified as closed string states in this paper turn out There are also massless bifundamental quarks, charged with respect to both non-
5o be baryons and look like monopole “necklaces,” see Fig. 1.
The presence of color-flavor locked group SU( N ) C + F is the reason for theformation of the non-Abelian vortex strings [8, 9, 10, 11] in our 4D SQCD.The most important feature of these vortices is the presence of the so-calledorientational zero modes.Let us briefly review the model emerging on the world sheet of thenon-Abelian critical string [2, 18, 19]. If N f = N the dynamics of theorientational zero modes of the non-Abelian vortex, which become orien-tational moduli fields on the world sheet, is described by two-dimensional N = (2 ,
2) supersymmetric CP( N −
1) model [14].If one adds extra quark flavors, non-Abelian vortices become semilocal.They acquire size moduli [29]. In particular, for the non-Abelian semilocalvortex at hand, in addition to the orientational zero modes n P ( P = 1 , ρ K ( K = 1 ,
2) [29, 8, 11, 30, 31, 32].The target space of the WCP(2 ,
2) sigma model on the string world sheet isdefined by the D -term condition | n P | − | ρ K | = β , (2.3)and a U(1) phase is gauged away.The total number of real bosonic degrees of freedom in this model issix, where we take into account the constraint (2.3) and the fact that oneU(1) phase is gauged away. As was already mentioned, these six internaldegrees of freedom are combined with four translational moduli to form aten-dimensional space needed for superstring to be critical.At weak coupling the world sheet coupling constant β in (2.3) is relatedto the 4D SU(2) gauge coupling as follows: g β ≈ πg , (2.4)see [14]. Note that the first (and the only) coefficient is the same for the 4DSQCD and the world-sheet model β functions. Both vanish at N f = 2 N .This ensures that our world-sheet theory is conformal. Abelian factors in (2.1). These are associated with the Higgs branch present in 4D QCD,see [14, 19] for details. N = (2 ,
2) inthe world sheet sigma model which is necessary to have N = 2 space-timesupersymmetry [33, 34]. Moreover, as was shown in [19], the string theoryof the non-Abelian critical vortex is type IIA.The global symmetry of the world-sheet sigma model isSU(2) × SU(2) × U(1) , (2.5)i.e. exactly the same as the unbroken global group in the 4D theory, cf.(2.1), at N = 2 and N f = 4. The fields n and ρ transform in the followingrepresentations: n : ( , , , ρ : (0 , , . (2.6) The coupling constant of 4D SQCD can be complexified τ ≡ i πg + θ D π , (2.7)where θ D is the four-dimensional θ angle. Note that SU( N ) version of thefour-dimensional SQCD at hand possesses a strong-weak coupling duality,namely, τ → − τ [20, 21]. This suggests that the self-dual point g = 4 π would be a natural candidate for a critical value g c , where our non-Abelianvortex string becomes thin. However, as was shown recently in [22], S -duality maps our U( N ) theory to a theory in which a different U(1) subgroupof the flavor group is gauged. In particular, in our U( N ) theory all quarkflavors have equal charges with respect to the U(1) subgroup of the U(2)gauge group, while in the S -dual version only one flavor is charged withrespect to the U(1) gauge group. As a result, the S -dual version supports adifferent type of non-Abelian strings [22].This means that S -duality is broken in our 4D theory by the choice ofthe U(1) subgroup which is gauged. We do not consider S -duality and itsconsequences here. We suggested this earlier in [18, 19]. We are grateful to E. Gerchkovitz and A. Karasik for pointing out to us this circum-stance. β can be naturally complexifiedtoo if we include the two-dimensional θ term, β → β + i θ D π . (2.8)The exact relation between the complexified 4D and 2D couplings is asfollows: exp ( − πβ ) = − h ( τ )[ h ( τ ) + 2] , (2.9)where the function h ( τ ) is a special modular function of τ defined in termsof the θ -functions, h ( τ ) = θ / ( θ − θ ) . This function enters the Seiberg-Witten curve in our 4D theory [20, 21].Equation (2.9) generalizes the quasiclassical relation (2.4). It can be derivedusing 2D-4D correspondence, namely, the match of the BPS spectra of the4D theory at ξ = 0 and the world-sheet theory on the non-Abelian string[35, 36, 10, 11]. Details of this derivation will be presented elsewhere. Notethat our result (2.9) differs from that obtained in [22] using the localizationtechnique.According to the hypothesis formulated in [2], our critical non-Abelianstring becomes infinitely thin at strong coupling at the critical point τ c (or g c ). Moreover, in [19] we conjectured that τ c corresponds to β = 0 in theworld-sheet theory via relation (2.9). Thus, we assume that m → ∞ at β = 0, which corresponds to g = g c in 4D SQCD.At the point β = 0 the non-Abelian string becomes infinitely thin, higherderivative terms can be neglected and the theory of the non-Abelian stringreduces to the WCP(2,2) model. The point β = 0 is a natural choice becauseat this point we have a regime change in the 2D sigma model. This is thepoint where the resolved conifold defined by the D term constraint (2.3)develops a conical singularity [17]. In this section we will briefly review the only 4D massless state associatedwith the deformation of the conifold complex structure. It was found in [19].As was already mentioned, all other modes arising from the massless 10D8raviton have non-normalizable wave functions over the conifold. In partic-ular, the 4D graviton is absent [19]. This result matches our expectationssince from the very beginning we started from N = 2 SQCD in the flatfour-dimensional space without gravity.The target space of the world sheet WCP(2,2) model is defined by the D -term condition (2.3). We can construct the U(1) gauge-invariant “mesonic”variables w P K = n P ρ K . (2.10)These variables are subject to the constraint det w P K = 0, or X α =1 w α = 0 , (2.11)where w P K ≡ σ P Kα w α , and the σ matrices above are (1 , − iτ a ), a = 1 , ,
3. Equation (2.11) de-fines the conifold Y . It has the K¨ahler Ricci-flat metric and represents anon-compact Calabi-Yau manifold [16, 37, 17]. It is a cone which can beparametrized by the non-compact radial coordinate e r = X α =1 | w α | (2.12)and five angles, see [16]. Its section at fixed e r is S × S .At β = 0 the conifold develops a conical singularity, so both S and S canshrink to zero. The conifold singularity can be smoothed out in two distinctways: by deforming the K¨ahler form or by deforming the complex structure.The first option is called the resolved conifold and amounts to introducing anon-zero β in (2.3). This resolution preserves the K¨ahler structure and Ricci-flatness of the metric. If we put ρ K = 0 in (2.3) we get the CP (1) modelwith the S target space (with the radius √ β ). The resolved conifold hasno normalizable zero modes. In particular, the modulus β which becomes ascalar field in four dimensions has non-normalizable wave function over the Y manifold [19].As explained in [38, 19], non-normalizable 4D modes can be interpretedas (frozen) coupling constants in the 4D theory. The β field is the moststraightforward example of this, since the 2D coupling β is related to the 4Dcoupling, see Eq. (2.9). 9f β = 0 another option exists, namely a deformation of the complexstructure [17]. It preserves the K¨ahler structure and Ricci-flatness of theconifold and is usually referred to as the deformed conifold . It is defined bydeformation of Eq. (2.11), namely, X α =1 w α = b , (2.13)where b is a complex number. Now the S can not shrink to zero, its minimalsize is determined by b .The modulus b becomes a 4D complex scalar field. The effective actionfor this field was calculated in [19] using the explicit metric on the deformedconifold [16, 39, 40], S ( b ) = T Z d x | ∂ µ b | log T L | b | , (2.14)where L is the size of R introduced as an infrared regularization of logarith-mically divergent b field norm. We see that the norm of the b modulus turns out to be logarithmicallydivergent in the infrared. The modes with the logarithmically divergent normare at the borderline between normalizable and non-normalizable modes.Usually such states are considered as “localized” on the string. We followthis rule. We can relate this logarithmic behavior to the marginal stabilityof the b state, see [19].The field b , being massless, can develop a VEV. Thus, we have a newHiggs branch in 4D N = 2 SQCD which is developed only for the criticalvalue of the coupling constant g c .The logarithmic metric in (2.14) in principle can receive both pertur-bative and non-perturbative quantum corrections in 1 /β , the sigma modelcoupling. However, in the N = 2 theory the non-renormalization theorem of[21] forbids the dependence of the Higgs branch metric on the 4D couplingconstant g . Since the 2D coupling β is related to g we expect that thelogarithmic metric in (2.14) will stay intact. This expectation is confirmedin [1]. The infrared regularization on the conifold e r max translates into the size L of the 4Dspace because the variables ρ in (2.12) have an interpretation of the vortex string sizes, e r max ∼ T L .
10n [19] the massless state b was interpreted as a baryon of 4D N =2 SQCD. Let us explain this. From Eq. (2.13) we see that the complexparameter b (which is promoted to a 4D scalar field) is singlet with respectto both SU(2) factors in (2.5), i.e. the global world-sheet group. Whatabout its baryonic charge?Since w α = 12 Tr (cid:2) (¯ σ α ) KP n P ρ K (cid:3) (2.15)we see that the b state transforms as(1 , , , (2.16)where we used (2.6) and (2.13). Three numbers above refer to the represen-tations of (2.5). In particular it has the baryon charge Q B ( b ) = 2.To conclude this section let us note that in our case of type IIA su-perstring the complex scalar associated with deformations of the complexstructure of the Calabi-Yau space enters as a component of a massless 4D N = 2 hypermultiplet, see [41] for a review. Instead, for type IIB super-string it would be a component of a vector BPS multiplet. Non-vanishingbaryonic charge of the b state confirms our conclusion that the string underconsideration is a type IIA. c = 1 string As was explained in Sec. 1, the critical string theory on the conifold ishard to use for calculating the spectrum of massive string modes becausethe supergravity approximation does not work. In this section we reviewthe results obtained in [1] based on the little string theory (LST) approach.Namely, in [1] we used the equivalent formulation of our theory as a non-critical c = 1 string theory with the Liouville field and a compact scalar atthe self-dual radius [24, 25]. We intend to use the same formulation in thispaper to analyze the 4D hypermultiplet structure of the massive states. c = 1 string theory Non-critical c = 1 string theory is formulated on the target space R × R φ × S , (3.1) Which is isomorphic to the 4D global group (2.1) at N = 2, N f = 4. R φ is a real line associated with the Liouville field φ and the theoryhas a linear in φ dilaton, such that string coupling is given by g s = e − Q φ . (3.2)We will determine Q in Eq. (3.7).Generically the above equivalence is formulated in the so called doublescaling limit between the critical string on non-compact Calabi-Yau spaceswith an isolated singularity on the one hand, and non-critical c = 1 stringwith the additional Ginzburg-Landau N = 2 superconformal system [24],on the other hand. Following [24] we assume the double scaling limit whenthe string coupling constant of the conifold theory g con and the deformationparameter of the conifold b simultaneously go to zero with the combination b Q /g con fixed. In this limit non-trivial physics is localized near the singu-larity of the Calabi-Yau manifold. In the conifold case the extra Ginzburg-Landau factor in (3.1) is absent [42].In [43, 24, 42] it was argued that non-critical string theories with thestring coupling exponentially falling off at φ → ∞ are holographic. Thestring coupling goes to zero in the bulk of the space-time and non-trivialdynamics (LST) is localized on the “boundary.” In our case the “boundary”is the four-dimensional space in which N = 2 SQCD is defined. (It is worthemphasizing that in our case the boundary 4D dynamics is the starting pointwhile the extra six dimensions represent an auxiliary mathematical construct.Perhaps, it can be referred to as a “reverse holography.”)In other words, holography for our non-Abelian vortex string theory ismost welcome and expected. We start with N = 2 SQCD in 4D space andstudy solitonic vortex strings. In our approach 10D space formed by 4D“actual” space and six internal moduli of the string is an artificial construc-tion needed to formulate the string theory of a special non-Abelian vortex.Clearly we expect that all non-trivial “actual” physics should be localizedexclusively on the 4D “boundary.” In other words, we expect that LST inour case is nothing but 4D N = 2 SQCD at the critical value of the gaugecoupling g c (in the hadronic description).The linear dilaton in (3.2) implies that the bosonic stress tensor of c = 1 The main example of this behavior is non-gravitational LST in the flat six-dimensionalspace formed by the world volume of parallel NS5 branes. T −− = − (cid:2) ( ∂ − φ ) + Q ∂ − φ + ( ∂ − Y ) (cid:3) , (3.3)where ∂ − = ∂ z The compact scalar Y represents c = 1 matter and satisfiesthe following condition: Y ∼ Y + 2 πQ . (3.4)Here we normalize the scalar fields in such a way that their propagators are h φ ( z ) , φ (0) i = − log z ¯ z, h Y ( z ) , Y (0) i = − log z ¯ z . (3.5)The central charge of the supersymmetrized c = 1 theory above is c SUSYφ + Y = 3 + 3 Q . (3.6)The criticality condition for the string on (3.1) implies that this central chargeshould be equal to 9. This gives Q = √ , (3.7)to be used in Eq. (3.2).Deformation of the conifold (2.13) translates into adding the Liouvilleinteraction to the world-sheet sigma model [24], δL = b Z d θ e − φ + iYQ . (3.8)The conifold singularity at b = 0 corresponds to the string coupling constantbecoming infinitely large at φ → −∞ , see (3.2). At b = 0 the Liouvilleinteraction regularize the behavior of the string coupling preventing the stringfrom propagating to the region of large negative φ .In fact the c = 1 non-critical string theory can also be described in termsof two-dimensional black hole [44], which is the SL(2 , R ) / U(1) coset WZNWtheory [45, 25, 46, 24] at level k = 2 Q . (3.9)In [47] it was shown that N = (2 ,
2) SL(2 , R ) / U(1) coset is a mirror descrip-tion of the c = 1 Liouville theory. The relation above implies in the case ofthe conifold ( Q = √
2) that k = 1 , (3.10)13here k is the total level of the Kaˇc-Moody algebra in the supersymmetricversion (the level of the bosonic part of the algebra is then k b = k + 2 = 3).The target space of this theory has the form of a semi-infinite cigar; thefield φ associated with the motion along the cigar cannot take large negativevalues due to semi-infinite geometry. In this description the string couplingconstant at the tip of the cigar is g s ∼ /b .In fact as was argued in [24] in the non-critical string theory by itself theparameter b does not have to be small. If we following [24] take b large thestring coupling at the tip of the cigar will be small and the string perturbationtheory becomes reliable, cf. [24, 26]. In particular, we can use the tree-levelapproximaion to obtain the string spectrum. Note also that as we alreadymentioned in the Introduction the SL (2 , R ) /U (1) WZNW model is exactlysolvable.In terms of 4D SQCD taking b large means moving along the Higgs branchfar away from the origin. Vertex operators for the string theory on (3.1) are constructed in [24], seealso [45, 42]. Primaries of the c = 1 part for large positive φ (where thetarget space becomes a cylinder R φ × S ) take the form V Lj,m L × V Rj,m R ≈ exp (cid:16) √ jφ + i √ m L Y L + m R Y R ) (cid:17) , (3.11)where we split φ and Y into left and right-moving parts, say φ = φ L + φ R .For the self-dual radius (3.7) (or k = 1) the parameter 2 m in Eq. (3.11) isinteger. For the left-moving sector 2 m L ≡ m is the total momentum plusthe winding number along the compact dimension Y . For the right-movingsector we introduce 2 m R which is the winding number minus momentum.We will see below that for our case type IIA string m R = − m , while for typeIIB string m R = m .The primary operator (3.11) is related to the wave function over “extradimensions” as follows: V j,m = g s Ψ j,m ( φ, Y ) . The string coupling (3.2) depends on φ . Thus,Ψ j,m ( φ, Y ) ∼ e √ j + ) φ + i √ mY . (3.12)14e will look for string states with normalizable wave functions over the “extradimensions” which we will interpret as hadrons in 4D N = 2 SQCD. Thecondition for the string states to have normalizable wave functions reducesto j ≤ − . (3.13)The scaling dimension of the primary operator (3.11) is∆ j,m = m − j ( j + 1) . (3.14)Unitarity implies that it should be positive,∆ j,m > . (3.15)Moreover, to ensure that conformal dimensions of left and right-moving partsof the vertex operator (3.11) are the same we impose that m R = ± m L .The spectrum of the allowed values of j and m in (3.11) was exactlydetermined by using the Kaˇc-Moody algebra for the coset SL(2 , R ) / U(1) in[48, 49, 50, 51, 45], see [52] for a review. Both discrete and continuous repre-sentations were found. Parameters j and m determine the global quadraticCasimir operator and the projection of the spin on the third axis, J | j, m i = − j ( j + 1) | j, m i , J | j, m i = m | j, m i (3.16)where J a ( a = 1 , ,
3) are the global SL(2 , R ) currents.We will focus on discrete representations with j = − , − , − , ..., m = ±{ j, j − , j − , ... } . (3.17)Discrete representations include the normalizable states localized nearthe tip of the cigar (see (3.13)), while the continuous representations containnon-normalizable states.Discrete representations contain states with negative norm. To excludethese ghost states a restriction for spin j is imposed [48, 49, 50, 51, 52] − k + 22 < j < . (3.18) We include the case j = − which is at the borderline between normalizable and non-normalizable states. In [1] it is shown that j = − corresponds to the norm logarithmicallydivergent in the infrared in much the same way as the norm of the b state, see (2.14) k = 1 we are left with only two allowed values of j , j = − , m = ± (cid:26) , , ... (cid:27) (3.19)and j = − , m = ±{ , , ... } . (3.20)Note that there are also continues (principal and exceptional) represen-tations of primaries of the c = 1 string theory [52], see also a brief review ofdiscrete and continues spectra in [1]. In particular, continues representationscorrespond to non-normalizable states in the Liouville direction. Moreover,in [1] we suggested an interpretation of these non-normalizable states: theycorresponds to decaying modes of normalizable 4D states. We also con-firm this interpretation showing that spectra of continues states start fromthresholds given by masses (3.24) and (3.27) of 4D states (see below). Still webelieve that the relation between discrete and continues states needs futureclarification. Four-dimensional spin-0 and spin-2 states were found in [1] using vertex op-erators ((3.11)). The 4D scalar vertices V S in the ( − , −
1) picture have theform [24] V S,Lj,m × V S,Rj, − m ( p µ ) = e − ϕ L − ϕ R e ip µ x µ V Lj,m × V Rj, − m , (3.21)where superscript S stands for scalar, ϕ L,R represents bosonized ghost in theleft and right-moving sectors, while p µ is the 4D momentum of the stringstate.The condition for the state (3.21) to be physical is12 + p µ p µ πT + m − j ( j + 1) = 1 , (3.22)where 1/2 comes from the ghost and we used (3.14). We note that theconformal dimension of the ghost operator exp ( qϕ ) is equal to − ( q + q / q is the picture number.The GSO projection restricts the integer 2 m for the operator in (3.21) tobe odd [53, 24] , m = 12 + Z . (3.23) We will demonstrate this in the next section. m we have only one possibility j = − , see (3.19). Thisdetermines the masses of the 4D scalars,( M Sm ) πT = − p µ p µ πT = m − , (3.24)where the Minkowski 4D metric with the diagonal entries ( − , , ,
1) is used.In particular, the state with m = ± / b , asso-ciated with deformations of the conifold complex structure [1], while stateswith m = ± (3 / , / , ... ) are massive 4D scalars.At the next level we consider 4D spin-2 states. The corresponding vertexoperators are given by (cid:0) V Lj,m × V Rj, − m ( p µ ) (cid:1) spin − = ξ µν ψ µL ψ νR e − ϕ L − ϕ R e ip µ x µ V Lj,m × V Rj, − m , (3.25)where ψ µL,R are the world-sheet superpartners to 4D coordinates x µ , while ξ µν is the polarization tensor.The condition for these states to be physical takes the form p µ p µ πT + m − j ( j + 1) = 0 . (3.26)The GSO projection selects now 2 m to be even, | m | = 0 , , , ... [24], thuswe are left with only one allowed value of j , j = − m = 0 is excluded. This leads to the following expression for themasses of spin-2 states:( M spin − m ) = 8 πT m , | m | = 1 , , .... (3.27)We see that all spin-2 states are massive. This confirms the result in [19]that no massless 4D graviton appears in our theory. It also matches the factthat our “boundary” theory, 4D N = 2 QCD, is defined in flat space withoutgravity.To determine baryonic charge of these states we note that U(1) B trans-formation of b in the Liouville interaction (3.8) is compensated by a shift of Y . The baryonic charge of b is two, see (2.16). Below we use the followingconvention: upon splitting Y into left and right-moving parts Y = Y L + Y R we define that only Y L is shifted under U(1) B transformation, b → e iθ b, Y L → Y L + 2 √ θ, Y R → Y R . (3.28)17
20 4 6 8−2−4−6−8 12345 QM /8 T π Figure 2:
Spectrum of spin-0 and spin-2 states as a function of the baryoniccharge. Closed and open circles denote spin-0 and spin-2 states, respectively.
This gives for the baryon charge of the vertex operator (3.11) Q B = 4 m. (3.29)We see that the momentum m in the compact Y direction is in fact the baryoncharge of a string state. All states we found above are baryons. Their massesas a function of the baryon charge are shown in Fig. 2.The momentum m in the compact dimension is also related to the R -charge. On the world sheet we can introduce the left and right R -chargesseparately. Normalizing charge of θ + , namely, R (2) L ( θ + ) = 1, we see that Y should be shifted under the R (2) L symmetry to make invariant the Liouvilleinteraction (3.8).This gives R (2) L ( V Lj,m ) = − m (3.30)for the R (2) L charge of the vertex (3.11) which is the bottom component ofthe world sheet supermultiplet. The R (2) R charge in the right-moving sectoris defined similarly. Here superscript (2) denotes the world sheet R -charge.18s was discussed above, the massless baryon b corresponds to j = − / m = ± /
2. Thus, the associated vertex V j,m has R (2) L = ± /
2, see (3.14). Therefore it satisfies the relation∆ = | R (2) L | b is a component of a short N = 2 BPS multiplet, namely hypermultiplet. The remainder of this paper is devoted to the study the supermultiplet struc-ture of the 4D string states described in the previous sections. Our strategyis as follows: we explicitly construct 4D supercharges and use them to gen-erate all components of a given multiplet starting from a scalar or spin-2representative shown in (3.21) or (3.25). We will generate supermultipletsoriginating from the lowest states with j = − / m = ± (1 / , /
2) and j = − m = ±
1. In this section we will start with the massless baryon b . First we bosonize world sheet fermions ψ µ , ψ φ and ψ Y , the superpartners of x µ , the Liouville field φ and the compact scalar Y , respectively. Followingthe standard rule we divide them into pairs ψ k = 1 √ ψ k − − iψ k ) , ¯ ψ k = 1 √ ψ k − + i ¯ ψ k ) , k = 1 , , (4.1) ψ = 1 √ ψ φ − iψ Y ) , ¯ ψ = 1 √ ψ φ + i ¯ ψ Y ) , (4.2)and define ψ k ¯ ψ k = i∂ − H k (no summation) , ψ ¯ ψ = i∂ − H, (4.3)where the bosons H k and H have the standard propagators h H k ( z ) , H l (0) i = − δ kl log z, h H ( z ) , H (0) i = − log z (4.4)19nd ψ k ∼ e iH k , ψ ∼ e iH . (4.5)The above formulas are written for the left-moving sector. In the right-moving sector bosonization is similar with the replacement z → ¯ z and ∂ z → ∂ ¯ z . As usual, we define spinors in terms of scalars H . Namely, S α = e P k is k H k , ¯ S ˙ α = e P k i ¯ s k H k (4.6)are 4D spinors, α = 1 ,
2, ˙ α = 1 ,
2. Moreover, S = e i H , ¯ S = e − i H (4.7)are spinors associated with ’“extra” dimensions φ and Y . Here s k = ± , k = 1 , s k are restricted by the GSOprojection, see below.Supercharges for non-critical string are defined in [53]. In our case four4D N = 1 supercharges Q α = 12 πi ¯ b | b | Z dz e − ϕ S α S exp (cid:18) i √ Y (cid:19) , ¯ Q ˙ α = 12 πi b | b | Z dz e − ϕ ¯ S ˙ α ¯ S exp (cid:18) − i √ Y (cid:19) (4.8)act in the left-moving sector, where we used the (cid:0) − (cid:1) picture. We have tomultiply these supercharges in the left-moving sector by the phase factors¯ b/ | b | and b/ | b | to make them neutral with respect to baryonic U(1) B . Otherfour supercharges of N = 2 4D supersymmetry are given by similar formulasand act in the right-moving sector. The action of the supercharge on a vertexis understood as an integral around the location of the vertex on the worldsheet.Supercharges (4.8) satisfy 4D space-time supersymmetry algebra { Q α , ¯ Q ˙ α } = 2 P µ σ µ , (4.9)while all other anti-commutators vanish. Note that P µ is the 4D momen-tum operator, the anti-commutator (4.9) does not produce translation in theLiouville direction. 20he GSO projection is the requirement of locality of a given vertex op-erator with respect to the supercharges (4.8).Let us start with Q α with s (0) k = (1 / , / s k = ± (cid:18) , (cid:19) , ¯ s k = ± (cid:18) , − (cid:19) (4.10)associated with four supercharges Q α and ¯ Q ˙ α .As an example, let us check the GSO selection rule (3.23) for 10D ’“tachyon”vertices (3.21). We have h Q α , V S,Ljm ( w ) i ∼ Z dz n ( z − w ) − ( − m ) + ... o (4.11)where dots stand for less singular OPE terms and 1 / ϕ . We see that locality requirement selects half-integer m as shown in (3.23).Note that an important feature of the supercharges (4.8) is the dependence onmomentum m in the compact direction Y . Without this dependence all 10D“tachyon” vertices (3.21) would be projected out as it happens for criticalstrings. Note also that none of the states (3.21) are tachyonic in 4D.Now we can introduce 4D space-time R -charges. We normalize them asfollows: R (4) = R (4) L + R (4) R , R (4) L ( Q α ) = − , R (4) L ( ¯ Q ˙ α ) = 1 , (4.12)and use the same normalizations for R (4) R . This definition ensures that for agiven vertex operator we have R (4) L = − m L , R (4) R = − m R . (4.13)Note that the scalars H are not shifted upon R (4) rotations, so the world-sheet fermions ψ k , ψ do not have R (4) charges. This is in contrast with theaction of the world sheet R (2) symmetry. To generate fermion vertex for the b state we apply supercharges (4.8) to theleft-moving part of the vertex (3.21) with j = − / m = ± /
2. To getthe fermion vertex in the standard ( − /
2) picture we have to convert the21ertex (3.21) from the ( −
1) to (0) picture. This is done in Appendix A usingthe BRST operator. The left-moving part of the scalar vertex (3.21) in the(0) picture has the form V (0) j,m ( p µ ) = (cid:20) √ jψ φ + imψ Y ) + i √ πT p µ ψ µ (cid:21) e ip µ x µ + √ jφ + i √ mY , (4.14)where we skip the subscripts L .Let us start with j = − / m = 1 /
2. The vertex (4.14) reduces to V (0) − , m = ( p µ ) = (cid:20) − ψ + i √ πT p µ ψ µ (cid:21) e ip µ x µ − φ √ + i Y √ . (4.15)Applying the supercharge Q α we find that correlation function does not con-tain pole contribution and hence gives zero. On the other hand ¯ Q ˙ α producesthe following fermion vertex¯ V ( − )˙ α = h ¯ Q ˙ α , V (0) − , m = ( p µ ) i∼ e − ϕ (cid:20) − ¯ S ˙ α S + ip µ √ πT (¯ σ µ ) ˙ αα S α ¯ S (cid:21) e ip µ x µ − φ √ (4.16)where we used h ψ ( z ) , ¯ S ( w ) i ∼ p ( z − w ) S , h e iY ( z ) √ , e − iY ( w ) √ i ∼ p ( z − w ) , h ψ µ ( z ) , ¯ S ( w ) ˙ α i ∼ p ( z − w ) (¯ σ µ ) ˙ αα S α . (4.17)Note that the momentum m along the compact direction is zero for thefermion vertex (4.16).As a check we can calculate the conformal dimension of the vertex (4.16).The condition for this vertex to be physical is38 + 38 + p µ p µ πT − j ( j + 1) = 1 , (4.18)22here the first and the second contributions come from the ghost ϕ andthe scalars H k and H , respectively. We see that for j = − / m = − / ψ → ¯ ψ and m = − /
2. Only the actionof Q α gives non-trivial fermion vertex. We get V α, ( − ) = h Q α , V (0) − , m = − ( p µ ) i∼ e − ϕ (cid:20) − S α ¯ S + ip µ √ πT ( σ µ ) α ˙ α ¯ S ˙ α S (cid:21) e ip µ x µ − φ √ . (4.19)To conclude this subsection we note that if we apply supercharges tothe fermion vertices (4.16) and (4.19) we do not generate new states. Forexample, acting on (4.16) with Q α gives (the left-moving part of) the scalarvertex (3.21), h Q α , ¯ V ( − )˙ α i ∼ p µ √ πT (¯ σ µ ) ˙ αα V S,L − , m = (4.20)in the picture ( − Q ˙ α produces the scalar vertex (3.21) with m = − / h Q ˙ α , ¯ V ( − )˙ β i ∼ ε ˙ α ˙ β V S,L − , m = − . (4.21) In this section we will use the bosonic and fermionic vertices obtained aboveto construct hypermultiplet of the massless b states. For simplicity in thissection and below we will consider only bosonic components of supermul-tiplets. As was already mentioned, in the case of type IIA superstring weshould consider the states with m R = − m L ≡ − m . We will prove thisstatement below, in this and the subsequent subsections.In the NS-NS sector we have one complex (or two real) scalars (3.21), b = V S,Lj = − , m × V S,Rj = − , − m (4.22)associated with m = ± / m is opposite in the left- andright- moving sectors, for the R-R states we get the product of fermion ver-tices (4.16) and (4.19), namely, V ˙ αα = ¯ V L ˙ α × V Rα , ¯ V α ˙ α = V Lα × ¯ V R ˙ α . (4.23)23he vertices above define a complex vector C µ via V ˙ αα = (¯ σ µ ) ˙ αα C µ . (4.24)However, as is usual for the massless R-R string states, the number of physicaldegrees of freedom reduces because the fermion vertices (4.16) and (4.19)satisfy the massless Dirac equations which translate into the Bianchi identityfor the associated form. For 1-form (vector) we have ∂ µ C ν − ∂ ν C µ = 0 , (4.25)which ensures that the complex vector reduces to a complex scalar, C µ = ∂ µ ˜ b . (4.26)Altogether we have two complex scalars, b and ˜ b , which form the bosonicpart of the hypermultiplet. As was already mentioned, deformations of thecomplex structure of a Calabi-Yau manifold gives a massless hypermultipletfor type IIA theory and massless vector multiplet for type IIB theory. Thederivation above shows that our choice m R = − m L corresponds to type IIAstring.We stress again that this massless hypermultiplet is a short BPS repre-sentation of N = 2 supersymmetry algebra in 4D and is characterized by thenon-zero baryonic charge Q B ( b ) = ± R (4) charge of thevertex operator (4.22) vanishes due to cancellation between left and right-moving sectors, see (4.13). For the vertex (4.23) it is also zero since both m L and m R are zero. Thus we conclude that b and ˜ b have the vanishing R (4) charge, as expected for the scalar components of a hypermultiplet. Our superstring is of type IIA. This is fixed by derivation of our string theoryas a description of non-Abelian vortex in 4D N = 2 SQCD, see [19]. In thissubsection we “forget” for a short while about this and consider superstringtheory on the manifold (3.1) on its own right. Then, as usual in string theory,we have two options for a closed string: type IIA and type IIB. We will showbelow that type IIB option corresponds to the choice m R = m L .24or this choice the massless state with j = − / a = V S,Lj = − , m × V S,Rj = − , m , (4.27)associated with m = ± /
2. In the R-R sector we now obtain V αβ = V Lα × V Rβ , ¯ V ˙ α ˙ β = ¯ V L ˙ α × ¯ V R ˙ β . (4.28)Expanding the complex vertex V αβ in the basis of σ matrices V αβ = F δ βα + ( σ µ ¯ σ ν ) αβ C µν (4.29)we get a complex scalar F and a complex 2-form C µν which can be expressedin terms of a real 2-form, C µν = F µν − iF ∗ µν , where F µν is real and F ∗ µν = ε µνρλ F ρλ . The Dirac equations for the fermion vertices (4.16) and (4.19)imply that F is a constant, while F µν satisfies the Bianchi identity. Thisensures that F µν can be constructed in terms of a real vector potential F µν = ∂ µ A ν − ∂ ν A µ . (4.30)We see that we get a massless N = 2 BPS vector multiplet with thebosonic components given by the complex scalar a and the gauge potential A µ . This is what we expect from deformation of the complex structure of aCalabi-Yau manifold for type IIB string.Let us note that R charges also match since the R (4) charge of a in (4.27)is R (4) = ± R (4) charge of (4.28) and A µ are zero asexpected.However, if we try to interpret this N = 2 vector multiplet as a state ofthe non-Abelian vortex in N = 2 SQCD we will get an inconsistency. Tosee this one can observe that our state has non-zero baryonic charge whichcannot be associated with a gauge multiplet. This confirms our conclusionthat the string theory for our non-Abelian vortex-string is of IIA type. j = − / Below we consider the supermultiplet structure of the lowest massive statesgiven by the vertex operators (3.21) and (3.25). In this section we start25ith the first excited state of the scalar vertex (3.21) with j = − / m = ± /
2. The mass of this state is (cid:16) M j = − ,m = ± / (cid:17) πT = 2 , (5.1)see (3.24). The left-moving part of the vertex operator in the (0) picture is given by(4.14). For m = 3 / V (0) − , ( p µ ) = (cid:20) − (2 ψ − ¯ ψ ) + i √ πT p µ ψ µ (cid:21) e ip µ x µ − φ √ + i √ Y . (5.2)In much the same way as for the b state, the supercharge Q acting on the ver-tex above gives zero while the supercharge ¯ Q produces the following fermionvertex in the picture (cid:0) − (cid:1) :¯ V ( − )˙ α = h ¯ Q ˙ α , V (0) − , m = ( p µ ) i ∼ e − ϕ (cid:2) − S ˙ α S + ip µ √ πT (¯ σ µ ) ˙ αα S α ¯ S (cid:21) ( ∂ − Y + ψ φ ψ Y ) e ip µ x µ − φ √ + i √ Y . (5.3)Note that the momentum m along the compact dimension is m = 1for this vertex. It is easy to check that the mass of this fermion is given by(5.1).In a similar manner, for m = − / ψ → ¯ ψ and m = − /
2. Action of supercharge Q gives the following fermionvertex: V α, ( − ) = h Q α , V (0) − , m = − ( p µ ) i ∼ e − ϕ (cid:2) − S α ¯ S + ip µ √ πT ( σ µ ) α ˙ α ¯ S ˙ α S (cid:21) ( ∂ − Y + ψ φ ψ Y ) e ip µ x µ − φ √ − i √ Y , (5.4)26ith m = − Q on (5.3) does not produce new states, while ¯ Q gives h ¯ Q ˙ α , ¯ V ( − )˙ β i ∼ ε ˙ α ˙ β V S, excited m = , (5.5)where the new excited scalar vertex in the picture ( −
1) with m = 1 / V S, excited m = = (cid:20) − ∂ − Y + ip µ √ πT ψ µ ¯ ψ ∂ − Y (cid:21) e − ϕ e ip µ x µ − φ √ + i Y √ . (5.6)The mass of this state is still given by (5.1). Action of supercharge Q on thefermion vertex (5.4) produces the conjugated scalar with m = − / Now we can use the vertices obtained in the previous subsection to constructsupermultiplets at the level (5.1). We have two scalar vertices with m = ± / m = ± /
2, see left-hand side of (3.21) and (5.6). Using these verticeswe can construct the scalar states in the NS-NS sector. Namely, we have onecomplex scalar V S,Lj = − , m = ± × V S,Rj = − , m = ∓ (5.7)formed by the m = ± / V S, excited ,Lj = − , m = ± × V S, excited ,Rj = − , m = ∓ (5.8)formed by the m = ± / V S,Lj = − , m = ± × V S, excited ,Rj = − , m = ∓ (5.9)and V S, excited ,Lj = − , m = ± × V S,Rj = − , m = ∓ (5.10)formed by products of two different vertices. Altogether in the NS-NS sectorwe observe four complex scalars.In the R-R sector we have V excited˙ αα = ¯ V L ˙ α × V Rα , ¯ V excited α ˙ α = V Lα × ¯ V R ˙ α , (5.11)27here now the fermion vertices are given by (5.3) and (5.4). Expanding thesevertices in the basis of σ matrices V excited˙ αα = (¯ σ µ ) ˙ αα B µ + (¯ σ µ σ ν ¯ σ ρ ) ˙ αα B µνρ (5.12)we arrive at the complex vector field B µ and the complex 3-form B µνρ .In four dimensions the massive 3-form is dual to a massive scalar [54]. Generically the rules of dualizing can be summarized as follows [54]. In D dimensions massless p -forms have c pD − = ( D − p !( D − − p )! (5.13)physical degrees of freedom. Therefore, the rule of dualizing of the massless p -form is p → ( D − − p ) . (5.14)In particular, 3-form in 4D has no degrees of freedom.For the massive p forms we have c pD − = ( D − p !( D − − p )! (5.15)physical degrees of freedom. The rule of dualizing now becomes p → ( D − − p ) . (5.16)Thus the massive 3-form in 4D is dual to a massive scalar. Explicitly theduality relation can be written as [54, 55] B µνρ ∼ ε µνρλ ∂ λ c . (5.17)We conclude in the R-R sector we obtained one complex scalar c andthe complex vector B µ . Altogether the bosonic part of the supermultipletwith mass (5.1) contains 5 scalars and a vector, all complex. This is exactlythe bosonic content of two real N = 2 long massive vector multiplets, eachcontaining 5 scalars and a vector, see Appendix B,( N = 2 ) vector = 1 vector + 5 scalar . (5.18) We did not include 3-form in the expansion (4.24) because in the massless case itcontains no physical degrees of freedom, see below. N = 2 massive vector multiplet can be realized as aresult of Higgsing of a U (1) massless gauge multiplet containing gauge fieldand a complex scalar (2 real scalars) by vacuum expectation values (VEVs)of a hypermultiplet which contains 4 real scalars. After Higgsing, one scalaris “eaten” by the Higgs mechanism, so we are left with massive vector fieldand 5 scalars. The number of degrees of freedom in this massive N = 2 longvector multiplet is 8=3+5, where 3 comes from the massive vector.Summarizing this section we present 4D R charges of the vector multipletcomponents. Due to cancellation of the R charges of the left and right-movingsectors, the R (4) charges of the R-R states (5.11) and two scalars (5.7), (5.8)of the NS-NS sector vanish, see (4.13). The R -charges of two scalars (5.9)and (5.10) are non-zero, R (4) = ±
2. These are exactly the R -charges of amassive N = 2 vector multiplet. This can be easily understood in termsof Higgsing of the massless gauge multiplet by hypermultiplet VEVs. Thegauge field and scalars from the hypermultiplet have the zero R charge whilethe R charges of two scalar superpartners of the gauge field in the masslessvector multiplet are indeed characterized by R (4) = ±
2, cf. Sec. 4.4. j = − In this section we consider the lowest spin-2 supermultiplet produced by thevertex operator (3.25). The mass of the state with j = − m = ± M j = − ,m = ± ) πT = 1 , (6.1)see (3.27).We will see below that the spin-2 state (3.25) is the highest componentof this supermultiplet. To simplify our discussion it is easier to start from ascalar component of this supermultiplet replacing the world-sheet fermions ψ L,Rµ by ψ L,Rφ and ψ L,RY . Thus, in the left-moving sector we start from thescalar vertex which, in the picture ( − V ( − j = − ,m =1 = ψ e − ϕ e ip µ x µ −√ φ + i √ mY (6.2)where we skip the superscripts L , while ψ is given by (4.2) and m = 1. For m = − ψ → ¯ ψ . The conformaldimension of this vertex is the same as that of the vertex in (3.25), so wehave a scalar state with mass (6.1). 29 .1 Action of supercharges To convert this vertex operator into the picture (0) we use the BRST oper-ator, see Appendix A. Then in the picture (0) we have V (0) j = − ,m =1 = (cid:20) √ ∂ − φ − i∂ − Y ) + ip µ √ πT ψ µ ψ (cid:21) e − ϕ e ip µ x µ −√ φ + i √ mY (6.3)for m = 1 and a similar vertex with ψ → ¯ ψ for m = − Q acts trivially on (6.3), while ¯ Q produces the following fermion vertex in thepicture ( − / V ( − )˙ α = h ¯ Q ˙ α , V (0) − , m =1 ( p µ ) i ∼ e − ϕ (cid:2) ¯ S ˙ α ¯ S + p µ √ πT (¯ σ µ ) ˙ αα S α S (cid:21) ( ∂ − Y + ψ φ ψ Y ) e ip µ x µ −√ φ + i Y √ , (6.4)This fermion vertex has m = 1 / Q to the scalar vertex V (0) j = − ,m = − we get a fermion vertex with m = − / V α ) ( − ) = h Q α , V (0) − , m = − ( p µ ) i ∼ e − ϕ [ S α S + p µ √ πT ( σ µ ) α ˙ α ¯ S ˙ α ¯ S (cid:21) ( ∂ − Y + ψ φ ψ Y ) e ip µ x µ −√ φ − i Y √ . (6.5)In order to generate new bosonic vertex operators with the same mass (6.1)we apply supercharges to the fermion vertices above. Supercharge Q actingon (6.4) gives the following bosonic vertices in the picture ( − h Q α , ¯ V ˙ α i ∼ σ α ˙ αµ (cid:18) ψ µ + p µ √ πT ψ (cid:19) e − ϕ e ip µ x µ −√ φ + i √ Y = σ α ˙ αµ (cid:18) V µj = − ,m =1 + p µ √ πT V ( − j = − ,m =1 (cid:19) (6.6)where V ( − j = − ,m =1 is the scalar vertex (6.2), while V µj = − ,m =1 = ψ µ e − ϕ e ip µ x µ −√ φ + i √ mY (6.7)30s a new vector vertex operator with m = 1. We recognize it as a left-movingpart of the spin-2 vertex (3.25). As was mentioned above, we obtained itby applying the supercharges to the scalar vertex (6.2). In a similar way wecan generate the complex-conjugated vector V µj = − ,m = − with m = − Q to the fermion vertex (6.5).We can also apply the supercharge ¯ Q to the fermion vertex (6.4). Thisgives h Q ˙ α , ¯ V ˙ β i ∼ δ ˙ α ˙ β V j = − ,m =0 , (6.8)where V ( − j = − ,m =0 = (cid:18) ¯ ψ + p µ √ πT ψ µ (cid:19) ∂ − Y e − ϕ e ip µ x µ −√ φ (6.9)is a new scalar vertex with m = 0 and mass (6.1). Similarly, the action of Q on the fermion vertex (6.5) gives a complex-conjugated scalar vertex withthe replacement ¯ ψ → ψ .Finally, instead of the scalar vertex (6.2) we can start from another scalarvertex, ˜ V ( − j = − ,m =1 = ¯ ψ e − ϕ e ip µ x µ −√ φ + i √ mY . (6.10)Note that this vertex is different from the one complex-conjugated to (6.2)because here we take m = 1. Conjugated to (6.10) is obtained by replacement¯ ψ → ψ and taking m = − j = − V ( − j = − ,m = ± , ˜ V ( − j = − ,m = ± , V ( − j = − ,m =0 , (6.11)given by (6.2), (6.10) and (6.9) respectively. Now we will use bosonic and fermionic vertex operators from the previoussubsection to construct the supermultiplet with j = − V j = − αα = ¯ V L ˙ α (cid:18) m = 12 (cid:19) × V Rα (cid:18) m = − (cid:19) , ¯ V j = − α ˙ α = V Lα (cid:18) m = − (cid:19) × ¯ V R ˙ α (cid:18) m = 12 (cid:19) , (6.12)where the fermion vertices are given by (6.4) and (6.5). Expanding V j = − αα and ¯ V j = − α ˙ α as in (5.12) we get a complex vector and a complex 3-form. Aswas discussed in Sec. 5.2, the massive 3-form dualizes into a massive scalar.Thus in the R-R sector we get one complex vector and one complex scalar.Now we pass to the NS-NS sector. The scalar vertices (6.11) give 3 × V Li ( m ≥ × V Rj ( m ≤ , (6.13)where V i ( m ), i = 1 , ,
3, are given by (6.2), (6.10) and (6.9), respectively.Changing the sign of m together with the replacement ψ → ¯ ψ gives ninecomplex conjugated scalars in addition to those in (6.13).Combining the vector vertex (6.7) with three scalar vertices (6.11) pro-vides us with six vectors of the form( V µj = − ,m =1 ) L × V Rj ( m ≤ ,V Li ( m ≥ × ( V µj = − ,m = − ) R ,i = 1 , , . (6.14)Again changing the sign of m together with the replacement ψ → ¯ ψ gives sixcomplex conjugated vectors to those.Finally we can combine two vector vertices (6.7) to produce a tensor( V µj = − ,m =1 ) L × ( V νj = − ,m = − ) R . (6.15)Changing the sign of m gives a complex conjugated tensor. In 4D a massivevector has ( D −
1) = 3 physical degrees of freedom. Therefore for the tensorstate (6.15) we get3 × ⇒ spin − + 1 vector + 1 scalar (6.16)massive degrees of freedom, where we show the expansion of the massivetensor into irreducible representations of SO( D − spin − + 8 vector + 11 scalar , (6.17)where we show the numbers of states with the given spin.How they split into 4D N = 2 supermultiplets? Long N = 2 spin-2multiplet contains [56]( N = 2 ) spin − = 1 spin − + 6 vector + 1 scalar (6.18)bosonic spin states while long N = 2 vector multiplet has( N = 2 ) vector = 1 vector + 5 scalar (6.19)bosonic spin states, see Appendix B and Eq. (5.18).We conclude that j = − j = −
1) states = 1 × ( N = 2 ) spin − + 2 × ( N = 2 ) vector (6.20)(one spin-2 and two vector) N = 2 long (non-BPS) supermultiplets, allcomplex. In this section we will show that all states we discussed in this paper (shownin Fig. 2) are the lowest states of the corresponding linear Regge trajectories.To construct these Regge trajectories we multiply the vertex operators (3.21)or (3.25) by derivatives of flat 4D coordinates. For example, for the scalarvertices (3.21) we construct a family of vertices n Y i =1 ∂ − x µ i ∂ + x ν i e − ϕ L − ϕ R e ip µ x µ V S,Lj = − ,m × V S,Rj = − , − m , (7.1)where n is n = 0 , , , ... . The hadronic states associated with these verticeshave at most spin 2 n . Their mass is (cid:16) M j = − ,m (cid:17) πT = m −
14 + n , n = 0 , , , ... (7.2)33e see that mass squared for these states depends linearly on the spin. Thislinear Regge dependence appears because we use the flat 4D part of the string σ model to construct the Regge trajectories.A similar construction can be developed for vertices (3.25). Masses ofthese states are ( M j = − ,m ) πT = m + n , n = 0 , , , ... (7.3)We have the same linear dependence with the same slope. In [2] we observed that a vortex string supported in N = 2 SQCD is criticalprovided the following conditions are met:(i) The gauge group of the model considered is U(2);(ii) The number of flavor hypermultiplets is N f = 2 N = 4;The 4D theory under consideration is not conformal because of the Fayet-Iliopoulos parameter ξ = 0. However, the gauge coupling β function vanishes;the Fayet-Iliopoulos parameter does not run either.In addition to four translational zero modes this string exhibits threeorientational and three size zero modes. Their geometry is described bya non-compact six-dimensional Calabi-Yau manifold, the so-called resolvedconifold Y . The target space takes the form R × Y . The emergence of sixextra zero modes on the string under consideration makes the target-spacemodel conformal, the overall Virasoro central charge (including the ghostcontribution) vanishes. Thus, this string is critical. The phenomenon weobserved could be called a “reverse holography.”The next question which was natural to address was the quantizationof this closed critical string and the derivation of the hadronic spectrum.The present paper completes the work started in [18, 19, 1]. We calculatedthe masses of the massive spin-0 and spin-2 states and constructed the 4Dsupermultiplets to which they belong. Our formulas match the previousresult for the massless states.The massive supermultiplets are shown to be long (non-BPS saturated).We also prove that the above states are the lowest states on the correspondingRegge trajectories which are linear and parallel.34 cknowledgments The authors are grateful to Edwin Ireson for useful discussions and to EfratGerchkovitz and Avner Karasik for helpful communications. This work issupported in part by DOE grant de-sc0011842. The work of A.Y. was sup-ported by William I. Fine Theoretical Physics Institute at the University ofMinnesota and by Russian Foundation for Basic Research Grant No. 18-02-00048. 35 ppendix A. BRST operator and vertices inthe picture (0)
To convert vertex operator in the picture ( −
1) into picture (0) we use theBRST operator as follows [57]: V (0) = (cid:10) Q BRST , ζ V ( − (cid:11) , (A.1)where Q BRST = 12 πi Z dz (cid:20) cT m + γG m + 12 ( cT gh + γG gh ) (cid:21) . (A.2)Here c and γ are the ghosts of fermionic ( b, c ) and bosonic ( β, γ ) systems,respectively, while T m , G m and T gh , G gh are the energy momentum tensorand the supercurrent for matter and ghosts. Below we will need the explicitexpression for the matter supercurrent, G m = i ( ψ µ ∂ − x µ + ψ φ ∂ − φ + ψ Y ∂ − Y ) . (A.3)The ghost system ( β, γ ) can be expressed in terms of fermions η , ζ , γ = e ϕ η, β = e − ϕ ∂ − ζ , (A.4)where the propagator of η , ζ is normalized as h η ( z ) , ζ (0) i = 1 z . (A.5)To convert the left-moving part of the scalar vertex (3.21) in the picture( −
1) into the picture (0) we use the rule (A.1). We arrive at the expression(4.14) with the help of (A.3).Similarly, for the j = − z Level 0 Level 2 Level 4 Sum1 0 1 0 10 1 4 1 6 − Structure of the vector multiplet. We show the numbers of states withthe given J z produced by the action of supercharges at each level and their sum. Appendix B. Long N = 2 vector and spin-2multiplets in 4D. In this Appendix we briefly review construction of N = 2 long massivesupermultiplets in four dimensions. For massive states in the rest framesupersymmetry generators Q αf and ¯ Q f ˙ α can be viewed as annihilation andcreation operators, where f = 1 , N = 1 supersymmetrieswhich constitute N = 2 . Assuming that the annihilation operators Q αf produce zero upon acting on a “ground state” | a i we can generate all statesof a given supermultiplet applying to | a i the creation operators ¯ Q f ˙ α .For simplicity we will consider only the bosonic states in a multiplet.Assuming that | a i is a bosonic state we have 6 possibilities (cid:8) ¯ Q ¯ Q , ¯ Q ¯ Q , ¯ Q ¯ Q , ¯ Q ¯ Q , ¯ Q ¯ Q , ¯ Q ¯ Q (cid:9) × | a i (B.1)at level 2 and only one possibility¯ Q ¯ Q ¯ Q ¯ Q × | a i (B.2)at level 4.First let us construct the long N = 2 massive vector supermultiplet. Inthis case we choose | a i to be a scalar with spin J = 0. The construction isshown in Table 1 where J z is the z -projection of spin and level 0 denotes thestate | a i itself. Here we used the fact that, say, in Eq. (B.1) the product¯ Q ¯ Q acting on | a i increases J z by one, four product operators of the type¯ Q f ˙1 ¯ Q g ˙2 ( f, g = 1 ,
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