1 Introduction

FTPI-MINN-18/09, UMN-TH-3718/18

May 24, 2018

Hadrons of Supersymmetric QCD in

[2mm] Four Dimensions from Little String Theory

M. Shifman and A. Yung

William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455

National Research Center “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300, Russia

St. Petersburg State University, Universitetskaya nab., St. Petersburg 199034, Russia

Abstract

It was recently shown that non-Abelian vortex strings supported in a version of four-dimensional supersymmetric QCD (SQCD) become critical superstrings. In addition to four translational moduli, non-Abelian strings under consideration have six orientational and size moduli. Together they form a ten-dimensional target space required for a superstring to be critical, namely, the product of the flat four-dimensional space and conifold – a non-compact Calabi-Yau threefold. In this paper we report on further studies of low-lying closed string states which emerge in four dimensions and identify them as hadrons of our four-dimensional SQCD. We use the approach based on “little string theory,” describing critical string on the conifold as a non-critical string with the Liouville field and a compact scalar at the self-dual radius. In addition to massless hypermultiplet found earlier we observe several massive vector multiplets and a massive spin-2 multiplet, all belonging to the long (non-BPS) representations of supersymmetry in four dimensions. All the above states are interpreted as baryons formed by a closed string with confined monopoles attached. Our construction presents an example of a “reverse holography.”

1 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 a solitonic vortex string under an appropriate choice of the coupling constant [2]. One of our main tasks is to analyze the structure of the 4D supermultiplets emerging from quantization of the closed string mentioned above. We will start though from a brief review of the setup.

The problem of understanding confining gauge theories splits into two different equally fundamental tasks. The first one is to understand the physical nature of confinement and describe the formation of confining strings. There was a great progress in this direction in supersymmetric gauge theories due to the breakthrough papers by Seiberg and Witten [3, 4] in which the monopole condensation was shown to occur in the monopole vacua of supersymmetric QCD (SQCD). This leads to the formation of Abelian Abrikosov-Nielsen-Olesen (ANO) vortices [5] which confine color electric charges. Attempts to find a non-Abelian generalization of this mechanism led to the discovery of the so called “instead-of-confinement” phase which occurs in the quark vacua of 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 is to quantize this string in four-dimensional (4D) theory outside the critical dimension to study the hadron spectrum. Most solitonic strings, such as the ANO strings, have a finite thickness manifesting itself in the presence of an infinite series of unknown higher-derivative corrections in the effective sigma model on the string world sheet. This makes the task of quantizing such a string virtually impossible.

Recent advances in this direction [2] demonstrated that the non-Abelian solitonic vortex in a particular version of 4D SQCD becomes a critical superstring. This particular 4D SQCD has the U(2) gauge group, four quark flavors and the Fayet-Iliopoulos (FI) [7] parameter .

Non-Abelian vortices were first discovered in SQCD with the U gauge group and flavors of quark hypermultiplets [8, 9, 10, 11]. In addition to four translational moduli characteristic of the ANO strings [5], the non-Abelian strings carry orientational moduli, as well as the size moduli if [8, 9, 10, 11] (see [12, 13, 14, 15] for reviews). If their dynamics are described by effective two-dimensional sigma model on the string world sheet with the target space

(1.1)

to which we will refer to as the weighted CP model (WCP).

For the model becomes conformal. Moreover, for the dimension of the orientational/size moduli space is six and they can be combined with four translational moduli to form a ten-dimensional space required for superstring to become critical.111The non-Abelian vortex string is 1/2 BPS saturated and, therefore, has supersymmetry on its world sheet. Thus, we actually deal with a superstring in the case at hand.

In this case the target space of the world sheet 2D theory on the non-Abelian vortex string is , where is a non-compact six-dimensional Calabi-Yau manifold, the so-called resolved conifold [16, 17].

Since non-Abelian vortex string on the conifold is critical it has a perfectly good UV behavior. This opens the possibility that it can become thin in a certain regime [2]. The string transverse size is given by , where is a typical mass scale of the four-dimensional fields forming the string. The string cannot be thin in a weakly coupled 4D theory because at weak coupling and is always small in the units of where is the tension. Here is the gauge coupling constant of the 4D QCD and is the string tension.

A conjecture was put forward in [2] that at strong coupling in the vicinity of a critical value of the non-Abelian string on the conifold becomes thin, and higher-derivative corrections in the action can be ignored. It is expected that the thin string produces linear Regge trajectories even for small spins [2]. The above conjecture implies 222At the beta function of the 4D QCD is zero, so the gauge coupling does not run. Note, however, that conformal invariance in the 4D theory is broken by the FI parameter which does not run either. that at .

A version of the string-gauge duality for 4D SQCD was proposed [2]: at weak coupling this theory is in the Higgs phase and can be described in terms of (s)quarks and Higgsed gauge bosons, while at strong coupling hadrons of this theory can be understood as string states formed by the non-Abelian vortex string.

The vortices in the U theories under consideration are topologically stable and cannot be broken. Therefore the finite-length strings are closed. Thus, we focus on the closed strings. The goal is to identify the closed string states with the hadrons of 4D SQCD.

The first step of this program, namely, identifying massless string states was carried out in [18, 19] using supergravity formalism. In particular, a single matter hypermultiplet associated with the deformation of the complex structure of the conifold was found as the only 4D massless mode of the string. Other states arising from the massless ten-dimensional graviton are not dynamical in four dimensions. In particular, the 4D graviton and unwanted vector multiplet associated with deformations of the Kähler form of the conifold are absent. This is due to non-compactness of the Calabi-Yau manifold we deal with and non-normalizability of the corresponding modes over six-dimensional space .

The next step was done in [1] where a number of massive states of the closed non-Abelian vortex string was found. This step required a change of strategy. The point is that the coupling constant of the world sheet WCP(2,2) is not small. Moreover tends to zero once the 4D coupling approaches the critical value we are interested in. At the resolved conifold develops a conical singularity. The supergravity approximation does not work for massive states.333This is in contradistinction to the massless states. For the latter, we can perform computations at large where the supergravity approximation is valid and then extrapolate to strong coupling. In the sigma-model language massless states corresponds to chiral primary operators. They are protected by world-sheet supersymmetry and their masses are not lifted by quantum corrections.

To analyze the massive states the little string theories (LST) approach (see [23] for a review) was used in [1]. Namely, we used the equivalence between the critical string on the conifold and non-critical string which contains 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 to identification 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 found earlier in [19, 1]. In addition to the massless BPS hypermultiplet associated with deformations of the complex structure of the conifold we identify several massive vector multiplets and a massive spin-2 multiplet, all belonging to long non-BPS representations of supersymmetry in four dimensions. We interpret all states we found as baryons formed by a closed string with confined monopoles attached.

The paper is organized as follows. In Sec. 2 we review the description of non-Abelian vortex as a critical superstring on a conifold and identify massless string state. In Sec. 3 we review LST approach in terms of non-critical string and the spectrum of massive states. In Sec. 4 we introduce 4D supercharges and construct massless BPS hypermultiplet. In Sec. 5 we consider the lowest massive string excitations and show that they forms a long vector supermultiplet. Section 6 deals with the construction of spin-2 stringy supermultiplet. In Sec. 7 we discuss linear Regge trajectories, while Section 8 summarizes our conclusions. In Appendix A we describe the BRST operator and transitions between different pictures. In Appendix B we review long supermultiplets in 4D.

2 Non-Abelian vortex string

2.1 Four-dimensional Sqcd

As was already mentioned non-Abelian vortex-strings were first found in 4D SQCD with the gauge group U and flavors (i.e. the quark hypermultiplets) supplemented by the FI term [8, 9, 10, 11], see for example [14] for a detailed review of this theory. Here we just mention that at weak coupling, , this theory is in the Higgs phase in which the scalar components of the quark multiplets (squarks) develop vacuum expectation values (VEVs). These VEVs breaks the U gauge group Higgsing all gauge bosons. The Higgsed gauge bosons combine with the screened quarks to form long multiplets with mass .

The global flavor SU is broken down to the so called color-flavor locked group. The resulting global symmetry is

(2.1)

see [14] for more details.

The unbroken global U(1) factor above is identified with a baryonic symmetry. Note that what is usually identified as the baryonic U(1) charge is a part of our 4D theory gauge group. “Our” U(1) is an unbroken by squark VEVs combination of two U(1) symmetries: the first is a subgroup of the flavor SU and the second is the global U(1) subgroup of U gauge symmetry.

As was already noted, we consider SQCD in the Higgs phase: squarks condense. Therefore, non-Abelian vortex strings confine monopoles. In the 4D theory these strings are 1/2 BPS-saturated; hence, their tension is determined exactly by the FI parameter,

(2.2)

However, the monopoles cannot be attached to the string endpoints. In fact, in the U theories confined monopoles are junctions of two distinct elementary non-Abelian strings [26, 10, 11] (see [14] for a review). As a result, in four-dimensional SQCD we have monopole-antimonopole mesons in which the monopole and antimonopole are connected by two confining strings. In addition, in the U gauge theory we can have baryons appearing as a closed “necklace” configurations of (integer) monopoles [14]. For the U(2) gauge group the lightest baryon presented by such a “necklace” configuration consists of two monopoles, see Fig. 1.

Figure 1: Examples of the monopole “necklace” baryons: Open circles denote monopoles.

Both stringy monopole-antimonopole mesons and monopole baryons with spins have mass determined by the string tension, and are heavier at weak coupling than perturbative states, which have mass . However, according to our conjecture, at strong coupling near the critical point , see [2] and Sec. 2.3 below. In this regime perturbative states decouple and we are left with hadrons formed by the closed string states.444There are also massless bifundamental quarks, charged with respect to both non-Abelian factors in (2.1). These are associated with the Higgs branch present in 4D QCD, see [14, 19] for details. All hadrons identified as closed string states in this paper turn out to be baryons and look like monopole “necklaces,” see Fig. 1.

2.2 World sheet sigma model

The presence of color-flavor locked group SU is the reason for the formation 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-called orientational zero modes.

Let us briefly review the model emerging on the world sheet of the non-Abelian critical string [2, 18, 19]. If the dynamics of the orientational zero modes of the non-Abelian vortex, which become orientational moduli fields on the world sheet, is described by two-dimensional supersymmetric model [14].

If one adds extra quark flavors, non-Abelian vortices become semilocal. They acquire size moduli [27]. In particular, for the non-Abelian semilocal vortex at hand, in addition to the orientational zero modes (), there are the so-called size moduli () [27, 8, 11, 28, 29, 30]. The target space of the sigma model on the string world sheet is defined by the -term condition

(2.3)

and a U(1) phase is gauged away.

The total number of real bosonic degrees of freedom in this model is six, where we take into account the constraint (2.3) and the fact that one U(1) phase is gauged away. As was already mentioned, these six internal degrees of freedom are combined with four translational moduli to form a ten-dimensional space needed for superstring to be critical.

At weak coupling the world sheet coupling constant in (2.3) is related to the 4D SU(2) gauge coupling as follows:

(2.4)

see [14]. Note that the first (and the only) coefficient is the same for the 4D SQCD and the world-sheet model functions. Both vanish at . This ensures that our world-sheet theory is conformal.

Since non-Abelian vortex string is 1/2 BPS it preserves in the world sheet sigma model which is necessary to have space-time supersymmetry [31, 32]. Moreover, as was shown in [19], the string theory of the non-Abelian critical vortex is type IIA.

The global symmetry of the world-sheet sigma model is

(2.5)

i.e. exactly the same as the unbroken global group in the 4D theory, cf. (2.1), at and . The fields and transform in the following representations:

(2.6)

2.3 Thin string regime

The coupling constant of 4D SQCD can be complexified

(2.7)

where is the four-dimensional angle. Note that SU version of the four-dimensional SQCD at hand possesses a strong-weak coupling duality, namely, [20, 21]. This suggests that the self-dual point would be a natural candidate for a critical value , where our non-Abelian vortex string becomes thin.555We suggested this earlier in [18, 19]. However, as was shown recently in [22], -duality maps our U theory to a theory in which a different U(1) subgroup of the flavor group is gauged. In particular, in our U theory all quark flavors have equal charges with respect to the U(1) subgroup of the U(2) gauge group, while in the -dual version only one flavor is charged with respect to the U(1) gauge group. As a result, the -dual version supports a different type of non-Abelian strings [22].

This means that -duality is broken in our 4D theory by the choice of the U(1) subgroup which is gauged.666We are grateful to E. Gerchkovitz and A. Karasik for pointing out to us this circumstance. We do not consider -duality and its consequences here.

The two-dimensional coupling constant can be naturally complexified too if we include the two-dimensional term,

(2.8)

The exact relation between the complexified 4D and 2D couplings is as follows:

(2.9)

where the function is a special modular function of defined in terms of the -functions,

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 derived using 2D-4D correspondence, namely, the match of the BPS spectra of the 4D theory at and the world-sheet theory on the non-Abelian string [33, 34, 10, 11]. Details of this derivation will be presented elsewhere. Note that our result (2.9) differs from that obtained in [22] using the localization technique.

According to the hypothesis formulated in [2], our critical non-Abelian string becomes infinitely thin at strong coupling at the critical point (or ). Moreover, in [19] we conjectured that corresponds to in the world-sheet theory via relation (2.9). Thus, we assume that at , which corresponds to in 4D SQCD.

At the point the non-Abelian string becomes infinitely thin, higher derivative terms can be neglected and the theory of the non-Abelian string reduces to the WCP(2,2) model. The point is a natural choice because at this point we have a regime change in the 2D sigma model. This is the point where the resolved conifold defined by the term constraint (2.3) develops a conical singularity [17].

2.4 Massless 4D baryon as deformation of the conifold complex structure

In this section we will briefly review the only 4D massless state associated with the deformation of the conifold complex structure. It was found in [19]. As was already mentioned, all other modes arising from the massless 10D graviton have non-normalizable wave functions over the conifold. In particular, the 4D graviton is absent [19]. This result matches our expectations since from the very beginning we started from SQCD in the flat four-dimensional space without gravity.

The target space of the world sheet WCP(2,2) model is defined by the -term condition (2.3). We can construct the U(1) gauge-invariant “mesonic” variables

(2.10)

These variables are subject to the constraint , or

(2.11)

where

and the matrices above are , . Equation (2.11) defines the conifold . It has the Kähler Ricci-flat metric and represents a non-compact Calabi-Yau manifold [16, 35, 17]. It is a cone which can be parametrized by the non-compact radial coordinate

(2.12)

and five angles, see [16]. Its section at fixed is .

At the conifold develops a conical singularity, so both and can shrink to zero. The conifold singularity can be smoothed out in two distinct ways: by deforming the Kähler form or by deforming the complex structure. The first option is called the resolved conifold and amounts to introducing a non-zero in (2.3). This resolution preserves the Kähler structure and Ricci-flatness of the metric. If we put in (2.3) we get the model with the target space (with the radius ). The resolved conifold has no normalizable zero modes. In particular, the modulus which becomes a scalar field in four dimensions has non-normalizable wave function over the manifold [19].

As explained in [36, 19], non-normalizable 4D modes can be interpreted as (frozen) coupling constants in the 4D theory. The field is the most straightforward example of this, since the 2D coupling is related to the 4D coupling, see Eq. (2.9).

If another option exists, namely a deformation of the complex structure [17]. It preserves the Kähler structure and Ricci-flatness of the conifold and is usually referred to as the deformed conifold. It is defined by deformation of Eq. (2.11), namely,

(2.13)

where is a complex number. Now the can not shrink to zero, its minimal size is determined by .

The modulus becomes a 4D complex scalar field. The effective action for this field was calculated in [19] using the explicit metric on the deformed conifold [16, 37, 38],

(2.14)

where is the size of introduced as an infrared regularization of logarithmically divergent field norm.777The infrared regularization on the conifold translates into the size of the 4D space because the variables in (2.12) have an interpretation of the vortex string sizes, .

We see that the norm of the modulus turns out to be logarithmically divergent in the infrared. The modes with the logarithmically divergent norm are at the borderline between normalizable and non-normalizable modes. Usually such states are considered as “localized” on the string. We follow this rule. We can relate this logarithmic behavior to the marginal stability of the state, see [19].

The field , being massless, can develop a VEV. Thus, we have a new Higgs branch in 4D SQCD which is developed only for the critical value of the coupling constant .

The logarithmic metric in (2.14) in principle can receive both perturbative and non-perturbative quantum corrections in , the sigma model coupling. However, in the theory the non-renormalization theorem of [21] forbids the dependence of the Higgs branch metric on the 4D coupling constant . Since the 2D coupling is related to we expect that the logarithmic metric in (2.14) will stay intact. This expectation is confirmed in [1].

In [19] the massless state was interpreted as a baryon of 4D SQCD. Let us explain this. From Eq. (2.13) we see that the complex parameter (which is promoted to a 4D scalar field) is singlet with respect to both SU(2) factors in (2.5), i.e. the global world-sheet group.888Which is isomorphic to the 4D global group (2.1) at , . What about its baryonic charge?

Since

(2.15)

we see that the state transforms as

(2.16)

where we used (2.6) and (2.13). Three numbers above refer to the representations of (2.5). In particular it has the baryon charge .

To conclude this section let us note that in our case of type IIA superstring the complex scalar associated with deformations of the complex structure of the Calabi-Yau space enters as a component of a massless 4D hypermultiplet, see [39] for a review. Instead, for type IIB superstring it would be a component of a vector BPS multiplet. Non-vanishing baryonic charge of the state confirms our conclusion that the string under consideration is a type IIA.

3 Massive states from non-critical string

As was explained in Sec. 1, the critical string theory on the conifold is hard to use for calculating the spectrum of massive string modes because the supergravity approximation does not work. In this section we review the 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 string theory with the Liouville field and a compact scalar at the self-dual radius [24, 25]. We intend to use the same formulation in this paper to analyze the 4D hypermultiplet structure of the massive states.

3.1 Non-critical string theory

Non-critical string theory is formulated on the target space

(3.1)

where is a real line associated with the Liouville field and the theory has a linear in dilaton, such that string coupling is given by

(3.2)

We will determine in Eq. (3.7).

Generically the above equivalence is formulated between the critical string on non-compact Calabi-Yau spaces with an isolated singularity on the one hand, and non-critical string with the additional Ginzburg-Landau superconformal system [24], on the other hand. In the conifold case this extra Ginzburg-Landau factor in (3.1) is absent [40].

In [41, 24, 40] it was argued that non-critical string theories with the string coupling exponentially falling off at are holographic. The string coupling goes to zero in the bulk of the space-time and non-trivial dynamics (LST) 999The main example of this behavior is non-gravitational LST in the flat six-dimensional space formed by the world volume of parallel NS5 branes. is localized on the “boundary.” In our case the “boundary” is the four-dimensional space in which SQCD is defined. (It is worth emphasizing that in our case the boundary 4D dynamics is the starting point while 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 is most welcome and expected. We start with SQCD in 4D space and study solitonic vortex strings. In our approach 10D space formed by 4D “actual” space and six internal moduli of the string is an artificial construction needed to formulate the string theory of a special non-Abelian vortex. Clearly we expect that all non-trivial “actual” physics should be localized exclusively on the 4D “boundary.” In other words, we expect that LST in our case is nothing but 4D SQCD at the critical value of the gauge coupling (in the hadronic description).

The linear dilaton in (3.2) implies that the bosonic stress tensor of matter coupled to 2D gravity is

(3.3)

where The compact scalar represents matter and satisfies the following condition:

(3.4)

Here we normalize the scalar fields in such a way that their propagators are

(3.5)

The central charge of the supersymmetrized theory above is

(3.6)

The criticality condition for the string on (3.1) implies that this central charge should be equal to 9. This gives

(3.7)

to be used in Eq. (3.2).

Deformation of the conifold (2.13) translates into adding the Liouville interaction to the world-sheet sigma model [24],

(3.8)

The conifold singularity at corresponds to the string coupling constant becoming infinitely large at , see (3.2). At the Liouville interaction regularize the behavior of the string coupling preventing the string from propagating to the region of large negative .

In fact the non-critical string theory can also be described in terms of two-dimensional black hole [42], which is the coset WZNW theory [43, 25, 44, 24] at level

(3.9)

In [45] it was shown that coset is a mirror description of the Liouville theory. The relation above implies in the case of the conifold () that

(3.10)

where is the total level of the Kač-Moody algebra in the supersymmetric version (the level of the bosonic part of the algebra is then ). The target space of this theory has the form of a semi-infinite cigar; the field associated with the motion along the cigar cannot take large negative values due to semi-infinite geometry. In this description the string coupling constant at the tip of the cigar is .

3.2 Vertex operators

Vertex operators for the string theory on (3.1) are constructed in [24], see also [43, 40]. Primaries of the part for large positive (where the target space becomes a cylinder ) take the form

(3.11)

where we split and into left and right-moving parts, say . For the self-dual radius (3.7) (or ) the parameter in Eq. (3.11) is integer. For the left-moving sector is the total momentum plus the winding number along the compact dimension . For the right-moving sector we introduce which is the winding number minus momentum. We will see below that for our case type IIA string , while for type IIB string .

The primary operator (3.11) is related to the wave function over “extra dimensions” as follows:

The string coupling (3.2) depends on . Thus,

(3.12)

We will look for string states with normalizable wave functions over the “extra dimensions” which we will interpret as hadrons in 4D SQCD. The condition for the string states to have normalizable wave functions reduces to 101010We include the case which is at the borderline between normalizable and non-normalizable states. In [1] it is shown that corresponds to the norm logarithmically divergent in the infrared in much the same way as the norm of the state, see (2.14)

(3.13)

The scaling dimension of the primary operator (3.11) is

(3.14)

Unitarity implies that it should be positive,

(3.15)

Moreover, to ensure that conformal dimensions of left and right-moving parts of the vertex operator (3.11) are the same we impose that .

The spectrum of the allowed values of and in (3.11) was exactly determined by using the Kač-Moody algebra for the coset in [46, 47, 48, 49, 43], see [50] for a review. Both discrete and continuous representations were found. Parameters and determine the global quadratic Casimir operator and the projection of the spin on the third axis,

(3.16)

where are the global currents.

We will focus on discrete representations with

(3.17)

Discrete representations include the normalizable states localized near the tip of the cigar (see (3.13)), while the continuous representations contain non-normalizable states.

Discrete representations contain states with negative norm. To exclude these ghost states a restriction for spin is imposed [46, 47, 48, 49, 50]

(3.18)

Thus, for our value we are left with only two allowed values of ,

(3.19)

and

(3.20)

3.3 Scalar and spin-2 states

Four-dimensional spin-0 and spin-2 states were found in [1] using vertex operators ((3.11)). The 4D scalar vertices in the picture have the form [24]

(3.21)

where superscript stands for scalar, represents bosonized ghost in the left and right-moving sectors, while is the 4D momentum of the string state.

The condition for the state (3.21) to be physical is

(3.22)

where 1/2 comes from the ghost and we used (3.14). We note that the conformal dimension of the ghost operator is equal to , where is the picture number.

The GSO projection restricts the integer for the operator in (3.21) to be odd [51, 24] 111111We will demonstrate this in the next section.,

(3.23)

For half-integer we have only one possibility , see (3.19). This determines the masses of the 4D scalars,

(3.24)

where the Minkowski 4D metric with the diagonal entries is used.

In particular, the state with is the massless baryon , associated with deformations of the conifold complex structure [1], while states with are massive 4D scalars.

At the next level we consider 4D spin-2 states. The corresponding vertex operators are given by

(3.25)

where are the world-sheet superpartners to 4D coordinates , while is the polarization tensor.

The condition for these states to be physical takes the form

(3.26)

The GSO projection selects now to be even, [24], thus we are left with only one allowed value of , in (3.20). Moreover, the value is excluded. This leads to the following expression for the masses of spin-2 states:

(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 fact that our “boundary” theory, 4D QCD, is defined in flat space without gravity.

To determine baryonic charge of these states we note that U(1) transformation of in the Liouville interaction (3.8) is compensated by a shift of . The baryonic charge of is two, see (2.16). Below we use the following convention: upon splitting into left and right-moving parts we define that only is shifted under U(1) transformation,

(3.28)

This gives for the baryon charge of the vertex operator (3.11)

(3.29)

We see that the momentum in the compact direction is in fact the baryon charge of a string state. All states we found above are baryons. Their masses as a function of the baryon charge are shown in Fig. 2.

Figure 2: Spectrum of spin-0 and spin-2 states as a function of the baryonic charge. Closed and open circles denote spin-0 and spin-2 states, respectively.

The momentum in the compact dimension is also related to the -charge. On the world sheet we can introduce the left and right -charges separately. Normalizing charge of , namely, , we see that should be shifted under the symmetry to make invariant the Liouville interaction (3.8).

This gives

(3.30)

for the charge of the vertex (3.11) which is the bottom component of the world sheet supermultiplet. The charge in the right-moving sector is defined similarly. Here superscript denotes the world sheet -charge.

As was discussed above, the massless baryon corresponds to , . Thus, the associated vertex has and conformal dimension , see (3.14). Therefore it satisfies the relation

(3.31)

as expected for the bottom component of a chiral primary operator, which defines the short representation of supersymmetry algebra (and similar relation in the right-moving sector). In 4D theory is a component of a short BPS multiplet, namely hypermultiplet.

4 Massless hypermultiplet

The remainder of this paper is devoted to the study the supermultiplet structure of the 4D string states described in the previous sections. Our strategy is as follows: we explicitly construct 4D supercharges and use them to generate all components of a given multiplet starting from a scalar or spin-2 representative shown in (3.21) or (3.25). We will generate supermultiplets originating from the lowest states with , and , . In this section we will start with the massless baryon .

4.1 4D supercharges

First we bosonize world sheet fermions , and , the superpartners of , the Liouville field and the compact scalar , respectively. Following the standard rule we divide them into pairs

(4.1)
(4.2)

and define

(4.3)

where the bosons and have the standard propagators

(4.4)

and

(4.5)

The above formulas are written for the left-moving sector. In the right-moving sector bosonization is similar with the replacement and .

As usual, we define spinors in terms of scalars . Namely,

(4.6)

are 4D spinors, , . Moreover,

(4.7)

are spinors associated with ’“extra” dimensions and . Here , and the choices of the allowed values of are restricted by the GSO projection, see below.

Supercharges for non-critical string are defined in [51]. In our case four 4D supercharges

(4.8)

act in the left-moving sector, where we used the picture. We have to multiply these supercharges in the left-moving sector by the phase factors and to make them neutral with respect to baryonic U(1). Other four supercharges of 4D supersymmetry are given by similar formulas and act in the right-moving sector. The action of the supercharge on a vertex is understood as an integral around the location of the vertex on the world sheet.

Supercharges (4.8) satisfy 4D space-time supersymmetry algebra

(4.9)

while all other anti-commutators vanish. Note that is the 4D momentum operator, the anti-commutator (4.9) does not produce translation in the Liouville direction.

The GSO projection is the requirement of locality of a given vertex operator with respect to the supercharges (4.8).

Let us start with with . Then mutual locality of the supercharges (4.8) selects polarizations

(4.10)

associated with four supercharges and .

As an example, let us check the GSO selection rule (3.23) for 10D ’“tachyon” vertices (3.21). We have

(4.11)

where dots stand for less singular OPE terms and comes from the ghost . We see that locality requirement selects half-integer as shown in (3.23). Note that an important feature of the supercharges (4.8) is the dependence on momentum in the compact direction . Without this dependence all 10D “tachyon” vertices (3.21) would be projected out as it happens for critical strings. Note also that none of the states (3.21) are tachyonic in 4D.

Now we can introduce 4D space-time -charges. We normalize them as follows:

(4.12)

and use the same normalizations for . This definition ensures that for a given vertex operator we have

(4.13)

Note that the scalars are not shifted upon rotations, so the world-sheet fermions , do not have charges. This is in contrast with the action of the world sheet symmetry.

4.2 Fermion vertex

To generate fermion vertex for the state we apply supercharges (4.8) to the left-moving part of the vertex (3.21) with and . To get the fermion vertex in the standard picture we have to convert the vertex (3.21) from the to (0) picture. This is done in Appendix A using the BRST operator. The left-moving part of the scalar vertex (3.21) in the (0) picture has the form

(4.14)

where we skip the subscripts .

Let us start with and . The vertex (4.14) reduces to

(4.15)

Applying the supercharge we find that correlation function does not contain pole contribution and hence gives zero. On the other hand produces the following fermion vertex

(4.16)

where we used

(4.17)

Note that the momentum along the compact direction is zero for the fermion 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 is

(4.18)

where the first and the second contributions come from the ghost and the scalars and , respectively. We see that for this state is massless, as expected.

By the same token, for we consider the action of the supercharges on the vertex in (4.15) with and . Only the action of gives non-trivial fermion vertex. We get

(4.19)

To conclude this subsection we note that if we apply supercharges to the fermion vertices (4.16) and (4.19) we do not generate new states. For example, acting on (4.16) with gives (the left-moving part of) the scalar vertex (3.21),

(4.20)

in the picture . This result is in full accord with supersymmetry algebra (4.9). Acting with produces the scalar vertex (3.21) with ,

(4.21)

4.3 Building the hypermultiplet

In this section we will use the bosonic and fermionic vertices obtained above to construct hypermultiplet of the massless states. For simplicity in this section and below we will consider only bosonic components of supermultiplets. As was already mentioned, in the case of type IIA superstring we should consider the states with . We will prove this statement below, in this and the subsequent subsections.

In the NS-NS sector we have one complex (or two real) scalars (3.21),

(4.22)

associated with .

Since for the scalar states the momentum is opposite in the left- and right- moving sectors, for the R-R states we get the product of fermion vertices (4.16) and (4.19), namely,

(4.23)

The vertices above define a complex vector via

(4.24)

However, as is usual for the massless R-R string states, the number of physical degrees of freedom reduces because the fermion vertices (4.16) and (4.19) satisfy the massless Dirac equations which translate into the Bianchi identity for the associated form. For 1-form (vector) we have

(4.25)

which ensures that the complex vector reduces to a complex scalar,

(4.26)

Altogether we have two complex scalars, and , which form the bosonic part of the hypermultiplet. As was already mentioned, deformations of the complex structure of a Calabi-Yau manifold gives a massless hypermultiplet for type IIA theory and massless vector multiplet for type IIB theory. The derivation above shows that our choice corresponds to type IIA string.

We stress again that this massless hypermultiplet is a short BPS representation of supersymmetry algebra in 4D and is characterized by the non-zero baryonic charge .

Let us also note that the four-dimensional space-time charge of the vertex 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 and are zero. Thus we conclude that and have the vanishing charge, as expected for the scalar components of a hypermultiplet.

4.4 What would we get for type IIB superstring?

Our superstring is of type IIA. This is fixed by derivation of our string theory as a description of non-Abelian vortex in 4D SQCD, see [19]. In this subsection we “forget” for a short while about this and consider superstring theory 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 show below that type IIB option corresponds to the choice .

For this choice the massless state with is described as follows. In the NS-NS sector we have one complex scalar,

(4.27)

associated with . In the R-R sector we now obtain

(4.28)

Expanding the complex vertex in the basis of matrices

(4.29)

we get a complex scalar and a complex 2-form which can be expressed in terms of a real 2-form, , where is real and . The Dirac equations for the fermion vertices (4.16) and (4.19) imply that is a constant, while satisfies the Bianchi identity. This ensures that can be constructed in terms of a real vector potential

(4.30)

We see that we get a massless BPS vector multiplet with the bosonic components given by the complex scalar and the gauge potential . This is what we expect from deformation of the complex structure of a Calabi-Yau manifold for type IIB string.

Let us note that charges also match since the charge of in (4.27) is (see (4.13)) while the charge of (4.28) and are zero as expected.

However, if we try to interpret this vector multiplet as a state of the non-Abelian vortex in SQCD we will get an inconsistency. To see this one can observe that our state has non-zero baryonic charge which cannot be associated with a gauge multiplet. This confirms our conclusion that the string theory for our non-Abelian vortex-string is of IIA type.

5 Exited state with

Below we consider the supermultiplet structure of the lowest massive states given by the vertex operators (3.21) and (3.25). In this section we start with the first excited state of the scalar vertex (3.21) with and . The mass of this state is

(5.1)

see (3.24).

5.1 Action of supercharges

The left-moving part of the vertex operator in the (0) picture is given by (4.14). For we obtain

(5.2)

In much the same way as for the state, the supercharge acting on the vertex above gives zero while the supercharge produces the following fermion vertex in the picture :

(5.3)

Note that the momentum along the compact dimension is

for this vertex. It is easy to check that the mass of this fermion is given by (5.1).

In a similar manner, for we use the bosonic vertex (5.2) with and . Action of supercharge gives the following fermion vertex:

(5.4)

with .

Now let us apply the supercharges to the fermion vertices (5.3) and (5.4). Action of on (5.3) does not produce new states, while gives