Critical Non-Abelian Vortex in Four Dimensions and Little String Theory

TL;DR

This paper explores the critical non-Abelian vortex strings in four-dimensional supersymmetric QCD, identifying their spectrum of states and relating them to hadrons and monopole configurations, thus connecting string theory with gauge theory phenomena.

Contribution

It demonstrates the emergence of a critical superstring from non-Abelian vortex strings in 4D ${ m N}=2$ SQCD and identifies massive states as monopole necklaces, extending the understanding of string-gauge dualities.

Findings

Identification of massive closed string states in 4D

Massless state as deformation of the conifold

Interpretation of states as monopole necklaces

Abstract

As was shown recently, non-Abelian vortex strings supported in four-dimensional ${\cal N}=2$ supersymmetric QCD with the U(2) gauge group and $N_f=4$ quark multiplets (flavors) become critical superstrings. In addition to the translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional target space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space and a Calabi-Yau non-compact threefold, namely, the conifold. We study closed string states which emerge in four dimensions (4D) and identify them with hadrons of four-dimensional ${\cal N}=2$ QCD. One massless state was found previously: it emerges as a massless hypermultiplet associated with the deformation of the complex structure of the conifold. In this paper we find a number of massive states. To this end we exploit the approach used in LST, "Little String Theory," namely equivalence between the critical string on the conifold and non-critical $c=1$ string with the Liouville field and a compact scalar at the self-dual radius. The states we find carry "baryonic" charge (its definition differs from standard). We interpret them as "monopole necklaces" formed (at strong coupling) by the closed string with confined monopoles attached.

Equations

$$\mathcal{O}(-1{)}_{{\mathbb{CP}}^1}^{\oplus (N_f-N)},$$

$$\mathrm{SU}(N{)}_{C+F}\times \mathrm{SU}(N_f-N)\times \mathrm{U}(1{)}_B,$$

$$\dim \mathcal{H}=4N(N_f-N)=16.$$

$$T=2\pi \xi .$$

$$S_0=\frac{T}{2}\int d^2\sigma \sqrt{h}{h}^{\alpha \beta }{\partial }_{\alpha }{x}^{\mu }{\partial }_{\beta }{x}_{\mu }+\text{mbox}fermions,$$

$$S_1$$

$${\nabla }_{\alpha }={\partial }_{\alpha }-i{A}_{\alpha },{\tilde{\nabla }}_{\alpha }={\partial }_{\alpha }+i{A}_{\alpha }$$

$$\beta \approx \frac{4\pi }{g^2}.$$

$$S=S_0+S_1.$$

$$\mathrm{SU}(2)\times \mathrm{SU}(2)\times \mathrm{U}(1),$$

$$n:(\text{2},0,0),\rho :(0,\text{2},1).$$

$$\tau \to {\tau }_D=-\frac{1}{\tau },\tau =i\frac{4\pi }{g^2}+\frac{{\theta }_{4D}}{2\pi },$$

$$\beta \to \beta +i\frac{{\theta }_{2D}}{2\pi }.$$

$$\exp (-2\pi \beta )=-h(\tau )[h(\tau )+2],$$

$$h(\tau )={\theta }_1^4/({\theta }_2^4-{\theta }_1^4).$$

$$m^{2}\to\xi\times\left\{\begin{array}[]{ccc}g^{2},&g^{2}\ll 1&\\[2.84526pt] \infty,&g^{2}\to 4\pi&\\[2.84526pt] 16\pi^{2}/g^{2},&g^{2}\gg 1&\\ \end{array},\right.$$

$$|n^P{|}^2-|{\rho }^K{|}^2=\beta .$$

$$w^{PK}=n^P{\rho }^K.$$

$$\sum _{\alpha =1}^4{w}_{\alpha }^2=0,$$

$$w^{PK}={\sigma }_{\alpha }^{PK}{w}_{\alpha },$$

$${\tilde{r}}^2=\sum _{\alpha =1}^4|w_{\alpha }{|}^2$$

$$\sum _{\alpha =1}^4{w}_{\alpha }^2=b,$$

$$S(b)=T\int d^{4}x|{\partial }_{\mu }b{|}^2\log \frac{T^2{L}^4}{|b|},$$

$$w_{\alpha }=\frac{1}{2}\mathrm{Tr}\left[({\bar{\sigma }}_{\alpha }{)}_{KP}{n}^P{\rho }^K\right]$$

$$(1,1,2),$$

$${\mathbb{R}}^4\times {\mathbb{R}}_{\varphi }\times S^1,$$

$$g_s=e^{-\frac{Q}{2}\varphi }.$$

$$T_{--}=-\frac{1}{2}\left[({\partial }_z\varphi {)}^2+Q{\partial }_z^2\varphi +({\partial }_{z}Y{)}^2\right].$$

$$Y\sim Y+2\pi Q.$$

$$⟨\varphi (z),\varphi (0)⟩=-\log z\bar{z},⟨Y(z),Y(0)⟩=-\log z\bar{z}.$$

Full text

FTPI-MINN-17/06, UMN-TH-3622/17

**Critical Non-Abelian Vortex in Four

Dimensions and 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

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

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

Abstract

As was shown recently, non-Abelian vortex strings supported in four-dimensional ${\mathcal{N}}=2\;$supersymmetric QCD with the U(2) gauge group and $N_{f}=4$ quark multiplets (flavors) become critical superstrings. In addition to the translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional target space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space and a Calabi-Yau non-compact threefold, namely, the conifold. We study closed string states which emerge in four dimensions (4D) and identify them with hadrons of four-dimensional ${\mathcal{N}}=2\;$QCD. One massless state was found previously: it emerges as a massless hypermultiplet associated with the deformation of the complex structure of the conifold. In this paper we find a number of massive states. To this end we exploit the approach used in LST, “Little String Theory,” namely equivalence between the critical string on the conifold and non-critical $c=1$ string with the Liouville field and a compact scalar at the self-dual radius. The states we find carry “baryonic” charge (its definition differs from standard). We interpret them as “monopole necklaces” formed (at strong coupling) by the closed string with confined monopoles attached.

1 Introduction

It was recently shown [1] that the non-Abelian solitonic vortex string in a certain 4D Yang-Mills theory becomes critical at strong coupling. This particular 4D theory in which the non-Abelian vortex is critical is ${\mathcal{N}}=2\;$supersymmetric QCD with the U(2) gauge group, four quark flavors and the Fayet-Iliopoulos (FI) [2] parameter $\xi$.

Non-Abelian vortices were first discovered in ${\mathcal{N}}=2\;$supersymmetric QCD with the gauge group U$(N)$ and $N_{f}\geq N$ flavors of quark hypermultiplets [3, 4, 5, 6]. The non-Abelian vortex string is 1/2 BPS saturated and, therefore, has ${\mathcal{N}}=(2,2)\;$supersymmetry on its world sheet. In addition to four translational moduli characteristic of the Abrikosov-Nielsen-Olesen (ANO) strings [7], the non-Abelian string carries orientational moduli, as well as the size moduli if $N_{f}>N$ [3, 4, 5, 6] (see [8, 9, 10, 11] for reviews). Their dynamics are described by effective two-dimensional sigma model on the string world sheet with the target space

[TABLE]

to which we will refer to as to the weighted CP model (WCP$(N,N_{f}-N)$). For $N_{f}=2N$ the model becomes conformal. Moreover for $N=2$ 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.111This set up corresponds to the Virasoro central charge in the WCP$(N,N_{f}-N)$ sector $c_{\rm WCP}=9$. In this case the target space of the world sheet 2D theory on the non-Abelian vortex string is $\mathbb{R}^{4}\times Y_{6}$, where $Y_{6}$ is a non-compact six dimensional Calabi-Yau manifold, the resolved conifold [12, 13].

The main obstacle in describing the solitonic vortex string as a critical string is that the solitonic strings are typically thick. Their transverse size is given by $1/m$, where $m$ is a typical mass scale of the four-dimensional fields forming the string. This leads to the presence of a series of higher derivative corrections to the low-energy sigma model action. The higher derivative corrections run in powers of $\partial/m$. They make the string world sheet “crumpled” [14] and the string does not produce linear Regge trajectories at small spins [1].

The higher derivative corrections on the non-critical string world sheet are needed to be properly accounted for in order to improve the ultra-violet (UV) behavior of the string theory [15]. Without them the low-energy world sheet sigma model does not lead to UV complete string theory. In particular, this means that, say, the ANO string in four dimensions [7] never becomes thin.

On the other hand, the non-Abelian vortex string on the conifold is critical and has a perfectly good UV behavior. This opens the possibility that it can become thin in a certain regime. This cannot happen in weakly coupled bulk theory because at weak coupling $m\sim g\sqrt{T}$ and is always small in the units of $\sqrt{T}$. Here $g$ is the gauge coupling constant of the four-dimensional ${\cal N}=2$ QCD and $T$ is the string tension.

A conjecture was put forward in [1] that at strong coupling in the vicinity of a critical value of $g_{c}^{2}\sim 1$ 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 for all spins. The above conjecture implies 222At $N_{f}=2N$ the beta function of the 4D ${\mathcal{N}}=2\;$QCD is zero, so the gauge coupling $g^{2}$ does not run. Note, however, that conformal invariance in the 4D theory is broken by the FI parameter $\xi$ which does not run either. that $m(g^{2})\to\infty$ at $g^{2}\to g_{c}^{2}$. Moreover, it was argued in [16, 17] that it is natural to expect that the critical point $g_{c}$ where the vortex string becomes thin is the self-dual point $g_{c}^{2}=4\pi$, see [18, 19].

A version of the string-gauge duality for 4D QCD was proposed [1]: 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. This hypothesis was further explored by studying string theory for the critical non-Abelian vortex in [16, 17].

The vortices in the U$(N)$ 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 closed string states with hadrons of the 4D ${\mathcal{N}}=2\;$QCD.

The first step of this program, namely, identifying massless string states was carried out in [16, 17] 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 $Y_{6}$.

Moreover, it was also discussed [16, 17] how the states seen in 4D ${\mathcal{N}}=2\;$QCD at weak coupling are related to what we obtain from the string theory at strong coupling. In particular, the hypermultiplet associated with the deformation of the complex structure of the conifold was interpreted as a monopole-monopole baryon [16, 17].

In this paper we make the next step and find a number of massive states of the closed non-Abelian vortex string which we interpret as hadrons of 4D ${\mathcal{N}}=2\;$QCD. However, to this end we cannot use our formulation of the critical string theory on the conifold. The point is that the coupling constant $1/\beta$ of the world sheet WCP(2,2) is not small. Moreover $\beta$ tends to zero once the 4D coupling $g^{2}$ approaches the self-dual value we are interested in. At $\beta\to 0$ 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 $\beta$ where the supergravity approximation is valid and then extrapolate to strong coupling. In the sigma-model language this procedure corresponds to chiral primary operators. They are protected by ${\mathcal{N}}=(2,2)\;$world-sheet supersymmetry and their masses are not lifted by quantum corrections.

To analyze the massive states we apply a different approach which was used for Little String Theories (LST), see [20] for a review. Namely, we use the equivalence between the critical string on the conifold and non-critical $c=1$ string which contains the Liouville field and a compact scalar at the self-dual radius [21, 22]. The latter theory (in the WZNW formulation) can be analyzed by virtue of algebraic methods. The spectrum can be computed exactly [23, 24, 25, 26, 27].

The states which we find carry a “baryonic” charge – we interpret them as “monopole necklaces” formed (at strong coupling) by the closed string with confined monopoles attached.

It is worth mentioning that the solitonic vortex describes only non-perturbative states. Perturbative states, in particular massless states associated with the Higgs branch in the original 4D Yang-Mills theory are present for all values of the gauge coupling and are not captured by the vortex string dynamics.

The paper is organized as follows. In Sec. 2 we review the world-sheet sigma model emerging on the critical non-Abelian vortex string. In Sec. 3 we briefly review conifold geometry and the massless state associated with deformations of the conifold complex structure. In Sec. 4 we describe the equivalent formulation in terms of the non-critical $c=1$ string, and in Sec. 5 we calculate its spectrum. Section 6 presents an interpretation of states we found in terms of the baryonic “monopole necklaces.” We summarize our conclusions in Sec. 7.

2 Non-Abelian vortex string

2.1 Four-dimensional ${\mathcal{N}}=2\;$QCD

As was already mentioned non-Abelian vortex-strings were first found in 4D ${\mathcal{N}}=2\;$supersymmetric QCD with the gauge group U$(N)$ and $N_{f}\geq N$ flavors of the quark hypermultiplets supplemented by the FI $D$ term $\xi$ [3, 4, 5, 6], see for example [10] for a detailed review of this theory. Here we just mention that at weak coupling $g^{2}\ll 1$ this theory is in the Higgs phase where scalar components of quark multiplets (squarks) develop vacuum expectation values (VEVs). These VEVs breaks the U$(N)$ gauge group Higgsing all gauge bosons, while the global flavor SU$(N_{f})$ is broken down to the so called color-flavor locked group. The resulting global symmetry is

[TABLE]

see [10] for more details. The unbroken global U(1)B 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)B is an unbroken by squark VEVs combination of two U(1) symmetries: the first is a subgroup of the flavor SU$(N_{f})$ and the second is the global U(1) subgroup of U$(N)$ gauge symmetry.

The 4D theory has a Higgs branch formed by massless quarks which are in the bifundamental representation of the global group (2.1) and carry baryonic charge, see [17] for more details. In the case $N=2$, $N_{f}=2N=4$ we will deal with here the dimension of this branch is

[TABLE]

The above Higgs branch is non-compact and is hyper-Kählerian [28, 19], therefore its metric cannot be modified by quantum corrections [19]. In particular, once the Higgs branch is present at weak coupling we can continue it all the way into strong coupling.

2.2 World-sheet sigma model

The presence of color-flavor locked group SU$(N)_{C+F}$ is the reason for the formation of the non-Abelian vortex strings [3, 4, 5, 6]. The most important feature of these vortices is the presence of the so-called orientational zero modes. In the ${\mathcal{N}}=2\;$4D theory these strings are 1/2 BPS-saturated; hence, their tension is determined exactly by the FI parameter,

[TABLE]

Let us briefly review the model emerging on the world sheet of the non-Abelian critical string [1, 16, 17].

The translational moduli fields (they decouple from all other moduli) in the Polyakov formulation [29] are given by the action

[TABLE]

where $\sigma^{\alpha}$ ($\alpha=1,2$) are the world-sheet coordinates, $x^{\mu}$ ($\mu=1,...,4$) describing the $\mathbb{R}^{4}$ part of the string target space and $h={\rm det}\,(h_{\alpha\beta})$, where $h_{\alpha\beta}$ is the world-sheet metric which is understood as an independent variable.

If $N_{f}=N$ 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 ${\mathcal{N}}=(2,2)\;$supersymmetric ${\rm CP}(N-1)$ model.

If one adds extra quark flavors, non-Abelian vortices become semilocal. They acquire size moduli [30]. In particular, for the non-Abelian semilocal vortex at hand, in addition to the orientational zero modes $n^{P}$ ($P=1,2$), there are the so-called size moduli $\rho^{K}$ ($K=1,2$) [30, 3, 6, 31, 32, 33].

The gauged formulation of the effective world sheet theory for the orientational and size moduli is as follows [34]. One introduces the U$(1)$ charges $\pm 1$, namely $+1$ for $n$’s and $-1$ for $\rho$’s,

[TABLE]

where

[TABLE]

and $A_{\alpha}$ is an auxiliary gauge field. The limit $e^{2}\to\infty$ is implied. Equation (2.5) represents the ${\rm WCP}(2,2)$ model.444Both the orientational and the size moduli have logarithmically divergent norms, see e.g. [31]. After an appropriate infrared regularization, logarithmically divergent norms can be absorbed into the definition of relevant two-dimensional fields [31]. In fact, the world-sheet theory on the semilocal non-Abelian string is not exactly the ${\rm WCP}(N,\widetilde{N})$ model [33], there are minor differences. The actual theory is called the $zn$ model. Nevertheless it has the same infrared physics as the model (2.5) [35].

The total number of real bosonic degrees of freedom in (2.5) is six, where we take into account the $D$-term constraint and the fact that one U(1) phase can be gauged away. As was already mentioned, these six internal degrees of freedom are combined with the four translational moduli from (2.4) to form a ten-dimensional space needed for superstring to be critical.

In the semiclassical approximation the coupling constant $\beta$ in (2.5) is related to the 4D SU(2) gauge coupling $g^{2}$ via [10]

[TABLE]

Note that the first (and the only) coefficient is the same for the 4D QCD and the world-sheet beta functions. Both vanish at $N_{f}=2N$. This ensures that our world-sheet theory is conformal.

The total bosonic world-sheet action is

[TABLE]

Since non-Abelian vortex string is 1/2 BPS it preserves ${\mathcal{N}}=(2,2)$ in the world sheet sigma model which is necessary to have ${\mathcal{N}}=2\;$space-time supersymmetry [36, 37]. Moreover, in [17] it is shown that the string theory of the non-Abelian critical vortex is type IIA.

The global symmetry of the world-sheet sigma model (2.5) is

[TABLE]

i.e. exactly the same as the unbroken global group in the 4D theory (2.1) at $N=2$ and $N_{f}=4$. The fields $n$ and $\rho$ transform in the following representations:

[TABLE]

2.3 Thin string regime

As well known [18, 19] the 4D Yang-Mills theory at hand possesses a strong-weak coupling duality, namely,

[TABLE]

where $\theta_{4D}$ is the four-dimensional $\theta$ angle.

The 2D coupling constant $\beta$ can be naturally complexified too if we include the $\theta$ term in the action of the model (2.5),

[TABLE]

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

[TABLE]

where the function $h(\tau)$ is a special modular function of $\tau$ defined in terms of the $\theta$-functions,

[TABLE]

This function enters the Seiberg-Witten curve for our 4D theory [18, 19]. The equation (2.12) generalizes the quasiclassical relation (2.7). Derivation of the relation (2.12) will be presented elsewhere [38, 39].

Note, that the 4D self-dual point $g^{2}=4\pi$ is mapped onto the 2D self-dual point $\beta=0$.

According to the hypothesis formulated in [1], our critical non-Abelian string becomes thin in the strong coupling limit in the self-dual point $\tau_{c}=i$ or $g^{2}_{c}=4\pi$ . This gives

[TABLE]

where the dependence of $m$ at small and large $g^{2}$ follows from the quasiclassical analysis [10] and duality (2.11), respectively.

Thus we expect that the singularity of mass $m$ lies at $\beta=0$. This is the point where the non-Abelian string becomes infinitely thin, higher derivative terms can be neglected and the theory of the non-Abelian string reduces to (2.8). The point $\beta=0$ is a natural choice because at this point we have a regime change in the 2D sigma model per se. This is the point where the resolved conifold defined by the $D$ term in (2.5) develops a conical singularity [13].

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

In this section we briefly review the only 4D massless state associated with the deformation of the conifold complex structure. It was found in [17]. As was already mentioned, all other modes arising from massless 10D graviton have non-normalizable wave functions over the conifold. In particular, 4D graviton is absent [17]. This result matches our expectations since we started with ${\mathcal{N}}=2\;$QCD in the flat four-dimensional space without gravity.

The target space of the sigma model (2.5) is defined by the $D$-term condition

[TABLE]

The U(1) phase is assumed to be gauged away. We can construct the U(1) gauge-invariant “mesonic” variables

[TABLE]

These variables are subject to the constraint ${\rm det}\,w^{PK}=0$, or

[TABLE]

where

[TABLE]

and the $\sigma$ matrices above are $(1,-i\tau^{a})$, $a=1,2,3$. Equation (3.3) defines the conifold $Y_{6}$. It has the Kähler Ricci-flat metric and represents a non-compact Calabi-Yau manifold [12, 34, 13]. It is a cone which can be parametrized by the non-compact radial coordinate

[TABLE]

and five angles, see [12]. Its section at fixed $\widetilde{r}$ is $S_{2}\times S_{3}$.

At $\beta=0$ the conifold develops a conical singularity, so both $S_{2}$ and $S_{3}$ 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 $\beta$ in (3.1). This resolution preserves the Kähler structure and Ricci-flatness of the metric. If we put $\rho^{K}=0$ in (2.5) we get the $CP(1)$ model with the $S_{2}$ target space (with the radius $\sqrt{\beta}$). The resolved conifold has no normalizable zero modes. In particular, the modulus $\beta$ which becomes a scalar field in four dimensions has non-normalizable wave function over the $Y_{6}$ [17].

As explained in [40, 17], non-normalizable 4D modes can be interpreted as (frozen) coupling constants in the 4D theory. The $\beta$ field is the most straightforward example of this, since the 2D coupling $\beta$ is related to the 4D coupling, see Eq. (2.12).

If $\beta=0$ another option exists, namely a deformation of the complex structure [13]. 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. (3.3), namely,

[TABLE]

where $b$ is a complex number. Now the $S_{3}$ can not shrink to zero, its minimal size is determined by $b$.

The modulus $b$ becomes a 4D complex scalar field. The effective action for this field was calculated in [17] using the explicit metric on the deformed conifold [12, 41, 42],

[TABLE]

where $L$ is the size of $\mathbb{R}^{4}$ introduced as an infrared regularization of logarithmically divergent $b$ field norm.555The infrared regularization on the conifold $\widetilde{r}_{\rm max}$ translates into the size $L$ of the 4D space because variables $\rho$ in (3.4) have an interpretation of the vortex string sizes, $\widetilde{r}_{\rm max}\sim TL^{2}$ .

We see that the norm of the $b$ 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 $b$ state, see [17]. This scalar mode is localized on the string in the same sense as the orientational and size zero modes are localized on the vortex-string solution.

The field $b$ being massless can develop a VEV. Thus, we have a new Higgs branch in 4D ${\mathcal{N}}=2\;$QCD which is developed only for the self-dual value of coupling constant $g^{2}=4\pi$.

The logarithmic metric in (3.6) in principle can receive both perturbative and non-perturbative quantum corrections in the sigma model coupling $1/\beta$. However, for ${\mathcal{N}}=2\;$theory the non-renormalization theorem of [19] forbids the dependence of the Higgs branch metric on the 4D coupling constant $g^{2}$. Since the 2D coupling $\beta$ is related to $g^{2}$ we expect that the logarithmic metric in (3.6) will stay intact. We confirm this expectation in the next section.

In [17] the massless state $b$ was interpreted as a baryon of 4D ${\mathcal{N}}=2\;$QCD. Let us explain this. From Eq. (3.5) we see that the complex parameter $b$ (which is promoted to a 4D scalar field) is singlet with respect to both SU(2) factors in (2.9), i.e. the global world-sheet group.666Which is isomorphic to the 4D global group (2.1) at $N=2$, $N_{f}=4$. What about its baryonic charge?

Since

[TABLE]

we see that the $b$ state transforms as

[TABLE]

where we used (2.10) and (3.5). In particular it has the baryon charge $Q_{B}(b)=2$.

To conclude this section let us note that in type IIA superstring the complex scalar associated with deformations of the complex structure of the Calabi-Yau space enters as a 4D ${\mathcal{N}}=2\;$hypermultiplet. Other components of this hypermultiplet can be restored by ${\mathcal{N}}=2\;$supersymmetry. In particular, 4D ${\mathcal{N}}=2\;$hypermultiplet should contain another complex scalar $\tilde{b}$ with baryon charge $Q_{B}(\tilde{b})=-2$. In the stringy description this scalar comes from ten-dimensional three-form, see [43] for a review.

4 Non-critical $c=1$ 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 paper we take a different route and use the equivalent formulation of our theory as a non-critical $c=1$ string theory with the Liouville field and a compact scalar at the self-dual radius [21, 22].

Non-critical $c=1$ string theory is formulated on the target space

[TABLE]

where $\mathbb{R}_{\phi}$ is a real line associated with the Liouville field $\phi$ and the theory has a linear in $\phi$ dilaton, such that string coupling is given by

[TABLE]

Generically the above equivalence is formulated between the critical string on non-compact Calabi-Yau spaces with isolated singularity on the one hand, and non-critical $c=1$ string with the additional Ginzburg-Landau ${\mathcal{N}}=2\;$superconformal system [21] on the other hand. In the conifold case this extra Ginzburg-Landau factor in (4.1) is absent [44].

In [45, 21, 44] it was argued that non-critical string theories with the string coupling exponentially falling off at $\phi\to\infty$ are holographic. The string coupling goes to zero in the bulk of the space-time and non-trivial dynamics (LST) 777The 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 ${\mathcal{N}}=2\;$QCD is defined.

The holography for our non-Abelian vortex string theory is most welcome and expected. We start with ${\mathcal{N}}=2\;$QCD in 4D space and study solitonic vortex string. In our approach 10D space formed by 4D “real” 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 “real” physics should be localized exclusively on the 4D “boundary.” In other words, we expect that LST in our case is nothing other than 4D ${\mathcal{N}}=2\;$supersymmetric QCD at the self-dual value of the gauge coupling $g^{2}=4\pi$ (in the hadronic description).

The linear dilaton in (4.2) implies that the bosonic stress tensor of $c=1$ matter coupled to 2D gravity is given by

[TABLE]

The compact scalar $Y$ represents $c=1$ matter and satisfies the following condition:

[TABLE]

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

[TABLE]

The central charge of the supersymmetrized $c=1$ theory above is

[TABLE]

The criticality condition for the string on (4.1) implies that this central charge should be equal to 9. This gives

[TABLE]

Deformation of the conifold (3.5) translates into adding the Liouville interaction to the world-sheet sigma model [21]

[TABLE]

The conifold singularity at $b=0$ corresponds to the string coupling constant becoming infinitely large at $\phi\to-\infty$, see (4.2). At $b\neq 0$ the Liouville interaction regularize the behavior of the string coupling preventing the string from propagating to the region of large negative $\phi$.

In fact the $c=1$ non-critical string theory can also be described in terms of two-dimensional black hole [46], which is the ${\rm SL}(2,R)/{\rm U}(1)$ coset WZNW theory [26, 22, 47, 21] at level

[TABLE]

This relation implies in the case of the conifold ($Q=\sqrt{2}$) that

[TABLE]

where $k$ is the total level of the Kač-Moody algebra in the supersymmetric version (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; the field $\phi$ 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 $g_{s}\sim 1/b$.

5 Vertex operators and the spectrum

In this section we consider vertex operators for the non-critical string theory on (4.1) and calculate the string spectrum.

5.1 Vertex operators

Vertex operators for the string theory on (4.1) are constructed in [21], see also [26, 44]. Primaries of the $c=1$ part for large positive $\phi$ (where the target space becomes a cylinder $\mathbb{R}_{\phi}\times S^{1}$) take the form

[TABLE]

For the self-dual radius (4.7) (or $k=1$) the parameter $2m$ in Eq. (5.1) is integer. In the left-moving sector $2m$ is the total momentum plus the winding number along the compact dimension $Y$. For the right-moving sector we introduce $2\bar{m}$ which is the winding number minus momentum; then we consider operators with $\bar{m}=\pm m$.

The primary operator (5.1) is related to the wave function over “extra dimensions” as follows:

[TABLE]

The string coupling (4.2) depends on $\phi$. Thus,

[TABLE]

We look for string states with normalizable wave functions over the “extra dimensions” which we will interpret as hadrons of 4D ${\mathcal{N}}=2\;$QCD. The condition for the string states to have normalizable wave functions reduces to [21]

[TABLE]

The scaling dimension of the primary operator (5.1) is

[TABLE]

Unitarity implies that it should be positive,

[TABLE]

The spectrum of the allowed values of $j$ and $m$ in (5.1) was determined exactly by using the Kac-Moody algebra for the coset ${\rm SL}(2,R)/{\rm U}(1)$ in [23, 24, 25, 27, 26], see [49] for a review. Both discrete and continuous representations were found. Parameters $j$ and $m$ determine the global quadratic Casimir operator and the projection of the spin on the 3-axis,

[TABLE]

where $J^{a}$ $(a=1,2,3)$ are global ${\rm SL}(2,R)$ currents. We have

(i) Discrete representations with

[TABLE]

(ii) Principal continuous representations with

[TABLE]

where $s$ is a real parameter and

(iii) Exceptional continuous representations with

[TABLE]

We see that discrete representations include the normalizable states localized near the tip of the cigar, while the continuous representations contain non-normalizable states,888We will discuss the case $j=-\frac{1}{2}$ which is on the borderline between normalizable and non-normalizable states in the next subsection. see (5.3). This nicely matches our qualitative expectations.

Discrete representations contain states with negative norm. To exclude the ghost states a restriction for spin $j$ is imposed [23, 24, 25, 27, 49]

[TABLE]

Thus, for our value $k=1$ we are left with only two allowed values of $j$,

[TABLE]

and

[TABLE]

Below in this section we will first consider normalizable string states from discrete representations and finally, in Subsec. 5.5, discuss physical interpretation of the continuous representations.

5.2 Massless baryon

Our first task now is to rederive the massless baryon $b$ associated with deformations of the conifold complex structure (see [17] and Sec. 3) within the framework of the non-critical Liouville string theory described above. To this end we consider vertex operators for 4D scalars. The 4D scalar vertices $V^{S}$ in the $(-1,-1)$ picture have the form [21]

[TABLE]

where $\varphi$ represents bosonized ghosts, and $p_{\mu}$ is the 4D momentum of the string state. These states are the lowest components of ${\mathcal{N}}=2\;$multiplets in four dimensions. Also, the GSO projection restricts the integer $2m$ for the operator in (5.13) to be odd [48, 21],

[TABLE]

The condition for the state (5.13) to be physical is

[TABLE]

where we used (5.4) and (5.14). This determines the masses of the 4D scalars,

[TABLE]

where the Minkowski 4D metric with the diagonal entries $(-1,1,1,1)$ is used.

Consider the states which are on the borderline between normalizable and non-normalizable, namely, the states with

[TABLE]

For $l=0$ ($m=\frac{1}{2}$) and $l=-1$ ($m=-\frac{1}{2}$) , Eq. (5.16) gives the lightest states with

[TABLE]

This is our massless baryon $b$ associated with deformations of the complex structure of the conifold (plus anti-baryon $\tilde{b}$). To confirm this let us show that it has a logarithmically normalizable wave function in terms of the conifold radial coordinate $\widetilde{r}$, see (3.6).

For $j=-\frac{1}{2}$ all states (5.13) have constant wave functions with respect to the Liouville coordinate $\phi$. Thus, the norm of these states is $(\phi_{\rm max}-\phi_{\rm min})$. To relate $\phi$ to $\widetilde{r}$ we note that $\phi_{\rm min}$ is determined by the Liouville interaction term (4.8) which becomes of the order of unity at this point. This gives

[TABLE]

where we used that $\widetilde{r}_{\rm min}^{\,\,2}=|b|$. In particular, this gives $\log{\widetilde{r}^{\,\,2}_{\rm max}/|b|}$ for the norm of the massless state with $l=0$, as expected (see (3.6)).

5.3 Massive 4D scalars

Now consider the states (5.13) with arbitrary values of $l$ in (5.14), still assuming $j=-\frac{1}{2}$. From (5.16) we obtain their masses

[TABLE]

All these states are logarithmically normalizable with respect to the conifold radial coordinate.

What are their quantum numbers with respect to the 4D global group (2.1)? They are all invariant with respect to SU(2) factors. To determine their baryon charge note that the U(1)B transformation of $b$ in the Liouville interaction (4.8) is compensated by a shift of $Y$. Therefore, $m$ in (5.1) is proportional to the baryon charge. Normalization is fixed by the massless baryon $b$ which has $Q_{B}(b)=2$ at $m=\frac{1}{2},\,\,l=0$. This implies

[TABLE]

We see that the momentum $m$ in the compact $Y$ direction is in fact the baryon charge of a string state. In particular, 4D scalar states (5.20) are all baryons for positive $l$ and anti-baryons for negative $l$ with

[TABLE]

The masses of 4D scalars as a function of the baryonic charge are shown in Fig. 1.

To conclude this subsection let us note that the second allowed value of $j$, $j=-1$ in (5.12), is excluded by the GSO projection which selects only half-integer values of $m$ for states (5.13), see (5.14).

Note also that the 4D scalar states found above are the lowest components of ${\mathcal{N}}=2\;$multiplets. Other components can be restored by virtue of 4D ${\mathcal{N}}=2\;$supersymmetry.

5.4 Spin-2 states

At the next level we consider 4D spin-2 states coming from space-time “gravitons.” The corresponding vertex operators are given by

[TABLE]

where $\psi^{\mu}_{L,R}$ are the world-sheet superpartners to 4D coordinates $x^{\mu}$; moreover, $\xi_{\mu\nu}$ is the polarization tensor.

The condition for these states to be physical takes the form

[TABLE]

The GSO projection selects now $2m$ to be even, $m=l$, $\,|l|=0,1,2,...$ [21], thus we are left with only one allowed value of $j$, $j=-1$ in (5.12). Moreover, the value $m=l=0$ is excluded. This leads to the following expression for the masses of spin-2 states:

[TABLE]

We see that all spin-2 states are massive. This confirms the result in [17] that no massless 4D graviton appears in our theory. It also matches the fact that our “boundary” theory, 4D ${\mathcal{N}}=2\;$QCD, is defined in flat space without gravity.

All states with masses (5.24) are baryons for $l>0$ and anti-baryons for $l<0$, with the baryon charge $Q_{B}=4m=4l$. The masses of 4D spin-2 states as a function of the baryonic charge are shown in Fig. 1.

We expect that all 4D states with a given baryon charge considered in this and previous subsections are the lowest states of the Regge trajectories linear in 4D spin.

5.5 Non-normalizable states

In [17] the continuum spectrum of non-normalizable states was interpreted as unstable string states. Since the Liouville coordinate $\phi$ is related to the radial coordinate $\widetilde{r}$ on the conifold (see (5.19)) the modes with $j>-\frac{1}{2}$ are power non-normalizable on the conifold. The conifold radial coordinate $\widetilde{r}$ has the physical interpretation of a distance from the string axis in 4D space, see (3.4) and (3.2). Therefore, the wave functions of the non-normalizable states are saturated at large distances from the vortex-string axis in 4D.

These states are not localized on the string. The infinite norm of these states should be interpreted as an instability. Namely, these states decay into massless bifundamental quarks inherent to the Higgs branch of our four-dimensional ${\mathcal{N}}=2\;$QCD. This instability is present already at the perturbative level, see Sec. 2 and [17].

Our vortex string has a conceptual difference compared to fundamental string. In compactifications of fundamental string all states present in four dimensions are string states. The string theory for the vortex strings of [1, 17] is different. The string states describe only non-perturbative physics at strong coupling, such as mesons and baryons. The perturbative massless moduli states seen at week coupling are not described by this theory. In particular, the Higgs branch (and associated massless bifundamental quarks) found at weak coupling can be continued to the strong coupling; they persist there. It can intersect other branches, but cannot disappear (for quarks with the vanishing mass terms) [19].

One class of non-normalizable unstable modes is given by the exceptional continuous representation (5.9). For $|m|=1,2,...$ the continuous spectra parametrized by $j$ in (5.9) start from the thresholds given by masses (5.24).

Another class of unstable string states corresponds to the principal representation given by complex values of $j$

[TABLE]

see (5.8). These states have continuous spectra parametrized by $s$. The parameter $s$ has a clear-cut interpretation of a momentum along the Liouville direction. Therefore we interpret these states as decaying modes of the string states interacting with the perturbative bifundamental quarks rather than the hadronic states of ${\mathcal{N}}=2\;$QCD.

Much in the same way as for the exceptional representation the principal contunuous spectra for half-integer $m$ start from thresholds given by masses (5.20). Using this picture we are led to conclude that the continuous spectra contain multi-particle states formed by a given baryon and a number of emitted bifundamental quarks with zero total baryonic charge. This issue needs a future clarification.

6 Physical interpretation of string states

In this section we reveal a physical interpretation of all baryonic states found in the previous section as monopole “necklaces.”

Consider first the weak coupling domain $g^{2}\ll 1$ in four-dimensional ${\mathcal{N}}=2\;$QCD. It is in the Higgs phase: $N$ squarks condense. Therefore, non-Abelian vortex strings confine monopoles. However, the monopoles cannot be attached to the string endpoints. In fact, in the U$(N)$ theories confined monopoles are junctions of two distinct elementary non-Abelian strings [50, 5, 6] (see [10] for a review). As a result, in four-dimensional ${\mathcal{N}}=2\;$QCD we have monopole-antimonopole mesons in which the monopole and antimonopole are connected by two confining strings. In addition, in the U$(N)$ gauge theory we can have baryons appearing as a closed “necklace” configurations of $N\times$(integer) monopoles [10]. For the U(2) gauge group the lightest baryon presented by such a “necklace” configuration consists of two monopoles, see Fig. 2.

Moreover, the monopoles acquire quantum numbers with respect to the global symmetry group (2.1). To see that this is the case note that in the world-sheet theory on the vortex string the confined monopole is seen as a kink interpolating between two distinct vacua (i.e. distinct elementary non-Abelian strings) in the corresponding 2D sigma model [50, 5, 6]. At the same time, we know that the sigma model kinks at strong coupling are described by the $n^{P}$ and $\rho^{K}$ fields [51, 52] (for the sigma model described by (2.5) it was shown in [53]) and therefore transform in the fundamental representations 999Strictly speaking, to make both bulk monopoles and world-sheet kinks well defined as localized objects one should introduce an infrared regularization, say, a small quark mass term. When we take the limit of the zero quark masses, the kinks become massless and smeared all over the closed string. However their global quantum numbers stay intact. of two non-Abelian factors in (2.1).

As a result, the monopole baryons in our case can be singlets, triplets, bifundamentals or form higher representations of both SU(2) global groups in (2.1). With respect to the baryonic U(1)B symmetry in (2.1) the monopole baryons can have charges

[TABLE]

see (2.10). In particular, non-zero baryonic charge is associated with the $\rho$ kinks. In the U(2) gauge theory the monopole “necklace” can be formed by even number of monopoles.

All these non-perturbative stringy states are heavy at weak coupling, with mass of the order of $\sqrt{\xi}$, and therefore can decay into screened quarks which are lighter and, eventually, into massless bifundamental screened quarks.

Now, we pass to the self-dual point $\beta=0$ in the strong coupling region. As was already discussed all string states found in Sec. 5 have non-zero baryon charge $Q_{B}=4m$, see Eq. (5.21). The lightest state (the massless $b$ state) has $Q_{B}=2$. It can be formed by minimum two monopoles, see Fig. 2. The spin-2 massive state (5.22) with $m=1$ can be formed by the monopole “necklace” with minimum four monopoles.

All stringy monopole “necklace” baryons found in Sec. 5 are singlets with respect to two SU(2) factors in (2.1). They are metastable and can decay into pairs of massless bifundamental quarks in the singlet channel with the same baryon charge. The metastability of stringy baryons on the string side is reflected in the logarithmic divergence of their norm and the presence of continuous spectra. Detailed studies of the non-perturbative Higgs branch formed by VEV of massless $b$ and interactions of stringy baryons with massless bifundamental quarks are left for future work.

7 Conclusions

Previously we observed that non-Abelian vortex strings supported in four-dimensional ${\mathcal{N}}=2\;$supersymmetric QCD with the U(2) gauge group and $N_{f}=4$ flavors of quark hypermultiplets can represent critical superstrings in ten-dimensional target space $\mathbb{R}^{4}\times Y_{6}$ where $Y_{6}$ is a noncompact Calabi-Yau manifold. This can be called “reverse holography.” Indeed, we start from a well-defined four-dimensional Yang-Mills theory and, analyzing the vortex strings it supports, add six extra dimensions – the moduli of the string world-sheet theory – which relates our construction (at the critical value of the coupling constant $g^{2}=4\pi$ corresponding to $\beta=0$) to a critical string theory on a six-dimensional conifold.

In this paper the mass spectrum of the string states is determined using equivalent formulation in terms of $c=1$ non-critical string theory with the Liouville field. The string states per se are identified with the hadronic states in the four-dimensional theory. Since the extra-dimensional space is not compact the above identification becomes possible because our string theory is holographic: non-trivial physics is “projected” to 4D “boundary”. This behavior is typical for LSTs. The reason for holography is that the string coupling constant exponentially falls-off at large values of the non-compact Liouville coordinate (see (4.2)), and the bulk physics becomes trivial and decouples.

Of course, the holography of our string theory is expected since our starting point was 4D ${\mathcal{N}}=2\;$QCD. Holography ensures the presence of normalizable string states localized in 4D space which we identified as hadrons of ${\mathcal{N}}=2\;$QCD. Also, we qualitatively interpret non-normalizable states as decay modes of the hadronic states.

The Higgs branch existing in our four-dimensional ${\mathcal{N}}=2\;$QCD leads to strictly massless quark hypermultiplets, which are evident at weak coupling and survive the transition to strong coupling. This implies a peculiar behavior in the infrared. In particular, the masses of the string states derived in Sec. 5.3 and Sec. 5.4 are in fact the endpoints of the branch cuts. Note, however, that exactly the same situation would take place in QCD with massless quarks giving rise to massless pions. Say, every lowest-lying state with the given baryon number would represent the beginning of a branch cut. The pion mass could be lifted by an arbitrarily small perturbation. Disentangling infrared effects of the Higgs branch from physics of the critical string under consideration will be the subject of a subsequent work.

Acknowledgments

The authors are grateful to Peter Koroteev and Xi Yin for very useful and stimulating discussions. This work is supported in part by DOE grant DE-SC0011842. The work of A.Y. was supported by William I. Fine Theoretical Physics Institute at the University of Minnesota, and by Russian State Grant for Scientific Schools RSGSS-657512010.2. The work of A.Y. was supported by Russian Scientific Foundation under Grant No. 14-22-00281.