From 3d dualities to hadron physics

When one of the space-time dimension is compactified on $S^1$, the QCD exhibits the chiral phase transition at some critical radius. When we further turn on a background $\theta$ term which depends on the $S^1$ compactified coordinate, a topological ordered phase appears at low energy via the winding of $\theta$. We discuss what kind of theories can describe the physics near the critical point by requiring the matching of topological field theories in the infrared. As one of the possibilities, we propose a scenario where the $\rho$ and $\omega$ mesons form a $U(N_f)$ gauge theory near the critical point. In the phase where the chiral symmetry is restored, they become the dual gauge boson of the gluon related by the level-rank duality between the three dimensional gauge theories, $SU(N)_{N_f}$ and $U(N_f)_{-N}$.


Introduction
The low energy limit of QCD is described by pions whose properties and interactions have information of the global symmetry of QCD. The lowest dimensional interaction terms can be determined once we know the coset space of which the pions are the coordinate. Also, the inconsistency in gauging a part of the global symmetry, i.e., the 't Hooft anomaly, is encoded in the Wess-Zumino (WZ) term in the low energy effective Lagrangian [1,2].
The long distance behavior is also important near the critical point of a phase transition.
If the phase transition is smooth enough, one can consider an effective theory of an order parameter which obtains a vacuum expectation value (VEV) in the broken phase. The global structure of the theory such as anomaly, should also be kept in the effective theory for consistency. Moreover, the 't Hooft anomaly results in a matching conditions which constrain the realization of the vacuum structure and the infrared degrees of freedom [3][4][5]. In the case of the finite temperature QCD above the QCD scale, such an anomaly matching is usually trivially satisfied. The finite temperature system can be regarded as an S 1 compactified QCD, and the effective theory is, therefore, a three dimensional theory that has no chiral anomaly.
The study of phase transitions in the three dimensional gauge theory has a long history [6][7][8][9][10][11][12][13][14][15][16] with some important recent developments [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Even though there is no chiral anomaly, there are topological orders in the low energy effective theories. For example, SU (N ) gauge theory can have the Chern-Simons (CS) term with an integer level, k. The CS term dominates the infrared physics, and it reduces to the CS theory, denoted as SU (N ) k , which has a gap, but the Wilson lines have non-trivial values depending on their topology. This non-trivial topological behavior should be matched when we discuss the effective description near the phase transition. Based on this discussion, the dualities between CS theories coupled to fermions and those coupled to bosons have been proposed and checked. The precise forms of the dualities are listed in Ref. [18]. Based on these dualities, the phase diagrams of the three dimensional QCD (QCD 3 ) have been discussed [24]. In particular, it has been conjectured that the SU (N ) k theory with N f (> 2|k|) fermions undergoes the symmetry breaking U (N f ) → U (N f /2 + k) × U (N f /2 − k) when the fermions masses are smaller than some critical value. There is a phase transition between the symmetry broken and unbroken phases as the fermion masses are varied. If the transition is of the second order, near the critical point, there is a dual description of the theory by the U (N f /2+k) −N or U (N f /2−k) N theory coupled to N f scalar fields whose Higgs phenomenon describes the phase transition.
In the unbroken phase in QCD 3 the CS theory describes the low energy physics. The fermionic theory flows to the SU (N ) ±N f /2+k theory while it is the U (N f /2 ± k) −N theory in the bosonic theory. These two theories are related by the level-rank duality and are known to give the same physics. It is quite not trivial that the matching of the low energy limit is realized in this way.
In this paper, motivated by the symmetry breaking and its dual description in QCD 3 , we discuss the low energy limits of the S 1 compactified QCD in four space-time dimensions [24, with the hope that the three dimensional duality may give us some hints of the four dimensional physics. Indeed for abelian gauge theories it has been pointed out that the three dimensional duality is lifted up to the S-duality in the four dimensional theory [65].
We consider a background θ term which depends on the coordinate of the S 1 direction. In particular, the function θ can have windings along the S 1 direction [66,67], which determines the CS level in the three dimensional effective theory. As the most interesting example, one can take the background with the winding number, N f , which is the number of quark flavors in four dimensions. For a small radius, one finds that the low energy theory is SU (N ) N f where the vacuum is gapped. At a large radius, the theory is better described by the chiral Lagrangian for pions with the WZ terms. Since the low energy limits are different, there must be a phase transition at some critical radius. The background θ induces a winding of the η meson (η in the three-flavor language), which gives rise to the non-trivial WZ term in the three dimensional theory. The two limits of the theory can be interpolated by the Higgs mechanism of the U (N f ) −N theory coupled to 2N f flavors of scalar fields. Here, again, in the unbroken phase, the theory is related by the level-rank duality. If this picture is correct, the natural candidate for this dual U (N f ) gauge bosons are the ρ and the ω mesons (in the spirit of Ref. [68]), which we know, phenomenologically, to be successfully described in terms of gauge fields [69][70][71][72][73][74][75]. See also [76][77][78] for the interpretation of the Seiberg duality as the gauge theory of vector mesons. The "gauge bosons" are massive anyway by the CS term even in the unbroken phase in the three dimensional effective theory. Therefore, they are not quite effective degrees of freedom, but their existence is important for having a non-trivial topological order required from the matching of the low energy physics.
We discuss the similarities between the QCD 3 for small k and QCD 4 , and propose an exotic possibility that the ρ and ω mesons continuously becoming a dual gauge boson of gluons related by the level-rank duality. This conjectured picture is at least consistent when the winding of θ is less than N f , especially, when it is zero at which case the system can be regarded as the finite temperature QCD.
The paper is organized as follows: In section 2 we discuss the pattern of the flavor symmetry breaking in QCD 3 with 2N f fermions and its breaking to the chiral symmetry group associated with QCD 4 . We identify the order parameter which leaves the correct Nambu-Goldstone bosons massless to match the one of QCD 4 . The corresponding effective low energy theory is written down with an emphasis on the CS terms and the related WZ terms, also in the presence of external gauge fields. In particular, we identify the term associated with baryon number. These terms are related to the flavor anomalies in QCD 4 .
They were recently discussed also by Komargodski in [28]. In section 3 we put QCD 4 on M 3 × S 1 including a θ angle which winds along S 1 and background gauge fields which depend on the S 1 coordinate. We explore the theory for small radius (Λ 4 R 1) and large radius (Λ 4 R 1) and argue for the existence of a phase transition at some critical R * . In section 4 we speculate/conjecture on the possible behavior of the hadronic vector mesons (ρ, ω, ...) near the critical point and the nature of the theory at the critical point. In particular, we put forward a scenario in which the hadronic vector mesons become gauge bosons and give rise to an U (N f ) gauge theory at the critical point. In section 5 we present holographic QCD-like models represented by quiver diagrams which capture this conjectured scenario of vector mesons as gauge bosons. In section 6 we comment briefly on the implications of our study on finite temperature QCD and possible scenarios for the nature of the phase transition at T * = 1/R * . Lattice simulations may decide between these various scenarios. Section 7 is devoted to discussion. There are three appendices. In Appendix A we write down the Lagrangian associated with the quiver diagram in section 5. Appendix B addresses the issue of the integration over S 1 when a winding θ term is present. In Appendix C we discuss the issue of the Baryon number and the (winding) configuration of η (η for N f > 2) by considering the WZ term and the associated anomaly of chiral U (1) A current, U (1) EM electromagnetic current and the U (1) baryon number. In particular we can consider the situation that η winding happens on a finite sheet and localized on S 1 . The sheet configuration is just the Hall droplet studied in [28]. Following [28] we can identify the Baryonic configuration which resides on the boundary of the finite region. It is "amusing" to note that one can also identify a non-local configuration which corresponds to the quark with its Wilson line going into the bulk. In this sense the quark appears as a "soliton" in the hadronic effective theory.

QCD 3 and chiral symmetry breaking
Based on the studies of the three dimensional dualities, it has been conjectured that the low energy theory of three dimensional SU (N ) 0 QCD with 2N f fermions is described by a non-linear sigma model with the target space, for small enough fermion masses [24,79]. There is an upper bound on N f although the precise location is unknown [80]. It is noted that an appropriate WZ term should be added in the Lagrangian. The same low energy theory is obtained by a linear sigma model with U (N f ) N or U (N f ) −N gauge theory coupled to 2N f scalar fields. Beyond the critical value of the fermion mass, both the fermionic and bosonic theories flow to topological field theories related by the We would like to discuss the relation between this symmetry breaking phenomena and the chiral symmetry breaking in QCD 4 . In QCD 4 with N f Dirac fermions (corresponding to 2N f fermions in QCD 3 ), the low energy theory is a non-linear sigma model with This coset space is a subspace of (1). One can reduce the space (1) to (2) by adding an explicit breaking terms in the Lagrangian.
By denoting ψ andψ as N f + N f flavors of the QCD 3 , one can, for example, introduce the explicit breaking term by coupling a massive adjoint scalar field a 3 to QCD 3 ,  [81,82]. As we will see later, this three dimensional model can be obtained as the low energy effective theory of the S 1 compactified QCD 4 with N f Dirac fermions, Ψ. The model with massless fermions can be realized at a meta-stable vacuum when N f < N . The periodic boundary condition is taken for Ψ. In this language, the former and the latter VEVs correspond to Ψ γ 3 Ψ and Ψ Ψ , respectively, where x 3 is the S 1 direction. When we consider modifying the model by adding ∂ 3 Ψ γ 3 Ψ to the Lagrangian with a real function , the vacuum energy density at the meta-stable vacuum is given by where E 0 is the vacuum energy density for = 0, and the VEV is the one at = 0. If Ψ γ 3 Ψ is non-vanishing, the vacuum energy density is locally shifted by O( ). On the other hand, one can eliminate the ∂ 3 Ψ γ 3 Ψ term in the Lagrangian by an appropriate redefinition of the field, The boundary condition of Ψ is maintained by taking a periodic function up to an 2π shift. In this basis, there is no shift of the vacuum energy density. Therefore, Ψ γ 3 Ψ = 0 is inconsistent, and thus the latter VEV, ψψ , should be chosen.
For ψψ = 0, the U (N f ) × U (N f ) chiral symmetry is broken to U (N f ), leaving a part of the Nambu-Goldstone modes massless and matches to the QCD 4 up to an anomalous axial U (1) A that can also be explicitly broken. Therefore, it is possible that the chiral symmetry breaking in QCD 4 has something to do with the phase transition in QCD 3 where there is a dual description by the Higgs mechanism of U (N f ) ±N gauge theory. In QCD 4 , it is wellknown that the physics of the vector mesons, ρ and ω, is nicely described by the color-flavor locked phase of U (N f ) gauge theory. The structure of the dual bosonic theory in three dimension is indeed of this type as we see below.
Let us look at the low energy effective theory. We first discuss the theory without the explicit breaking terms in Eq. (3). In this case, the non-zero VEV ψψ breaks U (2N f ) to The pions in this symmetry breaking have the WZ term as discussed in Ref. [24]. The WZ term can be obtained from the CS term in the bosonic U (N f ) ±N theory. The uneaten 2N 2 f Nambu-Goldstone fields in Eq. (1) are introduced as a 2N f × 2N f matrix: where T a are the generators of U (N f ) group. The Higgsed U (N f ) gauge field, b µ couples to ξ as In the low-energy limit, the gauge field b µ can be integrated out, since this gauge field is massive by the Higgs mechanism. The equation of motion for b µ gives where i, j = 1, · · · , N f run the first half of the 2N f indices. Substituting this into the CS term, one obtains the WZ terms.
We now introduce the explicit breaking term in Eq. (3). A half of pions obtain masses when we include the interaction to break the U (2N f ) symmetry to the chiral symmetry. For example, one can introduce a spurion field, and write down a U (2N f ) breaking term, This term gives a mass toπ while leaving π massless.
In the low-energy limit whereπ a are decoupled, one can setπ a = 0, and the WZ term among pions vanishes. However, one can trace the existence of the WZ term by turning on external gauge fields. Let us introduce, the background gauge fields for the unbroken U (N f ) global symmetry A µ , as the one which couples to the U (N f ) vector current,ψγ µ T a ψ + ψγ µ T aψ . The U (1) part is the baryon number. The equation of motion for b now gives, b = A + · · · , and thus we have In particular, we have a term where B is the baryon number normalized such that the quarks have the charge 1/N . The

QCD 4 on a circle
We discuss the four-dimensional QCD compactified on S 1 and look for a relation to the phase transition in QCD 3 . We start with the action of SU (N ) gauge theory coupled with massless The periodic boundary conditions are imposed on gauge fields. The Lorentz indices, M , N , P , Q, run from 0 to 3, where x 3 is the S 1 direction. P L,R are projection operators of chirality, . The boundary condition of the quarks are where 0 ≤ ν < 2π.
The S 1 compactification requires that θ and α are also valued on S 1 that allows where k and m i L,R are integers. The integral on S 1 should be properly defined as in Ref. [66] so that the partition function does not depend on the coordinate system on S 1 . (See Appendix B for the definition.) There is a redundancy due to the anomalous chiral symmetry, where with p integers to maintain the boundary conditions.
In the following discussion, we are particularly interested in the theory with since the vacuum structure looks the same as the three dimensional case discussed in the previous section. By the chiral rotations, this theory is equivalent to, for example, The physics should depend on the combination: We discuss the phase structure of the theory as a function of the radius R. The dynamical scale Λ 4 is defined as where Λ is an arbitrarily high scale. For Λ 4 R 1, the low energy dynamics is described by hadrons on an S 1 compactified background. In the other limit, Λ 4 R 1, the low energy description is a three dimensional gauge theory on M 3 via the Kaluza-Klein decomposition.
The gauge coupling constant in the three dimensional effective theory is given by and the dynamical scale in the three dimensional theory is defined as One can see that for a small enough Λ 4 R, there is an energy gap between the dynamical scale Λ 3 and the mass of the first Kaluza-Klein mode, 1/R.

Small radius, Λ 4 R 1
Let us use the basis with k = 0. The Kaluza-Klein expansion of the fermions can be done as with The three dimensional effective action is given by We have dropped the massive Kaluza-Klein modes of gauge fields. At this stage, one can see that the effects of m L and m R are to shift the Kaluza-Klein spectrum of the fermions by 1/R, and thus can be absorbed by the redefinition of n. However, in three dimensions, the signs of the fermion masses are important, and thus one cannot simply ignore m L,R .
The potential V (a 3 ) is generated at the one-loop level [47,83] and it depends on the boundary condition, ν. For example, by putting an ansatz, it is given by The shapes of the potential of ξ at ν = 0 (ν = π) is shown in the left (right) panel of Fig. 1 for N = 3 and N f = 2. In general, for N f < N , there are N minima at ξ = ±2pπ/N , (p = 0, · · · , [N/2]). (For even N , ξ = π and ξ = −π are equivalent.) At ν = 0, ξ = 0 is a local minimum, and the true minimum is at ξ = ±(N − 1)π/N for odd N and ξ = π for even N . For ν = π, the ξ = 0 point is the true vacuum.
The potential has a symmetry ν → ν + 2πn/N , n ∈ Z, together with ξ → ξ − 2πn/N that is the reflection of the fact that the action of Z N elements in U (1) B is the same as that of the gauge group SU (N ) [84]. For even N , ν = 0 and ξ = π is equivalent to ν = π and ξ = 0. For odd N , they are not equivalent. The ν = π point is equivalent to ν = π/N by an appropriate shift of ξ. There is a first order transition in between ν = 0 and ν = π/N .
In all the N minima, the SU (N ) gauge symmetry is unbroken as the Wilson loop along the x 3 direction is a phase times the unit matrix. The fermion masses for ψ n andψ n are, respectively, By following the global minimum of the potential, in the entire region of ν, the fermion masses are non-vanishing. Therefore, the low energy limit of the 4d QCD on pure gauge theories on M 3 for a small radius. There is a mass gap, but the low energy limit The CS level is consistent with the result in the basis of k = N f . The θ term can be expressed as Again, it is important that the integral on S 1 is properly defined [66]. Since dθ is single valued, one can naively use the right-hand side of the integral over S 1 , which reduces to the CS term with the level k for the lowest KK mode of the gauge field.
In summary, the low energy limit of QCD 4 on M 3 × S 1 with Eq. (21) for a small radius, Figure 1: The shapes of the potential at ν = 0 (left) and ν = π (right).

Large radius, Λ 4 R 1
One can also analyze the low energy limit of QCD 4 on M 3 × S 1 for a large radius as we know that the low energy theory is described by pions. We discuss the effect ofθ(x 3 ) in the low energy effective theory. In order to see theθ(x 3 ) dependence of the theory, one needs to introduce η meson (η in the three-flavor language) in addition to the massless pions.
The effective theory is given in terms of the N f ×N f unitary matrix U = e iπ a T a +iη . While we implicitly assume here that η behaves as the Nambu-Goldstone boson, we do not assume that we are in the large N limit. We treat η as a heavy meson as we see below. The field U transforms as it transforms as U → e 2iβ U . The effective action is given by The effect of the boundary condition, ν, can be taken into account by introducing a background gauge field for the baryon number, (ν/2πR)Ψγ 3 Ψ. The effect of this term appears in the WZ terms. The parameter ν couples to the topological current, i.e., the Skyrmions.
The equation of motion for η is For the periodicity ofθ, the potential for η has N f domains where the η −θ/N f is minimized at 2nπ/N f with n integers. Transition between two other domains requires a treatment beyond this effective theory. Once we restrict ourselves that the shape of the functionθ is not very rapid so that the effective theory can be used, η should stay in one of the domains to minimize the energy, which means η develops a winding under Eq. (21): This is consistent with the S 1 compactification. One should not be confused with the η winding as the jump of domains. We are treating η as a heavy field and we are working within the effective theory. Also, the argument of the winding of η depends on the basis. We can eliminateθ by the field redefinition of η. In that basis, η does not get winding while we obtain the same effects at low energy as we see below.
The three dimensional low energy effective theory is the non-linear sigma model with the coset in Eq. (2), but there are effects from the η winding. By turning on the external gauge field, A, which couple to the vector current as we discussed in the previous section, a part of the WZ term, to be η. In that bases, there is no winding of η, but the same WZ term appears by shifting η in Eq. (39). One can confirm the consistency of the appearance of the WZ term by comparing theθ → AA amplitude to that in QCD 4 .
In summary, we find that the low energy limit of QCD 4 with Eq. (21) is a topological field theory, SU (N ) N f , for small Λ 4 R and a non-linear sigma model with the WZ term for a large radius, Λ 4 R 1. There must be a phase transition between these two extreme regions.
It is interesting to find that the two limits are the same as the conjectured limit of the three dimensional SU (N ) 0 theory with 2N f fermions with large and small fermion masses. It is therefore possible to anticipate that the phase transition between a large and a small radius is For |k| < N f instead of Eq. (21), the η winding should be accompanied with the winding of π's. In order to satisfy the boundary condition, U (x 3 + 2πR) = U (x 3 ), the winding that can be realized as a configuration of π a as the phase factor is an element of SU (N f ). Although the description in terms of hadrons gets ineffective in this energy region, the low energy limit of the theory stays the same.
Since U (N f ) gauge group is spontaneously broken, there are vortex configurations in three dimensions made of ρ and ω, which carry magnetic and electric charges of [85]. The electric charge is a consequence of the CS term. In the color-flavor locked phase, the electric charge is identified as the baryon number. The vortex with the unit magnetic charge has B = 1. We will discuss how this vortex configuration extends to the S 1 direction in the next section.

Holographic model
The Of course, we are not aware if the phase transition is smooth enough that such an effective description exists. We here assume that is the case and look for a dual theory. In this sense, this is a construction of the Nambu-Jona-Lasino model or the Ginzburg-Landau model while taking into account the consistency with topology. Fork = 0 it was a trivial task since the low energy limit of symmetric phase is trivially gapped. But fork = 0, we need some gauge theory to survive to match the topological field theory.
where the right-hand side is external gauge fields which couple to chiral currents. The pions  They are all massive, but necessary to reproduce the CS term in the five dimensional theory.
Since the fermions are always massive even in the chiral symmetric phase as we see later, we can think of fermions as auxiliary degrees of freedom. We also gauge the Z N subgroup of The fermions in the upper and lower wings couple to A L and A R , respectively. The mass terms of fermions except for the one with the link Φ do not break the global symmetry, The combination of A L + A R couples to the conserved vector current.
By integrating out the massive fermions, one obtains the correct WZ terms. At this stage, we have not included the mass term of η. One can introduce it by writing a mass term to break the U (1) L−R gauge symmetry for Φ, such as The axial U (1) is now explicitly broken, and the η obtains the mass. In the background of Eq. (16), we have a term This term cause the winding of the trace part of Φ. The winding gives important effects through the WZ term as we discussed before. In addition to the WZ terms among external fields and Nambu-Goldstone modes, we also have where b is a gauge field of U (N f ) L+R part of the right most gauge sites. They are massive modes which correspond to the ρ and ω mesons.
One can modify the model by introducing Higgs fields, H L and H R , as in the right panel where N f i=1ξ i = 0 (mod 2π). The boundary condition of the fermions is where 0 ≤ ν < 2π. The KK spectra of the gauge fields and the fermions are, respectively, By using these, the one-loop effective potential is given by The first term in the right hand side is minimized whenξ 1 =ξ 2 = · · · =ξ N f =ξ. Furthermore, the second term is minimized atξ = ν − π. Therefore, for N f < N , the lowest minimum is at which gives the anti-periodic boundary conditions for the fermions. The U (N f ) group is unbroken at the minimum, and the 2N fermions get massive.
Next, we consider the one-loop effects of the gauged Z N which is the subgroup of U (1) B symmetry. The effective potential is given by where ξ = 2πm/N (m ∈ Z) is the VEV of the gauged Z N field. The different VEV corresponds to the sector of different boundary conditions twisted by the Z N elements. Since under the winding of Φ.
As in the three dimensional effective picture, there are vortex configurations made of ρ and ω. Under the winding of Φ, there is a winding θ term for the U (N f ) gauge group from Eq. (46). In the presence of this θ term (which we call itθ), the Abrikosov-Nielsen-Olesen (ANO) vortex string [87,88] which goes around S 1 cannot be connected since the magnetic flux obtains the electric charge as it goes around the S 1 direction by the Witten effect [89]. In order to have a stable string loop which goes around S 1 , one needs to have some non-trivial configuration which carries the electric charge, i.e., the baryon number.
In the background with a constant dθ, one can look for a static field configuration which is x 3 independent. The field equations are then the same as the CS case, and thus one finds the solution with a finite energy. The baryon number, B = 1, is indeed carried by the string through the electric charge of this solution. For a general background, this configuration will be relaxed to a solution of the field equations with a finite energy. We call it the B = 1 string.
The droplet is a configuration of the η field. It is a sheet with a boundary, and the value of  [90,91]. The presence of the monopole does not make the U (k) −N part to confine as the gauge bosons have a mass term from the CS term [92]. Therefore, we obtain the correct low energy limit, U (k) −N theory.
There is another possibility that the U (N f −k) part of the VEV is kept non-vanishing for H L,R while the chiral symmetry is restored by cutting the U (N f −k) part the link Φ. The U (N f −k) gauge group is kept in the Higgs phase, and the only U (k) −N part remains at low energy.
What happens for ρ and ω mesons is qualitatively different in the above two cases. When the S 1 radius is large, they are vector mesons which we are familiar with. As the radius approaches to the critical point, they behave as the gauge bosons in the Higgs phase. In particular, the meta-stable vortex strings made of ρ and ω appear. Beyond the critical radius, the chiral symmetry is restored, and the U (k) part of them goes into the topological phase, whereas the U (N f −k) part goes into either the confining phase or remains in the Higgs phase. The rest of them stays in the Higgs phase.

Finite temperature QCD
We discussed a somewhat exotic scenario for the chiral phase transition which happens at some critical radius, R * . Fork = 0 with the anti-periodic boundary condition for quarks, ν = π, one can think of this system as the finite temperature QCD by taking the Euclidean metric. At some critical temperature, T * = 1/R * , the chiral phase transition happens.
In The question of which picture is the most appropriate near the chiral phase transition should be able to be tested by the lattice simulations. By looking at the behavior of the two point functions of the vector currents, one may check if the ρ and the ω mesons get "massless." Although they have no mass term in the four dimensional Lagrangian, they have thermal masses and also masses from instantons (monopoles in four dimensions) in the actual spectrum. We will leave the study of these effects as well as that of actual methods in the lattice QCD to distinguish the scenarios.

Discussion
The three dimensional CS matter systems exhibit various non-trivial topological phases at low energy, and it has been conjectured that gauge theories with fermions and bosons describe We left the discussion of how to test the possibility of the vector mesons becoming gauge bosons. One of the natural frameworks to discuss this point is the holographic QCD where the vector mesons are already gauge bosons in the Higgs phase. The specific holographic model such as the Sakai-Sugimoto model [73] may be able to be used to study the dynamics of the phase transition in the winding θ background. Also, if the feature of gauge bosons getting "massless" in the four dimensional language remains in the trivial θ = 0 background, the lattice QCD may be able to directly test the scenario.
Another non-trivial prediction of the model is the existence of the monopoles which cut the string made of ρ and ω [94]. We are not sure what this objects to be identified in the hadron spectrum. Due to the color-flavor locking, the monopoles really carries the magnetic charge of QED while they are confined by the string. It is certainly interesting to look for the candidates of hadrons which are made of the monopole-string system.

A Lagrangian of quiver diagram
Here, we show the Lagrangian that is described by the quiver diagram in the left panel of Fig. 2: where n L and n R are the number of double circle nodes in the upper and lower lines, respectively.

B Winding θ term
Here, we review how to treat the θ term with a winding number [66]. When θ has winding number along a compact direction, the θ term is not well defined on one patch.
Let us consider an integral with S 1 dθ = 2πk, We would like to define the integral up to 2πm, (m ∈ Z) since the integral will be exponentiated in the path integral. In Ref. [66], a general prescription to define such an integral is discussed. By taking t, (0 ≤ t < 2π) as the coordinate on S 1 , the prescription gives Similarly, the integration ofθ(t)q(t) can be defined by The definitions of the integral in Eqs. (57) and (58) have the following desired features. The integral is invariant modulo 2π under the shifts, θ → θ + 2π and q → q + 2π. Also, the integral does not depend (modulo 2π) on the choice of the t = 0 point on S 1 .
These definitions are consistent with the integration by parts (modulo 2π), i.e., C Baryon number and the configuration of η We discuss a configuration to give the baryon number, B = 0, in QCD under a non-trivial background of η. We now take the space-time as the Minkowski space, M 4 . The WZ term in QCD contains the following term: where the N f × N f matrix A is the external gauge field which couples to the U (N f ) vector current, and η is the U (1) part of the Nambu-Goldstone mode, U = e iπ a T a +iη . The trace part of the gauge field normalized as, B = (N/N f )TrA, is the source for the baryon number.
The baryon charge density can be read off by differentiating with respect to B as Let us consider a configuration with η = 0 at x 3 = −∞ and η = 2π at x 3 = +∞. We also apply an external magnetic field of the (11) component of A in the x 3 direction, with i.e., putting a monopole and an anti-monopole at x 3 = ∓∞.
This configuration provides B = 1 as one can see from the baryon density in Eq. (61).
One can also understand this as the Witten effect. For the (11) component of A, there is an effective θ term from Eq. (60), which varies as a function of x 3 . Therefore, as we move a monopole from x 3 = −∞ to x 3 = +∞, the monopole obtains the electric charge, N , to couple to A (11) by the Witten effect. Since A (11) = B/N + · · · , the dyon carries the baryon number B = 1. Although the magnetic field is eliminated by this move, the baryon number remains.
Now we consider the situation that the change of the value of η happens in a finite region on the (x 1 , x 2 )-plane and at a localized location in the x 3 coordinate. This sheet-like configuration is called the Hall droplet in Ref. [28]. If a monopole goes through the Hall droplet, it becomes a dyon with B = 1 by the Witten effect. This means that if we put magnets on the both sides of the droplet, the magnetic lines cannot just go through the droplet. In order to let the monopole line to go through, one needs to throw in a baryon.
Conversely, starting from a configuration where the magnetic lines are penetrating the droplet, when the magnets are turned off or taken away, the system should relax to a state with a finite baryon number. It has been discussed in Ref. [28] that the excitation of the edge mode of the droplet corresponds to the baryon state with spin N/2. This state is a good candidate of the remnant of the system.
The flavor quantum numbers of systems can be read off as in the same way as the baryon number. Let us take the cases with N = 3 and N f = 2 as in real QCD, where A (11) couples to the current of the up quark. The configuration of the unit magnetic line of A (11) going through the droplet now has the quantum number of the operator uuu, i.e., it has the electric charge Q = 2. This is indeed the same as the baryon discussed in Ref. [28].
Let's consider another example where only one of the diagonal components of the Nambu-Goldstone field has the non-trivial configurations: (π a T a + η) , (π a T a + η) When the U (1) baryon (or QED as in the real world) is gauged, the minimal magnetic charge is dB N = dA (11) as in the well-known magnetic monopole in grand unified theories [95,96]. The Dirac quantization conditions for quarks are satisfied by taking into account the Z N magnetic flux of SU (N ) carried by the monopole [97,98]. The discussion here is closely related to the chiral soliton lattice studied in Ref. [100], where the pions get winding under strong magnetic fields and a chemical potential of baryons.
Microscopically, one baryon can be converted into a configuration of a Hall droplet with one unit of the magnetic flux penetrating through it.