Non-supersymmetric duality cascade of QCD(BF) via semiclassics on $\mathbb{R}^2\times T^2$ with the baryon-'t Hooft flux
Yui Hayashi Yuya Tanizaki Hiromasa Watanabe


Symmetry Breaking and Spatiotemporal Pattern Formation in Photonic
Time Crystals
    Yiming Pan
    April 29, 2024
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§ INTRODUCTION AND SUMMARY


Non-Abelian gauge theories in 4d are, typically, strongly coupled at low energies, which cause rich nonperturbative dynamics, such as color confinement and chiral symmetry breaking, and the θ angle serves as a useful tool to learn about the mechanism behind those phenomena. 
In this paper, we focus on a specific class of 4d gauge theories, Quantum ChromoDynamics with the Bi-Fundamental fermion (QCD(BF)), which is the 4d SU(N_1) × SU(N_2) gauge theory with a Dirac fermion Ψ in the bifundamental representation (N_1,N_2). 


This theory has two vacuum angles (θ_1,θ_2), and it is expected that there are various phase transitions in the (θ_1,θ_2) space. 
Indeed, the phase diagram of the QCD(BF) has been investigated from the anomaly matching as well as from some calculable limits <cit.> (See also Refs. <cit.> for recent developments of anomaly matching and global inconsistency), and these studies uncover its nontrivial phase diagram. 
In particular, when N_1≠N_2, the (θ_1,θ_2) phase diagram has different features between the massless fermion limit, m≈ 0, and the massive fermion limit, m→∞. 
Karasik and Komargodski <cit.> studied how the topological structure of the phase diagram is deformed as we change the fermion mass, and they proposed an interesting duality conjecture from their observations: 
There is an infrared (IR) duality between SU(N_1) × SU(N_2) QCD(BF) and SU(N_1) × SU(N_2-N_1) QCD(BF). 
This suggests the possibility of a “duality cascade” in non-supersymmetric theories.


The original idea of duality cascade was developed in studying the extension of the AdS/CFT correspondence to the branes at conical singularities <cit.>, where one can construct the gravity dual of 𝒩=1 supersymmetric QCD(BF). 
For 𝒩 = 1 supersymmetric SU(N_1) × SU(N_2) gauge theories, one can repeatedly use the Seiberg duality, reducing the rank of the gauge groups along with lowering the renormalization group scale <cit.>.
Karasik and Komargodski <cit.> conjectured the presence of a similar duality cascade[As detailed later, the meaning of duality cascade is slightly different from the original supersymmetric one.
First, the IR duality here means a correspondence between the vacuum structures after suitable reparametrization. Second, although the original supersymmetric duality cascade occurs with lowering the renormalization group scale, the non-supersymmetric duality cascade involves the change of the bifundamental fermion mass.] in non-supersymmetric SU(N_1) × SU(N_2) QCD(BF), so that one can easily understand the topology changing phenomena of the (θ_1,θ_2) phase diagram by repeating the duality transformation SU(N_1) × SU(N_2) → SU(N_1) × SU(N_2-N_1) (See Figure <ref>).
The equal-rank SU(N)× SU(N) QCD(BF) can then be regarded as the terminal of the duality cascade. 





In our previous paper <cit.>, we have developed a novel semiclassical method to reveal the vacuum structure of the equal-rank SU(N) × SU(N) QCD(BF).
Aside from the duality cascade, the equal-rank QCD(BF) is the daughter theory of the large-N orbifold equivalence from the 𝒩=1 SU(2N) super Yang-Mills (SYM) theory <cit.>. 
The nonperturbative justification of the orbifold equivalence requires uncovering the symmetry-breaking pattern of these theories <cit.>, and our semiclassical description gives its affirmative support by explicitly computing the vacuum phase diagram of equal-rank QCD(BF) <cit.> (see also <cit.>). 


In this paper, extending the preceding one <cit.>, we investigate the different-rank SU(N_1) × SU(N_2) QCD(BF) by employing the semiclassical approach with the 't Hooft flux T^2-compactification.
In this semiclassical approach, we realize the weak-coupling description of confinement in the four-dimensional gauge theories by putting them on ℝ^2 × T^2 incorporating the 't Hooft flux <cit.>. 
We can then perform the explicit computation of the confinement phenomena by using the dilute gas approximation of center vortices, so this setting gives an explicit realization for one of the prevalent understandings for the quark confinement <cit.>.
Grounded in the adiabatic continuity conjecture, we expect that the (θ_1,θ_2) phase diagram of the 4d QCD(BF) has the same structure with the weakly-coupled confining theory at small T^2.
It is noteworthy that the application of this approach has been successful in distinctly outlining the qualitative features of various confining gauge theories <cit.>, including SU(N) Yang-Mills theory, 𝒩 = 1 SU(N) SYM theory, QCD with fundamental quarks, QCD with 2-index quarks, and equal-rank QCD(BF).

Let us emphasize that it is not straightforward to extend the previous analysis of equal-rank QCD(BF) to different-rank QCD(BF).
In different-rank QCD(BF), the naive insertion of minimal 't Hooft flux for SU(N_1) × SU(N_2) violates the single-valuedness of the bifundamental matter.
We resolve this problem by introducing U(1) baryon magnetic flux at the same time as we have done for QCD with fundamental quarks <cit.>, but then the 2d semiclassical theory becomes the different one from what we obtained for the equal-rank QCD(BF) in <cit.>. 




 §.§ Summary and highlights



In Section <ref>, we explain some basics of QCD(BF) and also give a review of the duality conjecture <cit.>, which states that two different QCD(BF) have the same structure of the (θ_1,θ_2) phase diagram:

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) for m<m^* 
       ⟷  SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2), <ref>

where m^* denotes the (largest) fermion mass of the topology-changing point for the phase diagram.
In Ref. <cit.>, this duality is proposed based on observations of the vacuum branches with the large-gcd(N_1,N_2) argument. This duality should be understood as a matching between the meta-stable vacuum branches after a suitable reparametrization of the fermion mass and dynamical scales.
If this is true, it causes the non-supersymmetric “duality cascade” as the fermion mass is lowered, which is sketched in Figure <ref>.


In Section <ref>, we derive kinematic constraints on phase diagrams arising from the global symmetry by extending the results of previous works <cit.>. 
The vector-like symmetry of the SU(N_1) × SU(N_2) QCD(BF) is given by the baryon-number symmetry and the 1-form center symmetry, 

    G_vec=U(1)_V/ℤ_(N_1, N_2)×ℤ^[1]_(N_1,N_2), 
        <ref>

and its 4d symmetry-protected topological (SPT) states are characterized by the two discrete labels, (k̃,ℓ̃)∈ (ℤ_(N_1,N_2))^2. 
This model in the massless limit also enjoys the discrete chiral symmetry (ℤ_(N_1,N_2))_χ, and we see that it has to be spontaneously broken in the confinement state because of the mixed anomaly.
For the massive case, we show that the vector-like symmetry G_vec has the global inconsistency with the 2 π shifts of theta angles because θ_1 →θ_1 + 2π and θ_2 →θ_2 + 2π changes the SPT labels (k̃,ℓ̃) if gcd(N_1,N_2) ≠ 1.
These results impose restrictions on the possible vacuum structures.

Next, we develop the novel semiclassical approach on ℝ^2× T^2 with the baryon-'t Hooft flux to reveal the low-energy dynamics concretely. 
Through the semiclassical analysis shown in Section <ref>, we derive the following 2d effective theory:

    
  * 
    Let us write N_1 = gcd(N_1,N_2) N_1', N_2 = gcd(N_1,N_2) N_2', then the 2d effective theory is described by the 2 π N_1' N_2'-periodic scalar φ and discrete vacuum labels k, ℓ = 0,⋯, gcd(N_1,N_2)-1. 
    The effective potential is given by

    V[φ; k,ℓ]    = - m μcos (φ) - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π (N'_2 ℓ̃ + M_2 k̃) - θ_1/N_1) 
                 - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ
        + 2 π (N'_1 ℓ̃ - M_1 k̃) - θ_2/N_2), <ref>

    where M_1,M_2 is the Bezout coefficient, M_1N'_2+M_2N'_1=1, with the special choice satisfying M_1/N'_2, M_2/N'_1∈ℤ, μ is a dimensionful scale introduced by bosonization, and K^(i)^-S_I^(i)/N_i is the semiclassical strong scales for SU(N_i) (i=1,2).

The compact scalar describes the phase of the chiral condensate (Ψ1+γ_5/2Ψ)∼^φ, so its periodicity is originally given by φ∼φ+2π but is extended to φ∼φ+2π N'_1N'_2 when integrating out the gauge fields. 
This extension of the periodicity occurs in the same mechanism as that of η' periodicity in QCD with fundamental quarks <cit.>.
We show how the 4d anomaly constraints obtained in Section <ref> are satisfied in the semiclassical theory in Section <ref>, and the above discrete labels k̃,ℓ̃ in semiclassics are identified with the 4d SPT labels completely.
Under the assumption of adiabatic continuity, we expect this 2d effective theory predicts the vacuum structure of the original 4d SU(N_1) × SU(N_2) QCD(BF).


Let us highlight some of our results coming out of this 2d effective description:

    
  * The (θ_1,θ_2) phase diagram of the different-rank QCD(BF) is concretely determined for various values of the fermion mass m including the topology-changing point.
    
    As will be shown in Figure <ref>, the phase diagrams at the massless and large-mass limits have different topologies, and 
    we explicitly demonstrate the topology-changing phenomena by applying the above semiclassical description at intermediate masses. 
    We also numerically examine phase diagrams: Section <ref> presents the results of SU(2) × SU(3) QCD(BF) as an example of gcd(N_1,N_2) = 1 case, and Section <ref> gives that of SU(2) × SU(4) QCD(BF) as an example of gcd(N_1,N_2) ≠ 1 case.
    

    
  * We investigate the conjectured duality (<ref>) between SU(N_1) × SU(N_2) QCD(BF) and SU(N_1) × SU(N_2 - N_1) QCD(BF) based on the above semiclassical framework:


    
  * We construct the explicit duality map in the large-N_1,2 limit.  

    In the large-N_1,2 limit, the 2d effective theory is reduced to discrete vacuum branches.
    We then concretely construct the duality map of the semiclassical parameters (m μ, K^(1)^-S_I^(1)/N_1, K^(2)^-S_I^(2)/N_2), so that these vacuum branches match between the parent and daughter theories, in Sections <ref> and <ref>.
    
    
  * The duality also holds in the hierarchical limit Λ_1,m ≪Λ_2 without large-N_1,2.
    
    In the limit where the dynamical scales of SU(N_2) and SU(N_2 - N_1) gauge are extremely high, we can forget most of the SU(N_2) and SU(N_2 - N_1) dynamics, and it becomes possible to match the effective SU(N_1) gauge theories extracted from SU(N_1) × SU(N_2) QCD(BF) and SU(N_1) × SU(N_2 - N_1) QCD(BF).
    This matching is presented in Section <ref>.
    
    
  * The duality cascade is practically effective even away from those limits.
    
    We numerically study the phase diagrams of SU(2) × SU(5) QCD(BF) and SU(2) × SU(3) QCD(BF) in Section <ref> and Appendix <ref>.
    Choosing the semiclassical parameters (m μ, K^(1)^-S_I^(1)/N_1, K^(2)^-S_I^(2)/N_2) according to the dictionary in large-N_1,2 limit, we find a remarkable agreement of the phase diagrams between these theories.
    Quantitatively, the duality is violated due to finite-N_1,2 effects, but we argue it is just the O(N_2^-4) effect in Appendix <ref>.
    


    






§ MODEL: QCD(BF) WITH N1<=N2




In this section, we give a brief review on QCD(BF), including its symmetry and also some large-N properties.
In particular, the large-N argument predicts a nontrivial relation between the SU(N_1)× SU(N_2) QCD(BF) with the SU(N_1)× SU(N_2-N_1) QCD(BF), and this is called the “duality cascade,” proposed in Ref. <cit.>.



 §.§ Model and notation



Here we introduce the SU(N_1)× SU(N_2) QCD(BF) and notations. The field contents are 

    {
        a_1:   SU(N_1) gauge field with the field strength f_1= a_1+ a_1∧ a_1, 
    
        a_2:   SU(N_2) gauge field with the field strength f_2= a_2+ a_2∧ a_2, 
    Ψ:   N_1× N_2 matrix-valued Dirac fermion in the bifundamental representation.
    .

Under the SU(N_1)× SU(N_2) gauge transformations (U_1, U_2), they transform as 

    Ψ↦ U_1Ψ U_2^†,   
        a_i↦ U_i a_i U_i^† -   U_i U_i^†   (i=1,2),

and the covariant derivative for Ψ, denoted DΨ, is given by

    DΨ=γ^μ(∂_μΨ+  a_1μΨ- Ψ a_2μ).

The action of this model reads,

    S=   1/g_1^2∫  |f_1|^2 +  1/g_2^2∫ |f_2|^2 +∫trΨ(D+m)Ψ
       +θ_1/8π^2∫tr(f_1∧ f_1)+θ_2/8π^2∫tr (f_2∧ f_2),

where |f|^2 := tr(f ∧⋆ f) to represent the gauge kinetic terms. 

Without loss of generality, we can assume that the fermion mass is positive m>0 because the phase of m only results in the redefinition of (θ_1,θ_2) due to the chiral anomaly. 
Let us also assume N_2>N_1, and we will limit our consideration to theories that are asymptotically free, i.e., N_2 < 11/2 N_1. 
We define Λ_1 and Λ_2 as the dynamical scales for SU(N_1) and SU(N_2) sectors, respectively: Within the 1-loop renormalization group, 

    Λ_1 = μexp(-8π^2/(11/3N_1 - 2/3N_2) g_1^2(μ)),   Λ_2 = μexp(-8π^2/(11/3N_2 - 2/3N_1) g_2^2(μ)),

where μ is the renormalization scale. 

For later convenience, we define the coprime numbers (N_1',N_2') as

    N_1' := N_1/gcd(N_1,N_2),   N_2' := N_2/gcd(N_1,N_2).

Let us also introduce the Bezout coefficient (M_1,M_2) that satisfies

    N_1' M_2 + N_2' M_1 = 1,

or N_1 M_2+ N_2 M_1 = (N_1,N_2). The Bezout coefficient is not unique as the relation is invariant under (M_1,M_2)→ (M_1+N'_1, M_2-N'_2), so we pick one representative. 
Since (N'_1,N'_2)=1 implies ((N'_1)^2, (N'_2)^2)=1, we can require that 

    M_1∈ N_2' ℤ,   M_2∈ N_1' ℤ,

so we choose such a one.  
Note that M_1, M_2 satisfy gcd(M_1,N_1') = gcd(M_2,N_2') = 1.





 §.§ Symmetry



At general parameters, this model enjoys the vector-like global symmetry, 

    G_vec=U(1)_V/ℤ_(N_1, N_2)×ℤ^[1]_(N_1,N_2),

where U(1)_V acts on the bifundamental fermion as Ψ↦^αΨ, Ψ↦^-αΨ and ℤ_(N_1,N_2)^[1] is the 1-form symmetry acting on Wilson loops. 
At the massless point (m=0), the model acquires the discrete chiral symmetry, 

    G_massless=G_vec× (ℤ_(N_1,N_2))_χ,

where the chiral symmetry (ℤ_(N_1,N_2))_χ is generated by 

    Ψ↦^2π/(N_1,N_2)1+γ_5/2Ψ,   Ψ↦Ψ^-2π/(N_1,N_2)1-γ_5/2.


Let us first look at the vector-like symmetry in detail. We note that (U_1,U_2,^α)∈ SU(N_1)_gauge× SU(N_2)_gauge× U(1)_V acts on Ψ as 

    Ψ↦^α U_1 Ψ U_2^†.

This action has the redundancy that is generated by z_1=(^2π/N_11_N_1, 1_N_2, ^-2π/N_1) and z_2=(1_N_1, ^2π/N_21_N_2, ^2π/N_2), and they form the subgroup ℤ_N_1×ℤ_N_2. 
As a result, we can describe the structure group acting faithfully on the matter fields (including the gauge and global symmetries) as 

    G^gauge+global   = SU(N_1)_gauge× SU(N_2)_gauge× U(1)_V/ℤ_N_1×ℤ_N_2
        =  [(SU(N_1)× SU(N_2))_gauge/ℤ_(N_1,N_2) ] × U(1)_V/ℤ_(N_1,N_2).

To obtain the second line, it is convenient to change the basis for ℤ_N_1×ℤ_N_2 as 

    ℤ_N_1×ℤ_N_2≃ℤ_(N_1,N_2)×ℤ_(N_1,N_2),

where ℤ_(N_1,N_2) is generated by z_1^N_1'z_2^N_2'=(^2π/(N_1,N_2)1_N_1, ^2π/(N_1,N_2)1_N_2, 1) and ℤ_(N_1,N_2) by z_1^-M_1z_2^M_2=(^-2π M_1/N_11_N_1, ^2π M_2/N_21_N_2, ^2π/(N_1,N_2)).  
Therefore, the subgroup ℤ_gcd(N_1,N_2) of the quotients ℤ_N_1×ℤ_N_2 lives completely inside of the gauge group SU(N_1)_gauge× SU(N_2)_gauge, and we find the second line of (<ref>).

To get the global symmetry of the SU(N_1)× SU(N_2) QCD(BF), we can simply neglect the gauge group in (<ref>) for the 0-form symmetry and we find U(1)_V/ℤ_(N_1,N_2). 
In addition, the ℤ_(N_1,N_2) subgroup of the gauge group does nothing on the matter field, and it gives the 1-form symmetry acting on the loop operators, and we obtain (<ref>).


In the massless case (m = 0), the SU(N_1)× SU(N_2) QCD(BF) is classically invariant under the U(1)_χ chiral transformation,

    Ψ↦e^1+γ_5/2αΨ,     Ψ↦Ψe^-1-γ_5/2α.

Due to the Adler-Bell-Jackiw (ABJ) anomaly, this U(1)_χ transformation shifts the theta angles as

    θ_1 →θ_1 + N_2 α,  θ_2 →θ_2 + N_1 α.

When both shifts are quantized in the unit of 2π, it becomes the well-defined symmetry, so it becomes the ℤ_gcd(N_1,N_2) subgroup, which gives (<ref>).
Note that the physics at the massless point depends only on the particular combination of theta angles,

    θ_phys = N_1' θ_1 -N_2' θ_2,

due to the ABJ anomaly (<ref>). 




 §.§ Large-N argument and duality conjecture



In Ref. <cit.>, Karasik and Komargodski investigated phase diagrams using the large-N counting with N= gcd (N_1,N_2) and gave a conjecture that different QCD(BF) theories are related by non-supersymmetric dualities.
In this subsection, we review some features of phase diagrams predicted by the large-N argument.

At large N, the vacuum energy is the quadratic function of the theta angles in each vacuum branch <cit.>, so we expect:

    E_(k_1,k_2)(θ_1,θ_2)    = E_(0,0) (θ_1 - 2 π k_1, θ_1 - 2 π k_2), 
    
        E_(0,0) (θ_1 , θ_2 )    ≃ E_0 + a_1 θ_1^2  + a_2 θ_2^2 + b θ_1 θ_2.

Here, (k_1,k_2)∈ℤ×ℤ is introduced to restore the 2π-periodicities of θ_1,2, which describes the multi-branch structure of (quasi-)stable vacua at large N. 
The vacuum branch (k_1,k_2) gives the lowest energy at (θ_1,θ_2) = (2 π k_1,2 π k_2). 
As the fermion determinant is positive definite when m>0, the Vafa-Witten argument from the QCD inequalities claims that θ_1=θ_2=0 should be the lowest-energy state, which requires the positivity of the Hessian matrix of E_(0,0)(θ_1,θ_2). This gives the constraint that a_1, a_2 >0, b^2<4a_1 a_2 when m>0. 
At the massless point, one of the inequalities is saturated; b^2=4 a_1 a_2. 



  §.§.§ Phase diagram of massless QCD(BF)

Before introducing the duality conjecture, let us consider the massless case m = 0. 
We note that the large-N limit in this section is similar to the Veneziano-type large-N limit, not the 't Hooft-type, so the anomalously-broken U(1)_χ does not restore as N→∞. 
Due to the ABJ anomaly, the vacuum energy should depend only on θ_phys.=N'_1θ_1-N'_2θ_2, and we obtain

    E_(k_1,k_2) (θ_1 , θ_2 )    ≃ E_0 + a [ N_1' θ_1 -N_2' θ_2 - 2π(N_1' k_1 - N_2' k_2) ]^2.

For - π + 2 π n < θ_phys < π + 2 π n, the branches (k_1,k_2) with N_1' k_1 - N_2' k_2 = n are the ground states, and the solutions can be expressed using the Bezout coefficient (<ref>) as

    (k_1,k_2)∈ (n M_2, -n M_1)+(N'_2, N'_1)ℤ.

This implies that the system is infinitely degenerate.
For instance, the branches (k_1,k_2) = (0,0), ± (N_2',N_1'), ⋯ are all the true vacua for - π < θ_phys < π.
For finite (N_1,N_2), it would be natural to think that there are N=gcd(N_1,N_2) degenerate states on each sector from the chiral symmetry breaking.
An example of the phase diagram is shown in  Figure <ref>.






We can also speculate the phase diagram at small fermion mass m.
The gcd(N_1,N_2) degeneracy is resolved by the mass perturbation.
So we guess that the vacuum energy is,

    E_(k_1,k_2)(θ_1 , θ_2 )    = E_(0,0) (θ_1 - 2 π k_1, θ_1 - 2 π k_2) 
    
        E_(0,0) (θ_1 , θ_2 )    ≃ E_0 + a [ N_1' θ_1 -N_2' θ_2 ]^2 + ϵ_1 θ_1^2  + ϵ_2 θ_2^2 + ϵ_12θ_1 θ_2,

where ϵ_1,ϵ_2,ϵ_12 are small coefficients, which vanish as m → 0.
The positivity of the Hessian requires 4a_1 a_2-b^2=4a(N_1'N_2' ϵ_12+(N_1')^2 ϵ_2 +  (N_2')^2 ϵ_1)+O(ϵ^2)>0 for consistency with the QCD inequalities. 

To see how the mass perturbation affects the phase diagram, let us gradually increase the theta parameters along the almost flat direction, (θ_1,θ_2)→ (θ_1+2π N_2',θ_2+2π N'_1), so that θ_phys is unchanged. 
We note that (2π N'_2, 2π N'_1) is the minimal 2π periodicity along this direction because (N'_1, N'_2)=1, 
so let us calculate the phase boundary between (k_1,k_2) = (0,0) and (k_1,k_2) = (N_2',N_1') phases. 
The phase boundary between these phases is given by E_(N_2',N_1') (θ_1 , θ_2 ) = E_(0,0) (θ_1 , θ_2 ), which gives a linear equation

    (2 ϵ_1 N_2' + ϵ_12 N_1' ) θ_1 + (2 ϵ_2 N_1' + ϵ_12 N_2' ) θ_2 = 2 π( ϵ_1 (N_2')^2 + ϵ_2 (N_1')^2 + ϵ_12 N_1'  N_2' ).

This shows that the phase boundary between (k_1,k_2)=(0,0), (N'_2, N'_1) is the straight line that contains the halfway point (θ_1,θ_2) = (π N_2', π N_1'). 
The other details of the phase boundary depend on the specific form of the correction (ϵ_1 θ_1^2  + ϵ_2 θ_2^2 + ϵ_12θ_1 θ_2), and
an example of the phase diagram is shown in Figure <ref>.
One can see that the phase boundaries are consistent with the global inconsistency constraints, which will be discussed in Section <ref>.

When m=0, there are N(=(N_1,N_2)) degenerate vacua due to the discrete chiral symmetry breaking, and the mass perturbation lifts the degeneracy. 
This new phase boundary at nonzero m can be understood as the exchange of the different chiral broken vacua in the massless limit. 
Therefore, we can guess that there is no qualitative change between the massless and small-mass phase diagrams for the case (N_1,N_2)=1 because of the absence of discrete chiral symmetry, while this is outside the scope of the large-N discussion in this subsection.




  §.§.§ Duality cascade


Let us imagine that we dial the mass parameter m from almost the massless point to almost the infinitely large one. 
When we consider the limit m≫Λ_1, Λ_2, the bifundamental fermion decouples from the dynamics, and the QCD(BF) reduces to the product of SU(N_1) Yang-Mills theory and SU(N_2) Yang-Mills theory. 
Then, the (θ_1,θ_2) phase diagram looks like the checkerboard, where phase boundaries are on θ_i = π mod2 π (i=1,2). 
It is very nontrivial how the phase diagram at small mass (Figure <ref>) is deformed to the phase diagram at large mass because even the topologies of phase boundaries are different.
For instance, the phases (k_1,k_2) = (0,0) and (k_1,k_2) = (0,1) are adjacent in the large-mass limit, which is not the case in the small-mass limit (Figure <ref>). Therefore, as the fermion mass m increases, a “topology change” is expected to occur at some point. 

First, let us consider what the phase diagram looks like at this topology-changing point.
Following Ref. <cit.>, to observe the duality cascade, we here consider the phase boundary between  (k_1,k_2) = (0,0) and (k_1,k_2) = (1,1) phases in the example of N_1' = 1, N_2' = 2 (Figure <ref>).
The phase boundary is located on the line:

    E_(0,0)   (θ_1, θ_2) - E_(1,1) (θ_1, θ_2) 
        = (4π a_1 + 2π b) θ_1 + (4π a_2 + 2 π b) θ_2 - 4 π^2 a_1  - 4 π^2 a_2 - 4 π^2 b = 0.

This line goes through (θ_1,θ_2) = (π,π), and its slope depends on the parameters: - 2 a_1 + b/2 a_2 +  b.
We know that the slope is 1/2 in the small-mass limit m → +0, and also that the slope becomes negative[The negative slope at (θ_1,θ_2) = (π,π) is expected as well, from the perturbation theory at the decoupling limit <cit.> or semiclassical analysis (see below or Ref. <cit.>).] in the large-mass limit because a_1 ∼ O(Λ_1^4), a_2 ∼ O(Λ_2^4) and b ∼ 0 for m → +∞.
Thus, the slope vanishes (or diverges) at some mass m = m_*, which is actually the topology-changing point.

At m = m_*, where 2a_1 + b = 0 holds[We here assume that the slope vanishes 2a_1 + b = 0 at the topology-changing point, rather than 2a_2 + b = 0. This scenario is plausible for N_2 > N_1.], the vacuum energy reads,

    E_(k_1,k_2)(θ_1 , θ_2 )    = E_(0,0) (θ_1 - 2 π k_1, θ_1 - 2 π k_2) 
    
        E_(0,0) (θ_1 , θ_2 )    ≃ E_0 + a_1 θ_1^2  + a_2 θ_2^2 - 2 a_2 θ_1 θ_2 
       = a_1 (θ_1 - θ_2)^2 + (a_2-a_1) θ_2^2,

which is identical to the vacuum energy of the pure product-group Yang-Mills theory at (θ_1',θ_2') = (θ_1 - θ_2, θ_2). 



One can further guess the ranks of the gauge groups by matching the massless phase diagrams:

    θ_phys   = N_1' θ_1 - N_2' θ_2 = N_1'(θ_1 - θ_2) - (N_2' - N_1') θ_2.

Therefore, we have the following coincidence of the phase diagrams at the massless limit:

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) at m=0 
       ⟷  SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2) at m=0



Based on these observations, Karasik and Komargodski <cit.> conjectured a new infrared duality,

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) for m<m^* 
       ⟷   SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2),

where the fermion mass of the dual theory, m', should be a function of m that satisfies m'|_m=0=0 and m'|_m=m^*=∞. Here, the duality means the matching of the phase diagrams and meta-stable vacua between these theories.




Now, let us assume that the fermion mass m of the parent theory is set to be sufficiently small, then m' of the daughter theory would also be small enough and it is below the topology-changing point of SU(N_1)× SU(N_2-N_1) QCD(BF). 
If so, we can further apply the duality (<ref>) to obtain another daughter QCD(BF), and we can repeat this procedure, which is called the duality cascade in Ref. <cit.> (see Figure <ref>). 
Although this duality cascade involves changes in the fermion mass m, its structure resembles the duality cascade of the 𝒩=1 supersymmetric bifundamental gauge theories <cit.>.
Understanding this observation is quite valuable for extending non-perturbative techniques towards non-supersymmetric theories.

As an additional remark, it is worth mentioning that the asymptotic freedom of the daughter theory requires an additional constraint.
Indeed, an asymptotically free parent theory satisfies N_1 < N_2 < 11/2N_1.
Then, the asymptotic freedom for the SU(N_1) gauge, N_2 - N_1 < 11/2N_1, automatically holds, but the other condition N_1 < 11/2 (N_2 - N_1), i.e. N_1 < 11/13 N_2 gives an additional constraint.





§ ANOMALY AND GLOBAL INCONSISTENCY



In this section, we extend the anomaly and global inconsistency analysis for QCD(BF) <cit.> to the different-rank QCD(BF).
We first discuss the background gauge fields for the vector-like symmetry (<ref>) and classify its possible symmetry-protected topological (SPT) states. 
Using this knowledge, we derive the 't Hooft anomaly and global inconsistency for the theta periodicities and the discrete chiral symmetry. 




 §.§ Background gauging of the vector-like symmetry and its SPT classification


As discussed in Section <ref>, the vector-like symmetry G_vec of QCD(BF) consists of the baryon-number symmetry U(1)_V/ℤ_(N_1,N_2) and the 1-form symmetry ℤ_(N_1,N_2)^[1], and let us introduce its background gauge field. 
We have seen that the origins of the quotient in the baryon-number symmetry and of the 1-form symmetry both come from the quotient of the structure group G^gauge+global in (<ref>). 
This indicates that the background gauge fields for G_vec consist of 

    {
        A:   U(1)_V1-form gauge field,
    
        B_1:   ℤ_N_12-form gauge field, 
    
        B_2:   ℤ_N_22-form gauge field. 
        .

Following Ref. <cit.>, we describe the ℤ_N_1,2 2-form gauge fields as pairs of U(1) 2-form and 1-form gauge fields (B_i,C_i) satisfying

    N_i B_i= C_i,

for i = 1,2.
To reduce the pair of U(1) 2-form and 1-form gauge fields into the discrete 2-form gauge field, we postulate the 1-form gauge invariance:

    B_i↦ B_i+Λ_i,   C_i ↦ C_i+ N_i Λ_i,

where the 1-form gauge parameters Λ_1,2 are U(1) gauge fields. 

To represent the SU(N_1) × SU(N_2) gauge fields with the ℤ_N_1×ℤ_N_2 backgrounds, we locally promote the SU(N_1) × SU(N_2) gauge fields (a_1,a_2) to the U(N_1) × U(N_2) gauge fields (a_1,a_2), 

    a_i=a_i+1/N_i C_i 1_N_i   (locally).

The U(N_i) field strength is defined by f_i=a_i+a_i∧a_i, and its 1-form gauge-invariant combination is given by f_i = f_i - B_i.
To implement the quotient structure, we note that the (a_1 + A) combination (resp. a_2 - A combination) is kept under the ℤ_N_1 background (resp. ℤ_N_2 background) gauge transformations.
Hence, we locally modify the U(1)_V background gauge field A as,

    A =  A - 1/N_1 C_1 + 1/N_2 C_2    (locally),

and regard A as a proper U(1) gauge field (satisfying the cocycle condition).
Then, the combination f_1 +  A (resp. f_2 -  A) does not have a fractional ℤ_N_1 (resp. ℤ_N_2) flux.
Incidentally, the covariant derivative is not affected by the ℤ_N_1×ℤ_N_2 two-form backgrounds,

    DΨ⇒D[A]Ψ=γ^μ(∂_μΨ+ a_1 Ψ- Ψa_2+ AΨ).

To sum up, the introduction of the background gauge fields replaces the field strengths as

    f_1 ⇒f_1 - B_1 1_N_1, 
        f_2 ⇒f_2 - B_2 1_N_2,  
        A ⇒A + B_1-B_2,

where f_1, f_2, and A obey the standard Dirac quantization, and the backgrounds (B_1,B_2) introduce fractional fluxes.


For later purposes, it is convenient to rotate the basis of (B_1,B_2) in order to separate the ℤ_lcm(N_1,N_2) “quotient part” B_A for the U(1)_V/ℤ_(N_1,N_2) symmetry and the ℤ_gcd(N_1,N_2)^[1]-symmetry background B <cit.>: The translation rule turns out to be 

    {     B_A = -B_1 + B_2 , 
        B = - (M_2N_1' B_1 + M_1 N_2' B_2),
        .

where M_1,M_2 are Bezout coefficient (<ref>), and the inverse relation is given by

    {     B_1 = -  M_1 N_2' B_A - B , 
        B_2 = M_2N_1' B_A - B. .
 
Let us derive this result. The “quotient part” B_A can be easily found from the minimal coupling A ⇒A + B_1-B_2 =: A - B_A, and 

    B_A = - B_1 + B_2 = -N_2'  C_1 + N_1'  C_2/lcm(N_1,N_2),

is indeed a 2-form ℤ_lcm(N_1,N_2) background field since N_1' and N_2' are coprime integers.
In terms of a pair of U(1) gauge fields (C_A,B_A), the ℤ_lcm(N_1,N_2) background is

    lcm(N_1,N_2) B_A =  C_A,   
        C_A = -N_2' C_1 + N_1'  C_2 ,

with the 1-form gauge invariance

    C_A ∼ C_A + lcm(N_1,N_2) (- Λ_1 + Λ_2).

Next, we shall extract the ℤ_gcd(N_1,N_2)^[1] symmetry background B from (B_1,B_2), which should satisfy (N_1, N_2) B= C with some U(1) gauge field C. 
According to the basis rotation of the group (<ref>), this can be achieved as

    [ C_A    ;   C    ;     ]
    =
    [ -N_2'  N_1';  -M_2  -M_1;       ][ C_1; C_2;     ].

This transformation is invertible within integer coefficients as the transformation matrix has a unit determinant.
For C = -(M_2 C_1 + M_1 C_2), the 1-form gauge invariance becomes

    C ∼ C - gcd(N_1,N_2) ( M_2 N_1' Λ_1 + M_1 N_2' Λ_2).

Since the transformation matrix is invertible with integer coefficients, we can reparametrize the 1-form gauge transformations (<ref>) and (<ref>) as,

    C_A ∼ C_A + lcm(N_1,N_2) Λ_A,   C ∼ C + gcd(N_1,N_2) Λ,

where Λ and Λ_A are canonically normalized U(1) gauge fields.
Now, we can see the ℤ_gcd(N_1,N_2)^[1] symmetry background is expressed by the following pair of (B,C),

    gcd(N_1,N_2) B =  C,  C= -(M_2 C_1 + M_1 C_2).

To summarize, after the basis rotation, the 1-form gauge transformations for the background gauge fields are given by 

    {A↦   A+Λ_A, 
            B_A↦    B_A +Λ_A,      with   C_A↦ C_A + (N_1,N_2)Λ_A,
    
            B ↦    B + Λ,      with  C↦ C+(N_1,N_2)Λ.
        .

We note that “(N_1,N_2)(Ã-B_A)” gives the canonically quantized U(1) gauge field, which can be understood as the baryon-number gauge field <cit.>. 

The gauge-invariant local topological action (2π) of the background gauge fields is then given by 

    S^k,ℓ,ϕ_SPT [A, B_A, B]    = gcd(N_1,N_2) k/4π∫ B ∧ B + lcm(N_1,N_2)  ℓ/2π∫ B ∧ (A - B_A) 
         + (lcm(N_1,N_2))^2 ϕ/8π^2∫ (A - B_A)^2,

where k, ℓ∈ℤ_gcd(N_1,N_2) are discrete labels, and ϕ is continuous parameter with 2π periodicity. 
When the partition functions of the gapped vacua belong to different discrete labels (k,ℓ), there is no continuous deformation of the theory that connects those vacua. 
This means that they are discriminated as SPT states with the G_vec symmetry, and those states must be separated by some quantum phase transitions. 




 §.§ Anomaly for the theta1,2 periodicity and for the chiral symmetry (Zgcd(N1,N2))chiral


To find an 't Hooft anomaly of QCD(BF),  we define the partition function equipped with the gauge background A, B_A, B,

    Z_θ_1,θ_2 [A, B_A, B] := ∫𝒟a_1 𝒟a_2 𝒟Ψ𝒟Ψ ^-S[A, B_A, B] ,

where the gauged action is given by

    S[A, B_A, B]    :=   1/g_1^2∫  |f_1-B_1|^2 +  1/g_2^2∫ |f_2-B_2|^2 +∫trΨ(D[A]+m)Ψ
       +θ_1/8π^2∫tr[(f_1-B_1)^2]+θ_2/8π^2∫tr[(f_2-B_2)^2].

Here, (B_1,B_2) are given by (<ref>) in terms of (B_A, B).

Let us first discuss the generalized anomaly (or global inconsistency) related to the θ periodicities.
This partition function violates the usual θ angle periodicity, and it acquires the local counterterm; 

    Z_θ_1+2π,θ_2 [A, B_A, B]    = exp(  N_1/4 π∫ B_1 ∧ B_1 ) Z_θ_1,θ_2 [A, B_A, B] , 
    
        Z_θ_1,θ_2+2π [A, B_A, B]    =  exp(  N_2/4 π∫ B_2 ∧ B_2  ) Z_θ_1,θ_2 [A, B_A, B] .

We consider the implications of these global inconsistencies.
Let us suppose that the system has a trivial vacuum for some (θ_1,θ_2), 
then the partition function is described by the local topological action of the background fields in the low-energy limit:

    Z_θ_1,θ_2[A, B_A, B]    exp( S^k,ℓ,ϕ_SPT [A, B_A, B] ),

which has discrete labels k,ℓ∈ℤ_(N_1,N_2). 
For general 2π shifts of θ angles (θ_1,θ_2) ↦ (θ_1+2π n_1,θ_2+2π n_2), the relation (<ref>) with (<ref>) shows that 
the discrete labels of the effective action (<ref>) must jump as

    [ k; ℓ ]↦[ k; ℓ ]
        +
        [ N_1' N_2'; -M_1  M_2;      ][ n_1; n_2;     ].

As long as these shifts are nontrivial in ℤ_(N_1,N_2)×ℤ_(N_1,N_2), the two points (θ_1,θ_2) and (θ_1+2π n_1,θ_2+2π n_2) can be distinguished as SPT states.
Note that the above 2× 2 matrix has a unit determinant, so it is invertible with integer coefficients.
Therefore, there is a global inconsistency for the shift (θ_1,θ_2) ↦ (θ_1+2π n_1,θ_2+2π n_2) unless both n_1 and n_2 are multiples of gcd(N_1,N_2).
This global inconsistency rules out a trivially gapped phase connecting the two points (θ_1,θ_2) and (θ_1+2π n_1,θ_2+2π n_2).
One can interpret this obstruction as an anomaly between U(1)_V/ℤ_lcm(N_1,N_2), ℤ_gcd(N_1,N_2)^[1], and the θ_1 or θ_2 periodicity (see Refs. <cit.> for related anomalies), and one can check the large-N phase diagram (e.g. Figure <ref>) satisfies this constraint. 


In the massless case m=0, the SU(N_1) × SU(N_2) QCD(BF) additionally has the discrete chiral symmetry (ℤ_gcd(N_1,N_2))_χ, and let us discuss its mixed 't Hooft anomaly with G_vec.
Since the axial transformation shifts the theta angles, the calculation is parallel to that of global inconsistency (see also Refs. <cit.>).
The discrete chiral transformation (ℤ_gcd(N_1,N_2))_χ changes the fermion integration measure, 

    ΨΨ   ↦exp(2π/gcd(N_1,N_2)ind(D[A])) ΨΨ,

so we have to evaluate the index of the Dirac operator (<ref>),

    ind(D[A])
           =∫1/8π^2[(f_1⊗1_N_2-1_N_1⊗f_2+A 1_N_1⊗1_N_2)^2] 
       =∫1/8π^2[((f_1-B_1)⊗1_N_2-1_N_1⊗(f_2- B_2)+(A - B_A)1_N_1⊗1_N_2)^2]
       =∫1/8π^2(N_2 (f_1-B_1)^2 + N_1 (f_2-B_2)^2+N_1N_2(A - B_A)^2).

Note that the anomaly (<ref>) depends only on ind(D[A]) in mod (N_1,N_2). 
This modulo-gcd(N_1,N_2) index can be calculated as follows[We added A, which gives no contribution, in order to make the gauge-invariant combination A - B_A and align the expressions with (<ref>).]

    ind(D[A])
           = N_1 N_2/8π^2∫ (-B_1 ∧ B_1 - B_2 ∧ B_2 +  B_A ∧ B_A)   [mod gcd(N_1,N_2)] 
       = -2 N_1 N_2/8π^2∫ B ∧ B + 2 N_1 N_2 (M_1 N_2' - M_2 N_1' ) /8π^2∫  B ∧ (A - B_A) 
           + 2 M_1 M_2 (lcm (N_1,N_2))^2 /8π^2∫  (A - B_A)^2   [mod gcd(N_1,N_2)].

Here we have used N_1 N_2 (1- (M_1 N_2')^2 - (M_2 N_1')^2 ) = 2 M_1 M_2 (lcm (N_1,N_2))^2, which follows from the definition of (M_1,M_2), (<ref>).

This anomaly provides a constraint on possible phases, which rules out the trivially gapped phase. 
One of the minimal possibilities to satisfy the anomaly matching is to spontaneously break the discrete chiral symmetry completely. 
To illustrate this, we note that the labels of the topological action (<ref>) are shifted by the discrete chiral transformation as follows,

    (k,ℓ,ϕ)↦(k-2 N_1'N_2', ℓ + M_1N_2' - M_2 N_1', ϕ + 2π2 M_1 M_2/gcd(N_1,N_2)).

We can see that all the nontrivial discrete chiral transformations (ℤ_gcd(N_1,N_2))_χ change the discrete labels. 
To this end, let us rewrite the above transformation of the discrete labels as 

    [ k; ℓ ]↦[ k; ℓ ]
        -
        [ N_2' N_1';  M_2 -M_1 ][ N_1'; N_2' ],

which suggests that it is convenient to change the (k,ℓ) basis to another basis defined by 

    [ k̃; ℓ̃ ]
        =
        [ -N_2'  N_1';   M_2   M_1 ][ N_2' N_1';  M_2 -M_1 ]^-1[ k; ℓ ]
        =
        [ -N_2'  N_1';   M_2   M_1 ][   M_1  N_1';   M_2 -N_2' ][ k; ℓ ].

Then, the discrete chiral transformation acts as (k̃,ℓ̃)↦ (k̃,ℓ̃-1), and it is now evident that all (ℤ_gcd(N_1,N_2))_χ are anomalous under the G_vec background gauge field. 
Somewhat surprisingly, the discrete label (k̃,ℓ̃) naturally appears in the semiclassics as we shall discuss in Section <ref>. 
As a summary, the anomaly matching condition requires the complete chiral symmetry breaking, 

    (ℤ_gcd(N_1,N_2))_χ 1,

under the following assumptions: (1) The system is confining so that ℤ_gcd(N_1,N_2)^[1] is unbroken, (2) the vector-like U(1)_V/ℤ_lcm(N_1,N_2) symmetry is unbroken, and (3) there is no accidental massless mode or topological order in the low-energy theory.






§ SEMICLASSICS THROUGH T2 COMPACTIFICATION WITH THE BARYON-'T HOOFT FLUX



In this section, we develop a semiclassical framework for studying the dynamics of the different-rank QCD(BF). 
It is based on the novel semiclassical approach to the 4d gauge theories through the T^2-compactification with the 't Hooft flux <cit.>, which preserves the 4d 't Hooft anomalies in 2d effective theories and explains the confinement by the proliferation of center vortices or fractional instantons. 
The effective description of this T^2-compactified setup captures qualitative features of strongly-coupled 4d confining gauge theories, including pure SU(N) Yang-Mills theory, 𝒩=1 SU(N) SYM, QCD with fundamental quarks, 2-index quarks, and equal-rank QCD(BF) <cit.> (see also Refs. <cit.> for related studies on ℝ× T^3). 

Section <ref> describes the detailed setup of the T^2 compactification for the different-rank QCD(BF).
In Section <ref>, we derive the two-dimensional semiclassical effective theory, which serves as a foundational framework for the exploration of phase diagrams in subsequent sections under the assumption of the adiabatic continuity.




 §.§ T2 compactification with baryon-'t Hooft flux



We put the QCD(BF) on ℝ^2× T^2 with the nontrivial background gauge fields on T^2 so that the minimal 't Hooft fluxes are inserted for both SU(N_1) and SU(N_2) gauge groups: 

    ∫_T^2 B_1=2π/N_1,   ∫_T^2 B_2=2π/N_2.

For this purpose, we translate these conditions into those of the background fields for the global symmetry, U(1)_V/ℤ_(N_1,N_2)×ℤ_(N_1,N_2)^[1], according to (<ref>) (with setting A=0), 

    ∫_T^2(N_1,N_2)(A-B_A)=2π(N_2'-N_1'),   ∫_T^2 B = 2π-(M_1+ M_2)/(N_1, N_2).

This is the baryon-'t Hooft flux in our T^2-compactification. 

To obtain the 2d semiclassical description, it is convenient to realize the baryon-'t Hooft flux in terms of the transition functions. 
Let us denote the coordinates as x = (x_1,x_2,x_3,x_4) = (x, x_3, x_4) ∈ℝ^2 × T^2, and we identify (x_3,x_4) with both (x_3+L,x_4) and (x_3,x_4+L) to construct the torus T^2.
We call the transition functions on T^2 for SU(N_i) gauge group g_3^(i)(x_4) and g_4^(i)(x_3) (i = 1,2).
The 't Hooft fluxes require that the transition functions satisfy

    g_3^(i)(L)^† g_4^(i)(0)^† g_3^(i)(0) g_4^(i)(L) = ^2 π/N_i1_N_i.

Using the gauge transformations, we get the coordinate-independent transition functions,

    g_3^(i)(x_4) = S_i,    g_4^(i)(x_3) = C_i   (i=1,2),

where C_i and S_i are the SU(N_i) clock and shift matrices: C_i ∝diag(1,^2 π/N_i, ⋯,^2 π (N_i-1)/N_i) and (S)_mn∝δ_m+1,n.
For N_1 ≠ N_2, the wavefunction of the bifundamental field cannot be single-valued with these 't Hooft fluxes, and we also introduce the fractional U(1)_V flux in the torus T^2 to compensate it:

    A = ( 2 π/N_1 L^2 - 2 π/N_2 L^2) x_3  x_4

Note that this is equivalent to (<ref>) as A = A-B_A according to the local expression (<ref>). 
Consequently, the boundary conditions for the bifundamental fermion Ψ(x) are written as

    Ψ(x, x_3+L, x_4)    = ^( 2 π/N_1 L - 2 π/N_2 L) x_4 g_3^(1)(x_4)^†ψ(x, x_3, x_4) g_3^(2)(x_4)  
    Ψ(x, x_3, x_4+L)    =  g_4^(1)(x_3)^†Ψ(x, x_3, x_4) g_3^(2)(x_4).

Now, these twisted boundary conditions are consistent with the single-valuedness of the bifundamental fermion Ψ(x, x_3, x_4). 



 §.§ 2d effective description



The adiabatic continuity conjecture implies that the weakly-coupled theory on ℝ^2 × T^2 at small T^2 can provide insight into the qualitative aspects of the original strongly-coupled theory on ℝ^4. 
With the compactification described above, let us derive the 2d effective theory for the weakly-coupled theory on ℝ^2 × T^2 at small T^2.

Our approach involves the following two steps: 
First, we identify low-energy modes that remain perturbatively massless after the small T^2 compactification.
Subsequently, we incorporate center-vortex contributions together with the residual gauge symmetry after the Higgsing by the Polyakov loops along T^2.
 


  §.§.§ Perturbative analysis


To construct the 2d effective theory, let us first identify the low-energy modes of the 2d effective theory.
For SU(N_1) and SU(N_2) gluons, the boundary conditions admit no constant modes, and all excitations have O(1/NL) mass on the small torus T^2.
This can be understood as the adjoint Higgsing, SU(N_1)× SU(N_2) →ℤ_N_1×ℤ_N_2 (see Ref. <cit.> for details). 
The residual discrete gauge fields turn out to play an important role beyond perturbation theory, but we may neglect it for a while. 

For the bifundamental fermion, the low-energy modes can be found by solving the 2d zero-mode equation on T^2,

    [ γ_3 ∂_3 + γ_4 ( ∂_4 +  A_4) ] Ψ(x_3,x_4) = 0,

with the boundary conditions

    Ψ(x_3+L,x_4)    = ^( 2 π/N_1 L - 2 π/N_2 L) x_4 S_1^†Ψ(x_3,x_4) S_2, 
    Ψ(x_3,x_4+L)    = C_1^†Ψ(x_3,x_4) C_2.

We can easily count the number of such zero-modes by using the 2d index theorem,

    ind (D) = dim (Rep) ∫_T^2 A/2 π=N_2-N_1,

where the dimension of the representation is dim (Rep) = N_1N_2, and the flux is ∫_T^2F_U(1)/2 π = 1/N_1 - 1/N_2. 
As a result, there are (N_2 - N_1) of 2d massless Dirac fermions at the classical level.
Of course, we can obtain the same result by directly solving the zero-mode equation with the above boundary condition, and we demonstrate it in Appendix <ref>.


We note that most of these zero-modes are accidental. Recall that QCD(BF) does not have continuous chiral symmetries that can force massless fermions. 
To be more precise, we are now restricting ourselves to the perturbative analysis, so there still exists the U(1)_χ symmetry. 
This requires the presence of “one” massless fermion, but the other N_2-N_1-1 massless fermions are supposed to be gapped out by considering the interaction effect within the perturbation theory. 
To see it more clearly, let us perform the Abelian bosonization and we have N_2-N_1 compact bosonz, φ_1,…, φ_N_2-N_1. 
The U(1)_χ acts on them as the shift symmetry, φ_i↦φ_i + α. 
As the difference φ_i-φ_j is invariant under U(1)_χ, its mass term can be generated by the perturbative diagrams, in particular by the 1-loop diagrams <cit.>:

    Δ S
        =
        
        < g r a p h i c s >
    
        =∫ d^2 x ∑_i≠ j J_ijcos(φ_i-φ_j).

Note that this mass term for φ_i-φ_j is perturbatively generated: J_ij∼ O(g^4/(NL)^2), which is far larger than the (nonperturbatively small) dynamical scale ^-♯/g^2.
Therefore, the genuine low-energy mode for small T^2 is described by the diagonal one,

    φ := φ_1 = φ_2 = ⋯ = φ_N_2-N_1.

In what follows, we consider the 2d low-energy effective theory in terms of φ.
Since we obtain 2d free Dirac fermions, the action for the 2d effective theory in the lowest order would be,

    S_2d^(0)[φ] =  ∫^2x [ N_2-N_1/8π (∂_μφ)^2 - m μcosφ],

with a (scheme-dependent) dimensionful constant μ.




  §.§.§ Center vortex and the residual ZN1xZN2 gauge field



The perturbatively massless mode φ, analogous to the η' particle, acquires a nonperturbatively small mass induced by center vortices.
The second step to derive the 2d effective theory is to include contributions from center vortices.


Prior to moving on to the calculation, it would be pertinent to mention some aspects of center vortices in this setup.
Virtually, let us further compactify ℝ^2 into another T^2 with nontrivial 't Hooft flux n_12^(1) = 1 for SU(N_1).
Then, there exists a classical fractional instanton with

    (Q_top^(1), Q_top^(2)) = (1/N_1,0),

where Q_top^(i) denotes the instanton number for the SU(N_i) gauge field. 
Because of the perturbative gap, this classical solution should behave as a local vortex in the decompactified limit T^2 × T^2 →ℝ^2 × T^2. 
As suggested by numerical studies <cit.> (see also Refs. <cit.> for analytic studies), we further assume that this vortex solution saturates the BPS bound and the action is given by S = S_I^(1)/N_1, with the instanton action S_I^(1) = 8 π^2/g^2_1. 
In parallel, there also exists (Q_top^(1), Q_top^(2)) = (0,1/N_2) fractional instanton with the action S = S_I^(2)/N = 8 π^2/N_2 g^2_2 as a local vortex solution.
Within the framework of 2d effective theory, we can identify these fractional instantons as so-called center vortices.


Let us include the contributions from center vortices by the dilute gas approximation.
Using the U(1)_axial spurious symmetry,

    φ   →φ + 2 α,   θ_1 →θ_1 + 2 N_2α,   θ_2 →θ_2 + 2 N_1α,

the vertices of center vortices for SU(N_1) and SU(N_2) can be uniquely fixed:

    (± 1/N_1,0) vortex  𝒱_1(x) = K^(1)^-S_I^(1)/N_1 + θ_1 /N_1^-  N_2 φ /N_1
    𝒱_1^* (x) = K^(1)^-S_I^(1)/N_1 - θ_1 /N_1^ N_2 φ /N_1, 
    
    
    (0,± 1/N_2) vortex  𝒱_2(x) = K^(2)^-S_I^(2)/N_2 + θ_2 /N_2^-  N_1 φ /N_2
    𝒱_2^* (x) = K^(2)^-S_I^(2)/N_2 - θ_2 /N_2^ N_1 φ /N_2,

where K^(1) and K^(2) are some dimensionful positive constants of O(1/L^2). Using the 1-loop renormalization group, the magnitudes of these vertices can be roughly estimated as

    K^(1)^-S^(1)_I/N_1≃1/L^2(Λ_1 L)^(11-2 N_2/N_1)/3,   
        K^(2)^-S^(2)_I/N_2≃1/L^2(Λ_2 L)^(11-2 N_1/N_2)/3.


The operators ^-  N_2 φ /N_1 and ^-  N_1 φ /N_2 do not respect the 2π periodicity of φ and demand further refinement.
We should notice that there are residual ℤ_N_1×ℤ_N_2 gauge fields (a_1,a_2), which magnetically couple to φ:

    /2π∫ (θ_1-N_2 φ)a_1+/2π∫ (θ_2-N_1φ)a_2.

One can easily check consistency with the center-vortex vertices (<ref>).[Roughly speaking, the center vortices are the defects with fractional fluxes: a_1 ∼ 2π/N_1 and a_2 ∼ 2π/N_2. The above magnetic coupling (<ref>) reproduces the vertices (<ref>)]
The integration over the gauge field (a_1,a_2) restrict the compact boson to satisfy ∫φ∈ 2 π N_1' and ∫φ∈ 2 π N_2', which extends the periodicity of φ as φ∼φ + 2 π N_1' N_2', and ^-  N_2 φ /N_1 and ^-  N_1 φ /N_2 become well-defined.
Here, we still have N_1' N_2' ambiguity to lift the 2π-periodic scalar to a 2 π N_1' N_2'-periodic one, and this requires the discussion on the discrete vacuum labels. 

The discrete vacuum labels appear because the appropriate boundary condition for ℝ^2 constrain the total topological charges (Q_top^(1), Q_top^(2)) to be integers.
Indeed, the dilute gas approximation of fractional instantons with this constraint yields

    Z    = ∫𝒟φ ^-S_2d^(0)[φ] 
           ×[ ∑_n_1, n̅_1 ≥ 01/n_1! n̅_1 !( ∫^2x 𝒱_1(x) )^n_1( ∫^2x 𝒱_1^*(x) )^n̅_1δ_n_1-n̅_1 ∈ N_1 ℤ] 
           ×[ ∑_n_2, n̅_2 ≥ 01/n_2! n̅_2 !( ∫^2x 𝒱_2(x) )^n_2( ∫^2x 𝒱_2^*(x) )^n̅_2δ_n_2-n̅_2 ∈ N_2 ℤ]
        
       = 1/N_1 N_2∑_k_1 =0^N_1-1∑_k_2 = 0^N_2 -1∫𝒟φ ^-S_2d^(0)[φ] - S_center vortex^(k_1,k_2)[φ],

where k_1∈ℤ_N_1 and k_2∈ℤ_N_2 are the discrete vacuum labels and 

    S_center vortex^(k_1,k_2)[φ]    = ∫^2x {- 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π k_1 - θ_1/N_1). 
          . - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ  + 2 π k_2 - θ_2/N_2)}.

The periodicity of discrete labels, k_1∼ k_1+N_1 and k_2∼ k_2+N_2, are trivially realized, but the 2π shift of φ requires a nontrivial identification, 

    (φ+2π, k_1, k_2) ∼ (φ, k_1+N_2, k_2+N_1).

As ℤ_N_1×ℤ_N_2≃ℤ_(N_1,N_2)×ℤ_(N_1,N_2), we change the basis for the discrete labels as[Recall that our representative of the Bezout coefficient (<ref>) satisfies M_1∈ N'_2ℤ and M_2∈ N'_1 ℤ, and thus ℓ=M_1 k_1+M_2 k_2 is well-defined in ℤ_(N_1,N_2). ] 

    [ k̃; ℓ̃ ]
        =[  N_1' -N_2';   M_1   M_2 ][ k_1; k_2 ],

so that k̃∼k̃+(N_1, N_2) and ℓ̃∼ℓ̃+(N_1,N_2). In this basis, (<ref>) becomes  

    (φ+2π, k̃, ℓ̃)∼ (φ, k̃, ℓ̃+(N_1,N_2)).

Since (N_1,N_2)=(N_1,N_2)N_1'N_2', this shows that ℓ̃ also becomes the ℤ_(N_1,N_2) label by extending the periodicity of φ as φ∼φ+2π N_1' N_2'. 
Equivalently, we may regard k_1,k_2 are both the ℤ_(N_1,N_2) labels once φ is extended as φ∼φ+2π N_1' N_2'. 


Consequently, the low-energy effective theory is described by the extended periodic scalar φ, φ∼φ + 2π N_1' N_2', and the (ℤ_(N_1,N_2))^2 vacuum labels (k̃,ℓ̃) (or (k_1,k_2)), and its potential is given by 

    V[φ; k̃,ℓ̃]    = - m μcos (φ) - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π (N'_2 ℓ̃ + M_2 k̃) - θ_1/N_1) 
                 - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ
        + 2 π (N'_1 ℓ̃ - M_1 k̃) - θ_2/N_2),

where μ is some dimensionful constant introduced in the Abelian bosonization.[In Sections <ref> and <ref>, we are going to identify the vacua by finding the minima of this potential, which corresponds to the classical analysis. Strictly speaking, the 2d effective theory still has the renormalization-group (RG) flow, and it changes the coefficient of the effective potential. We note that (<ref>) contains the relevant perturbation, and thus the RG flow stops at a certain energy scale. Therefore, the classical analysis in Sections <ref> and <ref> is going to be justified by a suitable rescaling of these parameters. ]




 §.§ Realization of anomaly constraints in semiclassics



Here, we shall see how the anomaly constraints are realized within our semiclassical framework. 
In this T^2-compactification, we have introduced the background gauge fields (<ref>) along the compactified direction to preserve 4d anomalies in the 2d effective theory <cit.>.
As we have discussed in Section <ref>, there are the following constraints in the 4d setup:

    
  * Global inconsistency

    In the (θ_1,θ_2) plane, it is impossible to connect (θ_1,θ_2) and (θ_1 + 2 π n_1,θ_2 + 2 π n_2) with a trivially gapped phase, unless both n_1 and n_2 are multiples of gcd(N_1,N_2).


    
  * Mixed anomaly between (ℤ_gcd(N_1, N_2))_χ and U(1)_V/ℤ_lcm(N_1,N_2)×ℤ_gcd(N_1, N_2)^[1]
    
    In the massless case m = 0, the anomaly requires the complete spontaneous breaking of discrete chiral symmetry.



Let us first discuss the anomaly for the discrete chiral symmetry (ℤ_(N_1,N_2))_χ. 
As the compact boson is related to the chiral condensate as (Ψ1+γ_5/2Ψ)∼^φ, the discrete chiral symmetry acts as the shift symmetry of φ in the 2d effective description. 
Indeed, the potential (<ref>) at the massless point m=0 is invariant under the following transformation, 

    [  φ; k̃; ℓ̃ ]↦[ φ+2π/(N_1,N_2);             k̃;           ℓ̃-1 ]↦⋯↦[         φ+2π;           k̃; ℓ̃-(N_1,N_2) ]∼[  φ; k̃; ℓ̃ ],

where (<ref>) is used to see that this transformation forms ℤ_(N_1,N_2). 
Notably, the discrete chiral transformation shifts the discrete label, ℓ̃↦ℓ̃-1, so we can immediately conclude the (N_1,N_2) degenerate vacua associated with the chiral symmetry breaking. 
Therefore, the requirement of 4d 't Hooft anomaly is correctly realized in the 2d semiclassical framework. 


Similarly, we can understand the global inconsistency constraint when m≠0 from the 2d viewpoint. 
The 2π shifts of θ angles, (θ_1,θ_2)↦ (θ_1+2π n_1, θ_2+2π n_2), can be absorbed by the shifts of discrete labels (k_1,k_2)↦ (k_1+n_1, k_2+n_2), which is equivalent to 

    [ k̃; ℓ̃ ]↦[ k̃; ℓ̃ ]
        + 
        [  N'_1 -N'_2;   M_1   M_2 ][ n_1; n_2 ].

We should note that both k̃,ℓ̃ are the ℤ_(N_1,N_2) labels after the periodicity extension, φ∼φ+2π N'_1N'_2. 
Since the discrete labels are non-trivially shifted unless both n_1 and n_2 are multiples of gcd(N_1,N_2), the unique and trivially gapped vacuum is impossible to connect (θ_1,θ_2) and (θ_1+2π n_1, θ_2+2π n_2) without encountering quantum phase transitions. 
This is exactly the statement of the global inconsistency in 4d.


Lastly, we point out that these transformation properties (<ref>) and (<ref>) of (k̃,ℓ̃) turns out to be exactly the same with the ones for the label (k̃,ℓ̃) of (<ref>), so it would be quite natural to identify them. Then, (k̃,ℓ̃) in the semiclassics carries the information of the 4d SPT action completely even after the T^2 compactification with the baryon-'t Hooft flux, and this is a nontrivial evidence for the adiabatic continuity.  




§ PHASE DIAGRAMS OF QCD(BF) FOR GCD(N1,N2)=1



In this section, we explore the consequences of the semiclassical description for different-rank QCD(BF) with gcd(N_1,N_2) = 1. In this case, the discrete labels k_1, k_2 (or k̃,ℓ̃) are completely absorbed by the periodicity extension, φ∼φ+2π N_1 N_2, and the potential (<ref>) becomes 

    V[φ]    = - m μcos (φ) - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ - θ_1/N_1) - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ - θ_2/N_2).

In Section <ref>, we study the phase diagrams in the large-mass and small-mass limits by finding the minima of this effective potential, and we also provide a numerical demonstration of how these two limits are connected via the topology change. 
In Section <ref>, we establish the duality conjecture by Karasik and Komargodski <cit.> in the hierarchical and large-N limits within our semiclassical effective theory. Moreover, by constructing the explicit duality map as suggested by the combination of semiclassics and the large-N limit, we numerically justify its validity even for finite parameters. 




 §.§ Phase diagrams for gcd(N1,N2)=1 with semiclassics



We can analytically study the small-mass and large-mass limits m → +0,+∞. 
Let us consider the large-mass limit first, and then go to the small-mass limit.
We shall see that the semiclassical description supports the natural scenarios for both the small-mass and large-mass limits.
We then numerically explore how the intermediate-mass range connects these different limits and observe the topology change of the (θ_1,θ_2) phase diagram. 









  §.§.§ Large-mass limit m->infty



In the limit m→∞, we first need to minimize -m μcos(φ) in (<ref>), and the candidates of minima are given by φ=2πℓ̂ with ℓ̂∼ℓ̂+N_1N_2 at order O(m). 
    
We can decompose ℓ̂=M_1 k_1+M_2 k_2 with k_1∼ k_1+N_1 and k_2∼ k_2+N_2 as (N_1,N_2)=1. Since N_2 M_1+N_1 M_2=1 and M_1/N_2, M_2/N_1∈ℤ, the potential (<ref>) at O(m^0) becomes

    V[φ = 2 π (M_1k_1 + M_2 k_2)] 
       = - 2K^(1)^-S_I^(1)/N_1cos( 2 π N_2 M_1 k_1 - θ_1/N_1)
        - 2K^(2)^-S_I^(2)/N_2cos( 2 π N_1 M_2 k_2 - θ_2/N_2) 
       =- 2K^(1)^-S_I^(1)/N_1cos( 2 π k_1 - θ_1/N_1) 
        - 2K^(2)^-S_I^(2)/N_2cos( 2 π k_2 - θ_2/N_2).

This reproduces the phase diagram of the SU(N_1)× SU(N_2) pure Yang-Mills theory, where the phase boundaries are located on θ_1 = π (mod2π) and θ_2 = π (mod2π), and (k_1,k_2)∈ℤ_N_1×ℤ_N_2 describes the discrete vacuum label (see the left panel of Figure <ref>).



In the leading order, there are four-fold degenerate vacua at (θ_1, θ_2) = (π,π), but this degeneracy is partially resolved by including the next-to-leading-order corrections.
To see this, we add the small fluctuation φ = 2 π (M_1 k_1 + M_2 k_2) + δφ  (mod 2π N_1N_2), where δφ is supposed to be the O(m^-1) quantity. 
The order O(m^-1) corrections to the potential can be evaluated as follows:

    V    [2 π (M_1k_1 + M_2 k_2) + δφ] - V[2 π (M_1k_1 + M_2 k_2)] 
    
    
           ≃m μ/2 (δφ)^2  + [ 2K^(1)^-S_I^(1)/N_1/N_1/N_2sin( 2 π k_1 - θ_1/N_1) + 2K^(2)^-S_I^(2)/N_2/N_2/N_1sin( 2 π k_2 - θ_2/N_2) ] δφ
       = m μ/2(δφ+O(m^-1))^2 
       - 1/2 m μ[ 2K^(1)^-S_I^(1)/N_1/N_1/N_2sin( 2 π k_1 - θ_1/N_1) + 2K^(2)^-S_I^(2)/N_2/N_2/N_1sin( 2 π k_2 - θ_2/N_2) ]^2.

Therefore, the four-fold degeneracy at (θ_1, θ_2) = (π,π) is lifted to the two-fold degeneracy, and the lowest ground states φ are those of (k_1, k_2) = (0,0), (1,1). 
The phase boundary has a negative slope (Left panel of Figure <ref>), as also expected in the large-N argument.
This two-fold degeneracy at (θ_1,θ_2)=(π,π) is protected by the CP symmetry, which interchanges (k_1, k_2) = (0,0) and (k_1, k_2) = (1,1).



  §.§.§ Massless limit m->0 and small-mass perturbation

    
Next, let us consider the massless limit, m=0. 
For m=0, the potential (<ref>) depends only on θ_phys=N_1θ_1-N_2θ_2, and we can see it explicitly by shifting φ to φ+θ_2/N_1:

    V[φ+θ_2/N_1] 
           =  - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + N_2/N_1θ_2 - θ_1/N_1) - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ/N_2)
       =- 2K^(1)^-S_I^(1)/N_1cos( N_2 φ - θ_phys/N_1/N_1) - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ/N_2).

This potential exhibits a unique vacuum for φ∼φ+2π N_1 N_2 at generic values of θ_phys, and a two-fold degeneracy occurs when θ_phys=π (mod 2 π).[Here, we would like to note that this phase transition at θ_phys=π is “not” required by any kind of anomaly, as the anomaly and global inconsistency are absent for gcd(N_1,N_2) = 1.
Using the center-vortex vertices ^-N_2 φ - θ_1/N_1 and ^-N_1 φ - θ_2/N_2, we can construct an operator
( ^-N_2 φ - θ_1/N_1)^M_1/N_2( ^-N_1 φ - θ_2/N_2)^M_2/N_1=^-/N_1N_2(φ-(M_1θ_1+M_2θ_2). 
By shifting φ→φ+θ_2/N_1, this operator can be regarded as the following symmetric perturbation, 
cos( φ - M_1/N_1θ_phys/N_1 N_2), to the potential. When this perturbation is added and dominates V[φ], the ground state is always unique and behaves continuously as a function of θ_phys. ]
We note that the discrete chiral symmetry is absent for (N_1,N_2)=1, so the unique vacuum state is allowed.  
The two-fold degeneracy of θ_phys=π comes out of the 2π periodicity of θ_phys, which follows from the equivalence, θ_phys→θ_phys+2π ⟺ (θ_1, θ_2)→ (θ_1+2π M_2, θ_2-2π M_1). 
Indeed, the above potential is invariant under θ_phys→θ_phys+2π associated with φ→φ+2π M_1/N_1. 
As an example, the massless phase diagram with N_1 = 2 and N_2 = 3 is plotted in the right panel of Figure <ref>.
We note that the small mass perturbation does not introduce new transition lines when (N_1,N_2)=1, unlike Figure <ref>. 

Let us add some details to understand the structure of the massless phase diagram. 
First, we can readily confirm that the above potential at θ_phys=2π n has the unique vacuum at φ_*(n)=2π M_1/N_1n ( 2π N_1N_2). 
When we change the theta parameter gradually as θ_phys=2π n+δθ_phys with -π<δθ_phys<π, the vacuum configuration φ_*(n) also changes continuously and it lives inside 

    min(2π M_1/N_1n, 2π M_1/N_1n+δθ_phys/N_1N_2)
        <φ_*(n)<
        max(2π M_1/N_1n, 2π M_1/N_1n+δθ_phys/N_1N_2).

This is because φ=2π M_1/N_1n+δθ_phys/N_1N_2 minimizes the first term of (<ref>) and φ=2π M_1/N_1n minimizes the second one, and the true minimum should be in between them.
For θ_phys=π( =2π-π), there are two distinct configurations, φ_*(0) and φ_*(1), giving the same energy: The fact φ_*(0)≠φ_*(1) follows from the observation that they live in non-overlapping regions, 

    0<φ_*(0)<π/N_1N_2,  2π M_1/N_1-π/N_1N_2<φ_*(1)<2π M_1/N_1.

Therefore, the phase transition occurs at θ_phys = π for the semiclassical potential (<ref>).




  §.§.§ Topology change of the phase diagrams as a function of fermion mass







We have observed that the large-mass and small-mass limits yield strikingly different phase diagrams in the (θ_1,θ_2) space, and even their topological structures are distinct. 
It is quite nontrivial how the intermediate-mass range bridges these two limits. 
To obtain an overall picture, we numerically draw phase diagrams by varying the fermion mass mμ in the following setup; N_1 = 2, N_2 = 3, with 2K^(1)^-S_I^(1)/N_1 = 2K^(2)^-S_I^(2)/N_2( =1). We choose the Bezout coefficient as (M_1,M_2)=(3,-4) so that M_1N_2+M_2N_1=1 and M_1/N_2, M_2/N_1∈ℤ. 
The results of the phase diagrams at several mμ are shown in Figure <ref>. 

To illustrate the phase diagram, we plot the values of the global minimum φ of the potential V[φ] given in (<ref>).
In this figure, the minimum values φ/2π∈ [0,N_1N_2)=[0,6) are illustrated through a color gradient.
This allows us to clearly observe the phase transition, which is indicated by a sudden change in color.
Let us explain how the phase diagram changes as we increase the mass parameter mμ. 


    
  * The phase diagram at mμ=0.05 is shown in Figure <ref>. When the mass is sufficiently small, the phase diagram is qualitatively similar to that of the massless limit (right panel of Figure <ref>). Moreover, the jump of the φ values in neighboring phases is roughly given by Δφ/2π≃M_1/N_1=3/2, which is consistent with our analysis of the massless limit. 
    We note that phase boundaries are no longer parallel for nonzero mass. 

    
  * As we increase the mass, some of the non-parallel phase boundaries start to merge, and triple points (three-fold degenerate vacua) appear (see Figure <ref> for the phase diagram at m μ = 0.2). This is also a type of topology change, but it is a different one from what we have discussed in Section <ref>. 

    
  * As we increase the mass further, some phases are pushed out completely from the region 0<θ_1 < 2π and 0<θ_2 < 2π.
    During this process, the “topology change” discussed in Section <ref> happens in the vicinity of (θ_1,θ_2)=(0, π).
    Figure <ref> depicts the phase diagram at m μ = 0.3, slightly above the topology-changing mass.
    
    
  * After that, the phase diagram converges to the one in the large-mass limit. The phase diagram at mμ=0.4 is shown in Figure <ref>, and it is already qualitatively the same as the left panel of Figure <ref>. Also quantitatively, the vacuum values of φ are approximately given by φ_(k_1,k_2)/2π=3k_1-4k_2, which is consistent with our large-mass analysis. 






 §.§ On the duality cascade conjecture



As reviewed in Section <ref>, Karasik and Komargodski <cit.> proposed the infrared dualities between two QCD(BF)s in the large-N_1,2 limits: The duality claims that the following two theories,

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) for m<m^* 
       ⟷   SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2),  <ref>

have the same vacuum structures in the sense of congruence in both phase diagrams and meta-stable vacua. 
This is called the duality cascade, as we can repeatedly apply these dualities when the fermion mass is sufficiently small. 
The purpose of this subsection is to understand this duality within the semiclassical framework.

We first discuss two distinct limits, at which the above duality can be analytically shown within our semiclassical framework: 

    
  * The hierarchical limit m,Λ_1≪Λ_2 (Section <ref>), and 
    
  * The large-N_1,2 limit (Section <ref>). 

Away from these limits, it is nontrivial to what extent the duality is effective. 
Therefore, we construct the explicit duality map by combining the semiclassics and the large-N analysis, and 
we investigate the phase diagrams numerically using the corresponding parameters for (N_1,N_2)=(2,3) and (2,5) in Section <ref>. 
We find that the duality map works quite effectively, and it only has a very tiny quantitative violation. 
Hence, the proposed duality is useful to understand the topology-changing structure of the phase diagram even when neither of the above limits are not strictly taken. 


Again, we note that the asymptotic freedom of the SU(N_1)× SU(N_2-N_1) QCD(BF) imposes a constraint N_1 < 11/13N_2 for the duality (<ref>).
Similarly, because of the adiabatic continuity conjecture, we restrict our scope to cases where both parent and daughter theories have the confining QCD-like vacuum, e.g., we assume that the theory is not in the conformal window. 
This also imposes a further constraint on the possible numbers of colors (N_1,N_2).





  §.§.§ Duality in the hierarchical limit m,Lambda1<<Lambda2



In this subsection, let us consider the limit where the strong scale Λ_2 of SU(N_2) and of SU(N_2-N_1) is much larger than other scales in both theories:  m, Λ_1 ≪Λ_2. 
As we have chosen N_2>N_1, we should discriminate Λ_1≪Λ_2 and Λ_1≫Λ_2, and we consider the first case. 

If we put an extra assumption, N_2-N_1≳ N_1, 
the duality in this hierarchical limit can be shown directly in ℝ^4 by using the chiral Lagrangian. Let us have a quick look at it before working on the semiclassics. 
When Λ_1≪Λ_2, the SU(N_1) gauge field can be regarded as a background when discussing the dynamics of the SU(N_2) or SU(N_2-N_1) gauge field. 
Then, the theory enjoys the approximate SU(N_1)_L× SU(N_1)_R chiral symmetry as m≪Λ_2, and it is likely to be spontaneously broken in both sides of the dualities as N_2>N_1 and we have also assumed N_2-N_1≳ N_1. 
Then, the low-energy effective theory is described by the SU(N_1)-valued scalar field U coupled to the SU(N_1) gauge field a_1, and the SU(N_1) gauge transformation is given by U→ gUg^-1 and a_1→ g a_1 g^-1- g g^-1. 
Let us now obtain the effective action for SU(N_1)× SU(N_2) QCD(BF) at (θ_1,θ_2). To incorporate the information of θ_2 in the chiral Lagrangian, we perform the anomalous chiral transformation to put it into the phase of the fermion mass, m_1→ m_1 ^-θ_2/N_1. This anomalous transformation also affects θ_1 as 

    θ_1→θ_1-N_2/N_1θ_2=θ_phys/N_1,

and we obtain the following effective Lagrangian, 

    S_chiral∼∫ |D U|^2 - (m ^- θ_2/N_1tr U + c.c.)  + 1/g_1^2 |f_1|^2 + θ_phys/N_1/8 π^2tr f_1 ∧ f_1.

When repeating the same discussion for SU(N_1)× SU(N_2-N_1) QCD(BF) at (θ_1',θ_2), all the discussion is parallel except that the anomalous transformation on θ_1' is given by θ_1'→θ_1'-N_2-N_1/N_1θ_2. 
Therefore, we can find the same low-energy θ angle for SU(N_1) gauge group by setting θ_1'=θ_1-θ_2, and this suggests

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) 
       ⟷  SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2)

seems to hold in the limit m, Λ_1 ≪Λ_2.
However, this analysis with the chiral Lagrangian (<ref>) is actually incomplete.
For example, this chiral Lagrangian at m=0 has the apparent (ℤ_N_1)_chiral symmetry, U →^-2 π/N_1 U, while the correct chiral symmetry should be ℤ_(N_1,N_2). 
Such a subtle N_2 dependence is absent in the usual chiral Lagrangian, and we must incorporate the η' field with the extended periodicity, η'∼η'+2π N_2 <cit.>. 
Let us see how our semiclassical description takes care of this subtlety. 


For semiclassical parameters, the hierarchical limit corresponds to m, K^(1)^-S^(1)_I/N_1≪ K^(2)^-S^(2)_I/N_2, and thus the semiclassical vacua at the leading order minimizes the last term of (<ref>). 
On the SU(N_1) × SU(N_2) QCD(BF) side, this limit reduces the 2 π N_1 N_2-periodic scalar φ to the discrete one

    φ = θ_2 + 2 π N_2 ℓ/N_1,

with ℓ∈ℤ_N_1^2.
Then, there are N_1^2 candidates of vacua, and the energy of the ℓ-th vacuum at the subleading order O(Λ_2^0) becomes

    V_ℓ   = V [φ= θ_2 + 2 π N_2 ℓ/N_1] 
       = - m μcos(θ_2 + 2 π N_2 ℓ/N_1) - 2K^(1)^-S_I^(1)/N_1cos( 2 π N_2^2 ℓ - θ_phys/N_1^2).

Let us now discuss the dual side. For the SU(N_1) × SU(N_2 - N_1) QCD(BF), the same limit limit reduces the 2π N_1(N_2-N_1)-periodic field φ to the discrete one, 

    φ = θ_2' + 2 π (N_2 - N_1) ℓ'/N_1,

with ℓ' ∈ℤ_N_1^2. 
We write the theta angles in the dual theories by (θ_1', θ_2'), which are going to be set (θ_1'=θ_1 - θ_2, θ_2' = θ_2) according to the conjecture.
The energy of the ℓ'-th vacuum at the subleading order is,

    V_ℓ''    = - m μcos(θ_2' + 2 π (N_2-N_1) ℓ'/N_1) - 2K^(1)^-S_I^(1)/N_1cos( 2 π (N_2-N_1)^2 ℓ' - θ_phys/N_1^2) 
       = - m μcos(θ_2 + 2 π (N_2-N_1) ℓ'/N_1) - 2K^(1)^-S_I^(1)/N_1cos( 2 π (N_2-N_1)^2 ℓ' - θ_phys/N_1^2).


Here, let us consider the following one-to-one correspondence between ℤ_N_1^2 labels:

    (N_2-N_1)^2 ℓ' = N_2^2 ℓ.

This is one-to-one because gcd(N_1^2, (N_2-N_1)^2) = gcd(N_1^2, N_2^2) = 1.
Here we note that the Bezout coefficient M_1 satisfies both N_2 M_1 = 1  (mod N_1) and (N_2-N_1) M_1 = 1  (mod N_1).
With the identification ((N_2-N_1)^2 ℓ' = N_2^2 ℓ), the vacuum energy matches as follows,

    V_ℓ''    = - m μcos(θ_2 + 2 π M_2 (N_2-N_1)^2 ℓ'/N_1) - 2K^(1)^-S_I^(1)/N_1cos( 2 π (N_2-N_1)^2 ℓ' - θ_phys/N_1^2) 
       = - m μcos(θ_2 + 2 π M_2 N_2^2 ℓ/N_1) - 2K^(1)^-S_I^(1)/N_1cos( 2 π N_2^2 ℓ - θ_phys/N_1^2) = V_ℓ.

This establishes the duality (<ref>) in the hierarchical limit m, Λ_1 ≪Λ_2: there is an exact one-to-one correspondence between vacua of the two theories.
The generalization to the cases of gcd(N_1,N_2) ≠ 1 is straightforward.




  §.§.§ Large-N1,2 limit and the duality in semiclassics



In this section, we study the case when N_1,N_2≫ 1 with our semiclassical effective theory. 
We have to note that the perturbative mass gap for gluons is given by O(1/N_i L), and our semiclassical effective theory is derived with the assumption that this perturbative gap is well separated from the strong scale, N_i L Λ_i ≪ 1. 
Therefore, we have to make sure that the torus size L is sufficiently small to satisfy this criterion especially when N_i≫ 1. 


In Section <ref>, we discussed the vacuum structure described by (<ref>), which comes from the large-N counting argument: The vacuum structures in the large-N_1,2 limit have the multi-branch structure labeled by two integers k_1,k_2∈ℤ, and the ground-state energy of each branch is a quadratic function of θ_1,θ_2. 
By combining the large-N_1,2 limit with the semiclassics, we give not only the explicit derivation of this vacuum structure but also the concrete form of the coefficients, a_1,a_2, b, in the formula (<ref>). 
By applying this technique on both sides of the duality, we will show the concrete correspondence (see Figure <ref>). 




We first study the SU(N_1)× SU(N_2) QCD(BF) at (θ_1,θ_2). 
Let us rewrite the 2π N_1N_2-periodic field as
φ = 2 π (M_1 k_1 + M_2 k_2) + δφ  (mod 2 π N_1N_2), where -π < δφ < π, k_1 ∈ℤ_N_1, and k_2 ∈ℤ_N_2.
With this parametrization, the potential (<ref>) reads,

    V[φ]    = - m μcos (δφ)  - 2K^(1)^-S_I^(1)/N_1cos( 2 π k_1 - θ_1/N_1 + N_2/N_1δφ) 
          - 2K^(2)^-S_I^(2)/N_2cos( 2 π k_2 - θ_2/N_2 + N_1/N_2δφ).

In the large-(N_1, N_2) limit, the minimum point of δφ is around zero δφ≈ 0 for “vacuum branches” (k_1, k_2) that satisfy | θ_1 - 2 π k_1 /N_1| ≪ 1 and | θ_2 - 2 π k_2 /N_2| ≪ 1. 
We are going to consider the case θ_1,θ_2∼ O(1), and thus these constraints on the vacuum labels become |k_1/N_1|, |k_2/N_2|≪ 1. 
Due to this constraint, we may neglect the information of the periodicity and regard the vacuum labels as (k_1, k_2) ∈ℤ×ℤ in the large-(N_1, N_2) limit. 
Since the vacuum expectation value of δφ becomes O(1/N_1,2), we can solve for δφ with the quadratic expansion. Substituting the result, the ground-state energy of each branch becomes

    ℰ_(k_1, k_2) (θ_1, θ_2)
       = ℰ_(0, 0) (θ_1 -  2 π k_1, θ_2 -  2 π k_2) 
       ≃ℰ_0 + a_1 ( θ_1 - 2 π k_1 /N_1)^2 + a_2 ( θ_2 - 2 π k_2 /N_2)^2 + b ( θ_1 - 2 π k_1 /N_1) ( θ_2 - 2 π k_2 /N_2),

and each coefficient can be explicitly determined as

    ℰ_0    = - m μ - 2K^(1)^-S_I^(1)/N_1 - 2K^(2)^-S_I^(2)/N_2, 
    
        a_1    = (m μ + N_1^2/N_2^2 2K^(2)^-S_I^(2)/N_2) K^(1)^-S_I^(1)/N_1/m μ + N_2^2/N_1^2 2 K^(1)^-S_I^(1)/N_1 + N_1^2/N_2^2 2 K^(2)^-S_I^(2)/N_2, 
    
        a_2    = (m μ + N_2^2/N_1^2 2K^(1)^-S_I^(1)/N_1) K^(2)^-S_I^(2)/N_2/m μ + N_2^2/N_1^2 2 K^(1)^-S_I^(1)/N_1 + N_1^2/N_2^2 2 K^(2)^-S_I^(2)/N_2, 
    
        b    = - 4 K^(1) K^(2)^-S_I^(1)/N_1-S_I^(2)/N_2/m μ + N_2^2/N_1^2 2 K^(1)^-S_I^(1)/N_1 + N_1^2/N_2^2 2 K^(2)^-S_I^(2)/N_2.

We have the following consistency check of this result in the large-mass and massless limits:

    
  * In the large-mass limit m → +∞, the coefficients are a_1 ≃ K^(1)^-S_I^(1)/N_1, a_2 ≃ K^(2)^-S_I^(2)/N_2, and b → - 0 as expected from the SU(N_1)× SU(N_2) pure YM theory.
    
  * In the massless limit m→ 0, we find N_2^2/N_1^2a_1=N_1^2/N_2^2a_2=-b/2( =: N_1^2 N_2^2 a), and then the vacuum energy can be written as

    ℰ_(0, 0) (θ_1, θ_2)    ≃ℰ_0 + a ( N_1 θ_1 - N_2 θ_2 )^2 .

This is consistent with the fact that physics only depends on θ_phys at m=0.



We next study the dual side, SU(N_1)× SU(N_2-N_1) QCD(BF) at (θ'_1,θ'_2)=(θ_1-θ_2,θ_2) with the fermion mass m'. 
As above, since both N_1 and N_2-N_1 are large, we can effectively presume that the vacuum branches are labeled by integers (k_1', k_2') ∈ℤ×ℤ.
The energy density of each branch behaves as 

    ℰ_(k_1', k_2')' (θ_1 - θ_2, θ_2)
    
           ≃ℰ_0' + a_1' ( θ_1 - θ_2 - 2 π k_1' /N_1)^2 + a_2' ( θ_2 - 2 π k_2' /N_2-N_1)^2 + b' ( θ_1 - θ_2 - 2 π k_1' /N_1) ( θ_2 - 2 π k_2' /N_2-N_1),

where ℰ_0', a_1', a_2', and b' are counterparts of the coefficients ℰ_0, a_1, a_2, and b in the dual theory.
Explicitly, these coefficients are given by

    a_1'    = (m' μ + N_1^2/(N_2-N_1)^2 2K^(2')^-S_I^(2')/(N_2-N_1)) K^(1')^-S_I^(1')/N_1/m' μ + (N_2-N_1)^2/N_1^2 2 K^(1')^-S_I^(1')/N_1 + N_1^2/(N_2-N_1)^2 2 K^(2')^-S_I^(2')/(N_2-N_1), 
    
        a_2'    = (m' μ + (N_2-N_1)^2/N_1^2 2K^(1')^-S_I^(1')/N_1) K^(2')^-S_I^(2')/(N_2-N_1)/m' μ + (N_2-N_1)^2/N_1^2 2 K^(1')^-S_I^(1')/N_1 + N_1^2/(N_2-N_1)^2 2 K^(2')^-S_I^(2')/(N_2-N_1), 
    
        b'    = - 4 K^(1') K^(2')^-S_I^(1')/N_1-S_I^(2')/(N_2-N_1)/m' μ + (N_2-N_1)^2/N_1^2 2 K^(1')^-S_I^(1')/N_1 + N_1^2/(N_2-N_1)^2 2 K^(2')^-S_I^(2')/(N_2-N_1),

where K^(1')^-S_I^(1')/N_1 (resp. K^(2')^-S_I^(2')/(N_2-N_1)) is the center-vortex weight for SU(N_1) (resp. SU(N_2-N_1)) gauge field.

For the duality, we must have ℰ_(0, 0) (θ_1, θ_2) = ℰ_(0, 0)' (θ_1 - θ_2, θ_2)+const., which requires

    a_1    = a_1', 
    
        a_2    = N_2^2 ( a_2'/(N_2-N_1)^2 + a_1'/N_1^2 - b'/N_1 (N_2-N_1)) , 
    
        b    = N_2 ( -  2 a_1'/N_1 + b'/N_2-N_1) .

By a tedious calculation, the parameters (mμ,K^(1)^-S_I^(1)/N_1, K^(2)^-S_I^(2)/N_2) of the original theory are related to the ones of the dual theory as

    mμ   = m' μ/  1 + (N_2 - N_1)^2/2 N_1 N_2 K^(2')^-S_I^(2')/(N_2-N_1)m' μ , 
    
         K^(1)^-S_I^(1)/N_1   = K^(2')^-S_I^(2')/(N_2-N_1) N_1^3/K^(2')^-S_I^(2')/(N_2-N_1) N_1^3 - K^(1')^-S_I^(1')/N_1 (N_2 - N_1)^3 K^(1')^-S_I^(1')/N_1, 
    
         K^(2)^-S_I^(2)/N_2   = N_2^3/(N_2 - N_1)^3 K^(2')^-S_I^(2')/(N_2-N_1) .

This establishes the conjectured duality (<ref>) in the large-(N_1,N_2) limit between the SU(N_1)× SU(N_2) QCD(BF) on (θ_1,θ_2) and the SU(N_1)× SU(N_2-N_1) QCD(BF) on (θ_1-θ_2,θ_2). 
The above relation (<ref>) defines the duality map in the physical parameter range if the fermion mass of the original theory satisfies 

    mμ≤ m^*μ:= N_1 (N_2-N_1) /N_2^2· 2 K^(2)^-S_I^(2)/N_2.

In particular, the relationship of the fermion mass explicitly explains the duality at the massless point m=0 ⇔ m' = 0 and at the topology-changing point m=m^* ⇔ m' = +∞. 

We note that the positivity of K^(1)^-S_I^(1)/N_1 in (<ref>) puts the constraint on the strong scales of the dual theory as 

    K^(2')^-S_I^(2')/(N_2-N_1) 
        >(N_2 - N_1)^3 /N_1^3 K^(1')^-S_I^(1')/N_1.

Thus, the duality relation (<ref>) does not actually give the bijective relation but instead gives a map from the parent theory, SU(N_1)× SU(N_2) QCD(BF) with m<m^*, to the daughter theory, SU(N_1)× SU(N_2-N_1) QCD(BF), which is not surjective and the inverse map does not necessarily exists: 

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) for m<m^* 
       “⟶"   SU(N_1)× SU(N_2-N_1)  QCD(BF) on  (θ_1-θ_2,θ_2).

Let us emphasize that we can always find the semiclassical parameters for the daughter theory if the parent theory satisfies m≤ m^*, so this constraint is not a problem for applying the duality cascade.


We also note that the correspondence of parameters (<ref>) is consistent with the observation at the hierarchical limit Λ_2 →∞ discussed in  Section <ref>.
In our large-N formula (<ref>), we can see that K^(2)^-S_I^(2)/N_2→∞ is equivalent to K^(2')^-S_I^(2')/(N_2-N_1)→∞, so the hierarchical limit is taken on both the parent and daughter sides of the duality. Moreover, (<ref>) in this limit shows that the other parameters are kept unchanged, m = m' and K^(1')^-S_I^(1')/N_1 = K^(1)^-S_I^(1)/N_1, and this is exactly what we observed in Section <ref>.




  §.§.§ Numerical check of the duality conjecture



In this subsection, we numerically investigate the phase diagrams of SU(N_1) × SU(N_2) QCD(BF) and SU(N_1) × SU(N_2-N_1) QCD(BF) with (N_1,N_2) = (2,5), and examine the validity of duality when neither hierarchical nor large-N_1,2 limits are strictly taken. 
For the parent side, SU(2)× SU(5) QCD(BF), we set the strong scales as 

    2K^(1)^-S_I^(1)/N_1 = 2K^(2)^-S_I^(2)/N_2=1.

Numerical search of the topology changing point mμ=m^*μ gives m^* μ≈ 0.263, and it is comparable with the large-N_1,2 formula (<ref>), which suggests .m^*μ|_large-N=2· 3/5^2=0.24. 

When computing the daughter side, SU(2)× SU(3) QCD(BF), we need to set the strong scales; however, there is no clear criterion for their setting. 
Therefore, we shall use their values suggested by the large-N_1,2 formula (<ref>), 

    2K^(1')^-S^(1')_I/N_1   =2K^(1)^-S^(1)_I/N_1/1+N_2^3/N_1^32K^(1)^-S^(1)_I/N_1/2K^(2)^-S^(2)_I/N_2=8/133,
        
    
        2K^(2')^-S^(2')_I/(N_2-N_1)   =(N_2-N_1)^3/N_2^32K^(2)^-S^(2)_I/N_2=27/125.

We are going to compare the phase diagrams with several fermion masses, and we choose the fermion mass m'μ of the daughter side again from the large-N_1,2 formula (<ref>): When mμ<m^*μ, we shall set m'μ by

    m'μ= mμ/1-mμ/m^*μ=mμ/1-mμ/0.24.

We show the results of the phase diagrams in Figure <ref>. 








Let us explain the details of Figure <ref>. In the top panels, Figures <ref>-<ref>, we show the (θ_1, θ_2) phase diagram of the parent theory, SU(2)× SU(5) QCD(BF), for the range of the mass 0.02≤ m μ≤ 0.3, which includes the topology-changing point m^*μ≈ 0.263 shown in Figure <ref>. 
In the bottom panels, Figures <ref>-<ref>, we show the (θ'_1+θ'_2,θ'_2) phase diagram of the daughter theory, SU(2)× SU(3) QCD(BF), for the range of the mass 0.02≲ m'μ≤ 100. We can see that the figures of the parent theory for mμ≤ m^*μ, Figures <ref>-<ref>, and the counterparts of the daughter theory, Figures <ref>-<ref>, are showing the same vacuum structures, respectively, and the agreement is remarkable not only qualitatively but also quantitatively:


    
  * In Figure <ref>, we show the phase diagram of the parent theory for mμ=0.3>m^*μ. 
    We can see that the topological structure of the phase boundaries is the same with the large-mass limit (left panel of Figure <ref>). 
    
    
  * The phase diagram almost at mμ=m^*μ of the parent theory is shown in Figure <ref>, and it should be compared with the figure just below, Figure <ref> of the daughter theory. 
    In Figure <ref>, we set m'μ=100 as a numerical substitute of the limit m'μ→∞, so the phase diagram in the (θ'_1,θ'_2) basis looks as the checkerboard. 
    As the duality relates those theta angles to the parent one as (θ'_1,θ'_2)=(θ_1-θ_2,θ_2), we need to use the (θ'_1+θ'_2,θ'_2) basis (=(θ_1,θ_2)) when drawing the figures of the daughter theory. 
    
  * We make the fermion mass of the parent theory a little smaller than the topology-changing point, mμ=0.2<m^*μ, in Figure <ref>. 
    On the daughter side, this mass corresponds to m'μ=1.2 according to (<ref>), and we show its figure in Figure <ref>. 
    We note that m'μ=1.2 of the daughter theory is still quite heavy compared with its strong scales (<ref>), so Figure <ref> can be understood as the linear transformation of the large-mass limit (left panel of Figure <ref>) to the (θ'_1+θ'_2,θ'_2) basis. Indeed, the direct transition from the (k_1,k_2)=(0,0) state to the (k_1,k_2)=(1,1) state can be seen around (θ'_1+θ'_2,θ'_2)=(2π,π) in Figure <ref>, while its phase boundary is still tiny.  

    
  * As we lower the fermion mass further, the phase diagrams of the parent and daughter theories are deformed in the same manner. 
    At mμ=0.06, the parent theory approaches another topology-changing point as shown in Figure <ref>. 
    On the daughter side, it corresponds to m'μ=0.08 according to (<ref>), and the phase diagram with the (θ'_1+θ'_2,θ'_2) basis is shown in Figure <ref>. 
    The daughter side is a little far from its topology-changing point, but this quantitative mismatch would be acceptable recalling that our parameter setting is based on the large-N_1,2 formula.
    [
    About the parameter setting, we point out the issue coming out of the discrepancy between the large-N result of m^*μ|_large-N=0.24 and the numerical value m^*μ|_numerical≈ 0.263. In our numerical computation of the daughter theory, we use the large-N value for m^*μ to determine m'μ in (<ref>) for theoretical consistency. However, this causes a problem in that we cannot determine m'μ when m^*μ|_large-N<mμ<m^*μ|_numerical. We can circumvent this problem if we use m^*μ≈ 0.263 when determining m'μ in (<ref>), and then we get a bit lower values of m'μ compared with current ones. 
    If doing so, Figure <ref> would be a bit closer to the topology-changing point, and the apparent look of phase diagrams may become more similar between the parent and daughter theories. Anyway, we should keep in mind that our mapping of parameters has this kind of ambiguity. ] 
    
    
  * We also note that the behaviors after this topology changing point are also qualitatively the same between the parent and daughter theories as we can see in Figures <ref> and <ref>. 


Based on these observations, we are tempted to conclude that the duality map works so nicely even when neither hierarchical nor large-N_1,2 limit is strictly taken, and it is useful to understand the topology-changing behaviors for the (θ_1,θ_2) phase diagram of the parent theory. 
For example, when we are trying to study the topology-changing point mμ=0.06, Figure <ref>, of the parent theory, it would not be so easy to imagine its behavior since the phase diagram there already experienced the topology-changing phenomenon at mμ=m^*μ≈ 0.263. 
By switching to the picture of the daughter side, Figure <ref>, this topology-changing phenomenon is its first one, and we can understand it more easily as the deformation from the large-mass limit. 
Indeed, the topology change from Figure <ref> to <ref> is basically equivalent to the ones studied in Figure <ref>. 
The reason why their topology-changing scales, m'μ≈ 0.08 in Figure <ref> and mμ≈ 0.3 in Figure <ref>, are so different can be understood from the fact the duality map (<ref>) sets the strong scales of the daughter side small values. 

In this section, we have examined the validity of the duality for the case when the two strong scales of the parent theory are the same. 
It would be an interesting exercise to check the duality for the case when those strong scales are well separated, Λ_1≪Λ_2 or Λ_1≫Λ_2, and those cases are also numerically studied in Appendix <ref>: The duality works well again. 


Lastly, it might be useful to make a somewhat trivial remark that this duality at finite N_1,2 is not an exact one. 
As we will elaborate in Appendix <ref>, if the duality relation were exactly true, the phase boundaries at the topology-changing point should consist only of the completely straight lines. 
In Figure <ref>, we enlarge the phase diagram, Figure <ref>, of SU(2) × SU(5) QCD(BF) almost at mμ=m^*μ around θ_2≈π. 
The fact that the phase boundary is not a horizontal straight line indicates the non-exactness of the duality relation. 
Conversely, we may say that the violation of the duality relation is so mild, even away from the large-N limit (apart from the subtlety of the parameter mapping), and it is difficult to notice without a magnifying glass. 








§ PHASE DIAGRAMS OF QCD(BF) FOR GCD(N1,N2)>1



This section extends our analysis to the cases of gcd(N_1,N_2) ≠ 1. 
The discussion is almost parallel to the one in Section <ref>, but we should keep in mind that the 2d effective theory has the discrete vacuum labels (k̃,ℓ̃)∈ (ℤ_(N_1,N_2))^2 in addition to the extended compact scalar, φ∼φ+2π N'_1 N'_2. 
In Section <ref>, we consider the massless limit, small-mass limit (in the presence of small mass perturbation), and large-mass limit.
The semiclassical approach produces a phase diagram that aligns with expectations.
In Section <ref>, we examine the phase diagram of the simplest example, SU(2) × SU(4) QCD(BF), obtained numerically by the minimization of the effective potential.
We observe the topology-changing phenomenon as the fermion mass is varied, which bridges the small-mass phase diagram (Figure <ref>) and the large-mass checkerboard-like phase diagram.
In Section <ref>, we point out that the duality at large-N_1,2  (Section <ref>) can be easily generalized.
This section is concluded with comments on the domain wall and anomaly inflow, Section <ref>.






 §.§ Phase diagrams for gcd(N1,N2)!=1 with semiclassics



Let us determine the phase diagrams at the large- and small-mass limits based on the 2d semiclassical description. In Section <ref>, we have derived the following effective potential,

    V[φ; k,ℓ̃]    = - m μcos (φ) - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π (N'_2 ℓ̃ + M_2 k) - θ_1/N_1) 
                 - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ
        + 2 π (N'_1 ℓ̃ - M_1 k) - θ_2/N_2), <ref>

where φ is the extended compact variable φ∼φ + 2 π N_1' N_2', 
and we identify (φ;k,ℓ̃ + gcd(N_1,N_2)) ∼ (φ + 2 π; k,ℓ̃) and (φ; k + gcd(N_1,N_2),ℓ̃ ) ∼ (φ ; k,ℓ̃). 
Our choice of the Bezout coefficient (M_1,M_2) satisfies M_1/N_2', M_2/N_1'∈ℤ. 





  §.§.§ Large-mass limit m->infty



Let us briefly look at the phase diagram in the large-mass limit, Λ_1, Λ_2 ≪ m ( ≪ 1/NL).
For this purpose, instead of (k, ℓ), it is convenient to use the original notation for the discrete labels (k_1,k_2), by using (<ref>).
With these original labels, the potential is,

    V [φ]    = - m μcos (φ) - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π k_1 - θ_1/N_1) 
                 - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ
        + 2 π k_2 - θ_2/N_2),

These labels (k_1,k_2) obeys the identification (<ref>), from which

    (φ ; k_1 + gcd(N_1, N_2) , k_2)    ∼ (φ + 2 π M_1 ; k_1  , k_2), 
    
        (φ ; k_1 , k_2+ gcd(N_1, N_2) )    ∼ (φ + 2 π M_2 ; k_1  , k_2).

Thus, we can regard (k_1,k_2) as integers in { 0 ,⋯, gcd(N_1,N_2) -1 }.



In the m→∞ limit, we should first minimize -μ m cos (φ), and the candidates of minima are φ =0 (mod 2π).
As done in Section <ref>, we denote them as φ = 2 π (M_1 k_1' + M_2 k_2') with k_1' ∈ℤ_N_1' and k_2' ∈ℤ_N_2'. 
The potential of order O(m^0) is,

    V [φ]    = - 2K^(1)^-S_I^(1)/N_1cos( 2 π (gcd(N_1,N_2) k_1'  + k_1) - θ_1/N_1) 
        
          
        - 2K^(2)^-S_I^(2)/N_2cos( 2 π (gcd(N_1,N_2) k_2'  + k_2) - θ_2/N_2).

Then, we can combine the discrete labels (k_1', k_2', k_1,k_2) so that 
k̂_1 := gcd(N_1,N_2) k_1'  + k_1 ∈ℤ_N_1 and k̂_2 := gcd(N_1,N_2) k_2'  + k_2 ∈ℤ_N_2. These labels are nothing but the original ℤ_N_1×ℤ_N_2 vacuum labels that arise from the sum over center vortices in Section <ref>.
With these labels (k̂_1, k̂_2) ∈ℤ_N_1×ℤ_N_2, the vacuum energy V_k̂_1, k̂_2 = V [φ = 2 π (M_1 k_1' + M_2 k_2')] reads

    V_k̂_1, k̂_2   =  - 2K^(1)^-S_I^(1)/N_1cos( 2 πk̂_1 - θ_1/N_1)- 2K^(2)^-S_I^(2)/N_2cos( 2 πk̂_2 - θ_2/N_2).

The ℤ_N_1×ℤ_N_2 vacua with the above vacuum energy exactly reproduce the semiclassical picture of the SU(N_1)× SU(N_2) pure Yang-Mills theory, and the phase boundaries are located on θ_1 = π (mod2π) and θ_2 = π (mod2π). 
The analysis of the O(m^-1) contribution at (θ_1,θ_2)=(π,π) is exactly the same as what we have done in Section <ref>, and the large-mass phase diagram is again given by the left panel of Figure <ref>.

We note that the jump of φ in the above leading-order discussion occurs only when θ_1,2 are shifted by relatively large amounts, such as θ_1→θ_1+2π(N_1,N_2). 
When we restrict our attention to the region -2π<θ_1<2π and -2π<θ_2<2π, the leading-order vacuum configuration is always φ=0, and the phase transitions at θ_1=±π, θ_2=±π are described by the jump of discrete labels k_1,k_2. 
Thus, the discrete labels k_1,k_2 (or k̃, ℓ̃) play the primary role in the large-mass regime, and φ only affects sub-leading dynamics when (N_1,N_2)≠1. 



  §.§.§ Massless limit m->0 and small-mass perturbation



In the massless case, substituting φ+θ_2/N_1 into the effective potential yields,

    V[φ+θ_2/N_1; k,ℓ̃] 
           = - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ + 2 π (N'_2 ℓ̃ + M_2 k) - θ_phys/N_1/N_1) 
            - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ
        + 2 π (N'_1 ℓ̃ - M_1 k)/N_2).

Thus, the physics depends only on θ_phys=N_1θ_1-N_2 θ_2. 

When θ_phys=0, we can readily find the global minima, 

    φ=-2πℓ̃/(N_1,N_2),   k̃=0,

where ℓ̃∈ℤ_(N_1,N_2) due to the identification (<ref>).  
These minima represent the spontaneous breakdown of the discrete chiral symmetry (<ref>), and the vacua have (N_1,N_2)-degeneracy.
From this solution, we can construct the global minima for θ_phys=2π n. We note the equivalence between θ_phys→θ_phys+2π n and (θ_1,θ_2)→ (θ_1+2π M_2 n, θ_2-2π M_1 n), and the latter moves the vacuum label (k_1,k_2)→ (k_1+M_2n, k_2-M_1n). We can translate this result to (k̃,ℓ̃) by using (<ref>), which gives k̃→k̃+n without changing ℓ̃. 
Applying this transformation to (<ref>), we obtain the vacua for θ_phys=2π n as 

    φ=-2πℓ̃/(N_1,N_2),   k̃=n,

with ℓ∈ℤ_gcd(N_1,N_2).

When we adiabatically change θ_phys as θ_phys=2π n+δθ_phys with |δθ_phys|<π, the vacuum configurations also continuously change from (<ref>):

    φ=-2πℓ̃/(N_1,N_2)+δφ,   k̃=n.

We note that the first term of the effective potential is minimized if δφ=δθ_phys/N_1 N_2 and the second term is minimized if δφ=0. 
Thus, the true minima exists in the range |δφ|≤|δθ_phys|/N_1 N_2, so its deviation from (<ref>) is small. 
When δθ_phys reaches π, two branches with k̃=n and k̃=n+1 become degenerate, and the vacuum degeneracy becomes 2(N_1,N_2). 

We now get the result in the massless case. 
There are gcd(N_1,N_2) degenerate vacua for generic values of θ_phys. 
There are phase transitions described by the jump of k̃, and phase boundaries are located on θ_phys. = π (mod 2 π), where the vacuum degeneracy becomes 2 gcd(N_1,N_2).
The phase diagram is consistent with the large-N argument, shown in Figure <ref>.



Next, let us consider the small-mass perturbation. Since the change of θ_phys from 2π n only gives a tiny effect as we have seen above, let us simply set θ_phys=0. 
Recall that we substituted φ+θ_2/N_1 in the above discussion for the massless case, and then the mass perturbation gives the energy splitting for the chiral broken vacua (<ref>)  by 

    Δ E_ℓ   =-mμcos(θ_2/N_1-2πℓ̃/(N_1,N_2)) 
       =-mμcos(θ_2-2π N'_1 ℓ̃/N_1),

where ℓ̃=0,1,…, (N_1,N_2)-1. 
When we increase θ_2 from 0 to 2π N'_1, there is a level crossing at θ_2=π N'_1 from ℓ̃=0 to ℓ̃=1. As we have set θ_phys=0 in this argument, this phase transition point is given by (θ_1,θ_2)=π(N'_2, N'_1).  
The discrete label ℓ̃ has a jump at this point. Thus, this phase transition should form a curve separating the ℓ̃=0 and ℓ̃=1 states,[
We can determine this phase transition curve analytically in the small-mass limit: 
The phase boundary between ℓ = 0 and ℓ= 1 vacua is given by the curve in the (θ_1,θ_2) space, on which φ_*=π/(N_1,N_2) with ℓ̃=0, k̃=0 is the minimum of the masslesss potential before shifting θ_2/N_1. 
We can see this is actually sufficient by noticing that one of its chiral partners is φ'_*=-π/(N_1,N_2) with ℓ̃'=1. Then, the mass perturbation gives degenerate energy between them, -mμcos(φ_*)=-mμcos(φ'_*), which is what we want. 
Substituting φ=π/(N_1,N_2) and ℓ̃=k̃=0 to the saddle-point equation of the massless potential, we get 

    2 N_2 K^(1)^-S_I^(1)/N_1/N_1sin( π N_2' - θ_1/N_1) 
    + 2 N_1 K^(2)^-S_I^(2)/N_2/N_2sin( π N_1'  - θ_2/N_2) = 0,

and this curve goes through (θ_1,θ_2) = (π N_2', π N_1').
] as expected in the large-(N_1,N_2) argument (Figure <ref>).






  §.§.§ Topology change of the phase diagrams as a function of fermion mass




As we have done in Section <ref> for (N_1,N_2)=1, we perform the numerical calculation to see how the large-mass and small-mass phase diagrams are connected for (N_1,N_2) 1. 
Unlike the case of (N_1,N_2)=1, we have the discrete chiral symmetry in the massless limit, and the small-mass phase diagram has extra phase transition lines that exchange chiral broken vacua. 
Moreover, there are nontrivial constraints on the global structure for the phase diagram by anomaly matching conditions when (N_1,N_2)≠1, so it would be meaningful to observe the topology-changing phenomenon also in this case. 


In our computation, we consider the simplest setup: N_1 = 2, N_2 = 4, and we set their strong scales as 2K^(1)^-S_I^(1)/N_1 = 2K^(2)^-S_I^(2)/N_2=1. 
In this setup, N'_1=1 and N'_2=2, so we can take the Bezout coefficient as M_1=0, M_2=1, and this satisfies M_1/N'_2, M_2/N'_1∈ℤ. 
Then, the vacuum labels (<ref>) are simply given by k̃=k_1-2k_2=k_1 and ℓ̃=k_2 in 2, so there is no distinction between (k_1,k_2) and (k̃,ℓ̃) as ℤ_(2,4) labels. 





In our analysis of the large- and small-mass limits, we found that all the phase transitions are associated with the jump of discrete labels (k̃,ℓ̃)∈ (ℤ_(N_1,N_2))^2, and the phase transition lines are separating different 4d SPT states. 
This is a drastically different point compared with the (N_1,N_2)=1 QCD(BF). 
Of course, as a logical possibility, it would have been possible that some accidental phase transition appears in the intermediate mass range, at which only the continuous field φ jumps. 
As far as we numerically studied the case 0.05≤ mμ≤ 1.0, such an accidental transition does not happen. 


Figure <ref> depicts the phase diagrams on (θ_1,θ_2) plane with (N_1, N_2) = (2, 4) at several m μ.
In each panel, we determine the phases by the discrete labels (k_1,k_2) of the global minimum, and we omit the values of φ as its jump turns out to be always associated with the jump of (k_1,k_2). 


For a small-mass case (Figure <ref>), the phase diagram slightly alters from the small-mass limit depicted in Figure <ref>. When we change the (θ_1,θ_2) along the θ_phys direction, the phase transition occurs by the jump of k̃=k_1 as expected. When we change the (θ_1,θ_2) along the perpendicular direction to θ_phys, the jump of ℓ̃=k_2 occurs and the chiral-broken vacua are exchanged. 
When we increase m and it reaches a certain mass, the reconnection (topology change) occurs at (θ_1, θ_2) = (0, π), (2 π, π) (see from Figure <ref> to <ref>). 
After the reconnection, the phase diagram converges to that of the decoupled Yang-Mills theory (Figure <ref>).
Our semiclassical framework explains the topology-changing behavior speculated in Ref. <cit.>.






 §.§ On the duality of semiclassics in the large-N1,2 limit



In this section, we make a brief comment that the duality map of coupling constants (<ref>) is valid also for the case (N_1,N_2)≠1 in the large-N_1,2 limit. 

To see this, we parametrize the 2π N'_1N'_2-periodic scalar as φ = 2 π (M_1 k_1' + M_2 k_2') + δφ with -π< δφ<π and (k'_1,k'_2)∈ℤ_N'_1×ℤ_N'_2. 
Using the extension, ℤ_N'_i→ℤ_N_i→ℤ_(N_1,N_2), the discrete labels k'_i∈ℤ_N_i and k_i∈ℤ_(N_1,N_2) are combined into (k̂_1, k̂_2) ∈ℤ_N_1×ℤ_N_2, which is exactly what we have done in the discussion for the large-mass limit in Section <ref>.  
As a result, the potential can be expressed as 

    V    = - m μcos (δφ) - 2K^(1)^-S_I^(1)/N_1cos( 2 πk̂_1 - θ_1/N_1 + N_2/N_1δφ) 
            - 2K^(2)^-S_I^(2)/N_2cos( 2 πk̂_2 - θ_2/N_2 + N_1/N_2δφ).

The rest of the argument is completely the same as the one in Section <ref>.
One can find the minimum δφ∼ O(1/N_1,1/N_2) and have the identical expression for the quadratic form of the energy density with the coefficients (<ref>).
We can repeat the same discussion for the daughter side, SU(N_1)× SU(N_2-N_1) QCD(BF), and we find the concrete form of the duality map (<ref>).  


Lastly, let us give one cautionary remark. 
Assume that we are studying SU(N_1)× SU(N_2) QCD(BF) at sufficiently small fermion mass, then we can repeatedly apply the dualities and the gauge-group ranks changes following the Euclidean algorithm, (N_1,N_2)→ (N_1,N_2-N_1)→⋯. 
For N=gcd(N_1,N_2) ≠ 1, the last step of the duality cascade is the duality from SU(N) × SU(2N) QCD(BF) to SU(N) × SU(N) QCD(BF), and the latter theory is the equal-rank QCD(BF). 
In this paper, we assumed N_1≠N_2 to construct the 2d semiclassical theory, so we cannot immediately apply our duality map (<ref>) to the last step of the cascade, which requires an extra work. 
In our previous paper <cit.>, we construct the 2d semiclassical description for equal-rank QCD(BF). 
Although there is a difference in the renormalization scheme, the large-N expansion of the equal-rank QCD(BF) gives a parallel expression to (<ref>) as the functional form of the energy density.
Thus, it is possible to match these vacuum energies in a similar manner at least formally.
To gain a deep insight into the last step of the duality cascade, we need a more refined understanding of the semiclassical approach including the subtlety of renormalization scheme.




 §.§ Comments on the domain wall and the anomaly inflow



We here give the brief comment on properties of domain walls at the phase boundaries. 
In Section <ref>, we observed that all the phase transitions are characterized by the jump of the discrete vacuum labels (k̃,ℓ̃) when (N_1,N_2)≠1. 
These vacuum labels are identified with the labels (k,ℓ) for the 4d SPT states (<ref>) after a suitable linear transformation (<ref>), and thus the domain walls in the 4d setup should support nontrivial dynamics to cancel the anomaly inflow (see e.g. Refs. <cit.>). 
Let us uncover how this is realized in our T^2-compactified setup with the baryon-'t Hooft flux (see also Section 5 of Ref. <cit.>). 


We first note that the domain wall in 4d setup is the dynamical object, which would be spontaneously created if we perturb the θ parameters to the position-dependent one. 
In the 2d semiclassical theory, however, the vacua are specified by the discrete labels, which suggests that the relevant degrees of freedom for creating domain walls are heavy and already integrated out in this effective theory. 
Therefore, we need to introduce the loop operator in 2d that forces the jump of the discrete labels, and this is the counterpart of the 4d domain wall. 
We can see that this loop-operator insertion satisfies the requirement of anomaly inflow. 

To understand this, it is convenient to consider the decoupling limit m→∞, and then the vacuum labels (k̃,ℓ̃) are reduced to the original ones, (k_1,k_2)∈ℤ_N_1×ℤ_N_2, which constrain the total topological charges of the fractional-instanton gas. 
Since (k_1,k_2) specifies the 1-form symmetry charge of the SU(N_1)× SU(N_2) pure Yang-Mills theory, their Wilson loops W_1 and W_2 serve as domain walls connecting vacua with different (k_1,k_2). 
When the dynamics of bifundamental fermion Ψ is turned on with finite mass m, the part of 1-form symmetry loses its information as Ψ caries the same gauge charge of W_1 W_2^-1. 
The remnant charge of the 1-form symmetry is specified by the label k ∈ℤ_(N_1,N_2) in the 4d SPT action (<ref>). 
Importantly, Ψ carries the fractional charge of U(1)/ℤ_(N_1,N_2), and this also remains as the robust information described by the label ℓ in (<ref>). 

When two states are discriminated by the 1-form symmetry charge k, then we have to introduce the appropriate Wilson loop to have the domain wall in the 2d effective theory. 
When two states have the same 1-form charge k but different levels for ℓ, then the domain wall can be created either by insertion of the Wilson loops or by excitation of the Ψ quanta. 
Thus, the anomaly inflow to the domain wall in our T^2 compactified setup is realized by the fact that the domain wall becomes a non-dynamical object requiring the suitable amount of the loop-operator insertion. 




The authors thank Zohar Komargodski for discussion and useful comments on the early draft.
The work of Y. T. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant numbers, 22H01218, 23K22489, and by Center for Gravitational Physics and Quantum Information (CGPQI) at Yukawa Institute for Theoretical Physics.
Y. H. was supported by JSPS Research Fellowship for Young Scientists Grant No. 23KJ1161.








§ COUNTING THE LOW-ENERGY FERMIONS FOR QCD(BF) ON R2XT2




In the main text, we used the index theorem (<ref>) to count the number of low-energy modes arising from the bifundamental fermion under the T^2-compactification.
Here, we present an alternative derivation of the number of low-energy modes by directly solving the zero-mode equation on the torus,

    [ γ_3 ∂_3 + γ_4 ( ∂_4 +  A_V,4) ] Ψ(x_3,x_4) = 0,

where the boundary conditions are given by

    Ψ(x_3+L,x_4)    = ^( 2 π/N_1 L - 2 π/N_2 L) x_4 S_1^†Ψ(x_3,x_4) S_2, 
    Ψ(x_3,x_4+L)    = C_1^†Ψ(x_3,x_4) C_2.

Here, the U(1)_V background field is 

    A_V = ( 2 π/N_1 L^2 - 2 π/N_2 L^2) x_3 dx_4.



The normalizable solution to the zero-mode equation takes the form of

    Ψ_+(x_3,x_4) = ^- π/L^2( 1/N_1 - 1/N_2) x_3^2Ψ_+(x_3 -  x_4),

where Ψ_+ is a positive eigenmode[There is no normalizable zero-mode with a negative eigenvalue of γ_3 γ_4: γ_3 γ_4 Ψ_- = -Ψ_-. Indeed, if Ψ_- were a normalizable zero-mode, we would have

    0    = ∫_T^2^2x | [ ∂_3 + ( ∂_4 +  A_V,4) ] Ψ_-(x_3,x_4) |^2 
       = ∫_T^2^2x {|  ∂_3 Ψ_- |^2 + |  ( ∂_4 +  A_V,4) Ψ_- |^2 +  ( 2 π/N_1 L^2 - 2 π/N_2 L^2) | Ψ_- |^2},

and this implies Ψ- = 0.
] of γ_3 γ_4: γ_3 γ_4 Ψ_+ = +Ψ_+, and Ψ_+(x_3 -  x_4) is a holomorphic function in x_3 -  x_4.
The second boundary condition Ψ(x_3,x_4+L) = C_1^†Ψ(x_3,x_4) C_2 can be written as Ψ_i,j(x_3,x_4+L) = ^ - 2 π/N_1 L  (i - 1) + 2 π/N_2 L  (j - 1) Ψ_i,j(x_3,x_4), 
so we can expand the holomorphic function Ψ_+(x_3 -  x_4) with Fourier coefficients c_n^i,j,

    Ψ_+;i,j (x_3 -  x_4) = ^[ 2 π/N_1  (i - 1) - 2 π/N_2  (j - 1)] ( x_3 -  x_4/L) {∑_n ∈ℤ c_n ^i,j^2 π n/L (x_3 -  x_4) }.

The first boundary condition Ψ(x_3+L,x_4) = ^( 2 π/N_1 L - 2 π/N_2 L) x_4 S_1^†Ψ(x_3,x_4) S_2 relates Ψ_i,j to Ψ_i-1,j-1 as follows, 

    Ψ_+;i,j (x_3 -  x_4 + L) = ^π(1/N_1 - 1/N_2) + ( 2 π/N_1 L - 2 π/N_2 L) (x_3 -  x_4)Ψ_+;i-1,j-1 (x_3 -  x_4),

where i = 0 (resp. j=0) is understood as i = N_1 (resp. j=N_2).
This can be rewritten in terms of the Fourier coefficients c_n^i,j:
For 1<i ≤ N_1 and 1<j ≤ N_2,

    c_n^i,j =  ^π(1/N_1 - 1/N_2)^ -[ 2 π/N_1  (i - 1) - 2 π/N_2  (j - 1)]  ^ - 2 π n c_n^i-1,j-1.

The exceptions are listed as follows:

    c_n^1,j   =  ^π(1/N_1 - 1/N_2)^  - 2 π/N_2  (j - 1) ^ - 2 π n c_n-1^N_1,j-1    (1<j ≤ N_2), 
    
        c_n^i,1   =  ^π(1/N_1 - 1/N_2)^ -2 π/N_1  (i - 1)  ^ - 2 π n c_n+1^i-1,N_2    (1<i ≤ N_1), 
    
        c_n^1,1   = ^π(1/N_1 - 1/N_2)^ - 2 π n  c_n^N_1,N_2.


Now, let us count the number of independent solutions.
We first notice that the first boundary condition relates Ψ_i,j to Ψ_i-1,j-1, so we can decompose the N_1 × N_2 matrix into the gcd(N_1,N_2) classes.
For example, we look at the class including Ψ_1,1: {Ψ_1,1, Ψ_2,2, ⋯, }.
The periodicity of the sequence Ψ_1,1→Ψ_2,2→⋯ is
lcm(N_1,N_2), as the subscripts can be understood as those of ℤ_N_1×ℤ_N_2.
Thus, the N_1 × N_2 matrix {Ψ_i,j} can be indeed decomposed into gcd(N_1,N_2) classes, since each class consists of lcm(N_1,N_2) = N_1 N_2/gcd(N_1,N_2) elements.

We then count the number of independent solutions for each class.
For a sequence Ψ_i,j→Ψ_i+1,j+1→⋯, 
let us call

    
  * jump Ψ_N_1,j→Ψ_1,j+1: transition of the 1st index 
    
  * jump Ψ_i,N_2→Ψ_i+1,1: transition of the 2nd index 

Because the number of elements is N_1 N_2/gcd(N_1,N_2), the sequence Ψ_1,1→Ψ_2,2→⋯ undergoes N_2/gcd(N_1,N_2) transitions of the 1st index as well as N_1/gcd(N_1,N_2) transitions of the 2nd index.
The boundary conditions (<ref>)–(<ref>) relate the Fourier coefficients c_n^i,j as,

    c_n^i,j→ c_n^i+1,j+1→⋯→ c_n + N_2 - N_1/gcd(N_1,N_2)^i,j,

since the transition of 1st index relates c_n to c_n+1, and that of 2nd index relates c_n to c_n-1.
Therefore, each class includes N_2 - N_1/gcd(N_1,N_2) independent solutions.

The N_1 × N_2 matrix  {Ψ_i,j} has gcd(N_1,N_2) classes, and one class has N_2 - N_1/gcd(N_1,N_2) independent solutions.
In total, the number of zeromodes is,

    gcd(N_1,N_2) ×N_2 - N_1/gcd(N_1,N_2) = N_2 - N_1,

which reproduces the result from the index theorem (<ref>).
We therefore get N_2 - N_1 low-energy modes before taking the 4-fermi vertex into account.





§ MORE ON THE DUALITY BETWEEN SU(2)XSU(5) AND SU(2)XSU(3) QCD(BF)







 §.§ Quantifying the mild violation of the duality relation




We have numerically observed the validity of the duality relation in Section <ref>, and we found the surprisingly good agreement between the parent theory, SU(N_1)× SU(N_2) QCD(BF), and the daughter theory, SU(N_1)× SU(N_2-N_1) QCD(BF). 
In this appendix, we are going to uncover the reason why the duality relation works so well by quantifying the violation of the duality: We are going to demonstrate that the violation is the O(N_2^-4) effect. 

When we consider the duality map (<ref>) at m=m^*, we obtain 

    SU(N_1)   × SU(N_2)  QCD(BF) on  (θ_1,θ_2) at m=m^* 
       ⟶  SU(N_1)× SU(N_2-N_1)  pure YM on  (θ_1-θ_2,θ_2).

We note that the phase boundaries on the daughter side consist only of the straight lines, and, in particular,  they are given by θ_1-θ_2 = π (mod 2π) or by θ_2 = π (mod 2π). 
Therefore, if the duality relation (<ref>) were exact, the parent theory would also have the same phase boundaries described by those straight lines. 
By evaluating how the actual phase boundaries of the parent theory deviate from those lines, let us measure the non-exactness of the duality relation. 


We focus on the horizontal line θ_2=π, which goes through (θ_1, θ_2) = (π, π).
At (θ_1, θ_2) = (π, π), the potential V[φ] of the parent theory, given in (<ref>), is invariant under the CP symmetry transformation,

    φ→ -φ + 2 π (M_1+M_2),

because N_1 M_2 = 1  (mod N_2) and N_2 M_1= 1  (mod N_1).
Recall that the Bezout coefficients (M_1,M_2) are taken so that M_1/N_2', M_2/N_1'∈ℤ. 
Thus, we expect two-fold degenerate vacua  at (θ_1, θ_2) = (π, π): φ = φ_*, -  φ_* + 2 π  (M_1+M_2).


If the duality were exact, the phase boundary including (θ_1, θ_2) = (π, π) at mμ=m^*μ has to be a horizontal line θ_2 = π, and we can obtain φ_* under this assumption. 
As the two-fold degeneracy should be robust under the perturbation θ_1 = π + ϵ, we must have V[φ_*+O()]|_θ_1=π+,θ_2=π=V[2π(M_1+M_2)-φ_* + O()]|_θ_1=π+,θ_2=π. 
Expanding this equality at the lowest order of , this requirement constrains φ_* as,

    sin( N_2 φ_* - π/N_1) = 0.

Since one of the vacua is in the same phase as φ = 0 at (θ_1, θ_2) = (0, 0), we deduce φ_* = π/N_2.
To minimize the potential V[φ] of (<ref>) at φ_* = π/N_2, the following part

    V_mass+SU(N_2)[φ] = - m^* μcos (φ)  - 2K^(2)^-S_I^(2)/N_2cos( N_1 φ - π/N_2)

should have a (local) minimum at φ = π/N_2. In other words, we can estimate the topology-changing point m^*μ by this requirement, which gives 

    m^*μ = 
        N_1/N_2sin((N_2-N_1)π/N_2^2)/sin(π/N_2)· 2K^(2)^-S^(2)_I/N_2.

We note that this is consistent with the large-N formula (<ref>). 


Now, let us check if the doubly-degenerate vacua are preserved for general θ_1 at θ_2=π.
First, we look at the vacuum branch that takes φ = π/N_2 at θ_1 = π. Let us call this minimum the vacuum A and denote φ = φ_*^(A)(θ_1).
The potential consists of V_mass+SU(N_2)[φ] and - 2K^(1)^-S_I^(1)/N_1cos( N_2 φ - θ_1 /N_1).
The former term has a minimum at φ = π/N_2, and the latter term has a minimum at φ = θ_1/N_2.
The vacuum branch is constructed from the superposition of these minima, and φ_*^(A)(θ_1) is located between φ = π/N_2 and φ = θ_1/N_2 from these two contributions.

Next, we compare this vacuum with the other vacuum, which is φ = - π/N_2 + 2 π (M_1+M_2) at θ_1 = π.
We call this minimum the vacuum B and denote φ = φ_*^(B)(θ_1).
Similarly, the latter vacuum is located between 
φ = - π/N_2 + 2 π (M_1+M_2) (from V_mass+SU(N_2)[φ]) and φ = (θ_1 - 2 π)/N_2 + 2 π (M_1+M_2) (from the SU(N_1) part).
For a comparison, let us shift φ as φ→φ + 2π/N_2 - 2 π (M_1+M_2), and we write φ = φ̃ + 2π/N_2 - 2 π (M_1+M_2).
With this shift, the vacuum B, φ_*^(B)(θ_1), is located between φ̃ = π/N_2 and φ̃ = θ_1/N_2, which seems to be the same as the vacuum A φ = φ_*^(A)(θ_1).
However, this correspondence is not exact.
The vacuum A is the minimum around φ = π/N_2 of the potential

    V[φ]
            = V_mass+SU(N_2)[φ]  - 2 K^(1)^-S_I^(1)/N_1cos( N_2 φ - θ_1/N_1),

On the other hand, the potential in terms of the shifted  φ̃ is,

    V[φ   = φ̃ + 2π/N_2 - 2 π (M_1+M_2)] 
       = V_mass+SU(N_2)[-φ̃ + 2π/N_2]  - 2 K^(1)^-S_I^(1)/N_1cos( N_2 φ̃ - θ_1/N_1),

and the vacuum B is the minimum near φ̃ = π/N_2.
The two-fold degeneracy of these vacua, φ = φ_*^(A)(θ_1) and φ = φ_*^(B)(θ_1), is preserved if V_mass+SU(N_2)[φ] is symmetric under the reflection over the line φ = π/N_2: φ→ -φ  + 2π/N_2. 
However, this is not true: The potential can be expressed as

    V_mass+SU(N_2)[π/N_2 + δφ] 
       = [ - m^* μcos( π/N_2) cos( δφ)  - 2K^(2)^-S_I^(2)/N_2cos( (N_1 - N_2)π/N_2^2)  cos( N_1/N_2δφ)   ] 
        +[  m^* μsin( π/N_2) sin( δφ)  + 2K^(2)^-S_I^(2)/N_2sin( (N_1 - N_2)π/N_2^2)  sin( N_1/N_2δφ)   ],

where the first line is the δφ-even, and the last line is δφ-odd.
Although we have chosen m^*μ so that the O(δφ) term vanishes, the higher-order δφ-odd term O((δφ)^3) does not.

Thus, the statement at the topology-changing point does not hold for finite N_1 and N_2 in an exact sense.
However, we note that δφ =O(π/N_2), and thus the amount of the asymmetry in the above potential can be estimated as O(N_2^-4). 
This smallness would explain why the violation of the duality relation is so mild even for finite parameters. 




 §.§ Phase diagrams in the hierarchical limits: Lambda1<<Lambda2 and Lambda1>>Lambda2




In Section <ref>, we numerically examined the duality map from SU(2)× SU(5) QCD(BF) to SU(2)× SU(3) QCD(BF), when the strong scales of the parent theory satisfies Λ_1 ∼Λ_2. 
In this appendix, we expand our analysis to the case when these two scales of the parent theory are well separated: Λ_1≪Λ_2 and Λ_1≫Λ_2. 
In what follows, we restate the key points and show numerical results supporting the duality relation.



  §.§.§ Case: Λ_1 ≪Λ_2


Let us put the scale separation Λ_1≪Λ_2 in the parent theory, SU(2)× SU(5) QCD(BF). 
We have analytically shown that the duality approximately holds for m, Λ_1 ≪Λ_2 in Section <ref>, and our numerical analysis here serves also as a consistency check.
We set 2K^(1)^-S_I^(1)/N_1=1,  2K^(2)^-S_I^(2)/N_2=100 in the parent theory, and the large-N estimate of the topology-changing point is m^*μ=24. 
Exactly as we have done in Section <ref>, we determine the strong scales of the daughter theory by the large-N formula (<ref>), 

    2K^(1')^-S_I^(1')/N_1   =1/1+5^3/2^3· 100≈ 0.865, 
        
    
        2K^(2')^-S_I^(2')/(N_2-N_1)   =3^3/5^3· 100=21.6.

We show the result in Figure <ref>: 
The top panels represent the phase diagram of the parent theory on (θ_1, θ_2) plane.
The bottom panels represent the phase diagram of the corresponding daughter theory on (θ_1'+θ_2', θ_2') plane through the matching of parameters via the large-N indicated formula.
The fermion masses in the daughter theory are set through (<ref>) to compare the phase diagrams (Figures <ref>-<ref>) to those in the parent theory (Figures <ref>-<ref>).
Although we have not illustrated the phase diagram at the topology-changing point mμ = m^*μ, our rough numerical search indicates m^*μ≈ 25.0, which is slightly above but comparable with the large-N estimate.
The fact that the value of m^*μ is not simply 100 times the value at 2K^(2)^-S_I^(2)/N_2=1 reminds us that the topology-changing point in fact depends on both strong scales, m^*(Λ_1,Λ_2).
Similar to the case in  Figure <ref>, we can see the nice agreement between the parent theory for mμ≤ m^*μ≈ 25.0 and the daughter theory as we have expected.








    


  §.§.§ Case: Λ_2 ≪Λ_1

Let us next consider the opposite hierarchy, Λ_1≫Λ_2, and we are going to set 

    2K^(1)^-S^(1)/N_1=100,   
        2K^(1)^-S^(1)/N_1=1,

for the parent theory, SU(2)× SU(5) QCD(BF), in our numerical study. 
The large-N estimate of the topology changing point is m^*μ=0.24. 

Following the large-N duality map (<ref>), we set the strong scales of the daughter theory, SU(2)× SU(3) QCD(BF), as follows:

    2K^(1')^-S_I^(1')/N_1   =100/1+N_2^3/N_1^3100≈N_1^3/N_2^3=8/125, 
    
        2K^(2')^-S_I^(2')/(N_2-N_1)   =(N_2-N_1)^3/N_2^3=27/125.

We note that the hierarchy Λ_1≫Λ_2 does not cause the large hierarchy of the strong scales in the daughter theory. 
However, the above values almost saturate the inequality (<ref>), which suggests that this daughter theory lives almost at the boundary of the image of the duality map. In this sense, this opposite hierarchical limit provides a nontrivial testing ground for the duality relation (<ref>). 
We relate the fermion mass for mμ<m^*μ again by using the large-N indicated formula (<ref>). 

We show the result in Figure <ref>, and the duality works again nicely. 
Moreover, we can immediately notice that those figures are almost identical to those of Figure <ref>, including the location of the topology-changing point m^*μ≈ 0.263. 
As a result, the duality works so well also in the opposite hierarchical limit within our semiclassical framework. 
Let us consider why this is so. 
When mμ<m^*μ, we have the duality map to the daughter theory. When Λ_1≳Λ_2, the SU(N_1) strong scale of the daughter theory does not change so much according to the formula (<ref>). 
Even though the two setups Λ_1 ∼Λ_2 and Λ_1≫Λ_2 seem to be very different in the parent theory, the corresponding daughter theories have roughly the same parameters, and thus the phase diagrams in the small-mass range, mμ≲ m^*μ, are insensitive to the values of Λ_1. 
It seems not to be straightforward to understand this feature of the parent theory without relying on the duality relation. 










JHEP
 

