Inherited non-invertible duality symmetries in quiver SCFTs
Riccardo Argurio Andrés Collinucci Salvo Mancani Shani Meynet Louan Mol Valdo Tatitscheff


TOI-3568 b: a super-Neptune in the sub-Jovian desert
    
    E. Martioli<ref>,<ref> 
    R. P. Petrucci <ref>,<ref> 
    E. Jofré  <ref>,<ref> 
    G. Hébrard <ref>,<ref> 
    L. Ghezzi <ref> 
    Y. Gómez Maqueo Chew  <ref> 
    R. F. Díaz <ref> 
    H. D. Perottoni <ref> 
    L. H. Garcia <ref> 
    D. Rapetti  <ref>,<ref>  
    A. Lecavelier des Etangs <ref> 
    L. de Almeida <ref> 
    L. Arnold <ref> 
    É. Artigau <ref> 
    R. Basant <ref> 
    J. L. Bean <ref> 
    A. Bieryla <ref> 
    I. Boisse <ref> 
    X. Bonfils <ref>  
    M. Brady <ref> 
    C. Cadieux <ref> 
    A. Carmona <ref> 
    N. J. Cook <ref> 
    X. Delfosse <ref> 
    J.-F. Donati <ref> 
    R. Doyon <ref> 
    E. Furlan <ref> 
    S. B. Howell <ref> 
    J. M. Jenkins <ref> 
    D. Kasper <ref> 
    F. Kiefer <ref>,<ref> 
    D. W. Latham  <ref> 
    A. M. Levine <ref> 
    D. Lorenzo-Oliveira <ref> 
    R. Luque <ref> 
    K. K. McLeod <ref> 
    J. Melendez <ref> 
    C. Moutou <ref> 
    Y. Netto <ref> 
    T. A. Pritchard <ref>  
    P. Rowden <ref> 
    A. Seifahrt <ref> 
    G. Stefánsson <ref> 
    J. Stürmer <ref> 
    J D. Twicken <ref>,<ref>  
    
    Received xxxx ; accepted xxxx 
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§ INTRODUCTION


In recent years, symmetries in the context of Quantum Field Theory (QFT) have received a new paradigmatic formulation as topological defects <cit.>. One of the most noticeable consequences of this paradigm is the existence of non-invertible symmetries. Such symmetries had already been described in the context of d=2 Rational Conformal Field Theories (RCFTs) <cit.> and the d=3 Topological Quantum Field Theories (TQFTs) related to them <cit.>. Instead of having a group like structure, non-invertible symmetries enjoy a ring-like one, a × b = ∑_i c_i, and therefore not all elements admit an inverse. Topological operators enjoying such fusion rules, despite not being invertible symmetries, can still be used to study RG-flows and Ward identities of QFTs, putting constraints on  the dynamics of the system <cit.>.

In d=4, a particularly helpful playground to study these generalized symmetries is 𝒩=4 super Yang-Mills (SYM), which does in fact admit non-invertible symmetries. More precisely, 𝒩=4 SYM enjoys a duality group, SL(2,ℤ), that acts on the complexified gauge coupling τ_SYM via modular transformations. This is the so-called Montonen-Olive duality <cit.>. Remarkably, this SL(2,ℤ) action admits special values of τ_SYM, namely i and e^2π i/3, that are left invariant under a discrete subgroup: ℤ_2 and ℤ_3 respectively. At those fixed points of the conformal manifold, duality transformations become non-invertible symmetries of the theory <cit.>. The non-invertibility stems from the fact that the symmetry is actually the composition of a duality transformation with a topological manipulation that reverses the effect of the self-duality transformation on the gauge group, i.e. it relates the two different global variants <cit.> of the gauge algebra by gauging a 1-form symmetry. The literature on the subject is vast, for a sample see <cit.>. 

A richer set of theories, constrained enough to be reliably studied, are the so-called class 𝒮 theories of <cit.>.  
The data describing these theories is encoded in Riemann surfaces with marked points and they admit, as 𝒩=4 SYM, a duality group given by the Mapping Class Group (MCG) of the surface. As one might expect, these theories also admit non-invertible symmetries precisely when the duality group preserves the couplings of the theory while altering the global structure of the gauge group, see for instance <cit.> for a study of some class S theories. 

Symmetries, including non-invertible ones, are particularly interesting if they are preserved along an RG flow, since they can then constrain, or predict, some properties of the IR theory at the end of the flow. The simplest RG flows are those triggered by mass terms. In =2 (and =4) theories, the latter usually partially break supersymmetry to =1. If the =2 theory enjoys non-invertible symmetries, one can turn on mass terms that preserve them, and the IR theory is then expected to enjoy the same non-invertible symmetries. The analysis depends on whether the IR theory is gapped, or an SCFT. A class of gapped RG flows was considered in <cit.>. The case of the flows from some specific =2 class S theories to =1 SCFTs was also briefly considered in the same reference, see also <cit.>. 


One of the aims of this paper is to systematically discuss duality symmetries, and the mass deformations preserving them, in two broad classes of =2 SCFTs, namely the A_n-1 and the D_n quiver gauge theories. Such class S theories have appeared in string/M-theory constructions: they can be alternatively seen to arise from D3-branes at ℂ^2/Γ×ℂ singularities in type IIB <cit.>, from D4-branes suspended between NS5-branes in type IIA <cit.>, and from M5-branes wrapping complex surfaces in M-theory <cit.>. All such descriptions are related to each other by string dualities.

Starting from the A_n-1 quiver gauge theory, in <ref> we describe the duality group in terms of the mapping class group of a complex torus with n unordered marked points, revisiting the analysis of <cit.>. This is best understood from the M-theory uplift. Using this result, it is possible to classify point configurations that are invariant under the duality group, leading to non-invertible symmetry defects <cit.>. We then generalize this construction to the D_n quiver gauge theories, which has been mostly discussed in the type IIA setting <cit.>. The M-theory uplift of this theory consists of M5-branes inserted as marked points on a quotiented torus, i.e. a pillowcase. The duality group is the mapping class group of this object, which we determine. We then proceed to study the presence of non-invertible defects in these models as well, again by finding which elements of the MCG fix the modular parameter and the marked points. Finally, despite not having a Riemann surface describing the E_n quiver theories,[See <cit.> for an attempt towards this goal.] what we learned from the other cases allows us to make an educated guess of the structure for their duality groups. 

In <ref>, we turn our attention to mass deformations of A_n-1 and D_n quiver gauge theories, studying the action of the duality group on them. In both cases, an important distinction is made whether an overall mass parameter, the “global mass", is zero or not. In the former case, only permutations of points act on the masses, while in the latter, SL(2,ℤ) transformations act on them as well. This overall mass parameter plays an important role also in determining the moduli space of the SCFT that is supposed to exist in the IR of such RG flows, as we discuss in <ref>. Indeed, the moduli space of the starting theory is a three-fold algebraic variety given by the direct product of a Du Val type singular surface times ℂ. After the mass deformation, the flow brings the theory to another supersymmetric theory, whose moduli space is either a compound Du Val three-fold or a Du Val two-fold. The former (locally) is a non-trivial fibration of ℂ^2/Γ over ℂ, while the latter is just ℂ^2/Γ. A vanishing global mass triggers a deformation leading to three-dimensional moduli space, while a non-vanishing one leads to a two-dimensional one.

In <ref>, we finally turn our attention to mass deformations that preserve duality defects. We prove that such a deformation always exists for both the  A_n-1 and D_n. We find all the non-invertible defect-preserving mass deformations for those theories, discussing some selected examples. Our main result is thus a characterization of N=1 SCFTs which enjoy non-invertible duality symmetries inherited from their N=2 parent SCFTs.








§ 𝒩=2 QUIVER SCFTS, DUALITIES AND SYMMETRIES


In this section, we first review the brane/geometric construction that at low energy leads to an affine 4d 𝒩=2 ADE-type quiver SCFTs, and then also discuss the form of their duality groups. These theories are well known in the context of the AdS/CFT correspondence, where they are realized as the world volume theory of D3-branes probing Du Val surfaces, i.e. orbifolds of the form ℂ^2/Γ with Γ a finite subgroup of SU(2). Moreover, at least A_n-1 and D_n quivers also admit a class 𝒮 realization which is crucial for understanding their duality groups, as we will review shortly.

In the framework of theories of class 𝒮, 4d 𝒩=2 theories are engineered by wrapping M5-branes on a genus g Riemann surface with n marked points, to which we will also refer as punctures, with a partial topological twist <cit.>. Let Σ_g,n be the underlying smooth surface.[A smooth surface is a smooth manifold of real dimension two. We only consider surfaces of finite type. Smooth surfaces of finite type are entirely determined by their genus g and number of punctures n. A Riemann surface is a smooth surface endowed with a complex structure. In general there are many inequivalent complex structures with which a fixed smooth punctured surface can be endowed; more precisely, the real dimension of the Teichmüller space 𝒯(Σ_g,n) is 6g-6+2n. Given a Riemann surface in 𝒯(Σ_g,n), we interpret the punctures as marked points.] When the construction leads to an SCFT, the conformal manifold of the latter is the Teichmüller space 𝒯(Σ_g,n). By definition, 𝒯(Σ_g,n) is the space of all complex structures with which Σ_g,n can be endowed, up to diffeomorphims of Σ_g,n homotopic to the identity. The duality group of the theory is then embodied as the mapping class group MCG(Σ_g,n), which is the group of orientation-preserving diffeomorphims of Σ_g,n, modulo diffeomorphisms connected to the identity.[This can be described as the group of “large" diffeomorphisms modulo “small" ones, borrowing the usual gauge theory nomenclature for transformations that cannot or can, respectively, be deformed to the identity.] Indeed, the MCG does not change the physical properties of the configuration, but it acts non-trivially on 𝒯(Σ_g,n), relating different points of the conformal manifold <cit.>. We refer to <cit.> for a general introduction to class 𝒮 theories.

After computing the duality groups of A_n-1 and D_n quivers, and proposing a description for E_6,7,8 quivers, we discuss the interplay between dualities and 1-form symmetries, and compute the locus in the conformal manifold at which non-invertible symmetries are realized.



 §.§ A_n-1 quivers from class 𝒮


We first consider 4d 𝒩=2 quivers gauge theories shaped like affine A_n-1 Dynkin diagrams, which we will refer to as A_n-1 theories for convenience. These consist of n SU(k) gauge factors with bifundamental hypermultiplets, as shown in <ref>. We denote the gauge factors as SU(k)_1, …, SU(k)_n, and for each i=1,…,n modulo n there is an adjoint field ϕ_i and a pair of chiral multiplets X_i,i+1, X_i+1,i in the representation ( (1)_i, (1)_i+1), ( (1)_i+1, (1)_i) of SU(k)_i×SU(k)_i+1. The superpotential is the minimal one compatible with 𝒩=2 supersymmetry, that is:

    W_𝒩=2 = ∑_i=1^nϕ_i ( X_i,i+1X_i+1,i - X_i,i-1X_i-1,i)   .


These theories can be realized in type IIA string theory as the worldvolume theory of a stack of k D4-branes suspended between n NS5-branes along a circle: these are the elliptic models of <cit.>. More precisely, one considers type IIA string theory in ℝ^1,3×ℝ^2_4,5× S^1_6×ℝ^3_7,8,9, with n NS5-branes extending along ℝ^1,3×ℝ^2_4,5 and k D4-branes along ℝ^1,3× S^1_6, all at the same point in ℝ^3_7,8,9, as shown in <ref>.





A cartoon of this brane setup in  ℝ^2_4,5× S^1_6 is shown on the right hand side of <Ref>. 
A dual description in type IIB string theory of this configuration is obtained as the worldvolume theory of a stack of k D3-branes transverse to ℂ^2/ℤ_n×ℂ. The two descriptions are related by T-duality along x^6 and decompactification.

Such theories are superconformal at any value of the gauge couplings. The inverse gauge coupling squared of a given node is proportional to the distance between the corresponding NS5-branes along x^6 <cit.>. More precisely, if the circle x^6 has length 2π R_6 and g_s denotes the string coupling, then:

    1/g_i^2 = x_i+1^6-x_i^6/8π g_s R_6 .

We are now interested in the uplift of such configurations to M-theory, where the relationship between the elliptic brane model and the class 𝒮 construction is manifest.

Let 2π R_10 be the length of the M-theory circle S^1_10. As emphasized in <cit.>, the metric on S^1_6× S^1_10 is not necessarily the product metric: the shift x^6→ x^6+2π R_6 can in general be accompanied by a shift x^10→ x^10+θ R_10, where θ is some angle. Let:

    τ = θ/2π + i/8π g_s ,

so that the torus metric is the natural flat metric on the elliptic curve E_τ with modulus τ in the complex upper-half plane ℍ. 

In the uplift to M-theory both D4 and NS5-branes become M5-branes: the former correspond to M5-branes wrapping the elliptic curve E_τ, whereas the latter are interpreted as boundary conditions for the worldvolume theory on the stack of k M5-branes at marked points on E_τ. Thus, this set-up can be reformulated in the class 𝒮 framework, where the Riemann surface is the elliptic curve E_τ with n marked points, with underlying smooth surface  Σ_1,n. Let p_1,…p_n ∈ E_τ denote the positions of the marked points as in <ref>.




The mapping class group of Σ_1,n can be obtained from the one of the closed torus 𝕋^2 via the Birman exact sequence <cit.>, and it can be described explicitly as follows. We consider the universal cover of the curve E_τ together with the lifts of the marked points, as depicted on the right of <ref>. The elliptic curve E_τ can be presented as E_τ=ℂ/Λ, where Λ⊂ℂ is the lattice ℤ+τℤ. Let us fix a starting notation for the lifts of p_1,…p_n ∈ E_τ: p_1:=p_1^0,0,…,p_n:=p_n^0,0 denote the lifts in the fundamental parallelogram {0,1,τ,1+τ}, while those in the parallelogram {k+lτ,k+1+lτ,k+(l+1)τ,k+1+(l+1)τ} are denoted p_1^k,l,…,p_n^k,l. When the configuration of marked points on E_τ is generic, there is a way to label them that is suitable for the physical interpretation of the setup, which is such that Im(p_1)<Im(p_2)<…<Im(p_n). We will explain in which sense it is suitable for physics shortly. 

The generators of the mapping class group are of three types:


    
  * Mapping classes of the torus.  The modular group SL(2,ℤ) acts as a change of basis for the lattice Λ. The action of the standard generators T and S is the following: T:(1,τ)→ (1,τ+1), whereas S encodes the combined operation (1,τ)→ (τ,-1)≃ (1,-1/τ). Because of the rescaling by 1/τ, the generator S acts non-trivially on the marked points: if p∈ℂ is a lift of a puncture then S   p = p/τ. The action of S is depicted in <ref>.
    
    
  * Deck transformations.  They are defined as changing the choice of lifts of the marked points,[Here we use the standard expression “deck transformations" from the theory of coverings in a loose sense, as actual deck transformations would act on all marked points together. However, this terminology makes manifest the relation between the fundamental group of the Riemann surface and its braiding action on a given marked point.] and are generated by t_i^(1):p_i→ p_i+1 and t_i^(τ):p_i→ p_i+τ for i=1,…,n. The red arrows in <ref> depict the action of t_1^(τ). In other words after acting with a deck transformation the lifts denoted p_i, i=1,…,n, are not necessarily in the fundamental parallelogram {0,1,τ,1+τ} anymore.
                
    
  * Permutations of the punctures.  We describe these transformations in the universal cover of the torus; the generators are denoted s_i, i=1,…,n, where s_i for any i=1,…,n-1 exchanges p_i and p_i+1, whereas s_n exchanges p_n with p_1+τ = p_1^0,1. This specificity in the definition of s_n echoes in <ref> below.


Denoting p⃗ = (τ ;  p_1, …, p_n), the generators of the mapping class group act as 

    S   :  p⃗   ⟼( -1/τ; p_1/τ, p_2/τ, …, p_n/τ)   , 
    
        T   :  p⃗   ⟼( τ + 1; p_1, p_2, …, p_n )   , 
    
        t_i^(1)  :  p⃗   ⟼( τ; p_1, p_2, …, p_i + 1, …, p_n )   , 
    
        t_i^(τ)  :  p⃗   ⟼( τ; p_1, p_2, …, p_i + τ, …, p_n )   , 
    
        s_i   :  p⃗   ⟼( τ; p_1, p_2, …, p_i-1, p_i+1, p_i, p_i+2,…, p_n )   , 
    
        s_n   :  p⃗   ⟼( τ; p_n - τ , p_2, …, p_1 +τ)   ,




This transposes to the physical theory as follows. After lifting to M-theory, the complexified gauge couplings of each node of the quiver are recovered as differences in position of neighbouring marked points on E_τ <cit.>. Let

    τ_i = p_i+1-p_i  ,   i ≠ n 
    τ_n = τ + p_1 - p_n

where the seemingly special definition of τ_n follows from the fact that ∑τ_i = τ. The “physical" labelling of the punctures described above ensures that 4πIm(τ_i) = g_i^-2 >0 for all i, where g_i is the gauge coupling of the i-th gauge group of the quiver gauge theory.

Given <ref> and denoting τ⃗=(τ;τ_1,…,τ_n), one can recast the action of the generators of the MCG as

    S      :  τ⃗→( -1/τ; τ_1/τ, τ_2/τ, …, τ_n/τ - 1 - 1/τ)   , 
       T      :  τ⃗→( τ + 1; τ_1, τ_2, …, τ_n + 1 )   , 
       t_i^(1)  :  τ⃗→( τ; τ_1, τ_2, …, τ_i-1 + 1, τ_i - 1, …τ_n )   , 
       t_i^(τ)  :  τ⃗→( τ; τ_1, τ_2, …, τ_i-1 + τ, τ_i - τ, …, τ_n )   , 
       s_i      :  τ⃗→( τ; τ_1, τ_2, …, τ_i-1 + τ_i, -τ_i, τ_i+1 + τ_i, …, τ_n )   .

All s_i together generate the affine Weyl group of type A_n-1, as is usual for brane configurations on a circle <cit.>. Together with the transformations t_i^(τ), they generate the group of automorphisms of the A_n-1 root system, that is, the co-central extension of the affine Weyl group of type A_n-1 by the group of outer automorphisms of the affine Lie algebra of type A_n-1. We refer to <cit.> for more details on affine Lie algebras and Weyl groups, here we simply discuss how the correspondence is achieved. Let us consider the vertical band of fundamental parallelograms containing the vertices {0,1,τ,1+τ}. The differences p_i^0,a-p_j^0,b where i,j=1,…,n and a,b∈ℤ, define the affine root lattice of type A_n-1. The standard positive simple roots are the τ_i defined in <ref>, where τ_n = τ_0 is the affine simple root and the shift by τ embodies the single imaginary root of the affine root system. The whole group of automorphisms of the lattice is generated by the s_i and t_i^(τ), for i=1,…,n.[Let us note that there are relations between the generators of the MCG, for example t_i^1=S^-1(t_i^τ)^-1S. Moreover, the outer automorphism ω of the A_n-1 algebra can be expressed as ω = t_1^(τ)s_1s_2… s_n-1.]

The description of the MCG in terms of the automorphisms of an affine root system will be used in <ref> to argue the structure of the duality group for theories for which an explicit class 𝒮 construction is not known. 



 §.§ D_n quivers from class 𝒮


We now turn to the description of the duality group of D_n quiver gauge theories. An example of such a quiver is shown on the left of <ref>. The gauge group of the theory is a product of (n-3) copies of SU(2k) and 4 copies of SU(k), there is an adjoint field ϕ_i for each gauge factor and there are bifundamental hypermultiplets X_i,j as to make the corresponding affine quiver of type D_n, and the superpotential required by 𝒩=2 supersymmetry

    W_𝒩=2   = ∑_i=0,1ϕ_i X_i,2 X_2,i + ∑_j=n-1,nϕ_j X_j,n-2 X_n-2,j + ϕ_2 ( X_2,0 X_0,2 + X_2,1 X_1,2 + X_23 X_32) 
        + ϕ_n-2( X_n-2,n-1 X_n-1,n-2 + X_n-2,n X_n,n-2 - X_n-2,n-3X_n-3,n-2)  
        + ∑_l=3^n-3ϕ_l ( X_l,l+1X_l+1,l - X_l,l-1X_l-1,l)    ,

where we refer to <ref> for the index conventions of the fields.



Let us discuss the type IIA brane setups that realize these theories, following <cit.> (see also <cit.>). The relevant configurations of branes are described in <ref>: 2k D4-branes suspended between n NS5-branes along a segment with endpoints ONS5^--planes, these last ones being orientifold-like 5-plane magnetically charged under the Neveu–Schwarz B_2 (and not under the Ramond–Ramond field C_2).[ONS5^--planes are obtained from type IIA O4^--planes by uplift to M-theory and compactification back to type IIA string theory on a transverse orbifolded circle S^1/ℤ_2 <cit.>.] 




D4-branes end either on the NS5-brane closest to the ONS5^- plane or on its image, as shown in <ref>. In particular, the states corresponding to D4-branes stretching between this NS5-brane and its image are projected out by the orientifold <cit.>. This explains how one obtains the characteristic ends of the D_n quiver. Equivalently, one can bring an NS5-brane atop each ONS5^--plane; the worldvolume theory on this composite object is a 6d O(2)=ℤ_2⋉SO(2) gauge theory. D4-branes ending on such a composite object carry a charge ±1 for the O(2) gauge theory.[Another way to see this is discussed in <cit.>.] The superconformal configurations are those in which the stack of 2k D4-branes splits in two sub-stacks of k D4-branes at each half ONS5^--plane; equivalently half the D4-branes have charge + and the other half, charge -. This configuration realizes the D_n quiver theory as the worldvolume theory on the D4-branes.





Just as in the A_n-1 case, this quiver gauge theory can be obtained in Type IIB as the worldvolume theory of a stack of k D3-branes transverse to ℂ^2/Γ_D_n×ℂ, with Γ_D_n the corresponding finite subgroup of SU(2).



The uplift to M-theory of D4 and NS5-branes happens exactly as in the A_n-1 case, while the ONS5^- become OM5-planes inducing the involution I_C_3ℐ_5, where ℐ_5 acts by reversing the coordinates transverse to the OM5-plane and I_C_3 reverses the sign of the M-theory 3-form. The uplift yields M-theory on 

    ℝ^1,3×ℝ^2 ×(E_τ×ℝ^3)/ℤ_2 ,

where we have combined the x_6 direction and the M-theory circle S^1_10 into an elliptic curve E_τ as before, and with M5-branes either wrapping the torus (D4), becoming marked points (NS5) or OM5-planes located at the four fixed points on E_τ (ONS5^-) <cit.>.

As in the previous section, this construction allows to study the duality group of the theory in term of the MCG of the quotiented torus. To this end we now turn to the description of the quotient geometry, which will give us insight on how to construct the MCG.



  §.§.§ Quotient surface and its universal cover

        
The action z↦-z on the elliptic curve ℂ/Λ, where Λ = ℤ+τℤ, has four fixed points in the standard fundamental parallelogram {0,1,τ,1+τ}:

    ζ_A = 0,   ζ_B = 1/2,   ζ_C = τ/2    and   ζ_D = (1+τ)/2   .

The quotient space 𝕋^2/_2 is topologically a sphere with four ℤ_2-orbifold points ζ_A,ζ_B,ζ_C and ζ_D, often dubbed pillowcase and depicted in <ref>. Let us denote a,b,c and d the homotopy classes of small loops around ζ_A,ζ_B,ζ_C and ζ_D respectively. Since these are ℤ_2-orbifold points, one has a^2=b^2=c^2=d^2=1. The (orbifold) fundamental group of the pillowcase is given by:

    π_1 (𝕋^2/ℤ_2) = ⟨ a,b,c,d | a^2=b^2=c^2=d^2=1, ba=dc.⟩  .

This can be obtained as follows.



In the double cover 𝕋^2 of the space 𝕋^2/ℤ_2, we can define the reflections with respect to ζ_A, ζ_B, ζ_C and ζ_D respectively, as depicted in <ref>, which act as:

    R_A    : p↦ -p   ,
    
        R_B    : p↦ 1-p   ,
    
        R_C    : p↦τ-p   ,
    
        R_D    : p↦ (1+τ)-p   .



The fundamental group π_1(𝕋^2/ℤ_2) is generated by {R_A,R_B,R_C,R_D}, and one has R_B ∘ R_A (p) = R_D ∘ R_C (p) = p + 1 = t^(1)(p) and R_C ∘ R_A (p) = R_D ∘ R_B (p) = p + τ = t^(τ)(p). 

Correspondingly, π_1(𝕋^2) embeds in π_1(𝕋^2/ℤ_2) as a subgroup of order 2.





  §.§.§ The D_n-ality group


As in the A_n-1 case, we obtain the generators of the duality group in the D_n case by considering the lifts of the marked points on E_τ/ℤ_2, in the universal cover . Note that n marked points on E_τ/ℤ_2, none of which sits at an orbifold point, correspond to 2n marked points on E_τ consisting of n symmetric pairs with respect to the center of the fundamental cell. 

One can choose one half of the fundamental cell, for example the bottom one as in <ref>, and label the lifts of the marked points sitting in it p_1,…,p_n.[We assume the configuration of marked points to be generic.] The shifts of these lifts by elements of the lattice are as before labeled p_i^k,l = p_i+k+lτ, with k,l∈ℤ. Last, for all i,k,l we let q_i^k,l=-p_i^k,l be the image of p_i^k,l reflected about the origin. There is a labeling adapted to physics, which is such that Im(p_1)<…<Im(p_n). This setup is shown in <ref>.



As before, the generators of the duality group are of three types:


    
  * Modular group.  The modular group SL(2,ℤ) acts as in the A_n-1 case:
    
    𝒮   :(τ;p_1,…,p_n)↦(-1/τ;p_1/τ,…,p_n/τ)   ,
    𝒯   :(τ;p_1,…,p_n)↦(τ+1;p_1,…,p_n)   .

    
  * Deck transformations.  The group of deck transformations of →𝕋^2/_2 is generated by {R_A,R_B,R_C,R_D}, hence each marked point is transformed as:
    
    R_A,i   : (τ;p_1,…,p_n)↦(τ;p_1,…,-p_i,…,p_n)   ,
    
            R_B,i   : (τ;p_1,…,p_n)↦(τ;p_1,…,1-p_i,…,p_n)   ,
    
            R_C,i   : (τ;p_1,…,p_n)↦(τ;p_1,…,τ-p_i,…,p_n)   ,
    
            R_D,i   : (τ;p_1,…,p_n)↦(τ;p_1,…,(1+τ)-p_i,…,p_n)   ,

    with the same relations as the ones that {R_A,R_B,R_C,R_D} satisfy.
    
  * Permutation of the punctures.  Punctures can be permuted, with generators:
    
    s_i   : (τ;p_1,…, p_i, p_i+1,…,p_n)↦(τ;p_1,…,p_i+1,p_i,…,p_n)

    for i=1,…,n-1. One could add an additional generator s_n which would exchange p_n and q_n^-1,-1, however s_n=R_D,n which we have already taken into account.

        
Similarly to what we did in the A_n-1 case, we now turn to the “physical" picture in which we discuss complexified gauge couplings instead of punctures, defined as follows:

    τ_0    = p_2+p_1   , 
    τ_i   = p_i+1-p_i   ,      i=1,  …  , n-1 
    τ_n    =τ-p_n-p_n-1  .

Again, the physical labelling ensures that each of the τ_i has positive imaginary part. The couplings satisfy the following relation:

    τ_0 + τ_1 + τ_n-1 + τ_n + 2 ∑_i=2^n-2τ_i = τ.

Letting n_0=n_1=n_n-1=n_n=1 and n_i=2 for 1≤ i≤ n-1, the previous equations rewrites as

    ∑_i=0^n n_iτ_i = τ  .

These weights are nothing but the Dynkin labels of the affine D_n Dynkin graph, or equivalently, the ranks of the nodes in the McKay graph corresponding to dihedral groups.

Using the action in the marked point basis and <ref> and denoting as before τ⃗=(τ;τ_0,…,τ_n), one finds that the action of the duality group in the coupling basis reads

    S    : τ⃗⟼(-1/τ;τ_0/τ,…,τ_n-1/τ,τ_n/τ-1-1/τ) ,
    
        T    :  τ⃗⟼  (τ+1;τ_0,…,τ_n-1,τ_n+1) ,

and the action of a deck transformation acts as

    R_I,i : τ⃗⟼
            (τ; 2ζ_I+τ_1,τ_0-2ζ_I,…,τ_n)     (i=1),
    
            (τ; 2ζ_I-τ_1,2ζ_I-τ_0,τ_0+τ_1+τ_2-2ζ_I,…,τ_n)     (i=2),
    (τ; τ_0,…,2ζ_I-∑_k=0^i-1n_kτ_k+τ_i-1,-2ζ_I-∑_k=0^in_kτ_k+τ_i,…,τ_n)     (3≤ i≤ n-2),
    
            (τ; τ_0,…,2ζ_I+τ_n-2+τ_n-1+τ_n-τ, τ-τ_n-2ζ_I, τ-τ_n-1-2ζ_I)    (i=n-1),
    
            (τ; τ_0,…,2ζ_I+τ_n-τ, -2ζ_I+τ_n-1+τ)     (i=n),


for I=A,B,C,D. The structure of these transformations can be written more economically by using the partial sums

    P(i)=∑_k=0^in_kτ_k = 2 p_i+1  ,

in terms of which one obtains

    R_I,i : τ⃗⟼
            (τ;2ζ_I+P(1)-τ_0,-2ζ_I+P(1)-τ_1,τ_2,…,τ_n)     (i=1),
    
            (τ;2ζ_I-P(1)+τ_0,2ζ_I-P(1)+τ_1,-2ζ_I+P(2)-τ_2,…,τ_n)     (i=2),
    
            (τ;τ_0,…,2ζ_I-P(i-1)+τ_i-1,-2ζ_I+P(i)-τ_i,…,τ_n)     (3≤ i≤ n-2),
    
            (τ; τ_0,…,2ζ_I-P(n-2)+τ_n-2,-2ζ_I+P(n)-τ_n,-2ζ_I+P(n)-τ_n-1)    (i=n-1),
    
            (τ; τ_0,…,2ζ_I-P(n)+τ_n,-2ζ_I+P(n)+τ_n-1)     (i=n)   .
 

Finally, the permutations act on the couplings as


    s_i : τ⃗⟼
            (τ; τ_0 ,   - τ_1 ,  τ_2 + τ_1  ,  τ_3 ,  …  , τ_n)     (i=1),
    
            (τ;τ_0 + τ_2 ,  τ_1 + τ_2 ,   - τ_2 ,  τ_3 + τ_2 ,  τ_4 ,  …  , τ_n)     (i=2),
    
            (τ; τ_0,  …  , τ_i-1 + τ_i ,   - τ_i ,  τ_i+1 + τ_i ,  τ_i+2 ,  …τ_n)     (3≤ i≤ n-2),
    
            (τ; τ_0,  …  , τ_n-4 ,  τ_n-3 + τ_n-2 ,   - τ_n-2 ,  τ_n-1 + τ_n-2 ,  τ_n + τ_n-2 )    (i=n-2),
    
            (τ; τ_0,  …  , τ_n-3 ,  τ_n-2 + τ_n-1 ,   - τ_n-1 ,  τ_n )     (i=n-1)   .


The choice in <ref> makes manifest the relation between the couplings and the root lattice of the affine D_n algebra and indeed the above transformations generate the automorphisms of the affine root system. 
This can be checked explicitly by writing the deck transformations associated with π_1(𝕋^2) in terms of the R_I generators. The set of t^(τ)_i, s_i, R_A and R_D can be matched with the generators of the automorphism group of the affine D_n algebra, comprising the Weyl group, see for example <cit.>.




 §.§ Global variants and dualities


We now address higher form symmetries and duality symmetries that can arise in these quiver theories, discussing in particular how the mapping class group of the Riemann surface used to construct these theories plays a crucial role in both of these aspects.

Let us start by discussing the A_n-1 case. Recall that this theory can be obtained via a class 𝒮 construction, wrapping M5-branes on a torus with n punctures. If all the punctures are regular, one has a 1-form symmetry[More in general, the worldvolume theory of k M5-branes wrapping a Riemann surfaces Σ_g,n of genus g, with n regular punctures, has a ℤ_k^g 1-form symmetry.] <cit.>. This picture provides a  clear understanding of how the duality group acts on the 1-form symmetry, as follows.

Indeed, the generators of the 1-form symmetry correspond to non-trivial homology 1-cycles of the Riemann surface <cit.>. For example, if the Riemann surface is an elliptic curve with underlying smooth surface the torus, the two generators of H_1(T^2,ℤ) correspond to the symmetry operators capturing the 1-form “electric" and “magnetic" symmetries, which are mutually dual. Correspondingly, the usual A and B cycles of the torus form a symplectic basis of H_1(T^2,ℤ) with respect to the intersection pairing. Because of this, the mapping class group of the surface can act non-trivially on global variants: any transformation which shuffles the generators of H_1(T^2,ℤ), will, in general, also change the global variant of the theory one considers.
    
The simplest example of this is 𝒩=4 𝔰𝔲(k) SYM, obtained by compactifying the 6d 𝒩=(2,0) SCFT of type A_k-1 on an elliptic curve without punctures. Every global variant of the theory has a non-trivial 1-form symmetry <cit.> (see <cit.> for the class S perspective). For example, when the gauge group is SU(k) the 1-form symmetry is purely electric ℤ_k^(1), and the charged objects are the Wilson loops. On top of that, 𝒩=4 SYM enjoys Montonen-Olive duality, exchanging “electric" and “magnetic" degrees of freedom, most notably Wilson and 't Hooft loops. These two facts are captured in the class 𝒮 realization of the theory: the 1-form symmetry descends from the reduction of the 2-form symmetry of the 6d SCFT on one of the cycles of the torus, and Montonen-Olive duality is embodied as modular transformations of the elliptic curve, i.e. the MCG of the underlying torus. In particular, an S-duality transformation exchanges the two cycles of the torus, and maps for example 𝒩=4 SU(k) SYM at gauge coupling τ to 𝒩=4 PSU(k)_0 SYM at gauge coupling -1/τ, with the notation of <cit.>. This is one version of Langlands duality <cit.>.
    
It turns out that the presence of regular punctures on the Riemann surface of the class 𝒮 construction plays no role as far as the 1-form symmetry is concerned. This follows from the fact that the additional 1-cycle introduced by each puncture has trivial intersection pairing with any other 1-cycle <cit.>. This means that only the SL(2,ℤ) subgroup of the mapping class group of Σ_1,n, more precisely, its S transformation, can shuffle global variants of the theory, while the other mapping classes have no effect on the 1-form symmetry of the theory, despite inducing genuine dualities.

The D_n case deserves more attention. Indeed, the Riemann surface in this case is an orbifold, for which the notions of intersection pairing and integer 1-cycles are subtle. Even before establishing what is the 1-form symmetry, we can ask what are the dualities that can change the global structure of these theories. By analogy with the A_n-1 case, we assume that the change in global structure takes place only for the transformations for which τ→ -1/τ. The only element of the mapping class group that acts on τ in this way is S. We thus conclude that also in the D_n case only the SL(2,ℤ) subgroup of the mapping class group can change global variants.

In order to establish what is the 1-form symmetry of these theories, let us again start from the A_n-1 case. From field theory, it is clear that in the SU(k)^n global variant, the ℤ_k 1-form symmetry acts on the k non-trivial Wilson lines, which are obtained by identifying the Wilson lines of each SU(k) group due to the presence of dynamical bifundamental matter fields. Performing the S operation, the Wilson lines become k non-trivial 't Hooft lines, meaning that the new global variant is SU(k)^n/ℤ_k, which is alternatively obtained by gauging the 1-form symmetry. For the D_n quiver with SU gauge groups, the bifundamentals are such that again there are only k non-trivial independent Wilson lines, so that the 1-form symmetry is still ℤ_k.[More generally, any balanced 𝒩=2 quiver theory without flavor nodes has a ℤ_k 1-form symmetry, where SU(k) is the smallest gauge group in the quiver.] As argued above, the action of S on the global structure of the gauge group is then exactly as in the A_n-1 case, i.e. the same as gauging the ℤ_k^(1). 





  §.§.§ Duality symmetries and orbits of marked points



We now describe how dualities of the field theory for A_n-1 and D_n quiver theories can enhance to symmetries at specific points of the conformal manifold. This is because, as we have just seen, some dualities change the global structure of the gauge group, but the latter change can be undone by gauging the 1-form symmetry. Hence if the duality leaves the coupling invariant, the combination of duality and gauging becomes a symmetry of a specific theory. Since a gauging is involved, these duality symmetries are in general non-invertible <cit.>. 

For specific values of the τ_i, there might exist a non-trivial subgroup of the MCG leaving the field theory unchanged. The simplest example is the S transformation of 𝒩=4 𝔰𝔲(k) at τ=i, which leaves the local dynamics invariant, but changes the global variant. One can then recover the original theory by gauging the 1-form symmetry.

In the quiver theories of our interest, the same kind of symmetry transformation can be constructed. One starts by considering mapping classes which fix the configuration of marked points. When such operations act non-trivially on the global structure of the theory, one can compensate them by gauging (a subgroup of) the 1-form symmetry. As in the 𝒩=4 case, these two combined operations will, in general, lead to non-invertible symmetries.

The part of the MCG that acts non-trivially on the global variants of the theory is the modular one, moreover only the finite subgroups of SL(2,ℤ) that stabilize the coupling τ can lead to (non-invertible) symmetry defects. These subgroups are cyclic of order 2, 3, 4 or 6 and are generated by S^2, S^3T, S and ST respectively.[Another common choice for an element of order 3 is ST^-1; however, ST^-1 fixes exp(π i/3) rather than exp(2π i/3).] 
The transformation S^2 leaves τ unchanged, and therefore can be a symmetry for any choice of τ, depending on the location of the punctures. Meanwhile, S, S^3T and ST can be symmetries only for fixed values of τ: τ=i for S and τ=exp(2 π i / 3) for both ST and S^3T. These transformations will be actual symmetries of the theory, depending on the position of the punctures, as we will discuss shortly.

Before proceeding, a comment is due concerning the terminology. The term `duality' refers in general to the type of relations between theories that are the object of the present paper. However, more specifically, `duality' often refers to the particular action S on the theory space that fixes τ=i. Now, as just emphasized above, such action is actually of order 4, since while S^2 sends any τ to itself, it acts as charge conjugation on the spectrum. As we will see, it permutes the punctures in the cases of our interest. Similarly, `triality' usually refers to both ST and S^3T because they send any τ to itself after acting three times, but actually their action on the spectrum is respectively of order 6 and 3. Though we will not use `tetrality' instead of duality, since there is no distinction to be made, in the case of triality we will use the term `hexality' when it is important to stress that we are referring to the action which is of order 6 on the spectrum.




  
Non-invertible defects of 𝒩=2 A_n-1 quiver SCFTs.

Let us consider the standard fundamental cell ℱ in ℂ for the torus E_τ=i which is invariant under the action of S, i.e. the parallelogram {0,1,i,1+i}. A point p∈ℱ is mapped to p/i, which amounts in a clockwise π/2 rotation with respect to the origin. The point is now out of the fundamental cell ℱ, as in <ref>. We can then use the deck transformation t to bring it back to ℱ:

    p ⟶p/i+i = p-(1+i/2)/i+(1+i/2) .

The way in which the right-hand side is written makes explicit that t∘ S acts as a (-π/2)-rotation about the center (1+i)/2 of ℱ. 


The points in ℱ split in orbits under the action of t∘ S, where t here denotes the combination of the deck transformations t for all the marked points. Generic orbits consist of four points; an example is the orbit {2,3,5,6} shown in <ref>, in which the points are permuted by t∘ S as 2→ 5 → 6 → 3 → 2, which in standard cycle notation reads (2 5 6 3). Apart from the generic orbits, there is one orbit of size two depicted by the purple rhombi in <ref>–the points 1 and 4 form an orbit of size two denoted (1 4)–and two orbits of size one, depicted as red squares. This way to represent points with non-trivial stabilisers is standard in the theory of wallpaper groups; in the present case, configurations of marked points in E_i invariant under t∘ S have as group of symmetries the wallpaper group denoted p4 (in crystallographic notation).




With the labeling on the left of <ref>, the position of the marked points 3, 5 and 6 is determined by the one of 2 and the requirement that the configuration is invariant under t∘ S:

    p_3 = i p_2 + 1   ,      p_5 = -i p_2 + i   ,      p_6 = -p_2 + i + 1   .


    
The combined operation t ∘ S permutes the punctures accordingly to their orbits under the (-π/2)-rotation. One can recover the original configuration by a suitable permutation σ; for example in <Ref> one has σ = (2 3 6 5)(1 4), or in terms of s_i it reads σ = s_3 s_2 s_1 s_4 s_3 s_2 s_3 s_5. Therefore, the combination 𝒟 = σ∘ t^(i)∘ S maps the original theory to itself up to a discrete gauging of the 1-form symmetry acting on the global variants of the theory. In this way one constructs non-invertible duality defects of 𝒩=2 A_n-1 quivers SCFTs akin to those of <cit.>, for each configuration of punctures invariant under t∘ S. Such configurations of punctures necessarily split in orbits of size four, two and one. Discrete gauging of the one-form symmetry ensures that this duality enhances to a non-invertible symmetry of the field theory. 

One can repeat the reasoning replacing S by 

    ST = ([  0 -1;  1  1 ])   or   S^3T = ([  0  1; -1 -1 ]) ,
 
which are respectively of order 6 and 3 in SL(2,ℤ). Both ST and S^3T fix τ=exp(2iπ/3), and they act on the marked points as

    ST : p⟼ pexp(-iπ/3)   and    S^3T : p⟼ pexp(2iπ/3) ,

that is, as a rotations by -π/3 or 2π/3, respectively. As before, one can compose ST and S^3T with appropriate deck transformations, to ensure that points in the standard fundamental cell of E_τ–the parallelogram {0,1,τ,τ+1}–are mapped to points of the same fundamental cell. One finds that the generic orbits for ST are of order 6, and that there is one non-generic orbit of size 3, one of size 2 and one of size 1. This is depicted on the left of <ref>, where as before purple rhombi depict the points whose stabilizers are of order 2, whereas blue triangles and green hexagons correspond to those whose stabilisers are of order 3 and 6, respectively. The corresponding wallpaper group is p6. The generic orbit of size six shown on the left of <ref> is permuted by ST as the cycle (1 3 2 6 4 5). 

Conversely, generic orbits for S^3T are of size three, and there are three non-generic orbits of size one and with stabilizer of order 3, depicted as blue triangles on the right of <ref>. The corresponding wallpaper group is p3. The configuration of points shown there splits in two regular orbits: (1 4 2)(3 6 5).



Again combining the action of ST (resp. S^3T) composed with a deck transformation, with an appropriate permutation and a discrete gauging of the one-form symmetry, one ends-up with a comprehensive description of hexality (resp. triality) non-invertible defects for 𝒩=2 A_n-1 quivers SCFTs. 



  
Non-invertible defects of 𝒩=2 D_n quiver SCFTs.

In the case of 𝒩=2 D_n quiver SCFTs, we can apply readily the same method to determine non-invertible defects. Let us discuss the specific example of the 𝒩=2 D_4 quiver SCFT at τ=i, whose conformal manifold is described by configurations of four punctures in the lower half of the fundamental cell E_i, i.e. Im(p_i) ≤ 1/2.

Let p_1,p_2,p_3 and p_4 denote the position of the marked points 1,2,3 and 4 respectively, on the left of <ref>. We gather them in the tuple (p_1,p_2,p_3,p_4). Recall that the position of the i'-th image is determined by the position of the i-th marked point:

    p_i' := q_i^-1,-1 = -p_i+1+i .


Under t∘ S one has

    (p_1,p_2,p_3,p_4) ⟶ (-ip_1+i,-ip_2+i,-ip_3+i,-ip_4+i) = (p_3,p_4',p_1',p_2) .


We now apply R_D,2 and R_D,3 in order to have all marked points in the desired region of the fundamental cell

    R_D,2 R_D,3(p_3,p_4',p_1',p_2) = (p_3,p_4,p_1,p_2) .

Lastly, via a combination of the permutations s_i, one can restore the starting configuration of punctures. In the example of <Ref>, one can take:

    s_2s_1s_3s_2: (p_3,p_4,p_1,p_2) ⟶ (p_1,p_2,p_3,p_4) .

All in all, the non-invertible duality defect is obtained as the operation s_2s_1s_3s_2 ∘ R_D,2∘ R_D,3∘ t ∘ S combined with an appropriate discrete gauging of the one-form symmetry.



This procedure generalizes to all 𝒩=2 D_n quiver SCFTs and duality defects, with corresponding wallpaper group p4, or triality defects, with corresponding wallpaper group p6. Note that by construction, S^2 is a symmetry of any D_n configuration of punctures, which implies in particular that triality defects are necessarily of order 6 and not 3.




 §.§ Duality group of E_n quiver SCFTs


We have seen that the duality group of 𝒩=2 A_n-1 and D_n quiver SCFTs contains the group of automorphisms of the corresponding affine root system: each coupling constant is naturally associated to a positive simple root of the corresponding affine Lie algebra. The global coupling τ is defined as[Here τ_i are the coupling constants of the single nodes and n_i are the ranks of the nodes in the McKay graph.]

    τ = ∑_i n_i τ_i

and corresponds to the imaginary root δ of the affine root system.

The full duality group is generated by the automorphism group of the affine root system together with a copy of SL(2,ℤ) acting on τ as:

    S    : (τ,τ_0,…,τ_n) ↦(-1/τ,τ_0/τ,…,τ_n-1/τ,τ_n/τ-1-1/τ) ,
    
         T    :  (τ,τ_0,…,τ_n) ↦  (τ+1,τ_0,…,τ_n-1,τ_n+1) .

Such modular transformations of the parameter τ were argued to exist from the underlying class S construction.

It is natural to conjecture that this analysis extends to the E_6,7,8 quiver gauge theories. These can be constructed as worldvolume theories of D3-branes transverse to ℂ^2/Γ_E_n×ℂ singularities, where Γ_E_n is the corresponding finite subgroup of SU(2).
More precisely, we conjecture that the duality group of these theories is the semi-direct product of SL(2,ℤ) with the affine Weyl group, further centrally coextended by the automorphisms of the affine Lie algebra, acting on τ as in <ref>. From this one can in principle derive the action of the duality group on the global variants of the theory, and hence construct non-invertible defects for suitable configurations of couplings.

Unlike in previous cases, there is no known class 
𝒮 realization of these theories—at least to our knowledge—so we lack direct methods to test our arguments. This analysis might actually pave the way for discovering explicit class 𝒮 realizations, or generalizations thereof, of E_n quiver theories.






§ DUALITY GROUP OF THE MASS DEFORMED THEORY


In the previous section, we have constructed the duality group of 𝒩=2 A_n-1 and D_n quiver gauge theories from class 𝒮 arguments. We devote this section to the above quiver theories mass deformed to 𝒩=1, exploiting the class 𝒮 setup while close in spirit  to <cit.>.



 §.§ Duality group of mass deformed A_n-1
 

In the following, we revisit the duality group of 𝒩=1 theories obtained as mass deformations of 𝒩=2 A_n-1 quiver gauge theories with gauge group SU(k)^n, along the lines of <cit.>. We will consider mass deformations of the form

    Δ W = ∑_i=1^nm_i/2ϕ_i^2   ,

which lead to 𝒩=1 SCFTs <cit.>, whose duality groups are induced by the original 𝒩=2 theory. 

The mass deformed theory is specified by the masses (m_1, m_2, …, m_n). As in <ref>, the strategy to construct the duality group consists in uplifting the associated type IIA elliptic model to M-theory, where the theory can be fully described geometrically.

Recall from <ref> that the starting type IIA setup consists of k D4-branes on a circle of radius R_6 intersecting n NS5-branes along the transverse direction. The low energy theory on the D4-branes is a 4d 𝒩=2 A_n-1 quiver gauge theory, where the VEVs of the complex adjoint scalars in 𝒩=2 vector multiplets parameterize the position of the D4-branes along (x^4,x^5). Let: 

    u := x^4 + i x^5   , 
    
        v := x^7 + i x^8   .

In this setup, the mass deformation we are interested in can be induced by tilting the NS5-branes relatively to each other in the complex (u,v)-plane. More precisely, if two adjacent NS5-branes are not parallel then any displacement of the center of mass of a D4-brane stretched between them changes the minimal length of the D4-segment, therefore the D4 are no longer free to move along the NS5's. From the point of view of the field theory, this means that some flat direction has been lifted. 
This lifting can be achieved, in first approximation, by adding  mass terms to the adjoint scalars <cit.>.
One can see that, when the relative angle between two adjacent NS5-branes is small, the mass is directly proportional to the angle <cit.>. In this limit, to which we will refer as the limit of small masses, one can identify the small mass with the ones in <ref> <cit.>.

We now proceed with the uplift to M-theory, where we introduce the elliptic curve E_τ, parameterized by the complex coordinate w = x^10 + i x^6:

    w ∼ w + q + τℓ ,   q, ℓ∈ℤ ,

as outlined in <ref>. The k D4-branes become M5-branes wrapping E_τ, whereas the NS5-branes lift to marked points. However, since each NS5-brane corresponds to a specific complex line in the (u,v)-plane, we can assign to them the point [u_i:v_i] ∈ℂℙ^1 which corresponds to the latter.[For example, the original 𝒩=2 case in which all NS5-branes extend along u is given by [u_i:v_i]=[1:0] ∀ i.] Thus, the M-theory uplift is encoded in the data of (p_i,[u_i:v_i])∈ E_τ×ℂℙ^1.

In the limit of small masses, the lines of homogeneous coordinates [u_i:v_i] are all close to [1:0], i.e. they are all in the complex chart ℂ≃{u≠ 0}⊂ℂℙ^1 and one can always rewrite [u_i:v_i]=[1:z_i] with z_i∈ℂ, where now the small mass limit can be formally expressed as z_i → 0 ∀ i. As a consequence of supersymmetry, since the superpotential needs to be holomorphic in the fields, the mass parameters m_i must be holomorphic in z_i and vanish when all branes are parallel, i.e. when z_i approaches z_i+1 <cit.>. This implies:

    m_i = z_i+1 - z_i   ,    z_i ∼ z_i+n  .


Tilting the n NS5-branes in the type IIA setup yields only n-1 relative angles, which is at odds with the n mass parameters one can define in field theory. Equivalently, given the definition in <ref>, the masses satisfy ∑ m_i = 0. This apparent paradox can be tackled similarly to what is done in <cit.> by considering a non-trivial ℂℙ^1-fibration over the elliptic curve E_τ. Indeed, as we will see shortly, if we consider the z_i as sections of a non-trivial fibration over E_τ, we can recover the missing mass deformation in term of a “global mass", m = ∑ m_i, which vanishes precisely when the fibration trivializes. 

Let us denote R ("Right") and U ("Up") the topologically non-trivial cycles of E_τ corresponding to w→ w+1 and w→ w+τ, respectively. Saying that ℂℙ^1 is fibered non-trivially over E_τ means that there can be non-trivial (projective) monodromies along R and U, in the form of matrices in GL(2,ℂ) acting projectively on the fiber coordinate:

    z →a z + b/c z + d  .


In order for the fibration to preserve 𝒩=1 supersymmetry, the monodromy along either cycles of the torus must preserve the holomorphic 3-form Ω = du ∧ dv ∧ dw, and this implies that the monodromies actually live in SL(2,ℂ): 

    M = ( 
        
            a     b 
    
            c     d
        )   ,    a d - b c = 1   .

In field theory the mass deformation can be continuously turned off, which implies that the fibration must be topologically trivial. It is is then entirely described by the monodromies M_R and M_U associated to the cycles R and U of E_τ, which provide a representation of the fundamental group of E_τ. Since π_1(E_τ) is abelian, the matrices M_R and M_U must commute, as we would have intuitively expected. 

In the limit of small masses, one can approximate the projective bundle with an affine fibration over E_τ. From field theory one expects that the monodromies in SL(2,ℂ) act as shifts z_i → z_i + b in this limit. Note that when z is small:

    a z + b/c z + d≃a d - b c/d^2 z + b/d + 𝒪(z^2) = 1/d^2 z + b/d + 𝒪(z^2)   ,

thus it must be the case that d=1. The affine shift is non-trivial when b ≠ 0, which we now assume. The monodromies M ∈SL(2,ℂ) must therefore be of the form:

    M = ( 
        
            a     b 
    a-1/b    1
        )   .

One can check that two such generic matrices commute if and only if a=1, thus we define:

    M_R = ( 
                
                    1     b_R 
    
                    0     1
                )   ,   
                M_U = ( 
                
                    1     b_U 
    
                    0     1
                )   ,

where b_R and b_U are complex numbers characterizing the fibration.[Our solution is slightly different from the one in <cit.>, but it makes more explicit the relation between M_R,U and the shifts z→ z+constant.] The monodromies induce the following transformations:

    (τ, b_R, b_U, p_i, z_i)  R⟶  (τ, b_R, b_U, p_i + 1, z_i + b_R)   , 
       (τ, b_R, b_U, p_i, z_i)  U⟶  (τ, b_R, b_U, p_i + τ, z_i + b_U)   .

Allowing the affine fibration over E_τ to be non-trivial introduces the freedom to change the fiber coordinate z by (p,z)→ (p, z + λ p), such that the shifts of the monodromies in <ref> are preserved. This “gauge" symmetry acts[This action is recovered by asking the action of M_R,U on the parameter to match before and after the gauge fixing.] on the parameters of the M-theory setup as: 

    (τ, b_R, b_U, p_i, z_i) f⟶ (τ, b_R + λ, b_U + τλ, p_i, z_i + λ p_i)   .

One can fix the gauge by imposing M_R to be trivial, i.e. b_R=-λ and we can define

    b_U - b_R τ = m

to be the global mass.

To conclude, in M-theory the setup is fully specified by the tuple

    ( τ, 0, m, p_i, z_i)   ,

where m and the z_i can be traded for n masses m_i:

    m_i    = z_i+i - z_i  ,      i = 1, … , n-1   , 
    
        m_n     = m + z_1 - z_n  , 
    
        m    = ∑_i=0^n-1 m_i   .

The above equations show how a non-trivial fibration allows for a tilted configuration with only one mass term, say m=z_n. This can also be understood as a consequence of <ref>, where we defined m_n = z_1-z_n. When we have a non-trivial fibration, the difference between z_1 and z_n is computed across the fundamental cell of the torus, thus we should consider not z_1, but M_U(z_1)=z_1+m and hence the definition in <ref>. 

We see that the mass deformed theory is now fully specified by the vector m⃗=( m;   m_1,  …  , m_n). However, this set of variables depends of the gauge fixing and it is not preserved by S, which exchanges the R and U cycles and consequently b_R and b_U. Explicitly:

    (τ, 0, m, p_i, z_i) S⟶(-1/τ, m, 0, p_i/τ, z_i )   .

Therefore the action of S needs to be followed by another f-gauge fixing with λ = - m, leading to:

    (τ, 0, m, p_i, z_i) f ∘ S⟶(-1/τ, 0, m/τ, p_i/τ, z_i - m p_i/τ)   .

From now on we assume that S is always post-composed with a suitable f-gauge fixing. 

We are now set to describe the duality group of these 𝒩=1 theories. By construction, the action of T and t_i^(1) is trivial on the z_i, whereas they act on p_i and τ as in <ref>. The generator t_i^(τ) moves the punctures along U, thus:

    (τ, 0, m, p_i, z_i) t_i^(τ)⟶ (τ, 0, m, p_i + τ, z_i + m)   .

Last, permutations s_i exchange z_i and z_i+1 as well as p_i and p_i+1. 

With the masses defined as in <ref>, we can write the action of the generators { S, T, t_i^(1), t_i^(τ) , s_i } of the duality group on the masses as

    S     :  m⃗ → ( m/τ ;   m_1 - m τ_1/τ,  …  , m_i - m τ_i/τ , … , m_n - m τ_n/τ + m/τ)   , 
    
                t_i^(τ)    :  m⃗ →  ( m;   m_1 ,  …  , m_i-1 + m , m_i - m, m_i+1,  …  , m_n)   , 
    
                s_i     :  m⃗ →  ( m;   m_1,  …  , m_i-1 + m_i , - m_i,  m_i+1 + m_i , … , m_n)   ,

while the remaining generators T and t_i^(1) act trivially on m⃗.

We conclude this section by remarking that the duality group of the mass-deformed theory can be presented similarly to the duality group of the original 𝒩=2 theory. The masses behave as roots of the affine A_n-1 algebra, with the global mass playing the role of the imaginary root. Therefore, the duality group is the extension of the automorphism group of the A_n-1 algebra by the modular group SL(2,ℤ), acting as in <ref>, while acting as well on the couplings τ_i as in <ref>.





 §.§ Duality group of mass deformed D_n


We now address the mass deformations of the D_n theory by a superpotential of the form <ref>, with the masses defining the deformation gathered in a vector m⃗=(m;   m_0,…,m_n). As reviewed in <ref>, this theory admits a type IIA construction in terms of D4s suspended between NS5s, in presence of orientifold ONS5^--planes, <ref>. In terms of branes, mass deformations are obtained as in <ref> by tilting the NS5 branes in the ^2_u,v plane. The lift to M-theory then fully unveils the duality group of the deformed theory. 
    
    Each NS5-brane corresponds to a complex line in the ^2_u,v plane, and hence to a point [u_i : v_i] ∈ℂℙ^1. In the limit of small angles, one can assume that the slope of these lines is close to [1:0], thus one can set [u_i : v_i]=[1 : z_i] where  z_i ∈ℂ. The masses can then be expressed in term of the z_i, as in <ref>. The main difference with respect to A_n-1 theories comes from the orientifold projection: since it maps x^7,8 to -x^7,8, tilting an NS5 brane by a complex number z amounts to tilting its image with respect to the ONS5^--planes by -z, at least when the ^2_u,v plane is trivially fibered over the x^6-segment; this is depicted in <ref> (which builds on the previous <ref>) for the brane closest to the leftmost ONS5^--plane, and its image.
    
    
    
    Similarly to what was done above in A_n-1 theories, the masses are expressed in terms of the z_i as:
    
    m_0    = z_2-(-z_1)   ,
    
            m_i   = z_i+1-z_i  ,    i=1,  …  , n-1 ,
    
            m_n    = +(-z_n)-z_n-1  ,

    where we kept the - signs coming from the orientifold projection since they will be relevant in the following. This definition is consistent with the requirement of holomorphy in the z_i's and vanishing of all masses when the branes are parallel.

    The definition in <ref> leads to a vanishing total mass:
    
    m=m_0+m_1+m_n-1+m_n+2 ∑_i=2^n-2 m_i=0 ,
 
    hence as in A_n-1 theories it naively seems that there is a missing mass parameter in the brane setup, as compared to the adjoint masses appearing of field theory. Here again, the mismatch is resolved by considering slightly more general brane setups in which ℂℙ^1_u,v is allowed to fiber non-trivially over the M-theory pillowcase E_τ/ℤ_2.
    
    Such a fibration is specified by a representation of the fundamental group π_1(E_τ/ℤ_2) into GL(2,ℂ). Because of the way the orientifolds act on the coordinates x^7,8, in order to preserve 𝒩=1 supersymmetry the generators R_A, R_B, R_C and R_D must correspond to matrices of determinant -1. Moreover, in the limit of small z_i, one expects the monodromies corresponding to R_l, l=A,B,C,D, to act as z_i→ -z_i + b_l, where the b_l are i-independent complex numbers. Recalling that the R_l are involutions, the form of such elementary monodromies is constrained to be: 
        
    M_l = (
                    
                        a_l     b_l 
    1-a_l^2/b_l    -a_l
                    )   ,    b_l ≠ 0   ,    l = A, B, C, D ,

    and the other relations eventually yield:
    
    M_l = (
                    
                        -1     b_l 
    
                        0     1
                    )   ,    b_l ≠ 0   ,    l = A, B, C, D 
        b_B - b_A = b_D - b_C  .

    
    As in <ref>, allowing non-trivial fibrations over E_τ/ℤ_2 introduces an additional freedom in the choice of fiber coordinate. One can do the redefinition z→ z+f(p) where p∈ E_τ/ℤ_2 and with f a holomorphic function, however only affine functions f(p)=λ p +κ preserve the form of the monodromies. Such a coordinate change induces the following transformation:
    
    (τ, b_A,b_B,b_C,b_D, p_i, z_i) f⟶ (τ, b_A , b_B + λ ,b_C + τλ ,b_D + (1+τ)λ, p_i, z_i + λ p_i)   .

    
    With the physical interpretation in mind, we can fix λ in such a way that:
        
    b_B - b_A + λ = b_D - b_C + λ = 0   ,

    This makes clear that up to redefinition of the fiber coordinate, the fibration depends only on the two parameters b_A and b_E = b_C + (b_A - b_B) τ. With the notation of above, the fibration is defined by the data:
    
    (τ, b_A , b_E , p_i , z_i )  ,

    on which the monodromies act as follows:
    
    (τ, b_A, b_E, p_i , z_i ) R_A⟶ (τ, b_A, b_E, -p_i, - z_i + b_A )   , 
       (τ, b_A, b_E, p_i , z_i ) R_B⟶ (τ, b_A, b_E, - p_i + 1 , - z_i + b_A )   , 
       (τ, b_A, b_E, p_i , z_i ) R_C⟶ (τ, b_A, b_E, - p_i + τ , - z_i + b_E )   , 
       (τ, b_A, b_E, p_i , z_i ) R_D⟶ (τ, b_A, b_E, - p_i + τ + 1 , - z_i + b_E )   .

    
    In <ref>, we stressed that the tilting of a brane and its image with respect to an orientifold plane are not independent. In M-theory, this amounts to saying that given the tilting z_i of a brane, the tilting of its image with respect to a fixed point of the involution ℤ_2 is encoded in the image of z_i by the monodromy M_l around that fixed point. In general, the fibration is not trivial, and:
    
    M_l(z_i) = - z_i + b_l   .

    This leads to the following generalization of <ref>:
    
    m_0 = z_2 - M_A/B(z_1) = z_2 + z_1 - b_A   ,
       m_i = z_i+1-z_i  ,      i=1, …  , n-1
       m_n = M_C/D(z_n)-z_n-1 = - z_n -z_n-1 + b_E   .

    <Ref> corresponds to a trivial fibration, for which b_A=b_E=0.
    
    The corresponding global mass reads
    
    m = ∑_i n_i m_i = b_E - b_A

    with the n_i the Dynkin labels of affine D_n, as in <ref>. Note that as expected, the global mass m vanishes when the fibration is trivial. 

    We have thus shown that the theories obtained by deforming 𝒩=2 D_n quiver SCFTs by 𝒩=1 preserving masses are fully determined by the set of gauge couplings τ_i satisfying τ=∑_i n_i τ_i, and the set of adjoint masses m_i with m=∑_i n_i m_i. Though the set of couplings and the set of masses play very similar roles in the geometric description of the theories we are interested in, there is an important difference in the way the mapping class group acts on them. Its action on the couplings is given in <ref>, whereas the one on the masses is described as follows. 
    
    First of all one can note that the action of T is trivial, whereas S[Composed with a redefinition of the fiber coordinate for the same reason as in <ref>.] acts as
    
    m⃗S⟶( m/τ ;   m_0 - m τ_0/τ,  …  , m_i - m τ_i/τ ,  …  ,  m_n - m τ_n/τ + m/τ)   .


    Deck transformations act on the masses as in <ref> with the m_i in place of the τ_i, and with the shifts of the z_i due to the non-trivial fibration taken into account. For example, R_I,i maps 
    m⃗ to:

    
    (m;m_1+δ_C|D,Im,m_0+δ_C|D,Im,m_2,…,m_n)     (i=1),
    
            (m;-m_1+δ_C|D,Im,-m_0+δ_C|D,Im,P(2)-m_2-δ_C|D,Im,…,m_n)     (i=2),
    
            (m;m_0,…,-P(i-2)-m_i-1+δ_C|D,Im,P(i-1)+m_i-δ_C|D,Im,…,m_n)     (3≤ i≤ n-2),
    
            (m; m_0,…,-P(n-2)+m_n-2+δ_C|D,Im,δ_A|B,Im-m_n,δ_A|B,Im-m_n-1)    (i=n-1),
    
            (m; m_0,…,m_n+δ_A|B,Im,m_n-1+δ_A|B,Im)     (i=n)   ,

where I=A,B,C,D, where C|D in δ_C|D,I means either C or D, and with: 

    P(i)=∑_k=0^in_k m_k .


Finally the transposition s_i maps the mass vector m⃗ to:

    (m; m_0 ,   - m_1 ,   m_2 + m_1  ,   m_3 ,  …  , m_n)     (i=1),
    
            (m;m_0 + m_2 ,   m_1 + m_2 ,   - m_2 ,   m_3 + m_2 ,   m_4 ,  …  , m_n)     (i=2),
    
            (m; m_0,  …  , m_i-1 + m_i ,   - m_i ,   m_i+1 + m_i ,   m_i+2 ,  …  m_n)     (3≤ i≤ n-3),
    
            (m; m_0,  …  , m_n-4 ,   m_n-3 + m_n-2 ,   - m_n-2 ,    m_n-1 + m_n-2 ,   m_n + m_n-2 )    (i=n-2),
    
            (m; m_0,  …  , m_n-3 ,   m_n-2 + m_n-1 ,   - m_n-1 ,   m_n )     (i=n-1).

This concludes our analysis of the duality group's action on the masses that define the deformation of D_n quivers.



§ MODULI SPACE OF THE MASS DEFORMED THEORY


We have seen in the previous section how the respective MCGs of the A_n-1 and D_n N=2 SCFTs act on the mass parameters that one can turn on. Such relevant mass deformations break supersymmetry to N=1 and trigger an RG flow. In the present section, we ask in all generality what is the moduli space of the theory that is the result of this RG flow. As we will see, such moduli spaces describe geometries which are often a non-trivial fibrations of the geometry described by the N=2 moduli space.




 §.§ Moduli space of mass deformed A_n-1
 

We are interested in A_n-1 quiver theories deformed by 𝒩=1 preserving masses, that is:

    Δ W = ∑_i=1^n m_i/2ϕ_i^2   ,

where the ϕ_i denote the adjoint scalars in the 𝒩=2 vector superfields. 

On general grounds, one expects that the deformed theories flow to interacting 𝒩=1 SCFTs <cit.>. One of the simplest examples is the conifold field theory, which is the mass deformation of the 𝒩=2 A_1 quiver gauge theory with mass parameters (m_1,m_2)=(m,-m) <cit.>. Other SCFTs of interest can be obtained from other choices of masses; for example, the Pilch–Warner (PW) point <cit.> is also obtained from the 𝒩=2 A_1 quiver gauge theory, though with the choice of deformation parameters (m_1,m_2)=(m,m). The moduli space of the former is given by the locus xy=zw in ℂ^4, while the latter's one is the two-fold ℂ^2/ℤ_2. The general description of the moduli space of 𝒩=1 deformations of 𝒩=2 quiver gauge theories that we are going to present, will in some cases allow us to argue directly that these theories flow to interacting 𝒩=1 SCFTs.

We first consider general mass deformations of the A_n-1 quiver gauge theory, with gauge group U(1)^n.[In the general case where the gauge group is SU(k)^n for some k, the moduli space is generically the k-th symmetric product of the abelian one, hence the customary simplification when discussing the moduli space.] The deformed superpotential reads:

    W_𝒩=1 = ∑_i=1^nϕ_i ( X_i,i+1X_i+1,i - X_i,i-1X_i-1,i) + ∑_i=1^nm_i/2ϕ_i^2   ,

where i is understood modulo n. Let

    x = ∏_i=1^n X_i,i+1  ,    y =  ∏_i=1^n X_i,i-1  , 
       w_i =    X_i,i-1X_i-1,i  ∀  i   , 
       u_i =   ϕ_i   ∀  i   ,

be the elementary gauge invariant operators. They are constrained by F-terms equations, which read

    X_i,i+1X_i+1,i - X_i,i-1X_i-1,i + m_i ϕ_i = 0   , 
       X_i+1,i ϕ_i - ϕ_i+1  X_i+1,i = 0   ,

for all i=1,…,n. These lead to the relations: 

    x y = ∏_k=1^n w_k   , 
    
        u_i = u   , 
     
        w_i+1 - w_i = - m_i u   ,

again for all i=1,…,n.
The w_i's can be written recursively as

    w_i = w_1 - ( ∑_k=1^i-1 m_k) u = w_1 - t_i u    ,

and, since by definition w_n+1 = w_1, we have the constraint

    ( ∑_i=1^n m_i ) u = m   u = 0   ,

where m denotes the “global mass".

Therefore, denoting w_1 = w, the moduli space of the deformed theory is defined by the equations

    x y = ∏_k=1^n( w - t_k u)  
    
            m u = 0   .

Note that the second equation imposes either u=0 or m=0. 

If m≠ 0, then u=0 and the moduli space is defined by

    w_i = w   ∀  i   ,    x y = w^n   ,

i.e. it is the 2-fold ℂ^2/ℤ_n. This generalizes the case of the PW fixed point. If rather m=0, the moduli space is a 3-fold determined by the partial sums t_k. This is analogous to the case for the conifold theory.

From this analysis, we see that the moduli space of the deformed theory is either a two- or a three-fold singularity. In particular, the former is a Du Val singularity of type A_n-1, while the latter is a compound Du Val, again of type A_n-1, i.e. xy=w^n + u g(w,u),[In general, a compound Du Val three-fold is given by the equation f_Du Val(x,y,w) + u g(w,u)=0 in ℂ^4.] for some polynomial g(w,u). The two-fold case is less explored in the literature, only for specific examples there are results showing that the mass deformation we are interested in leads to an interacting SCFTs <cit.>. On the other hand, in the three-fold case, it has been proven that the deformations under consideration always lead to interacting SCFTs <cit.>.


Finally, let us give a different perspective on the IR moduli space in <ref>. In <cit.>, a graphical tool called “bug calculus” is exploited in order to deform the algebraic curves of ADE singularities with FI terms b_i, associated to each node of the extended Dynkin diagram. The singularity is deformed by a versal deformation that depends on the FI parameters b_i. In the case of A_n-1 quiver, one finds

    x y = ∏_k=1^n[ w - ( ∑_j=1^k+1 b_j ) ]   ,

with the condition b_1+…+b_n = 0, which closely resembles <ref>. 

The F-terms equations, after mass deformation, have the same form of the gauge invariants constructed in <cit.>, provided the correspondence b_i ↔ m_i ϕ_i. A formal correspondence can be established if we deform the 𝒩=2 superpotential with complex FI terms,

    W_𝒩=1 = ∑_i=1^nϕ_i ( X_i,i+1X_i+1,i - X_i,i-1X_i-1,i) + ∑_i=1^n b_i ϕ_i   .

After applying the “bug calculus" procedure, we can now trade the b_i for  m_i ϕ_i to get to

    b_i    ↦ m_i ϕ_i   , 
    ∑_i=1^n b_i = 0    ↦∑_i=1^n m_i ϕ_i = 0   .

As discussed above, the F-terms requires ϕ_i = u, thereby obtaining the condition m u=0. 

This approach will be used in the next section to get the moduli space of deformed D_n quiver theories.
    


 §.§ Moduli space of mass deformed D_n
 


The analysis of the previous section can be repeated for the D_n theory.[The same can also be done for E_n-quivers, the resulting moduli space is either ℂ^2/Γ_E_n, for non-vanishing global mass, or a compound Du Val of type E.] Let us start by considering the superpotential for the D_n theory with the addition of masses for the adjoint fields 

    W_𝒩=1   = ∑_i=0,1ϕ_i X_i,2 X_2,i + ∑_j=n-1,nϕ_j X_j,n-2 X_n-2,j + ϕ_2 ( X_2,0 X_0,2 + X_2,1 X_1,2 + X_23 X_32) 
        + ϕ_n-2( X_n-2,n-1 X_n-1,n-2 + X_n-2,n X_n,n-2 - X_n-2,n-3X_n-3,n-2)  
        + ∑_l=3^n-3ϕ_l ( X_l,l+1X_l+1,l - X_l,l-1X_l-1,l) +∑_i=0^nm_i/2ϕ_i^2   ,

where we refer to <ref> for the index conventions of the fields. 



As before, we start by considering a theory with abelian gauge factors for the external nodes and U(2) for the internal ones. Analogously to the previous section, the moduli space of the non-abelian theory can be recovered from the symmetry product of k copies the this moduli space.
The F-terms for the chiral fields are

    ϕ_2 X_2,i = - X_2,iϕ_i   ,    i = 0,   1    ,  
        X_i,2ϕ_2 = - ϕ_i X_i,2  ,    i = 0,   1    ,  
       ϕ_n-2 X_n-2,j = - X_n-2,jϕ_j  ,    j = n-1,   n   ,  
        X_j,n-2ϕ_n-2 = - ϕ_j X_j,n-2  ,    j = n-1,   n    ,  
       ϕ_l X_l,l+1 = X_l,l+1ϕ_l+1  ,    l = 2,  …  , n-2   , 
        X_l+1,lϕ_l = ϕ_l+1 X_l+1,l  ,    l = 2,  …  , n-2   ,

while for the adjoint fields we have

    X_i,2 X_2,i = - m_i ϕ_i   ,    i = 0,   1    ,  
        X_j,n-2 X_n-2,j = - m_j ϕ_j   ,    j = n-1,   n    ,   
        X_2,0 X_0,2 + X_2,1 X_1,2 + X_23X_32 = - m_2 ϕ_2   ,  
        X_n-2,n-1 X_n-1,n-2 + X_n-2,n X_n,n-2 - X_n-2,n-3X_n-3,n-2 = - m_n-2ϕ_n-2  ,  
        X_l,l+1X_l+1,l - X_l,l-1X_l-1,l = - m_l ϕ_l   ,    l = 3,  …  , n-3   .


One can solve the F-terms or employ the graphical computational technique described in <cit.>. The detailed computation is given in <ref>, whereas here we limit ourselves to a summary of the salient points.

First of all, as a consequence of <ref>, we have that ϕ_0,1,n-1,n = u, for all external nodes and ϕ_i=2,…,n-2=diag(-u,-u), after using the gauge freedom to diagonalize the fields of the internal nodes. Second, from <ref>, dubbing w_i,j=(X_i,jX_j,i)=w_j,i, we have the following constraints

    w_0,2 = - m_0 u   ,    w_1,2 = - m_1 u   ,     w_n-1,n-2 = - m_n-1 u   ,    w_n,n-2 = - m_n u    , 
        w_2,3 = (2 m_2 + m_0 + m_1)u    , 
        w_n-2,n-3 = - (2 m_n-2 + m_n-1 + m_n)u    , 
        w_l,l+1 - w_l,l-1 = 2 m_l u   ,    l = 3,  …  , n-3   .

Finally, by taking the sum of <ref>, one then gets the following condition

    u ( m_0 + m_1 + ∑_l=2^n-2 2 m_l + m_n-1 + m_n ) = u m = 0   ,

while the other gauge invariants constructed out of the chiral fields lead to the algebraic curve describing the moduli space, see <cit.>.

As in the A_n-1 case, we see that there are two possibilities: either the global mass does not vanish, leading to a 2-fold moduli space, or it does and the moduli space is a 3-fold. In the former case, the moduli space is just the Du Val singularity corresponding to D_n, while in the latter it is a compound Du Val described by the following equation in ℂ^4

    x^2 + y^2 w + β  y   u^n = w^-1[∏_k=1^n(w + u^2 t_k^2 ) - ∏_k=1^n u^2 t_k^2]   ,

where the t_k are given by

    t_1 = 1/2( m_0 - m_1 )   ,    
        t_2 = 1/2( m_n - m_n-1)   ,   
        t_3 = 1/2( m_0 + m_1 )   ,  
    
        t_4 = 1/2( m_0 + m_1 ) - m_2   ,   …,     t_n = 1/2( m_0 + m_1 ) - ∑_l=2^n-2 m_l   ,

and

    β = - 2 ∏_k=1^n t_k   .


Contrary to the previous case, less is known about the existence of local CY metrics on these spaces and thus only the field theory analysis is accessible to study the conformality of the deformed theory. While we leave a detailed analysis to future works, but assuming that the deformed theory flows to an interacting SCFT, we can still analyse the duality group inherited by the deformed theory from the parent one. In particular, we will show that requiring to preserve some duality symmetries of the starting theory, constrains the moduli space of the deformed one.[One could investigate whether the deformations of Du Val singularities that preserve non-invertible duality defects can be characterized geometrically. Exploring the interplay with deformations that maintain toricity in the A_n-1 case could lead to deep insights, given the extensive techniques available for studying toric affine Calabi–Yau threefolds and their corresponding 𝒩=1 gauge theories.]




§ MASS DEFORMATIONS PRESERVING NON-INVERTIBLE SYMMETRIES


In <ref> we have characterized the locus in the conformal manifolds of 𝒩=2 A_n and D_n quiver SCFTs at which these theories admit non-invertible duality defects: it is the locus which corresponds to symmetric configurations of punctures on the M-theory torus. The study of relevant deformations preserving 𝒩=1 supersymmetry and such non-invertible defects has recently been pioneered in <cit.>. Our goal here is to provide a general method to characterize which mass deformations of 𝒩=2 A_n and D_n quivers preserve non-invertible duality, triality or hexality defects. In particular, we derive the dimension of the space of mass deformations which preserve non-invertible defects. In some cases, such deformations lead to known 𝒩=1 SCFTs with moduli spaces Calabi–Yau threefolds, generalizing the deformation of the 𝒩=2 A_1 quiver to the conifold SCFT.

The supercharges Q of 4d 𝒩=4 theories transform non-trivially under the SL(2,ℤ) duality group <cit.>. More precisely, they transform as Q →exp(-i β) Q. At self-dual values of the coupling τ, it turns out that β = π i / q, where q is the order of the stabilizer of τ in SL(2,ℤ). In 𝒩=1 superspace, a mass deformation takes the form

    Δ S_W = ∫d^2 θ m/2Φ_i^2   .

Such a deformation preserves the duality symmetry if and only if the transformation of the measure d^2 θ→exp(-2i β)d^2 θ can be reabsorbed by a transformation of the chiral fields Φ_i. This can be done using the R-symmetry <cit.>, if the masses (m_1,…,m_n) defining the deformation satisfy

    (m_1',…,m_n') = e^iα (m_1,…,m_n) ,

where (m_1',…,m_n') is the image of (m_1,…,m_n) under the duality transformation at hand.

We have seen that dualities sometimes act non trivially on the deformation masses, which makes the analysis more involved. This happens when the global mass does not vanish; therefore, in what follows we distinguish the cases with vanishing global mass with the ones where the global mass is non-zero.



 §.§ Vanishing global mass


When the global mass m vanishes, the action of SL(2,ℤ) and deck transformations on the z_i's is trivial. This means that, while the punctures p_i are rotated to p_i', cf. <ref>, the image p_i' of each p_i is still associated with the original z_i. Consider a configuration of n punctures invariant under a duality transformation of order q, as discussed in <ref>, schematically 𝒟 = σ∘ t ∘ S/T is a composition of permutation, deck and SL(2,ℤ) operations. 𝒟 acts trivially on p⃗ by construction, while only σ acts on z⃗ 

    𝒟((p_1,z_1),(p_2,z_2),…,(p_n,z_n)) = ((p_1,z_σ(1)),(p_2,z_σ(2)),…,(p_n,z_σ(n))) .

Therefore, for the masses to preserve the duality symmetry, they must solve the eigenvalue problem

    𝒟 m⃗ = e^i α m⃗  ,

where 𝒟 acts on the masses only through the subgroup of permutations in the whole duality group. The n `decorated' punctures (p_i,z_i) split into orbits under 𝒟 of size a divisor of the order of 𝒟. For example, under 𝒟 = σ∘ t ∘ S at τ=i, the punctures always split into O_4 orbits of size 4, O_2≤ 1 orbits of size 2 and O_1≤ 2 orbits of size 1,[Note that some configurations require that one or more gauge groups have infinite coupling even if punctures are distinct, for example when O_2=O_1=1.] with a total number of distinct orbits O_tot = O_1 + O_2 + O_4. 

The total number of solutions to <ref> can be explicitly given in terms of the number of orbits. Let q be the order of 𝒟, thus 𝒟^q=1 implies that the phase e^i α is a q^th root of unity. Then, the total number of independent mass deformations satisfying <ref> is given by

    
  * If e^iα≠ 1, there is one deformation for each orbit of size k such that e^ikα=1. Hence the total of independent deformations is ∑_k O_k.
    
    
  * If e^iα=1, then the number of independent deformations is O_tot-1.

We refer the reader to <ref> for a complete and detailed proof of this result. 

The space of solutions represent all of the possible relevant deformations of the 𝒩=2 theory that preserve the non-invertible duality symmetry, and distinct solutions flow in principle to distinct 𝒩=1 SCFTs. The moduli space of the IR theories is then specified by each mass deformation, as discussed in <ref>. 

We dedicate the rest of this section to explicit examples. 




  
Mass deformed A_3 quiver theory

We consider the configuration shown in <ref>. At τ=i, 𝒟 = σ∘ t ∘ S defines a non-invertible duality defect, where t=∏_i t^τ_i and σ= s_2 s_1 s_3. If one then turns on mass deformations in such a way that the global mass vanishes, 𝒟 acts on the masses as

    (m_1',m_2',m_3',m_4')    =(z_2'-z_1',z_3'-z_2',z_4'-z_3',z_1'-z_4') 
       =(z_4-z_2,z_1-z_4,z_3-z_1,z_2-z_3) 
       =(m_2+m_3,m_4,m_1+m_2,-m_2) .



The condition on the masses to preserve the non-invertible duality defect reads

    (m_1',m_2',m_3',m_4')=e^i α(m_1,m_2,m_3,m_4) ,

with solutions:
    
    (m_1, …, m_4) = (1,0,-1,0) m_1   for α=π  , 
       (m_1, …, m_4) = (-i-1/2,1,-i-1/2,i ) m_2   for α=π/2  , 
       (m_1, …, m_4) = (i-1/2,1,i-1/2,-i ) m_2    for α=-π/2  .

In other words, inside the (complex) 3-dimensional space of mass deformations for the 𝒩=2 A_3 SCFT, the subspace of deformations preserving non-invertible duality defects is one-dimensional. More precisely, it is the union of three lines.

As discussed in <ref>, these three mass eigenvectors determine the moduli space of the IR SCFT at the end of the RG flow. For α=π we have

    xy = w^2 ( w - m_1 u )^2   ,

which is the equation of the toric L^2,2,2 singularity. In contrast, for α=±π/2 the moduli space is defined by

    xy = w^4 - ( m_2 u/2)^4   .





  
Mass deformed A_5 quiver theory We consider the S-self dual configuration of 6 punctures on the M-theory torus E_i shown in <ref>. Note that the six punctures split into a generic orbit of size four (2 5 6 3), and a non-generic one of size two (1 4). The combination 𝒟=σ∘ t ∘ S, where 

    t=∏_i=1^6 t_i^(τ)   and   σ = s_3 s_2 s_1 s_4 s_3 s_2 s_3 s_5 ,

acts on the masses as 

    (m_1,m_2,m_3,m_4,m_5,m_6) → (-m_3, m_3 + m_4 + m_5, m_6, m_1, m_2 + m_3 + m_4, -m_4)   .




Solving the eigenvalue equation 𝒟m⃗ = e^iαm⃗ leads to the following five solutions

    [                         α=π         (1,0,1,-1,0,-1) m_1 xy = w^4 [ w^2 - (u m_1)^2];                         α=π         (0,1,0,0,-1,0 ) m_2     xy = w^3 ( w - u m_2)^3;                       α=π/2    (1,i-1,-i,-i,i-1,1 ) m_1 xy = w^2 [ w^4 - (u m_1)^4];                      α=-π/2    (1,-i-1,i,i,-i-1,1 ) m_1 xy = w^2 [ w^4 - (u m_1)^4];                         α=0         (1,0,-1,1,0,-1) m_1 xy = w^2 ( w - u m_1)^4   . ]

In accordance with the general analysis, we find that there are five independent mass deformation, where only two of them can be turned on simultaneously, i.e. the ones associated to α=π. The moduli space of the second and last solutions are toric singularities usually referred to as L^3,3,3 and L^2,4,2, respectively. 

We give more examples and details in <ref>.




  
Mass deformed D_4 quiver theory The same analysis applies to the D_n case. For instance, let us consider D_4, which requires 8 marked points to be placed on the torus 𝕋^2, and organize them in two orbits of size four. In <ref>, we showed that this configuration leads to a theory with a duality defect 𝒟 = R_D,4∘ R_D,2∘σ∘ t^(i)∘ S, with σ = s_1 s_3 and[Recall that for D_n, t_k^(τ) = R_C,k∘ R_A,k, acting on both marked points and images.] t^(i) = ∏_k=1^4 t_k^(i). Under 𝒟, the masses transform as 

    𝒟 : (m_0,m_1,m_2,m_3,m_4) → (m_1, -m_0, m_0 + m_2 + m_3, m_4, -m_3)   .

The solutions of the eigenvalues equation 𝒟m⃗ = e^i αm⃗ for vanishing global mass are 

    [                                               α = π/2                                     (i,-1,0,-i,1) m_4               x^2 + y^2 w = w^3 + 1/2(m_4 u )^4 (y+w);                                               α = π/2                                  (i-1,-i-1,1,0,0) m_2     x^2 + y^2 w = [ w^2 - (m_2 u)^4 ] ( w + 4 m_2 u );                                              α = -π/2                                     (-i,-1,0,i,1) m_4               x^2 + y^2 w = w^3 + 1/2(m_4 u )^4 (y+w);                                              α = -π/2                                  (-i-1,i-1,1,0,0) m_2 x^2 + y^2 w = [ w^2 - (m_2 u)^4 ] ( w + 4 m_2 u )   . ]

    



    


 §.§ Non-vanishing global mass


The case of non-vanishing global mass is more involved, since now 𝒟 acts non-trivially on the z_i. However, the punchline is the same. One applies the transformation 𝒟 to the masses using <ref> and looks for an eigenvector of masses with the further constraints that the global mass change as m' =  m/τ, which forces the phase to be e^iα=1/τ. 

We can again prove in this case that a solution to the eigenvalue problem 𝒟m⃗ = e^i αm⃗ always exists. To this end, we need to prove that there is at least one eigenvector with eigenvalue 1/τ, as required by the transformation properties of the global mass. One can check explicitly that in this case the Dynkin vector n⃗ is a left eigenvector with eigenvalue 1/τ.[The vector n⃗ is a left eigenvector with eigenvalue 1 for both the t_i^τ/R_I,i and s_i transformations, and it has eigenvalue 1/τ for the modular transformation S.] Since right and left eigenvectors of an automorphism form a basis for a vector space and its dual respectively, we have that at least one right eigenvector m⃗ exists, with eigenvalue 1/τ, such that m = n⃗·m⃗≠ 0. This proves there is always a mass deformation, with non-vanishing global mass, that preserves the duality symmetry. Moreover, for each other right eigenvector with eigenvalue 1/τ, we have an extra dimension in the space of solutions to <ref>.[Contrary to the case of non-vanishing global mass, we do not have a general formula for the dimension of this space.]

As an example, consider A_3 and the configuration of <ref>, but this time in the presence of a global mass. In order to the duality defect 𝒟 = σ∘ t ∘ S, the masses need to transform as
    
    m_2 + m_3 - m τ_1    = -i m_1   , 
    
            m_4 - m [ τ_1(i-1) + 1 ]    = - i m_2    , 
    
            m_1 + m_2 - m τ_1    = -i m_3   , 
    
            -m_2 + m [ τ_1 (i+1)-i ]    = -i m_4   ,

and there are two independent mass deformations with 𝒟:
    
    ( m_1 , m_2, m_3, m_4  ) = ( 0 , τ_1/1 - τ_1, 0, 1 ) m_4   ,      m = m_4/1-τ_1  , 
       ( m_1 , m_2, m_3, m_4  ) = ( 1, 2 τ_1 - i + 1/1 - τ_1, 1, 0 ) m_1   ,      m = m_1 1-i/1-τ_1  ,

and the moduli space of these IR SCFTs is a 2-fold, as discussed in <ref>.

In the mass deformed A_5 theory with global mass, the condition to preserve 𝒟 is
    
    -m_3     = -i m_1 + m (τ_1 - i)   , 
    
            m_3 + m_4 + m_5     = - i m_2 + m [ τ_1 (i-1) + i+1/2]   , 
    
            m_6     = -i m_3 - m i τ_1   , 
    
            m_1     = -i m_4 - m i τ_1   , 
    
            m_2 + m_3 + m_4     = -i m_5 + m [ τ_1 (i-1) + i+1/2]   , 
    
            - m_4     = - i m_6 + m ( τ_1 + i )   ,

    which has two eigenvectors with eigenvalue α=-π/2 
    
    ( 2τ_1 -i/2 τ_1 + i +2 ,   0 ,  -2τ_1 + 2i - 1/2 τ_1 + i +2 ,  1 - 2τ_1/2 τ_1 + i +2 ,   0,   1 ) m_6   ,      m = 2 + 2i/2 τ_1 + i +2m_6 
       ( -2/2 τ_1 + i +2 ,   1 ,  -2τ_1 /2 τ_1 + i +2 ,  - 2τ_1 - 2i/2 τ_1 + i +2 ,   1,   0 ) m_2   ,      m = 2 + 2i/2 τ_1 + i +2m_2

    and in both cases the IR theory has moduli space given by ℂ^2/ℤ_5, from the discussion in <ref>.

As a D_n example, consider the case with n=4 and two orbits of size four, for which we find that there are two mass configurations that preserve 𝒟 = R_D,4∘ R_D,2∘σ∘ t^(i)∘ S with α = π/2

    ( m_0 ,  …  , m_4 ) = (-i+1/2( τ_0 + τ_3 - 2/τ_3), i+1/2( τ_0 + i τ_3 - i - 1/τ_3), 0, 0, 1 ) m_4   , 
       ( m_0 ,  …  , m_4 ) = (-i+1/2( τ_0 - i τ_3 - 2/τ_3), - -i+1/2( τ_0 + τ_3 - i - 1/τ_3), 0, 1, 0 ) m_3   ,

with global mass m=m_4/τ_3 and m=i m_3/τ_3 respectively. From the discussion in <ref>, in both cases the moduli space is simply the 2-fold Du Val singularity of type D.

More examples can be found in <ref>.









§ ACKNOWLEDGEMENTS


The authors would like to thank Simone Giacomelli, Azeem Hasan, Elias Riedel Gårding and Luigi Tizzano for useful comments and clarifying discussions. 
R.A. and A.C. are respectively a Research Director and a Senior Research Associate of the F.R.S.-FNRS (Belgium). 
The work of S.M. is supported by “Fondazione Angelo Della Riccia” and by funds from the Solvay Family. S.N.M. acknowledges the support from the Simons Foundation (grant #888984, Simons Collaboration on Global Categorical Symmetries).
V.T. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster). This research is further supported by IISN-Belgium (convention 4.4503.15) and through an ARC advanced project.





§ DETAILS ON THE F-TERMS OF MASS DEFORMED D_N


In this section we want to identify the moduli space of the 𝒩=1 theory obtained by mass deformation of the D_n-shaped quiver gauge theory with gauge group SU(k)^4 ×SU(2k)^n-3. 

We will use the following convention for fields: denote the adjoint fields with ϕ_i, the chiral fields that transform in the fundamental representation of an external node of the quiver with A and in their anti-fundamental representation with B, while the remaining field transforming in the bifundamental representation of SU(2k) gauge factors with X_i,i+1 so that

    A_i=( (1)_i, (1)_2 )   ,    i = 0,  1   , 
        B_i=( (1)_2, (1)_i )   ,    i = 0,  1   , 
        A_j=( (1)_j, (1)_n-2)   ,    j = n-1,   n   , 
        B_j=( (1)_n-2, (1)_j )   ,    j = n-1,   n   , 
        X_l,l+1=( (1)_l , (1)_l+1)   ,    l = 2,  …  , n-3   ,

and accordingly for X_l+1,l. The generic superpotential deformed with mass terms for adjoints reads

    W_𝒩=1   = ∑_i=0,1ϕ_i B_i A_i + ∑_j=n-1,nϕ_j B_j A_j
       + ϕ_2 ( A_0 B_0 + A_1 B_1 + X_23 X_32) + ϕ_n-2( A_n-1 B_n-1 + A_n B_n - X_n-2,n-3 X_n-3,n-2) 
       + ∑_l=3^n-3ϕ_l ( X_l,l+1X_l+1,l - X_l,l-1X_l-1,l) + ∑_i=0^nm_i/2ϕ_i^2   .

We need to solve the F-terms, in order to find the equation that defines the moduli space and how it is affected by the choice of the masses. For the first goal, we rely on the computation carried out in <cit.> via the bug calculus graphical approach. In the following, we explicitly show how the masses affect the value of ϕ_i. The F-terms for the chiral fields are

    ϕ_2 A_i = - A_i ϕ_i   ,    i = 0,   1    , 
        B_i ϕ_2 = - ϕ_i B_i   ,    i = 0,   1    , 
       ϕ_n-2 A_j = - A_j ϕ_j  ,    j = n-1,   n   , 
        B_j ϕ_n-2 = - ϕ_j B_j   ,    j = n-1,   n    , 
       ϕ_l X_l,l+1 = X_l,l+1ϕ_l+1  ,    l = 2,  …  , n-2   , 
        X_l+1,lϕ_l = ϕ_l+1 X_l+1,l  ,    l = 2,  …  , n-2   ,   ,

while for the adjoint fields 

    B_i A_i = - m_i ϕ_i   ,    i = 0,   1    , 
        B_j A_j = - m_j ϕ_j   ,    j = n-1,   n    , 
        A_0 B_0 + A_1 B_1 + X_23X_32 = - m_2 ϕ_2   , 
        A_n-1 B_n-1 + A_n B_n - X_n-2,n-3X_n-3,n-2 = - m_n-2ϕ_n-2  , 
        X_l,l+1X_l+1,l - X_l,l-1X_l-1,l = - m_l ϕ_l   ,    l = 3,  …  , n-3   .

As we did for the A_n-1, we consider the moduli space of the theory with gauge group U(1)^4 ×U(2)^n-3, and the generic case will be given by the k-th symmetric product of this space. 
Let us proceed in steps. First, we show that -ϕ_0 and -ϕ_1, which are complex numbers, are the eigenvalues of ϕ_2, which is a 2 × 2 matrix. <Ref> and <ref> have the form of a right and left eigenvalue equation for ϕ_2, and there are two eigenvectors A_0, B_0^T with eigenvalue -ϕ_0, and two A_1, B_1^T with eigenvalue -ϕ_1. Assume for now that none of them is a null vector, otherwise either m_i=0 or ϕ_i=0. By <ref> we see that A_i and B_i are not orthogonal, and there is no relation between A_0 and A_1. The way to accommodate them is that B_0^T and B_1^T are proportional to A_0 and A_1, respectively, and the latter are linearly independent. So it exists a matrix V_2, whose columns are the two eigenvectors A_0 and A_1, that diagonalizes ϕ_2. On the other hand, if we lift the non-null assumption, and consider that, say, B_0 = 0 = B_1, and m_0 = m_1 = 0 and again we are left with A_0 and A_1 as the two eigenvectors of ϕ_2. A similar reasoning holds for ϕ_n-2, whose eigenvalues are -ϕ_n-1 and -ϕ_n with eigenvectors A_n-1 and A_n. Hence

    ϕ_2^d = V_2^-1ϕ_2 V_2 = ( 
        [ - ϕ_0     0;     0 - ϕ_1 ])  
       ϕ_n-2^d = V_n-2^-1ϕ_n-2 V_n-2 = ( 
        [ - ϕ_n-1       0;       0   - ϕ_n ])   .

As a second step, we show that all of these eigenvalues are equal. Consider

    B_i ϕ_2 X_23 X_34… X_l,l+1… X_n-3,n-2 A_j   ,    i=0,1   ,   j=n-1,n   ,

and by <ref> and <ref> we can write it in two equivalent ways

    - ϕ_i B_i X_23 X_34… X_l,l+1… X_n-3,n-2 A_j = B_i X_23ϕ_3 X_34… X_l,l+1… X_n-3,n-2 A_j   ,      i=0,1   , 
       j=n-1,n   .

Using recursively <ref> and <ref> we can move the adjoint until the end 

    B_i X_23 X_34… X_l,l+1… X_n-3,n-2ϕ_n-2 A_j   ,    i=0,1   ,   j=n-1,n   ,

where we can use <ref> to write 

    -B_i X_23 X_34… X_l,l+1… X_n-3,n-2 A_j ϕ_j   ,    i=0,1   ,   j=n-1,n   ,

and comparing with <ref> we obtain

    ϕ_i = ϕ_j := u   ,    i=0,1   ,   j=n-1,n   , 
       ϕ_2^d = ϕ_n-2^d = - u  1_2× 2  .


As a third step, we show that all 2-dimensional matrices have the same eigenvalues. From <ref>, consider l=2, diagonalize ϕ_2 and use <ref>

    ϕ_2 X_23 = X_23ϕ_3 = V_2 V_2^-1ϕ_2 V_2 V_2^-1X_23 = V_2  ϕ_2^d   V_2^-1 X_23 = -u V_2 1_2×2 V_2^-1 X_23 = - u X_23  ,

and from second and last step this is now a left eigenvalue equation for ϕ_3, with eigenvalue ϕ associated to X_23. The same reasoning can be repeated for X_32ϕ_2 = ϕ_3 X_32 from <ref>, obtaining that X_23 and X_32 are the eigenvectors of ϕ_3 with eigenvalue -u. We get that ϕ_2^d = ϕ_3^d. We can recursively repeat the argument for all l, obtaining

    ϕ_l = - u  1_2 × 2  ,    l = 2,  …  n-2   .

Note that the same reasoning holds in case of all vanishing masses, so that u is the variable that parametrizes the ℂ factor in the moduli space of the 𝒩=2 theory.

As a fourth step, we construct the X_l,l+1X_l+1,l. Taking the trace of <ref> and <ref>, and using <ref>-<ref> and the fact that ϕ_2 =  ϕ_n-2 = - 2 u, we get

    X_23X_32 = u ( m_0 + m_1 + 2m_2 )   , 
    
        -     X_n-2,n-3X_n-3,n-2 = u ( m_n-1 + m_n + 2m_n-2)   .

Similarly, from <ref> we find that

    X_l,l+1X_l+1,l =   X_l,l-1X_l-1,l - m_l  ϕ_l =   X_l,l-1X_l-1,l + 2 m_l u   ,    l = 3,  …  , n-3   ,

and using it recursively we get that 

    X_n-3,n-2X_n-2,n-3 =   X_23X_32 + u ∑_l=3^n-3 2 m_l   .

By inserting <ref> in <ref> and summing with <ref> we obtain

    u ( m_0 + m_1 + ∑_l=2^n-2 2 m_l + m_n-1 + m_n ) = u m = 0   .

Similarly to what happens in the A_n-1 case, the global mass and the value of the adjoint fields are related: when the global mass is zero, u can be non-zero, while when the global mass m≠ 0, it is forced u=0. 

Finding the form of moduli space for m=0 and u≠0 by solving directly the F-terms is quite involved. As for A_n-1, in <cit.> they deform the quiver gauge theory by FI terms b_i at each node and they carry out this computation exploiting the graphical tool of bug calculus. By comparison of the F-terms in <ref> : <ref> with the graphical representation in <cit.>, we can identify

    b_i ↔ - m_i  ϕ_i  ,   ∀ i   ,

where all FI-terms are subject to the condition 

    b_0 + b_1 + b_n-1 + b_n = 2 ∑_i=2^n-2 b_i   ,

which translates in the trace of the sum of m_i ϕ_i, i.e.  

    ( m_0 + m_1 + ∑_l=2^n-2 2 m_l + m_n-1 + m_n ) u = 0   .





§ ON 𝒩=1 MASS DEFORMATIONS PRESERVING NON-INVERTIBLE SYMMETRIES


We systematically study the solutions to the eigenvalue problem

    𝒟 m⃗ = e^i α m⃗

when the global mass vanishes. We consider the action of the permutation first on the z_i, and then on the masses m_i. Let 𝒵 be the vector space spanned by the z_i's in A_n-1 configurations. 

Note that 𝒟 can be block diagonalized in 𝒵 according to the orbit decomposition discussed in <ref>. Each block is a finite order matrix and hence can be diagonalized. Moreover, the minimal polynomial of each block is x^n-1, where n is the order of the orbit, and it divides the characteristic polynomial of the block, which is of the same order. Therefore, each orbit contributes n eigenvalues, specifically n districts roots of unity. Now, the masses m_i span a codimension one subspace ℳ of 𝒵. The direction orthogonal to ℳ in 𝒵 is generated by the vector of Dynkin labels n⃗ since m⃗·n⃗ = 0, and it is associated with an eigenvector of the permutation matrix with eigenvalue 1. Thus, the defect 𝒟 acting on ℳ retains all the eigenvectors but the one dual to n⃗. 

In conclusion, within the space of mass deformations solving <ref>, those corresponding to the eigenvalue e^i α where (e^i α)^k=1 and e^i α≠ 1, span a subspace of dimension the number of orbits of order a multiple of k. Mass deformations solving <ref> with eigenvalue e^i α=1 rather span a subspace of dimension the total number of orbits minus 1. This reasoning applies both for duality and triality symmetries of A_n-1 quivers. Since every D_n configuration of marked points can be seen as special A_2n configuration, where the tiltings associated to a puncture and its image under R_D satisfy z_i'=-z_i, the logic also applies to D_n quivers provided one restricts to the subspace of 𝒵 (and ℳ) satisfying this additional constraint.

This rationale translates into an efficient method for computing which mass deformations preserve non-invertible duality defects in general cases. While the system of <ref> can in principle always be explicitly solved by brute force, in practice it becomes rapidly cumbersome as the number of punctures grows. However, the underlying orbit structure allows the advertised more efficient calculation.

The main point is to trade the `physical' basis of ℳ for another basis adapted to the orbit decomposition under 𝒟. For example, in A_3 as studied in <ref>, a convenient choice[Since n_1+n_2+n_3+n_4=0, here “basis" is to be understood as “generating set".] is:

    (n_1,n_2,n_3,n_4)=(z_3-z_1,z_4-z_3,z_2-z_4,z_1-z_2) .

It satisfies the appreciable property that

    (n_1',n_2',n_3',n_4')=(n_4,n_1,n_2,n_3) ,

which in turn simplifies the analysis of the condition (n_1',n_2',n_3',n_4') =α(n_1,n_2,n_3,n_4): e^i α must satisfy e^4 i α=1 and e^i α≠ 1 (so that the global mass vanishes). This result is equivalent to the one of <ref>, as

    n_1 = m_1+m_2 ,   n_2 = m_3 ,   n_3=-m_2-m_3 ,   n_4 = -m_1 ,

is invertible. 
    
To analyse a general configuration one needs to group the punctures into orbits under t∘ S. An example is shown in <Ref>, which displays a configuration consisting of two orbits of size four and one orbit of size two under σ∘ t∘ S on E_i.
    

    
The 9-uplet (m_1,…,m_9) is a basis of ℳ and we also show n_1=-(m_3+m_4+m_5) and n_2=-(m_7+m_8+m_9) in <Ref> for symmetry. Under σ∘ t∘ S:

    (m_1',m_2',m_3',m_4',m_5',m_6',m_7',m_8',m_9') = (-m_1,m_2-n_1,n_1,m_3,m_4,n_1+m_6-n_2,n_2,m_7,m_8) .

Imposing m_i'=α m_i ∀ i with α an i-independent phase yields conditions which also split into orbits:
    
    -m_1    =e^i α m_1 ,
    
            (n_1,m_3,m_4,m_5)    =e^i α(m_3,m_4,m_5,n_1) ,
    
            (n_2,m_7,m_8,m_9)    =e^i α(m_7,m_8,m_9,n_2) ,
    
            m_2-n_1    = e^i α m_2 ,
    
            n_1+m_6-n_1    = e^i α m_6.

If m_1≠ 0 then e^i α=-1, and all remaining masses are determined by, say, m_1,m_3 and m_7. If m_1=0 and at least one of n_1,m_3,m_4,m_5,n_2,m_7,m_8 or m_9 is non-zero, then e^4 i α=1 and e^i α≠ 1. As before, all masses can be expressed in terms of, say, m_3 and m_7. Last, if only m_2 and m_6 which connect different orbits are non-zero, then e^i α=1, and m_2,m_6 are free parameters. This generalizes to any configuration of punctures, leading to the count of deformation parameters preserving non-invertible symmetries written in <Ref>.

The same strategy applies to triality defects of order 6 and 3, as well as to duality and triality symmetries of D_n quivers, with the additional constraint evoked above.




§ EXAMPLES WITH MASS DEFORMATIONS




 §.§ Duality defects




  §.§.§ Duality symmetry for D_4 with vanishing global mass


The configuration for D_4 requires 8 marked points to be placed on the torus 𝕋^2, and we can organize them in either two orbits of size four, or one orbit of size four and one orbit of size 2, where the latter requires two points to be placed on top of ℤ_2 fixed points, consequently their ℤ_2 images sits on the same position. Despite two marked points on the same location would lead to inconsistency in the A_n-1 case, in the D_n case a marked point and its orientifold image can indeed sit on the same point. This is consistent with the construction in <ref> as well as with the definition of the τ_i, <ref>, where none of them vanish in the present configuration. Since the first configuration is discussed in <ref>, here we examine the second one. To be precise, we place the points as

    p_1 = 1/2  ,      p_3 = i p_2 + 1   ,      p_4 = i/2  ,

where p_2 is free to be placed with 0 < Re (p_2) < Im (p_2) < 1/2. Starting with this configuration, we can define a non-invertible duality defect as 𝒟 = R_D,3∘σ∘ t ∘ S, with σ = s_3 s_2 s_3 s_1 s_2 s_3 and t = t_1^(i) t_2^(i) t_3^(i). On the masses, we have that

    𝒟 : (m_0,m_1,m_2,m_3,m_4) →  (-m_4, -m_3, -m_2, -m_0, -m_1)   ,

and the solutions of the eigenvalue equation 𝒟m⃗ = e^i αm⃗ are

    ( m_0 ,  …  , m_4 ) = (i, -i, 0, -1, 1 ) m_4   ,     for α = π/2  , 
       ( m_0 ,  …  , m_4 ) = (-i, i, 0, -1, 1 ) m_4   ,     for α = - π/2  , 
       ( m_0 ,  …  , m_4 ) = (-1, -1, 0, 1, 1 ) m_4   ,     for α = 0   .

The deformed theory's moduli space is given by

    x^2 + y^2 w = w ( w^2 - v^4 ),

for both the first and the second solution and with v=m_4 u, while the others lead to

    x^2 + y^2 w = w ( w + v^2 )^2   .






 §.§ Triality defects


We provide some examples with duality symmetries that involve an ST transformation, whose action is discussed around <ref> and that leaves τ = ρ = e^2 π i/3 invariant. The allowed orbits are the following. Orbits of size one are given be the vertices of the fundamental cell. The single orbit of size two consists of points denoted by C_1 and C_2 with coordinates,
    
    C_1 = 1/2 + i √(3)/6 = ρ+2/3  ,      C_2 = i√(3)/3 = 2 ρ + 1/3  ,

The orbit of size three is realized with the three points
    
    q_1 = 1/2  ,      q_2 = ρ/2  ,      q_3 = ρ+1/2  ,

while the orbits of size six are given by the points placed at
    
    p_1 = α  ,    p_2 = ρ^2 α - ρ^2   ,    p_3 = - ρα + ρ  , 
       p_4 = 1 + ρα  ,    p_5 = - ρ^2 α  ,    p_6 = - ρ^2 - α  ,

    with α in the triangle (0,   1,  ρ +1 ). 



 §.§ Mass deformed A_1 quiver theory




  
Vanishing global mass

Consider the theory A_1 at τ = ρ = e^i 2 π/3 with an orbit of size 2, and global mass m = 0. This configuration preserves a defect 𝒟 = s_1 ∘ t_1^(ρ)∘ S T. The unique solution reads

    [             α = - π/2          (m_1, - m_1) xy = w (w - um_1)   , ]

which flows to the conifold.



  
Non-vanishing global mass

In the case of A_1 with global mass m ≠ 0, the masses transform as

    - m_1 - m ρ (τ_1 + 1)     = - ρ m_1   , 
    
         m_1 + m ρτ_1      = - ρ m_2   ,

whose solution is (1/2,1) m_2.



 §.§ A_7 with vanishing global mass


Let us compute the mass deformations of the A_7 quiver gauge theory which preserve the non-invertible triality symmetry of order 6. The eight corresponding punctures at τ= e^2iπ/3 necessarily split into an orbit of size 6 and an orbit of size two as displayed in <ref>.



We apply the strategy outline in <ref> and consider the masses shown in <ref>. Under the action of ST composed with deck transformations and a permutation, the mass deformation preserves the non-invertible triality symmetry if and only if they solve:

    (n,m_1,m_2,m_3,m_4,m_5)    = e^iα(m_1,m_2,m_3,m_4,m_5,n) 
    
        -m_6    = e^iαm_6 
    
        m_1 + m_7 + m_6    = e^iαm_7


We can then read directly that

    
  * If (e^iα)^6=1 and e^iα≠±1, then all masses are determined by m_1,
    
  * If e^iα=-1 then all masses are determined by m_6 and m_1,
    
  * If e^iα=1 then m_7 is the only free parameter, as all other masses have to be set to zero.

This result translates in any other mass basis, for example the physical one.



 §.§ Mass deformed D_4 quiver theory




  
Vanishing global mass

Consider the theory D_4 at τ = ρ = e^i 2 π/3 with an orbit of size 6 and one orbit of size 2. The configuration preserves a defect 𝒟 = R_D,1∘ R_D,4∘ s_2 s_1 ∘ t_1^(ρ)t_2^(ρ)t_4^(ρ)∘ ST, which transforms the masses as

    𝒟 : (m_0, m_1, m_2, m_3, m_4) → (m_2, m_0 + m_1 + m_2, - m_1 - m_2, m_1 + m_2 + m_4, m_1 + m_2 + m_3)   .

The solutions are 

    [                                                       α = π                                             (1,0,-1,0,1)m_0 x^2 + y^2 w = w^3 + 12 t^2 w^2 + 30 t^4w + 6 t^4 y + 28 t^6;                                                       α = π                                             (1,0,-1,1,0)m_0 x^2 + y^2 w = w^3 + 12 t^2 w^2 + 30 t^4w + 6 t^4 y + 28 t^6;                                                     α = π/3                                        (-1,i √(3),ρ,1,1)m_3                          x^2 + y^2 w = w^3 + ws^4 - w^2 s^2;                                                    α = 4π/3                                       (-1,-i √(3),ρ,1,1)m_3              x^2 + y^2 w = w^3 + 4 ws^4 - 4 w^2 s^2 - 3 s^6 ]

where t = u m_0/2 and s=u m_3.



  
Non-vanishing global mass
In the case of D_4 with global mass m ≠ 0, the masses transform as 

    m_2 + m (1 - τ_1)     = - ρ m_0   , 
    
        m_0 + m_1 + m_2 + m (ρ^2 - ρ)( 1 - τ_1)     = - ρ m_1   , 
    
        - m_1 - m_2 - m ρ^2 (1 - τ_1)     = - ρ m_2   , 
    
        m_1 + m_2 + m_4 + m [ - 1/3ρ( ρ - 1 ) + ( ρ + 1 ) ( - ρτ_1 + ρ - 1 ) ]     = - ρ m_3   , 
    
        m_1 + m_2 + m_3 + m [ 2/3( ρ^2 - 1 ) +  τ_1 ]     = - ρ m_4   ,

whose unique solution is

    ( √(3)τ_1 - 1/√(3) (3 - τ_1) + 2 i  ,  3(τ_1 - 1)(i + 2 √(3))/√(3)(τ_1 - 3) - 2i  ,  1/2(τ_1 - 1)(3i + 7 √(3))/√(3)(3 - τ_1) + 2i  ,   1 + i + √(3)/√(3)(τ_1 - 3) - 2i  ,   1 ) m_4   ,

with global mass

    m = 45 i + 21 √(3)/23 √(3) - 3 (2i + 3 √(3) )τ_1 + 36 i  .



JHEP


