Deformations of $W_{A,D,E}$ SCFTs

We discuss aspects of theories with superpotentials given by Arnold's $A,D,E$ singularities, particularly the novelties that arise when the fields are matrices. We focus on 4d ${\cal N}=1$ variants of susy QCD, with $U(N_c)$ or $SU(N_c)$ gauge group, $N_f$ fundamental flavors, and adjoint matter fields $X$ and $Y$ appearing in $W_{A,D,E}(X,Y)$ superpotentials. Many of our considerations also apply in other possible contexts for matrix-variable $W_{A,D,E}$. The 4d $W_{A,D,E}$ SQCD-type theories RG flow to superconformal field theories, and there are proposed duals in the literature for the $W_{A_k}$, $W_{D_k}$, and $W_{E_7}$ cases. As we review, the $W_{D_\text{even}}$ and $W_{E_7}$ duals rely on a conjectural, quantum truncation of the chiral ring. We explore these issues by considering various deformations of the $W_{A,D,E}$ superpotentials, and the resulting RG flows and IR theories. Rather than finding supporting evidence for the quantum truncation and $W_{D_\text{even}}$ and $W_{E_7}$ duals, we note some challenging evidence to the contrary.


Introduction
The simply-laced Lie groups, A k , D k , and E 6 , E 7 , and E 8 ("ADE") relate to, and classify, far-flung things in physical mathematics. The Platonic solids are classified by the discrete subgroups Γ G ⊂ SU (2)-cyclic, dihedral, tetrahedral, octahedral, and icosahedral-which connect to the ADE Lie algebras via the McKay correspondence 1 . Another connection is in Arnold's simple surface singularities, which follow an ADE classification [7]: (1.1) These have resolutions, via lower order deformations, associated with the corresponding ADE Cartan, with the adjacency of the singularities that of the ADE Cartan matrix.
In two dimensions, the ADE groups arise in the classification of minimal models and their partition functions [8]. The 2d N = 2 minimal models with c < 1 are given by Landau-Ginzburg theories with the W G=A,D,E superpotentials (1.1) [9], [10]. The chiral ring of the W G 2d N = 2 SCFT is related to the ADE group's Cartan, with r G = rank(G) chiral primary operators. Deforming the theory by adding these chiral ring elements to · · · · · · D k 1 +k 2 +2 −→ · · · · · · A k 2 D k 1 +2 This gives the vacua associated with 1d representations of the F -terms.
the superpotential, W → W + ∆W , the deformation parameters can be associated with expectation values in the adjoint of G. The deformation leads to multiple vacua, where the ADE group breaks into a subgroup. This breaking pattern is in accord with adjoint Higgsing, preserving the rank r G and corresponding to deleting a node from the extended Dynkin diagram, e.g.
The generic deformation gives G → r G A 1 , giving Tr(−1) F = r G susy vacua. The solitons of the integrable ∆W deformations also exhibit the ADE structure, e.g. [11].
A related connection with ADE groups is via local Calabi-Yau geometries: when the defining hypersurface has a singularity (1.1), there are (collapsed) cycles corresponding to the ADE Dynkin diagram nodes, with intersections given by the group's Cartan matrix. String theory on these backgrounds can yield the corresponding ADE gauge groups in spacetime [12].
In this context, the geometric resolutions of the local singularities corresponding to ∆W deformations lead to adjoint Higgsing of the corresponding group.
1.1. The chiral ring of W A,D,E (X, Y ) for matrix fields X and Y We are interested in an ADE classification that arises in the context of a family of 4d N = 1 SCFTs [13]. Before delving into specifics, we highlight a difference in comparison with (1.1): now X and Y are matrices, with W A k = Tr(X k+1 + Y 2 ), W D k+2 = Tr(X k+1 + XY 2 ), (1. 3) The fact that matrices allow non-zero, nilpotent solutions to the equations of motion, and can have [X, Y ] = 0, makes for important differences-even classically.
Recall that in theories with four supercharges, chiral primary operators have dimension proportional to their U (1) R charge, which is hence additive, and their OPEs yield the chiral ring. In terms of a microscopic, Lagrangian description, the chiral ring consists of gauge-invariant composites formed from the microscopic chiral superfields. Superpotentials lead to chiral ring relations, since ∂ X W ∼ Q 2 ∂ X K is not a primary, and is thus set to zero in the ring; for instance the LG theories (1.1) then have r G=A,D,E elements.
We are here interested in aspects of the chiral rings for theories with matrix X and 3), whose F -term chiral ring relations, ∂ X W = ∂ Y W = 0, are given by Y is massive and could be integrated out, setting Y = 0; we merely included it here to make the ADE cases in (1.3) more uniform. For k = 1 and any N c , X is also massive, and there is a unique supersymmetric vacuum at X = Y = 0. For k > 1 and N c = 1, (1.4) gives isolated vacua at X = 0, and resolving the singularity by lower order ∆W shows that there are Tr(−1) F = r G = k such vacua. For both k > 1 and N c > 1, on the other hand, X k = 0 has a non-compact moduli space of flat direction solutions with nilpotent X; for example, X could contain a block v(σ 1 + iσ 2 ) for arbitrary complex v.
In our context, SU (N c ) or U (N c ) is gauged, and the nilpotent matrix solutions of (1. The "other matter field contributions" are for example the contributions from N f fundamentals and anti-fundamentals Q, Q in variants of SQCD, which we need not consider for the moment; i.e. we consider the theory at Q = Q = 0. For the A k case, (1.5) gives [X, X † ] = 0, implying X and X † can be simultaneously diagonalized; then nilpotent solutions are eliminated, and (1.4) implies that the vacua are all at X = 0.
The D and E cases, with N c > 1, have more matrix-related novelties since generally [X, Y ] = 0. For the D-series, the F -terms in the undeformed case are The 1d representations are the same as in the N c = 1 case, giving r D k+2 = k + 2 chiral ring elements. For matrices X and Y , the chiral-ring relations (1.6) lead to a qualitative difference between k odd and k even. For k odd, (1.6) imply that Y 3 ∼ Y X k ∼ −Y X k = 0, and thus there are 3k independent chiral ring elements formed from X and Y , given by k odd : Θ j = X −1 Y j−1 , = 1, . . . , k; j = 1, 2, 3. (1.7) For k even, Y m≥3 = 0 in the ring, so there are chiral ring elements with allowed values of j that do not truncate, i.e. they do not have a maximum value independent of N c .
Likewise, for W E 6 the chiral ring relations allow for r E 6 = 6 chiral ring elements with 1d representations, {1, X, Y , X 2 , XY , X 2 Y }.
For N c > 1, one can form, for example, Tr(XY ) with arbitrary as independent chiral ring elements, so the ring does not truncate. Similarly, for W E 7 , the chiral ring relations (1.9) lead to r E 7 = 7 chiral ring elements when N c = 1, while for N c > 1 the classical chiral ring is not truncated. For W E 8 , the chiral ring relations lead to r E 8 = 8 chiral ring elements for 1d representations (X −1 Y j−1 for = 1, . . . , 4 and j = 1, 2), but the classical chiral ring does not truncate for matrix representations.

W A,D,E in 4d SQCD with fundamental plus adjoint matter
We consider ADE superpotentials in the context of 4d N = 1 SCFTs, with gauge group SU (N c ) or U (N c ), X and Y adjoint chiral superfields, and N f (anti)fundamental flavors Q (andQ). The possible interacting SCFTs were classified in [13] as along with (1.3). The reappearance of Arnold's ADE classification in this context [13] was unexpected. Some interesting ideas and conjectures for a geometric explanation of the W A,D,E in this context appeared in [14], in connection with matrix models and the construction of [15]. We will not further explore these interesting ideas here.
The IR phase of the theory depends on N f and N c . It is convenient to consider these theories in the Veneziano limit of large N c and N f , with the ratio held fixed; the IR phase then only depends on x. The O theory is (or is not) asymptotically free for x > 1 (or for x ≤ 1), and RG flows to an interacting (or free electric) theory.
Larger x values means that the theory is more asymptotically free, and hence the original "electric" description is more strongly coupled in the IR. The asymptotically free theories are expected 2 to be in the interacting SCFT conformal phase for all N f < 2N c (i.e. x > 1 2 ) for the A cases, and for all N f < N c (i.e. x > 1) for the O, D and E cases. For the W A,D,E theories (1.3), on the other hand, there are more possible IR phases.
In the W A 1 case, the adjoints are massive and can be integrated out. The resulting IR theory is SQCD, which has the duality [17], with "magnetic" gauge group SU (N f −N c ). The dual reveals the bottom of the conformal window, and the existence of the IR-free magnetic phase for 2 3 ≤ x ≤ 1; for x > 1, the theory generates a dynamical superpotential [18]. The W A k>1 theories were considered in [1,2], where a duality was proposed and checked.
Following [6,19] we write the W A k duality in a way that will generalize to some cases: (1.13) Superpotential deformations of W A k were considered in [3], where the fact that α A k = k was shown to tie in with the fact that upon a generic ∆W deformation, Arnold's A k singularity is resolved as since the low-energy theory in each of the k vacua has the SU (n i ) ↔ SU (N f − n i ) duality of [17]. The IR phases and relevance of the W A k theories were clarified in [4] using a-maximization [20], including accounting for accidental symmetries.
A duality of the form (1.13) for the case of two adjoint chiral superfields X and Y , with W D k+2 as in (1.3), was proposed in [5], with The IR phases and relevance of the superpotential terms were clarified in [13], where it was also noted how the α D k+2 value (1.15) can be understood / derived from ∆W deformations; this will be discussed much further, and clarified, in the present paper.
More recently, a duality for the case of W E 7 was proposed in [6], with The value (1.16) was moreover shown in [19] to be compatible with the superconformal index in the Veneziano limit 3 , and it was argued [6,19]

W A,D,E + ∆W RG flows
Possible flows between these fixed points are illustrated in Figure 2, taken from [13].
We here emphasize that this figure is somewhat incomplete: the ∆W superpotential deformations give additional vacua, with a richer IR structure than is indicated in the figure. Indeed, even the 1d (N c = 1) representations of the chiral ring of the deformed The exact matching of the electric and magnetic indices beyond this limit requires mathematical identities which have only been demonstrated explicitly for the W A1 SQCD duality case [21]; the needed identities are conjectural for the A k>1 , D k+2 , and E 7 dualities. 4 We use the standard notation for the floor and ceiling functions, x and x , respectively. So, for k odd, The 1d and 2d refers to the dimension of the representation of the (deformed) chiral ring.
The higher-dimensional representations of the chiral ring are the new elements of the matrix-variable superpotentials. The gauge group is then broken as [13] For k odd, the low-energy theory is SQCD for each factor, with N f flavors for the U (n i ) groups and 2N f flavors for the U (n 2d j ) groups, and then the duality of [17] in each factor fits with the value (1.15) [13].
We will discuss even vs odd D k+2 and the duality of [5] in much more detail in what follows. We will also report on our attempt to understand the duality [19]-and the value (1.16)-by considering various ∆W deformations, similar to (1.17) and (1.18).

W A,D,E flat direction flows
The W A,D,E theories can also be deformed by moving away from the origin, on the moduli space of supersymmetric vacua. There are fundamental matter flat directions associated with expectation values for the Q and Q matter fields (e.g. Q N f = Q N f = 0), and adjoint flat directions associated with expectation values X and/or Y , as well as mixed directions where both fundamentals and adjoints receive expectation values. We will here primarily focus on the purely adjoint flat directions.
For X and Y adjoints of SU (N c ) gauge group, there are certain flat directions which exist for special values of N c which do not exist for the U (N c ) case. For example, for W A k there are flat directions when N c = kn for integer n; along such flat directions, (1.19) where in the low-energy theory each U (n) factor is a decoupled copy of SQCD with N f flavors. As we will review in Section 3.4, this gives another check of α A k = k in the duality (1.13). We will discuss similar checks of α D k+2 = 3k, for the case of k odd. As we will emphasize, the D even case is quite different from D odd ; similar series of flat directions for D even and E 7 have a more subtle story.
For the cases where the classical chiral ring does not truncate-namely, W D k+2 for k even and W E 7 -we show that there are classically unlifted flat directions given by matrix solutions to the F -and D-terms of the undeformed theories. We argue that these flat directions are not lifted or removed by any dynamics, and they thus present a possible challenge for the proposed duals for these theories.

Outline
The outline of the rest of this paper is as follows. In Section 2 we review some technical details, including a review of the known and conjectured dualities for the 4d W A,D,E SCFTs, and a discussion of their moduli spaces of vacua-especially with respect to higherdimensional vacua. In Section 3 we review some aspects of the WÂ and W A k theories to set the stage for subsequent analysis.
In Section 4, we consider the W D k+2 theories: First, we study a matrix-related classical moduli space of supersymmetric vacua present for the D even theory, which poses a puzzle for duality for D even , and argue that these flat directions are not lifted by quantum effects.
We demonstrate that these flat directions seem to violate the a-theorem, and discuss possible resolutions to this puzzle. We then study SU (N c )-specific flat directions of the D k+2 theories, reviewing that such flat directions provide a nontrivial check of the D odd duality, and then showing that they lead to puzzles for the conjectured D even duality. Next, we study RG flows from the W D k+2 SCFTs via relevant superpotential deformations, again finding nontrivial checks of duality for D odd , and more hurdles for D even . We conclude Section 4 with comments on hints as to how these puzzles might be resolved.
In Section 5, we similarly analyze the W E 7 SCFT. We study matrix-related flat directions and SU (N c )-specific flat directions of the E 7 theory, which turn out to be analogous to the puzzling D even flat directions. We then study some ∆W RG flows from the W E 7 SCFT, noting some features in the resulting higher-dimensional vacuum structure that are new to the E-series. Finally, we conclude in Section 6 with comments on future directions, and some discussion of how the present work might be applied to the W E 6 and W E 8 SCFTs. In an appendix, we explore additional E-series RG flows.

Technical review
2.1. The a-theorem, and a-maximization The 4d a-theorem [23] implies that the endpoints of all RG flows must satisfy In superconformal theories, a is related to the 't Hooft anomalies for the superconformal U (1) R symmetry as [24] (we rescale to a convenient normalization): In cases where U (1) R can mix with U (1) F global flavor symmetries, the exact superconformal R-symmetry is determined by a-maximization [20], by locally maximizing (2.2) over all possible U (1) R symmetries. Cases with accidental symmetries or irrelevant interactions require special care: one then maximizes (2.2) over R-symmetries that are not obvious from the original description. One situation where such enhanced symmetries are evident is when a gauge invariant operator saturates, or seemingly violates, an SCFT unitarity bound, e.g.
for scalar chiral primary operators O: The inequality is saturated for free chiral superfields, and apparent violations instead actually saturate the inequality, with an accidental symmetry U (1) O which only acts on the IR-free-field composite operator. See [4] for how a-maximization is modified in such cases, and its application to the A SCFTs. See [13] for additional applications to the other theories in Fig. 2, and additional discussion.
The a-theorem (2.1) requires, for example, that a decreases when a fundamental flavor is given a mass and integrated out, where SCFT refers to any of the SCFTs in Fig. 2. In the Veneziano limit, (2.4) for this Upon computing a(x) for the SCFTs in Fig 2, it is verified that x −2 a(x)/N 2 f is indeed monotonically decreasing for small x, but then flattens out when x is sufficiently large, e.g. at x ≈ 13.8 for the W E 6 SCFTs [13]. The a-theorem implies that some new dynamical effect must kick in for x at or before the problematic range where (2.5) is violated.
One such effect, for sufficiently large x, is that a dynamical superpotential could be generated, and the theory is no longer conformal; this is referred to as the stability bound.
For W A k theories, the stability bound is x < x stability = k [1,3,18]. Another effect, which can occur for x < x stability , is that the theory could develop non-obvious accidental symmetries.
In cases with known duals, such accidental symmetries can be evident in the dual description, where it is seen that some superpotential terms-or the dual gauge interaction-become irrelevant when x elec is too large (x mag is too small). It is satisfying that the a-theorem condition (2.5) is indeed satisfied in the W A k theories [4] and the W D k+2 theories [13] upon taking such accidental symmetries into account.

Duality for the 4d SCFTs
Recall that the chiral ring consists of gauge-invariant composites, e.g. meson, baryon, and glueball operators, formed from the microscopic chiral superfields: here X and Y , the fundamentals and anti-fundamentals Q, Q, and the gauge field strength fermionic chiral superfields W α , subject to classical and quantum relations. Such theories, with adjoint(s) X (or X and Y , or similarly, other two-index representations, e.g. in the examples in [25]) only have a known dual if the chiral ring of products of the adjoint(s) truncates. Here, truncate means that the number of independent elements in the ring is independent 5 of N c .
An example of an untruncated case is the A theory, for which a basis of adjoint-valued products is given by Θ j (X) = X j−1 , for j = 1, . . . , N c ; such theories do not have a known 5 There is a classical chiral ring relation that the adjoint-valued operator X Nc can be expressed in terms of products of lower powers X <Nc and the u j ≡ TrX j . To see this, write the characteristic polynomial . . , and note that P (x, u j )| x=X = 0. Thus one can write any gauge invariant TrX = P (u 1 , . . . , u Nc ) for some polynomial P . As shown in [26], such relations can be modified by instantons for sufficiently large . See e.g. [26,27], and references therein, for examples of chiral ring relations involving the adjoint-valued gaugino and gauge field chiral superfield W α , including the glueball operator S ∼ TrW α W α and generalizations. Relations involving W α and S will not be discussed in this current work. dual. A truncated case is W A k , where Θ j (X) = X j−1 , for j = 1, . . . , k.
More generally, suppose that a truncated case has a basis of elements Θ j (X, Y ), with j = 1, . . . , α; these are holomorphic products without traces, so gauge-invariant chiral ring elements are formed by taking traces or contracting with Q andQ. One can form dressed quarks Q (j) ≡ Θ j (X, Y )Q, which can then be used to construct gauge-invariant operators, such as the αN 2 f mesonic operators For SU (N c ) there are also baryonic operators, built out of the dressed quarks: As shown in [6], the many constraints on any possible dual-including matching of the The truncation of the ring to α generators is a necessary ingredient for these classes of conjectured dualities. The chiral ring of the electric theory truncates classically in the A k and D odd cases, and has been conjectured to truncate quantum mechanically in the D even [5] and E 7 [6] cases.
The . Generally, as long as the gauge coupling is asymptotically free, its negative contribution to anomalous dimensions drives the cubic superpotential terms to be relevant. The A → A k , and D → D k+2 , and E → E r flows with non-cubic terms in W (X, Y ) only occur if x > x min , such that the negative anomalous dimension from the gauge interactions is large enough to drive the W (X, Y ) terms relevant; the values of x min were obtained using a-maximization for W A k in [4] and in [13] for the other W G=A,D,E theories. Duality, if it is known and applicable, clarifies the IR phase structure of the theories for x > x min , where the magnetic dual becomes more weakly coupled. The fixed point theories whose duals are known or conjectured all have a similar phase structure [4, 6, 13]: x ≤ 1 free electric (see for instance [28]). When W = 0, one also imposes the F -term chiral ring relations.
The quantum moduli space M qu can be (fully or partially) lifted if W dyn is generated, or deformed for a specific N f as in [29] or variants 6 ; the constraints of symmetries and holomorphy often exactly determine the form of such effects, and with sufficient matter (e.g. sufficiently small x) this implies that W dyn = 0 and M cl ∼ = M qu .
We will here focus on vacua with Q =Q = 0, with non-zero expectation values for the adjoints, X and Y ; such vacua preserve the SU (N f ) L × SU (N f ) R global flavor symmetry.
The N c × N c matrices X and Y are decomposed into multiple copies of a set of basic, irreducible solutions of the D-and F -flatness conditions. We refer to such a basic vacuum solution representation as being d-dimensional if X and Y are represented as d × d matrices, which cannot be decomposed into smaller matrices. 6 There are exotic examples of classical flat directions that are lifted by, for example, confinement (see e.g. [30]); this can only occur if a gauge group remains unbroken and strong there. It turns out that, if there are n copies of a d-dimensional vacuum, there will be an unbroken U (n) D ⊂ U (N c ), where U (n) D can be regarded as coming from breaking a so the diagonally embedded U (n) D has dN f flavors. If both adjoints receive a mass from the superpotential F -terms, the low-energy U (n) D will then be SQCD with dN f flavors.
This factor then has a dual gauge group U (dN f − n) D , with dN f flavors (with SU (N f ) L,R enhanced to SU (dN f ) L,R as an accidental symmetry in the IR limit). By the dual analog of the electric Higgsing, this low-energy U (dN f − n) D can be embedded in a U (d 2 N f − dn) with N f flavors. For example, consider the case of n copies of a 2d vacuum, with X breaking U (2n) → U (n) × U (n), and then Y in the bifundamental breaking to U (n) D .
Duality maps this process as follows: The low-energy theory for such a vacuum is denoted as A 2d 1 if all the adjoints are massive, where the 2d superscript indicates that it comes from a 2d representation, and thus has 2N f (or more generally, dN f ) flavors. Applying such considerations for all d i vacua in (2.10) suggests that the dual theory has α given by This relation indeed works for the A k and the D odd theories, but not for D even or E 6,7,8 .
For W D k+2 , the generic deformation has k + 2 1d vacuum solutions, and 1 2 (k − 1) 2d representations. If there are n i copies of the i'th 1d solution, and n 2d j copies of the j'th 2d solution, then U (N c ) is broken as in (1.18). For odd k, (2.12) indeed gives α = 3k.

A → A k flow and A k duality
Consider SU (N c ) SQCD with N f chiral superfields Q(Q) in the (anti)fundamental of the gauge group, and adjoint chiral superfields X and Y with superpotential The Y field is massive and can be integrated out; this is the O → A RG flow in Fig. 2. The t k coupling, if relevant, drives the A → A k RG flow in Fig. 2; if irrelevant, the IR theory is instead an A SCFT. For k = 1, t k = m X is an X mass term and is always relevant; then both X and Y can be integrated out and the IR A 1 theory is ordinary SQCD. For k = 2, t k is marginally relevant as long as the matter content is within the asymptotically free range, thanks to the gauge coupling. For k > 2, the t k coupling is relevant only if x > x min k [4].
The chiral ring of the A k theory truncates classically, and we may write the k generators There are then kN 2 f meson operators (2.6), with α A k = k, and baryonic operators (2.7). The A theory (t k = 0) does not have a known dual description. The magnetic description The dual has N f (anti)fundamentals q(q), adjointsX,Ŷ , and k gauge singlets M j We can rescale X andX to set t k =t k = 1, and µ is a scale that appears in the scale matching of the electric and magnetic theories. The and the 't Hooft anomalies properly match [1][2][3].
The A theories have a quantum moduli space of vacua, W dyn = 0, for all N f and N c . [18], giving a QQ → ∞ runaway instability for massless flavors or Tr(−1) F = N c gapped susy vacua for massive flavors. We are here interested in cases with massless flavors and W dyn (M j ) = 0, so we restrict to kN f > N c , i.e. x < x stability = k; this is the vacuum stability bound [1,2]. For kN f < N c , the quantum theory A k has a moduli space of vacua, where the M j mesons have expectation values. The classical constraints on this moduli space, e.g. rank(M k ) ≤ N c , are recovered in the magnetic dual description from its stability bound,x < k, since the M k expectation value gives masses via (3.3) to the dual quarks q, q.

W A k + ∆W deformations and A k → A k <k RG flows
The A k theories of different k are connected by RG flows upon resolving the A k singularity (3.1) by lower order ∆W deformations. The generic deformation, for instance by a mass term ∆W = 1 2 m X TrX 2 , leads to an RG flow with k vacuum solutions for X , with X massive in each, hence k copies of SQCD in the IR-i.e. A k → kA 1 .
We now consider a partial resolution, by tuning the superpotential couplings such that some of the eigenvalues coincide. We first consider the U (N c ) case, which is simpler because we don't have to worry about imposing the tracelessness of X. Consider the deformation in such a vacuum, the electric gauge group is broken as The subscripts denote the low-energy theory, obtained by expanding (3.4) around the corresponding vacuum, X = X + δX. The vacua at X = 0 have the most relevant term in (3.4) given by W low ∼ Tr(δX) k +1 = W A k . The vacua at X = 0 have a mass term for the low-energy adjoint, W low ∼ Tr(δX) 2 = W A 1 . We write this breaking pattern as By further tuning the t i parameters in the deformation (3.4), we could cause some or all of the (k − k ) SQCD vacua to coincide, e.g leading to Quantum mechanically, the vacuum stability condition-needed to have W dyn = 0-requires In the magnetic dual, we deform by the dual analog of the perturbations in and the stability bound in the electric theory ensures that kN f − n > 0. The theories on the UV and IR sides of (3.5) thus map in the dual as For the case in (3.7) the map is The two sides of the RG flow arrow in (3.9) properly fit together as a dual description of the flow associated with the ∆W deformation, This demonstrates that the value α A k = k (see Section 2.2) ties in with the fact that the A k deformation breaking patterns (e.g. as in (3.6)) have matching sum on the two sides. This matching gives a check on the duality [3] -a perspective which we utilize throughout the present work.
As an aside, we note that the a-theorem (2.1) applies for any choice of the IR vacuum; i.e. for any fixed choice of how to distribute the N c eigenvalues of X among solutions to W (X) = 0 (subject to the stability bounds). Regarding a as counting a suitably defined "number of degrees of freedom" of the QFT, one might wonder if a hypothetical stronger statement holds: if a U V is also larger than the sum i a IR,i over all IR vacua. These examples show that the hypothetical stronger statement is false. There are so many vacua from the many partitions of N c that it is straightforward to explicitly verify that i a IR,i can be larger than a U V .

Comments on
It is standard that the local 8 dynamics of 4d U (N c ) and SU (N c ) are the same: the overall U (1) factor in U (N c ) is IR-free anyway in 4d (although that is not the case in 3d and lower). The original dualities of [1-3, 5, 17] etc. were written in terms of SU (N c ), with U (1) B as a global symmetry. Since U (1) B is anomaly free, one can gauge it on both sides of the duality, leading to U (N c ) → U (αN f − N c ) dualities. For the theories with adjoint matter, the U (N c ) version of the theories are simpler, in that we do not need to impose the tracelessness of the adjoints. The adjoints X of the SU (N c ) vs U (N c ) theories are related by X U (Nc) = X SU (Nc) + X 0 1 Nc , where TrX SU (Nc) = 0 and X 0 is an SU (N c ) singlet. In the purely SU (N c ) theory, it is standard to eliminate X 0 by including a Lagrange multiplier Then λ x pairs up with X 0 , giving it a mass, and the vacua have X 0 = 0. The W X = 0 chiral ring relation here gives X k = λ x 1 Nc .
Upon deforming W A,D,E → W A,D,E + ∆W , the TrX SU (Nc) = TrY SU (Nc) = 0 constraints complicate the SU (N c ) theories compared with U (N c ). This is particularly the case if we are interested in ∆W flows as in Fig. 2 which have some X and Y dynamics remaining in the IR, rather than flowing all the way down to just decoupled copies of SQCD. We can enforce TrX SU (Nc) = TrY SU (Nc) = 0 via Lagrange multipliers, which shifts the eigenvalues of X and Y along the flow away from the preferred U (N c ) origin at X = Y = 0. Such a shift will induce the more general, relevant ∆W deformations which were tuned to zero for the U (N c ) case, unless the reintroduced ∆W terms are subtracted off by a tuned choice of coefficients in the initial ∆W . We will see that there are subtleties-especially for the D and E cases-from the d > 1 dimensional vacuum representations.

Consider for example the flow
For the SU (N c ) version of this flow, we add the Lagrange multiplier λ x to eliminate X 0 , shifting the X eigenvalues. But simply doing this shift in (3.10) would induce the TrX 2 term, giving instead A 3 → 3A 1 . To get A 3 → A 2 + A 1 , we must add to (3.10) the remaining t m<2 terms in (3.4), with t 1 tuned in terms of the multiplicities n 0 , n 1 of eigenvalues in the A 2 and A 1 solutions.
For fixed t 1 , vacua with other partitions N c = n 0 + n 1 will instead have 3A 1 in the IR.
It is not immediately apparent if this procedure works in the D and E cases to shift higher-dimensional representations in just the right way to be able to map any U (N c ) deformation into a corresponding SU (N c ) one. The chiral ring algebra that determines how one labels the higher-dimensional vacua is sensitive to additional deformation terms

SU (N c ) flat direction deformations
The ADE SCFTs, for SU (N c ) gauge group and special values of N c , have flat directions that are not present for U (N c ). These are discussed for the A k case in [3]. Adding a Lagrange multiplier term λ x TrX to (3.1), there is a flat direction of supersymmetric vacua when N c = km for integer m, labeled by arbitrary complex λ x : where ω = e 2πi/k is a k'th root of unity and the off-diagonals are zero. This flat direction In each vacuum the adjoints are massive, so in the IR we end up with k copies of SQCD. The magnetic A k theory has an analogous flat direction, along which the low-energy theory matches to that of the k copies of SQCD via Seiberg duality: This gives yet another check that the A k duality hasÑ c = αN f − N c , with α = k.

The W D k+2 fixed points and flows
The W D k+2 SCFTs are the IR endpoints of the RG flow from theD SCFT, and correspond to the superpotential (with Y normalized to set the coefficient of the first term to 1) Such theories were first studied in [5]. The TrXY 2 term in (4.1) is always relevant and drives the RG flow O → D, while the second term in (4.1) gives the D → D k+2 RG flow. x min D k+2 was determined via a-maximization in [13], For relevant t k , we can normalize X to set t k = 1 at the IR D k+2 SCFT. For x < x min D k+2 , t k → 0 in the IR and the theory stays at the D SCFT. We will here assume that x > x min D k+2 . The F-terms of the undeformed D k+2 superpotential (4.1) are given by For k odd, it follows from (4.3) (as explained after (1.6)) that the chiral ring classically truncates to the 3k generators (1.7). As in the A k case, there is a stability bound: we must require x < x stability in order to avoid W dyn , which would lead to a runaway potential for the generalized mesons. For x < x stability , there is instead a moduli space of supersymmetric vacua with W dyn = 0. As we will review (at least for odd k) x stability = 3k, which is related to the fact that the chiral ring has 3k elements.

Proposed dualities for W D k+2 [5]
A dual description of the D k+2 theories was proposed in [5], and many of the usual, non-trivial checks were verified-for instance matching of the global symmetries, 't Hooft anomaly matching, and mapping of the chiral ring operators. As reviewed in Section 2.2, the conjectured duals have gauge group SU (α D k+2 N f − N c ) with α D k+2 = 3k (1.15), and matter content consisting of N f (anti)fundamentals q(q), adjointsX,Ŷ , and 3k gauge singlet mesons M jl which map to the composite meson operators of the electric theory as The dual theory has superpotential A detailed calculation, via a-maximization, is needed to determine thex min (2.8) values for the various non-cubic terms in (4.5) to be relevant rather than irrelevant [13].
The above dual, with α D k+2 = 3k mesonic operators (4.4), requires the chiral ring truncation (1.7), which is only evident from the classical F -terms for k odd. It was conjectured in [5] that quantum effects make the even k theories similar to odd k, with a quantum truncation of the chiral ring, in order for the duality to hold for both even and odd k. It is as-yet unknown if and how such a quantum truncation occurs for the even k case, and thus the status of the duality remains uncertain for even k. The fact that e.g.
the 't Hooft anomaly matching checks work irrespective of whether k is even or odd can be viewed as evidence that the duality also applies for D even , or perhaps just a coincidence following merely from the fact that these checks are meaningful for odd k.
In addition to the usual checks of duality, the proposed chiral ring truncation and duality for D even were used in [5] to predict a duality for an SU (N c ) × SU (N c ) quiver gauge theory with (anti)fundamentals and an adjoint for each node, and (anti)bifundamentals between.
This latter duality was later re-derived, and confirmed, by considering deformations of the more solid, odd k D k+2 theories [31]. But it was also noted in [31] that the D k+2 duality implies some other dualities that are clearly only applicable for k odd, with fields appearing in the superpotentials with powers like X (k+1)/2 . The fractional power for k-even suggests an incomplete description, which is missing some additional degrees of freedom. The status of the D even duality thus remained (and it still remains) inconclusive.
A powerful, more recent check of dualities is to verify that the superconformal indices of the electric and magnetic theories match; see e.g. [21]. In [19], the superconformal indices for the electric and magnetic dual D k+2 theories are verified to indeed match in the Veneziano limit for both even and odd k. The matching beyond the Veneziano limit provides a physical basis for a conjectural mathematical identity. It was moreover noted in [19] that the conjectural quantum truncation of the k even chiral ring should be verifiable via the the index, by expanding it to the appropriate order in the fugacities and checking if the contributions from operators that are eliminated by the quantum constraints are indeed cancelled by those of other operators. It was noted, however, that this check is complicated by the fact that there are many possible contributing operators, so it was not yet completed.
One of the original arguments for the D even quantum truncation is based on the fact that one can RG flow from D odd → D even via appropriate ∆W deformation, e.g. D k+2 → D k+1 + A 1 . Another, similar argument [13] uses the connection between the stability bound and the chiral ring truncation. The duality suggests that the original electric theory has an instability, e.g. via W dyn = 0 leading to a runaway vacuum instability, when 3kN f − N c < 0, i.e. for x > 3k, and we expect RG flows to reduce the stability bound in the IR. Flowing, for instance, from D k+2 → D k+1 + A 1 for k odd, the UV D k+2 theory has a truncated chiral ring and stability bound, which suggests that the IR (even) D k+1 theory should also have a stability bound, and hence chiral ring truncation. We will analyze such RG flows in detail here, and show that there are subtleties.
In summary, the evidence that the duality holds for D odd is compelling, while the evidence for D even is mixed, with aspects that are not understood. Our analysis here fails to find evidence for the quantum truncation of the chiral ring for D even , and instead points out additional hurdles for the conjectured duality.
3) would then imply that X is also a Casimir, so there can not be a non-trivial d > 1 dimensional representation.
For D even , on the other hand, the F -terms, D-terms, and Casimir conditions are solved by the 2-dimensional solutions This gives a moduli space of supersymmetric vacua, passing through the origin. Modding out by gauge transformations, which take x → −x and y → −y, the moduli space can be labeled by x 2 and y 2 satisfying (4.6), which allows for an additional Z k/2 phase for x 2 . Since Consider the Higgsing in stages: first, X breaks U (2n) → U (n) × U (n), and then Y breaks U (n) × U (n) → U (n) D , the diagonally embedded subgroup (for simplicity, we write the gauge groups as U (m), and corresponding expressions apply if we work in terms of SU (m) groups). This breaking pattern leaves five uneaten U (n) D adjoints from X and Y , four of which get a mass from the W D k+2 superpotential (4.1). The low-energy U (n) D along this moduli space has a massless adjoint matter field and W low = 0; i.e. it is a U (n) D A theory. Giving general expectation values to the adjoint matter field of the low-energy A theory corresponds to unequal expectation values of the x i in the n copies of the 2d vacuum We have not found a mechanism for this classical moduli space to be lifted by a dynamical superpotential or removed by quantum effects. The low-energy U ( N c /2 ) A theory with 2N f flavors clearly has W dyn = 0, and unmodified quantum moduli space.
The original theory can have additional effects e.g. from instantons in the broken part of the group (see [32] for discussion and examples), from the last step of the breaking (4.8). Indeed, for x above the stability bound, there can be a W dyn which leads to runaway expectation values for the mesonic operators.
But holomorphy, the U (1) R symmetry, and the condition that W dyn must lead to a potential that, by asymptotic freedom, goes to zero far from the origin of the moduli space, precludes any W dyn that only lifts the 2d flat directions (4.6) without generating a runaway W dyn for the mesonic operators. As usual, the low-energy theory along the flat direction is less asymptotically free than the theory at the origin, and the theory is more weakly coupled for vacua farther from the origin on the moduli space. The original D even theory at the origin is asymptotically free for N f < N c , while the low-energy U ( N c /2 ) A theory far along the flat direction is IR-free if N f > (N c /2), i.e. if x < 2. In that case, the IR spectrum consists of the IR-free U ( N c /2 ) A gauge fields and matter.
We now consider if this D even flat direction is compatible with the conjectural, dual As in the electric theory, we do not yet see a mechanism for quantum effects to modify the classical dimensions of these moduli spaces. Note that the low- which is non-overlapping with the range x < 2 where the corresponding electric theory is IR-free; this at least avoids an immediate, sharp contradiction with the duality, since two theories cannot have a different IR-free spectrum in the same region of the moduli space. As a concrete example, consider the case k = 2, i.e. W D 4 , and take N c even. The electric W D 4 superpotential (4.1) is relevant as long as the gauge group is asymptotically free, for x > 1. The stability bound suggested by the conjectural The supersymmetric flat direction discussed in the previous subsection has another puzzle, independent of the conjectured duality: it leads to naive violations of the a-theorem (2.1) for sufficiently large x. The exact a SCF T is evaluated by using the relation (2.2) between a and the 't Hooft anomalies for the superconformal U (1) R symmetry, along with a-maximization (when needed) and accounting for all accidental symmetries. The values of a SCF T for the W D k+2 theories were analyzed in [13], following the W A k analysis in [4] with regard to the crucial role of including the effect of accidental symmetries in a-maximization. One type of accidental symmetry, when gauge invariant chiral operators hit the unitarity bound and decouple, is readily apparent in the electric theory. Dualities reveal other types of accidental symmetries, e.g. those where the analog of the t k coupling in (4.1) for the magnetic dual is irrelevant, or where the magnetic gauge coupling is irrelevant (the free-magnetic phase); such accidental symmetries are-as far as we know-unseen without knowing the dual.
We consider ∆a for the RG flow associated with the flat direction in (4.8). We compute a U V (x) corresponding to the D k+2 theory with gauge group SU (N c ) and N f flavors as in [13], and a IR (x) corresponding to an A theory with gauge group SU (N c /2) and 2N f flavors as in [4], including as there the effects of all mesons hitting the unitarity bound and becoming IR-free. We plot the results for the cases k = 2 and k = 4, working in the Veneziano limit of large N c and N f , with x fixed. (U (N c ) vs SU (N c ) is a subleading difference in this limit.) plotted for x in the respective conformal windows. The W D 4 theory is IR-free for x ≤ 1. The W D 6 theory requires x > 3.14 for the TrX 5 term in W D 6 to be relevant, while the corresponding term in the Brodie dual is relevant if x < 8.93. The A theory is asymptotically free in both plotted domains.
As we can see in Figures 3a and 3b, both the k = 2 and the k = 4 flat direction RG flows seem to violate the a-theorem for sufficiently large x. For the W D 4 case, the conformal window where both the electric and magnetic theories are asymptotically free is 1 < x < 5, and the cubic t k term in (4.1), or its magnetic analog, is relevant in this entire x range.
As seen in Fig. 3a, the a-theorem is seemingly violated for x ≥ 4, within the conformal window. For W D 6 , the situation is plotted in Fig. 3b: the flat direction seemingly violates the a-theorem for x ≥ 8.31. This is within the expected W D 6 conformal window (i.e. below 3k −x min ≈ 8.93 beyond which duality suggests that the theory is instead in the D mag phase, and below x = 11, where duality suggests the IR-free magnetic phase).
Of course, we do not believe that there will be violations of the a-theorem, so the puzzle of these apparent violations must somehow be resolved. We also note that the apparent violations first occur for x still below the values where mesons involving Y 3 would first hit the unitarity bound (this occurs first at x = 5 for k = 2, and at x = 9.33 for k = 4). Thus, the calculation of a U V is not affected by the issue of whether or not such mesons should be included-we've removed them in the plots above, which would be correct if Brodie duality is correct for D even and the quantum truncation indeed occurs.
We see two possible resolutions to the puzzle of the apparent a-theorem violations. 1) These classical flat directions are somehow lifted by quantum effects, in a way that we do not yet understand. 2) Some additional degrees of freedom make the calculation of a wrong, e.g. giving a larger value for a U V for the W Deven theory. We do not yet know the resolution.
Option 1) could also resolve the conflict with Brodie-duality, discussed in the previous subsection. As we discussed there, asymptotic freedom, along with holomorphy and the R-symmetry, suggests that W exact = 0, but perhaps another mechanism could remove the flat directions-at least for x large enough to be in the problematic range. The existence of the classical flat direction fits with the classically untruncated chiral ring, and it sharpens the issue of if, and how, the chiral ring for the D even theory is quantumly truncated.

Additional evidence that the W Deven → A flat directions aren't lifted
We here present additional arguments against any quantum barrier to the W Deven → W A flat directions. The idea is to explore more of the full moduli space of supersymmetric vacua, going along Q-flat directions, until the low-energy theory is IR-free.
Consider an even D k+2 theory at the origin, with N f < N c such that the theory is asymptotically free. Going along a Q-flat direction by giving a vev to a flavor, Q f = (v, 0, ..., 0) = Q f , gives a low-energy theory that is less asymptotically free. The gauge group is Higgsed SU (N c ) → SU (N c −1), under which the adjoints decompose X →X +F x +F x +s x forX an adjoint and s x a singlet (and likewise for Y ). Then, the number of light flavors in the low-energy theory is N f − 1 + 2 = N f + 1, where the −1 is for the eaten flavor and the +2 is from additional light flavors, F x,y . Expanding the superpotential under this decomposition gives, for instance for W D 4 , an IR superpotential of the form (4.9) Along the above flat direction, the 1-loop beta function coefficient changes by , as usual, the low-energy theory is less asymptotically free.
We iterate this procedure, giving expectation values to n flavors of Q andQ, and thus reducing N c → N c − n, with N f → N f + n and b 1 → b 1 − 2n, until the low-energy theory is no longer asymptotically free, i.e. n > (N c − N f )/2. Then X decomposes as By taking m sufficiently small and n sufficiently large, the low-energy SU (m) D W A theory will have a 1-loop beta function of non-asymptotically free sign, so the theory will be IR-free and thus weakly coupled. Because every interaction is IR-free in this region of the moduli space, quantum effects from the intermediate or low-energy theory cannot lift or remove the W Deven → W A flat direction. As remarked earlier, any possible effects from the Higgsed, original gauge theory at the origin (e.g. instantons in the broken part of the group) must moreover slope to zero for vacua farther from the origin on the classical moduli space (4.6).
In sum, as illustrated in Fig. 4 the full, quantum theory. As discussed in the previous subsection, there would then have to be some missing contribution to a for the D even theory to avoid the apparent a-theorem violation along this moduli space for sufficiently large x.

SU (N c )-specific (as opposed to U (N c )) flat directions
For SU (N c ), one includes Lagrange multipliers λ x , λ y to impose TrX = TrY = 0: For D odd , and N c = 2m + kn for m, n integers, there is a flat direction labeled by λ x [5] where ω = e 2πi/k . The gauge group is Higgsed as SU (2m + kn) → SU (m) 2 × SU (n) k × U (1) k+1 . The SU (n) k theories are, in the IR, k decoupled copies of SQCD, each with N f flavors. The low-energy SU (m) 2 sector includes SQCD, with N f massless flavors, along with bifundamentals F andF coming from the adjoint X of the original theory at the origin, with a low-energy superpotential W low ∼ Tr(FF ) (k+1)/2 . All other components from X and Y are either eaten in the Higgsing, or get a mass from the superpotential (4.10) along the flat direction (4.11). This low-energy theory is depicted in Fig. 5, where as usual adjoints are arrows that start and end on the same node of the quiver diagram, and dotted adjoints depict those that get a mass term from the superpotential. Brodie duality along this flat direction is then compatible with a duality in [25] (see Section 8 there) for the SU (m) 2 factor, and with Seiberg duality for the SU (n) factors: where horizontal arrows are the flat direction (4.11) and vertical arrows are the duality.  We now consider the analogous flat directions (4.11) for the puzzling D even cases, which again exist for N c = 2m + kn and are parameterized by arbitrary λ x . As in the D odd case, the gauge group is Higgsed SU (2m + kn) → SU (m) 2 × SU (n) k × U (1) k+1 , where the SU (m) 2 and SU (n) k decouple from each other at low energies. But the D even case differs from D odd in two respects. First, the SU (m) 2 sector has massless bifundamentals F andF , with W low (FF ) = 0. Similarly, the SU (n) k sector reduces at low-energy to k/2 decoupled copies of SU (n) 2 which each have, in addition to N f flavors, massless bifundamentals with W low (FF ) = 0. For example, for k = 2, N c = 2(m + n), and expanding (4.10) along the flat direction (4.11) gives where the low-energy SU (m) D is an A theory, with massless adjoint X (coming fromF ) and 2N f fundamentals, with W low (X ) = 0.
More generally, for even k > 2, since ω = e 2πi/k in (4.11), there will be k/2 massless bifundamental pairs. The low-energy SU (n) k theory then reduces to k/2 decoupled SU (n) 2 quiver gauge theories, where the i'th node couples to the (k/2 + i)'th node via a pair of massless bifundamental fields. Each SU (n) 2 theory has a flat direction to an SU (n) D A theory. The low-energy theories along these flat direction are as depicted in Figure 6.
The conclusion is that, for D even , we end up with (k/2 + 1) A theories corresponding to nodes with 2N f flavors 9 . The A theories along the flat direction are puzzling, as in Section 4.2.1: we have not found a quantum mechanism for lifting these flat directions, and have not found how to make these flat directions compatible with Brodie's proposed duality. 9 For N c = 2m, there is a similar generalization of these flat directions parameterized by both λ x and λ y , with X ∝ Y ∝  (1) n n

D k+2 RG flows from relevant ∆W deformations
In this subsection, we consider RG flows from the W D k+2 SCFTs upon deforming by relevant ∆W . As in the previous subsections, we find that cases involving only D odd are nicely compatible with the duality of [5], while those involving D even exhibit subtleties. For simplicity, we mostly consider U (N c ), with brief discussion of the more complicated SU (N c ) version in Section 4.5.
We begin with the class of ∆W deformation RG flows D k+2 → D k +2 , which is relevant for k < k (taking x > x min D k+2 > x min D k +2 as in (4.2)): which yields the F-terms The solution X = Y = 0 corresponds to the D k +2 theory at the origin. There are also (k−k ) 1d solutions with non-zero X-eigenvalue, corresponding to A 1 's. The representation theory of (4.15) was discussed in [13,22]. Taking X and Y to be matrices, it follows from (4.15) that X 2 and Y 2 are Casimirs (proportional to the unit matrix), so we may rewrite the first F-term as (y 2 + Q k/2 (x 2 ))1 + P (k−1)/2 (x 2 )X = 0, where the subscripts on P and Q denote the degrees of the polynomials in x 2 . There are 2d representations of the second F-term, taking X = xσ 2 , Y = yσ 1 ; then a non-zero solution for X requires P (k−1)/2 (x 2 ) = 0. Hence, there are (k − 1)/2 independent such solutions for x 2 , and then y 2 is uniquely fixed 10 . If X and Y have n j copies of such a vacuum, where j = 1, . . . , (k − 1)/2 labels the value of x 2 j , then the non-zero X and Y values break SU (2n j ) → SU (n j ) × SU (n j ) → SU (n j ) D , where the low-energy SU (n j ) D theory has 2N f flavors. Expanding W (X, Y ) in such vacua, the X and Y adjoints have mass terms and the low-energy theory is SQCD; we label such vacua as A 2d 1 . In sum, the ∆W deformation (4.14) leads to vacua The N c × N c matrices X and Y are decomposed into blocks, distributed among these vacua, with n 0 eigenvalues at the origin, n i at the i'th A 1 node, and n 2d j in the j'th A 2d 1 node, with N c = n 0 + i n i + 2 j n 2d j . The gauge group is Higgsed in the electric and dual magnetic descriptions (for x in the conformal window) as: We now consider the RG flow D k+2 → A k , by adding ∆W = m Y 2 TrY 2 to the superpotential in (4.14). There is then a low-energy A k theory at the origin, X = Y = 0, along with (k − k ) A 1 's corresponding to the 1d solutions of the vacuum equations with eigenvalues As always, these 1d solutions of the F -term equations match the rank of the ADE group: k +2 in the UV matches the IR sum k +(k −k )+2. In addition, there are 2d representations of the D-and F -terms, with Casimirs Y 2 = y 2 1 and k i=k t i X i = f (x)1. The 2d vacua may thus be parameterized as X = − v 2 1 + x 1 σ 1 , Y = yσ 3 , and the F -terms have (k − 1)/2 solutions for x 1 , each of which determines f (x 1 ) and specifies the 2d vacuum. In each such vacuum, the low-energy theory is SQCD (both X and Y have mass terms) with the X and Y expectation values breaking SU (2n 2d j ) → SU (n j ) × SU (n j ) → SU (n j ) D , with 2N f flavors in the low-energy theory. In sum, the full (classical) structure of the vacua from such deformations is 2n 2d j , the deformation results in the following Higgsing in the electric and magnetic descriptions: . i=k t i X i+1 as in (4.16), and then deforming by v 2 TrY 2 as in (4.18), which gives  These extra terms are needed in order to re-tune, to zero, the corresponding ∆W relevant deformations which would be generated by adding the Lagrange multiplier constraint terms, and which would generically further deform the RG flow to merely multiple A 1 vacua. For flows starting at D k+2 as in (4.1), the needed deformations are included in For generic couplings in (4.21), the RG flow leads to vacua as which is the same for SU (N c ) and U (N c ). One can now tune the couplings in (4.21) to enhance to an A k or D k +2 singularity, and then the flow involves Higgsing as in e.g. are complicated, and depend on how many eigenvalues n 0 are in the enhanced D k +2 or A k vacua. We have verified that, despite these technical complications, the vacuum structure is qualitatively similar to that of the U (N c ) case, replacing U (n) → SU (n) everywhere in Section 4.4.
Interestingly, there can be several options in performing the wanted shift, and these can result in different Casimirs along the flow. We illustrate this for the example D 5 → D 3 , and note that there are similar versions for other D flows. The first way to enhance to D 3 is via a tuned addition of the {m Y , λ x } deformations to (4.14), where the needed shift of these couplings depends on the {t 3 , t 2 , t 1 } couplings in (4.14), as well as the multiplicities of the eigenvalues in the vacua. The Casimirs along the flow are then Y 2 and t 3 X 3 + t 2 X 2 + t 1 X.
Much as in (4.16), we indeed find one A 2d 1 vacuum. Another option for D 5 → D 3 + . . . is to add only the u 1 2 TrX 2 Y deformation in (4.21), with the other ∆W couplings set to zero. Then X 2 and Y 2 are Casimirs, but X and Y no longer anticommute as they did in the U (N c ) case, and so a 2d solution is now of the form We again find one 2d representation of the F -and D-terms, which reduces to the U (N c ) 2d solution as u 1 → 0. Different sets of lower order deformations in the chiral ring lead to different Casimirs along the flow, but nevertheless non-trivially give the same counting for the higher dimensional vacua.
4.6. The D odd → D even RG flow and the hypothetical D even theory As discussed in the previous subsections, the D even theories have some puzzles, whereas the D odd theories appear to be under control. This suggests trying to understand the D even theories via RG flows from the understood UV case: D odd → D even . Indeed, the idea of embedding D even in D odd was the basis for the original conjecture [5] that quantum effects somehow make the troubling D even theories similar to the nice D odd theories. In this subsection, we examine the D odd → D even RG flow more carefully, and note that this flow has its own subtleties.
As seen in (4.16), the ∆W RG flow from D k+2 → D k +2 comes with jumping number of A 2d 1 representations, from the floor and ceiling functions, which is only straightforward for the D odd → D odd' cases. We here further discuss the relation and difference between D odd → D odd vs D odd → D even . Consider starting from the D k+2 SCFT, with k odd, and deforming by ∆W . To simplify the discussion, we consider U (N c ) (as opposed to SU (N c )) and start with the ∆W deformation considered in (4.16) with k = k − 2: D k+2 → D k + 2A 1 + A 2d 1 . The low-energy D k theory is at X = Y = 0, the 2A 1 theories are at X having eigenvalues x ± with Y = 0, and the A 2d 1 theory has (X, Y ) values at (x 2d , y 2d ) given by: If we start at the D k+2 theory (as opposed to D), we can set t k = 1, and t k−1 and t k−2 are the ∆W deformation parameters.
We now try to tune the superpotential couplings to collide the D k singularity with an A 1 singularity, to get an enhanced D k+1 singularity. This can be accomplished by tuning t k−2 → 0 in (4.22), which brings one of the A 1 singularities (x + or x − ) to the origin.
Note that t k−2 → 0 also brings x 2d and y 2d to the origin. We denote this enhancement as where the prime distinguishes the theory from the even D k+1 theory that one would obtain by flowing directly from the D theory. We can formally obtain that latter theory, D k+1 , directly from the D fixed point, by taking t k → 0 along with t k−2 → 0 in (4.22); this brings one of the x ± to the origin and the other to infinity, and then the last equation in (4.22) gives the line of A 2d 1 solutions (4.6) where D even → A, since (4.22) is satisfied for all x 2d when t k = 0. The two procedures are indicated in the Figure 7.
(a) The 2d vacuum and an A 1 both collapse to the origin.  The two procedures suggest that perhaps there are actually two types of D even theories.
One is the D even theory of Figure 7a, which can actually be obtained from the RG flow D odd → D even , and which therefore inherits the simpler properties of D odd . The other is the mysterious D even theory of Figure 7b, which actually is not obtained from RG flow from D odd , but instead only from D → D even , since it requires t k = 0 and the D odd theory had t k = 1. The latter, D even theory has the puzzles, discussed in the previous subsections, associated with the D even → A moduli space of vacua and the non-truncated chiral ring.
We have thus considered the possibility that Brodie duality actually only applies to the simpler D even theory, which inherits the truncated chiral ring from D odd , and does not apply to the D even theory. However, this scenario also has challenges. If we take seriously the idea that a D k+1 (for k odd) theory is made by bringing together D k + A 1 + A 2d 1 , this seems to suggest that the chiral ring of the D k+1 theory contains (3k − 1)N 2 f mesonic operators, where the −N 2 f are those in the A 1 singularity, which decouples from D k+1 in the IR. On the other hand, assuming that Brodie duality applies to D k+1 , we would have expected 3(k − 1)N 2 f mesonic operators. The D k+1 theory has an extra 2N 2 f mesonic operators. Perhaps then, in collapsing the A 1 and A 2d 1 theories to the D k theory at the origin, a slightly modified version of Brodie duality applies, with α D k+2 = 3k + 2. We have also tried to cure the apparent a-theorem violations of Section 4.2.2 by adding the 2N 2 f mesons to the UV D even theory. But the results did not look promising: the extra operators seem to become free at too large x to cure the apparent wrong sign of ∆a. It is still possible that some modified version of Brodie duality resolves these puzzles, and we invite the interested reader to try.

The W E 7 fixed point and flows
The W E 7 SCFT arises as the IR limit of a relevant superpotential deformation to the E SCFT, with corresponding superpotential The TrY X 3 term is a relevant deformation to the E fixed point for x > x min E 7 ≈ 4.12, where x min E 7 was determined via a-maximization in [13]; here we will assume that x > x min E 7 . The F-terms of the undeformed E 7 superpotential in (5.1) are given by from which it follows that the chiral ring does not truncate classically. We may write the generators of the classical chiral ring in a basis Θ (1,n) = X n , Θ (2,n) = Y X n , Θ (3,n) = XY X n , Θ (4,n) = Y XY X n ; n = 0, 1, ...

Proposed dualities for W E 7 [6]
It was noted in [13] that the W E 6 apparently violates the a-theorem condition (2.5) for x 13.8, and thus some new dynamics must enter for x ∼ 13.8 (or less) to ensure that the a-theorem is satisfied. In [6], it was pointed out that for the W E 7 theories the condition (2.5) is violated for x 27, so some new dynamics is needed there, or at smaller x. The dual theory proposed in [6] resolves this apparent a-theorem violation, since it implies different IR phases for larger x 26.11 [6]. The duality of [6] requires that the chiral ring truncates, similar to the conjecture in [5] for D even , as Y X 6 + bXY X 5 = 0 in the chiral ring In addition to the usual tests of duality-'t Hooft anomaly matching, that the charge assignment for the magnetic fields under the global symmetry is consistent with the duality map-it was verified in [19] that the superconformal index of the dual theories agrees, at least in the Veneziano limit (away from that limit, the duality and agreement of their superconformal indices suggests new mathematical identities).
As we discuss in the following subsections, we find similar puzzles for the E 7 theories as with the D even theories. In the following, we mirror our analysis of the W D k+2 theories for W E 7 ; as such, we will be brief when analysis or discussion is similar to what has already 11 As in [6], we scale the factors of µ to unity.
been discussed in Section 4. Much as we found for D even , we fail to find evidence for this truncation, and point out additional hurdles for the conjectured duality.

Matrix-related flat directions at the origin
We consider the moduli space of vacuum solutions of (5.2) with D-term constraints (1.5), setting Q =Q = 0. The only 1d solution corresponds to the E 7 singularity at the origin.
The first F -term in (5.2) shows that Y 2 and X 3 are Casimirs, yielding Casimir conditions There is a line of d = 2 solutions to these conditions analogous to (4.6), for ω = e 2πi/3 . As X and Y are not traceless, this flat direction is present for only U (N c ) 12 .
In general, E 7 has vacua with multiple copies of the solution (5.6), with the remaining eigenvalues of X and Y at the origin, giving a moduli space of supersymmetric vacua labeled by y 2 i and x 3 i satisfying (5.6), for i = 1, . . . , N c /2 . These vacua Higgs the gauge group in a way that turns out to be analogous to the D even case discussed in 4.2.1. In particular, for N c = 2n with n copies of the 2d vacuum (4.6) and unequal expectation values of the y 2 i , x 3 i , the resulting breaking pattern is U (2n) → U (n) D → U (1) n . In summary, much as in (4.8), there is a (classical) flat direction: If we assume that Kutasov-Lin's duality [6] holds, then as in the D even case we are led to a puzzle similar to that of the D even theories: Independent of the conjectured duality [6], the deformation (5.7) seemingly violates the a-theorem (2.1) for sufficiently large x. As in Section 4.2.2, we compute a U V (x) for the 12 For special cases of (5.6) there will be SU (N c ) flat directions; for example, when there are equal multiplicities of X, ωX, and ω 2 X along the line given in (5.6). In that case, one could check the proposed SU (N c ) duality along the corresponding flat directions. W E 7 theory, with gauge group U (N c ) and N f flavors, as in [13]. Likewise, a IR (x) for the A theory, with gauge group U (N c /2) and 2N f flavors, is computed as in [4]. We include the effects of all mesons hitting the unitarity bound assuming that the chiral ring is quantumly truncated, such that all the operators listed in Appendix A are taken into account, and work in the Veneziano limit. We plot until the bottom of the conformal window-which occurs before the electric E 7 theory's stability bound, x < 30 as predicted by duality-such that we expect the a-theorem to hold in the whole range plotted.
in the UV and A in the IR. The E 7 deformation term in the UV theory is relevant for x 4.12, while the corresponding term in Kutasov-Lin dual is relevant if x 26.11. TheÂ theory is UV-free in this whole range.
As seen Figure 8, this flat direction seems to violate the a-theorem in the conformal window for x 23.39. Unlike the D even case, this violation occurs for x larger than the value where the mesons removed by the proposed quantum constraint (5.4) would hit the unitarity bound and become free; the first such meson that would be nonzero involves the operator Y X 6 , which would become free at x = 21. To understand the effect that these would-be mesons would have on the computation of a for this flat direction, we have performed the same check as in Figure 8, but without imposing the proposed constraint. It turns out that this is not enough; the effect of including these operators in the ring is only to push the range of the apparent a-theorem violation to x 23.44.
The apparent violation of the a-theorem for these flat directions must of course be somehow resolved. As in the discussion in Section 4.2.2, either these flat directions are lifted in a way we don't understand, or some additional degrees of freedom make the calculation of a incorrect-perhaps in the UV W E 7 theory. The arguments made in Section 4.2.3 would also apply here, and suggest that the former is not the solution. Since the calculation of a in Figure 8 already took into account the proposed W E 7 duality, we are left with a puzzle.

SU (N c )-specific (as opposed to U (N c )) flat directions
We now study SU (N c ) flat directions of the W E 7 theory, imposing the tracelessness of the adjoints with Lagrange multipliers λ x , λ y : When N c = 2m + 3n for m, n integers, there is a flat direction labeled by λ y X = λ y s 1 where ω = e 2πi/3 and off-diagonals are zero. (5.9) is the special case of k = 3 in (4.11).
Along this flat direction, the gauge group is Higgsed SU (2m + 3n) → SU (m) 2 × SU (n) 3 × U (1) 4 . The low-energy SU (m) 2 sector includes N f massless flavors, along with bifundamentals F,F and adjoints A 1 , A 2 coming from the adjoint X of the original theory at the origin, with a low-energy superpotential that is cubic in the massless fields (written in Figure 9a). Thus, each SU (m) node corresponds to a W A 2 theory plus extra flavors from the bifundamentals. The low-energy SU (n) 3 sector includes N f massless flavors along with three pairs of bifundamentals F 12 , F 23 , F 13 , and their conjugates, coming from the adjoint Y of the original theory at the origin. There is an IR superpotential for these fields W low Tr(F 12 F 23F13 +F 12F23 F 13 ), which corresponds to making a loop around the quiver diagram shown in Figure 9b. All other components from X and Y are either eaten in the Higgsing, or get a mass from the superpotential (5.8), such that the SU (m) 2 and SU (n) 3 sectors decouple from each other at low energies. These low-energy theories are summarized in the left-most quiver diagrams in Figure 9.
We can then go along a further flat direction of the low-energy SU (m) 2 theory, where we give an arbitrary vev to the massless F , such that SU (m) 1 × SU (m) 2 breaks to the diagonal subgroup SU (m) D . The low-energy SU (m) D has an adjoint that remains massless, 3 sector, Higgses to SQCD.
To summarize, these flat directions pose puzzles for the proposed W E 7 duality, both in the A theory of 9a, and in the SQCD theory of 9b. We have not found a quantum mechanism for lifting these flat directions. We begin with the RG flow E 7 → A 2 flow for gauge group U (N c ), taking x > x min E 7 :

Case studies in
which yields the F -terms There are seven 1d solutions to (5.11): two coincident at X = Y = 0, corresponding to the A 2 theory, and five solutions with nonzero X and Y eigenvalues, corresponding to A 1 theories; as always, the 1d solutions correspond, as in Arnold's ADE singularity resolutions, to adjoint Higgsing of the G = ADE, and preserving r G . Taking X and Y to be matrices, it follows from (5.11) that X 3 Y 2 are Casimirs along the flow, so that we may write There is a 2d as well as a 3d representation that solve the F -terms, D-terms (1.5), and Casimir conditions, The (+ . . .?) indicate that there might be additional d > 3 dimensional vacuum solutions, beyond the ones that we found here 13 . In the following we will assume that there are no such additional vacua in (5.13), but we do not have a proof that this is the case.
If there are n 0 eigenvalues at the origin, n i in the i'th A 1 node, n 2d in the A 2d 1 node, and n 3d in the A 3d 1 node, such that N c = n 0 + 5 i=1 n i + 2n 2d + 3n 3d , then the gauge group is Higgsed in the electric and proposed magnetic descriptions (for x in the conformal window): .
(5.14) 13 We use the SU (N ) or U (N ) symmetry to gauge fix one real adjoint's worth of components in X, X † , Y , and Y † , and the remaining entries are constrained by the D-and F -terms, along with any Casimir conditions.
We did not find an analytic way to construct, or exclude, higher-dimensional solutions beyond scanning computationally. Even gauge-fixing, scanning the solution space is harder for larger d, and so in (5.13) we only completed the scan for d ≤ 3.
The down arrows are Kutasov-Lin duality for the E 7 U (N c ) theory in the UV, and Kutasov or Seiberg duality for the approximately decoupled low-energy gauge group factors in the IR.
Comparing the UV and IR of the dual theories of the lower row of (5.14) as we did for the To recover the SU (N c ) version of this flow, we must deform the superpotential (5.10) by the operators TrY 2 , TrXY, TrX, TrY , (the latter two with Lagrange multipliers) whose coefficients are shifted appropriately. The 2d representation for the deformed superpotential smoothly matches onto the U (N c ) solution in (5.12) upon taking the coefficients of the lower order deformations to zero. The analogous check for the 3d representations in (5.12) turns out to be technically challenging, and while we expect that it also matches, such that the SU (N c ) version of the flow will match onto (5.14), we have not verified this. (For reasons that will become apparent in Section 5.4.2, this can be a subtle issue in the E-series.)

E 7 → D 5 : Disappearing vacua?
We here consider the flow E 7 → D 5 for U (N c ) gauge group, which corresponds to the superpotential (normalizing the couplings in the UV E 7 theory to one) The F -terms of (5.15) are given by The 1d vacuum structure along this flow consists of the D 5 theory at X = Y = 0, and 2 A 1 's away from the origin. To study for higher-dimensional vacua we look for Casimirs, but note that there are no simple Casimirs of (5.16), except of course the F -terms (5.16) themselves. There is a 2d solution to the F -terms (5.16) and D-terms (1.5) of the form .
For the flow (5.23), the eigenvalues corresponding to the 1d and 2d A 1 singularities in (5.17) come together, enhancing to an A 2 singularity. Labeling the multiplicities of X and Y 's eigenvalues as in (5.18), then for the enhancement (5.23) the eigenvalues rearrange such that the electric theory is Higgsed U (N c ) → U (n 0 ) × U (n 1 + n 2 + 2n 2d ). For the case (5.24), the eigenvalues corresponding to the A 2d 1 theory in (5.17) match onto copies of the eigenvalues corresponding to the 1d A 1 theories, such that in the IR the vacua are D 5 + 2A 1 . In this case, the eigenvalues in the electric version of the flow rearrange such This feature that a 2d representation can "go away" is also present in the SU (N c ) version of the flow (5.15). As was the case for the D-series flows discussed in Section 4.5, there are multiple sets of deformations ∆W that one can add to (5.15) to recover the same 1d vacuum structure as in (5.17) for SU (N c ) gauge group 14 . For instance, one possibility is is not satisfied: here it is because α E 7 = α D 5 + 2α A 1 + 3 × 2 2 α A 1 , i.e. 30 = 9 + 2 + 12.
Analogously to the U (N c ) flow (5.17), one of the 2d vacua in (5.22) reduces to 1d vacua in special cases. The difference here is that the other two 2d vacua in (5.22) remain:  [33], and this formalism is applied in [34] to several of the resolutions of present interest to us. We used an adaption of this construction to obtain the deformations of this section.
We here consider the ∆W deformation which leads to the RG flow E 7 → A 6 . This is given ∆W ∼ TrX 7 . At first glance, this ∆W seems irrelevant at the W E 7 SCFT, since it scales with a higher U (1) R charge than the terms in (5.1), but we know that such a flow should be possible (for instance, we can cut the E 7 Dynkin diagram to recover the A 6 diagram, as demonstrated for other cases in Figure 1). The resolution to this puzzle is that only a special shift of the deformation couplings will recover the A 6 singularity in the IR-even for the U (N c ) case. The clearest way to see the enhancement of the A 6 singularity is through a change of variables. Since the change of variables is already complicated in the U (N c ) case, we will only consider this flow for U (N c ) gauge group here. We analyze other E-series flows whose ∆W deformations seem irrelevant in Appendix B.
We start with W E 7 plus ∆W deformations, It follows from the F -terms of (5.25) that there are seven 1d vacua in the IR, corresponding to seven A 1 theories (we will discuss higher-dimensional vacua below). It is useful to next linearly shift the fields X → X + n, Y → Y + m, where we choose m and n as functions of the couplings in (5.25) to cancel in linear terms in X and Y which result from the change of variables. Dropping constants, the superpotential can then be rewritten as where the t i 's are defined in terms of the couplings in (5.25) and m, n. We then implement the following change of variables for all t 1 = 0: (5.28) Studying the F -terms of this superpotential and expanding (5.28) in the vacua, there is one vacuum at the origin corresponding to the A 6 theory, and one away from the origin corresponding to an A 1 theory. Thus, we have recovered the desired flow.
We have also studied the 2d vacuum structure of this RG flow 15 . For generic values of the couplings in (5.26), there are nine 2d vacua which we can parameterize as X = x 0 1 + x 3 σ 3 , Y = y 0 1 + y 3 σ 3 , such that the generic ∆W deformations lead to the vacua However, all of these 2d vacua "go away" in the enhancement to the A 6 theory, in the sense described in Section 5.4.2. In particular, of the 18 eigenvalue pairs corresponding to the A 2d 1 's in (5.29), 15 come to the origin to form the A 6 theory in the shift to (5.28), while the remaining 3 become copies of the shifted A 1 theory. Thus the 1 and 2d vacuum structure of this flow appears to be where the multiplicities of the eigenvalues corresponding to the higher-dimensional vacua of (5.29) have redistributed appropriately. The ADE SCFTs have a rich structure of vacua, and deformations. The fact that the fields X and Y are matrices introduces many novelties, as we have here illustrated-but not yet fully understood. It is natural to expect that the higher-dimensional representations of the F -and D-terms have dimensions d i given by some G = A, D, E group theory quantities, e.g. the Dynkin indices n i as with the McKay correspondence. But we find that 15 We have not as of writing attempted to find d > 2 dimensional vacua for this flow. d i = n i in general, and we do not yet know how to analytically find the d i and associated representations.
Our analysis of the E-series shows that even associating a fixed set of representations with the deformation flow can be subtle. For example, the case studies of Section 5.4 give the following puzzle: we can RG flow from the W E 7 SCFT via different ∆W deformations, to decoupled copies of SQCD (A 1 ) at low energies, and for different routes seemingly get different numbers of higher-dimensional representations in the IR. It will be interesting to understand how the proposed duality [6] fits in with this picture. The present work has raised several additional hurdles for the conjectured D even and E 7 dualities, and it will be interesting to see how all of these puzzles are resolved.
The TrX 4 and TrX 5 terms are relevant for x E 6 min ≈ 2.44 and x E 8 min ≈ 7.28, respectively [13]. As reviewed in Section 1.1, the chiral rings of these theories do not classically truncate, and are especially rich since X and Y decouple in the F -terms (1.8) and (1.10). As shown in [6,19], the W E 6,8 theories cannot have a dual of the form reviewed following (1.13). It is unknown if there is a dual of some different form.
The a-theorem condition (2.5) is violated for sufficiently large x for both theories [13], showing that some new quantum effects must arise for large x. One possibility is that a W dyn is generated, and the theory is no-longer conformal, for some x > x stability . Another possibility is that there is some unknown dual description which becomes IR-free for large x.
There are other reasons to expect that there might be some description of the IR physics of (6.1) and (6.2) in terms of dual variables: we can flow, for instance, E 6 → D 5 , and we expect that the stability bound is reduced x max E 6 > x max D 5 along RG flow. It is also pointed out in [6,19] that in E 6 the number of operators at a given value of R grows with R-charge, but somehow the theory must find a way to preserve unitarity.
We have studied a few aspects of the moduli space and ∆W deformations of the W E 6 and W E 8 SCFTs, looking for clues in formulating a dual description of the theories, but finding puzzles (similar to D even and W E 7 ). We here briefly report on some of our findings.
The undeformed W E 6 and W E 8 theories have a variety of flat directions similar to those discussed for the W Deven and W E 7 theories in Sections 4.2.1 and 5.2. In particular, both have 2d and 3d nilpotent flat directions (of course, a flat direction of E 6 is also a flat direction of E 8 , since X 3 = 0 ⇒ X 4 = 0). The 2d vacuum solutions are of the form X 2d = x(σ 3 + iσ 1 ), Y 2d = −x(iσ 3 + σ 1 ) where arbitrary complex x labels the flat direction.
There are several 3d flat directions of these theories, again labeled by x, for instance  As with the D even and E 7 cases, these (classical) flat directions are surely related to the classical nontruncation of the ring. We expect, as with those cases, that some dynamics must alter these flat directions, at least for sufficiently large x, to avoid apparent violations of the a-theorem. It would be interesting to understand this further.
For SU (N c ), as opposed to U (N c ), upon imposing the tracelessness of the adjoints by adding Lagrange multiplier terms to (6.1) and (6.2), these theories have SU (N c ) flat directions for particular values of N c , similar to those discussed in Section 3.4, 4.3, and 5.3. The W E 6 theory has a flat direction for N c = 3m and/or N c = 2n, while E 8 has a flat direction for N c = 2n, for integer m and n. We expect low-energy A theories along these classical flat directions; it would be interesting if one can obtain insights about the theory at the origin from these flat directions.
We now briefly comment on the RG flows from some ∆W deformations of the W E 6 and W E 8 SCFTs. Consider e.g. the flow E 6 → D 5 , obtained via adding ∆W = TrXY 2 to (6.1). The 1d vacua correspond to the D 5 theory at the origin, and an A 1 theory away from the origin. The F -terms imply that [Y 2 , X] = 0, and [X 2 , Y ] = [X 3 , Y ] = 0, so that d > 1 dimensional solutions to the F -terms must actually satisfy X 2 = 0. It is then straightforward to show that there are no 2d or 3d solutions that satisfy the F -terms and D-terms, so that the vacua along the flow are just the 1d vacua (up to possible d > 3 representations, again as in the discussion around (5.13)) E 6 → D 5 + A 1 (+ . . .?). Perhaps understanding the IR limits of such flows will yield hints pointing towards a dual description of the W E 6 ,W E 8 theories. We invite the interested reader to try. Some additional comments on E-series flows are provided in Appendix B.
Appendix B. RG Flows whose deformations seem irrelevant We briefly consider (as in Section 5.4.3) some cases where the ∆W s, corresponding to some ADE adjoint Higgsing pattern, are not immediately apparent. We focus on recovering the desired 1d vacuum structure for U (N c ) flows, leaving a full analysis of the higher-dimensional structure for future work. The cases studied in Sections B.1 and B.2 are analogous to singularity resolutions studied in [34].
We start with the deformed E 6 superpotential, s , for nonzero s, the two eigenvalues on the last line of (B.3) collapse to the origin to enhance the A 3 singularity. This is more clearly seen by changing variables Y = s t 1 (Z − X 2 ). Then, for the special value of t 2 = t 2 1 s , (B.1) rewritten in terms of the X, Z fields gives the A 5 theory at the origin from the TrX 6 term in The F -terms of (B.4) then yields the 1d vacua A 5 + A 1 .
In sum, the flow (B.1) has the following 1d and 2d vacua: Here, we start with the E 7 superpotential deformed by the D-series term TrXY 2 , There are two sets of 1-dimensional vacuum solutions for X and Y , corresponding to the eigenvalues (x = 0, y = 0), and (x = 5t 2 9s , y = − 25t 3 27s ). Expanding near the origin appears to just give W low TrXY 2 = W D . Consider though the following sequence of variable changes: We've organized the terms in (B.9) by increasing relevance from the perspective of the UV fixed point. The most relevant terms in the IR limit of the flow are those in the last parentheses, such that the D 6 theory resides at the origin. There is a 1d vacuum solution to the F -terms of (B.9) corresponding to an A 1 theory, such for all t = 0 that we recover the 1d vacua: We start by deforming the E 8 theory with a D-series deformation and E 7 deformation, W = 1 3 TrY 3 + s 5 TrX 5 + t 1 TrX 3 Y + t 2 TrXY 2 . (B.11) From the 1d F -terms of this superpotential, there is eigenvalue pair at the origin and two away from the origin. As in the previous subsection, there is naively some ambiguity in identifying the solution at the origin, since each of TrX 3 Y , TrX 5 , and TrXY 2 appear to be marginal deformations of the UV theory, but the eigenvalue decomposition suggests that the theory at the origin corresponds to a D 6 . Then, the 1d vacua of (B.11) are D 6 + 2A 1 .
There is a particular shift of the coefficients t 2 = TrU 2 X. (B.12) The 1d F -terms of (B.12) still yield one zero eigenvalue pair and two nonzero eigenvalue pairs, but if we now shift t 2 = t * , then the D 6 theory at the origin is enhanced to a D 7 theory, while only one nonzero (1d) vacuum remains, in which both X and U receive masses.
In some, the shift t 2 = t * results in the 1d vacua D 7 + A 1 .
We now study higher-dimensional representations of vacuum solutions to the F -terms of (B.11) and D-terms (1.5). For generic values of the couplings, there is a 2d vacuum (letting s = 1) X = x 0 1 + x 3 σ 3 , Y = y 0 1 + y 3 σ 3 , x 0 = t 1 (− 9 2 t 2 1 + 4t 2 ), x 3 = 1 2 (9t 2 1 − 4t 2 ) 1/2 (3t 2 1 − 2t 2 ), In all cases above, the (+. . . ?) refers to d > 2 dimensional vacua. The special case (B.15) corresponds precisely to the shift t * already discussed, in which the D 6 singularity is enhanced to a D 7 singularity. In this case, one of the two eigenvalues corresponding to an A 2d 1 in (B.13) goes the origin, and the other becomes a copy of the eigenvalues corresponding to the remaining A 1 theory. In (B.16), the eigenvalues corresponding to the A 2d 1 theories in (B.13) become copies of the eigenvalues corresponding to the 1d A 1 theories. For the shift in (B.17), the two A 1 theories as well as the A 2d 1 theory in (B.13) are enhanced to an A 2 theory.