Testing our understanding of SCFTs: a catalogue of rank-2 \mathcal{N}=2 theories in four dimensions
aa r X i v : . [ h e p - t h ] F e b Testing our understanding of SCFTs: acatalogue of rank-2 N = 2 theories in fourdimensions Mario Martone , C. N. Yang Institute for Theoretical Physics, Stony Brook University,Stony Brook, NY11794-3840, USA Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3840, USA
E-mail: [email protected]
Abstract:
In this paper we begin mapping out the space of rank-2 N = 2 supercon-formal field theories (SCFTs) in four dimensions. This represents an ideal set of theorieswhich can be potentially classified using purely quantum field-theoretic tools, thus pro-viding a precious case study to probe the completeness of the current understandingof SCFTs, primarily derived from string theory constructions. Here, we collect andsystematize a large amount of field theoretic data characterizing each theory. We alsoprovide a detailed description of each case and determine the theories’ Coulomb, Higgsand Mixed branch stratification. The theories naturally organize themselves into seriesconnected by RG flows but which have gaps suggesting that our current understandingis not complete. ontents N = 2 moduli space 10 e − so (20) series 224.2 sp (12) − sp (8) − f series 394.3 su (6) series 484.4 sp (14) series 564.5 su (5) series 604.6 sp (12) series 644.7 Other series 674.7.1 sp (8) − su (2) series 674.7.2 g series 694.7.3 su (3) series 714.7.4 su (2) series 714.8 Isolated theories 72 A S and T theories 77B Theories with enhanced supersymmetry 82C Flavor structure along the Higgsing and generalized free fields VOA 84D Reading tools 86 – 1 – Introduction
In this paper, we initiate the bottom-up analysis of rank-2 N = 2 Superconformal FieldTheories in four dimensions. We compile a catalogue of all rank-2 N = 2 SCFTs cur-rently known and determine the full moduli space stratification for each theory heavilyleveraging the recent advancements of [1, 2]. This remarkable amount of informationalready shows interesting patterns suggesting that our current understanding is notcomplete (for more details see section 2). Our results are summarized in table 1, 2,3, 6, 7 and 8 which are organized in sets of theories mutually connected by massdeformations. Table 3 and 8 lists in particular the “isolated” ones for which no massdeformation connecting them to any other rank-2 SCFT is known. This hints at theexistence of new theories as it will be further argued below though a detailed discussionof RG-flows among rank-2 theories will be postponed to [3]. It is also worth noticethat some gaps in our understanding of the detailed structure of some theory’s modulispace remain (mostly those in the su (6) and su (5) series).The study of SCFTs has been at the center stage of mathematical physics forover two decades. By now, we have a remarkably large amount of information on thespace of the allowed theories in different spacetime dimensions and a great deal ofunderstanding of the interconnections among them. This achievement is largely dueto string theory which has provided, and still does, absolutely amazing and essentialtools to study theories with large amount of supersymmetry, particularly in dimensionslarger than four (see below). These results are so inextricably linked with string theorythat is often challenging to be sure that our current understanding reflects propertiesof quantum field theory rather than string theory itself. Assessing the completenessof the string theoretic picture it is then an important priority and the work in thismanuscript is a step in this direction.One way to make this assessment is to develop tools to answer a basic question: canall consistent SCFTs, with perhaps appropriate caveat like d ≥ d > It is my intention to constantly update the list on known rank-2 theories as our understandingimproves. I would therefore be extremely grateful if the reader who is aware of any rank-2 N = 2Superconformal Field Theory (SCFT) not appearing in the tables, could readily communicate thisinformation to me. – 2 –n six dimension is fairly clear with a belief that maximally symmetric (2,0) theoriesare completely classified by an ADE classification [5, 12, 13] and growing evidence ofa complete story in the (1,0) case [14–17] . In five dimensions our understanding isless settled but extraordinary progress has taken place recently leading to some initialattempt at classifying N = 1 SCFTs in [19–26] . A similar classification of SCFTswith eight (or more) supercharges in is instead wide open. A large majority of N = 2 SCFTs belong to the so called class- S set [27, 28] which directly descendsfrom the compactification of (2,0) theories in . Thus again much of our currentunderstanding in is derived from string theoretic constructions.Despite its richness, the situation is qualitatively different from the higher di-mensional one. There are in fact SCFTs which have no known string theory realization[50] and the lowest rank at which they appear is precisely two ( i.e. sp (4) + ).Furthermore in this case a variety of tools [51–54] which are constraining enough toconceive a bottom-up approach, are available. Potentially, this could give a way toprobe the completeness of the string theoretic description of the space of supersym-metric field theories or, at least, expand it. The approach most dear to the author isprimarily focused on the systematic study of the consistency of moduli space geometries[55].This philosophy came already to fruition after completing the classification of thesimplest set of four dimensional N = 2 SCFTs, the so-called rank-1 theories, [47,48, 56, 57]. This work highlighted the incompleteness of our understanding at thetime and led to the discovery of many new theories [35, 46] as well as new insights intostring constructions [43–45, 58] and compactification of higher dimensional SCFTs [40].Thanks in particular to [40, 43], the question of whether all rank-1 N = 2 SCFTsare realizable in string theory was settled with an affirmative answer. Neverthelessthere are too many simplifications which take place in the rank-1 case ( e.g. all scaleinvariant N = 2 rank-1 CB geometries are flat) and it is therefore unwise to extrapolatethis positive result to all ranks.It is perhaps appropriate to comment on the hopes of achieving a bottom-up com-plete classification for the rank-2 case. The methods used in [47, 48, 56, 57] are certainlyinsufficient. The main roadblock currently being the absence of a rank-2 equivalent ofthe Kodaira classification [59, 60] which provided a list of consistent geometries inter-pretable as CBs of rank-1 SCFTs . At rank- r this task entails determining the possible For a recent review of SCFTs in six dimensions and many more references see [18]. The literature on the subject is too vast so I apologize in advance for not providing fair credit tothose who deserve it. This statement requires some clarification. Strictly speaking the analog of the Kodaira classifica-tion at rank-2, i.e. the study of the possible degeneration of a genus two curve over a one dimensional – 3 – able of Rank-2 theories: CFT & Coulomb branch data I
Moduli Space Flavor and central charges u,v T knot T u T v d HB h f a c Comments e − so (20) series1 . { , } S (1) ∅ , ∅ T (1) E ,
59 1 [ e ] × su (2)
263 161 T (2) E , . { , } [ I , ∅ ] ∅ [ I ∗ , so (20)] 46 0 so (20)
202 124 3 rd entry at page 25 of [29]3 . { , } [ I , ∅ ] ∅ T (1) E ,
46 0 [ e ]
202 124 D ( E ) [30]4 . { , } S (1) ∅ , ∅ T (1) E ,
35 1 [ e ] × su (2)
167 101 T (2) E , . { , } [ I , ∅ ] [ I , su (2)] [ I ∗ , so (16)] 30 0 su (2) × so (16)
138 84 3 rd entry at page 26 of [31]6 . { , } [ I , ∅ ] ∅ [ I , su (10)] 26 0 su (10)
122 74 S : 3 rd entry of page 30 of [32]7 . { , } S (1) ∅ , ∅ T (1) E ,
23 1 [ e ] × su (2)
119 71 T (2) E , . { , } [ I , ∅ ] ∅ [ I ∗ , so (14)] 22 0 so (14) × u (1) 106 64 R , [32]9 . { , } [ I , ∅ ] [ I , su (2)] [ I , su (8)] 18 0 su (2) × su (8)
90 54 R , [32]10 . { , } [ I , ∅ ] ∅ [ I ∗ , so (12)] 14 0 so (12)
74 44 sp (4) + 6 F . { , } S (1) ∅ , ∅ T (1) D ,
11 1 so (8) × su (2)
75 42 sp (4) + 4 F + V or T (2) D , . { , } [ I , ∅ ] ∅ [ I , su (6)] 10 0 u (6)
58 34 su (3) + 6 F . { , } [ I , su (2)] [ I , su (2)] ∅ su (2)
42 24 2 F + su (2) − su (2) + 2 F . { , } S (1) ∅ , ∅ T (1) A , su (3) × su (2)
47 26 T (2) A , . { , } [ I , ∅ ] ∅ [ I , su (5)] 6 0 su (5)
42 24 D (cid:0) SU (5) (cid:1) [33]16 . { , } S (1) ∅ , ∅ T (1) A , su (2) × su (2)
39 21 T (2) A , . { , } [ I , ∅ ] ∅ [ I , su (2)] 2 0 su (2)
26 14 ( A , D ) AD Theory18 . { , } S (1) ∅ , ∅ T (1) ∅ , su (2) T (2) ∅ , . { , } [ I , ∅ ] ∅ [ I , su (2)] 1 0 su (2)
12 ( A , D ) AD Theory20 . { , } [ I , ∅ ] ∅ ∅ u (1) 22 ( A , A ) AD Theory21 . { , } [ I , ∅ ] ∅ ∅ ∅ ( A , A ) AD Theory sp (12) − sp (8) − f series22 . { , } [ I , ∅ ] [ I , su (12)] Z ∅
22 0 sp (12)
130 76 66 th entry at page 49 of [34]23 . { , } [ I , ∅ ] [ I , su (8)] Z [ I ∗ , sp (4)] 20 2 sp (4) × sp (8)
128 74 5 th / th entry at page 29 of [31]24 . { , } [ S (1) ∅ , ] [ T (1) E , ] Z ∅
24 2 su (2) × [ f ]
156 90 T (2) E , [35]25 . { , } [ I , ∅ ] [ I , su (8)] Z [ I , su (2)] 12 0 su (2) × sp (8)
84 48 6 th entry of page 61 of [36]26 . { , } [ I , ∅ ] [ I , su (6)] Z S (1) ∅ ,
11 1 su (2) × sp (6) × u (1) 83 47 2 nd entry of page 61 of [36]27 . { , } [ S (1) ∅ , ] [ T (1) D , ] Z ∅
12 2 su (2) × so (7)
96 54 T (2) D , [35]28 . { , } [ I , ∅ ] ∅ [ T (1) E , ] Z
16 0 [ f ] × u (1) 112 64 b T E , [37]29 . { , } [ I , ∅ ] [ I , su (6)] Z ∅ sp (6) × u (1) 61 34 e T E , [38]30 . { , } [ S (1) ∅ , ] [ T (1) A , ] Z ∅ su (3) × su (2)
66 36 T (2) A , [35]31 . { , } [ I , ∅ ] [ I , su (4)] Z ∅ sp (4)
38 20 su (2) − su (2)32 . { , } S (1) ∅ , S (1) ∅ , ∅ su (2)
36 18 N = 4 su (2) × su (2) Table 1 : First part of the list of rank-2 theories organized by series . Theories in eachseries are mutually connected by mass deformations. The second column lists the scalingdimension of the Coulomb Branch (CB) parameters. Columns 3, 4 and 5 instead list therank-1 theories describing the massless states on the connected components of the CB singularlocus, see table 4 for how to read these entries. The Z ℓ subscript indicates discrete gauging.Continuing, d HB indicates the quaternionic dimension of the Higgs Branch (HB) while h thatof the Enhanced Coulomb Branch (ECB). Column 8 indicates the flavor symmetry of theory,along with the level, followed by the a and c central charges. base, was already performed in [61, 62]. Since the rank-2 SCFT “lives” at the origin of the modulispace, which is complex co-dimension two, we need here to understand the possible degenerations ofgenus two curves on a two-dimensional base also satisfying special K¨ahler constraint ( i.e. exists SWdifferential). – 4 – able of Rank-2 theories: CFT & Coulomb branch data II Moduli Space Flavor and central charges u,v T knot T u T v d HB h f a c Comments su (6) series33 . { , } [ I , ∅ ] ∅ [ I ∗ , so (12) × su (2)] Z
23 1 su (6) × su (2)
179 101 33 th entry at page 16 of [39]34 . { , } [ I , ∅ ] [ I , su (2)] [ I ∗ , so (8) × su (2)] Z
13 1 su (4) × su (2) × u (1) 121 67 10 th entry at page 41 of [34]35 . { , } [ I , ∅ ] ∅ ⋆ w / b = 7 11 0 su (3) × su (3) × u (1) 107 59 SCF T on S | Z [38]36 . { , } [ I , ∅ ] ∅ [ I ∗ , so (6) × su (2)] 8 1 su (3) × su (2) × u (1) 92 50 SCF T on S | Z [38]37 . { , } [ I , ∅ ] [ I , su (2)] ⋆ w / b = 5 6 0 su (2) × su (2) × u (1)
78 42 SCF T on S | Z [38]38 . { , } [ I , ∅ ] ∅ [ I , su (2)] 2 0 u (1) × u (1) 49 25 su (3) + F + S sp (14) series39 . { , } [ I , ∅ ] ∅ [ I ∗ , sp (14)] 29 7 sp (14)
185 107 min ( D , D ) on T | Z [40]40 . { , } [ I , ∅ ] [ I , su (2)] [ I ∗ , sp (10)] 17 5 su (2) × sp (10)
125 71 15 th entry at page 53 of [34]41 . { , } [ I , ∅ ] ∅ [ I ∗ , sp (8)] 11 4 sp (8) × u (1) 95 53 R , [41]42 . { , } [ I , ∅ ] ∅ [ I ∗ , sp (6)] 6 3 sp (6)
65 35 sp (4) + 3 V su (5) series43 . { , } [ I , ∅ ] ∅ [ I ∗ , so (12)] Z
19 0 su (5)
170 92 T on S | Z [38]44 . { , } [ I , ∅ ] [ I , su (2)] [ T (1) D , ] Z su (3) × u (1) 114 60 page 39 of [38]45 . { , } [ I , ∅ ] ∅ [ I ∗ ⋆ , so (6)] Z su (2) × u (1) 86 44 Unpublished [3] sp (12) series46 . { , } [ I , ∅ ] ∅ [ I ∗ , sp (12)] 32 6 sp (12)
188 110 2 nd entry at page 29 of [42]47 . { , } [ I , su (2)] ∅ [ I ∗ , sp (4)] 8 2 sp (4) × so (4)
68 38 sp (4) + 2 F + 2 V . { , } [ I , ∅ ] ∅ [ I ∗ , sp (8)] 14 4 sp (8)
98 56 g + 4 F sp (8) − su (2) series49 . { , } S (1) ∅ , ∅ S (1) E ,
28 6 sp (8) × su (2)
232 130 S (2) E , [43]50 . { , } S (1) ∅ , ∅ S (1) D ,
14 4 sp (4) × su (2) × su (2)
146 80 S (2) D , [43]51 . { , } S (1) ∅ , ∅ S (1) A , su (2) × su (2) × u (1) 103 55 S (2) A , [43]52 . { , } [ S (1) ∅ , ] [ T (1) A , ] Z ∅ su (2) × su (2)
102 54 T (2) A , [35]53 . { , } [ I , ∅ ] [ T (1) A , ] Z ∅ su (2)
67 34 b T A , [44]54 . { , } S (1) ∅ , ∅ S (1) ∅ , su (2)
60 30 N = 4 sp (4) g series55 . { , } S (1) ∅ , [ T (1) D , ] Z ∅
12 2 [ g ] × su (2)
120 66 T (2) D , [35]56 . { , } S (1) ∅ , [ T (1) A , ] Z ∅ su (2) × su (2)
72 38 T (2) A , [35]57 . { , } [ I , ∅ ] [ T (1) D , ] Z ∅ g ]
82 44 b T D , [44]58 . { , } S (1) ∅ , ∅ ∅ su (2)
48 24 N = 4 su (3) su (3) series59 . { , } S (1) ∅ , ∅ S (1) D ,
15 5 su (3) × u (1) 219 117 S (2) D , [43]60 . { , } S (1) ∅ , ∅ S (1) A , u (1)
137 71 S (2) A , [43]61 . { , } S (1) ∅ , ∅ S (1) ∅ , u (1) 96 48 N = 3 G (3 , ,
2) [45] su (2) series62 . { , } S (1) ∅ , ∅ S (1) A , su (2) × u (1) 212 110 S (2) A , [43]63 . { , } S (1) ∅ , ∅ S (1) ∅ , u (1) 132 66 N = 3 G (4 , ,
2) [45]
Table 2 : Second part of the list of rank-2 theories organized by series . Theories in each seriesare mutually connected by mass deformations. The second column lists the scaling dimensionof the CB parameters. Columns 3, 4 and 5 instead list the rank-1 theories describing themassless states on the connected components of the CB singular locus, see table 4 for howto read these entries. The Z ℓ subscript indicates discrete gauging. Continuing, d HB indicatesthe quaternionic dimension of the HB while h that of the ECB. Column 8 indicates the flavorsymmetry of theory, along with the level, followed by the a and c central charges. – 5 – ank-2: CFT & Coulomb branch data III (Isolated theories) Moduli Space Flavor and central charges u,v T knot T u T v d HB h f a c Comments64 . { , } [ I , ∅ ] S (1) ∅ , [ I ∗ , sp (8)] 15 5 su (2) × sp (8)
123 69 5 th entry at page 51 of [34]65 . { , } [ I , sp (4)] [ I , su (2)] ∅
10 4 sp (4) × su (2)
118 64 17 th entry at page 7 of [46]66 . { , } S (1) ∅ , [ T (1) ∅ , ] Z ∅ su (2)
14 4565 2345 T (2) ∅ , [44]67 . { , } S (1) ∅ , ∅ S (1) ∅ , su (2)
84 42 N = 4 g Theory with no known string theory realization68 . { , } [ I , ∅ ] ∅ [ I , ∅ ] 0 0 ∅
58 28 sp (4) w / Table 3 : Third part of the list of rank-2 theories. In this table we list the isolated theories,those for which no mass deformation connecting them to other rank-2 SCFTs is known. Thesecond column lists the scaling dimension of the CB parameters. Columns 3, 4 and 5 insteadlist the rank-1 theories describing the massless states on the connected components of the CBsingular locus, see table 4 for how to read these entries. The Z ℓ subscript indicates discretegauging. Continuing, d HB indicates the quaternionic dimension of the HB while h that of theECB. Column 8 indicates the flavor symmetry of theory, along with the level, followed by the a and c central charges. ways a rank- r abelian variety can be fibered over a r complex dimensional base com-patibly with the constraints from scale invariant special K¨ahler geometry [63]. Anenormous simplification takes place for r = 2 when all polarized abelian varieties canbe written as Jacobian tori of (possibly singular) hyperelliptic curves and therefore CBgeometries can be expressed in a simple algebraic form: y = f ( x, u, v ) (1.1)where u and v are the (globally defined) coordinate of the CB and f ( x, u, v ) is a eithera six or fifth order polynomial in x with coefficient meromorphic in ( u, v ). An attemptto perform a study along these lines was made over a decade ago [64, 65] producingmany new geometries but falling short of providing a complete picture.A second obstacle is to develop appropriate tools which can translate the geometricinformation into field theory data. For example in [64, 65] only a very limited amountof physical information was provided regarding the SCFTs realizing the many newgeometries making it hard to make clear predictions testable with other methods. Therecent results in [1, 2] as well as the techniques implemented here, de-facto overcomethis problem altogether. We will report on the progress on the Kodaira classificationat rank-2 in a separate publication [66].The paper is organized as follows. In the next section we will present an overviewof the current status of rank-2 theories. In section 3 we will provide some useful back-ground on the geometry of the moduli space of vacua of N = 2 theories highlighting in– 6 – ummary of rank-1 theories T i supported on S i Name 12 c ∆ u h R h b f k f Comments T (1) E ,
62 6 0 e
12 [ II ∗ , e ] in [47] S C F T s T (1) E ,
38 4 0 e III ∗ , e ] in [47] T (1) E ,
26 3 0 e IV ∗ , e ] in [47] T (1) D ,
14 2 0 so (8) 4 [ I ∗ , so (8)] in [47] T (1) A , su (3) 3 [ IV, su (3)] in [47] T (1) A , su (2) [ III, su (2)] in [47] T (1) ∅ , ∅ − [ II, ∅ ] in [47] S (1) E ,
49 6 5 sp (10) 7 [ II ∗ , sp (10)] in [47] S (1) D ,
29 4 3 ( , ) 6 sp (6) × su (2) (5 ,
8) [
III ∗ , sp (6) × su (2)] in [47] S (1) A ,
19 3 2 sp (4) × u (1) (4 , ⋆ ) [ IV ∗ , sp (4) × u (1)] in [47] S (1) ∅ , su (2) 3 N = 4 su (2) S (1) D ,
42 6 4 ⊕ su (4) 14 [ II ∗ , su (4)] in [47] S (1) A ,
24 4 3 + ⊕ − su (2) × u (1) (10 , ⋆ ) [ III ∗ , su (2) × u (1)] in [47] S (1) ∅ ,
15 3 1 + ⊕ − u (1) ⋆ N = 3 S - f old [45] S (1) A ,
38 6 3 ⊕ su (3) 14 [ IV ∗ , su (3)] in [47] S (1) ∅ ,
21 4 1 + ⊕ − u (1) ⋆ N = 3 S - f old [45][ I , ∅ ] 3 1 0 u (1) 1 u (1) Theory w / I R - f r ee [ I n , su ( n )] n + 2 1 0 0 n su ( n ) × u (1) 2 u (1) Theory w / n hyper[ I n , su (2 n )] Z n + 2 1 0 0 2 n sp (2 n ) 2 [ u (1) Theory w /
2n hyper] Z [ I ∗ n , so (2 n + 8)] 9 + n n + 6 so (2 n + 8) 4 su (2) w / ( n + 4) [ I ∗ n , sp (2 n + 2)] 9 + n n + 1 n + n + 3 sp (2 n + 2) 3 su (2) w / ( n + 1) ♣♣ = the beta function is renormalized by , see discussion in [48, Section 4.2] . Table 4 : For the convenience of the reader, we list the properties of the rank-1 theorieswhich can describe the low energy physics on the CB co-dimension one singular locus as wellas that of rank-decreasing HB strata. The list of Z n gauging of SCFTs, and their CFT data,can be found in [49]. The discretely gauged theories appearing on the CB singular locusof theories in the su (6) series are instead discussed explicitly in the text in the appropriatesections. particular the notion of their stratification, the de-facto non-perturbative generalizationof partial higgsing, which plays a central role in our analysis. In section 4 we will delveinstead into the detailed description of the rank-2 theories listed in table 1, 2 and 3. Wekept the discussion of each theory as self-contained as possible. The paper also has fourappendices. In appendix A we will summarize the properties of the recently discovered T and S theories which will not be discussed individually. Appendix B summarizes theproperties of theories with enhanced supersymmetry which again will not be discussedin much detail individually. In appendix C we provide some rudimental information– 7 –n the generalized free field constructions of the VOA of the rank-2 theories. Finallyappendix D is dedicated to a glossary, a list of acronyms and symbols which appearthroughout the manuscript . Before delving into a detailed description of the structure of the moduli space of vacuaof N = 2 theories, let us provide an update on the current state of the classification ofrank-2 theories.First, nearly all the theories in table 1, 2 and 3 descend from theories, thusnearly all have a string theory construction. The lone exception being the lagrangiantheory sp (4) + which is one of the numerous theories (the only one with rank-2!) found in [50] and which still lack a string theory realization, we’ll discuss thistheory more below. This is perhaps not surprising. As I discussed at length in theprevious section, most of our current understanding descends from higher dimensions.This knowledge can also provide extremely useful insights in developing the tools for abottom-up classification directly in . For example, overwhelmingly SCFTs haveat least one IR-free gauge theory description associated to it, namely (one of) the gaugetheory of which they are the infinite coupling limit. This allows to easily study thespace of mass deformations of a given theory. If the realization of SCFTs from isunderstood, this information can also be extended to and the RG-flows trajectoriesmapped. This approach, which has numerous subtleties, will be further developedin [3] where a detailed discussion of the RG-flows among the rank-2 theories will bepresented.With this knowledge in mind, the structure of the currently known theories canbe nicely organized by RG-flows, and in particular mass deformations. We will call aset of theories which are interconnected by mass deformations a series . This allowsus to see patterns naturally generalizing what observed in rank-1. Each series hasa (not unique) top theory, from which the rest can be obtained, and a (again notunique) bottom theory, which cannot be mass deformed to any interacting rank-2SCFT. We name each series by the largest flavor symmetry factor of the top theories( e.g. the top theory of the e − so (20) series have flavor symmetries [ e ] × su (2) and so (20) ). Overwhelmingly the bottom theories are either lagrangian or N = 3theories [3]. An analogous hierarchy exists at rank-1. An important difference betweenrank-1 and rank-2 is that for the latter we have little understanding of how the scaling I would like to thank Jason Pollack, Patrick Rall and Andrea Rocchetto, for the stimulating weeklydiscussions on the quantum error correcting code interpretation of holography. It is these interactionsthat inspired the idea of including a glossary. – 8 –imensions of the CB parameters change along mass deformations (it is not clear at allthat a definite rule even exists). Conversely, the extremely constrained structure of theallowed geometries in rank-1 made this unambiguous and it was of tremendous help inperforming a systematic analysis.Of course for some of the lagrangian N = 2 SCFTs there exist special mass de-formations which land you on an Argyres-Douglas (AD) theory and allow to continuemass deforming beyond a lagrangian theory. It is curious that the current list of knownrank-2 AD theory only descends from two of the ten N = 2 lagrangian SCFTs (namely su (3) + 6 F and sp (4) + 4 F + 1 V ) both belonging to the same, e − so (20) series. Theappearance of a AD theory seems instead a completely generic feature of theories witha large enough flavor symmetry. It is therefore not unconceivable that more theoriesare awaiting to be discovered.To further support this point, it is worth noticing the following. Special K¨ahlergeometry, and in particular the fact that particular monodromies which correspond tospecial paths encircling the singualr locus on the CB, have to be elliptic ( i.e. theireigenvalues have to all lie on the unit circle) provides extremely strong constraints onthe allowed scaling dimensions for CB parameters. So much so, that there is a finiteset of permitted values at any given rank and there is a closed formula to computethem [67–69]. Of course there is no guarantee that all allowed scaling dimensions haveto be realized but it is interesting that this happens at rank-1. At rank-2 all nineinteger CB scaling dimensions are also realized by theories in our tables. Among the 15fractional ones instead, there are six ( , , , , , ) which appear nowhere and fourof those are smaller than two. At general rank, the number of allowed fractional scalingdimensions dramatically exceeds the integer ones yet in our current understanding of N = 2 SCFTs, theories with only integer scaling dimensions dramatically exceed thenumber of AD theories. It is then tempting to suggest that rather than a fundamentalfeature of SCFTs this is to be blamed on the techniques currently available toconstruct such theories. In particular AD theories are harder to construct from higherdimension and impossible for untwisted class- S with regular punctures which, thanks tothe Herculaneum effort of the tinkertoys program [29, 31, 32, 42, 70, 71], has provideda large chunk of currently known N = 2 theories,.There is one more reason which make it plausible that there might be new SCFTs ofthe AD type “hiding” on the CB of some of the lagrangian theories, those highlightedin yellow in the tables. And that is that a systematic search of all loci where non-mutually local particles could coincide is prohibitive at rank-2 and a search of this kindnecessarily makes some initial assumptions introducing biases which limit the scope ofthe search itself.Let’s now also comment on the list of isolated theories in table 3. Looking at the– 9 –igher dimensional construction, it is plausible that two of these theories (entry 66 and68) are indeed isolated. For the remaining theories, it is not unconceivable that theremight be other theories which connect via mass deformation and which are not yetknown. In particular the somewhat curious lagrangian theory sp (4) + has no flavorsymmetry nor HB and it is also the only lagrangian theory which does not appearas a bottom of an RG-flow. Coincidentally, it is also the only theory with no knownrealization in string theory. It is again tentative to speculate that there might exista tower of new N = 2 SCFTs which can be then mass deformed to this lagrangiantheory.There is of course the possibility of the existence of entire new series which don’tflow to any of the known theories. But perhaps the most concrete indication of theincompleteness of our current understanding of N = 2 theories at rank-2, is provided bythe fact that many (depending on the degree of optimism from three to nine) seeminglyconsistent rank-2 N = 3 theories [72] have yet to find a physical realization. Giventhe organizational structure described above, each new N = 3 might be the bottomcomponent of a series and therefore bring along many new theories. N = 2 moduli space Before moving to a detailed discussion of the each rank-2 theory individually, let usstart from a quick review of the general structure of the moduli space of vacua of N = 2 field theories; see, e.g. , [47, 73], focusing on the various branches and on theirstratification as either Special K¨ahler or Hyperkhaler varieties. The presence of supersymmetry allows in general for ground state configurations parame-triezed by a set of continuous variables which can in turn be interpreted as coordinatesof a space called the moduli space of vacua. The relation between the structure ofthe operator algebra and the moduli space of vacua of four dimensional SCFTs seemsspecial. In particular the problem of when a given operator can acquire a vev can beconjecturally formulated in terms of a set of precise conditions on the operator algebrain the four dimensional case. These heavily rely on the complex structure that thesemoduli spaces inherit by virtue of supersymmetry, and, relatedly, on the shorteningconditions satisfied by the BPS operators whose vevs parametrize the space.We define the moduli space of vacua of an N = 2 SCFT, which henceforth we willgenerically label as T , as the space of vevs of Lorentz scalar chiral primaries of BPSoperators. Depending on the su (2) R × u (1) r R-symmetry charges of these operators,their vevs are interpreted as complex coordinates of various branches of the moduli– 10 –pace. Specifically, labeling as R and r their su (2) R and u (1) r charges respectively,we have a Coulomb branch , which will be indicated as C , if R = 0, a Higgs branch ,indicated as H , if r = 0, or a mixed branch , indicated as M , if R r = 0. We also havea projection from the Mixed Branch (MB) into CB (HB) by simply setting to zero allthe vevs of operators with R = 0 ( r = 0).Supersymmetry induces different structures on the various branches. The CB isSpecial K¨ahler [63] and its complex dimension is called the rank of the SCFT whilethe HB is a hyperk¨ahler cone [74] and therefore a symplectic form on its smooth locus(more below). A MB intersects the CB along an in general singular special K¨ahler sub-variety. It can likewise intersect a HB, along an, again in general singular, hyperk¨ahlersubvariety. (Also, MBs can intersect each other in both special K¨ahler and hyperk¨ahlerdirections.)The operators whose vevs parametrize each branch, form a corresponding chiralring which are therefore called Coulomb , Higgs and mixed chiral rings . Even though weare not going to use it, it might be useful to connect with the nomenclature introducedin [75]. The BB multiplets with general R are the as Higgs branch operators andtheir OPEs contain the Higgs branch chiral ring. The Coulomb branch chiral ring isgenerated by those scalar LB chiral multiplet primaries ϕ [0 , a with R = 0. Finally themixed branch chiral ring is generated by scalar primaries of the LB chiral multipletwith R = 0. Explicit examples of chiral rings of theories containing mixed brancheswere worked out in [47].There is a special case of MB which deserves a separate discussion and will play aimportant role in our construction. This is the case when the projection of the MB intothe CB described above, gives back the entire CB. In this case the CB is a subvarietyof the MB and for this reason we will call such MB an Enhanced Coulomb branch andlabel its quaternionic dimensionality as h .Not all directions in the moduli space of vacua are equivalent. In fact there arespecial Higgsings which do not Higgs completely the theory but perhaps take the theoryto another one of close enough complexity. If the theory has a weakly coupled gaugedescription, these Higgsings correspond to those vacuum expectation values of themicroscopic fields for which the gauge group is minimally broken. Iterating this process,we see an interesting pattern of partial Higgsing, which can be characterised by thevarious subspaces of the CB or HB. These subspaces are naturally partially orderedby inclusion of their closures, and as such can be arranged into a Hasse diagram. AHasse diagram simply represents a finite partially ordered set, in the form of a drawingof its transitive reduction.The majority of N = 2 SCFTs do not have such a gauge description and a fun-damental understanding on how to reformulate the problem of Higgsing in the general– 11 –ase is still lacking. What helps is that the presence of charged massless states makesthe metric on the moduli space of vacua singular on the loci where the low-energy the-ory is not described by free-fields. The moduli space of vacua is therefore in general asingular space and studying the singular locus, which we will label S , gives insights intointeresting Higgsing directions. This singular structure induces on S a stratification .The type of this stratification, depends on the specific branch we focus on. The studyof the various branches of the moduli space of vacua as stratified spaces is extremelyhelpful to characterize theories and to extend the notion of minimal Higgsings to theo-ries with no weakly coupled lagrangian description. This will play a central role in theanalysis below.Many of the properties just outlined apply more generally to any N = 2 fieldtheory. But one of the key properties which distinguish apart the conformal case, isthat they carry a C ∗ action which arises as the combination of the spontaneously broken R + dilatation transformation and the (also spontaneously broken) u (1) r , on the CB,or the Cartan of the su (2) R , on the HB. We will commonly refer to this action as thescaling action. The geometry of the moduli space transforms homogenously under thistransformation and we will refer to this property as scale invariance of the modulispace. Note on color coding
Throughout the manuscript we will adopt the color codingproposed in [76] and use the color blue for HB related quantities
H → H and red forCB ones,
C → C . Since MBs can be seen as either extension of the CB or of the HB,depending on the context, we will use both color
M → MM . Let us start now with a summary of the structure of the CB of N = 2 SCFTs and inparticular a review of the notion of CB stratification [1] which is particularly effectivefor rank-2 theories. For more details on CB geometry see for example [1, 48, 51, 52],or more pedagogical reviews [77–80].The low-energy theory on a generic point of the CB C is almost as boring as itgets; a free N = 2 supersymmetric u (1) r gauge theory with no massless charged states. r is called the rank of the theory and coincides with the complex dimensionality of C ,dim C C = r ; we will indicate the global collective coordinates of C as u . As we saidabove, C is a singular space and its singular locus, which is a closed subset of C , willbe denoted as S . The CB singularities can be of two types. The metric singularitiesarise when charged states become massless while singularities of the complex structure The double coloring was introduced at an earlier stage of the draft. At the end, we made limitedto no use of it, nevertheless we left it to show off our coding abilities. – 12 –enote non-trivial relations among CB chiral ring generators [81–83]. In what followswe will always make the simplifying assumption that C is a non singular complex varietyand thus we are only interested in the singularities of the first kind. This in turn impliesthat C is topologically C r .The smooth part of the CB is C reg := C \ S and thus C reg is an open subsetof C . When the N = 2 theory is superconformal the symmetry group includes an R + × u (1) R (we are neglecting the su (2) R factor as it acts trivially on C ) which is ingeneral spontaneously broken and combines to give a C ∗ action on the CB. The entirestructure of C has to be compatible with this C ∗ action and in particular S and C reg haveto be closed under it and the CB coordinates u have definite scaling dimension, whichwill be label by the letter ∆ u i with the subscript indicating the specific coordinate werefer to. From here onwards we will only focus on the rank-2 case, in which case scaleinvariance is particularly constraining [67] as there will be only two type of singularloci which we will call knotted stratum and unknotted stratum. These will be definedshortly. Henceforth we will use the following convention u := ( u, v ), where u has thelowest scaling dimension of the two CB coordinates.General argument on the physical interpretation of the CB singularities [51, 52, 67],show that S has to be a complex co-dimension one algebraic subvariety of C . Thus S can be defined as the zero locus of a single polynomial in u and v . The fact that S has to be closed under the scaling action implies that the polyonomial has to behomogenous, which in turn implies that it can always be brought to the form: S := n ( u, v ) ∈ C (cid:12)(cid:12)(cid:12) P ( u, v ) = 0 o , P ( u, v ) = u · v · Y i ∈ I ( u p + λ i v q ) (3.1)where λ i ∈ C , and p and q are integers fixed by the relative scaling dimension of u and v and by requiring that gcd( p, q ) = 1. Each factor in (3.1), P i ( u, v ), identifies aconnected component of S (which is then the union of a bunch of disconnected pieces).The homogenous dimension of the P i ( u, v ) identifying each connected component playsan important role in what follows. We will label it as:∆ sing i := ∆ (cid:16) P i ( u, v ) (cid:17) . (3.2)We also adopt the following nomenclature:– 13 –) u = 0: u unknotted stratum or S u . ∆ sing u = ∆ u .2) v = 0: v unknotted stratum or S v . ∆ sing v = ∆ v .3) u p + v q = 0: knotted stratum or S u p + v q . ∆ sing u p + v q = p ∆ u = q ∆ v .The nomenclature knotted/unknotted is explained in detail in [67]It can be proven that the following facts apply [1, 69]: Fact 1.
A four dimensional rank 2 N = 2 SCFT which cannot be decomposedinto the product of two rank-1 theories has at least one knotted stratum.
Fact 2.
A two complex dimensional CB C , parametrized by CB coordinates ( u, v ) ,only admits a stratum corresponding to the C ∗ orbit S v ( S u ) if ∆ u ( ∆ v ) is ascaling dimension allowed at rank 1. The corresponding CB parameter is calledHiggsable CB parameter. Both statements can be relatively straightforwardly generalized to higher ranks.Now comes one of the key point. Each connected stratum in (3.1) supports aneither IR-free or superconformal rank-1 low energy theory which describes preciselythe charged states which are becoming massless there . Understanding these rank-1theories is a central piece of our analysis. So let’s formalize this point a bit more.Again, call the rank-2 theory at the superconformal vacuum T and call T u thelow-energy effective description of T at the generic point of the CB u . For example wehave: T u ≡ free N =2 u (1) , u ∈ C reg . (3.3)If instead u ∈ S , extra charged states become massless and the effective theory inthe IR is no longer u (1) but, rather, an either IR-free or superconfrmal rank-1 theory. T u is identified precisely with this theory describing the low-energy degrees of freedom Technically this statement is incorrect as the origin of each stratum is the origin of the modulispace where our rank-2 SCFT is supported. Thus a rank-1 theory is supported on a dense open subsetof the stratum which is called the component associated to the stratum [1]. In order to keep things asintuitive as possible, we will be sloppy and not make this distinction here. – 14 –hich plays a special role in what follows. We will call: T i ≡ T u , for ( u, v ) ∈ S u T v , for ( u, v ) ∈ S v T u p + v q , for ( u, v ) ∈ S u p + v q (3.4)and the quantities indexed by i ∈ I , ( c i , k i , h i ), label the central charges of theserank-1 theories T i and will be used to compute the central charges of the SCFT atthe superconformal vacuum T (see below). We also use u i to label the coordinateparametrizing the one complex dimensional CB of T i and define:∆ i := ∆( u i ) (3.5)which defines the last quantity entering the central charge formulae which we willshortly define. If this discussion is a bit too abstract, many many explicit examplescan be found in [1] or below in section 4.Before introducing the central charge formulae [2] let’s discuss another very con-straining property of CB geometries which was proven in the same paper and whichwill be extremely useful in our analysis below. The UV-IR simple flavor conditionstates that simple flavor factors of SCFTs of arbitrary rank, and thus in particular ofrank-2 SCFTs, act on the massless BPS spectrum which arise on singular complex co-dimension one strata of the CB. In other words any simple flavor factor f of an SCFT T , is realized (with possible rank-preserving enhancement) as the flavor symmetries of(at least one) rank-1 theory T i defined in (3.4). This observation allows to then studythe structure of the HB from the CB perspective and gain new insights on allowedHiggsing. This point will be clarified in explicit examples below but will be furtherleveraged and discussed in [84].The rank-1 theories T i carry more information than just the flavor symmetry ofthe SCFT. In fact, generalizing [85], it is possible to derive explicit formulae expressingthe central charges of an arbitrary N = 2 SCFT in terms of corresponding quantitiesof the rank-1 theories T i ’s [2]:24 a = 5 r + h + 6 r X ℓ =1 ∆ u ℓ − ! + X i ∈ I ∆ sing i b i , (3.6a)12 c = 2 r + h + X i ∈ I ∆ sing i b i , (3.6b) k f = X i ∈ I f ∆ sing i d i ∆ i (cid:0) k i − T ( h i ) (cid:1) + T ( h ) . (3.6c)– 15 –ere, r is the rank of the SCFT, h is the quaternionic dimension of the theory’s ECBand ∆ u ℓ is the scaling dimension of the theory’s ℓ -th component of the CB coordinatevector u . The sums indexed by i are performed over all the singular strata S i and the b i are defined to be: b i := 12 c i − − h i ∆ i (3.7)where ∆ sing i and ∆ i are defined in (3.2) and (3.5), all the remaining quantities indexedby i (except d i ) refer to corresponding quantities of T i defined in (3.4). Finally d i isthe embedding index of the flavor symmetry. We call these formulae central chargeformulae and their great service is that they allow to re-write the SCFT data of arank- r SCFT in terms of easily accessible geometric data (e.g. the scaling dimensionof their CB parameter or dimension of its ECB) and the SCFT data of rank-1 theorieswhich have been fully classified.A warm-up example, which will also enable us to derive a rule which will turn veryhandy in what follows, is to compute the level k f for a simple flavor factor realized on astratum identified by a polynomial of dimension ˜∆ by any of the entry in table 4 with h =0. Plugging the appropriate values in (3.6c) we derive the following: Doubling rule
Any entry in table 4 with h =0 realizes a flavor symmetry factorwith k f = 2 ˜∆ (3.8)where ˜∆ is the homogenous dimension of the polynomial identifying the singularstratum ( ˜∆ = ∆ u / ∆ v is the theory is supported on an unknotted stratum or˜∆ = p ∆ u = q ∆ v if is supported on a knotted stratum). The converse is nearlyalways true as well.The stratification of the CB singular locus is even richer than what is discussedabove; in fact, the strata themselves have to be scale invariant special K¨ahler varieties.But we won’t review this here and refer the interested reader to the original paper formore details [1].Let’s conclude with a remark which will be used in a few cases below. As discussedin [2], if a form of the Seiberg-Witten (SW) curve is known where the curve is writtenas a fibration of an hyperelliptic curve over C , that is in the form y = f ( u, v, x ) (3.9)where f ( u, v, x ) is at most of degree six in x with meromorphic coefficient in ( u, v ),– 16 –here is an easy way to gain many information about the singular locus of the CB bytaking the x discriminant of f ( u, v, x ). This is called the quantum discriminant of thegeometry [2] which allows in most cases to characterize the entire Hasse diagram, notjust S . The relation between the quantum discriminant and the CB stratification willbe further investigated in [66]. A wonderful recent discussion of the structure of HBs of SCFTs with eight supercharges(thus in particular N = 2 in ) was recently presented in [86] where many lagrangianexamples are also explicitly discussed. It is hard to do a better job and in fact wemost likely won’t. But to keep the paper as self-contained as possible we neverthelesspresent a brief discussion of the HB stratitification.Supeconformal invariance implies that the HB, is a hyper-K¨ahler cone [74] whichin particular implies that it is a symplectic singularity [87]. Like the CB, the HB is alsoa singular space. In analogy with what we did in the previous section, we call H reg theset of points where the symplectic structure is non-degenerate. This symplectic formnaturally induces a symplectic structure on the singular points which we will label as S H [88, 89, 92]. A powerful and general result is that symplectic singularities admit afinite stratification [93]: H ≡ n G i =0 H i (3.10)where F indicates the disjoint union, the H i ’s are irreducible and connected and arecalled symplectic leaves . The normalization of their closure are symplectic singularities.Importantly symplectic leaves are partially ordered by the operation of inclusion in theclosure of other symplectic leaves and they can be represented by a Hasse diagram. Aleaf H a covers H b and notated a ⋗ b iff a > b and there is no H c such that a > c > b , wewill also say that H a and H b are neighboring leaves. In addition each pair of symplecticleaves, ( H a , H b ), defines a subvariety, S ( a,b ) , which is transverse in the sense of [94] andwhose dimension is equal to the codimension of the smaller leaf into the closure of thelarger one. We will call S ( a,b ) the transverse slice of H a into H b if a < b . A transverseslice between two neighboring leaves is called an elementary slice and the edges of theHasse diagram are precisely labeled by those. Standard examples of Hasse diagramrepresentations of symplectic singularities stratification are given by the Kraft-Procesitransition between nilpotent orbits [90, 91]. A full list of possible elementary slices isstill unknown and it is unclear whether there is an answer for this question. Initiallythere was a hope that these could be restricted to minimal nilpotent orbits of classicaland exceptional Lie algebras (see table 5) and Du Val or Kleinian singularities [86].– 17 –ut as our understanding of HB of SCFTs with eight of more supercharges improves,new elementary slices are discovered [117]. In this work we will also conjecture theexistence of new mysterious ones.The structure of symplectic singularities is illuminated by its physical interpre-tation. The smallest and largest symplectic leaves are naturally identified with theorigin of the moduli space, H = { } , and the non-singular part of the HB, H n = H reg respectively. The other leaves H i , since singularities of moduli spaces corresponds toloci where extra interacting degrees of freedom become massless, precisely identify thesubvarieties of H where a SCFT T i is supported. For a lagrangian SCFT, the H i arespanned by pattern of partial Higgsing where subgroups of lower and lower rank, as i increases, are left unbroken. In the non-lagrangian set up, moving from one leaf to theother involves turning on vevs of some HB operator. Nevertheless the physical intuitionremains the same and therefore we will henceforth refer to the action of moving fromone symplectic leaf to neighboring ones as partial Higgsing . If the higgsing associatedto a given leaf is of generalized highest weight Higgsing (gHW) type, then there is anextremely useful relation which we will be used extensively below to reconstruct theHB structure of SCFTs [35, 84, 95]: GHW central charge formula
For a leaf associated to a higgsing of gHWtype, the difference between the c central charge of an SCFT T supported atthe origin of the (closure of the) leaf and that of SCFT, T gHW supported on ageneric point of the leaf12 c T − c T gHW = 2 (cid:18) k f − (cid:19) + δ dim H H − δ dim H H is the variation in HB dimension induced by the Higgsing or thequaternionic dimension of the leaf. In this section we will delve into the details of the results reported in table 1, 2,3, 6, 7 and 8. The first three tables collect the CFT data of the N = 2 rank-2SCFTs currently known to the author along with some information about the CBstratification, while the latter three specifically list information about the theories’HBs and information about how the flavor symmetry is realized along the varioushiggsings . We organize the known rank-2 theories in ten series of theories which are This is also useful to guess the basics of the generalized free field construction of their VOAs[96, 97] which will be sketched in appendix C. – 18 – inimal nilpotent orbit of Lie algebras f dim H f ♮ π R I f ♮ ֒ → f a N N a N − ⊕ C ( N − ) + ⊕ ( N − ) − (1 , b N N − a ⊕ b N − ( , − ) (1 , c N N c N − (1 , d N N − a ⊕ d N − ( , − ) (1 , e a e d e e g a f c ′ Table 5 : For the convenience of the reader, we summarize the properties of minimalnilpotent orbit for all classical and exceptional Lie algebras which appear copiously aselementary slices on the HB. This table is almost verbatim taken from [96].mutually connected by mass deformations and five isolated ones. A more detailedstudy of RG-flows of rank-2 theories will be presented elsewhere [3]. We will do ourbest in referencing the literature and clarify the results which have appeared elsewhereand apologize in advance for those who don’t get the credit they certainly deserve.For example, it is worth mentioning that the class- S construction [27, 28] of N = 2SCFTs gives often remarkable information about the HB structure of the theory, thusparts of the HB Hasse diagrams of the theories below could be derived this way and itwas certainly known before. We won’t follow this path and use instead primarily thetechniques in [1, 2]. At the cost of being slightly repetitive, we will write the descriptionof each theory in such a way that it can be read somewhat independently from the rest.The upshot of our analysis is that most rank-2 theories have been completelyunderstood and reveal general patterns; one, or multiple, knotted stratum supportingeither a [ I , ∅ ] or a S (1) ∅ , , and unknotted stratum supporting one of the rank-1 theoriesin table 4 (see below). But some entries don’t quite fit these patterns. Specifically,Th. 65 is the only theory with an IR free theory with a non-trivial HB, and a non-trivial ECB, supported over a knotted stratum. The ECB structure of this theory isalso particularly complicated and involved. Also theories in the su (6) and su (5) serieshave IR-free theories supported on CB strata with semi-simple flavor symmetries and aHB stratification presenting mysterious elementary transitions which are labeled witha question mark and a blue dashed line in the corresponding Hasse diagrams.– 19 – ank-2: Higgs branch data I f S u ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T u ([ f IR ] k IR , I f IR ֒ → f UV ) S v ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T v ([ f IR ] k IR , I f IR ֒ → f UV ) e − so (20) series1 . [ e ] × su (2) ∅ - - - - e ([ e ] × su (2) , (1 , , ) T (1) E , × H ([ e ] × su (2) × su (2) , (1 , , . so (20) ∅ - - - - d ( su (2) × so (16) , (1 , , ) T (1) E , ( so (16) , . [ e ] ∅ - - - - e ([ e ] , T (1) E , ([ e ] , . [ e ] × su (2) ∅ - - - - e ( so (12) × su (2) , (1 , , ) T (1) E , × H ( so (12) × su (2) × su (2) , (1 , , . su (2) × so (16) a ( so (16) , − T (1) E , ( so (16) , d ( su (2) × su (2) × so (12) , (1 , , , ) T (1) E , ( su (2) × so (12) , (1 , . su (10) ∅ - - - - a ( su (8) × u (1) , ⊕ T (1) E , ( su (8) , . [ e ] × su (2) ∅ - - - - e ( su (6) × su (2) , (1 , , ) T (1) E , × H ( su (6) × su (2) × su (2) , (1 , , . so (14) × u (1) ∅ - - - - d ( su (2) × so (10) , (1 , , ) T (1) E , ( so (10) × u (1) , . su (2) × su (8) a ( su (8) , − T (1) E , ( su (8) , a ( su (6) × su (2) × u (1) , (1 , ⊕ T (1) E , ( su (6) × su (2) , (1 , . so (12) ∅ - - - - d ( su (2) × so (8) , (1 , , ) T (1) D , ( so (8) , . so (8) × su (2) ∅ - - - - d ( su (2) × su (2) , (1 , , ) T (1) D , × H ( su (2) × su (2) , (1 , . u (6) ∅ - - - - a ( su (4) × u (1) , ⊕ T (1) D , ( su (4) × u (1) , . su (2) a ( su (2) , − T (1) D , ( su (2) , ∅ - - - -14 . su (3) × su (2) ∅ - - - - a ( u (1) × su (2) , ( − , + ⊕ − T (1) A , × H ( su (2) × su (2) , (1 , . su (5) ∅ - - - - a ( su (3) × u (1) , ⊕ T (1) A , ( su (3) , . su (2) × su (2) ∅ - - - - a ( su (2) , T (1) A , × H ( su (2) , su (2) , (1 , . su (2) ∅ - - - - a ( ∅ , -) − T (1) A , u (1)18 . su (2) ∅ - - - - ∅ - - - -19 . u (1) ∅ - - - - ∅ - - - -20 . su (2) ∅ - - - - a ( ∅ , -) − T (1) ∅ , ∅ . ∅ ∅ - - - - ∅ - - - - sp (12) − sp (8) − f series22 . sp (12) c ( sp (10) , S (1) E , ( sp (10) , ∅ - - - -23 . sp (4) × sp (8) c ( sp (4) × sp (6) , (1 , S (1) E , ( sp (4) × sp (6) , (1 , c ( su (2) × sp (8) , (1 , Th . su (2) × sp (8) , (1 , . [ f ] × su (2) f ( sp (6) × su (2) , (1 , ′ , ) S (1) E , ( sp (6) × su (2) , (1 , ∅ - - - -25 . su (2) × sp (8) c ( sp (6) × su (2) , (1 , S (1) D , ( sp (6) × su (2) , (1 , a ( sp (8) , T (1) E , ( sp (8) , . su (2) × sp (6) × u (1) c ( sp (4) × su (2) × u (1) , (1 , S (1) D , ( sp (4) × su (2) × u (1) , (1 , a ( sp (6) , − su (3) + 6 F ( sp (6) , . so (7) × su (2) b ( su (2) × su (2) × su (2) , (1 , , , , ) S (1) D , ( su (2) × su (2) , (1 , ∅ - - - -28 . [ f ] × u (1) ∅ - - - - f ( sp (6) × u (1) , ′ S (1) D , ( sp (6) × u (1) , . sp (6) × u (1) c ( sp (4) × u (1) , S (1) A , ( sp (4) , ∅ - - - -30 . su (3) × su (2) a ( u (1) × su (2) , ( − , + ⊕ − S (1) A , ( su (2) × u (1) , (1 , -)) ∅ - - - -31 . sp (4) c ( su (2) , S (1) ∅ , ( su (2) , ∅ - - - -32 . su (2) a ( su (2) , − S (1) ∅ , ( su (2) , ∅ - - - - Table 6 : This table summarizes the HB data for the theories in table 1. The second column lists the flavor symmetry of the SCFT,column three to seven, lists the information of the higgsing of the flavor symmetry realized on the CB unknotted stratum u = 0, whilethe last four columns present the same information for v = 0. For more details and an explanation of the connection with generalized freefields realization of the theory’s VOA, see appendix C. u (1) factors in this table will be mostly omitted. –20– ank-2: Higgs branch data II f S u ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T u ([ f IR ] k IR , I f IR ֒ → f UV ) S v ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T v ([ f IR ] k IR , I f IR ֒ → f UV ) su (6) series33 . su (6) × su (2) ∅ - - - - a ( su (6) ,
1) - Th. 22 ( su (6) , . su (4) × su (2) × u (1) ? ? ? ? ? a ( su (4) × u (1) ,
1) - Th. 25 ( su (2) × su (4) , (1 , . su (3) × su (3) × u (1) ∅ - - - - h , su (3) × su (2) ⊕ S (1) D , ( su (3) × su (2) , (2 , . su (3) × su (2) × u (1) ∅ - - - - a ( su (3) × u (1) ,
1) - e T E , ( su (3) × u (1) , . su (2) × su (2) × u (1) A ( su (2) × su (2) , (1 , T (1) D , ( su (2) × su (2) , (2 , h , su (2) × u (1) - S (1) A , ( su (2) × u (1) , . u (1) × u (1) ∅ - - - - a ( ∅ , -) - S (1) ∅ , u (1) sp (14) series39 . sp (14) ∅ - - - - c ( sp (12) , Th .
22 ( sp (12) , . su (2) × sp (10) a ( sp (10) , − S (1) E , ( sp (10) , c ( sp (8) × su (2) , (1 , Th .
25 ( sp (8) × su (2) , (1 , . sp (8) × u (1) ∅ - - - - c ( sp (6) × u (1) , e T E , ( sp (6) × u (1) , . sp (6) ∅ - - - - c ( sp (4) , su (2)- su (2) ( sp (4) , su (5) series43 . su (5) ∅ - - - - h , su (4) ⊕ S (1) D , ( su (4) , . su (3) × u (1) ? ? - S (1) A , ( su (3) , h , ( su (2) × u (1) , ⊕ S (1) A , ( su (2) × u (1) , . su (2) u (1) ∅ - - - - h , su (4) ⊕ S (1) D , ( su (4) , sp (12) series46 . sp (12) ∅ - - - - c ( sp (10) , S ( sp (10) , . sp (4) × so (4) ∅ - - - - c ( su (2) × so (4) , (1 , F + su (2) − su (2) + F ( su (2) × so (4) × u (1) , (1 , . sp (8) ∅ - - - - c ( sp (6) , su (3) + 6 F ( sp (6) , sp (8) − su (2) series49 . sp (8) × su (2) ∅ - - - - c ( sp (6) × su (2) , (1 , , ) T (2) E , ( sp (6) × su (2) × su (2) , (1 , , . sp (4) × su (2) × su (2) ∅ - - - - c ( su (2) × su (2) , (1 , , ) T (2) D , ( su (2) × su (2) × su (2) , (2 , , . su (2) × su (2) × u (1) ∅ - - - - a ( su (2) , T (2) A , ( su (2) × su (2) × u (1) , (1 , , -)52 . su (2) × su (2) a ( su (2) , S (1) A , u (1) × u (1) ∅ - - - -53 . su (2) a ( ∅ , -) − S (1) ∅ , ( u (1)) ∅ - - - -54 . su (2) ∅ - - - - a ( su (2) ,
1) - S (1) ∅ , × u (1) ( su (2) , g series55 . [ g ] × su (2) g ( su (2) × su (2) , (3 , , ) S (1) D , ( su (2) × su (2) × u (1) , (1 , ∅ - - - -56 . su (2) × su (2) a ( su (2) , S (1) A , ( su (2) × u (1) , ∅ - - - -57 . [ g ] g ( su (2) , S (1) A , ( su (2) , ∅ - - - -58 . su (2) ∅ - - - - a ( su (2) ,
1) - S (1) ∅ , × u (1) ( su (2) , su (3) series59 . su (3) × u (1) ∅ - - - - h , ( su (2) , ⊕ T (2) D , ( su (2) × su (2) , (3 , . u (1) ∅ - - - - h , u (1) - T (2) A , ( su (2) × su (2) , (1 , . u (1) ∅ - - - - A - - S (1) ∅ , × u (1) ( su (2) , su (2) series62 . su (2) × u (1) ∅ - - - h , u (1) - T (2) A , ( su (2) × su (2) , (1 , . u (1) ∅ - - - - A - - S (1) ∅ , × u (1) ( su (2) , Table 7 : This table summarizes the HB data for the theories in table 2. The second column lists the flavor symmetry of the SCFT,column three to seven, lists the information of the higgsing of the flavor symmetry realized on the CB unknotted stratum u = 0, while thelast four columns present the same information for v = 0. The entries in red are uncertain as discussed in the corresponding sections. Formore details and an explanation of the connection with generalized free fields realization of the theory’s VOA, see appendix C. u (1) factorsin this table will be mostly omitted. –21– ank-2: Higgs branch data III (Isolated) f S u ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T u ([ f IR ] k IR , I f IR ֒ → f UV ) S v ([ f ♮ ] k ♮ , I f ♮ ֒ → f ) π R T v ([ f IR ] k IR , I f IR ֒ → f UV )64 . su (2) × sp (8) a ( sp (8) , − g + 4 F ( sp (8) , c ( su (2) × sp (6) , (1 , Th .
26 ( su (2) × sp (6) , (1 , . sp (4) × su (2) a ( sp (4) , − S (1) D , ( sp (4) , ∅ - - - -66 . su (2) a - - T (2) ∅ , - ∅ - - - -67 . su (2) ∅ - - - - a ( su (2) ,
1) - S (1) ∅ , × u (1) ( su (2) , . ∅ ∅ - - - - ∅ - - - - Table 8 : This table summarizes the HB data for the isolated theories. The second columnlists the flavor symmetry of the SCFT, column three to seven, lists the information of thehiggsing of the flavor symmetry realized on the CB unknotted stratum u = 0, while the lastfour columns present the same information for v = 0. For more details and an explanation ofthe connection with generalized free fields realization of the theory’s VOA, see appendix C. u (1) factors in this table will be mostly omitted. A final note is that the discussion of the S and T theories, as well as the theorieswith extended supersymmetry, will be far less detailed than the rest. The former havebeen studied in depth recently and it would be redundant to present the same resultshere, only less eloquently. The latter are instead extremely constrained so there is alimited number of moving parts. We therefore made the choice of collecting the mainresults for theories in these two classes, as well as their CFT data, in Appendix A andB. e − so (20) series This is the largest series with two SCFTs at the top from which descend a total oftwenty one theories. This series includes all the AD theory known to the author. It isalso worth mentioning that no discretely gauged rank-1 theories appear on CB singularstrata. T (2) E , This is the rank-2 theory with the largest HB and central charges and sits at the topof the e − so (20) series. This theory can be engineered in type II B string theory asthe worldvolume theory of two D3 branes probing an E E Minahan-Nemeschansky (MN) theory[102, 103]. Recently this theory has been shown to belong to a larger class of N = 2theories dubbed T -theories which have been studied in detail and their properties aresummarized in appendix A. We collect the relevant CFT data as well the stratificationin table 44. – 22 – h. 2 This theory is the other top theory of the series and can be obtained, for example, inthe untwisted E class- S series [29]. This study allows to fill in most of the CFT datareported in table 1b which will be used below to fill in all the details of the full modulispace of vacua which we now discuss. Th. 2 [ I , ∅ ][ I ∗ , so (20)] (cid:2) u + v = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB e T (1) E , d Th. 2 (a) The Coulomb and Higgs stratification of Th. 2.
Th. 2(∆ u , ∆ v ) (6 , a c f k so (20) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 1 : Information about the Th. 2.The flavor symmetry of this theory is simple, expectedly so given it has only oneHiggsable CB parameter, v . Because of the UV-IR simple flavor condition, the so (20)factor must be realized on the CB as the flavor symmetry of a rank-1 theory supportedon a singular stratum, and since the only allowed unknotted stratum is v = 0, we canstart with such an option. An encouraging fact is that the level of the so (20) preciselydoubles ∆ v so we can use the doubling rule. A quick look at table 4 makes it obviousthat the right guess is T v ≡ [ I ∗ , so (20)] where this latter theory is nothing but an N = 2 su (2) gauge theory with ten fundamental hypers. Using (3.6c), it is immediateto check that this guess does reproduce the correct level k so (20) = 16. Since this N = 2gauge theory has no ECB, we also conclude that h =0. That the guess we just made iscorrect, can be also checked by reproducing the a and c central charges of this theory,shown in table 1b, plugging the b i for the [ I ∗ , so (20)] (which the reader can check tobe 12) in (3.6a)-(3.6b). With T u + v ≡ [ I , ∅ ], everything works beautifully.The Hasse diagram of the HB is linear and it involves only two transitions, the firstone being associated with the HB of the theory supported on the CB which has a d as its HB. To identify the (rank-1) theory supported on d we can use the propertythat the total HB dimension of the rank-2 theory is 46 from which, subtracting the17 dimension of the d , we obtain a prediction for the dimension of the HB of therank-1 theory: 19. This immediately singles out T d ≡ T (1) E , . There is another way– 23 –f going about determining the theory supported on various strata and which will beused copiously below. Using (3.11) we can directly predict the central charge of thetheory after higgsing. This formula only applies to higgings of gHW type. If the CBrealization of the flavor symmetry which is getting spontaneously broken is known, itis easy to assess whether or not a given higgsing has this property. For the case of the d transition this is indeed the case. (3.11) then gives 12 c T d = 62 which matches ourprevious guess. As it is explained in section C, in this case the matching of the momentmap along the higgsing works in a non-trivial and somewhat interesting way. D ( E ) This theory was discussed in [30] where it is also pointed out that it can be obtainedin the E class- S [42]. The nomenclature D ( E ) was introduced in [33, 104] wherethe geometric engineering realization of this theory in type II B string theory is alsodiscussed. Most of the CFT data in table 2b is taken from [30] and leveraged here tocomplete the study of the full moduli space. D ( E ) [ I , ∅ ] T (1) E , (cid:2) u + v = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB e T (1) E , e D ( E ) (a) The Coulomb and Higgs stratification of D ( E ) D ( E )(∆ u , ∆ v ) (4 , a c f k e (8) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 2 : Information about the D ( E )Since the theory has a single Higgsable CB parameter we expect a relatively simplestructure. The fact that the flavor symmetry is simple, and furthermore exceptional,makes our life quite easy. In fact the only natural guess for the realization of the e onthe CB is T v ≡ T (1) E , . This is further confirmed from the fact that the level of the e flavor factor is indeed double of ∆ v . This guess can be checked in two ways. Firstly,as we have done in the previous case, we can apply (3.6a)-(3.6b) to match the centralcharges of this theory. This works well and in turns allows to determine the theoryon the knotted stratum: T u + v ≡ [ I , ∅ ]. The second approach is insightful. This isone of the few lucky cases where the CB geometry is known in terms of a hyperelleptic– 24 –bration of a two dimensional base [64, 65]. Therefore we have a way to extract theCB stratification by studying the discriminant locus of the fibration as discussed at theend of section 3.2. This philosophy is described in more details, for example, in [2].From [65] the CB geometry of this theory can be written as: y = x + ( ux + v ) . (4.1)Taking the discriminant of the right hand side we obtain: D x ∼ v ( c u + c v ) (4.2)where c , are irrelevant numerical factor. This result implies that the CB geometryis only singular at v = 0 and u + v = 0, which matches nicely with our previousguess. But this is not all. In fact the order of the zero of the discriminant carries extrainformation which can be used to characterize the theory supported on two singularstrata. Performing the analysis we find that the v = 0 singularity (we are takingimplicitly u = 0) is a II ∗ singularity while the one at u + v = 0 is an I . Thisperfectly match with what we find using the UV-IR simple flavor condition and thecentral charge formulae.We are now ready to study the HB which we expect to be linear. Furthermore thistheory has h =0 and therefore to identify the theory supported on the e stratum sufficesto impose the constraint that the total dimension of the HB should add up to 46. Thissingles out T e ≡ T (1) E , . As we did before, we can confirm this guess recognizing thatthe spontaneous breaking of the e gives a higgsing of gHW type and apply (3.11) tofind 12 c T e = 38. T (2) E , This theory can be engineered in type II B string theory as the worldvolume theory oftwo D3 branes probing an E brane exceptional singularity [98–101]. It is also com-monly known as the rank-2 E MN theory [102, 103]. It is well-known that this theorycan be obtained by mass deforming the T (2) E , and the mass deformation is geometric inthe sense that it corresponds to “peel” away a D7 brane which makes the E branesingularity a E one. This theory has been shown to belong to a larger class of N = 2theories dubbed T -theories which have been studied in detail and their properties aresummarized in appendix A. We collect the relevant CFT data as well the stratificationin table 44. – 25 – h. 5 This theory appears in numerous class- S constructions, one example is the untwisted D [31]. This is where most of the CFT data in table 3b is taken from. Th. 5[ I , su (2)] [ I ∗ , so (16)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3)(cid:2) u = 0 (cid:3) H d HB Th. 5 T (1) E , T (1) E , e e d a (a) The Coulomb and Higgs stratification ofTh. 5. Th. 5(∆ u , ∆ v ) (4 , a c f k su (2) × so (16) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 3 : Information about the Th. 5 theory.This is the first case we encounter of a totally higgsable theory. This property isalso reflected in the fact that the flavor symmetry has two simple flavor factors. Giventhe value of the levels, we can use the doubling rule to determine how both factors arerealized on the CB. The so (16) is easily identified as the flavor symmetry of a N = 2 su (2) gauge theory with eight fundamental flavors, therefore leading to the identification T v ≡ [ I ∗ , so (16)]. The su (2) is more ambiguous as it might be the isometry of an ECBbut again the fact that the level is twice ∆ u convincingly suggests that T u ≡ [ I , su (2)].This conclusions are confirmed by the computing the a and c central charges of thetheory using (3.6a)-(3.6b) which also allows to fix the last ambiguity T u + v ≡ [ I , ∅ ]thus concluding our analysis of the CB.The CB perspective indicates that h =0 which also implies that the same constraintsapplies for the rank-1 theories supported on the a and d higgsings [84]. This infor-mation, along with the Ricci-flatness of the HB and the constraint that the total HBdimension should add up to 30, is sufficient to make the identification T a ≡ T (1) E , and T d ≡ T (1) E , . This guess can be checked by exploiting the fact that both higgsings areof gHW type and that (3.11) applied to these cases gives 12 c T a = 62 and 12 c T d = 38.Thus concluding our analysis. S This a rank-2 theories, belongs to an infinite series of N = 2 SCFTs discussed in [32].The generic S N theories have flavor symmetry su ( N + 2) N × su (3) × u (1) and precisely– 26 –or N = 5 there is a possibility of an enhancement to su (10) which in fact happens.The S-duality property of these theories for any N are discussed in the original paperalong with the computation of many CFT data which, for N = 5 are reported in table4b. Let’s start with the analysis of the full moduli space. S [ I , su (10)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB S T (1) E , e a (a) The Coulomb and Higgs stratification of Th. 14.. S (∆ u , ∆ v ) (4 , a c f k su (10) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 4 : Information about the S theory.The fact that the flavor symmetry is simple is consistent with the fact that this the-ory has a single Higgsable CB parameter, v . Furthermore, since the level of the flavorsymmetry is precisely doubled the scaling dimension of the Higgsable CB parameter,the doubling rule immediately suggests the identifcation T v = [ I , su (10)]. The rest ofthe CB stratification can be easily filled in by matching the central charge using (3.6b)and we therefore conclude that T u + v = [ I , ∅ ].The analysis of the HB is also straightforward; the a strata is mandated by theCB analysis which also shows that this higgsing is of gHW type. Since h =0, the theorysupported there is a rank-1 theory which could be immediately identified from matchingthe unbroken flavor symmetry along the Higgsing and imposing that the total HB ofthe theory matches what is found in the original class- S construction. This leads usto the conclusion that T a = T (1) E , . It is a useful exercise to check that the result from(3.11) are consistent with this identification. T (2) E , This theory can be engineered in type II B string theory as the worldvolume theoryof two D3 branes probing an E brane exceptional singularity [98–101]. It is alsocommonly known as the rank-2 E MN theory [102, 103]. Again, it is well-known thatthis theory can be obtained by mass deforming the T (2) E , and the mass deformation isagain geometric corresponding to making the E brane singularity a E one. This– 27 –heory has been shown to belong to a larger class of N = 2 theories dubbed T -theorieswhich have been studied in detail and their properties are summarized in appendix A.We collect the relevant CFT data as well the stratification in table 44. R , This theory was first introduced in the context of Z twisted E class- S [39]. Theclass- S construction gives access to most of the CFT data reported in figure 5b whichwe will leverage here to fully solve the moduli space structure of the theory. R , [ I , ∅ ] [ I ∗ , so (14)] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB R , T (1) E , e d (a) The Coulomb and Higgs stratification of R , . R , (∆ u , ∆ v ) (3 , a c f k so (14) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 5 : Information about the R , theory.Since the theory has a single simple flavor symmetry factor, we expect an easyHB structure. This is a reflection that the theory is not totally higgsable and only v is a Higgsable CB parameter. The CB realization of the so (14) can be easily andreadily identified as T v ≡ [ I ∗ , so (14)]. This identification predicts a d transition onthe HB side of things but imposes no constraints on the subsequent transitions whilerestricting the ECB of the theory supported over it to be zero. As a further check thatthe CB identification which we just made is correct, we can check that, plugging the b i corresponding to [ I ∗ , so (14)] supported over v = 0 and a [ I , ∅ ] on the knotted stratumin (3.6a)-(3.6b), we perfectly reproduce the a and c central charges of the theory.Completing the HB analysis is straightforward. From the CB analysis, we couldnotice that the spontaneous breaking of the so (14) factor is of gHW type and use (3.11)to compute the central charge of the theory supported over d . But a possibly evensimpler way to complete the study of the HB stratification is to notice that the theory T d has to have no ECB (coming from the CB analysis) and an 11 dimensional HB.Using the Ricci-flatness of the HB we are left with only one possibility: T d ≡ T (1) E , .– 28 –e leave it for the reader to check that the prediction from (3.11) indeed perfectlymatch with the value of the central charge of the rank-1 MN E theory. R , This theory was first introduced in the context of twisted Z A class- S theories [36].In the original paper interesting S-dualities of this theory are discussed as well as mostof the CFT data reported in table 6b computed.The analysis will be similar to the previous cases. The theory is totally higgsableand the two simple factors of the flavor symmetry show that the HB contains twodisconnected transitions. It is easy to argue that the su (8) is realized as the flavorsymmetry of an [ I , su (8)] supported on an unknotted v = 0 stratum leading to theidentification T v ≡ [ I , su (8)]. We therefore expect that one of the two HB transition isan a . The su (2) instead could potentially give rise to an ECB but a more careful lookat the level of this symmetry, which precisely doubles ∆ u , suggests that it should berealized as the flavor symmetry of a [ I , su (2)] on a v = 0 stratum. Thus T v ≡ [ I , su (2)]and h =0. As usual the calculus of the c central charge via (3.6b), inputting the known b i s of the already identified CB components, allow us to both check that the thoseare indeed correct, and determine the theory supported on the unknotted strata. Thiscomplets the analysis of the stratification of the CB. R , [ I , su (2)] [ I , su (8)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3)(cid:2) u = 0 (cid:3) H d HB R , T (1) E , T (1) E , e e a a (a) The Coulomb and Higgs stratification of R , . R , (∆ u , ∆ v ) (3 , a c f k su (2) × su (8) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 6 : Information about the R , theory.Let’s move now to the analysis of the HB. The absence of an ECB and the factthat both Higgsing are of gHW type, make things fairly easy to work out. Indeed using(3.11) we can right away identify the theories supported on the a and a as the rank-1 MN E ( T (1) E , ) and E ( T (1) E , ) theory (notice that the difference in HB dimensionsamong these two theories precisely makes up for the difference in dimension of the– 29 –trata over which they are supported to give rise to a total HB of dimension 12). Thisis enough to reproduce the HB stratification in figure 6b. sp (4) + 6 F Let’s now discuss the first lagrangian case. We will be somewhat brief since mostof this, is standard material. The huge advantage of the lagrangian case is that todetermine the CB singular structure we can directly study the masses induced by thevev of the adjoint vector multiplet scalar for the various hypers present in the theory.The extra charged states which can become massless are either W-bosons (where thereis an unbroken su (2) gauge factors) or charged matter, i.e. specific components of thehypermultiplets which become massless.In the sp (4) case there are two inequivalent directions, up to Weyl tranformation,where an su (2) is left unbroken (corresponding to the long and short simple roots) andwhich therefore give surely rise to singularities. In one case each hypermultiplet in the contributes a massless flavor while in the other it contributes no massless matter.It therefore implies that along these two interesting directions we find in one case an N = 2 su (2) theory with N f = 6 and in the other a pure su (2) theory. The latter theoryis asymptotically free and the result of strong coupling is to “split” the singularity intotwo knotted strata each supporting a [ I , ∅ ]. The other low-energy theory is insteadIR-free and contributes a knotted stratum supporting a [ I ∗ , so (12)] reproducing the CBstratification in figure 7a. This theory has no ECB.This result can be confirmed both by reproducing the central charges of this theoryfrom (3.6a)-(3.6b) and by studying the discriminant locus of the Seiberg-Witten curvewhich has been worked out explicitly [105]. We leave both checks as an exercise for thereader.The analysis of the HB can also be performed in a straightforward manner byanalyzing the possible vevs of the hypermultiplets. Rather than performing the grouptheory analysis, it is quicker to impose our “non-lagrangian” constraints. In fact the d Higgsing should support a rank-1 theory with a five dimensional HB and no ECB. Thisleads to the only consistent guess T d ≡ T (1) D , which can also be checked by carefullyturning a minimal vev for the mesons of this theory. sp (4) + 4 F + AS or T (2) D , This is a lagrangian theory which belong to an infinite series of sp (2 n )+4 F +1 AS gaugetheories with which can be engineered in type II B string theory as the worldvolumetheory of two D3 branes probing an exceptional D brane singularity [98–101]. For n = 2 sp (4) ∼ = so (5) and thus we label the two indices traceless antisymmetric of sp (4) simply as V. Because of the string theoretic realization, it is well-known that– 30 – p (4) + 6 F [ I , ∅ ] [ I ∗ , so (12)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB d T (1) D , d sp (4) + 6 F (a) The Hasse diagram for the CB and the HBof the sp (4) gauge theory with six hypermul-tiplets in the . sp (4) + 6 F (∆ u , ∆ v ) (2,4)24 a c f k so (12) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 7 : Information about the sp (4) N = 2 theory with six hypermultiplet in thefundamental.this theory can be obtained by mass deforming the T (2) E , with the mass deformationgeometric realized as moving a D7 away from the E exceptional brane to make it a D one. This theory has been shown to belong to a larger class of N = 2 theories dubbed T -theories which have been studied in detail and their properties are summarized inappendix A. We collect the relevant CFT data as well the stratification in table 44. su (3) + 6 F The CB stratification of this theory is discussed explicitly, for example, in [1, 67] whilethe HB in [86]. Let us simply discuss a few interesting points. Firstly the doubleknotted stratum, is the “mark” of the dyon-monopole singularity of the pure N = 2 su (2) CB solution [67]. Secondly the a and c central charges in table 8a can be readilymatched using (3.6a)-(3.6b). Thirdly the a transition, is immediately associated withthe HB of T v and it is a useful exercise to check that (3.11) precisely reproduces thecentral charge of the N = 2 su (2) theory with N f = 4. Our more abstract way of goingabout characterizing the full moduli space structure, perfectly reproduces the resultexpected from the lagrangian analysis and does it perhaps even more straightforwardlythan the standard way of working with gauge variant fields and equations of motions.2 F + su (2) − su (2) + 2 F Let’s start the analysis of this theory from the CB perspective. Here there are threehiggsing directions which we need to consider. We can turn on a vev for the scalarcomponent of the vector multiplet corresponding to each separate su (2) or the twocombined. In the first case, all the components of the ( , ) are massive while the other– 31 – u (3) + 6 F [ I , ∅ ] [ I , su (6)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB su (3) + 6 F T (1) D , d a (a) The Hasse diagram for the CB and HB ofthe su (3) gauge theory with 6 s. su (3) + 6 F (∆ u , ∆ v ) (2,3)24 a c f k su (6) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 8 : Information about the su (3) N = 2 theory with 6 hypermultiplets in the .hypermultiplets contribute two flavors for each separate su (2). Therefore, each one ofthese Higgsings gives rise to a su (2) with N f = 2 which is reflected each by two stratasupporting a [ I , su (2)] (which realize the so (4) symmetry of the gauge theory on theCB). The other higgsing instead, breaks su (2) × su (2) → u (1) × u (1), makes massive allthe components of the hypers in the ( , ) ⊕ ( , ), but the hyper in the bifundamentalcontributes two hypers with charge one under an appropriate linear combination ofthe u (1) which survives. This contributes a singular locus supporting yet another[ I , su (2)]. This analysis reproduces the intricate CB stratification depicted in figure9a and immediately implies that h =0. F + su (2) − su (2) + 2 F [ I , su (2)] [ I , su (2)][ I , su (2)][ I , su (2)][ I , su (2)] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB d T (1) D , a F + su (2) − su (2) + 2 F (a) The Hasse diagram for the CB and the HB ofthe 2 F + su (2) − su (2) + 2 F . F + su (2) − su (2)+2 F (∆ u , ∆ v ) (2,2)24 a c f k su (2) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 9 : Information about the su (2) N = 2 theory with two hypermultiplets in the( , ) ⊕ ( , ) and one in the ( , ) – 32 –o analyze the HB we only consider one of the five possible Higgsing, since thereis a symmetry among all of them and will give rise to the same structure (in theHasse diagram in the figure we only depict one of the five transitions). Rather than ex-plicitly doing the calculation, let’s use shortcut to identify the rest of the HB Hasse diagram.From the CB analysis we concluded that the theory supported on a should be a rank-1theory with no ECB. Reproducing the total HB dimension of the theory readily impliesthat the HB of the rank-1 theory should be five quaternionic dimension, which in turnimmediately singles out the N = 2 su (2) with N f = 4 completing our analysis. T (2) A , This is an AD theory which can be engineered in type II B string theory as the world-volume theory of two D3 branes probing an A brane singularity [98–101]. It is alsocommonly known as the rank-2 H theory. It is well-known that this theory can beobtained by mass deforming the T (2) D , and the mass deformation is geometric and cor-responds to moving away two D7 branes to make the D brane singularity a A one.This theory has been shown to belong to a larger class of N = 2 theories dubbed T -theories which have been studied in detail and their properties are summarized inappendix A. We collect the relevant CFT data as well the stratification in table 44. D ( su (5)) This theory appears on the CB of su (3) theory with N f = 5 by appropriately tuningtheir mass parameters [106] and is the rank-2 entry of an infinite series D ( su (2 N + 1)),see below. A useful expression for its SW curve was derived in [65] but the derivationof its central charges were discussed in [33, 104] in the context of geometric engineeringwere the name was also coined. Recently [107], this theory was shown to also arise intwisted A class- S (while the twisted A N engineers the D ( su (2 N + 1)).Since the hyperelliptic form of the SW curve is known in this case, we use thisroute to characterize the moduli space structure: y = x + ( ux + v ) (4.3)taking the x discriminant of the RHS is it is straightforward to identify its CB strati-fication which is depicted in figure 10a while the CFT data is reported in 10b.As said, this theory is part of an infinite series dubbed D (cid:2) su (2 N + 1) (cid:3) . This seriesis characterized by the following properties: • It appears on the CB of a su ( N ) N = 2 gauge theory with N f = 2 N + 1 flavors. • The D (cid:2) su (2 N +1) (cid:3) has rank N and CB scaling dimension ∆ i = i +12 , i = 1 , ..., N .– 33 – (cid:2) su (5) (cid:3) [ I , ∅ ] [ I , su (5)] (cid:2) v = 0 (cid:3)(cid:2) u + v = 0 (cid:3) H d HB D (cid:2) su (5) (cid:3) T (1) A , a a (a) The Hasse diagram for the CB and HB ofthe D (cid:2) su (5) (cid:3) theory. D (cid:2) su (5) (cid:3) (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (5) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 10 : Information about the D (cid:2) su (5) (cid:3) AD theory. • The c and a central charges are given by 24 a = 7 N ( N + 1) and 12 c = 4 N ( N + 1). • The flavor symmetry is a su (2 N + 1) N +1 . • The associated vertex operator algebra is conjectured to be the affine currentalgebra [108]: V h D (cid:2) su (2 N + 1) (cid:3)i = \ su (2 N + 1) − N +12 (4.4) T (2) A , This is an AD theory which can be engineered in type II B string theory as the world-volume theory of two D3 branes probing an A brane singularity [98–101]. It is alsocommonly known as the rank-2 H theory. This theory can be obtained by mass de-forming the T (2) A , by moving away a single D7 brane to make the A brane singularitya A one. The mass deformation is thus geometric. This theory has been shown tobelong to a larger class of N = 2 theories dubbed T -theories which have been studiedin detail and their properties are summarized in appendix A. We collect the relevantCFT data as well the stratification in table 44. ( A , D ) This theory belongs to an infinite series of rank- n AD theory: ( A , D n +2 ). We describetheir somewhat involved HB structure below. This theory was first found on the CBof a pure so (12) N = 2 theory (as we will further discuss below, in general an ( A , G )theory, where G is a simply laced Lie algebra, arise on special loci of pure G N = 2gauge theory) and it can be engineered both in class- S and type II B on a Calabi-Yauthreefold hypersurface singularity. – 34 –he stratification of the CB singular locus can again be read off straightforwardlyfrom the expression of the SW curve reported in [64]: SW curve : y = x + x ( ux + v ) ⇒ D Λ x = v (cid:0) u + 3125 v (cid:1) , (4.5)and then we conclude that there is a (4,5) knotted stratum ( T u + v ≡ [ I , ∅ ]) as wellas an unknotted stratum at v = 0. (4.5) leads to the identification T v ≡ [ I , su (2)].The analysis which leads to the reproduction of the central charges as well as the fullcharacterization of the Hasse diagram in figure 11a, is largely similar to the one abovetherefore we won’t discuss it and instead focus on discussing the HB of this theory. ( A , D )[ I , ∅ ] [ I , su (2)] (cid:2) v = 0 (cid:3)(cid:2) u + v = 0 (cid:3) H d HB ( A , D ) T (1) A , a a (a) The Hasse diagram for the CB and HB ofthe ( A , D ) AD theory. ( A , D )(∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 11 : Information about the ( A , D ) AD theory.The HB of the ( A , D n +2 ) can be elegantly written as the intersection of symplecticvarieties [109] O [ n +1 , ∩ S [ n, , , (4.6)where O [ n +1 , is the subregular nilpotent orbit of sl ( n + 2) and S [ n, , is the Slodowyslice of the nilpotent orbit associated to the [ n, ,
1] partition. Adapting the notationthat we have used to label the stratification on the CB to the stratification of nilpotentorbits: S [ n, , ∼ = T ( O [ n, , , O [ n +2] ), that is, S [ n, , can be identified as the transverseslice of the nilpotent orbit associated to the [ n, ,
1] into the principal nilpotent orbitof sl ( n + 2). As n increases this space can be quite complicated but for n = 2 isrelatively simple. It is two quaternionic dimensional, and it has only three strata withelementary slices C / Z (see figure 11a).From the CB analysis, it is obvious that the higgsing is of gHW type and thereforewe can use (3.11) to compute the central charge of the theory supported on the secondHB stratum finding 12 c T a = 6 which immediately singles out the T (1) A , as depicted in– 35 –gure 11a. And indeed the stratification that we find is compatible with the fact thelower leaf of the HB extends into the MB of this theory and the low-energy theoryliving on the second stratum of the HB is the rank-1 AD theory ( A , D ) which is thesame as ( A , A ). T (2) ∅ , This AD theory is the first of three bottom theories of the e − so (20) series. It canbe engineered in type II B string theory as the worldvolume theory of two D3 branesprobing two D7 branes with mutually non-local charges [98–101]. It is also commonlyknown as the rank-2 H theory. This theory can be obtained by mass deforming the T (2) A , by moving away a single D7 brane from the A brane singularity. The massdeformation is thus geometric. This theory has been shown to belong to a larger classof N = 2 theories dubbed T -theories which have been studied in detail and theirproperties are summarized in appendix A. We collect the relevant CFT data as wellthe stratification in table 44. ( A , A ) This theory, appears on a special locus of the pure N = 2 su (6) gauge theory. It can beengineered in class- S and type II B string theory like other AD theory. It also belongsto an infinite series of rank- n AD theory, called ( A , A n +1 ), whose HB can be writtensomewhat homogeneously as C / Z n +1 . The stratification of the CB singular locus canbe again read off straightforwardly from the expression of the SW curve reported in[64]: SW curve : y = x + ux + v ⇒ D Λ x = (cid:0) u − v (cid:1) , (4.7)and again it implies that T u + v ≡ [ I , ∅ ]. Performing an analysis analogous to theone above, the rest of the Hasse diagram, shown in figure 12a, can be completely char-acterized and the central ( a, c ) correctly reproduced via the central charge formulae.The theory supported on the CB singular locus has no HB but the rank-2 theory atthe origin has a non-trivial HB. This in turn implies that the low-energy theory on thegeneric point of the MB, is trivial. ( A , D ) This theory appears on a special locus of a N = 2 so (10) pure gauge theory [106, 110]and it belongs instead to an infinite series of AD theory with a C / Z HB: ( A , D n +1 ).This is the second bottom theory of the e − so (20) series. These can be geometricallyengineered in type II B on a Calabi-Yau three-fold hypersurface singularity specified bythe two simply laced Lie algebras A and D n +1 [111]. It can also be obtained in A N – 36 – A , A )[ I , ∅ ] (cid:2) u + v = 0 (cid:3) ( A , A ) H a (a) The Hasse diagram for the CB and HB ofthe ( A , A ) AD theory. ( A , A )(∆ u , ∆ v ) (cid:0) , (cid:1) a c f k u (1) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 12 : Information about the ( A , A ) AD theory.class- S with a full puncture and a I , irregular puncture [112] (in general ( A , D N )theories can be obtained in A N class- S with a full puncture and an irregular I ,N − ir-regular puncture). Although the class- S description gives a formulation for the theory’sCB geometry, we instead read off the stratification of the CB singular locus, followingthe remarks at the end of section 3.2, using a different form for the SW curve reportedin [65], SW curve : y = x + x ( ux + v ) ⇒ D Λ x = v (cid:0) u − v (cid:1) . (4.8)This implies that there is a (3,4) knotted stratum as well as an unknotted stratum at v = 0. ( A , D )[ I , ∅ ] [ I , su (2)] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) ( A , D ) T (1) ∅ , a (a) The Hasse diagram for the Coulomb andHiggs branch of the ( A , D ) AD theory. ( A , D )(∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 13 : Information about the ( A , D ) AD theory.– 37 –s we did in other cases, we can use the extra information provided by the orderof the zeros of the discriminant to further characterize the CB. As before we infer that T u + v ≡ [ I , ∅ ]. There is now an ambiguity in identifying T v as both an [ I , su (2)]and a T (1) ∅ , would be compatible with the discriminant (4.8). This ambiguity can beresolved using the UV-IR simple flavor condition which implies that the su (2) flavorsymmetry has to be realized as a flavor symmetry of a rank-1 theory on the CB and wetherefore conclude that T v ≡ [ I , su (2)]. This perfectly reproduces the level k su (2) = (via (3.6c)) and reproduces the CB stratification shown in figure 13a.We observe that this CB stratification has two implications which are also con-sistent with known facts about this theory [96]. First, the entire HB of the ( A , D )extends over its CB and it is therefore a MB, and secondly the low-energy theory onthe generic point of the MB, of the theory is the rank-1 ( A , A ). ( A , A ) Finally the last bottom theory of the e − so (20) series. This theory is the rank-2 entryof an infinite series of rank- n N = 2 SCFTs with trivial HB, which is often labeledas ( A , A n ). It appears on a special locus on the CB of a pure su (5) N = 2 gaugetheory [106, 110]. It can also be geometrically engineered in type II B string theory ona Calabi-Yau 3-fold hypersurface singularity [111]. This is where the name ( A , A n )comes from as these two labels uniquely identify the polynomial cutting the 3-foldsingularity. Finally, this theory can also be obtained in class- S compactifying an 6d A (2,0) theory on a sphere with a single, type I, irregular puncture [112]. Its VOA isconjectured to be the (2,5) Virasoro minimal model [109, 113].The stratification of the CB can be read off directly from the discriminant of thecurve presented in [65], as remarked at the end of section 3.2. For the ( A , A ) we have: SW curve : y = x + ux + v ⇒ D Λ x = 256 u + 3125 v , (4.9)and therefore we readily conclude that T u + v ≡ [ I , ∅ ]. It is an instructive exercise tocheck that with this information we can reproduce the a and c central charges in table14b using our central charge formulae.A curious phenomenon is that the knotted stratum is an irregular geometry ( S u + v ≡ I ∗ (3)0 ) as it can be read off by analyzing more closely the special K¨ahler structure of theknotted stratum. The uniformizing parameter [1] for this hypersurface is t ( A ,A ) ∼ ( u ) ∼ ( v ) ⇒ ∆ t = 27 . (4.10)For more details on irregular geometries and how to determine the uniformizing pa-– 38 – [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (a) The Hasse diagram for the CB and the,trivial, HB of the ( A , A ) AD theory. ( A , A )(∆ u , ∆ v ) (cid:0) , (cid:1) a c f k ∅ d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 14 : Information about the ( A , A ) AD theory.rameter for a one-complex dimensional stratum see [1, 81]. ◦ sp (12) − sp (8) − f series This is the second largest series with again multiple top theories (three) to whichconnect a total of eleven N = 2 SCFTs. This series does not contain any AD theory. Th. 22
This theory is one of the theories sitting at the top of the sp (12) − sp (8) − f series andcan be obtained, for example, in the Z twisted D class- S [34] where most of the CFTdata reported below is computed.This theory is totally higgsable and therefore we would expect an at least semi-simple flavor symmetry. Surprisingly f is instead simple and its HB linear and it istherefore reasonable to guess that one of the two allowed unknotted stratum associatedwith each Higgsable CB parameter, simply supports no low-energy rank-1 theory. Bylooking at the level of the sp (12) flavor symmetry, the natural guess is that T u ≡ [ I , su (12)] Z , where the subscript refer to a Z discrete gauging, while T v ≡ I , whichis another way of saying that there are no charged states becoming massless at v = 0.The fact that discretely gauged theories can appear on singular strata of the CB, wasalready noticed in [1] but a closer analysis reveals that when this happens the CBanalysis is particularly constrained. As it will be discussed in more detail in [84] youcan for example derive, in this case, not only that h =0 but also that the theory T c supported on the c transition associated to the HB of this discretely gauged theory,should support a rank-1 theory with h =5. Before confirming this fact directly with– 39 – h. 22[ I , ∅ ] [ I , su (12)] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) H d HB e T (1) E , c S (1) E , c Th. 22 (a) The Coulomb and Higgs stratification of Th. 22.
Th. 22(∆ u , ∆ v ) (4 , a c f k sp (12) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 15 : Information about the Th. 22.a HB analysis, let’s confirm that the CB analysis we just performed is correct. Forthat we can as usual apply the central charge formulae (3.6a)-(3.6b) which works outnicely and furthermore suggests that there are two knotted strata each supportinga [ I , ∅ ]. Unfortunately there is another possible solution which is compatible witheverything that we know, i.e. a single knotted stratum with T u + v = [ II, ∅ ]. In figure15a, we pick the former but for not better reason than symmetries with the otherunknotted stratum, also of I n type. It is important to clarify, that these two choicesare indistinguishable as far as the analysis we are performing here goes but there is ofcourse an easy way to distinguish them as the two rank-1 theories are associated withdifferent monodromies. In fact one is parabolic (the I s) and the other is elliptic. Thuswe expect that a more careful analysis of the global structure of the CB could verylikely distinguish the two cases.With all this information at hand, the HB is straightforward. In fact the predictionof the higgsing from the CB are enough to uniquely identify T c ≡ S (1) E , from which wecan determine the rest of the Higgsings and thus the full HB stratification. We leaveit up to the reader to check that this identification is indeed consistent with (3.11) butit reassuring that the entire structure holds up together so well. Th. 23
This theory is also at top of the series and was initially obtained in the untwisted D class- S series [31] where most of the CFT data reported in figure 16b was initially com-puted. In the original paper, various dual description of this theory are also discussed.This theory is totally higgsable, we therefore expect and interesting moduli space– 40 – h. 23 [ I , ∅ ][ I , su (8)] Z [ I ∗ , sp (4)] (cid:2) u + v = 0 (cid:3)(cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB e e T (1) E , T (1) E , c a a R , S (1) E , c c Th. 23 (a) The Coulomb and Higgs stratification of Th. 23
Th. 23(∆ u , ∆ v ) (4 , a c f k sp (4) × sp (8) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 16 : Information about the Th. 23.structure. First notice that we can immediately identify how one of the two sp factor,the sp (8), is realized on the CB. In fact we notice that it has a level which doubles∆ u and using the doubling rule directly leads to the identification T u ≡ [ I , su (8)] Z .On the other hand the relation between k f sp (4) and ∆ v suggests that the sp (4) factoris instead realized as isometry of an ECB leading to T v ≡ [ I ∗ , sp (4)]. As usual wecan cross-check this identification by plugging the corresponding b i into (3.6a)-(3.6b)to correctly reproduce the values of the a and c central charges in table 16b. This alsofixes the theory supported on the knotted strata thus completing our analysis of theCB stratification.Let’s start our analysis of the HB from the rank decreasing transition. This is a c and the CB analysis suggests that this stratum should support a theory with a fivedimensional ECB [84]. The theory supported on this stratum has to be a rank-1 theory,thus the ECB information just derived uniquely identify T c ≡ S (1) E , . This identificationcan be of course cross-checked by applying (3.11) and reproducing 12 c T c = 49. Thehiggsing corresponding to the ECB is also of gHW type and therefore (3.11) can beused to derive 12 c T c = 54. Unfortunately this is information is not enough to identifythe rank-2 theory supported on the stratum. As it is often the case, the degeneracy canbe lifted by imposing the simple condition that the total HB should be 20 dimensional.This leads to the unique identification T c ≡ R , . The subsequent Higgsings canbe reproduced by studying the HB of T c and T c which give rise to the intricateHasse diagram in figure 16a. – 41 – (2) E , This theory is one of the top theories of the sp (12) − sp (8) − f series. It can be obtainedin class- S , for example, in the untwisted D [31]. It was also recently shown to beobtainable by higgsing the N = 2 S -fold S (1) E , or by a twisted compactification of a (1,0) theory [35] or as a wordvolume theory of two D3 branes probing an exceptional E brane singularity in the presence of an S -fold without flux [44]. Theories probingan exceptional brane plus an S -fold without fluxes are more generally dubbed N = 2 T -theories and their properties are summarized in appendix A. The CFT properties aswell as the stratification of this particular theory can be found in table 45. Th. 25
This theory can be obtained by mass deforming both theory Th. 22 and Th. 23 [3]. Itwas introduced for the first time in the context of twisted Z A class- S theories [36].In the original paper interesting S-dualities of this theory are discussed as well as mostof the CFT data reported in table 17b computed. This information will be leveragedhere to fully understand the moduli space structure of this theory.This theory is again characterized by having two simple flavor symmetry factorswhich reflect the fact that the theory is totally higgsable. This in turn implies thatthe HB is not linear. Looking at the levels we can use the doubling rule to identifyboth theories realizing the two simple flavor symmetry factors. In this particular casethis would imply that the su (2) is realized by T u ≡ [ I , su (2)] while the sp (8) as a T v ≡ [ I , su (8)] Z . We can immediately support this guess by checking that with an[ I , ∅ ] supported on the unknotted strata, the c central charge of the theory is perfectlyreproduced by (3.6b).The CB analysis implies that: a ) all the Higgsing of this theory decrease the rank,therefore h =0 b ) both Higgsing are of gHW type c ) we expect that the theory supportedon the c , which arises because of the [ I , su (8)] Z , to have a three dimensional ECB[84]. This latter property is immediately verified by using (3.11) which singles out the S (1) D , theory as the one supported on c . The other Higgsing instead leads to the rank-1MN E ( T (1) E , ) as it can be verified easily using again (3.11). Th. 26
This theory was first introduced in the context of the Z twisted A [36]. The originalreference also discusses interesting S-duality of this theory.The presence of two simple flavor factors signals that the HB should have twotransitions stemming out of the superconformal vacuum. This is expected as the theoryis totally higgsable. Let’s study them in turn. First the sp (6) factor. Since the level is– 42 – h. 25[ I , su (2)] [ I , su (8)] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB Th. 25 T (1) D , S (1) D , T (1) E , e d c c a (a) The Coulomb and Higgs stratification of Th. 25. Th. 25(∆ u , ∆ v ) (3 , a c f k su (2) × sp (8) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 17 : Information about the Th.25 theory.twice the u scaling dimension, it is tempting to use the doubling rule and associate thisfactor to a theory supported on the unknotted stratum u = 0 and make the followingidentification T u ≡ [ I , su (6)] Z . This in turn implies that one of the first HB transitionis a c . The level of the su (2) factor suggests instead that the su (2) is realized onthe other unknotted stratum by T v ≡ S (1) ∅ , . This in turn implies two things: 1) thatthe theory should has a one dimensional ECB and 2) that the theory supported onthe c stratum has to have a three dimensional ECB [84]. These two facts turn out tobe completely consistent with what we find from the HB analysis. But before turningto that, we can plug the b i corresponding to the T u and T v into (3.6b) and solve forthe theory supported on the knotted stratum which completes our analysis of the CBstratification in figure 18aTo identify the theories supported on the two strata which stem out of the su-perconformal vacuum we notice that both of these Higgsings are of gHW type andtherefore we can compute the corresponding central charges using (3.11). Performingthis calculation we find that the theory supported on the c stratum is the S (1) D , , thus T c ≡ S (1) D , . This rank-1 theory has indeed h =3 compatibly with the prediction arisingfrom the CB analysis. To identify the rank-2 theory supported on the ECB we can usethe extra information that the total HB of the SCFT in exam is known (that is 11) andtherefore this singles out the lagrangian theory su (3) with N f = 6 as our candidate.The rest of the HB is determined by following the higgsings of both T c and T a andthe final result is summarized in figure 18a.– 43 – h. 26 S (1) ∅ , [ I , su (6)] Z [ I , ∅ ] (cid:2) u + v =0 (cid:3) (cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB Th. 26 T (1) D , su (3) + 6 F S (1) D , d a c a c (a) The Coulomb and Higgs stratification of Th.26 Th. 26(∆ u , ∆ v ) (3 , a c f k su (2) × sp (6) × u (1) d HB h T ( h ) 1 (b) Central charges, CB parameters and ECBdimension. Figure 18 : Information about the Th. 26 theory. T (2) D , This theory can be straightforwardly obtained by mass deformation of the T (1) E , . Itwas one of the recently discovered T theories. It can be obtained by higgsing the N = 2 S -fold S (1) A , or by a twisted compactification of a (1,0) theory [35]. It canalso be realized in type II B string theory as a wordvolume theory of two D3 branesprobing an exceptional E brane singularity in the presence of an S -fold without flux[44]. Finally, it can also be obtained in class- S , for example in the untwisted A case[35]. The properties of N = 2 T -theories are summarized in appendix A. The CFTproperties as well as the stratification of this particular theory can be found in table45. b T E , This theory was first constructed in twisted E class- S [39] but in the initial referencethe flavor symmetry symmetry was wrongly identified as so (9) × u (1). The f enhance-ment was instead realized in [37]. Finally this theory can also be obtained as massdeformation of the T (2) E , [44] which can most easily seen from the 5 d construction. T (2) E , can be obtained as a Z twisted compactification of a 5 d SCFT which UV completesboth a su (4) + 2 AS + 6 F and 1 F − su (2) − su (2) − su (2) − F , where the latter de-scription also includes two fundamentals for the middle su (2) . The b T E , theory can beobtained as a Z twisted compactification of the N = 1 5 d SCFT which UV completes It is customary to call two 5 d gauge theories which have the same UV-completion as UV dual . Webelieve that this terminology is misleading therefore we will refrain from using it. – 44 – u (2) − su (2) − su (2) with the θ angle of the two edge su (2)s set to 0 [44] (where themiddle su (2) flavor factor still has two flavors attached to it). b T E , [ T (1) E , ] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) d H d HB b T E , S (1) D , c T (1) D , f (a) The Coulomb and Higgs stratification of b T E , . b T E , (∆ u , ∆ v ) (4 , a c f k [ f ] × u (1) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 19 : Information about the b T E , . theory.Since ∆ = 5 is not an allowed CB scaling dimension at rank-1, the theory is nottotally higgsable which simplifies our lives. The theory has in fact a single simple flavorfactor and the HB is linear. Identifying which theory on the CB realizes an exceptionalflavor symmetry is straightforward and this leads to the following T v = [ T (1) E , ] Z whichis also consistent with the level of the f flavor symmetry being doubled ∆ v . TheHB structure of this rank-1 theory readily implies that h =0 and that the total HBof the rank-2 SCFT should start with a f stratum followed by a rank preserving c transition. It is straightforward to complete the CB Hasse diagram by using (3.6a)-(3.6b) to reproduce the a and c central charges listed in table 19b. This exercise impliesthat T v + u ≡ [ I , ∅ ].Leveraging all the information which we have gathered from the CB analysis willimmediately allow us to characterize the HB side. In fact the rank-1 theory supportedon the f stratum is uniquely fixed by requiring that such theory has a eight dimensionalHB (to reproduce the total dimension of the HB of b T E , ) and h =3 (to reproduce thesubsequent rank-preserving c transition). Thus we conclude that T f ≡ S (1) D , . Thisresult can be also checked by matching the central charge of T f with what we obtainfrom (3.11). e T E , This theory was first discussed in [38] in the context of twisted compactification of 5dSCFTs and then further analyzed in [44]. It can be obtained via mass deformation of– 45 –he Th.25 which was discussed above. In particular the Z twisted compactification ofa 5 d SCFT which completes a su (4) + 1 AS + 6 F , where the subscript indicates theChern-Simons level of the 5 d gauge theory, gives the e T E , [38, 44]. e T E , [ I , ∅ ] [ I , su (6)] Z (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) H d HB e T E , T (1) A , S (1) A , a c c (a) The Coulomb and Higgs stratification of e T E , e T E , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k sp (6) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 20 : Information about the e T E , theory.The moduli space of this theory is fairly straightforward to analyze. In fact thetheory is not totally higgsable and it only has a single Higgsable CB parameter ( u ). Asusual this is reflected in the fact that there is a single simple flavor factor which in turnimplies that the HB is linear, see figure 20a. The rank-1 theories which realizes the sp (6) can be easily identified as the [ I , su (6)] Z and which, by using the doubling ruleto match the level via (3.6c), it has to be supported on the unknotted stratum u =0 which leads to the identification T u ≡ [ I , su (6)] Z and readily implies h =0. Todetermine the theory supported on the knotted stratum it is enough to match the c central charge via (3.6b).Moving on the HB side, the HB of the T u should give rise to a c followed bya c transition [84]. The central charge of the rank-1 theory supported on the c stratum can be easily identified noticing that this Higgsing is of gHW type, thus using(3.11) singles out S (1) A , . This also reproduces the subsequent c Higgsing. The restof the Hasse diagram is obtained straightforwardly from the Higgsings of S (1) A , . Theway in which the flavor symmetry of the theory is reproduced from the Higgsing isstraightforward using the properties of c summarized in table 5. T (2) A , This theory can be obtained mass deforming the T (1) D , and belongs to the recentlydiscovered T theories. It can be obtained in class- S , for example in the recent study– 46 –f twisted A [107] , as well as by higgsing the N = 2 S -fold S (1) A , . As usual, the T -theories also allow a twisted construction from (1,0) theories [35] and they ariseas a wordvolume theory of two D3 branes probing an exceptional E brane singularityin the presence of an S -fold without flux [44]. The properties of N = 2 T -theories aresummarized in appendix A. The CFT properties as well as the stratification of thisparticular theory can be found in table 45. su (2) − su (2) su (2) − su (2))[ I , ∅ ][ I , ∅ ] [ I , ∅ ] [ I , su (4)] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) u (1) a S (1) ∅ , c su (2) − su (2) (a) The Hasse diagram for the CB of the su (2) N = 2 theory with two hypermultiplets in the( , ). su (2) − su (2)(∆ u , ∆ v ) (2,2)24 a c f k sp (4) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 21 : Information about the su (2) N = 2 theory with two hypermultiplets inthe ( , )Let’s start from the analysis of the CB. Two obvious strata correspond to turn-ing on separately a CB vev for each su (2). This makes all the hypers massive, andsemi-classically leaves a pure su (2) which breaks quantum mechanically into a dyonand monopole giving rise to a total of four knotted singularities each supporting an[ I , su (2)]. There is also another semi-classical locus where massless charged matterarises. I can in fact tune the vevs of the two su (2) and make them equal. This ofcourse will break su (2) × su (2) → u (1) × u (1) but each bifundamental hypermultipletwill contribute two massless hyper with charge 1 giving rise to an effective u (1) theorywith four massless hypers. Since the [ I , su (4)] is discretely gauged, we expect a c tran-sition on the HB side of things which supports a rank-1 theory with a one dimensionalECB.Let’s now focus on the HB. The information that we gathered from the CB analysisare already enough to make the identification T c ≡ S (1) ∅ , (the su (2) N = 4 theory). In [107] T (2) A , was named ˜ T . – 47 –t is possible to use (3.11) to explicitly confirm this guess or else perform the higgsingexplicitly solving for both the F and D term conditions. su (2) × su (2)This is an N = 4 theory. The moduli space of these theories is extremely constrainedand it is basically entirely specified by the Weyl group of the gauge algebra which in thiscase is Z × Z . More details on the moduli space structure of theories with extendedsupersymmetry can be found in appendix B. The CFT data of this theory, as well asthe explicit Hasse diagram of both the CB and HB stratification are depicted in figure47. ◦ su (6) series All the theories but one (the lagrangian entry) in this series present some intriguingmysteries. Specifically it appears that unknown elementary slice appear on the HBwhich seem to be realized in highly non-trivial way on the CB. In the Hasse diagramsbelow there are many question marks, which characterize our current ignorance on thedetails of these theories.
Th. 33
This theory, which sits at the top of this series, was first realized in [39] but it can alsobe realized by a twisted Z compactification with non commuting holonomies of a 6 d SCFT completing the su (4) + 1 AS + 12 F [40]. Its CFT data is summarized in table22. Immediately we are faced with a puzzle. In fact the flavor symmetry is semi-simple but the theory is not totally higgsable. A closer look at the levels of the flavorsymmetries present a possible resolution: k su (6) = 2∆ v and k su (2) = ∆ v + 1 suggestingthat both flavor symmetries are realized on the v = 0 stratum. A further indicationcomes from reproducing the central charges in 22b from the central charge formulae.This instructive exercise does not provide a unique solution but the most reasonableone assigns a b = 1 to knotted stratum and b = 9 to the v = 0 unknotted one.We are therefore led to the following identification: T v ≡ [ I ∗ , su (2) × so (12)] Z and T u + v ≡ [ I , ∅ ]. The latter is now standard but let’s discuss the former.[ I ∗ , su (2) × so (12)] is nothing but an N = 2 su (2) + 6 F + 1 adj gauge theory. Thistheory has a twelve dimensional HB of which one dimension is an ECB. Clearly theUV-IR simple flavor condition tells us that its flavor symmetry is too large for beingthe correct identification but so (12) has a Z (inner) automorphism whose commutant– 48 –s su (6), for more details on automorphisms of Lie algebras see [114, Theorem 8.6] orfor a more physical discussion [115, Sec. 3.3]. We therefore claim that the theory thatcorrectly realizes the flavor symmetry of this theory is a Z discretely gauged version ofthis gauge theory, [ I ∗ , su (2) × so (12)] Z , where the Z acts trivially on the adjoint hyperand therefore leaves the su (2) untouched and implies that h =1. Th. 33[ I ∗ , su (2) × so (12)] Z [ I , ∅ ] (cid:2) v = 0 (cid:3) (cid:2) u + v = 0 (cid:3) H d HB e T (1) E , c S (1) E , c Th. 22 a ?Th. 33 (a) The Coulomb and Higgs stratification of Th. 33. Th. 33(∆ u , ∆ v ) (6 , a c f k su (6) × su (2) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 22 : Information about the Th. 33We can now analyze the HB of this theory. Great insights on the overall structureof this 29 dimensional symplectic variety can be gained by first analyzing the ECB ofthe theory. From our CB analysis we learned that this corresponds to higgsing theadjoint multiplet and therefore this higgsing is of gHW type. Using the (3.11) wereadily obtain that the central charge of the theory supported on this a stratum is12 c T a = 76 which by itself leads to the identification T a ≡ Th .
30. The HB of thistheory, see figure 15a, determines most of the remaining structure. To complete theanalysis we need to understand what is the transition carrying the su (6) action andwhich corresponds to giving a vev to the fundamentals of the T v . This transition isdepicted by a dashed line in figure 22a. Because of the structure of the Hasse diagram,this cannot correspond to an elementary slice. A possibility to answer this question isto directly analyze the magnetic quiver of the theory which can be derived from thehigher dimensional realization of the theory. Unfortunately, “subtracting off” the e and the c transition we are left with an unknown transition. Perhaps the discretelygauged realization which we identified on the CB side will provide interesting insightsallowing to resolve this puzzle. We will further investigate this question elsewhere [116]– 49 – h. 34 This theory was first realized in class- S within the twisted D -series [34]. But it canalso be realized as Z compactification of a brane web which is obtained after massdeformation of the circle compactification of the 6 d SCFT completing a su (4) + 1 AS +12 F . This implies that this theory can be obtained by mass deforming the theorydiscussed in the previous section. Most of the CFT data summarized in figure 22 wascomputed in the original class- S realization. Th. 34[ I ∗ , su (2) × so (8)] Z [ I , ∅ ][ I , su (2)] (cid:2) v = 0 (cid:3) (cid:2) u = 0 (cid:3) (cid:2) u + v = 0 (cid:3) H d HB T (1) D , S (1) D , T (1) E , e d c c a Th. 25 a ??Th. 34 (a) The Coulomb and Higgs stratification of Th. 34. Th. 34(∆ u , ∆ v ) (4 , a c f k su (4) × su (2) × u (1) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 23 : Information about the Th. 34.The analysis of this case is largely analogous to the previous one and thus quiteinvolved. The theory is totally higgsable but again the level of the two simple flavorsymmetry factors are both compatible to be realized by a theory living on the sameunknotted stratum. Indeed k su (2) = ∆ v + 1 and k su (4) = 2∆ v . This again suggeststhat the low energy theory T v will have a semi-simple flavor symmetry. To makeprogress in this identification we leverage the central charge formulae to reproducethe c and a central charges in table 23b. This exercise does not produce a singlesolution but again the most reasonable one implies b u + v = 1, b u = 2 and b v = 7.This immediately suggests the identification T u + v ≡ [ I , ∅ ], T u ≡ [ I , su (2)] and T v ≡ [ I ∗ , su (2) × so (8)] Z which is nothing but a N = 2 su (2) + 1 adj + 4 F gauge theory.This solution is satisfactory in some senses and puzzling in others. Let’s elaborate. The author is deeply thankful to Gabi Zafrir for his patient and clarifying explanation of the braneweb deformations which lead to this and the following three theories. – 50 –he Z gauging that we conjecture is implemented on the v = 0 unknotted stratum,only acts on the fundamental hypers as a inner automorphism of so (8) with commutant su (4). This implies in turn that the one dimensional ECB obtained by turning on avev for the adjoint hyper is left unchanged and can be leverage to great extent to learnabout the HB of this theory. Running this anlaysis with our usual tools, which includeusing (3.11), we find T a ≡ T h.
25. This identification allows to fill in the left sideof the Hasse diagram in figure 23a. The higgsing of the fundamental is as usual morecomplicated. Indeed we are not able to identify the stratum which is acted upon bythe su (4). We can only conclude that this cannot be and elementary slice. Since the realization of the theory is known, so is the magnetic quiver of the theory. Wehope that an in-depth study of this object might help identifying the “question marktransition” in the figure. But there is one more puzzle in this case. And that is thatthe T u also has a Higgs branch and thus we expect another branch of the total HB.The fact that the su (2) realizes the u (1) with an IR enhancement, suggests that thereis an extra discrete identification at the origin of the moduli space acting on the a .But even allowing for that, our analysis is not powerful enough to say anything moreabout this branch nor even conclusively determine whether it is actually there. Thisis reflected in the figure by another dashed branch which connects nowhere. Again anin-depth study of the magnetic quiver might help resolve this puzzle. Th. 35
Th. 35[ ⋆ w /b = 7] [ I , ∅ ] (cid:2) v = 0 (cid:3) (cid:2) u + v = 0 (cid:3) H d HB d T (1) D , c c S (1) D , S (1) D , h , h , Th. 35 (a) The Coulomb and Higgs stratification of Th. 35.
Th. 35(∆ u , ∆ v ) (4 , a c f k su (3) × su (3) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 24 : Information about the Th. 35.This theory can be obtained by the twisted Z compactification of the braneweb with three D5 and three NS5 all intersection at one point described in [38].– 51 –ur understanding of this theory suffers of many of the same problems encoun-tered in the previous cases. The theory is not totally higgsable and v is the onlyHiggsable CB parameter. Furthermore even though the flavor symmetry is not simple,the semi-simple component has the right level to be interpreted as flavor symmetry ofan IR-free rank-1 gauge theory supported on v = 0. To gain a better feeling about whatthis theory could be, it is useful to trying in reproducing the a and c central chargesin table 24b using the central charge formulae. Performing this exercise we don’t geta unique answer but the most reasonable one predicts a T u + v ≡ [ I , ∅ ] while the T v is predicted to have b = 7. It is likely that the correct interpretation will involve adiscrete gauging of an IR-free theory but we don’t have a sharp guess yet.Without a complete knowledge of the CB stratification it is hard to perform acomplete analysis on the HB side using purely field theoretic methods. Thankfullythe information of the magnetic quiver, which can be derived from the realization,allows to reproduce the full HB Hasse diagram which is depicted in figure 24a . Thecareful reader will have notice that a new elementary slice appeared which is labeledas h , . These are new elementary slices which will be introduced in [117] and appearin the affine Grassmanian of sp (2 n ). These slices have been known for quite some timein the math literature with a different name h n, = a c n [118–120] the details will be discussed elsewhere [116]. Th. 36
This theory can be obtained as twisted Z compactification of a SCFT obtainedfrom a mass deforming previous brane webs, which in particular implies that this 4 d SCFT comes from deforming Th. 34 [3, 38].The analysis of the moduli space of vacua of this theory resembles the previouscases. First notice that despite having a single Higgsable CB parameter ( v ) the flavorsymmetry of the theory is semi-simple. A careful look at the levels reveals that itthe entire su (3) × su (2) factor should be realized on the v = 0 stratum by a discretelygauged su (2) N = 2 gauge theory. The experience built in the analysis of previoustheories suggests T v ≡ [ I ∗ , su (2) × so (6)] Z . This identification is perfectly confirmedby the exercise of reproducing the a and c central charges in table 24b using thecentral charge formulae. The Z that we gauge does not act on the su (2) factor, thus We thank Julius Grimminger for performing the quiver subtraction which led to this HBHasse diagram. We are grateful to Antoine Bourget and Julius Grimminger for the computation of many of theHasse diagrams in this section, for sharing with me unpublished results of (one of) their upcomingpaper(s) and for providing the list of math references on elementary slices in the affine Grassmanian. – 52 –mplying h =1, and instead has only a component in the inner automorphism group of so (6) with commutant su (3). Th. 36[ I ∗ , su (2) × so (6)] Z [ I , ∅ ] (cid:2) v = 0 (cid:3) (cid:2) u + v = 0 (cid:3) H d HB T (1) A , S (1) A , a c c e T E , a ?Th. 36 (a) The Coulomb and Higgs stratification of Th. 36. Th. 36(∆ u , ∆ v ) (3 , a c f k su (3) × su (2) × u (1) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 25 : Information about the Th. 36.The analysis of the HB proceeds also in a similar manner as previous examples.First let’s analyze the ECB. Using (3.11) we obtain that 12 c a = 34. This information,by itself, it is not enough to identify the theory supported on a yet adding the con-straints that the total HB should be eight dimensional does obtaining T a ≡ e T E , . TheHB of the latter allows to fill in most of the remaining part of the HB Hasse diagram infigure 25a and it remains to characterize the higgs direction corresponding to turningon a vev for the fundantamental hypers on the gauge theory supported on the CBunknotted stratum. Even with the help of the magnetic quiver, we are unable to deter-mine the precise structure of this four dimensional component, which is indicated witha dashed line and a question mark in figure 25a, and leave this task for future work. Th. 37
This theory can be obtained by Z twisted compactification of the brane web withtwo D5 , two NS5 and one (1,-1) brane all intersection at one point [38]. This is thusconnected by mass deformation to the theory Th. 35 discussed previously.The state of our understanding of this theory is only marginally better than theoryTh. 35. In this case the theory is totally higgsable but a careful analysis of the levelof the two simple flavor symmetry factors, suggest that they are both realized as an– 53 – h. 37[ I , ∅ ] [ I , su (2)][ ⋆ w /b = 5] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB a T (1) A , d T (1) D , A c c S (1) A , S (1) A , h , h , Th. 37 (a) The Coulomb and Higgs stratification of Th. 37.
Th. 37(∆ u , ∆ v ) (3 , a c f k su (2) × su (2) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 26 : Information about the Th. 37.IR-free rank-1 gauge theory supported on v = 0. To gain a better feeling about the CBstratification, it is useful leverage the central charge formulae to reproduce the a and c central charges in table 26b. Performing this exercise we again don’t get a uniqueanswer but the most reasonable one predicts a T u + v ≡ [ I , ∅ ], T u ≡ [ I , su (2)] (whichenhances one of the u (1) factors) while the T v is predicted to have b = 5.Again it is hard to perform a complete analysis of the HB without a completeknowledge of the CB stratification. Luckly in this case, the magnetic quiver, which canbe readily obtained from the realization, can be used to obtain the full Hasse diagramwhich is reproduced in figure 26a. Again we point out the appearance of the newelementary slices h , [117]. su (3) + F + S This case was discussed before [121] and more recently in [1, 67].Since this theory has two hypers in two different complex representations, we expectthe flavor symmetry of the theory to be u (1) × u (1). The CB stratification depicted infigure 27a follows from the lagrangian analysis where again the two knotted stratumsupporting a [ I , ∅ ] correspond to the CB locus where only a pure su (2) N = 2 theoryarises. In this case, the T v ≡ [ I , su (2)] corresponds to a u (1) theory with three masslesshypers, two of which of charge one and the other of charge two. Notice two facts: 1)in this case there is an enhancement of the flavor symmetry on the CB: u (1) → u (2) 2)the T v has a one quaternionic dimensional HB with a free massless hyper on its genericpoint. Thus we expect the full theory to have a non-trivial HB with a one quaternionicdimensional first transition supporting a theory with a h =0. Let’s see how this works.– 54 – u (3) + F + S [ I , ∅ ] [ I , su (2)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) su (3) + F + S S (1) ∅ , a A (a) The Hasse diagram for the CB and the HBof the su (3) gauge theory with one and one . su (3) + F + S (∆ u , ∆ v ) (2,3)24 a c f k u (1) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 27 : Information about the su (3) N = 2 theory with one hypermultiplet in the and one in the .An su (3) theory with 1 flavor and one symmetric has the following gauge invariantoperators: M = Q ˜ Q, M = X ˜ X (4.11) B = XQ , B = X (4.12)˜ B = ˜ X ˜ Q , ˜ B = ˜ X , (4.13)where we have suppressed the gauge indices, the X and ˜ X are the component of thesymmetric hypermultiplet, Q and ˜ Q are the fundamental flavors and the subscript of B i labels the power of X . Also in labeling the representations we chosen the convention of[122] where the has Dynkin label (0 , B and ˜ B (which have no fundamentalcomponents) are turned on the theory flows precisely to S (1) ∅ , , that is N = 4 su (2) gaugetheory. The structure of the first stratum of the HB is easily obtained once the F-termconditions are taken into account. For Q = 0, we find the following relation: B ˜ B = M ⇒ Sf u (1) = C / Z ≡ A . (4.14)therefore concluding that T A = S (1) ∅ , matching our expected h =1.It is interesting to notice that this is an example where the transition on the HBdiffers from the generic transition into the MB. This is though expected because of theflavor symmetry enhancement. The full HB of the theory is reported in figure 27a. ◦ – 55 – .4 sp (14) series This series has a single top theory and four entry total. The moduli space structure ofthe theories in this series don’t present particular difficulties and have been completelysolved.
Th. 39
Th. 39 [ I , ∅ ][ I ∗ , sp (14)] (cid:2) u + v = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB e T (1) E , c S (1) E , c Th. 22 c Th. 39 (a) The Coulomb and Higgs stratification of Th. 39
Th. 39(∆ u , ∆ v ) (6 , a c f k sp (14) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 28 : Information about the Th. 39This specific theory was first discussed in [1] but it can be constructed using thegeneral construction presented in [40]. Specifically it can be obtained as a T compacti-fication of the minimal ( D , D ) conformal matter [14] with a Z valued non-commutingholonomy . This theory can also be engineered in twisted D class- S as it is describedin the analysis of [40].This theory is not totally higgsable with v being the only Higgsable CB parameter.Not surprisingly the flavor symmetry is simple and by noticing that k sp (14) = ∆ v + 1we immediately can identify the theory realizing the flavor symmetry on the CB: T v ≡ [ I ∗ , sp (14)]. This in turn implies that the theory has h =7 and the theory sup-ported on the c stratum of the HB should have no ECB. Imposing that the a and c central charges in table 28b can be reproduced by the central charge formulae allows As explained [40], for ( D n +1 , D n +1 ) there are two possible such Z compactifications, one thatpresevers the full enhanced sp (4 n + 2) flavor symmetry, and one that only preserves a subgroup. Weare here interested in the former. – 56 –o complete the analysis of the CB stratification by identifying the theory supportedon the knotted stratum as a [ I , ∅ ].The analysis of the HB is fairly easy. The fact that the flavor symmetry is simple itimmediately implies that the Hasse diagram is linear. Thus the symplectic stratificationis basically determined once the theory supported on the first stratum is identified.This identification can be readily performed by noticing that the sp (14) higgsing is ofgHW type thus we can use (3.11) and readily derive 12 c T c = 76. This information isenough to identify T c ≡ Th .
30. Following the subsequent higgsings reproduces theHasse diagram in figure 28a completing our analysis.
Th. 40
This theory was initially obtained in the Z twisted D class- S series [34] where mostof the CFT data reported in figure 29b. As usual we will be using this information tofill in the detailed structure of the full moduli space. And as usual we will start ouranalysis from the CB side of things. Th. 40 [ I , ∅ ][ I , su (2)][ I ∗ , sp (10)] (cid:2) u + v = 0 (cid:3)(cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB d e T (1) D , c T (1) E , S (1) D , c a c Th. 25 S (1) E , a c Th. 40 (a) The Coulomb and Higgs stratification of Th. 40.
Th. 40(∆ u , ∆ v ) (4 , a c f k su (2) × sp (10) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 29 : Information about the Th. 40.First notice that both of the CB parameter are Higgsable CB parameter, thereforewe expect a non-linear HB Hasse diagram and a somewhat involved HB structure.This in fact matches with the fact that the flavor symmetry of the theory contains twosimple flavor factors. Giving its level, we can use the doubling rule to identify the theoryrealizing the su (2) factor while the level of the sp (10) is k sp (10) = ∆ v +1. Then there is a– 57 – , [ I ∗ , sp (8)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB e T E , T (1) A , S (1) A , a c c c R , (a) The Coulomb and Higgs stratification of R , . R , (∆ u , ∆ v ) (3 , a c f k sp (8) × u (1) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 30 : Information about the R , theory.natural guess for how these two simple factors are realized on the CB: T u = [ I , su (2)]and T v = [ I ∗ , sp (10)]. This immediately suggests that this rank-2 theory has h = 4and a c transition followed by a c one. As a check that this identification is indeedcorrect, the reader can easily match the a and c central charges using (3.6a)-(3.6b). Thisexercise also fixes the theory supported on the knotted stratum to be T u + v ≡ [ I , ∅ ].Now let’s analyze the HB. Start first from the a transition which is a regular HBstratum and as such should support a rank-1 theory. Noticing that the higgsing is ofgHW type and using (3.11) immediately accomplish the task of making the identifica-tion T a ≡ S (1) E , . To identify the theory supported on the five dimensional ECB we canperform a similar analysis. Again the c higgsing is of gHW type and we can use (3.11)to compute the c central charge of the theory. This doesn’t quite specify the theoryuniquely but either the constraint derived from the CB analysis that the c stratumshould be followed by a c or simply imposing that the total HB dimension, is enoughto make the identification T c ≡ R , and conclude our analysis. R , The R , is the rank-2 entry of an infinite family of N = 2 SCFTs discussed in [41].The R , N is a rank N SCFT with sp (4 N ) N +2 × u (1) flavor symmetry whose sp (2 N )gauging is S-dual to an N = 2 su (2 N + 1) gauge theory with one symmetric and oneantisymmetric with R , ≡ S (1) A , , for more details we refer to the original paper.– 58 –his theory again is not totally higgsable and, relatedly, has a single simple flavorfactor. This signals that the HB is linear. Since the level of the sp (8) flavor factordiffers precisely by one from the only Higgsable CB parameter ( v in this case), it isimmediate to realize the sp (8) as the flavor symmetry of T v ≡ [ I ∗ , sp (8)]. This in turnimplies that h =4 and that the c stratum on the HB should be followed by a c [84].The analysis of the HB is straightforward. In fact the CB analysis signals thatthe sp (8) higgsing is of gHW type and thus we can use (3.11) to compute the centralcharge of the theory supported on the c finding 12 c T c = 34. Since we are higgsingalong an ECB direction, T c has to be rank-2, thus the c central charge we find doesnot uniquely specifies it. We can use then either the information coming from theCB analysis of the presence of a subsequent c Higgsing or the constraint coming fromthe total dimension of the HB of R , . Either lift the degeneracy and lead us to theconclusion that T c = ˜ T E , . This R , → ˜ T E , can also be seen directly from the 5drealization [44]. The rest of the HB Hasse diagram in figure 30a can be inferred byfollowing the subsequent higgsing of e T E , which are depicted in figure 20a. sp (4) + 3 V As we discussed in the analysis of other sp (4) theories, there are two interesting di-rections on the CB leaving an su (2) unbroken. Each V ( ) contributes a masslesshypermultiplet in the for one su (2) and none for the other with the result that weexpect a knotted stratum with an effective su (2) with 3 adjoint hypermultiplets in thelow-energy. The other knotted stratum “splits” into two, each supporting a [ I , ∅ ], thuswe conclude that h =3 and furthermore we expect that the rank-2 theory supported onthe ECB has either a a or c HB transition [84]. This analysis, which is depicted infigure 31a, can be confirmed both by reproducing the central charges of the theoriesusing our central charge formulae or explicitly studying the discrimant locus of theSeiberg-Witten curve which is known explicitly [105].Let’s move to the analysis of the HB. Since we have an explicit lagrangian descrip-tion available, we could solve for the higgsging by standard methods [73] but we willfind it quicker to leverage the geometric constraints which can be derived from theconsistency of the full moduli space. In fact from our CB analysis we know that thetheory supported on the three dimensional ECB should have an either c or a transi-tion. Furthermore, to reproduce the total dimension of the HB (which can be computedboth from anomaly matching or standard Hyperk¨ahler quotient), implies this rank-2theory should furthermore have a three dimensional HB. This enough information toconclude that T c ≡ su (2) − su (2). ◦ – 59 – p (4) + 3 V [ I , ∅ ] [ I ∗ , sp (6)][ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB a S (1) ∅ , c su (2)- su (2) c sp (4) + 3 V (a) The Hasse diagram for the CB of the sp (4)gauge theory with three hypermultiplets in the . sp (4) + 3 V (∆ u , ∆ v ) (2,4)24 a c f k sp (6) d HB h T ( h ) 1 (b) Central charges, CB parameters and ECBdimension. Figure 31 : Information about the sp (4) N = 2 theory with three hypermultiplet inthe su (5) series This series has a single top theory from which we can reach the remaining three theories.The moduli space structure of these theories is fairly involved and some open questionson the details remain and are explained in the text below.
Th. 43
Th. 43[ I , ∅ ] [ I ∗ , so (12)] Z (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB d T (1) D , h , S (1) D , h , Th. 43 (a) The Coulomb and Higgs stratification of Th. 40.
Th. 43(∆ u , ∆ v ) (6 , a c f k su (5) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 32 : Information about the Th. 43.This theory was first proposed in [38] where it was constructed as circle compact-– 60 –fication of the 5 d T SCFT with a Z twist along the circle. Much of CFT data wascomputed a few years later in [40], here we will present an analysis of its moduli spacestructure.In this case we will start from the analysis of the HB. Part of its structure wasalready worked out in the original paper where it was pointed out that the first tran-sition of the HB is five quaternionic dimensional and that leads to the rank-1 S (1) A , theory. This information, determines the rest of the HB Hasse diagram but we stillneed to characterize the first stratum which is five quaternionic dimensional. This canbe done by using the magnetic quiver of this theory which corresponds to the n = 3entry of the A n +1 series in table 11 of [123] . Using the technique of quiver subtraction [124] it is possible to determine that this five dimensional stratum is a h , which isa new elementary slice which further generalizes the h ,n series which appeared in the su (6) series. These elementary slices, labeled as h n, , do not in general appear in theaffine Grasmannian of any lie algebra unless n = 2 where they appear in the affineGrassmanian of g and we have the identification h , = a g . In order to furthercharacterize them, let’s now turn to the analysis of the CB.The theory is not totally higgsable with v being the only Higgsable CB parameter.This is reflected by the fact that the flavor symmetry is simple. Since k su (5) = 2∆ v one might be tempted to realize the flavor symmetry on the CB with a [ I , su (5)]. Asign that this cannot be the case is the fact that this identification would imply theexistence of a a transition which is four and not five dimensional. Another evidence isthat this identification (which implies a b = 5) is not compatible with reproducing the a and c in table 25b using the central charge formulae. This exercise is the one thatprovides some hint on how to realize the su (5) factor on the CB, indeed we find thatthe b i of the theories on the CB strata should be one for T u + v and eight for T v . Theformer immediately leads to the identification T u + v ≡ [ I , ∅ ], let’s discuss insteadhow to interpret the latter.First let’s go back to our HB analysis. An extra hint which helps in making theright guess of what theory realizes the su (5) on the CB is provided by the fact thatthe rank-1 theory supported on the h , has h =4. This implies that the theory we areafter must likely have a nine dimensional HB and it is likely a discretely gauged versionof a rank-1 theory. With a bit more thinking, the only reasonable guess with theseproperties is T v ≡ [ I ∗ , so (12)] Z where the Z acts as an inner automorphism with noouter component and whose commutant has su (5) as its simple component. Thereforewe conjecture that modding by this Z action transforms the d (which, from table 5 We thank Gabi Zafrir for pointing this out. Again, we are grateful to Julius Grimminger for the explanation of this point. – 61 –s indeed nine dimensional) to a symplectic.variety with three strata, and a h , and a h , transition. Th. 44
Th. 44[ I , ∅ ] [ I , su (2)] [ T (1) D , ] Z (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB a a T (1) A , T (1) A , h , h , S (1) A , S (1) A , h , A ? Th. 44 (a) The Coulomb and Higgs stratification of Th. 44.
Th. 44(∆ u , ∆ v ) (4 , a c f k su (3) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 33 : Information about the Th. 44.This theory was first proposed in [38] as a theory obtained by mass deforming thetheory described in the previous section. As we will see, lot of the strange features ofthe previous analysis carry over to this one.We will start again from the HB which is more directly accessible from the 5 d analysis and therefore some of its properties were already pointed out in the originalpaper. Namely that one of the first HB strata is three quaternionic dimensional with a S (1) A , supported over it. This identification readily allows to determine the rest of rightside of the HB Hasse diagram yet it leaves the characterization of the three dimensionalstratum, as well as the left part, unresolved. The study of the magnetic quiver of thistheory which corresponds to the n = 3 entry in the A n − × u (1) series in table 11 of[123] shows that this stratum should be a h , . It is curious to notice that the magneticquiver does not give any evidence of the left branch in figure 33a.Now let’s start with the CB analysis. This theory is totally higgsable, despite thatthere is a single simple flavor factor. The relation k su (3) = 2∆ v it is again suggestivethat this su (3) could be realized on the CB by a [ I , su (3)] supported on the v = 0stratum. The study of the HB already shows that this identification is not consistentsince [ I , su (3)] has a a as its HB and thus would give rise to a two dimensionaltransition and not three which is instead the case we find here. To resolve this puzzlewe can again rely on the central charge formulae which tells us that b u = 2, b v = 6 and– 62 – u + v = 1. The last immediately leads to the identification T u + v ≡ [ I , ∅ ], the first to T u ≡ [ I , su (2)] while the middle one T v ≡ [ T (1) D , ] Z . Let’s conclude with a discussionof these last two identifications.First the higgsing of the [ I , su (2)] is what leads to the left branch of Hasse diagramin figure 33a. Since we see en enhancement of the flavor symmetry on the CB weconjecture that there is a further identification acting on the a at the origin. Thequestion mark in figure 33a, signals that we are not able to conclusively determinewhat this identification is. Conversely the higgsing of the [ T (1) D , ] Z leads to the branchon the right. The Z that we are gauging is the one which has commutant su (3) andwe conjecture that under this modding the d (which is indeed five dimensional, seetable 5) becomes a symplectic variety with three strata, with a h , transition followedby a h , . Th. 45
This theory was is obtained as a mass deformation of the theory studied in the previoussection but its explicit higher dimensional construction has not appeared anywhere. Apossible construction from will be discussed in [3]. The CFT data in table 34b isderived using the construction and checked for self-consistency below. Th. 45[ I , ∅ ] [ I ∗ ⋆ , so (6)] Z (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) × H A S (1) u (1) , h , Th. 45 (a) The Coulomb and Higgs stratification of Th. 45.
Th. 45(∆ u , ∆ v ) (3 , a c f k su (2) × u (1) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 34 : Information about the Th. 45.This theory is not totally higgsable and we expect that the simple factor of theflavor symmetry to be realized on the v = 0 unknotted stratum. To gain more insightwe go through the usual exercise of matching the a and c central charges in table 34awith the central charge formulae. The most reasonable solution assigns b v = 5 and b u + v = 1. The latter lead immediately to the identification T u + v ≡ [ I , ∅ ] but theformer one is a bit more problematic. In fact a natural guess would be a [ I ∗ ⋆ , so (6)] Z – 63 –ith the Z acting as inner automorphism of so (6) ∼ = su (4) with commutant su (2).And there is a theory which almost has the right properties, that is an su (2) + 3 F gauge theory. This theory has indeed b = 5 and the right flavor symmetry. The onlyshortcoming of this identification is that the theory in question is asymptotically free,thus there is no notion that a single singularity on the CB could support all the degreeof freedom of that theory. In fact in [125] the geometry of the CB of the su (2) + 3 F is worked out and it presents two singualrities separated by a distance proportional tothe strong coupling scale. We don’t have at the moment a resolution of this puzzlewhich makes the evidence for the existence of this theory less solid than other cases.The HB depicted in figure 34a can be inferred exploiting the mass deformation fromother theories but again further checks will be left for the future. ◦ sp (12) series This series contains a unique top theory and two lagrangian ones.
Th. 46
This theory has appeared for the first time in [42] in the E class- S and has oneHiggsable CB parameter and therefore we expect the HB to linear. Th. 46 [ I , ∅ ][ I ∗ , sp (12)] (cid:2) u + v = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB e T (1) E , a S c Th. 46 (a) The Coulomb and Higgs stratification of Th. 46.
Th. 46(∆ u , ∆ v ) (4 , a c f k sp (12) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 35 : Information about the Th. 46.The CB stratification can be readily obtained by noticing that the simple flavorsymmetry of the theory has a level which is off by one from the scaling dimension ofthe only Higgsable CB parameter ( v ). It is therefore reasonable to make the following– 64 –dentification T v ≡ [ I ∗ , sp (12)]. This immediately suggests that h =6 and that therank-2 theory supported on the c stratum should have a a rank-decreasing transition(though this is not necessarily the case, see a discussion in [84]). Matching the a and c central charges using (3.6a) and (3.6b) confirms the validity of our initial guess andin turns fixes the last info needed to complete the CB picture: T u + v ≡ [ I , ∅ ].To completely specify the HB structure we will employ our usual tactics. Thehiggsing of T v is of gHW type and therefore we can use (3.11) to compute the c centralcharge of the rank-2 theory supported on c . This, along with the constraint that thistheory should have a 26 dimensional HB to account to the remaining dimension, leadsto the final identification: T c ≡ S . The HB Hasse diagram in figure 4a shows thatthe constraint that we guessed from the CB structure is in fact satisfied providing afully consistent picture. sp (4) + 2 F + 2 V The analysis of this theory follows the same line as other sp (4) theories. As we men-tioned earlier, there are two inequivalent su (2)s (corresponding to a long and a shortroot) which can be left unbroken by turning on a vev for the scalar component of the sp (4) vector multiplet. For one su (2) each hypermultiplet in the contributes a mass-less fundamental flavor while those in the none. Viceversa for the other su (2) thehyper in the has no massless component while those in the contribute a hyper inthe adjoint of su (2). The result is that we expect three knotted strata, two supportinga [ I , su (2)] (which together carry the so (4) flavor symmetry of an su (2) gauge theorywith N f = 2) and the other knotted stratum supports an N = 2 su (2) gauge theorywith two adjoints. We therefore also conclude that the theory has h =2. It is possible tocheck that this stratification perfectly reproduces the a and c central charges in table36b once the appropriate b i are plugged into (3.6a)-(3.6b).The analysis of the HB is even more straightforward. In fact to identify the rank-2 theory supported on c we can use the constraint on the dimension of the HB aswell as the central charge obtained using (3.11) (the CB analysis clarifies that thisHiggsing is indeed of gHW type). These two conditions lead to the identification T c ≡ F + su (2) − su (2) + 2 F which can also be checked by solving the equation ofmotions and working directly with the vevs of the hypermultiplets. g + 4 F The analysis of the CB of this theory follows closely what we have done before. Asin the case of sp (4), G has two inequivalent su (2) factors which can remain unbrokenby turning on the vev of the scalar component of the vector multiplet. Each hyper in– 65 – p (4) + 2 F + V [ I , su (2)] [ I ∗ , sp (4)][ I , su (2)] I I (cid:2) u + v = 0 (cid:3) I ∗ (cid:2) v = 0 (cid:3) H d HB d T (1) D , a F + su (2) − su (2) + F c sp (4) + 2 F + V (a) The Hasse diagram for the CB and the HB ofthe sp (4) gauge theory with two hypermultipletsin the ⊕ . sp (4) + 2 F + V (∆ u , ∆ v ) (2,4)24 a c f k sp (4) × so (4) d HB h T ( h ) 1 (b) Central charges, CB parameters and ECBdimension. Figure 36 : Information about the sp (4) N = 2 theory with two hypermultiplets inthe ⊕ g + 4 F [ I , ∅ ] × H [ I ∗ , sp (8)][ I , ∅ ] × H (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB d T (1) D , a su (3) + 6 F c g + 4 F (a) The Hasse diagram for the CB and the HBof the G gauge theory with 4 s. g + 4 F (∆ u , ∆ v ) (2,6)24 a c f k sp (8) d HB h T ( h ) 1 (b) Central charges, CB parameters and ECBdimension. Figure 37 : Information about the g N = 2 theory with 4 hypermultiplets in the .the contributes a massless hyper in the for one su (2) and none for the other. Thisreadily gives the stratification depicted in figure 37a and sets h =4.The analysis of the HB of this theory has been performed, for example, in [86]therefore we won’t reproduce it here. It is a useful exercise to see that our geometricconstraints are perfectly reproduced by the Hasse diagram in figure 37a. ◦ – 66 – .7 Other series We here summarize the rest of the series, many of which had been understood in gooddetail already in [43, 44]. sp (8) − su (2) series This series contain two top theories and a total of six theories and all but one massdeformation can be seen geometrically from the brane realization of the correspondingtheories. The remaining one was discussed in [44]. S (2) E , This theory, which is the top theory of the sp (8) series, can be obtained inclass- S , for example in the untwisted D case [31, 35]. More recently, it was shown tobelong to an infinite set of theories called N = 2 S -fold [43], or S theories for short.This specific case, can be engineered as worldvolume theory of 2 D3 branes probingan exceptional E brane singularity in the presence of a Z S -fold [45] with flux.This theory can also be realized as the compactification of a (1,0) theory [35]. Asummary of the properties of S -theories can be found in appendix A and the CFTdata and depiction of the Hasse diagrams of both the CB and HB stratification, canbe found in figure 46. S (2) D , This theory can be obtained mass deforming the previous one. Specificallythis mass deformation is realized in the brane picture as moving away two D7 branesfrom the E exceptional brane singularity. The S D , arises probing the remaining D singularity in the presence of a fluxfull S -fold by two D3 branes. It can also beobtained in class- S , for example in the Z twisted A case and as the compactificationof a (1,0) theory [35]. A summary of the properties of S -theories can be found inappendix A and the CFT data and depiction of the Hasse diagrams of both the CBand HB stratification, can be found in figure 46. S (2) A , A similar mass deformation works in this as well. In fact this theory can beobtained by moving away another two D7 branes from the E exceptional branesingularity, or just two from the D , and probing the remaining A singularity plus S -fold with flux by two D3 branes. It can also be obtained as compactification of a(1,0) theory [35]. Currently the author is unaware of any class- S realization. Asummary of the properties of S -theories can be found in appendix A and the CFTdata and depiction of the Hasse diagrams of both the CB and HB stratification, canbe found in figure 46. T (2) A , It was one of the recently discovered T theories. It can be obtained higgsinga S (2) A , N = 2 S -fold [35, 43] or as a wordvolume theory of two D3 branes probing an– 67 – T A , [ T (1) A , ] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) H d HB b T A , S (1) ∅ , a a (a) The Coulomb and Higgs stratification of b T A , . b T A , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 38 : Information about the b T A , . theory.exceptional A brane singularity in the presence of a Z S -fold without flux [44]. Thistheory cannot be obtained by mass deformation of the previous three and it is thereforea top theory of the series. Currently the author is not aware of any class- S realizationof this theory. The properties of N = 2 T -theories are summarized in appendix A.The CFT properties as well as the stratification of this particular theory can be foundin table 45. b T A , This theory was initially found in [44] as a mass deformation of the T (2) A , . Thismass deformation is apparent from the 5 d perspective. Indeed this latter theory canbe obtained as a twisted Z compactification of a 5 d SCFT whose brane web is known[44]. The brane web gives access to the mass deformation that preserves the Z andthe twisted compactification of the resulting 5 d SCFT is what gives b T A , .Let’s move now to the analysis of the full moduli space structure. The theoryhas a single simple flavor factor compatibly with the lone Higgsable CB parameter ( u ).The level of the su (2) is double ∆ u and therefore a possible realization of the su (2) isvia a [ I , su (2)] supported on a u = 0 unknotted stratum. This identification wouldimmediately imply that h =0 and that the theory supported on the a stratum also hasno ECB. Right now we cannot exclude this possibility. Though the structure of the T (2) A , moduli space, along with general behavior of the moduli space under mass deformation[3] suggests an alternative realization with T u ≡ [ T (1) A , ] Z . This latter identification alsopredicts a su (2) and h =0 but now the theory supported on the a stratum should havea one dimensional ECB [84]. To check the validity of this working assumption, we canuse (3.6a)-(3.6b) to match the a and c central charges in figure 38b to then complete theanalysis of the CB stratification and reproduce the Hasse diagram in figure. Note thatthe [ I , su (2)] and [ T (1) A , ] Z have different b i and contirbute differently to (3.6a)-(3.6b).– 68 –t is an instructive exercise to check that the former choice simply gives no consistentsolution and we cannot reproduce the central charges from the CB analysis.Now let’s turn to the analysis of the HB. From the previous analysis we expectthat the HB would start with a a transition and we have also derived that the rank-1theory supported there should have h =1. To reproduce the total dimension of the b T A , , we immediately obtain that the entire one dimensional HB should be an ECB,which suggests that the theory must have N ≥
3. The CB analysis tells us that the a higgsing is of gHW type. Applying (3.11) to this stratum then implies 12 c T a = 21which readily implies that T a ≡ S (1) ∅ , . This concludes our anlaysis. sp (4) This is an N = 4 theory. The moduli space of these theories is extremelyconstrained and it is basically entirely specified by the Weyl group of the gauge algebrawhich in this case is D , the dihedral group of order eight. More details on the modulispace structure of theories with extended supersymmetry can be found in appendix B.The CFT data of this theory, as well as the explicit Hasse diagram of both the CB andHB stratification are depicted in figure 47. ◦ g series This series was basically already discussed in [43, 44]. In fact all the theories here belongto the same N = 2 S -fold class which can be engineered in type II B string theory, andall mass deformations connecting the theories in the series are geometrically realizedby motion of D7 branes. T (2) D , This theory, which sits at the top of the g series, can be obtained in class- S by compactifying a D (2,0) theory with a Z twist [46]. It was also recently recognizeto belong to the infinite series of T theories [35]. Specifically it can be realized as theworldvolume theory of two D3 branes probing an exceptional D brane singularity inthe presence of a fluxless Z S -fold. General properties of T -theories are summarizedin appendix A and the CFT properties as well as the stratification of this particulartheory can be found in table 45. T (2) A , This theory can be obtained by moving away three D7 branes from the excep-tional D brane singularity plus a Z fluxless S -fold and probing the remaining A singularity, in the presence of the S -fold, with two D3 branes. This action in the branesystem corresponds to a mass deformation. It belongs to the recently discovered infi-nite series of T theories [35] and it can be obtained by mass deforming T (2) D , . Generalproperties of T -theories are summarized in appendix A and the CFT properties as wellas the stratification of this particular theory can be found in table 45– 69 – T D , This theory was initially found in [44] as a mass deformation of the T (2) D , . The T (2) D , can be obtained as a twisted Z compactification of a 5 d SCFT with a known braneweb [44]. The brane web then has a deformation which preserves the Z symmetry andthe twisted compactification of the resulting 5 d SCFT is what gives b T D , . Let’s moveto the analysis of the moduli space which is anyway performed already in [44]. b T D , [ T (1) D , ] Z [ I , ∅ ] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) a H d HB b T D , S (1) A , T (1) A , g (a) The Coulomb and Higgs stratification of b T D , . b T D , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k [ g ] d HB h T ( h ) 0 (b) Central charges, CB parame-ters and ECB dimension. Figure 39 : Information about the b T D , . theory.The theory is not totally higgsable with u representing the only Higgsable CB parameter.The exceptional flavor symmetry makes it easy to identify the CB structure: T u ≡ [ T (1) D , ] Z which is also compatible with the fact that k g = 2∆ u . This immediately pre-dicts that h =0, that the HB has a g transition and that the rank-1 theory supportedon this stratum has a two dimensional ECB. We can conclude the analysis of the CB byreproducing the a and c central charges in figure 39b using (3.6a)-(3.6b) which imposesthat T u + v ≡ [ I , ∅ ].The analysis of the HB is straightforward. The constraints coming from the CBanalysis as well as the demand that the total HB being six quaternionic dimensionsuniquely identifies the rank-1 theory and we conclude that T g ≡ S (1) A , . This identi-fication is further confirmed by using (3.11) which can be applied to g and predictsprecisely the central charge of S (1) A , . su (3) This is an N = 4 theory. The moduli space of these theories is extremelyconstrained and it is basically entirely specified by the Weyl group of the gauge algebrawhich in this case is S , the symmetric group of order six. More details on the modulispace structure of theories with extended supersymmetry can be found in appendix B.– 70 –he CFT data of this theory, as well as the explicit Hasse diagram of both the CB andHB stratification are depicted in figure 47. ◦ su (3) series This series was basically already discussed in [43]. In fact all the theories here belongto the same N = 2 S -fold class which can be engineered in type II B string theory, andall mass deformations connecting the theories in the series are geometrically realizedby motion of D7 branes. S (2) D , This theory is the top theory of the su (3) series and can be obtained by probingan exceptional D brane singularity in the presence of a fluxfull Z S -fold by two D3 branes. It can also be obtained as compactification of a (1,0) theory [35].Currently the author is unaware of any class- S realization. A summary of the propertiesof S -theories can be found in appendix A and the CFT data and depiction of theHasse diagrams of both the CB and HB stratification, can be found in figure 46. S (2) A , This theory can be obtained by moving away three D7 branes from the D exceptional brane singularity and probing the remaining A singularity plus a Z S -fold with flux by two D3 branes. The motion of the three D7 brane correspondsto a mass deformation in the N = 2 theory. This theory can also be obtained ascompactification of a (1,0) theory [35]. Again, the author is unaware of any class- S realization. A summary of the properties of S -theories can be found in appendixA and the CFT data and depiction of the Hasse diagrams of both the CB and HBstratification, can be found in figure 46. G (3 , , This is an N = 3 theory. The moduli space of these theories is as con-strained as in the N = 4 case and it is basically entirely specified by a crystallographic complex reflection group (CCRG)which in this case is G (3 , , ◦ su (2) series As it was the case in the previous few series, this set of RG-flows was basically alreadydiscussed in [43]. Again, all the theories belong to the same N = 2 S -fold class and allmass deformations are geometrically realized by motion of D7 branes.– 71 – (2) A , This theory, which sits at the top of the two theories su (2) series, can beobtained by probing the A exceptional brane singularity with two D3 branes but inthe presence of a Z S -fold with fluxes. It can also be obtained as compactificationof a (1,0) theory [35]. Currently the author is unaware of any class- S realization.A summary of the properties of S -theories can be found in appendix A and the CFTdata and depiction of the Hasse diagrams of both the CB and HB stratification, canbe found in figure 46. G (4 , , This is an N = 3 theory. The moduli space of these theories is extremelyconstrained and it is basically entirely specified by a CCRG which in this case is G (4 , , ◦ All theories discussed in this section do not belong to any series. While for those whichcan be realized by brane constructions there are reasons to expect that they are indeednot connected by mass deformations to any other N = 2 theory, for the rest therearen’t really strong argument in this direction. Thus one could speculate that theremight be theories which are connected to them by RG-flows, awaiting to be discovered.The case of the lagrangian sp (4) + is particularly interesting as it currently has nostring theory realizations . Th. 64
This theory was initially obtained in the Z twisted D class- S series [34] and again theCFT data reported in figure 40b is for the most part derived from this initial analysis.We start our analysis of the full moduli space structure from the CB. The factthat this theory has two Higgsable CB parameter is reflected in the fact that the flavorsymmetry has two simple flavor factor. As we will see shortly, this will give rise to aninvolved HB Hasse diagram, expectedly. The identification of how the flavor symmetryis realized on the CB is straightforward since the level of both simple flavor factorsdiffer by one from from the CB parameters. This observation then readily leads to thefollowing identifications: T u ≡ S (1) ∅ , and T v ≡ [ I ∗ , sp (8)] which in turn imply that for This is not the only lagrangian case with no known string theory realization. Many more examplesare described in [50]. We thank Yuji Tachikawa for pointing this out. – 72 – h. 64 [ I , ∅ ] S (1) ∅ , [ I ∗ , sp (8)] (cid:2) u + v = 0 (cid:3)(cid:2) u = 0 (cid:3)(cid:2) v = 0 (cid:3) H d HB d T (1) D , a c su (3) + 6 F S (1) D , c a c Th. 26 g + 4 F a c Th. 64 (a) The Coulomb and Higgs stratification of Th. 64.
Th. 64(∆ u , ∆ v ) (4 , a c f k su (2) × sp (8) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 40 : Information about the Th. 64.this theory h = 5 and it furthermore implies that the c Higgsing supports a theory witha one dimensional ECB and a rank decreasing c transition while the a component ofthe ECB supports a theory with a four dimensional ECB [84]. This guess can be checkedby plugging in the b i for these theories in (3.6a)-(3.6b) and reproducing the a and c reported in table 40b. This also fixes the theory supported on the knotted stratumwhich is found to be our usual [ I , ∅ ].Let’s turn now to the HB. Both ECB arise from lagrangian theories supported onthe CB and are therefore of the gHW type. This makes our analysis fairly easy. Weleave it up to the careful reader to check that the results of (3.11) applied separatelyto a and c uniquely lead to the identification T a ≡ g + 4 F and T a ≡ Th .
26. Therest of the Hasse diagram in figure 40a can be filled in by following the HBs of thesetwo theories which are worked out in figure 37a and 18a respectively giving rise to thefairly intricate diagram in figure 40a.
Th. 65
This theory can be obtained, for example, in the Z twisted D class- S [46] where mostof the CFT data reported below is computed.This theory is totally higgsable with a semi-simple flavor symmetry. At first onemight be tempted to guess that the two simple flavor symmetry factors are realizedeach on an allowed unknotted stratum, u = 0 and v = 0. But a more careful lookat the value of the levels immediately reveal that the situation cannot be as simple.– 73 – h. 65[ I ∗ , sp (4)] [ I , su (2)] (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) H d HB d T (1) D , S (1) D , a F + su (2) − su (2) + 2 F c sp (4) + 2 F + 2 V c a Th. 65 (a) The Coulomb and Higgs stratification of Th. 65.
Th. 65(∆ u , ∆ v ) (4 , a c f k sp (4) × su (2) d HB h T ( h ) 2 (b) Central charges, CB parame-ters and ECB dimension. Figure 41 : Information about the Th. 65.The level of the su (2) does not create much problem. It is indeed double ∆ u thus themost natural guess is the identification T u ≡ [ I , su (2)]. The sp (4) is instead puzzling.Since the u = 0 is already “occupied” the two other options are either the v = 0 whichhas ∆ v = 8 or the knotted stratum which instead has ∆ knot = 12. The insight comesfrom the fact that h =4 while this theory would naturally support a h =2 acted upon bythe sp (4) factor (and for example realized by a [ I ∗ , sp (4)]). The resolution is that thefour quaternionic (eight complex) dimensional ECB transforms as ⊕ of the flavorsymmetry realized on the knotted stratum by a T u + v ≡ [ I ∗ , sp (4)]. This perfectlyreproduces the level but since is a new situation let’s do things explicitly. Recall (3.6c): k f = X i ∈ I f ∆ sing i d i ∆ i (cid:0) k i − T ( h i ) (cid:1) + T ( h ) . (4.15)In our case, ∆ sing i = ∆ knot = 12. The fact that the theory supported on the stratum isa [ I ∗ , sp (4)] implies d i = 1, ∆ i = 2, k i = 3 and T ( h i ≡ ) = 1. But because of ourprevious observation on the dimensionality of the ECB, T ( h ≡ ⊕ ) = 2 and thuswe reproduce the correct level of the sp (4) factor. This identification also perfectlyreproduces the a and c central charges in figure 41b using (3.6a) and (3.6b).Let’s now move to analyze the HB, this analysis will confirm the validity of ourprevious guesses. Let us first focus on the stratum associated with the CB higgsing ofthe [ I , su (2)]. This is of gHW type and thus we can use (3.11) to predict the rank-1– 74 –heory supported there which leads to T a ≡ S (1) D , (this guess could have also beenmade by observing that to match the total HB dimension, the rank-1 theory supportedon a had to have a nine dimensional HB). The other “side” of the Hasse diagram istrickier. Since we have a single sp (4) factor we expect a single c transition, yet we knowfrom the CB analysis that the ECB does not transform irreducibly under the flavorsymmetry, signaling that the theory that is supported on the c should itself have a h =2.By matching both the total dimensionality of the HB and using the (3.11) it is possibleto identify that the rank-2 theory supported on the c stratum is a sp (4) + 2 F + 2 V ,which indeed has a two quaternionic dimensional ECB as expected (this higgsing canalso be guessed from the class- S construction and can be then checked independently).Following the subsequent higgsings of the two theories supported on the first two strata,we can readily reproduce the full Hasse diagram in figure 41a. T (2) ∅ , T (2) ∅ , S (1) ∅ , [ T (1) ∅ , ] Z (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) T (1) ∅ , ×T (1) ∅ , a T (2) ∅ , k T (2) ∅ , (a) The Coulomb and Higgs stratification of T (2) ∅ , . T (2) ∅ , (∆ u , ∆ v ) (cid:0) . (cid:1) a c f k su (2) d HB h T ( h ) 1 (b) Central charges, CB parame-ters and ECB dimension. Figure 42 : Information about T (2) ∅ , .This theory is curious. It in fact belongs to an infinite series of theories firstconjectured in [40] and then shown to be consistent in [44] which naturally sit in theinfinite series of T theories [35]. It is curious since the construction using the twistedcompactification of (1,0) theories suggests a brane realization as all other T theories.Yet this would imply the presence of a Z S-fold which doesn’t seem to be allowed [45]for the simple reason that the latter are specified by finite subgroup of SL (2 , Z ) and Z simply isn’t one. For a more detailed discussion see [44].The structure of the moduli space of vacua has very much the same features of theremaining T -theories, which are again discussed in appendix A. But given the strange– 75 –ature of the case we will present the CFT data and the depiction of the moduli spacestratification separately. All the relevant info are summarized in figure 42. sp (4) w/ The existence of this theory is pointed out in [50]. It is important to stress once againthat to the author’s knowledge, no string theory realization of this theory is known.Let’s discuss here how to derive the result depicted in figure 43a.The analysis of the HB is obvious. Since there is a single half-hypermultiplet nogauge invariant operator can be made from the hypermultiplets and the HB is trivial.The analysis of the CB is instead more involved. sp (4) w/ [ I , ∅ ] [ I , ∅ ][ I , ∅ ] [ I , ∅ ][ I , ∅ ] (cid:2) u + v = 0 (cid:3) sp (4) w/ (a) The Hasse diagram for the CB and the HBof the sp (4) gauge theory with a half . sp (4) w/ (∆ u , ∆ v ) (2,4)24 a c f k ∅ d HB h T ( h ) 0 (b) Central charges, CB parameters and ECBdimension. Figure 43 : Information about the sp (4) N = 2 theory with a-half hypermultiplet inthe .As mentioned in the discussion of other sp (4) cases, there are two inequivalent su (2).By carefully decomposing the we obtain that the half-hypermultiplet contributesmassless matter only to one of these two su (2)s. Thus along one direction the low energytheory is effectively a pure su (2) N = 2 gauge theory while along the other is an su (2)with a single hypermultiplet in the . Both of these theories are asymptotically freeand therefore they flow to strong coupling in the IR which causes the knotted stratumto spilt. The pure su (2) theory contributes two [ I , ∅ ] while the other has an extra[ I , ∅ ] coming from the massless hyper. Finally there is an extra [ I , ∅ ] which arisesby tuning the CB vev but with an u (1) commutant. To check that the six knottedsingularities each supporting an [ I , ∅ ] are correct, we can plug things in the centralcharge formulae (3.6a)-(3.6b) and perfectly reproduce the expected values which arereported in table 43b. – 76 – This is an N = 4 theory. The moduli space of these theories is extremely constrainedand it is basically entirely specified by the Weyl group of the gauge algebra which inthis case is D , the dihedral group of order twelve. More details on the moduli spacestructure of theories with extended supersymmetry can be found in appendix B. TheCFT data of this theory, as well as the explicit Hasse diagram of both the CB and HBstratification are depicted in figure 47.Also, this theory was shown to be realizable as worldvolume theory of two D3 branes probing a fluxfull Z S -fold [45]. Since no exceptional seven brane singularityis compatible with the presence of an Z S -fold [43], it reasonable that this theory isindeed isolated. Acknowledgments
I would like to thank G. Zafrir for a very enjoyable and fruitful correspondence duringwhich he clarified countless issues for me, many of which made it into this manuscript.I would also like to thank P. Argyres, A. Bourget, J. Distler, J. Grimminger, A. Roc-chetto, S. Schafer-Nameki, Y. Tachikawa and G. Zafrir for comments on the manuscript.Finally I benefited tremendously from many exchanges with C. Beem, A. Bourget,J. Distler, S. Giacomelli, J. Grimminger, A. Hanany, C. Meneghelli, W. Peelaers, L.Rastelli, S. Schafer-Nameki and Y. Tachikawa. I am extremely grateful for these in-teractions which illuminated many details of the constructions which made this paperpossible. M.M. gratefully acknowledges the Simons Foundation (Simons Collaborationon the Non-perturbative Bootstrap) grants 488647 and 397411, for the support of hiswork. A S and T theories This set of theories has been introduced recently by generalizing the N = 3 S -foldset up [45, 126] to the N = 2 case [35, 43] as well as generalizing the “classic” F-theory N = 2 theories [98–101]. This construction thus involves considering the D3 brane worldvolume theory probing an exceptional 7brane in the presence of an S -fold. Depending on whether the S -fold has fluxes turned on or off, we get S or T theories respectively [44]. For a given exceptional 7brane, there is a restricted set of S -folds which are allowed. We won’t review this discussion here and instead refer theinterested reader to the original paper [43]. It is important to notice that both the S and T theories can be also obtained as compactification of 6 d (1,0) theories with non-commuting holonomies [35, 40]. Since the (1 ,
0) theories are most naturally constructed– 77 –n M -theory, the construction of S and T theories suggest potentially interesting dualitybetween F and M theory [44].Both the S and T theories have been studied in depth, particularly recently [1,1, 35, 43, 44, 58, 127, 128]. Rather than literally reproducing here results from otherpapers, we make the choice of simply refer to the relevant literature. Namely:1) For a discussion of the general structure of the moduli space see [1, 35, 43]. TheHB Hasse diagram has instead been worked out in detail and for general ranksin [127].2) For a discussion of the CB stratification see [1, 44] (one of the two references alsocontain a discussion of the mass deformations among these theories).3) For a discussion of the generalized free-field VOA construction see [35, 97].Below we will then simply summarize the results with various tables and the explicitstratification. To better organize the presentation we will collect the relevant CFT datain three figures: a ) In figure 44 we will collect all the info for the T (2) G, . Those correspond to the well-known 4d theories rank-2 theories arising on a worldvolume of two D3 branesprobing an exceptional G brane singularity [98–101]. These theories they areall connected to one another by mass deformations. b ) In figure 45 we collect the information for the T theories which arise when the D3 probe a G brane singularity plus a fluxless Z ℓ S-fold: T (2) G,ℓ> . For fixed ℓ , thesetheories are connected to one another by mass deformation. They also eventuallyflow to N = 4 theories with gauge algebra su (2) × su (2) for ℓ = 2, su (3) for ℓ = 3and sp (4) for ℓ = 4 [44]. c ) Finally figure 46 collects the information for the S theories which arise insteadwhen two D3 probe a G brane singularity plus a fluxfull Z ℓ S-fold [43]. Againthese theories are connected with one another for a given ℓ and they flow to N = 4 sp (4) for ℓ = 2 and N = 3 G (3 , ,
2) and G (4 , ,
2) theories for ℓ = 3 and ℓ = 4respectively.We will separately discuss the case of T (2) ∅ , in the text because its moduli spacestructure it does not perfectly fit in the homogenous analysis of the remaining theories.This would also allow us to point out a few interesting features of this case. T (2) G, theories correspond to the well-known 4d theories rank-2 theories arising in type II B byprobing and exceptional G – 78 – FT data of T (2) G, theories T (2) E , (∆ u , ∆ v ) (6 , a c f k [ e ] × su (2) d HB h T ( h ) 1 (a) T (2) E , (∆ u , ∆ v ) (4 , a c f k [ e ] × su (2) d HB h T ( h ) 1 (b) T (2) E , (∆ u , ∆ v ) (3 , a c f k [ e ] × su (2) d HB h T ( h ) 1 (c) T (2) D , / sp (4) + 4 F + V (∆ u , ∆ v ) (2 , a c f k so (8) × su (2) d HB h T ( h ) 1 (d) T (2) A , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (3) × su (2) d HB h T ( h ) 1 (e) T (2) A , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) × su (2) d HB h T ( h ) 1 (f) T (2) ∅ , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) d HB h T ( h ) 1 (g) Moduli space structrure T (2) G, S (1) ∅ , T (1) G, (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB g T (1) G, g g T (1) G, ×T (1) G, T (1) G, × H a × a g T (2) G, Figure 44 : In this figure we report the relevant CFT data for the T (2) G, theories.– 79 – FT data of T (2) G,ℓ> theories T (2) D , (∆ u , ∆ v ) (4 . a c f k [ g ] × su (2) d HB h T ( h ) 1 (a) T (2) A , (∆ u , ∆ v ) (cid:0) , (cid:1) a c f k su (2) × su (2) d HB h T ( h ) 1 (b) T (2) E , (∆ u , ∆ v ) (6 , a c f k [ f ] × su (2) d HB h T ( h ) 2 (c) T (2) D , (∆ u , ∆ v ) (4 , a c f k so (7) × su (2) d HB h T ( h ) 2 (d) T (2) A , (∆ u , ∆ v ) (3 , a c f k su (3) × su (2) d HB h T ( h ) 2 (e) T (2) A , (∆ u , ∆ v ) (3 , a c f k su (2) × su (2) d HB h T ( h ) 2 (f) Moduli space structrure ℓ = 3 T (2) G, odd S (1) ∅ , [ T (1) G,ℓ ] Z ℓ (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) ℓ = 2 , T (2) G, even S (1) ∅ , S (1) ∅ , [ T (1) G,ℓ ] Z ℓ (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB e T (1) G,ℓ g h m,ℓ T (1) G,ℓ ×T (1) G,ℓ k ℓ S (1) G,ℓ T (2) G,ℓ f a T (2) G,ℓ
Figure 45 : In this figure we report the relevant CFT data for the T (2) G,ℓ> theorieswhere the m in h m,ℓ is equal to 4, 3, 2, 2, 1, 1 for ( E , D , D , A , A ,
2) and ( A ,
3) respectively and the k ℓ slice is defined, for example, in [127, Eq.(C.6)]. – 80 – FT data of S (2) G,ℓ theories S (2) E , (∆ u , ∆ v ) (6 , a c f k sp (8) × su (2) d HB h T ( h ) 3 (a) S (2) D , (∆ u , ∆ v ) (4 , a c f k sp (4) × su (2) × su (2) d HB h T ( h ) 3 (b) S (2) A , (∆ u , ∆ v ) (3 , a c f k su (2) × su (2) × u (1) d HB h T ( h ) 3 (c) S (2) D , (∆ u , ∆ v ) (6 , a c f k su (3) × u (1) d HB h T ( h ) 2 (d) S (2) A , (∆ u , ∆ v ) (4 , a c f k u (1) d HB h T ( h ) - (e) S (2) A , (∆ u , ∆ v ) (6 , a c f k su (2) × u (1) d HB h T ( h ) - (f) Moduli space structrure S (2) G,ℓ S (1) ∅ , S (1) G,ℓ (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) H d HB g T (1) G,ℓ g h m +1 ,ℓ T (1) G,ℓ ×T (1) G,ℓ k ℓ S (1) G,ℓ T (2) G,ℓ g g k ℓ T (2) G,ℓ S (1) G,ℓ ×T (1) G,ℓ a h m,ℓ S (2) G,ℓ
Figure 46 : In this figure we report the relevant CFT data for the S (2) G,ℓ theories wherethe m in h m,ℓ is equal to 4, 3, 2, 2, 1, 1 for ( E , D , D , A , A ,
2) and( A ,
3) respectively and the k ℓ slice is defined, for example, in [127, Eq. (C.6)]. – 81 – Theories with enhanced supersymmetry
Theories with enhanced (
N ≥
3) supersymmetry have a much tighter moduli spacestructure. The N = 4 case has been discussed for many decades, e.g. [129], while N = 3 theories have been constructed considerably more recently [45, 126, 130]. Thetwo cases bear many similarities; the metric on the entire moduli space is flat (see e.g. [131]) and the R -symmetry group enhancement ties in the structure of the CB and theHB of these theories giving rise to a mathematical structure on the entire moduli spacewhich has been deemed triple special K¨ahler (TSK) [132]. In particular it implies thatall the theories which appear on singular starta have to be themselves N ≥
3. Wewon’t delve further into the details of this construction here and only mention that allknown
N ≥ M = C r / Γ (B.1)where r is the rank of the theory and Γ is a CCRG [133, 134] which preserves a principalpolarization [68]. The singular locus of orbifold moduli spaces of vacua (B.1) can beeasily determined by studying the fix locus of the Γ action. The case in which Γ isa real reflection group gives rise to the N = 4 case with a Lie algebra g where Γ isnaturally associated to the Weyl group of the g .It is important to notice that imposing the condition of N ≥ N = 4 case [135], the structure of the moduli spaceof vacua alone does not uniquely specify a theory.Already at rank-2 the situation is rich. A systematic analysis of the allowed orbifoldTSK geometries at rank-2 was performed not long ago [72] with the result that only asmall subset of allowed geometries has been realized as N = 3 geometries. Since theCCRG largely determines the full structure of the moduli space of vacua of N ≥ N = 4 and in figure 48 for N = 3 theories.– 82 – FT data of N = 4 theories Γ = S N = 4 su (3) S (1) ∅ , (cid:2) u + v = 0 (cid:3) u (1) × u (1) a S (1) ∅ , N = 4 su (3) N = 4 su (3)(∆ u , ∆ v ) (2 , a c f k su (2) d HB h T ( h ) 2 ◦ N = 4 su (2) × su (2)(∆ u , ∆ v ) (2 , a c f k su (2) d HB h T ( h ) 2 N = 4 su (2) × su (2) S (1) ∅ , S (1) ∅ , (cid:2) u + v = 0 (cid:3) (cid:2) u = 0 (cid:3) u (1) × u (1) a a S (1) ∅ , a S (1) ∅ , a N = 4 su (2) × su (2) Γ = Z × Z ◦ Γ = D N = 4 sp (4) S (1) ∅ , S (1) ∅ , (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) × u (1) a a S (1) ∅ , a S (1) ∅ , A N = 4 sp (4) N = 4 sp (4)(∆ u , ∆ v ) (2 , a c f k su (2) d HB h T ( h ) 2 ◦ N = 4 g (∆ u , ∆ v ) (2 , a c f k su (2) d HB h T ( h ) 2 N = 4 g S (1) ∅ , S (1) ∅ , (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) × u (1) a a S (1) ∅ , a S (1) ∅ , A N = 4 g Γ = D Figure 47 : CFT data for N = 4 rank-2 theories.– 83 – FT data of N = 3 theories Γ = G (3 , , S (2) ∅ , S (1) ∅ , S (1) ∅ , (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) × u (1) a a S (1) ∅ , a S (1) ∅ , A S (2) ∅ , N = 3 S (2) ∅ , (∆ u , ∆ v ) (2 , a c f k su (2) d HB h T ( h ) 2 ◦ N = 3 S (2) ∅ , (∆ u , ∆ v ) (4 , a c f k u (1) d HB h T ( h ) 2 S (2) ∅ , S (1) ∅ , S (1) ∅ , (cid:2) u + v = 0 (cid:3) (cid:2) v = 0 (cid:3) u (1) × u (1) a a S (1) ∅ , a S (1) ∅ , A S (2) ∅ , Γ = G (4 , , Figure 48 : CFT data for N = 3 rank-2 theories. C Flavor structure along the Higgsing and generalized freefields VOA
In this section we quickly sketch how the information in table 6, 7 and 8 can be leveragedto determine the VOA of the various entries. It is important to stress that we havenot performed any in-depth calculation and the presentation in this appendix shouldbe intended as a sketch and not as an actual costruction.Let’s start from the basics. To any four-dimensional N = 2 SCFT T , one cancanonically associate a two-dimensional VOA [54], χ : 4d N = 2 SCFT −→ VOA . (C.1)The VOA χ [ T ] arises as a cohomological reduction of the full local OPE algebra of afour-dimensional theory T with respect to a certain nilpotent supercharge. We won’treview any of the details here but mention that there are numerous indications that χ [ T ] is deeply connected with the physics of the HB H . The full extent of the connec-tion remains somewhat elusive but a remarkable fact, observed in many examples and– 84 –onjectured to be universally true [109], is that H can be recovered directly from χ [ T ] H = X χ [ T ] , (C.2)where X V denotes a symplectic variety canonically associated to V called the associatedvariety of a VOA V [136].In [96, 97], striking evidence was provided that data associated with the Higgsbranch physics of a theory may be sufficient to determine the full VOA by studyingthe theory on a higgs stratum S i which supports a non-trivial SCFT T i . The gist ofthis construction is that the VOA of the initial theory T can be written in terms of freefields which parametrize S i and the VOA generators of χ [ T i ], the VOA of T i . Thisconstruction was deemed a generalized free-field construction in [96, 97] where theprescription to build χ [ T ] from the information about the effective field theory (EFT) T i was explained. The general picture can be summarized as follows: χ [ T ] ⊂ χ [ T i ] ⊗ V i free [ S i ] , T i = T ≀ x, x ∈ S i (C.3)Here V i free [ S i ] is a free field VOA for which X V i free [ S i ] = S i . Also we have adopted thenotation introduced in [137], where ≀ signifies “supported on”.These generalized free-field realizations are remarkable and handy in many ways, e.g. , they realize the simple quotient of χ [ T ], that is null vectors vanish on the nosewhen expressed in terms of free fields. They have also been used to characterize many N = 2 SCFTs as they give a tool to “invert the higgsing” [35]. For example the flavorlevel of a simple factor of the SCFT T can be related to properties of the stratum S f which arises by spontaneous breaking of the f , thus the subscript, and the theorysupported over it T f . The basic formula reads: k f = T ( R ) + I f IR ֒ → f UV k f IR I f ♮ ֒ → f (C.4)this of course requires some explanation. f is obvious. I h ֒ → h indicates the index ofembedding of h into h , f ♮ is what is left unbroken on the generic point of S f of f UV ,the full flavor symmetry of T , f IR is the subgroup of the T f flavor symmetry realized onthe component and T ( R ) is Dynkin index of the f ♮ representation of the goldstoneboson associated to the higgsing.This data for each one of the theories discussed in this paper, is reported in table There are simple cases where T i is itself a theory of free fields, in which case the constructionprovides an actual free-fields realization. We use a somewhat unusual normalization where the n of su ( n ) has T ( n ) = 1. – 85 –, 7 and 8. Here we will not provide a careful realization of the VOAs of these theoriesbut only discuss a couple of examples to explain how to check that (C.4) applies in allcases using the information in tables. We take this as a suggestion that a generalizedfree field realization exists for all theories discussed in this paper. D ( E ) Let’s start from a simple case. For this theory there is a single simple factor, e , and the stratum associated with it is its minimal nilpotent orbit, S e ≡ e . Fromtable 5 we readily obtain that f ♮ = e which will have k e = 20 and the goldstone bosonstransform in the of e with T ( ) = 12, as in fact is reported in the correspondingentry in table 6. The index of embedding of e inside e is one as it can be computed bythe decomposition of any e irreducible representations in e ⊕ su (2) ones (for example → ( , ) ⊕ ( , ) ⊕ ( , )). The theory supported on this stratum is T e ≡ T (1) E , and thus k f IR = 8. This information is enough for the reader to check that (C.4) isindeed satisfied. Th. 2
The matching of the level is not always a simple as the previous example.Consider now another theory of the e − so (20) series, namely theory Th. 2. As discussedin the corresponding section, the HB stratum associated to the simple flavor factoris the minimal nilpotent orbit of so (20) : S so (10) ≡ d . From table 5 we obtainthat f ♮ = so (16) × su (2), in particular f ♮ is semi-simple, and that the goldstone bosonstransform in the ( , ). On the other hand the theory supported on this stratum is T so (20) ≡ T (1) E , which has flavor symmetry e and we noticed that so (16) is a maximalsubalgebra of e , thus f IR = so (16) at level k so (16) = 12. Using the fact that I so (16) ֒ → e = 1and that in our normalization T ( ) = 2, we can immediately reproduce the result weare after: k so (20) = 16. But what about the extra su (2)? It arises from the breakingof the initial so (20), so we expect the level of this su (2) to be also 16. Since we have“used up” all the moment maps of the rank-1 SCFT on the stratum, our only hope isthat the goldstone bosons alone can make up for that. It is a nice surprise to noticethat indeed the goldstone transform as 16 copies of the fundamental of this su (2) andwe are working in a normalization such that T ( ) = 1.In a similar manner, the data in tables 6, 7 and 8 can be leveraged to show theconsistency of (C.4) of the theories discussed in this paper. D Reading tools
To ease the reader accessibility to the content of the paper, here we will collect aGlossary and list of Acronyms and Symbols:– 86 – cronyms
AD theory
Argyres-Douglas theory.
Glossary:
Argyres-Douglas theory, 9, 22, 33, 34,36, 39, 88 CB Coulomb Branch.
Glossary:
CB, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19,20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 82, 88, 89, 90
CCRG crystallographic complex reflection group.
Glossary:
CCRG, 71, 72, 82
ECB
Enhanced Coulomb Branch.
Glossary:
ECB, 4, 5, 6, 16, 19, 23, 24, 26, 27, 28,29, 30, 31, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53,54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 68, 69, 70, 73, 74, 75, 76, 90 gHW generalized highest weight Higgsing.
Glossary: gHW, 18, 24, 25, 26, 27, 28, 29,35, 41, 42, 43, 46, 49, 57, 58, 59, 65, 69, 73, 74 HB Higgs Branch.
Glossary:
HB, 4, 5, 6, 7, 10, 11, 12, 15, 17, 18, 19, 20, 21, 22, 23,25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47,48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73,74, 75, 76, 77, 78, 82, 84, 86, 89, 90 MB Mixed Branch.
Glossary:
MB, 11, 12, 36, 38, 55 MN Minahan-Nemeschansky. 22, 25, 27
SCFT
Superconformal Field Theory. 2, 3, 4, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19,20, 21, 22, 26, 38, 39, 43, 44, 45, 46, 48, 50, 52, 58, 61, 68, 70, 84, 85, 86, 88, 89,90 SW Seiberg-Witten. 16, 33, 35, 36, 37, 38
TSK triple special K¨ahler [132]. 82 – 87 – lossary adj hypermultiplet in the adjoint representation. 48, 50
Argyres-Douglas theory
An AD theory is characterized by having at least one CBparameter of dimension ∆ < perspective, it makes little qualitative difference whether a given scaling dimen-sion is integer or fractional, whereas the existence of a chiral relevant deformationdoes. 9 AS hypermultiplet in the antisymmetric representation. 30, 44, 46, 48, 50 CB The Coulomb Branch (CB) of a N = 2 SCFT is a branch of the moduli spacewhere only the u (1) r component of the theory’s R-symmetry is spontaneouslybroken. 4 CCRG a crystallographic (point) complex reflection group is a finite group which isgenerated by complex reflections and which acts on a lattice. 71 central charge formulae the formulae (3.6a)-(3.6c) which allow to compute the a , c and k f central charges of an arbitrary SCFT from CB stratification data. 15, 25,36, 38, 40, 48, 50, 52, 54, 56, 61, 62, 63 ECB
The Enhanced Coulomb Branch (ECB) is a branch of the moduli space wherethe SCFT T is Higgsed to a theory of the same rank. 4 elementary slice The transverse slice among two adjacent symplectic leaves. Ele-mentary slices represent the individual transition depicted in the Hasse diagramand correspond to minimal higgsings. 17, 18, 19, 35, 48, 49, 51, 52, 61 F hypermultiplet in the fundamental representation. 9, 20, 30, 31, 32, 44, 46, 48, 50,54, 55, 64, 65, 66, 73, 74, 75, 79 gHW We conjecture that these are the Higgsings which are realized on the CB ashighest weight Higgsings. A generalized highest weight Higgsing is a Higgsingwhich respects (3.11).. 18 – 88 – asse diagram
A Hasse diagram is a graphical depiction of a partially ordered set(poset). A point is drawn for each element of the poset, and line segments aredrawn between these points according to the following two rules:1. If x < y in the poset, then the point corresponding to x appears lower inthe drawing than the point corresponding to y .2. The line segment between the points corresponding to any two elements x and y of the poset is included in the drawing iff x covers y or y covers x .. 11, 17, 19, 23, 31, 32, 33, 34, 35, 36, 37, 39, 41, 45, 46, 47, 48, 49, 51, 52, 53,54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78 HB The Higgs Branch (HB) of a N = 2 SCFT is a branch of the moduli space whereonly the su (2) R component of the theory’s R-symmetry is spontaneously broken.4 Higgsable CB parameter
A CB parameter u i is higgsable if it can support a non-trivial (either IR-free or SCFT) rank-1 theory T u i giving rise to an unknottedstratum u i = 0 of the CB stratification. 14, 23, 24, 27, 28, 39, 46, 52, 56, 57, 59,61, 64, 68, 70, 72, 90 knotted stratum The scaling action on the CB imposes that the singular locus S hasto be closed under it. This in turn tremendously constraints the type of complexco-dimension one singularity which can appear at rank-2. A knotted stratum isone such connected complex co-dimension one stratum which algebraically can bewritten as u p + v q = 0. Where p, q ∈ Z are completely fixed by scaling invariance( i.e. homogeneity of the polynomial) and gcd( p, q ) = 1. Any interacting rank-2SCFT must possess at least a knotted stratum. 13, 14, 16, 19, 24, 28, 30, 31, 35,37, 38, 40, 43, 46, 48, 54, 57, 58, 59, 65, 73, 74, 76 linear Symplectic leaves of a symplectic singularities ( e.g.
HBs of N = 2 SCFTs) forma partially order set, were the ordering is given by the inclusion of their closures,and therefore can be represented by a Hasse diagram. When the symplectic leavesare instead totally ordered the corresponding Hasse diagram can be depicted asa straight line. In this case we call the HB a linear HB. 23, 25, 39, 42, 45, 46,57, 59, 64 MB The Mixed Branch (MB) of a N = 2 SCFT is a branch of the moduli space wherethe whole theory’s R-symmetry is spontaneously broken.. 11– 89 – hypermultiplet in the symmetric representation. 54, 55 scaling action C ∗ action defined on the moduli space of vacua, arising from the spon-taneous breaking of the R + × u (1) ∗ symmetry, with u (1) ∗ = u (1) r or the cartanof the su (2) R depending on whether we are on the CB or the HB. 12, 13, 89, 90 totally higgsable An SCFT is totally higgsable if all its CB parameters are Higgsable CB parameter.Becauseof the UV-IR simple flavor condition, totally higgsable SCFTs have in generalnon-simple flavor factors and intricate HB Hasse diagrams. 26, 28, 29, 39, 40, 42,45, 46, 48, 50, 52, 53, 56, 59, 61, 62, 63, 70, 73 unknotted stratum
The scaling action on the CB imposes that the singular locus S has to be closed under it. This in turn tremendously constraints the type ofcomplex co-dimension one singularity which can appear at rank-2. A unknottedstratum is one such connected complex co-dimension one stratum which alge-braically can be written either as u = 0 or v = 0. Where u ( v ) must be aHiggsable CB parameter. For any interacting rank-2 SCFT there can be at mostone such unknotted stratum for each Higgsable CB parameter. 13, 14, 16, 19, 20,21, 22, 23, 35, 37, 39, 40, 43, 46, 50, 51, 53, 63, 68, 73 UV-IR simple flavor condition
The UV-IR simple flavor condition [2], or F-conditionfor brevity, states that simple flavor factors f of SCFTs of arbitrary rank, are re-alized (with possible rank-preserving enhancement) as the flavor symmetries of(at least one) rank-1 theory T i . 15, 23, 25, 38, 48, 88, 90 V hypermultiplet in the vector representation. 9, 30, 59, 60, 65, 66, 74, 75, 79 Symbols T i Rank-1 theory supported on a CB stratum of co-dimension 1. 15, 16, 90 b i Contribution of a T i to the central charge formulae. 16, 23, 28, 29, 41, 43, 61, 65,68, 73 h Total quaternionic dimension of the ECB of the theory. 4, 5, 6, 11, 16, 23, 25, 26,27, 29, 32, 39, 42, 43, 45, 46, 49, 53, 54, 55, 56, 58, 59, 61, 65, 66, 68, 69, 70, 73,74, 75 – 90 – eferences [1] P. C. Argyres and M. Martone,
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