Tinkertoys for the Twisted $E_6$ Theory

2.4 MB·English

by  Oscar Chacaltana

UTTG–XX–15
TCC–XXX–15
ICTP–SAIFR/2015–001
January 1, 2015
Tinkertoys for the Twisted E Theory
6
5
1
0
2

n Oscar Chacaltana a, Jacques Distler b and Anderson Trimm b
a
J

2
a ICTP South American Institute for b Theory Group and

] Fundamental Research, Texas Cosmology Center
h
Instituto de F´ısica Te´orica, Department of Physics,
t
-
Universidade Estadual Paulista, University of Texas at Austin,
p
e 01140-070 S˜ao Paulo, SP, Brazil Austin, TX 78712, USA
h
[
Email: [email protected] Email: [email protected]

1
Email: [email protected]
v
7
5
3
0
0
.
1 Abstract
0
5
We study 4D N = 2 superconformal field theories that arise as the compactification of the
1
: six-dimensional (2,0) theory of type E on a punctured Riemann surface in the presence of
v 6
i Z outer-automorphism twists. We explicitly carry out the classification of these theories
X 2
in terms of three-punctured spheres and cylinders, and provide tables of properties of the
r
a Z -twisted punctures. An expression is given for the superconformal index of a fixture with
2
twisted punctures of type E , which we use to check our identifications. Several of our
6
fixtures have Higgs branches which are isomorphic to instanton moduli spaces, and we find
thatS-dualitiesinvolvingthesefixturesimplyinterestingisomorphismsbetweenhyperKa¨hler
quotients of these spaces. Additionally, we find families of fixtures for which the Sommers-
Achar group, which was previously a Coulomb branch concept, acts non-trivially on the
Higgs branch operators.
Contents
1 Introduction 1
2 The twisted E theory 1
6
2.1 The Hitchin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 k-differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Tinkertoys 3
3.1 Twisted punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Free-field fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Interacting fixtures with one irregular puncture . . . . . . . . . . . . . . . . 6
3.4 Interacting fixtures with enhanced global symmetry . . . . . . . . . . . . . . 6
3.5 Mixed fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.6 Gauge theory fixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Global symmetries and the superconformal index 17
4.1 Superconformal index for twisted fixtures . . . . . . . . . . . . . . . . . . . . 17
4.2 Higher-order expansion of the index . . . . . . . . . . . . . . . . . . . . . . . 17
5 Enhanced global symmetries: the Sommers-Achar group on the Higgs
branch 18
6 R 22
2,5
7 Product SCFTs 24
8 Instanton moduli spaces 28
(cid:16)
8.1 M(E ,2)///SU(3) (cid:39) M(E ,1)×M(E ,1)×H7(cid:1)///G . . . . . . . . . . . . 29
6 6 6 2
(cid:16) (cid:17)
8.2 M(E ,2)×H ///SU(2) (cid:39) M(E ,1)///SU(3) . . . . . . . . . . . . . . . . . 30
6 8
(cid:16) (cid:16) (cid:17)
8.3 M(E ,3)///Spin(8) (cid:39) M(E ,1)3×H26(cid:1)///F and M(E ,2)×M(E ,1) ///Spin(8) (cid:39)
7 7 4 7 7
(cid:16)
M(E ,1)3 ×H9(cid:1)///Spin(9) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7
(cid:16)
8.4 (M(E ,2)×H32)///Spin(12) (cid:39) M(E ,1)×M(E ,1)×H45(cid:1)///Spin(13) . . 35
8 8 8
8.5 Semi-simple quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8.6 More isomorphisms among hyperKa¨hler quotients . . . . . . . . . . . . . . . 38
9 Instanton moduli spaces as affine algebraic varieties 41
Acknowledgements 45
Appendix A Constraints 46
Appendix B Appendix: Embeddings of SU(2) in F 47
4
Appendix C Projection matrices 48
1. Introduction
Inrecentyears, remarkableprogresshasbeenmadeinthestudyof4D N = 2superconformal
field theories by realizing them as partially-twisted compactifications of 6D (2,0) theories of
type j = A,D,E on a punctured Riemann surface, C [1,2,3,4,5,6,7]. In addition to ordinary
N = 2 gauge theories, this class of theories (sometimes called “class S”) contains many
strongly-interacting SCFTs, with no known Lagrangian description [8,9].
An even larger class of theories can be constructed by including punctures which carry
a non-trivial action of the outer-automorphism group of j [10,11]. In a series of papers
[12,13,11,14,15,16,17], we have presented a method of classification of these theories. By
listing the allowed three-punctured spheres (“fixtures”), and the cylinders connecting them,
whichcanoccurinapantsdecompositionofC, andgivingtherulesforgluingthesetogether,
one can build up an arbitrary theory in this class. Different pants decompositions of the
same surface C give different weakly-coupled presentations of the same theory, related by
S-duality.
In this paper, we turn our attention to the theories obtained by compactifying the (2,0)
theory of type E in the presence of punctures twisted by a Z outer-automorphism. (The
6 2
analysis of the untwisted theories of type E can be found in [16].) The twisted punctures
6
are in 1-1 correspondence with embeddings ρ : su(2) (cid:44)→ f , and we label them by the Bala-
4
Carter label of the corresponding nilpotent orbit. For a given puncture, we compute all the
local properties which contribute to determining the 4D N = 2 SCFT and record them in
Table 3.1. We also determine a projection matrix implementing the branching rule under
each embedding, which we use to compute the expansion of the superconformal index. These
can be found in Appendix C.
2. The twisted E theory
6
2.1. The Hitchin system
For a choice of Riemann surface C, the compactification of the 6D (2,0) theory of type
E on R3,1 × C yields a 4D N = 2 theory on R3,1. The (2,0) theory of type E has an
6 6
outer-automorphism group which is isomorphic to Z . This allows us to introduce a class of
2
“twisted” punctures, around which the fields on C undergo a monodromy by a non-trivial
element of the outer-automorphism group1. The properties of these punctures are listed in
table 3.1.
In [16] we studied the theories that arise from compactifying the E (2,0) on a Riemann
6
surface with untwisted punctures. These punctures are classified by nilpotent orbits in the
complexified Lie algebra e , and obey a Hitchin boundary condition of the form
6
A
Φ(z) = +e
6
z
where Φ is the Higgs field, z is a local coordinate on C such that the puncture is at z = 0,
1We also allow for the fields on C to undergo a monodromy upon traversing a homologically non-trivial
cycle.
1
A is a nilpotent element in e , and e in the boundary condition above denotes a generic
6 6
element of e (or a regular function of z taking values in e ).
6 6
By contrast, twisted punctures are classified by nilpotent orbits in the complexified Lie
algebra f , and obey a twisted boundary condition,
4
A o
−1
Φ(z) = + +f
z z1/2 4
Here, we have split e into eigenspaces under the action of the Z outer-automorphism, as
6 2
e = f ⊕o , where f (o ) is the even (odd) eigenspace. Also, A is a nilpotent element in
6 4 −1 4 −1
f , and o and f above represent generic elements in the respective spaces.
4 −1 4
2.2. k-differentials
We use the basis of E Casimir k-differentials {φ ,φ ,φ ,φ ,φ ,φ } of our previous paper
6 2 5 6 8 9 12
[16]. For the reader’s convenience, we repeat here how to construct this basis in terms of the
trace invariants P = Tr(Φk) for Φ in the adjoint representation of E .
k 6
φ = 1 P
2 48 2
(cid:0) (cid:1)
φ = 1 P − 7 (P )3
6 24 6 4608 2
(cid:0) (cid:1)
φ = 1 P − 2P P + 155 (P )4
8 30 8 9 6 2 663552 2
(cid:0) (cid:1)
φ =− 1 P − 17P P + 77 P (P )2 − 427 (P )5
10 105 10 96 8 2 6912 6 2 63700992 2
(cid:0) (cid:1)
φ = 1 P − 107P P + 515 P (P )2 − 41 (P )2 + 295 P (P )3 − 5669 (P )6
12 155 12 504 10 2 32256 8 2 108 6 497664 6 2 9172942848 2
(cid:0)
φ = 1 P − 3479 P P + 61391 P (P )2 − 539 P P − 139733 P (P )3
14 4389 14 14880 12 2 3214080 10 2 2160 8 6 617103360 8 2
(cid:1)
+ 165781 (P )2P − 3488947 P (P )4 + 19596907 (P )7
4821120 6 2 44431441920 6 2 409480168734720 2
These relations define all the Casimirs except φ and φ . These can be computed from φ
5 9 10
and φ , which factorize as
14
φ = φ2,
10 5
φ = φ φ .
14 5 9
Notice that the choice of sign of φ determines also the sign of φ . This is precisely the
5 9
action of the Z outer-automorphism of E on the Casimir k-differentials,
2 6
φ (cid:55)→ −φ
5 5
φ (cid:55)→ −φ
9 9
φ (cid:55)→ φ , k = 2,6,8,12
k k
So, we can expect that the leading pole orders of the φ for twisted punctures will be half-
k
integer for k = 5,9, and integer for k = 2,6,8,12, corresponding to the orders of the Casimirs
of F .
4
2
3. Tinkertoys
We find 2078 fixtures with 3 regular punctures, two twisted and one untwisted, which cor-
respond to either an interacting SCFT, a mix of an interacting SCFT and free hypers, or
a gauge theory. Of these, we find 1757 SCFTs without global symmetry enhancement, 122
SCFTs with enhanced global symmetry, 32 mixed fixtures, and 167 gauge theory fixtures.
Additionally, there are 23 fixtures with one irregular puncture: 15 free-field fixtures, 6
interacting fixtures, 1 mixed fixture, and 1 gauge theory fixture.
Below, we give tables of the twisted punctures and their properties, as well as tables
of the twisted fixtures. For the mixed fixtures, we list {d } and (n ,n ) of the interacting
k h v
SCFT,andtherepresentationofthefreehypermultiplets. Wedonotlistthefixtureswithout
global symmetry enhancement, as their properties can be readily computed from the tables
of punctures. Tables of untwisted punctures and fixtures can be found in our previous paper
[16]. Following the conventions of that paper, in the tables we denote the Bala-Carter labels
of twisted punctures by underlining them; in the figures, twisted punctures are denoted in
gray.
3.1. Twisted punctures
Twisted punctures in the E theory are labeled by nilpotent orbits in f , which we denote by
6 4
the corresponding Bala-Carter label. As discussed in [16], the Bala-Carter notation provides
a systematic way to label nilpotent orbits in any exceptional semisimple Lie algebra, and
a concise review can be found in appendix A of [16]. Here we merely add that for the f
4
nilpotent orbits, components of the Levi subalgebra in the Bala-Carter label with (without)
a tilde are constructed from the short (long) roots of f . (So, e.g., A + A(cid:101) and A(cid:101) + A
4 2 1 2 1
represent different orbits.)
The pole structure of the k-differentials is denoted by {p ,p ,p ,p ,p ,p }, and, for
2 5 6 8 9 12
twisted punctures, p are p are half-integer. The contributions to the graded Coulomb
5 9
branch dimensions are denoted by {d ,d ,d ,d ,d ,d ,d ,d }, allowing for new Coulomb
2 3 4 5 6 8 9 12
branchparameters(introducedbya-constraints)ofdimensions3and4, whicharenotdegrees
of E Casimirs. The constraints are shown separately in Appendix A.
6
Nahm Hitchin Coulomb branch
orbit orbit Pole structure contributions Flavour group (δn ,δn )
h v
0 F {1,9,5,7,17,11} {1,0,0,9,5,7,17,11} (F ) (624,601)
4 2 2 2 2 4 18
A (F (a ),Z ) {1,9,5,7,15,11} {1,0,0,9,5,7,15,11} Sp(3) (599,584)
1 4 1 2 2 2 2 2 13
A(cid:101)1 F4(a1) {1,92,5,7,125,11} {1,0,0,29,6,7,125,10} SU(4)12 (584,572)
A1+A(cid:101)1 F4(a2) {1,92,5,7,125,10} {1,0,0,29,5,7,125,10} SU(2)64×SU(2)10 (570,561)
A B {1,7,5,7,15,10} {1,0,0,7,5,7,15,10} SU(3) (560,552)
2 3 2 2 2 2 16
3
Nahm Hitchin Coulomb branch
orbit orbit Pole structure contributions Flavour group (δn ,δn )
h v
A(cid:101)2 C3 {1,92,5,7,125,10} {1,1,0,29,5,6,125,9} (G2)10 (536,528)
A2+A(cid:101)1 (F4(a3),S4) {1,72,5,6,125,10} {1,0,0,27,5,6,125,10} SU(2)39 (543,537)
B (F (a ),Z ×Z ) {1,7,5,6,15,10} {1,1,0,7,6,6,13,9} SU(2)2 (518,513)
2 4 3 2 2 2 2 2 2 7
A(cid:101)2+A1 (F4(a3),S3) {1,72,5,6,125,10} {1,1,0,27,5,6,125,9} SU(2)20 (524,519)
C (a ) (F (a ),Z ) {1,7,5,6,15,10} {1,2,0,7,5,6,13,9} SU(2) (511,507)
3 1 4 3 2 2 2 2 2 7
F (a ) F (a ) {1,7,5,6,15,10} {1,3,0,7,4,6,13,9} − (504,501)
4 3 4 3 2 2 2 2
B A {1,7,4,6,13,9} {1,0,1,7,4,5,11,8} SU(2) (440,438)
3 2 2 2 2 2 24
C3 A(cid:101)2 {1,72,5,6,125,10} {1,1,1,27,4,5,121,7} SU(2)6 (422,420)
F4(a2) A1+A(cid:101)1 {1,72,4,6,123,9} {1,0,1,27,4,5,121,7} − (416,415)
F4(a1) A(cid:101)1 {1,72,4,6,123,9} {2,1,0,52,4,4,92,6} − (352,352)
F 0 {1,5,3,4,9,6} {1,1,0,3,2,2,5,3} − (184,185)
4 2 2 2 2
There is a special piece consisting of five nilpotent orbits,
˜ ˜
A +A , A +A , B , C (a ), F (a ).
2 1 2 1 2 3 1 4 3
The corresponding Hitchin boundary conditions are (F (a ),Γ), where the Sommers-Achar
4 3
group, Γ, is a subgroup of S . The leading pole coefficients,
4
(cid:16) (cid:17)
c(6) = − 6a2 +3a(cid:48)2 +a(cid:48)(cid:48)2
5
(cid:16) (cid:17)
c(9) = 1(a+a(cid:48)) (2a−a(cid:48))2 −a(cid:48)(cid:48)2 (1)
15/2 3
(cid:16) (cid:17)2
c(12) = 3a(cid:48)2(4a+a(cid:48))2 +2(8a2 −12aa(cid:48) +a(cid:48)2)a(cid:48)(cid:48)2 + 1a(cid:48)(cid:48)4 − 4 c(6)
10 3 3 5
(cid:16) a (cid:17) (cid:16) a (cid:17)
are invariant under the S action, a(cid:48) (cid:55)→ γ a(cid:48) , generated by
4
a(cid:48)(cid:48) a(cid:48)(cid:48)
     
−1 2 0 2 0 0 1 0 0
1 1
σ12 =  4 1 0, σ23 = 0 −1 1, σ34 = 0 1 0 ,
3 2
0 0 3 0 3 1 0 0 −1
• For the special orbit, F (a ), the Sommers-Achar group is trivial, and a, a(cid:48), a(cid:48)(cid:48) are
4 3
invariants.
4
• For C (a ), the Sommers-Achar group is the Z generated by σ and the invariants
3 1 2 34
are a, a(cid:48), a(cid:48)(cid:48)2.
• ForB ,theSommers-AchargroupistheZ ×Z generatedbyσ , σ andtheinvariants
2 2 2 12 34
are a+2a(cid:48), a(cid:48)(cid:48)2, 2a2 +a(cid:48)2.
˜
• ForA +A ,theSommers-AchargroupistheS generatedbyσ , σ andtheinvariants
2 1 3 23 34
are a, 3a(cid:48)2 +a(cid:48)(cid:48)2, c(9) .
15/2
˜
• Finally, for A + A , the Sommers-Achar group is the full S , and the invariants are
2 1 4
c(6), c(9) , c(12).
5 15/2 10
In §5, we will discover an action of this S group on the Higgs branch of certain fixtures
4
obtained by varying one of the punctures over this special piece.
3.2. Free-field fixtures
# Fixture n Representation
h
F
4
1 (D ,SU(3) ) 0 empty
4 0
F
4
F
2 4 (A ,SU(3)2) 0 empty
2 0
F (a )
4 1
F
3 4 (A ,SU(6) ) 10 1(20)
1 6 2
F (a )
4 2
F
4
4 (0,SU(6) ) 0 empty
0
B
3
F
4
5 (F (a ),) 0 empty
4 1
E (a )
6 1
F
6 4 (C (a ),SU(2) ) 1 1(2)
3 1 1 2
E (a )
6 3
F
7 4 (B ,SU(2) ) 1 1(2)
2 1 2
A
5
F
4
8 (A(cid:101) ,SU(3) ) 1 1(3)
2 2
D (a )
5 1
F
4
9 (C ,SU(2) ) 2 1(2)
3 2
D
5
5
# Fixture n Representation
h
F
4
10 (A(cid:101) ,SU(3) ) 0 empty
1 0
A +A
4 1
F
4
11 (A(cid:101) ,SU(4) ) 8 (2,4)
1 4
A
4
F (a )
12 4 1 (B ,SU(2)2) 2 1(2,1)+ 1(1,2)
2 1 2 2
E (a )
6 1
F (a )
4 2
13 (A(cid:101) ,Sp(2) ) 0 empty
1 0
E (a )
6 1
C
14 3 (A(cid:101) ,SU(4) ) 6 1(2,6)
1 4 2
E (a )
6 1
B
15 3 (A ,Sp(3) ) 9 1(3,6)
1 3 2
E (a )
6 1
3.3. Interacting fixtures with one irregular puncture
# Fixture (n ,n ,n ,n ,n ,n ,n ,n ) (n ,n ) Theory
2 3 4 5 6 8 9 12 h v
F
4
1 (A(cid:101) ,G ) (0,1,0,0,0,0,0,0) (16,5) (E ) SCFT
2 2 6 6
D
4
F
4
2 (0,Spin(8)) (0,1,0,0,0,0,0,0) (16,5) (E ) SCFT
6 6
D (a )
4 1
F
4
3 (A ,SU(6) ) (0,1,0,0,0,0,0,0) (16,5) (E ) SCFT
1 6 6 6
C
3
F
4
4 (0,Spin(9)) (0,1,0,1,0,0,0,0) (36,14) Spin(14) ×U(1) SCFT
10
A
3
C
3
5 (0,Spin(9)) (0,1,1,1,0,0,0,0) (38,21) Spin(9) ×SU(2) ×U(1)
10 6
D
5
F (a )
4 2
6 (0,Spin(9)) (0,0,1,1,0,0,0,0) (32,16) Spin(9) ×U(1)
10
D
5
3.4. Interacting fixtures with enhanced global symmetry
6
# Fixture (n ,n ,n ,n ,n ,n ,n ,n ) (n ,n ) G
2 3 4 5 6 8 9 12 h v k
F
1 4 F (a ) (0,4,0,0,0,0,0,0) (64,20) [(E ) SCFT]4
4 3 6 6
0
F
2 4 C (a ) (0,3,0,0,1,0,0,0) (71,26) [(E ) SCFT]2×[(E ) ×SU(2) SCFT]
3 1 6 6 6 12 7
0
F
4
3 A(cid:101)2+A1 (0,2,0,0,1,0,1,0) (84,38) [(E6)6 SCFT]×[(E6)18×SU(2)20 SCFT]
0
F
4 4 B (0,2,0,0,2,0,0,0) (78,32) [(E ) ×SU(2) SCFT]2
2 6 12 7
0
F
4
5 A(cid:101)2 (0,2,0,1,1,0,1,0) (96,47) [(E6)6 SCFT]×[(E6)18×(G2)10 SCFT]
0
F
4
6 A (0,1,0,0,1,1,0,0) (64,31) Spin(13) ×U(1)
2 16
2A
1
F
4
7 A (0,1,0,0,1,1,1,0) (86,48) (G ) ×SU(6)
2 2 16 18
A
1
F
4
8 A1+A(cid:101)1 (0,1,0,1,1,1,0,0) (74,40) Spin(10)16×SU(2)10×SU(2)32×U(1)
2A
1
F
4
9 A1+A(cid:101)1 (0,1,0,1,1,1,1,0) (96,57) SU(6)18×SU(2)64−k×SU(2)k×SU(2)10
A
1
F
10 4 A(cid:101)1 (0,1,0,1,2,1,0,0) (88,51) Spin(8)16×SU(4)12×U(1)2
2A
1
F
4
11 A(cid:101)1 (0,1,0,1,2,1,1,0) (110,68) SU(6)18×SU(4)12×U(1)
A
1
F
4
12 A (0,1,0,1,1,0,0,1) (84,48) Sp(4) ×SU(3)
1 13 24
3A
1
F
4
13 0 (0,1,0,1,0,0,1,0) (70,31) (E ) ×U(1)
7 18
A +2A
2 1
F
4
14 0 (0,1,0,0,1,0,0,0) (56,16) [(E ) SCFT]×[(E ) SCFT]
8 12 6 6
2A
2
F
4
15 0 (0,1,0,1,1,0,1,0) (83,42) (E ) ×SU(3) ×U(1)
6 18 12
A +A
2 1
F
16 4 0 (0,1,0,1,2,0,1,0) (96,53) (E ) ×SU(3)2
6 18 12
A
2
F (a )
4 2
17 F (a ) (0,0,2,2,1,1,1,0) (94,75) SU(2) ×SU(2) ×U(1)
4 2 54−k k
A +2A
2 1
F (a )
4 2
18 F (a ) (0,0,2,1,2,1,0,0) (80,60) Spin(7) ×U(1)
4 2 12
2A
2
7
# Fixture (n ,n ,n ,n ,n ,n ,n ,n ) (n ,n ) G
2 3 4 5 6 8 9 12 h v k
F (a )
19 4 2 F (a ) (0,0,2,2,2,1,1,0) (107,86) SU(3) ×U(1)2
4 2 12
A +A
2 1
F (a )
20 4 2 F (a ) (0,0,2,2,3,1,1,0) (120,97) SU(3)2 ×U(1)
4 2 12
A
2
F (a )
4 2
21 C (0,1,2,2,1,1,1,0) (100,80) SU(2) ×SU(2) ×SU(2) ×U(1)
3 36 18 6
A +2A
2 1
F (a )
4 2
22 C (0,1,2,1,2,1,0,0) (86,65) Spin(7) ×SU(2) ×U(1)
3 12 6
2A
2
F (a )
23 4 2 C (0,1,2,2,2,1,1,0) (113,91) SU(3) ×SU(2) ×U(1)2
3 12 6
A +A
2 1
F (a )
24 4 2 C (0,1,2,2,3,1,1,0) (126,102) SU(3)2 ×SU(2) ×U(1)
3 12 6
A
2
F (a )
25 4 2 B (0,0,4,1,1,0,0,0) (64,48) SU(2)3×U(1)2
3 8
D (a )
4 1
F (a )
4 2
26 B (0,0,3,1,1,1,0,0) (73,56) SU(2) ×SU(2) ×SU(2) ×U(1)
3 16 8 9
A +A
3 1
F (a )
4 2
27 B (0,0,3,2,1,1,0,0) (84,65) SU(2) ×SU(2) ×Sp(2) ×U(1)
3 16 8 10
A
3
F (a )
28 4 2 C (a ) (0,2,1,1,2,1,0,0) (79,63) SU(2) ×U(1)3
3 1 7
A +A
4 1
F (a )
29 4 2 C (a ) (0,2,2,1,2,1,0,0) (87,70) SU(2) ×SU(2) ×U(1)3
3 1 8 7
A
4
F (a )
30 4 2 A(cid:101)2+A1 (0,1,1,1,2,1,1,0) (92,75) SU(2)20×U(1)2
A +A
4 1
F (a )
31 4 2 A(cid:101)2+A1 (0,1,2,1,2,1,1,0) (100,82) SU(2)20×SU(2)8×U(1)2
A
4
F (a )
32 4 2 B (0,1,1,1,3,1,0,0) (86,69) SU(2)2×U(1)2
2 7
A +A
4 1
F (a )
33 4 2 B (0,1,2,1,3,1,0,0) (94,76) SU(2) ×SU(2)2×U(1)2
2 8 7
A
4
F (a )
34 4 2 A(cid:101)2 (0,1,1,2,2,1,1,0) (104,84) (G2)10×U(1)2
A +A
4 1
F (a )
35 4 2 A(cid:101)2 (0,1,2,2,2,1,1,0) (112,91) (G2)10×SU(2)8×U(1)2
A
4
F (a )
4 2
36 A (0,1,1,0,1,1,0,0) (56,38) Spin(7) ×U(1)
2 16
E (a )
6 3
8