#### Tinkertoys for the E 6 theory

Received: May
E6 theory
Oscar Chacaltana 0 1 2 4 5
Jacques Distler 0 1 2 3 5
Anderson Trimm 0 1 2 3 5
0 Department of Physics, University of Texas at Austin
1 01140-070 S ̃ao Paulo , SP , Brazil
2 Instituto de F ́ısica Te ́orica, Universidade Estadual Paulista
3 Theory Group and Texas Cosmology Center
4 ICTP South American Institute for Fundamental Research
5 Bala-Carter f Projection Matrix
Compactifying the 6-dimensional (2,0) superconformal field theory, of type ADE, on a Riemann surface, C, with codimension-2 defect operators at points on C, yields a 4-dimensional N = 2 superconformal field theory. An outstanding problem is to classify the 4D theories one obtains, in this way, and to understand their properties. In this paper, we turn our attention to the E6 (2,0) theory, which (unlike the A- and D-series) has no realization in terms of M5-branes. Classifying the 4D theories amounts to classifying all of the 3-punctured spheres (“fixtures”), and the cylinders that connect them, that can occur in a pants-decomposition of C. We find 904 fixtures: 19 corresponding to free hypermultiplets, 825 corresponding to isolated interacting SCFTs (with no known Lagrangian description) and 60 “mixed fixtures”, corresponding to a combination of free hypermultiplets and an interacting SCFT. Of the 825 interacting fixtures, we list only the 139 “interesting” ones. As an application, we study the strong coupling limits of the Lagrangian field theories: E6 with 4 hypermultiplets in the 27 and F4 with 3 hypermultiplets in the 26.
Supersymmetric gauge theory; Supersymmetry and Duality; Extended Su-
1 Introduction 2
The E6 theory
The Hitchin system
ALE geometry
Puncture properties
Flavour groups
Pole structures
Puncture collisions
Global symmetries and the superconformal index
Levels of enhanced global symmetry groups
Cataloguing fixtures using the superconformal index
Computing the expansion of the index
Regular punctures
Free-field fixtures
3.3 Interacting fixtures with one irregular puncture 3.4 Interacting fixtures with enhanced global symmetry 3.5 Mixed fixtures
A detour through the twisted sector
E6 and F4 gauge theory
Adding (E8)12 SCFTs
E6 + 4(27)
F4 + 3(26)
n = 3
n = 2
n = 1
n = 0
Connections with F-theory
6 Isomorphic theories A Bala-Carter labels – 1 – 2
B Projection matrices
superconformal theories, realizing them as compactifications of the 6D (2,0) theories on
a punctured Riemann surface. Because of its six-dimensional origin, this class of theories
(sometimes called class “S”) is endowed with a powerful set of tools [1–11] for studying its
physical properties. For a theory in this class, it is trivial to write its low-energy
Seibergglobal symmetry group, graded dimensions of the Coulomb branch, etc.
Equally interesting, the understanding of the theories in this class expands our
knowlbut also isolated interacting fixed points of the renormalization group with no known
Lagrangian description. Gaiotto’s construction generates infinitely many new examples of
these interacting theories. Thus, while it seems unlikely that this construction covers the
the construction can themselves be classified. The (2,0) theories are classified by a choice
of simply-laced Lie algebra j [14]; (a class of) punctures on the Riemann surface are labeled
by nilpotent orbits in j [15, 16], and degenerating Riemann surfaces can be decomposed
into a collection of three-punctured spheres connected by cylinders. The set of theories
becomes larger if we allow for the (2,0) theory to be “twisted” by an outer-automorphism
of j, when traversing a nontrivial cycle of the punctured Riemann surface, C. In particular,
a twist when circling a puncture introduces a new class of defects, called “twisted
punctures,” which are classified by nilpotent orbits in non-simply-laced Lie algebras obtained
by dividing j by the action of the outer automorphism [9, 17]. In [4], we extended the
classification of [3] to the (untwisted) DN series, and in [10] and [11] we incorporated Z2
In this paper, we extend our classification program to the (2,0) theory of type E6.
We leave the study of this theory in the presence of Z2 outer automorphism twists for
another publication. There is no known construction of the E6 theory as a low-energy
theory of a stack of M5 branes, as was the case for the A- and D-series. Rather, the only
known construction is as a compactification of IIB string theory on a K3 manifold at an E6
of the E6 theory is controlled by a Hitchin system [2] with gauge group E6.
As a byproduct, we realize E6 gauge theory with matter in the 4(27), as well as F4
gauge theory with matter in the 3(26), as compactifications of the E6 (2,0) theory on a
27s, appeared first in [18]. Our solution to the superconformal F4 gauge theory is new.
The E6 theory
The Hitchin system
6D (2,0) theory of type E6 on a Riemann surface C is described by the Hitchin equations
on C with complexified gauge group E6 [2]. We may also include codimension-two defects
of the (2,0) theory localized at points on C; we refer to these as “punctures”. A class
of punctures is classified by nilpotent orbits (or, equivalently, by embeddings of sl(2)) in
the complexified Lie algebra e6 [9]. One of the main points of the construction is that a
properties of the nilpotent orbits that label the punctures on C, without any detailed
knowledge of the (2,0) theory.
a local boundary condition for the Higgs field,
representation of the gauge group, X is a representative of the nilpotent orbit d(O) in e6,
and . . . represents a generic regular function of z taking values in e6. Here, d(O) is the
image of O under the Lusztig-Spaltenstein map d [4, 7, 9]. Representatives of all nilpotent
orbits in e6 can be found in [19], and a diagram specifying the action of d, as well as other
properties of the e6 orbits, are collected in appendix C of [9] (taken from [20, 21]) When d
is not injective, we distinguish different punctures with the same d(O) by their
Sommersunder the action of C(O).
As in our previous papers, we call O, which labels the puncture, the Nahm pole,
and d(O), which appears in the Hitchin system boundary condition, the Hitchin pole.
The physical properties of a puncture labeled by O will be directly related to geometric
properties of the orbits O and d(O), and the discrete group C(O).
Unlike classical Lie algebras, there is no natural parameterization of the nilpotent
orbits of exceptional Lie algebras in terms of partitions or Young diagrams. Instead, the
notation due to Bala and Carter is standard in the representation theory literature. This
notation has been briefly discussed in previous works [9, 16, 22], but, for completeness, we
review it in appendix A, and discuss how to extract relevant information from it.
which is given by the spectral curve of the Hitchin system, i.e., by the characteristic
polyextract from them is the same.
(s0 and s1 are trivial — they are 1 and 0, respectively.) The sk(z) are holomorphic
kdifferentials on C (with poles at the punctures), and can be expressed as polynomials in
the representation R.
Lie algebra g, the number of Casimirs is equal to the rank of g, and their scaling dimensions
are the exponents (minus 1) of g. Unlike the sk or the Pk, the Casimirs encode the
non
In our previous papers [3, 4, 10, 11], the Lie algebra was of classical type, and R was
the vector for DN ). In such cases, the coefficients sk directly provide a basis for the Casimirs
These dimensions match precisely those of the non-trivial coefficients sk if R is chosen
to be the fundamental representation. Thus, the sk can be taken to be the Casimirs of
is the vector representation, the sk with k odd vanish, and the non-trivial coefficients are
for large enough N , the sk would not have given directly the Casimirs, but instead a lot
of redundant information. For example, for A5, there are five Casimirs, with dimensions
2, 3, 4, 5, 6. However, we have 34 non-trivial coefficients sk, with dimensions 2, 3, 4, . . . , 35.
These sk are polynomials in the five Casimirs.
R to be the adjoint representation of e6, as it is readily available in the form of structure
constants; we used those from the computer algebra system GAP 4 [24]. Instead of trying
to compute the 78 coefficients sk, we focus directly on the trace invariants Pk for values of k
only as large as needed to extract the Casimirs. For the adjoint representation of e6, the Pk
From the Pk, one can construct a less-redundant basis,
P6 − 4608
P8 − 9 P6P2 +
32256 P8(P2)2 − 108
P14 − 14880 P12P2 +
(P6)2P2 − 44431441920 P6(P2)4 +
409480168734720
This basis was constructed so that it reduces the constraints in our punctures to a
min497664 P6(P2)3 − 9172942848
flips under the Z2 outer automorphism of E6). So, we can declare the k-differentials
As for the Seiberg-Witten curve, to write it explicitly, we need to know how the 78
using the Casimirs [25, 26]. Let us briefly review that construction.
ALE geometry
J (where J is of A-D-E type) on the Riemann surface C can also be obtained, in a dual
ALE fibration over C of type J [25, 26]. For e6, the threefold is realized as the hypersurface
⊂ tot(KC6 ⊕ KC4 ⊕ KC3 )
2 =
5 =
6 =
8 =
9 =
12 =
144 (−3φ24 + 4φ2φ6 − φ8)
(4φ12 + 6φ26 − 12φ23φ6 + 4φ62 + 3φ22φ8)
where the k(z) are k-differentials on C [22] (in the “Katz-Morrison basis” [27]), related to
The Seiberg-Witten solution is obtained by computing the periods of the holomorphic
of a 2-sphere in the fiber times a curve on C. In the conformal case (which will be our
focus in this paper), many of these cycles will necessarily be noncompact (the curve on C
being a open curve, stretching between punctures). But, precisely for the parabolic case
slightly singular nature of X~u itself.
Puncture properties
We describe below how to compute the properties of a puncture. There is a systematic
way to compute every property of the puncture, except for the constraints, so it is easiest
to compute the other properties first, and use them to guess the constraints. Below, let O
be the Nahm nilpotent orbit that labels a given puncture, and su(2)O the associated su(2)
embedding in e6.
Flavour groups
the centralizer of su(2)O in e6. A list of the centralizers for each O can be found in table
14 of [9], taken originally from [28].
Let fi be the simple, nonabelian factors in f. The level ki of fi is the central charge
of the fi current algebra. (See, e.g., [12] for a review.) Following [9], the ki can be
computed from the decomposition of the adjoint, 78, of e6 under the subalgebra su(2)O × f.
These decompositions can, in turn, be deduced from the Bala-Carter label for O, and are
summarized in the table in appendix A.
R1 = 8 + 3(1),
R2 = 2(8),
R3 = 8 + 1,
R4 = 2(1),
Let the decomposition of the 78 be
e6 =
M Vn ⊗ Rn,i ,
where Vn is the n-dimensional irrep of su(2) (denoted by “n” in the table) and Rn,i is the
corresponding (reducible) representation of fi. Let ln,i be the index of Rn,i. Then, the level
of fi is ki = Pn ln,i.
in appendix A, we have, for f1 = su(3),
R1 = 3 + 8(1),
R2 = 8(2),
R3 = 9(1),
R4 = 2,
The conformal anomaly central charges of the 4D theory, a and c, can be conveniently
written in terms of effective numbers of hyper- and vector multiplets, nh and nv, as
the contributions from the punctures plus certain global terms.
The formulas in eq.
eigenspaces of the Cartan element of su(2)O.
Here, let us recast those formulas in terms of the weighted Dynkin diagram for O,
which can be found in table 14 of [9]. Let ~x be the six-dimensional vector consisting of
roots. The “Weyl vector” is W~ = 12 P
(The dot product is Euclidean.) In this notation, the formulas in eq. (3.19) of [9] are:
h∨(E6) dim(E6) − 2
h∨(E6) dim(E6) − 2
1 W~ · ~x
1 W~ · ~x
− 2 n0
rank(E6), and n1/2(~0) = 0. Thus,
2 h∨(E6) dim(E6) = 624
(dim(E6) − rank(E6)) = 588
(manifold) dimension dimC(O) of O:
(dim(E6) − rank(E6)) − 2
dimC(O).
The dimensions of the nilpotent orbits of E6 are listed in table 14 of [9].
For a non-trivial example, consider the puncture 2A1, which has weighted Dynkin
Pole structures
The “pole structure” is the set of leading pole orders {p2, p5, p6, p8, p9, p12} in the expansion
To compute the pole structure, we need a representative of the Hitchin nilpotent orbit
d(O). A table of representatives of all nilpotent orbits of E6 can be found in table 2 of [19].
In this table, a nilpotent representative is given by a sum of weighted Dynkin diagrams,
and each weighted Dynkin diagram represents an element in the root vector space of e6 for
by the labels of the Dynkin diagram. The nilpotent representative is the sum of these
rootvector space elements. This procedure is most easily understood in terms of an example.
sum of five elements [19],
root vector spaces for the positive roots
Fortunately, GAP4 provides a Chevalley basis for the adjoint representation of e6, so it is
pole structure {1, 3, 4, 6, 6, 9} for the D4(a1) puncture. (Actually, there are three orbits,
D4(a1), A3 + A1 and 2A2 + A1, that map under Spaltenstein to D4(a1), so we have three
are different.)
The constraints for some E6 punctures are, in some cases, much less obvious than those
and ungraded local contributions to the Coulomb branch.
Let us be specific. Let z be a local coordinate on C centered at the puncture, and let
c(k) be the coefficient of z−l in the expansion of φk = φk(z) in z. Recall that, in the notation
l
of our previous papers, a “c-constraint” is a polynomial relation among coefficients cl(k) (of
homogeneous bi-degree in both k and l). On the other hand, an “a-constraint” is a relation
that defines a new quantity, al
(k), of dimension k, in terms of the cl(k). Only the cl(k) with
l > 0 parameterize the Hitchin nilpotent orbit [16]. In the absence of constraints, all the
c(k) with 0 < l ≤ pk are independent, so their total number, P pk, should be equal to
l
A c-constraint reduces the total number of independent parameters by one, whereas an
a-constraint does not affect this number. So, one should have:
X pk − (number of c-constraints) = d
Hence, d tells us how many c-constraints exist. On the other hand, the graded sum of
the presence of “a”-constraints, k is not restricted to the degrees of the Casimirs), should
be equal to nv. An a-constraint replaces a parameter of a certain degree k by another
one of a different degree k0 < k. So, to get precisely nv, one must take into account all
a-constraints and c-constraints.
For example, consider the Higgs field near a D5 puncture, and the corresponding
We find the pole structure {1, 2, 3, 4, 4, 6}, and thus,
cube of a single quantity, which we call a
(24), of scaling dimension 4 and pole order 2:
c(8) = 3 a(4) 2
c(12) =
Since there is no Casimir of E6 of dimension four, a(4) cannot be a coefficient in one
2
parameter. It is not hard to discover additional relations by inspection, and we see that
they, too, involve the a(24) parameter:
c(36) =
c(49) = − 41 a(24)c(25),
We can take one of the relations linear in a(24), say the one for c(36), as the a-constraint
(that is, the relation that defines a(24) unambiguously in terms of the cl(k)), and the other
the formula in section 2.4.2. We conclude that we have found the correct set of constraints
for this puncture.
As we will see next, when two punctures collide, we get a new puncture. Thus, working
with the k-differentials, we must be able to reproduce the pole structure and constraints
of the new puncture from those of the colliding punctures. We can regard this expectation
as another test of the correctness of our expressions.
Puncture collisions
Suppose we have two punctures on a plane, so the Higgs field has two simple poles with
X1 +X2 (by the residue theorem applied to the sphere that bubbles off), which corresponds
to a new puncture. Generically, X will be mass deformed. The mass deformations are
interpreted as VEVs of the scalars in the gauge multiplet associate to the factor in the gauge
group which becomes weakly coupled in the collision limit. One can also study this
degen
Alternatively, one can bypass the Higgs field, and study the collision directly with the
(given by their pole structures), and imposing at each pole the constraints of the
corresponding puncture. Then, taking the collision limit, the pole structure and constraints of
the resulting puncture on the plane arise naturally.
As an example, let us see that the collision of two D5 punctures on a plane produces
an Sp(2) gauge group, gauged off an A3 puncture. Let us write generic Casimirs for the
u2 + zv2 + z(z − x)P2(z)
z(z − x)
u5 + zv5 + z(z − x)w5 + z2(z − x)P5(z)
u6 + zv6 + z(z − x)w6 + z2(z − x)P6(z)
z2(z − x)2
z3(z − x)3
z4(z − x)4
z4(z − x)4
z6(z − x)6
where P2(z), P5(z), . . . , P12(z) denote regular functions in z. To solve the constraints at
each D5 puncture, we introduce new parameters s4 and t4 of dimension four, and write:
u6 =
u8 = 3s42,
u9 = − 4
u12 =
w12 = (3s4t42 + s4w8 + 2t43x),
v6 = 2 (t4u2 + s4v2 + t4v2x),
v8 = 3(2s4t4 + t42x),
v9 = − 4 (t4u5 + s4v5 + t4v5x),
v12 = 2 t4(3s42 + 3s4t4x + t42x2),
the Casimirs in this limit is:
3(3s4t24 + s4w8)
3(t34 − t4w8 − s4y8) + . . . ,
where the . . . indicate less singular terms in z. So, u2 and s4 can be interpreted as the
VEVs of Coulomb branch parameters (of degree two and four) of the gauge group (which,
with a little more work, can be checked to be Sp(2)). In the limit u2, s4 → 0, we obtain the
Casimirs for the massless puncture, with pole orders {1, 4, 4, 6, 7, 9}, and with constraints
c(79) =
Cataloguing fixtures using the superconformal index
For the E6 theory, we find 880 fixtures with three regular punctures which correspond
SCFTs has a manifest global symmetry group, which is given by the product of the flavor
symmetry groups of the three punctures. This global symmetry group may, in general,
become enhanced to a larger group.
To determine the global symmetry group and number of free hypermultiplets for each
of these fixtures, we use the superconformal index [29–33]. The superconformal index
of E-type class S theories has not yet been systematically studied. However, since the
methods used for A- and D-type theories generalize to any root system, we assume the
superconformal index1 for a fixture in the E6 theory takes the usual form
resentations of e6.
w∈W
where R+ denotes the set of positive roots, W the Weyl group, and `(w) the length of the
Weyl group element w.
flavor symmetry of the ith puncture. atriv denotes the set of fugacities dual to the
Cartan of the embedded su(2) of the trivial puncture.
• The K-factors are discussed in [32–35]. We will not need their detailed form for our
Consider a fixture corresponding to an interacting SCFT, with global symmetry Gglobal,
plus free hypermultiplets transforming in a representation R of a flavor symmetry F . Let
1In what follows we will consider the Hall-Littlewood limit of the index [32], which depends on one
number of free hypers in the fixture and the global symmetry of the fixture can be read off
from the first two non-trivial terms in the Taylor expansion of the index. Schematically,
this is given by
I = 1 + χFRτ + χaGdfijxt τ 2 + . . .
of Gfixt, where both of these representations are viewed as reducible representations of the
the Plethystic exponential) and removing the contribution of the free hypermultiplets
in (2.5), we arrive at
ISCFT = I/Ifree
= 1 + χaGdgjlobal τ 2 + . . .
from which we can read off the global symmetry of the interacting SCFT.
Computing the expansion of the index
In (2.4) the term in the sum coming from the trivial representation of e6 gives, to second
I = 1 + χaGdmj anifest τ 2 + · · ·
encoding the manifest global symmetry group. The global symmetry group of the fixture
I = 1 + χaGdmj anifest τ 2 +
To compute (2.6), we consider each e6 representation in the sum to be a reducible
representation of su(2) × f and plug in the corresponding character expansion, where the embedded
representations can be obtained using the projection matrices listed in appendix B.
Of the 881 fixtures involving three regular punctures, we find that 1 is a free-field
fixture, 60 are mixed fixtures and another 134 are interacting fixtures with an enhanced
fixtures, the global symmetry group is the manifest one.
used the fact that P λ = χλ + O(τ 2).
2Since the theories considered here are all “good” or “ugly” (in the sense of [8]), the lowest possible
As an example, consider the fixture
come from the sum over the 27, 27, 78, 351, 351, 3510, 3510, and 650 of e6. The expansion
of the superconformal index is given by3
in the fundamental representation of E6. The index of these free hypers is given by
The index of the underlying SCFT is then
ISCFT = I/Ifree
Computing the other numerical invariants of the fixture, we find that this is the (E8)12
theory of Minahan and Nemeschansky [36] with 27 additional free hypermultiplets.
Levels of enhanced global symmetry groups
Since the superconformal index gives the branching rule for the adjoint representation of
Gglobal under the subgroup Gmanifest, it most cases it is straightforward to determine the
level of each factor in Gglobal from those of Gmanifest: if Hk0 is a subgroup of Gk, then k is
given by [12]
where IH,→G is the index of the embedding of H in G.
There are two cases which require a little more work. The first is when a manifest U(1)
becomes enhanced to SU(2). Since we do not know how to assign a level to a U(1) flavor
3For simplicity, we write the dimension to stand for the character of the corresponding representation.
The subscript is the U(1) weight.
k =
symmetry (which would require a precise understanding of how the generator is
normalized), we cannot immediately determine the level of the enhanced SU(2) from the index.
The second case is when some factor Hk in Gmanifest is embedded diagonally as
Hk ,→ Hk1 × Hk2 .
Since the only embedding of H in itself has index one, in this case, all we know is that
k1 + k2 = k.
If any of these remain as factors in Gglobal (that is, if they do not combine with some
other factor, with known level, to enhance Gglobal), we cannot determine their levels from
sphere for which the SCFT appears in some degeneration, with Hki in the centralizer of
subgroup of Gglobal being weakly gauged.
Unfortunately, there are a few such fixtures for which no puncture can be gauged (some
of these can still be gauged in the twisted sector, which will be discussed in section 4). For
these, we do not have a way to determine the levels. In the end, there are two interacting
fixtures whose levels we cannot completely determine.
Tinkertoys
Regular punctures
terms in the Laurent expansion of the k-differentials.
{1, 4, 5, 7, 8, 11}
{1, 4, 5, 7, 8, 10}
{1, 4, 5, 7, 7, 10}
(E6(a3), Z2) {1, 4, 5, 6, 7, 10}
SU(3)24 × SU(2)13 (549, 533)
{1, 4, 5, 6, 7, 10} c(1102) = − c(56) 2 + a(56) 2
Sp(2)10 × U(1)
c(79) =
Spin(7)16 × U(1)
SU(3)12 × U(1)
SU(2)54 × U(1)
c(68) = 3 a(34) 2
c(69) =
c(58) = − 4c(46)c(2) + 4c(35)a(3)
1 2
{1,3,4,5,6,8}
2A2+A1 (ns) (D4(a1),S3) {1,3,4,6,6,9}
A3 + A1 (ns) (D4(a1),Z2) {1,3,4,6,6,9} c(12) = a(34) 16 a(4) 2 − c(68)
9 9 3
SU(2)9 × U(1) (465,457)
{1,3,4,6,6,9}
{1,3,4,6,6,9}
SU(2)8 × U(1) (408,402)
A2 + 2A1 {1,3,4,5,5,7}
{1,2,4,4,4,6} c(46) = a(23) 2
c(46) = −8
{1, 2, 3, 4, 4, 6}
c(36) =
c(48) = 3 a(24) 2
Note that there is a special piece, consisting of three punctures: 2A2 + A1, A3 + A1 and
the special puncture D4(a1). For 2A2 + A1, the Sommers-Achar group is the nonabelian
group, S3. It acts on a(4), a0(4) as
0 −1
−1 −3
1 −1
−1 3 !
−1 −1
−1 −3
−1 1
−1 3
For A3 + A1, the Sommers-Achar group is the Z2 subgroup of S3, generated by a0(4) →
survive as Coulomb branch parameters.
Free-field fixtures
three punctures. For the free-field fixtures, one of the punctures is an irregular puncture4
(in the sense used in our previous papers), which we denote5 by the pair, (O, Gk), where
O is the regular puncture obtained as the OPE of the two regular punctures which collide,
and this fixture is attached to the rest of the surface via a cylinder
(O, Gk) ←−−−→ O
4Or, in the case of fixture 13, a full puncture, corresponding to the trivial orbit, 0.
5For brevity, we will often omit the level, k, when denoting an irregular puncture.
A4 + A1
A3 + A1
2A2 + A1
A2 + 2A1
(A5, SU(2)1)
(A4, SU(2)0)
(2A2, SU(3)0)
(2A2, (G2)4)
(A2 + A1, SU(3)0)
(2A1, (G2)0)
(A2, SU(3)0)
(2A1, Spin(7)4)
(0, Spin(8)0)
(0, Spin(9)4)
(0, (F4)12)
(0, Spin(10)8)
54 (2, 27)
(2A1, SU(4)0)
Interacting fixtures with one irregular puncture
In the tables below, nd is the number of Coulomb branch parameters of degree d. The
total Coulomb branch dimension is P
nv = Pd(2d − 1)nd.
d nd and the effective number of vector multiplets is
(n2,n3,n4,n5,n6,n8,n9,n12) (nh, nv)
(0, (F4)12)
(0, 0, 0, 0, 1, 0, 0, 0)
(0, Spin(10)8)
(0, 0, 1, 0, 0, 0, 0, 0)
(0, (F4)12)
(0, (F4)12)
(0, 2, 0, 0, 0, 0, 0, 0)
(0, 1, 0, 0, 1, 0, 0, 0)
(0, (F4)12)
(0, 0, 0, 0, 2, 0, 0, 0)
(E8)12 SCFT
(E7)8 SCFT
[(E6)6 SCFT]2
(E6)12 × SU(2)7 SCFT
(F4)12 × SU(2)72 SCFT
The (E6)12 × SU(2)7 and (F4)12 × SU(2)72 first appeared in [4], as fixtures in the
untwisted D4 theory.
Interacting fixtures with enhanced global symmetry
(n2,n3,n4,n5,n6,n8,n9,n12) (nh, nv)
A4 + A1
(A1, SU(5)2)
(0, Spin(10)8)
(A1, SU(6)6)
(0, 0, 0, 0, 2, 0, 0, 0)
[(E8)12 SCFT]2
(0, 0, 0, 0, 1, 0, 0, 1)
(E8)24 × SU(2)13
(0, 0, 0, 0, 1, 1, 0, 1)
(112, 49) (E7)24 × Spin(7)16
(0, 0, 0, 0, 1, 1, 1, 0)
(100, 43) SU(12)18
the SU(2)6 is manifest). The 4-punctured sphere
has global symmetry
The S-dual
(0, SU(6)) SU(6)
(E6)24−k × (E6)k × SU(3)122 SCFT
F = SU(3)122 × SU(2)24−k × SU(2)k
(E6)6 SCFT
(G2)10 × SU(3)12 × SU(2)18 × SU(2)6 SCFT
Similarly, for
(0, SU(6)) SU(6)
the global symmetry group is
Now there are two S-dual presentations of the theory:
empty (E6)18 × (E6)6 × SU(3)12 × U(1) SCFT
F = SU(3)12 × SU(2)24−k × SU(2)k × U(1)
(G2)10 × SU(2)18 × SU(2)6 × U(1) SCFT
( A˜2, SU(3)) SU(3)
(G2)10 × SU(3)12 × SU(2)18 × SU(2)6 SCFT
Again, the fact that one of the SU(2) levels is manifest suffices to determine the other.
As another example, consider the pair of fixtures
In each case, only the diagonal SU(2)54 subgroup, of the indicated SU(2)s, is manifest.
Moreover, these fixtures are not gaugeable within the untwisted theory. So there is no
obvious way to determine the individual SU(2) levels. Fortunately, the twisted sector
provides the empty fixture
(D4, SU(3))
which allows us to gauge the SU(3)12 symmetry of each of these fixtures:
From the S-duals
(D4, SU(3)) SU(3)
(D4, SU(3)) SU(3)
(E7)18 × U(1) SCFT
(E6)6 SCFT
Spin(8) (0, Spin(8))
Spin(9) (0, Spin(9))
(E7)18 × U(1) SCFT
1(9) + (E6)6 SCFT
and the Lie-algebra embeddings
(e7)k ⊃ (f4)k ⊕ su(2)3k
(e7)k ⊃ so(9)k ⊕ su(2)2k ⊕ su(2)k
(e7)k ⊃ so(8)k ⊕ su(2)k ⊕ su(2)k ⊕ su(2)k
Finally, let us turn to the mixed fixture
1(1, 2) +
SU(3)12 symmetry of the D4 puncture, as before, we find that the S-dual is a Spin(8) gauge
theory, with matter in the 1(8v) + 1(8s) + 1(8c) + 2(1), coupled to two copies of the (E6)6
(0, Spin(8)) Spin(8)
(E6)6 SCFT
1(26) + (E6)6 SCFT
Applications
E6 and F4 gauge theory
E6 + 4(27)
as the 4-punctured sphere
E6 gauge theory, with four fundamental hypermultiplets, is superconformal. It is realized
The S-dual theory is an SU(2) gauging of the SU(4)54 × SU(2)7 × U(1) SCFT, with an
additional half-hypermultiplet in the fundamental.
(A5, SU(2))
SU(4)54 × SU(2)7 × U(1) SCFT
The k-differentials, which determine the Seiberg-Witten solution, are
(z − z1)(z − z2)(z − z3)(z − z4)
(z − z1)(z − z2)(z − z3)4(z − z4)4
u6 z122z344 (dz)6
u8 z122z364 (dz)8
u9 z122z374 (dz)9
(z − z1)2(z − z2)2(z − z3)4(z − z4)4
(z − z1)2(z − z2)2(z − z3)6(z − z4)6
(z − z1)2(z − z2)2(z − z3)7(z − z4)7
u12 z132z394 (dz)12
(z − z1)3(z − z2)3(z − z3)9(z − z4)9
and, for calculational purposes, it is usually convenient to use SL(2, C) to fix
(z1, z2, z3, z4) = (0, ∞, f (τ ), 1) in (5.1).
found in [18].
(A5, SU(2))
matter in the 12 (2) + 2(1).
The S-dual theory is an SU(2) gauging of the Sp(3)26 × SU(2)7 SCFT, with additional
2(26) + 2(1)
Sp(3)26 × SU(2)7 SCFT + 2(1)
F4 gauge theory, with three fundamentals, is also superconformal. It is realized as
of the E6 + 4(27).
Adding (E8)12 SCFTs
this theory is obtained by Higgsing E6 → F4, using one of the hypermultiplets in the 27.
In practice, given the solution to E6 +4(27), the solution to F4 +3(26)+2(1) is obtained
by noting that
• The Coulomb branch geometry of F4 + 3(26) + 2(1) is the geometry of the fixed-locus
Starting with the E6 +4(27) Lagrangian field theory, we can start replacing hypermultiplets
the flavour symmetry group of the theory is
F = SU(3)142−n
× SU(n)54 ×
SU(2)7 × U(1) SCFT, with an additional half-hypermultiplet in the fundamental (the U(1)
is absent for n = 0).
5.2.1 n = 3
With one copy of the (E8)12 SCFT,
is dual to
(A5, SU(2))
(A5, SU(2))
1(27) + (E8)12 SCFT
SU(3)54 × SU(3)12 × SU(2)7 × U(1) SCFT
5.2.2 n = 2
With two copies of the (E8)12 SCFT, there are two possible realizations. Either
1(27) + (E8)12 SCFT
1(27) + (E8)12 SCFT
SU(3)122 × SU(2)54 × SU(2)7 × U(1) SCFT
(A5, SU(2))
SU(3)122 × SU(2)54 × SU(2)7 × U(1) SCFT
These give two, apparently distinct, realizations of the SU(3)122 × SU(2)54 × SU(2)7 × U(1)
n = 1
With three copies of the (E8)12 SCFT, we have
1(27) + (E8)12 SCFT
(A5, SU(2))
SU(3)132 × SU(2)7 × U(1) SCFT
Finally, the E6 gauging of four copies of the (E8)12 SCFT,
is dual to
(A5, SU(2))
SU(3)142 × SU(2)7 SCFT
Connections with F-theory
are, respectively, the (E6)6, (E7)8 and (E8)12 superconformal field theories of Minahan and
Nemenschansky [36]. For higher n, the properties of these SCFTs were computed in [39].
The results may be summarized as follows
III∗
II∗
Flavour symmetry
Graded Coulomb
branch dimensions
(E6)6n × SU(2)(n−1)(3n+1)
n3l = 1, l = 1, 2, . . . , n
3n2 + 14n − 1, n(3n + 2)
(E7)8n × SU(2)(n−1)(4n+1)
n4l = 1, l = 1, 2, . . . , n
4n2 + 21n − 1, n(4n + 3)
(E8)12n × SU(2)(n−1)(6n+1)
n6l = 1, l = 1, 2, . . . , n
6n2 + 35n − 1, n(6n + 5)
Manifest flavour symmetry
III∗
II∗
[n2, n − 1, 1]
[n3, n − 1, 1]
[(3n)2] [(2n)3]
[n5, n − 1, 1]
SU(3)62n × SU(2)6n × U(1)2
(E6)6n × SU(2)k + 12 (2)
SU(2)8n × SU(4)8n × SU(3)8n × U(1)2
(E7)8n × SU(2)k + 12 (2)
SU(2)12n × SU(3)12n × SU(5)12n × U(1)2
(E8)12n × SU(2)k + 12 (2)
the predicted level (given that the hypermultiplet transforms as 12 (2) under the SU(2)).
of the SU(2).
examples of the higher-n theories
product SCFT in fixture 39 of section 3.4. It also appeared as an interacting fixture
in the D4 theory in [4].
product SCFT in fixture 6 of section 3.4.
Further examples can be found in the Z2-twisted sector. Notably, the fixtures
Isomorphic theories
In our table of interacting fixtures with enhanced global symmetry in section 3.4, we find
can be checked by various dualities. Some, however, cannot and we list them below.
SU(3)12×SU(2)36×SU(2)18×SU(2)9×U(1)2 SCFT
SU(3)122 × SU(2)9 × U(1)3 SCFT
SU(3)122 × Sp(2)10 × U(1)3 SCFT
It would be nice to check these conjectured isomorphisms by comparing the expansions
Acknowledgments
We would like to thank S. Katz, D. Morrison, A. Neitzke, R. Plesser and Y. Tachikawa
for helpful discussions. J. D. and O. C. would like to thank the Aspen Center for Physics
(supported, in part, by the National Science Foundation under Grant PHY-1066293)
for their hospitality when this work was initiated. O. C. would further like to thank
the Simons Foundation for partial support in Aspen. The work of J. D. and A. T. was
supported in part by the National Science Foundation under Grant PHY-1316033. The
work of O. C. was supported in part by the INCT-Matem´atica and the ICTP-SAIFR in
Brazil through a Capes postdoctoral fellowship.
Bala-Carter labels
In the twisted and untwisted sectors of the A and D series, punctures were in
one-toone correspondence with certain classes of partitions [1, 9, 10, 40]. The partition denotes
how the fundamental representation (vector representation, in the case of so(N )) of g6
read off the centralizer, f, of su(2) inside g, as well as the decomposition of the fundamental
representation of g under su(2) × f, from the partition (see (2.7) in [9]). The decomposition
under su(2) × f for each puncture is precisely the information needed to compute the
flavour group levels in section 2.4.1, as well as the expansion of the superconformal index
in section 2.5. In what follows, we will explain how these decompositions are obtained for
the punctures in the e6 theory.
In contrast to classical g, nilpotent orbits in the exceptional Lie algebras, which label
our punctures, are not naturally classified by partitions. Here, we recall the classification
of Bala and Carter [41, 42], following the exposition in [43]. Their theorem states that
there is a one-to-one correspondence between nilpotent orbits in g and (conjugacy classes
of) pairs (l, Ol) where l is a Levi subalgebra7 of g and Ol is a distinguished 8 nilpotent orbit
in l. By the Jacobson-Morozov theorem, any representative X of Ol embeds in a standard
l =
k∈Z
parabolic subalgebra of l, with explicit Levi decomposition into a Levi subalgebra l0 and
A nilpotent orbit in g is then given the label XN (ai), called the Bala-Carter label,
where XN is the Cartan type of the semisimple part of l, and i is the number of simple
in l, which is always distinguished.
There are 16 conjugacy classes of Levi subalgebras of E6. These are specified by their
semisimple parts: 0, A1, 2A1, 3A1, A2, A2 + A1, 2A2, A2, A2 + 2A1, A3 + A1, D4, A4,
A4 + A1, A5, D5, and E6. Here, kAN denotes the direct sum of k copies of AN . The
label 0 denotes the Cartan subalgebra, for which the only distinguished orbit is the zero
orbit. For l of classical type, distinguished orbits in l are easily specified in terms of their
partition: for l of type A, the only distinguished orbit is the principal orbit (which, for
which the partition has no repeated parts. It was found by Bala and Carter that, for l of
type G2, F4, E6, E7, and E8, there are 2, 4, 3, 6, and 11 distinguished orbits, respectively.
The distinguished orbits in the Levi subalgebras listed above give rise to 21 nilpotent
orbits in e6. We list these in the table below, along with the centralizer, f, and the
de6For untwisted (twisted) punctures in the A and D series, g is of type A (B) and D (C), respectively.
section 3.8 of [43] for an introduction.
8A nilpotent orbit, O, in g is distinguished if and only if the only Levi subalgebra of g, containing O, is
9Any su(2) subalgebra of g is spanned by a standard triple {H, X, Y } of nonzero elements of g satisfying
composition of the 27 and 78 of e6 under su(2) × f.10 But, before that, let us give a few
examples of how to obtain the decomposition of the 27 for various embeddings.
[7,1] and [5,3], corresponding to nilpotent orbits D4 and D4(a1), respectively, in e6. The
first has centralizer su(3) and the second, u(1)2. We can obtain the decomposition of the 27
The 27 of e6 decomposes under so(10) × u(1) as
The 10 and 16 of so(10) decompose under so(8) × u(1) as
e6 ⊃ so(10) × u(1)
27 = 1−4 + 102 + 16−1
so(10) ⊃ so(8) × u(1)
10 = 12 + 1−2 + (8v)0
16 = (8s)1 + (8c)−1
e6 ⊃ so(8) × u(1) × u(1)
For D4(a1), we embed su(2) in so(8) by taking
so we have
so(8) ⊃ su(2)
8v,s,c = 5 + 3
so(8) ⊃ su(2)
8v,s,c = 7 + 1
e6 ⊃ su(2) × u(1) × u(1)
For D4, we embed su(2) in so(8) by taking
e6 ⊃ su(2) × u(1) × u(1)
10The decomposition of the 27 determines a projection matrix, which can be used to obtain the
decompoThe decomposition of the 78 determines the levels of the flavor groups, as described in section 2.4.1.
For this embedding, the u(1)2 centralizer enhances to su(3). To see this, we can make a
change of basis so that the two u(1) charges are given in terms of the old ones by
q10 =
q20 =
(q1 + q2)
(q1 − q2)
Then the decomposition becomes
e6 ⊃ su(2) × u(1) × u(1)
where we recognize these u(1)2 charges as the weights (in the Dynkin basis) of the 6 and
3 of su(3). Thus, the decomposition of the 27 is given by
e6 ⊃ su(2) × su(3)
27 = (1, 6) + (7, 3)
orbits E6, E6(a1), and E6(a3). The decomposition of the 27 for each of these can be obtained
by taking the inner product of the Cartan element H of the embedded su(2) (which can
be read off from the weighted Dynkin diagram) into the weight vectors of the 27.
We conclude this appendix with a summary of the nilpotent orbits in e6 and the
corresponding decompositions of the 27 and the 78 under su(2) × f.
(1; 1) + (3; 7) + (5; 1)
A2 + 2A1
2A2 + A1
so(7) × u(1)
su(3) × su(2)
su(3) × su(3)
su(3) × u(1)
su(2) × u(1)
sp(2) × u(1)
(1; 15) + (2; 6)
(1;72 + 1−4)
+ (2; 8−1) + (3; 12)
(1;6, 1) + (2; 3, 2) + (3; 3, 1)
(1;3, 3) + (3; 1, 3) + (3; 3, 1)
(1; 32) + (2; 3−1 + 11)
+ (3; 30 + 1−2) + (4; 11)
(1;12 + 1−4) + (2; 4−1)
+ (3; 32) + (4; 2−1)
(1;5−2 + 14) + (4; 41)
+ (5; 1−2)
(1; 1) + (2; 2) + (3; 3)
+ (4; 2) + (5; 1)
(1; 35) + (2; 20) + (3; 1)
(1;10 + 210)
+ (2; 83 + 8−3) + (3; 70 + 10)
(1;8, 1) + (1; 1, 3) + (2; 8, 2)
+ (3; 1, 1) + (3; 8, 1) + (4; 1, 2)
(1;8, 1) + (1; 1, 8) + (3; 1, 1)
+ (3; 3, 3) + (3; 3, 3) + (5; 1, 1)
(1; 80 + 10) + (2; 31 + 3−1 + 1−3 + 13)
+ (3; 3−2 + 32 + 10 + 10)
+ (4; 31 + 3−1) + (5; 10)
(1; 14) + (3; 7 + 1) + (5; 7 + 1)
(1;10 + 30) + (2; 43 + 4−3)
+ (3; 10 + 30 + 50) + (4; 23 + 2−3)
(1;100 + 10) + (3; 10)
+ (4; 43 + 4−3) + (5; 50) + (7; 10)
(1;3) + (2; 4 + 2) + (3; 3 + 1 + 1)
+ (4; 2 + 2) + (5; 3 + 1) + (6; 2)
su(2) × u(1)
u(1) × u(1)
su(2) × u(1)
(1;14 + 1−2) + (2; 2−2)
+ (3; 11) + (4; 21)
+ (5; 11 + 1−2)
12,2 + 10,−4 + 1−2,2
+ 31,−1 + 30,2 + 3−1,−1
+ 51,−1 + 50,2 + 5−1,−1
(1;2−5) + (3; 1−2)
+ (5; 21 + 14) + (7; 1−2)
(1; 6) + (7; 3)
2−5 + 3−2 + 41
+ 54 + 61 + 7−2
1−4 + 2−1 + 32
+ 6−1 + 72 + 8−1
(1; 1) + (5; 1) + (6; 2) + (9; 1)
1 + 5 + 5 + 7 + 9
12 + 1−4 + 5−1 + 92 + 11−1
5 + 9 + 13
1 + 9 + 17
(1;10 + 30) + (2; 2)0
+ (3; 13 + 1−3 + 10 + 10)
+ (4; 23 + 2−3 + 20)
+ (5; 13 + 10 + 1−3)
+ (6; 2)0 + (7; 10)
10,0 + 10,0 + 30,0
+ 32,0 + 31,3 + 31,−3 + 30,0
+ 3−2,0 + 3−1,−3 + 3−1,3 + 30,0
+ 52,0 + 51,3 + 51,−3 + 50,0
+ 5−2,0 + 5−1,−3 + 5−1,3
+ 70,0 + 70,0
(1;30 + 10) + (3; 23 + 2−3 + 10)
+ (5; 16 + 10 + 1−6)
+ (7; 23 + 2−3 + 10) + (9; 10)
(1; 8) + (3; 1) + (7; 8) + (11; 1)
10 + 23 + 2−3 + 30 + 30 + 43 + 4−3
+ 56 + 50 + 5−6 + 6−3 + 63
+ 70 + 8−3 + 83 + 90
10 + 23 + 2−3 + 30 + 30 + 50
+ 63 + 6−3 + 70 + 70
+ 83 + 8−3 + 90 + 110
(1; 3) + (3; 1) + (4; 2) + (5; 1) + (6; 2)
+ (7; 1) + (9; 1) + (10; 2) + (11; 1)
3 + 3 + 3 + 5 + 5 + 5
+ 7 + 7 + 9 + 9 + 11 + 11
10 + 30 + 53 + 5−3 + 70 + 90
+ 113 + 110 + 11−3 + 150
3 + 5 + 7 + 9 + 11 + 11 + 15 + 17
3 + 9 + 11 + 15 + 17 + 23
Projection matrices
Our classification of interacting and mixed fixtures using the superconformal index,
carried out in section 2.5, required that we know the decomposition of a number of
higherdimensional e6 representations (and not just the 27 and the 78) under su(2) × f. These
are trivial to obtain using LieART [44], provided we know a projection matrix for each
embedding [44, 45].
From the decomposition of the 27, listed in the table above, one obtains a projection
matrix simply by defining a 6 × rk (su(2) × f) matrix, M , such that the LieART command
gives the corresponding su(2) × f weights. This projection matrix can then be used to
obtain the decomposition of any e6 irrep under su(2) × f.
Below, we list a projection matrix for each embedding, following the conventions of
su(3)×su(3) 0 0 0 0 1 0
−1 −2 −3 −2 −1 −2
A2+A1 su(3)×u(1)
1 1 −1 −1 −1 −1
2A2 + A1
−2 −1 0 1 2 0
A3 + A1
A4 + A1
−2 −4 −6 −2 2 −3
−1 −2 0 2 1 0
8 14 18 14 8 8
−1 −2 0 2 1 0
12 22 30 22 12 16
16 30 42 30 16 22
As an example, let’s work out the decomposition of the 51975 for the orbit 2A2.
Running LieART, we obtain the decomposition with the following two lines of code:
+ 4(11, 1) + 25(7, 7) + 9(1, 14) + 17(9, 7) + 16(3, 14) + 6(11, 7) + 22(5, 14)
+ 2(13, 7) + 15(7, 14) + 10(9, 14) + 3(11, 14) + (13, 14) + 6(1, 27) + 25(3, 27)
+ 23(5, 27) + 21(7, 27) + 9(9, 27) + 4(11, 27) + 5(1, 64) + 12(3, 64) + 13(5, 64)
+ 9(7, 64) + 4(9, 64) + (11, 64) + 4(1, 77) + 6(3, 77) + 2(3, 770) + 8(5, 77)
+ (5, 770) + 4(7, 77) + (7, 770) + 2(9, 77) + (3, 182) + (1, 189) + 2(3, 189)
+ 2(5, 189) + (7, 189)
This works for all of the orbits above, except for D4(a1), as the LieART command
“DecomposeIrrep” does not seem to work when the target subalgebra has more than one
u(1) factor. In this case, getting the decomposition is only slightly more complicated. For
example, we obtain the decomposition of the 27 of E6 as follows:
In[1]= ProjectionMatrix[D5,ProductAlgebra[D4,U1]]=0 0 1 1 0 ;
0 1 0 0 0
0 0 1 0 1
Irrep[E6][1, 0, 0, 0, 0, 0], ProductAlgebra[D5, U1]],
ProductAlgebra[D4, U1], 1], ProductAlgebra[A1], 1]
+(3)(-1)(-1)+(5)(1)(-1)+(5)(0)(2)+(5)(-1)(-1)
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approximation, arXiv:0907.3987 [INSPIRE].
Riemann surface with finite area, Prog. Theor. Exp. Phys. 2013 (2013) 013B03
[INSPIRE].
[11] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the twisted D-series, JHEP 04
[16] S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) 87
Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action,
Encyclopaedia Math. Sci. 131 (2002) 159, Springer, Germany (2002).
[23] E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl.
[24] GAP Group collaboration, GAP — Groups, Algorithms, and Programming, Version 4.7.4,
supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
[INSPIRE].
surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
[INSPIRE].
[1] D. Gaiotto , N = 2 dualities, JHEP 08 ( 2012 ) 034 [arXiv:0904.2715] [INSPIRE].
[2] D. Gaiotto , G.W. Moore and A. Neitzke , Wall-crossing, Hitchin systems and the WKB [3] O. Chacaltana and J. Distler , Tinkertoys for Gaiotto duality, JHEP 11 ( 2010 ) 099 [4] O. Chacaltana and J. Distler , Tinkertoys for the DN series, JHEP 02 ( 2013 ) 110 [5] D. Nanopoulos and D. Xie , Hitchin equation, singularity and N = 2 superconformal field [6] D. Nanopoulos and D. Xie , N = 2 generalized superconformal quiver gauge theory , JHEP 09 [7] F. Benini , Y. Tachikawa and D. Xie , Mirrors of 3d Sicilian theories , JHEP 09 ( 2010 ) 063 [8] D. Gaiotto , G.W. Moore and Y. Tachikawa , On 6d N = ( 2 , 0) theory compactified on a [9] O. Chacaltana , J. Distler and Y. Tachikawa , Nilpotent orbits and codimension-two defects of [10] O. Chacaltana , J. Distler and Y. Tachikawa , Gaiotto duality for the twisted A2N−1 series , [13] L. Bhardwaj and Y. Tachikawa , Classification of 4d N = 2 gauge theories , JHEP 12 ( 2013 ) [14] E. Witten , Some comments on string dynamics , hep-th/ 9507121 [INSPIRE].
[15] S. Gukov and E. Witten , Gauge theory, ramification, and the geometric Langlands program , [17] Y. Tachikawa , N = 2 S-duality via outer-automorphism twists , J. Phys. A 44 (2011) 182001 [18] S. Terashima and S.-K. Yang , Exceptional Seiberg-Witten geometry with massive fundamental matters , Phys. Lett . B 430 ( 1998 ) 102 [hep-th /9803014] [INSPIRE].
[19] R. Lawther and D.M. Testerman , Centres of centralizers of unipotent elements in simple algebraic groups , Mem. Amer. Math. Soc . 210 ( 2011 ) no. 988, vi+ 188 .
[20] W.M. McGovern , The adjoint representation and the adjoint action , in Algebraic Quotients.
[21] N. Spaltenstein , Classes unipotentes et sous-groupes de Borel , Lect. Notes Math. 946 , [22] Y. Tachikawa and S. Terashima , Seiberg-Witten geometries revisited, JHEP 09 ( 2011 ) 010 [25] A. Klemm , W. Lerche , P. Mayr , C. Vafa and N.P. Warner , Selfdual strings and N = 2 [26] W. Lerche and N.P. Warner , Exceptional SW geometry from ALE fibrations, Phys . Lett . B [27] S. Katz and D.R. Morrison , Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups , J. Alg. Geom . 1 ( 1992 ) 449 [alg -geom/9202002].
[28] R.W. Carter , Finite groups of Lie type - conjugacy classes and complex characters , Pure and Applied Mathematics, John Wiley & Sons, U.S.A. ( 1985 ).
[29] J. Kinney , J.M. Maldacena , S. Minwalla and S. Raju , An index for 4 dimensional super conformal theories, Commun . Math. Phys. 275 ( 2007 ) 209 [hep-th /0510251] [INSPIRE].
[30] A. Gadde , E. Pomoni , L. Rastelli and S.S. Razamat , S-duality and 2d topological QFT, [31] A. Gadde , L. Rastelli , S.S. Razamat and W. Yan , The 4d superconformal index from q-deformed 2d Yang-Mills , Phys. Rev. Lett . 106 ( 2011 ) 241602 [arXiv:1104.3850] [INSPIRE].
[32] A. Gadde , L. Rastelli , S.S. Razamat and W. Yan , Gauge theories and Macdonald polynomials, Commun . Math. Phys. 319 ( 2013 ) 147 [arXiv:1110.3740] [INSPIRE].
[33] M. Lemos , W. Peelaers and L. Rastelli , The superconformal index of class S theories of type [34] D. Gaiotto and S.S. Razamat , Exceptional indices, JHEP 05 ( 2012 ) 145 [arXiv:1203.5517] [35] D. Gaiotto , L. Rastelli and S.S. Razamat , Bootstrapping the superconformal index with [36] J.A. Minahan and D. Nemeschansky , Superconformal fixed points with EN global symmetry , [37] K. Dasgupta and S. Mukhi , F theory at constant coupling , Phys. Lett . B 385 ( 1996 ) 125 [38] T. Banks , M.R. Douglas and N. Seiberg , Probing F-theory with branes , Phys. Lett. B 387 [39] O. Aharony and Y. Tachikawa , A holographic computation of the central charges of D = 4 , [40] Y. Tachikawa , Six-dimensional DN theory and four-dimensional SO-USp quivers , JHEP 07 [41] P. Bala and R. Carter , Classes of unipotent elements in simple algebraic groups I , Math.
[42] P. Bala and R. Carter , Classes of unipotent elements in simple algebraic groups II , Math.
[43] D.H. Collingwood and W.M. McGovern , Nilpotent orbits in semisimple Lie algebras , Van [44] R. Feger and T.W. Kephart , LieART - a Mathematica application for Lie algebras and representation theory, Comput . Phys. Commun . 192 ( 2015 ) 166 [arXiv:1206.6379] [45] R. Slansky , Group theory for unified model building , Phys. Rept . 79 ( 1981 ) 1 [INSPIRE].