Tinkertoys for the $E_8$ Theory
Oscar Chacaltana Jacques Distler Anderson Trimm Yinan Zhu




UTTG–05–18













We construct the 4D 𝒩=2 SCFTs of class-S, which stem from the E_8 (2,0) theory. There are 49,836 isolated SCFTs which arise as 3-punctured spheres. Of these, 149 are “mixed" (contain free hypermultiplets accompanying the interacting SCFT) and 775 have enhanced global symmetries (beyond the manifest global symmetry associated to the punctures). Among the 49,836 3-punctured spheres we find (after removing any free hypermultiplets which may be present) 29 that are product SCFTs.  Turning to 4-punctured spheres, we find 1,025,438 4D SCFTs arising as a gauging (with a simple gauge group) of a pair of 3-punctured spheres. We discuss a number of applications, including recovering several known 4D SCFTs. Our full set of results can be accessed on the Web at  https://golem.ph.utexas.edu/class-S/E8/https://golem.ph.utexas.edu/class-S/E8/.


Tinkertoys for the E_8 Theory
    Oscar Chacaltana
    
    Howard Community College

    10901 Little Patuxent Pkwy

    Columbia, MD 21044, USA

     

    ochacaltanaalarcon@howardcc.edu

    ,
    Jacques Distler
     
     Theory Group

     Department of Physics,

     University of Texas at Austin,

     Austin, TX 78712, USA 

      

      distler@golem.ph.utexas.edu

      zhuyinan@physics.utexas.edu
	,
	Anderson Trimm
	
	Department of Mathematics, 

     Gwinnett School of Mathematics,

     Science and Technology,

     970 McElvaney Lane,

     Lawrenceville, GA 30044, USA

      

     anderson.trimm@gsmst.org
     
     and Yinan Zhu ^b

    December 30, 2023
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§ INTRODUCTION


In the Wilsonian approach, a continuum quantum field theory is a relevant perturbation of a conformal theory. A first step, then, in understanding all quantum field theories, is to understand the conformal theories. With extended supersymmetry, this seems like an achievable goal and tremendous progress has been made in recent years.

With 𝒩=2 supersymmetry, the class-S construction <cit.> has proven a very fruitful tool. These theories arise as compactifications of the 6D (2,0) theory on a punctured Riemann surface, with a partial topological twist along the fibers. The punctures are the locations of (4D spacetime-filling) codimension-2 defect operators which are labeled by nilpotent orbits in the ADE Lie algebra of the (2,0) theory. The 4D SCFT, thus obtained, has a space of exactly marginal deformations which is (roughly) ℳ_g,n, the moduli space of complex structures for the punctured Riemann surface, C. At the boundaries of ℳ_g,n, where C develops a node, the SCFT decomposes into a product of free vector multiplets and the SCFTs corresponding to each of the components of the normalization of C. Moving away from the boundary corresponds to turning on a marginal (complex) gauge coupling for a weakly-gauged subgroup of the flavour symmetry of the theory. Each pants-decomposition of C gives rise to a different (“S-dual”) presentation of this family of theories as a gauging of a product of SCFTs corresponding to 3-punctured spheres. To understand the theories that can arise in this way, we need to classify the SCFTs corresponding to 3-punctured spheres and their allowed gaugings. Arbitrarily complicated 4D SCFTs can then be assembled, in tinkertoy fashion.

In previous papers, we have carried out this program for the (2,0) theories of A <cit.> and D <cit.> types, and for E_6 <cit.> and E_7 <cit.> (as well as for the twisted A_2N-1 <cit.>, twisted D_N <cit.> and twisted E_6 <cit.> theories). Here, we complete the classification by presenting the tinkertoy catalogue for E_8.

There are 70 nilpotent orbits in 𝔢_8. Excluding the regular orbit (which corresponds to the trivial defect), that gives 69 codimension-2 defect operators (“regular punctures”) in the E_8 (2,0) theory. Given a triple of defects, we can compactify the (2,0) theory on a 3-punctured sphere (fixture). Naively, that would give rise to 57,155 different fixtures. Of these, 7,319 are “bad” (do not lead to well-defined 4D theories)[A simple diagnostic for when a fixture is bad will be given in <ref>]. But that still leaves 49,836 new 4D SCFTs, arising as spheres with 3 regular punctures.

To these, we must supplement a set of “irregular fixtures” which result from the collision of a pair of punctures (conformally-equivalent to bubbling off a 3-punctured sphere) where the 3-punctured sphere that would have resulted from this degeneration of C (with the third puncture being the full puncture) would have been bad. There are 50 irregular fixtures in the E_8 theory. These never occur in isolation; they only appear when C degenerates in such a way that one of the components of its normalization has genus-0 and 6 or fewer punctures.

Unlike all of the previous cases, where there was at least one regular fixture consisting of free hypermultiplets, all of the regular fixtures in the E_8 theory are interacting SCFTs (or “mixed” fixtures, consisting of an interacting SCFT plus some number of free hypermultiplets). As a consequence, none of the 4D SCFTs obtained via the class-S construction for E_8 has an interpretation as a Lagrangian field theory in any S-duality frame.

Still, there are a vast number of them. If we consider the next-simplest case, a 4-punctured sphere, we have a gauge theory with a simple gauge group, coupled to some appropriate matter content (consisting of interacting SCFTs and perhaps some additional hypermultiplets) which ensures the vanishing of the β-function. There are 1,025,438 distinct possibilities (after eliminating the bad ones).

Since these numbers are vastly too large to list in a paper, we have created an interactive web application [https://golem.ph.utexas.edu/class-S/E8/https://golem.ph.utexas.edu/class-S/E8/ where the interested reader can explore them.

Most of these results are a straightforward application of the ideas of <cit.>, in that much of the physics is captured by the local properties of the punctures.

Three parts of the calculation are particularly arduous, though.



  * Determining which fixtures have enhanced global symmetries (enhanced over the manifest global symmetry group, which is the product of the global symmetries associated to each of the punctures) and which fixtures contain free hypermultiplets requires a computations of the Hall-Littlewood index <cit.> to order τ^2. This is tedious because the index receives contributions from E_8 representations of very high dimensions.

  * Determining which fixtures are product SCFTs, using the methods of <cit.> is even more arduous, as it requires computing the Hall-Littlewood index to order τ^4 and the Schur index to the same order. Fortunately, we have found a simpler diagnostic which gives a sufficient (it is unknown whether it is necessary) condition for a fixture to be a product SCFT. <cit.> proved a lower bound on the level of the current algebra for any simple factor in the global symmetry group. When this bound is violated, an assumption that went into the proof — namely that the SCFT has only a single stress-tensor multiplet — is violated, and the theory must be a product SCFT. We found 22 fixtures which are product SCFTs and where we were able to identify the factors in the product. As discussed in <ref>,<ref>, there are 7 more fixtures that we know to be product SCFTs, but were not able to definitively identify the factors in the product. 

  * Finally, to determine the Coulomb branch geometry of any of these theories, we need the detailed form of the contraints on the pole coefficients of the meromorphic k-differentials which have poles at the locations of the punctures. This is, by far, the most tedious part of the calculation, and one that we have not yet completed. So far, we have only determined the constraints associated to half of the punctures (and have partial results for the rest). Nevertheless, this was enough to bootstrap our knowledge of each puncture's contribution to the graded Coulomb branch dimensions. The full set of constraints will be presented elsewhere <cit.>.




§ THE E_8 THEORY




 §.§ Puncture Properties


As usual, we use Bala-Carter notation <cit.> to label the nilpotent orbits, where O_N=0 is the full puncture and O_N=E_8(a_1) is the simple puncture.

The flavour symmetry algebra, 𝔣, associated to a puncture is the centralizer of ρ_N:𝔰𝔲(2)→𝔢_8. For the distinguished orbits, this is trivial, whereas for the 0 orbit, it is all of 𝔢_8. The level of each simple factor, 𝔣_i⊂𝔣 is determined from the decomposition of the adjoint representation under the embedding 𝔢_8⊃𝔰𝔲(2)×𝔣_i. Writing this decomposition as


    𝔢_8 = ⊕_n V_n⊗ R_n,i

where V_n is the n-dimensional irreducible representation of 𝔰𝔲(2) and R_n,i is the corresponding (in general, reducible) representation of 𝔣_i, the level of 𝔣_i is given by


k_i = ∑_n l_n,i

 where l_n,i is the Dynkin index of the representation R_n,i. The normalization of the levels is such that vanishing β-function for a gauging of F_i is achieved when the total level of all of the matter charged under F_i is equal to 4κ_𝔣_i, where κ is the dual Coxeter number. There are 10 distinguished punctures, whose flavour symmetry algebra is trivial.

For 𝔣_i = 𝔲(1), the level is computed as above, where l_n,i is the 𝔲(1) charge squared. We normalize the 𝔲(1) generators so that the free hypermultiplets in the mixed fixtures have charge 1. However, the only puncture with a 𝔲(1) factor in its flavor symmetry algebra with free hypers charged under that 𝔲(1) is A_3+A_2. Since we don't know how to normalize the other 𝔲(1) generators, we don't compute their levels.

At a puncture Φ(z) has a simple pole with nilpotent residue,


Φ(z) = E/z + regular

where E is a representative of the “Hitchin” Nilpotent orbit which is Spaltenstein-dual to the Nahm orbit (which we use to label our punctures)


O_H = d (O_N).

Let P_k(z)= 1k Tr(Φ^k), where the trace is in the adjoint representation. It is more convenient to work in the Katz-Morrison basis <cit.>, related to the P_k(z) by


    ϕ_2 =    12^3× 3× 5P_2
    ϕ_8=   -12^7× 3^2× 5(P_8-132^8× 3^3× 5^4 P_2^4)
    ϕ_12=   12^6× 3^4× 5× 7(P_12-3^22^7× 5^2 P_8 P_2^2+7× 1012^15× 3^5× 5^6 P_2^6)
    ϕ_14=   12^6× 3^2× 5^2× 7× 11(P_14-712^4× 3^2× 5 P_12 P_2+11× 1032^11× 3^3× 5^3 P_8 P_2^3-11× 25312^19× 3^7× 5^7 P_2^7)
    ϕ_18=   -12^8× 3^4× 5^2× 7× 13(P_18-15232^6× 3× 5^3× 11 P_14 P_2^2+13× 44512^10× 3^3× 5^4× 11 P_12 P_2^3.
       .-13× 3312^11× 3^2× 5^2 P_8^2 P_2-13× 47× 1012^18× 3^5× 5^5 P_8 P_2^5+13× 263992^27× 3^7× 5^9 P_2^9)
    ϕ_20 =     -12^7× 3^2× 5^2× 11× 17× 41(P_20 - 2366272^5× 3^3× 5^2× 7× 13 P_18P_2+17× 67× 173892^11× 3^4× 5^5× 7× 13 P_14P_2^3.
       .-17× 1272^7×  3× 5× 7P_12 P_8-17× 1910712^11× 3^6× 5^6× 7 P_12 P_2^4+11× 17× 3233712^16× 3^5× 5^4× 7 P_8^2 P_2^2.
       .+11× 17× 907× 33012^23× 3^8× 5^7× 7 P_8 P_2^6-11× 13^2× 17× 606492^32× 3^11× 5^10× 7 P_2^10)
    ϕ_24 =   12^8× 3^3× 5× 7× 11× 19× 199(P_24-7^2× 951892^7× 3^2× 5^4× 17× 41 P_20 P_2^2+19× 101× 151× 368212^12× 3^3× 5^6× 13× 17× 41P_18 P_2^3.
       .-19× 31× 7512^10× 3^3× 5^4 P_14 P_8 P_2-19× 349× 262804272^17× 3^6× 5^9× 13× 41 P_14 P_2^5-19× 1932^6× 3× 5^2× 7P_12^2.
       .+19× 269× 8705932^13× 3^5× 5^5× 7× 41P_12 P_8 P_2^2+19× 199× 379× 6022972^22× 3^8× 5^10× 41P_12 P_2^6-11× 19× 5932^14× 3^2× 5^3 P_8^3.
       .-11× 19^2× 8913513472^23× 3^6× 5^8× 7× 41 P_8^2 P_2^4-11× 19× 1489× 140875992^29× 3^10× 5^11× 41 P_8 P_2^8+19× 331× 49871× 5008812^39× 3^13× 5^15× 41 P_2^12)
    ϕ_30=   -12^7× 3^4× 5^5× 7× 11× 13× 61(P_30-13× 389× 548292^13× 3^2× 5^3× 11× 19× 199  P_24P_2^3-7^3× 13× 14832^11× 3^2× 5^2× 11× 41P_20 P_8 P_2.
       . +7× 13× 9787× 2206235412^20× 3^5× 5^7× 11× 19× 41× 199P_20 P_2^5 -25212^8× 3^2× 7 P_18P_12 +83× 126668772^15× 3^5× 5^2× 7× 11× 41 P_18P_8 P_2^2.
       . -1873× 1118261× 74421172^25× 3^8× 5^9× 11× 41× 199 P_18P_2^6 -17× 16765692^11× 3^2× 5^3× 7× 11^2  P_14^2 P_2+19× 99530232^14× 3^4× 5^3× 7× 11^2 P_14 P_12 P_2^2.
       . -13× 12232^14× 3^2× 5P_14 P_8^2 +34423329381709932^23× 3^6× 5^7× 7× 11× 41× 199 P_14 P_8 P_2^4+23× 710658077568509232^29× 3^9× 5^12× 7× 11× 41× 199 P_14 P_2^8.
       . +13× 89× 71497612^18× 3^7× 5^2× 7× 11^2× 199 P_12^2 P_2^3 +13× 1381× 1447732^19× 3× 5^3× 7× 11× 41 P_12 P_8^2 P_2 -13× 31× 90659× 1333462312^23× 3^7× 5^8× 7× 11× 41× 199 P_12 P_8 P_2^5.
       . -13× 9059× 3294013× 41147412^34× 3^12× 5^13× 7× 11× 41 P_12 P_2^9 -13× 1609× 1399875681112^27× 3^8× 5^6× 7× 41× 199 P_8^3 P_2^3 +13× 17× 2273× 4259× 3251885032^36× 3^10× 5^11× 7× 199 P_8^2 P_2^7.
       . +13× 9662050433528942872^39× 3^13× 5^14× 7× 41× 199 P_8 P_2^11 -29× 71× 10477× 43777× 410181089832^52× 3^16× 5^18× 7× 11× 41× 199 P_2^15)

The ϕ_k(z) have poles at the punctures of order at most p_k:


ϕ_k(z) = ∑_j=1^p_k c^(k)_j/z^j + regular

The c^(k)_j are gauge-invariant parameters which contribute to the (graded) dimension(s) of the Coulomb branch. Were they all independent, the contribution would be p_k. Instead, the c^(k)_j typically obey an elaborate set of polynomial relations. When the special piece of O_N has more than one element, we have an additional quotient by a finite group (the “Sommers-Achar group”) acting on the coefficients.

The detailed form of the constraints will be presented elsewhere <cit.>. For present purposes, it will suffice to list the net contribution n_k to the graded Coulomb branch dimension, which results after solving the constraints.

For each puncture, we also list its contribution to n_h=4(5c-4a) and n_v=4(2a-c).



 §.§ Regular Punctures


Here, we list the pole structures, {p_2,p_8,p_12,p_14,p_18,p_20,p_24,p_30} of the 69 regular punctures of the E_8 theory, their graded Coulomb branch dimensions,{n_2,n_3,n_4,n_5,n_6,n_8,n_9,n_10,n_12,n_14,n_15,n_18,n_20,n_24,n_30}, flavour symmetries and (n_h,n_v).





|c|c|c|c|c|c|
      Pole structure   Graded Coulomb Branch Dimensions   Flavour group   (δ n_h,δ n_v)

0   E_8   {1, 7, 11, 13, 17, 19, 23, 29}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 17, 19, 23, 29}   (E_8)_60   (4960,4840)

A_1   E_8(a_1)   {1, 7, 11, 13, 17, 19, 23, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 17, 19, 23, 28}   (E_7)_48   (4872,4781)

2A_1   E_8(a_2)   {1, 7, 11, 13, 17, 19, 22, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 17, 19, 22, 28}   SO(13)_40   (4808,4734)

3A_1   (E_8(a_3),ℤ_2)   {1, 7, 11, 13, 17, 18, 22, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 17, 18, 22, 28}   (F_4)_36× SU(2)_31   (4759,4695)
A_2   E_8(a_3)   {1, 7, 11, 13, 17, 18, 22, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 1, 17, 18, 22, 27}   (E_6)_36   (4728,4665)
4A_1   (E_8(a_4),ℤ_2)   {1, 7, 11, 13, 16, 18, 22, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 16, 18, 22, 28}   Sp(4)_31   (4716,4660)
A_2+A_1   E_8(a_4)   {1, 7, 11, 13, 16, 18, 22, 28}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 1, 16, 18, 22, 27}   SU(6)_30   (4682,4630)
A_2+2A_1   E_8(b_4)   {1, 7, 11, 13, 16, 18, 22, 27}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 13, 0, 16, 18, 22, 27}   SO(7)_28×SU(2)_144   (4648,4601)
A_2+3A_1   (E_8(a_5),ℤ_2)   {1,7,11,12,16,18,22,27}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 12, 0, 16, 18, 22, 27}   (G_2)_48×SU(2)_25   (4617,4574)
2A_2   E_8(a_5)   {1, 7, 11, 12, 16, 18, 22, 27}   {1, 0, 0, 0, 0, 7, 0, 0, 12, 12, 0, 16, 18, 21, 27}   (G_2)^2_24   (4592,4550)
A_3   E_7(a_1)   {1, 7, 11, 13, 16, 18, 22, 27}   {1, 0, 0, 0, 0, 7, 0, 1, 11, 13, 0, 16, 18, 21, 26}   SO(11)_28   (4560,4514)
2A_2+A_1   (E_8(b_5),S_3)   {1, 7, 11, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 0, 0, 11, 12, 0, 16, 18, 21, 27}   (G_2)_24×SU(2)_62   (4566,4527)
A_3+A_1   (E_8(b_5),ℤ_2)   {1, 7, 11, 12, 16, 18,21,27}   {1, 0, 0, 0, 0, 7, 0, 1, 11, 12, 0, 16, 18, 21, 26}   SO(7)_24×SU(2)_21   (4525,4487)
D_4(a_1)   E_8(b_5)   {1, 7, 11, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 0, 2, 11, 12, 0, 16, 17, 21, 26}   SO(8)_24   (4504,4467)
2A_2+2A_1   (E_8(a_6),S_3)   {1, 7, 10, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 0, 0, 10, 12, 0, 16, 18, 21, 27}   Sp(2)_62   (4540,4504)
A_3+2A_1   (E_8(a_6),ℤ_2)   {1, 7, 10, 12,16,18,21,27}   {1, 0, 0, 0, 0, 7, 0, 1, 10, 12, 0, 16, 18, 21, 26}   Sp(2)_21×SU(2)_40   (4498,4464)
D_4(a_1)+A_1   E_8(a_6)   {1, 7, 10, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 0, 2, 10, 12, 0, 16, 17, 21, 26}   SU(2)_20^3   (4476,4444)
D_4   E_6   {1, 7, 11, 12, 16, 17, 21, 26}   {1, 0, 0, 0, 1, 7, 0, 0, 11, 12, 0, 16, 16, 20, 24}   (F_4)_24   (4272,4236)
A_3+A_2   D_7(a_1)   {1, 7, 10, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 0, 1, 10, 12, 0, 16, 17, 21, 26}   Sp(2)_20× U(1)_48   (4456,4425)
A_4   E_7(a_3)   {1, 7, 10, 12, 16, 18, 21, 27}   {1, 0, 0, 0, 0, 7, 1, 1, 10, 12, 1, 15, 17, 20, 25}   SU(5)_20   (4360,4330)
A_3+A_2+A_1   (E_8(b_6),ℤ_2)   {1,7,10,12,16,17,21,26}   {1, 0, 0, 0, 0, 7, 0, 0, 10, 12, 0, 16, 17, 21, 26}   SU(2)_384× SU(2)_19   (4435,4406)
D_4(a_1)+A_2   E_8(b_6)   {1, 7, 10, 12, 16, 17, 21, 26}   {1, 0, 0, 0, 0, 7, 1, 0, 10, 12, 0, 15, 17, 21, 26}   SU(3)_96   (4416,4388)
A_4+A_1   E_6(a_1)+A_1   {1, 7, 10, 12, 16, 17, 21, 26}   {1, 0, 0, 0, 0, 7, 1, 0, 10, 12, 1, 15, 17, 20, 25}   SU(3)_18× U(1)   (4337,4311)
D_4+A_1   (E_6(a_1),ℤ_2)   {1, 7, 10, 12, 16,17,21,26}   {1, 0, 0, 0, 1, 7, 0, 0, 10, 12, 0, 16, 16, 20, 24}   Sp(3)_19   (4241,4213)
D_5(a_1)   E_6(a_1)   {1, 7, 10, 12, 16, 17, 21, 26}   {1, 0, 0, 0, 1, 7, 1, 0, 10, 12, 0, 15, 16, 20, 24}   SU(4)_18   (4220,4195)
2A_3   (D_7(a_2),ℤ_2)   {1, 7, 10, 12, 15, 17, 20, 26}   {1, 0, 0, 0, 0, 7, 0, 0, 10, 12, 0, 15, 17, 20, 26}   Sp(2)_31   (4350,4324)
A_4+2A_1   D_7(a_2)   {1, 7, 10, 12, 15, 17, 20, 26}   {1, 0, 0, 0, 0, 7, 0, 0, 10, 12, 1, 15, 17, 20, 25}   SU(2)_30× U(1)   (4318,4294)
A_4+A_2   D_5+A_2   {1, 7, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 0, 7, 0, 0, 10, 12, 0, 15, 17, 20, 25}   SU(2)_16× SU(2)_200   (4288,4265)
D_5(a_1)+A_1   E_7(a_4)   {1, 7, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 1, 7, 0, 0, 10, 12, 0, 15, 16, 20, 24}   SU(2)_16× SU(2)_112   (4200,4178)
A_4+A_2+A_1   A_6+A_1   {1, 6, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 0, 6, 0, 0, 10, 12, 0, 15, 17, 20, 25}   SU(2)_200   (4272,4250)
D_4+A_2   A_6   {1, 6, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 1, 6, 0, 0, 10, 12, 0, 15, 16, 20, 24}   SU(3)_28   (4184,4163)
A_5   (D_6(a_1),ℤ_2)   {1, 7, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 1, 7, 0, 0, 11, 11, 0, 15, 16, 19, 24}   (G_2)_16×SU(2)_13   (4149,4127)
E_6(a_3)   D_6(a_1)   {1, 7, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 2, 7, 0, 0, 10, 11, 0, 15, 16, 19, 24}   (G_2)_16   (4136,4115)
D_5   D_5   {1, 7, 10, 12, 15, 17, 20, 25}   {1, 0, 0, 0, 1, 7, 0, 1, 10, 11, 0, 14, 15, 18, 22}   SO(7)_16   (3904,3884)
A_4+A_3   (E_8(a_7),S_5)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 0, 6, 0, 0, 10, 11, 0, 15, 16, 20, 25}   SU(2)_124   (4204,4184)
A_5+A_1   (E_8(a_7),S_3×ℤ_2)   {1,6,10,11,15,16,20,25}   {1, 0, 0, 0, 2, 6, 0, 0, 10, 11, 0, 14, 16, 20, 24}   SU(2)_38× SU(2)_13   (4131,4112)
D_5(a_1)+A_2   (E_8(a_7),S_4)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 1, 6, 0, 0, 10, 11, 0, 15, 16, 20, 24}   SU(2)_75   (4155,4136)
D_6(a_2)   (E_8(a_7),ℤ_2×ℤ_2)   {1,6,10,11,15,16,20,25}   {1, 0, 0, 0, 3, 6, 0, 0, 9, 11, 0, 15, 16, 19, 24}   SU(2)_13^2   (4106,4088)
E_6(a_3)+A_1   (E_8(a_7),S_3)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 2, 6, 0, 0, 10, 11, 0, 15, 16, 19, 24}   SU(2)_38   (4118,4100)
E_7(a_5)   (E_8(a_7),ℤ_2)   {1, 6, 10, 11,15,16,20,25}   {1, 0, 0, 0, 3, 6, 0, 0, 10, 11, 0, 14, 16, 19, 24}   SU(2)_13   (4093,4076)
E_8(a_7)   E_8(a_7)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 4, 6, 0, 0,  9, 11, 0, 14, 16, 19, 24}   -   (4080,4064)
A_6   D_4+A_2   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 1, 6, 0, 1,  9, 11, 0, 14, 15, 18, 23}   SU(2)_12× SU(2)_60   (3920,3905)
D_5+A_1   (E_6(a_3),ℤ_2)   {1, 6, 10, 11, 15,16,20,25}   {1, 0, 0, 0, 1, 6, 0, 1, 10, 11, 0, 14, 15, 18, 22}   SU(2)_13× SU(2)_24   (3885,3869)
D_6(a_1)   E_6(a_3)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 2, 6, 0, 1, 9, 11, 0, 14, 15, 18, 22}   SU(2)_12^2   (3872,3857)
A_6+A_1   A_4+A_2+A_1   {1, 6,  9, 11, 14, 16, 19, 24}   {1, 0, 0, 0, 0, 6, 0, 1, 9, 11, 0, 14, 15, 18, 23}   SU(2)_60   (3908,3894)
E_7(a_4)   D_5(a_1)+A_1   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 1, 6, 0, 1, 9, 11, 0, 14, 15, 18, 22}   SU(2)_12   (3860,3846)
E_6(a_1)   D_5(a_1)   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 1, 1, 6, 1, 0, 9, 10, 1, 13, 14, 17, 21}   SU(3)_12   (3688,3675)
D_5+A_2   A_4+A_2   {1, 6,  9, 11, 14, 16, 19, 24}   {1, 0, 0, 0, 0, 6, 0, 1, 9, 11, 0, 14, 15, 18, 22}   U(1)   (3848,3835)
E_6   D_4   {1, 6, 10, 11, 15, 16, 20, 25}   {1, 0, 0, 0, 2, 6, 0, 0, 9,  9, 0, 12, 12, 15, 18}   (G_2)_12   (3232,3220)
D_7(a_2)   A_4+2A_1   {1, 6, 9, 11, 14, 16, 19, 24}   {1, 0, 0, 1, 0, 6, 1, 0, 9, 10, 0, 13, 15, 18, 22}   U(1)   (3792,3780)
E_6(a_1)+A_1   A_4+A_1   {1, 6, 9, 11, 14, 16, 19, 24}   {1, 0, 0, 1, 0, 6, 1, 0, 9, 10, 1, 13, 14, 17, 21}   U(1)   (3675,3664)
D_6   (A_4,ℤ_2)   {1, 6, 9, 11, 14, 16, 19, 24}   {1, 0, 1, 0, 0, 6, 0, 2, 8, 10, 0, 12, 14, 16, 20}   Sp(2)_11   (3502,3490)
E_7(a_3)   A_4   {1, 6, 9, 11, 14, 16, 19, 24}   {1, 0, 1, 1, 0, 6, 0, 1, 8, 10, 0, 12, 14, 16, 20}   SU(2)_10   (3490,3480)
A_7   (D_4(a_1)+A_2,ℤ_2)   {1, 6, 9, 10, 13,14,17,22}   {1, 0, 0, 0, 0, 6, 0, 0, 9, 10, 0, 13, 14, 17, 22}   SU(2)_31   (3679,3668)
E_8(b_6)   D_4(a_1)+A_2   {1, 6, 9, 10, 13, 14, 17, 22}   {1, 0, 0, 0, 0, 6, 0, 0, 9, 10, 1, 13, 14, 17, 21}   -   (3648,3638)
D_7(a_1)   A_3+A_2   {1, 6, 9, 10, 13, 14, 17, 21}   {1, 0, 1, 0, 0, 6, 0, 1, 8, 10, 0, 12, 14, 16, 20}   U(1)   (3480,3471)
E_8(a_6)   D_4(a_1)+A_1   {1, 6, 9, 10, 13, 14, 17, 21}   {1, 0, 2, 0, 0, 5, 0, 2, 8, 9, 0, 12, 13, 16, 20}   -   (3424,3416)
E_6+A_1   (D_4(a_1),S_3)   {1, 6, 9, 10, 13, 14, 17, 21}   {1, 0, 0, 0, 1, 6, 0, 0, 9, 9, 0, 12, 12, 15, 18}   SU(2)_26   (3218,3209)
E_7(a_2)   (D_4(a_1),ℤ_2)   {1, 6, 9, 10,13,14,17,21}   {1, 0, 1, 0, 1, 6, 0, 0, 8, 9, 0, 12, 12, 15, 18}   SU(2)_9   (3201,3193)
E_8(b_5)   D_4(a_1)   {1, 6, 9, 10, 13, 14, 17, 21}   {1, 0, 2, 0, 1, 5, 0, 0, 8, 9, 0, 12, 12, 15, 18}   -   (3192,3185)
D_7   (2A_2,ℤ_2)   {1, 5, 8,  9, 12, 13, 16, 20}   {1, 0, 0, 0, 1, 5, 0, 0, 8, 8, 0, 11, 12, 14, 18}   SU(2)_13   (3069,3062)
E_8(a_5)   2A_2   {1, 5, 8,  9, 12, 13, 16, 20}   {1, 0, 0, 0, 2, 5, 0, 0, 7, 8, 0, 11, 12, 14, 18}   -   (3056,3050)
E_7(a_1)   A_3   {1, 6, 9, 10, 13, 14, 17, 21}   {1, 0, 1, 0, 1, 5, 0, 1, 7, 8, 0, 10, 11, 13, 16}   SU(2)_8   (2832,2826)
E_8(b_4)   A_2+2A_1   {1, 5, 8, 9, 12, 13, 16, 20}   {1, 0, 0, 0, 1, 5, 0, 1, 7, 8, 0, 10, 11, 13, 16}   -   (2824,2819)
E_8(a_4)   A_2+A_1   {1, 5, 8, 9, 12, 13, 16, 20}   {1, 1, 0, 1, 0, 4, 1, 0, 6, 7, 1, 9, 10, 12, 15}   -   (2608,2604)
E_7   (A_2,ℤ_2)   {1, 5, 8, 9, 12, 13, 16, 20}   {1, 0, 0, 0, 2, 4, 0, 0, 6, 6, 0, 8, 8, 10, 12}   SU(2)_7   (2159,2155)
E_8(a_3)   A_2   {1, 5, 8, 9, 12, 13, 16, 20}   {1, 1, 0, 0, 1, 4, 0, 0, 6, 6, 0, 8, 8, 10, 12}   -   (2152,2149)
E_8(a_2)   2A_1   {1, 4, 6, 7, 9, 10, 12, 15}   {1, 0, 1, 0, 0, 3, 0, 1, 4, 5, 0, 6, 7, 8, 10}   -   (1752,1750)
E_8(a_1)   A_1   {1, 3, 4, 5, 6, 7, 8, 10}   {1, 0, 0, 0, 1, 2, 0, 0, 3, 3, 0, 4, 4, 5, 6}   -   (1080,1079)





 §.§ Irregular Punctures and Irregular Fixtures


An irregular puncture, in the sense of <cit.>, is a pair: (𝒪, H_k'), where 𝒪 is a regular puncture, H_k' is a simple subgroup of the global symmetry group F(𝒪), and 0≤ k' = 4κ_H - k where κ_H is the dual Coxeter number.

For the E_8 theory, there are 30 irregular punctures, all but 3 of which actually appear in the irregular fixtures below. The remaining ones, (A_1, Spin(12)_8), (A_1, Spin(11)_4), and (A_1, Spin(10)_0), appear to play no role in the theory.

To each irregular puncture, (𝒪, H_k'), we assign



n_h    = 9920 -n_h(𝒪)

n_v    = 9928 -n_v(𝒪)- (H).



An irregular fixture


 
    < g r a p h i c s >


consists of



  * a pair of regular punctures, 𝒪_1 and 𝒪_2 for which
 
    < g r a p h i c s >


would be bad.



  * an SCFT, X, (possibly free hypers, an isolated interacting SCFT, or a mixture) whose flavour symmetry
F(X)×cent(H_k⊂F(𝒪_3))⊃F(𝒪_1)×F(𝒪_2)×H_k'

and whose central charges satisfy



n_h(X)   = n_h(𝒪_1)+n_h(𝒪_2)-n_h(𝒪_3)

n_v(X)   = n_v(𝒪_1)+n_v(𝒪_2)-n_v(𝒪_3)-(H).





In fact, we have something stronger than the relation on n_v. A similar relation must hold on the graded Coulomb dimensions[Since n_v = ∑_k (2k-1) n_k, (<ref>) implies the relation on n_v.] 


    n_k(X) =
    
    n_k(𝒪_1)+n_k(𝒪_2)-n_k(𝒪_3)-1    (k-1) an exponent of 𝔥
    
    n_k(𝒪_1)+n_k(𝒪_2)-n_k(𝒪_3)   otherwise.

Going through the list of pairs 𝒪_1,𝒪_2, we readily identify the 50 irregular fixtures in the E_8 theory. In fact, to determine (𝒪_3,H_k'), it suffices to know that the relation on n_v (rather than the more refined relations on the n_k) are satisfied. Given the n_k(X) (in particular, for a free-field fixture, all the n_k=0), we thus obtain predictions for the n_k(𝒪_i). These predictions are of assistance in computing the constraints on the pole-coefficients at the punctures – by far the most tedious part of the computation. Since we won't compute any Seiberg-Witten solutions here, we will leave the detailed form of the constraints for a separate publication <cit.>, and report only the graded Coulomb branch dimensions here.



  §.§.§ Free-field Fixtures


While there is no regular fixture consisting of free hypermultiplets (there are no Lagrangian field theories arising as the compactification of the E_8 theory), there are 29 fixtures, consisting of two regular punctures and an irregular puncture, which correspond to free hypermultiplets.



|c|c|c|c|
#   Fixture   Number of hypers   Representation

1   E_8(a_1)
 E_8(a_1)   (E_7,SU(2)_1)   1   12(2)

2   E_8(a_1)
 E_8(a_2)   (E_7(a_1),SU(2)_0)   0   empty

3   E_8(a_1)
 E_8(a_3)   (E_6,SU(3)_0)   0   empty

4   E_8(a_1)
 E_7   (E_6,(G_2)_4)   7   1(7)

5   E_8(a_1)
 E_8(a_4)   (E_6(a_1),SU(3)_0)   0   empty

6   E_8(a_1)
 E_8(b_4)   (D_5,(G_2)_0)   0   empty

7   E_8(a_1)
 E_7(a_1)   (D_5,Spin(7)_4)   8   1(8)

8   E_8(a_1)
 E_8(a_5)   (E_6(a_3),(G_2)_0)   0   empty

9   E_8(a_1)
 D_7   (A_5,(G_2)_0)   0   empty

10   E_8(a_1)
 E_8(b_5)   (D_4,Spin(8)_0)   0   empty

11   E_8(a_1)
 E_7(a_2)   (D_4,Spin(9)_4)   9   1(9)

12   E_8(a_1)
 E_6+A_1   (D_4,(F_4)_12)   26   1(26)

13   E_8(a_1)
 E_8(a_6)   (D_4(a_1),Spin(8)_0)   0   empty

14   E_8(a_1)
 D_7(a_1)   (A_3,Spin(9)_0)   0   empty

15   E_8(a_1)
 E_8(b_6)   (A_2,(F_4)_0)   0   empty

16   E_8(a_1)
 A_7   (3A_1,(F_4)_0)   0   empty

17   E_8(a_1)
 E_7(a_3)   (A_3,Spin(10)_4)   10   1(10)

18   E_8(a_1)
 D_6   (A_3,Spin(11)_8)   22   2(11)

19   E_8(a_1)
 E_6(a_1)+A_1   (A_2,(E_6)_12)   27   1(27)

20   E_8(a_1)
 D_7(a_2)   (A_1,(E_6)_0)   0   empty

21   E_8(a_1)
 D_5+A_2   (A_1,(E_7)_24)   56   1(56)

22   E_8(a_1)
 A_6+A_1   (0,(E_7)_12)   28   12(56)

23   E_8(a_2)
 E_8(a_2)   (D_6,Sp(2)_1)   2   12(4)

24   E_8(a_2)
 E_8(a_3)   (D_5,SU(4)_0)   0   empty

25   E_8(a_2)
 E_7   (D_5,Spin(7)_4)   7   1(7)

26   E_8(a_2)
 E_8(a_4)   (A_4,SU(5)_0)   0   empty

27   E_8(a_2)
 E_8(b_4)   (A_3,Spin(11)_8)   16   12(32)

28   E_8(a_2)
 E_8(a_5)   (2A_1,Spin(12)_0)   0   empty

29   E_8(a_2)
 D_7   (2A_1,Spin(13)_4)   13   1(13)






  §.§.§ Interacting Fixtures with one Irregular Puncture


There are 19 fixtures, consisting of two regular punctures and an irregular puncture, which correspond to interacting SCFTs. Most are (combinations of) Minahan-Nemeschansky theories; but the (F_4)_12×SU(2)_7^2 SCFT (which first appeared in the D_4 theory <cit.>) and the (E_7)_24 SCFT, the (E_7)_24×SU(2)_7 SCFT, the (E_7)_24×SU(2)_8 SCFT, the (E_7)_24×SU(2)_8×SU(2)_7 SCFT, the (E_7)_24×Spin(7)_16 SCFT (all of which first appeared in the E_6 theory <cit.>) and the (E_7)_24×(G_2)_12 (which https://golem.ph.utexas.edu/class-S/E7/fixtures/select?puncture1[id]=1   puncture2[id]=23   puncture3[id]=43appeared in the E_7 theory <cit.>) also make appearances.

Again, the graded Coulomb branch dimensions are (n_2,n_3,n_4,n_6,n_8,n_12,n_14,n_18,n_20,n_24,n_30).






|c|l|c|c|l|
#   Fixture   Graded Coulomb branch dims   (n_h,n_v)   Theory

1   E_8(a_1)
 E_6   (D_4,(F_4)_12)   (0,0,0,1,0,0,0,0,0,0,0)   (40,11)   (E_8)_12 SCFT
 2   E_8(a_1)
 E_6(a_1)   (A_2,(E_6)_12)   (0,0,0,1,0,0,0,0,0,0,0)   (40,11)   (E_8)_12 SCFT
 3   E_8(a_1)
 D_6(a_1)   (A_1,(E_7)_24)   (0,0,0,2,0,0,0,0,0,0,0)   (80,22)   [(E_8)_12 SCFT]^2
 4   E_8(a_1)
 D_5+A_1   (A_1,(E_7)_24)   (0,0,0,1,0,1,0,0,0,0,0)   (93,34)   (E_8)_24×SU(2)_13 SCFT
 5   E_8(a_1)
 A_6   (0,(E_7)_12)   (0,0,0,1,0,0,0,0,0,0,0)   (40,11)   (E_8)_12 SCFT
 6   E_8(a_1)
 D_5   (A_1,(E_7)_24)   (0,0,0,1,1,1,0,0,0,0,0)   (112,49)   (E_7)_24×Spin(7)_16 SCFT
 7   E_8(a_2)
 E_7(a_1)   (A_3,Spin(11)_8)   (0,0,1,0,0,0,0,0,0,0,0)   (24,7)   (E_7)_8 SCFT
 8   E_8(a_2)
 E_8(b_5)   (A_1,(E_7)_24)   (0,0,3,0,0,0,0,0,0,0,0)   (72,21)   [(E_7)_8 SCFT]^3
 9   E_8(a_2)
 E_7(a_2)   (A_1,(E_7)_24)   (0,0,2,0,1,0,0,0,0,0,0)   (81,29)   [(E_7)_8 SCFT]
×
 [(E_7)_16×SU(2)_9 SCFT]
 10   E_8(a_2)
 E_6+A_1   (A_1,(E_7)_24)   (0,0,1,0,1,1,0,0,0,0,0)   (98,45)   (E_7)_24×SU(2)_26 SCFT
 11   E_8(a_2)
 E_6   (A_1,(E_7)_24)   (0,0,1,1,1,1,0,0,0,0,0)   (112,56)   (E_7)_24×(G_2)_12 SCFT
 12   E_8(a_3)
 E_8(a_3)   (D_4,(F_4)_12)   (0,2,0,0,0,0,0,0,0,0,0)   (32,10)   [(E_6)_6 SCFT]^2
 13   E_8(a_3)
 E_7   (D_4,(F_4)_12)   (0,1,0,1,0,0,0,0,0,0,0)   (39,16)   (E_6)_12×SU(2)_7 SCFT
 14   E_8(a_3)
 E_8(a_4)   (A_2,(E_6)_12)   (0,2,0,0,0,0,0,0,0,0,0)   (32,10)   [(E_6)_6 SCFT]^2
 15   E_8(a_3)
 E_8(b_4)   (A_1,(E_7)_24)   (0,1,0,1,1,1,0,0,0,0,0)   (104,54)   (E_7)_24 SCFT
 16   E_8(a_3)
 E_7(a_1)   (A_1,(E_7)_24)   (0,1,1,1,1,1,0,0,0,0,0)   (112,61)   (E_7)_24×SU(2)_8 SCFT
 17   E_7
 E_7   (D_4,(F_4)_12)   (0,0,0,2,0,0,0,0,0,0,0)   (46,22)   (F_4)_12×SU(2)_7^2 SCFT
 18   E_7
 E_8(a_4)   (A_2,(E_6)_12)   (0,1,0,1,0,0,0,0,0,0,0)   (39,16)   (E_6)_12×SU(2)_7 SCFT
 19   E_7
 E_8(b_4)   (A_1,(E_7)_24)   (0,0,0,2,1,1,0,0,0,0,0)   (111,60)   (E_7)_24×SU(2)_7 SCFT
 20   E_7
 E_7(a_1)   (A_1,(E_7)_24)   (0,0,1,2,1,1,0,0,0,0,0)   (119,67)   (E_7)_24×SU(2)_8
×SU(2)_7 SCFT
 




  §.§.§ Mixed Fixtures with one Irregular Puncture


There is one mixed fixture with two regular and one irregular puncture. The value of n_h, listed, is the one associated to the SCFT, after subtracting the contribution of the free hypermultiplets.



|c|l|c|c|c|
#   Fixture   Graded Coulomb Branch dims   (n_h,n_v)   Theory

1   E_8(a_1)
 E_7(a_4)   (A_1,(E_7)_24)   (0,0,0,1,0,0,0,0,0,0,0)   (40,11)   (E_8)_12 SCFT + 12(56)






 §.§ Fixtures


The basic building block of the theories of class-S is the compactification of the (2,0) theory on a 3-punctured sphere (“fixture”). Given the 69 punctures of the E_8 theory, there are 57,155 possible fixtures. Of these, 7,319 are bad (do not give rise to a well-defined 4D SCFT). These are “replaced” by the 50 irregular fixtures listed above. The remaining 49,836 fixtures are too numerous to list here, but they can be explored in our https://golem.ph.utexas.edu/class-S/E8/fixtures/online application.

Our main task is to determine which of the 49,836 are mixed fixtures (containing free hypermultiplets + an interacting isolated SCFT) and which are isolated interacting SCFTs with an enhanced global symmetry



   F    ⊃F_manifest

   F_manifest    = F(𝒪_1)×F(𝒪_2)×F(𝒪_3).


To this end, we compute the (refined) Hall-Littlewood index up to order τ^2. The O(τ) contribution counts the number of free hypermultiplets. Removing the free hypermultiplets, the O(τ^2) contribution is the character χ(adj(F)) decomposed as a representation of F_manifest. <cit.>

We find 149 mixed fixtures and an additional 775 fixtures with enhanced global symmetries. The remaining 48,912 fixtures have global symmetry F_manifest.



 §.§ Computing the Hall-Littlewood Superconformal Index


To classify the fixtures, as described in <ref>, we need to calculate their superconformal index <cit.>. Up to second order, the superconformal index of a fixture of type G simplifies to:


    I=1+χ_F_manifest^adjτ^2+.(∑_λ'∏_i=1^3χ^λ_i(a_i|τ)/χ^λ_0)|_τ^2

where χ^λ(a_i|τ) is the Weyl character coming from puncture i associated with the irreducible representation of G whose highest weight is λ. τ is the fugacity of SU(2) symmetry while a_i are the fugacities of global symmetry f of the puncture. The last term is an infinite sum over all nontrivial irreducible representation of G. The detailed form of χ depends on the branching rule of each puncture. However, due to the symmetric form of the character, it is easy to see that χ_i can always be written as


1/τ^w_i(F_i(a_i)+O(τ))

where w is the highest weight of the SU(2)_R representation appearing in the branching rule of each puncture. F(a_i) is a function of the fugacities of global symmetry. So we have


(∑_λ∏_i=1^3χ^λ_i(a_i|τ)/χ^λ_0)=∑_λτ^w_0^λ-w_1^λ-w_2^λ-w_3^λ(F_1^λ(a_1)F_2^λ(a_2)F_3^λ(a_3)/F_0^λ+O(τ))

If we can compute W^λ≡ w_0^λ-w_1^λ-w_2^λ-w_3^λ for a representation λ, we will know whether we need to include it in the sum at order τ^2. Clearly we need to include λ in the sum if and only if W^λ≤ 2.

Now observe that the projection matrix p for each puncture projects the weights of an irreducible representation of G onto the weights of a subgroup of G. For SU(2), the weights are all just numbers and the highest weight is nothing but the largest of them. Let V be the weight space of irreducible representation with highest weight λ, then:


w_i^λ=max_v∈V(p_i·v)

Because every v can be represented as λ minus a linear combination of simple positive roots r of G. If we can choose p such that p· r is nonnegative for all simple positive roots[A choice which does the trick is to take p to be the weighted Dynkin diagram for the puncture times the inverse of the Cartan matrix.] , we will be able to compute w and W easily:



w_i^λ   =p_i·λ

W^λ   =(p_0-p_1-p_2-p_3)·λ

Let P=p_0-p_1-p_2-p_3. The vector is determined only by the punctures and can be easily computed for all fixtures. We have four cases for P:



  * If any entry of P is non-positive, then the SCI diverges and the fixture is bad. For example, consider a representation with λ=(0… 0 10… 0) where the position of “1" matches the position of the non-positive entry of P, this will lead to a term with non-positive power in τ.

  * If all entries of P are greater than 2, then the fixture is an SCFT with no symmetry enhancements or free hypermultiplets.

  * If all entries of P are positive and at least one of them is 1 (and some other elements might be 2), the SCFT will have free hypermultiplets and possibly will have enhanced symmetry. To compute this, we need to consider the λ=(0… 010… 0), (0… 020… 0) and (0… 010… 010… 0) where the position of the nonzero elements matches the 1's in P. We also need to consider λ=(0… 010… 0) where the position of 1 matches the 2's in P.

  * If all elements of P are greater than 1 and at least one of them is 2, the SCFT will have enhanced symmetry and no free hypermultiplets. We need to consider λ=(0… 010… 0) where the position of 1 matches the 2s in P.


Note that all representations we need to consider are of the form (0… 010… 0), (0… 020… 0) or (0… 010… 010… 0). Therefore, despite the appearance, (<ref>) is actually a finite sum over λ.

As an example, consider a fixture of E_7 theory: < g r a p h i c s >.


P_(3A_1)', A_2+3A_1, E_7(a_1)=p_E_7-p_(3A_1)'-p_A_2+3A_1-p_E_7(a_1)=(1, 2, 4, 3, 2, 1, 1)

By the rule proposed above, the Dynkin labels of all the representations we need to compute the index to O(τ^2) are (1000000), (2000000), (0100000), (0000100), (0000010), (0000020), (0000001), (0000002), (1000010), (1000001) and (0000011). These representations have dimensions 133, 7371, 8645, 1539, 56, 1463, 912, 253935, 6480, 86184, 40755 respectively. This is in agreement with the result shown in <cit.>.

For the E_8 theory, the largest representation contributing to the SCI of a fixture is 4881384, and this occurs in two fixtures:


 
    < g r a p h i c s >

  and   
    < g r a p h i c s >


From


P_A_2+3A_1,2A_3,E_8(a_1)=(1,3,6,4,3,2,1,3)

we see that the index for the first fixture receives a contribution from the representation (20000000)=4881384, and similarly for the second. However, the majority of fixtures require far less computation.



§ APPLICATIONS




 §.§ A Very Special Piece


Consider the fixture


 
    < g r a p h i c s >


where 𝒪 is drawn from the Special Piece of E_8(a_7), i.e., 𝒪 is one of

  ∙     A_4+A_3               (C(A_4+A_3)=S_5)         

∙     A_5+A_1               (C(A_5+A_1)=S_3×ℤ_2)         

∙     D_5(a_1)+A_2      (C(D_5(a_1))=S_4)         

∙     D_6(a_2)              (C(D_6(a_2))=ℤ_2×ℤ_2)         

∙     E_6(a_3)+A_1      (C(E_6(a_3)+A_1)=S_3)         

∙     E_7(a_5)              (C(E_7(a_5))=ℤ_2)         

∙     E_8(a_7).

Luzstig's canonical quotient, A(E_8(a_7))=S_5. So the Sommers-Achar groups, C(𝒪), associated to the 𝒪s are various subgroups of S_5.

Choosing 𝒪=E_8(a_7), we get 5 decoupled copies of the (E_8)_12 Minahan-Nemeschansky theory, whose Coulomb branch is 1-dimensional and whose Higgs branch is the 1-instanton moduli space for E_8. At the other end, choosing 𝒪=A_4+A_3 yields the k=5 Minahan-Nemeschansky,theory, (E_8)_60×SU(2)_124, whose Coulomb branch is the S_5 quotient of the 𝒪=E_8(a_7) case, and whose Higgs branch is the 5-instanton moduli space for E_8.

The intermediate cases lie somewhere in between. For instance, 𝒪=A_5+A_1 (Sommers-Achar group =S_3×ℤ_2) yields the product of one copy of the k=3, (E_8)_36×SU(2)_38, Minahan-Nemeschansky theory with one copy of the k=2, (E_8)_24×SU(2)_13, Minahan-Nemeschansky theory.

If we take one of the above fixtures, and gauge an E_7 subgroup of the diagonal E_8 (introducing an additional 12(56) to make the β-function vanish),


 
    < g r a p h i c s >


the global symmetry is the product of the SU(2)_k_i which centralize the E_7⊂∏_i (E_8)_k_i stemming from the enhanced (E_8)_60 of the full puncture (and an additional SU(2) when 𝒪 is non-special). S-dualizing, we obtain


 
    < g r a p h i c s >


where the global symmetry of the interacting fixture on the left is the same product of SU(2)s (times an additional SU(2)_7, which we gauge). We can, thus, fill in the levels of the SU(2)_k_i.

Replacing E_8(a_1) by another distinguished puncture (E_8(a_2), E_8(a_3), E_8(a_4), E_8(a_5), E_8(a_6), E_8(a_7), E_8(b_4) or E_8(b_6)), the (E_8)_60 global symmetry associated to the 0 puncture is not enhanced to a product of E_8s, so those fixtures don't follow the same pattern of product SCFTs that this example did.

However, the same pattern arises for the fixture

 
    < g r a p h i c s >


by letting the puncture 𝒪 vary over the same special piece. When 𝒪=E_8(a_7), the SU(2)_60 of the puncture A_6+A_1 enhances to SU(2)^5. This is the global symmetry enhancement F_kn→ (F_k)^n discussed in 5 of <cit.>: as 𝒪 varies over the special piece, the enhanced factor of the global symmetry group is the subgroup of (F_k)^n which is invariant under the action of C(𝒪), which permutes the n copies of F_k.





|c|c|c|c|
𝒪   C(𝒪)   Enhanced Global symmetry   Levels Determined

E_8(a_7)   1   SU(2)_60-k_1-k_2-k_3-k_4× SU(2)_k_1× SU(2)_k_2× SU(2)_k_3× SU(2)_k_4   SU(2)_12^5

E_7(a_5)   𝐙_2   SU(2)_13×SU(2)_60-k_1-k_2-k_3× SU(2)_k_1× SU(2)_k_2× SU(2)_k_3   SU(2)_13×SU(2)_24×SU(2)_12^3

E_6(a_3)+A_1   S_3   SU(2)_38×SU(2)_60-k_1-k_2× SU(2)_k_1× SU(2)_k_2   SU(2)_38×SU(2)_36×SU(2)_12^2

D_6(a_2)   𝐙_2 ×𝐙_2   SU(2)_13^2×SU(2)_60-k_1-k_2× SU(2)_k_1× SU(2)_k_2   SU(2)_13^2×SU(2)_24^2× SU(2)_12

D_5(a_1)+A_2   S_4   SU(2)_75×SU(2)_60-k× SU(2)_k   SU(2)_75×SU(2)_48× SU(2)_12

A_5+A_1   S_3 ×𝐙_2   SU(2)_38× SU(2)_13×SU(2)_60-k× SU(2)_k   SU(2)_38× SU(2)_13×SU(2)_36× SU(2)_24

A_4+A_3   S_5   SU(2)_124× SU(2)_60   SU(2)_124× SU(2)_60




As in our previous papers <cit.>, in this case we can use the action of the Sommers-Achar group on the Higgs branch to fill the missing levels, in the last column of the table.

We have used this action of the Sommers-Achar group on the Higgs branch to fill in the levels for various other families of fixtures which undergo this pattern of global symmetry enhancement.



 §.§ Fun with Minahan-Nemeschansky Theories


There are no purely Lagrangian field theories among the Class-S theories of type E_8. All of the S-dualities relate gaugings of one non-Lagrangian SCFT to another. The next-best thing would be to replace free hypermultiplets with Minahan-Nemeschansky theories. For instance, we could take two of the the fixtures discussed in <ref>, where 𝒪_1,𝒪_2 are drawn from the special piece of E_8(a_7), and gauge the diagonal E_8. The S-dual theory


 
    < g r a p h i c s >


is an SU(2) gauge theory with a half-hypermultiplet in the fundamental, gauging the SU(2)_7 subgroup of the global symmetry of the fixture on the left. The remaining global symmetry is the product of the SU(2)s that centralized the E_8 global symmetry of the Minahan-Nemeschansky theory.

As another example, we can consider a Spin(13) gauge theory, with hypermultiplets in the 2(13)+1(1) and the rest of the β-function condition saturated by coupling to the rank-3 (E_8)_36×SU(2)_38 Minahan-Nemeschansky theory.


 
    < g r a p h i c s >


which is S-dual to


 
    < g r a p h i c s >


and


 
    < g r a p h i c s >


Using one of the hypermultiplets in the (13) to Higgs Spin(13) to Spin(12), we obtain a Spin(12) gauge theory, with hypermultiplets in the 1(12)+2(1), coupling to the same rank-3 (E_8)_36×SU(2)_38 Minahan-Nemeschansky theory.


 
    < g r a p h i c s >


This is S-dual to


 
    < g r a p h i c s >


and


 
    < g r a p h i c s >




 §.§ The (E_8)_20 Theory


Recently, Giacomelli <cit.> gave a string theory construction of a rank-2 𝒩=2 SCFT with global symmetry (E_8)_20, n_4=n_10=1 and (n_h,n_v)=(72,26). This SCFT is realized in the A_9 theory <cit.> as the fixture


 
    < g r a p h i c s >


and in the D_6 theory <cit.> as the fixture


 
    < g r a p h i c s >


The same SCFT appeared at a factor in product theories in the E_7 theory <cit.>.


 
    < g r a p h i c s >

           
    < g r a p h i c s >



[(E_8)_20  SCFT]

×

[(E_7)_16×SU(2)_9 SCFT]
          

[(E_8)_20  SCFT]

×

[(E_7)_8 SCFT]^2



It also appears here in the E_8 theory, again as a product, but this time as three copies of itself.


 
    < g r a p h i c s >


While the Coulomb branch is 2-dimensional, the Higgs branch of the (E_8)_20 theory is also interesting: it has dim_ℍ=46 and is conjectured[Note that c=(2n_v+n_h)/12 saturates the unitarity bound(<ref>)
which implies the absence of flavour-singlet B̂_2 multiplets in the spectrum <cit.>.]  to be the (closure of the) 2A_1 nilpotent orbit of E_8.



 §.§ Theories with E_8 Global Symmetry


As already mentioned, a fixture with a pair of distinguished punctures and one full puncture has a manifest (E_8)_60 global symmetry. There are 32 fixtures where this is, in fact, the global symmetry. There are 5 more where the symmetry is enhanced.

We have already discussed the case of (E_8(a_7),E_8(a_1)), where the enhanced symmetry is (E_8)_12^5 and the fixture is actually 5 copies of the (E_8)_12 Minahan-Nemeschansky theory, and the case of (E_8(a_6),E_8(a_2)), where we get three copies of the (E_8)_20 SCFT.

The remaining three cases are also interesting.


 
    < g r a p h i c s >


is the rank-5 (E_8)_36 theory (first found in 4.3.3 of <cit.>) plus two copies of the (E_8)_12 theory. The fixture


    < g r a p h i c s >

has an enhanced (E_8)_60-k×(E_8)_k global symmetry and (n_h,n_v)=(256,120). This gives c=124/3. If the theory were indecomposable (possessing a single stress tensor), then this would violate the unitarity bound <cit.>


    k_i ≥24 κ_F_ic/(F_i)+12 c

for one or the other of the two E_8s. The resolution is that (<ref>) must be a product of two SCFTs, with global symmetries F_i = (E_8)_k_i and Weyl anomaly coefficients c_i satisfying k_1+k_2=60 and c_1+c_2=124/3. The fact that (<ref>) has graded Coulomb branch dimensions n_3=n_5=n_9=n_15=2 excludes obvious candidates, like the (E_8)_12, the (E_8)_20 and the (E_8)_36 theories, from being one of the factors in the product. Instead, it strongly suggests[A very similar phenomenon occurs with the fixture


 
    < g r a p h i c s >


in the E_7 theory. For the same reason, this must be a product SCFT and, in fact, is two copies of a rank-3 theory with global symmetry (E_7)_18× U(1), (n_h,n_v)=(70, 31) and n_3=n_5=n_9=1.]  that (<ref>) is the product of two copies of a rank-4 SCFT, with global symmetry (E_8)_30, (n_h,n_v)=(128,60) and graded Coulomb branch dimensions n_3=n_5=n_9=n_15=1. This choice saturates the bound (<ref>) and (3,5,9,15) is an allowed 4-tuple of graded Coulomb branch dimensions <cit.>.

This (E_8)_30 SCFT is realized in the A_14 theory as the fixture


 
    < g r a p h i c s >


where only an SU(5)_30×SU(3)_30×SU(2)_30× U(1) subgroup of the global symmetry is manifest.

Finally, we turn to


 
    < g r a p h i c s >


Again, the unitarity bound (<ref>) would be violated, unless this is a product theory. But here, we are not so lucky in identifying candidate factors for the product. One possibility is that one of the factors is the (E_8)_12 SCFT, and the other is a rank-11 SCFT with global symmetry (E_8)_48, (n_h,n_v)=(344, 235) and n_3=n_6=n_8=n_14=n_24=1, n_4=n_12=n_18=2. But there are several others, with no good way to decide between them.



 §.§ More Product Theories from Enhanced (E_8)_60


There are 8 more theories, where the (E_8)_60 symmetry of the full puncture is enhanced to (E_8)_60-k×(E_8)_k. Again, in all these cases, the unitarity bound (<ref>) requires that these are actually product SCFTs.





|c|c|c|c|c|
#   Fixture   Graded Coulomb Branch Dims   (n_h,n_v)   Global Symmetry

1   < g r a p h i c s >   {0,1,0,0,2,2,0,0,3,1,0,2,0,1,0}   (410, 270)   (E_8)_60-k×(E_8)_k×SU(2)_26

2   < g r a p h i c s >   {0,1,1,0,2,2,0,0,2,1,0,2,0,1,0}   (393, 254)   (E_8)_60-k×(E_8)_k×SU(2)_9

3   < g r a p h i c s >   {0,0,2,0,0,1,0,3,0,1,0,0,1,0,0}   (294, 152)   (E_8)_60-k×(E_8)_k×Sp(2)_11

4   < g r a p h i c s >   {0,0,2,1,0,1,0,2,0,1,0,0,1,0,0}   (282, 142)   (E_8)_60-k×(E_8)_k×SU(2)_10

5   < g r a p h i c s >   {0,0,2,0,0,1,0,2,0,1,0,0,1,0,0}   (272, 133)   (E_8)_60-k×(E_8)_k× U(1)

6   < g r a p h i c s >   {0,0,0,0,2,1,0,0,1,1,0,2,0,1,0}   (361, 204)   (E_8)_60-k×(E_8)_k×Sp(3)_19

7   < g r a p h i c s >   {0,0,0,0,2,1,1,0,1,1,0,1,0,1,0}   (340, 186)   (E_8)_60-k×(E_8)_k×SU(4)_18

8   < g r a p h i c s >   {0,0,0,0,2,1,0,0,1,1,0,1,0,1,0}   (320, 169)   (E_8)_60-k×(E_8)_k
×SU(2)_16×SU(2)_112



with Graded Coulomb Branch dimensions {n_2,n_3,n_4,n_5,n_6,n_8,n_9,n_10,n_12,n_14,n_15,n_18,n_20,n_24,n_30}.

Some of these have obvious candidates for the factors in the product. #7 is the product of the (E_8)_12 Minahan-Nemeschansky SCFT with a rank-7 (E_8)_48×SU(4)_18 SCFT with (n_h,n_v)=(300, 175) which is realized in the A_23 theory as the fixture


 
    < g r a p h i c s >


#4 is the product of the (E_8)_20 SCFT discussed above with a rank-5 (E_8)_40×SU(2)_10 SCFT, with (n_h,n_v)=(210, 116), which is realized in the A_19 theory as the fixture


 
    < g r a p h i c s >


It is likely that #s 3,5 are products of the (E_8)_20 theory with another SCFT and that the rest are products of the (E_8)_12 SCFT with another theory. But we have not found suitable candidates for alternate realizations of the other factor in the product for these cases.

A similar phenomenon occurs <cit.> in the E_7 theory: fixtures where the (E_7)_36 symmetry of the full puncture is enhanced to (E_7)_36-k×(E_7)_k and hence (by the unitarity bound) must be product SCFTs.

This seems to be a fruitful problem for further investigation: given that we know an SCFT is a product, how to determine the factors in the product. One piece of very useful information (which usually suffices when the rank is low) is knowing what are the allowed n-tuples of Coulomb branch scaling dimensions <cit.>. That certainly narrows the possibilities but is not — at least in the present cases — quite sufficient to nail things down. 



 §.§ Compactifications of 6D Conformal Matter Theories


Another source of 4D 𝒩=2 SCFTs is the compactification of 6D (1, 0) SCFTs on a torus <cit.>. The compactification of the (E_8,G_2) conformal matter theory <cit.> yields a 4D 𝒩=2 SCFT with (E_8)_36× (G_2)_16 global symmetry which was realized <cit.> as the E_7 fixture


 
    < g r a p h i c s >


It also appeared in <cit.> as D_2^20(E_8). Here, we see it arising as a factor in a product theory, for two fixtures[The two fixtures are related by the usual Sommers-Achar action on the E_8 Minahan-Nemeschansky theories. A_5 and E_6(a_3) lie in the same special piece, with d(A_5)=(D_6(a_1),ℤ_2) and d(E_6(a_3))=D_6(a_1); so the fixture with the A_5 puncture contains the rank-2 Minahan-Nemeschansky theory, where the fixture with the E_6(a_3) puncture contains two copies of the rank-1 Minahan-Nemeschansky theory.] :


 
    < g r a p h i c s >

           
    < g r a p h i c s >



[(E_8)_36×(G_2)_16  SCFT]

×

[(E_8)_24×SU(2)_13 SCFT]
          

[(E_8)_36×(G_2)_16  SCFT]

×

[(E_8)_12 SCFT]^2



Similarly, the compactification of the (E_7,SO(7)) conformal matter theory on a torus yields a 4D 𝒩=2 SCFT with (E_7)_24× Spin(7)_16 global symmetry. It is realized in the E_8 theory as the mixed fixture (accompanied by 15 hypermultiplets)


 
    < g r a p h i c s >


and also as the irregular fixture


 
    < g r a p h i c s >





§ ACKNOWLEDGEMENTS

We would like to thank Mario Martone, Philip Argyres and Sergio Cecotti for helpful discussions. The work of J.D. and Y.Z. was supported in part by the National Science Foundation under Grant No. PHY-1620610. J.D. would also like to thank the organizers of the Pollica Summer Workshop 2017, partly supported by the ERC STG grant 306260, and the Simons Center for Geometry and Physics for their gracious hospitality during an early stage of this work.







§ APPENDIX: EMBEDDINGS OF SU(2) IN E_8





|c|c|l|
Bala-Carter   𝔣   248

A_1   𝔢_7   (3;1)+(2;56)+(1;133)

2A_1   𝔰𝔬(13)   (1;78)+(2;64)+(3;1)+(3;13)

3A_1   𝔣_4×𝔰𝔲(2)   (1;1,3)+(1;52,1)+(2;26,2)+(3;1,1)+(3;26,1)+(4;1,2)

A_2   𝔢_6   (1;78)+(3;1)+(3;27)+(3;27)+(5;1)

4A_1   𝔰𝔭(4)   (1;36)+(2;48)+(3;1)+(3;27)+(4;8)

A_2+A_1   𝔰𝔲(6)   (1;35)+(2;6)+(2;6)+(2;20)+2(3;1)+(3;15)+(3;15)+(4;6)
   +(4;6)+(5;1)

A_2+2A_1   𝔰𝔬(7)×𝔰𝔲(2)   (1;1,3)+(1;21,1)+(2;8,4)+(3;7,3)+(3;1,5)+(3;1,1)+(4;8,2)
   +(5;1,3)

A_3   𝔰𝔬(11)   (1;55)+(3;1)+(4;32)+(5;11)+(7;1)

A_2+3A_1   𝔤_2×𝔰𝔲(2)   (1;14,1)+(1;1,3)+(2;14,2)+(3;27,1)+(3;1,1)+(4;7,2)+(5;7,1)

2A_2   (𝔤_2)^2   (1;14,1)+(1;1,14)+(3;7,7)+(3;1,1)+(5;1,7)+(5;7,1)

2A_2+A_1   𝔤_2×𝔰𝔲(2)   (1;1,3)+(1;14,1)+(2;1,4)+(2;7,2)+2(3;1,1)+(3;7,3)
   +(4;1,2)+(4;7,2)+(5;1,3)+(5;7,1)+(6;1,2)

A_3+A_1   𝔰𝔬(7)×𝔰𝔲(2)   (1;1,3)+(1;21,1)+(2;7,2)+2(3;1,1)+(3;8,1)+(4;8,2)
   +(4;1,2)+(5;8,1)+(5;7,1)+(6;1,2)+(7;1,1)

D_4(a_1)   𝔰𝔬(8)   (1;28)+3(3;1)+(3;8_v)+(3;8_c)+(3;8_s)+(5;1)
   +(5;8_v)+(5;8_c)+(5;8_s)+2(7;1)

D_4   𝔣_4   (1;52)+(3;1)+(7;26)+(11;1)

2A_2+2A_1   𝔰𝔭(2)   (1;10)+(2;20)+(3;1)+(3;5)+(3;14)+(4;16)+(5;10)+(6;4)

A_3+2A_1   𝔰𝔭(2)×𝔰𝔲(2)   
(1;10,1)+(1;1,3)+(2;4,3)+(2;1,2)+2(3;1,1)
   +(3;5,1)+(3;4,2)+(4;1,2)+(4;4,1)+(4;5,2)
   +(5;4,2)+(5;1,3)+(6;4,1)+(6;1,2)+(7;1,1)


D_4(a_1)+A_1   𝔰𝔲(2)^3   (1;1,1,3)+(1;1,3,1)+(1;3,1,1)+(2;2,2,2)+(2;2,1,1)
   +(2;1,1,2)+(2;1,2,1)+4(3;1,1,1)+(3;1,2,2)+(3;2,2,1)
   +(3;2,1,2)+2(4;2,1,1)+2(4;1,2,1)+2(4;1,1,2)+(5;1,1,1)
   +(5;2,2,1)+(5;1,2,2)+(5;2,1,2)+(6;2,1,1)+(6;1,2,1)
   +(6;1,1,2)+2(7;1,1,1)

A_3+A_2   𝔰𝔭(2)×𝔲(1)   (1;10)_0+(1;1)_0+(2;4)_-1/2+(2;4)_1/2+(3;1)_-2+(3;1)_2
   +2(3;1)_0+(3;5)_1+(3;5)_-1+(3;1)_1+(3;1)_-1+(4;4)_3/2
   +(4;4)_1/2+(4;4)_-1/2+(4;4)_-3/2+(5;1)_0+(5;5)_0+(5;1)_1
   +(5;1)_-1+(6;4)_-1/2+(6;4)_1/2+(7;1)_0+(7;1)_1+(7;1)_-1

A_4   𝔰𝔲(5)   (1;24)+(3;1)+(3;5)+(3;5)+(5;1)+(5;10)
   +(5;10)+(7;1)+(7;5)+(7;5)+(9;1)

A_3+A_2+A_1   𝔰𝔲(2)^2   (7;5,1)+(5;7,1)+(5;3,1)+(6;3,2)+(4;7,2)+(3;9,1)
   +(3;5,1)+(3;1,1)+(2;5,2)+(1;3,1)+(1;1,3)

D_4+A_1   𝔰𝔭(3)   (1;21)+(2;14')+2(3;1)+(6;6)+(7;14)+(8;6)+(11;1)

D_4(a_1)+A_2   𝔰𝔲(3)   (1;8)+(3;1)+(3;27)+(5;10)+(5;10)+(7;8)

A_4+A_1   𝔰𝔲(3)×𝔲(1)   (1;1)_0+(1;8)_0+(2;3)_-5+(2;3)_5+(2;1)_3+(2;1)_-3+2(3;1)_0
   +(3;3)_-2+(3;3)_2+(4;1)_3+(4;1)_-3+(4;3)_1+(4;3)_-1+(5;1)_0
   +(5;1)_6+(5;1)_-6+(5;3)_4+(5;3)_-4+(6;3)_1+(6;3)_-1+(6;1)_3
   +(6;1)_-3+(7;1)_0+(7;3)_-2+(7;3)_2+(8;1)_3+(8;1)_-3+(9;1)_0

2A_3   𝔰𝔭(2)   (8;4)+(7;5)+(7;1)+(6;4)+(5;10)+(4;16)+(3;5)+(3;1)+(2;4)
   +(1;10)

D_5(a_1)   𝔰𝔲(4)   (1;15)+(2;4)+(2;4)+2(3;1)+(3;6)+(5;1)+(6;4)+(6;4)
   +2(7;1)+(7;6)+(8;4)+(8;4)+(9;1)+(11;1)

A_4+2A_1   𝔰𝔲(2)×𝔲(1)   (9;1)_0+(8;2)_1+(8;2)_-1+(7;1)_0+(7;1)_4+(7;1)_-4+(7;1)_2+(7;1)_-2
   +(6;2)_1+(6;2)_-1+(6;2)_3+(6;2)_-3+(5;1)_0+(5;1)_2+(5;1)_-2
   +(5;3)_2+(5;3)_-2+(4;2)_1+(4;2)_-1+(4;2)_3+(4;2)_-3+(3;1)_2
   +(3;1)_-2+2(3;1)_0+(3;1)_4+(3;1)_-4+(3;3)_0+(2;2)_5+(2;2)_-5
   +(2;2)_1+(2;2)_-1+(1;1)_0+(1;3)_0

A_4+A_2   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+(3;1,1)+(3;2,6)+(3;1,5)+(5;1,7)+(5;1,3)
   (5;2,2)+(7;1,5)+(7;2,4)+(9;1,3)

A_5   𝔤_2×𝔰𝔲(2)   (1;1,3)+(1;14,1)+(3;1,1)+(4;1,2)+(5;7,1)
   +(6;7,2)+(7;1,1)+(9;7,1)+(10;1,2)+(11;1,1)

D_5(a_1)+A_1   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+(2;2,5)+(3;1,5)+2(3;1,1)+(4;2,1)+(5;1,3)
   +(6;2,3)+(7;1,5)+(7;1,3)+(8;2,3)+(9;1,3)+(11;1,1)

A_4+A_2+A_1   𝔰𝔲(2)   (9;3)+(8;4)+(7;5)+(6;2)+(6;4)+(5;3)+(5;7)
   +(4;2)+(4;6)+2(3;1)+(3;5)+(2;6)+(1;3)

D_4+A_2   𝔰𝔲(3)   (1;8)+2(3;1)+(3;6)+(3;6)+(5;1)+(5;3)+(5;3)+(7;3)
   +(7;3)+(7;8)+(9;3)+(9;3)+(11;1)

E_6(a_3)   𝔤_2   (1;14)+3(3;1)+(5;1)+2(5;7)+(7;1)+(7;7)+(9;1)+(9;7)+2(11;1)

D_5   𝔰𝔬(7)   (1;21)+(3;1)+(5;8)+(7;1)+(9;7)+(11;1)+(11;8)+(15;1)

A_4+A_3   𝔰𝔲(2)   (10;2)+(9;3)+(8;4)+(7;1)+(7;5)+(6;2)+(6;4)
   +2(5;3)+(4;2)+(4;6)+(3;5)+(3;1)+(2;4)+(1;3)

A_5+A_1   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+(2;4,1)+2(3;1,1)+(4;1,2)+(4;2,1)
   +(5;3,1)+(5;2,2)+(6;2,1)+(6;3,2)+(7;1,1)+(7;2,2)
   +(8;2,1)+(9;3,1)+(10;1,2)+(10;2,1)+(11;1,1)

D_5(a_1)+A_2   𝔰𝔲(2)   (11;1)+(10;2)+(9;3)+(8;4)+(8;2)+(7;3)+(7;1)
   +(6;4)+(6;2)+2(5;3)+2(4;2)+2(3;1)+(3;5)+(2;4)+(1;3)

D_6(a_2)   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+3(3;1,1)+2(4;2,1)+2(4;1,2)
   +(5;1,1)+(5;2,2)+(6;1,2)+(6;2,1)+3(7;1,1)+(7;2,2)
   +(8;1,2)+(8;2,1)+(9;1,1)+(10;1,2)+(10;2,1)+2(11;1,1)

E_6(a_3)+A_1   𝔰𝔲(2)   (1;3)+(2;4)+4(3;1)+2(4;2)+(5;1)+2(5;3)+3(6;2)
   +(7;1)+(7;3)+2(8;2)+(9;1)+(9;3)+(10;2)+2(11;1)

E_7(a_5)   𝔰𝔲(2)   (1;3)+6(3;1)+3(4;2)+4(5;1)+3(6;2)+5(7;1)
   +2(8;2)+3(9;1)+(10;2)+3(11;1)

D_5+A_1   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+(2;2,3)+2(3;1,1)+(4;1,2)+(5;2,2)
   +(6;1,2)+(7;1,1)+(8;2,1)+(9;1,3)+(10;1,2)+(10;2,1)
   +(11;1,1)+(11;2,2)+(12;1,2)+(15;1,1)

E_8(a_7)   -   4(11)+6(9)+10(7)+10(5)+10(3)

A_6   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+(3;1,1)+(3;2,2)+(5;1,3)+(7;1,5)
   +(7;2,4)+(9;1,3)+(11;1,1)+(11;2,2)+(13;1,3)

D_6(a_1)   𝔰𝔲(2)^2   (1;1,3)+(1;3,1)+2(3;1,1)+(3;2,2)+(4;1,2)+(4;2,1)
   +(6;1,2)+(6;2,1)+2(7;1,1)+(9;1,1)+(9;2,2)+(10;1,2)
   +(10;2,1)+2(11;1,1)+(12;1,2)+(12;2,1)+(15;1,1)

A_6+A_1   𝔰𝔲(2)   (1;3)+(2;2)+2(3;1)+(4;2)+(5;3)+(6;4)+(7;5)
   +(8;4)+(9;3)+(10;2)+(11;1)+(12;2)+(13;3)

E_7(a_4)   𝔰𝔲(2)   (1;3)+(2;2)+4(3;1)+2(4;2)+2(5;1)+(6;2)+3(7;1)
   +(8;2)+2(9;1)+2(10;2)+4(11;1)+(12;2)+(13;1)+(15;1)

E_6(a_1)   𝔰𝔲(3)   (1;8)+(3;1)+(5;1)+(5;3)+(5;3)+(7;1)+(9;1)
   +(9;3)+(9;3)+2(11;1)+(13;3)+(13;3)+(15;1)+(17;1)

D_5+A_2   𝔲(1)   15_0+13_1+13_-1+11_0+11_1+11_-1+11_2+11_-2+11_3+11_-3
   +9_0+9_1+9_-1+9_2+9_-2+7_0+7_1+7_-1+7_2+7_-2+5_0
   +5_1+5_-1+5_3+5_-3+2(3_0)+3_1+3_-1+3_2+3_-2+3_4+3_-4+1_0

D_6   𝔰𝔭(2)   (1;10)+(3;1)+(6;4)+(7;1)+(10;4)+(11;1)
   +(11;5)+(15;1)+(16;4)+(19;1)

E_6   𝔤_2   (1;14)+(3;1)+(9;7)+(11;1)+(15;1)+(17;7)+(23;1)

D_7(a_2)   𝔲(1)   1_0+2_1+2_-1+2(3_0)+4_1+4_-1+5_0+5_2+5_-2
   +6_1+6_-1+3(7_0)+2(8_1)+2(8_-1)+9_0+9_2+9_-2
   +10_1+10_-1+2(11_0)+12_1+12_-1+13_0+14_1+14_-1+15_0

A_7   𝔰𝔲(2)   (1;3)+(3;1)+(4;2)+(5;3)+(6;2)+(7;1)+(8;4)+(9;3)+(10;2)
   +(11;1)+(12;2)+(13;3)+(15;1)+(16;2)

E_6(a_1)+A_1   𝔲(1)   1_0+2_3+2_-3+2(3_0)+4_1+4_-1+5_0+5_2+5_-2
   +6_1+6_-1+7_0+8_1+8_-1+9_0+9_2+9_-2+10_1+10_-1
   +2(11_0)+12_1+12_-1+13_2+13_-2+14_1+14_-1+15_0+17_0

E_7(a_3)   𝔰𝔲(2)   (1;3)+(2;2)+2(3;1)+(5;1)+(6;2)+2(7;1)+(9;1)+2(10;2)
   +3(11;1)+(12;2)+2(15;1)+(16;2)+(17;1)+(19;1)

E_8(b_6)   -   1(17)+3(15)+2(13)+6(11)+3(9)+5(7)+4(5)+4(3)

D_7(a_1)   𝔲(1)   1_0+2(3_0)+3_2+3_-2+5_1+5_-1+7_0+7_1+7_-1+9_0+9_1+9_-1+2(11_0)
   +11_1+11_-1+11_2+11_-2+13_0+15_0+15_1+15_-1+17_1+17_-1+19_0

E_6+A_1   𝔰𝔲(2)   (1;3)+(2;4)+2(3;1)+(8;2)+(9;3)+(10;2)
   +(11;1)+(15;1)+(16;2)+(17;3)+(18;2)+(23;1)

E_7(a_2)   𝔰𝔲(2)   (1;3)+2(3;1)+(4;2)+(7;1)+(8;2)+(9;1)+(10;2)+2(11;1)
   +2(15;1)+(16;2)+(17;1)+(18;2)+(19;1)+(23;1)

E_8(a_6)   -   2(19)+1(17)+3(15)+3(13)+3(11)+3(9)+5(7)+1(5)+3(3)

D_7   𝔰𝔲(2)   (23;1)+(22;2)+(19;1)+(16;2)+(15;1)+(13;3)+(12;2)+(11;1)
   +(10;2)+(7;1)+(4;2)+(3;1)+(1;3)

E_8(b_5)   -   1(23)+2(19)+3(17)+3(15)+3(11)+3(9)+2(7)+1(5)+4(3)

E_7(a_1)   𝔰𝔲(2)   (1;3)+(3;1)+(6;2)+(7;1)+2(11;1)+(12;2)+(15;1)
   +(16;2)+(17;1)+(19;1)+(22;2)+(23;1)+(27;1)

E_8(a_5)   -   2(23)+1(21)+1(19)+1(17)+3(15)+2(13)+4(11)+1(9)+1(7)
   +1(5)+3(3)

E_8(b_4)   -   1(27)+2(23)+1(21)+1(19)+2(17)+2(15)
   +1(13)+3(11)+2(7)+1(5)+2(3)

E_7   𝔰𝔲(2)   (1;3)+(3;1)+(10;2)+(11;1)+(15;1)+(18;2)
   +(19;1)+(23;1)+(27;1)+(28;2)+(35;1)

E_8(a_4)   -   1(29)+1(27)+2(23)+2(19)+1(17)
   +3(15)+2(11)+1(9)+1(7)+1(5)+1(3)

E_8(a_3)   -   1(35)+1(29)+2(27)+1(23)+2(19)+1(17)+1(15)+2(11)+1(9)+2(3)

E_8(a_2)   -   1(39)+1(35)+1(29)+1(27)+2(23)+1(19)+1(17)+1(15)+1(11)
   +1(7)+1(3)

E_8(a_1)   -   1(47)+1(39)+1(35)+1(29)+1(27)+1(23)+1(19)+1(15)+1(11)+1(3)

E_8   -   1(59)+1(47)+1(39)+1(35)+1(27)+1(23)+1(15)+1(3)






§ PROJECTION MATRICES





|c|c|c|
Bala-Carter   𝔣   Projection Matrix

A_1   𝔢_7   [ 2 4 6 5 4 3 2 3; 1 0 0 0 0 0 0 0; 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 0 1 ]

2A_1   𝔰𝔬(13)   [  4  7 10  8  6  4  2  5;  0  0  0  0  0  0  1  0;  0  0  0  0  0  1  0  0;  0  0  0  0  1  0  0  0;  0  0  0  1  0  0  0  0;  0  0  1  0  0  0  0  0;  0  1  0  0  0  0  0  1 ]

3A_1   𝔣_4×𝔰𝔲(2)   [  4  8 12 10  8  6  3  6;  0  0  0  0  0  0  0  1;  0  0  1  0  0  0  0  0;  0  1  0  1  0  0  0  0;  1  0  0  0  1  0  0  0;  0  0  0  0  0  0  1  0 ]

A_2   𝔢_6   [  4  8 12 10  8  6  4  6;  0  0  0  0  1  1  0  0;  0  0  1  1  0  0  0  0;  0  1  0  0  0  0  0  0;  0  0  1  0  0  0  0  1;  0  0  0  1  1  0  0  0;  1  0  0  0  0  0  0  0 ]

4A_1   𝔰𝔭(4)   [  5 10 15 12  9  6  3  8;  1  0  0  0  0  0  1  0;  0  1  0  0  0  1  0  0;  0  0  1  0  1  0  0  0;  0  0  0  1  0  0  0  0 ]

A_2+A_1   𝔰𝔲(6)   [  6 11 16 13 10  7  4  8;  0  0  0  0  1  1  0  0;  0  0  1  1  0  0  0  0;  0  1  0  0  0  0  0  0;  0  0  1  0  0  0  0  1;  0  0  0  1  1  0  0  0 ]

A_2+2A_1   𝔰𝔬(7)×𝔰𝔲(2)   [  6 12 18 15 12  8  4  9;  1  0  0  0  0  0  0  0;  0  1  0  0  0  0  0  0;  0  0  2  1  0  0  0  1;  0  0  0  1  0  2  2  1 ]

A_3   𝔰𝔬(11)   [  8 15 22 18 14 10  6 11;  0  0  0  0  1  0  0  0;  0  0  0  1  0  0  0  0;  0  0  1  0  0  0  0  0;  0  0  0  0  0  0  0  1;  0  1  0  0  0  0  0 -1 ]

A_2+3A_1   𝔤_2×𝔰𝔲(2)   [  7 14 20 16 12  8  4 10;  0  0  0  2  2  0  1  1;  0  0  1  0  0  1  0  0;  1  0  0  0  0  0  0  0 ]

2A_2   𝔤_2^2   [  8 14 20 16 12  8  4 10;  0  1  2  0  0  1  1  1;  0  0  0  1  1  0  0  0;  0  1  0  0  1  1  0  1;  0  0  1  1  0  0  0  0 ]

2A_2+A_1   𝔤_2 ×𝔰𝔲(2)   [  8 15 22 18 14 10  5 11;  0  1  2  0  1  0  0  1;  0  0  0  1  0  0  0  0;  0  1  0  0  0  0  1  1 ]

A_3+A_1   𝔰𝔬(7)×𝔰𝔲(2)   [  8 16 24 20 16 11  6 12;  0  0  0  1  0  0  0  0;  0  0  1  0  0  0  0  0;  1  1  0  0  0  0  0  1;  0  0  0  0  0  1  0  0 ]

D_4(a_1)   𝔰𝔬(8)   [  8 16 24 20 16 12  6 12;  0  0  0  0  1  0  0  0;  0  0  0  1  0  0  0  0;  0  0  1  0  0  0  0  0;  1  1  1  0  0  0  0  1 ]

D_4   𝔣_4   [ 12 24 36 30 24 18 10 18;  0  0  0  0  0  0  0  1;  0  0  1  0  0  0  0  0;  0  1  0  1  0  0  0  0;  1  0  0  0  1  0  0  0 ]

2A_2+2A_1   𝔰𝔭(2)   [  8 16 24 20 15 10  5 12;  0  2  2  0  1  0  1  0;  1  0  0  0  0  1  0  1 ]

A_3+2A_1   𝔰𝔭(2)×𝔰𝔲(2)   [  9 18 26 21 16 11  6 13;  0  0  0  1  0  0  0  1;  0  0  1  0  0  0  0  0;  1  0  0  0  0  1  0  0 ]

D_4(a_1)+A_1   𝔰𝔲(2)^3   [  9 18 27 22 17 12  6 14;  1  0  0  0  0  0  0  0;  0  0  1  0  0  0  0  0;  0  0  0  0  1  0  0  0 ]

A_3+A_2   𝔰𝔭(2)×𝔲(1)   [  10  19  28  23  18  12   6  14;   0   1   0   1   0   0   0   0;   0   0   1   0   0   0   0   1;   0 1/2   1 1/2   0   1   1   0 ]

A_4   𝔰𝔲(5)   [ 12 22 32 26 20 14  8 16;  0  0  0  0  0  0  0  1;  0  0  1  0  0  0  0  0;  0  0  0  1  0  0  0  0;  0  0  0  0  1  0  0  0 ]

A_3+A_2+A_1   𝔰𝔲(2)^2   [ 10 20 30 24 18 12  6 15;  2  2  0  4  6  6  4  0;  0  0  0  0  0  0  0  1 ]

D_4+A_1   𝔰𝔭(3)   [ 13 26 39 32 25 18 10 20;  1  0  0  0  1  0  0  0;  0  1  0  1  0  0  0  0;  0  0  1  0  0  0  0  0 ]

D_4(a_1)+A_2   𝔰𝔲(3)   [ 10 20 30 24 18 12  6 16;  1  0  2  2  2  0  1  0;  1  3  2  2  2  3  1  0 ]

A_4+A_1   𝔰𝔲(3)×𝔲(1)   [ 12 23 34 28 22 15  8 17;  0  0  0  0  0  0  0  1;  0  0  1  0  0  0  0  0;  0  3  2  0  0  3  0  1 ]

2A_3   𝔰𝔭(2)   [ 12 23 34 28 21 14  7 17;  0  1  0  0  1  0  1  1;  0  0  1  0  0  1  0  0 ]

D_5(a_1)   𝔰𝔲(4)   [ 14 27 40 33 26 18 10 20;  0  0  1  1  0  0  0  0;  0  0  0  0  0  0  0  1;  0  1  1  0  0  0  0  0 ]

A_4+2A_1   𝔰𝔲(2)×𝔲(1)   [ 12 24 36 29 22 15  8 18;  0  0  0  1  0  1  0  0;  2  0  0  1  2  1  0  2 ]

A_4+A_2   𝔰𝔲(2)^2   [ 12 24 36 30 24 16  8 18;  1  2  2  1  0  0  0  1;  1  0  4  3  0  2  2  3 ]

A_5   𝔤_2×𝔰𝔲(2)   [ 16 30 44 36 28 19 10 22;  0  1  1  0  0  0  0  2;  0  0  0  0  0  0  0 -1;  0  0  0  0  0  1  0  0 ]

D_5(a_1)+A_1   𝔰𝔲(2)^2   [ 14 28 42 34 26 18 10 21;  0  0  0  0  0  0  0  1;  2  0  0  2  0  0  0  0 ]

A_4+A_2+A_1   𝔰𝔲(2)   [ 13 26 38 31 24 16  8 19;  1  0  0  1  0  2  0  3 ]

D_4+A_2   𝔰𝔲(3)   [ 14 28 42 34 26 18 10 22;  0  1  0  1  0  0  0  0;  1  0  0  0  1  0  0  0 ]

E_6(a_3)   𝔤_2   [ 16 30 44 36 28 20 10 22;  0  1  1  1  1  0  0  2;  0  0  0  0  0  0  0 -1 ]

D_5   𝔰𝔬(7)   [ 20 38 56 46 36 26 14 28;  0  0  0  0  0  0  0  1;  0  0  1  0  0  0  0  0;  0  1  0  1  0  0  0  0 ]

A_4+A_3   𝔰𝔲(2)   [ 14 28 42 34 26 18  9 21;  2  0  0  2  0  0  1  1 ]

A_5+A_1   𝔰𝔲(2)^2   [ 16 31 46 37 28 19 10 23;  0  1  0  0  0  1  0  1;  0  0  0  1  0  0  0  0 ]

D_5(a_1)+A_2   𝔰𝔲(2)   [ 15 30 44 36 28 19 10 22;  1  0  2  0  0  1  0  0 ]

D_6(a_2)   𝔰𝔲(2)^2   [ 16 32 47 38 29 20 10 24;  0  0  0  0  1  0  0  0;  0  0  1  0  0  0  0  0 ]

E_6(a_3)+A_1   𝔰𝔲(2)   [ 16 31 46 38 29 20 10 23;  0  1  0  0  1  0  0  1 ]

E_7(a_5)   𝔰𝔲(2)   [ 16 32 48 39 30 20 10 24;  0  0  0  1  0  0  0  0 ]

D_5+A_1   𝔰𝔲(2)^2   [ 20 39 58 48 37 26 14 29;  0  0  1  0  0  0  0  0;  0  1  1  0  1  0  0  1 ]

E_8(a_7)   -   [ 16 32 48 40 30 20 10 24 ]

A_6   𝔰𝔲(2)^2   [ 20 38 56 46 36 24 12 28;  0  0  0  0  0  0  0  1;  0  0  2  2  0  2  2  1 ]

D_6(a_1)   𝔰𝔲(2)^2   [ 20 40 59 48 37 26 14 30;  0  0  0  0  1  0  0  0;  0  0  1  0  0  0  0  0 ]

A_6+A_1   𝔰𝔲(2)   [ 20 39 58 47 36 24 12 29;  0  1  0  1  0  2  2  1 ]

E_7(a_4)   𝔰𝔲(2)   [ 20 40 60 49 38 26 14 30;  0  0  0  1  0  0  0  0 ]

E_6(a_1)   𝔰𝔲(3)   [ 24 46 68 56 44 30 16 34;  0  0  0  1  0  0  0  0;  0  0  1  0  0  0  0  0 ]

D_5+A_2   𝔲(1)   [ 20 40 60 50 38 26 14 30;  2  0  0  0  1  0  0  1 ]

D_6   𝔰𝔭(2)   [ 28 54 79 64 49 34 18 40;  0  0  1  0  1  0  0  0;  0  0  0  1  0  0  0  0 ]

E_6   𝔤_2   [ 32 62 92 76 60 42 22 46;  0  1  0  1  0  0  0  1;  0  0  1  0  0  0  0  0 ]

D_7(a_2)   𝔲(1)   [ 22 43 64 52 40 27 14 32;  0  1  0  0  0  1  0  0 ]

A_7   𝔰𝔲(2)   [ 24 47 70 57 44 30 15 35;  0  1  0  1  0  0  1  1 ]

E_6(a_1)+A_1   𝔲(1)   [ 24 47 70 57 44 30 16 35;  0  1  0  1  0  0  0  1 ]

E_7(a_3)   𝔰𝔲(2)   [ 28 54 80 65 50 34 18 40;  0  0  0  1  0  0  0  0 ]

E_8(b_6)   -   [ 24 48 72 58 44 30 16 36 ]

D_7(a_1)   𝔲(1)   [ 28 54 80 66 50 34 18 40;  0  0  0  0  0  1  0  1 ]

E_6+A_1   𝔰𝔲(2)   [ 32 63 94 77 60 42 22 47;  0  1  0  1  0  0  0  1 ]

E_7(a_2)   𝔰𝔲(2)   [ 32 64 95 78 60 42 22 48;  0  0  1  0  0  0  0  0 ]

E_8(a_6)   -   [ 28 56 84 68 52 36 18 42 ]

D_7   𝔰𝔲(2)   [  36  70 103  84  64  43  22  52;   0   0   1   0   0   1   0   0 ]

E_8(b_5)   -   [ 32 64 96 78 60 42 22 48 ]

E_7(a_1)   𝔰𝔲(2)   [  40  78 115  94  72  50  26  58;   0   0   1   0   0   0   0   0 ]

E_8(a_5)   -   [  36  70 104  84  64  44  22  52 ]

E_8(b_4)   -   [  40  78 116  94  72  50  26  58 ]

E_7   𝔰𝔲(2)   [  52 102 151 124  96  66  34  76;   0   0   1   0   0   0   0   0 ]

E_8(a_4)   -   [  44  86 128 104  80  54  28  64 ]

E_8(a_3)   -   [  52 102 152 124  96  66  34  76 ]

E_8(a_2)   -   [  60 118 174 142 108  74  38  88 ]

E_8(a_1)   -   [  72 142 210 172 132  90  46 106 ]

E_8   -   [  92 182 270 220 168 114  58 136 ]






utphys





