]>NUPHB14533S05503213(18)30362610.1016/j.nuclphysb.2018.12.022The AuthorsHigh Energy Physics – TheoryFig. 1A Series linear and balanced quiver types. In the canonical linear quiver, the unitary gauge nodes (blue/round) are in descending order with decrements in nonincreasing order. In the balanced quiver, the unitary gauge nodes are all balanced by their attached gauge nodes (blue/round) and flavour nodes (red/square).Fig. 1Fig. 2Quivers for A1 to A4 Slodowy slices. The Higgs quivers are of type BA(Nf(ρ)) and the Coulomb quivers are of type LA(ρ).Fig. 2Fig. 3Quivers for A5 Slodowy slices. The Higgs quivers are of type BA(Nf(ρ)) and the Coulomb quivers are of type LA(ρ).Fig. 3Fig. 4BCD linear and balanced quiver types. In the linear quivers LBC, LCD and LDC, the ranks and fundamental dimensions of the gauge nodes (blue/round) are in nonincreasing order L to R and the quivers are in the form of alternating B − C or D − C chains. In the balanced quivers, BB/C/D, the gauge nodes (blue/round) inherit their balance, taking account of attached gauge and flavour nodes (red/square), from a quiver for the nilpotent cone. Nodes labelled Cr represent the group USp(2r). Nodes labelled Br and Dr represent SO/O(2r + 1) and SO/O(2r) respectively. Nodes labelled BC, BD or DC indicate a group of one of the two types, subject to the alternation rule and to balance.Fig. 4Fig. 5Quivers for B1 to B3 Slodowy Slices. The Higgs quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LCD(dBV(ρ)TCD). Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 5Fig. 6Quivers for B4 Slodowy slices. The Higgs quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LCD(dBV(ρ)TCD). Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 6Fig. 7Quivers for C1 to C3 Slodowy slices. The Higgs quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LBC(dBV(ρ)TBC). Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 7Fig. 8Quivers for C4 Slodowy slices. The Higgs quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LBC(dBV(ρ)TBC). Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 8Fig. 9Quivers for D2 to D3 Slodowy slices. The Higgs balanced quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LDC(dBV(ρ)TDC). The Dynkin type quivers DD(Nf′) are identified by A series isomorphisms. Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 9Fig. 10Quivers for D4 Slodowy slices. The Higgs balanced quivers are of type BB/C/D(Nf(ρ)) and the Coulomb quivers are of type LDC(dBV(ρ)TDC). The Dynkin quivers of type DD(Nf([dBV(ρ)])), are those that have Higgs branches matching the balanced quivers. Gauge nodes of B or D type are evaluated as O nodes on the Higgs branch and SO nodes on the Coulomb branch. Δ = 0 indicates a diagram for which the monopole formula contains zero conformal dimension.Fig. 10Fig. 11A Series 3d Mirror Symmetry. All constructions give refined Hilbert series for a partition ρ and its dual ρT under the Lusztig–Spaltenstein map.Fig. 11Fig. 12BCD Series 3d Mirror Symmetry. Solid arrows indicate Higgs branches which give refined Hilbert series for a partition ρ. Dashed arrows indicate Higgs branches which give refined Hilbert series for the Barbasch–Vogan dual partition dBV(ρ) of a special nilpotent orbit. Dotted arrows indicate Coulomb branches which give unrefined Hilbert series for those special nilpotent orbits whose quivers have positive conformal dimension.Fig. 12Table 1Subregular Slodowy slices of classical groups.Table 1GroupSingularityDimensionHilbert series
ArAˆr≡C2/Zr+12PE[2tr+1+t2−t2r+2]
BrAˆ2r−1≡C2/Z2r2PE[2t2r+t2−t4r]
Cr>1Dˆr+1≡C2/Dicr−12PE[t2r−2+t2r+t4−t4r]
Dr>2Dˆr≡C2/Dicr−22PE[t2r−4+t2r−2+t4−t4r−4]
The dicyclic group of order 4k is denoted as Dick.Table 2Hilbert series for Slodowy slices of A1, A2, A3 and A4.Table 2Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[0]2A1[2]t2 − t4(1−t4)(1−t2)3
[2]0∅01

[00]6A2[1,1]t2 − t4 − t6(1−t4)(1−t6)(1−t2)8
[11]2D1t2+(1)q1/q2t3−t6(1−t6)(1−t2)(1−t3)2
[22]0∅01

[000]12A3[1,0,1]t2 − t4 − t6 − t8(1−t4)(1−t6)(1−t8)(1−t2)15
[101]6A1 ⊗ D1t2+[2]t2+[1](1)q1/q2t3−t6−t8(1−t6)(1−t8)(1−t2)4(1−t3)4
[020]4A1[2]t2 + [2]t4 − t6 − t8(1−t6)(1−t8)(1−t2)3(1−t4)3
[202]2D1t2+(1)q1/q3t4−t8(1−t8)(1−t2)(1−t4)2
[222]0∅01

[0000]20A4[1,0,0,1]t2 − t4 − t6 − t8 − t10(1−t4)(1−t6)(1−t8)(1−t10)(1−t2)24
[1001]12A2 ⊗ U(1)t2 + [1,1]t2 + [1,0]q1/q2t3 + [0,1]q2/q1t3 − t6 − t8 − t10(1−t6)(1−t8)(1−t10)(1−t2)9(1−t3)6
[0110]8A1 ⊗ D1t2+[2]t2+[2]t4+[1](1)q1/q2t3−t6−t8−t10(1−t6)(1−t8)(1−t10)(1−t2)4(1−t3)4(1−t4)3
[2002]6A1 ⊗ D1t2+[2]t2+[1](1)q1/q3t4−t8−t10(1−t8)(1−t10)(1−t2)4(1−t4)4
[1111]4D1t2+(1)t3+t4+(1)q2/q3t5−t8−t10(1−t8)(1−t10)(1−t2)(1−t3)2(1−t4)(1−t5)2
[2112]2D1t2+(1)q1/q4t5−t10(1−t10)(1−t2)(1−t5)2
[2222]0∅01
N.B. (n)q denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).Table 3Hilbert series for Slodowy slices of A5.Table 3Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[00000]30A5[1,0,0,0,1]t2 − t4 − t6 − t8 − t10 − t12(1−t4)(1−t6)(1−t8)(1−t10)(1−t12)(1−t2)35
[10001]20A3 ⊗ U(1)t2+[1,0,1]t2+([1,0,0]q1/q2+[0,0,1]q2/q1)t3−t6−t8−t10−t12(1−t6)(1−t8)(1−t10)(1−t12)(1−t2)16(1−t3)8
[01010]14A1 ⊗ A1 ⊗ D1t2+[2][0]t2+[0][2]t2+[1][1](1)q1/q2t3+[2][0]t4−t6−t8−t10−t12(1−t6)(1−t8)(1−t10)(1−t12)(1−t2)7(1−t3)8(1−t4)3
[00200]12A2[1,1]t2 + [1,1]t4 − t6 − t8 − t10 − t12(1−t6)(1−t8)(1−t10)(1−t12)(1−t2)8(1−t4)8
[20002]12A2 ⊗ U(1)t2+[1,1]t2+[1,0]q1/q3t4+[0,1]q3/q1t4−t8−t10−t12(1−t8)(1−t10)(1−t12)(1−t2)9(1−t4)6
[11011]8U(1)⊗U(1)2t2+((1)q1/q2+(1)q2/q3))t3+t4+(1)q1/q3t4+(1)q2/q3t5−t8−t10−t12(1−t8)(1−t10)(1−t12)(1−t2)2(1−t3)4(1−t4)3(1−t5)2
[02020]6A1[2]t2 + [2]t4 + [2]t6 − t8 − t10 − t12(1−t8)(1−t10)(1−t12)(1−t2)3(1−t4)3(1−t6)3
[21012]6A1 ⊗ D1t2+[2]t2+[1](1)q1/q4t5−t10−t12(1−t10)(1−t12)(1−t2)4(1−t5)4
[20202]4D1t2+t4+(1)q2/q4t4+(1)q2/q4t6−t10−t12(1−t10)(1−t12)(1−t2)(1−t4)3(1−t6)2
[22022]2D1t2+(1)q1/q5t6−t121−t12(1−t2)(1−t6)2
[22222]0∅01
N.B. (n)q denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).Table 4Generators for Slodowy slice to A[101].Table 4Pij(a)Quiver pathGenerator
P1,1(1)Image 3[2]t2
P2,2(1)Image 4t2
P1,2(1)Image 5[1]qt3
P2,1(1)Image 6[1]q−1t3
P2,2(2)Image 7t4
Table 5Generators for Slodowy slice to A[1111].Table 5Pij(a)Quiver pathGenerator
P2,2(1)Image 9t2
P3,3(1)Image 10t2
P2,3(1)Image 11q2/q3t3
P3,2(1)Image 12q3/q2t3
P2,2(2)Image 13t4
P3,3(2)Image 14t4
P2,3(2)Image 15q2/q3t5
P3,2(2)Image 16q3/q2t5
P3,3(3)Image 17t6
Table 6A1, A2 and A3 varieties, generated by complex matrices M, A and B and their relations, with Hilbert series calculated by Macaulay2 to match Slodowy slices SN,ρ. Note that SN,(12) has two alternative descriptions, one as the nilpotent cone and one as the subregular Kleinian singularity.Table 6OrbitPartitionDimensionGenerators; DegreeRelations
[0](12)2M2×2;2tr(M)=0tr(M2)=0
M1×1;2A1×1;2B1×1;2tr(M2)=AB
[2](2)0––

[00](13)6M3×3;2tr(M)=0tr(M2)=0tr(M3)=0
[11](2,1)2M1×1;2A1×1;3B1×1;3tr(M3)=AB
[22](3)0––

[000](14)12M4×4;2tr(M)=0tr(M2)=0tr(M3)=0tr(M4)=0
[101](2,12)6M2×2;2A1×2;3B2×1;3tr(M3)=ABtr(M4)=AMB
[020](22)4M2×2;2N2×2;4tr(M)=0tr(N)=0tr(M3)=tr(MN)tr(M4)=tr(N2)
[202](3,1)2M1×1;2A1×1;4B1×1;4tr(M4)=AB
[222](4)0––
Table 7A4 varieties, generated by complex matrices M, A and B and their relations, with Hilbert series calculated by Macaulay2 to match Slodowy slices SN,ρ.Table 7OrbitPartitionDimensionGenerators; DegreeRelations
[0000](15)20M5×5;2tr(M)=0tr(M2)=0tr(M3)=0tr(M4)=0tr(M5)=0
[1001](2,13)12M3×3;2A1×3;3B3×1;3tr(M3)=ABtr(M4)=AMBtr(M5)=AM2B
[0110](22,1)8M2×2;2A1×2;3B2×1;3N2×2;4tr(M3)=ABtr(M4)+tr(N2)=AMBtr(M5)=A(M2+N)Btr(N)=0
[2002](3,12)6M2×2;2A1×2;4B2×1;4tr(M4)=ABtr(M5)=AMB
[1111](3,2)4M1×1;2A1×1;3B1×1;3N1×1;4C1×1;5D1×1;5tr(M4)+tr(N2)=AMB+AD+BCtr(M5)=CD
[2112](4,1)2M1×1;2A1×1;5B1×1;5tr(M5)=AB
[2222](5)0––
Table 8Higgs branch quivers for nilpotent cones.Table 8GroupCanonical linear quiver for NGauge node balance
AImage 190 for all
BImage 200 for all
CImage 21{USp:+2O(even):−2
DImage 22{USp:+2O(even):−2
Table 9Coulomb branch quiver candidates for Slodowy slices.Table 9GroupO¯ρTransformationSN,ρ
AHiggs[LA(ρT)]ρ=(ρT)TCoulomb[LA(ρ)]
BHiggs[LB(ρT)]σ≡(((ρT)N→N−1)C)TCDCoulomb[LCD(σ)]
CHiggs[LC(ρT)]σ≡(((ρT)N→N+1)B)TBCCoulomb[LBC(σ)]
DHiggs[LD(ρT)]σ=((ρT)D)TDCCoulomb[LDC(σ)]
Table 10Monopole and Dynkin label lattices.Table 10GroupMonopole latticeBasis transformationsDynkin labels [n1,…,nr]
U(r)∞ > q1 ⩾ …qi ⩾ …qr > −∞qi≡∑j=irnj{∞>ni<r⩾0∞>nr>−∞
Ar∞ > q1 ⩾ …qi ⩾ …qr ⩾ 0qi≡∑j=irnj∞ > ni ⩾ 0
Br∞ > o1 ⩾ …oi ⩾ …or ⩾ 0oi≡∑j=ir−1nj+nr/2{∞>ni⩾0nr=2k
Cr∞ > s1 ⩾ …si ⩾ …sr ⩾ 0si≡∑j=irnj∞ > ni ⩾ 0
Dr∞>o1⩾…oi⩾…or⩾0{oi<r≡∑j=ir−2nj+(nr−1+nr)/2or=(−nr−1+nr)/2{∞>ni⩾0nr+1+nr=2k
Table 11Vector/fundamental characters.Table 11GroupMonopole basis vector/FundamentalWeight space basis vector/Fundamental
U(r)∑i=1ryix1+∑i=2rxi/xi−1
Ar∑i=1ryi+∏i=1r1/yix1+∑i=2rxi/xi−1+1/xr
Br1+∑i=1ryi+∑i=1r1/yi1+1/x1+x1+∑i=2r−1(xi−1/xi+xi/xi−1)+xr−1/xr2+xr2/xr−1
Cr∑i=1ryi+∑i=1r1/yi1/x1+x1+∑i=2r(xi−1/xi+xi/xi−1)
Dr∑i=1ryi+∑i=1r1/yi1/x1+x1+∑i=2r−2(xi−1/xi+xi/xi−1)+xr−2/(xr−1xr)+xr−1/xr+xr/xr−1+(xr−1xr)/xr−2
Table 12Adjoint characters.Table 12GroupMonopole basis
U(r)r + ∑i≠jyi/yj
Arr+∏i=1r1/yi(∑j=1r1/yj)+∏i=1ryi(∑j=1ryi)+∑i≠jyi/yj
Brr+∑i=1r(yi+1/yi)+∑i<j(yiyj+yi/yj+yj/yi+1/(yiyj))
Crr+∑i=1r(yi2+1/yi2)+∑i<j(yiyj+yi/yj+yj/yi+1/(yiyj))
Drr+∑i<j(yiyj+yi/yj+yj/yi+1/(yiyj))
Table 13Gauge node conformal dimensions.Table 13Gauge groupΔ(Node)
U(r)−∑1⩽i<j⩽rqi − qj
Br−∑i=1roi−∑1⩽i<j⩽roi±oj
Cr−2∑i=1rsi−∑1⩽i<j⩽rsi±sj
Dr−∑1⩽i<j⩽roi ± oj
Table 14Bifundamental conformal dimensions.Table 14Gauge groupsΔ(Bifundamental)
U(r1)−U(r2)12∑i=1r1∑j=1r2q1,i−q2,j
Br1−Cr212∑j=1r2sj+12∑i=1r1∑j=1r2oi±sj
Dr1−Cr212∑i=1r1∑j=1r2oi±sj
Table 15Quiver chain unit conformal dimensions.Table 15Gauge group chainΔr(1,0…0)
Cr1−Dr−Cr2r1 + r2 − 2r + 2
Cr1−Br−Cr2r1 + r2 − 2r + 1
Br1−Cr−Br2r1 + r2 − 2r + 1
Br1−Cr−Dr2r1 + r2 − 2r + 1/2
Dr1−Cr−Dr2r1 + r2 − 2r
Table 16Hilbert series for Slodowy slices of B1, B2, and B3.Table 16Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[0]2B1[2]t2 − t4(1−t4)(1−t2)3
[2]0∅01

[00]8B2[0,2]t2 − t4 − t8(1−t4)(1−t8)(1−t2)10
[01]4C1 ⊗ B0[2]t2 + [1]kt3 − t8(1−t8)(1−t2)3(1−t3)2
[20]2D1 ⊗ B0t2 + (1)kt4 − t8(1−t8)(1−t2)(1−t4)2
[22]0∅01

[000]18B3[0,1,0]t2 − t4 − t8 − t12(1−t4)(1−t8)(1−t12)(1−t2)21
[010]10B1 ⊗ C1[2]Bt2 + [2]Ct2 + [2]B[1]Ct3 − t8 − t12(1−t8)(1−t12)(1−t2)6(1−t3)6
[200]8D2 ⊗ B0[2,0]t2 + [0,2]t2 + [1,1]kt4 − t8 − t12(1−t8)(1−t12)(1−t2)6(1−t4)4
[101]6C1 ⊗ B0[2]t2 + [1]kt3 + t4 + [1]kt5 − t8 − t12(1−t8)(1−t12)(1−t2)3(1−t3)2(1−t4)(1−t5)2
[020]4D1 ⊗ B0t2 + (1)kt4 + (2)t4 + t6 − t8 − t12(1−t8)(1−t12)(1−t2)(1−t4)4(1−t6)
[220]2D1 ⊗ B0t2 + (1)kt6 − t12(1−t12)(1−t2)(1−t6)2
[222]0∅01
N.B. (n) denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).k denotes the character ±1 if B0→O(1) or 1 if B0→SO(1).Table 17Hilbert series for Slodowy slices of B4.Table 17Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[0000]32B4[0,1,0,0]t2 − t4 − t8 − t12 − t16(1−t4)(1−t8)(1−t12)(1−t16)(1−t2)36
[0100]20B2 ⊗ C1[0,2]t2 + [2]t2 + [1,0][1]t3 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)13(1−t3)10
[2000]18D3 ⊗ B0[0,1,1]t2 + [1,0,0]kt4 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)15(1−t4)6
[0001]16C2 ⊗ B0[2,0]t2 + [1,0]kt3 + [0,1]t4 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)10(1−t3)4(1−t4)5
[1010]12C2 ⊗ D1 ⊗ B0[2]t2 + [1](1)t3 + [1]kt3 + (1)kt4 + t4 + [1]kt5 + t2 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)4(1−t3)6(1−t4)3(1−t5)2
[0200]10B1 ⊗ D1[2]t2 + (2)t4 + [2](1)t4 + t2 + t6 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)4(1−t4)8(1−t6)
[0020]8B1[2]t2 + [4]t4 + [2]t6 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)3(1−t4)5(1−t6)3
[2200]8D2 ⊗ B0[2,0]t2 + [0,2]t2 + [1,1]kt6 − t12 − t16(1−t12)(1−t16)(1−t2)6(1−t6)4
[0201]6C1 ⊗ B0[2]t2 + [1]kt5 + [2]t6 − t12 − t16(1−t12)(1−t16)(1−t2)3(1−t5)2(1−t6)3
[2101]6C1 ⊗ B0[2]t2 + [1]kt5 + [1]kt7 + t4 − t12 − t16(1−t12)(1−t16)(1−t2)3(1−t4)(1−t5)2(1−t7)2
[2020]4B0 ⊗ B0 ⊗ B0(k1k3 + k3k5)t4 + (k1k5 + k3k5)t6 + k3k5t8 + t4 − t12 − t16(1−t12)(1−t16)(1−t4)3(1−t6)2(1−t8)
[2220]2D1 ⊗ B0(1)kt8 + t2 − t161−t16(1−t2)(1−t8)2
[2222]0∅01
N.B. (n) denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).k denotes the character ±1 if B0→O(1) or 1 if B0→SO(1).Table 18Hilbert series for Slodowy slices of C1, C2, and C3.Table 18Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[0]2C1[2]t2 − t4(1−t4)(1−t2)3
[2]0∅01

[00]8C2[2,0]t2 − t4 − t8(1−t4)(1−t8)(1−t2)10
[10]4C1 ⊗ B0[2]t2 + [1]kt3 − t8(1−t8)(1−t2)3(1−t3)2
[02]2D1t2 + (1)t4 − t8(1−t8)(1−t2)(1−t4)2
[22]0∅01

[000]18C3[2,0,0]t2 − t4 − t8 − t12(1−t4)(1−t8)(1−t12)(1−t2)21
[100]12C2 ⊗ B0[2,0]t2 + [1,0]kt3 − t8 − t12(1−t8)(1−t12)(1−t2)10(1−t3)4
[010]8C1 ⊗ D1[2]t2 + [1](1)t3 + (2)t4 + t2 − t8 − t12(1−t8)(1−t12)(1−t2)4(1−t3)4(1−t4)2
[002]6B1[2]t2 + [4]t4 − t8 − t12(1−t8)(1−t12)(1−t2)3(1−t4)5
[020]4C1[2]t2 + [2]t6 − t8 − t12(1−t8)(1−t12)(1−t2)3(1−t6)3
[210]4C1 ⊗ B0[2]t2 + [1]kt5 − t121−t12(1−t2)3(1−t5)2
[202]2B0 ⊗ B0t4 + k2k4t4 + k2k4t6 − t121−t12(1−t4)2(1−t6)
[222]0∅01
N.B. (n) denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).k denotes the character ±1 if B0→O(1) or 1 if B0→SO(1).Table 19Hilbert series for Slodowy slices of C4.Table 19Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[0000]32C4[2,0,0,0]t2 − t4 − t8 − t12 − t16(1−t4)(1−t8)(1−t12)(1−t16)(1−t2)36
[1000]24C3 ⊗ B0[2,0,0]t2 + [1,0,0]kt3 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)21(1−t3)6
[0100]18C2 ⊗ D1[2,0]t2 + [1,0](1)t3 + (2)t4 + t2 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)11(1−t3)8(1−t4)2
[0010]14C1 ⊗ B1[2]Bt2 + [2]Ct2 + [2]B[1]Ct3 + [4]Bt4 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)6(1−t3)6(1−t4)5
[0002]12D2[2,0]t2 + [0,2]t2 + [2,2]t4 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)6(1−t4)9
[2100]12C2 ⊗ B0[2,0]t2 + [1,0]kt5 − t12 − t16(1−t12)(1−t16)(1−t2)10(1−t5)4
[0200]10C1 ⊗ C1[2]t2 + [2]t2 + [1][1]t4 + [2]t6 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)6(1−t4)4(1−t6)3
[0110]8C1 ⊗ B0[2]t2 + [1]kt3 + [1]kt5 + [2]t6 + t4 − t8 − t12 − t16(1−t8)(1−t12)(1−t16)(1−t2)3(1−t3)2(1−t4)(1−t5)2(1−t6)3
[2010]8C1 ⊗ B0 ⊗ B0[2]t2 + [1]k2t3 + [1]k4t5 + k2k4t4 + k2k4t6 + t4 − t12 − t16(1−t12)(1−t16)(1−t2)3(1−t3)2(1−t4)2(1−t5)2(1−t6)
[2002]6D1 ⊗ B0(1)kt4 + (2)t4 + (1)kt6 + t2 + t4 − t12 − t16(1−t12)(1−t16)(1−t2)(1−t4)5(1−t6)2
[0202]4D1(2)t4 + (2)t8 + t6 − t12 − t16(1−t12)(1−t16)(1−t2)(1−t4)2(1−t6)(1−t8)2
[2210]4C1 ⊗ B0[2]t2 + [1]kt7 − t161−t16(1−t2)3(1−t7)2
[2202]2B0 ⊗ B0k2k6t6 + k2k6t8 + t4 − t161−t16(1−t4)(1−t6)(1−t8)
[2222]0∅01
N.B. (n) denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).k denotes the character ±1 if B0→O(1) or 1 if B0→SO(1).Table 20Hilbert series for Slodowy slices of D2, D3, and D4.Table 20Nilpotent orbitDimension SN,ρSymmetry FGenerators of HS ≡ PL[HS]Unrefined HS
[00]4D2[2,0]t2 + [0,2]t2 − 2t4(1−t4)2(1−t2)6
[20]2C1≅A1[2]t2 − t41−t4(1−t2)3
[02]2C1≅A1[2]t2 − t41−t4(1−t2)3
[22]0∅01

[000]12D3[0,1,1]t2 − t4 − t6 − t8(1−t4)(1−t6)(1−t8)(1−t2)15
[011]6C1 ⊗ D1[2]t2 + [1](1)t3 + t2 − t6 − t8(1−t6)(1−t8)(1−t2)4(1−t3)4
[200]4B1 ⊗ B0[2]t2 + [2]kt4 − kt6 − t8(1−t6)(1−t8)(1−t2)3(1−t4)3
[022]2D1(2)t4 + t2 − t81−t8(1−t2)(1−t4)2
[222]0∅01

[0000]24D4[0,1,0,0]t2 − t4 − 2t8 − t12(1−t4)(1−t8)2(1−t12)(1−t2)28
[0100]14D2 ⊗ C1[2,0]t2 + [0,2]t2 + [2]t2 + [1,1][1]t3 − 2t8 − t12(1−t8)2(1−t12)(1−t2)9(1−t3)8
[0002]12C2[2,0]t2 + [0,1]t4 − 2t8 − t12(1−t8)2(1−t12)(1−t2)10(1−t4)5
[0020]12C2[2,0]t2 + [0,1]t4 − 2t8 − t12(1−t8)2(1−t12)(1−t2)10(1−t4)5
[2000]12B2 ⊗ B0[0,2]t2 + [1,0]kt4 − kt8 − t8 − t12(1−t8)2(1−t12)(1−t2)10(1−t4)5
[1011]8C1 ⊗ B0 ⊗ B0[2]t2 + [1](k1 + k3)t3 + [1]k1t5 + t4 + k1k3t4 − k1k3t8 − t8 − t12(1−t8)2(1−t12)(1−t2)3(1−t3)4(1−t4)2(1−t5)2
[0200]6D1 ⊗ D12t2 + (1)(1)t4 + (2)t4 + t6 − 2t8 − t12(1−t8)2(1−t12)(1−t2)2(1−t4)6(1−t6)
[0202]4C1[2]t2 + [2]t6 − t8 − t12(1−t8)(1−t12)(1−t2)3(1−t6)3
[0220]4C1[2]t2 + [2]t6 − t8 − t12(1−t8)(1−t12)(1−t2)3(1−t6)3
[2200]4B1 ⊗ B0[2]t2 + [2]kt6 − kt8 − kt12(1−t8)(1−t12)(1−t2)3(1−t6)3
[2022]2B0 ⊗ B0k3k5t4 + t4 + k3k5t6 − t121−t12(1−t4)2(1−t6)
[2222]2∅01
N.B. (n) denotes the character of the D1≡SO(2) reducible representation qn+q−n of U(1).k denotes the character ±1 if B0→O(1) or 1 if B0→SO(1).Table 21Generators for Slodowy slice to D[0100].Table 21Pij(a)Quiver pathGenerator
P11(1)Image 24Λ2([1,1]t)=[2,0]t2 + [0,2]t2
P2,2(1)Image 25Sym2([1]t)=[2]t2
P2,2(2)Image 26Λ2([1]t2)=[0]t4
P1,2(1)Image 27[1,1][1]t3
Table 22B1, B2 and B3 varieties, generated by complex matrices M, N, O, A and B and their relations, which have Hilbert series matching Slodowy slices SN,ρ. The matrices M = −MT and O = −OT are antisymmetric, N = NT is symmetric and Ω represents a square matrix that is antisymmetric and invariant under the action of USp(2n).Table 22OrbitPartitionDim.Generators; DegreeRelations
B[0](13)2M3×3;2tr(M2)=0
B[2](3)0––

B[00](15)8M5×5;2tr(M2)=0tr(M4)=0
B[01](22,1)4N2×2;2A2×1;3tr((NΩ)4)=ATΩNΩA
B[20](3,12)2M2×2;2A2×1;4tr(M4)=ATA
B[22](5)0––

B[000](17)18M7×7;2tr(M2)=0tr(M4)=0tr(M6)=0
B[010](22,13)10M3×3;2N2×2;2A3×2;3tr(M4)+tr((NΩ)4)=tr(AΩATM)+tr(ATAΩNΩ)tr(M6)+tr((NΩ)6)=tr((AΩAT)2)
B[200](3,14)8M4×4;2A4×1;4tr(M4)=ATAtr(M6)=ATM2A
B[101](3,22)6N2×2;2A2×1;3M2×2;4B2×1;5tr((NΩ)4+(MΩ)2)=BTΩAtr((NΩ)6+(MΩ)3)=BTΩMΩA
B[020](32,1)4M2×2;2A2×1;4N2×2;4O2×2;6tr(N)=0tr(M4+N2)=ATAtr(M6+N3+O2)=ATNA
B[220](5,12)2M2×2;2A2×1;6tr(M6)=ATA
B[222](7)0––
Table 23C1, C2 and C3 varieties, generated by complex matrices M, N, O, P, A and B and their relations, which have Hilbert series matching Slodowy slices SN,ρ. The matrices M = −MT and O = −OT are antisymmetric, N = NT and P = PT are symmetric and Ω represents a square matrix that is antisymmetric and invariant under the action of USp(2n).Table 23OrbitPartitionDim.Generators; DegreeRelations
C[0](12)2N2×2;2tr(N2)=0
C[2](2)0––

C[00](14)8N4×4;2tr((NΩ)2)=0tr((NΩ)4)=0
C[10](2,12)4N2×2;2A2×1;3tr((NΩ)4)=ATΩNΩA
C[02](22)2M2×2;2N2×2;4tr(N)=0tr(M4)=tr(N2)
C[22](4)0––

C[000](16)18N6×6;2tr((NΩ)2)=0tr((NΩ)4)=0tr((NΩ)6)=0
C[100](2,14)12N4×4;2A4×1;3tr((NΩ)4)=ATΩNΩAtr((NΩ)6)=ATΩ(NΩ)3A
C[010](22,12)8N2×2;2M2×2;2A2×2;3P2×2;4tr(P)=0tr((NΩ)4+M4+P2)=tr(ATΩNΩA)tr((NΩ)6+M6+P3)=tr(ATΩ(NΩ)2ΩA)
C[002](23)6M3×3;2N3×3;4tr(N)=0tr(M4)=tr(N2)tr(M6)=tr(N3)
C[020](32)4N2×2;2M2×2;4P2×2;6tr(MΩ)=0tr(PΩ)=0tr((NΩ)4+(MΩ)2)=0tr((NΩ)6+(MΩ)3+(PΩ)2)=0
C[210](4,12)4N2×2;2A2×1;5tr((NΩ)6)=ATΩNΩA
C[202](4,2)2N1×1;4A1×1;4P1×1;6tr(NA2)=tr(P2)
C[222](6)0––
Table 24D2 and D3 varieties, generated by complex matrices M, N, and A and their relations, which have Hilbert series matching Slodowy slices SN,ρ. The matrix M = −MT is antisymmetric, N = NT is symmetric and Ω represents a square matrix that is antisymmetric and invariant under the action of USp(2n). pf() denotes the Pfaffian.Table 24OrbitPartitionDim.Generators; DegreeRelations
D[00](14)4M4×4;2tr(M2)=0pf(M)=0
D[20](22)2N2×2;2tr((NΩ)2)=0
D[02](22)2N2×2;2tr((NΩ)2)=0
D[22](4)0––

D[000](16)12M6×6;2tr(M2)=0tr(M4)=0pf(M)=0
D[011](22,12)6M2×2;2N2×2;2A2×2;3tr(AΩATΩ)=0tr(M4)+tr((NΩ)4)=tr(AΩATM+AΩNΩAT)
D[200](3,13)4M3×3;2A3×1;4ϵijkMijAk=0tr(M4)=ATA
D[022](32)2M2×2;2N2×2;4tr(N)=0tr(M4)=tr(N2)
D[222](5,1)0––
Table 25Types of generating function.Table 25Generating functionNotationDefinition
Refined HS (Weight coordinates)gHSG(x,t)∑n=0∞an(x)tn
Refined HS (Simple root coordinates)gHSG(z,t)∑n=0∞an(z)tn
Unrefined HSgHSG(t)∑n=0∞antn≡∑n=0∞an(1)tn
Quiver theories and formulae for Slodowy slices of classical algebrasSantiagoCabrerasantiago.cabrera13@imperial.ac.ukAmihayHananya.hanany@imperial.ac.ukRudolphKalveks⁎rudolph.kalveks09@imperial.ac.ukTheoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United KingdomTheoretical Physics GroupThe Blackett LaboratoryImperial College LondonPrince Consort RoadLondonSW7 2AZUnited Kingdom⁎Corresponding author.Editor: Clay CórdovaAbstractWe utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone S∩N of g. We calculate refined Hilbert series for Classical algebras up to rank 4 (and A5), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections S∩N; such dual quiver theories exist for the Slodowy slices of A algebras, but are limited to a subset of the Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3d mirror symmetry within the class of SCFTs known as Tσρ(G) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the SU(2) embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.1IntroductionThe relationships between supersymmetric (“SUSY”) quiver gauge theories, the Hilbert series (“HS”) of their Higgs and Coulomb branches, and the nilpotent orbits (“NO”) of simple Lie algebras g were analysed in two recent papers [1,2]. Closures of classical nilpotent orbits appear as Higgs branches on N=2 quiver theories in 4d, and also as Coulomb branches on N=4 quiver theories in 2+1 dimensions. Both these types of theory have 8 supercharges.The aim herein is to examine systematically the relationships between these SUSY quiver gauge theories and the spaces transverse to nilpotent orbits, known as Slodowy slices. The focus herein is the Slodowy slices of the nilpotent orbits of Classical algebras, which are associated with a rich array of 3dN=4 quiver theories and dualities. The relationships between SUSY quiver gauge theories and the Slodowy slices of nilpotent orbits of Exceptional algebras remain to be treated.The mathematical study of Slodowy slices has its roots in [3], which built on earlier work by Brieskorn [4], Grothendieck and Dynkin [5]. This showed that each nilpotent orbit Oρ of a Lie algebra g of a Classical group G has a transverse slice, or Slodowy slice Sρ, lying within the algebra g.11ρ identifies the embedding of su(2) into g that defines the nilpotent orbit. There is a variety defined by the intersection between the Slodowy slice and the nilpotent cone N of the algebra: SN,ρ≡N∩Sρ. In this paper, we deal almost entirely with these intersections SN,ρ and refer to them loosely as Slodowy slices (except where the context requires otherwise). Each such slice is a singularity that can be characterised by a subalgebra f of g that commutes with (or stabilises) the su(2). In the case of the subregular nilpotent orbit SN,ρ is a Kleinian singularity of type ADE.22For general background on nilpotent orbits the reader is referred to [6].The connection between nilpotent orbits and their Slodowy slices, and instanton moduli spaces, i.e. the solutions of self dual Yang–Mills equations, was made in [7]. The use of Dynkin diagrams and quiver varieties to define instantons on ALE spaces was discussed in [8]. The relevance of nilpotent orbits and Slodowy slices to 3d N=4 quiver theories was later explored in detail in [9] and [10]. In this context, they appear as effective gauge theories describing the brane dynamics of a system in Type IIB string theory. Brane systems of the type of [11] are relevant for the A series and systems with three dimensional orientifold planes [12] for the BCD series.33Note that these brane systems can explicitly realize the transverse slices developed by Brieskorn and Slodowy [3,4]. A systematic analysis of transverse slices was carried out by Kraft and Procesi [13] and the physics realization was studied in [14,15]. The concept of transverse slices can be further extended as an operation of subtractions between two quivers [16].In the course of these latter papers, a class of superconformal field theories (“SCFT”) was proposed, with moduli spaces defined by the intersections between Slodowy slices and nilpotent orbits. These are termed Tσρ(G) theories, where G is a Lie group. Several types of Classical quiver theories were identified, along with associated brane configurations, including theories whose Higgs or Coulomb branches correspond to certain varieties SN,ρ, and a relationship between Sduality and dualities arising from the 3d mirror symmetry [17] of Classical quiver theories was conjectured.44In the case of nilpotent orbits of C and D type, the precise match between quivers and orbits was given in [18]. Subsequently, [19] described the relation for B type and unified all classical cases via the introduction of the Barbasch–Vogan map [20].For example, in the case of an A series nilpotent orbit Oρ, where ρ describes the embedding of su(2) into su(n) that defines the nilpotent orbit, and ρ=(1N) corresponds to the trivial nilpotent orbit, these dualities entail that the Higgs branch of a linear quiver based on a partition ρT, yields the closure of the nilpotent orbit O¯ρ, while the Coulomb branch of a linear quiver based on the partition ρ gives its Slodowy slice SN,ρ. The application of 3d mirror symmetry to this pair of linear quivers yields a pair of “balanced” quivers, with the Coulomb branch of the former yielding O¯ρ and the Higgs branch of the latter yielding SN,ρ.55The notation in the Literature regarding partitions and their dual maps has changed a great deal; see [15, sec. 4] for a summary of the different maps that are relevant to our study and an explicit review of the different conventions used in mathematics and physics.More recently, in [19] and [21], nilpotent orbits and Slodowy slices have been used in the study of 6d N=(2,0) theories on Riemann surfaces. Relationships between diagram automorphisms of quiver varieties and Slodowy slices are explored in [22]. In [23] the algebras of polynomial functions on Slodowy slices were shown to be related to classical (finite and affine) Walgebras.Each Slodowy slice of a subalgebra f of g has a ring of holomorphic functions transforming in irreps of the subgroup F of G. Our approach is to compute the Hilbert series of these rings. Presented in refined form, such Hilbert series faithfully encode the class function content of Slodowy slices, and can be subjected to further analysis using the tools of the Plethystics Program [24–26].Importantly, following a result in [3], the Hilbert series of Slodowy slices SN,ρ are always complete intersections, i.e. quotients of geometric series. It was shown in [27] how the HS of the Slodowy slices of A series and certain BCD series algebras can be calculated from the Coulomb branches of linear quivers (or from the Higgs branches of their 3d mirror duals). [27] also identified a relationship between Slodowy slices and the (modified) Hall Littlewood polynomials of g, under the mapping g→su(2)⊗f.Methods of calculating Hilbert series for Tσρ(G) theories with multiflavoured quivers of unitary or alternating O/USp type were developed in [28], using both Coulomb branch and Higgs branch methods. As elaborated in [29], the calculation of Coulomb branches of quivers of O/USp type requires close attention to the distinction between SO and O groups.This paper builds systematically on such methods to calculate the Hilbert series of Slodowy slices of closures of nilpotent orbits of low rank Classical Lie algebras and to identify relevant generalisations to arbitrary rank.In Section 2 we summarise some facts about nilpotent orbits and review the relationship between a Slodowy slice SN,ρ and the homomorphism ρ defining the embedding of su(2) into g (and thus of the mapping of irreps of G into the irreps of SU(2)) associated with a nilpotent orbit Oρ. We give some simple representation theoretic formulae for calculating the dimensions and Hilbert series of a Slodowy slice, given a homomorphism ρ.In Section 3 we treat A series Slodowy slices, summarising the relevant Higgs branch and Coulomb branch formulae, describing the quivers upon which they act, and tabulating the commutant global symmetry group and the Hilbert series of Slodowy slices for all nilpotent orbits up to rank 5. We also build upon the language of Tσρ(SU(N)) theories to summarise the known exact A series dualities between quiver theories for Slodowy slices and nilpotent orbits. We find matrix formulations for the generators of A series Slodowy slices and their relations, which explicate the mechanism of symmetry breaking and the residual symmetries of the parent group.In Section 4 we extend this analysis to Slodowy slices of BCD series algebras up to rank 4. We find a complete set of refined Hilbert series, by working with the Higgs branches of multiflavoured alternating O/USp quivers with appropriately balanced gauge nodes. As in the case of BCD nilpotent orbits [1], calculation of these Higgs branches requires taking Z2 averages over the SO and O− forms of O group characters. We also identify a limited set of Higgs branch constructions based on D series Dynkin diagrams. We summarise the restricted set of Coulomb branch monopole constructions that are available for SN,ρ, which are based on alternating SO/USp linear quivers. We highlight apparent restrictions on 3d mirror symmetry between Higgs and Coulomb branches of BCD quiver theories; these include the requirements that the nilpotent orbit Oρ should be special, and that the O/USp quivers should not be “bad” [10] due to containing monopole operators with zero conformal dimension. We find matrix formulations for the Higgs branch generators of BCD series Slodowy slices, and their relations, which explicate the mechanism of symmetry breaking and the residual symmetries of the parent group.Taken together with other recent studies [1,29], this analysis of Hilbert series is relevant for the understanding of Tσρ(G) theories of type BCD. It highlights the difference between orthogonal O(n) and special orthogonal SO(n) nodes and the surrounding problems associated with 3d mirror symmetry between orthosymplectic quivers.The final Section summarises conclusions, discusses open questions and identifies areas for further work. Some notational conventions are detailed in Appendix A.2Slodowy slices2.1Relationship to nilpotent orbitsEach nilpotent orbit Oρ of a Lie algebra g is defined by the conjugacy class gX of nilpotent elements X∈g under the group action [6]. Each nilpotent element X forms part of a standard su(2) triple {X,Y,H} and, following the Jacobson Morozov theorem, the conjugacy classes are in one to one correspondence with the equivalence classes of embeddings of su(2) into g, described by some homomorphism ρ. The closure of each orbit O¯ρ, can, as discussed in [1,2], be described as a moduli space, by a refined Hilbert series of representations of G, graded according to the degree of symmetrisation of the underlying nilpotent element.The closures O¯ρ of the nilpotent orbits of g form a poset, ordered according to their inclusion relations.66See for example the Hasse diagrams in [13]. The closure of the maximal (also termed principal or regular) nilpotent orbit is called the nilpotent cone N; it contains all the orbits Oρ and has dimension N equal to that of the rootspace of g. The poset of nilpotent orbits contains a number of canonical orbits. These include the trivial nilpotent orbit (described by the Hilbert series 1 with dimension zero), a minimal (lowest dimensioned nontrivial) nilpotent orbit, a subregular orbit of dimension N−2 and the maximal nilpotent orbit:(2.1){0}=Otrivial⊂O¯minimal…⊂O¯sub−regular⊂O¯maximal=N. All nilpotent orbits have an even (complex) dimension and are HyperKähler cones.The closure of the minimal nilpotent orbit of g corresponds to the reduced single Ginstanton moduli space [7,30]. As discussed in [1], the Hilbert series of the nilpotent cone has a simple expression in terms of the symmetrisations of the adjoint representation of G, modulo Casimir operators, or equivalently in terms of (modified) Hall Littlewood polynomials:(2.2)gHSN=PE[χadjointGt2−∑i=1rt2di],gHSN=mHLsingletG[t2], where t is a counting fugacity, χadjointG is the character of the adjoint representation and {d1,…,dr} are the degrees of the symmetric Casimirs of G, which is of rank r.Slodowy slices are defined as slices Sρ⊆g that are transverse in the sense of [3] to the orbit Oρ. The varieties SN,ρ that concern the present study are slices inside the nilpotent cone N. They can be constructed as:(2.3)SN,ρ≡Sρ∩N.Naturally, the slice SN,ρ transverse to the trivial nilpotent orbit is the entire nilpotent cone N and the slice SN,ρ transverse to the maximal nilpotent orbit is trivial. In between these limiting cases, however, the Slodowy slices do not match any nilpotent orbit. Consequently we have a complementary poset of Slodowy slices:(2.4)N=Strivial>Sminimal…>Ssub−regular>Smaximal={0}.2.2Dimensions and symmetry groupsThe dimensions of a Slodowy slice SN,ρ plus those of the nilpotent orbit Oρ combine to the dimensions of the nilpotent cone N:(2.5)SN,ρ+Oρ=N=g−rank[g]. The elements of the Slodowy slice SN,ρ lie in a subalgebra f, which is the centraliser of the nilpotent element X∈g, so that [X,f]=0, and f is often termed the commutant of su(2) in g. The structure of f and the dimensions of SN,ρ and Oρ can be determined by analysing the embedding of su(2)→g.Following [5], a homomorphism ρ can be described by a root space map from g to su(2), and this is conveniently encoded in a Characteristic set of Dynkin labels.77A Characteristic G[…] is distinct from highest weight Dynkin labels […,…]G. The Characteristic [q1…qr] provides a map from the simple root fugacities {z1,…,zr} of g to the simple root fugacity {z} of su(2):(2.6)ρ[q1…qr]:{z1,…,zr}→{zq12,…,zqr2}, where the labels qi∈{0,1,2}. This induces corresponding weight space maps under which each representation of G of dimension N branches to representations [n] of SU(2) at some multiplicity mn. This branching is conveniently described using partition notation, ([N−1]mN−1,…,[n]mn,…,1m0), which lists the dimensions of the SU(2) irreps, using exponents to track multiplicities. These partitions are tabulated in [1] for the key irreps of Classical groups up to rank 5, identifying each nilpotent orbit by its Characteristic.For example, the homomorphism ρ with Characteristic [202], which generates the 10 dimensional nilpotent orbit of A3, induces the following maps:(2.7)ρ[202]:{z1,z2,z3}→{z,1,z},ρ[202]:[1,0,1]→[4]+3⊗[2]+[0]⇔χadjointA3→(5,33,1),ρ[202]:[1,0,0]→[2]+[0]⇔χfundamentalA3→(3,1). These SU(2) partitions are invariant under the symmetry group F⊆G of the Slodowy slice and hence the multiplicities encode representations of F.Under the branching, the adjoint representation of G decomposes to representations of the product group SU(2)⊗F with branching coefficients anm:(2.8)χadjointG→⨁[n][m]anm(χ[n]SU(2)⨂χ[m]F). Other than for the trivial nilpotent orbit (in which the adjoint of G branches to itself times an SU(2) singlet), the adjoint of SU(2) and the adjoint (if any) of F each appear separately in the decomposition, so that rank[G]⩾rank[F]⩾0. Along with the requirement that any multiplicities mn appearing in a partition must be dimensions of representations of F, this often makes it possible to determine the Lie algebra f of the Slodowy slice directly from the partition data. In Example 2.7 the presence of a single SU(2) singlet in the partition of the adjoint of A3 entails that the symmetry group of the Slodowy slice to the [202] orbit is simply U(1).The adjoint partition data also permits direct calculation of the complex dimensions of a Slodowy slice or nilpotent orbit, by summing multiplicities of SU(2) irreps or, equivalently, dimensions of F irreps:(2.9)Sρ=∑[n][m]anmχ[m]F,Oρ=G−Sρ,SN,ρ=Sρ−rank[G].2.3Hilbert seriesThe branching of the adjoint representation of G determines the generators of the Slodowy slice. If the decomposition (2.8) is known, the Hilbert series for the Slodowy slice can be derived from the HS of the nilpotent cone by substitution under a particular choice of grading. Consider the map ρ˜ of the adjoint that is obtained from (2.8) by the replacement of SU(2) irreps by their highest weight fugacities χ[n]SU(2)→tn, taking the particular choice of t from (2.2) as the counting variable:(2.10)ρ˜:χadjointG→⨁[n][m]anmχ[m]Ftn. When the adjoint map (2.10) is applied to the generators of the nilpotent cone (2.2), the replacement of the SU(2) representations [n] by the counting fugacity tn entails that the resulting Hilbert series only transforms in the symmetry group of the Slodowy slice. Thus, gHSSN,ρ=gHSNρ˜, or, written more explicitly:(2.11)gHSSN,ρ(x,t)=PE[χadjointGρ˜t2−∑i=1rt2di]=PE[⨁[n][m]anmχ[m]Ftn+2−∑i=1rt2di]. The expression (2.11) gives the refined Hilbert series of the Slodowy slice in terms of its generators, which are representations of the Slodowy slice symmetry group F, at some counting degree in t, less its relations, which are set by the degrees of the Casimirs of G.88This construction for Slodowy slices is simpler, but equivalent to the Hall Littlewood method presented in [28].Importantly, an unrefined Hilbert series, with representations of F replaced by their dimensions, mn=∑manmχ[m]F, can be calculated directly from the adjoint partition under ρ, without knowledge of the precise details of the embedding (2.8):(2.12)gHSSN,ρ(1,t)=PE[∑nmntn+2−∑i=1rt2di]. Finally, the freely generated Hilbert Series for the canonical Slodowy slices Sρ are related to those of their nilpotent intersections SN,ρ simply by the exclusion of the Casimir relations:(2.13)gHSSρ(x,t)≡gHSSN,ρ(x,t)PE[∑i=1rt2di]=PE[⨁[n][m]anmχ[m]Ftn+2].In Sections 3 and 4 we set out the quiver constructions that provide a comprehensive method for identifying the decomposition (2.8) and for calculating the refined Hilbert series of the Slodowy slices SN,ρ.2.4Subregular singularitiesAs shown in [3,4], the Slodowy slices of subregular orbits SN,subregular take the form of ADE type singularities, C2/Γ, where Γ is a finite group of type ADE. Under the nilpotent orbit grading by t2 used herein, these take the forms in Table 1. The intersection SN,subregular is an example of a transverse slice between adjacent nilpotent orbits; all such transverse slices of Classical algebras were classified by Kraft and Procesi in [13].This known pattern of singularities amongst the Slodowy slices of subregular orbits, along with the known forms of trivial and maximal Slodowy slices and dimensions, provide consistency checks on the grading methods and constructions given herein.3A series quiver constructions3.1Quiver typesThe constructions for the Slodowy slices of A series nilpotent orbits draw upon the same two quiver types as the constructions for the closures of the nilpotent orbits. These are shown in Fig. 1:1.Linear quivers based on partitions. These quivers LA(ρ) consist of a SU(N0) flavour node connected to a linear chain of U(Ni) gauge nodes, where the decrements between nodes, ρi=Ni−1−Ni, constitute an ordered partition of N0, ρ≡{ρ1,…,ρk}, where ρi⩾ρi+1 and ∑i=1kρi=N0.2.Balanced quivers based on Dynkin diagrams. These quivers BA(Nf) consist of a linear chain of U(Ni) gauge nodes (in the form of an A series Dynkin diagram), with each gauge node connected to a flavour node of rank Nfi, where Nfi⩾0. The ranks of the gauge nodes are chosen such that each gauge node is balanced (as explained below), after taking account of any attached flavour nodes. On the Higgs branch, the flavour nodes of both types of quiver define an overall S(⊗iUNfi) global symmetry, while on the Coulomb branch, the global symmetry group follows from the Dynkin diagram formed by any balanced gauge nodes in the quiver.99The concept of balance was used in [9], in order to distinguish between (a) those Coulomb branch monopole operators that are “good”, with unit conformal dimension and act as root space operators, (b) those that are “ugly” with halfinteger conformal dimension and act as weight space operators, and (c) those that are “bad” with zero or negative conformal dimension, which lead to divergences.The balance of a unitary gauge node is defined as the sum of the ranks of its adjacent gauge nodes, plus the number of attached flavours, less twice its rank:(3.1)Balance(i)≡Nfi+∑j=i±1Nj−2Ni. For the A series balanced theories, the balance condition is B≡{Balance(i)}=0, and (3.1) can be simplified as:(3.2)Nf=A⋅N, where the flavour and gauge nodes have been written as vectors Nf≡(Nf1,…,Nfk) and N≡(N1,…,Nk), and A is the Cartan matrix of Ak.A series nilpotent orbits are in bijective correspondence with the partitions of N, and the linear quivers provide a complete set of Higgs branch constructions. The balanced quivers also provide a complete set of Coulomb branch constructions under the unitary monopole formula. Both types of quiver are thus in bijective correspondence with A series orbits and can be related by 3d mirror symmetry [17].For Slodowy slices, the roles of these quiver types are reversed: the linear A series quivers provide a complete set of Coulomb branch constructions, while the balanced A series quivers provide a complete set of Higgs branch constructions.When quivers of linear type are used to calculate Slodowy slices, via their Coulomb branches, the lack of balance of such quivers generally breaks the symmetry of SU(N0) to a subgroup, which becomes the isometry group of the Slodowy slice; this subgroup is in turn defined by the Dynkin diagram of the subset of gauge nodes in the linear quiver that remain balanced.The identification of quivers for Slodowy slices follows directly from the partition data discussed in section 2.2. For the A series, it is convenient to write the SU(2) partition of the fundamental representation under ρ as:(3.3)ρ[1,0,…]A=(NNfN,…,nNfn,…,1Nf1), so that the multiplicities of partition elements, which may be zero, are mapped to the flavour vector Nf. The linear quiver LA(ρ) can be extracted simply by writing ρ[fund.] in long form. The ranks N of the gauge nodes of the balanced quiver BA(Nf(ρ)) can be found from Nf by inverting (3.2). Alternatively, the balanced quivers BA(Nf(ρ)) can be obtained by applying 3d mirror symmetry transformations to the linear quivers LA(ρ), and vice versa.We can use the notation above to express the key relationships and dualities involving A series quivers for the Slodowy slices of nilpotent orbits:(3.4)SN,ρ=Higgs[BA(Nf(ρ))]=Coulomb[LA(ρ)],O¯ρ=Higgs[LA(ρT)]=Coulomb[BA(Nf(ρT))], or, taking the transpose of ρ:(3.5)SN,ρT=Higgs[BA(Nf(ρT))]=Coulomb[LA(ρT)],O¯ρT=Higgs[LA(ρ)]=Coulomb[BA(Nf(ρ))]. The quivers for A series slices are related to the quivers for the underlying orbits simply by the transpose of the partition ρ, combined with exchange of Coulomb and Higgs branches. This transposition of partitions, which is an order reversing involution on the poset of A series orbits, is known as the Lusztig–Spaltenstein map [31].These linear or balanced quiver types correspond to the limiting cases of Tσρ(SU(N)) theories [10,28], where one of the partitions σ or ρ is taken as the trivial partition:(3.6)LA(ρ)⇔Tρ(1,1,…,1)(SU(N)),BA(Nf(ρ))⇔T(1,1,…,1)ρ(SU(N)).Those quivers, whose Higgs or Coulomb branches yield Slodowy slices of A series groups up to rank 5, are tabulated in Figs. 2 and 3, labelled by the nilpotent orbit, giving the partition ρ of the fundamental, the dimensions of the Slodowy slice, and the residual symmetry group.1010We describe a U(1) symmetry as D1 if the characters qn of U(1) irreps always appear paired with their conjugates in representations (qn+q−n). The balanced quivers used in the Higgs branch construction always have gauge nodes equal in number to the rank of G=SU(N), while the linear quivers used in the Coulomb branch constructions always have a number of flavours equal to the fundamental dimension of G=SU(N). The quivers LA((1N)) and BA(Nf(1N)) for the Higgs and Coulomb branch constructions of the Slodowy slice to the trivial nilpotent orbit are identical.3.2Higgs branch constructionsThe calculation of Higgs branch Hilbert series from the balanced quivers draws on similar methods to those used in the calculation of the Higgs branches of the linear quivers for A series nilpotent orbits, as elaborated in [1]. Pairs of bifundamental fields (and their complex conjugates) connect adjacent gauge nodes and, in addition, each nontrivial flavour node gives rise to a pair of bifundamental fields connected to its respective gauge node. The characters of all these fields are included in the PE symmetrisations. A HyperKähler quotient is taken once for each gauge node, exactly as in the case of a linear quiver, and the Weyl integrations are then carried out over the gauge groups. The order of Weyl integrations can be chosen to facilitate computation.The general Higgs branch formula for A series Slodowy slices is:(3.7)gHSHiggs[BA(Nf(ρ))]=∮U(N1)⊗…U(Nk)dμ∏n=1kPE[[fund.]U(Nn)⊗[anti.]U(Nfn)+[anti.]U(Nn)⊗[fund.]U(Nfn),t]PE[[adjoint]U(Nn),t2]×∏n=1k−1PE[[fund.]U(Nn)⊗[anti.]U(Nn+1)+[anti.]U(Nn)⊗[fund.]U(Nn+1),t], where dμ is the Haar measure for the U(N1)⊗…U(Nk) product group. Note that the bifundamental fields are symmetrised with the fugacity t, while the HyperKähler quotient (“HKQ”) is symmetrised with t2.The Higgs branch formula can be simplified, by drawing on the dimensions of the bifundamentals and the gauge groups, to give a rule for the dimensions of an A series Slodowy slice, and this can be simplified further by the balance condition (3.2):(3.8)gHSHiggs[BA(Nf(ρ))]=2Nf(ρ)⋅N(ρ)−N(ρ)⋅A⋅N(ρ)=Nf(ρ)⋅N(ρ). For further details of the calculation methodology the reader is referred to the Plethystics Program Literature. The same Hilbert series can in principle also be obtained algebraically by working with matrix generators and relations, as in section 3.5.3.3Coulomb branch constructionsThe monopole formula, which was introduced in [32], provides a systematic method for the construction of the Coulomb branches of particular SUSY quiver theories, being N=4 superconformal gauge theories in 2+1 dimensions. The Coulomb branches of these theories are HyperKähler manifolds. The formula draws upon a lattice of monopole charges, defined by the linked system of gauge and flavour nodes in a quiver diagram.Each gauge node carries adjoint valued fields from the SUSY vector multiplet and the links between nodes correspond to complex bifundamental scalars within SUSY hypermultiplets. The monopole formula generates the Coulomb branch of the quiver by projecting charge configurations from the monopole lattice into the root space lattice of G, according to the monopole flux over each gauge node, under a grading determined by the conformal dimension of each overall monopole flux q.The conformal dimension (equivalent to Rcharge or the highest weight of the SU(2)R global symmetry) of a monopole flux is given by applying the following general schema [10] to the quiver diagram:(3.9)Δ(q)=12∑i=1r∑ρi∈Riρi(q)︸contributionofN=4hypermultiplets−∑α∈Φ+α(q)︸contributionofN=4vectormultiplets. The positive Rcharge contribution in the first term comes from the bifundamental matter fields that link adjacent nodes in the quiver diagram. The second term captures a negative Rcharge contribution from the vector multiplets, which arises due to symmetry breaking, whenever the monopole flux q over a gauge node contains a number of different charges.The calculation of Hilbert series for Coulomb branches of A type quivers draws on the unitary monopole formula, which follows from specialising (3.9) to unitary gauge groups. Each U(Ni) gauge node carries a monopole flux qi≡(qi,1,…,qi,Ni) comprising one or more monopole charges qi,m. The fluxes are assigned the collective coordinate q≡(q1,…,qr). Each flavour node carries Nfi flavours of zero charge.1111Flavour nodes may also carry nonzero charges, although these are not required by the Slodowy slice (or nilpotent orbit) constructions.With these specialisations, the conformal dimension Δ(q) associated with a flux q yields the formula:(3.10)Δ(q)=12∑j>i⩾1r∑m,nqi,mAij−qj,nAji︸gauge  gauge hypers+12∑i∑mNfiqi,m︸gauge  flavour hypers−∑i=1r∑m>nqi,m−qi,n︸gauge vplets, where (i) the summations are taken over all the monopole charges within the flux q and (ii) the linking pattern between nodes is defined by the Aij offdiagonal Ar Cartan matrix terms, which are only nonzero for linked nodes.1212For theories with simply laced quivers of ADE type, Aij=0 or −1, for i≠j.With conformal dimension defined as above, the unitary monopole formula for a Coulomb branch HS is given by the schema [32]:(3.11)gHSCoulomb(z,t2)≡∑qPqU(N)(t2)zqt2Δ(q), where:1.The limits of summation for the monopole charges are ∞⩾qi,1⩾…qi,m⩾…qi,Ni⩾−∞ for i=1,…r.2.The monopole flux over the gauge nodes is counted by the fugacity z≡(z1,…,zr), where the zi are fugacities for the simple roots of Ar.3.The monomial zq combines the monopole fluxes qi into total charges for each zi and is expanded as zq≡∏i=1rzi∑m=1Niqi,m.4.The term PqU(N) encodes the degrees di,j of the Casimirs of the residual U(N) symmetries that remain at the gauge nodes under a monopole flux q:(3.12)PqU(N)(t2)≡∏i,j1(1−t2di,j(q))=∏i=1r∏j=1Ni∏k=1λij(qi)11−t2k.Recalling that a U(N) group has Casimirs of degrees 1 through N, the residual symmetries can be determined as in [32].1313We construct a partition of Ni for each node, which counts how many of the charges qi,m are equal, such that λ(qi)=(λi,1,…,λi,Ni), where ∑m=1Niλi,m=Ni and λi,m⩾λi,m+1⩾0. The nonzero terms λi,j in the partition give the ranks of the residual U(N) symmetries associated with each node, so that it is a straightforward matter to compound the terms in the degrees of Casimirs. For example, if qi,m=qi,n for all m,n, then {di,1,…di,Ni}={1,…,Ni} and if qi,m≠qi,n for all m,n, then {di,1,…di,Ni}={1,…,1}. Alternatively, the residual symmetries for a flux qi can be fixed from the subgroup of U(Ni) identified by the Dynkin diagram formed by those monopole charges that equal their successors {qi,m:qi,m=qi,m+1}, (or equivalently, correspond to zero Dynkin labels). The exact calculation of a Coulomb branch HS can be carried out by evaluating (3.11) as a geometric series over each sublattice of monopole charges q, for which both conformal dimension Δ(q) and the symmetry factors PqU(N) are linear (rather than piecewise or step) functions, and then summing the many resulting polynomial quotients. These sublattices of monopole charges form a hypersurface and care needs to be taken to avoid duplications at edges and intersections.3.4Hilbert seriesThe Hilbert series of the Slodowy slices of algebras A1 to A4, calculated as above, are summarised in Table 2, and those of A5 are summarised in Table 3. Both the Higgs and Coulomb branch calculations lead to identical refined Hilbert series, up to choice of CSA coordinates or fugacities.The Hilbert series are presented in terms of their generators, or PL[HS], using character notation [n1,…,nr] to label Ar irreps. Symmetrisation of these generators using the PE recovers the refined Hilbert series. The underlying adjoint maps (2.10) can readily be recovered from the generators by inverting (2.11). The HS can be unrefined by replacing representations of the global symmetry groups by their dimensions.Several observations can be made about the Hilbert series.1.As expected, (i) the Slodowy slice to the trivial nilpotent orbit SN,(1N) has the same Hilbert series as the nilpotent cone, (ii) the slice to the subregular orbit SN,(N−1,1) has the Hilbert series of a Kleinian singularity of type AˆN−1, and (iii) the slice to the maximal nilpotent orbit SN,(N) is trivial.2.As expected, the Slodowy slices SN,ρ are all complete intersections.3.The global symmetry groups of the Slodowy slice generators include mixed SU and unitary groups, and descend in rank as the dimension of the Slodowy slice reduces. Sometimes different Slodowy slices share the same symmetry group, with inequivalent embeddings of F into G.4.Complex representations always appear combined with their conjugates to give real representations.5.The adjoint maps often contain singlet generators at even powers of t up to the (twice the) degree of the highest Casimir of g; these generators may be cancelled by one or more Casimir relations. Many of these observations have counterparts amongst the Slodowy slices of BCD series, although these also raise several new intricacies, as will be seen in section 4.3.5Matrix generators for unitary quiversA Hilbert series over the class functions of a Classical group can be viewed in terms of matrix generators (or operators), and this perspective makes it possible to identify the generators of a Slodowy slice directly from the partition data or its Higgs branch quiver.3.5.1Fundamental decompositionFrom (3.3), it follows that the character of the fundamental representation of G decomposes into fundamental representations of a unitary product group:(3.13)ρ:χfund.G→⨁[n][n]ρχ[fund.]SUNfn+1qn+1, where [n]ρ are irreps of the SU(2) associated with the nilpotent orbit embedding ρ, and the U(1) charges qi on the flavour nodes satisfy the overall gauge condition ∏i=1kiNfiqi=1.1414This corresponds to viewing the fields in a centre of mass frame. Once this decomposition has been identified, the mapping of the adjoint of G into matrix generators follows, by taking the product of the fundamental and antifundamental characters, and eliminating a singlet. This can be checked against the adjoint partition ρ:χadjointG.3.5.2Generators from quiver pathsAlternatively the operators can be read from a quiver of type BA(Nf(ρ)), following the prescription:1.Draw the chiral multiplets explicitly as arrows in the quiver:(3.14)2.Every path in the quiver that starts and ends on a flavor node corresponds to an operator in the chiral ring of the Higgs branch.3.There is a one to one correspondence between paths that appear as generators in the PL[HS] of the Higgs branch and the paths of the type Pij(a), defined as below.4.The operator Pij(a) transforms under the fundamental representation of U(Nfi) and the antifundamental representation of U(Nfj) and sits on an irrep of SU(2)R with spin s=A/2, where A is the number of arrows in the path that defines Pij(a). This means that it appears in the plethystic logarithm of the refined Hilbert series as the character of the corresponding representation multiplied by the fugacity tA.5.Therefore, there is a one to one correspondence between operators Pij(a) and irreducible representations in the decomposition of the adjoint representation of Ak in (2.10).Definition Pij(a)Let Pij(a) be an operator Pij(a) with i,j∈{1,2,…,k} and a∈{1,2,…,min(i,j)}. Pij(1) is defined as the operator formed by products of operators represented by arrows in the shortest possible path that starts at node Nfi and ends at node Nfj (note that i and j could be equal). Pij(2) represents a path that differs from Pij(1) only in that it has been extended to incorporate the arrows between the gauge nodes Nmin(i,j) and Nmin(i,j)−1. Pij(3) differs from Pij(2) in that it also includes arrows between the gauge nodes Nmin(i,j)−1 and Nmin(i,j)−2. In this way Pij(a) is defined recursively as an extension of Pij(a−1).Example 1Let us start with the balanced A3 quiver based on the fundamental partition ρ=(2,12), whose Higgs branch is the Slodowy slice SN,(2,12) to the nilpotent orbit A[101]. The quiver is:(3.15) From Table 2, the Hilbert series is:(3.16)gHSHiggs[BA(Nf(2,12))]=PE[t2+[2]t2+[1](q+1/q)t3−t6−t8]. To obtain this using the prescription in section 3.5.1, we first identify the fugacity map for the group decomposition using (3.13):(3.17)SU(4)→SU(2)ρ⊗SU(2)⊗U(1),[1,0,0]→[1]ρq1/2+[1]q−1/2,[0,0,1]→[1]ρq−1/2+[1]q1/2,[1,0,1]→([2]+1)[0]ρ+[1](q+1/q)[1]ρ+[2]ρ. Next the irreps [n]ρ of SU(2)ρ are mapped to the fugacity tn+2, giving the generators:(3.18)[1,0,1]→[2]t2+t2+[1](q+1/q)t3+t4. Subtracting the relations −t4−t6−t8, corresponding to Casimirs of A3, we obtain:(3.19)PL[gHSHiggs[BA(Nf(2,12))]]=[2]t2+t2+[1](q+1/q)t3−t6−t8. The generators in (3.19) can be understood as operators from paths in the quiver (3.15) (Table 4).The irrep of each generator corresponds with the flavor nodes where the path starts and ends. The U(1) fugacity q≡q1/q2. The exponent of the fugacity t corresponds to the length of the path.Example 2Now consider the balanced quiver based on the A4 partition (3,2), whose Higgs branch is the Slodowy slice SN,(3,2) to the nilpotent orbit A[1111]:(3.20) The group decomposition is:(3.21)SU(5)→SU(2)ρ⊗S(U(1)⊗U(1)). The paths in the quiver can be used to predict the generators in Table 5. Subtracting relations −∑i=15t2i, corresponding to the special condition in (3.21), which eliminates one of the U(1) symmetries, and the Casimirs of A4, and substituting q for q2/q3 gives the expected PL[HS]:(3.22)PL[gHSHiggs[BA(Nf(3,2))]]=t2+(q+1q)t3+t4+(q+1q)t5−t8−t10, in accordance with Table 2.3.5.3Matrices and relationsIn this section we offer a reinterpretation of the previous results for Slodowy slices SN,ρ as sets of matrices that satisfy specific relations. The aim of this analysis is to build a bridge between the algebraic definition of the nilpotent cone SN,(1N)=N and that of the Kleinian singularity SN,(N−1,1)=C2/ZN.First, let us remember that the Kleinian singularity SN,(N−1,1)=C2/ZN can be defined as the set of points parametrized by three complex variables x,y,z∈C, subject to one relation:(3.23)xN=yz.Secondly, the nilpotent cone SN,(1N)=N can be defined as a set of complex variables arranged in a N×N matrix M∈CN×N, subject to the following relations:(3.24)tr(Mp)=0∀p=1,2,…,N.We want to show that a Slodowy slice SN,ρ can be viewed as an intermediate case between these two descriptions. In order to do this we build examples of varieties described by sets of complex matrices, choose relations among them and compute the (unrefined) Hilbert series of their coordinate rings, utilizing the algebraic software Macaulay2 [33]. These Hilbert series can be checked to be the same as those in Table 2.The specific matrices that generate the coordinate rings are chosen according to the operators Pij(a) found in the balanced quivers BA(Nf(ρ)). For example, let us study the balanced quiver whose Higgs branch is the Slodowy slice SN,(2,13):(3.25) One can assemble the generators Pij(a) into three different complex matrices M, A and B of dimensions 3×3, 3×1 and 1×3 respectively. Let us show how this can be done explicitly. There are five paths of the form Pij(a): P11(1), P22(1), P22(2), P12(1), P21(1). Out of these five sets of operators P22(1) can be removed by the relation −t2 that removes the centre of mass and P22(2) by the first Casimir invariant relation −t4. This means that there is a remaining set of generators transforming in the following irreps:(3.26)P11(1)→([1,1]+[0,0])t2,P12(1)→([1,0]q)t3,P21(1)→([0,1]1q)t3.One can now assemble these generators in three complex matrices that transform in the usual way under the global symmetry U(3):(3.27)([1,1]+[0,0])t2→M3×3,([1,0]q)t3→A1×3,([0,1]1q)t3→B3×1.The chiral ring is then parametrized by the set of all matrices {M,A,B}, subject to one relation at order t6, another relation at order t8 and a final relation at order t10. These relations are invariant under the global U(3) symmetry. One can choose the following set of relations:(3.28)tr(M3)=AB,(3.29)tr(M4)=AMB,(3.30)tr(M5)=AM2B. Note that these look like corrections to the equations of the nilpotent cone (3.24). The Hilbert series of the coordinate ring is then computed using Macaulay2 to be:(3.31)HS=(1−t6)(1−t8)(1−t10)(1−t2)9(1−t3)6. This is the same Hilbert series as that of the variety SN,(2,13) computed in Table 2.In Tables 6 and 7 we provide a set of algebraic varieties described by matrices such that their HS have been computed to be identical to those of the corresponding Slodowy slices SN,ρ. Note that we rewrite the Kleinian singularity in terms of 1×1 matrices, to clarify the connection with the algebraic description of the other Slodowy slices.4BCD series quiver constructions4.1Quiver typesThe constructions for the Slodowy slices of BCD algebras draw upon a different set of quiver types to the A series.1.Linear orthosymplectic quivers. These quivers LB/C/D(σ) consist of a B, C or D series flavour node of vector irrep dimension N0 connected to an alternating linear chain of (S)O/USp(Ni) gauge nodes of nonincreasing vector dimension. For a subset of these linear quivers, the decrements, σi=Ni−1−Ni, between nodes constitute an ordered partition of N0, σ≡{σ1,…,σk}, where σi⩾σi+1 and ∑i=1kσi=N0. More generally, however, the σi form a sequence of nonnegative integers, subject to ∑i=1kσi=N0, and to selection rules, such that USp nodes of odd dimension do not arise.2.Balanced orthosymplectic quivers. These quivers BB/C/D(Nf) consist of an alternating linear chain of O/USp(Ni) nodes, with each gauge node connected to a flavour node O/USp(Nfi), where Nfi⩾0. The ranks of the gauge nodes are chosen such that, taking account of any attached flavour nodes, each gauge node inherits its balance B (via (4.2)) from that of a canonical quiver (as defined below).3.Dynkin diagram quivers. These quivers DG(Nf) consist of a chain of U(Ni) gauge nodes in the form of a simply laced Dynkin diagram, with each gauge node connected to Nfi flavours, where Nfi⩾0. Nf matches the Characteristic G[…] of a nilpotent orbit, and the ranks of the gauge nodes are chosen such that each is balanced (similarly to the A series quivers in section 3.1). These constructions are limited to certain Slodowy slices of ADE algebras, as the Higgs branch construction is not available on nonsimply laced Dynkin diagrams. Recall, the nilpotent orbits of a BCD algebra correspond to a subset of the partitions ρ of N, once these have been subjected to selection rules,1515In a valid B or D partition ρ each even integer appears at an even multiplicity; in a valid C partition each odd integer appears at an even multiplicity [6]. and linear quivers LB/C/D(ρT) provide a complete set of Higgs branch constructions. Also, balanced quivers BB/C/D(Nf) provide Coulomb branch constructions, using the O/USp monopole formula, for the unrefined Hilbert series of certain nilpotent orbits of orthogonal groups, as discussed in [29]. The linear and balanced quivers can partially be related by 3d mirror symmetry, as discussed further in section 5. Many of these linear quivers have “Higgs equivalent” quivers, LB/C/D(σ), with a different choice of orthogonal gauge node dimensions, but the same Higgs branches; these are generally described by sequences σ rather than partitions ρT: a USp−O−USp subchain with the subpartition (…,n,n,…) has the Higgs equivalent sequence (…,σi,σi+1,…)=(…,n−1,n+1,…), in which the vector dimension of the central O node is increased by 1 [1].Returning to Slodowy slices, the roles of these quiver types are essentially reversed: balanced quivers BB/C/D provide a complete set of Higgs branch refined Hilbert series constructions, while linear quivers LB/C/D provide Coulomb branch constructions for the unrefined HS of certain Slodowy slices. Within the general classes of linear and balanced quiver types, those that are most relevant to the construction of Slodowy slices are shown in Fig. 4.We refer to the quivers of type LBC, LCD or LDC, which contain pure B−C, C−D or D−C chains, as canonical linear quivers. On the Higgs branch, the flavour nodes (of either type of quiver) identify the overall global symmetry, although it is necessary to distinguish within the B and D series between O and SO groups. However, it is not easy to identify the global symmetry of the Coulomb branch of a O/USp quiver.It is important to explain how the specific quivers used in the construction of the Hilbert series for BCD Slodowy slices arise from the partition of the vector representation of G under the homomorphism ρ.The balanced quivers BB/C/D(Nf(ρ)) are found via a modification of the A series method explained in section 3.1. Firstly, the SU(2) partition of a BCD series vector representation under ρ can be used to define a vector Nf(ρ) of alternating O/USp flavour nodes, similarly to (3.3):(4.1)ρ[1,0,…]B/C/D=(NNfN,…,nNfn,…,1Nf1). Next, consider linear quivers, whose Higgs branches match the nilpotent cone N. In the case of BCD groups, these quivers can be chosen, using Higgs equivalences, to be of canonical type. The balances B of their gauge nodes can be calculated by applying (3.1) to vectors Nf and N defined from the vector/fundamental dimensions of the fields, as shown in Table 8.These canonical quivers obey the generalisation of (3.2):(4.2)Nf=A⋅N+B, where B=(0,…,0) for the A and B series canonical quivers, B=(−2,2,…,−2) for the C series and B=(2,−2,…,−2) for the D series canonical quivers.By fixing B, the gauge node balance condition (4.2) can be extended from N to general Slodowy slices SN,ρ, permitting the calculation of each gauge node vector N from its flavour node vector Nf. In effect, the quivers BB/C/D(Nf(ρ)) descend from the canonical linear quivers for N, through a series of transitions that leave the balance vector B invariant. These canonically balanced quivers provide Higgs branch constructions for BCD Slodowy slices. They are tabulated in Figs. 5–10, along with the partitions of the fundamental, the dimensions of the Slodowy slices, and their residual symmetry groups.1616Note that other quivers whose Higgs branches match N could be taken to define B; each leads to a different family of quivers, whose Higgs branches match the Slodowy slice Hilbert series. The canonical choice, however, best illustrates the Higgs–Coulomb quiver dualities.On the other hand, the identification of linear quivers LB/C/D(σ) for Coulomb branch constructions of BCD series Slodowy slices poses a number of complications.1.There is no bijection between partitions of N and nilpotent orbits of O(N) or USp(N). So the quiver LB/C/D(ρ) is valid only for partitions ρ of special nilpotent orbits; in the other cases LB/C/D(ρ) (unlike LB/C/D(ρT)) would contain USp(N) vectors of odd dimension N.2.In the case of Coulomb branch constructions, GNO duality [34] is relevant. This indicates that, since the nonsimply laced B and C groups are GNO dual to each other, partitions of B type will be necessary to produce quivers whose Coulomb branches generate Slodowy slices of C algebras, and vice versa.3.A quiver LB/C/D(ρT) may have several Higgs equivalent quivers LB/C/D(σ), in which σ is a sequence of nonnegative integers, rather than an ordered partition. Such quivers have the same Higgs branch refined HS, but generally have different ranks of gauge groups, and therefore different Coulomb branch dimensions.4.Any candidate quiver for a Slodowy slice must have the correct Hilbert series dimension. Since the Coulomb branch monopole construction leads to a HS with complex dimension equal to twice the sum of the gauge group ranks in the quiver, this limits the candidates amongst Higgs equivalent quivers.5.The Coulomb branches of quivers with O gauge groups differ from those with SO gauge groups; a correct choice of orthogonal gauge groups needs to be made [29].6.When the orthosymplectic Coulomb branch monopole formula is applied to a quiver, the conformal dimension of all monopole operators must be positive for the Hilbert series to be well formed.Leaving the discussion of conformal dimension to section 4.3, it is remarkable that a procedure exists for a partial resolution of these complexities, and indeed forms the basis for Coulomb branch constructions for the unrefined Hilbert series of nilpotent orbits of special orthogonal groups in [29]. The method draws on the Barbasch–Vogan map1717A particularly clear description of this map is given in equation (5) of [35]. A summary of dual maps between partitions and their appearance in the literature can be found in [15, sec. 4]. dBV(ρ) [20], which provides a bijection between the partitions of real vector representations associated with B series special nilpotent orbits and those of pseudoreal vector representations associated with C series special nilpotent orbits. By making use of Higgs equivalences, to select canonical linear quivers of type LBC, LCD or LDC, which can be done for all special nilpotent orbits, the dBV(ρ) map can be extended to identify candidates for Coulomb branch constructions of Hilbert series of BCD Slodowy slices, in each case starting from a homomorphism ρ.The specific transformations from the partitions ρT to the sequences σ are summarised in Table 9. Within these; ρT indicates the transpose of a partition; ρN→N±1 indicates incrementing (decrementing) the first (last) term of a partition by 1; ρB, ρC, or ρD indicates collapsing a partition to a lower partition that is a valid B, C, or D partition [6]; BC or CD indicates shifting D or B nodes in a linear quiver to a ‘Higgs equivalent’ quiver that consists purely of B−C or of C−D pairs of nodes. The transformations can be written more concisely as σ=dBV(ρ)Tcanonical. The resulting linear quivers, LCD(σ), LBC(σ) and LDC(σ), whose Coulomb branches are candidates for Slodowy slices of BCD groups up to rank 4, are included in Figs. 5 through 10.The quivers LDC((1N)) and BD(Nf(1N)) for the Higgs and Coulomb branch constructions of the Slodowy slice to the trivial nilpotent orbit are the same. These tables also include identified quivers of type DG(Nf([dBV(ρ)])), whose Higgs branch Hilbert series match those of BB/C/D(Nf(ρ)).Type IIB string theory brane systems Note that all the resulting quivers, presented in Figs. 5 through 10 represent 3dN=4 gauge theories that admit an embedding in Type IIB string theory. They correspond to the effective gauge theory living on the worldvolume of D3branes suspended along one spatial direction between NS5branes and D5branes. This is achieved by taking the construction of [11] and introducing O3planes [12]. This type of system was further explored in [10] where the Coulomb branches and Higgs branches were described in terms of nilpotent orbits and Slodowy slices, and the label Tσρ(G) was introduced to denote the SCFT at the superconformal fixed point. These brane systems and 3d quivers were also studied in [15], finding the physical realization of transverse slices between closures of nilpotent orbits that are adjacent in their corresponding Hasse diagrams. This phenomenon has been named the Kraft–Procesi transition.4.2Higgs branch constructionsIn the case of the balanced unitary quivers DD(Nf), based on D series Dynkin diagrams, the calculation of Higgs branch Hilbert series proceeds similarly to the A algebras. This leads to a Higgs branch formula that is comparable to (3.7), modified to include the connection of three pairs of bifundamental fields to the central node. The dimension formula (3.8) remains unchanged.In the case of orthosymplectic quivers of type BB/C/D(Nf), modifications to the A series Higgs branch formula are required. The O/USp alternating chains are taken to comprise bifundamental (half) hypermultiplet fields that transform in vector representations [1,0,…,0]B/D⊗[1,0,…,0]C. Also, it is necessary to average the integrations over the disconnected SO and O− components of the O gauge groups; this requires precise choices both of the character for the vector representation of O− and of the HKQ associated with the integration over O−, as explained in [1].1818The effect of nonconnected O group components is also discussed in [36].In other respects, the calculation of the Higgs branch of a balanced BCD quiver follows a similar Weyl integration to the A series. The general Higgs branch formula for BCD series Slodowy slices is:(4.3)gHSHiggs[BB/C/D(Nf(ρ))]=12#O∑O±∮G1(N1)⊗…Gk(Nk)dμ∏n=1kPE[[vector]Gn(Nn)⊗[vector]Gfn(Nfn),t]HKQ[Gn(Nn),t]×∏n=1k−1PE[[vector]Gn(Nn)⊗[vector]Gn+1(Nn+1),t]. In (4.3), Gn alternates between O(N) and USp(N), dμ is the Haar measure for the G1(N1)⊗…Gk(Nk) product group, HKQ[Gn(Nn),t] is the HyperKähler quotient for a gauge node, and the summation indicates that the calculation is averaged over the nonconnected SO and O− components of O gauge groups [1].The character [vector]O(2r)−=[vector]O(2)−⊕[vector]USp(2r−2), where [vector]O(2)− is (the trace of) a diagonal matrix with eigenvalues {1,−1}. The HKQ is given by HKQ[Gn(Nn),t]=PE[[adjoint]Gn,t2], where for the O(2r)− component of an O(2r) group, [adjoint]O(2r)−≡Λ2[[vector]O(2r)−].1919For further detail, see [1].The structure of the Higgs branch formula can be used to identify the dimensions of the Hilbert series. In essence, each bifundamental field contributes HS generators according to its dimensions (being the product of the dimensions of the O and USp vectors), and each gauge group offsets the generators by HS relations numbering twice the dimension of the gauge group (once for the Weyl integration and once for the HKQ). This gives a rule for the dimensions of a Slodowy slice calculated from a balanced BB/C/D(Nf(ρ)) quiver:(4.4)gHSHiggs[B(Nf(ρ))]=Nf(ρ)⋅N(ρ)−12N(ρ)⋅A⋅N(ρ)+N(ρ)⋅K, where Kn={+1 ifGn=B/D−1 ifGn=C and (4.2) is used to calculate N(ρ).4.3Coulomb branch constructionsWhile the O/USp version of the monopole formula (4.5) derives from (3.9) by following similar general principles to the unitary monopole formula (3.11), there are several aspects and subtleties that require discussion:(4.5)gHSCoulomb(f,t2)≡∑o,sPo/sG(t2)fo/st2Δ(o,s).1.Monopole lattice. The lattice of monopole charges depends on the symmetry group. For SU and USp groups, points in the monopole charge lattice correspond to sets of ordered integers and are in bijective correspondence with highest weight Dynkin labels. However, the monopole charge lattices of orthogonal groups only span the vector sublattices and exclude weight space states whose spinor Dynkin labels sum to an odd number. Labelling monopole charges as q≡(q1,…,qr) for unitary nodes, s≡(s1,…,sr) for symplectic nodes and o≡(o1,…,or) for orthogonal nodes, the relationships between monopole and integer weight space lattices can be summarised as in Table 10.2.Characters. The definition of conformal dimension draws on the characters of the bifundamental scalar fields in the hyper multiplets and of the adjoint scalars in the vector multiplets: the weights of the fugacities in the characters become coefficients of the monopole charges q in Δ(q). These characters take a relatively simple form in the monopole lattice basis, compared with the weight space integer (Dynkin label) basis, as shown in Tables 11 and 12. CSA fugacities are taken as {x1,…,xr} in the weight space integer (Dynkin label) basis, or {y1,…,yr} in the monopole basis.3.Conformal dimension. The contributions to conformal dimension of the O/USp bifundamental fields linking gauge or flavour nodes, and of the O/USp gauge nodes, follow from (3.9) in a similar manner to the unitary Case (3.10), starting from the relevant characters: the coefficients {0,±1,±2} of the monopole charges {o,s} in the conformal dimension formula match the weights (exponents) of the y fugacities in the characters of the respective bifundamental or adjoint representations in the monopole basis. Tables 13 and 14 show the resulting contributions from the various types of gauge node and bifundamental field.4.Symmetry factors. The residual symmetries for a flux (whether o, s, or q) over a gauge node can be fixed from the subgroup of the O/USp/U gauge group identified by the Dynkin diagram formed by those monopole charges that equal their successors (or, equivalently, correspond to zero Dynkin labels). Note that the symmetry factors may belong to a subgroup from a different series to the gauge node.5.O vs SO gauge nodes. Both the characters of vector irreps and symmetry factors depend on whether a D series gauge node is taken as SO or as O. As noted in [28], the Casimirs of an O(2n) symmetry group are the same as those of SO(2n+1), due to the absence of a Pfaffian in O(2n) (since the determinant of representation matrices can be of either sign). The Coulomb branch calculations for Slodowy slices herein are based entirely on SO gauge nodes. This is a choice consistent with the results in [29]. When these results are translated to the brane configurations, the action of the Lusztig's Canonical Quotient A¯(O) related to each quiver can be seen in terms of collapse transitions [15] performed in the branes. Each time a collapse transition moves two half D5branes away from each other all magnetic lattices of the orthogonal gauge nodes in between are acted upon by a diagonal Z2 action. The brane configurations [10,12,15] for linear quivers LB/C/D(σ) do not present this effect, and therefore all gauge nodes are SO.6.Fugacities. In the unitary monopole formula, z in (3.11) can be treated as a fugacity for the simple roots of the group for which the quiver is a balanced Dynkin diagram. As discussed in [27], such a treatment cannot be extended to the O/USp monopole formula due to the nonunitary gauge groups involved. Thus, while it can be helpful to include fugacities f≡(f1,…,fr) during the calculation of Coulomb branches, their interpretation is unclear. Such issues do not affect the validity of the unrefined Hilbert series ultimately obtained by setting ∀fi:fi→1.In order for a Coulomb branch Hilbert series not to lead to divergences when the fugacities f are set to unity, it is necessary that no sublattice of the monopole lattice (other than the origin) should have a conformal dimension of zero (to ensure that the fugacities f only appear as generators when coupled with tk, where k>0). A necessary (albeit not always sufficient) condition on O/USp quivers can be formulated by examining unit shifts away from the origin of the monopole lattice. This is similar to the “good or ugly, but not bad” balance condition on unitary quivers [10].In Table 15 we examine the unit conformal dimensions that result, based on Tables 13 and 14, from setting a single monopole charge (o1, or s1) on a central gauge node in a chain of three nodes to unity, depending on the ranks of the nodes involved. We can use this table to check that no gauge node in a quiver is necessarily “bad”. For example, the central gauge node in the chain D2−C1−D1 is assigned a unit conformal dimension of 1 and is a “good” node. Quivers with zero conformal dimension are identified as such in Figs. 5 through 10. Their Hilbert series clearly do not match those of the Higgs branch constructions for Slodowy slices, and are not tabulated here.Providing (i) a nilpotent orbit Oρ is special (so that the Barbasch–Vogan map can be uniquely applied), and (ii) that the quiver LBC/CD/DC(σ(ρ)) does not suffer from zero conformal dimension, the O/USp monopole formula (4.5) can be used to calculate unrefined Hilbert series for Slodowy slices; these match those calculated on the Higgs branch of BB/C/D(Nf(ρ)) using (3.7).4.4Hilbert seriesThe Hilbert series of the Slodowy slices of algebras B1 to B4, C1 to C4 and D2 to D4, calculated as above, are summarised in Tables 16, 17, 18, 19 and 20. The refined Hilbert series are based on the Higgs branches of the balanced quivers BB/C/D(Nf(ρ)).Whenever the flavour symmetry groups are from the B or the D series, a choice has to be made between the characters of SO(N) or O(N)−. In the tables, B/D flavour nodes have been taken as SO type, with the exception of B0 where the O(1) fugacity ki=±1 has been used (with indices dropped where no ambiguity arises).2020Note that if one wishes to read the generators of the chiral ring from the quiver as described in Section 4.5.2, then all fugacities ki need to be set to 1.The Hilbert series are presented in terms of their generators, or PL[HS], using character notation [n1,…,nr]G to label irreps. Symmetrisation of these generators using the PE recovers the refined Hilbert series. The underlying adjoint maps (2.10) can readily be recovered from the generators by inverting (2.11). The HS can be unrefined by replacing irreps of the global symmetry groups by their dimensions.Many observations can be made about these Hilbert series.1.As expected, (i) the Slodowy slice to the trivial nilpotent orbit SN,(1N) has the same Hilbert series as the nilpotent cone, (ii) the slice to the subregular orbit has the Hilbert series of a Kleinian singularity of type Aˆ2r−1 for the B series, Dˆr+1 for the C series, and Dˆr for the D series, and (iii) the slice to the maximal nilpotent orbit is trivial.2.The Slodowy slices SN,ρ are all complete intersections, giving a good answer to the question posed in [37].3.The adjoint maps can contain singlet generators at even powers of t up to (twice) the degree of the highest Casimir of G; these generators may be cancelled by one or more Casimir relations.4.The global symmetry groups of the Slodowy slice generators include mixed BCD Lie groups (or A series isomorphisms), as well as finite groups of type B0, and descend in rank as the dimension of the Slodowy slice reduces. Different Slodowy slices may share the same symmetry group, while having inequivalent embeddings into G.5.The subregular Slodowy slices of nonsimply laced algebras match those of specific simply laced algebras, in accordance with their Kleinian singularities, as listed in Table 1. In the case of Slodowy slices of Cn nilpotent orbits with vector partitions of type (2n−k,k), it was identified in [22] that these isomorphisms with Dn+1 extend further down the Hasse diagram: SN,C(2n−k,k)≡SN,D(2n−k+1,k+1). This occurs due to matching chains of Kraft–Procesi transitions [13] within such slices.6.We have not attempted an exhaustive analysis of Z2 factors associated with the choice of SO vs O flavour groups and the ensuing subtleties.For example, the slices SN,B[20] and SN,C[02] have the global symmetries D1⊗B0 and D1, respectively, with the B series Slodowy slice having an extra B0 fugacity (k=±1), notwithstanding the isomorphism between the B and C Lie algebras.Similarly, in the case of D4, the spinor pair slices, respectively SN,D[0020]/SN,D[0002] or SN,D[0220]/SN,D[0202], only carry a C1 or 2 series symmetry, while the corresponding vector slices of the same dimension, SN,D[2000] or SN,D[2200], carry a B0⊗B1 or 2 symmetry.Whilst Higgs branch constructions based on the balanced quivers of type BB/C/D(Nf(ρ)) are available for all Slodowy slices, Coulomb branch constructions based on LBC/CD/DC quivers or Higgs branch constructions based on the quivers of type DG(Nf) are not generally available:1.In the cases calculated, the slice to a subregular nilpotent orbit always has a Coulomb branch construction.2.Many BCD Slodowy slices do not have Coulomb branch constructions as LBC/CD/DC quivers, either because their underlying nilpotent orbits are not special, or due to zero conformal dimension problems under the O/USp monopole formula. While the issue of zero conformal dimension (Δ=0) is less prevalent for low dimension Slodowy slices, the problem is inherent in maximal Br−Cr−Br−1 subchains, and so affects many C series Slodowy slices; certain other quivers are also problematic.3.Other than A series isomorphisms, the quivers of type DG(Nf) only provide Higgs branch constructions for D series Slodowy slices of low dimension. The nilpotent orbits underlying these Slodowy slices are dual, under the Barbasch–Vogan map, to (minimal or neartominimal) nilpotent orbits of Characteristic height 2, for which Coulomb branch constructions using the unitary monopole formula are known [2], plus some others, such as SN,D[0200]. These Dynkin diagram quivers have S(U⊗…U) flavour nodes and their refined Hilbert series may not replicate all the possible combinations of orthogonal group characters.These matters are discussed further in the concluding section.4.5Matrix generators for orthosymplectic quiversIn the case of BCD series, prescriptions are similarly available for obtaining the generators of the chiral ring corresponding to a Slodowy slice directly from the partition data or from the Higgs branch quiver.4.5.1Vector decompositionFrom (4.1) and the alternating nature of the quiver, it follows that the character of the vector representation of G decomposes into vector representations of an O/USp product group, tensored with the SU(2) embedding:(4.6)ρ:χvectorO(N)→⊕[n]bosonic[n]ρχvectorO(Nfn+1)⊕[n]fermionic[n]ρχvectorUSp(Nfn+1),ρ:χvectorUSp(N)→⊕[n]bosonic[n]ρχvectorUSp(Nfn+1)⊕[n]fermionic[n]ρχvectorO(Nfn+1), where [n]ρ are bosonic (odd dimension) or fermionic (even dimension) irreps of the SU(2) associated with the nilpotent orbit embedding ρ. The requirement that the partition ρ obeys the BCD selection rules ensures that the USp irreps are all of even dimension. Once this decomposition has been identified, the mapping of the adjoint of G into matrix generators (2.8) follows, either by symmetrising the USp vector, or by antisymmetrising the O vector. This can be checked against the adjoint partition ρ:χadjointG. Note that a choice can be made whether to use the SO form of orthogonal group characters or the O− form.4.5.2Generators from quiver pathsFor orthosymplectic quivers, the method in section 3.5.2 can be applied, with a few changes. An operator Pij(a) formed from a path in the quiver is defined identically. However, for orthosymplectic quivers, Pij(a)=Pji(a)T, and a path yields only one generator when i≠j. Other differences follow from the irreducible representations of the operators Pij(a) and the gauge group invariants. There are two cases:1.i≠j. The operator transforms in the defining representation of the initial flavour group and the defining representation of the final flavour group. For example, if the flavour node at position i is O(7) and the flavour node at position j is USp(4), Pij(a) transforms in the irrep of dimension 7×4.2.i=j. The operator has two indices that transform under the flavour group at position i. They are symmetrized if the gauge node at the mid point of the path is of Otype, or antisymmetrized if the gauge node is of USptype. The set of operators Pij(a) gives us all the generators of the chiral ring. The relations are inherited from those of the nilpotent cone N, and for SN,ρ are always the Casimir invariants of G.Now, an O(Nfi) flavour node (of rank >0) always contributes (at least) a path Pii(1) of length 2 that starts at O(Nfi), goes to the gauge node USp(Ni) and comes back to O(Nfi). Since the gauge node in the middle of the path is USp, the operator transforms in the second antisymmetrization Λ2[fund.]O=[adjoint]O. Similarly, a USp(Nfi) flavour node always contributes (at least) a path Pii(1) of length 2 that starts at USp(Nfi), goes to the gauge node O(Ni) and comes back to USp(Nfi). Since the gauge node in the middle of the path is O, the operator transforms in the second symmetrization Sym2[fund.]USp=[adjoint]USp. Consequently, the adjoint of every flavour group appears as a generator at path length 2.ExampleConsider the balanced quiver based on the partition (22,14), whose Higgs branch is the Slodowy slice SN,(4,2) to the nilpotent orbit D[0100]:(4.7) The decomposition of G to SU(2)⊗F is:(4.8)SO(8)→SU(2)ρ⊗O(4)⊗USp(2). The Hilbert series of the chiral ring of operators in the Higgs branch has generators Pij(a) given by the quiver paths in Table 21. For D4 the Casimirs give relations, −t4−2t8−t12, therefore, the PL[HS] read directly from the quiver is:(4.9)PL[gHSHiggs[BD(Nf(22,14))]]=[2,0]t2+[0,2]t2+[2]t2+[1,1][1]t3−2t8−t12.4.5.3Matrices and relationsFinally, in Tables 22–24 we provide a set of algebraic varieties described by matrices such that their HS have been computed to be identical to those of the corresponding Slodowy slices SN,ρ of B1 to B3 nilpotent orbits. The analysis can in principle be continued to higher rank.5Discussion and conclusionsHiggs branch We have presented constructions for quivers whose Higgs branches yield Hilbert series corresponding to the Slodowy slices of the nilpotent orbits of A1 to A5 plus BCD algebras up to rank 4. There are essentially two families of quivers, the balanced unitary type {BA=DA,DD} and the canonically balanced orthosymplectic type {BB/C/D}. The balanced unitary quivers have gauge nodes in the pattern of the parent algebra Dynkin diagram and yield constructions for Slodowy slices of simply laced algebras, including all A series slices and D series slices of low dimension. The orthosymplectic quivers yield constructions of all BCD Slodowy slices.The global symmetry F of a Slodowy slice descends from that of the parent group G (in the case of the slice to the trivial nilpotent orbit), via subgroups of G, down to Z2 symmetries (for the slices of some near maximal nilpotent orbits). The grading of the Hilbert series is such that (i) the sets of Slodowy slices and nilpotent orbits intersect at the nilpotent cone and at the origin and (ii) the subregular slices match the known singularities [3,4,15]. In between, we have shown how the Slodowy slice symmetry groups and mappings of G representations to SU(2)⊗F follow, via the Higgs branch formula, from the SU(2) homomorphisms into G of the associated nilpotent orbits.We anticipate that these results generalise to Classical groups of arbitrary rank.Coulomb branch As is known, in the case of the A series, the existence of a bijection between partitions and their transposes (the Lusztig–Spaltenstein map) leads to a complete set of Coulomb branch constructions for Slodowy slices; these yield the same set of Hilbert series as the Higgs branch constructions. The Coulomb branch constructions are based on applying the unitary monopole formula to linear quivers LA, which are not generally balanced.In the case of the BCD series, however, other than for accidental isomorphisms with the A series, this study has clarified that (i) the existence of suitable linear orthosymplectic quivers {LBC,LCD,LDC} is limited to the Slodowy slices of special nilpotent orbits, (ii) within these, the applicability of the Coulomb branch orthosymplectic monopole formula is restricted to those quivers that have positive conformal dimension, and (iii) the resulting Hilbert series are only available in unrefined form.Slodowy slice formula The refined Hilbert series of a Slodowy slice can also be obtained directly from the mapping of the adjoint representation of G into SU(2)⊗F, using (2.11). This mapping follows from the decomposition of the fundamental/vector of G→SU(2)⊗F under (3.13) or (4.6).Dualities and 3d mirror symmetry The A series findings verify the known 3d mirror symmetry relations (3.4) and (3.5). Under these, linear or balanced quivers based on partitions ρ can be used either for Higgs branch or Coulomb branch constructions; one combination yields a Slodowy slice and the other combination yields a (generally different) dual nilpotent orbit under the Lusztig–Spaltenstein map ρT, as illustrated in Fig. 11.The analysis of BCD series quivers shows, however, that such a picture of dualities [10] does not extend to the BCD series, other than in a limited way, due to the various restrictions on Coulomb branch constructions, discussed above. The refined (i.e. faithful) HS relationships for nilpotent orbits of the BCD series can be summarised:(5.1)SN,ρ=Higgs[BB/C/D(Nf(ρ))],O¯ρ=Higgs[LB/C/D(ρT)], and, for D series Dynkin type quivers of Characteristic height 2:(5.2)O¯ρ=Coulomb[DD([ρ])],SN,dBV(ρ)=Higgs[DD([ρ])], where dBV(ρ) is the dual partition to ρ under the D series Barbasch–Vogan map.If we restrict ourselves (i) to special nilpotent orbits, (ii) to quivers with positive conformal dimension, and (iii) to unrefined Hilbert series, then we can summarise the more limited 3d mirror symmetry for the BCD series as in Fig. 12.Note that even for these cases there is a further obstruction: the difference between SO and O nodes in the quiver [28,29]. For the A series, 3d mirror symmetry involves a pair of quivers for which the Coulomb branch and Higgs branch are swapped. In the BCD series however, once the gauge algebra of the quiver is specified there is still the question of whether the gauge groups are orthogonal or special orthogonal. As shown in Fig. 12 a different choice needs to be made depending on the branch of the quiver. This is not quite the same as 3d mirror symmetry.On the other hand, there is a pair of SCFTs, Tσρ(G) and Tρσ(G∨) [10,18,19], which are predicted to have precisely the two different gauge algebras depicted in one of the diagrams of Fig. 12: if Tσρ(G) corresponds to quiver LBC/CD/DC(ρT), then Tρσ(G∨) has the quiver BB/C/D(Nf(dBV(ρ))), along with the Higgs and Coulomb branches depicted in the same diagram. However, the present results, together with [1,28,29], show that this cannot be the case, since there are factors of Z2 in the gauge group of the quiver for Tσρ(G) that differ depending on the branch being computed. This is a very intriguing point that needs to be addressed in future studies, especially since it has consequences for the way effective gauge theories can be employed to understand the dynamics of Dpbranes in the presence of Opplanes.Thus, it is the Higgs branch that provides the means to conduct a refined analysis of the HS of BCD series nilpotent orbits and Slodowy slices. These represent only a subset of the BCD series moduli spaces, Sρ1,ρ2≡O¯ρ1∩Sρ2, which include nilpotent orbits Sρ,trivial and Slodowy slices SN,ρ as limiting cases.2121Such BCD series moduli spaces Sρ1,ρ2 generalise naturally to any pair of nilpotent orbits (unlike Tσρ(O/USp) theories, which are restricted to special orbits). The indications are that Higgs branch methods should provide a fruitful means of analysing such spaces.Further work Besides a study of quivers for Sρ1,ρ2 moduli spaces, it would be interesting to extend this analysis to the Slodowy slices of Exceptional groups. While Higgs branch quiver constructions are not available for nilpotent orbits of Exceptional groups, a limited number of Coulomb branch quiver constructions are known. For Slodowy slices, where the situation is somewhat reversed by dualities, some Higgs branch constructions should be available, based, for example, on Dynkin diagrams of the E series.With respect to the Coulomb branch, it would be interesting to understand (i) whether some nonlinear fugacity map can be developed for the orthosymplectic monopole formula in order to obtain refined Hilbert series, and (ii) whether a modified monopole formula can be found that avoids the zero conformal dimension problem associated with many orthosymplectic quivers. A recent advance has been made on this front in [38], where Coulomb branches of bad quivers with a single Cr gauge node have been computed. A case that also appears in our study is the quiver [D2r]−(Cr), where the expected Slodowy slices are formed in quite a surprising way.2222[38] computes that there are two most singular points in this Coulomb branch, related by a Z2 action. Crucially, at each point, an SCFT denoted TUSp(2r),2r has a Coulomb branch identical to the expected Slodowy slice (identified in [38] as the Higgs branch of the corresponding DG(Nf) quiver). It remains a challenge to develop such techniques to obtain Coulomb branch calculations for the Slodowy slices of the other quivers with Δ=0 in our tables.More generally, the family of transverse spaces and symmetry breaking associated with Slodowy slices provides a rich basis set of quivers that can be extended or used as building blocks to understand the relationships between a wide array of quiver theories and their Higgs and/or Coulomb branches.AcknowledgementsWe would like to thank Stefano Cremonesi and Benjamin Assel for helpful conversations during the development of this project. S.C. is supported by an EPSRC DTP studentship EP/M507878/1. A.H. is supported by STFC Consolidated Grant ST/J0003533/1, and EPSRC Programme Grant EP/K034456/1.Appendix ANotation and terminologyWe refer to Slodowy slices and nilpotent orbits either by their Lie algebras g, or by the Lie groups G in which they transform. While such references are relatively interchangeable for USp groups, with Lie algebras of C type, it can be important to distinguish between O and SO forms of orthogonal groups, which may share the same B or D type Lie algebra, but whose representations have different characters. We have sought to highlight those areas where this distinction is important in the text.We freely use the terminology and concepts of the Plethystics Program, including the Plethystic Exponential (“PE”), its inverse, the Plethystic Logarithm (“PL”), the Fermionic Plethystic Exponential (“PEF”) and, its inverse, the Fermionic Plethystic Logarithm (“PFL”). For our purposes:(A.1)PE[∑i=1dAi,t]≡∏i=1d1(1−Ait),PE[−∑i=1dAi,t]≡∏i=1d(1−Ait),PE[∑i=1dAi,−t]≡∏i=1d1(1+Ait),PE[−∑i=1dAi,−t]≡PEF[∑i=1dAi,t]≡∏i=1d(1+Ait), where Ai are monomials in weight or root coordinates or fugacities. The reader is referred to [24] or [26] for further detail.We present the characters of a group G either in the generic form XG(xi), or as [irrep]G, or using Dynkin labels as [n1,…,nr]G, where r is the rank of G. We often represent singlet irreps implicitly via their character 1. We typically label unimodular Cartan subalgebra (“CSA”) coordinates for weights within characters by x≡(x1…xr) and simple root coordinates by z≡(z1…zr), dropping subscripts if no ambiguities arise. The Cartan matrix Aij mediates the canonical relationship between simple root and CSA coordinates as zi=∏jxjAij and xi=∏jzjA−1ij.We label field (or Rcharge) counting variables with t, adding subscripts if necessary. Under the conventions in this paper, the fugacity t corresponds to an Rcharge of 1/2 and t2 corresponds to an Rcharge of 1. We may refer to series, such as 1+f+f2+…, by their generating functions 1/(1−f). Different types of generating function are indicated in Table 25; amongst these, the refined HS faithfully encode the group theoretic information about a moduli space.References[1]A.HananyR.KalveksQuiver theories for moduli spaces of classical group nilpotent orbitsJ. High Energy Phys.062016130arXiv:1601.04020[2]A.HananyR.KalveksQuiver theories and formulae for nilpotent orbits of exceptional algebrasJ. High Energy Phys.112017126arXiv:1709.05818[3]P.SlodowySimple Singularities and Simple Algebraic Groups1980Springer Verlag[4]E.BrieskornSingular elements of semisimple algebraic groupsActes du Congres International des MathématiciensNice, 1970vol. 21970279284[5]E.DynkinSemisimple subalgebras of semisimple Lie algebrasTrans. Am. Math. Soc.61957111[6]D.H.CollingwoodW.M.McGovernNilpotent Orbits in Semisimple Lie Algebra: An Introduction1993CRC Press[7]P.B.KronheimerInstantons and the geometry of the nilpotent varietyJ. Differ. Geom.321990473[8]H.NakajimaInstantons on ALE spaces, quiver varieties, and Kac–Moody algebrasDuke Math. J.761994365[9]D.GaiottoE.WittenSupersymmetric boundary conditions in N=4 super Yang–Mills theoryJ. Stat. Phys.1352009789arXiv:0804.2902[10]D.GaiottoE.WittenSduality of boundary conditions in N=4 super Yang–Mills theoryAdv. Theor. Math. Phys.132009721arXiv:0807.3720[11]A.HananyE.WittenType IIB superstrings, BPS monopoles, and threedimensional gauge dynamicsNucl. Phys. B4921997152arXiv:hepth/9611230[12]B.FengA.HananyMirror symmetry by O3 planesJ. High Energy Phys.112000033arXiv:hepth/0004092[13]H.KraftC.ProcesiOn the geometry of conjugacy classes in classical groupsComment. Math. Helv.571982539[14]S.CabreraA.HananyBranes and the Kraft–Procesi transitionJ. High Energy Phys.112016175arXiv:1609.07798[15]S.CabreraA.HananyBranes and the Kraft–Procesi transition: classical caseJ. High Energy Phys.042018127arXiv:1711.02378[16]S.CabreraA.HananyQuiver subtractionsJ. High Energy Phys.092018008arXiv:1803.11205[17]K.A.IntriligatorN.SeibergMirror symmetry in threedimensional gauge theoriesPhys. Lett. B3871996513arXiv:hepth/9607207[18]F.BeniniY.TachikawaD.XieMirrors of 3d Sicilian theoriesJ. High Energy Phys.092010063arXiv:1007.0992[19]O.ChacaltanaJ.DistlerY.TachikawaNilpotent orbits and codimensiontwo defects of 6d N=(2,0) theoriesInt. J. Mod. Phys. A2820131340006arXiv:1203.2930[20]D.BarbaschD.A.VoganUnipotent representations of complex semisimple groupsAnn. Math.121198541[21]N.MekareeyaJ.SongY.Tachikawa2d TQFT structure of the superconformal indices with outerautomorphism twistsJ. High Energy Phys.032013171arXiv:1212.0545[22]A.HendersonA.LicataDiagram automorphisms of quiver varietiesAdv. Math.2672014225[23]A.D.SoleV.KacD.ValeriStructure of classical (finite and affine) walgebrasJ. Eur. Math. Soc.1820161873[24]S.BenvenutiB.FengA.HananyY.H.HeCounting BPS operators in gauge theories: quivers, syzygies and plethysticsJ. High Energy Phys.07112007050arXiv:hepth/0608050[25]B.FengA.HananyY.H.HeCounting gauge invariants: the plethystic programJ. High Energy Phys.07032007090arXiv:hepth/0701063[26]A.HananyR.KalveksHighest weight generating functions for Hilbert seriesJ. High Energy Phys.102014152arXiv:1408.4690[27]S.CremonesiA.HananyN.MekareeyaA.ZaffaroniCoulomb branch Hilbert series and Hall–Littlewood polynomialsJ. High Energy Phys.092014178arXiv:1403.0585[28]S.CremonesiA.HananyN.MekareeyaA.ZaffaroniTρσ (G) theories and their Hilbert seriesJ. High Energy Phys.012015150arXiv:1410.1548[29]S.CabreraA.HananyZ.ZhongNilpotent orbits and the Coulomb branch of Tσ(G) theories: special orthogonal vs orthogonal gauge group factorsJ. High Energy Phys.112017079arXiv:1707.06941[30]S.BenvenutiA.HananyN.MekareeyaThe Hilbert series of the one instanton moduli spaceJ. High Energy Phys.10062010100arXiv:1005.3026[31]B.FuD.JuteauP.LevyE.SommersGeneric singularities of nilpotent orbit closuresAdv. Math.30520171[32]S.CremonesiA.HananyA.ZaffaroniMonopole operators and Hilbert series of Coulomb branches of 3d N=4 gauge theoriesJ. High Energy Phys.14012014005arXiv:1309.2657[33]D.R.GraysonM.E.StillmanMacaulay2, a software system for research in algebraic geometryavailable athttps://faculty.math.illinois.edu/Macaulay2/[34]P.GoddardJ.NuytsD.I.OliveGauge theories and magnetic chargeNucl. Phys. B12519771[35]P.AcharAn orderreversing duality map for conjugacy classes in Lusztig's canonical quotientArXiv Mathematics eprintsarXiv:math/02030822002[36]A.BourgetA.PiniNonconnected gauge groups and the plethystic programJ. High Energy Phys.102017033arXiv:1706.03781[37]A.HananyN.MekareeyaComplete intersection moduli spaces in N=4 gauge theories in three dimensionsJ. High Energy Phys.012012079arXiv:1110.6203[38]B.AsselS.CremonesiThe infrared fixed points of 3d N=4 USp(2N) SQCD theoriesSciPost Phys.52018015arXiv:1802.04285