PPreprint typeset in JHEP style - HYPER VERSION
UTTG-15-11TCC-017-11
Tinkertoys for the D N series Oscar Chacaltana and Jacques Distler
Theory Group andTexas Cosmology CenterDepartment of Physics,University of Texas at Austin,Austin, TX 78712, USA [email protected]@golem.ph.utexas.edu
Abstract:
We describe a procedure for classifying 4D N = 2 superconformal theories ofthe type introduced by Davide Gaiotto. Any punctured curve, C , on which the 6D (2 , C correspond to different S-duality frames for the same underlying family of4D N = 2 SCFTs. In a previous work [1], we developed such a classification for the A N − series of 6D (2 ,
0) theories. In the present paper, we extend this to the D N series. Weoutline the procedure for general D N , and construct, in detail, the classification through D . We discuss the implications for S-duality in Spin (8) and
Spin (7) gauge theory, andrecover many of the dualities conjectured by Argyres and Wittig [2]. a r X i v : . [ h e p - t h ] S e p ontents
1. Introduction 12. The D N Series 3 D ) 172.4.1 D D D D
3. The D theory 21 Sp (4) × Sp (2) and Sp (5) SCFTs 303.3.1 Sp (4) × Sp (2) SCFT 303.3.2 Sp (5) SCFT 32 Spin (8)
Gauge Theory 35 s ) + 2(8 c ) + 2(8 v ) 354.2 3(8 s ) + 2(8 c ) + 8 v s ) + 3(8 c ) 384.4 4(8 s ) + 2(8 c ) 394.5 4(8 s ) + 8 c + 8 v Spin (7)
Gauge Theory 41
6. Other Interesting Examples 42 D example: Spin (10) gauge theory 45– 1 – . Appendix: Nilpotent orbits in so (2 N ) . Introduction Gaiotto duality [3, 4, 5, 6, 7, 1, 8, 9, 10] identifies a large class of 4D N = 2 SCFTs withcompactifications of the 6D N = (2 ,
0) SCFT on a punctured Riemann surface, C . Themoduli space, M g,n , parametrizes the family of exactly-marginal deformations of the SCFT.For every pants-decomposition of C , there is an N = 2 gauge-theoretic interpetation, inwhich each cylinder represents the vector multiplets for some (simple) gauge group, andthe 3-punctured spheres represent some sort of “matter”, charged under the gauge groupsof the attached cylinders. In particular, this construction identifies the boundaries of themoduli space, M g,n , with limits in which some, or all, of the gauge couplings become weak.Different degenerations correspond to different, S-dual, realizations of the same family ofSCFTs.Classifying the theories that arise, in this way, comes down to specifying (for a given 6D(2,0) theory) the 3-punctured spheres, the gauge groups associated with the cylinders thatconnect them, and the rules for gluing these ingredients together. Arbitrarily complicated4D N = 2 SCFTs can be constructed, in “tinkertoy” fashion, by connecting together thesebasic ingredients.For a given (2,0) theory, this is a finite task. In our previous paper [1], we carried outthis program for theories that are obtained from a compactification of the (2,0) theoriesof type A N − . In so-doing, we identified a multitude of new interacting, non-LagrangianSCFTs (generalizing [11]), corresponding to compactifications of the A N − theory on cer-tain 3-punctured spheres. Their appearance, in the context of Gaiotto duality, is a vastgeneralization of the classic examples of non-Lagrangian SCFTs appearing in the S-dualdescription of more-familiar N = 2 gauge theories, discovered by Argyres and Seiberg [12].While Gaiotto’s original arguments relied on the realization of the 6D theory as thelow-energy theory of N M5-branes, which necessarily implied working with a 6D theory of A N − type, the idea can be straightforwardly generalized to the case of N M5 branes inthe presence of an orientifold, whose low-energy limit is the 6D theory of type D N . (Thereis, by contrast, no known realization of the 6D theories of type E as a low-energy theoryof M5 branes.) The class of 4D SCFTs arising from the compactification of the D N A N − analogue.As for the A N − theories, the Seiberg-Witten curve of 4D theories arising from the D N theories can be written in Gaiotto’s form, as a polynomial equation in the Seiberg-Wittendifferential (a 1-form on T ∗ C ), whose coefficients are (the pullbacks of) differentials on C .The differentials descend from protected operators of the 6D theory, and so their degreesare equal to the exponents of Spin (2 N ).Just as Gaiotto used the well-known SU ( n ) linear quivers to test his arguments forthe A N − theory, Tachikawa [8, 9] studied the SO-Sp linear quivers [13, 14] to find the polestructure and flavour symmetry group for punctures in the D N theory, and discovered afew examples of S-duality. Unfortunately, the SO-Sp linear quivers, that arise from theorientifold construction, live in a theory slightly larger than the one we are interested in.The A N − , D N and E theories have a Z outer-automorphism group (which gets enhanced– 1 –o S in the case of D ), and we can consider compactifications of the (2,0) theory, wheregoing around a homologically-nontrivial cycle on C (circumnavigating a handle, or circlinga puncture) is accompanied by an outer-automorphism twist.A proper discussion of the incorporation of outer-automorphism twists should treat the A N − , D N and E (2,0) theories in tandem, as all of these Dynkin diagrams have a Z outerautomorphism. We will leave that discussion to future work . Instead, in this paper, wewill study the compactifications of the D N theory, without outer-automorphism twists, anddevelop a classification precisely analogous to the one we developed for the A N − theory(also without outer automorphism twists). Nonetheless, at a crucial point, we will haverecourse to Tachikawa’s linear quiver tail analysis which, strictly speaking, embeds the D N theories without outer automorphism twists in the larger class of D N theories which do include outer automorphism twists.The analysis in the D N case introduces several new complications, not seen in the A N − case. In the A N − theory, each puncture corresponded to a choice of partition of N (equivalently, to an N -box Young diagram, or a nilpotent orbit in the complexified Lie al-gebra, sl ( N )). The chosen partition determined the “flavour symmetry” group (essentially,the isometry group of the Higgs branch) associated to a given puncture. At the same time,it (or, more accurately, its transpose ) determined the singular behaviour of the Hitchinsystem at the puncture which, in turn, gave the geometry of the Coulomb branch.In the present case, that relationship is more complicated. As in the A N − case, theflavour symmetry group (geometry of the Higgs branch) is determined by a “D-partition”of 2N. Such partitions also label nilpotent orbits in so (2 N ). However, only for a subsetof these, the “special” D-partitions [16], is the behaviour of the Hitchin system at thepuncture given by (the “Spaltenstein dual”) nilpotent orbit.The Coulomb branch of the theory comprises the degrees of freedom associated to aset of meromorphic k -differentials on the Riemann surface which are allowed to have polesof certain orders (determined by the choice of partition) at the punctures. A new feature,of the D N case, is that the coefficients of the leading poles of these differentials obeycertain polynomial constraints. The “true” Coulomb branch is obtained, after imposingthe constraints.These constraints were derived by Tachikawa [8], by considerations involving linearquiver tails. We will present a slightly different, more intrinsic, viewpoint on the originof these constraints. For the special partitions, we will see that the constraints pop outnaturally from requiring that the Higgs field have a simple pole with residue lying in theSpaltenstein-dual nilpotent orbit. For the non-special partitions, we will content ourselveswith determining the pole structure of the k -differentials at the puncture, and the associatedconstraints, using the linear quiver tail analysis. We refer to [17] for a full discussion of theboundary condition of the Hitchin system for non-special punctures.A further peculiar feature of the non-special punctures is that the global symmetrygroup of the puncture contains Sp ( l ) k factors, with k odd. This level for the current algebrais that which would be induced by an odd number of half-hypermultiplets in the funda- See [15] for a treatment of the Z -twisted A N − series. – 2 –ental 2 l -dimensional representation. In other words, this symmetry is subject to Witten’sglobal anomaly [18] and (in the absence of additional matter) could not be consistentlygauged.Even after having dealt with these new complexities, simply enumerating the results in the D N case is considerably more tedious than it was in the A N − case. The number offixtures (3-punctured spheres), and the number of cylinders that connect them, proliferatemuch more rapidly with N .We will restrict ourselves to presenting a complete catalogue only for D . As a measureof the complexity, there are 99 3-punctured spheres for D ; we will list all of those. Thereare 785 4-punctured spheres — theories with a single gauge group factor — it would beprohibitive to list all of those.Nevertheless, D is an interesting case to study. As we said before, the outer automor-phism group is enhanced to S . This group is a symmetry of the D (2,0) theory, and soacts on the set of punctures/fixtures/cylinders, which are naturally organized into multi-plets, permuted by the outer automorphisms. As already mentioned, we will not considerthe inclusion of outer-automorphism twists .For the D and D theories, we will present tables of the regular punctures and theirproperties, but will refrain from presenting a complete catalogue of fixtures and cylinders.As in the A N − series, we discover several new interacting SCFTs — non-Lagrangianfixed points of the renormalization group — and we realize a number of S-dualities pre-dicted by Argyres and Wittig [2]. We also provide formulæ for the conformal-anomalycentral charges a, c , and explain how to compute the flavour current-algebra charges k , forinteracting SCFTs.
2. The D N Series
Much of the construction is well-reviewed in previous works [3, 4, 8, 9, 7, 6, 19, 10], sowe will be somewhat brief, concentrating on the novelties which arise in the D N case.We consider a 6D (2,0) theory compactified on a Riemann surface C of genus g with n punctures (complex codimension-1 defect operators) located at points y i ∈ C, i = 1 , . . . , n .In the A N − case, the Seiberg-Witten curve, Σ ⊂ T ∗ C of the 4D low-energy theory isgiven by 0 = λ N + ( − N N (cid:88) k =2 λ N − k φ k ( y ) , (2.1)where λ is the Seiberg-Witten differential, and the φ k ( y ) are k -differentials on C (pulledback to T ∗ C ). The φ k are allowed to have poles of various orders at the y i .The theory possesses a set of relevant operators, whose vacuum expectation valuesparametrize the Coulomb branch of the theory. At a generic point on the Coulomb branch,the theory is infrared-free; at the origin, it is superconformal. The tangent space at theorigin of the Coulomb branch is a graded vector space,– 3 – = N (cid:77) k =2 V k . (2.2)where V k = H (cid:16) C, K k (cid:16)(cid:80) ni =1 p ( i ) k y i (cid:17)(cid:17) is the vector space of meromorphic of k -differentials, φ k , with poles of order at most p ik at the punctures y i .As we vary the gauge couplings, the graded vector spaces, V , fit together to form thefibers of a graded vector bundle over the moduli space, M g,n , of marginal-deformations.Our main guiding principle is that this vector bundle should extend to the boundary of M g,n . What naturally extends, over M g,n , are the virtual bundles whose fibers are H (cid:16) C, K k (cid:0) n (cid:88) i =1 p ( i ) k y i (cid:1)(cid:17) (cid:9) H (cid:16) C, K k (cid:0) n (cid:88) i =1 p ( i ) k y i (cid:1)(cid:17) . We will arrange for the H s to vanish, so that the virtual bundle is an honest bundle,which extends to the boundary. At the boundary, the Coulomb branch has componentsassociated to the irreducible components of C and components associated to the gaugegroups on the degenerating cylinders.For the D N series of (2 ,
0) theories, the story is superficially similar. The Seiberg-Witten curve takes the form0 = λ N + N − (cid:88) k =1 λ N − k ) φ k ( y ) + ˜ φ ( y ) (2.3)Again, the φ k and ˜ φ are meromorphic differentials on C , with poles of up to the prescribedorders at the punctures. ( ˜ φ is the Pfaffian, i.e., an N -differential.)However, there are some crucial differences between the A N − and D N theories. Whilein the A N − case, the coefficients in the Seiberg-Witten equation (2.1) were just linearfunctions of the Coulomb branch (2.2), in the D N case, the coefficients in Seiberg-Wittenequation (2.3) are, in general, polynomial expressions when expressed in terms of thenatural linear coordinates at the origin of the Coulomb branch. We see that, already,in the fact that the Seiberg-Witten equation depends quadratically on ˜ φ . But there arefurther polynomial constraints on the coefficients in the φ k , which need to be solved beforeone sees the natural linear structure.While the constraints are polynomial, they are always linear in (at least) one of thevariables. Moreover, they are of homogeneous degree in the aforementioned grading. Sothe space of solutions of the constraints is always smooth at the origin of the Coulombbranch, and hence the tangent space at the origin has the desired structure of a gradedvector space.The other complication in the D N theories is that, whereas the differentials in the D N theory have degrees 2 , , , . . . , N − N , the Coulomb branch has components inother degrees. For instance, in D , there is a component of degree 3, in addition to the“expected” components of degrees 2 , ,
6. In general, the Coulomb branch takes the form– 4 – ⊂ V where V = N − (cid:77) k =1 H (cid:32) C, K k (cid:0) n (cid:88) i =1 p ( k ) i y i (cid:1)(cid:33) ⊕ N − (cid:77) k =3 W k ⊕ H (cid:32) C, K N (cid:0) n (cid:88) i =1 ˜ p i y i (cid:1)(cid:33) Here the W k are vector spaces of degree k and E is the subvariety satifying the collectionof polynomial constraints (linear in at least one variable, and of homogeneous degree).If we denote the coefficient of l th -order pole of φ k , at one of the punctures, by c ( k ) l , theconstraints can roughly be divided into • polynomials (of homogeneous degree in both k and l ) in the c ( k ) l • polynomials (again, of appropriately homogeneous degree) involving both the c ( k ) l and a basis a ( k ) for the vector spaces, W k In the case of D , there is just W , and dim ( W ) = n o , the number of punctures, on C , corresponding to a particular special D-partition. At each such puncture, there is aconstraint c (6)4 = (cid:0) a (3) (cid:1) , which says that the coefficient of the leading singularity of φ isthe square of a gauge-invariant quantity, a (3) , of scaling dimension three. In the A N − series, punctures are labeled by partitions of N . To each such partition,[ h , h , . . . h p ], with h ≥ h ≥ · · · ≥ h p , p (cid:88) i =1 h i = N, we associated a Young diagram, whose i th column has height h i . The corresponding flavoursymmetry group is G = S (cid:32)(cid:89) h U ( n ( h ) ) (cid:33) , (2.4)where n ( h ) is the number of columns of height h . We call the partition [ h , h , . . . h p ], whichlabels the puncture, the Nahm partition for the puncture. We emphasize that when werepresent a Nahm partition by a Young diagram, we will always take its parts to be the column-heights of the Young diagram.Of course, a Young diagram with column-heights [ h , h , . . . , h p ] determines a secondpartition of N , given by the row-lengths, [ r , r , . . . , r q ]. The two partitions are said to be transposes of each other. This nomenclature is justified in [17]. – 5 –his second partition determines a nilpotent orbit [16], O [ r ,r ,...,r q ] ⊂ sl ( N ), whichdetermines the pole structure of the φ k ( y ) at the puncture. Specifically, the Higgs field, ϕ ,of the Hitchin system on C has a simple pole at the puncture, with residue X lying on thenilpotent orbit O [ r ,r ,...,r q ] [20, 4, 7], ϕ ( y ) = Xy + generic , (2.5)where y is a local coordinate on C such that the puncture is at y = 0, and we allow for ageneric element (a regular function of y ) in sl ( N ).We call the partition [ r , r , . . . , r q ], which determines the boundary condition for theHitchin system, the Hitchin partition of the puncture. When we want to represent a Hitchinpartition by a Young diagram, we will always take its parts to be the row-lengths . For apuncture in the A N − series, the Hitchin partition is simply the transpose of the Nahmpartition, and both are represented by the same Young diagram .There is a fairly simple algorithm for choosing the nilpotent representative X in termsof the Hitchin partition: • Let X be a block-diagonal matrix, where the i th block is r i × r i . • Within each block, let X be strictly upper-triangular.The characteristic equation for ϕ ,det( ϕ ( y ) − q ) = ( − q ) N + N (cid:88) k =2 q N − k φ k ( y ) , (2.6)which yields the Seiberg-Witten equation (2.1), determines also the allowed pole orders ofthe φ k . The resulting list of pole orders is easily expressed in terms of the Young diagram: • Starting with 0 in the first box, number the boxes in the first row with successivepositive integers. • When you get to the end of a row, repeat that integer as the number assigned tothe first box of the succeeding row. Continue numbering the boxes of that row withsuccessive integers. • The integers inscribed in boxes 2 , . . . , N are, respectively, the pole orders of φ , . . . , φ N .For the D N series, punctures are labeled by partitions of 2 N . However, not all partitionsare allowed. Let us remark that any partition of N corresponds to a nilpotent orbit of sl ( N ) [16], so, in particular,the Nahm partition of a puncture also corresponds to a nilpotent orbit. However, it is the nilpotent orbitassociated to the Hitchin partition that is relevant to the Hitchin system boundary condition (2.5). Also,the fact that both the Nahm and the Hitchin partitions can be represented by the same Young diagram isa peculiarity of the A N − series, and does not extend to the D N series, as we will soon see. – 6 – Even integers must occur with even multiplicity. • When all the integers in the partition are even (such a partition is called “very even”),we get two punctures. Such partitions only occur for N even. These two puncturesare exchanged by the Z outer automorphism of D N which exchanges the two spinorrepresentations. We will colour the corresponding Young diagrams red and blue, todistinguish them.Such a partition is called a “D-partition of 2 N .” So, we say that a puncture in the D N series is labeled uniquely by a Nahm
D-partition of 2 N , except in the case of a “very-even”Nahm D-partition, which corresponds to two punctures, and so requires an additional labelto distinguish them. As before, if we wish to represent a Nahm D-partition by a Youngdiagram, its parts give the column-heights of the Young diagram. For very-even Nahm D-partitions, we will colour the Young diagram in red/blue, to distinguish the two punctureslabeled by it.On the other hand, it is known [16] that a D-partition of 2 N labels nilpotent orbits in so (2 N ), except in the case of a very-even D-partition, which corresponds to two nilpotentorbits, and, again, an additional label is needed to distinguish them. So, if we wish,punctures in the D N series are labeled uniquely by nilpotent orbits in so (2 N ), which wecall Nahm nilpotent orbits in so (2 N ).From the Young diagram corresponding to the Nahm D-partition of a puncture, wereconstruct the flavour symmetry group, associated to the puncture, G = (cid:89) h odd Spin (cid:16) n ( h ) (cid:17) × (cid:89) h even Sp (cid:16) n ( h ) (cid:17) . (2.7)From this, the necessity of the the rule that n ( h ) be even, for even h , is obvious. The originof the additional rule (which arises for N even) — that “very even” D-partitions occurtwice — has a more subtle origin.For N odd, the irreducible spinor representation of D N is complex, and the right-handed spinor representation is the complex-conjugate of the left-handed one. So a “hy-permultiplet in the spinor” contains fields transforming as spinors of both chiralities.For N even, the irreducible spinor representation is real ( N = 4 l ) or pseudoreal( N = 4 l + 2), and the left- and right-handed spinor representations are inequivalent. So a“hypermultiplet in the left-handed spinor representation” is different from a “hypermulti-plet in the right-handed spinor representation.” When we discuss fixtures, we will need tokeep track of this distinction. Exchanging “red” and “blue” punctures will exchange theroles of left- and right-handed spinors.Understanding the singularities of the φ k at the puncture is somewhat more involvedthan in the A N − case.As in the A N − case, we might expect to associate a D-partition of 2 N (or, equivalently,a nilpotent orbit in so (2 N )) to the rows of the Nahm Young diagram. Unfortunately, whenthe columns of a 2 N -box Young diagram form a D-partition, the rows typically do not. Inother words, the transpose does not map D-partitions to D-partitions. Nevertheless, there– 7 –s a simple modification of the transpose map, called the “Spaltenstein map” which does map D-partitions to D-partitions.This procedure may be described as (row) “D-collapse”: • Given a Nahm Young diagram (that is, one whose column-heights form a D-partition),take the longest even row, which occurs with odd multiplicity (if the multiplicity isgreater than 1, take the last row of that length), and remove the last box. Place thebox at the end of the next available row, such that the result is a Young diagram. • Repeat the process with next longest even row, which occurs with odd multiplicity. • This process eventually terminates, and the result is a “corrected” Young diagram(which we call
Hitchin
Young diagram), whose row-lengths form a
Hitchin
D-partition.Conversely, starting with a Hitchin Young diagram (i.e., whose rows form a D-partition) ,we can define a process of column D-collapse , which yields a Nahm Young diagram (whosecolumns form a D-partition).In the A N − case, the Spaltenstein map was given by the transpose. In the D N case,the Spaltenstein map is defined as the composition of the transpose with the D-collapse.Unfortunately, unlike the transpose, the Spaltenstein map is not an involution of the setof D-partitions; in general, it is neither 1-1 nor onto. The set of partitions in the imageof the Spaltenstein map are called “special”, and the Spaltenstein map, restricted to thespecial partitions, is an involution.More formally, let s be the Spaltenstein map, and let p be a D-partition. p is called“special” if s ( p ) = p . In the A N − case, all partitions were special (( p t ) t = p ). That isnot the case for D N . Instead, we have the theorem Theorem 1. ([16] Corollary 6.36 and Proposition 6.3.7)1. For any D-partition, p , s ( p ) is a special D-partition.2. A D-partition, p , is special, if and only if p t is a C-partition. (A C-partition of N is a partition with the property that odd integers occur with even multiplicity.) The Hitchin system boundary conditions for punctures labeled by special
D-partitionsare determined as in the A N − case. Let f be the Nahm D-partition, and let o = s ( f )be the Hitchin nilpotent orbit, that is, the image of f under the Spaltenstein map. If f is special (which was always the case for A N − ), then the Higgs field ϕ ( y ) has a simplepole, with residue X ∈ o , exactly as in (2.5), except that the generic element now lives in so (2 N ). Under the obvious embedding so (2 N ) (cid:44) → sl (2 N ), the characteristic equationdet( ϕ ( y ) − q ) = q N + N − (cid:88) k =1 q N − k ) φ k ( y ) + ( ˜ φ ( y )) , (2.8)– 8 –hich reproduces the Seiberg-Witten curve (2.3), yields the pole orders of the k -differentials .These can be read off from the Hitchin Young diagram, just as if it were a Young diagramfor A N − . (See the rule above.) Because ϕ ( y ) lies in the so (2 N ) subalgebra, the φ k vanishfor odd k , and φ N ( y ) = ( ˜ φ ( y )) . That, however, does not quite exhaust the constraintson the polar parts of the k -differentials, which follow from restricting to so (2 N ) ⊂ sl (2 N ).There are additional polynomial constraints among the coefficients of the leading-orderpoles of the various k -differentials.These additional constraints were previously found by Tachikawa [8] by applying therestrictions, imposed by M-theory orientifolds [21], to SO-Sp linear quiver tails. As alreadymentioned, the SO-Sp quivers naturally live in the larger theory, with outer-automorphismtwists. From our present perspective it is better to think of the constraints as comingdirectly from putting the polar part of ϕ ( y ) in a special nilpotent orbit of so (2 N ). (Forour explicit conventions on nilpotent orbits in so (2 N ), see § A.)As a simple example, consider the minimal D puncture, which has special NahmYoung diagram . To find its pole structure, we put the polar part of the Higgs field inthe nilpotent orbit of the Spaltenstein dual, corresponding to the Hitchin Young diagramWe write ϕ ( y ) = Xy + M , where X = X − , is the canonical nilpotent element in this orbit(see § A for our conventions), and M is a generic matrix in so (2 N ), of the form (A.1). Thedifferentials are thus of the form φ = 2 ay + . . . , φ = a y + . . . , ˜ φ = by + . . . (2.9)Hence, the pole structure is { ,
2; 1 } , with a constraint c (4)2 = (cid:16) c (2)1 (cid:17) . This pole structureand constraint were computed in [8] from the SO-Sp linear quiver tail for this puncture.That takes care of the special punctures, that is, those labeled by a special Nahm D-partition. What about non-special punctures, i.e., the ones labeled by non-special NahmD-partitions? Here the situation is a bit more awkward. The Spaltenstein map is not aninvolution. When applied to a non-special partition, the image is a special partition, andthere are several Nahm partitions that map to the same (special) Hitchin D-partition. Tofaithfully preserve the information of the original Nahm partition, one needs to supplement Since it will be important for us to keep track of the sign of ˜ φ ( y ), it is best to compute it separately.In the antisymmetric basis of so (2 N ), we have˜ φ ( y ) = 12 N N ! (cid:88) π ∈ S N sgn( π ) N (cid:89) i =1 ( ϕ ( y )) π (2 i − ,π (2 i ) , where S n are all permutations of { , . . . , N } . This puncture is “minimal” in the sense that its Spaltenstein dual is the smallest non-trivial (Hitchin)nilpotent orbit. This nomenclature agrees with that of the existing D N -series literature (e.g., [8, 9]). – 9 –he Spaltenstein map by some additional discrete data. We leave the details of this problemto [17]. The effect on the pole structure of the k -differentials, however, is easy to find (say,from the linear quiver tail analysis), and amounts to the following. Given a non-specialNahm D-partition, f , f s = s ( f ) is a special Nahm D-partition. The pole structure ofthe φ k ( y ) for the non-special puncture f is precisely that one would find for the specialpuncture f s . However, f s has a series of constraints of the form c (2 k )2 l = (cid:0) a ( k ) (cid:1) on theleading pole coefficients. For the non-special puncture, f , some (or all) of these constraintsare relaxed .To see which constraint(s) are relaxed, notice that the Nahm Young diagram for f s can be obtained from that for f by a process of (row) C-collapse. That is, we removethe last box from a row of odd length (which occurred with odd multiplicity) and place itlower-down on the Young diagram. The box we removed was an odd-numbered box (callit 2 k + 1). By removing it, an even-numbered box (box 2 k ) becomes the last box in thatrow. The puncture, f s , had a constraint of the form c (2 k )2 l = (cid:0) a ( k ) (cid:1) . For each (2 k ) th box,thus exposed, we relax the corresponding constraint of f s .For instance, for D , there is just one non-special puncture and, correspondingly, justone constraint that gets relaxed. Plenty of other examples can be seen in the tables ofSec. 2.4.Finally, let us elaborate on our conventions for “very even” punctures. When N iseven, the Pfaffian, ˜ φ has the same degree as φ N . The outer-automorphism of D N , whichexchanges the roles of the two spinor representations, takes˜ φ (cid:55)→ − ˜ φφ k (cid:55)→ φ k , k = 1 , . . . , N − unique Coulomb branch param-eter (the coefficient c (2 k ) l of the highest-order pole of one of the φ k ) which appears linearly.We can then take c (2 k ) l to be the variable eliminated by the constraint, so for the purposeof counting the graded dimension of the Coulomb branch, it is as if we simply reduced theallowed pole-order, p k , for φ k by 1.Certain red/blue punctures are an exception. At these punctures, both ˜ φ and φ N areallowed to have poles of some order (say, l ), but a linear combination of the coefficients, c ( N ) l ± c l , is the variable that appears linearly in the associated constraints, which are ofthe form c ( N ) l ± c l = . . . , (2.11)where the ellipsis stands for additional terms. The signs above may correspond to red orblue, depending on the case. At any rate, because of (2.10), the full sets of constraints forred and blue punctures with the same Nahm D-partition are related by ˜ c l → − ˜ c l .As an example, let us look at the punctures with Nahm Young diagrams and ,which are the same as their Hitchin Young diagrams . The canonical nilpotent elements We refrain from arguing whether a Nahm red/blue D-partition should map to a Hitchin red or – 10 –see § A) are X ( r ) = X − , + X − , and X ( b ) = X − , + X +3 , , respectively. After writing ϕ ( y ) = X (r/b) y + M for the Higgs field, with M a generic so (2 N ) matrix, we find for thedifferentials, φ = 2 ay + . . .φ = a ∓ by + . . .φ = ∓ aby + . . . ˜ φ = by + . . . (2.12)with the top sign for the red and the lower sign for the blue puncture. So the pole structurefor these punctures is { , ,
3; 2 } , with constraints c (4)2 ± c = ( c (2)1 ) and c (6)3 = ∓ ˜ c c (2)1 .The Z outer automorphism acts as b (cid:55)→ − b , and it exchanges the red and blue constraints.In the presence of red/blue punctures with constraints of the form (2.11), a little extracare must be taken in computing the graded Coulomb branch dimensions. Too large anexcess, of one or the other, over-constrains the differentials and would lead to a differencebetween the virtual and actual dimension of the Coulomb branch. The dimension of thedegree- N component is dim( V N ) = d N + ˜ d − n r − n b , (2.13)where d N and ˜ d are the dimensions we would obtain from applying Riemann-Roch (suitably-adjusted for the other constraints) to φ N and ˜ φ , and n r,b are the number of constraintsof the form c ( N ) ± c l = . . . for red and blue punctures, respectively. In order that theconstraints not be over-determined, it suffices to ensure that either d N − n r ≥ , ˜ d − n b ≥ d N − n b ≥ , ˜ d − n r ≥ V N ) ≥
0, but is slightly stronger.For instance, there is no 3-punctured sphere with three punctures. The con-straints would overconstrain (imply a negative virtual dimension for) the space of sectionsof the differential φ (4) + 2 ˜ φ . In addition to regular punctures, we will, again, need to introduce a class of “irregular”punctures, which admit higher-order poles. Ignoring, for the moment, the question of blue D-partition. While [16] contends that it is natural to define the D k Spaltenstein map to take { red , blue } (cid:55)→ { red , blue } for k even, and { red , blue } (cid:55)→ { blue , red } for k odd, it is possible that thephysical map Nahm → Hitchin be the Spaltenstein map composed with the Z -action that exchanges redand blue. So, in this paper, we simply assume that a Nahm red (blue) puncture maps to a Hitchin red(blue) puncture. – 11 –onstraints, the class of irregular punctures is the one we introduced in [1] for the A N − series . • Each irregular puncture is associated to a simple subgroup G ⊂ Spin (2 N ). • From the pole structure { p k } , of the irregular puncture, we construct the “conjugatepole structure,” { p (cid:48) k } – p (cid:48) k = p k = k − k is an exponent of G . – p (cid:48) k + p k = 2 k − • We demand that the conjugate pole structure be that of a regular puncture, andwe denote the irregular puncture, thus constructed, by the Young diagram of theconjugate regular puncture, with one or more “ ∗ ”s appended.Incorporating the constraints simply amounts to “correcting” which values of k correspondto exponents of G .For example, the D puncture, * , has as its conjugate puncture the maxi-mal puncture, . Its pole structure, { , ,
5; 4 } , allows for a quartic, rather thanmerely a cubic pole for ˜ φ . Thus, the corresponding symmetry group is a Spin (7) subgroupof
Spin (8). There are three inequivalent embeddings of
Spin (7) (cid:44) → Spin (8) (dependingon which eight-dimensional representation decomposes as the 7 + 1). Thus, we also have * and * , which are exchanged by the usual Z outer automor-phism. These latter have pole structure { , ,
5; 4 } , and impose, respectively, a constraint c (4)4 ∓ c = 0. This constraint is consequence of using φ (4) , ˜ φ as our basis of 4-differentials(rather than the linear combination that appears more naturally at a red/blue puncture).Similarly, the puncture ** corresponds to an SU (4) subgroup of Spin (8),and has poles { , ,
6; 4 } . There are again blue and red versions of this puncture corre-sponding to the other two embeddings of SU (4) related by triality to the green one. Theexponent 3 in SU (4) (as opposed to 6) means that we need a constraint c (6)6 = − ( a (3) ) that appropriately corrects the dimensions of the Coulomb branch. In a free-field fixture,e.g., ** Our use of the term “irregular puncture”, in this paper and in [1], differs from the conventional one ofthe Hitchin system literature (e.g., [4]). – 12 –he constraint c (6)6 = − ( a (3) ) from ** offsets the constraint c (6)6 = ( a (3) ) from , so the virtual dimension of the Coulomb branch is indeed equal to its actualdimension (zero).The red and blue versions of this puncture, ** and ** , havepoles { , ,
6; 4 } , and have the same constraint as the green one, c (6)6 = − ( a (3) ) , plus anadditional constraint c (4)4 ∓ c = 0 as usual.Finally, we can assign a level, k , to the G symmetry of the irregular puncture. It issimply defined such that the G gauge group on the cylinder, p G ←−−−→ p (cid:48) between p andits conjugate regular puncture p (cid:48) , is conformal. The conformal-anomaly coefficients, a and c , defined via the trace anomaly in a curvedbackground [22], T µµ = c π (Weyl) − a π (Euler) , (2.16)are useful invariants, characterizing 4D conformal field theories. Along with the flavourcurrent-algebra central charges [12], k i , they are among the few readily computable invari-ants of interacting SCFTs. For the N = 2 SCFTs, under discussion, these invariants areconstant [23] over the whole family of SCFTs parametrized by M g,n .The central charge, k , for each simple factor in the flavour symmetry group associatedto a regular puncture can be computed directly from the Nahm Young diagram. Denotethe length of the i th row by r i . In the A N − case, the flavour symmetry group was givenby (2.4) and each SU ( r i − r i +1 ) factor had level k = 2 i (cid:88) j =1 r j (2.17)For the D N case, the flavour symmetry group is given by (2.7), and • For i odd, this gives a Spin ( r i − r i +1 ) k factor in the flavour symmetry group, where k = (cid:16)(cid:80) ij =1 r j (cid:17) − r i − r i +1 ≥ (cid:16)(cid:80) ij =1 r j (cid:17) − r i − r i +1 = 3 (2.18a) • For i even, this gives an Sp (cid:16) r i − r i +1 (cid:17) k in the flavour symmetry group, where k = i (cid:88) j =1 r j (2.18b)From Theorem 1, a non-special puncture corresponds to a 2 N -box Nahm Young dia-gram, whose columns form a D-partition, with at least one (in fact, at least two) odd-length– 13 –ow(s) which appears with odd multiplicity. With a little more work, one can show that atleast one of these rows is an even-numbered row. By (2.18b), this gives an Sp ( l ) k factor, inthe flavour symmetry group, with k odd. As mentioned in the introduction, this poses anobstruction to gauging: without additional matter to cancel the anomaly, the Sp ( l ) gaugetheory would suffer from Witten’s global anomaly [18].The trace anomaly coefficients, a and c , of the SCFT, can be computed (as we did [1],for the A N − series) from two auxiliary quantities: the effective number of hypermultiplets, n h , and the effective number of vector multiplets, n v , a = n v + n h c = n v + n h . (2.19)In [1] we gave formulæ to compute n h and n v for regular and irregular punctures in the A N − series. As before, n h and n v are the actual number of hypermultiplets and vectormultiplets in a Lagrangian
S-duality frame of the theory, provided such frame exists. As aconsequence, the n h of a free-field fixture (for which n v = 0) is equal to the number of freehypermultiplets in this fixture.To compute n v for a D N theory on a curve of genus g , one should first calculate thegraded dimensions of the Coulomb branch. Then n v = (cid:88) k (2 k − d k = N − (cid:88) k =1 (4 k − d k + [ N −
12 ] (cid:88) k =1 (4 k + 1) d k +1 . (2.20)For example, in the D theory, the possible non-zero Coulomb branch dimensions are d , d , d , d , while in the D theory, they are d , d , d , d , d , d . The odd-degree compo-nents of the Coulomb branch of the D N theory appear only up to degree 2[ N − ] + 1. Wewill discuss below how to compute the d k and d k +1 , but we will treat the case of d N separately, since it involves the pole orders of the Pfaffian ˜ φ .As we saw before, the even-degree sectors of the Coulomb branch, with dimensions d k (2 k (cid:54) = N ), arise from 2 k -differentials, and so d k = (1 − k )(1 − g ) + (cid:88) α ( p α k − s α k + t α k ) (2.21)where α runs over the punctures on the curve, p α k is the pole order of φ k at the α th puncture, s α k is the number of constraints of homogeneous degree 2 k (i.e., polynomial con-straints of the form c (2 k ) l = . . . ), and t α k is the number of a (2 k ) parameters (i.e., parametersarising from constraints of the form c (4 k ) l = ( a (2 k ) ) ) that the α th puncture contributes.On the other hand, since there are no φ k +1 differentials (except for the Pfaffian,when N is odd), these odd-degree sectors of the Coulomb branch receive contributions only from the a (2 k +1) parameters (i.e., parameters arising from constraints of the form c (4 k +2) l = ( a (2 k +1) ) ). We write – 14 – k +1 = (cid:88) α t α k +1 , (2.22)Notice that this expression is independent of the genus (in contrast to the contributions,to the d k , from the Riemann-Roch Theorem).As for d N , if N is even, then d N gets a contribution from both φ N and from the Pfaffian˜ φ . The formula for d N is almost the same as for the d k case, d N = 2(1 − N )(1 − g ) + (cid:88) α ( p αN − s αN ) + ˜ p α . (2.23)Notice that there is no t αN term, since we do not have a 2 N -differential.Similarly, if N is odd, only the Pfaffian (the unique odd-degree differential) contributesto d N , and so, d N = (1 − N )(1 − g ) + (cid:88) α ˜ p α . (2.24)Adding up the global, genus-dependent contribution from the 2 k -differentials and the Pfaf-fian, we obtain n v = − (1 − g ) N (16 N − N + 11) + (cid:88) α δn ( α ) v , (2.25)where α runs over the punctures on the curve, and the contribution δn ( α ) v of the α th puncture to n v is δn ( α ) v = N − (cid:88) k =1 (4 k − p α k − s α k + t α k ) + [ N −
12 ] (cid:88) k =1 (4 k + 1) t α k +1 + (2 N − p α (2.26)Let us see a few examples of how to compute δn v . First, consider the maximal D puncture,which has poles { ,
3; 2 } , and no constraints. One gets δn v = 3(1) + 7(3) + 5(2) = 34 . (2.27)Next, consider the D puncture, . The poles are { , ,
4; 3 } and there is one constraint( c (4)3 + 2˜ c = 0), so s = 1. We then have δn v = 3(1) + 7(3 −
1) + 11(4) + 7(3) = 82 . (2.28)Now consider the D puncture . The poles are { , ,
4; 2 } and there is one constraint( c (6)4 = (cid:0) a (3) (cid:1) ), so s = 1 and t = 1. Thus, δn v = 3(1) + 7(2) + 11(4 −
1) + 7(2) + 5(1) = 69 . (2.29)– 15 –ow look at the non-special D puncture . Its poles are { , ,
4; 2 } , and it has noconstraints. This means that δn v = 3(1) + 7(2) + 11(4) + 7(2) = 75 . (2.30)Finally, let us look at the D puncture (2.31)which has poles { , , ,
5; 3 } . The two constraints ( c (6)4 = ( a (3) ) and c (8)5 = 2 a (3) ˜ c ) implythat t = 1, t = 1, and s = 1. Hence, δn v = 3(1) + 7(2) + 11(4 −
1) + 15(5 −
1) + 9(3) + 5(1) = 142 . (2.32)Let us now go on to discuss n h . Just like n v , n h is a sum of a global piece and contributionsfrom each puncture, n h = − (1 − g ) N ( N − N −
1) + (cid:88) α δn ( α ) h (2.33)where α runs over the punctures, and δn ( α ) h = δn ( α ) v + f ( α ) (2.34)is the contribution of the α th puncture to n h . We will see below how to compute f ( α ) forregular and irregular punctures.For a regular puncture, f ( α ) can be found from the row-lengths r ≥ r ≥ . . . of theNahm Young diagram, f (reg) = 14 (cid:88) r i − (cid:88) r odd , (2.35)where the first sum is over all rows, and the second is restricted to odd-numbered rows( r , r , r , r , . . . ).For example, the D puncture, , has f = [4 + 3 + 1 ] − [4 + 1] = 4. Sincewe previously computed n v = 75 for this puncture, we have n h = 79.The f (irreg) for an irregular puncture, p , follows from consistency with degeneration, f (irreg) = − N + dim G − f (reg) , (2.36)where f (reg) is the contribution of the regular puncture, p (cid:48) , conjugate to p . G is the flavoursymmetry group we ascribe to the irregular puncture, p (equivalently, the gauge group onthe cylinder p G ←−−−→ p (cid:48) ). The contribution f (reg) = δn h − δn v of a regular puncture can be computed from the associated SO-Splinear quiver tail (as done in [6] for the A N − series), and (2.35) turns out to be, essentially, the dimension[16] of the Nahm ( not Hitchin) nilpotent orbit. More intrinsically, the individual δn h and δn v , rather thantheir difference, can also be computed from the Nahm nilpotent orbit, as explained in [17]. – 16 – .4 Regular Punctures (up through D ) We list below the properties of regular punctures for D , D , D , and D . As explainedpreviously, a puncture in the D N series is labeled by a Nahm Young diagram, whosecolumn-heights are the parts of a (Nahm) D-partition. On the other hand, the Higgs fieldboundary condition for the puncture (from which one extracts the pole structure and theconstraints), is determined by a Hitchin Young diagram, whose row-lengths are the partsof a (Hitchin) D-partition.As in the A N − case, there is a trivial puncture, with Nahm D-partition [2 N − ,
1] andHitchin D-partition [1 N ] (the zero nilpotent orbit), which corresponds to a non-singularpoint on the curve C , so we exclude it from our discussion.Also, as already mentioned, for D N , we have red and blue punctures for each very-even D-partition. The constraints for red/blue punctures may differ by a sign. In everycase, the top (bottom) sign corresponds to the red (blue) Hitchin D-partition.Finally, in writing down the global symmetry groups, we find it convenient to use theisomorphisms Spin (2) (cid:39) U (1) Spin (3) (cid:39) Sp (1) (cid:39) SU (2) Spin (4) (cid:39) SU (2) Spin (5) (cid:39) Sp (2) Spin (6) (cid:39) SU (4) (2.37) D Since D (cid:39) A , the results for D were already reported in our previous paper. However,as a warm-up, it will be convenient to repeat them here, recast in the notation we will usefor the higher entries in the D N series.NahmYD HitchinYD Polestructure Constraints A NahmYD FlavourSymmetry ( δn h , δn v ) { ,
3; 2 } − SU (4) (40 , { ,
2; 2 } − SU (2) × U (1) (30 , { ,
2; 1 } − SU (2) (24 , { ,
2; 1 } c (4)2 = (cid:16) c (2)1 (cid:17) U (1) (16 , D description, the quartic differential is allowed to have a doublepole at the minimal puncture, instead of only a simple pole (as in the A description).However, the coefficient of the double pole is constrained, so that the Coulomb branch hasthe same graded dimension as before. – 17 – .4.2 D For D , the outer automorphism group is enhanced from Z to S . Hence, the pairs of punc-tures, which were related by exchanging 8 s ↔ c , are actually organized into triples, underpermutations of 8 s , c , v . We indicate this by colouring the Young diagram, correspondingto the other puncture in the triple, green.The fact that the nilpotent orbits in a triple are related by triality becomes particularlyclear if one looks at their weighted Dynkin diagrams [16]. More practical evidence comesfrom the fact that the punctures in a triple exhibit the same flavour group and ( δn h , δn v ).In this table, and in the D , D tables below, we have shaded each non-special NahmYoung diagram and the (special) Hitchin Young diagram which is its image under theSpaltenstein map.NahmYD HitchinYD Polestructure Constraints FlavourSymmetry ( δn h , δn v ) { , ,
5; 3 } −
Spin (8) (112 , { , ,
4; 3 } − SU (2) (96 , { , ,
4; 2 } − Sp (2) (88 , , , { , ,
4; 3 } c (4)3 ± c = 0 Sp (2) (88 , { , ,
4; 2 } c (6)4 = (cid:0) a (3) (cid:1) U (1) (72 , { , ,
4; 2 } − SU (2) (79 , { , ,
2; 1 } − SU (2) (48 , , , { , ,
3; 2 } c (4)2 ± c = (cid:16) c (2)1 (cid:17) c (6)3 = ∓ ˜ c c (2)1 SU (2) (48 , { , ,
2; 1 } c (4)2 = (cid:16) c (2)1 (cid:17) none (40 , D NahmYD HitchinYD Polestructure Constraints FlavourSymmetry ( δn h , δn v ) { , , ,
7; 4 } −
Spin (10) (240 , { , , ,
6; 4 } − SU (4) × SU (2) (218 , δn h , δn v ) { , , ,
6; 3 } −
Spin (7) (208 , { , , ,
6; 4 } − Sp (2) × U (1) (204 , { , , ,
6; 3 } c (8)6 = (cid:0) a (4) (cid:1) SU (2) × U (1) (184 , { , , ,
6; 3 } − SU (2) × SU (2) (193 , { , , ,
6; 3 } c (8)6 = (cid:16) c (4)3 (cid:17) SU (2) × U (1) (176 , { , , ,
5; 3 } − SU (2) (168 , { , , ,
4; 2 } − Sp (2) (152 , { , , ,
5; 3 } c (6)4 = (cid:0) a (3) (cid:1) c (8)5 = 2 a (3) ˜ c SU (2) × U (1) (146 , { , , ,
4; 2 } c (6)4 = (cid:0) a (3) (cid:1) U (1) (136 , { , , ,
4; 2 } − SU (2) (143 , { , , ,
4; 2 } c (cid:48) (4)2 ≡ c (4)2 − (cid:0) c (2)1 (cid:1) c (6)3 = c (2)1 c (cid:48) (4)2 c (8)4 = (cid:0) c (cid:48) (4)2 (cid:1) U (1) (104 , { , , ,
2; 1 } − SU (2) (80 , { , , ,
2; 1 } c (4)2 = (cid:0) c (2)1 (cid:1) none (72 , D NahmYD HitchinYD Pole structure Constraints FlavourSymmetry ( δn h , δn v ) { , , , ,
9; 5 } −
Spin (12) (440 , { , , , ,
8; 5 } −
Spin (8) × SU (2) (412 , { , , , ,
8; 4 } −
Spin (9) (400 , δn h , δn v ) { , , , ,
8; 5 } − Sp (2) × SU (2) (392 , , , { , , , ,
8; 5 } c (6)5 ± c = 0 Sp (3) (380 , { , , , ,
8; 4 } c (10)8 = ( a (5) ) SU (4) × U (1) (368 , { , , , ,
8; 4 } − Sp (2) × SU (2) (379 , { , , , ,
8; 4 } c (10)8 = ( a (5) ) SU (2) × U (1) (354 , { , , , ,
8; 4 } − Sp (2) (366 , { , , , ,
7; 4 } − SU (2) × SU (2) (344 , { , , , ,
7; 4 } c (8)6 = ( c (4)3 ) SU (2) (328 , { , , , ,
6; 3 } −
Spin (7) (328 , { , , , ,
7; 4 } c (8)6 = ( a (4) ) c (10)7 = a (4) ˜ c SU (2) × SU (2) (316 , , , { , , , ,
7; 4 } c (8)6 = ( c (4)3 ) c (10)7 = ± ˜ c c (4)3 SU (2) × SU (2) (308 , { , , , ,
6; 3 } c (8)6 = (cid:0) a (4) (cid:1) SU (2) (304 , { , , , ,
6; 3 } − SU (2) × SU (2) (313 , { , , , ,
6; 4 } − SU (2) (300 , { , , , ,
6; 3 } c (8)6 = ( c (4)3 ) SU (2) (296 , { , , , ,
6; 3 } − U (1) (288 , { , , , ,
6; 3 } c (6)4 = (cid:16) a (3) (cid:17) c (10)6 = (cid:16) a (5) (cid:17) c (8)4 = 2 a (3) a (5) U (1) (256 , { , , , ,
4; 2 } − Sp (2) (232 , { , , , ,
4; 2 } c (6)4 = (cid:0) a (3) (cid:1) U (1) (216 , δn h , δn v ) { , , , ,
4; 2 } − SU (2) (223 , , , { , , , ,
5; 3 } c (cid:48) (4)2 ≡ c (4)2 − (cid:0) c (2)1 (cid:1) c (6)3 ∓ c = c (2)1 c (cid:48) (4)2 c (8)4 = (cid:16) c (cid:48) (4)2 (cid:17) ± ˜ c c (2)1 c (10)5 = ± ˜ c c (cid:48) (4)2 SU (2) (196 , { , , , ,
4; 2 } c (cid:48) (4)2 ≡ c (4)2 − (cid:0) c (2)1 (cid:1) c (6)3 = c (2)1 c (cid:48) (4)2 c (8)4 = (cid:16) c (cid:48) (4)2 (cid:17) none (184 , { , , , ,
2; 1 } − SU (2) (120 , { , , , ,
2; 1 } c (4)2 = (cid:16) c (2)1 (cid:17) none (112 ,
3. The D theory In this section, we will develop the complete “tinkertoy” catalogue for the D theory. Theregular punctures are listed in § A N − series) the full list of irregular punctures,cylinders and fixtures by considering the degenerations of all 4-punctured spheres that are“good” (i.e., that have non-negative Coulomb branch dimensions [24, 25]). In the end, ourfixtures that include an irregular puncture are “ugly” (and typically include a number offree hypers), while those which do not are “good”. It is possible that at least some “bad”punctured Riemann surfaces possess a sensible 4D N = 2 low-energy interpretation, asstressed in [25], but we do not attempt to cover them in this paper. For irregular punctures, we show the Nahm Young diagram of their conjugate regularpuncture. The number of stars accompanying the Nahm Young diagram simply serves toenumerate the distinct irregular punctures with the same conjugate regular puncture.– 21 –ahm YD Pole structure Constraints Flavour Symmetry ( δn h , δn v ) * { , ,
5; 4 } −
Spin (7) (112 , * , * { , ,
5; 4 } c (4)4 ∓ c = 0 Spin (7) (112 , ** { , ,
6; 4 } c (6)6 = − ( a (3) ) SU (4) (112 , ** , ** { , ,
6; 4 } c (4)4 ∓ c = 0 c (6)6 = − ( a (3) ) SU (4) (112 , *** { , ,
5; 4 } − ( G ) (112 , **** { , ,
6; 4 } c (6)6 = − ( a (3) ) SU (3) (112 , * { , ,
7; 4 } − SU (2) (128 , * { , ,
7; 5 } − Sp (2) (136 , * , * { , ,
7; 5 } c (4)5 ∓ ˜ c = 0 c (4)4 ∓ ˜ c = 0 Sp (2) (136 , ** { , ,
7; 5 } − SU (2) (136 , ** , ** { , ,
7; 5 } c (4)5 ∓ ˜ c = 0 SU (2) (136 , * { , ,
7; 5 } − SU (2) (145 , D theory are Spin (8) ←−−−−−−−−−−→
Spin (7) ←−−−−−−−−−−→ * Spin (7) ←−−−−−−−−−−→ * Spin (7) ←−−−−−−−−−−→ * SU (4) ←−−−−−−−−−→ ** SU (4) ←−−−−−−−−−→ ** SU (4) ←−−−−−−−−−→ ** G ←−−−−−−−→ *** * G ←−−−−−−−→ ** G ←−−−−−−−→ ** G ←−−−−−−−→ * – 22 – SU (3) ←−−−−−−−−−→ ** * SU (3) ←−−−−−−−−−→ ** * SU (3) ←−−−−−−−−−→ ** * SU (3) ←−−−−−−−−−→ ** * SU (3) ←−−−−−−−−−→ ** * SU (3) ←−−−−−−−−−→ ** SU (3) ←−−−−−−−−−→ **** Sp (2) ←−−−−−−−−−→ * Sp (2) ←−−−−−−−−−→ * Sp (2) ←−−−−−−−−−→ * SU (2) ←−−−−−−−−−→ ** SU (2) ←−−−−−−−−−→ ** SU (2) ←−−−−−−−−−→ ** SU (2) ←−−−−−−−−−→ * SU (2) ←−−−−−−−−−→ * Note that some of the irregular punctures have level k = 0. Appropriately, these willappear, below, on “empty” fixtures, with zero hypermultiplets. Also, note that each of thecylinders, p G ←−−−→ p (cid:48) , satisfies δn h + δn h (cid:48) − N ( N − N − / δn v + δn v (cid:48) − N (16 N − N + 11) / dim ( G ) k + k (cid:48) = k critical (3.1)where k critical = 2 (cid:96) adj is the value of k which gives vanishing β -function for G . Whilethis was true (by construction) when p (cid:48) is the conjugate regular puncture to p , it is notautomatically-satisfied for cylinders between two irregular punctures. In essence, theseconditions determine which cylinders between pairs of irregular punctures are allowed. Here, we list all of the 3-punctured spheres. There are a lot of them, but fortunately, the– 23 –rofusion is partially tamed by the fact that they are organized into multiplets under theouter automorphism group.
Free-field fixtures are either empty, or contain only free matter hypermultiplets, in somerepresentation of the global symmetry group for the fixture. Below, we show the matterrepresentations only for the non-Abelian part of the global symmetry group.Fixture Numberof Hypers Representation * , * , * (2 , , * , * , * , , ,, , (1 , , u ) + (2 , , d ),where 8 u/d = 8 v , s , or 8 c depending on whether theupper/lower left-handpuncture is colouredgreen, red, or blue.24 (2 , , , v )+ (1 , , , s )+ (1 , , , c ) * , * , * (4 , * , * , * (2 , ,
8) + (1 , , ** , ** , ** , ** , ** , ** *** (2 , * (2) **** SU (2) and SU (2) , respectively) to the regular punctures on them. However, they are attached to therest of the surface by an SU (2) cylinder, which gauges an SU (2) subgroup of the globalsymmetry group of the attaching puncture. The centralizer of that SU (2) is, respectively SU (2) or SU (2) . That centralizer is what is detected by the punctures on the ostensibly“empty” fixture. Similar remarks applied to the analogous fixtures that we saw in the D and A N − cases, studied in [1]. Interacting fixtures are those that contain a non-Lagrangian SCFT (e.g., the Minahan-Nemeschansky E n theories [11]), and no accompanying free hypermultiplets.– 25 –ixture ( d , d , d , d , d ) ( a, c ) ( G global ) k Theory * , * , * (0 , , , ,
0) ( , ) ( E ) The E SCFT(0 , , , ,
1) ( , ) ( E ) The E SCFT , , (0 , , , ,
1) ( , Spin (16) × SU (2) (0 , , , ,
1) ( , ) Sp (6) , , (0 , , , ,
2) ( , ) Spin (9) × Sp (2) × SU (2) , , (0 , , , ,
2) ( , ) Spin (9) × Sp (2) , , (0 , , , ,
2) ( , ) Spin (8) × Sp (2) , , (0 , , , ,
1) ( , ) Spin (10) × Sp (2) × U (1) , , (0 , , , ,
2) ( , ) Spin (8) × Sp (2) × SU (2) , , (0 , , , ,
3) ( , ) Spin (8) × Sp (2) – 26 –ixture ( d , d , d , d , d ) ( a, c ) ( G global ) k Theory(0 , , , ,
2) ( , ) ( F ) × SU (2) (0 , , , ,
1) ( , ) ( E ) × SU (2) (0 , , , ,
2) ( , ) Spin (8) × SU (2) × SU (2) (0 , , , ,
3) ( , ) Spin (8) × SU (2) (0 , , , ,
0) ( , ) ( E ) Two copiesof the E SCFT(0 , , , ,
1) ( , ) Spin (8) × SU (2) × U (1) (0 , , , ,
2) ( , ) Spin (8) × U (1) (0 , , , ,
1) ( , ) SU (2) (0 , , , ,
2) ( , Spin (8) × SU (2) (0 , , , ,
3) ( , ) Spin (8) × SU (2) – 27 –ixture ( d , d , d , d , d ) ( a, c ) ( G global ) k Theory(0 , , , ,
4) ( , ) Spin (8) , , (0 , , , ,
1) ( , ) Sp (3) × SU (2) (0 , , , ,
1) ( , ) Sp (2) × SU (2) , , (0 , , , ,
1) (9 , Sp (2) × SU (2) (0 , , , ,
0) ( , ) SU (4) T Mixed fixtures are those that include an interacting SCFT, plus a number of free hyper-multiplets. Fixture ( d , d , d , d , d ) ( a, c ) SCFT , , , ,
0) (2 , ) ( E ) (2; 1 , , , , , , , , , , (0 , , , ,
0) ( , ) ( E ) ( d , d , d , d , d ) ( a, c ) SCFT , , (0 , , , ,
0) ( , ) ( E ) (1; 2 , ,
1; 8 v ), (1; 1 , ,
1; 8 s )or (1; 1 , ,
2; 8 c ),depending onthe colour of thegreen/red/bluepuncture(0 , , , ,
1) ( , ) Sp (5) (1; 1; 2 , , (1; 1; 1 , , (1; 1; 1 , , , , (0 , , , ,
1) ( , ) Sp (5) (1; 4; 1)+ (4; 1; 1) , , (0 , , , ,
0) ( , ) SU (2) × SU (8) , , , (1; 1 , , , or (1; 2 , , , , ,, , (0 , , , ,
1) ( , ) Sp (4) × Sp (2) (1; 1; 4) , , (0 , , , ,
1) ( , ) Sp (4) × Sp (2) (1; 1; 2 , , , (1; 1; 1 , , (1; 1; 1 , , , , (0 , , , ,
1) ( , ) Sp (2) × SU (2) (1; 1; 2 , , , (1; 1; 1 , , (1; 1; 1 , , .3 The Sp (4) × Sp (2) and Sp (5) SCFTs
A couple of SCFTs make a somewhat unusual appearance in the above list of mixed fixtures.Usually, the mixed fixtures contain SCFTs which have previously appeared elsewhere (with-out the additional hypermultiplets). Indeed, ( E ) , ( E ) and SU (2) × SU (8) SCFTs(the latter was called the “ R , theory” in [1]) have all appeared previously. In the presentcase, we find two new ones, which do not appear to arise in the absence of accompanyinghypermultiplets. Sp (4) × Sp (2) SCFT
One is the Sp (4) × Sp (2) SCFT. It has ( a, c ) = (cid:0) , (cid:1) , and graded Coulomb branchdimension ( d , d , d , d , d ) = (0 , , , , G X = Sp (4) × Sp (2) It appears in our table, accompanied by either 1 hypermultiplet (3 fixtures) or 2 hyper-multiplets (6 fixtures).Let us look a couple of examples of its appearance.Consider a
Spin (7) gauge theory, with matter in the 3(8) + 2(7) + 1.
Spin(7) * This theory has two distinct strong-coupling points. One, G * * is a G gauge theory, with matter in the 2(7) + 1, coupled to the ( E ) SCFT. Aside fromthe addition of the free hypermultiplet, this was example 10 of Argyres and Wittig [2].The other strong coupling point of this theory,– 30 –
U(2) * is an SU (2) gauge theory coupled to the Sp (4) × Sp (2) SCFT. The fixture on the rightis empty; the mixed-fixture on the left provides both the SCFT and an additional freehypermultiplet.As a second example, consider
Spin(7) * This is a
Spin (7) gauge theory, with matter in the 4(8) + (7) + (1). The S-dual theory
SU(2) * is an SU (2) gauge theory. The fixture on the right contributes a half-hypermultiplet inthe fundamental. The fixture on the left is the Sp (4) × Sp (2) SCFT plus a single free hypermultiplet. We weakly gauge an SU (2) subgroup of Sp (2) ⊂ G X . From both pointsof view, we reproduce G global = Sp (4) × SU (2) + 1 free hypermultipletA third example is provided by the S-dual of Spin (8) gauge theory with matter in the4(8 s ) + 2(8 c ). This is discussed in section § .3.2 Sp (5) SCFT
The other “new” SCFT is the Sp (5) SCFT. It has ( a, c ) = (cid:0) , (cid:1) and a Coulomb branchof graded dimension ( d , . . . , d ) = (0 , , , , Sp (5) .The Sp (5) SCFT appears twice on our list, once accompanied accompanied by 3hypermultiplets (transforming as the (1; 1; 2 , ,
1) + (1; 1; 1 , ,
1) + (1; 1; 1 , ,
2) of themanifest SU (2) × SU (2) × SU (2) associated to the punctures), and once (3 fixtures)accompanied by 4 hypermultiplets (transforming as the (1; 4; 1)+ (4; 1; 1) of the manifest Sp (2) × Sp (2) × SU (2) associated to the punctures).Let us look at some examples of the Sp (5) SCFT. Consider the 4-punctured sphere G * * Both fixtures provide 2 hypers in the 7 of G , plus 2 free hypers, so the 4-punctured sphererepresents the G theory with 4 hypers in the 7, plus 4 free hypers. G global = Sp (4) + 4 free hypersAside from the 4 free hypers, this is example 4 of Argyres-Wittig [2].The S-dual theory is SU(2) * The fixture on the left is the Sp (5) SCFT, with 4 free hypers. The fixture on the rightcontributes a half-hyper in the fundamental of SU (2). Gauging an SU (2) ⊂ Sp (5) , yieldsthe expected Sp (4) global symmetry group of the S-dual of G with 4 fundamentals.As another example, consider the 4-punctured sphere– 32 – *** Here the fixture on the left represents 3 hypers in the 7 of G plus 3 free hypers, and thefixture on the right represents 1 hyper in the 7. Notice that the G cylinder in this exampleis different from the one in the previous example.S-dualizing, we obtain SU(2) * The fixture on the left is the Sp (5) SCFT, where we gauge an SU (2) ⊂ Sp (5), accompaniedby 3 free hypers. The fixture on the right contributes 1 fundamental half-hyper.A third example, also involving G , is G * * This is G with 4 fundamentals and two free hypermultiplets.The S-dual is – 33 – U(2) * The fixture on the right is empty. The fixture on the left is, again the Sp (5) SCFT, withone hypermultiplet transforming as a half-hyper in the fundamental of SU (2) and two freehypermultiplets.For a non- G -related example, consider Spin(7) * The fixture on the left contributes hypermultiplets in the 2(7) + 1. The fixture on the rightis an 8 s + 2(8 c ), considered as a representation of Spin (8). Under the chosen embeddingof
Spin (7), the 8 s decomposes as 7 + 1, which the 8 c (and also the 8 v ) decomposes as the8. So, all-in-all, this is a Spin (7) gauge theory, with matter in the 3(7) + 2(8) + 2(1), so G global = Sp (3) × Sp (2) + 2 free hypersThe S-dual theory is Sp(2) * The fixture on the right contribute 2 hypermultiplets in the fundamental of Sp (2). Thefixture on the left is the Sp (5) SCFT, accompanied by 4 hypermultiplets, two of whichform an additional half-hypermultiplet in the fundamental of Sp (2) and two of whichare free. Altogether, there are 5 half-hypermultiplets in the fundamental, yielding the– 34 – pin (5) = Sp (2) factor in G global . Gauging the Sp (2) ⊂ Sp (5) yields the remaining Sp (3) . This is example 5 of Argyres and Wittig [2]. Spin (8)
Gauge Theory
Spin (8) gauge theory — with n s hypermultiplets in the 8 s , n c hypermultiplets in the 8 c and n v hypermultiplets in the 8 v — has vanishing β -function for n s + n c + n v = 6. Theglobal symmetry group is G global = Sp ( n s ) × Sp ( n c ) × Sp ( n v ) In the D theory, all of the cases, with n s,c,v ≤
4, are realized on the 4-punctured sphere.Up to
Spin (8) triality, this yields five different cases. We will discuss each of them, in turn,and give the strong-coupling behaviour in each case.For the cases of ( n s , n c , n v ) = (3 , ,
1) and (3 , , two distinct strong-couplinglimits. In each case, the conjecture of Argyres and Wittig corresponds to one of the twostrong-coupling limits, that we find. s ) + 2(8 c ) + 2(8 v )The dual of Spin (8), with matter in the 2(8 s ) + 2(8 c ) + 2(8 v ), is an SU (2) gauge theory,coupled to a half-hypermultiplet in the fundamental, and to the Sp (2) × SU (2) SCFT.One realization is
Spin(8)
Each fixture contributes one (8 v + 8 s + 8 c ). The S-dual theory is SU(2) * where the fixture on the right is a half-hypermultiplet in the fundamental of SU (2), andthe fixture on the left is the Sp (2) × SU (2) SCFT.– 35 –nother realization of the same theory is
Spin(8)
Here, the fixture on the left contributes 8 s + 2(8 c ), and the fixture on the right contributes8 s + 2(8 v ). The S-dual is SU(2) * The fixture on the right is empty; the fixture on the left is the Sp (2) × SU (2) SCFT plusa half-hypermultiplet in the fundamental of SU (2). s ) + 2(8 c ) + 8 v Spin (8) gauge theory, with matter in the 3(8 s )+2(8 c )+8 v , has two distinct strong-couplinglimits. One is a Spin (7) gauge theory, with matter in the 3(8), coupled to the ( E ) SCFT.The other strong coupling limit is an SU (2) gauging of the Sp (3) × SU (2) .One realization is Spin(8)
The fixture one the left contributes 2(8 s ) + 8 v , and the fixture on the right contributes8 s + 2(8 c ).One of the corresponding strong-coupling points is given by– 36 – pin(7) * The fixture on the right yields matter in 3 copies of the 8; the fixture on the left is the( E ) SCFT.The other strong coupling point is
SU(2) * The fixture on the right is empty, while the fixture on the left is the Sp (3) × SU (2) SCFT,where we gauge an SU (2) ⊂ Sp (3) .Another realization of the same theory is Spin(8)
One strong coupling point is given by
Spin(7) * – 37 –he fixture on the right contribute 2 hypermultiplets in the 8 of Spin (7). The fixture onthe left is the ( E ) SCFT plus an additional hypermultiplet in the 8.The other strong coupling point is
SU(2) ** The fixture on the right is empty; the fixture on the left is, again, the Sp (3) × SU (2) SCFT. s ) + 3(8 c ) Spin (8) gauge theory, with matter in the 3(8 s ) + 3(8 c ) also has two distinct strong couplingpoints. One is G gauge theory, coupled to two copies of the ( E ) SCFT. The other is an SU (2) gauging of the Sp (3) × SU (2) SCFT.This is realized via
Spin(8)
The fixture on the left yields 2(8 s ) + 8 c , while the figure on the right yields 8 S + 2(8 c ).One strong-coupling point is given by G * * Here, each fixture is a copy of the ( E ) SCFT.The other strong coupling point is – 38 –
U(2) * The fixture on the right is empty. The fixture on the left is the Sp (3) × SU (2) SCFTwhere, this time, we gauge the SU (2) . s ) + 2(8 c ) Spin (8), with matter in the 4(8 s ) + 2(8 c ) has, as its S-dual, an Sp (2) gauge theory, with 5half-hypermultiplets in the fundamental, coupled to the Sp (4) × Sp (2) SCFT.
Spin(8) yields a
Spin (8) gauge theory, with matter in the 4(8 s ) + 2(8 c ).The S-dual theory is Sp(2) * The fixture on the right contributes two hypermultiplets in the fundamental. The fixtureon the left is the Sp (4) × Sp (2) with an additional half-hypermultiplet in the fundamentalof Sp (2). Since there are, in total, five half-hypermultiplets in the fundamental, the flavoursymmetry associated to the matter is Spin (5) = Sp (2) ; the rest of G global comes from the Sp (4) ⊂ Sp (4) × Sp (2) . – 39 – .5 s ) + 8 c + 8 v Finally,
Spin (8) gauge theory, with matter in the 4(8 s ) + 8 c + 8 v has, as its S-dual, an Sp (2)gauge theory, with 2 hypermultiplets in the fundamental, coupled to the Sp (6) SCFT.The
Spin (8) gauge theory can be realized as
Spin(8) where the fixture on the left gives matter in the 2(8 s ) + 8 c and the fixture on the rightgives matter in the 2(8 s ) + 8 v .The S-dual is Sp(2) * The fixture on the right is 2 fundamental hypermultiplets of Sp (2), which contribute the Spin (4) = SU (2) factor to the global symmetry group. The fixture on the left is the Sp (6) SCFT.
It is straightforward to compute the Seiberg-Witten curves, associated to any of thesetheories, in the form (2.3) 0 = λ + (cid:88) k =1 λ − k φ k ( y ) + ˜ φ ( y )For instance, for Spin (8) gauge theory, with hypermultiplets in the 3(8 v ) + 3(8 s ),– 40 –mposing the constraints, at each of the punctures, yields φ ( y ) = u ( dy ) ( y − y )( y − y )( y − y )( y − y ) φ ( y ) = [ u ( y − y )( y − y ) − u ( y − y )( y − y ) + u ( y − y )( y − y ) / dy ) ( y − y ) ( y − y ) ( y − y ) ( y − y ) φ ( y ) = [ u ( y − y ) + u ˜ u ( y − y )]( dy ) ( y − y ) ( y − y ) ( y − y ) ( y − y ) ˜ φ ( y ) = ˜ u ( dy ) ( y − y ) ( y − y ) ( y − y ) ( y − y ) (4.1)Here u , u , u and ˜ u are the Coulomb branch parameters. The obvious SL (2 , C )symmetry means that the physics depends only on the cross-ratio e ( τ ) = ( y − y )( y − y )( y − y )( y − y )The e ( τ ) → Spin (8) gauge theory; e ( τ ) → ∞ is the weakly-coupled SU (2) gauge theory and e ( τ ) → G gauge theory.The other cases are equally-easy to write down. It would be interesting to comparethese results with the Seiberg-Witten curves obtained in [26, 27]. Spin (7)
Gauge Theory
Spin (7), with n hypermultiplets in the 8 and (5 − n ) in the 7, also has vanishing β -function. Perhaps with the addition of some free hypermultiplets, we can realize the cases n = 2 , , , D theory. § • One is a G gauge theory, with two hypermultiplets in the 7, coupled to the ( E ) SCFT. • The other is an SU (2) gauge theory coupled to the Sp (4) × Sp (2) SCFT. § Sp (2) gauge theory with 5 half-hypermultiplets in the 4, coupled to the Sp (5) SCFT. § SU (2) gauge theory with a half-hypermultiplet in the 2, coupledto the Sp (4) × Sp (2) SCFT. – 41 – .4 Spin(7) * Spin(7) * are Spin (7) gauge theories with matter in the 5(8). The fixture on the left contributes2(8); the fixture on the right contributes 3(8).The other degeneration,
SU(2) ** is an SU (2) gauge theory coupled to the Sp (6) SCFT (the fixture on the right is empty).
6. Other Interesting Examples
Let us take the Sp (2) × SU (2) SCFT and gauge an SU (2) subgroup (the fixture on theright is empty): – 42 – U(2) * The S-dual theory is
Spin(7) * The fixture on the right contributes hypermultiplets in the 7 + 8. The fixture on the left isthe ( E ) SCFT with matter in the 8 c of Spin (8). Under the given embedding of
Spin (7),this matter transforms as an additional 8. So the matter contributes an Sp (2) × SU (2) to the global symmetry group of the theory. The rest, Sp (2) × SU (2) , is the centralizerof Spin (7) ⊂ E .As another example of our methods, let us consider various gaugings of the Sp (2) × SU (2) SCFT. We can gauge an Sp (2) subgroup, Sp(2) * where the fixture on the right provides two hypermultiplets in the fundamental of Sp (2).The S-dual theory, – 43 – pin(8) is a Spin (8) gauge theory, with matter in the 2(8 s ), coupled to two copies of the ( E ) SCFT.Instead, we can gauge an SU (2) subgroup SU(2) ** where the fixture on the right is empty. The S-dual Spin(8) is a
Spin (8) gauge theory, with matter in the 2(8 s ) + 8 c + 8 v , coupled to one copy of the( E ) SCFT.A different SU (2) gauging of the Sp (2) × SU (2) SCFT
SU(2) * – 44 –as two distinct strong-coupling points. One, Spin(7) * is a Spin (7) gauge theory, with matter in the 8, coupled to two copies of the ( E ) SCFT.The other,
Spin(8) is a
Spin (8) gauge theory, with matter in the 2(8 s ) + 2(8 c ), coupled to a single copy of the( E ) SCFT. D example: Spin (10) gauge theory
To further illustrate our methods, let us study one example from the D theory, involvinga Spin (10) gauge theory with matter in the 3(16) + 2(10).Start with the 4-punctured sphere
Spin(10)
This is a Spin(10) Lagrangian field theory with matter in the 3(16) + 2(10) representation.The left fixture provides 32 free hypermultiplets in the (16 ,
2) of
Spin (10) × SU (2), andthe right fixture, 36 free hypermultiplets in the (16 ,
1) + (10 ,
4) of
Spin (10) × Sp (2).The global symmetry group of the theory is, thus,– 45 – global = SU (3) × Sp (2) × U (1) , This theory has two distinct strong coupling cusp points. One appears in the degeneration
SU(2) ** Here the left fixture is empty. The ** irregular puncture has pole structure { , , ,
10; 6 } ,and imposes the constraint c (8)10 = ( c (4)5 ) . The right fixture is an interacting SCFT withgraded Coulomb branch dimension d = (0 , , , , , ,
1) and global symmetry group G SCFT = Sp (2) × SU (3) × SU (2) × U (1) , and we gauge the SU (2) subgroup.The second strong coupling point appears in the remaining degeneration, G ** Here the fixture on the left is an SCFT with graded Coulomb branch dimension d =(0 , , , , , ,
1) and global symmetry group G SCFT = ( E ) × Sp (2) × U (1) , and the fixture on the right is empty. The ** irregular puncture has polestructure { , , ,
8; 5 } . Under the decomposition ( E ) k ⊃ ( G ) k × SU (3) k , we gauge a( G ) ⊂ ( E ) . Acknowledgements
The research of the authors is based upon work supported by the National Science Founda-tion under Grant No. PHY-0969020. The work of J. D. was also supported by the United– 46 –tates-Israel Binational Science Foundation under Grant D theory, we had misidentified the global symmetry group of the SCFT. We wouldlike to thank Simone Giacomelli and Yuji Tachikawa for pointing out the enhanced globalsymmetry for some of those entries, which led us to uncover the rest. We would also liketo thank Andy Trimm for checking some of those calculations.– 47 – . Appendix: Nilpotent orbits in so (2 N ) Here we lay out our conventions for nilpotent orbits in so (2 N ). For more details, see [16].We take so (2 N ) to consist of block matrices of the form (cid:32) A BC − A t (cid:33) (A.1)where A, B, C are N × N matrices and B t = − B , C t = − C . Nilpotent orbits are in 1-1correspondence with embeddings ρ : sl (2) (cid:44) → so (2 N ), up to conjugation. Here, sl (2) isgenerated by { H, X, Y } satisfying[ H, X ] = 2 X, [ H, Y ] = − Y, [ X, Y ] = H (A.2)and we take ρ ( X ) (which we will, henceforth, simply denote by X ) as our representativeelement of the nilpotent orbit.As noted in the text, a nilpotent orbit, in so (2 N ), is specified by a D-partition of2 N . Here, we will give our convention for assigning a triple of matrices of the form (A.1),satisfying (A.2), to such a partition.Let e , e , . . . e n be the standard basis for C N . Let E i,j be the 2 N × N matrix with a1 in the ( i, j ) th position and zeroes everywhere else. To the root, e i − e j , assign the matrix,of the form (A.1), X − i,j = E i,j − E j + N,i + N To the root e i + e j (for i < j ), assign X + i,j = E i,j + N − E j,i + N , i < j Also, let H i = E i,i − E i + N,i + N • Take the D-partition, [ r , r , . . . ], and divide it into pairs of the form [ r, r ] and [2 s +1 , t + 1] ( s > t ). This is not quite unique: the D partition, [3 , , , , ,
1] can bedivided into [3 , , [2 , , [1 ,
1] or into [2 , , [3 , , [3 , • To each pair of the form [ r, r ], assign a block of r consecutive basis vectors of C N .We will denote those by ( e , e , . . . , e r ), but they might be, say, ( e , e , . . . , e r ).To each pair of the form [2 s + 1 , t + 1], assign a block of s + t consecutive basisvectors of C N . The blocks, thus assigned, must be non-overlapping, and will exhaust e , . . . , e N . • For each pair of the form [ r, r ], let – 48 – = r (cid:88) k =1 ( r + 1 − k ) H k X = r − (cid:88) k =1 (cid:112) k ( r − k ) X − k,k +1 Y = X t • For pairs of the form [2 s + 1 , t + 1], the general formula can be found in [16]. Wewill need just the first few, for small values of t . – For pairs of the form [2 s + 1 , H = s (cid:88) k =1 s + 1 − k ) H k X = s − (cid:88) k =1 (cid:112) k (2 s + 1 − k ) X − k,k +1 + (cid:112) s ( s + 1) / (cid:16) X − s,s +1 + X + s,s +1 (cid:17) Y = X t – For pairs of the form [2 s + 1 , H = s (cid:88) k =1 s + 1 − k ) H k + 2 H s +1 X = s − (cid:88) k =1 (cid:112) k (2 s + 1 − k ) X − k,k +1 + (cid:112) ( s − s + 2) X − s − ,s + (cid:112) s ( s + 1) / (cid:16) X − s,s +2 + X + s,s +2 (cid:17) + (cid:16) X − s +1 ,s +2 − X + s +1 ,s +2 (cid:17) Y = X t • Add up the contributions to
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