Little String Defects and Bala-Carter Theory
LLITTLE STRING DEFECTS AND BALA–CARTER THEORY
NATHAN HAOUZICHRISTIAN SCHMID
Center for Theoretical PhysicsUniversity of California, Berkeley, USA
Abstract.
We give a physical realization of the Bala–Carter labels that classify nilpotentorbits of semi-simple Lie algebras, for the case g = A, D, E . We start from type IIB stringtheory compactified on an
ADE singularity and study the six-dimensional (2,0) g -typelittle string on a Riemann surface with punctures. The defects are introduced as D-braneswrapping the 2-cycles of the singularity. At low energies, the little string becomes the (2,0)conformal field theory of type g . As an application, we derive the full list of E n little stringdefects, and their Bala–Carter label in the CFT limit. Furthermore, we investigate newrelations between the quiver gauge theory describing the D-brane defects at low energies,and the weighted Dynkin diagrams of g . We also give a physical version of the dimensionformula of a nilpotent orbit based on its weighted Dynkin diagram. E-mail addresses : [email protected], [email protected] . a r X i v : . [ h e p - t h ] D ec LITTLE STRING DEFECTS AND BALA–CARTER THEORY
Contents
1. Introduction 32. Review: Little String Defects and Parabolic Subalgebras 52.1. Brane Defects of the Little String 52.2. Polarized Defects 82.3. Unpolarized Defects 113. Bala–Carter Classification 133.1. Bala–Carter Labeling of Nilpotent Orbits 133.2. Physical Origin of Bala–Carter Labels 154. Weighted Dynkin Diagrams 174.1. Mathematical construction 174.2. From Weighted Dynkin Diagrams to Little String Defects 184.3. Dimension Formula 21Acknowledgments 22Appendix A. E n little string defects 24Appendix B. Zero Weight Multiplicity 49References 50 ITTLE STRING DEFECTS AND BALA–CARTER THEORY 3 Introduction
String theory predicts the existence of a six-dimensional conformal field theory with (2 , g of ADE type. One can construct it from string theory by taking twolimits: Type IIB string theory is first put on an
ADE surface X . Then, the string coupling g s is sent to zero; this gives a six-dimensional string theory, called the (2 ,
0) little string.Afterwards, the string mass m s is taken to infinity (or equivalently, the string length tozero), while keeping the various moduli of the (2 ,
0) string theory fixed. As a result, thetheory doesn’t have any more scale parameters, and we end up with the (2 ,
0) CFT.However, it proves useful to study the (2 ,
0) little string proper, at finite m s , instead of itsCFT limit. Namely, it is a powerful tool to analyze supersymmetric gauge theories in lowerdimensions. For instance, in [6], the (2 ,
0) little string is studied on a Riemann surface C ,with defects. Specifically, one introduces codimension-two defects as D5 branes that arepoints on C and wrap non-compact 2-cycles of X . In the setup of [7], which we will useagain here for consistency of notations, one further compactifies the little string on a torus T ; the defects are then equivalently described as D3 branes at points on T , by T-duality.The (2 ,
0) little string theory with the D3 brane defects turns out to be described by thetheory on the branes themselves (except at the singular origin of the moduli space). At lowenergies, it is given by a 2d N = (4 ,
4) quiver gauge theory; the shape of the quiver is theDynkin diagram of g .In [7], we fully classified the defects of the (2 ,
0) little string on C . The defects are describedby a set of weights of g , subject to some conditions. From this data, one can extract aquiver gauge theory. After taking the string mass m s to infinity, to get a defect of the (2 , How-ever, in that limit, one is then able to extract a parabolic subalgebra of g . This subalgebracharacterizes a defect of the (2 ,
0) CFT which agrees with the known classification in termsof nilpotent orbits [4, 8]. In other words, a given set of D3 branes wrapping 2-cycles of X determines a unique parabolic subalgebra of g at low energies. In hindsight, the fact thatparabolic subalgebras make an appearance in the CFT limit could have been anticipated.Indeed, in the limit, the brane defects of the little string become the surface defects of N = 4 SYM studied by Gukov and Witten [9]. As they explain, these can be described asthe sigma model T ∗ ( G/P ), with P a parabolic subgroup of the Lie group G . A main resultof [7] is that the space T ∗ ( G/P ) should be identified with the Coulomb branch of the brane Indeed, the D-brane inverse gauge coupling τ must go to zero, because τ m s is a modulus of the (2 , LITTLE STRING DEFECTS AND BALA–CARTER THEORY defect theory, when m s → ∞ .In this paper, we elucidate further the connection between codimension 2 defects of thelittle string and nilpotent orbits. In the CFT limit, we find that defects realize physicallythe classification of nilpotent orbits derived by Bala and Carter [10]. In this note, we willlimit our analysis to the case g = A, D, E , but note that the Bala–Carter classification isin fact applicable to all semi-simple Lie algebras. We derive the Bala–Carter labels for thenilpotent orbits from the weight data that defines the low energy theory on the D3 branesin the little string.This characterization of defects is recovered in the context of 2d Toda CFT on C . Indeed,the AGT correspondence [11] relates four-dimensional N = 2 theories compactified on aRiemann surface to a two-dimensional g -Toda conformal field theory on the surface. TheBala–Carter labels can be read off directly from null state conditions at level 1 in the TodaCFT [12].There exists yet another way to classify nilpotent orbits of g , which turns out to be relatedto the above defect classification: by the so-called weighted Dynkin diagrams. These ariseas a consequence of the Jacobson–Morozov theorem and we will point out an unexpectedconnection to the quivers arising in the little string theory context.Indeed, any nilpotent orbit O has a representative X that fits into an sl triple. This meansthat there exists a nilpotent Y and a semi-simple H with[ H, X ] = 2 X, [ H, Y ] = − Y, [ X, Y ] = H. (1.1)By the Jacobson–Morozov theorem, this triple is unique up to conjugation. One then con-structs a quiver of Dynkin shape, with the integer entries 0, 1, or 2, from the semi-simpleelement H . In this way, every nilpotent orbit is denoted by a distinct diagram, called aweighted Dynkin diagram. Surprisingly, it turns out that these diagrams can be understoodas physical little string defect quivers; the integers 0, 1, or 2 of the weighted Dynkin diagramshould then be understood as the rank of unitary flavor symmetry groups. We do not havean interpretation for this observation, but we nonetheless explore some of its consequences.The paper is organized as follows. In section 2, we review the construction of [7], and theclassification of surface defects of the (2 ,
0) little string theory; we further recall how oneextracts a parabolic subalgebra from a defect, in the CFT limit m s → ∞ . In section 3,we show that this characterization of defects is precisely what is called the Bala–Carterclassification of nilpotent orbit in the mathematics literature [10]. In section 4, we reviewhow nilpotent orbits of g are also classified by weighted Dynkin diagrams, and we point outa connection to the quivers arising in the little string theory. In appendix A, we give theexplicit list of defects of g = E n little string and its CFT limit, as an application of theBala–Carter theory introduced in section 3. ITTLE STRING DEFECTS AND BALA–CARTER THEORY 5 Review: Little String Defects and Parabolic Subalgebras
The (2 , ADE little string is a six dimensional theory, and therefore has 16 supercharges.Its discovery dates back to [13, 14, 15] (see [16] for a review), and it has been studied morerecently in [17, 18, 19, 20, 21, 6, 22, 23]. It is obtained by considering type IIB string theoryon an ADE surface X and sending the string coupling g s to zero; this decouples the bulkmodes of the full type IIB string theory, and keeps only the degrees of freedom supportednear X . The space X is a hyperk¨ahler manifold, obtained by resolving a C / Γ singularity.Here Γ is a discrete subgroup of SU (2) that is related to g by the McKay correspondence[24].Since the strings have a finite tension m s , the little string theory is not a local QFT. Atenergies below the string scale m s , it reduces to a (2 ,
0) 6d CFT.2.1.
Brane Defects of the Little String.
We compactify the (2 ,
0) little string theoryon a Riemann surface C , which we take to be the complex plane. Then, X × C is a solutionof the full type IIB string theory. Codimension-two defects are introduced as branes thatare points on C and fill the four remaining directions C . These are D5 branes in IIB stringtheory, wrapping non-compact 2-cycles in X and C [6]. Their tension remains finite inthe little string limit. Equivalently, after further compactifying the theory on T and usingT-duality, the defects are described by D3 branes at points on T . This viewpoint has theadvantage of giving the little string theory origin of the surface defects studied by Gukov andWitten [9]. Indeed, after T compactification and at energies far below m s , the little stringtheory becomes 4d N = 4 SYM [25], and the D3 branes become codimension two defects.In particular, S-duality of N = 4 SYM with surface defects originates from T -duality ofthe little string. For details, see [7].The dynamics of the (2 ,
0) little string theory on
C × C × T , with an arbitrary collection ofD3 brane defects at points on C × T , is captured by the theory on the branes themselves.Because C has a flat metric, the theory on the D3 branes at low energies is a two-dimensionalquiver gauge theory, of shape the Dynkin diagram of g [26]. The theory has 2d N =(4 ,
4) supersymmetry, since the D3 branes break half the supersymmetry. As in [6, 7],we are interested in the class of D3 branes that retain some conformal invariance in theresulting low energy theory (the terminology here is borrowed from four dimensions, before T compactification). This corresponds to a very specific choice of non-compact 2 cycles of X wrapped by the D3 branes. The moduli of the 6d (2 ,
0) theory become gauge couplingsof the 2d theory. The positions of the D3 branes on C wrapping non-compact two-cyclesof X become the masses of fundamental hypermultiplets. Finally, the positions of the D3branes on C wrapping compact two-cycles of X become Coulomb moduli.In the rest of this paper, we study the quiver gauge theory that describes this low energy limitof D3 branes that wrap 2-cycles of the ALE surface X times C – we will call this theory T d . LITTLE STRING DEFECTS AND BALA–CARTER THEORY
To specify the charge of D3 branes wrapping non-compact two-cycles of X , we pick aclass [ S ∗ ] in the relative homology H ( X, ∂X ; Z ) = Λ ∗ ; by the McKay correspondence, thishomology group is identified with the (co-)weight lattice of g :[ S ∗ ] = − n (cid:88) a =1 m a w a ∈ Λ ∗ , (2.2)with w a the fundamental weights, m a non-negative integers, and n = rank( g ). As explained,we want to preserve conformal invariance in the low energy quiver theory (in a 4d sense).This is equivalent to a vanishing D3 brane flux at infinity. We therefore need to addadditional D3 branes wrapping a compact homology class [ S ] in H ( X, Z ) = Λ. Thishomology group is identified with the root lattice of g :[ S ] = n (cid:88) a =1 d a α a ∈ Λ . (2.3)Here, α a are the simple positive roots of g , and d a are non-negative integers such that[ S + S ∗ ] = 0 . (2.4)If S + S ∗ vanishes in homology, then S a ∩ ( S + S ∗ )) vanishes as well, for all a . We thereforerewrite (2.4) as n (cid:88) b =1 C ab d b = m a , (2.5)with C ab the Cartan matrix of g .Going to the Higgs branch of the low energy quiver theory, the gauge group (cid:81) na =1 U ( d a )breaks to (cid:81) na =1 U (1). There, all the D3 branes recombine and form a configuration of D3branes wrapping a set of non-compact cycles S ∗ i ; their homology classes ω i live in the weightlattice Λ ∗ = H ( X, ∂X ; Z ): ω i = [ S ∗ i ] ∈ Λ ∗ . (2.6)In what follows, the weights ω i will all belong to fundamental representations of g . Differentconfigurations of the weights ω i end up classifying the defects of the little string [7]. Everyone of these ω i ’s comes from one of the non-compact D3 branes on S ∗ . For the branes tobind, the position of each compact brane has to coincide with the position of one of thenon-compact D3 branes on C . As we mentioned, the positions of non-compact D3 branesare mass parameters of the quiver gauge theory, while the positions of compact D3 braneson C are Coulomb moduli. Whenever a compact brane and a non-compact brane meet on C , the corresponding fundamental hypermultiplet becomes massless and can get a vacuumexpectation value. These vevs describe the root of the Higgs branch.Thus, on the root of the Higgs branch, we can write the weights ω i in the following form: These are the weights that are represented by unit vectors in the Dynkin basis.
ITTLE STRING DEFECTS AND BALA–CARTER THEORY 7 ω i = − w a + n (cid:88) b =1 k ib α b , where − w a is a negative fundamental weight, the k ib are non-negative integers, and the α b are positive simple roots (from bound compact D3 branes). The collection of weights W S = { ω i } (2.7)we get must be such that it accounts for all the D3 brane charges in [ S ∗ ] and in [ S ]. Onesimple consequence is that the number of ω i ’s is the total rank of the 2d flavor group, (cid:80) na =1 m a . Because the net D3 brane charge is zero, [ S + S ∗ ] = 0, so that (cid:88) ω i ∈W S ω i = 0 , which is equivalent to (2.5). In this note, we take the set W S to have size at most n + 1, as a n + 1 weights are sufficient to describe the most generic defect of the g -type little string. Weemphasize that all weights of W S we consider will be taken in fundamental representations.As an example, the full puncture of g = D is featured below. Example 2.1.
Let W S be the following set of weights of D : ω = [ 1 , , ,
0] = − w + 2 α + 2 α + α + α ,ω = [ − , , ,
0] = − w + α + 2 α + α + α ,ω = [ 0 , − , ,
1] = − w + α + α + α + α ,ω = [ 0 , , − ,
0] = − w ,ω = [ 0 , , , −
1] = − w . Note that these weights add up to zero, as they should. Written as above, these weightsdefine a 2d quiver gauge theory, shown in Figure 1.4 5 333 11
Figure 1.
The quiver describing a full puncture for g = D . Unitary gaugegroups are circles, while unitary flavor groups are squares.This quiver gauge theory description is only valid at finite m s . After taking the m s → ∞ limit, the (2 ,
0) little string becomes the (2 ,
0) CFT of type g ; in general, this results inthe loss of the Lagrangian description of the defect. In particular, the Coulomb branch LITTLE STRING DEFECTS AND BALA–CARTER THEORY dimension, previously equal to (cid:80) na =1 d a , generically decreases for g = D n , E n . This loss ofCoulomb moduli is not unexpected, since in the limit, the theory loses degrees of freedom.We denote the resulting theory by T dm s →∞ .In this setup, it turns out there is a remarkably simple classification of defects, divided intotwo categories, both for the little string defect T d and for its (2 ,
0) CFT limit T dm s →∞ . Wenow review it.2.2. Polarized Defects.
Pick a weight ω in the a -th fundamental representation of g , la-beled by the fundamental weight w a . Suppose that ω is in the Weyl group orbit of − w a ;then if all weights of W S satisfy this condition, we call the resulting 2d theory T d on theD3 branes polarized . In the CFT limit, such a theory T dm s →∞ distinguishes a parabolic sub-algebra.First remember that a Borel subalgebra of g is a maximal solvable subalgebra of g . It canalways be written as b = h ⊕ m , with h a Cartan subalgebra of g and m = (cid:80) α ∈ Φ + g α , where g α are the root spaces associated to a set of positive roots Φ + . In what follows, we fix thisset of positive roots, or equivalently, the Borel subalgebra b , for each g .A parabolic subalgebra p Θ is a subalgebra of g that can be written as p Θ = l Θ ⊕ n Θ . Here, Θ is a subset of the set of simple positive roots of g . Here, l Θ = h ⊕ (cid:80) α ∈(cid:104) Θ (cid:105) g α iscalled a Levi subalgebra, and n Θ = (cid:80) α ∈ Φ + \(cid:104) Θ (cid:105) + g α , is called the nilradical of p Θ . (cid:104) Θ (cid:105) is thesubroot system of g generated by Θ, and (cid:104) Θ (cid:105) + is made of the positive roots of (cid:104) Θ (cid:105) .Note that this implies n Θ ∼ = g / p Θ . See below for an example: Example 2.2.
Consider g = A in the first fundamental representation, which is the definingrepresentation of sl . The root space g α associated to a root α = h i − h j is C E ij , where E ij is a matrix that has a 1 in the i -th row and j -th column, and zeros everywhere else. (SeeTable 1 below for an illustration.)Θ p Θ l Θ n Θ ∅ ∗ * *0 ∗ *0 0 ∗ ∗ ∗
00 0 ∗ ∗ ∗ ∗ : α : ( α + α ): α Table 1.
The colored boxes label the root spaces associated to the positiveroots of g , as indicated. ITTLE STRING DEFECTS AND BALA–CARTER THEORY 9
The Borel subalgebra we choose is b = ∗ ∗ ∗ ∗ ∗ ∗ . Then the Cartan subalgebra has the form h = ∗ ∗
00 0 ∗ . (2.8)The full list of all A parabolic subalgebras is: p ∅ = b = ∗ ∗ ∗ ∗ ∗ ∗ , p { α } = ∗ ∗ ∗∗ ∗ ∗ ∗ , p { α } = ∗ ∗ ∗ ∗ ∗ ∗ ∗ , p { α ,α } = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ , with Levi decomposition: p ∅ = ∗ ∗ ∗ ∗ ∗ ∗ = ∗ ∗
00 0 ∗ ⊕ ∗ ∗ ∗ = l ∅ ⊕ n ∅ , (2.9) p { α } = ∗ ∗ ∗∗ ∗ ∗ ∗ = ∗ ∗ ∗ ∗
00 0 ∗ ⊕ ∗ ∗ = l { α } ⊕ n { α } , (2.10) p { α } = ∗ ∗ ∗ ∗ ∗ ∗ ∗ = ∗ ∗ ∗ ∗ ∗ ⊕ ∗ ∗ = l { α } ⊕ n { α } , (2.11) p { α ,α } = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ⊕ = l { α ,α } ⊕ n { α ,α } . (2.12)Parabolic subalgebras of g are recovered from D3 brane defects in the following way [7]:Given a set of weights W S = { ω i } , we want to construct a set Θ as the subset of all simple roots that have a vanishing innerproducts with all the weights ω i . To achieve this, we have to choose a set in the Weyl grouporbit of W S for which | Θ | is maximal. Such a set of weights is called distinguished .Furthermore, the nilradical n Θ = g / p Θ specifies the Coulomb branch of T dm s →∞ . It is givenby the direct sum of the root spaces associated to the positive roots e γ of g for which: The The Weyl group acts on all weights of W S simultaneously. positive roots e γ that satisfy (cid:104) e γ , ω i (cid:105) < (2.13)for at least one weight ω i ∈ W S . The inner product (cid:104)· , ·(cid:105) is the Killing form of g . Thedimension of the Coulomb branch of T dm s →∞ can therefore be conveniently recovered fromthis prescription. In the little string context, at finite m s , the dimension of the Coulombbranch of T d is instead given by (cid:88) (cid:104) e γ ,ω i (cid:105) < |(cid:104) e γ , ω i (cid:105)| , where the sum runs over all positive roots e γ and all weights ω i of W S satisfying (cid:104) e γ , ω i (cid:105) < Example 2.3.
We will calculate the Coulomb branch dimension of the full puncture theoryof example 2.1, both for T d and T dm s →∞ .The positive roots Φ + of D are( h + h , h + h , h + h , h + h , h − h , h + h ,h − h , h − h , h − h , h + h , h − h , h − h )The negative inner products of each of these roots with the weights in W S are given in thefollowing table, where all positive inner products were replaced by 0: (cid:104) Φ + , ω (cid:105) = ( 0 , , , , , , , , , , , , (cid:104) Φ + , ω (cid:105) = ( 0 , , , , , , , , , , , − , (cid:104) Φ + , ω (cid:105) = ( 0 , , , , , , , − , − , , , , (cid:104) Φ + , ω (cid:105) = ( − , − , − , , − , , − , , , , − , , (cid:104) Φ + , ω (cid:105) = ( − , − , − , − , , − , , , , − , , . Adding the absolute value of all these entries gives 15, the dimension of the Coulomb branchof T d . Comparing this to the quiver in Figure 1, this is indeed correct.Furthermore, we can see that all 12 positive roots have a negative inner product with atleast one of the weights. Thus, the Coulomb branch of T dm s →∞ has (complex) dimension 12.The set W S is distinguished, and one can see immediately that Θ = ∅ . So the parabolicsubalgebra associated to this defect is all of D .The parabolic subalgebra p Θ also makes an appearance in the context of the AGT corre-spondence [11]. This duality predicts that codimension-two defects of the g -type 6d (2 , C should be classified from the point of view of a 2d conformal field theory on C , called g -Toda conformal field theory. This CFT has an extended conformal symmetry,called a W ( g )-algebra symmetry. Defects of the 6d (2 ,
0) conformal field theory correspond equivalently, (cid:104) e γ , ω i (cid:105) > ITTLE STRING DEFECTS AND BALA–CARTER THEORY 11 to certain vertex operator insertions at points on C in the 2d Toda CFT.In fact, the elements of the Cartan subalgebra h ⊂ g define the highest weight states | (cid:126)β (cid:105) ofthe W ( g )-algebra. In terms of the weights ω i of the algebra g , we write (cid:126)β = (cid:80) |W S | i =1 β i ω i .Polarized defects then turn out to be characterized in Toda theory by level 1 null states[12]. Explicitly, a highest weight state | (cid:126)β (cid:105) generating a degenerate representation of W ( g )satisfies (cid:126)β · (cid:126)α i = 0 ∀ (cid:126)α i ∈ Θfor a subset of simple roots Θ. Then the parabolic subalgebra associated to (cid:126)β is just p Θ ,which defines a theory T dm s →∞ . In particular, note that a set of positive simple roots char-acterizes a level 1 null state of Toda.2.3. Unpolarized Defects.
All the fundamental representations of A n are minuscule, soall A n codimension-two defects are polarized. However, in the D n and E n cases, it canalso happen that some weight of W S fails to satistfy the conditions to produce a polarizeddefect. Namely, two things can happen: First, W S can turn out to be the set containingthe zero weight only (possibly multiple times, as will occur first for g = E , see AppendixB). Second, W S can contain a nonzero weight ω in the representation generated by (minus)a fundamental weight − w a without being in the Weyl orbit of − w a . Either way, additionaldata is needed beyond simply specifying the set W S . Therefore, weights that make up anunpolarized defect need to be given a subscript denoting (minus) the representation theyare taken in . In the CFT limit m s → ∞ , no parabolic subalgebra of g is singled out. Whythis is so will be explained in the next section. the ”minus” here is because every weight we consider is written as ω = − w a + . . . ω ω ω : [0 , , ,
0] = − w + α + 2 α + α + α Figure 2.
The zero weight [0 , , ,
0] of the D algebra is the simplest ex-ample of how one constructs an unpolarized defect of the little string; on theleft is pictured the type IIB brane engineering of the weight. NS5 branes arevertical black lines, D5 branes are red crosses, and D3 branes are horizontalred lines. The green dotted line produces a Z -orbifold of an A theory, realiz-ing the D theory. The resulting defect will be unpolarized because [0 , , , , , ,
0] representation, but is not in the Weyl group orbit ofthat weight. ω ω ω ω ω : [ − , , , ,
0] = − w ω : [ − , , , , = − w + α +2 α + α + α Figure 3.
The weight [ − , , , ,
0] of D , with the corresponding type IIBbrane engineering on the left. [ − , , , ,
0] can be written in two ways. First,by placing a D5 brane between the two leftmost NS5 branes (top), the weightis written appropriately to characterize a polarized defect. This is so because[ − , , , ,
0] not only belongs in the [1 , , , ,
0] representation, it is also inthe Weyl group orbit of that weight. By placing the D5 brane between a differ-ent set of NS5 branes (bottom), we will obtain instead an unpolarized defect.This is so because [ − , , , ,
0] belongs in the [0 , , , ,
0] representation, butis not in the Weyl group orbit of that weight. An additional subscript is addedto the weight in this case, denoting (minus) the representation it belongs in.
ITTLE STRING DEFECTS AND BALA–CARTER THEORY 13
Finally, let us note that the labeling of defects we presented also has implications in thecontext of the triality worked out in [27, 28, 6]: the partition function of the (2 , g = ADE little string on C with polarized brane defects is equal to a q -deformation of the g -Toda CFTconformal block on C , with vertex operators determined by positions and the weights. Forunpolarized brane defects, the relation to q -deformed Toda fails, and other methods haveto be used to recover the duality [7].Having reviewed the classification of little string defects and their CFT limit, we now pro-ceed to show that their characterization, along with their organization into two classes,polarized and unpolarized, is precisely the Bala–Carter labeling of nilpotent orbits that ap-pears in mathematics [10]. 3. Bala–Carter Classification
Defects of the 6d (2 , g -type CFT have been studied in the literature [4] in terms of nilpotent orbits of the algebra: an element X ∈ g is called nilpotent if the matrix representing X is nilpotent. If X is nilpotent, every element in the G -adjoint orbit O X is nilpotent – thisis called a nilpotent orbit. Nilpotent orbits are directly related to the parabolic subalgebraswe have been considering. Indeed, given a parabolic subalgebra with Levi decomposition p = l ⊕ n , the nilpotent orbit O l associated to p is the maximal orbit containing a representative X ∈ O l for which X ∈ n .Many of the interesting properties of nilpotent orbits are related to the existence of a dualitymap: The Spaltenstein map [29] sends the set of nilpotent orbits of a simply-laced Lie algebra g to itself, and reorganizes them.For g = A n , nilpotent orbits are in one-to-one correspondence with integer partitions of n + 1 (or Young diagrams); for g = D n , they can also be labeled by Young diagrams with2 n boxes that satisfy certain conditions (cf. the textbook [30] for more details.) Howeverno such classification in terms of Young diagrams exists for the E n algebras.It proves fruitful instead to ignore Young diagrams altogether and resort to the classificationof Bala and Carter [10], which is valid for any semi-simple Lie algebra. We will see nextthat this is the natural language to describe the D3 brane defects in the low energy limit.3.1. Bala–Carter Labeling of Nilpotent Orbits.
Since there are only finitely manyorbits in g , we want to find a convenient way of classifying them. One such classificationscheme uses Levi subalgebras of g :Recall that a Levi subalgebra l of g is a subalgebra of g that can be written as: We assume that a fixed faithful representation is chosen for g . l = h ⊕ (cid:77) α ∈(cid:104) Θ (cid:105) g α , where h is a Cartan subalgebra of g , Θ is an arbitrary subset of the simple roots of g , and g α is the root space associated to a root in the additive closure of Θ.Then, the idea of the Bala–Carter [10] classification of nilpotent orbits is to label an orbit O by the smallest Levi subalgebra that contains a representative of O . This is always uniqueif g = A n , but for other algebras, two different nilpotent orbits can be associated to thesame minimal Levi subalgebra.In general, the following result holds: A nilpotent orbit O is uniquely specified by a Levisubalgebra l ⊂ g and a certain (distinguished) parabolic subalgebra of [ l , l ]. These twoalgebras give the Bala–Carter label of O . A parabolic subalgebra p = l (cid:48) ⊕ u , with nilradical u and Levi part l (cid:48) is distinguished if dim l (cid:48) = dim ( u / [ u , u ]). One such distinguished parabolicsubalgebra is the Borel subalgebra of l . The nilpotent orbit associated to it is called the principal nilpotent orbit of l .Whenever the minimal Levi subalgebra associated to O only contains one distinguishedparabolic subalgebra (so when l uniquely specifies O ) we call the orbit O polarized. Forsimplicity of notation, the Bala–Carter label for such an orbit is just l . For an unpolarizedorbit, it is given by l and an additional label . Example 3.1.
For g = A , consider the orbit of the element X = . The algebra sl has five different (conjugacy classes of) Levi subalgebras, correspondingto the five integer partitions of 4. X itself obviously is an element of the Levi subalgebra l { α ,α } : l { α ,α } = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . This algebra contains l { α } = ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ . The additional label specifies the number of simple roots that live in a Levi subalgebra of p . ITTLE STRING DEFECTS AND BALA–CARTER THEORY 15
Since every element in any conjugacy class of l { α } has at most one non-trivial Jordan block, X can never be contained in any of them; thus, the orbit of X is associated to l { α ,α } andhas the Bala–Carter label 2 A .3.2. Physical Origin of Bala–Carter Labels.
The relation to the little string defectclassification of section 2 is immediate: since polarized defects of the little string distinguishthe parabolic subalgebra p Θ of g in the CFT limit, we simply identify the set of simpleroots Θ with the Bala–Carter label of the defect. Namely, the union of all the elementsof Θ forms a subquiver of g , which denotes the Bala–Carter label for the defect. Thecorresponding nilpotent orbit is the principal nilpotent orbit of l Θ , defined in the previoussection. Equivalently, the Bala–Carter label is given by the union of the simple roots α i inthe Toda level 1 null state condition: (cid:126)β · (cid:126)α i = 0 ∀ (cid:126)α i ∈ Θ , for some highest weight state | (cid:126)β (cid:105) of the W ( g )-algebra. See figure 4 below.If the polarized theory T d is described by the Bala–Carter label denoting a nilpotent orbit O , its Coulomb branch is a resolution of the Spaltenstein dual of O . g Bala–Carter Classification and Polarized Little String Defects A Θ = { α , α } ω : [ 1 , , ω : [ − , ,
0] 1 1 11 1Bala–Carter label: A D Θ = { α , α , α } ω : [ 1 , , , ω : [ − , , ,
0] 2 2 112Bala–Carter label: A E Θ = { α , α , α , α } ω : [ 0 , , , , − , ω : [ − , , , , , ω : [ 1 , , , , ,
0] 2 4 6 5 43 3Bala–Carter label: D Figure 4.
From a distinguished set of weights W S defining a polarized littlestring defect theory T d , one can extract a quiver gauge theory, shown on theright, and a parabolic subalgebra p Θ in the m s → ∞ limit, shown on the left.The set Θ defines a Bala–Carter label, also shown in red as a subquiver of g on the right. Note (cid:126)β · (cid:126)α i = 0 for all (cid:126)α i ∈ Θ, which defines a null state at level1 in g -Toda. g Bala–Carter Classification and Unpolarized Little String defects D No parabolicsubalgebra! ω : [ 0 , , , D ( a ) E No parabolicsubalgebra! ω : [0 , , , , − , ω : [0 , , , , , D ( a ) Figure 5.
From a distinguished set of weights W S defining an unpolarizedlittle string defect theory T d , one can extract a quiver gauge theory, shownon the right, but no parabolic subalgebra p Θ in the m s → ∞ limit, as shownon the left. We added a subscript denoting which representation the weights ω i belong in to fully specify the Bala–Carter label. The additional simple rootdata of the Bala–Carter label is written as a i , where i is a number of simpleroots.Concerning unpolarized defects of the little string, recall that they are characterized asfollows: either W S is the set containing the zero weight ω = [0 , , . . . ,
0] only (possiblymultiple times), or W S contains a nonzero weight ω in the representation generated by(minus) a fundamental weight − w a without being in the Weyl orbit of − w a .Either way, additional data is needed to characterize such defects: in the end, it is sufficientto specify the representation ω belongs in. This prescription is in one-to-one correspondencewith specifying a set of additional simple roots next to the Bala–Carter label of a non-principal nilpotent orbit, as we explained in section 3.1. To our knowledge, this extrasimple root label unfortunately does not have a nice geometric interpretation for the defect.See figure 5 for examples.At any rate, note that an unpolarized defect will still satisfy the relation (cid:126)β · (cid:126)α i = 0 ∀ (cid:126)α i ∈ Θ , for some subset of positive simple roots Θ, with (cid:126)β = (cid:80) |W S | i =1 β i ω i . This is the same level1 null state condition of g -type Toda satisfied by polarized defects. There is however onecrucial difference: the above constraint is no longer sufficient to characterize the defect, andone should specify the representation each ω i belongs in. Example 3.2.
In the case of the single zero weight ω = [0 , , . . . , (cid:126)β = 0 and thenull state condition is of course trivially satisfied; however, one should also specify whichrepresentation the weight ω is taken in, since for a given algebra g , ω belongs in general tomany representations. This corresponds to specifying additional simple roots next to the ITTLE STRING DEFECTS AND BALA–CARTER THEORY 17
Bala–Carter label g . Note the Bala–Carter label g without any extra simple roots specifieddenotes the trivial nilpotent orbit, that is to say the absence of a defect.In this way, one derives the full Bala–Carter classification of nilpotent orbits simply froma distinguished set W S of weights defining a little string defect. It would be interestingto extend the analysis to the non-simply laced semi-simple Lie algebras, for which a Bala–Carter classification is also available. We leave the study of these defects to future work.4. Weighted Dynkin Diagrams
There is yet another way to classify nilpotent orbits of g , known as the so-called weightedDynkin diagrams. We now show how to derive them.4.1.
Mathematical construction.
Weighted Dynkin diagrams are vectors of integers r i ∈ { , , } , where i = 1 , . . . , rk g ; thus, we get one number for each node in the Dynkindiagram of g . We can associate such a vector to each nilpotent orbit of g , and each nilpotentorbit has a unique weighted Dynkin diagram. Note, however, that not all such labellings ofthe Dynkin diagram also have a nilpotent orbit corresponding to it.To construct such a weighted Dynkin diagram, we use the following theorem by Jacobsonand Morozov [31].Remember that sl is the algebra generated by X, Y and H with the relations[ H, X ] = 2 X, [ H, Y ] = − Y, [ X, Y ] = H. (4.14)Every nilpotent orbit in g arises as the orbit of the image of X in an embedding ρ : sl → g .In other words, for any embedding ρ : sl → g , the element ρ ( X ) always is a nilpotentelement of g . The Jacobson–Morozov theorem tells us that any nilpotent orbit uniquelyarises (up to conjugation) as the orbit of such an element.This means in particular that any nilpotent orbit also determines an element ρ ( H ), which issemi-simple (we assume it to be diagonal). For simplicity, we’ll just write ρ ( H ) as H . The(diagonal) entries of H are always integers, and allow us to read off the weighted Dynkindiagram; the entry of the i -th node is defined to be r i = α i ( H ), where α i is the i -th simpleroot of g . It turns out that these numbers are always 0, 1 or 2. Example 4.1.
We illustrate the above construction for the nilpotent orbit of X = in sl . One first constructs H ; we won’t do this explicitly here (see [30] for details), but theresult is H = − − . The next step is to reorder the elements in the diagonal of H in a monotonically decreasingorder. The quadruple we get is ( h , h , h , h ) = (1 , , − , − r i = h i − h i +1 . This gives us ( r , r , r ) = (0 , , One can generalize the above construction to all semi-simple Lie algebras, with minor mod-ifications.4.2.
From Weighted Dynkin Diagrams to Little String Defects.
We make the fol-lowing observations:All weighted Dynkin diagrams can be interpreted as physical quiver theories: the label oneach node of the weighted Dynkin diagram should be understood as the rank of a flavorsymmetry group in a quiver. The quivers one reads in this way are always superconformal(in a 4d sense), and the flavor symmetry on each node is either nothing, a U (1) group, ora U (2) group. For instance, the full puncture, or maximal nilpotent orbit, denoted by theweighted Dynkin diagram (2 , , . . . , , U (2) flavor attached to each node, for all semi-simple Lie algebras (see also [32]). Pushingthis idea further, we find, surprisingly, that these quivers are little string defect theories T d , at finite m s .In the case of g = A n , this correspondence between weighted Dynkin diagrams and defecttheories T d can be made explicit. Indeed, all A n weighted Dynkin diagrams are invariantunder the Z outer automorphism action of the algebra; in other words, the quivers are allsymmetric. For low dimensional defects, these quivers are precisely the little string quivers T d studied in this note. For instance, consider the simple puncture of A n , generated bythe set of weights W S = { [1 , , . . . , , [ − , , . . . , } , with Bala-Carter label A n − ; theweighted Dynkin diagram with this Bala-Carter label can be shown to be (1 , , . . . , , T d for the simple puncture! Ithas a U (1) flavor symmetry on the first node, and a U (1) flavor symmetry on the last node,as it should. See figure 6.Many of the little string quivers T d of A n , however, are not weighted Dynkin diagrams.They are the quivers not invariant under Z reflection. We claim that such theories T d can ITTLE STRING DEFECTS AND BALA–CARTER THEORY 19 (1 , , , (0 , , , (2 , , , H. W.
H. W. (1 , , , H. W. (2 , , , H. W. (2 , , , Figure 6.
Either directly, or after flowing on the Higgs branch by Hanany-Witten transition to symmetrize the theories T d , the little string quivers(left) are precisely the weighted Dynkin diagrams of g (right); the integers0 , , g = A .however uniquely be turned into the correct weighted Dynkin diagrams, by moving on theHiggs branch of the theories.Such a flow on the Higgs branch translates to a weight addition procedure in the algebra:this uses the fact that a weight in a fundamental representation can always be written asthe sum of new weights in possibly different fundamental representations. Each of themshould be in the orbit of some fundamental weight (possibly different orbits), while obeyingthe rule that no subset adds up to zero. In the context of brane engineering, this weightaddition procedure agrees with what is referred to as Hanany–Witten transitions [33]. See[7] for details, and figure 7 below for an example. ω ω ω (cid:48)(cid:48) ω (cid:48) ω H. W. ω : [ 0 , ,
0] = − w + α + 2 α + α ω : [ 0 , − ,
0] = − w ω : [ 0 , ,
0] = − w + α + 2 α + α ω (cid:48) : [ − , ,
0] = − w ω (cid:48)(cid:48) : [ 1 , − ,
0] = − w + α + Figure 7.
Writing a weight in a fundamental representation of g as a sumof several weights in (possibly different) fundamental representations corre-sponds to flowing on the Higgs branch of T d . In the context of brane en-gineering, when g = A n , this is the familiar Hanany–Witten transition [33].In this example, we rewrite [0 , − ,
0] as the sum [ − , ,
0] + [1 , − , α on the right is frozen to the value ofthe mass parameters denoted by ω (cid:48) (and ω (cid:48)(cid:48) ). [ − , , , , − , , , , − , , , , − , , , H. W. [0 , , , − , , ,
1] [ − , , ,
0] + [0 , − , , , , ,
0] + [0 , − , , , , ,
0] + [0 , , − , H. W. (2 , , , Figure 8.
An example of how one symmetrizes a little string quiver of A n using Hanany-Witten transitions, to end up with a weighted Dynkin diagram.The Coulomb parameters in red are frozen, and therefore do not increase theCoulomb branch dimension. In this example, no matter what the details ofthe transition are, the resulting symmetric quiver is always (2,2,2,2), the fullpuncture. Note some of the masses are equal to each other in the resultingquiver, as they should after Higgs flow.Then, it turns out that all A n little string quivers that are not symmetric under Z reflectioncan be uniquely written after Higgs flow as weighted Dynkin diagrams with correct Bala–Carter label. For instance, one can show that the full puncture of A n , with Bala–Carter ITTLE STRING DEFECTS AND BALA–CARTER THEORY 21 label ∅ , can be symmetrized uniquely to give the weighted Dynkin diagram (2 , , . . . , , g = A n , but many-to-one for the other algebras, as a large number of different little stringquivers typically describe one and the same defect in those cases. Nevertheless, the mapalways exists.We now come to another result about weighted Dynkin diagrams, motivated by their ap-parent connection to little string defects we have pointed out: the dimension of a nilpotentorbit can be easily computed from its weighted Dynkin diagram.4.3. Dimension Formula.
Recall that the “flavor symmetry rank” of a weighted Dynkindiagram never exceeds 2 (as the flavor symmetry is always a product of U (1) and U (2)groups only). This is a claim about the hypermultiplets of the quiver theory. There existsa “vector multiplet” counterpart to this statement, which is given by the following mathe-matical statement:We interpret the weighted Dynkin diagram of a nilpotent orbit O as a weight ω , writtendown in the Dynkin basis. We then compute the sum of the inner products of all the positiveroots of g with this weight. This gives a vector of non-negative integers. Truncating theentries of this vector at 2 and taking the sum of the entries gives the (real) dimension of O .This result can be derived from the following dimension formula for nilpotent orbits (cf.for instance [30]): dim O = dim g − dim g − dim g , (4.15)where g i = { Z ∈ g | [ H, Z ] = i · Z } , (4.16)and where H is the semisimple element in the sl triple corresponding to O .Note that whenever Z ∈ g β for a root β , [ H, Z ] = β ( H ) Z . So g i = (cid:77) β ∈ Φ ,β ( H )= i g β . On the other hand, if g is simply laced, then the inner product of the weighted Dynkindiagram weight ω with a root β is just (cid:42) n (cid:88) i =1 α i ( H ) ω i , β (cid:43) = n (cid:88) i =1 α i ( H ) (cid:104) ω i , β (cid:105) = β ( H ) , We thank Axel Kleinschmidt for pointing out this proof to us. where α i and ω i are the simple roots and fundamental weights of g , respectively.Thus, if g is simply laced, the above inner products just give us the grading 4.16.The prescription we give is therefore equivalent to the dimension formula 4.15; namely,dim( g ) + 2 (cid:88) i ≥ dim( g i ) = dim( g ) + (cid:88) i ≥ dim( g i ) + (cid:88) i ≤− dim( g i )= dim( g ) + dim( g ) − (cid:88) − ≤ i ≤ dim( g i )= dim( g ) + dim( g ) − g ) − dim( g )= dim( g ) − dim( g ) − dim( g ) . (4.17) Example 4.2.
Let us take the example of the weighted Dynkin diagram (2,1,1,2) in thealgebra g = A . We write ω = [2 , , ,
2] as a weight in Dynkin basis.The positive roots Φ + of A are( h − h , h − h , h − h , h − h , h − h , h − h , h − h , h − h , h − h , h − h )Calculating the inner product of all of these positive roots with ω gives the numbers (cid:104) Φ + , ω (cid:105) = (6 , , , , , , , , , . Truncating at multiplicity 2, the sum of the inner products is 2 × × , , , T d we have been studying can be interpreted as 3d N = 4 theories.It is then interesting to compare this formula to the dimension of the Coulomb branch ofa 3d N = 4 quiver theory [34], which is given by a slice in the affine Grassmannian [35].In that setup, the dimension can be calculated by the exact same procedure, coming froma monopole formula [36], but without truncating the inner products at the value 2. Forconformal theories, this is simply the sum of the ranks of the gauge groups.Lastly, we want to emphasize that the above formula we gave does not compute the Coulombbranch dimension of the defect theory T dm s →∞ denoted by the weighted Dynkin diagram.Instead, the Coulomb branch dimension is given by the dimension of the diagram’s imageunder the Spaltenstein map. Note that not all nilpotent orbits are in the image of theSpaltenstein map, so in many cases, it is unclear what the physical interpretation of thedimension formula should be. Acknowledgments
We want to thank Mina Aganagic, Tudor Dimofte, Jacques Distler, Ori Ganor, AmihayHanany, Alex Takeda, Yuji Tachikawa and Michael Viscardi for helpful discussions and
ITTLE STRING DEFECTS AND BALA–CARTER THEORY 23 comments. Furthermore, we want thank Axel Kleinschmidt for explaining the derivation ofthe dimension formula in section 4.3 to us.The research of N. H. and C. S. is supported in part by the Berkeley Center for Theo-retical Physics, by the National Science Foundation (award PHY-1521446) and by the USDepartment of Energy under Contract DE-AC02-05CH11231. ppendix A. E n little string defects As an application of the Bala–Carter classification, we now present a table of the defects of the E n little string. Unpolarizeddefects are shaded in yellow. For each defect type, we give a set W S of weights, along with the low energy 2d quiver gaugetheory T d on the D3 branes that results from it. The Bala–Carter label that designates the nilpotent orbit in the CFT limit m s → ∞ is written in the left column. Each set W S is a distinguished set, in the sense of section 2; in particular, the weights ω i of W S satisfy (cid:126)β · (cid:126)α i = 0 ∀ (cid:126)α i ∈ Θ , with (cid:126)β = (cid:80) |W S | i =1 β i ω i . This constraint has an interpretation as a level 1 null state condition of g -Toda. For unpolarizeddefects, a subscript is added to the weights, specifying the representation they are taken in. This corresponds to giving theadditional simple root label a i in the Bala–Carter picture. For polarized defects, no subscript is needed for the weights.The dual orbit is the orbit describing the Coulomb branch of T dm s →∞ ; for polarized defects, this is given by the Spaltensteindual of the Bala–Carter label. For unpolarized defects, these dual orbits had to be conjectured based on other approaches,such as dimension counting. The dimension of this dual orbit describing the Coulomb branch is given by d .Note the quivers are either literally the weighted Dynkin diagrams as given in the literature, or are quivers that can be madeto be weighted Dynkin diagrams after Higgs flow. Table 2.
Results for E Orbit Θ Weights Quiver Dual orbit d ∅ [ 0 , , , , , − − , , , , − , , , , − , , , − , , , , , , − , , , , , , , , − , , , , , − E able 2. Results for E Orbit Θ Weights Quiver Dual orbit dA { α } [ 0 , , , , − , , , , − , , , , , − , , , , , , , − , , − , , , , − , , , , E ( a ) 702 A { α , α } [ 0 , , , , − , − , , , , , − , − , , , , , , , − , , , , , , , D A { α , α , α } [ 0 , − , , , , , , , − , , − , − , , , , , , , , , E ( a ) 66 A { α , α } [ 0 , , , − , , − , , , , − , , , , − , , , , − , , − , , , , , , E ( a ) 66 able 2. Results for E Orbit Θ Weights Quiver Dual orbit dA + A { α , α , α } [ 0 , , , , , − , , , , − , , , − , , , , , , , , − D ( a ) 642 A { α , α , α , α } [ 0 , , , , , − , , − , , , , , , , , − D A + 2 A { α , α , α , α } [ 0 , , , , , , , − , , , , , , , − , A + A A { α , α , α } [ 0 , , , , − , , , , − , , , , , , , − , , , , , A A + A { α , α , α , α , α } [ 0 , , − , , , , , , , , D ( a ) 58 able 2. Results for E Orbit Θ Weights Quiver Dual orbit dA + A { α , α , α , α } [ 0 , , , − , , , , , , , − , , , , , D ( a ) 58 D ( a ) { α , α , α , α } [ 0 , , , , , [ − , , , , − , [ 1 , , , , , A + A A { α , α , α , α } [ 0 , , , , , − , , , , − , , , , , − , A D { α , α , α , α } [ 0 , , , , − , − , , , , , , , , , , A A + A { α , α , α , α , α } [ 0 , , , , , , , , − , , A + 2 A A { α , α , α , α , α } [ 0 , , , , , − , , , , , A able 2. Results for E Orbit Θ Weights Quiver Dual orbit dD ( a ) { α , α , α , α , α } [ 0 , , , , , [ 0 , , , , − , A + A E ( a ) { α , α , α , α , α , α } [ 0 , , , , , A D { α , α , α , α , α } [ 0 , , , , , , , , , − , A E ( a ) { α , α , α , α , α , α } [ 0 , , , , , A able 3. Results for E Orbit Θ Weights Quiver Dual orbit d ∅ [ − , , , , , , , , , − , , , , − , , , , − , − , , , , , − , , , , , , , , , − , , , , , , − , , − , , , , , − , , , −
13 22 31 24 17 10164 31 E A { α } [ − , , , , , , , , , − , , , , − , , , , − , − , , , , , − , , , , , , , , , − , , , , , , , , , , −
10 17 24 19 14 9123 4 E ( a ) 1242 A { α , α } [ − , , , , , , , , , − , , , , − , , , , − , − , , , , , − , , , − , , , , , , , , , , E ( a ) 1223 A b { α , α , α } [ 0 , , , , − , , , , − , , , , − , , , , , , , − , , , , , , , − , , , ,
10 20 30 25 20 1015 5 E A a { α , α , α } [ 0 , , , , − , , , , − , , , , , − , , , , − , , − , , , − , , , , − , , , ,
10 20 30 25 20 1015 5 E ( a ) 120 able 3. Results for E Orbit Θ Weights Quiver Dual orbit dA { α , α } [ 0 , , , , , − , , , , , − , , , , , , , , − , , − , , , , , , , − , , , , , , , − , , E ( a ) 1204 A { α , α , α , α } [ 0 , , − , , , , , , , , − , , , , − , , , , − , , , , , ,
16 32 48 36 24 12244 E ( a ) 118 A + A { α , α , α } [ 0 , , , − , , − , , , − , , , , , , − , , − , , , , , − , , − , , , , , , ,
12 23 34 27 20 10171 1 3 E ( a ) 118 A + 2 A { α , α , α , α } [ 0 , , , , , − , , , − , , , , , , , , , , , , , , − , , E ( a ) 116 A { α , α , α } [ 0 , , , , − , , − , , , − , , , , , , , − , , , , , , , − , , , , , , , D ( a ) 114 able 3. Results for E Orbit Θ Weights Quiver Dual orbit d A { α , α , α , α } [ 0 , , − , , , , , , , , , , − , , , − , , , , , , , , , − D + A A + 3 A { α , α , α , α , α } [ 0 , , , , − , , , , − , , , , , , , , − , , A A b + A b { α , α , α , α } [ − , , , , , , , , , , − , , , − , , , , , − , , , , , , D A + A { α , α , α , α , α } [ 0 , , − , , , , , , , , , − , , , , , , , E ( a ) 112 A a + A a { α , α , α , α } [ 0 , − , , , , , , , , , − , , , , , , , − , , , , , − , , E ( a ) 112 able 3. Results for E Orbit Θ Weights Quiver Dual orbit dD ( a ) { α , α , α , α } [ 1 , , , , , , [ − , , , , − , , [ 0 , , , , , , [ 0 , , , , , − , D ( a ) 110 A + 2 A { α , α , α , α , α } [ 0 , , − , , , , , − , , , , , , , , , , , E ( a ) 110 D { α , α , α , α } [ 0 , , , , , − , − , , , , , , , , , , − , , , , , , , − , A b D ( a ) + A { α , α , α , α , α } [ 1 , , , , , , [ − , , , , − , , [ 0 , , , , , , A a A + A { α , α , α , α , α } [ 0 , , , − , , , , , , , , , − , , , , , , D ( a ) + A able 3. Results for E Orbit Θ Weights Quiver Dual orbit dA { α , α , α , α } [ 0 , , , , , − , , , , , − , , , , , , , , − , , , , , , D ( a ) 106 A + A + A { α , α , α , α , α , α } [ 0 , , − , , , , , , , , , , A + A A b { α , α , α , α , α } [ − , , , , , , , − , , , , , − , , , , , , D D + A { α , α , α , α , α } [ − , , , , , , , , , , − , , , , , , , , A A + A { α , α , α , α , α } [ 0 , , , , − , , , , , , , , − , , , , , , A + A D ( a ) { α , α , α , α , α } [ 1 , , , , , , [ − , , , , , − , [ 0 , , , , , , A able 3. Results for E Orbit Θ Weights Quiver Dual orbit dA + A { α , α , α , α , α , α } [ 0 , , , − , , , , , , , , , A + A + A A a { α , α , α , α , α } [ 0 , , , , , , − , , , , , − , , , , , , , D ( a ) + A A + A { α , α , α , α , α , α } [ 0 , − , , , , , , , , , , , D ( a ) 94 D ( a ) + A { α , α , α , α , α , α } [ 0 , , , , − , , [ 0 , , , , , , A + A D ( a ) { α , α , α , α , α , α } [ 1 , , , , , , [ − , , , , , , A a + A a E ( a ) { α , α , α , α , α , α } [ 0 , , , , , − , [ 0 , , , , , , A + 2 A able 3. Results for E Orbit Θ Weights Quiver Dual orbit dD { α , α , α , α , α } [ 0 , , , , − , , , , , , , − , , , , , , , A b + A b E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , [ 0 , , , , , , A + A A { α , α , α , α , α , α } [ 0 , , , , , , − , , , , , , A + 3 A D + A { α , α , α , α , α , α } [ 0 , , , , − , , , , , , , , A D ( a ) { α , α , α , α , α , α } [ 1 , , , , , , [ − , , , , , , A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , A + 2 A able 3. Results for E Orbit Θ Weights Quiver Dual orbit dD { α , α , α , α , α , α } [ − , , , , , , , , , , , , A E ( a ) { α , α , α , α , α , α } [ 0 , , , , , , [ 0 , , , , , − , A E { α , α , α , α , α , α } [ 0 , , , , , − , , , , , , , A b E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , A a E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , A able 4. Results for E Orbit Θ Weights Quiver Dual orbit d ∅ [ − , , , , , , , , − , , − , , , , , , − , , , − , , − , − , , − , , , − , , , − , , , , , − , , , , − , , , − , − , , , , , , , , , − , , , , − , , − , , , , ,
46 89 130 106 80 54 28653 2 2 2 E A { α } [ − , , , , , , , , − , , − , , , , , , − , , , − , , , , − , , , , , − , , , , − , , , − , − , , , , , , , , , − , , , , − , , − , , , , ,
33 63 92 75 57 39 21463 1 1 3 E ( a ) 2382 A { α , α } [ − , , , , , , , , − , , − , , , , , , − , , , − , , , , − , , , , , , − , , , , , , , , , − , , , , − , , − , , , , ,
20 37 54 44 34 24 14273 4 E ( a ) 2363 A { α , α , α } [ − , , , , , , , , − , , − , , , , , , − , , , − , , , , − , , , , , , − , , , , , , , , , − , , , ,
14 27 40 33 26 19 12201 5 E ( a ) 234 able 4. Results for E Orbit Θ Weights Quiver Dual orbit dA { α , α } [ 2 , , , − , , − , , , , , , , − , , − , , , , − , , − , , , , , , − , , − , , , , − , , , , , , , , , , − , , , , , , ,
30 58 85 69 53 37 19432 1 2 11 E ( a ) 2344 A { α , α , α , α } [ 0 , − , , , , − , , , − , , , , − , , , , , − , , , , − , − , , , , , , , , , , , , ,
29 58 87 72 57 38 1944 41 E ( a ) 232 A + A { α , α , α } [ 0 , , , − , , , , , , , , − , , , − , , , , , − , , , , , , − , , , , , , , , , , − , , , , , , ,
24 48 72 59 45 31 1737 1 32 E ( a ) 232 A + 2 A { α , α , α , α } [ 0 , , , , − , , , , , , , , , − , , , , , , , , − , , − , , , , , , , , , , , ,
20 40 60 49 38 27 15301 1 3 E ( b ) 230 A { α , α , α } [ 2 , , , − , , − , , , , , , , − , , − , , , , − , , − , , , , , , − , , , , , , , , , − , , , , , , ,
24 48 71 58 45 32 17361 2 21 E ( a ) 228 able 4. Results for E Orbit Θ Weights Quiver Dual orbit dA + 3 A { α , α , α , α , α } [ 0 , , , , , − , , , , , − , , , , , , , , , − , , , − , , , , , ,
14 28 42 35 28 21 1121 3 1 E ( a ) 2282 A { α , α , α , α } [ 0 , , , , , , , − , , , , , , , − , , − , , , , − , , , , , , − , , , , , , , , , −
25 50 75 60 45 30 15405 E ( a ) 2282 A + A { α , α , α , α , α } [ 0 , , − , , , , , , , , , , , , , , , , , − , , − , , , , , − , ,
14 28 42 35 28 20 1121 1 1 2 E ( b ) 226 A + A { α , α , α , α } [ 0 , , , , , , − , , , , , , , , − , , , , , , , − , , , , , − , , , , , − , , , − ,
19 38 57 46 35 24 1330 23 E ( b ) 226 D ( a ) { α , α , α , α } [ 0 , , , , − , , , [ 0 , , , , , , , [ 0 , , , , , , − , [ 1 , , , , , − , , [ − , , , , , − , ,
18 36 54 44 34 24 14271 4 E ( b ) 226 D { α , α , α , α } [ 0 , , , , , , − , , , , , , − , , − , , , , , , , , , , , − , , , , , , , , − , ,
10 20 30 25 20 15 1015 5 E able 4. Results for E Orbit Θ Weights Quiver Dual orbit d A + 2 A { α , α , α , α , α , α } [ 0 , , − , , , , , , , , , , − , , , , , , , , ,
14 28 42 35 28 20 1021 1 2 E ( a ) 224 A + 2 A { α , α , α , α , α } [ 0 , , , , , , , − , , , , , − , , , , , − , , , , , , , , , , ,
15 29 43 35 27 19 10221 1 11 E ( a ) 224 D ( a ) + A { α , α , α , α , α } [ 0 , , , , − , , , [ 0 , , , , , , , [ 1 , , , , , − , , [ − , , , , , − , ,
18 36 54 44 34 24 13271 1 2 E ( a ) 224 A + A { α , α , α , α , α } [ 0 , , , − , , , , , , , , , , , − , , , , , , − , , , , , , , ,
14 25 36 29 22 15 8183 1 D ( a ) 222 A { α , α , α , α } [ 0 , , , , , , , − , , , , , , , − , , , , , − , − , , , , , , − , , , , , , − , , − ,
25 50 75 60 45 30 15405 E ( a ) 220 A + A + A { α , α , α , α , α , α } [ 0 , − , , , , , , , , , − , , , , , − , , , , , ,
15 30 44 36 28 19 10221 1 1 E ( b ) 220 able 4. Results for E Orbit Θ Weights Quiver Dual orbit dD + A { α , α , α , α , α } [ − , , , , , , , , , , , − , , , , , , , , , − , , , , , , , ,
14 26 38 31 24 17 9192 1 1 E ( a ) 214 D ( a ) + A { α , α , α , α , α , α } [ 1 , , , , , , , [ − , , , , , , , [ 0 , , , , − , , ,
16 30 44 36 27 18 9222 1 A A + A { α , α , α , α , α } [ 0 , , , , − , , , , , , , , , − , , , , , , , , − , , , , , , ,
13 26 39 32 25 18 1020 1 21 E ( a ) + A A { α , α , α , α , α , α } [ 0 , , , − , , , , , , , , , , , − , , , , , , ,
13 24 35 28 21 14 7182 1 D ( a ) 216 D ( a ) { α , α , α , α , α } [ 0 , , , , , , , [ 0 , , , , − , , − , [ 0 , , , , , − , − , [ 0 , , , , , − , ,
13 26 38 31 24 17 10191 3 E ( a ) 214 A + 2 A { α , α , α , α , α , α } [ 0 , , − , , , , , , − , , , , , , , , , , , , ,
21 41 60 48 36 24 12301 1 1 D ( a ) 216 able 4. Results for E Orbit Θ Weights Quiver Dual orbit dA + A { α , α , α , α , α , α } [ 0 , , , , , , , − , , − , , , , , , , , , , , ,
12 24 36 29 22 15 819 12 D + A A { α , α , α , α , α } [ 0 , , , , , , − , , , , , , − , , , , , , , , , − , , , , , , ,
11 22 33 27 21 15 917 31 D ( a ) 210 D ( a ) + A { α , α , α , α , α , α } [ 0 , , , , − , , , [ 0 , , , , , , , [ 0 , , , , , , − ,
18 36 54 44 34 23 12271 1 1 E ( a ) 212 A + A + A { α , α , α , α , α , α , α } [ 0 , , − , , , , , , , , , , , ,
20 40 60 48 36 24 12302 A + A D + A { α , α , α , α , α , α } [ − , , , , , , , , , , , − , , , , , , , , , ,
14 26 38 31 24 16 8192 1 A able 4. Results for E Orbit Θ Weights Quiver Dual orbit dE ( a ) { α , α , α , α , α , α } [ 0 , , , , , − , − , [ 0 , , , , , , , [ 0 , , , , , , ,
14 28 42 35 27 19 1021 1 1 1 D + A D { α , α , α , α , α } [ 0 , , , , , , − , , , , , , − , , , , , , − , , , , , , , , , − , D A + A { α , α , α , α , α , α , α } [ 0 , , , − , , , , , , , , , , ,
16 32 48 40 30 20 1024 2 E ( a ) 208 A + A { α , α , α , α , α , α } [ 0 , , , , , − , , , , , , , , , − , , , , , , ,
11 22 33 27 21 15 817 1 11 E ( a ) 208 D ( a ) + A { α , α , α , α , α , α , α } [ 0 , , , , − , , , [ 0 , , , , , , ,
16 32 48 39 30 20 10241 1 E ( a ) 206 D ( a ) { α , α , α , α , α , α } [ 1 , , , , , , , [ − , , , , , , − , [ 0 , , , , , , ,
14 27 40 33 25 17 9201 1 1 D ( a ) + A able 4. Results for E Orbit Θ Weights Quiver Dual orbit dE ( a ) + A { α , α , α , α , α , α , α } [ 0 , , , , , − , , [ 0 , , , , , , ,
14 28 42 34 26 18 9211 1 A + A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , ,
12 24 36 29 22 15 8181 1 A + A D + A { α , α , α , α , α , α } [ 0 , , , , − , , , , , , , , , − , , , , , , , ,
12 24 36 30 24 17 918 1 1 1 E ( a ) 198 E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , [ 0 , , , , , , ,
16 32 48 40 30 20 1024 2 E ( a ) 208 A { α , α , α , α , α , α } [ 0 , , , , , , , − , , , , , , − , , , , , , , ,
11 21 31 25 19 13 7161 11 D + A D ( a ) { α , α , α , α , α , α } [ 1 , , , , , , , [ − , , , , , , − , [ 0 , , , , , , ,
12 23 34 28 22 15 8171 1 1 A able 4. Results for E Orbit Θ Weights Quiver Dual orbit dA + A { α , α , α , α , α , α , α } [ 0 , − , , , , , , , , , , , , ,
14 28 40 32 24 16 8202 A + A + A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , ,
10 20 30 25 19 13 715 1 1 D ( a ) + A E ( a ) { α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , , [ 0 , , , , , − , ,
13 26 38 31 24 17 9191 1 1 D + A D + A { α , α , α , α , α , α , α } [ 0 , , , , − , , , , , , , , , ,
12 24 36 30 24 16 818 2 A + A D { α , α , α , α , α , α } [ − , , , , , , , , , , , , , − , , , , , , , , A E { α , α , α , α , α , α } [ 0 , , , , , − , , , , , , , , − , , , , , , , , D D ( a ) { α , α , α , α , α , α , α } [ 1 , , , , , , , [ − , , , , , , ,
12 23 34 28 21 14 7171 1 A able 4. Results for E Orbit Θ Weights Quiver Dual orbit dA { α , α , α , α , α , α , α } [ 0 , , , , , , , − , , , , , , ,
10 20 30 24 18 12 6162 D ( a ) + A E ( a ) + A { α , α , α , α , α , α , α } [ 0 , , , , , − , , [ 0 , , , , , , ,
10 20 30 25 20 14 715 1 1 A + A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , , A E ( b ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , ,
10 20 30 24 18 12 6151 A + A + A D ( a ) { α , α , α , α , α , α , α } [ 1 , , , , , , , [ − , , , , , , ,
10 19 28 23 18 12 6141 1 A + A E + A { α , α , α , α , α , α , α } [ 0 , , , , , − , , , , , , , , , D ( A ) 166 able 4. Results for E Orbit Θ Weights Quiver Dual orbit dE ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , , A + A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A + 2 A D { α , α , α , α , α , α , α } [ − , , , , , , , , , , , , , , A E ( b ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , [ 0 , , , , , , , A + A E ( a ) { α , α , α , α , α , α , α } [ 0 , , , , , , − , [ 0 , , , , , , , A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A + 3 A E ( b ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A + 2 A able 4. Results for E Orbit Θ Weights Quiver Dual orbit dE { α , α , α , α , α , α , α } [ 0 , , , , , , − , , , , , , , , A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A E ( a ) { α , α , α , α , α , α , α , α } [ 0 , , , , , , , A ITTLE STRING DEFECTS AND BALA–CARTER THEORY 49
Appendix B. Zero Weight Multiplicity
We make a comment about the multiplicity of the zero weight in unpolarized defects. Thisis relevant for two of the defects analyzed in appendix A: one has Bala–Carter label E ( a )in g = E , and the other has Bala–Carter label E ( b ) in g = E . For both of these, W S isthe set of the zero weight only, but appearing twice . In the little string, at finite m s , defectsusually add up in a linear fashion [28, 6]. If a subset of weights in W S adds up to zero,then one is simply describing more than one elementary defect. In the case of polarizeddefects, where a direct Toda interpretation is available, we would refer to this situation as ahigher-than-three point function on the sphere. We note here that for the two unpolarizeddefects we mentioned, this is not the case. For both cases, the zero weight is requiredto appear twice and does characterize a single exotic defect, with Bala–Carter label givenabove. In particular, E ( a ) and E ( b ) are not engineered in the little string as the sum oftwo elementary defects with a single zero weight. See Figure 9 for the example of E ( a ). ω : [0 , , , , , , ω : [0 , , , , , , Coulomb dim. of T dm s →∞ : 45Bala–Carter label: E ( a ) ω : [0 , , , , , , Bala–Carter label: E ( a ) Coulomb dim. of T dm s →∞ : 32 ω : [0 , , , , , , Bala–Carter label: E ( a ) Coulomb dim. of T dm s →∞ : 26 Figure 9.
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