Little String Origin of Surface Defects
LLittle String Origin of Surface Defects
Nathan Haouzi and Christian Schmid Center for Theoretical PhysicsUniversity of California, Berkeley, USA
Abstract
We derive the codimension-two defects of 4d N = 4 Super Yang-Mills (SYM) theoryfrom the (2 ,
0) little string. The origin of the little string is type IIB theory compactifiedon an
ADE singularity. The defects are D-branes wrapping the 2-cycles of thesingularity. We use this construction to make contact with the description of SYMdefects due to Gukov and Witten [1]. Furthermore, we derive from a geometricperspective the complete nilpotent orbit classification of codimension-two defects, andthe connection to
ADE -type Toda CFT. The only data needed to specify the defectsis a set of weights of the algebra obeying certain constraints, which we give explicitly.We highlight the differences between the defect classification in the little string theoryand its (2 ,
0) CFT limit. Email: [email protected] Email: [email protected] a r X i v : . [ h e p - t h ] D ec ontents N = 4 SYM . . . . . . . . . . . . . . . . . 52.2 Little String on a Riemann surface and D5 Branes . . . . . . . . . . . . . . . 62.3 Little String Theory origin of SYM surface defects and S-duality . . . . . . . 10 W ( g ) -algebras 27 W ( g )-algebras . . . . . . . . . . . . . . . . . . . 295.3 Null state relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 T d and Higgs flow as Weight Addition . . . . . . . . . . . . . . . . . . . . . 336.2 Brane Engineering and Weights . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Polarized and Unpolarized Punctures of the Little String . . . . . . . . . . . 366.4 All Codimension-Two Defects of the (2,0) Little String . . . . . . . . . . . . 39 A n Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.1.1 Maximal (“full”) Puncture . . . . . . . . . . . . . . . . . . . . . . . . 437.1.2 Next to Maximal Puncture . . . . . . . . . . . . . . . . . . . . . . . . 457.1.3 Minimal (“simple”) Puncture . . . . . . . . . . . . . . . . . . . . . . 487.2 D n Examples: Polarized Theories . . . . . . . . . . . . . . . . . . . . . . . . 527.2.1 Examples for Arbitrary n . . . . . . . . . . . . . . . . . . . . . . . . 527.2.2 Complete D Classification . . . . . . . . . . . . . . . . . . . . . . . . 547.3 E n Examples: Polarized Theories . . . . . . . . . . . . . . . . . . . . . . . . 577.4 Unpolarized Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
Introduction
String theory has been an invaluable tool to study the properties of quantum field theories invarious dimensions. In particular, it predicts the existence of the so-called (2 ,
0) conformalfield theory in six dimensions; this CFT has become of great interest in recent years ([2, 3, 4]).Most importantly, it gave new insights into lower-dimensional supersymmetric gauge theories,for example in four dimensions (see [5, 6]). The (2 ,
0) CFT is labeled by an
ADE
Lie algebra g , and arises in string theory after taking a double limit. One sends the string coupling tozero in type IIB string theory on an ADE surface X to decouple the bulk modes; this givesa six-dimensional string theory, called the (2 ,
0) little string theory. Second, one takes thestring mass m s to infinity, while keeping the moduli of the (2 ,
0) string fixed.After taking these limits, there is no scale left in the theory, and we are left with the (2 , m s finiteand study the (2 ,
0) little string. In [7], the (2 ,
0) little string was studied on a Riemannsurface C with defects. Specifically, the (2 ,
0) little string is placed on C , and codimension-twodefects are introduced as D5 branes at points on C and wrapping non-compact 2-cycles ofthe ADE surface X .In a different context, Gukov and Witten analyze the surface defects of 4d N = 4 SYMfrom a gauge theory perspective, by studying the singular behavior of the gauge and Higgsfields in SYM near the defect [1]. In this paper, we explain the origin of these defects usingthe (2 ,
0) little string on C , compactified on an additional torus T . At energies below theKaluza–Klein scale of compactification and the string scale, this becomes the 4d N = 4 SYMtheory. The defects come from D5 branes wrapping the T , or equivalently, from D3 branesat points on T . In particular, the S-duality of 4d SYM theory with defects is realized inlittle string theory as T-duality on the torus; the case without defects was studied alreadysome time ago in [8]. In the CFT limit, the resolution [9] of the (singular) Coulomb branchof the theory on the D3 branes is in fact described by the cotangent bundle T ∗ ( G/ P ), where P is a parabolic subgroup of the gauge group G . This space already appeared in [1] as analternate way to describe surface defects. It comes about as a moduli space of solutions toHitchin’s equations, which are precisely the equations obeyed by the brane defects in the lowenergy limit. We will see here that the space T ∗ ( G/ P ) arises from the geometry; indeed, agiven set of D3 branes wrapping 2-cycles of X will determine a unique parabolic subalgebraof g at low energies.Everywhere except at the origin of the moduli space, which is singular, the (2 ,
0) littlestring theory with the D5 brane defects is in effect described by the theory on the branesthemselves. Specifically, it has a description at low energy as a 4d N = 2 superconformalquiver gauge theory of Dynkin shape. In particular, the gauge theory description of the3o-called T N theory (when viewed as a 5d N = 1 theory), or full puncture, was derived forany simply-laced Lie algebra g = A, D, E in [7] (see also [10, 11] for the A n case). In thispaper, we give the full classification of punctures of the (2 ,
0) little string theory on C . Each“class” of defects will be given by a collection of certain weights of g , from which one canread off a superconformal quiver gauge theory. Taking the string mass m s to infinity, wegenerically lose the low energy quiver gauge theory description of the defects. Indeed, if τ isthe D-brane gauge coupling, it must go to zero, as the combination τ m s is a modulus of the(2 ,
0) theory, kept fixed in the limit. Nonetheless, we obtain the full list of punctures givenin the literature in terms of nilpotent orbits [5] (see also [12] for an M-theory approach inthe specific case of D n ).Finally, the AGT correspondence [13] relates 4d N = 2 theories compactified on a Riemannsurface to 2d Toda conformal field theory on the surface. In the little string setup, the precisestatement is that the partition function of the (2 ,
0) little string on C with brane defects is infact equal to a q -deformation of the Toda CFT conformal block on C . The vertex operatorsare determined by positions and types of defects. So in particular, the codimension-twodefects of the 6d (2 ,
0) CFT are expected to be classified from the perspective of the 2dToda theory. This can be done by studying the Seiberg-Witten curve of the 4d N = 2 quivergauge theory on the D5 branes, or equivalently, after T compactification, the curve of the2d N = (4 ,
4) theory on the D3 branes. At the root of the Higgs branch, and in the m s toinfinity limit, the curve develops poles at the puncture locations. The residues at each poleobey relations which describe the level 1 null states of the Toda CFT; this was previouslystudied in the A n case in [14]. We argue that this characterization of defects as null states ofthe CFT naturally gives the same parabolic subalgebra classification obtained in this note,for g = A, D, E .The paper is organized as follows. In section 2, we review the description of surface defectsof N = 4 SYM given in [1], and give its (2 ,
0) little string theory origin. We further derivethe action of S-duality from T-duality on T . In section 3, we explain how to extract aparabolic subalgebra and characterize the sigma model T ∗ ( G/ P ) from the D3 brane defectdata of the little string. In section 4, we make contact with the nilpotent orbit classificationof defects given in the literature [5]. In section 5, we explain how the parabolic subalgebrasdetermined in section 3 can also be recovered from null states of the g -type Toda CFT,and how this is related to the nilpotent orbit classification. In section 6, we explain thedifferences between the defects of the little string proper, and its (2 ,
0) CFT limit. In orderto give the exhaustive list of defects of the little string, we will need to extend our definitionof defects to characterize punctures on C that do not specify a definite parabolic subalgebra.In section 7, we provide a plethora of detailed examples for g = A, D, E, and illustrate allthe statements made in the rest of the paper.4
SYM Surface Defects From Little String Theory
In this section, we begin by recalling the description of two-dimensional surface defects in 4d N = 4 SYM given by Gukov and Witten in [1]. We then review the analysis of little stringtheory on a Riemann surface [7], use it to describe these surface defects and derive theirS-duality transformation. N = 4 SYM
Surface defects of N = 4 SYM are -BPS operators; to describe them, one starts with afour-dimensional manifold M , which is locally M = D × D (cid:48) , where D is two-dimensional,and D (cid:48) is a fiber to the normal bundle to D . Surface defects are then codimension-twoobjects living on D , and located at a point on D (cid:48) ; they are introduced by specifying thesingular behavior of the gauge field near this defect. A surface operator naturally breaks thegauge group G to a subgroup L ⊂ G , called a Levi subgroup.The story so far is in fact valid for N = 2 SUSY, but N = 4 SUSY has additional parameters (cid:126)β and (cid:126)γ , which describe the singular behavior of the Higgs field φ near the surface operator;choosing D (cid:48) = C with coordinate z = re iθ = x + ix , we have: A = (cid:126)αdθ + . . . , (2.1) φ = 12 (cid:16) (cid:126)β + i(cid:126)γ (cid:17) dzz + . . . , (2.2)which solve the Hitchin equations [15]: F = [ φ, φ ] , (2.3) D z φ = 0 = D z φ. (2.4)As written above, we have chosen a complex structure which depends holomorphically on β + iγ , while the K¨ahler structure depends on α . Quantum mechanics also requires theconsideration of the Theta angle, denoted by η ; by supersymmetry, it will complexify theK¨ahler parameter α .S-duality is the statement that this theory is equivalent to N = 4 gauge theory with a dualgauge group and coupling constant g (cid:48) d = 1 /g d . The action of S-duality on the surface defect parameters is a rescaling of the Higgs field5esidue ( β, γ ) → (cid:18) πg d (cid:19) ( β, γ ) , (2.5)and an exchange of the gauge field and Theta angle parameters [1]( α, η ) → ( η, − α ) . (2.6)The analysis of [1] gives a second description of the surface operators of N = 4 SYM, whichwill be of great relevance to us; one couples the 4d theory to a 2d non-linear sigma modelon D . In the N = 4 case, the 2d theory is a sigma model to T ∗ ( G/ P ), where P ⊂ G is aparabolic subgroup of the gauge group. The quotient describes a partial flag manifold whenthe Lie algebra g is A n . In the case of a general Lie algebra, the quotient is a generalized flagvariety. This target space is in fact the moduli space of solutions to the Hitchin equations (2.3).Then, to describe a surface operator, one can either specify the parameters ( β, γ, α ) forthe singular Higgs and gauge fields, or spell out the sigma model T ∗ ( G/ P ). It turns outthat both of these descriptions have an origin in string theory, and we will now show thisexplicitly; our starting point will be the (2 ,
0) little string theory.
We first review some basic facts about the little string ([16, 17, 18]; see [19] for a review),and discuss the role of D5 branes in the theory. –(2,0)
ADE
Little String Theory–
The
ADE little string theory with (2 ,
0) supersymmetry is a six dimensional string theory,and therefore has 16 supercharges. It is obtained by sending the string coupling g s to zeroin type IIB string theory on an ADE surface X ; this has the effect of decoupling the bulkmodes of the full type IIB string theory. X is a hyperk¨ahler manifold, obtained by resolvinga C / Γ singularity where Γ is a discrete subgroup of SU (2), related to g by the McKaycorrespondence [20]. The little string is not a local QFT, as the strings have a tension m s .The (2 ,
0) little string reduces to a (2 ,
0) 6d conformal field theory at energies well belowthe string scale m s . The moduli space of the little string is ( R × S )rk ( g ) /W , with W theWeyl group of g . The scalars parametrizing this moduli space come from the periods of theNS B-field m s /g s (cid:82) S a B NS , the RR B-field m s (cid:82) S a B RR , and a triplet of self-dual two-formsobtained from deformations of the metric on X , m s /g s (cid:82) S a ω I,J,K . Here, S a are two-cyclesgenerating the homology group H ( X, Z ). The ( S )rk ( g ) have radius m s and are parametrizedby the periods of B RR . When g s is sent to zero, we keep the above periods fixed in that6imit. We set for all a ’s (cid:90) S a ω J,K = 0 , (cid:90) S a B NS = 0 , (2.7)and let τ a = (cid:90) S a ( m s ω I /g s + i B RR ) (2.8)be arbitrary complex numbers with Re( τ a ) > ,
0) little string theory on a fixed Riemann surface C , whichis chosen to have a flat metric. This guarantees X × C to be a solution of type IIB stringtheory. We want to introduce codimension-two defects in the little string, at points on C and filling the four remaining directions C . These correspond to D5 branes in IIB stringtheory, wrapping non-compact 2-cycles in X and C [7]. Their tension remains finite in thelittle string limit, so they are the correct objects to study (D3 branes also keep finite tension,but they do not describe the codimension-two defects we are after; other objects of type IIBeither decouple or get infinite tension when g s → ,
0) little string theory on
C × C , with anarbitrary collection of D5 brane defects at points on C , is captured by the theory on thebranes themselves. Because the Riemann surface C , which is transverse to the D5 branes,has a flat metric, the theory on the D5 branes is four dimensional at low energies. In fact, ithas 4d N = 2 super Poincare invariance, since the D5 branes break half the supersymmetry.We will focus specifically on the class of D5 branes that retain some conformal invariance inthe resulting low energy 4d theory. This corresponds to a very specific choice of non-compact2 cycles of X wrapped by the D5 branes, which we review here. –D5 Branes and ADE quiver gauge theories–
For definiteness, we will choose the Riemann surface C to be the complex plane in whatfollows (one could equally choose to work on the cylinder as in [7], or on the torus.) Thefour-dimensional theory on the D5 branes is a quiver gauge theory, of shape the Dynkindiagram of g [21]. The 4d gauge couplings are the τ a defined in equation (2.8), which arethe moduli of the (2 ,
0) theory in 6d. The masses of fundamental hypermultiplets are thepositions of the D5 branes on C wrapping non-compact two-cycles of X . Finally, the Coulombmoduli are the positions of the D5 branes on C wrapping compact two-cycles of X .In order to specify a defect D5 brane charge, we pick a class [ S ∗ ] corresponding to non-compact two-cycles in the relative homology H ( X, ∂X ; Z ) = Λ ∗ , which we identify with the7co-)weight lattice of g : [ S ∗ ] = − n (cid:88) a =1 m a w a ∈ Λ ∗ (2.9)with non-negative integers m a and fundamental weights w a . A necessary condition forconformal invariance in 4d is that the net D5 brane flux vanishes at infinity. This constrainsthe form of the coefficients m a . To satisfy the condition, we add D5 branes that wrap acompact homology class [ S ] in H ( X, Z ) = Λ, which we identify with the root lattice of g :[ S ] = n (cid:88) a =1 d a α a ∈ Λ (2.10)with non-negative integers d a and the simple roots α a , such that[ S + S ∗ ] = 0 . (2.11)The vanishing of S + S ∗ in homology is equivalent to vanishing of S a ∩ ( S + S ∗ )) for all a .We can therefore rewrite (2.11) as n (cid:88) b =1 C ab d b = m a (2.12)where C ab is the Cartan matrix of g . On the Higgs branch of the low energy gauge theory,the gauge group (cid:81) na =1 U ( d a ) is broken to its U (1) centers, one for each node. There, the D5branes wrapping the compact cycles S and the non-compact cycles S ∗ recombine to form D5branes wrapping a collection of non-compact cycles S ∗ i , whose homology classes are elements ω i of the weight lattice Λ ∗ = H ( X, ∂X ; Z ): ω i = [ S ∗ i ] ∈ Λ ∗ . (2.13)It is these weights ω i that will classify the defects of the little string. Each of the ω i ’s comesfrom one of the non-compact D5 branes on S ∗ . For the branes to bind, the positions on C ofthe compact branes must coincide with the positions of one of the non-compact D5 branes.Recall that the positions of non-compact D5 branes are mass parameters of the quiver gaugetheory, while the positions of compact D5 branes on C are Coulomb moduli; when a Coulombmodulus coincides with one of the masses, the corresponding fundamental hypermultipletbecomes massless and can get expectation values, describing the root of the Higgs branch.One can reasonably worry that the binding of the D5 branes will break supersymmetry, butit is in fact preserved when one turns on the FI terms, which are the periods (cid:82) S a ω J,K , (cid:82) S a B NS .8hen, the ω i ’s can always be written as a negative fundamental weight − w a plus the sumof positive simple roots α a , from bound compact branes. Not any such combination willcorrespond to truly bound branes: a sufficient condition is that ω i is in the Weyl orbit of − w a = [ S ∗ a ] (we will relax this condition in section 6.3 and end up with a new class of defectsof the little string). Furthermore, the collection of weights W S = { ω i } (2.14)we get must be such that it accounts for all the D5 brane charges in [ S ∗ ] and in [ S ]. Onesimple consequence is that the number of ω i ’s is the total rank of the 4d flavor group, (cid:80) na =1 m a . The fact that the net D5 charge is zero, [ S + S ∗ ] = 0, implies that (cid:88) ω i ∈W S ω i = 0 , which is equivalent to (2.12).The most canonical type of defect is the one analyzed in [7], which makes use of the factthat the weight lattice of a Lie algebra of rank n is n -dimensional. Then we can constructa set W S by picking any n + 1 weight vectors which lie in the Weyl group orbits of thefundamental weights − w a such that they sum up to zero and n of them span Λ ∗ . This leadsto a full puncture defect of the (2 ,
0) little string on C . The example below features g = A . Example 2.1.
Let us look at the set of all the weights in the antifundamental representation of A ; these weights all add up to 0, and all weights are in the orbit of (minus) the fundamentalweight [ − , , W S . Writing w i forthe i -th fundamental weight, we note that: ω = [ − , ,
0] = − w ,ω = [ 1 , − ,
0] = − w + α ,ω = [ 0 , , −
1] = − w + α + α ,ω = [ 0 , ,
1] = − w + α + α + α . Written in this fashion, the set W S defines a 4d superconformal quiver gauge theory, shownin Figure 1. This is called the full puncture.The full classification of defects for simply-laced g is obtained by constructing the set W S to have size n + 1 or less . As we will explain in later sections, this is where the richstructure of the parabolic subalgebras of g will emerge, and it will be our main object of study.9 2 14Figure 1: The quiver theory describing a full puncture for g = A When the string scale m s is taken to infinity, the (2 ,
0) little string reduces to the (2 ,
0) CFTof type g compactified on C ; we lose the Lagrangian description in general, and the Coulombbranch dimension, previously equal to (cid:80) na =1 d a , generically decreases. This loss of Coulombmoduli is expected, as the theory loses degrees of freedom in the limit. This distinction incounting Coulomb moduli between the little string and CFT cases will be important to keepin mind throughout the rest of our discussion. –Hitchin System and Higgs Field Data– In the little string theory, the brane defects we are studying are solutions to Bogomolnyequations on C times an extra circle S ( R ) [7]: Dφ = ∗ F. (2.15)Little string theory enjoys T-duality, so in particular, the (2 , ADE
Little String of typeIIB compatified on S ( R ) is dual to the (1,1) ADE
Little String of type IIA compatifiedon S ( ˆ R ) of radius ˆ R = 1 /m s R . The defects are then D4 branes after T-dualizing, andare points on C × S ( ˆ R ). These are monopoles, magnetically charged under the gauge fieldcoming from the (1,1) little string. The n scalars are φ a = (cid:82) S a m s ω I /g (cid:48) s , where g (cid:48) s is the IIAstring coupling, related to the IIB one by 1 /g (cid:48) s = R m s /g s . F is the curvature of the gaugefield coming from the (1,1) little string.If we want to recover the original description of the defects as D5 branes, we can take thedual circle size ˆ R to be very small; the upshot is that the Bogomolny equations simplifyand we recover the Hitchin equations (2.3) we considered previously: F = [ φ, φ ] , (2.16) D z φ = 0 = D z φ. (2.17)10 subtlety here is that the field φ got complexified in passing from D4 branes back to D5branes. The imaginary part of φ is the holonomy of the (1,1) gauge field around S ( ˆ R );this comes from the fact that the D4 branes are magnetically charged under the RR 3-form: R (cid:82) S a × S ( R ) m s C (3) RR . In type IIB language, after T-duality, the D5 branes are chargedunder the RR 2-form instead: 1 / ˆ R (cid:82) S a B RR . All in all, the Higgs field is then written in IIBvariables as φ a = ( α a , φ ) = 1 / ˆ R (cid:90) S a ( m s ω I /g s + iB RR ) = τ a / ˆ R . (2.18)The Seiberg-Witten curve of the quiver gauge theory on the D5 branes arises as the spectralcurve of the Higgs field φ , taken in some representation R of g :det R ( e ˆ R φ − e ˆ R p ) = 0 . (2.19)In the absence of monopoles, φ is constant: the vacuum expectation value of the Higgs fieldis ˆ R φ = τ .By construction, then, the Coulomb branch of the ADE quiver theory on the D5 branesis the moduli space of monopoles on
C × S ( ˆ R ). As we described in the previous section,we ultimately want to go on the Higgs branch of the theory, where we get a description ofquiver theories as a fixed set of weights W S in g ; there, all the non-abelian monopoles reduceto Dirac monopoles. The effect on φ of adding a Dirac monopole of charge ω ∨ i , at a point x i = ˆ R ˆ β i on C , is to shift: e ˆ R φ → e ˆ R φ · (1 − z e − ˆ R ˆ β i ) − w ∨ i . (2.20)Here, z is the complex coordinate on C = C . Thus, the Higgs field solving the Hitchinequations at the point where the Higgs and the Coulomb branches meet is e ˆ R φ ( x ) = e τ (cid:89) ω Vi ∈W S (1 − z e − ˆ R ˆ β i ) − ω ∨ i . (2.21)To take the string mass m s to infinity, we relabel e ˆ R ˆ β i = z P e ˆ R β i, P . We can then safely takethe limit ˆ R →
0; the imaginary part of φ decompactifies, and equation (2.19) becomes thespectral curve of the Hitchin integrable system [3]:det R ( φ − p ) = 0 . (2.22)11n this limit, the Higgs field near a puncture of C has a pole of order one, and takes the form φ ( z ) = β z + (cid:88) P (cid:88) ω i ∈W P β i, P ω ∨ i z P − z , (2.23)with β = τ / ˆ R and P the set of punctures. Therefore, in the (2 ,
0) CFT, we have poles on C at z = z P , with residues β P = (cid:88) ω i ∈W P β i, P ω ∨ i . These residues are what we called β + iγ in the N = 4 SYM setup of eq. (2.2). – 4d S-duality is T-duality of the Little String– To provide evidence that the surface defects of N = 4 SYM really are branes at points on C in the (2 ,
0) little string, we now derive four-dimensional S-duality from T-duality of thestring theory, compactified on an additional torus T . Here, T is the product of two S ’s,one from each of the two complex planes C . We label those circles as S ( R ) and S ( R ),of radius R and R respectively.(1 ,
1) string on S ( ˆ R ) × S ( R ) × R × C with (D4 , D4) branes T -duality(2 ,
0) string on S ( R ) × S ( R ) × R × C with (D3 , D5) branes T -duality(1 ,
1) string on S ( R ) × S ( ˆ R ) × R × C with (D4 , D4) branesFigure 2: One starts with the (1 ,
1) little string theory on T × R × C . After doing twoT-dualities in the torus directions, we get the (1 ,
1) little string theory on the T-dual torus;in the low energy limit, the pair of (1 ,
1) theories gives an S-dual pair of N = 4 SYMtheories. D3 branes at a point on T map to D4 branes in either (1 ,
1) theory, while D5branes wrapping T map to another set of D4 branes.First, without any D5 branes, S-duality was derived in [8], and the line of reasoning went asfollows: suppose we first compactify on, say, S ( R ); this is what we just did in the previoussection to make contact with D4 branes as magnetic monopoles. Then we are equivalentlystudying the (1,1) little string on S ( ˆ R ). Compactifying further on S ( R ), this theory isthe same as the (1,1) little string on S ( R ) × S ( ˆ R ), by T -duality. 4d SYM S-duality thennaturally follows from the T -duality of this pair of (1,1) theories. Indeed, at low energies,12oth (1 ,
1) little string theories become the maximally supersymmetric 6d SYM, with gaugegroup dictated by g and gauge coupling 1 /g d = m s . We wish to take the string scale m s toinfinity; in the case of the (1 ,
1) string on S ( ˆ R ), since m s ˆ R = 1 /R , the radius ˆ R goesto 0 in that limit. The theory then becomes 5d N = 2 SYM, with inverse gauge coupling1 /g d = 1 /R . After the further compactification on S ( R ), we obtain at low energies 4d N = 4 SYM, with inverse gauge coupling 1 /g d = R /g d = R /R .Now, the same reasoning applied to the T -dual theory S ( R ) × S ( ˆ R ) gives 4d N = 4SYM in the m s to infinity limit, with inverse gauge coupling 1 /g (cid:48) d = R /R .Note that 1 /g (cid:48) d = g d . This is just the action of S-duality on the gauge coupling of N = 4SYM. Writing R /R ≡ Im( τ (cid:48) ), with τ (cid:48) the modular parameter of the T , we see thatS-duality is a consequence of T -duality for the pair of (1 ,
1) little string theories. Anillustration of the dualities is shown in Figure 2.Now, we extend this argument and introduce the D5 brane defects; since the D5 braneswere initially wrapping T × C , note that we can equivalently consider the defects to be D3branes at a point on T . We now argue that the S-duality action on the half BPS surfacedefects of SYM has its origin in the same T -duality of (1 ,
1) theories we presented in theprevious paragraph.First, recall that after S ( R ) compactification, the D5 branes are charged magnetically,with period: φ a = 1 / ˆ R (cid:90) S a ( m s ω I /g s + iB RR ) . In type IIB variables, we call this period β + iγ . By T-dualizing along S ( R ) we obtainD4 branes wrapping S ( R ) in the (1 ,
1) little string. Now suppose we T-dualize the D5branes along S ( R ) instead; then we have D4 branes wrapping S ( R ), in the T -dual (1 , ,
1) theories are proportional to each other,with factor R /R . But then ( β, γ ) → R /R ( β, γ ) after T -duality. The D4 branes arethen heavy, magnetic objects in one (1 ,
1) theory, while they are light, electric objects in theother. In the m s → ∞ limit, ( β, γ ) are the parameters of the Higgs field in 4d SYM. This isprecisely the action of S-duality for the Higgs field data: ( β, γ ) → Im( τ (cid:48) )( β, γ ) (2.5).Second, after T compactification, the D3 branes, which are points on T , are charged underthe RR 4-form: (cid:82) S a × (cid:102) S × S ( R ) C (4) RR , where (cid:102) S is a circle around the point defect on C . Asbefore, S ( R ) is one of the 1-cycles of T , and S a is a compact 2-cycle in the ALE space X . We call this period α . The D3 branes are also charged under (cid:82) S a × (cid:102) S × S ( R ) C (4) RR , where S ( R ) is the other 1-cycle of T ; we call this period η .Suppose we T-dualize in the S ( R ) direction. Then α becomes the period of the RR 3-formon S a × (cid:102) S ; this period is in fact an electric coupling for the holonomy of the (1 ,
1) gauge13eld around (cid:102) S . Also, η becomes the period of the RR 5-form on S a × (cid:102) S × S ( R ) × S ( ˆ R );this period is in fact a magnetic coupling for the holonomy of the (1 ,
1) gauge field around (cid:102) S . T-dualizing on S ( R ) instead, we reach the T -dual (1 ,
1) theory. We see that α getsmapped to η , while η gets mapped to − α (the minus sign arises because the 5-form isantisymmetric). So in the end, under T -duality, the periods change as ( α, η ) → ( η, − α ).Note that because the 1-cycles generating the T appear explicitly in the definition of theseperiods, T -duality does not amount to a simple rescaling of ( α, η ), as was the case for( β, γ ). In the low energy limit, we recover the S-duality of the gauge field and Theta angleparameters of 4d SYM α and η in the presence of a defect (2.6). – T ∗ ( G/ P ) sigma model and Coulomb branch of the Defect Theory– We made contact with the surface defects of Gukov and Witten after compactifying the(2 ,
0) little string on T and T-dualizing the D5 branes to D3 branes. In this process, as longas m s is kept finite, the 4d ADE quiver gauge theory that describes the D5 branes at lowenergies simply becomes a two-dimensional quiver theory for the D3 branes, with the samegauge gauge groups and fundamental matter. In the rest of this paper, we will label thislow energy
ADE quiver theory on the D3 branes, together with the set of weights W S thatspecified it, as T d . In the CFT limit, we will label the theory as T dm s →∞ . As we mentionedalready, unlike T d , the theory T dm s →∞ generically has no Lagrangian description.Now, Gukov and Witten showed that surface operators of N = 4 SYM can also be describedby a 2d sigma model T ∗ ( G/ P ), which is a moduli space of solutions to the Hitchin equations(2.3). After taking the CFT limit of the little string theory, we saw that this moduli space isalso the Coulomb branch of the (2 ,
0) CFT theory on the Riemann surface C times a circle S ( R ) (the radius R here being very big). As an algebraic variety, this Coulomb branchis singular, while T ∗ ( G/ P ) is smooth. The statement is then that the (resolution of the)Coulomb branch of the 2d ADE quiver gauge theories on the D3 branes we presented, in theappropriate m s to infinity limit, is expected to be the sigma model to T ∗ ( G/ P ). In otherterms, the Coulomb branch of T dm s →∞ can be identified with T ∗ ( G/ P ).A natural question arises: how do parabolic subgroups P in T ∗ ( G/ P ) arise from the pointof view of the defects of the (2 ,
0) little string?We will now see that to every
ADE theory T d on the D3 branes, we will be able to associatea unique parabolic subalgebra from the geometry (specifically, the non-compact 2-cycles of X ), or equivalently, from the representation theory of g (the Higgs field we introduced isvalued in the Lie algebra g , so we will speak of parabolic subalgebras rather than parabolicsubgroups); in particular, after taking the CFT limit, we will be able to read it from the14ata of the weight system W S that defines the theory T dm s →∞ .As a side note, it is known ([5, 22, 23, 24]) that T ∗ ( G/ P ) is the resolution of the Higgsbranch of different theories from the ones we have been considering. In the little stringsetup, as we reviewed, the moduli space of monopoles naturally arises as a Coulomb branchinstead of a Higgs branch. A natural guess is that those two descriptions could be related bymirror symmetry, and this is indeed the case in all the cases we could explicitly check (alldefects in the A n case, and some low rank defects in the D n case; see also [25]). We will notinvestigate this point further here, but it would be important to get a clear understanding ofthe mirror map. We now explain how the
ADE quiver theories T d determine the parabolic subalgebras of g . Because they will be so crucial to our story, we review here the mathematics of parabolicand Levi subalgebras of a Lie algebra g .A Borel subalgebra of g is a maximal solvable subalgebra of g , and always has the form b = h ⊕ m , where h is a Cartan subalgebra of g and m = (cid:80) α ∈ Φ + g α for some choice ofpositive roots Φ + . A parabolic subalgebra p is defined to be a subalgebra of g that contains aBorel subalgebra b , so b ⊆ p ⊆ g .There are many different choices of Borel subalgebras of g , but we will choose one for each g and keep it fixed. Since the Borel subalgebra is the sum of all the positive root spaces, wecan get any p by adding the root spaces associated to any closed system of negative roots.Let us extend our notations to differentiate between distinct parabolic subalgebras: Wedenote the set of positive simple roots by ∆. Take an arbitrary subset Θ ⊂ ∆. We define p Θ to be the subalgebra of g generated by b and all of the root spaces g α , with α ∈ ∆or − α ∈ Θ . Then p Θ is a parabolic subalgebra of g containing b , and every parabolicsubalgebra of g containing b is of the form p Θ for some Θ ⊂ ∆. In fact, every parabolic sub-algebra of g is conjugate to one of the form p Θ for some Θ ⊂ ∆. We state the important result:15et (cid:104) Θ (cid:105) denote the subroot system generated by Θ and write (cid:104) Θ (cid:105) + = (cid:104) Θ (cid:105) ∩ Φ + . There is a direct sum decomposition p Θ = l Θ ⊕ n Θ , where l Θ = h ⊕ (cid:80) α ∈(cid:104) Θ (cid:105) g α is a reductive subalgebra(a reductive Lie algebra is a direct sum of a semi-simple and an abelian Lie algebra), called aLevi subalgebra, and n Θ = (cid:80) α ∈ Φ + \(cid:104) Θ (cid:105) + g α , is called the nilradical of p Θ . Here, α ∈ Φ + \(cid:104) Θ (cid:105) + means that α is a positive root not in (cid:104) Θ (cid:105) + . Note that n Θ ∼ = (cid:80) α ∈ Φ − \(cid:104) Θ (cid:105) − g α ∼ = g / p Θ .Furthermore, all Levi subalgebras of a given parabolic subalgebra are conjugate to eachother [26]. We illustrate the above statements in the examples below: Example 3.1.
Consider g = A in the fundamental, three-dimensional representation. Thenthe elements in the Cartan subalgebra have the form h = ∗ ∗
00 0 ∗ . (3.24)We associate to a root α ij = h i − h j the space C E ij , where E ij is the matrix that has a +1in the i -th row and j -th column, and zeroes everywhere else. Thus, we see that b = ∗ ∗ ∗ ∗ ∗ ∗ , (3.25)and the parabolic subalgebras are p ∅ = b = ∗ ∗ ∗ ∗ ∗ ∗ , (3.26) p { α } = ∗ ∗ ∗∗ ∗ ∗ ∗ , (3.27) p { α } = ∗ ∗ ∗ ∗ ∗ ∗ ∗ , (3.28) p { α ,α } = g = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ . (3.29)Let us look at the Levi decompositions of the above:16 xample 3.2. For g = A , we get the following decompositions: p ∅ = ∗ ∗ ∗ ∗ ∗ ∗ = ∗ ∗
00 0 ∗ ⊕ ∗ ∗ ∗ = l ∅ ⊕ n ∅ , (3.30) p { α } = ∗ ∗ ∗∗ ∗ ∗ ∗ = ∗ ∗ ∗ ∗
00 0 ∗ ⊕ ∗ ∗ = l { α } ⊕ n { α } , (3.31) p { α } = ∗ ∗ ∗ ∗ ∗ ∗ ∗ = ∗ ∗ ∗ ∗ ∗ ⊕ ∗ ∗ = l { α } ⊕ n { α } , (3.32) p { α ,α } = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ = ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ⊕ = l { α ,α } ⊕ n { α ,α } . (3.33) Example 3.3.
In the table below, we show the root spaces that the Borel subalgebra of A ismade of: Θ p Θ l Θ n Θ ∅ ∗ * * *0 ∗ * *0 0 ∗ *0 0 0 ∗ ∗ ∗ ∗
00 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ : α : ( α + α ) :( α + α + α ): α : ( α + α ): α Table 1: This table illustrates the Levi decomposition of p Θ , when Θ is the empty set and g = A . p Θ consists of all the matrices in A with zeroes in the indicated places and theother entries are arbitrary. The color code shows which positive root is denoted by whichnonzero entry. We reviewed in section 2 how we could specify a defect of the little string from a set ofweights W S = { ω i } , (3.34)17ll in the orbit of some (possibly different) fundamental weights, and adding up to 0. Wemake the claim that to each set W S we can associate a parabolic subalgebra p . This map isnot injective, as many different sets of weights will typically determine the same p .As reviewed in the last section, all parabolic subalgebras of g are determined by a subset Θof the simple positive roots ∆ of g . Thus, our strategy will be to extract such a set Θ fromthe weights in W S .We do so by first computing the inner product (cid:104) α i , ω j (cid:105) , for all weights ω j in W S , and for allpositive simple roots α i of g . Then all the α i which satisfy (cid:104) α i , ω j (cid:105) = 0 (3.35)for all weights ω j in W S will make up the set Θ. There is one caveat to the above procedure:The set Θ defined as such is not invariant under the global action of the Weyl group on W S .Thus, we modify the above prescription and define Θ as the maximal such set in the Weylgroup orbit of W S . Moreover, the positive roots e γ for which (cid:104) e γ , ω i (cid:105) < , (3.36)for at least one ω i ∈ W S , form a nilradical n ; this nilradical specifies the Coulomb branchof T dm s →∞ .This n can always be obtained from the Levi decomposition p Θ = l Θ ⊕ n Θ of the parabolicsubalgebra p Θ .As mentioned already, we emphasize here that the Coulomb branch of T d is genericallybigger than the Coulomb branch of T dm s →∞ . In the little string case, the Coulomb branch of T d has dimension (cid:80) na =1 d a , where d a are the ranks of the gauge groups (here, we includethe U (1) centers of the U ( d a ) gauge groups). In the CFT limit, the space X × C can bereinterpreted as a Calabi–Yau manifold. Thus, one can use the techniques of complexgeometry to count the Coulomb moduli of T dm s →∞ as the complex structure deformationsof this Calabi–Yau [27]. For instance, for C a sphere with 3 full punctures, meaning theresidues of the Higgs fields φ ( z ) are generic, the dimension of the Coulomb branch of T dm s →∞ is the number | Φ + | of positive roots of g . Note that for A n , the full puncture of T d hasCoulomb branch dimension (cid:80) na =1 d a = | Φ + | , so in that specific case the CFT counting is thesame as the little string counting. This is generally not so for g = D n and E n . Note that the Weyl group acts on all the weights in W S simultaneously. Or equivalently, (cid:104) e γ , ω i (cid:105) > T dm s →∞ can be conveniently recovered from the representation theory of g . Indeed, by just keeping track of which positive roots satisfy (cid:104) e γ , ω i (cid:105) < ω i ∈ W S ,and not recording the actual value of the inner product, the positive roots are countingCoulomb moduli of the defect theory in the CFT limit. This point is crucial in the D n and E n cases, where higher positive root multiplicity has to be ignored to identify a nilradical of g . Example 3.4.
From Table 1 above, we will read off a nilradical from a set of weights W S for an A theory. We choose W S to be the set of all four weights in the antifundamentalrepresentation (note that they add up to 0, as they should); they make up the full punctureof A . Next, we note the following:[ − , ,
0] has a negative inner product with h − h , h − h , h − h .[ 1 , − ,
0] has a negative inner product with h − h , h − h . [ 0 , , −
1] has a negative inner product with h − h .[ 0 , ,
1] has no negative inner product with any of the positive roots.We see that all positive roots of g are accounted for, so the nilradical n Θ is constructedusing all the positive roots, and thus, Θ = ∅ . From the Levi decomposition, we thereforeidentify the parabolic subalgebra as p ∅ . This is consistent with the fact that no simple root α i has a vanishing inner product (cid:104) α i , ω j (cid:105) with all the weights ω j in W S . The discussion issummarized in Figure 3 below.Θ = ∅ ω : [ − , , ω : [ 1 , − , ω : [ 0 , , − ω : [ 0 , ,
1] 3 2 14Simple rootsubset of T dm s →∞ Weights 2d Gauge TheoryFigure 3: From the set of weights W S , we read off the parabolic subalgebra p ∅ of A (in thiscase, the choice of weights is unique up to global Z action on the set). Reinterpreting eachweight as a sum of “minus a fundamental weight and simple roots,” we obtain the 2d quivergauge theory shown on the right. The white arrow implies we take the CFT limit. Example 3.5.
As a nontrivial example, let us first study the set at the top of Figure 4 for g = D : W S = { [ − , , , , [1 , − , , , [0 , , , } . Except for the two simple roots α and α , all the other positive roots e γ satisfy (cid:104) e γ , ω i (cid:105) < ω i ∈ W S . Indeed, it is19 = { α , α } ω : [ − , , , ω : [ 1 , − , , ω : [ 0 , , ,
0] 3 4 222 1 ω : [ 1 , , , ω : [ 1 , , , ω : [ − , , , ω : [ 0 , − , ,
0] 4 6 332 2Simple rootsubset of T dm s →∞ Weights 2d Gauge TheoryFigure 4: From the two sets of weights W S , we read off the parabolic subalgebra p { α ,α } of D . Reinterpreting each weight as a sum of “minus a fundamental weight and simple roots,”we obtain two different 2d quiver gauge theories shown on the right. The white arrows implywe take the CFT limit.easy to check that (cid:104) α , ω i (cid:105) = 0 = (cid:104) α , ω i (cid:105) for all the ω i ∈ W S ; the set of positive roots weobtain defines the nilradical n { α ,α } . We then conclude from the Levi decomposition that W S characterizes the parabolic subalgebra p { α ,α } .Now, in this example, we could have very well studied a different set: W S = { [1 , , , , [1 , , , , [ − , , , , [0 , − , , } , shown at the bottom of Figure 4. It is an easy exercise to show that one identifies the samenilradical n { α ,α } as previously, so the same parabolic subalgebra p { α ,α } . This illustratesthat theories T d that have different quiver descriptions can end up determining the sameparabolic subalgebra after taking m s to infinity.In particular, the two 2d theories of Figure 4 have different Coulomb branch dimensions.In the CFT limit, we lose the quiver description of the theories, and the complex Coulombbranch dimension of both theories reduces to 10, which is the dimension of n { α ,α } .We want to emphasize that throughout this discussion, it really is the set of weights W S ,not the resulting quiver, that characterizes a defect, since two different defects in the CFTlimit can have the same quiver origin in the little string; see Figure 5 for an illustration.20 = ∅ ω : [ 0 , , , ω : [ 1 , − , , ω : [ − , , , − ω : [ 0 , , − , { α , α } ω : [ 0 , , , ω : [ 0 , − , , ω : [ 0 , , − , ω : [ 0 , , − ,
0] 3 6 342 2Simple rootsubset of T dm s →∞ Weights 2d Gauge TheoryFigure 5: Two sets of weights W S which spell out the same quiver, but denote two differentdefects; we see it is really the weights, and not quivers, that define a defect. This is clear inthe CFT limit, where two distinct parabolic subalgebras are distinguished.Note that the case of g = A n is special, in that one can start from a parabolic subalgebra of A n and obtain a 2d quiver theory from it, without any explicit reference to a set of weights;see Figure 6 for an illustration. All the resulting quivers obey equation 2.12, as a consequenceof the Levi decomposition. Indeed, the nilradical gives the exact Coulomb moduli of thequiver, read in a diagonal fashion from a matrix representative. The masses are read offfrom the Levi subalgebra, since the latter specifies a partition.Lastly, there are extra “special” punctures which cannot be obtained from the sets of weights W S as defined so far. They are special in the sense that they do not determine a parabolicsubalgebra of g . We defer the analysis of these extra theories to section 6.3. The characterization of defects so far has relied on identifying a subset of simple roots Θof the algebra g . There is yet another way the above classification can be recovered, whichrelies on identifying a Levi subalgebra of g instead. This Levi subalgebra appears in the Levidecomposition of p Θ as p Θ = l Θ ⊕ n Θ . Either way, we obtain the same parabolic subalgebra p Θ . Let us derive this explicitly.Recall that the Seiberg–Witten curve of the quiver gauge theory on the D5 branes is thespectral curve of the Higgs field φ , taken in some representation R of g ([28, 29, 30]). We21 l Θ ∅ ∗ ∗ ∗
00 0 0 ∗ { α } ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ { α } ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ { α } ∗ ∗ ∗ ∗ ∗ ∗ { α , α } ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ { α , α } ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ { α , α } ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ { α , α , α } ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ [1,1,1,1][2,1,1][3,1][2,2] n Θ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Figure 6: How to read off A n quiver theories directly from the Levi decomposition of aparabolic subalgebra; here we show n = 3. The matter content is written as a partition,specified by the Levi subalgebra. The nilradical, read off in diagonal fashion in the uppertriangular matrix, gives the Coulomb content. Note the resulting quivers automatically obeythe condition 2.12. This way of reading off a quiver gauge theory directly from a parabolicsubalgebra is a peculiarity of the g = A n case.22escribed the m s to infinity limit after which the Seiberg–Witten curve of the theory becomesthe spectral curve of the Hitchin integrable systemdet R ( φ − p ) = 0 . After T compactification, the same equation is solved by D3 branes instead, so we can saythat the above spectral curve is the Seiberg–Witten curve of the two-dimensional theory T dm s →∞ . At the root of the Higgs branch, where the Coulomb and Higgs branches meet, thisexpression simplifies: the Higgs field near a puncture of C has a pole of order one. Aftershifting this pole to z = 0, we get0 = det (cid:32) p · − (cid:80) ω i ∈W S ˆ β i ω i z + reg. (cid:33) , (3.37)where W S is the set of weights introduced in section 2. The ˆ β i are mass parameters of thegauge theory, which correspond to insertion points of the D3 branes on C .Thus, the residue at the pole diagonalizes, and the diagonal entries can be interpreted ashypermultiplet masses. So at the root of the Higgs branch, the Higgs field is described by anhonest semi-simple element of g . From this semi-simple element, we can once again recovera parabolic subalgebra p . Indeed, given a semi-simple (diagonalizable) element S (in ourcases, we’ll always have S ∈ h ), its centralizer g S := { X ∈ g (cid:12)(cid:12) [ X, S ] = 0 } (3.38)is reductive and is in fact a Levi subalgebra l S of some parabolic subalgebra p S .Since the Higgs field at a puncture of C has a pole with semi-simple residue, we can use thisconstruction to associate a Levi subalgebra l to a defect. The smallest parabolic subalgebracontaining l is then the parabolic subalgebra defining the theory. Thus, we achieved our goalof building a parabolic subalgebra, starting from a given Higgs field of a quiver theory T d . Example 3.6.
For g = A , assume that the Higgs field has a pole with semi-simple residue φ = Sz near z = 0. In the fundamental representation of sl , a possible choice for S is S = β β
00 0 − β . (3.39)The Levi subalgebra of sl associated to this semi-simple element is the centralizer of S ,23hich has the form g S = ∗ ∗ ∗ ∗
00 0 ∗ = l { α } (3.40)The parabolic subalgebra associated to this S is then p { α } from example 3.2. We now explain how the classification of surface defects presented here is connected to theclassification of codimension-two defects via nilpotent orbits.
The characterization of a puncture as studied in the 6d (2 ,
0) CFT literature [5] is givenin terms of a nilpotent orbit of the algebra: An element X ∈ g is nilpotent if the matrixrepresentative (in some faithful representation) is a nilpotent matrix. If X is nilpotent,then the whole orbit O X of X under the adjoint action of G is nilpotent – we call this anilpotent orbit. For readers interested in details and applications, the textbook [31] servesas an excellent introduction.For a simple Lie algebra, the number of such nilpotent orbits is finite, and studying theirproperties leads to many connections to different branches of representation theory. For g = A n , these orbits are labeled by Young diagrams with n + 1 boxes; for g = D n , theyare classified by Young diagrams with 2 n boxes which satisfy some conditions (see [31] fordetails.)An important fact is that for any nilpotent orbit O , the closure O is always a union ofnilpotent orbits. Furthermore, there is a maximal orbit O max whose union contains all othernilpotent orbits of g . This allows us to define an ordering on these orbits:Given two nilpotent orbits O , O ⊂ g , we define the relation O (cid:22) O : ⇔ O ⊆ O , (4.41)where O is the closure in the Zariski topology. This turns the set of all nilpotent orbits intoa partially ordered set.For A n and D n , this order corresponds to the dominance order of the Young diagrams usedto label the orbits. Example 4.1.
For an A n nilpotent orbit labeled by a partition [ d , . . . , d k ], a matrix represen-tative is given by k Jordan blocks of size d i × d i . Taking the example of n = 3, there are24ve different nilpotent orbits. Their Hasse diagram can be found below in Figure 7. Forinstance, the sub-dominant diagram [3 ,
1] labels the orbit of X [3 , = . (4.42)Figure 7: This diagram represents the inclusion relations between the nilpotent orbits of A .In [5], boundary conditions of the 6d (2 ,
0) CFT are determined by solutions to Nahm’sequations. These equations admit singular solutions near a puncture which are labeled byembeddings ρ : sl → g . Since σ + ∈ sl is nilpotent, its image ρ ( σ + ) is as well, and defines anilpotent orbit. By the Jacobson–Morozov theorem, this gives a one-to-one correspondencebetween such embeddings and nilpotent orbits. Thus, by dimensional reduction, -BPSsurface defects of 4d N = 4 super Yang–Mills are typically labeled by nilpotent orbits. Since we now have two different constructions of surface defects, we should explain how wecan relate them (a related discussion can be found in [5]):Given a parabolic subalgebra p = l ⊕ n , the nilpotent orbit O p associated to it is the maximalorbit that has a representative X ∈ O p for which X ∈ n . This induced orbit agrees withwhat is referred to as the Richardson orbit of p .25f g is A n or D n , this map can be most easily described using the semi-simple pole of theHiggs field. We represent the pole in the first fundamental representation, and assign aYoung diagram (with n + 1 or 2 n boxes, respectively) to it by counting the multiplicities ofthe eigenvalues. For A n , these Young diagrams are given by the sizes of the blocks makingup the Levi subalgebra l (see Figure 6).To this Young diagram, we can apply the so-called Spaltenstein map [32], which gives anotherYoung diagram of the same size [31]. For A n , this map is just the transposition.This Young diagram labels the nilpotent orbit describing a defect according to [5]; addingthis nilpotent element to the Higgs field describes a Coulomb deformation of the theory T dm s →∞ , meaning we are moving away from the root of the Higgs branch.Young diagrams are not available for exceptional Lie algebras, but this correspondence canbe described at any rate by using the so-called Bala-Carter labels [33, 34].Thus, we get a map which associates one of the theories in [5] to the 2d theory T dm s →∞ . Thiswas checked explicitly by comparing to the data in [35, 36, 37]. Furthermore, we will revisitthis correspondence when considering the Seiberg–Witten curves of our theories in section5.3. Example 4.2.
Let us show how to get the nilpotent orbits of A in Figure 7 from parabolicsubalgebras. To assign the right nilpotent orbit to them, we take the transpose of thepartition describing the Levi subalgebra. The resulting Young diagram labels a nilpotentorbit, which describes a Coulomb deformation of the theory. Since this partition is the sameone that is assigned to the pole of the Higgs field (in the first fundamental representation),we can also directly get the nilpotent orbit from the Higgs field data.The correspondence we get can be read off from Table 2 below.Θ O ∅ [4] { α i } i =1 , , [3,1] { α , α } [2,2] { α , α } [2,1,1] { α , α , α } [1,1,1,1]Table 2: In this table, we read off which parabolic subalgebras of A (labelled by a subset Θof positive simple roots) induce which nilpotent orbits O (labelled by Young diagrams).26 Surface Defect classification and W ( g ) -algebras In [7], the partition function of the (2 , g = ADE little string on C with certain D5 branedefects is shown to be equal to a q -deformation of the g -Toda CFT conformal block on C , with vertex operators determined by positions and types of defects. In this section, weanalyze the previous classification of defects of the little string and its relation to parabolicsubalgebras from the point of view of the dual g -type Toda CFT. Strictly speaking, thetheory dual to the little string is a q -deformation of g -type Toda, which has a deformed W ( g )-algebra symmetry, and is therefore not a CFT [38]; for an analysis in this deformedsetting, see [39]. For our purposes, it will be enough to turn off that deformation and workwith the usual Toda CFT and its W ( g )-algebra symmetry; this is the counterpart to the m s to infinity limit in the (2 ,
0) little string description, which gives the (2 ,
0) 6d CFT.
In free field formalism, the
ADE
Toda field theory can be written in terms of n = rk( g ) freebosons in two dimensions with a background charge contribution and the Toda potentialthat couples them: S T oda = (cid:90) dzd ¯ z √ g g z ¯ z [( ∂ z (cid:126)ϕ · ∂ ¯ z (cid:126)ϕ ) + ( (cid:126)ρ · (cid:126)ϕ ) QR + n (cid:88) a =1 e (cid:126)α a · (cid:126)ϕ/b ] . (5.43)The field ϕ is a vector in the n -dimensional (co-)weight space, the inner product is theKilling form on the Cartan subalgebra of g , (cid:126)ρ is the Weyl vector, and Q = b + 1 /b . The (cid:126)α a label the simple positive roots.The Toda CFT has an extended conformal symmetry, a W ( g )-algebra symmetry. The ele-ments of the Cartan subalgeba h ⊂ g define the highest weight states | (cid:126)β (cid:105) of the W ( g )-algebra.It turns out that null states of this algebra play a crucial role in classifying the defects wehave identified from the gauge theory perspective. Indeed, as shown in [14] for g = A n ,punctures can be classified via level 1 null states of the Toda CFT. This is also true for D n and E n ; in this section, we will review how to construct these null states, and we will see thatthey distinguish the same parabolic subalgebras p Θ of g we encountered before. As we will ex-plain, the set of simple roots Θ plays a very clear role in the W ( g )-algebra null state condition.We can use the vertex operators to construct highest weight states | (cid:126)β (cid:105) of the W ( g )-algebraby acting on the vacuum, | (cid:126)β (cid:105) = lim z → e (cid:126)β · (cid:126)φ ( z ) | (cid:105) . These give rise to a Verma module over | (cid:126)β (cid:105) by acting with W ( g )-algebra generators. For some of the | (cid:126)β (cid:105) , these representations aredegenerate, because they contain a null state; we say that | χ (cid:105) , in the Verma module over27 (cid:126)β (cid:105) , is a level k null state of the W ( g )-algebra if for all spins s : W ( s ) n | χ (cid:105) = 0 , ∀ n > , (5.44) W (2)0 | χ (cid:105) = ( E β + k ) | χ (cid:105) , (5.45)where W (2)0 | (cid:126)β (cid:105) = E β | (cid:126)β (cid:105) .The Verma module over | (cid:126)β (cid:105) contains such a null state at level k if the Kaˇc determinant atlevel k vanishes. For any semi-simple g , this determinant at level k is a non-zero factor times (cid:89) (cid:126)α ∈ Φ m,n ≤ k (cid:16) ( (cid:126)β + α + (cid:126)ρ + α − (cid:126)ρ ∨ ) · (cid:126)α − ( (cid:126)α mα + + nα − ) (cid:17) p N ( k − mn ) , (5.46)where p N ( l ) counts the partitions of l with N colours and Φ is the set of all roots of g [40].For us, ( α + , α − ) = ( b, /b ).Note that this determinant is invariant only under the shifted action of the Weyl group, (cid:126)β (cid:55)→ w ( (cid:126)β + α + (cid:126)ρ + α − (cid:126)ρ ∨ ) − ( α + (cid:126)ρ + α − (cid:126)ρ ∨ ) , (5.47)where w is the ordinary Weyl action.If g is simply laced, and (cid:126)α = (cid:126)α i is a simple root, the condition that this determinant vanishescan be phrased as (cid:126)β · (cid:126)α i = (1 − m ) α + + (1 − n ) α − . (5.48)We see that any (cid:126)β with (cid:126)β · (cid:126)α i = 0 for a simple root (cid:126)α i gives rise to a level 1 null state, and if Q := ( α + + α − ) →
0, a null state at level 1 occurs if (cid:126)β · (cid:126)α = 0 for any (cid:126)α ∈ Φ. Furthermore,in this limit, the shift in the Weyl group action disappears. It is enough to work in this“semi-classical” limit for our purposes, so we will set Q to 0 in what follows.We can explicitly construct these null states: Consider the screening charge operators Q ± i = (cid:73) dz πi exp( iα ± (cid:126)α i · (cid:126)φ ) (5.49)and observe that [ W ( k ) n , Q ± i ] = 0 . (5.50)The level 1 null state is then S + i | (cid:126)β − α + (cid:126)α i (cid:105) . (5.51)Explicit forms of these null states for g = A n or D n are shown in the examples of section 7.28he relation to the parabolic subalgebras introduced in section 3 is immediate: we simplyassociate a generic null state | (cid:126)β (cid:105) satisfying (cid:126)β · (cid:126)α i = 0 ∀ (cid:126)α i ∈ Θwith the parabolic subalgebra p Θ .We also note that this (cid:126)β defines a semi-simple element in g ; this is just the residue of theHiggs field at the puncture, as explained in section 3.3.We show next that these these null states induce relations in the Seiberg–Witten curve of thetheory T dm s →∞ . Indeed, the Seiberg–Witten curve of T dm s →∞ (3.37) can be obtained from afree field realization of the W ( g )-algebra. We will simply read off the null states as relationsbetween the curve coefficients. Generically, these relations only involve semi-simple elementsof the algebra g . In 5.3, we will see these relations are still preserved when one additionallyintroduces certain nilpotent deformations.When working in the q -deformed setting, the formula for the Kaˇc determinant is an expo-nentiated version of (5.46) [41]. This implies that the null states can be defined analogouslyfor the q -deformed W ( g )-algebra. W ( g ) -algebras As we reviewed previously, the Seiberg–Witten curve of T dm s →∞ is the spectral curve equationdet R ( φ − p ) = 0 . (5.52)In our case, φ has a simple pole such that the residue is a semi-simple element of g , whichwe can write as (cid:126)β = (cid:88) ω i ∈ W S ˆ β i ω i . (5.53)To find the curve near the pole, which we assume to be at z = 0, we can just choose some conve-nient representation R , where the residue of φ is diagonal, and given by diag( β , β , . . . ) =: M .Then φ = Mz + A , with A a generic element in g .We now expand eq. (5.52) and write the curve as0 = det (cid:18) − p · + Mz + A (cid:19) = ( − p ) dim( R ) + (cid:88) s p dim( R ) − s ϕ ( s ) , (5.54)29here ϕ ( s ) is a meromorphic differential, i.e. ϕ ( s ) = s (cid:80) k =0 ϕ ( s ) k z k , where the ϕ ( s ) k are regularfunctions of β i and a ij (the entries of A ).Since M is diagonal, this determinant just picks up the diagonal terms a ii of A , which weidentify with the gauge couplings of the quiver theory.Now, we can also construct the Seiberg–Witten curve of T dm s →∞ from the W ( g )-algebra[14, 42]: For this, we need to perform a Drinfeld–Sokolov reduction to obtain explicit W ( g )-algebra generators in the free field realization . Setting Q = 0 gives us a direct connectionto the two dimensional quiver defined by the semi-simple element (cid:126)β ∈ g (cf. section 3.3):We can identify the poles of the Seiberg-Witten differentials with expectation values of these W ( g ) algebra generators in the state | (cid:126)β (cid:105) : ϕ ( s ) = (cid:104) (cid:126)β | W ( s ) | (cid:126)β (cid:105) . (5.55)We checked this relation explicitly for A n and D n theories. Example 5.1.
Let us look at the curve describing the full puncture for g = A :Take the fundamental three-dimensional representation of sl and write M = β β
00 0 − β − β , A = a a a a a a a a − a − a . (5.56)Then the curve can be expanded, and we read off the differentials. For example, ϕ (2) , thecoefficient multiplying p , has the form ϕ (2) = ϕ (2)2 z + ϕ (2)1 z + ϕ (2)0 , (5.57)where ϕ (2)2 = 12 (cid:0) β + β + ( − β − β ) (cid:1) := 12 ( (cid:126)β ) , (5.58) ϕ (2)1 = a (2 β + β ) + a ( β + 2 β ) . (5.59)Furthermore, ϕ (3)3 = − β β − β β ,ϕ (3)2 = a ( − β β − β ) + a ( − β β − β ) . (5.60) We thank Kris Thielemans for sending us his
OPEDefs.m package [43], which allowed us to do thesecalculations g = A , define X j = i∂φ j . In the fundamental representation, X + X + X = 0. Then the generators are just the energy momentum tensor T ( z ) = W (2) ( z ) = 13 (: X X : + : X X : + : X X : − : X X : − : X X : − : X X :)and the spin 3 operator W (3) ( z ) = : (cid:18) X − X − X (cid:19) · (cid:18) − X + 23 X − X (cid:19) ·· (cid:18) − X − X + 23 X (cid:19) : . For the full puncture, we find at once that (cid:104) (cid:126)β | L | (cid:126)β (cid:105) is equal to ϕ (2)2 from above, while (cid:104) (cid:126)β | W (3)0 | (cid:126)β (cid:105) is equal to ϕ (3)3 , as expected. For the level 1 modes, one finds (cid:104) (cid:126)β | W (2) − | (cid:126)β (cid:105) = (2 β + β ) (cid:104) (cid:126)β | j − | (cid:126)β (cid:105) + ( β + 2 β ) (cid:104) (cid:126)β | j − | (cid:126)β (cid:105) , (5.61) (cid:104) (cid:126)β | W (3) − | (cid:126)β (cid:105) = ( − β β − β ) (cid:104) (cid:126)β | j − | (cid:126)β (cid:105) + ( − β − β β ) (cid:104) (cid:126)β | j − | (cid:126)β (cid:105) , (5.62)where j ik denotes the k -th mode of X i .Observe that this has the form (5.60) if we identify (cid:104) (cid:126)β | j i − | (cid:126)β (cid:105) with the i -th gauge couplingconstant.For more complicated defects, the W ( g )-algebra generators will have terms that are derivativesof X — these are set to zero in the semiclassical Q → Punctures that are not fully generic are determined by semi-simple elements (cid:126)β ∈ g whoseVerma modules contain null states at level one. Since the eigenvalues of the level one W ( g )-algebra generators appear as coefficients in the curve, the existence of these null statesinduces some relations between these coefficients.For g = A n and D n in the fundamental representation, the pattern is easy to see. Thecondition (cid:126)β · (cid:126)α = 0 for some positive root (cid:126)α will cause some of the entries of M =diag( β , β , . . . ) to be equal to each other; if the entry β i occurs k times, we get null statesby letting the operator (cid:88) s β si W (dim( R ) − s ) − , (5.63)and its k − β i , act on | (cid:126)β (cid:105) . Thus, each theory induces somecharacteristic null state relations which are realized in the Seiberg–Witten curve.31e now use this observation to connect these curves to nilpotent orbits: note that all thecurves considered so far were written asdet (cid:18) − p · + Mz + A (cid:19) = 0 (5.64)for some diagonal M and a generic A in g . In the nilpotent orbit literature, the curvesconsidered in [36, 35, 37] have the formdet (cid:18) − p · + Xz + A (cid:19) = 0 , (5.65)where, again, A is a generic element in g , and X is a representative of a nilpotent orbit O X .We can now simply combine these two poles and form a curve of the formdet (cid:18) − p · + e Xz + Mz + A (cid:19) = 0 , (5.66)where M is semi-simple, X ∈ O X is nilpotent and e is a parameter. We will test thecorrespondence between theories defined by nilpotent orbits and theories defined by semi-simple elements from this vantage point. Recall from section 4 that the semi-simple element M ∈ g induces a nilpotent orbit O . We observe the following facts: • Whenever an orbit O (cid:48) (cid:22) O , it is always possible to find an X ∈ O (cid:48) such that all thenull state relations of the curve (5.64) are still satisfied by the curve (5.66). • Whenever an orbit O (cid:48) (cid:14) O , it is never possible find an X ∈ O (cid:48) such that all the nullstate relations of the curve (5.64) are still satisfied by the curve (5.66).This gives a prescription for allowed deformations; from the perspective of the theory T dm s →∞ ,this corresponds to leaving the root of the Higgs branch by turning on certain Coulombmoduli. Example 5.2.
For g = A , the only interesting state is (cid:126)β = ( β , β , − β ); we can get thelevel one coefficients of the curve by setting β = β in example 5.1: φ (2)1 = (cid:104) W (2) − (cid:105) = 3 β ( a + a ) ,φ (3)2 = (cid:104) W (3) − (cid:105) = − β ( a + a ) , (5.67)so we see that (cid:104) W (3) − (cid:105) + β (cid:104) W (2) − (cid:105) = 0 . (5.68)32f we now add the nilpotent element X = , then φ (2)1 = 3 β ( a + a ) + e a ,φ (3)2 = − β ( a + a ) − e β a , (5.69)and the null state relation (5.68) is still satisfied. Up until now, we have been been using the little string theory as a tool to derive codimension-two defects of the (2 ,
0) CFT, and in particular exhibit the parabolic subalgebras that arisein that limit. In this section, we keep m s finite, and comment on the classification of defectsof the (2 ,
0) little string proper. In particular, as we have emphasized in section 3, when wework with the little string and not its conformal field theory limit, parabolic subalgebras arein general not visible (exceptions are when g = A n , as we had illustrated in Figure 6, and ina few low rank cases when g = D n and g = E n .)We also address a question that was not answered so far: certain nilpotent orbits of g arenot induced from any parabolic subalgebra. The simplest example would be the minimalnilpotent orbit of D . These denote nontrivial defects of the (2 ,
0) CFT, so one should askif they arise at all from the little string, since so far all the quiver theories we constructeddistinguished a parabolic subalgebra. We will see that these exotic defects do indeed originatefrom the little string. To properly analyze them, we must first understand how flowing onthe Higgs branch of a defect is realized from representation theory. T d and Higgs flow as Weight Addition In this section, we describe an effective and purely group-theoretical way to flow on theHiggs branch of different 2d quiver theories T d , for any simple g . We show that in the A n case, this agrees with standard brane engineering and Hanany-Witten transitions [44]. Asan application, this procedure will be used to analyze the punctures that fall outside of theparabolic subalgebra classification we have spelled out so far.Our setup will be the usual one in this paper: we consider the quiver gauge theory T d thatdescribes the low energy limit of D3 branes wrapping 2-cycles of the ALE space X times C .33he D3 branes are points on the Riemann surface C and on the torus T .We claim that moving on the Higgs branch of T d translates to a weight addition procedure inthe algebra: this makes use of the fact that a weight belonging to a fundamental representationcan always be written as the sum of new weights. Each of them should be in the orbit ofsome fundamental weight (the two orbits do not have to be the same, here), while obeyingthe rule that no subset adds up to the zero weight.After moving on the Higgs branch of T d , we obtain a new 2d theory T d (cid:48) with a new setof weights, but the same curve. When the gauge theory can be engineered using branes,going from T d to T d (cid:48) is called a Hanany–Witten transition [44]. There, as we will see, aD5 brane passing an NS5 brane creates or removes D3 branes stretching between the two.When a brane construction is not available, the weight description we give is still valid, foran arbitrary simply laced Lie algebra..Note that this weight addition formalism also gives a generalization of the S-configuration[44]: No weight in a fundamental representation can ever be written as the sum of twoidentical weights.In the A n case, where we have a brane picture, this statement translates immediately to theS-rule, which is then automatically satisfied. This argument is however applicable to D n and E n theories as well, so this gives an ADE -type S-rule.
For A n theories and D n theories obtainable by an orbifolding procedure, the above discussioncan be realized by brane engineering of the theory. We can conveniently represent the weightsof the algebra, and in particular, their Dynkin labels, using a configuration of D3 branesstretching between NS5’s and D5 branes. To see how this works, let us focus on the i-thDynkin label of a weight: • A D3 brane coming from the left ending on the i − th NS5 contributes − • A D3 brane coming from the right ending on the i -th NS5 contributes +1 to theweight’s i -th label. • A D3 brane coming from the left ending on the i + 1-th NS5 contributes +1 to theweight’s i -th label. • A D3 brane coming from the right ending on the i + 1-th NS5 contributes − i -th label. 34 − , , − w + α + α [0 , − , , − w [ − , − w [ − , − w [0 , − , , − − w + α [0 , − , , − w + α + α Figure 8: How to read off weights from a system of D3, D5, and NS5 branes. • Finally, a D5 brane present between the i -th and i + 1-th NS5’s contributes − i -th label.All in all, a D3 brane stretching between a D5 brane and an NS5 brane (while possibly goingthrough some other NS5 branes) produces a weight, whose Dynkin labels are a combinationof 1’s, − T d we would write based on the weight data W S . See Figure 9 for someexamples. 35 , − , ,
0] + [ − , , , − w + α − w [0 , − , , −
1] + [0 , , − , − w + α − w + α [0 , − , , − w [0 , − , ,
0] + [0 , , − , − w + α + α − w [0 , − , ,
1] + [0 , , , − − w + α + α − w Figure 9: Flowing on the Higgs branch of T d : starting from the theory in the middle, theseare all the theories one can obtain by replacing the weight on node 2 by a sum of two weights.The top picture shows the detailed brane picture for each of the quivers. These all have alow-energy 2d quiver gauge theory description (the ones shown below). At the root of theHiggs branch, the partition functions of all 5 theories are equal. We finally come to the description of defects in the little string that happen to fall outsidethe parabolic subalgebra classification we have spelled out so far.Suppose we pick a weight in the i -th fundamental representation. Unless it is the null weight,36 ω ω ω ω : [ − , , , ,
0] = − w ω : [ − , , , ,
0] = − w + α +2 α + α + α − , , , ,
0] of D . We started with A theory andperformed a Z orbifold to obtain the picture. This weight can be written in two ways: by placingthe D5 brane between the first two NS5 branes (top), the weight is written in an “appropriate way”. By placing the D5 brane between the “wrong” set of NS5 branes (bottom), the resulting quiverwill be unpolarized and will not distinguish a parabolic subalgebra. ω ω ω : [0 , , ,
0] = − w + α + 2 α + α + α , , ,
0] of D , realized here with branes. Westarted with A theory and performed a Z orbifold to obtain the picture. This theory does notdistinguish a parabolic subalgebra. it is in the orbit of one and only one fundamental weight, say the j -th one. In our entirediscussion so far, and in all the examples of [7], we had i = j . In terms of the gauge theory,if all weights are chosen so that i = j , then T dm s →∞ distinguishes a parabolic subalgebra, asexplained in section 3.2. We call such a 2d theory polarized . However, in the D n and E n cases, it can also happen that i (cid:54) = j , or that the weight we pickis the null weight. See Figures 10.a and 10.b. In terms of the gauge theory, if at least one of the weights in W S falls under this category, the theory T dm s →∞ does not distinguish a The terminology here comes from the fact that the parabolic subgroup P in T ∗ ( G/ P ) is often called apolarization of some nilpotent orbit O , through the resolution map T ∗ ( G/ P ) → O , with O the closure of O . unpolarized .We saw in section 6.1 that if we start with a polarized theory T d , then after flowing on theHiggs branch, we still end up with a polarized theory T d (cid:48) . What happens to unpolarizedtheories? If we start with a such a theory T d , then after moving on the Higgs branch, it is infact always possible to end up with a theory T d (cid:48) that is polarized. This resulting polarizedtheory T d (cid:48) is of course highly specialized, since some masses have to be set equal to eachother as a result of the Higgs flow.This is the viewpoint we take to analyze all unpolarized theories: we will flow on theHiggs branch until they transition to polarized theories. In practice, it means that every“problematic” weight in an unpolarized theory can be written as a sum of weights to give apolarized theory. Note that for A n , every quiver theory T d is polarized, while this is notthe case for D n and E n . An illustration of how one can start with an unpolarized theoryand arrive at a polarized theory is shown in Figure 10 below. ω (cid:48) ω (cid:48)(cid:48) ω (cid:48) ω (cid:48)(cid:48) ω ω UnpolarizedPolarized ω (cid:48) : [ − , , ,
0] = − w ω (cid:48)(cid:48) : [ 1 , , ,
0] = − w +2 α + 2 α + α + α ω : [0 , , ,
0] = − w + α + 2 α + α + α Figure 10: The brane picture for the zero weight of D (top of the figure), which makes upan unpolarized theory at low energies. It is obtained after Z orbifolding of A . The D5branes sit on top of the D3 branes, and all the D3 branes are stacked together. After flowingon the Higgs branch, we end up with a polarized theory, with the two masses equal to eachother.One should ask what happens to unpolarized defects in the context of Toda CFT. As wehave seen in section 5, polarized defects are described by momenta that obey null state38elations of the corresponding W ( g )-algebra. This is consistent with what can be found inthe class S literature [5]; for instance, for the minimal puncture of D from Figure 10, thedefect in the CFT limit is predicted to have no flavor symmetry. In particular, it is unclearwhat vertex operator one would write in D -Toda; indeed, in the little string formalism, thedefect is the null weight, which suggests a trivial conformal block with no vertex operatorinsertion! To investigate this issue more carefully, it is useful to keep m s finite and workin the little string proper; there, a computation in the spirit of [45, 46, 7], shows that thepartition function of T d is in fact not a q -conformal block of D Toda, due to subtleties ofcertain non-cancelling fugacities. In other words, the claim that the partition function of T d is a q -conformal block of g -type Toda fails precisely when T d is an unpolarized defect,and only for those cases. From the considerations above, we get a complete list of the D3 brane defects of the (2 , C × T , and which preserve conformality (in a 4d sense, before T compactification). These are the polarized and unpolarized punctures we presented. Eachof them is characterized by a set of weights in g , which produce a quiver gauge theory atlow energies, satisfying 2.12. Enumerating the (2 ,
0) little string defects, for a given g , isthen a finite counting problem.For D n and E n , we find that the number of resulting theories T d one obtains from specifyinga set of weights, although finite, far exceeds the number of the CFT defects as enumarated in[5]. What is happening is that in the CFT limit, many distinct defect theories T d typicallycoalesce to one and the same defect theory T dm s →∞ . The discussion in Figure 5 illustratesthis phenomenon. See also Figure 11 for the example of all theories T d describing a genericfull puncture of the D little string.An important point is that even though we focused on the case of a sphere with two fullpunctures and an additional arbitrary puncture, the formalism we developed is automaticallysuited to study a sphere with an arbitrary number of defects.Simply choose a set of weights W S , as done before. If there are k subsets of weights whichadd up to zero in W S , then the little string is in fact compactified on a sphere with k + 2punctures. This just follows from linearity of equation (2.12). In particular, for the caseof the sphere with 3 punctures we have been analyzing, there are then no proper subsetof weights in W S that add up to zero. An immediate consequence is that not all quivertheories characterize a sphere with two full punctures and a third arbitrary one: some quiversrepresent composite arbitrary defects (and two full punctures). See Figure 12.39 5 333 11 4 7 441 2 11 3 6 441 223 5 341 13 3 5 431 31 4 6 342 1 2 4 6 432 1 2Figure 11: All D
2d quiver theories one obtains from a set W S of 5 weights, and which alldenote full punctures. In the CFT limit, all these theories produce the same full puncture,denoted by the parabolic subalgebra p ∅ . In particular, the Coulomb branch of T dm s →∞ forall these theories has dimension twelve.2 2 22 2 3 6 9 6 363 ω : [ − , , ω : [ 1 , , ω : [ 1 , − , ω : [ − , , ω : [ 0 , , , , , ω : [ 0 , , , , , − ω : [ 0 , , , , , A , with two maximal (full) punctures andtwo minimal (simple) punctures, both denoted by the parabolic subalgebra p { α ,α } . Thetwo simple punctures indicate that there are two subsets of weights in W S that add up tozero. In this specific example, the fact that the weights [1 , − ,
0] and [ − , ,
0] denote asimple puncture can easily be seen by applying a Weyl reflection about the first simple rootof A . Right: a four-punctured sphere of E , with two maximal punctures and two otherpunctures; the first of these is the minimal puncture, denoted by the zero weight in the6-th fundamental representation, and is unpolarized. The second puncture is polarized, anddistinguishes the parabolic subalgebra p { α ,α ,α ,α ,α } .40s a final remark, let us mention that the techniques we used in this note to study codimension-two defects of the little string can also be applied to analyze codimension-four defects; thesedefects do not originate as D5 branes in the (2 ,
0) little string, but as D3 branes instead,before considering any T compactification. 41 Examples A n Examples n n − n − ∅ n + 1 n − n − n − { α i } , i = 1 , . . . , n . n − \ { α } orΘ = ∆ \ { α n } .1 1 Figure 13: The top quiver is the full puncture, denoted by the partition [1 n +1 ]. The middlequiver is the next to maximal puncture, with partition [2 , n − ]. The bottom quiver is thesimple puncture. It is denoted by the partition [ n, p ∆ \{ α } and p ∆ \{ α n } .We can explicitly write the parabolic subalgebras in some representation; for A n , it iscustomary to do so in the fundamental representation. Therefore, in what follows, thematrices are valued in sl ( n + 1); a star ∗ i denotes a nonzero complex number, and the label“ i ” stands for the positive root e i . A star ∗ − i denotes a nonzero complex number, and thelabel “ − i ” stands for the negative root − e i . Unless specified otherwise, a partition refers toa semi-simple element denoting the Higgs field structure of the theory. These partitions arerelated to the nilpotent element partitions from section 4 by transposition in the A n case,and more generally by the Spaltenstein map (cf. [31]).42 .1.1 Maximal (“full”) Puncture We start with the set W S of all weights in the n -th fundamental representation (antifun-damental). Writing w i for the highest weight of the i -th fundamental representation, theweights can be written as: ω = − w ω = − w + α ω = − w + α + α ... = ... ω n +1 = − w + α + α + . . . + α n , from which we read the top 2d quiver in Figure 13. This is called the full puncture. Wecompute the inner product of the weights with the positive roots: ω ≡ [ − , , , . . . ,
0] has a negative inner product with: α , α + α , . . . , α + α + . . . + α n ω ≡ [1 , − , , . . . ,
0] has a negative inner product with: α , α + α , . . . , α + α + . . . + α n ω ≡ [0 , , − , , . . . ,
0] has a negative inner product with: α , α + α , . . . , α + α + . . . + α n ... ... ω n +1 ≡ [0 , . . . , , ,
1] has no negative inner product with any of the positive roots.Since all of the positive roots of g have a negative inner product with some weight, theydefine the nilradical n ∅ . The parabolic subalgebra is p ∅ . It is denoted by the partition [1 n +1 ],which is immediately readable from the Levi subalgebra with symmetry S ( U (1) n +1 ).43he Levi decomposition gives: p ∅ = ∗ ∗ ∗ · · · ∗ ... +( n − ∗ ... + n ∗ ∗ · · · · · · ∗ ... + n ... . . . . . . . . . ... ...... . . . . . . ∗ ( n − ∗ ( n − n ... . . . ∗ ∗ n · · · · · · · · · ∗ , with p ∅ = l ∅ ⊕ n ∅ , where l ∅ = ∗ · · · · · · · · · ∗ . . . ...... . . . . . . . . . ...... . . . . . . . . . ...... . . . ∗ · · · · · · · · · ∗ and n ∅ = ∗ ∗ · · · ∗ ... +( n − ∗ ... + n ∗ · · · · · · ∗ ... + n ... . . . . . . . . . ... ...... . . . . . . ∗ ( n − ∗ ( n − n ... . . . 0 ∗ n · · · · · · · · · . We see explicitly that the nonzero inner products (cid:104) e γ , ω i (cid:105) make up the i -th line of thenilradical n ∅ .In this example, there is in fact one other set W S that singles out the nilradical n ∅ ; it is theset of all weights in the first fundamental representation of A n . The resulting 2d quiver isagain the top one in Figure 13, but with reversed orientation.44ow we analyze this defect from the Toda CFT perspective: starting from our set W S andrecalling that β = (cid:80) |W S | i =1 ˆ β i w i , W S defines the Toda momentum vector β . We can writethis momentum β explicitly as the semi-simple element diag( β , β , . . . , β n +1 ), where all theentries add up to 0. One checks at once that the commutant of this element is the Levisubalgebra l ∅ written above.The flag manifold T ∗ ( G/ P ) associated to this defect also appears as the resolution of theHiggs branch of the same quiver, n n − n + 1 which is an instance of mirror symmetry, since the complete flag is self-mirror. Furthermore,it is easy to see from the method of section 4.2 that the nilpotent orbit associated to thistheory is the maximal nilpotent orbit of A n , denoted by the partition [ n + 1]. We start by constructing the set W S : Consider all the n + 1 weights of the n -th fundamentalrepresentation. For each 1 ≤ i ≤ n , the set contains two unique weights ω i and ω i +1 suchthat α i = ω i − ω i +1 , with α i the i -th simple root. Remove ω i and ω i +1 from the set, andreplace them with the single weight ω (cid:48) ≡ ω i + ω i +1 . ω (cid:48) is always a weight in the n − A n . Therefore, the set we consider is made of n − n -th fundamental representation, and the weight ω (cid:48) in the n − n weightsdefine a valid set W S . The weights once again define a 2d quiver gauge theory T d ; it isshown in the middle of Figure 13. All of the positive roots except the i -th simple root α i have a negative inner product with at least one weight ω i ∈ W S , so these positive rootsdefine the nilradical n { α i } .For a given simple root α i , the parabolic subalgebra is then p { α i } . It is denoted by thepartition [2 , n − ], which is immediately readable from the Levi subalgebra with symmetry S ( U (2) × U (1) n − ). 45he Levi decomposition gives: p { α i } = ∗ ∗ ∗ · · · · · · · · · · · · · · · ∗ ... +( n − ∗ ... + n ∗ ∗ · · · · · · · · · · · · · · · · · · ∗ ... + n ... . . . . . . . . . ... ...... . . . . . . . . . ... ...... 0 ∗ ∗ i ... ...... ∗ − i ∗ . . . ... ...... 0 . . . . . . ... ...... . . . . . . ∗ ( n − ∗ ( n − n ... . . . ∗ ∗ n · · · · · · · · · · · · · · · · · · · · · ∗ , with p { α i } = l { α i } ⊕ n { α i } , where l { α i } = ∗ · · · · · · · · · · · · · · · ∗ . . . ...... . . . . . . 0 ...... 0 ∗ ∗ i ...... ∗ − i ∗ ∗ · · · · · · · · · · · · · · · ∗ n { α i } = ∗ ∗ · · · · · · · · · · · · · · · ∗ ... +( n − ∗ ... + n ∗ · · · · · · · · · · · · · · · · · · ∗ ... + n ... . . . . . . . . . ... ...... . . . . . . ∗ i − ... ...... . . . 0 0 ... ...... 0 0 ∗ i +1 ... ...... . . . . . . . . . ... ...... . . . . . . ∗ ( n − ∗ ( n − n ... . . . 0 ∗ n · · · · · · · · · · · · · · · · · · · · · . There is in fact another set W S that spells out the nilradical n { α i } for fixed α i ; just as forthe full puncture, the corresponding 2d quiver would be the middle one in Figure 13, butagain with reversed orientation.Now we rederive this result from the Toda CFT perspective: consider once again the set W S . We define the momentum vector β from β = (cid:80) |W S | i =1 ˆ β i ω i . It is easy to check that (cid:104) β, α i (cid:105) = 0for the simple root α i , since β has a unique 0 as its i -th Dynkin label. This defines a nullstate at level 1 in the CFT. One can easily check that there is only one other set W S suchthat (cid:104) β, α i (cid:105) = 0; this alternate choice gives the reflection of our 2d quiver. Also note thatthe commutant of the semi-simple element β is the Levi subalgebra l α i written above in thefundamental representation.We make the following important observations: • This puncture is in fact described by many sets W S . To obtain them, one simplyconsiders all possible Weyl group actions that preserve the root sign: w ( α i ) must bea positive root. Then all possible momenta are given by β (cid:48) = w ( β ). Note that thecondition (cid:104) β, α i (cid:105) = 0 is Weyl invariant: (cid:104) β, α i (cid:105) = (cid:104) β (cid:48) , w ( α i ) (cid:105) . Therefore, from the CFTperspective, the momentum of this different theory satisfies instead: (cid:104) β (cid:48) , w ( α i ) (cid:105) = 0 . w ( α i ) is a positive non -simple root, this is strictly speaking a higher thanlevel-1 null state condition of A n -Toda. As explained in section 5.1, this higher leveldistinction is not relevant in the semi-classical limit (cid:126) → Q → W S can then be written at level 1, it is (cid:16) W ( n +1) − + β i W ( n ) − + β i W ( n − − + · · · + β n − i W (2) − (cid:17) | (cid:126)β (cid:105) . (7.70)Here, W ( j ) − is the mode − j generator, and β i is the i -th entry of β , writtenin the fundamental representation, where i labels the singled-out simple root α i . Theeigenvalues of the W ( j )0 modes are then functions of all the entries of β . • All of the many different sets W S mentioned above give rise to the same 2d quivergauge theory, in the middle of Figure 13. • The definition of the weight ω (cid:48) ≡ ω i + ω i +1 above is an illustration of the weightaddition rule from section 6.1. This corresponds to moving on the Higgs branch, andtransitioning from the top quiver to the middle quiver in Figure 13. In gauge theoryterms, when the hypermultiplet masses for ω i and ω i +1 of the full puncture are setequal, one can transition from the top 2d theory to the middle 2d theory, which has asingle hypermultiplet mass for ω (cid:48) instead. • The nilpotent orbit associated to this puncture is the unique subregular nilpotent orbitof A n , with partition [ n, T ∗ ( G/ P ) also appears as the resolution of the Higgs branch of the quiver n − n − n + 1 which is again mirror to ours. We start by constructing the set W S . Writing w i for the highest weight of the i -th fundamentalrepresentation, we define W S as: ω = − w n ,ω = − w + α + α + . . . + α n . ω ≡ [0 , , . . . , , −
1] has a negative inner product with: α n , α n + α n − , . . . , α n + α n − + . . . + α ω ≡ [0 , , . . . , ,
1] has no negative inner product with any of the positive roots.So the only positive roots of g that have a negative inner product with some weight ω i ∈ W S are α n , α n + α n − , . . . , α n + α n − + . . . + α , and they define the nilradical n ∆ \{ α n } . Theparabolic subalgebra is then p ∆ \{ α n } . It is denoted by the partition [ n, S ( U ( n ) × U (1)). The Levi decompositiongives: p ∆ \{ α n } = ∗ ∗ ∗ · · · · · · ∗ ... +( n − ∗ ... + n ∗ − ∗ ∗ · · · · · · ∗ ... +( n − ∗ ... + n ∗ − (1+2) ∗ − ∗ · · · · · · ∗ ... +( n − ∗ ... + n ... ... ... . . . · · · ... ...... ... ... · · · . . . ∗ ( n − ∗ ( n − n ∗ − (1+ ... +( n − ∗ − (2+ ... +( n − ∗ − (3+ ... +( n − · · · ∗ − ( n − ∗ ∗ n · · · ∗ , with p ∆ \{ α n } = l ∆ \{ α n } ⊕ n ∆ \{ α n } , where l ∆ \{ α n } = ∗ ∗ ∗ · · · · · · ∗ ... +( n − ∗ − ∗ ∗ · · · · · · ∗ ... +( n − ∗ − (1+2) ∗ − ∗ · · · · · · ∗ ... +( n − · · · ... ...... ... ... · · · . . . ∗ ( n − ∗ − (1+ ... +( n − ∗ − (2+ ... +( n − ∗ − (3+ ... +( n − · · · ∗ − ( n − ∗
00 0 0 · · · ∗ n ∆ \{ α n } = · · · · · · · · · · · · ∗ ... + n ... . . . ... ∗ ... + n ... . . . ... ∗ ... + n ... . . . ... ...... . . . ... ∗ ( n − n · · · · · · · · · · · · ∗ n · · · · · · · · · · · · . We see explicitly that the non-zero inner products (cid:104) e γ , ω i (cid:105) give the last column of thenilradical n ∆ \{ α n } .Now we rederive this result from the CFT perspective: consider once again the set W S . Wedefine the momentum vector β from β = (cid:80) |W S | i =1 ˆ β i ω i . It is easy to check that (cid:104) β, α i (cid:105) = 0 , i = 1 , , . . . , n − β has a 0 as its i -th Dynkin label for i = 1 , , . . . , n −
1. This defines many level 1 nullstates in the CFT. One can easily check that no other set W S satisfies the above vanishinginner product conditions. Also note that the commutant of the semi-simple element β is theLevi subalgebra l ∆ \{ α n } written above in the fundamental representation.We make the following important observations: • This puncture is in fact described by many sets W S . To obtain them, one simplyconsiders all possible Weyl group actions that preserve the root sign: w ( α i ) must be apositive root; the details are in the previous example. The upshot is once again thatthe explicit null states for all these 2d theories can be written at level 1; they are: (cid:16) W ( n +1) − + βW ( n ) − + β W ( n − − + . . . + β n − W (2) − (cid:17) | (cid:126)β (cid:105) , (7.71)50nd the n − β : (cid:16) W ( n ) − + 2 βW ( n − − + . . . + ( n − β n − W (2) − (cid:17) | (cid:126)β (cid:105) (cid:16) W ( n − − + . . . + ( n − n − β n − W (2) − (cid:17) | (cid:126)β (cid:105) ... W (2) − | (cid:126)β (cid:105) Here, W ( j ) − is the mode − j generator, and (cid:126)β =diag( β, β, . . . , β, − nβ ),written in the fundamental representation. The eigenvalues of the W ( j )0 modes areagain functions of β . • All the many different sets W S mentioned above give rise to the same 2d quiver gaugetheory, in the bottom of Figure 13, and they all characterize the parabolic subalgebra p ∆ \{ α n } , even if not directly readable from the positive root inner products with theWeyl reflected weights. • Once again, we can use the weight addition procedure to move on the Higgs branch,and transition from the top quiver to the bottom quiver in Figure 13. In gauge theoryterms, when the hypermultiplet masses for ω , ω , . . . , ω n of the full puncture are setequal, one can transition from the top 2d theory to the bottom 2d theory, which has asingle hypermultiplet mass for the single weight ω + ω + . . . + ω n instead. Explicitly,[ − , , , . . . ,
0] + [1 , − , , . . . ,
0] + . . . + [0 , . . . , , , −
1] = [0 , , . . . , , − . • The nilpotent orbit for this theory is the minimal non-trivial orbit of A n , with partition[1 n +1 ].The flag manifold T ∗ ( G/ P ) associated to this defect also appears as the resolution of theHiggs branch of the quiver n + 1 which is the Grassmanian G (1 , n + 1). Note this is again precisely mirror to our quivertheory T d . 51 .2 D n Examples: Polarized Theories n n − n − n −
11 1 11
Θ = { α , α , . . . , α n − } Θ = ∆ \ { α } Figure 14: The top quiver is a nontrivial puncture characterized by the parabolic subalgebra p { α ,α ,...,α n − } . It is denoted by the partition [( n − , ] in the fundamental representation.The bottom quiver is the simple puncture of D n , characterized by the parabolic subalgebra p ∆ \{ α } . It is denoted by the partition [2 n − , ].Here, we give two nontrivial D n examples. We proceed as in the A n case and start byconstructing a valid set of weights W S : ω ≡ [ 1 , , . . . , , , ,ω ≡ [ 0 , , . . . , , − , ,ω ≡ [ 0 , , . . . , , , − ,ω ≡ [ − , , . . . , , , . W S . Now note that: ω = − w + 2 α + 2 α + . . . + 2 α n − + α n − + α n ω = − w n − + α + 3 α + 5 α . . . + (2 n − α n − + ( n − α n − + ( n − α n ω = − w n − ω = − w n This defines the 2d quiver gauge theory T d shown on top of Figure 14. Computing (cid:104) e γ , ω i (cid:105) for all positive roots e γ , we identify the nilradical n { α ,α ,...,α n − } . Therefore, we associate to W S the parabolic subalgebra p { α ,α ,...,α n − } from the Levi decomposition.Now we rederive this result from the CFT perspective: consider once again the set W S . Wedefine the momentum vector β from β = (cid:80) |W S | i =1 ˆ β i ω i . It is easy to check that (cid:104) β, α i (cid:105) = 0 , i = 2 , , . . . , n − β has a 0 as its i -th Dynkin label for i = 2 , , . . . , n −
2. This defines many level 1 nullstates in the CFT. One can easily check that no other set W S satisfies (cid:104) β, α i (cid:105) = 0. Also notethat the commutant of the semi-simple element β is the Levi subalgebra l { α ,α ,...,α n − } .We make the following important observations: • This puncture features the instance of a new phenomenon: there are in fact many 2dquivers associated to the parabolic subalgebra p { α ,α ,...,α n − } . We just exhibited onepossible 2d quiver among many valid others. • Just as in the A n case, there are many different sets W S for each 2d quiver, which donot directly allow us to read off the parabolic subalgebra. The upshot is once againthat the explicit null states for all these sets W S can be written at level 1; they aregiven by: (cid:16) ( ˜ W ( n ) ) − + β W (2 n − − + β W (2 n − − + · · · + β n − W (2) − (cid:17) | (cid:126)β (cid:105) (7.72)and derivatives of this equation with respect to β . Here, W ( j ) − is the mode − j generator. In the split representation of so (2 n ), a generic semi-simple element is (cid:126)β = diag( β , β , . . . , β n , − β , − β , . . . , − β n ). The puncture we study sets n − β i equal to each other; call them β (and so n − − β i become − β ). It is thisparameter β that appears in the null state (7.72). • We can also identify the nilpotent orbit corresponding to this theory: for even n ,53t is given by the partition [5 , , n − ], and for odd n , the orbit has the partition[5 , , n − , ,
1] (this agrees with the results of [35].)We now turn to the second example. We start with the set of weights: ω ≡ [ 1 , , , . . . ,
0] = − w + 2 α + 2 α + . . . + 2 α n − + α n − + α n ω ≡ [ − , , , . . . ,
0] = − w These weights obviously add up to 0, so they define a valid set W S . Written as above, theyspell out a 2d quiver theory T d shown at the bottom of Figure 14. Computing (cid:104) e γ , ω i (cid:105) forall positive roots e γ , we identify the nilradical n ∆ \{ α } . So we associate to W S the parabolicsubalgebra p ∆ \{ α } from the Levi decomposition. Unlike the previous example, the 2d quivertheory associated to this puncture is unique. All other possible sets W S are then obtainedby Weyl reflection.The nilpotent orbit corresponding to this theory is the minimal non-trivial orbit in D n , withpartition [3 , n − ].The corresponding space T ∗ ( G/ P ) also appears as the resolution of the Higgs branch of thequiver U Sp (2) SO (2 n ) Note that this quiver theory is again mirror to ours. D Classification
In Figure 15 we give the full classification of surface defects for D : the left column shows arepresentative quiver T d from [7] that describes each puncture. The middle column showsthe subset of simple roots Θ which defines the parabolic subalgebra associated to T dm s →∞ .The right column features all the nilpotent orbits, in the notation of [35], as Hitchin Youngdiagrams. Note that lines 2 to 5 on the left denote one and the same nilpotent orbit, butdifferent parabolic subalgebras. More subtle is the fact that lines 2 and 3 on the right featurethe same Young diagram, but that is just an unfortunate misfortune in the notation: theyreally denote distinct nilpotent orbits and parabolic subalgebras; the Levi decompositionsindeed yield two distinct nilradicals. An asterisk is written down to differentiate those twopunctures. In order to specify which of the three parabolic subalgebras the left 2d quiver ofline 6 is associated to, one would need to specify explicitly the set W S that defines it. We54mitted writing W S for brevity.The nilpotent orbit classification of punctures has a disadvantage: two distinct puncturescan be associated to one and the same Hitchin Young diagram (see for instance lines 2 and3 on the right in Figure 15), so extra data is needed to differentiate them. Classifying theCFT defects from the little string perspective, on the other hand, every polarized puncturein the classification is associated to a distinct parabolic subalgebra. Unpolarized punctures,however, have to be added separately. For D , there is exactly one such unpolarized puncture:the one featuring the null weight [0 , , , T d in thesection 7.4. It is interesting to note that special and non-special punctures in the classificationof [5] are treated on an equal footing in the little string formalism.55 d Quiver Theory Θ Nilpotent orbit 2d Quiver Theory Θ Nilpotent orbit4 5 333 11 Θ = ∅ { α , α } { α } { α i , α j } ( i,j )=(1 , , (2 , , (2 , (cid:70) { α } { α , α , α } { α } { α , α , α } { α } { α , α , α } { α , α } { α , α , α } { α , α } Figure 15: Surface defects of D . 2d quiver theories from the Little String are shown inthe left column. Parabolic subalgebras that arise in the CFT limit T dm s →∞ are shown in themiddle column. Nilpotent orbits from the defect classification of [35] are shown in the rightcolumn. We omitted writing down an explicit set of weights W S for each defect for brevity.The minimal nilpotent orbit is analyzed separately in section 7.4.56 .3 E n Examples: Polarized Theories
Here, we give the quivers of E n with the smallest number of Coulomb moduli that describea polarized puncture.2 3 4 3 22 Θ = ∆ \ { α } orΘ = ∆ \ { α } Θ = ∆ \ { α }
24 8 12 10 8 6 46 2
Θ = ∆ \ { α } Figure 16: The top, middle, and bottom quivers are E , E , and E
2d theories respectively.The associated parabolic subalgebras are p ∆ \{ α } , p ∆ \{ α } , and p ∆ \{ α } respectively. Thesepunctures all have Bala-Carter label 2 A in the classification of [5].For E , we start with the set W S : ω ≡ [ 1 , , , , ,
0] = − w + 2 α + 3 α + 4 α + 2 α + α + 2 α ω ≡ [ − , , , , ,
0] = − w This defines a 2d theory (shown in Figure 16). One checks at once from the positive rootsthat W S characterizes the nilradical n ∆ \{ α } , so the associated parabolic subalgebra is p ∆ \{ α } .In fact, no other set W S is associated to this parabolic subalgebra. The level 1 null statecondition in the E -Toda CFT is: (cid:104) β, α i (cid:105) = 0 , i = 2 , . . . , W S : ω ≡ [0 , , , , ,
0] = − w + 2 α + 3 α + 4 α + 2 α + α + 2 α ,ω ≡ [0 , , , , − ,
0] = − w , produces the same 2d quiver as above, but the associated parabolic subalgebra is instead p ∆ \{ α } , and the level 1 null state condition is: (cid:104) β, α i (cid:105) = 0 , i = 1 , , , , . All the other possible sets W S associated to p ∆ \{ α } are obtained by Weyl reflection on thetwo weights (and the same is true about p ∆ \{ α } ).For E , we start with the set W S : ω ≡ [0 , , , , , ,
0] = − w + 2 α + 4 α + 6 α + 5 α + 4 α + 3 α + 3 α ω ≡ [0 , , , , , − ,
0] = − w This defines a 2d theory (shown in the middle of Figure 16). One checks at once fromthe positive roots that W S characterizes the nilradical n ∆ \{ α } , so the associated parabolicsubalgebra is p ∆ \{ α } . In fact, no other set W S is associated to this parabolic subalgebra.The level 1 null state condition in the E -Toda CFT is: (cid:104) β, α i (cid:105) = 0 , i = 1 , , , , , . All the other possible sets W S associated to p ∆ \{ α } are obtained by Weyl reflection on thetwo weights.For E , we start with the set W S : ω ≡ [0 , , , , , , ,
0] = − w + 4 α + 8 α + 12 α + 10 α + 8 α + 6 α + 4 α + 6 α ω ≡ [0 , , , , , , − ,
0] = − w This defines a 2d theory (shown at the bottom of Figure 16). One checks at once fromthe positive roots that W S characterizes the nilradical n ∆ \{ α } , so the associated parabolicsubalgebra is p ∆ \{ α } . In fact, no other set W S is associated to this parabolic subalgebra.The level 1 null state condition in the E -Toda CFT is: (cid:104) β, α i (cid:105) = 0 , i = 1 , , , , , , . W S that are associated to p ∆ \{ α } are obtained by Weyl reflectionon the two weights. Here we give some examples of unpolarized theories for D n and E n only, since there is nosuch theory for A n . D D D Figure 17: Exhaustive list of unpolarized quiver gauge theories T d for D , D , and D . Thenilpotent orbit in the classification of [35] is also written for reference.The simplest case of an unpolarized quiver gauge theory arises when only a single fundamen-tal hypermultiplet is present, so there is only one mass. The corresponding weight is thenthe null weight, which is obviously not in the orbit of any fundamental weight. For instance,such a scenario occurs for the unique unpolarized theory of D , where the weight [0 , , , D n g : E B.-C.-label: A g : E B.-C.-label: A g : E B.-C.-label: A Figure 18: Examples of unpolarized quiver gauge theories for E n . The ones shown here havethe smallest Coulomb branch dimension. The Bala Carter label A in the defect classificationof [5] is also written for reference.case, and Figure 18 for examples in the E n case.As explained in section 6.3, unpolarized theories can also have more than one weight: forexample, looking at D , it is possible to choose weights in the third fundamental representationthat actually belong to the orbit of the first fundamental weight instead. One can thenconstruct the bottom D quiver of Figure 17. An example of two weights that make up sucha quiver is [1 , , , , − , , , , Acknowledgements
We first want to thank Mina Aganagic for her guidance and insights throughout this project.We also thank Aswin Balasubramanian, Oscar Chacaltana, Sergey Cherkis, Jacques Distler,Amihay Hanany, Peter Koroteev, Noppadol Mekareeya, Hiraku Nakajima, Shamil Shakirovand Alex Takeda for their time to discuss various points and their willingness to answer ourquestions. The research of N. H. and C. S. is supported in part by the Berkeley Center forTheoretical Physics, by the National Science Foundation (award PHY-1521446) and by theUS Department of Energy under Contract DE-AC02-05CH11231.60 eferences [1] S. Gukov and E. Witten, “Gauge Theory, Ramification, And The Geometric LanglandsProgram,” arXiv:hep-th/0612073 [hep-th] .[2] D. Gaiotto, “ N = 2 dualities,” JHEP (2012) 034, arXiv:0904.2715 [hep-th] .[3] D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin Systems, and theWKB Approximation,” arXiv:0907.3987 [hep-th] .[4] E. Witten, “Geometric Langlands From Six Dimensions,” arXiv:0905.2720[hep-th] .[5] O. Chacaltana, J. Distler, and Y. Tachikawa, “Nilpotent orbits and codimension-twodefects of 6d N = (2 ,
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