JHEP05(2015)017 series, theory n q 0 Springer A , May 5, 2015 2 April 7, 2015 : p : , for a specific February 6, 2015 g : Published Accepted d 10.1007/JHEP05(2015)017 Received , supersymmetry are shown q doi: 0 , algebra of type 2 p W SCFTs. This construction has likely Published for SISSA by q 0 , 2 p and Balt C. van Rees is precisely the g a,b,c to those obtained using the holographic dual description n [email protected] , . 3 Leonardo Rastelli a 1404.1079 The Authors. Extended Supersymmetry, Conformal and W Symmetry, Field Theories in c

Six-dimensional conformal field theories with , [email protected] symmetry in six dimensions [email protected] C.N. Yang Institute forStony Theoretical Physics, Brook, Stony NY Brook 11794-3840, University, U.S.A. Theory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland E-mail: Institute for Advanced Study, Einstein Dr., Princeton, NJ 08540,Kavli U.S.A. Institute for Theoretical Physics,Kohn University Hall, of Santa California, Barbara, CA 93106-4030, U.S.A. b c d a Open Access Article funded by SCOAP ArXiv ePrint: implications for understanding the microscopic origin of the AGTKeywords: correspondence. Higher Dimensions, Conformal Field Models in String Theory labelled by the simply-laced Lie algebra value of the centralthis charge. correspondence are Simple the examples three-pointwe functions of compare of observables our half-BPS results that operators. at areand For large the made find perfect accessible agreement. by worldvolumes We of further codimension find protected two chiral defects algebras in that appear on the Abstract: to possess a protecteddimensional sector chiral of algebra. operators We and argue observables that that the chiral are algebra isomorphic associated to to a a two- Christopher Beem, W JHEP05(2015)017 16 25 26 31 q 29 4 19 | ‹ 8 p 12 osp 6 14 theories q – 1 – 22 0 , n 9 2 p 1 22 10 21 28 27 q 4 cohomology | ˚ Q 8 32 p 23 osp superconformal algebra q 0 , 2 ]. The existence of such a sector leads to a wide variety of insights, including new p C.2.1 Null statesC.2.2 and supercharge combinations Contribution to the unrefined superconformal index 1 -chiral operators Q C.1 Long representations C.2 Short representations A.1 Oscillator representation A.2 Subalgebras 4.2 Three-point couplings at large 5.1 The critical character 2.1 Elements of the 4.1 Testing with the superconformal index unitarity bounds and powerful organizingerators principles that of underlie such the theories. spectrumtractable Furthermore, of for the BPS correlation constraints op- functions of in crossing thislem subsector, symmetry associated and are with solving these eminently the constraints “mini-bootstrap”ing is prob- an the important preliminary full step numerical towards implement- bootstrap program for unprotected correlation functions in such 1 Introduction and summary It has recently beentended supersymmetry observed has that a protected any sectoralgebra that four-dimensional [ is conformal isomorphic to field a two-dimensional theory chiral with ex- D C Characters of A The B Unitarity irreducible representations of 5 The chiral algebras of codimension-two defects 3 The free tensor multiplet 4 Chiral algebras of interacting Contents 1 Introduction and summary 2 Chiral symmetry in a protected sector JHEP05(2015)017 ] q g 2 | 3 are W , 1 sub- 2 (1.1) , q ¯ 1 q SCFT z 4. p SCFTs 1 and the q theories , z, q 0 ‰ su p g 1 q , 0 p 0 d , 2 , p 2 su 2 p p theories at large n ] relates instanton A 7 , 6 SCFTs more generally. q 2. 0 , “ 2 . The space-time dependence p -closed operator is 6 d Q ] to Toda correlators, suggesting R 8 in , [ Ă q q ¯ z S 4 2 ]. In this work we explore the third. , 1 R z, 4 p I ]. This makes the presence of a solvable “ p O 5 , q ¯ 4 z N p I u – 2 – and “ : q 6. 4 q , ¯ “ z 0 4. 4. d z, p theory labelled by the simply laced Lie algebra . The requirement that a local operator be annihilated “ “ SCFTs and chiral algebras. More precisely, there should “ p in q Q O q algebra! In this context, the generating currents of d d 0 q 0 , N 0 , g 2 , 2 p 2 p W 2 in 4 in “ “ “ p N N N : “Small” : : : ], we identify a privileged set of BPS operators that is closed under the 1 q q q q . However, the microscopic origin of this symmetry has so far remained 2 2 4 4 | | | g | ‹ 1 2 2 , , , 8 W p 1 2 2 p p p is precisely the su su su osp g -closed operator within the fixed plane is also slightly unusual: its orientation in Q Our analysis involves a few technicalities, but the essential argument is not difficult to The appearance of chiral algebras in the context of the six-dimensional The six-dimensional case holds particular interest since six-dimensional The arguments presented in the four-dimensional case were fairly general, with the I II by means of the AdS/CFT correspondence [ IV III by this supercharge restricts itof to lie a on a fixedR-symmetry plane space is correlatedcomplex coordinate with on its the position plane, on the schematic the form plane. of a Concretely, if correspondence. summarize. As in [ operator product expansion. Thesea operators certain are nilpotent defined by supercharge passing to the cohomology of chiral algebra unclear. Our main result isof that type the protected chiralarise algebra very associated concretely to as the cohomologyobservation classes seems a of likely half-BPS starting local point operators for in a the truly SCFT. microscopic understanding This of the AGT does not come aspartition a functions of complete four-dimensional theories surprise. ofa class The deep connection AGT between correspondencebe [ a connection between the remain quite mysterious. To thecomputed best in of these our theories knowledge, exceptn no in correlation the functions free have case, been orsubsector indirectly all for the the more interesting,some as clues the about structure of the the right computable language correlators with may which hold to describe The first two entries on this list were the subject of [ subsector will consequently existcludes in such any a theory subalgebra. forleads to which A a the quick rather superconformal survey short of algebra list the in- of available possibilities: superconformal algebras [ produced in superconformal field theories (SCFTs) in spacetimes ofexistence of dimension a protected chiralsuperconformal algebra subalgebra following entirely of from the thealgebra full existence acts of superconformal as an algebra anti-holomorphic for M¨obiustransformations on which some the fixed plane. A similar theories. An obvious question that presents itself is whether such a structure can be re- JHEP05(2015)017 n A (1.2) . The k 2 traceless subgroup k “ — in other R q g chiral algebra 3 p g so W SCFT. Since chiral q theory contains a sin- 0 q , by an 0 2 , p z 2 theory. Precisely this limit p q theory sit in rank 0 , q 2 0 p , . 2 . (The precise form is dictated by q p z z is isomorphic to the p u O n ,E ]) is that this ring is freely generated by a n

10 , and have conformal dimension ∆ ,D , – 3 – . For example, each n R 9 i Q q The protected chiral algebra of the six-dimensional k A 5 qs p ¯ z representations form a ring. Our starting postulate “ t so z, -exact, meaning that its cohomology class depends on R p g M¨obiussymmetry acting on ¯ q Q O 5 q r p 2. This is the superconformal primary of the stress-tensor 2 p so are simple functions of ¯ “ sl theories by leveraging a minimal amount of physical data as q k ¯ z q p ], where a simple expression was proposed for the case of the 0 I , u 2 12 p , . The cohomological construction maps each of these generators to a g of the generators of the half-BPS ring coincide with the orders of the 11 u i k t is a conventional local operator that obeys a suitable BPS condition. The q ¯ z z, p set of generators of the chiral algebra. This guess passes the following non-trivial .) The crucial point of this construction is that the anti-holomorphic position- I runs over the components of a finite-dimensional irreducible representation of the R q O I R-symmetry, and 5 superconformal theory of type p All in all, we are led to the following conjecture: A natural conjecture is that the generators arising from the half-BPS ring are the This formal construction associates a chiral algebra to any q q 0 5 so p , 2 theory. That expression takes preciselywhich the the half-BPS form generators one are would the expect only for generators. a chiral algebra for Conjecture 1 (Bulk chiralp algebra) complete test. On general grounds, onea can certain argue limit that of the thehas character superconformal been of studied index the in of chiral [ algebra the is parent equal to ring map to higher-spinis currents not of hard the to chiraltwo-dimensional recover algebra. stress is tensor. Another the This piece central can ofpoint charge be information of function read off that the of from Virasoro half-BPS thein symmetry operators, appropriately the associated normalized which six-dimensional two- with in Weyl the turn anomaly. is proportional to certain coefficients Casimir invariants of generator of the chiral algebragle with half-BPS spin operator with multiplet. This operator isa mapped holomorphic to stress-tensor a in spin-two the chiral chiral operator, algebra. which Higher-rank plays generators the of role the of half-BPS for the analysis.symmetric Recall tensor that representations half-BPS of operatorshighest-weight of states a of these (well-motivated from several viewpointsset [ of elements inwords one-to-one the correspondence ranks with the Casimir invariants of algebra. algebras are very rigidsociated structures, to we the known mayinput. hope In to particular, completely the characterize spectrum the of ones half-BPS as- operators provides a useful starting point the insertion point meromorphically, Consequently, correlation functions of theseof twisted the operators insertion are points, meromorphic and functions as such they inherit the structure of a two-dimensional chiral index so “twisting” the right-moving of dependence of such an operator is where JHEP05(2015)017 : 1 ρ . ]) is left n 21 4 SCFTs in “ N algebra computes algebras with the g W W ] for superalgebra. For any limit (for fixed operator 16 q 2 n | SCFTs will be the subject , which correspond to the 2 2 q , 0 2 , ,A p 1 2 algebra with null states present p su A W ] allows the correlators in question to be “ 20 . 4 supersymmetric Yang-Mills theory in four ]. In particular, the global symmetry g 1 g “ r N case, it is known that the solution is not ` theories, these defects create “punctures” 4 _ g , we take the large A S h n g A d – 4 – 4 “ algebras respectively, it is easy to prove that the , the associativity constraints may admit multiple “ g 3 d 2 W c ]. For the theories, along the lines taken in [ ]. 14 A priori q , 0 17 . , 13 g 2 p W . g . Specializing to ] for g 15 qq Ă theories [ 2 . p 5 n In the construction of class sl A ] — wherein a non-renormalization theorem [ p 2 ρ 19 ]. , 8 and [ 18 ], but the additional solution corresponds to a 3 g 14 A theories. For example, it predicts that the OPE of two half-BPS operators must The algebraic analysis underlying the existence of a protected chiral algebra in the With a view towards a microscopic derivation of the AGT correspondence, we also As an illustration of the kind of information that can be extracted from the chiral This conjecture has immediate implications for the spectrum and interactions of the Even after making the above assumptions, our argument falls short of a general proof q q Ñ Note that in contrast to the analogous comparison for The additional structure associated with outer automorphism twists around the defect (cf. [ 0 1 2 2 , p , there exists a family of such defects whose members are labelled by embeddings 2 dimensions [ computed at weak ’tthere Hooft has coupling heretofore and been compared no to independent the calculation supergravity of computationfor these at future three strong work. point coupling functions — even at large sl on the UV curve,centralizer with of each such defect carrying a globaltheory living symmetry on group these equal defects is to identical the to that of [ More importantly, we now haveexactly a procedure at for finite computing half-BPS three-point functions consider superconformal defect theoriesg that preserve an algebra, we consider three-point functions ofthem half-BPS exactly, operators. for any The dimensions) of the three-pointthem couplings with calculated from the the holographicWe find chiral prediction an algebra, exact computed and match. using compare involved The calculations eleven-dimensional is agreement quite of supergravity. miraculous these and two constitutes completely strong different, evidence technically for very our conjecture. with calculable OPE coefficients.bootstrap This program is [ essential information infour setting dimensions. up The the application conformal ofof bootstrap a methods forthcoming to publication [ for generic values of the centralcharacter charge, with which the would superconformal conflict with index. the match of the vacuum p contain an infinite tower of protected multiplets obeying certain semi-shortening conditions, identical sets of generators.Virasoro In and simple to cases thecrossing such Zamolodchikov as symmetry relations admitfor a the unique solution.unique [ This is also known to be the case of this conjecture, assame we set are of not generators awaresolutions as of for a the uniqueness theorem singular for OPE coefficients, leading to inequivalent chiral algebras with with central charge JHEP05(2015)017 ]. 24 (1.4) (1.5) (1.3) should b symmetry of q 4 p so , the protected chiral g ) in the bulk case and theory compactified on 2  q 0 and spans two of the four , “ is isomorphic to the quantum 2 C p 1 ρ  , the partition functions display , C _ g theory of type h g q . ] applies to the same compactification 1 would break the d 0 1 (i.e., , 2 2 1 . 24 2 b ‰ ˙ p deformation in the remaining four non- “ b _ b 1 ´ 2 h 2 , b _ ` affine Lie algebra at the critical level, 1  h b g In the “ ´ – 5 – ˆ d 2 “ ´ ` k g d r 2 of the k “ ρ d 2 c ] and of its generalization to include surface defects [ ] symmetry with central charge 7 , g 25 6 would imply an effective compactification of the plane transverse W ‰ 8 and subjected to the Ω 2 0) in the presence of defects. That these particular values of b C . In the presence of defects wrapped on “ 2  2 {  1  ]). We are led to the following natural conjecture, which brings the potential expect the chiral algebra associated to defects to include a meromorphic stress “ : (i.e., 23 invariance at level [ , 2 b g 22 Our claims regarding the bulk and defect chiral algebras are strongly reminiscent of do not Ñ 8 2 in the resulting defect theory.stress In tensor the construction is considered certainly here,is absent, such at and a such four-dimensional one a sensiblythe level discovers that chiral that algebras it the contains described defectin no in chiral greater meromorphic this algebra detail stress paper in tensor. future and work. The the connection AGT between correspondence will be pursued we reproduce these symmetriesb for the case arise is somewhat reasonable.the In transverse the space, bulk whereas case, theIn construction the defect used case, in thisto paper the respects defect. that In symmetry. such a scenario, one would expect to find a four-dimensional stress tensor affine Our construction, on thetures other the hand, six-dimensional does theory not in involve flat an space. explicit Nevertheless, Ω at deformation the and level fea- of chiral algebras, partition functions enjoy where Strictly speaking, the AGTa correspondence complex applies curve tocompact directions. the Similarly, the constructionaccompanied of [ by a codimensionnon-compact directions. two In defect the that story also with no wraps defects wrapping the UV curve, the resulting Drinfeld-Sokolov reduction of type the AGT correspondence [ two-dimensional conformal field theory ande.g., the [ geometric Langlands correspondenceconnection (see, to the geometric Langlands into sharpConjecture relief: 2 (Defect chiralalgebra algebra) of a codimension two defect labelled by the embedding the two-dimensional context. Howeverwe in contrast to thetensor, purely since four-dimensional such setting, anwill operator be is absent from associatedfrom the with a defect a physicist’s theories. four-dimensional perspective, Though stress such this tensor, algebras is play which a a somewhat major strange role characteristic in the connection between of these defects implies the existence of affine currents for the same symmetry algebra in JHEP05(2015)017 ] , 1 R- 5 q (2.1) 4 p . subalgebra usp conformal q R 2 q q , 2 2 4 p p . In section p su sl n D q ˆ q Ă 2 the maximal bosonic 2 p , . Our construction is per- 3 sl 2 q p 4 | ‹ D , we recall the logic of [ 8 p ˆ 2 superconformal algebra. We 1 to the osp superconformal representation q q theory at large 4 q 2 q on which our chiral algebra will | , we consider the simple case of p 2 n ‹ 0 , 6 , 8 3 A su 6 theories. Our approach is a direct p 2 R p algebra of rotations in the transverse superalgebra, p q conformal algebra times a 2 q 0 so superconformal algebra is reviewed in Ă q osp q ˆ 2 , q 2 q 2 , 2 2 2 p p 0 p 4 , R p , sl 6 su 2 p D M¨obiustransformations from the p discusses the construction of the irreducible ˆ Ă q breaks so 1 as the bosonic subalgebra of the superalgebra , we review what is known about the half-BPS 2 q C ¯ p 6 – 6 – q 2 superconformal algebra is a subalgebra of the 2 q 4 sl p 4 R 4 p p “ su Ă usp N usp 2 ] that was used to uncover a similar structure in four- q – R ˆ 1 ˆ 4 2 p 2 q q 2 so 2 p p su su chiral algebra plane which may be useful in future work. q superalgebra. The relevant real form is 4 q ˆ q ˆ | 2 SCFTs and motivate the bulk chiral algebra conjecture. We show ‹ 2 superconformal algebra. The six-dimensional case turns out to be p p 8 q . It is isomorphic to the p sl q sl 0 0 ´ , ] for a detailed description of the analogous four-dimensional case. A , 2 SCFTs. That such a generalization should exist is made apparent by theory, which is a free theory and completely tractable. This serves to 2 1 osp 2 , so all told we are concerned with the embeddings p q p q “ q ˆ 0 2 2 , , complexified p 2 4 N 1 p p R-symmetry that will be introduced shortly. su D q 4 p q ˆ q Ă usp 2 2 p We first select a fixed The organization of this paper is as follows. In section , This is the sl 2 . We can regard 3 p 4 haps most naturally phrasedthe in distinction terms between of complexifiedmnemonically complexified algebras helpful, algebras, e.g., and but in their weof distinguishing real will the the not forms. be Using overly the concerned with natural real forms can be algebra on the plane, times the R D live. The first orderconformal of algebra business that is fixes todetail this determine in the plane. appendix maximal The subalgebrasubalgebra of of the full which super- issymmetry algebra. the product The of choice the of observing that the four-dimensional six-dimensional somewhat richer, however, duetheory. to We will the keep intricaciesearly the of sections exposition of relatively [ brief. We refer the interested reader to the In this section we set upof the general a algebraic machinery protected that is chiralgeneralization responsible algebra for of the of the existence six-dimensional approachdimensional of [ several appendices. In particular,characters appendix of the 2 Chiral symmetry in a protected sector case of the interacting theories.spectrum In of section the that this conjecture passesindex, a and at number the ofwe level checks, address of the both three-point case of functions at half-BPS foralgebra defect the operators the conjecture. level and motivate Various of the technical above-stated the details defect chiral and superconformal useful points of reference are included in and provide the specificsfurther of characterize the its local application operatorsof that to may six-dimensional the play superconformal a role theories.the in abelian the In protected chiral section algebra illustrate some aspects of the correspondence that will prove useful in the more abstract JHEP05(2015)017 , q q 2 2 u “ q 2 2 are 2 p p 2 ix p , but µ µ sl sl (2.4) (2.5) (2.2) (2.3) q , x exact ` v direc- 2 1 su , , 1 x u µ 4 x 6 t q ˆ p x and their 2 D p “ , x and u generators. vector 5 sl that is A z 1 q R-symmetry. q q q Ă 2 q r 6 , x Q 1 2 p 2 . p 2 4 , , p q . p , sl 2 so 2 A 3 p A , x sl p su 3 12 34 S Q D . x t su , P P t “ 3 3 : q ˆ . The , , , for our conventions): L L 1 “ “ q ˆ , A 2 2 1 1 4 5 subalgebra. 2 r ` ´ transform as doublets of A S p p 1 1 r r q Q Q Q Q 2 2 2 ´ ´ q D | ]: 4. Finally the fermionic gen- L L ¯ ´ ` ´ ` 1 L usp A 1 , r 4 4 3 3 S 1 , p r r , S S S S ” ” 4 su A A ,..., 2 2 4 4 S interesting nilpotent supercharges S 1 p “ “ “ “ : : : : ,L, M M “ -symmetry twist of “ : , respectively. 21 43 R ´ ´ 3 : 1 : 2 : 3 : 4 B four , A and have an K K , 1 1 3 3 2 Q Q Q Q , that generate rotations in the q 1 4 A M M h “ “ A not 3 , r 2 Q , 1 1 , , , 1 ,S ` ` with – 7 – 2 1 4 3 A does L 3 ¯ L q A Q 1 , , , , M M p subalgebra, which is a necessary condition for the 2. AB , Q , transforming under a 3 3 4 4 2 ? q p q R r r S S S S 2 , , “ q{ | D : A 2 1 2 3 4 1 r ´ ` ´ ` S ,L, iv , A , 1 1 2 2 1 q q ] to go through. M M r ´ p A Q r r 1 1 1 Q Q Q Q 1 S L L v su p ` ´ “ “ “ “ “ p : : : : : : , H H 4 p p 1 2 l.w. 1 2 3 4 A q q v supercharges, we construct 1 2 1 2 planes, with eigenvalues 2 2 Q Q Q Q superalgebra is the supersymmetrization of the right-moving subalgebra comprise eight Poincar´esupercharges Q p p q u q 2 q 6 “ “ 2 su su “ , 2 contains a : , 2 . The generators of the two-dimensional conformal algebra , 2 and , x 2 q 0 0 p 2 2 5 A p ? 2 p ¯ L L x D , ix , we take the chiral algebra plane to have complex coordinates q{ Q D t D 2 6 2 p ´ R iv generators are denoted by D 1 q ` x 4 1 p , and v u “ 6 for . 4 2 “ p z usp From the Let us describe this embedding more explicitly. Introducing coordinates q An inequivalent choice of maximal subalgebra preserving theOur plane conventions is are such that the highest- and lowest-weight components of an , x 2 4 5 3 p ,..., x h.w. The key point, as in four dimensions, is to define an this is not relevant for our purposes since v Because the conformal algebra, all of these supercharges commute with the left-moving and their conjugates, generalizing the two that appeared in [ The embedding of these supercharges into the six-dimensional superalgebra is given by The identification makes itsu clear that The erators of the special conformal conjugates t subalgebras correspond to self-dual andtions, anti-self with dual rotations generators in the We have introduced Cartan generators cohomological construction of [ 1 and ¯ that acts on this plane can be identified as follows (see appendix Crucially, JHEP05(2015)017 i : i Q , Q R then (2.9) (2.6) (2.7) (2.8) q (2.10) (2.11) 2 6 , p ), which R i , su q 0 on any 4 u necessarily p 3 . ě , q ˆ r Q u u 2 0 , usp : 4 . 1 p 4 p L Q sl ` . Q Ă ,S , 1 4 4 R r ´ q coincide. Moreover, Q Q 1 . Operators obeying 1 ´ i p ‘ 1 u Q will have the structure 1 ` and their conjugates ` ˆ . ¯ i L r , r . 1 i 2 2 Q R 1 q ‘ “ Q ´ u “ t : (separately for each 2 verbatim ´ ` p 0 L 1 p ¯ 1 z : i 3 r 4 4 4 4 u “ ´ t S u “ t ` ´ 3 , Q : 3 su 1 3 p L M M Q ´ . , we have Q , ,Q u 3 3 ´ ´ zL 0 j Q Q ´ 3 3 3 3 . Q e “ , , 0 q i M M 3 0 , define the same cohomology, and so the R Q , h 1 “ cohomology classes are allowed to form i “ “ 0 ´ decomposes as p ´ ´ , Q R i r r p L “ t q 0 4 3 2 O 2 2 : u “ t Q 4 0, i.e., if it has quantum numbers satisfying ¯ 1 ´ , L p 4 ´ ij ´ ` r 0 M – 8 – 1 Q p u “ t u “ ´ t L “ 3 3 Z usp , and its conjugate h ¯ : : 2 z 2 2 ] can be carried over i L L qs “ ` 2 “ “ Q 0 of 1 ´ 1 Q 0 , h , ,S Q p L p ´ ´ ´ 2 2 5 0 E 34 24 2 2 . commutators as follows, Q Q O zL 1 “ , Z Z L L i e q 0 r 2 Q , and so must satisfy the additional relations p , L p 2 r q . Defining q “ ´ i su 2 ¯ z “ “ ´ “ “ p can be defined as the diagonal subalgebra of R Q inserted at the origin is a “harmonic representative” of a u “ t z, q 2 su p ` q 2 r u “ ´ t S u “ t 0 p 1 12 13 14 23 O 4 , p : 1 z ´ sl 1 Z Z Z Z Q O and ¯ . If we take the maximal subalgebra L Q need not be zero, so , ,Q i 1 1 r Q q “ 3 Q Q : 1 h ) can be translated away from the origin (within the chiral algebra plane) p 1 u ´ 2.9 “ p L ) we can deduce that a state obeying the above condition is “ t “ t “ ´ t 2 1 1 0 2.8 h ´ ` p L , p p L L 2 theory. In principle, the cohomology of any one of the At this point, the construction of [ A local operator We are now in a position to define the protected chiral algebra of a six-dimensional q eigenstate, with the equality saturated if and only if all This is the subalgebra under which the 0 6 , 0 2 p the condition ( by means of the twisted momentum operator invariant under A priori non-trivial representations of It follows that, asfrom stated above, eq. the cohomology ( classes of all four of a chiral algebra.structure of It interest turns is out theus simultaneous that cohomology outline all of the four all main four points nilpotent of supercharges. the Let construction. cohomology class if it is annihilatedhappens by if and only if it obeys p annihilate the state.mutual Additional commutators interesting of the bosonic generators are those that appear in In any unitary superconformalL representation one will necessarily have such a twisted algebra The twisted generators occur as with respect to the JHEP05(2015)017 (2.14) (2.15) (2.16) (2.12) (2.13) , a pair s quantum 3 q , c 4 2 2. Then the p . , , c highest weight 1 1 . 1 . usp c q R r “ 0 ¯ z q , 2 i p 1 “ , , I and SCFT have the right 3 3 3 q su , 3 h h h “ p q 3 : , of the superconformal 0 h , h “ “ “ q , E 2 ¯ z 2 2 2 2 ` , with p p q h h h , h 2 I k ] and is discussed in detail in 1 2 h I ě “ “ h p 28 ¨¨¨ Dynkin labels, “ 1 1 1 – 1 . q 3 I , and thanks to the second line of eigenvalue, violating unitarity. We p 6 26 q i p 0 z O Q p p , u L so , c , q O as 2 ¯ z 6 h 0 is necessarily an R z, ` q ´ p 2 q r

1 r . p k 2 2 2 qs “ h I su 0 of the Cartan. Indeed, if this were not the case, – 9 – ` p “ ¨¨¨ “ Q ,, h h 1 R O 2 R 2 I quantum numbers qs 4 2 p , 2 ¯ z 0 q O ` ` . The results are summarized in table 6 p ` L z, p r q r r p , c 3 ¯ z D 2 2 , and the twisted-translated operator at any other point , and the scaling dimension, so 3 h p O q s r k h 2 r , d 0 ` ` 2 ´ p would have negative I -exact. It follows that the cohomology class of the twisted- r , h 1 ´ i and , d 2 ´ R R u 2 1 ¨¨¨ 2 R h 2 2 Q d C R h cohomology r 11 ` ` ` ` ) only appear in a select subset of these representations. The “ representation of O “ 1 1 1 Q R q ¨ ¨ ¨ 1 h h h 2 ¯ 1 z k p c d 2.9 1 ) is “ “ “ “ I u E E E E 2.9 : : : : “ : introduced above, the Dynkin labels can be written as A B C D q dependence is q ¯ z z . Let us summarize the salient points here. Dynkin labels, z, p R, r p B R O q ) its ¯ 4 p superconformal algebra has been worked out in [ 2.7 A generic representation is specified by a set of q 0 usp , 2 Operators satisfying ( complete list, along with the locationis within determined the full in representation appendices of the relevant operator, satisfied. In particular,quantum numbers (semi-)short introduced representations here satisfy come the in following conditions, four series, for which the Shortening conditions arise when certain linear relations for these quantum numbers are p appendix of primary operator. In terms ofnumbers the algebra. 2.1 Elements ofThe the next step isproperties to to determine play precisely a role which in operators the in protected a chiral algebra. The representation theory of the Operators constructed infunctions this of manner the insertion have points,the correlation and cohomology. functions enjoy well-defined These that meromorphic are OPEs are precisely at meromorphic the the level ingredients of that define a two-dimensional chiral By construction, such aneq. operator ( is annihilated by translated operator defines a purely meromorphic operator, state, carrying the maximum eigenvalue states with greater values of denote the whole spin operator obeying ( is given by A local operator at the origin with JHEP05(2015)017 are s 1 , 1 0). It is d . 0; theories are “ , 2 q 0 2 0 , selection rules, d , 0 series. In these r q 2 p 2 D p D 0 1 2 2 3 4 4 su Level s s s s s s s 0 0 0 0 0 0 0 , , , , , , , 0 1 2 1 2 2 2 , which is a generalization of ` ` ` ` ` ` ` 1 1 1 1 1 1 1 d d d d d d d series representations is necessary r r r r r r r ; B s ; ; ; s s s 0 0 0 0 , -chiral , , , 2 Q 0 1 2 ; ; ; , , ` s s s 2 1 ` 0 0 0 2 theories. , , , half-BPS ring 2 ` ` q 0 0 1 , c are more exotic representations that satisfy 0 , , , , c 1 1 1 , 0 1 0 c c 0 c 1 2 r r r r r r r – 10 – p series multiplets in the known -chiral operators. Representations labelled with a star -chiral operators are displayed, along with the level in Q D s s s s s s s Q 0 1 2 0 1 0 0 , , , , , , , 1 1 1 1 1 1 1 d d d d d d d r r r r r r r ; s ; ; ; s s s ; ; ; 0 s s s 0 0 0 , 0 0 0 , , , 2 , , , 2 0 0 Primary 0 0 0 , , , c , , , , c 1 1 1 ]. , and for this reason representations of type 0 0 0 c c 0 c s r r r r r r r 0 10 , 1 r p‹q p‹q p‹q p‹q series appearing in table C C B B D D D C Series that comprises a scalar, two Weyl fermions, and a two-form with self-dual quantum numbers. Using this fact in conjunction with s and 0 R , q B 2 . Summary of superconformal representations that contain chiral algebra currents. The p 0; 1 theory. This is the theory of a free tensor multiplet, and so the chiral algebra can , su 0 The The half-BPS operators form a ring, the Of the representations listed, the most familiar are those in the q , 0 0 , r 2 Having established this basicp machinery, let us considerbe the constructed chiral explicitly. algebra The tensor ofD multiplet the lies abelian in anfield ultra-short representation strength. of The type quantum numbers of these fields are summarized in table we will see in the restand of natural this from paper that the the point presence of of view of the protected3 chiral algebra. The free tensor multiplet ring can never appearthe as half-BPS a ring normal arestructurally ordered identical necessarily result product. mapped to to the Thisso-called fact generators “Hall-Littlewood means that of chiral that in ring” the the the are four-dimensional chiral generators mapped case, algebra. to of generators of chiral This the algebrasemi-shortening conditions generators. is at level a two or greater. Although these are somewhat unfamiliar, the chiral ring in four-dimensionalhalf-BPS supersymmetric operators theories. is An that important theytheir property are of the the operators withone the quickly lowest possible sees dimension that given the chiral algebra operator associated to a generator of the half-BPS representations the superconformal primaryinteresting to is note quarter-BPS that in (half-BPSbelieved practice, to if all transform inmany representations copies that of appear the inexpected the to be tensor absent product [ of sufficiently quantum numbers of the primarythe representation and where the one mayare find those the that seem to be absent from actual Table 1 JHEP05(2015)017 (3.3) (3.4) (3.5) (3.6) (3.1) (3.2) . Q  . affine current qq ¯ z q 1 qqq p q z, 1 p u p . These map in the 2 q u J Φ i 3 highest-weight compo- h.w. k h.w. 1 2 1 0 q ψ pB 4 p q ´ q p 0 ¯ 1 z p p L u z, usp J p ) are just the normal ordered 2 1 k Φ 2.9 p pB , . q R 1 2 4 2 p q . 2 | ¯ z u q 4 . ? ) is the y p J 1 4 5 2 w IJ q 1 can be constructed using twisted trans- ´ δ Φ 1 k z 2.9 i p ´ q ` 2 usp x q pB | ¯ z 1 z , leading to the following cohomology class, ` h.w. ? p p u z, 1 2

p J Φ : q „ 3 Φ OPE follows directly from the free field OPE y q q „ q – 11 – Φ

p “ 6 ¯ z ˆ w J theory is therefore precisely a q p 4 1 p 10 h.w. z Φ current in the chiral algebra, e.g., q p Φ so q 0 Φ q q h.w. x Φ , 3 qq ` 1 z p k 12 Φ 2 p ¯ p z I p u Φ Φ z, qpB p 5 2 3 2 2 ∆ Φ h.w. i . Field content of the abelian tensor multiplet. Φ affine current, 2 k 12 descendants of the scalar have strictly positive eigenvalues under q q ` 1 ¯ z s R p qpB q I A u ab z, a 4 ` r Φ p p Table 2 λ ω 1 h.w. Φ Φ Operator usp p 1 k 12 2 1 pB ? : „ “ : q z p It is worthwhile to take a moment to understand the appearance of Virasoro symmetry. The other operators in the free theory that obey ( Of these basic fields, the only one satisfying ( Φ . The meromorphic operator associated to Φ 0 p algebra. In four dimensions, Virasoro symmetry ofstress the tensor chiral in algebra four followed from dimensions.comes the for In presence free six of with a dimensions, the we six-dimensional again stress find tensor that multiplet. Virasoro symmetry The six-dimensional stress products of holomorphic derivatives inobvious the way chiral to algebra composites plane of of the Φ The chiral algebra of the abelian This is the OPE of a then the resulting chiral algebra OPE takes a familiar form, The singular part of theof meromorphic the Φ scalarcanonical fields. OPEs, Specifically, if we normalize the six-dimensional operators to have Other fields and L lation in the plane as was described in section nent of the scalar multiplet. In our conventions, this is the field JHEP05(2015)017 ]. 30 (3.7) (3.8) (3.9) 1. As (3.10) (3.11) “ . Having g d 2 of the Weyl and any one c . i d c q is non-abelian. 2 z c theory takes the g qp q q 1 0 , p , u 2 ), and upon mapping J p q 1 p 2.9 u , then direct computation q J z , p q S “ p 0 1 2 : obeys ( p , in which the stress tensor is a z s q q T “ 0 : z B , 14 p q p IJ . z S ` current algebra, namely O scheme dependent p . : 0; 2 q q q , T

0 J ` 1 0 2 p p q , 3 Φ 1 theories z tens i u I T I z 0 p c r p 3 2 q q c 0 D :Φ “ ` 14 , ` p the six-dimensional stress tensor itself. q 2 “ 2 – 12 – q 2 4 : p O g { I are certain Weyl invariants whose precise form is g z p q 1 p 2 d : i 3 not c 2 , 14 c 2 c “ p IJ , ` 1 q „ O I 1 Q 0 I p qs 1 T c ¯ z q theories are fixed in terms of the coefficients z ` z, p p q 6 q 0 T , 14 2 aE p IJ p O “ q ¯ z d p 6 J . In the case of the free theory, this primary is the symmetric traceless A q u q 4 ¯ p z p I u usp r theory. We have the general result is the Euler density and of theories is always positive. that follows from supersymmetry and therefore holds for any choice of q 6 ] q i 0 0 E 14 c , 29 , 2 affine current algebra, 2 p This evaluation of the chiral algebra central charge for the abelian theory is useful p q 1 p the 4 Chiral algebras ofA interacting direct analysis in theNevertheless, style we find of compelling the evidence previous in section favor is of not the possible bulk chiral when algebra conjecture put This relation will prove useful inthat, the in discussion contrast of with the the non-abelian four-dimensional theories case, to the come. central Notice charge of the chiral algebra of invariants, and theseThere constants will in therefore turn exist aof have universal the their constant of ratios proportionalitydetermined fixed between this by constant supersymmetry inany the [ abelian theory, the same result will necessarily hold for form [ where unimportant for our purposes.tensor multiplets The in ratios the of the two- and three-point functions of stress chiral algebra arises froman the operator stress in tensor that multiplet multiplet in which is six dimensions,since it it corresponds determines to for ussix-dimensional the central constant charges. of proportionality Recall between that the the two-dimensional Weyl and anomaly of a leads to the standard OPE of a holomorphic stress tensor in two dimensions, where the Virasoro centralwas charge the is case in that four of dimensions, a we see that although the holomorphic stress tensor in the to the chiral algebrau this is identified with the un-normalized Sugawara operator in the If we further canonically normalize this operator as level-four descendant. The superconformal primaryin is the a dimension four scalarbilinear transforming of scalar fields, As a half-BPS operator, the highest weight state of tensor lies in a short representation of type JHEP05(2015)017 ] S 10 (4.3) (4.4) (4.1) (4.2) algebra 1, or in theories, ], and the “ 1 W ´ ). The six- 2 33 n ] for class b theory of type A is a 7 , that correspond q 3.11 6 g is the orbifold 0 g , , and we will find is the degree of the g g 2 i p k W ] for the , . . . , r . The generalization to theory takes a suggestive 1 31 1 , “ ´ Toda CFT for ]). Indeed, our conjecture is n i . . 10 1 2 , , . . . , n A , 32 { i ´ 3 9 1 g theory of type . n , O q tens i theory of type 2 . 1 A M c 0 2 q q ,  ´ . “ 0 . 1 2 , p multiplets where g n “ g k 2 theories has been studied in, e.g., [ ´ s 3 p 1 W 0 W q  n theory is precisely { , { ´ 0 3 i g g , r 3 g r k q 2 ´ n 5 p C 0; 4 3 ], for , R n “ 6 0 has been determined explicitly in [ : – 13 – 4 , 1 q “ 2 0 “ p { r 1 ´ 1 g g ´ n D n q “ p A M 1 M A , ´ p “ g n d 2 g A c . The degree of each generator is equal to the degree of M p i g the Weyl group of the Lie algebra. Let us further define the c that appears in the AGT correspondence [ g g W W has exactly such a structure (see, e.g., [ is straightforward. g we saw that the meromorphic currents associated to generators of the g ]). These papers found confirmation of a folk theorem that states that the 2 31 , 9 is the rank and g r Before moving on to specific checks, we can determine the central charge of the In section This ring can be given a simple description using the Harish-Chandra isomorphism. Recall that the moduli space of vacua for the is isomorphic to the holomorphic polynomial ring on ’th Casimir invariant. This was understood explicitly in [ Consequently, the central charge of theform, chiral algebra of the This is precisely thethe value language of of the the central AGT charge correspondence of [ the non-abelian chiral algebradimensional independent Weyl of anomaly any for guessworkrelevant anomaly by coefficients using obey the eq. following ( relation, that the chiraltheories algebra of type that the protected chiralcompelling algebra evidence in of favor the of type this claim. other choices of half-BPS ring areguess necessarily would then generators be ofgenerated that the by the chiral precisely associated algebra these for chiral currents. the algebra. This guess A is minimal made more appealing upon noting It is freely generated,to with generators the given Casimir by elements invariantsthe of invariant. Ingenerators of the the half-BPS language ring live ofi in superconformal representations,where this this means is that a the single Casimir invariant of degree ring of BPS operatorson of (an an appropriate SCFT subspace isamounts of to isomorphic the) the to moduli statement the space thatg ring of the of the ring theory. holomorphic of polynomials In half-BPS the operators present in case, the this following complex subspace of The spectrum of half-BPS(see operators also [ in the motivate our claim. where forward in the introduction. Let us first make some elementary observations that serve to JHEP05(2015)017 ) ]. 4 1 12 (4.8) (4.9) (4.5) (4.6) (4.7) Q , (4.10) 11 -symmetry R , respectively. studied in the g . 3 h ` 2 h . . s r -chiral operators defined 0 0 . ´ Q 1. 3 R “ “ t h that was defined for four- r 3 3 ` “ 2 the h h 2 unrefined index 2 ` h 3 b s ` ´ h , as well, and we are left with an R theories have been computed via 2 2 ´ ´ 2 p h h , n h E tens i g series and with the structure of p ´ ` q c A r exactly u q n R 1 1 4 g 4 ` ´ A h h r S , E 4 _ g Schur index ]. ´ ´ q ` 1 h u r r Q g 4 4 _ g . In this case, the resulting partition function 35 t 2 2 , d S 3 , h β 7 4 2 4 g e 34 ´ ` 1 [ – 14 – d F Q Q 4 q g R R t q “ 1 2 2 β g e p ´ ´ p´ d F q “ p 2 q g c E E 1 Tr p i ]. c “ p´ : 36 u “ u “ q 3 4 Tr 3 4 S S “ , , q, s : 3 4 p 2 1 q I Q Q t t are the dimension, dual Coxeter number, and rank of g r ] 2 SCFTs in [ p, q, s, t p 28 “ I , and N . As a consequence, this index can be reinterpreted as a Witten index of the _ g h 2 , g d 1, whereupon the index becomes independent of The operators that contribute to this index are For the sake of completeness, we can also derive the result for the more general case. Although we are not aware of this result appearing explicitly in the literature, this is the unique 7 “ in section expression compatible with the knownanomaly central charge polynomials, of which the are known for any index that depends on only two fugacities This is thedimensional six-dimensional analogue of the The relevant index withaforementioned this property paper. is equivalent Int to our the conventions, the unrefined index is recovered by setting The index undergoes acombinations radical of simplification Cartan when generators theadditional that fugacities supercharge. are appear In specified particular, in we so theenhanced may that choose supersymmetry exponents the with the all fugacities respect so commute to thatwill with the only index some receive has contributions an from operators that obey the additional shortening condition The states that contribute to this index obey a shortening condition, Certain limits of thesupersymmetric localization superconformal in index five-dimensional supersymmetric for Yang-MillsThe theory the most [ general superconformal indextakes (defined the here with form respect [ to the supercharge which again matches the relevant Toda central charge4.1 for Testing with the superconformal index Here the anomaly coefficients take the form where The prediction for the central charge of the chiral algebra is then JHEP05(2015)017 3 us c 1 , ` 0 spins, 1 (4.13) (4.14) (4.15) (4.11) (4.12) global t c , 1 q 1 q 2 d 2 p p 0; su , . 2 ff , c n 0 , . In terms of the q r q ff -chiral operator, it B n q n p Q of the chiral algebra, q χ ´ H 1 , includes an SU ` ¨ ¨ ¨ ` 1 q coincident M5 branes was j for a . In conjunction with the 2 2 2 s q n ´ s 1 . ` ¨ ¨ ¨ ` 0 « j 1 . L 2 2 -chiral operators transforming ff 8 q q E q Q . F “ . P m ` q (with no null states appearing in 1 x m 1 q 0 p “ c L « f p´ . q plays the role of (twice) the Cartan of q q m 1 partition function E n n 3 . ` “ p p 8 k ÿ h χ χ P , . . . , n m 1 q 3 H H ` « , “ ´ 2 2 Tr Tr 1 h m exp “ “ theory, it contains an extra factor corresponding 0 ` – 15 – : k “ q q “ 8 s q “ 1 q ź q n : m p p 2 ‰ 1 ´ — namely the only superconformal representations q, s q n “ rotations transverse to the chiral algebra plane in six ´ 1 p ź k x n 1 d 1 p 0 A c 2 q f “ “ I I 2 -chiral operators necessarily occur in representations with 8 “ ź p . m q q Q 1 E n su . ; 1 ; n q “ series theories there are no “ ź P k ) has precisely the form of the vacuum character of a chiral n q, s q, s that actually make an appearance will be q, s p A p p q q q “ 2 0 I 0 4.14 . The spin-statistics theorem in six dimensions implies that , , . Consequently all operators contributing to the unrefined index are n s 2 2 ; p p 0 us , I I 2 2 five-dimensional SYM. The resulting expression is relatively simple, , q, s 0 p , c q t 1 I q “ n , is half-integral for fermionic operators appearing in the index and integral c p 1 r 1 d q, s j p denotes plethystic exponentiation, 0; 1 . , ], the unrefined index of the worldvolume theory of ´ 0 E n . , 12 A 0 r I D The index in eq. ( In [ This is a notable feature that, in particular, does not hold for the Schur index in four dimensions. 8 algebra generated by currents of spins In that setting, therecontribute can to the be index. “accidental” cancellations between protected operators that individually do from the list inand section bosonic, and the index can be reinterpreted as the Note that this indexabove argument is for actually the independent absencethis of of implies cancellation the between that fugacity states inin with the the representations same with non-zero Since the calculation was done into the the U index of the freeIn tensor multiplet other that words, describes for the center the of interacting mass degrees theory of we freedom. have where P for bosonic operators.fermionic operators Consequently that there contribute can to be the unrefined no index. cancellationcomputed between in U bosonic and where we have denoted thesix-dimensional Lorentz Hilbert group, space ofDynkin the labels chiral algebra by is equal to fermionfollows that number (mod 2). Now because symmetry inherited from the dimensions. In particular, thethis combination flavor symmetry, and we have associated chiral algebra. Recall that the construction of section JHEP05(2015)017 10 . 4 : g S , and . The (4.16) (4.17) 2 is very { R ˆ 3 q 1 7 5 ´ . Such an ´ p n n comprise a k A 2 so n AdS , the result at W . “ g , ff y 3 ) for general I 30 C q and the higher-spin series). Turning the 2 , I 1 n ` 4.14 C ´ ff 1 A 24 I L 18 q C q x ` ` , , 312 q 20 form an orthonormal basis of 3 ff ff ∆ 31 14 q q x i q 2 , k 12 I ` 2 q ´ C ` 231 n 18 , k 2 ∆ 23 q ` 12 1 q q x denotes the unique scalar contraction 9 q k ´ is precisely such a chiral algebra, and q p is the only solution to crossing symmetry with ` y q 123 (at least for the with scaling dimension ∆ 1 ` 3 g 1 C ` I ∆ 12 . The g 14 ´ ´ C ) then the vacuum module can be seen to k W 8 10 8 x q n q q 2 1 q q W I A ∆ ` ¨ ¨ ¨ ` ` C 4.4 ´ ´ ` ` 1 4 W ´ I ]. these three-point couplings scale as 1 1 6 8 12 q ,..., qy “ limit. While there is no way aside from our chiral C – 16 – n j q q q 3 3 x n 14 ` , n x , ∆ ` ` ` 2 p 2 q 13 5 6 8 q 3 ` “ q q q 3 k and i p I q k ` p ` ` ` ∆ O 5 q 2 n 2 2 p “ 2 q q q q : x so « « « « there is an additional solution for which the vacuum module contains p . . . . theories there is no obvious sense in which gauge-invariant op- ijk q 4 2 -fold symmetric traceless tensor representation of q E E E E 2 k . . . . cases the corresponding k A 0 p I for each , P P P P 5 q O 2 A k solution of crossing symmetry with no nulls (up to the choice of q p p 1 series in the large and ∆ O x q “ q “ q “ q “ 1 p j q ´ , and x 1 n 3 theory is the corresponding q, s q, s q, s q, s 1 k This provides compelling support for the claim that the protected chiral p p p p p I A A ´ q unique 7 8 6 n , ]. In fact, given the stated spectrum of generating currents, 9 i 0 O 2 E E E D x , x I I I A I 2 32 , p 1 “ case can presumably be treated similarly. : A n ij is accessible holographically using eleven-dimensional supergravity in D x n For the The 9 10 of the three tensors. For large values of the given generators, whilesingular for vectors for arbitrary central charge [ where traceless symmetric tensors of erators are constructed from elementary matrix-valuederators fields. are The therefore single the trace generators half-BPSsingle of op- scalar the operator half-BPS chiral ring,operator which at transforms large in the three-point functions of these operators are required by symmetry to take the general form operators for the algebraic approach to computelarge half-BPS three-point functions forIn general particular, the three-pointputed. functions The of notion of “single-trace” a half-BPS single-tracefield operator theory, operators since that in is can the applicable be here is com- that of generalized free 4.2 Three-point couplings atAs large a more refined check of our claim, we can compute the three-point functions of half-BPS algebra of a logic around, we have a simple prediction for the generalization of ( indeed when the centralcontain charge no is null set states asequivalents) (aside in [ from ( those obtainedlikely by to acting with be the central charge). the vacuum Verma module). The algebra JHEP05(2015)017 , µ . µ 312 (4.21) (4.22) (4.18) (4.19) (4.23) (4.20) k q 2 1 3 34 z , , k 2 231 ˛ ‚ k q , k 2 1 23 ¯ 1 1 z 1 k p ´ ` 2 123 3 C k 312 k 2 1 are limits of a one- 12 k 2 z p q ´ Γ Γ Cl q by their twisted coun- p n 1 ¯ ]. For generic values of qy “ q 1 W ´ 3 will be exactly reproduced x ` z 42 p 2 2 . algebra will not be the well- p q q – k . q three-point couplings in the 231 expansion. . k 3 3 i 2 p I k k 39 k 1 p d i W [ p , k n 2 ´ O 2 k Γ q 2 j 2 s c q | Γ k Ñ W { W ]. Instead to analyze this limit we IJ µ w 1 i . y q , k r δ ¯ k 3 2 1 d s ] using the supergravity description. δ 1 ´ ´ ij 8 2 z k n ´ 38 , while at positive integer values of n ), whereupon the resulting correlation p δ ` c , p 1 4 z 5 . The operators for which this formula q 2 x , W p q k 8 2 | C “ 4 37 123 3 k 2 p k p k , 2.12 µ r ´ Γ . These are nonlinear Poisson algebras that W ` q qy “ 8 q ,..., Γ qy “ a 2 ), 1 y w Cl k z W p p p Ñ 8 g ¨ ˝ p q q – 17 – algebras q j j ` d 1 “ k W 4.17 ˙ k 2 ) as in ( k p J 1 p algebra (with appropriately normalization for the q p q 3 W . Moreover, the structure constants of the quantum k z O k g p Cl n W p q W p ] n 1 2 q q , c x ` 5 W k z W p p p W 2 q “ 2 q i : k i k W k p I p α ` Ñ 8 O 1 qy ùñ x x W n k 3 x x and ˆ algebras, p q k q Γ 3 k 3 k 3 2 2 p I Cl q p n ´ ´ O α limit. j q 2 W πn 2 k ). We should replace the operators 2 p n x algebras agree at leading order in the 1 p ` q i 2 4.19 k 2 k q “ W p I have classical counterparts 3 algebra of Pope, Shen, and Romans [ “ O g : ), ( , k q 2 8 1 W x ijk , k W p 4.18 k 1 q 1 k 1 k p We further make use of the fact that the Poisson algebras Note that because of the double scaling, the limiting First, recall how the chiral algebra correlation functions are obtained p I C O x parameter family of universal classical this algebra has oneit generator truncates each to of the spin 2 will take advantage of thealgebras fact that in the limitcan of be large described central in terms charge, oflimits the the of quantum Poisson chiral the brackets of generating a currentsand set of classical of generators that are the classical currents) in the double scaling limit, known Our claim is thusby that the the structure three-point constants couplings of the amounts to a simple transformation of ( Making the same replacement leads to canonical normalizations for the chiral operators, appropriate large from ( terparts (which we shall denote function will be meromorphic and interpretable as a chiral algebra correlator. This holds are canonically normalized, with two-point couplings given by This is the result that should be compared to the They were found to take the form [ where the leading order terms have been computed in [ JHEP05(2015)017 11 1. ´ ]. We (4.27) (4.29) (4.30) (4.25) (4.26) (4.28) (4.24) . 43 here i ! q “ 1 !! q 1 ´ there ´ i ijk ]. For us, the indices k . . kij p ..., k ! q q 43 p q . 6 n ` !! 2 p ´ q u i q k k 0 1 k k 1 ´ k k t 2 , k j j p ´ u ´ k n j k i k p k k k ] and are quite complicated. 2 W jki t ` O β q s i k qp , j 43 n k p 1 ` ˆ p q q !! , k , k d q 4 q w i ´ k 2 in [ d 1 d j ´ k c 2 k i q 2 k r ´ ; q c i n i k ´ c k q q k 2 k n n ; C z p 2 n ! 1 p p p β n p n p ijk i q k i p q 2 k ! 1 k W ´ ! k j d q ` N ˆ α q k 2 1 k 6 ´ N 1 c ˆp α !! 2 ij ; a k i q p ´ 2 1 δ ´ k n k 3 b q i ´ ´ 2 p i q “ d j j k q k k d 2 k ´ ´ k q 2 k p 2 c !! i w “ ´ p – 18 – ´ j ´ are given by c k q 6 ; : k α ) holds differ from the canonically normalized i i p k ; k 1 k 1 k n q ´ q 2 n p a ` d ` ` n Ă i p p z ` W i ´ 2 k k p i k p k 4.24 !! i i k c k j k q q k α ; k 2 n k in a primary basis are known in closed form [ Ñ 1 α 3 p n 2 N q s p ˆ p i a ´ k µ k 2 q „ are defined for general r p i α ´ k j u w 8 k 0 p 2 W q “ ´ q “ q “ p´ t q p ´ , 2 W n j k i u n p k k p k p ˆ k ` i , k k k k k t j ! j k s !! W q k q j N q q i , k 2 3 i z 1 k , k p k β i q ´ p ´ i Ñ8 k lim k k n r k C p k “p´ k C u W 0 ` ` t , j j u k k k k t ` ` s i j i k k p p , k i The relevant terms in the chiral algebra are the linear terms, which take the form, k ˆ Here we are using slightly different indexing conventions from those used in [ r are equal to the spin of the current, while in the reference the convention was that correlators. The calculation itself is tedious and we found the form of the structure 11 C computing, i, j which leads to the following prediction for the three-point couplings that we should be Note that thecurrents currents with which for we should whichimplement compare the ( the canonical supergravity self-OPE results. according to They can be rescaled to The coefficients In the scaling limit,to the the generic non-vanishing case. structure With constants a simplify significant dramatically amount of relative massaging they can be put into the form In the scaling limit, the functions while the three-point functions take the form where in terms of the functions defined in the reference, the two-point functions are given by can take the double scalingn limit of these resultsconstants explicitly to in be order to notscaled determine limit illuminating the of large in some general, of the so functions we that display make an here appearance. the appropriate double- The structure constants of JHEP05(2015)017 This 12 , then its )! C three-point 4.18 in the double scaling (corresponding to the s superconformal invari- ]). Nevertheless, using theory will flow to an , then it gives rise to a µ g q 4 r q 44 2 non-extremal carries a global symmetry R | 0 8 , 2 ρ , 2 Ă W p 2 2 p R su , a C algebra, and by making a redefinition theories. In principle, this reduces the W q 0 . , g 2 p , for example, one may obtain an equally good q in ) it is necessary to make careful choices of the signs for j , and is thus located at a point on k – 19 – ρ 4 ´ i 4.18 R k p ]. The defect labelled by -point function of half-BPS operators is completely 8 n [ W q ] themselves) are only valid in the non-extremal case, g j and occupies a subspace k expansion. 5 p , C d 4 2 q Ñ W ). With some work, these choices can be made systematically. i c 2 { p λ extremal and similarly for permutations of the indices. Strictly speaking, sl 4.30 ` : q k i k k ρ , there is an important class of half-BPS codimension-two defects algebra structure constants and supergravity correlation functions p any g ă W j W k Ñ ` q i i k k p W . The codimension-two defects appear in two different roles in this context: S that is the centralizer of the image presence changes the four-dimensional theory. Thesymmetries resulting of SCFT the inherits the defect. global codimension-two defect of the four-dimensional theory. 2 superconformal field theory in four dimensions. These are four-dimensional SCFTs If instead a defect wraps If a defect fills the non-compact ]. Consider the maximal defect operator in the theory of type These defects play a fundamental role in bridging six- and four-dimensional physics. Finally, we should remark that there is in principle some freedom in the choice of gen- g In order to recover exact agreement with ( 1 “ (i) 12 Ă (ii) The four-dimensional worldvolume of such aance, defect and enjoys consequently comes within a [ protected chiral algebra of exactly the sort discussed the square roots appearing in ( Upon twisted compactificationN on a Riemannof surface class Finally, we come toIn the the subject theory of of chirallabelled type algebras by embeddings associated toh codimension-two defects. functions, and in fact determined by the corresponding chiral algebra correlation function. 5 The chiral algebras of codimension-two defects seen to only affect subleadingfunctions, terms i.e., as long asour we check are (and looking at thedue results to of subtleties of [ the operator chiral mixing algebra in construction extremal there correlators is (cf. no [ obstruction to computing extremal three-point better defined problem of determininglimit the order quantization by of order the in the 1 erators in terms of whichof one the chooses form to express a set of generators. In the double scaling limit considered here, such a redefinition can be recovers precisely the supergravityagreement three-point between functions displayedis in an eq. important ( confirmationidentified of a our chiral claim, algebraic andproblem structure of goes in finding to the arbitrarily demonstrate many corrections the to power these of three-point having couplings to the much Remarkably, upon plugging in the above expressions and further massaging the result, one JHEP05(2015)017 , g ρ just that (5.1) (5.2) q a . (This C q, there is a p and of the ρ ρ must be the q ˆ K q , once suitably a is the quantum q q, a ρ p ρ q, ], one sees that the p ˆ 1 K ρ ˆ K ]) to scenario (ii), we further theories has been completely de- 25 , . What’s more, there are no other S g 21 Ă should be an intrinsic property of the is the presence of factors q h . , a _ 13 _ q, ], the Schur index of the SCFT corresponds to the h h p 1 2 ρ ˆ . By leveraging generalized S-duality and the K “ . Using the dictionary of [ index “ ´ S ] (see also [ – 20 – global symmetry with flavor central charge given index that depend purely on local properties of g d d 4 S 2 g 24 k S k of the current algebra at the critical level. ρ . We may therefore suspect that ρ theory and of its various defect operators. A salient feature at the critical level. q 0 g , 2 -symmetry current). This dovetails nicely with the absence of a of the associated chiral algebra. p , which is a function of the superconformal fugacity 0 q R , for which the Sugawara construction becomes singular, and con- L a ]: the stress tensor of the protected chiral algebra arises from the id), which carries q 1 F ]. We may then hope to use this detailed knowledge to infer some q, q “ p 1 ρ of the global symmetry algebra ρ ˆ 46 . According to the dictionary of [ K p´ , q a 45 , critical level 36 is the dual Coxeter number of _ symmetry current of the four-dimensional theory, which in turn belongs to the same h ] R A quantitative check of our conjecture comes from an analysis of the protected spec- q 21 For our purposes we should focus on the Schur limit of the index, which depends on a single super- 2 p 13 character of the irreducible vacuumthe module current of algebra the of quantum type Drinfeld-Sokolov reduction of conformal fugacity graded character Tr building blocks appearing inthe the punctures. class This stronglycodimension-two suggests defect that of type normalized (see below), iswhich in the turn is Schur equal indextion, to of the our character defect the of chiral associated SCFT algebra the conjecture living chiral leads algebra. on to Under the a this sharp defect assump- prediction: of type of of the generalare formulae naturally for associated the class tois the codimension-two configuration defects (i) localizedpuncture mentioned factor, at above). punctures Indeed on flavor for fugacities each puncture of type trum of four-dimensional SCFTsexistence of of class specialthe limits general where form some for thetermined of superconformal [ these index theories of class properties admit of Lagrangian the mother descriptions, stress tensor in the currentcorrespondence algebra and at its the critical generalizationconjecture level. [ that Taking the inspiration chiral fromDrinfeld-Sokolov the algebra reduction AGT of associated type to the defect labelled by in the dictionary ofsu [ superconformal multiplet as thedefect, four-dimensional however, stress is not tensor. expectedtry, The to there have SCFT should a be living local no on stress the tensor (and hence, by supersymme- This is the sequently the current algebrais is that without the stressthe chiral tensor. current algebra algebra associated The at most to the economical the critical possibility SCFT level. living An on immediate the check maximal comes from defect another is entry by [ where chiral algebra supported oncurrent this algebra defect at will level necessarily include as a subalgebra an affine trivial embedding JHEP05(2015)017 . n ]. q n are D p 48 (5.6) (5.5) (5.4) (5.3) λ , sl ]. 47 49 and satisfy 1 , ´ n ˙ up to an overall . ], one finds that q A q q 36 a ´ ˛ ‚ theories the embed- y . 1 q, _ theories. For the ] for a detailed expla- 1 p ¯ α q ´ ρ n α 50 ρ, n ˆ x q ` K ´ A q A a e p ` boxes. We will be concerned ´ ¯ ∆ adj ÿ P 1. 1 n ¯ χ α ρ p ´ q re ` q´ ` 1 ρ ∆ . In the q ´ P q q ´ k ` ), we should consider its normaliza- α 1 a a λ . p j ´ rq ¯ ˆ ś a q, w q 5.4 1 . . From eq. (6.9) of [ p q e q 1 a ρ q y q E “ . p ¯ ˆ _ q, K w t ¯ P α q ´ p p n j,k Λ a Λ ρ, ˆ p ... K “ ř ˆ ` K λ sign adj ¸ – 21 – “ ¯ ´x χ q q “ W q q 1 P 0 q a “ ¯ ´ k p w L ´ ´ q a ´ j ř 1 1 adj ch 1 p a χ leads to the final form, ` ¨ ˝ ˜ ¯ correspond to the orthonormal basis of . ∆ r . P . i E E ¯ α . g . , so the precise version of our claim is a t P q ` P ś ¯ α 1 ] ) to a suitably normalized puncture factor requires some rewrit- ´ “ “ n e q “ 50 ź λ j,k a 5.3 are labelled by Young tableaux with ` L q, q ¯ α p e ch n q “ 0 takes the form p p a L 1 ` sl ¯ ´ q, ∆ is the highest weight of the module, which we will take to be trivial, but ch p n P ¯ Λ α λ A flavor fugacities 0 we can use the Weyl denominator formula to simplify the numerator, and we ˆ q Ñ K ř 1. To obtain the above expression we have used the fact that the adjoint 2 n p “ “ sl “ theories a completely analogous proof is possible using the results of [ λ : i n adj ρ a χ E 1 Before we can compare this expression to eq. ( Let us now turn to the consider the factors A comparison eq. ( Below, we prove this statement for maximal defects in the “ n i -dependent multiplicative factor. Our interpretation suggests a natural normalization, be- cause the index of theto trivial 1. defect The (i.e., trivial notableau defect defect is corresponds the whatsover) should dual to be tableau the identically Λ principal equal embedding, whose associated Young ś character for tion. Reasoning based onq the superconformal index only defines where the The corresponding Young tableau is Λ This equation is valid for any dings with the trivial embedding, in which case the flavor symmetry is maximal and equal to ing. For can also write outthat the product of the real positive roots in the denominator. Recognizing The notation here is standardnation. in the Here mathematical literature,the see formula [ is actuallynon-negative valid integers. for all cases where all but the zeroth Dynkin labels of The irreducible vacuum charactershown of to take an the affine form [ Lie algebra at the critical level has been and For non-trivial embeddings, oneDrinfeld-Sokolov reduction needs on the to character of takeexpressions the chiral again into algebra. agree account A with proof the the that the corresponding effects resulting puncture of5.1 factors the can be quantum The found in critical [ character JHEP05(2015)017 (5.7) (5.8) (A.1) (A.2) . The indices A α 1 and other unrelated . “ ˙ 23 i , q Ω q , 2 “ AB ´ , “ A n i i Ω 1 α ]. q 14 ř a 1 ˜ c 23 “ s “ q n i q ´ “ B : ř a ´ p , α a 1 Lie algebras. A ´ A S adj 1 α χ ´ r n q n q , is again easily determined from the results A q q “ ´ , q – 22 – y p 1 t A , _ ¯ Λ ˆ α b α ]). We define a set of four fermionic oscillators with . a ˆ a ρ, K δ x c E 51 . q “ P ` : ¯ u “ ∆ a b “ ÿ P ) we see that the flavor fugacity dependence matches per- ˜ c ¯ α A q , q a Q 5.4 a q -dependent terms requires the relation c p q q, t t p Λ , along with four (symplectic) bosonic oscillators Λ ˆ q K ˆ K a ˜ c ) and ( 14 , a 5.7 c superconformal algebra p q 0 algebra. This is in fact a well-known result, for example it is an essential ingredient in the , n 2 p W ) makes it clear that the null states that are subtracted from the Verma module are precisely run from one to four, and the oscillators satisfy (anti-)commutation relations 5.7 A ]. We find that The fermionic generators of the superconformal algebra are fermionic bilinears of the Eq. ( 36 and 14 basic oscillators, those of the construction of Hecke eigensheaves using conformal field theory [ where Ω is the skew-symmetric symplecticentries matrix equal with to Ω zero. A.1 Oscillator representation In order to establish conventions for thelize the an six-dimensional oscillator superconformal representation algebra, we (cf.their uti- [ conjugates, a for its hospitality duringin the part initial by phase the of National this Science work. Foundation under This Grant research No.A was NSF also PHY11-25915. supported The V. Schomerus, D. Simmons-Duffin,Kavli and Institute E. for Witten. Theoreticalvironment Physics The during for authors providing the would a late likefrom hospitable to stages NSF and thank grant of enjoyable PHY-1314311. the this workSimons L. project. en- Foundation R. and gratefully of C. acknowledges the the B. Solomon generous gratefully Guggenheim support acknowledges Foundation, of and support the the IAS, Princeton, Acknowledgments The authors haveglas, benefited D. Gaiotto, from T. discussions Hartman, with K. Intriligator, N. J. Bobev, Maldacena, S. T. Minwalla, Dimofte, S. Razamat, M. Dou- fectly. Matching the extra which is indeed a simple fact of life for the of [ Upon comparing ( The factor for the trivial puncture, JHEP05(2015)017 q ˆ 2 , . 6 , p c A , c C b S so S C b a BC Q . δ c a Ω δ 1 4 ` AC ´ ` , c A a R . First of all, let us b C c C S C q , S S 2 c a BD Q , 1 2 H c δ AC , 4 Ω b p , δ 0 Ω c a ` AB D Ω b s “ s “ s “ ´ s “ ´ s “ M BC a d c c c c c C C C C C δ b R δ S S S S S 1 2 , , , , , , b ´ a ab ab c ` H AD , r ˜ d c AB b , P b K Ω c a r r ab M R c r ab r ` P M b M d c K a a c d δ δ , δ c AD 1 4 . AB , δ d 1 2 ´ a 1 2 a R Ω c B ´ , , ˜ c b ` α b b b , , ` a M ´ ˜ ˜ c c c BC b c ac – 23 – M A c ab , a a a ab Ω bd d δ 1 2 P ˜ c c α c c a AB P K d , K δ ` b A ` a c c R “ “ “ “ “ δ : : : : : δ b b Q C AB AB ` c a b ´ BC , d δ Ω Ω a Q ´ , a AB b ab BC ab M b R bc C a d , ad ab K R M H P Ω δ S P a M a d K 4 1 u “ u “ u “ c K δ AC ab c ` b b b a c b b B B δ P Ω δ δ δ c B ´ S S B , , , ´ a Q a c , a Q C A a C s “ s “ ´ s “ s “ s “ s “ s “ ´ a A C S d d d S Q A cd c c c ab ab t b b Q AC Q , c c CD t P Q K K 1 2 Ω δ 0 δ , t M M M , , R , , , , H ab b H r r ab a ab s “ s “ s “ s “ s “ P c c c c c r P K AB r r M C C C C C r R r Q Q Q Q Q , , , , , b ab ab a H r AB P K r r M , R r r q The fermionic anti-commutators are as follows 4 p A.2 Subalgebras It will be convenient tofix explicitly our define conventions various for subalgebras the of generators of various maximal and Cartan subalgebras of the Finally, the fermionic chargessubalgebra, have the following commutation relations with the bosonic while the non-vanishing commutationselves relations are of given the by bosonic generators amongst them- Repeated indices are summed over. usp whereas the bosonic bilinears make up the generators of the bosonic subalgebra JHEP05(2015)017 . 1 ´ 1 (A.3) (A.4) , with ‘ q 4 1 p ` . 1 usp q ‘ q 2 23 Ă 0 , 4 3 2 1 4 3 2 1 3 r R 2 q ˜ ˜ ˜ ˜ p Q Q Q Q Q Q Q Q 1 ´ , and are related to Ñ p D q u 5 5 14 p ˆ R so p R q 1 2 q 2 2 | p 2 “ is given by the generators of , , 9 9 9 9 : 2 ` 1 ` 2 ´ 1 ´ 2 ´ 1 ´ 2 ` 1 ` , q q su q 2 r r r r 4 4 4 4 Q Q Q Q Q Q Q Q 6 p p M M su , r so r . q 2 ´ ´ decomposes as planes, while the two-dimensional chiral 3 3 3 3 23 “ r ´ ` ´ ´ ´ ` ` ´ ` ` ´ ´ ` ` ´ ` q 3 R 2 4 M M h p ` ´ ` R ´ ´ ` ´ ´ ` ´ ` ` ´ ` ´ ` ´ ` ` 2 2 2 2 , usp 14 and 6 2 R q R M M of q q q q q q q q p h 2 – 24 – r , d 0 0 0 0 0 0 0 0 1 2 `q `q `q ´q `q ´q ´q ´q ` ´ 5 , , , , , , , , , j , , , , , , , , ´ 1 1 1 1 1 0 0 0 0 0 0 0 0 j p p p p p´ p p p p p´ p´ p` p´ p` p` p` p R “ ´ M M according to : p p “ 2 1 2 1 2 3 d 1 d , h “ “ : : 2 ,R 1 2 and , h 34 L L 1 ´ ` ´ ´ ` ´ ´ ` ´ ´ ´ ` ` ´ ´ ´ ` ´ ´ ´ ` ´ ´ ` ´ ´ ` ` ´ ´ ` ´ ´ ` ` ` ` ´ ´ ` ` ` ` ` ` ` ` ` 1 that we can take to be given by d h R r define the orthogonal basis of weights for plane. “ : r a 1 A h ´ , and Q 3 3 3 4 2 3 4 4 4 2 2 1 2 1 1 1 and u 4 3 2 4 4 1 3 2 1 3 2 4 1 3 2 1 ,R R ,R Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ˘ 12 R R t Charge Dynkin weights “ q . Supercharge summary. All orthogonal basis quantum numbers have magnitude one half. 5 ` p R The orthogonal basis for the Cartan subalgebra of so rotations in the three orthogonal planes in The generators the generators This is the subalgebra under which the Table 3 The four-dimensional subalgebra actssubalgebras on act in the the bosonic symmetry groups. There is a maximal subalgebra JHEP05(2015)017 q 6 p with alge- so (B.1) (A.5) super- s q can be 2 of 2 q q , 4 , d 4 | 2 2 | 1 ‹ p 2 d 8 , p D and the four- 2 0; representation p . , 2 . q 0 osp q 3 6 su , 2 0 c p 1 p 1 2 c r “ so , , according to su C 3 3 3 ´ q . In the text, a par- ˆ h h h , q 1 4 s p c 2 are written, the last one 1 0 , q “ “ “ . These orthogonal basis 1 2 i , q y 4 su 2 c 2 2 2 p p 4 . The precise map between . ψ | h h h . s | of ‹ D i 0; 0 1 su 3 “ ´ q -type bound for its dimension, , s h 8 L ě “ “ 4 4 2 plane, which is a q p 3 3 A 1 1 1 1 , c u . p , c M s 1 2 2 u y “ c 0 , osp ` r , h , ψ , c r , x in all cases are only restricted to be | 1 3 3 q 3 B 1 i c c 1 , 2 x r , L p 0; 2 1 2 t 1 M 6 , u d r 0 2 contain conserved currents or free fields. ` ´ ` , 2 superconformal algebra 0 1 2 2 r ď ]. There are four linear relations at the level r c 2 diag 2 1 2 “ D M 28 d 0, and the highest weight state of the ` – ˆ – 25 – , h , h “ ´ ` N “ R R 26 4 2 1 1 2 1 q 2 r d 2 ` ` p M ` p r r 3 su , h 2 2 1 2 h 3 with c 1 and ` ` ´ “ s 1 2 : . In all cases, when some of the r , h 2 2 R R “ 2 3 4 2 h 2 ` , d L R 1 ` 2 ` ` ` d c . 1 1 1 q R 0; 1 2 h 2 h h ` has , , 1 q 0 1 2 “ “ “ “ c , 4 , p 1 2 0 1 2 r E E E E : : : : “ D usp 1 C D A B -symmetry group is h of q “ p 3 R 5 , h 1, and 2 ď , h 1 2 h d This structure of null states makes the decomposition rules for long multiplets trans- For a representation in any one of the classes listed above, the structure of null states in There are a number of superconformal subalgebras of p ` 1 non-negative integers. Thed multiplets of theIn type particular, the stress tensor multiplet is parent. Starting with a generic multiplet approaching the the Verma module built on the superconformalof primary that depends primary. on the Every shortadditional null representation states possesses being a obtained by single theprimary. primary action Different of null locations additional state, for raising with the operators primarywhich the on null the we state null summarize lead in to different table is multiplet structures, necessarily non-zero. The quantum numbers Note that we are using conventionsstate for of the the orthogonal Cartans suchhas that the highest weight We recall the classification ofalgebra. unitarity These irreducible have representations beenof of described quantum the in numbers [ that,tation, if guarantee satisfied that by the the resultingnotation superconformal representation for is primary labelling (semi-)short. state these in We relations: adopt a the represen- following dimensional the supercharges for these two embeddings is shown in table B Unitarity irreducible representations of ticularly important role isanti-holomorphic played M¨obiustransformations by in the the maximalbra. supersymmetrization of In the addition,embedded algebra the such of four-dimensional that the four-dimensional rotation group is We denote the eigenvalues ofquantum numbers these are generators related by to the Dynkin basis JHEP05(2015)017 q 6 p so (B.2) (B.3) , , s , s 1 s 2 3 ` ` . 2 ` 1 s 2 , d 4 1 , d 1 , d d ` 1 d 2 1; d 0; , , d 0; ´ 1 , 1 3 d 0 [ 0, 0, -1 ; 0, 1 ] [ 0, -1, 0 ; 0, 2 ] [ -1, 0, 0 ; 0, 3 ] [ 0, 0, 0 ; 0, 4 ] [ 0, -1, 1 ; 0, 1 ] [ -1, 0, 1 ; 0, 2 ] [ 0, 0, 1 ; 0, 3 ] [ -1, 1, 0 ; 0, 1 ] [ 0, 1, 0 ; 0, 2 ] [ 1, 0, 0 ; 0, 1 ] , ´ , c 0; 1 2 2 , 0 ´ , c , c in the main text. , 1 1 1 0 c c c 1 r r r . r A B C D , us , 3 0 0 0 0 0 0 0 0 0 0 us , us 2 2 “ “ “ “ “ “ “ “ “ “ , s ‘ 1 , s ‘ 2 , 1 1 s ‘ 2 ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ , 0 , s ‘ 2 , d t 0 4 4 4 4 3 3 3 2 2 1 2 0 , d 1 , t 1 1 1 1 1 1 1 1 1 1 t , d 1 1 , d , , d 1 Q Q Q Q Q Q Q Q Q Q d 1 ; d 1 1 d 3 3 3 2 2 1 3 d d d 1 1 1 1 1 1 0; 0; – 26 – 0; , c , , 0; 0; 0; Q Q Q Q Q Q , 2 2 2 , , 2 2 1 , 0 0 1 1 1 0 0 , c , c , , c , , , 1 1 1 1 1 Q Q Q 0 c c c 0 c c 1 r r r r r r r 1 B C D A A A A Q q 0 0 0 0 4 | Ñ Ñ Ñ Ñ ˚ δ δ δ δ 8 s ÝÑ -chiral operators given in table p s ÝÑ 2 Q s ÝÑ 2 s ÝÑ 2 , d 2 , d 1 osp ] ] ] ] ] ] ] ] ] ] , d 1 d 2 2 2 2 2 2 2 2 2 2 , d 1 d ; 1 d d d d d d d d d d d 3 , , , , , , , , , , d 0; 1 1 1 1 1 1 1 1 1 1 0; , , c -chiral operators that give rise to currents of the protected chiral algebra d d d d d d d d d d 0; , ; 2 2 , Q 3 0 0 c , c , , c , , 0 ; , 0 ; , 1 1 1 discussed in the previous appendix. We will then use these characters to 2 2 2 c c c c c c ; ; ; 0 ; , , , 0 , 0 ; , , 0 , 0 ; , 0 , 0 ; q δ δ δ δ 1 1 1 1 1 1 4 | c c c c c c [ [ [ [ 0 , 0 , 0 ; [ [ [ 0 , 0 , 0 ; [ [ 0 , 0 , 0 ; [ 0 , 0 , 0 ; ˚ ` ` ` ` . The primary null state for each of the shortened multiplets, expressed in terms of a C C B B B 8 A A A A D ˚ ˚ ˚ ˚ p E E E E r r r r osp ψ ψ ψ ψ C Characters of In this appendixof we discuss aenumerate method the to full set compute of the characters for the various UIRs Amusingly, the are all selected from among these non-recombinant representations. There is a relatively short list of multiplets that can never appear in a recombination rule: of supercharges. Noticeidentical R that symmetry the indices Lorentz on each indices supercharge. are implicitly antisymmetrizedthe because following decompositions of take the placerepresentation (which of decomposition occurs the depends long on multiplet): the Table 4 combination of supercharges actingcolumn on is the superconformal schematic, primary. sincedescendants. the The The expression actual rightmost in null column the contains state second the be Dynkin a labels linear corresponding combination to of the this combination state with other JHEP05(2015)017 s q , 2 b ¯ p and ´ s 2 , (C.6) 2 char- 3 a (C.3) (C.4) (C.5) (C.1) (C.2) 2 1 ` 1 a ,d r q a 1 5 R d r p ´ χ 1 so . 1 ¯ ´ and 3 ´ and 2 2 a q 1 a qq q a a µ , 4 p q p s P ´ 3 p qq , q 1 su ,c a χ 2 b p , whose highest weight , ,c ´ s ¯ ´ w 1 a 2 c p 3 . p 1 raising operators r 1 a p q 2 a . χ , d Q q . qq a 1 M ∆ q 1 5 ¯ 3 6 “ b q 2 2 1 p d c 2 2 2 1 ź p ´ µ , q b b ; b b q 2 3 so d 2 3 1 a w 1 p b p a ´ , c are just the p 1 q “ weights of a state in the Dynkin P 1 2 ¯ ´ q “ d 1 R q q w 3 2 2 b 2 ` 2 q , q 2 a a , c b b d 3 c ¯ ´ 5 1 p p a q c 3 R q p s s 1 2 2 c p 2 ´ p 2 b b a 2 2 r b a P 1 so 2 p ,d ,d L p c 2 ´ 1 1 , w d d a w 1 r r ¯ ´ 1 1 1 d 3 W χ χ c 1 c 1 q and a q q 1 ¯ ´ a 1 1 1 a p 2 2 a b q 1 p b a p – 27 – s 4 R p and 3 p ´ w ´ w ,c , χ q q Tr 1 2 ) in a form that is useful to describe the short rep- su q 4 1 qq a p 2 1 2 2 4 ,c a p p b b 1 s ¯ ´ c A su usp ¯ ´ 3 2 2 1 q “ C.2 r ÿ 1 a a ,c ÿ 1 W W Q b χ 2 , q P P p q “ ,c ∆ ´ to denote a monomial of the fugacities associated to an w w b χ ´ 1 ` 1 q , c 1 are the q 1 r a ´ ` R ´ then represent the action of the supercharges and the deriva- χ p s p O 1 are the denominators of the Verma module characters, obtained q “ q “ 2 p q “ q R χ “ p “ a q d χ χ 1 p , d p , q ,a , q 1 s b s 1 ´ 3 p b 2 ´ b ź A ˘ d . As an example, consider the two , ˘ ,c , q r ,d q R q 2 a 1 a ´ i 4 p ´ j d ,c p | r 1 q “ L ˚ c R Q M χ r 8 p and χ p and , q p χ χ χ q d s a , 3 ´ ´ p and osp a 1 1 p , c q ` and therefore ` M 2 of Q 1 1 , q s 4 6 , c “ “ 2 a ź ź i j 1 O , p c 1 r P q “ q “ r´ We will now rewrite equation ( We will below write The characters are defined as corresponding to the positive simple roots. Their respective Dynkin labels are a b p p ` 2 R M The factors from a product over all negative roots, resentations below. To this end,can be we notice written that as the orbits characters over the Weyl group terms tives and are defined as In this expression the terms acters of the irreducible highest weight representation with the given Dynkin labels. The C.1 Long representations The character for ahas generic scaling dimension long ∆, representation is easily constructed. It takes the form element R and where basis, respectively, and ∆representation. is its scaling dimension. The trace runs over all states in the JHEP05(2015)017 ] W 53 , < = (C.9) (C.7) (C.8) 1 ´ 52 , ¸ 2 { 28 1 3 q q a 2 . b , q q ` W b . The recipe then 1 9 p 3 q ˜ a q , w { , q 2 q 2 d , q b { a q 1 , p b 2 q a , b 2 w p p a p b p Q w Q q 1 ) are to simply remove from Q q d q q “ , q q , q 4 1 a , q q C.8 1 b p a a p p and obtained in this way expres- Q p P w p q P w 3 χ q b p c p q b P 3 p R are invariant under the Weyl group we a q qq p Y q a q b shows that for a short multiplet of type p w p a 2 , q p c w 4 M b q p . Notice that the additional factor effectively 2 X , 2 d 2 2 2 R a a b d 2 d – 28 – p Mathematica p 1 b qq 0 the only supercharge that generates a primary 2 d 1 1 w Q a b d 1 1 p ą ` 3 c b ) as c 3 q 3 w 1 , the Weyl group of the maximal compact bosonic 2 c 3 1 a p q d d 2 a and a 5 c 2 2 p 2 p C.2 M c 2 q a so w 1 a ` the superconformal primary state is annihilated by a subset c 1 ˆ , q 1 qˆ c 1 2 a W q a 4 { a P p p 4 0 and ∆ ÿ 3 | w ∆ q su c P ˚ q ą 1 3 8 ∆ W p q not only the primary null state but also all the states obtained 3 4 5 ` c . In the following we will denote the Weyl symmetrizer sum as q q “ those combinations of supercharges and momentum operators that “ 2 q“ q osp c and, in the case of free fields or conserved currents, of the momentum 4 q “ , q q | , q , q b ˚ ` b 15 , q W b , q with , 8 , , 2 b a p a s to which we associate the monomial a p a { , p 2 p p 1 a 4 s P c L Q p 1 2 , d χ L 1 ,d Q ` 1 d χ d ; 6 ; and 3 3 q ,c “ , c 2 so that we may write 2 , q ,c 1 b , c W c We have implemented the recipe in , r w 1 The exact null state is generically a linear combination of states obtained by acting with the supercharges a c A 15 r p χ ... obtained match known resultswhen and we satisfy compute the the superconformal correctin index recombination from an rules. these appropriate characters manner Furthermore, by dialing we the find fugacities the expectedand other form lowering operators. where This only distinction the is however “ground irrelevant for states” the in discussion in this appendix. removes from from it by thewe action say of that further we supercharges ‘remove’ - a this certain is combination always ofsions what supercharges. for we the have in irreducible characters mind of when all the shortened representations. The characters so leads to with ∆ Q annihilate the highest weightsuperconformal state, and primary. to For dialA example, ∆ to table the correctnull scaling state dimension is of the C.2 Short representations For shortened UIRs of of the supercharges operators as well. Into that compute case the there character: is the a only remarkable changes (but required conjectural) in recipe ( [ As we will shortly see, this form of the character extends most easily to short multiplets. where we defined subgroup of OSp v Kac character formula they actfor on the the irreducible highest weight charactersformula. (in are a Since however shifted the a way).may factors direct also Our write expressions rewriting the of full character the ( Weyl-Kac character Notice that the elements of the Weyl group act on the fugacities, whereas in the usual Weyl- JHEP05(2015)017 , W (C.12) (C.10) (C.11) < = 1 multiplet ´ s ¸ 2 2 { 3 , d 1 a 1 q 2 d 1 b ; lowering operator b 3 q also annihilates the , c ` 5 . 2 p s 4 1 0 2 , , c so ˜ 1 1 Q d 1 c r r ´ ψ a a A 4 1 3 ¸ 2 2 Q Q { Q 0 (C.13) 1 3 ´ 2 ´ 2 ´ 1 q a R R 2 R “ “ b s s 0 2 , ` 1 ,d lowering operators on the null states, d 1 r 1 a a d q ; 2 4 0 to find that ψ 3 5 ˜ s p Q Q ,c q 4 “ 2 1 so ,c , q s 1 Q 0 b c , . In the remainder of this appendix we dis- , r or , 1 q ´ 2 d ψ a 3 q r – 29 – p k . As in the previous example, such additional a 4 2 R ψ a 2 p Q 4 1 A b 4 q 1 su “ r Q Q {p Q , q ´ 2 ´ 1 2 s { a 0 , 1 ... p M M 1 q 1 d P 1 r a q b 1 0. In that case we can act with an ψ ´ 3 b 4 p Q 4 1 3 1 “ M R A A A 2 Q q “ q d 4 ´ 2 a Q Q Q 2 p R Q q “ M p “ 0 2 χ b , q 0 1 b d 1 , b a 3 c 3 p s a 0 2 , c 2 1 d a ; 1 3 0 but now with c 1 ,c a 2 ą ∆ ,c , we find the following rule: if 1 3 q c c 4 r Let us first consider new null states appearing from the action of the Lorentz generators. Notice that the recipe requires a precise enumeration of all the different combinations 4 5 A χ Using the first diagramtable given above, and the specific pattern of shortening conditions in at least for lowthe values of lowering operators the on Dynkin the labels individual of supercharges the is superconformal as primary. follows: The action of We would like toprimary find state, combinations besides ofthe those supercharges primary obtained that null from annihilatecombinations states the the arise listed action superconformal from of in the action zero table of or more supercharges on where we used that cuss how to systematicallyadditional enumerate terms all in the the supercharge character combinations formula. that leadC.2.1 to such Null states and supercharge combinations on the primary nullsuperconformal state primary condition state, The correct character therefore becomes expressions to be correct. of the supercharges thatsubtle. annihilate To illustrate the the superconformalwith general primary, idea, which consider once is more in the fact rather the cohomology of a particular supercharge contribute. We therefore believe the resulting JHEP05(2015)017 1 3 in 1 . ą Q ´ i 0 2 (C.17) (C.19) . The d M “ q s multiplet 3 , q , s 1 b d 3 , , 0; a , 1 p 0 d , 0 Q r 0; then each ψ , 4 0 ´ i 1 0 (C.14) , 0 Q M r ‰ D k 0 (C.16) in the character formula. b , 1 q 0 “ Q ψ ψ ψ ψ ψ , q s s shows that the primary null 4 4 “ 0 ss 4 q 1 , b sss 4 s 4 1 0 ... , 4 ´ 2 , 3 1 Q , 1 a 1 1 Q 1 3 c b p R Q r 3 d 1 1 p Q 3 1 Q 0; ψ 4 3 1 , 0 (C.18) Q Q 4 1 1 0 Q 2 0 (C.15) , 1 : Q 2 1 0 Q “ Q 2 r 1 Q 3 2 “ 1 Q ψ 3 1 1 ψ Q s 1 . Therefore if 3 1 Q ss “ 4 0 1 1 , Q 1 Q ´ i 1 2 k 1 0 Q 2 1 1 , a Q Q 1 1 1 Q 1 Q 4 M c Q 3 , q r Q Q , both of which annihilate the superconformal , Q 2 ´ 2 1 ´ 2 , ψ multiplet. Table , ´ 2 2 ´ ) it suffices to realize that one may rewrite this 1 3 Q – 30 – ´ 2 s 0 0 R R 1 2 ... 3 let us consider the type r R 2 Q p 2 M R r 3 that the superconformal primary is killed by 1 Q “ “ r , q a , d ď 2 Q , C.18 s s 4 1 1 2 ´ ´ 2 1 1 0 3 2 ´ 2 , , d 2 Q d 0 1 R Q R , , r R d p Q 1 0; r 2 c 0; m , r q , ´ i ´ 2 0 0 , ψ ´ 2 , 0 R 4 1 M r r R 1 r c p ψ ´ 2 r Q 2 B 2 1 R 1 Q ,..., Q 1 ´ i ) is non-zero for a specific combination of the , M 0 r C.14 and then further with “ s ´ 3 3 , 1 d M 0; , 0 . To see what happens for , q 0 r , q ψ 1 b 1 Next we consider the R symmetry quantum numbers. By carefully matching how . Altogether this leads to , ´ a a Q p and therefore that thischaracter formula. combination of In order superchargesexpression to also as show a needs ( linear to combination be of the removed following from terms: the Let us now demonstratetional that relation: the highest weight state in this multiplet satisfies the addi- the above analysis suffices andQ there are no additional termsas that an need example. to be Weand, removed obtain from in from agreement table with theM rule given above, we find three more shortenings by acting with one finds that if ( this combination annihilates the superconformal primary and the resultmany follows. states should be removeduncovers at the a following given slightly level more against involved the pattern. number First of of states all, available one we find that for primary, we find the additional null states These combinations therefore alsoexplanation behind need our to general rule be tracks the removed logic from of this the example: factor from a direct analysis For example, consider the state is given by Acting with all possible actions of the lowering operators then the resulting combination of superchargesso annihilates the the corresponding superconformal term primary and needs to be subtracted from then the additional shortenings with the same R symmetry indices are obtained by taking JHEP05(2015)017 q 5 p so -chiral (C.20) (C.21) Q 3. type acted “ s q 2 2 3, we need to s d p , d 1 1 ą χ d has on the shortening 1 1 d d q 0; 1 1 q q ψ , ` d 1 , respectively. Then, ` 2 0 ´ d { ` u 1 ´ ´ , 3 3 . These are the q d 1 0 1 D q 1 q q λ r , ´ 1 : : : 0 and on the combination states A C s s s , R 4 1 0 2 p ą q B , , , , 1 1 1 ´ 2 2 d d d d A R t . We have not implemented this p 0; 0; 0; q ´ , , , 1 0 0 0 , q ´ X for , , , a λ 1 0 0 0 ,..., p 3. In that case one should also remove u ´ r r r 2 ´ obtained using the above procedure for s 4 P s D D D ` , ď s 2 with 3 ´ 2 d ´ 2 , q 1 s d 2 s , d 2 ´ 2 ` , 1 λ ` 1 R d , d s q p 1 ; – 31 – 1 q ), and the last one because s , q 3 d s d p 1 s p ; 1 “ t p , c 1 ` 3 q “ ` 2 c 2 2 indices, but it does not matter which term. 1 s C.17 d ` , c c χ X p q precisely those combinations of the supercharges that q , c 1 2 2 5 λ χ c 1 c p ´ 2 q c 2 χ , c χ { ` 16 so 1 R then the expressions simplify drastically. We find a non- . , q 1 2 2 qr p c s { { c q 2 b r 1 1 2 q ` d c c , q 1 ´ X 2.1 a ` ` , d d ´ p 1 1 ´ ´ 1 1 ` d d 1 d 2 1 Q 1 ` ` { ; d 4 2 7 3 q q q ´ , c ) is precisely the shortening condition of : : : 2 s s s X 0 0 1 , c p , , , 1 . This is indicative of the following general pattern. Let us enumerate C.18 1 1 1 c 4 d d d q qr 2 ´ 2 0; 0; 0; d , , , indices to be symmetrized. The precise statement is that one has to remove precisely one R 2 0 0 p q ´ ] to obtain expressions that match our expectations. , , 5 , c 1 1 p character corresponding to the irrep with highest weight 1 c c 52 X r r c q p so r 2 C C 3, as well as all the supercharge combinations obtained from all the other p B 3, the superconformal primary has Finally, when the multiplet contains conserved currents one should further remove In a similar vein one finds that further factors may need to be removed if, in addition to Notice that ( su Notice that the Lorentz indices in the shortening conditions are always antisymmetrized so one may ă ď 16 2 2 the operators described in the main text. take the term in the sum that symmetrizes the with zero contribution only for the following six cases: The explicit form of the charactersto is reproduce obviously them rather here, involved. but theyhand, It are does we available can from not use the seem these authors wortwhile upon charactersof to request. each compute shortened On multiplet. the the contribution If other to weindex the in as superconformal addition described index take the in unrefined section limit of the superconformal the action of certainin momentum detail, operators relying from algebra instead [ on the known formC.2.2 of short representations Contribution of to the the unrefined conformal superconformal index the supercharge combinationsconditions obtained of from type thed action of lowering operators acting on this combination. in a short multipletadditionally of remove type from one obtains from theof action type of d of the states listed abovethat is vanishes null: due the to first the four shortening because the ( commutators evaluateupon to a with term the shortening types by an integer with coefficients that are easily determined but unimportant for our analysis. Now, each JHEP05(2015)017 , s 3 ,c 2 (D.1) (D.2) ,c -chiral 1 . If we c q r Q 4 ψ p su Ă 1 q -chiral state must 2 p Q su -chiral state representation. Indeed, Q q , 5 p 0 so -chiral operators in shortened “ , , , s Q s s . 1 R , 2 2 -chiral operators are of the form on this state will be as follows, . We may conclude that within a s ´ s ´ ` 3 3 2 1 Q ´ q 2 2 q ,c ,c ` 4 acting on a . ,d ,d 1 -chirality conditions, and indeed the 2 1 2 3 p 2 1 q p h ´ ` ,c Q 0 4 ` ´ 2 2 1 , p 2 1 1 su ´ usp ,c ,c ´ d d “ 2 1 r r 2 su y y E ` ´ ,c h.w. 1 1 1 ψ ψ y c c c | | r r r 1 2 ψ ψ ψ ψ | λ λ -chirality conditions and has Dynkin weights k , r q Q „ „ „ – 32 – “ “ ùñ 3 ´ 1 s s s h s s -chiral operator satisfies the following defining set 3 3 3 2 2 M ,c ,c ,c Q were non-zero, one can easily see that the resulting ,d ,d “ p 2 2 2 0 1 1 , the only potential ,c ,c ,c 2 d d λ q r r 1 1 1 h c c c y y 5 r r r p ψ ψ qs “ ψ ψ ψ | | 0 so ` 3 ` 1 ` 2 ` 1 ` 2 p O R R M M M , q ˆ 0 4 that obeys the p ˆ L . If either r s 2 representations is not quite as simple. This is because the super- su 0 , , 1 highest weight. 1 q λ d 4 q r p y 5 p ψ su | so . The action of the positive simple roots of -chiral operators s 2 Q The story of Now the important question for us is where such states may appear in a UIR of the six- As explained in the main text, a , d 1 d The second and third of thesehand, states could will violate be unitarity an if allowed non-zero. statefirst that The represents also first, the satisfies on action the the of other given the representation raising of operator of have negative eigenvalue when actingnecessarily be on a that state. Consequently, a charges involved in these argumentsconsider all the commute action with of the the positive subgroup simple roots of for some coefficients state would violate unitarity in the sense that sums of squares of some supercharges would regarding the placement of suchsee an that state such a in state a mustconsider six-dimensional be represenation. a in the First, state highest we weightr can state of its By unitary, an operator satisfyingtions this condition will necessarily obey the additional rela- dimensional superconformal algebra. Let us start by determining some general properties of conditions for its quantum numbers: In this appendixoperators. we The provide derivation presented an belowand alternative may does give and a not more require intuitive more the pictureUIRs. behind computation direct the of The presence derivation characters of results ofappendix. that the we obtain are in complete agreement with those of the previous D JHEP05(2015)017 , , r scp ψ 2 -chiral 2 -chiral, Q Q Q 1 symmetry 1 r Q . , representation -chiral operator q us scp 5 Q 2 that are p ψ , q 1 12 so multiplet. 5 , p 0 Q 1 1 t q q ˆ so 1 , 2 4 1 p p Q d highest weight transforms. su q ˆ su 0; 4 q , p . 4 0 p s -chiral operators reside, it will , su . 0 , 0 Q su . Indeed, one can check that such s 1 r ... , and one may at first think that those 0 , ` 0 D 1 2 ˆ ` L d ` 1 0; , , d ψ , s.c.h.w. 0 . Thus we are actually only interested , 4 2 0; 0 , 2 ψ n 2 us -chiral operators. 2 ˆ 2 L ` 1 Q 2 s.c.h.w. 1 ` Q , . For such a multiplet, there are potential c 3 1 2 s 0 ψ r Q n 2 ˇ ˇ -chiral operator, which is as follows: ,c t 0 1 1 Q , , ˘ Q 2 Q c in which the 1 r 1 2 2 ψ ˇ ˇ – 33 – d Q d n 1 1 ¨ ¨ ¨ 2 ψ . In this case there is a potential q b s may be non-zero, in which case such a 4 2 Q 0; 0, which would violate unitarity. 0; 2 Q 0 1 1 , , p 2 b Q 1 , n 1 ‰ 0 0 b 2 c 1 Q , , “ su r 1 Q d 1 1 Q multiplet of c c ` Q r r 0; . It turns out that only the first of these actually appears q , C h.w. C -chiral operators are those which the superconformal pri- 2 2 p ψ scp 0 and Q , c ψ , “ 1 2 s c 0 2 eigenvalue of any superconformal primary state can be lowered 0 r ˆ L , -chiral operators searching of states of this form with the correct Q B 0 1 1 Q ˆ d L 2 Q 0; , 2 -chiral operator is therefore two. Consequently, the types of multiplets Q , or , c -chiral operator existed that included an action of such a supercharge, there would be 1 Q c scp r 17 ψ eigenvalue less than or equal to two, along with some additional B must have positive eigenvalues on any physical state, and there are only four 1. 2 0 , 1 0 0 ˆ L ˆ Q L 1 “ 2 -chiral operators including terms of the following forms: i operator will lie in an SU Q Q in the highest weight component of a a state does existhighest (it weight is state not of excluded the following by projection: the shortening conditions), and it isNote in that the generally speaking including a term of the form n The next consideration is A multiplet of type The most that the We can do this as follows. The highest weight state of any i There are also supercharges that do not shift the value of 17 ii(a) could be included inmeaning the that action if as a another well. operator However, present those with supercharges will necessarily shift the value of Let us consider these options in order. before reaching a that may conceivably contain mary has constraints. The possible cases are easily enumerated to be the following: fact that supercharges whose action reducesin the operators value of of the form with one of which will takehighest the weight form state: of up to sixteen supercharges acting on the superconformal Thus, we can search for quantum numbers. The possible operators of this type are immediately restricted by the For the purpose oftherefore identifying be representations sufficient in toand look which subsequently for include highest any weight additional states states of in the relevant appearing in the superconformal multiplet will be a linear combination of states, at least which fill out the representation of JHEP05(2015)017 , s 0 , scp , 3 ψ 1 2 d “ 1 -chiral, 0; Q 336 D Q , 0 , 0 r D -chiral operators (1978) 149 (1999) 085011 (1977) 31 Q 53 . For . Again, only the first D D 60 (1999) 61 B 135 scp , ψ s . 0 2 , . 2 s 2 0 -chiral operators in this case. s , ` Commun. Math. Phys. Q B 467 0 2 1 , Q 2 , d 1 ` 1 1 0; ` -chiral states include the term , Phys. Rev. d Q 1 1 , , d 1 Q 0; Nucl. Phys. , 1 1 0; , 1 , , ` . 0 Q 0 1 , r c s 1 limit ˇ ˇ r r 0 ˇ ˇ , ˇ ˇ ˘ Phys. Lett. 1 s ˘ ˘ N d 2 , and s Commun. Math. Phys. s , 1 1 0; multiplet of 1 , , , , N d 1 . In this case the potential 1 0 q , d s scp d 0; 0 2 , -chiral operator, which is as follows: 1 r 0; ψ 0; p 0 , , , , ]. Q 2 0 ψ 0 – 34 – 0 1 , , 2 r 1 Correlation functions of operators and Wilson -chiral states include terms of the form 0 d c r ψ Q r Q ψ 2 0; ψ 1 b , SPIRE 0 b 2 b Q , IN multiplet again. 1 b 1 3 Q Three point functions of chiral primary operators in 2 , the only possible c q ][ b ` Q s r SCFT at large- ), which permits any use, distribution and reproduction in 2 Q ` 2 theory in the large- p C Q q , ]. ]. ` 0 q 1 , 2 d 2 , 0 0; p , “ p , , the possible SPIRE SPIRE 0 s 6 , IN IN 1 N 0 , “ CC-BY 4.0 r , ][ ][ 1 6 Infinite chiral symmetry in four dimensions d D D . The second of these is in the same multiplet as the first, which gives This article is distributed under the terms of the Creative Commons “ , which occurs in the following projection: 0; arXiv:1312.5344 , scp [ D scp Supersymmetries and their representations 0 A sketch of Lie superalgebra theory ψ , ψ 2 ]. ]. 0 2 -chiral highest weight state in the following projection: 2 r 2 and Q Q D Q 8 2 1 “ SPIRE SPIRE Q and us a We have a non-trivial SU these are actually half BPS states, and the superconformal primary itself is include terms of the form of these appears in a highest weight There will always be a non-trivial SU hep-th/9902153 hep-th/9907047 IN IN N [ surfaces in the [ [ [ (2015) 1359 For Finally, we consider the (at least) quarter BPS states of type We also consider the case Finally, for R. Corrado, B. Florea and R. McNees, F. Bastianelli and R. Zucchini, V.G. Kac, W. Nahm, C. 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