JHEP12(2019)039 = 2 -string N k Springer , August 21, 2019 December 4, 2019 : November 9, 2019 : : Received [email protected] Accepted Published , b,e Published for SISSA by https://doi.org/10.1007/JHEP12(2019)039 SCFTs, as a generalization of Del Zotto- G and Xin Wang k H d 0) SCFTs include certain rank one theories with . In this paper, we propose a universal recursion , 8 , 7 , 6 ,E 4 ,F Kaiwen Sun [email protected] , b,c SO(8) , . 3 theories. We also observe an intriguing relation between the G 1905.00864 H The Authors. = SU(3) c Conformal Field Models in String Theory, Field Theories in Higher Dimen-

G The building blocks of 6d (1 , [email protected] Albrecht Klemm, a Vivatsgasse 7, D-53111 Bonn, Germany E-mail: [email protected] D-53115 Bonn, Germany Hausdorff Center for Mathematics,D-53115 Universit¨atBonn, Bonn, Germany Scuola Internazionale Superiore divia Studi Bonomea Avanzati 265, (SISSA), 34136, Trieste,Max Italy Planck Institute for Mathematics, ´preetdeD´epartement Physique Th´eoriqueet Section deGen`eve,CH-1211 Math´ematiques,Universit´ede Gen`eve, Switzerland Bethe Center for Theoretical Physics, Universit¨atBonn, b c e d a Open Access Article funded by SCOAP Keywords: sions, Solitons Monopoles and Instantons, Topological Strings ArXiv ePrint: superconformal elliptic genus and the SchurLockhart’s indices conjecture of at rank the ranknon-Higgsable one clusters cases. with In matters. a subsequent paper, we deal with all other gauge group formula for the elliptic generablowup of equations all introduced such theories. ingenera This our and formula previous is refined paper. solvedstring BPS from theory, the invariants, modular We elliptic which explicitly bootstrap, recover compute Hilbert all the series, 2d elliptic previous quiver results gauge from theories and topological 4d Abstract: Jie Gu, Elliptic blowup equations forExceptional 6d cases SCFTs. Part II. JHEP12(2019)039 38 41 45 48 51 7 77 ) 29 23 k ( 57 G 32 H 20 26 19 29 – i – 54 35 10 7 27 29 13 18 22 0) minimal SCFTs 69 76 , 73 52 fields 34 r 16 = SU(3) and SO(8) 81 theories G G 60 1 6 7 8 4 k H F E E E = = = = G 4.1.2 Symmetric4.1.3 product approximation Symmetries G G G 4.1.1 Universal expansion 3.3.1 Recursion3.3.2 formulas for elliptic genera Uniqueness of recursion formulas 5.3 Rank one:5.4 Del Zotto-Lockhart’s Rank conjecture two 5.5 Rank three and higher 4.6 5.1 Rank 5.2 Hall-Littlewood and Schur indices 4.2 Revisiting 4.3 4.4 4.5 3.4 Vanishing blowup equations 4.1 Universal behaviors of elliptic genera 3.1 Modularity of3.2 elliptic blowup equations Universality of3.3 elliptic blowup equations Unity blowup equations 2.1 Geometry of2.2 elliptic fibrations Semiclassical partition2.3 functions Determination of 2.4 De-affinisation A Lie algebraic convention B Mirror symmetry for elliptic non-compactC Calabi-Yau three-folds Geometric data 6 Conclusion and outlook 5 On the relation with 4d SCFTs of type 4 Elliptic genera for 6d (1 3 Elliptic blowup equations Contents 1 Introduction 2 Elliptic non-compact CY 3-folds and generalised blowup equations JHEP12(2019)039 ] N 2 ∈ r ) label ,j l r β j , j N l j The refinement of 1 and two Ω background parameters t and describe them in more detail in ]. Such a classification in 6d is highly 7 – X 5 ]. Solving the topological string partition little group of the 5d Poincar´egroup, which 8 , r 3 – 1 – ) is the degree and the half integers ( Z SU(2) 84 X, × ( isometry a refined topological string partition function 2 l H R ] and the relation to the refined BPS invariants ∈ 94 10 β , ]. Limits of non-perturbative string theory compactifications [ 9 1 ] and lead recently to a complete classification of geometrically engi- 82 4 87 ]. Here , 3 12 , 11 . ), which depends on the K¨ahlerparameters 2 2.1 ,  elliptic non-compact Calabi-Yau geometries is defined as generating function of refined stable pair invariants. 1 The full topological string partition function on these elliptic non-compact CY geome- The 6d geometry is the one of an — in general desingularised — elliptic fibration with 2 In this section we underline a symbol, if it is a vector. After the introduction section, we drop the ,  t 1 ,  ( 1 was given in [ a spin representation incan the SU(2) be identified with Lefschetz actions on theunderline moduli when space there is of no D2 risk of and confusion. D0-branes. On function on compact Calabi-Yau manifoldsCalabi-Yau is spaces currently an with open an problem.Z U(1) On non-compact  the stable pair invariants [ short section tries has received muchstates attention of as the it 6d contains superconformal important theories information [ about protected curves in the base. Insurfaces the decoupling is limit the scaled volume outside toof of the infinite complex configuration size, of desingularised elliptic leaving ellipticCalabi-Yau us surfaces threefold. with that an, Because canare be in compact sometimes general contracted components called within reducible, can local a configuration be non-compact Calabi-Yau contracted spaces. such geometries We will call the above specific ones for the first examples [ neered 6d superconformal quantum fielddesirable, theories [ as it mightsuperconformal lead theories. by further compactifications, to an exhaustive classificationa of contractable configuration of desingularised elliptic surfaces fibred over a configuration of 1 Introduction Six is the highest dimension informal which quantum representation theories theory [ allows forand interacting in supercon- particular the decoupling of gravity in F-theory compactifications to 6d provided E Relation with modular ansatz F Elliptic genera G Refined BPS invariants D Useful identities JHEP12(2019)039 = 2 N ] up to . 8 2 ) and all  2 1  ) can be ef- ,  − 2 1  = ,  expansions fail, 1 2 , i.e. the classical λ 4) gauge theories ,  , t are identified with N X ( ], the integration of 0) 6d SCFT is non- d and Z , of the base [ Z )) containing the informa- 11 2 ]. 2 t  ] for the calculation of exact in ( d ( ]. ]. − g 13 25 d F 19 = 2 18 – − 1 g  2 16 λ =0 ∞ g of the desingularisations and even- P a ] leading to Jeffrey-Kirwan integrals. ] and elliptic non-compact CY 3-folds studied and for each 15 21 torus localisation [ d , to consider ever more complicated quiver ] and a recursive solution of blowup equa- ) = exp( 3 20 d ], ) below. at different winding , λ 14 t ( of the elliptic fibre class becomes the modular 13 d 2.3 – 2 – Z Z τ the classical topological data of 5 ) transform as a Jacobi form and are identified with 2 ,  1 ) for the latter can often be recovered [ techniques [ 2 ,  ,  1 N are so constrained by modularity, the pole — as well as the , m ,  t d ( ]. τ, a Z in the elliptic fibration become elliptic parameters. The refined Z ( = 1 supersymmetric gauge theories [ isometry, and can be viewed as the borderline case for calculating d 24 ) as well as K¨ahlerparameters – 2 m E R to the blowup equations, which can be easily calculated from the intersection and ] the world-sheet elliptic genus of dual 2d quiver (0 N ,  8 22 0 [ 1 . The Z  4 d ) that is further based on a specialisation of the generalized blowup equation ). Since the techniques based on toric localisation and large ] generalized from K-theoretic the blowup G¨ottsche-Nakajima-Yoshioka equations 2 2 ,  ,  15 ] to the elliptic non-compact Calabi-Yau geometries. The main advantage of this 1 1 ]. For the elliptic non-compact CY 3-folds associated to exceptional gauge symmetry studied in this In the 2d approach one needs for higher The class of elliptic non-compact Calabi-Yau relevant for the (1 15 The more supersymmetric caseThis of holds the for M-strings all has non-compact been CY pioneered 3-folds in studied [ in [ The holomorphic all genus partition function Strictly speaking the refined topological vertex applies directly only to geometries which engineer ,  ,  25 t t 4 5 2 3 ( ( transitions of gauge theoriesgauge geometries theory with interpretation, Chern-Simons terms to geometries which have no immediate in [ paper, we also input the multi-covering of isolated rational curves, see ( other K¨ahlerparameters. tion of all genus Gromov-Witten invariants is obtained asgauge specialisation theories, as these have the required preferred direction in the torus action. In blow downs and in [ approach is that it needs astriple input intersection numbers only as well asthe the Chow evaluation group, of the and Chern yieldspartition classes with function on a the iteratively non elements in of ambiguous the efficient base recursive degree procedure the string gauge theories, while inplicated the modular rings approach of onethe weak has recursive to Jacobi approach deal forms. based withZ more on For and the this more elliptic com- reason blowup equations we [ further develop in this paper modular bootstrap approach wheremeromorphic different Jacobi winding forms contributions with weightthe zero and base an degree index, whichrefined depends BPS quadratically structure on ofin the many topological examples string [ that they can be completely reconstructed the topological string partitioncertain prefactor. function The K¨ahlerparameter parameter while ( tual further sections holomorphic anomaly equations and other B-model techniques also apply and lead to a toric, but has aZ U(1) two related new methodsone can have calculate been developed.with supersymmetric Similar localisation as techniquesThese in [ elliptic heterotic/Type genera II duality ficiently calculated by large the refined holomorphic anomalytions equations [ [ in the context of 5d toric non-compact Calabi-Yau spaces the refined partition function JHEP12(2019)039 ] . 1 ], 2 i P 15 P D (1.1) — a ] on those 2 is the rank b P 27 ] respectively. N ]. If the geom- parametrized by ) is related with 18 26 , 2 2 ] in the context of Φ) on b P ,  17 ) theories without or 1 ) and provides an al- 28 isometry and the (re- 2 E, = 0 to the one on N ,  on R t ,  ( k ∗ 1 ) at all relevant fix loci is Z C ] and [ ,  t ( × 16 . Then one defines vector ∗ Z X . A general feature of this cal- C i N, k, n ] ( of 1 P c c 2 M [ , , ) 2 ], and leads to blowup equations for the and refined topological string partition / E ( r 1 , . . . , b 18 X c + , =1 n −h 16 · j = 2 supergravity prepotential, while the latter = – 3 – ], C j k N ) on non-compact Calabi-Yau engineering super- = 2 ], which do engineer supersymmetric gauge theories. n ,  and R 1 ) gauge theory, the full correspondence states that the 17 , i ) framed gauge instanton calculus, by studying invari- ,  ] ). Here Φ is the framing automorphism, N t ) of framed torsion free sheaves ( 2 16 ( N b P [ Z can be also directly specialised for , ) 2 which parametrise the shift of the K¨ahlerparameters. With N, k, n E b P N, k, n ( ( ]. The K-theoretic blowup equation for U( c 4 ( 2 1 b c c M Z c M 1 18 , − N ∈ and Coulomb branch (in type IIA normalisable K¨ahler)parameters. , which arise due to the action of the 2 on N 16 1 = 2 or Seiberg-Witten prepotential, which is related to the former in a ) be the intersection matrix between compact divisor classes [ 2 r , P N ij 1 −  N C ) Gl and , the generalized blowup equations can be cast in the following form [ E — via the Atiyah-Bott localization formalism w.r.t. an induced toric action i ( × ] to 5d and 4d supersymmetric gauge theories as c 4 2 n = ( 2 and compact curve classes [Σ ∗ b c 26 P h c 4 Z C c 4 =1 . Upon evaluation of the Atiyah-Bott localization formula the two contributions b i C 2 = × ∈  = 2 supersymmetric SU( at shifted P ∗ n n C 2 , Let Nakajima and Yoshioka derived the original blowup equations [ Let us first give a short summary of the structure behind the blowup equations in four, , . . . , b = N P and E = | =1 1 n i with | relevant to the calculationsymmetric of gauge theories. The blowupgeometric equations data can of then the befunction non-compact reformulated as Calabi-Yau in follows. terms of the The partition function on The identification of the twofunction results of the gives 4d rise supersymmetric toapplies theory a to on the the finite Omega K-theoretic set background. instanton of calculuspartition A equations [ function similar for mechanism of the the partition 5d SYM on the Omega background. The latter setup is directly always a product ofof contributions the from exceptional two fixed points at the northyield and the — south up poles toon calculable factors — a sum of products of the original partition function ants on moduliblowup space of T of culation is that the Euler class of the tangent space of K-theoretic partition function of theternative latter definition is [ identified with with Chern-Simons terms has been rigorously established in [ 4d In the geometric engineering approach, givenworld-sheet mirror instantons symmetry, it and is space physicallycorrect obvious time that the instantons topological are related. stringcorrect theory Simply the or rigid because thewell former defined limit in theetry engineers B-model, five that dimensional decouples U( gravity as decribed in [ five and six dimensions. Non-compactfined) Calabi-Yau topological spaces string with partition U(1) function featureapproach prominently in [ the geometric engineering the K-theoretic extension ofnon-compact Nekrasov’s Calabi-Yau 4d spaces gauge [ theory instanton partition [ JHEP12(2019)039 . , 12 v u , ) of ) re- 8 2 (1.2) (1.3) ] this 2 , ∈S ∈S ,  6 SO(8). r ,  25 1 , , . These 1 c 4 5 to denote ,  , ) ,  t 2 4 t t ( ( , ,  Z Z . 1 , . . . , b = 3 m ,  = SU(3) t ( n = 1 ] that the classical G Z ) k 15 , r ], . ]. k . For ) determine 1 2 6 , m D , P 2 5 ,  -fields in these two sets are 1 r ,  1 → symmetry so that the refined ,   ) r , r i R n 0 Λ( ] that these equations, called the π − ( . The ( + 25 u , t O S mod 2 (1.4) ( )= ) have to be only considered modulo 0) SCFTs [ 2 , 23 from the K¨ahlerparameters , inst ,  1.2 β 2 and Z · 15 m  ) v r 2 − S 1 ,  0) SCFTs with gauge group ≡ 1 correspond to the residues of the meromorphic , ,  – 4 – n ,  -field, is consistent with the checkerboard pattern t m r + 1 R ( -fields in ( 2 r  r j cls ]. At the technical level the precise non-trivial claim + ) and the method of proof in the gauge theory con- Z t and give a universal description for all minimal building ( 12 + 2 , ]) 8 l b 1.2 , Z j ) = 7 ) , in other words, they satisfy , 11 29 2 2 1 r 6 means the K¨ahlermoduli in the instanton partition function ) and the  ,j c 4 ,  , should hold for the refined partition functions -fields. It is important that Λ, whose form is known, depends l β b Z j 1 − ,E r 4 2 N ,  = F t ,  ] and [ ( . The set of -fields in the two classes and the mass parameters 1 g r b = r Z mentioned above and the genus zero sector 23 ,j ,  unity l , which we call the β j n G r X R N 1 and  and no matter contains base surface should be non-empty. In addition it was observed in [ + only on t u G ( . The hat over 2 , b λ ], it seems reasonable to conjecture [ Z 1 | ∪ S  n | v 18 , 1) S vanishing , which leaves two classes of finite sets − 16 ( n The elliptic non-compact Calabi-Yau geometry corresponding to minimal SCFTs with Given the simple form of ( c 4 · b correspond to mass parameters in the gauge theory context, while the other K¨ahler Z C ∈ X tional gauge groups blocks without matter in the classification of 6d (1 gauge group which are of interest in this paper, the Kodaire singularity of the elliptic fibration gives is that topological data of cursively, and many examples were alreadyapproach checked in was great used detail. to In computeYau particular geometries the in associated [ refined to minimal BPS 6d invariantsIn (1 of this elliptic paper non-compact Calabi- we extend this approach to the remaining minimal 6d SCFTs with excep- text [ generalised blowup equations all non-compact Calabi-Yau threefoldsinvariants with or a equivalently global the U(1) correspondingΩ BPS background index can for be the defined space [ time theory with an for non-vanishing 2 called beside on The integral vector of refined BPS invariants mirror curve (of genus differential have already been shifted those curve classes thatm do not intersect withparameters compact correspond divisors to [ Coulomb branch parameters,ters. thus If are a also local called mirrorK¨ahlerparameters “true” are curve parame- exists, mapped e.g. to for the non-compact complex toric Calabi-Yau structure spaces, parameters the of “true” the (hyperelliptic) n Here we have separated the K¨ahlerparameters (see also section 8 of [ JHEP12(2019)039 . |  2, in 0) 0)
∨ / , , d ω ) as 1 Q · (1.5) E : − m 1.5 into the P n − | 2 n k/  G ) 2 = of d , and predicts a + ∨ E 2 ) ) of the 6d (1 Q 2 || 2 ω . The subscript of , ,  || 2 12 which are of main 1 G  1 2 , ,  8 − , . t , ). The blowup equations 1 ) for the definition. Due +( ( 6 N N 1 , , Z  6∈ ∈ 3.5 ) ω 3.32 1 2 mass parameters. d  = 5 ]. Moreover, the unity elliptic ) in particular the elliptic genus + − . Since the number of different 2 n 2 λ not 30 ,  || , φ , d 1 ) ω τ,m 2 16 ( ,  || t 2 1 2 , . For ( d and 1 ( 1 E Z , a ) n 1 functions, see (  − respectively. We find that for these geome- ) τ,m − η 2 8 ( is even, and the characteristic 2 ,  d 7 – 5 – , , + n E 6 1 · 1 associated to gauge group and  , )) ,E 6 1 2 4 ω θ  2)( 1  ,F + − = 1 supergravity or more precisely M-theory play for the 1 of the corresponding 6d SCFT: − n previous partial results. Since we made many checks of is an embedding of the coroot lattice  ( d N λ τ, E 2)( all φ τ,m n SO(8) -field is implicit in ). The Coulomb parameters are (  , ] − r 1 a d n [ i 1.2 is odd and 3 if ( ). In particular it is possible to derive from the structure of the θ E | 2 . In fact, they can be regarded as the natural elliptic lift of the ) ) ) ultimately lead to a complete solution of the elliptic genera n τ, in ( ω ,  , the total number of non-equivalent blowup equations is n ( m | 1 ( 1 ( 1.5 m ] ∨ − λ ω a ,  = SU(3) , d [ ) is composed of i φ t Q | 0 θ A ( : m is 4 if 1. Besides, G 1) ( ( × Z . Here the i ω P − − | = P ( A ). The recursive structure also helps clarify the form of the denominator of d n N . ∈ = 2 2 , 1.5 d 1 + are the Coulomb parameters ,d ,..., 1 ) 1 d 3.3.1 ∨ , m X + Q 2 ( Since the elliptic blowup equations determine In the program of classifying superconformal field theories in various dimensions the 6d || λ Do not confuse with completely, we could make many detailed and indeed successful checks on our results. ω 6 φ = 0 || d ∈ 1 2 ω We summarise the currentminimal status SCFTs of from the various knowledgeinterest approaches in on in the table the current ellipticblowup paper, genera equations our of reproduces complete all recursive 6d solution (1 for the elliptic genera from limits as well asminimal suitable SCFTs expansion relate of totheories. the the partition protected function quantities in lower dimensionalE supersymmetric many non-trivial relations among thesesection Jacobi forms, one particular of which is provenSCFTs in play a similar role as 11d classification of supergravity in lower dimensions. For this reason we expect that various are not only effectivewith tools the to general calculate constraintsthe the from structure refined modularity of partition and functions, BPSblowup but structure equations the shed also index some together and new theblocks weight light in of on ( the Jacobi formselliptic that genus constitute in the the building modular boostrap approach as discussed in appendix to the Jacobi formelliptic nature blowup of equations everyK-theoretic component blowup equations of for theblowup 5d equations above gauge in equations, ( theories we [ terms call of an ( universal recursion formula, as will be shown in ( theta functions k weight lattice embeddings is The function Here tries, the generalized blowup equationsrelations can of be the uniformly elliptic written genera as the following recursive the gauge group JHEP12(2019)039 ] ] 56 51 [ ] 1 2 51 [ - 7 0 + 38 k → - - , ? 7 q k E 4 SCFT, E E 37 , 2 4 , 5 F , we explicitly 8 4 = 1 ? 12 ? E ] ] 8 N ]. 48 7 ] [ 8 E 31 1 ] - 0) minimal SCFTs, the 45 E , ] , 43 with fugacities off [ – 44 ] , current paper ] 30 [ [ k 1 41 6 0 40 [ 0 E E 47 [ 6 genus zero [ , E → 0 → ], and the de-affinisation pro- 0 q k q k → E → 46 E q k , 15 q k ] E E 8 37 [ the latter two occur in a quite non- [ ] 4 k 1 ] 5 37 E SO(8) E , 25 , we review the geometric construction [ ] 24 k [ 2 ] to obtain elliptic blowup equations for . We also prove two important properties 34 E 8 1 [ , ? ? 3 25 7 E , k 6 SU(3) E are quite obvious as for example the 5d limit ,E – 6 – 1 ] ] ] ] 4 19 19 49 52 ,F [ [ [ [ - - - ? 2 k k k k E E E E , we discuss both unity and vanishing elliptic blowup ] M-strings 3 SO(8) ] 36 , ] ] , 50 ] [ 23 35 [ 52 ] , trivial 3 k 33 E , ]. We will push the study on such relation for all rank two and 39 E [ 23 ] - - - 1 , 32 37 k [ 49 15 [ E = SU(3) k [ E-strings 2 E , k G 1 low genus [ E E is the modular parameter. - means the method does not apply, and ? means τ i the one- and two-string elliptic genera computed from our universal recur- π 2 G theories in [ = e theories G τ G H n Q H . Known results on the elliptic genera of 6d minimal (1,0) SCFTs from various approaches. = features B-model HL index 2d quiver q This paper is organized as follows: in section Part of these checks indicated in table domain walls Hilbert series = SU(3) and SO(8). In section topological vertex modular bootstrap twisted 5d blowup equations 6d blowup equations G equations in detail, andminimal derive SCFTs with a universal recursionof the formula elliptic for blowup the equations,show i.e. elliptic for modularity genera each and universality. of In all section description in which other methods are quite difficult to carryof through. elliptic non-compact Calabi-Yau threefoldsbasic that properties engineer of 6d the (1 cedure generalized which blowup was equations essentially in already [ used in [ are relatively easy to compute.trivial As explained manner in in section the expansionsurprising of the relation elliptic between genera elliptic thatrank we genera one can and efficiently calculate. superconformaleven This indices some was rank found three forbut cases. the also This allows not to only calculate sheds them light efficiently on for the example structure in of theories these with objects, no Lagrangian exciting connections to the protected quantitiesone in of lower the dimensional most theories. importantduality For tools example in for the four analysis dimensional ofwhich SCFTs the count are spectra and operators the phenomena superconformal invarious like indices limits Seiberg the such for chiral as Macdonald rings indices, of Hall-Littlewood these indices and theories. Schur indices These which indices have in turn the elliptic blowup equationsinsights, based we provide on the extensive results calculations, of these which calculations might on yield agives further webpage a [ good confirmation of our results. Others are highly non-trivial and indicate new Table 1 Here possible applicable but results not yet attained. HL means Hall-Littlewood. JHEP12(2019)039 ]. ]) 12 , 25 29 8 , 6 , ] for 5d 5 ) and its , , and it is ]. We find G 4 ] and [ 0 18 , we discuss , , we describe 37 calculated in 23 6 16 isometry which = 3 ns i 2.1 b R n in the base, while the B has rank rk( ] and summarised in [ G and the generalised blowup 53 curve Σ 2.3 n 0) SCFTs with , − ) + 2 linearly independent compact G ] (see also section 8 of [ . s resulting from resolution of the singular , we discuss a surprising relation between 15 1 G 5 P = rk( c 2 b – 7 – rk) are + 1 curves intersect with each other according to r One of them is the proposed in [ ) as rk to lighten the notation. ,..., 7 theories, as was revealed for rank one in [ 1 G , G , where each node corresponds to a rational curve of self- -fields. In section g H r = 0 . These B I , then there are ( g I , will be discussed in full detail in the next section. ]. 15 fields can be determined in subsection r 2 and two curves intersect with intersection number 1 if the corresponding , the − 2.2 = 2 superconformal N generalised blowup equations We discuss some generic features of these Calabi-Yau threefolds, in particular the We then consider the expansion of the partition function in the base degrees and From now on, we simply denote rk( 7 elliptic fiber fibered overthe Σ affine Dynkin diagramintersection of nodes are linked. We denote the curve corresponding to the affine node by Σ compact divisors in theseLet geometries us are go best overassociated illustrated them Lie in here algebra [ quickly. is curves in Suppose the Calabi-Yau the threefold. remaining gauge rk+1 group curves Σ In this paper we are specificallyon interested which in F-theory the compactification non compact yields ellipticso minimal Calabi-Yau 6d that threefolds (1 the bulk theory has a pure gaugecompact group curves and the compact divisors in the Calabi-Yau. The compact curves and describe how to recastfunctional the equations generalised of blowup ellipticelliptic equations, blowup genera with equations some of additional the input, 6d as SCFTs.2.1 The latter, which Geometry we of call elliptic fibrations the minimal 6d SCFTs withsubsection pure gauge bulkequations theory. can With be this expandedBPS input to invariants as and extract in BPS [ constraints that allow for solution of refined The generalise the K-theoreticSYM blowup theories to equations all of non-compactmay Calabi-Yau Nakajima geometries or that and may have a not Yoshioka U(1) the engineer [ 5d geometric supersymmetric data field of theories. the In non-compact subsection elliptic Calabi-Yau threefolds associated to the convention, some technical details, andBPS invariants collect too more lengthy results to on be elliptic put genera in and the main refined text. 2 Elliptic non-compact CY 3-folds and generalised blowup equations intersection numbers and the the elliptic genera of 6dof minimal 4d SCFTs and the Hall-Littlewoodanalogous indices relation and indeed Schur indices existvarious for possible rank application two and and future directions. higher. In Finally, a in series section of appendices we explain our sion formula and also some relevant information for the blowup equations, such as triple JHEP12(2019)039 ]. 53 over ). (2.3) (2.2) I are the (rk + 1) 4.43 I × by ( .Note that the B B 2 . Σ are fibrations of Σ B 1 t are the degrees of its normal in a double section. Globally k 1 b is given by = e F ,... F Σ 1 1 I Σ B of , . 4 -1 D 1 Q -1 -6 ! in figure kF = 0 meets d B 4 = 5 local elliptic Calabi-Yau from [ + 6 type sphere tree over Σ I 0 and F Q 6 Σ ! F d S n ] (2.1) 4 respectively. 0 B = δ Z -8 F 1 I = for t -3 0 G I ,Σ =1 ∞ b S ..., A d I X 3 that corresponds to an outer automorphism of ] = [ D Σ 0 labelled by I − or and , 3 B 8 n B -5 [Σ . We illustrate compact curves and compact 1 + S B – 8 – I − t G

a 0

B Σ Z by =0 rk = I X . The linear combination 4 I cls Σ B C Z = = 5 model with and Σ n Z B B ] to be Hirzebruch surfaces of various degrees, and the Σ 53 is homologous to the generic elliptic fiber. We denote the complex- 2) in the corresponding direction. ˆ g monodromy encirceling Σ w.r.t. the base curve Σ − 2 2) on each rational section d n Z ( − ⊕O n, n ) which intersects with Σ − n = rk + 1 vertical compact divisors 1 − encoding the intersections between Σ c 4 is the affine Cartan matrix of ( b P . Compact curves and compact divisors in the plays the role of a Mori cone generator and is related to the base curve Σ the marks of O C b b A Dynkin diagram and folds its sphere tree to an I 6 a The topological string partition function on elliptic fibrations can be expanded in terms The E . They are argued in [ fibers of these Hirzebruch surfaces. It is then easy to deduce that the (rk + 2) B 1 divisors in the example of the of the base degree matrix where with ified K¨ahlerparameters of Σ Σ P curve Σ the only Figure 1 The indices ( bundles the fibration has a the JHEP12(2019)039 . , 2 2 B , 1 T  (2.7) (2.4) (2.5) (2.6) × in the taking 4 0 actually R m I d> ], which is base of the Σ Z , 26 1 and nodes in B P . i m rk measures the volume by the ] as well as the general , ,..., τ 11 B rk , and we find, by looking = 1 q t into a single vector ,..., ! . The coefficients i , i i d ell t m = 1 Q = d with 1-loop i E Z , i rk ) gets contributions from the rational τ . to infinity. In the resulting 5d theory, by i and replace Σ =1 + C 2 ∞ = I 0 d X I t ,  2 , m ). These contribution of isolated rational 0 , m − = , t i − − I Z n B t C α for rational curves [ 1 + t I ,  h h 2.36 I a −

t in the expansion of the BPS partition function = – 9 – ( ell 0 i intersect with no compact divisor, it is a mass pa- =0 rk d t I X , . . . r Z δ E 1-loop = . = Z q = 0 , m t cls i I 2.2 ω Z i τ , τ = m = 1, 2 Z ] ], and since rk =1 2 I − i 1 2 X δ , . and send its volume [Σ n ), which leads to ( 0 = 4 0 N − m 3.21 , we can collect the K¨ahlermoduli B rk), which are related to t ell is sometimes identified as g are now interpreted as the Wilson loops of the vector multiplets in t i 0 = Z m ) using = e ell ,..., . t 2 ell 2.2 T = 1 Q in the elliptic fibre that form the affine Dynkin diagram. These can be directly i in ( ( I ˆ i A . We discuss them in section comes from the degree zero maps and depends hence on the classical topological data We finally comment that in light of the correspondence between Note that the 6d SCFT can be reduced to a 5d pure SYM with the same gauge group The topological string partition function is identical with the BPS partition function of At this point a clarification of subtlety is in order. The curve classes Σ is defined such that the coefficients , τ, m X if we decompactify Σ . In the latter point of view, it is more natural to use another set of K¨ahlerparameters cls 2 ell ell Dynkin diagram of value in the complexified Cartan subalgebra G the only mass parameterfor is curve the class instanton not counting intersecting parameter with divisors, that are elliptic genera of theof the self-dual generic strings elliptic present fiber in [ rameter the of 6d the SCFT. theory. Inof addition, the it torus is also identified with the complex structure modulus t in terms of the corresponding 6d SCFT inT the tensor branch, putt on the Omega background generators. To remedyHirzebruch this, surface one with should thelinear lowest keep degree combinations Σ of in other thepoint chain, curves in so with example that non-negative section it coefficients. cannot We be will expressed illustrate as this curves can be alsothe calculated reason as that oneexpansion, loop on correction the to other the hand,and encode gauge are the coupling rather BPS difficult [ invariants that to do compute. wrap the basedo curve not Σ give a good basis for computing the BPS invariants, as they are not all Mori cone of curves Σ calculated from thematrix geometry reflecting the affinemulti cover group formula structure, ( i.e. the intersection Z JHEP12(2019)039 ) = n ) = − G 2 (2.9) (2.8) ( 1 ), and ,  P ). The 1 ). In the 2.8 . t,  1.5 2.9 ( → O i t cls It has 7 Mori X NS i Z with the second b 9 i ]. c 2 8 J the natural pairing b =1 8 i X Poincar´edual to the i [ with a single section, 2 , i ) h n 2 J ) 2 F 2   1 +  ) ( t 1 , and , on the other hand, do not over (  g ( NS i 1) ]. Let us look at an example in ˆ , X b 8 cls − (0 i 7 (We will use the same notation F . t 2 ) A of the minimal 6d SCFTs with 2 GV i . 2  b ,...,  by geometric means. , not all the affine Dynkin diagams have 1 c 2 NS i + 8  b =1 i X 1 = 0 , b can be also predicted from blowup equations by GV i = 1  ,E b ( + 7 i aided by the nontrivial automorphism of the NS i GV i i NS b b − ! ,E , b G k ) 6 – 10 – + the simple roots of t t j ( i GV i ) for t b ,E ) i are intersections of the divisors 0) α i , 4 GV i t ( n ]. b l cls 5) is the zero loci of the section of the anti-canonical F (1 ijk 55 F − GV i = κ ( b = ( 1 . P ) G =1 i ) + i c 2 , whose toric data is given in table t ( t b l ( , we need to embed the Calabi-Yau threefold X ∆ i,j,k 0) P , → O GV i 6 1 cls (0 b ] we argued that X F 25 : 2 2   term, the numbers 1 1 π 1 1 and are the triple intersection numbers of divisors   τ . This allows a reformulation of the generalised blowup equations in ∗ C ijk = ijk h κ ], which are difficult to apply here. The Nekrasov partition function of a with volumes ) = κ )) can be written as 2 i ) to be true and only compute 2 54 ,  , ,  1 and 2.9 1 14 C t,  are the fundamental weights, t,  h ( ( i can be realised as a hypersurface in a toric variety [ SO(8) can be computed by uplifting the semiclassical Nekrasov partition function cls ω cls , ˆ F X F To compute Up to a irrelevant See also the geometric description in [ 8 9 bundle of the toriccone variety generators with charges requiring the consistency ofvalues BPS computed invariants. from For the all method the we minimal will SCFTs, describe these later. predictions agree with the in a compact Calabi-Yau, forand instance first the compute elliptic these fibration intersection numbersYau in the compact geomtry. The compactdetail. Calabi- The threefold of the 5d pureaffine SYM Dynkin with diagrams, and the we same foundremaining these gauge minimal coefficients also SCFTs satisfy with the relationa ( non-trivial automorphism, andwe the assume method ( of uplifting does not always work. Instead the linear coefficients are subject to the relation In our previous paperSU(3) [ curve classes Σ Chern class of thehave Calabi-Yau a threefold. geometric meaning, The theyequations coefficients are [ usually computed by the5d refined pure holomorphic SYM anomaly also has a semiclassical contribution which takes the same form as ( The coefficients We summarise the computationthe of minimal the initial semiclassical datageneralised partition one blowup functions equations. needs here. inexp( First These order of are to all extract the refined semiclassical BPS contribution invariants from the between terms of Lieconvention algebraic of Lie data, algebra which we use we is use given2.2 in heavily appendix in Semiclassical the partition uniform functions formula ( where JHEP12(2019)039 2 for ) 1 1 4 ,D b I ( F 1 − − l (2.10) (2.11) D 5), and 2 2 0 4 is associ- − factorises (4) F ) and the − − l 0 B B . D 2 1 4 (3) F − l (7) . l ) ) + 3 = 7 of these ) 2 1 0 0 ) = Σ 4 = should actually be 0 k 4 (2) F ( n − 0 l 4 l S F ) ( S is the zero section at Σ j , and the intersection ( π n 2 0 0 2 0 4 i l , ˆ (1) F 4, J 0 − K l . (6) are identified (up to linear 5) curve (i.e. Σ l 4 X n> , this difference disappears. 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 F − (0) F 000 X ,..., = − − l 1 = ) 3 , ,S k 4. Only rk( ( 0 00 Σ G l 2 (7) ) , l j = 0 − ,S ( 0 for l 4 can also be identified. I ( ,..., (1) l 2 1 ∆ 2 ,D (6) P ijk − l 3 of the divisors = for = 1 ˆ κ 2 D i I 2 0 0 is written in the Weierstrass form, Σ ijk jk (5) D − X l κ , ˆ for X 1 2 i 1 0 1 0 1 0 0 0 (7) – 11 – l (4) = 0 respectively, while = D and its dual 2-form. − − l i i are identified as the ( ˆ y ˆ J J + 2 2 0 1 0 0 11 ] (3) ∧ − (5) l . When (6) ) 56 l , l ∆ ˆ 2 1 2 0 0 0 0 1 X . The toric data of P ( + (4) (2) − − l l 2 c are specified, there are standard techniques in toric geometry = 0 and (2) ˆ l 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 2 0 0 1 0 1 2 0 0 0 0 0 3 0 0 0 1 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 X while the other five toric curves combine linearly into Σ ∆ x (1) Z − − l P = 10 Table 2 , 1 = 5 Σ 5 3 2 1 2 1 F in the base. Since the elliptic fibration over , − − − − − − GV i b ˆ 5 ) are given by [ (3) 1 . The remaining four divisors F ˆ l 24 X 0 are respectively the vertical divisors pulled back from the (0), ( − ( ] ∗ i − 2 = D 8 c 0 ,S 1 0 0 0 4 [ Σ − F,S 2 3 1 0 2 3 0 1 2 3 2 3 0 0 2 3 0 2 3 0 2 3 0 1 2 0 0 1 0 0 1 0 0 0 0 0 0 0 ,..., . − 1 , 0 4 3 2 1 0 00 Once the toric data of 000 Strictly speaking, the (0) curve should be corrected byWe use certain linear the combination same of notation other for curves. the But divisor S F F D ν K S S D D D D D S → ∞ 10 11 = 0 combination) as the exceptional divisors divisors are linearly independent. to compute the triplenumbers with intersection numbers ˆ after decompactifying this curve to arrive at the non-compact threefold correspond to the divisors x the (5) curves in to an intersecting treeidentified of with Hirzebruch surfaces (0) curve in theI base The toric divisors listedated in the to first the column canonical of bundle table of for both the curves(rk and + their 3). toric Among charges). these Note curves the number of Mori cone generators is JHEP12(2019)039 and ) 12 i (2.16) (2.15) (2.12) (2.14) ( l 1) cycles direction. , τ goes to infinity, . ]) in the direction of 0) , ijk 57 ˆ κ (0 X F derivative of the genus associated to the curve ∂ ∂τ 0 two B-periods diverge, τ 5 z → . ; 0) (2.13) dc z =0 z ( i of the compact Calabi-Yau. Similarly 0 ρ . | , i ˆ i ω ) ˆ t ˆ F / ρ ,B, rk ; discussed in the previous section. At , (2) i z rk ( ˆ B . Π 0 =0 i ˆ Σ ω ,..., 0) ρ = , k , | i 1 r ρ ) i ˆ t ,..., , (0 ˆ ∂ ρ Σ 1 F ˆ ∂ j F ; , ρ , ∂ z = 0 ∂ ( – 12 – 0 = 0 = ,..., ˆ ; 0) ijk ω uniquely fixes the coefficients ˆ i 0 i z ˆ κ ˆ ρ ( F 1 2 ,J ∂ 0 0) is obtained by decompactifying , ) ˆ ω ,I model visited above, this is (0 J / ) as a holomorphic function of the Batyrev coordinates ) X ˆ F F I ) = ) = ρ 4 t ( 1) cycles to infinite size, keeping the other (1 (1) i ; z z F ( to infinity in A-model or taking the corresponding complex 0 of linear combinations of , ( ( z ˆ Π ( = 0 dc → (1) i (2) i t = ω ˆ ˆ Π Π dc i G ˆ t z 0 of proper linear combinations of to zero in B-model. In A-model, this limit can be understood as . 0 the affine A-period associated to the (0)-curve diverges, while the → 0) , dc term, we study the normalization scheme of z dc (0 → z can be computed by the special geometry relation. For a compact . The affine A- and B-periods defined by 3 F i ˆ τ dc J ∂ ∂τ z ijk κ unfixed, which is very important for the refined BPS invariants in the 3 τ In the example of the The non-compact Calabi-Yau They are the limit . In the limit defined below are the limit and compute the homogeneous A- and B-periods (see for instance [ 12 . We can then integrate over the new periods to get the triple intersection numbers of J i (5) (0)-curve become infinite in volume.and only Therefore rk in + the 1 limt affine B-periods remain finite. We choose a basis F which correspond to the curvethe classes same Σ time, both the zero section of the elliptic fibration and the vertical disivor of the term To determine the zero free energy l other rk + 2 affine A-periods remain finite. We can choose a basis of the latter to be finite. The periodssome B-periods of go the towhile geometry infinity. the will In corresponding be our A-periodτ current rearranged remains case, finite, so and the that becomes B-period the only the non-compact elliptic one geometry. fiber A-period However, parameter this and method will always have a integration constant the (0)-curve in the base.the decompactified In direction practise, thisstructure corresponds parameter to taking thetaking K¨ahlerparameter in some of the compact (1 satisfy the relation The existence of the prepotential They are interpreted as thethe masses dual of divisors the D2-, D4-branes supported on the curves Calabi-Yau threefold realised asformed a fundamental hypersurface period in ˆ z a toric variety one can define the de- Alternatively, ˆ JHEP12(2019)039 ) ) 2.18 2.11 (2.19) (2.17) (2.18) . See Let us C we will h discussed ) up to a 13 B t 4 ( F ). The iden- 0) , ... (0 = ], even without 2.2 + F can not be fixed 15 G 3 q t . τττ − ... κ 2 , 3 ) q 2) rk t ) + rk. The special geometry − + n τ ( ,..., n m, m ,..., 1 2) ( 6 ), we use the same formula ( , 1 . In any case, for all the minimal as in table τ , − X ) 2 can always be factored out of the 4 4 ( b n ( F = 0 2 4 3 − c (( = 0 , l τ n . In the example of 4 I − (4) F B − ,J , l Σ  ) parameter of the elliptic fiber. are the true K¨ahlerparameters. 2 , 4 0) i r (3) F , τ i I with t Σ ) is the invariant bilinear form on (0 , l , dictates ) computed in this way reduces correctly to the ∂t I 4 m, m t – 13 – (2) F ( α, m ( ∂F D h X ) respectively. We recognise the sum over positive , l ell 0) ,..., , ∨ G t JI 4 q 0 cls (1 h (1) t F 1 2 C 2.6 2 , where F , l + ,B 0) 4 , + . Here ( (0) F rk τ i l (0 3 2 ell i m F ,..., X t ) and ( j n 6 fields =0 1 2 ∂t 2 I i ∂ 2.7 α, m r ∂t h = = ; in other words, the intersection number ) must be quasi-modular of weight zero under the Siegel modular  J 3 = 0) , + τ F 1.2 cls (0 ∆ ij 3 is a mass parameter, the term ∈ τ X F τ parameter is different from the α τ ) and the intersection numbers with fields associated to a non-compact Calabi-Yau can be determined by the by ( ij t − τ ( r ]. Here we give a brief description of it. As proved in [ 0) , ) = cls (1 are defined in ( 15 for our convention. t are the components of the divisor-curve intersection matrix ( which is irrelevant, although it can be fixed by the same normalisation scheme ( F q and cubic terms in replaced by the triple intersection numbers of the non-compact Calabi-Yau and ) allows the computation of the semi-classical components of τ defined A 0) , JI ). Since 3 cls (0 τ C m, t ijk F 2.17 κ 2.17 As in the case of the prepotential, one cannot determine the pure mass term propor- As for replaced by the toric charges of Σ Note that this ) 13 i ( method in [ any assumption or constraint putthe on Λ Λ as (for instancetransformations it of can depend on all K¨ahlermoduli), if one wishes. We havesemiclassical checked Nekrasov that partition function when the 6d SCFT is reduced to2.3 the 5d pure SYM. Determination of In general, the with ˆ l above, one has the toric charges tional to up to appendix where roots is from theinto Nekrasov a prepotenital more of suggestive the form 5d pure SYM. We can also massage ( adopt this normalisation scheme in6d the SCFTs example with section a purewe gauge find theory that in up the to bulk which can be reduced to a 5d pure SYM, tity ( term proportional to by ( blowup equations and itintroduce is a not of normalisation importance scheme to which us. fixes Nevertheless, such in a appendix term in a reasonable way, and we relation of the non-compact Calabi-Yau where which correspond to the divisor classes JHEP12(2019)039 ), 14 k , t n are i ( t r   ], the ell (2.21) (2.22) (2.20) j defines f 25 R i C R fields we are ijk κ . We can first r τ -field, and these c 2 =1 r b X . i,j r 1 2 fields. In particular, . The leading terms, − i r t   ) for true K¨ahlerparame- c 2 =1 n b , ( k X . Then the weak consistency ) can only take value in the k k t B ). As also shown in [ f ) )+ n -field, this power series with all 2 , r ( r  2.22 k rk f 2.21 + e or an incorrect ) . The intersection matrix 1 n .  r (  ( 0 ) f 2 rk+1 , . . . , r   e i = 0 | 1 Z , n 1 | I R , r  ) r = 0 ( , and the minimum of the associated 1) I r c 4 a NS i b − cls ell b ( t Z =0 – 14 – rk − /Z I ∈I X ∈ ) X n 2 GV i that minimize all the n = fields that satisfy ( , b ∀ τ ( 2 n r ) = r  condition, which we will prove shortly c 2 ), already suffices to fix all the 2 , b =1 − for i n k X ,  ( t 1 1 )  n + ( ell · n k f in the direction of the elliptic fiber must vanish. Recall that ( fields with the condition ( . The τ,  cls R k into components C r j r f Z . The strong constraint then means that for the r Λ( ) R τ c 2 i 1 rk+2 =1 b  k X R Z − admissibility , read 2 ijk . It implies that Λ can be expanded as a well-defined power series in )+ i → , 2 after subtracting mass parameters. If the minimal values for one κ cls t , modulo 2  1 Z  r =1 + c 2 ( 1 b rk+1 X  parameter is always some combinations of K¨ahlerparameters in the fiber irrelevant K¨ahlerparameter cls i,j,k Z )( Z τ τ 1 6 n  ( , we decompose − t 0 f -fields we are interested in. is the set of integral vectors log   r I =: ∼ = simultaneously In the case of elliptic non-compact Calabi-Yau threefolds associated to minimal 6d In our current cases of elliptic non-compact Calabi-Yau threefolds, there is only one This assumption is the most natural consideration for the generalized blowup equations compared with . Now let us assume that for appropriate choice of the 14 i t logΛ It means the component of we only consider the injection the initial Nakajima-Yoshioka blowup equations. similar to condition implies the SCFTs, the direction and it willconsider be the a little bitand subtle then to check subtract the theexistence solved mass of parameter the minimum of where ters not zero simultaneously, then it must be a vanishing mass parameter which is interested in only the lowestdefine order Λ of K¨ahlerparameters contributes. Then we can simply the instanton contributions takenwhich into actually account implies truncates that the atstated Λ finite does in orders not the depend for introduction, on all sinceto any otherwise true say, of this it the true assumption can K¨ahlerparameters as not putsare be very the strong of modular constraint on weight zero. the choice Needless of the which are linear in e which come from expand this Λ in terms of all the exponentiated K¨ahlermoduli e JHEP12(2019)039 8 , in 7 , 6 β . (2.29) (2.26) (2.27) (2.28) (2.24) (2.25) λ ,E 4 φ F . = 3 G and the following m C θ , 0 n + ], any rational curve , λ ) 25 rk) by , and they are thus finite in n . + ( =0 ∨ i G ∨ τ α . r P α i , | ,..., , n − i r = 1 = 2) must have mod 2 (2.23) rk+2 rk =1 ) =: 1 2 i X i i Z mod 2 ( ) due to its particular importance, then − ω + = n i β ). An important ingredient of the blowup n ( R,R J · ( ! ∨ m n ( ) mod 2 G i n r 1 2 n r P 2.22 ≡ , 1 2 i,J -fields giving rise to the same embedding ≡ 0 ⊕ O + r , α C ) + – 15 – τ ) of the intersection matrix i with Γ = n + 1 ω J =0 R + 1 rk i r ] and for the remaining models with − X J n 2.2 ,..., r r j ( 3 4 5 6 8 12 3 4 1 3 2 1 ell j 25 (0 = i,J R O rk+1 rk =1 ˆ i | i + 2 1 n X A Z ≡ + 2 l ∨ / R ) 9 16 5 18 16 12 1 2 l j r =0 r Q j rk 2 n X J 2 ) = : = n #( − ( λ P |

ell f . We collect them into a single vector just like of this paper. In all these examples we checked that they satisfy rk =1 i Z X 4 = SO(8) models in [ ), which we also denote by R , -field always satisfies n r ( ell ) are the shifts of K¨ahlermoduli f . Numbers and sizes of groups of , but also provides a guideline on how to determine them. We find all of them 1.2 r take value in = SU(3) J G n Table 3 Let us prove the admissibility condition ( The function has the form where we have used theLie generic algebraic notation form ( where for the in the example section the stronger consistency condition. Here we summarise theirequations numbers in ( table This implies the for minimal 6d (1,0)admissible SCFTs. The argument above not only establishes the finiteness of down without computing anythe BPS Calabi-Yau invariants. with As normal bundle argued in [ dimension zero quotient latticenumber. Γ Finally we impose the BPS checkerboard pattern condition to remove half of them. We comment that the checkerboard condition can be written JHEP12(2019)039 , # τ (2.35) (2.36) (2.30) (2.31) (2.32) (2.34) (2.33) 16 Q ) ] 1 , all the − 25 1 4 2.22  i α,m −h . The reader may ... e due to ( τ 3 R Q . ) + + I ∨ J i r ]. I , α rk) are even integers. It is 25 a I α,m h b ω e ( =0 rk I X  J λ ,..., . n + 1 2 ) in terms of K¨ahlermoduli and I # . If we demand that this function + ∆ have nice interpretation as weight r ) ∈ n = X = 1 ∨ n , 1.2 α . R τ i x =0 α ) (
( r rk ) r X 2 f − 1 2 i I,J / 1 n r 1 1 2 P − = 2 R,R =1 ) = q ∞ ( = #( τ X n − ∨ | 1 2 − " ) = 0. The number of inequivalent embeddings α ) must have a common zero. Multiplying each into ∨ 2 terms which will decouple from the blowup equations. ( ∨ J -fields are known, we can start solving refined 0 / 1 n λ r Q 1 τ ( 2 ∨ – 16 – n − R P ,R ) = q φ , α : G q . Q ∨ I n P ell
( + P → 3 α I 7→ , ) if r 2 )] = exp ( G / n R given respectively by ∨ · | 1 2 x ∨ ∨ 1 J q ∂ P ( n Q − n α 1 f Q ( + , τ I and the q as well as the shift : λ 0 n eff − r φ PE[ λ t n cls 2 ] by an intuitive geometric argument for minimal theories with a gauge φ n / =0 , i.e. it is pure Z 1 1 rk − 25 X q m I,J = 1 2 and adding them up, we immediately arrive at the admissibility − ) is nothing else but half the norm square of ell I " a n R ) = , which also happens to be the order of the automorphism group of ( | Let us make some remarks here. As we will see in section n G ∨ ( P associated to G Q 15 ) = PE P : λ 2 defines an embedding of ). are the shifts of , P λ τ | 1 -fields are such that the components . And r 2.22 g ,R denote the terms that are independent from m, ell ( of the ADE type. Here we prove it for all gauge groups including the non-ADE types. piece is rather easy to compute for the minimal 6d SCFTs, which reads [ takes value in the image ... R G R Here PE is the plethystic exponent operator defined as This condition was found in [ 1-loop 16 15 1-loop Z depend on the gaugeThus parameter we do not consider them here. See more discussion in section 2.4 in [ group Note we only consider thetensor contribution multiplets from actually vector multiplets also to contribute the to one-loop partition one-loop function partition here. function, The but their contribution does not 2.4 De-affinisation Once the semiclassical piece BPS invariants by expandingextracting the constraint blowup equations equations of ( Z BPS invariants at each order. Alternatively since the the associated Dynkin diagram.notice the We curious also relation list these numbers in table whose meaning will be clear in section Furthermore and is the index admissible clear then that vectors of where have a minimum, all the derivatives of them with comark condition ( A little algebraic manipulation leads to where JHEP12(2019)039 ) ) ), ) 2 2 ell t ,  , 2.31 2 1 (2.40) (2.42) (2.43) (2.37) (2.38) (2.39) (2.41) ), this  − 2.4 ! ( 1 ) τ,m, ], while the n ) that char- ( ,  τ ( . Due to the ) n 25 ( ) in two steps. n ell Z ). With ( ( G R ]. In the second 1-loop ∈ ), a step we call m P ) with 2 2.37 rk 2.37 25 k R  /Z I 2 ) 2 t  2 d , + + 2 , over 1  . In addition, the one- , . . . , n  only appears in the shifts r 2 rk 1 − cls , 1 d 0 τ, m n , n ( , 1 2 D ) +( d n ,..., I , n E ( t 1 ) ) . , m 1 n  Z R ( 2 I = (0 d . = 0  − ∈ f 2 n + = =0 rk k ,  2 I X 1 ∨ ,I d ), and the relation of τ,m a I ,  ) determined by ( n , )+ and only sum k , k + ) r ( ∨ 2 ) + I ∨ I n  1 1 2 2 2 Q ( a n d the dependence on ,d ( + , f ,  1 1-loop m – 17 – + λ ) 1 d 1 + d = ˆ  Z φ R n J E I E ) ) + ( 1 n n )( 1 n 0  , n  n (ˆ f ), one can show that the polynomial ( I,J + − G ). As a result, we can decompose the summation index 0 → and , and obtain recursion relations of elliptic genera τ, m,  ˆ 2 A f P ( I fixed to zero, i.e. ˆ 1-loop ell is invariant under the translation d ,

2.22 n 1 ). These equations are the highlights of the next section. =0 E ) as the summation index, which as we argued before is rk Q 2.40 D τ, m that satisfy n n ∨ X J , ( n ) 1-loop 1 (ˆ n Q exp d − n ( Z ( R = Λ λ E ell = m r φ also turns out to be a quotient of theta functions [ ) and ˆ I R d 1 2 − R  2 1-loop d , d + 1-loop D 1 + 1 d D d cls , τ,m 0 ( +( | n D | we can treat ) only in terms of d 1) = 1-loop n 2 1.2 − d Z + = 1 = ( d X )+ cls , n 0 ( 1-loop G D D P elliptic genera themselves are meromorphicblowup Jacobi equations forms. can Therefore be in presentedthe the as end, shifted beautiful the coroot equations elliptic of lattice We Jacobi will forms present with these a equations sum inproperties over the and beginning how of to the solve next elliptic section, genera and then from discuss them. their step, we sum over Instead of ˆ now interpreted as aloop contribution weight vector in with the zeroth component“de-affinisation”, of and perform ˆ the infiniteIn sum the on first the step, leftquadratic we hand nature side factor of of out ( thefirst polynomials summation in fact produces a theta function with characteristics [ which are also invariant under ( and the admissibility condition ( acterises the summation index Besides, in the components and is the contribution from thebe one-loop put partition in function. an The elegant elliptic form blowup by equations partially can resumming the left hand side of ( which allows the solution of the elliptic genera in compact formulas. Here collects contributions from the semiclassical partition function (as well as the shift of equation ( we can also plug in this piece of information, expand the partition functions in the blowup JHEP12(2019)039  0)
, ω (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) m elliptic − = (1 2 ,  ) ) N 2 2  d + ) 2 and the natural 2 + 1) || ∗ , ω n , . h 2 ||  N N 1 2 + ( or − ∈ 6∈ , we derive the 1 1 d d +( h  η 1  ω, 2.4 . ) 2 1 ∨ i  + 1) d is even, and the characteristic fixed fixed α . − + ∨ , i i m n 2 ) β || ω + ( τ,m , ω · α, β rk =1 ( ) i h z || X 2 2 ( α 1 2 d 1 ) is given by ( = , = − E θ 1 · , m β ∨ 2 n ) α α . ( · 1 β − · 0 ω −  m R ) α ≥ ( A α − 2 ≤ ˘ θ , and discuss their two interesting properties, τ,m, is odd and 3 if  2 Y n ( 8 or = + m,n + n d + , with , ∆ 2 1 1 – 18 – m E ,E )  ∈ Y · || 7 ) α α , 2 )) ) non-vanishing or identically vanishing, and discuss ω, 2)( || 2 ,E 1 ∨ is 4 if i α, β  n 6  i − β ) = + n − + ( ,E 1 = ( m , m, r rk =1  1 4 ( i 2 X 1. The factor τ, β ω η ∗ n · τ,m ,  ,F 2)( m = A − (  h 1 ] | α 1  − a n d + [ i ∨ n θ E z β ( · 0, · | ( and | ) SO(8) ) 1 τ, representing the K¨ahlerparamters associated to a Lie algebra, we , ω h ≥ θ ,..., n ( m 1 ( 1 1 ( , we also define ] m , R − λ ω a ∨ − 0 , [ i for short. φ | R θ 0 A Q ≥ as ≤ = 0 α 1) ( × Y n ∈ · k m,n − = SU(3) + = ∨ ( m m d N β 2, G ∈ = / = 2 2 1 , d 1 ) = α + ,d − 1 ) m d n ∨ z, R X + Q ( 2 ( Following the de-affinisation procedure described in section Let us first fix some conventions. In the following whenever there is no risk of confusion k/ ˘ || θ λ ω φ = || ∈ 2 1 ω where we denote for Here the subscript of theta functions a Besides, for a vector denote blowup equations For a coroot inner product between We define the norm square these two cases invanishing detail. Λ In allow particular, usgenera, the to thus blowup write offering equations down a in complete an solution the exact to first and the case universal ellipticwe with recursion genera. will non- formula use for the elliptic dot to denote both the invariant bilinear form on In this section, weSCFTs first with present thethe elliptic modularity blowup equations and for thesupport 6d universality. for minimal the The validity firstdistinguish two of property cases with the in Λ( elliptic particular blowup serves equations as to a arbitrary strong degrees. Then we 3 Elliptic blowup equations JHEP12(2019)039 , . ) ) a ω | ∨ ∨ A Q Q (3.9) (3.7) (3.8) z, R ( ( (3.10) : λ 3.3.1 ˘ θ φ ), which P | ∈ ω . The basic 2.28 a . 2 as both monodromies / 2 T  ω ) by showing that m . If we choose basis . and 3.4 a , which explains the ) is guaranteed to be ) through ( | − 2 S are already known, for ). The remaining com- ∨ 2  P  3.5 d Q   E 2. Let us denote ∈ : 2.22 2 1 . in ( controls the characteristic  λ ) has weight 1 n − N/ 1 P |  ω ]. In general the blowup equa- ell n n 3.4 · − r z ( 58 − n α 1 2 θ ) d 2  + ,..., ), the corresponding components of the ). See more about the origin of . 1 4 0  , 2 η d ) match the predictions for the index and z 2.4 || ) is of index  ( z, R ω 2. The identification of the modular indices 1 ( = 0 3.4 / θ || + ˘ θ . Note the ], we can plug in their expressions, and verify ) 1 2 , and the summation index vector 1 2 1 ,   1 P – 19 –  ) is related to the choice of 34 =  , , k Nτ, Nz  2 0 8 ( → rk as in ( on the left hand side has modular index , 3.4 k d z θ →− n ] − . The number of different embeddings is 2 ( 2 ∨ , a n 1 1 [ i , n R  θ 2 Q − θ | ] and more generally [ ) ) − ,..., ) : n − z z rk. The first component has weight 1 22 ( ( η R λ 1 , 1 1 i − φ d = 1 − θ θ θ -field in ( i = expansion. + z, r ,..., ( a 0 ), , see [ τ ˘ ). θ i d 1) = 2) = 0) = 1 Q X SO(8) models [  = 1 ,   . Let us show some examples here: z, i  2.34 ( ) = rk always correspond to a weight vector z, z, ˘ , τ, m θ 2 n D ), ( ( i ˘ ˘ , thus the latter can take any of the following values ell θ θ vanishes due to the admissibility condition ( t = n z, R , r τ ( 2 0 τ ,..., r ˘ θ / G d = SU(3) , r ell 0, ). In the fortuitous cases where the expressions of = 1 r G ell Ind r i ≤ = , i 3.48 a r R are of weight zero and d E It is easy to see that each term in the summation of ( The dependence on the -field are ( It is easy to see the theta function these identities by small and requires a bit ofidea computation, is which to is repeatedly independent use from the the fact that characteristic erties of the refined topologicalof the string Calabi-Yau partition space functiontions under the give interesting identitiessee for also Jacobi ( forms,instance one the example is proven in section In this section, we providethe evidence components for of the the elliptic elliptic blowupThis blowup equations equations is ( transforms established correctly by as showing weaknomial, that Jacobi of forms. the the weight corresponding and componentsweight the made in index, from ( in the general 2d and a the quadratic 6d poly- anomaly polynomial or from the transformation prop- in turn induces the embedding plays the role ofThe the total shift number vector numerology of found different in blowup ( equations is3.1 then Modularity of elliptic blowup equations through The component ponents of K¨ahlermoduli ( r an integer, as requested forfunction in the appendix definition of and for JHEP12(2019)039 ! ) − d m , 1 · 3.4 (3.15) (3.12) (3.13) (3.16) (3.11) (3.14) m of the ω,  blowup n 2 2 a  d n − − . 2 ,  ]  1 2   ) τ, m m . 2 37 ( ω , ) · 2 2 d ω  ω 24 1 + 2) E  − dm m where the right hand n ) 2 n . We claim that if two − 2 m N − n  1 2 ( / m n 6∈ 2 ) and ( + d − 1 d ( d 1  1 so that the right hand side 2 n d  d  17 − n 1 − N + )(  2 . 2 2 n − ) ) ,  ∈ ,  2 2 fields, i.e. three different characteristics ) 2 ) 2  1 − 1  ) 2 r d differ by 1 + 2)   2 + + ) can be computed as  n a 1 ω,  − 2 − + 2  m 1  + − 2 ( ( 1  0 ). 1   1 ) ω  d d ( (  − n ) all terms share the same modular index, 1 n ( 2 A n d  + d ( 2 3.15 1 2 1 d 3.4 d + (  2) 1 2 τ, m 2)) n can be ordered by the characteristics 2 + ( 2 )( − m 1 – 20 – λ + − n d · − n n + 2) φ 2 2 n ( E n ) − n ( m 2 0  = − − d 2)( individually but only depends on their combination 2 1 + 2 2 n + G Λ − d d 2 0 1  − d n n n ) is known to have the modular index [ Ind , d ( n 2 2 ω 1 ( 2)( − + ( + ( ,  m 0 2 1 2 1 2 2 d − 2 n = ω, d   , we find that the four components on the left hand side of ( 2 1 2 n ) indeed sum up to ( 2 − − ( d d d G d d = 2) 2) + 1 2 τ, m,  3.12 1 ( G A − − − d d +Ind n n E 1 = ( ( Ind G d 2+ − − G d /   2 ) with a fixed embedding || 1 2 1 2 Ind +Ind ), the modular index polynomial of ω ) vanishes, this is the end of the story. If 3.4 G A || = = 3.4 1 2 = G d G d D.16 ) is non-vanishing, we still need to show the right hand side also shares the same d +Ind ) can be computed respectively as Ind Ind 0 2 3.4 The following argument assumes the form of refined BPS expansion. Thus the universality here does not G d ,  17 , one is able to use the blowup equations to solve out the elliptic genus. 2 consecutive unity blowup equations are valid,We the call other this equations the must universality hold of automatically. the elliptic blowup equations. contradict with our statement thata choosing arbitrary three unity 3.2 Universality of ellipticWe demonstrate blowup here an equations interesting property ofequations the elliptic ( blowup equations. The theta functions, where for two consecutive equations index polynomial. The index polynomial of Λ is simply, which together with ( This highly nontrivial factmeans guarantees in summation the of modularity thethus of left transform hand elliptic as side whole blowup of Jacobi equation,side form ( which together! of In ( the caseof that ( has total modular index polynomial as Ind which is independent from Using Thus the modular index polynomials of The elliptic genus Using ( JHEP12(2019)039 . τ 4 B r , n j l · , t β j I wβ N t (3.19) (3.20) (3.21) (3.23) (3.18) (3.24) (3.17) ) Q r j ) 2 + l in section j w/ l 2( q − 2 1) , therefore is q ≡ − Z and check how ( x − )  , τ ∈ , . β 2 w r r q q Q Z Z ) is invariant up to ( w/ 2 by 1 2 2 r ≡ q j ∈ ∈ − 1 . The shift integer v m, n ,  )( χ 2  τ 1 ) 2 m 2 n ), w l q · 2 w/ q 1 2 ( , m − 1 , l n − ) q j . 1 τ, m,  χ ) ( + m − 1 r rk+2 . k mτ q 1 2 ) , m r 1 1 2 E , as we have argued, is equivalent + Z ) − m r w/ 1 τ +integer j m q + q 2 , t 2 (mod 2) (3.22) l τ 1 ( 2 − mτ 2 + − m . We will also use l − + 1, w q 1 2 2 q ,  q − τ, z α q r 1 β 1  n − ,j − + q − · l ( β 1 j + 1 4 ]. √ r q  → θ τ, m +1 1 N ) mod (2 ( j )  ≡ τ, z n 1 38 ≡ 2 m 1 r =1 n ( ,  n j r 2 q θ z , ( i q + 37 0 ∞ 3 π l , ,m Y → – 21 – j θ 2 1 + 1 + ) n 2( 2 ) = m r m n /q q j 4 1 z , t m i 1) ( q ,..., 2 l/r τ π j i j − p + ,  (0 π χ + 2 ( − 1 2 n + l l/r = ≡  ,β 4 j Y j ≡ m n ( 1 l m 2 2 τ q l/r i i r ≥ X π π m w 0 =0 ) change. ≥ . 2 and ∞ r ) the instanton partition function reads Y , n l/r is equivalent to shifting the characteristic of these theta function ) = e ) = e ,j X j l 3.4 / j τ 2 = 1 β 3.19 τ, z τ, z Y − j n n by ( ( ) = ) ] ] j 2 z n n ) = by m m 2 2 2 ,  [ [ 4 3 ) and the checkerboard pattern identity -vector for the minimal 6d SCFTs in terms of the K¨ahlermoduli πi a 1 θ θ , r ): the elliptic parameter changes by 18 1 3.19 Q,  exp(2 ( ): it is a product of factors like Q, τ, . . . ( ≡ n m ( inst j ( : we first argue that under the shift ] q inst ω a k F [ i . Then starting from one unity blowup equation, let us shift Z Since the have components for non-vanishing BPS invariants one can showis that the invariant refined under BPS the partition function combined transformation Using ( where Exponentiating ( an exponential factor. Theenergy refined as BPS are defined from the topological string free to shifting A invariant under this shift up to anE exponential factor. θ can be removedperiodicity at of the the theta expense function, while of the shift an additional exponential factor due to quasi- n to make contact with the literature such as [ / Theta functions with characteristics have the following properties Here • • • 5 18 and by 1 various Jacobi forms in ( Therefore, shifting JHEP12(2019)039 ) 
its ... ∨ ) by τ β (3.26) (3.25) + of the + m for the 3.4 2 1 ) should a  − k  2 · E  ) is quasi- 3.4 ) 2 → 2 ) 1 2 d ,  and they take  1 , + + 1, at most up λ 2 ) in ( 2 19  1 2 ) is quasi-periodic ||  ]. 2 ∨ − + integer ,  1 1 β ,  24 τ, m,  1 → . 1 ( ||  , k ) 1 1 2 · ∨ . Now if we forget for  2 . E β τ ,  2 +( N τ, m,  1  1 + ( τ, m,   ∈ in the coroot lattice. Then k ( − ) 2 . This immediately indicates , k is unique, the number of unity 1 E n 1 ∨ integer d E  / ∨ β means that upon ), 2 type following the nomenclature τ, τ,m τ, m,  + Q 2 ( ( ( k 2 − 1 2 → , d E || ,  E d θ ) , should be identical. Thus all the k 2 ∨ 2 E n 2 ) =  , β ) times a weight vector, which can also 1 ∨ unity 1 ,  || 1) , this implies −  1 3.1 τ 1 2 k Q − 1 ( ( − E as vector λ 2 n φ ω ω,  , up to an exponential factor. The shift on the − 2 1 ) = ( τ,m, )  → ( 2 , τ 2 , d – 22 –  ,  or not. We first consider the former case where ∨ ∨ − + E · β + 1 Q ∨ 1 1  and in addition multiplying each term in ( )) + 1  is equivalent to shifting the characteristic  Q ) is invariant under the shift 2 τ, m 1 2  n − → ( τ / 2)( ∈ k + ,  1 2 1 1  E + − λ ] that the elliptic genera of 6d SCFTs with a pure gauge bulk theory  in the first instance of − τ,m n 1 ( ( ), 24  τ, m,  1 ω 1 ( 2)( d 1 . Similarly, one can show that τ,  times a weight vector. k ], which are composed of  τ n E − τ, m,  . We can denote → E τ ) by )  − ( n ] ∨ − + k ( 34 a 1 2 m [ , i 1 Q  ( E  m θ 8 τ, ,  | ∨ n τ, . . . 1 as well. As examples one could inspect the expressions of ∨ β are always nonnegative integers, and as a result the right hand side ( n β → ] | ( τ A a is odd. In fact d ] [ i 1 ω,  a 1)  θ × [ i n 1 + θ  − 2 = (  d − = 4 models [ N 2 → , ∈ d 2 2 , +  τ, m 1 1 ( d = 3 ,d 2 and thus k ) does not vanish ∨ + ) coincides with / X already be invariant under mass parameters elliptic parameter is inbe addition removed shifted at by the expense of an additional exponential factor [ for n and therefore arethe indeed moment quasi-periodic the for shiftE on the mass parameters, the three instances of elliptic genera to a sign if Together with theperiodic modular for property of the elliptic genus 2 ]. Since the number of embedding 2 ∨ Q || || ∈ 3.4 ∨ It is established in section 4 of [ Q 15 ∨ ∨ ( β 19 β || λ β 2 1 We say these ellipticin blowup [ equations are of the in fact consist ofelliptic special parameter Weyl is invariant Jacobi shifted forms, by which, among other things, are quasi-periodic if the different forms depending onφ if || of ( that starting from twounity consecutive blowup equations, unity hence blowup the equations, universality property. we can3.3 obtain all Unity the blowup other equations The elliptic blowup equations depend on the choice of the weight vector theta functions an exponential factor. Theseof exponential each factors term, are determined whichexponential by factors as the can a index be polynomial consequenceblowup factored equation of out where section and the removed, characteristic and is we shifted are by left again with a unity In summary, the shift JHEP12(2019)039 . ) 2 = 0. , ∨ 2 (3.32) (3.27) (3.29) (3.30) (3.31) (3.28) 2 = 0.  α / − 2 1 0) SCFTs , || , , ∨ since ∨ )) α ∨ 2 α ||  α 2 = (1 ) these equations  . We also point 2 = ) n − d 2 N 20 0 m . Since three copies . + d ,  6, . ( d 2 / 2 ) 0  d )     d ∨ G 1 , if we have three such E } } }

1 E θ − ) ) α α ) − 2 2 E } α  2 A A } 0 m | | , , 1 − ( , , m 0 1 ∨ ∨ 1 e , 1 3 1 ( ∨ ∨ , α α 1 ∨ 1 | | 1 D α α D { . In particular, for ADE type algebras, the − α { θ = 1 pure SYM theory [ A 0 limit, β 1) 1) m, | D D m, are the short notations for α ( · 1 ∨ ( )(1 − − D 1 1 N α 1 D ( ( → + | =2  ∨ D E m E 2 τ − } ∆ ∆ 1) =4 =2 } || 2 e 2 2 1 ∈ ∈ , ∨ , X Q = − =0 1 3 || || X =0 α 1 2 α denotes the set of long roots which is the same with coroots with , α ( − , ∨ and ∨ ∨ 0 X X ∨ D || β 0 l α { D α α α { = =2 (1 α || || m 2 D = D 1 || D  + + = 1 ∨ X E Furthermore, the two-string elliptic genus is given by In the Let us look at some examples. The one-string elliptic genus is given by = α 1 E = || 2 Z 3 E + E Note in the bracketIn of the the later second section, line(1,0) we of use minimal ( this SCFTs formula with todown the compute universal the formula two-string for elliptic three-string genus elliptic of genus all as 6d where ∆ Here function of 5d simplified due to the identificationthe of following roots universal and formula coroots. Indeed, for where JHEP12(2019)039 z i ). 0. π 2 = 1, → 3.32 1, the (3.39) (3.38) d τ = e Q y d > encodes the by recursion , and , by assuming d are computed, n k τ 2 E i d ) π y SU(3) 2 SU(3) 1 E − E ]. For all = e = SU(3) with ( 2 n q 34 k 4 G runs over values of the ] works perfectly for the q 3) splits to two 5d unity ) , , in [ and there may be certain k n 0 59 n k i , 6= 0. We have checked that , (2 2 a 0 r d y X 30 , = 2 n D SU(3) 1 k k 4 , E q = ) = (0 n l = 1 r y is the difference of two contributions (2 j ) -string elliptic genus r r 1 y X d = 1 + D k , one can actually use one single unity − =0 nl 0 Y j ( -field n l = Z r q contradict with the fact that the universal ) thus allow us to reproduce the genus zero l l y and arbitrary degrees along other directions ) 1) r ↔ 3.32 not d − + – 25 – ( functions nl Z ( as well as multi-wrapping contributions from lower l θ ∈ l q X d = 0 order term of n Z r 2 i ∈ of 5d pure SYM theories [ τ l X z y 1 Q 1 2 n n r r 2 2 Z − =0 q such that only two of the three unity blowup equations are j X y n ], and to compute the refined BPS invariants for the first ) in the second sums, mean that r i n 2 n 1 vanish. This is a special situation since for the model with 4 r 4 π 24 m E q ( 1 . We claim that if r D Z 12 [ = 0 here despite its simple form does not seem so trivial. In fact, , ) = ) = e = ) works well for SU(3). respectively, both of which remain finite and identical. Nevertheless, ∈ 8 , with base degree l 2 k and 6 r r SU(3) , 1 3.32 nτ, z nτ, z } ,j ( ( l 0 β , D ] ] j 0 , and the leading n n , and 2 = 5 2 2 N 1 r 1 α { r/ r/ ), practically one can use the exact formula n r [ [ 3 4 D , for any θ θ and ml 3.36 1 r + ]. If further assuming the knowledge on r The identity Let us make a remark here concerning the validity of the recursion formula ( From the topological string point of view, the 25 = SU(3) there are only three choices of the characteristics -fields where the notations form we find it is awe special define case the of following double the indexed following series of identities. With blowup equation to solvein out [ all the refinedblowup BPS equations invariants, to which solve is outformula what all ( we refined have BPS done invariants.recursion To formula compute ( What happens in this limitr is that the 6dassociated unity to it should be emphasize that thoughthe recursion refined formula does BPS not expansion, work one for can still use two unity blowup equations and one vanishing where both G symmetry enhancement for linearly independent. Note thatone-instanton partition this function does SU(3) theory which is recovered from one-string elliptic genus in the 5d limit with SCFTs with time in the literature. Obviously, the recursion formula isthis only is indeed well-defined true when for all the minimal models except for the model of BPS invariants base degree curves. Onceall all the the elliptic BPS genera invariants upcan up be to to certain extracted. base base The degree Gopakumar-Vafa recurison degree invariants formulas ( for the Calabi-Yau threefolds associated to the minimal 6d JHEP12(2019)039 ) /n (1) ) and 1 1 1 2 1 3.32 − − (3.42) (3.40) (3.41) n − n n k n = | k . = 3 case ¯ k 1 the terms 1 + − n ∗ . . . y i − n , k k n 0 ... k 0 y    7→ + ) ) ) 1 . . . y i 0 − k 0 2 n    2 d d | Q 6∈ || E ∨ + D ω : Q ||    is in a lowest dimen- : 1 2 ) ) ) , λ P | ω P , moduli space. In fact, θ + 7 2 Sym n C ,Q + G θ 2 ], which we will review and extend 8 C 1 χ θ , 2 v 2 7 − )+ k G , Sym 37 6 )+ χ )+ G nθ in the Omega background, and can be χ G 2 Sym ( G χ G 2 6 + v, x 2 (  ( ,E χ  ( v 3 2 2 4 B ) =0 +
∞ χ B + B ) n k X θ + 2 ( ,F k,G . We can further compute the θ + θ The exceptions of SU(3) and SO(8) can be + χ v G g 2 and x θ ( θ χ ]  . From now on we use SU(2) 2 + 2 6 1 θ ) = x 5 Alt χ i 2  C 41 – 30 – 29 Alt . χ ) = χ revealed by its intriguing relation with the Schur m + i − SO(8) χ Alt θ + 1 θ 5 , m 3 + 3 + θ χ E 3 3 χ θ v, Q ( v 3 + χ ( 3 2 is ) = 2 + 2 Sym χ θ i χ χ 2 χ ,G ) is the character of the adjoint representation of + +2 θ m (0) 1 χ v, x, Q n θ θ χ ( g k,G )+ ( 2 2 θ )+ 28 + θ g ) θ = SU(3) + χ χ χ which respect Weyl symmetries of both groups. For example, 2 2 θ n 2 v, Q ( k,G SO(8), χ χ ( v χ +( +( , G g G ) + 4 6 + 3 ,G θ θ (2) χ χ 1 χ 2 1+ 2 ) ) g θ θ  + has the expansion has the expansion [ = SU(3) 2 2 ) are rational functions. In particular, θ i Sym Sym + χ χ G Alt Alt G G m χ θ ( ( = SU(3) χ χ 5 χ 3 χ + + χ G θ θ + 2 2 + gives the Hilbert series of the reduced θ ; in particular θ χ χ ) = 1+( v, x, Q θ i χ χ ( 7 5 ) m . +( +( ) = 1+ ... χ ) are characters of some representations for which we do not find any universal is the character of the representation whose highest weight is n χ i 6 8 and characters of ( k,G function where the coefficients are finite sum of products between the characters 5 + χ χ G m + g x ( 9 θ θ kθ ) 7 2 n χ χ χ χ χ ( k,G v,x,Q    B ( g v, Q = 1 + Furthermore, we find the Hilbert series of reduced two-instanton modulis space for any It is known that the Hilbert series of the reduced one-instanton moduli space for any ( + + + + From now on, to shortenThe bold formulas, numbers we mean do the not character of explicitly representations write with dimension of such number. Note different ,G (2) 0 ,G 28 29 g (1) (0) 1 k,G of SO(8). Torecovered lighten by taking the into notation, account we the symmetry do of not Dynkin distinguish diagrams. them in the table. Nevertheless, they can be representations can have the same dimension sometimes, for instance, the representations simple gauge group expressions, and we listexplained by them the in higher table structuresindices of of certain rankin one section 4d SCFTs discovered in [ while except for Here longest root particular this iscontribution true to for one-string elliptic genus.find As that for except sub- for and subsub-leading contributions,g we simple gauge group where symmetry to stress the associated fugacityeach is of SU(2) g we find plenty of universal coefficients for the first a few order One obvious symmetry for all which comes from the symmetryunderstood between as the Weyl symmetry of SU(2) Here all JHEP12(2019)039 v G  . 2 78 (4.9) 35 ) χ 7 26 · (4.10) (4.11) + . v ) ( + 1539 3 2 v O +1)) v 650 + ( ) ) + 3 θ + 2 = SU(3), 1 324 + χ O 6 4 v + + + G , SO(7), Sp(4), v 8 2  2 ]. Note the first + 1) +2(  θ v C 40755 2 G 2 functions. θ 5824  34 350 · χ + ) +1 3 4096 n ... θ + Sym v ( k,G +1) 2 + g . The expansion coef- + ( χ θ + + 2  x 3 2 χ 3 + Sym χ 300 χ χ θ ( ) 3 χ · θ +(( 2 + 2 4 +9) 3 + 3) Sym θ χ χ χ θ χ Sym , θ ]. In particular it is true for χ χ + ) χ +1) 5 3 42 +6 θ + + v 7 θ χ 567 ( 2 5 χ , although we have not checked for 2 C + (2 ( · -expansion: except for χ O  5 (3) G 3 v  779247 147250 5 +1) Sym h χ + θ v + 1) + ) are characters of certain representations χ E 2 4  3 θ G + ( θ v v ], and we further push the observation up ( χ 2 ) 6 +2 +4)+ 28 ) v 7 0 Sym 2 2 θ · C 30 2 χ χ χ 2 ... + ( χ Sym ,C θ – 31 – ) θ χ +1) χ + +3 2 +2 G θ θ θ 5 χ ( + is the leading contribution to the two-string elliptic 2 + 1) χ 2 +2 6 4 χ χ 4 θ 2 35 1 27 8 χ C ,G + ( χ χ · (0) 2 B Sym +(( 4 +( g 2 5 χ +( + 3) 5 χ 2 + ( χ θ χ + +1) v 3 θ 3 θ χ ) χ θ -dimensional representation of SU(2) χ χ χ + 1) +4) when n χ θ θ + 8 instanton moduli space in [ +3 + (2 + , χ θ χ θ +2) 7 +(2 7 3 3 , 7 θ χ 6 6 χ χ (2 χ G χ χ χ 2  +(2   ,E 7 +( 4 + + ( χ + + 5 = SU(3) and SO(8),  χ ,F were already observed in [ ( + G ) = 1 + . Certain representations appearing in the expansion of θ ) = 1+( i ], and against the three-string elliptic genus for SU(3) [ 6 i 2 v m χ m ) = i 60 , SO(8) m , due to the complexity of computation, still we propose the following universal 43 . As for the subleading and subsubleading contribution to the two-string elliptic is the character of Table 5 θ v, x, Q v,x,Q 5 8 27 64 G 28 300 1925 4096 52 1053 12376 29172 273 10829 8424 78 2430 43758 105600 650 34749 34749 ( n ( 133 7371 238602 573440 1463 152152 150822 248 27000 1763125 4096000 for which universal expressions are not found. They are collected for individual χ . We have checked this expression to be consistent with all the results on Hilbert v,x,Q ,G ,G For the reduced three string elliptic genus ( 8 (2) (3) 1 0 4 v g 2 6 7 8 g 4 G = SU(3) ,G G χ F (2) E E E A 2 D g We have checked this against the three-instantonSp(6) Hilbert in series for [ SU(2), all six expansion: while except for of in table genera, we find there exists the following universal ficients up to to series of reducedG two genera. Note that in this expression, Here JHEP12(2019)039 ] ) ) τ 1 2 v Q 43 ( . − ) ,a ) n ,G 2 ∨ G 4 ( (4.16) (4.15) (4.12) (4.13) (4.14) 2 v h v ]). For ( P ( = 1 in is large. ,G 45 O (1) 0 n m g + ) Q . .  ) and turn out to be ) ) 2 v v,a 3 ( v (  when x ) ( v x ) n ( ,G ,G ( v 2 n 1 ,G (1) 0 1 ( O 2 v g P ) P + − 2 2 ×  1 ) for the minimal SCFTs 1) to still store meaning v x ) v 1 2 2 ( 8 − ) v n E ∨ G , a ( v, x, k,G − h + 2) 2 ( 2 v P θ ) ) -instanton Hilbert series can be ( 3 7 v 2 k χ )(1 ) ( n,G G v 1 E ,G 2 g (1) 0 +  1) are very useful as they encode χ g . + θ ) 2 6 vx v 4 + E -instantons as approximation: v − 2 v, x, + 1) ( ( Sym G  θ = 1 in (1 , ) O χ 1 (1 + 4 χ ) k -expansion. Indeed, when the fugacities of (  x n,G + G + F v Hall-Littlewood index ((A.14) in [ v b g 3 ( 2 + 2  ) + ( vx 6 1  ) ) 1 n ,G 3 ) 4 ( v 3 E 1 − χ – 32 – χ P ,a 1 v,a 3 to lighten the notation. It also was noticed in [ ) ( × are presented in the following example subsections v (1 + ( + 4) G ,G 8 1) 1) (1) 0 θ m v, a ,G g , such rational functions with generic − − ( (1) 0 χ ,E Q ] that the reduced two 1 n g ∨ G ∨ G + 1) + ( 7 ,G h h  3 (1) θ 0 3 6 11 16 26 46 45 =  2( 2( g or χ 4 except for SU(3). The numerators ,E + (3 1 ) ) 1 vx a x  SU(3) SO(8) 6 2 2 5 k 3 2 + − − v v  ) are both polynomials and their maximum degrees are v χ 3 1 ,E v 1 2 3 -instanton Hilbert series can be realized as certain symmetric 4 ( − − b − ] and [ χ G  / 1 G are turned off, we find + (  ,G ∨ G ,F (0) 42 (1 (1 -instantons as approximation: 2 1 6 x h ) = + G ) = ( . ,P respectively. The explicit expressions of i ) ) = ) = ) = order, again except SU(3), we propose are given by = 5 v m F G v v ( SO(8) τ b ( ( -expansion coefficients can be easily obtained by setting v, x, a G b , ) ) ( v ,G b Q ,G ,G (0) n n 1 ( ( 1 2 ,G v,x,a P ). Thus the rational functions g g (2) 0 ( 1)+2 v, x, Q g m ( ,G 1 − (3) 0  ,G ∨ G g (3) 1 = SU(3) 1 h g vx 1  G v, x, Q − All above ( 1 vx ) 1 k − ( n,G 1 product of three one Here we adopt their notation that the reduced three and also appendix 4.1.2 Symmetric productIt approximation was noticed both inrealized [ as certain symmetric product of two one We notice that are palindromic Laurent polynomials.Nevertheless They have negativeand powers 2(2 of with The exponents information on arbitrary orderflavor coefficients as of well as SU(2) subleading order. g most information. For large too lengthy. One can take the unrefined limit the subleading As in rank one and two cases, for SU(3), the higher contributions begin to merge in at two terms also agree with the rank three JHEP12(2019)039 2 of  ) 4 2  v (4.19) (4.21) (4.17) (4.18) (4.20) (4.22) ,a 2 ... ) ,x 2 τ 2 ) + . v 2 (  ,Q , 2 ) ,G Q , x (1) 0  ) 2 ) e , a g ), the first term 4 2 2+1  , v ) begins to appear. / 2 2 instanton are far ,a , x ∨ G a 4 4.16 , 2 h ( v, x, a − ) )+3 v ( − τ ,x 4.11 4 2 ( ,G expansion of symmetric k 4 (1) 0 v Q ,G (1) v e g ,a ( ( 0 (1) G v ( 2 2 . h e e g O O − ) E ,G  ,x k 4 (1) ) 0 2 + v τ  e g v ( ) + ) 2 PE ( / -expansion coefficients. For ar- O v ∨ G ,G ≡ . (1) h )+6 0 ) + 3 e g  − τ 4 ) 2 3 v, x, a v, x, a v ,a Q ( (  v ( ..., ) 3 ) 1 v, x, a, Q ) ( ,G 2 v, a O G, (1) 0 ,x + ( χ e g (1) G 3 + θ k M  h ,G e v  3 χ -instanton Hilbert series can be realized as ) (1) 0 ( 2 v,x,a k v v, x, a E 3 τ g ( ) ( C + ,G 2 1 (1) 0 ,G ,G 4 k G ,Q χ (1) + 3 0  e g (3) 0 + 3 θ – 33 – ) χ e g g 3 k χ )  vx 1 , a Q ) = Sym  τ 3  + + ( ) − , the interaction among instantons will contribute. ∨ G 4 +6 2 4 instantons are far from each other, while the second h 3, the leading coefficients in , x 4 v,x,a 1 vx χ v 3 v 3 (  )  v k ) v ≥ θ v, x, a − ,G ( +( ( v, x, a (1) χ 0 order and the first four 1 k 2 ) ( e g +1, one can see the difference with ( (1) G ,G k v v, x, a, Q τ into the elliptic genus to write down the above symmetric ( + 0 h e θ ) ,G ( From = PE 2 (1) g θ 0 v,x,a 3 E Q ) +8 ) = ( e g 1 (1) G χ k ,G k χ τ ( 0 h  e  -instantons as approximation: ,G ... Sym Q + g (1) 0 + 2 E  ) 3 χ e g ! vx 1  χ 1  k + k G 1 6  − θ ) can be obtained from generating function 2 = 1 + ( 1 1 24 v, x, a χ ) = v, x, a, Q ( ) = ( 1 = 1+( + τ ), we obtain 3 ) = G, G (3) v, x, a h e χ ( k M θ 1 v, x, a E ( χ 4.16 k G, 1 v,x,a + ( G, -strings elliptic genus, it is better to use Hecke transformation. Neglecting the Sym 5 1 v, x, a, Q k ( χ k M G, =1 ∞ k X (3) G as 4 M We can also include In fact, it is reasonable that arbitrary h e Sym E ,G (3) 0 Sym which holds for thebitrary leading product approximation. For example in the reduced three-string elliptic genus, since combining ( Here the first termterm represents represents all two instantonsfrom sit them at and the each other same site and the rest It is not hard toproduct find that are for the all same: For example, symmetric product of where Sym represents the configuration where three instantonsrepresents are far the from configuration each where other, the twofar second instantons from term sit them, on while the the samesit third site on term and the represents the the same third configuration site.g one where Note is all the three triple instantons symmetric product would give the coefficient of The above formulas have clear physical meaning. For example in ( JHEP12(2019)039 ) τ , ).
Q τ goes . 4.27 ): # τ 3.12 ), one (4.27) (4.26) (4.23) (4.24) (4.25) ] for a τ Q  ) can be 37 G τ v,x,Q 4.23 ) k v, x, Q ( G , c m ( h . i ) . k ) E ( G . ) 6 h τ τ / , c  = 1 in ( +4 v, x, a, Q E Q 3) b ∨ G ( ) should be seen as d . Similar situation also ) − τ k kh + v, Q ( G ∨ G d ( Q v ( kh 4.27 expansion of the elliptic ( τ , (1) G c τ ). Note symmetry ( Sym O h ) τ Q  E E m Q 2 D.6 +1 / (1) G ) + 2 h order is enough high, for one order e k ,Q = SU(3) and SO(8), the above 1) τ ), ( − E τ − ). Also take 6+1 ) 3 G ∨ G / Q d ) satisfies an additional symmetry / G h ,Q ∨ G τ ∨ G D.5 1 1) h kh − − ( τ , m This means ( 6 in Omega background, it was found X i k − τ ( v,Q (mod Q ) actually computes the NS-R elliptic k 2 ( b Q 30 -string elliptic genus  v, x 3) − 0 τ,  ( ( n / k (1) G ). ( x ) = h ∨ G O k 4.26 ( G h X (1) G cd E c,d> +1 h and h e − 2 4.26 E k 1 n E ) + 1 + 2(1 τ  n v ∨ G – 34 – -strings elliptic genus Q h ) = k 0 +1 ) and the modular anomaly of elliptic genera ( 5) m ) where there is no modularity. Finally, we obtain n ≥ − 6 X n k ( 1) -expansion coefficient. This means all subleading ,Q k " v 3.25 τ ) to arbitrary − 4.18 v, x, a, Q v ( ) +1 k ( G k 4.26 n ) = ( = exp τ v, x, Q is implicit. This symmetry was later interpreted as a spectral 1) Sym ( k ) − E G k Q ( G order goes down for 3. Thus, if one naively does the transformation for the ) expansion, one would get infinite negative order of h m τ /v, Q = ( τ v E 2 ) = / Q ! τ ]. The left hand side of ( 1 τ τ Q 38 ( ) in ,Q 2 (1) G / even for its leading v, x, a, Q h 1 τ order goes down for 2, which only result in finite negative order of 4.27 x ( E ) ) v Q v, x, a, Q k ,G k G ( ( , ( 1 ) 2 g k / ( G 1 τ v Sym h e Q E E goes up, the leading

] that the reduced one-string elliptic genus =0 ) We extend the symmetry ( τ orders k X k Practically, we find that for the two-string elliptic genus, when ( G 37 Q h 30 τ E genus, except for the one-string case that is ( of left hand side of ( happens for three-string ellipticup, genus. the leading But for one-string elliptic genus, luckily for one order of which can be derived by combining ( For the situation where 2d quiver descriptionsymmetry is can known, also i.e. be obtainedtion by with looking the into quasi-periodicity the of transformation Jacobiis of theta a function integrand nonperturbative ( of symmetry, which localiza- can not be seen from the Here the dependence on flow symmetry in [ genus, which should belack equal of to chiral the fermions R-R indetailed elliptic the discussion. genus minimal on SCFT the in right consideration. hand See side section due 6.4 to of the [ which comes from thein [ symmetry between Q orders involves interaction among strings. 4.1.3 Symmetries Besides the obvious symmetry Note this relies on the Jacobiwill form go nature back of to instanton formula ( As we have checked this symmetric product approximation does not give exact subleading generated from interaction among strings, the resulting JHEP12(2019)039 ]. 25 =10. (4.30) (4.28) (4.29) n . 16 ) is a true v v ( + 2 , ) ,A ] using just the (0) which induce the ··· 2 v =1) up to ( P i 25 2 + 2 a ) 9 m , n ,A v ( ) ]. For the subleading ]. They are equivalent 2 i P 2 0) minimal SCFTs for 25 m 42 , 4) 2d CFT, as suggested ω × , 2 1 v, x, Q +110 + ( 5 ω ω . 8 ) 2 1 v ) 2 7 ,A ω n v, x, Q v ( 2 ( g 2 + ) and ,A n +112 v ( 2 , we obtain 7 6 g x fundamental weights v n τ fields of 6d (1 Q (1 + = SU(3) were computed using Jeffrey- 6 r 1) 1) 1) 1) 1) 1) 1) ) =0 1 ∞ 1) 3) +110 , , , , , , G v X n − , , 6 2 2 0 0 2 2 r , 6 0 0 , , , , v / − − 0 , , 0 2 2 2 5 , , , 0 0 , , old 2 − − 0 0 − τ , , (1 + 2 2 , , , , , +92 – 35 – Q 10 − − 5 5 ) v v v shift. n − · 1) ( 3) (0 5) (2 1) ( 3) (0 5) (2 ) = +52 5) (0 3) (0 1) (0 , , , , , , i fields for SU(3) and SO(8) found in [ ), the reduced two-string elliptic genus for SU(3) model (1 4 , , , C 2 2 2 0 0 0 -fields are from our previous paper [ v 0) SCFT with r 0 0 0 , , , , , , r , , , , 4.3 r , m 0 0 0 2 2 2 τ 0 0 0 , , , , , , = 3 model and the fundamental weights of , , , 2 2 2 2 2 2 +31 n 3 − − − − − − (0 (0 (0 ( ( ( ( ( ( = 1) = v i v, x, Q m ( 2 ], and were checked to satisfy the elliptic blowup equations [ . The old +17 (2) A 2 = SU(3) and SO(8) ,Q P ) are rational functions. We computed h 34 v i ] and fundamental weights in tables E G m → , = 1 ) are palindromic Laurent polynomials, in which only unity 25 +6 v ∨ v -fields of the ( vanishing vanishing r 2 Q ) v, x : n ( ,A v, x, Q ( 2 2 ( φ ) 2 in the second column by 2 P ,A n ) ( 2 ) = 1+ ,A r n . The g ( 2 v g ( ]. 2 , we now can reproduce all The elliptic genera of 6d (1 ,A 37 G (0) 2 fields given in [ P polynomial: Here the ellipsis isthe Hilbert completed series by of reduced making two the SU(3)-instanton moduli expression space palindromic. in [ This agrees with Let us turn off the fugacities of both SU(3) and SU(2) where all can be written as where fundamental weights of ther Lie algebras. We summarize the correspondence between the Kirwan residue inFollowing [ the general proposal ( the symmetry of the chiral algebrain associated [ to the underlying (0 4.2 Revisiting With the new understandingall on the structure of Table 6 embedding with the JHEP12(2019)039 . 5 5 v
10 2 v (4.31) (4.32) (4.33) χ ) 8 2 +1777 4 9 v ) v +Alt 8 which induce the 2 27 4 . Let us also show d , c 1 2 3 +( ) +1121 ]. They are equivalent F 4 7 ω ω ω ω 8 +Sym v χ v 25 ( 3 8 χ O ) from all the other minimal + 1 + +601 4 6 + ). 7 v v
v 8 fundamental weights
8 1 4.8 2 . +( −
5 different 2) 2) 2) 2) 2) 8 +238 χ 2) 0) 0) 0) 2) 0) 0) 0) 24 , , , , , 3 2) 4) 0) 6 , , , , , , , v 2 2 2 2 2 − , , , v 0 0 0 2 0 0 0 , , , , , . +( , +Sym 0 0 0 + , , , , , , , 0 2 2 2 0 ) 3 0 , , , 3 0 2 0 0 2 2 0 r 6 , , v , 0 0 0 − − − , , , , , , , χ 0 2 v ··· ) 0 , , , , , , +55 0 2 2 2 0 0 2 ( , +Sym 2 8 , 0 0 0 2 0 0 − 5 , , , , , , + 2 3 old 0 − χ , , , , , , O v 2 0 0 2 0 2 + , which is − , , χ 0 0 0 − 2 0 2 8 13 , , , , 5 v , ) + 0 0 − − , , , , , 2 2 2 2 v 0 − − χ 5 , , , , + 2 2 – 36 – , , , − − − − 4 v +13 27 − − 1 can be found in appendix ) 4 χ + 2 + shift. v 3 χ 8 v n +( 2 8 +2262 2 · 2 2 n > 0) ( 2) ( 4) (0 6) (0 0) ( 2) ( 4) (0 6) (0 0) ( 2) ( 4) (0 6) (0 χ v +18 12 2)4) (0 6) (0 (0 0) (0 , , , , , , , , , , , , C ) 8 3 v -fields are from our previous paper [ , , , , 0 0 0 0 0 0 0 0 0 0 0 0 8 v r 0 0 0 0 , , , , , , , , , , , , + , , , , 0 0 0 0 0 0 0 0 2 2 2 2 + 2 = 4 model and the fundamental weights of 0 0 0 0 , , , , , , , , , , , , +(Sym 3 +Sym v , , , , r 0 0 0 0 2 2 2 2 0 0 0 0 ) 5 4 ) with +11 χ n 0 0 0 0 , , , , , , , , , , , , 3 v 2 χ +2424 χ , , , , 2 2 2 2 0 0 0 0 0 0 0 0 ( χ v 8 8 0 0 0 0 , , , , , , , , , , , , +( 2 11 ) , , , , 2 2 2 2 2 2 2 2 2 2 2 2 + v + v + n ,A 2 ( +8 8 7 6 2 (0 (0 (0 (0 − − − − − − − − − − − − . The old χ ( ( ( ( ( ( ( ( ( ( ( ( v χ χ P P + +( → , +2262 1+3 ∨ -fields of the 4 ) = 1+ ) = 1+( r i i Q − ) starts with negative power of : v v unity ( φ 2 Indeed, ) agrees with our universal expansion formula ( in the second column by 2 vanishing vanishing vanishing ) = ,A r (1) v,x,m v,x,m v . The 2 31 ( ( ( P 4.32 2 2 2 ,A ,A ,A More results on (1) (2) (2) 1 0 This phenomenon as also occurring in one-string elliptic genus, will be discussed in detail in section 2 g g P 31 4 v Note ( some results with generic fugacities, for example, order, SCFTs. embedding with the Table 7 JHEP12(2019)039 11 13 v 9 7 v ) can v (4.38) (4.40) (4.41) (4.34) (4.36) (4.37) (4.39) (4.35) v v 16 1, they ( , v 9 2 ]. More ) , v 15 n ) ,A ) is a true 42 +2986 ( = 1) up to , 3 v v v = 0 +1926 ( 2 agree with i ( P 10 32 2 , 8 2 ) . n v v m 1 . v +242492 n +6522178 ,A ) ,A , (
+87437 (0) 3 + 6 3 v 12 +29700 v ( P 32 P 15 v 4 8 ... v ) = 0 v . v × +964 , n +359742 ,D fugacity, we have . v, x, Q + +2079 ) ( ) + 7 n 2 ) ( i 6 i 9 14 v x 2 ) 17 P v m 38 ) v ··· 2 v ,A n v for v × +97961 ( 3 + 5 4 g + +64960 + +14290 v 17 11 +456 v, x, m +4789888 7 ) v ). More higher v 6 14 ... ( (2) n,D +1260 2 v, x, Q v 4 v v +6940 g 11 ( 8 v + , we obtain ) 2 . v v ,D n x = SO(8) were computed using +293226 16 ) . ( 4.11 2 20 + ,A n v (1 + F g v ( 3 13 34 v 5 +34415 n τ G g +242 v v ) 4 +6116 5 n τ 2 Q +772 6 v + v +44954 v 7 Q v +9248825 v =0 (1 + 13 ∞ +7334 ··· 16 X n v =0 ∞ 1 +3205774 v 12 + 6 X 15 n 1 ) +121046 (1 + / +114 v 3 10 v 18 4 +220408 / v 11 +388 19 +10582 +2435 10 v 4 v 6 ) v 3 − τ 5 12 − τ ). Turning off the SU(2) v v v v v Q Q – 37 – +28038 (1 + = 6. In particular, 8 +62 4.9 +6940 11 v 12 22 3 v 0) SCFT with n ) +9668450 14 v v , (1 + +213 +841 v ), ( v 5 15 4 +2913 16 +406014 ) = +1958185 v ) +125368 2 v ) = v − i i 9 4.8 17 v +24 v +152749 18 v v 2 (1 v 11 , m − v , m +82 τ +16317 +6384 v τ 4 +254 (1 1 can be found in appendix ], and they were checked to satisfy elliptic blowup equations v 3 +697 +8 11 13 = 0) up to v v v v v 24 i +9248825 = 0) = +40 n > v, x, Q +421960 v, x, Q ) are rational functions. We computed ( 3 i . +65 14 +121046 +1085137 i ( +96519 2 agrees with the SO(8) two-instanton Hilbert series in [ v 2 16 8 v 4 m = 1) = 17 (2) A )(1+2 F v 10 +8448 +5226 v v (2) D 32+90 , m 4 2 h i v v, x, m v h v ( 10 12 ) E m ,D E +14 4 2 v v (2) 0 ) with ) are palindromic Laurent polynomials, in which only 2 = 1 +20 v g v v v ,Q v ( v, x, Q (2) n,D (1+ ( 4 ( 4 g 2 ) 2 ) v, x n − ) ) are palindromic Laurent polynomials. In particular, only for ,D n +540749 +8110633 +406014 +55947 = 1 ,A ( ( 2 ,A ( n v v +4225 +106636 +4226 3 agrees with our universal expansion formula ( 4 ( 3 ( P ) P g 4 2 ,D ) = 1+ ) = (1+ n ) ( ) = ) = 1+6 2 v, x v v n ,D ,A ( ( g v v ( ( 2 (0) 3 2 ( ( 4 4 g ]. Let us write the reduced two-string elliptic genus as ) P 2 2 ,A n The elliptic genera for the 6d (1 Similarly, the reduced three-string elliptic genus for SU(3) model can be written as ,D ,D ( 3 (0) (1) ,A ,A 25 2 2 (1) (0) g 3 3 = 6. Turning off the fugacities of both SU(3) and SU(2) P P P P our universal expansion formulas ( We computed Note the above results on Here are true polynomials: in [ Note be found in appendix Jeffrey-Kirwan residue in [ where all polynomial: where all n JHEP12(2019)039 . . 1 1 i B P P n and (see fibers 1 6 (4.42) (4.43) (4.44) (4.45) ). We F F 1 denoted 1 ]. There P 3 53 F and the associ- I ]. The 61 . See the illustration 1 of various degrees F i does not follow exactly the n 8 F C F (4:2:1) normal to their respective . . ) following [ 1 32 ] , the linear combination ∨ I I 1 P          a B 6 2 surface that intersects with by D t . : F 5) curve in the base denoted by Σ − 1 1 B I t − (3:4:2) F 2 1 a ], which we reproduce in figure ] + [Σ and and τ , : + 0 − 53 b I 0 I = 5 geometry are explained in [ 1 0 0 2 2 0 = t t Σ , I − − n 1 t I F I + 3 1 2 1 0 0 a ] + 3[Σ b and the ( – 38 – (2:3:3) − − . The arrow with double line means the b t 1 I 5 =0 2 1 0 0 0 , which is why the matrix 5). They intersect with each other like the affine base of the P = I X 0 1 0 1 0 00 1 0 0 1 D B − 1 B → P t          ] = [Σ 1 ,..., F B = 1 (3) , (1:2:2) [Σ C , and it is related to Σ b = 0 with the notation ( ⊕ O instead of Σ ∨ I I b 3 5) ( , is the volume of the elliptic fiber. Since we will be interested F /a − 4 I of the divisors f ( I (0:1:1) a I D O which is the double cover of the (+1) curve in 4 f 1 4), as well as the P fibers Σ 4 ,..., F 1 1 -matrix of intersection between Σ . There are six linearly independent curves, which we choose for the moment , ). P C = the marks of 1 2.2 I = 0 G a I The ( Note that here we use Σ intersect at a 32 I 6 pattern ( denote the latter curve by Σ This implies the relation between their K¨ahlermoduli with in the extractionidentify of among BPS the invariants compactΣ from curves the the Mori partition conethe function, generators. Dynkin we They diagram include would above the and like the to figure also 7 in [ to be the Denoting their complexified K¨ahlermoduli by ated marks/comarks by a dashed circle correspondswith to normal the bundle affine nodeF and it intersects within the figure base at the where each nodethe corresponds corresponding to a Hirzebruchfibers. Hirzebruch surfaces surface, intersect In and at the two a diagram nodes above are we connected also if give the ordering of the nodes The divisors and curvesare of five the compact non-compact divisors,We all denote of them which by dynkin are diagram Hirzebruch of surfaces 4.3 JHEP12(2019)039 . ), 2 4 m 2.4 . Ac- (4.47) (4.49) (4.50) (4.51) (4.46) (4.48) +2 8 4 3 t m 2 3 5 ). Therefore t 2 3 4 m t 2 5 + t which induce the 2.19 +6 2 4 6 t 4 4 5 f 4 + m 3 4 2 . We obtain + t . 2 3 ... 15 m t B b 5 16 t 6 + +4 5 ) + + 36 3 3 4) 4 2 2 t 5 t 6 + m 10 3 1 4 + m, m t , + ( ,..., m , 3 2 2 1 τ 4 t 5 10 t t 3 4 + 8 3 +2 = 1 + t + 3 5 2 3 i 2 t − , 3 1 1 , ( t 4 t t 15 ) m 5 6 i 0 ∨ i F t ω + ω ). Furthermore, using the relations ( + i + 12 +6 2 1 + 1) mod 2 4 t 3 fundamental weights ) and the BPS checkerboard pattern condi- 2 t , m, m m 2 0 0 t 10 2 3 ( t 5 t 0 2.18 m t 10 3 , 2 4 ell 9 =1 + 0 i t 2.22 X + 6 9) 7) 5) 3) 1)

– 39 – , m 1 + 1 2 , , , , , 1 t 0 + = 0 0 0 0 0 t 4 2 0 , t 2 b 10 , , , , , t + 0 +8 5 t -fields and all the fundamental weights induce the same 2 2 5 0 0 0 0 0 3 , 16 m 3 t r τ , , , , , , we should classify them according to the embedding r  0 0 0 0 0 + (0 m + 4 2 ell 3 + , , , , , 5 1 3 0 t t 3 -fields, and we list their representatives in table 0 0 0 0 0 0 t ≡ t t 10 r , , , , , m 3 1 + 2 2 5 10 r , which agrees with the universal formula ( t 3 5 (0 (0 (0 (0 (0 i = 5 model and the fundamental weights of + 18 2 t 5 b +4 = 2 m t n . All the 2 2 + = 0) + ! 2 4 P , . goes through. Here for m t 4 2 4 0) , (0 3 f t can be more succinctly written as t 5 t 10 → 3 cls 5 , 3 (1 F unity with the normalisation scheme in appendix +3 t 2.3 0) F , 6 12 2 for ∨ + − cls (0 1 Q m + + 5 2.2 P t 1 F : 4 2 4 t t + = m φ ), 2 2 5 5 -fields of the t t 0 ∨ r t 2 4 10 Q +3  4.42 + + 2 1 m = ), which specialises to and terms cubic in . The 0) , 3 cls τ ) = (0 2.25 ) and ( Imposing the admissibility condition ( F − m, m 4.44 tion ( there are only fivecording inequivalent to the discussion in section ( We also find and scription in section ( up to the analysis in section Table 8 same embedding embedding as The semiclassical components of the partition function can be computed using the pre- which is consist with the universal formula ( JHEP12(2019)039 14 = 5 v , we x n (4.52) (4.57) (4.55) (4.56) (4.53) (4.54) 19 v ) reduces , 8
v , 12 3.33 , v . 16 ) 27 v v reads 16 +167106519 ( v v + 4 23 τ ) 13 +53 , v n ,F v 14 Q ( +444325 n τ 2 10 +3790004228 also induce the same v 7 v P Q i , v 18 ) n τ ω i v +1484 × , +36 Q m ) ). In the case of the ) ··· 17 i 12 v ) ( +1478 v + m 2 4 8 0 limit. Furthermore, higher ) 8 v 2.28 v n +74321701 ,F v +125355 ( +30731161887 1 v, x, Q + 6 → v, Q ( 12 0), which induces the unique em- P ( 26 v 4 +341 +16505926853 v v , ) 4 τ v ) ,F n 0 × 10 22 ( ,F 2 n Q , ( v 1 v +9419 g +2284606168 0 g 16 6 . Its expansion in , (1 + ) 17 v x +909400 . =0 2 ∞ v =0 6 X n 22 ∞ +32316 v 1 defined in ( 56 1 X n ) v 6 5 v = (0 v 3 / +1208 − v λ / 8 + λ 17 4 +30913094 ) in the v (1 − τ − τ 11 +18036 ··· (1 + 4 Q Q v – 40 – 3.35 8 +28442119825 + v 34 +7503 17 v ) +663716 4 25 29 v -vector +12392104012 v 4 +1820 v r v v v 6 21 = 1) = +1300152932 − ) = v v i ) = i 16 +9419 i . As a consequence, this model has no blowup equation (1 m 2 v ∨ , m v ]. +1560 , m τ +11989241 Q 3 τ agrees with the Hilbert series of the reduced moduli space of 37 v, Q +1208 v 10 ( +241608 = 4 4 v ) to compute the one-string and two-string elliptic genera. The 4 2 v, Q ) v ,F ( v ,F (0) n = 1) = P 1 ( 1 4 g i (1) F g v, x, Q +287 3.36 +24994963144 +30731161887 53+1478 h ( ) is completed by making the expression palindromic. Here the ], which is not surprising since the one-string formula ( m → 2 , +8818128233 4 v +341 E v 24 28 +697557618 2 ) are rational functions. Turning of flavor fugacities and SU(2) (2) F ( 41 ∨ 2 20 v v i ) ,Q 4 h v 15 2 v Q m ) are rational functions. Turning off all flavor fugacities, E ,F +4322993 v v i (2) : 1 +48 ) and ( 9 = 1 m v P v -fields have the same reduced 0) , r induced by the reduced 0 , 3.33 expansion of the two-string elliptic genus reads v, x 0 v, x, Q v, Q , ( P ( ( τ 4 4 ) = 1+36 ) = (1+ ) = 1484+36252 4 (0 ) ) ) v v v Q ,F φ ,F n ,F n n → ( ( ( , ( ) = 1+5 2 ( 2 ( 1 4 4 4 g v g g -instanton in [ ∨ ( ,F ,F ,F +1443572 +352245510 +5943020899 +20851379873 +31533797982 4 (1) (2) (0) 4 1 1 1 Q The We use ( F P P P ,F : (0) 2 λ P where where find one to the one-instanton partitionorder function expressions ( agree with [ The ellipsis in leading order expression where one-string elliptic genus does not depend on SU(2) where model, all the bedding of the vanishing type. Weembedding notice (which that is all not the the fundamental case weights in all the other models.) φ JHEP12(2019)039 ) ) ]. I a 24 : D.8 . We (4.59) (4.58) i I n ), ( ]. There 27 6 v . For this v 53 23 D.7 r v ,j l 19 β j v N can be found in 11 v 15 n v 4) associated to the +2266151 5 v ,..., 4 with the notation ( 1 . They display the proper of various degrees F +1094033908456 I ,..., , (6:1) ]). In the diagram above we i +711294838217 1 a G 26 n , ) with the help of ( 22 v +777049966 53 +192278945658 F = 0 v +529286 10 18 I 4 = 0 ( v v +20492637094 3.33 v I 2 ) of higher order 14 F v , t v (5:2) ( 1) are b , k t 4 , ) 0 n ,F ( , 2 = 1 0 (0:1) (1:2) = 6 geometry is explained in [ +110327 P 3 1) n , v 4 2 0 , k, +286304088 0 , F F F 0 9 +557641624668 +1065069893896 to extract the BPS invariants 0 , (4:3) v – 41 – 2 21 25 ,k, +120240591034 0 v v , E 17 ,k , +10017235514 (0 0 v 1 +20295 1), (0 13 2 and the associated marks E N , . v v ) 0 2 I , = F 6). They intersect with each other like the affine dynkin 56 0 (3:2) v , 1) +97433532 , agrees with the Hilbert series of the reduced moduli space 0 8 0 + , v 4 +3217 0 ,..., , ··· v 1 , k, 0 ,F (0) , 2 ]. Some polynomials + ,k, g 4 +413676858801 +982638991174 ,k 42 (0 denoted by a dashed circle corresponds to the affine node and it 28 F 0 +70871529676 = (0 = 0 20 24 v (2:1) +4581942186 4 N 16 v v I β F 12 v ( v I )(56+386 +30479449 2 7 D v v . 6 F ]. The 6 E e 61 -instanton in [ ). = 4 ) = (1+ F v G ( +859008747683 +1065069893896 +8718327 +1956035588 +39315499928 +290168035137 D.9 4 We also use the expressions of ,F (1) 2 P where each node correspondscorresponding to Hirzebruch a surfaces Hirzebruch intersect surface (seealso figure and give 5 two the in ordering nodes [ of arefollowing the connected nodes [ if the denote them by diagram of which in fact can be provedand from ( the one-string formula ( 4.4 The divisors and curvesare of seven the compact non-compact divisors, which are Hirzebruch surfaces At base degree one,for we the also curve notice classes a pattern that the only non-vanishing BPS invariants of two the appendix purpose, we need to useMori instead cone the generators. K¨ahlermoduli Thecheckerboard results pattern, are and reproduce tabulated the in known appendix genus 0 Gopakumar-Vafa invariants [ and the ellipses areleading order again expression completed by making the expressions palindromic. Here the JHEP12(2019)039 0 ), 6) F , 6 4.62 of the (4.66) (4.65) (4.63) (4.64) (4.60) (4.61) (4.62) t 3 ,..., 2 5 I t ), ( + = 1 2 6 by 2.4 t i 5 3 base of the B ( t fibers Σ 2 1 i . There are eight P 1 + 1 P 3 P t 2 2 3 . We obtain , τ, m t → 2 ell B ... t + (4) 2 3 t ) + 3 . 2 t ⊕O . Denoting their complexified b               + t B . 2 6) m, m is 2 1 ] (  t − 1 − 3 I 0 τ 6), as well as the 6 ( t t 2 1 3 . D 5 O , and it is related to Σ ). Using the relations ( − t 1 − + b . ) t 1 + 2 1 0 2 0 0 t ,..., ∨ i τ and 2 0 ] + 2[Σ 3 3 1 2.19 − − ω t t + 2 0 , i 3 b 2 2 = 2 b 0 2 1 0 0 m, m t t Σ t m ( I + , − t = 0 +  I ell 3 6 I 6 t 1 2 1 0 0 0 t =1 6 9 I a + 4 i t X t ( 4 12 1 2 – 42 – 3 ] + 4[Σ − 0 b b t t I 6 =0 = + + 2 0 0 1 0 0 + I X 3 5 5 + = 6 t t − τ 9 m 2 6 3 2 t B 2 ell 2 1 0 0 0 0 0 + t t 0 0 0 0 1 0 0 0 00 1 10 0 00 1 0 0 0 0 1 0 0 1 ] = [Σ + 4 + − 4 with normal bundle t B 3 3 1 2 5 fibers Σ 12 3 t 9               t 1 ) in terms of the K¨ahlermoduli , where + 2 [Σ 1 t i P 3 + ( = = P 6 t + m 2 3 0) 3 , the linear combination t 3 2 , 0) C , + t 9 B (0 4 t + (0 2 t 12 F 2 2 6) curve in the base denoted by Σ 3 F , is the volume of the elliptic fiber. We also identify the Mori cone + t with the normalisation scheme in appendix 6 3 1 + − e − t + 9 and 1 6 2 t 2 1 I 3 2.2 t t + + + 3 0 0 t 9 2 0 t 12 3 4 t   + and the ( = -matrix of intersection between Σ I and terms cubic in C 0) D , the marks of 3 cls (0 I τ F a ), we can express The − 4.60 up to which is consistent with( the universal formulaand ( find scription in section The semiclassical components of the partition function can be computed using the pre- This implies the following relation of their K¨ahlermoduli with generators. They include the surface in the center. We denote the last curve by Σ linearly independent curves, which wedivisors choose for the moment toK¨ahlermoduli by be the intersects with the base at the JHEP12(2019)039 λ 2.3 . We (4.67) (4.68) (4.69) (4.70) 9 -field -fields or r r ). 4.1 4 5 6 which induce the , ω , ω , ω 1 2 3 6 . e ω ω ω . The expansion in b t x , n τ + 8 Q , ) 6 ) i t fundamental weights v m 9 2 ( 6 ) , + n ,E ( 1 2) 6) 2) 6) v, Q 5 10) 10) 2) 6) ( induced by the reduced t , , , , P 10) , , 6 , , 0 0 2 2 , ) 0 2 0 0 , , , , P 0 × ,E , , n , , 0 0 0 0 + 5 ( 1 , 0 0 0 0 , , , , ), and thus the analysis in section g 0 4 , , , , 22 → 0 0 0 0 t , , 0 0 ) 0 0 , , , , 0 , , 3 2 2 =0 , , ∞ 0 0 0 0 ∨ , 0) mod 2 0 0 X n v 0 0 , , , , 2.19 1 0 , , , Q , , + 6 2 2 0 0 , 0 2 0 / − 0 0 : , , , , ) and the BPS checkerboard pattern condi- 3 0 , , , , , t 0 0 0 0 , 11 0 0 0 λ 0 0 , , , , (1 0 , , , − τ , , φ 2 2 2 2 , 0 2 2 2.22 + 5 – 43 – , Q − − − − − − 0 2 11 t , r v 0 9 2 , = 1) = 0 + 0) ( 4) ( 8) ( 0) ( 4) ( 8) ( , -fields, and we list their representatives in table i ) = 0) (0 4) (0 8) (0 , , , , , , 1 i r , , , m t 0 0 0 2 2 2 (0 0 0 0 , , , , , , , , , 0 0 0 0 0 0 , m ≡ 0 0 0 , , , , , , = 6 model and the fundamental weights of + 5 τ v, Q , , , r 0 0 0 0 0 0 ( 0 0 0 0 n , , , , , , ) to compute the one-string and two-string elliptic genera. The t 6 , , , 0 0 0 0 0 0 . They can be divided into three groups; inside each group ) 9 2 0 0 0 v, Q , , , , , , ,E n P ( ( , , , 1 2 2 2 0 0 0 6 g 0 0 0 , , , , , , 3.36 = ). We also list in the table the fundamental weights which induce (1) E , , , 0 0 0 0 0 0 → , h 0 0 0 , , , , , , 0) , , , , ∨ 2 2 2 2 2 2 E cls (1 2.33 Q ) are rational functions. Turning off all flavor fugacities (0 (0 (0 − − − − − − i F ( ( ( ( ( ( : ) and ( m φ ), ( -fields of the r 3.33 v, Q 2.28 ( 6 ), which specialises to ) ,E n . The unity ( 1 g vanishing vanishing 2.25 The reduced one-string elliptic genus does not depend on SU(2) We use ( Imposing the admissibility condition ( reads τ where results are again presented in terms of the reducedQ elliptic genera defined in ( there are in totalclassify 18 them inequivalent according todefined the embeddings in ( the same embedding. tion ( It is in agreement withgoes the through. universal expression We ( also find Table 9 same embedding fundamental weights induce the same embedding. JHEP12(2019)039 ) ]. 1), 37 . 42 , v D.8 22 0 33 v ]. At (4.75) (4.71) (4.72) (4.74) (4.73) v ,..., 29 ), ( 2 24 v , k, , 0 , D.7 agrees with 0 +3239 25 , = 1 6 8 v 0 ··· . For this pur- , v , ,E r , + 0 (0) 1 22 ,j ) , l g v β 21 10 j , k v v ( . v N 6 13 ) n τ v +79 n ,E = 1 ( Q 2 +2464163123099051 1), (0 ) ··· +4036298 , i P 17 1) 36 , +1340924322289616 7 0 v + m 0 v +392614934492341 v ], while higher order con- , , × 32 0 0 10 28 v , 41 , v -instanton moduli space [ v 23 ,k, +42223058 0 2 6 ) with the help of ( ) 0 22 +60399840583490 8 / , 6) associated to the Mori cone 2 v v E 0 , v, x, Q 24 v , k, +1 0 ( + v , +2807705547 0 6 +874485 3.33 ,k +4695878015170 + ) (0 , . 6 0 ··· 12 ,E n . They display the proper checker- 0 ,..., v v 20 ( v 2 N , + 1 v 78 +1862806 g , v 8 G 10 = +173919980352 v v + =0 ∞ (1 + X n 16 +25393522 1) = 0 , 6 v ··· 1), (0 6 0 32 v / I 1 , ) +173895 +2241216639463322 + ( 0 ,k, 5 v 23 , 0 35 39 b v +261500117384417 +1046126546231772 , τ − 2 +43989 0 v v 0 +866242068 / , , t 8 , 27 31 – 44 – Q 1 0 +1067286 I v , 0 v v to extract the BPS invariants +34066083855696 11 6 t − 0 (1 + , 23 , , we obtain v v 2 0 23 ,k x v 46 (0 , 0 v E +2214670938701 +8798601 ) +31432 0 , 4 -instanton moduli space [ v N 4 19 1 v 6 v v ) = +67304396959 E +23815 i − E = , k, 6 15 v v +337457 1) (1 , m , 4 0 τ , v +5121 = (0 0 +250943933 , 3 0 v 10 β +1603334 2 +6776 v 2 / ,k, 4 ]. +1962268356880815 +2658213934310966 1 0 v v , = 1) = +167232472484542 +784947220008032 v, x, Q − 0 34 38 ( , 37 +54206 i +739 26 30 v v ,k 6 +18398716114730 2 (0 2 0 m v v (2) E +996158535441 v v 22 h +945 N 18 v +24690503239 2 E v v 14 = 1) are +94 +68039474 v agrees with the Hilbert series of reduced two , v 9 v, x, Q 1) 0 6 , ( v , 0 expansion of the two-string elliptic genus reads 6 , ,E 0 ) 0 (0) 2 , τ ,E n 0 g ( 2 , 2 Q ) = 1+56 ) = 79+3774 ) = 3239+130034 ). g 0 , k, , / v v v ) = 1+9 0 0 1 ( ( ( v , 6 6 6 ( − D.9 ,k, 0 6 ,E ,E ,E ,k We also use the expressions of The , (0) (1) (2) (0 1 1 1 0 +1653587141756229 +2608327634962043 +8569454706 +426790882149 +9507297417908 +102628223553496 +566271723784347 +17200367 ,E 0 (0) 2 P P P , N P which in fact can be provedand from ( the one-string formula ( (0 pose, we need to usegenerators. the K¨ahlermoduli The resultsboard are pattern, tabulated and inbase reproduce appendix degree the one, we known noticeants genus the for interesting 0 pattern the Gopakumar-Vafa that curve the invariants classes only [ non-vanishing BPS invari- Note that where the first few orders are Turning off flavor fugacities and SU(2) The ellipses are completedthe by Hilbert making the series expression oftributions palindromic. reduced agree one Here with [ where the first few orders are JHEP12(2019)039 . 0 1 1 F P P . We (4.77) (4.79) (4.76) (4.78) i → n ]. There (6) 53 by B base of the ⊕ O 1 6 . Denoting their 8) P F B (6:1) − ( O 4 and the associated marks . of various degrees F . I (5:2) i                 ] n 2 2 F − is . 2 0 I 2 7), as well as the t 2 − , which is related to Σ D F ] + 2[Σ b 2 1 0 1 (4:3) + 2 − ,..., τ 1 and 1 t 2 1 0 0 , b (7:2) = 8 geometry is explained in [ with normal bundle = − Σ 1 I n + 4 ] + 4[Σ , 2 0 0 0 0 2 1 0 0 1 t 2 0 = 0 0 P I 0 I F F − − t I a (3:4) ( – 45 – 2 1 0 0 0 0 , the linear combination I 7 =0 + 6 − B I X 8) curve in the base denoted by Σ t b denoted by a dashed circle corresponds to the affine ] + 6[Σ t 2 1 0 0 0 0 0 − b 6 − = 2 F and F 2 1 0 0 0 0 0 0 7). They intersect with each other like the affine dynkin B 1 0 1 0 00 1 00 0 00 1 0 00 0 0 00 1 0 0 0 0 1 0 1 0 0 0 (2:3) fibers Σ I t − t 1 ] = [Σ                 P B ,..., and the ( = ). The 1 [Σ I , I 4 a C , is the volume of the elliptic fiber. We identify the Mori cone F D : 7 (1:2) = 0 e I I ( I D 6 F 7 (0:1) 7 E e -matrix of intersection between Σ of the divisors C = the marks of I I G a The with the notation ( I Their K¨ahlermoduli are consequently related by with generators. They include the surface in the middle. We denote the last curve by Σ fibers Σ complexified K¨ahlermoduli by In the diagram abovea we also give thenode ordering and of it the intersectsThere nodes with are the nine base linearly at independent curves, the which we choose for the moment to be the The divisors and curvesare of eight the compact non-compact divisors,denote which them are by Hirzebruch surfaces diagram of 4.5 JHEP12(2019)039 ) λ 7) 2 6 . We 4.78 reads t 8 (4.86) (4.83) (4.84) (4.85) (4.81) (4.82) (4.80) 3 τ -field ,..., 10 r 2 6 ), ( + 3 5 Q t t 3 2 5 5 2 2 2 = 1 t t 2.4 3 i + + ( . 3 4 + 2 4 b 4 i ). We also find t t 8 t 2 6 3 t + 4 4 3.4 4 . We obtain 11 t + 3 2 4 t 3 , τ, m 2 2 B + t 8 , + + ell 3 6 n τ t ... 2 5 t 3 1 t t 3 + Q , 4 8 2 ) t 2 1 51 ) 2 i t ) + v + + m ( + + 3 0 2 2 7 , t 4 5 t 2 0 2 t 3 1 t 8 (1) t m, m n,E v, Q 3 ( 4 ( induced by the reduced + 35 P 7 + τ b  ) t 2 3 2 P ,E × n + t + ), and therefore the analysis in sec-  ( 1 ). Using the relations ( 1 2 b 4 g t 6 − t t 34 . → t 0 , 2 ) ) t 5  0) mod 2 8 ∨ i 2 =0 t 2.19 ∞ 57 ∨ 2.19 , 7 X n + v ω 8 t 1 0 i Q + 2 2 6 + , t / − : ) and the BPS checkerboard pattern condi- 6 + 4 m, m m 0 0 t 3 ( 4 , t 17 t λ 4 6 (1 0 t t , 16 τ − 7 3 2 φ ell =1 , + 6 i t 2.22 X t – 46 – Q + 0 2 1 + 1 2 + 5 t , 5 5 t = 17 0 8 0 t 2 t 4 t 2 v t + , 4 t t 0 m + + τ 8 , + = 1) = 57 2 4 + 6 0 t i ell 2 t ) = ) in terms of the moduli K¨ahler -fields, and we list their representatives in table t , 2 16 t i 2 0 t 4 3 + r 2 5 m t t ( 2 t 1 (0 1 1 3 + 16 t , where 0) t , m , + i 3 τ + ≡ 4 t + v, Q 6 4 (0 = 35 m t 1 ( ) to compute the one-string and two-string elliptic genera and r 2 4 t 2 4 7 F + t t 2 0 ) 4 0) 2 0 + , v, Q 2 with the normalisation scheme in appendix t ,E n t t ( ( 1 3 0 + (0 16 3.36 7 3 t g + 5 ). We also list in the table the fundamental weights which induce (1) E F + t 2.2 1 8 h + 2 2 4 51 3 7 t 6 t 2 − 1 t E 0 8 t t 2.33 ) are rational functions. When all flavor fugacities are turned off = + + i + 2 ) and ( + 3 6 t m 0) t 0 , 4 ), ( 2 7 2 1 4 t t 3 16 t cls (1 3.33  F + + + v, Q 2.28 ( = 7 ), which specialises to ) ), we can express and terms cubic in ,E 0) n , ( 1 goes through, which then leads to the elliptic blowup equations ( 3 cls g (0 τ 2.25 4.76 F We use ( Imposing the admissibility condition ( The semiclassical components of the partition function can be computed using the 2.3 − where the same embedding. convert them to reduced versions. The one string elliptic genus when expanded in there are in total 16classify inequivalent them according todefined the embeddings in ( tion ( It is in agreementtion with the universal expression ( and ( and find up to which is consistent with the universal formula ( prescription in section JHEP12(2019)039 7 , ,E ) (0) 2 g 32 10 (4.89) (4.87) (4.88) v v 12 -fields or 5 + v r 7 16 . For this , ω ··· 8 r v 3 ]. , ω v ,j + l 6 β j , ω 42 18 2 . N , ω which induce the v ) 4 , ω 7 v ω . Indeed, our 1 e +7664780 . ( +182614170 7 ω n τ F 8 10 v v Q (2) n,E ) i fundamental weights P , m ) +1645739978 +43009574252 × 6 32 14 v v 2) 6) 35 v +1014596958 7) associated to the Mori cone 10) 14) agrees with the Hilbert series 2) 6) ) , , 10) 14) , , + , , 2 0 0 16 7 , , 0 0 v, x, Q +2421286 0 0 , , v v 0 0 ( +38850151 , , 8 , , ··· ,E 2 2 7 , , 6 2 2 (0) 1 v 0 0 , , ) + ,..., 0 0 , , v + g , , ,E 0 0 n 1 , , 0 0 v ( 2 0 0 , , , 0 0 , , 18 g , , 0 0 , , 0 0 v 0 0 , , . They display the proper checkerboard 0 0 , , , , 0 0 =0 , , = 0 ∞ +231525774 0 0 0 0 , , ], while higher order contributions agree (1 + X n G 0 0 , , 4 , , 0 0 I , , 6 0 0 -instanton moduli space in [ v 0 0 +468612 , , ( 52 41 / 0 0 7 , , +35350906224 1 , , 6 0 0 ) b , , 0 0 +4931707 0 0 , , 35 E v v 0 0 12 , , +1252490096 4 , , , t 2 2 − τ , , to extract the BPS invariants 2 2 v v I 14 t − − – 47 – Q − − 2 v +15203076 (1 + 35 E , r 16 v 1 which we put in the appendix . 70 1 ) v , +53305 ) +18210733 E 4 v 32 0) ( 4) ( 8) ( 2 12) ( v ) = 0) (0 4) (0 8) (0 v , , , +345521 v i 12) (0 = 0 , − , , , 0 0 0 2 , 0 + 0 0 0 , , , n 0 v , , , , (1 2 2 2 , m , = 8 model and the fundamental weights of 2 0 0 0 , , , ··· τ 0 , , , , 0 0 0 n +19507715662 , 0 +3312 0 0 0 , , , 18 +1014596958 0 , 2 . They can be divided into two groups; inside each group , , , ) for 0 0 0 10 v , +19086400 0 v 0 0 0 , , , v 12 P v 0 , , , , ( 0 0 0 14 v , 0 7 0 0 0 , , , v, x, Q = 1) = v 0 , → , , , ( , 0 0 0 , i 0 7 0 0 0 , , , (2) 0 n,E , ∨ , , , (2) E m 0 0 0 , 0 and flavours, we have P h 0 0 0 , , , Q 0 )(1+98 )(134+11593 )(9179+693316 , , , , -instanton moduli space in [ 2 2 2 E x , 2 2 2 : 2 7 v v v -fields of the (0 − (0 − (0 − φ (0 − E ( ( ( r ( v, x, Q ( expansion of the two-string elliptic genus reads 7 +15203076 +536726278 +7093827388 +35350906224 ) τ ,E n ( 2 . The Q ) = (1+ ) = (1+ ) = (1+ g ]. v v v ( ( ( 37 7 7 7 unity We also use the expressions of The ,E ,E ,E (2) (0) (1) 1 1 1 vanishing P P P Turning off SU(2) We have computed agrees with the Hibert series of reduced two purpose, we need to use thegenerators. K¨ahlermoduli The results are tabulated in appendix of reduced one with [ where the ellipses are completed by palindrome. Here Table 10 same embedding fundamental weights induce the same embedding. where the leading order contributions are JHEP12(2019)039 . ) 1 8 I , 1 e a P 1), P and D.8 ,... I (4.90) (4.93) (4.91) (4.92) → , k, 2 t , 0 ), ( , 0 surface in , = 1 (10) ]. At base D.7 4 0 0 ]. There are F , . Let F 24 0 (7:2) B 53 , ⊕ O 0 , , , k 0 12) , , = 1 ] − 2 ( 4 F 1) , O (6:4) 0 , base of the 1), (0 0 , , 1 0 by . 0 , (8:3) P , 0 4 and the associated marks ] + 2[Σ 2 0 B t 3 ) with the help of ( / ,k, , I 2 0 0 0 , F F +1 0 + 2 , (5:6) ,k 3.33 3 (0 0 , k, t 0 N ] + 4[Σ , 2 0 + 4 = , 2 0 τ 2 t 1) , F ,k, (4:5) = 0 with normal bundle , + 6 ] + 6[Σ I 0 8) as well as the = 12 geometry is explained in [ 1 , t 1 1 0 I t 1), (0 , n P 2 0 a , / , – 48 – 0 1 0 , , + 8 ,..., 8 4 − 0 =0 12) curve in the base denoted by Σ , 0 1 I X F 0 , which is related to Σ ,k , ] + 8[Σ , (0 − t 0 b (3:4) 0 0 denoted by a dashed circle corresponds to the affine , N 0 = 0 10 , = + 10 0 F I b ( 1) t , 6 0 I , , k, F ] + 10[Σ 0 = and the ( , 0 b (2:3) 0 , I B t ,k, 2 ). The D 0 / , I 1 0 , = (0 a ] = [Σ fibers Σ − 0 , : 8 ,k β B 1 (0 8). They intersect with each other like the affine dynkin diagram of 0 F , is the volume of the elliptic fiber. We identify the Mori cone genera- I P (1:2) 8 [Σ 1) are N e , 0 = ,..., , 1 0 1) 8 , , , 0 , 0 E 0 10 , , F of the divisors 0 0 = 0 , (0:1) ). = 0 marks of , I I 2 0 ( / I , k, G 1 D.9 ,k, a 0 I − 0 , , be their complexified K¨ahlermoduli. The linear combination D 0 ,k (0 0 , B N which implies with tors. They include the center. We denote the latter by Σ fibers Σ t In the diagram above wewith also the give the notation ordering ( ofnode the and nodes it intersects withThere the are base ten at linearly the independent curves, which we choose for the moment to be the The divisors and curves of thenine non-compact compact divisors, which areby Hirzebruch surfaces of various degrees. We denote them which in fact can be provedand from ( the one-string formula ( 4.6 degree one, we noticefor the the interesting curve pattern classes that(0 the only non-vanishing BPS invariants pattern, and reproduce the known genus zero Gopakumar-Vafa invariants [ JHEP12(2019)039 4 t 3 3 t 3 t 0 2 (4.98) (4.95) (4.96) (4.97) (4.94) t 4 t t 2 1 2 0 6 t t + 5 4 . t +4 b + 2 3 t t t 3 4 0 t 2 2 t t 6 t 2 0 0 3 t t + 5 + 10 ), we can express . We obtain 4 +3 + t 8 + 1 3 t 3 B t t t ). We also find 2 3 0 2 1 3 0 t 4.91 4 t t 11 t ... 2 0 2 8 t 4 3 3.4 . 5 + + ), ( + + , 2 ) + 7 2                    + t 3 t 7 2 t t 0 2 t 1 2 1 9 2 3 t 2 t 2 6 4.93 t t − t 8) and find 2 0 0 3 t + + m, m t 2 0 + is 1 ( +4 ), ( 6 t 10 4 ), and therefore the analysis in − t b τ 2 7 I t 0 3 t +2 t t 5 2 3 ,..., + 2 1 0 2.4 6 t 3 3 D 2  2 b t 3 8 t 2 + 5 t 6 − 2.19 7 − t t 2 1 . t + 5 t = 1 3 3 )  6 2 0 0 0 2 1 0 1 t 0 2 8 + ∨ i + t 4 i 8 and t t 8 3 2 +2 ( t 3 7 − − 2 4 ω 8 t i b t 4 9 + i + t 4 3 2 1 0 0 0 + 3 + 4 Σ + t 3 2 7 m, m m t 3 , 4 t t 7 2 − ( 3 2 + 3 3 t t t I 1 12 t 8 3 6 t 1 ell 2 1 0 0 0 0 =1 + + 2 t i t 4 t 9 , τ, m X 4 + 45 – 49 – 2 6 0 2 2 4 − 3 1 2 t t t 6 + ell 6 + 2 = t t 2 2 1 0 0 0 0 0 + + 4 4 t + + t t 3 4 3 2 +4 − + 2 2 4 m 2 t t τ + t t t 18 2 4 5 5 4 12 2 1 0 0 0 0 0 0 t t 5 1 2 4 6 2 ell t 0 t + t − 3 + t 1 + + 4 t + 17 3 3 t 1 4 2 1 0 0 0 0 0 0 0 2 3 t 3 24 +2 + 9 , where 1 t 0 1 0 00 1 00 0 00 1 0 00 0 0 00 1 0 0 00 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 t 2 3 i + 8 t ). Using the relations ( 24 − t 4 2 3 5 2 2 3 t t = m t + 2 3 0 + 4 3 3                    + + 75 2 t t 3 3 2 t 0) 3 2 2 4 2 t 2.19 t , 6 2 with the normalisation scheme in appendix = t t 4 2 + 3 (0 t 3 + + +2 1 + C + 4 2 2 t F + 2 3 + 2.2 2 t t 8 t 3 1 t 3 2 2 0 2 1 2 2 2 t 1 − 3 t 33 t t t 9 3 t 1 3 + 3 3 2 4 t 16 + 1 2 + + + t 12 + + 0 4 2 1 2 0 t 2 2 + 3 0 t t t t + 2 t 12 2 1 0 3 3 1 5 4 t t t 41 0 18 t 1 t 4 25 8 24  t -matrix of intersection between Σ goes through leading to the elliptic blowup equations ( =  +3 + + + + + and terms cubic in C 0) = 3 , ) in terms of the K¨ahlermoduli 2.3 t τ cls (1 ( 0) , F The semiclassical components of the partition function can be computed using the The 0) cls (0 , F (0 − It is insection agreement with the universal expression ( up to which is consistent withF ( prescription in section JHEP12(2019)039 . , ∨ 26 18 v v 58 Q . All v (4.99) = + (4.103) (4.100) (4.101) (4.102) 10 11 v P ··· 8) expansion + 18 → , τ v 30 Q ∨ 10 v ,..., v 24 Q which induce the v : 8 = 1 e λ i φ ( , +11332578013712 +79099752469353 i n τ +38985250263 ω 24 16 8 Q , v v ) v fundamental weights ) i v +220731150 m , ( +492718282457 8 8 , v ) 16 6) 2) 18) 22) 14) 10) (1) v, Q v , , 56 n,E , , , , ( 0 0 v P 8 0 0 0 0 , , ) , , , , +2759575276449180 + 0 0 ,E n × 0 0 0 0 ( , , 1 , , , , 28 +1372471431431505 0 0 g ··· 0 0 0 0 v 58 , , +2558096217 0) mod 2 , , , , 22 ) + 0 0 , 6 =0 0 0 0 0 2 ∞ v , , 0 +13722599 X v n , , , , v 30 0 0 +7692605013883 1 , 6 0 0 0 0 6 v , , +18281159666407 0 , , , , / v − 22 ) and the BPS checkerboard pattern condi- 0 0 , 0 0 0 0 , , 14 v 29 0 , , , , 0 0 v (1 , τ − 0 0 0 0 , , +113608689871 0 , , , , 2.22 0 0 Q – 50 – , 0 0 0 0 , , 14 0 , , , , 29 v , v 0 +562133 -fields and all the fundamental weights induce the same , 4 r +111587988 r = 1) = 0 4 v , ) = i v -fields, and we list their representatives in table i 0 which induces the same embedding r 8) (0 4) (0 0) (0 m , 16) (0 20) (0 12) (0 , , , +2759575276449180 λ , , , 0 0 0 , m (0 +678620928038790 0 0 0 = 12 model and the fundamental weights of , , , τ 26 +11332578013712 , , , v, Q 0 0 0 20 v ≡ +4022154098447 0 0 0 ( n +14080 ) to compute the one-string and two-string elliptic genera and , , , . All the 28 v , , , 8 +3197400818096 2 r 0 0 0 ) 20 v 0 0 0 P v, Q . v , , , ,E n 12 v , , , ( 8 +19517762786 ( 1 +2998484 0 0 0 e v 0 0 0 8 3.36 , , , g 2 → , , , 12 , (1) E 0 0 0 v 0 0 0 h v , , , for ∨ , , , E 0 0 0 0 0 0 Q , , , , , , P 0 0 0 : 0 0 0 )(1+189 , , , ) and ( , , , = 2 -fields of the 0 0 0 φ 0 0 0 v r , , , ∨ , , , Q (0 (0 (0 3.33 (0 (0 (0 are rational functions. Turning off flavor fugacities 8 ), which specialises to ) . The ,E +415090167480 +262872507223458 +2187800775100695 +1612836871168 +12891341012848 +2454952400 n ( 1 ) = 249+43435 ) = (1+ g v v 2.25 ( ( unity 8 8 We use ( Imposing the admissibility condition ( ,E ,E (1) (1) 1 0 P P reads where where the leading orders are there are in total 12of inequivalent them have the sameIn reduced this special case, there is no vanishing blowupconvert equations. them to reduced versions. The one-string reduced elliptic genus in tion ( Table 11 same embedding embedding as JHEP12(2019)039 ) , 1), D.8 ,... ]. At x 2 , k, , (4.107) (4.104) (4.105) (4.106) 0 24 ), ( , 26 0 v = 1 -instanton . For this , D.7 8 r 0 20 ,j 8 , E l v β 0 v j , , k . N for the minimal 0 , ) ) 0 k v = 1 , ( G , ( h 14 8 0 n τ . 1) ) , , v E n 0 ,E Q , 58 ( 2 ) 0 , v i P 0 . The simplest series of ) m + 8) associated to the Mori k ,k, × 1), (0 ( 0 G , , 2 ··· 0 / , H 0 59 +230109165319 0 ) with the help of ( , + , ) 6 indeed agrees with the Hilbert +1 0 ,..., 2 0 , v, x, Q v ) 30 1 ,k +129037047832755990 v 8 ( (0 0 , k v 8 . They display the proper checker- 3.33 ( G ,E 24 ) , k, +16738824444898167 + N (0) 1 ]. Higher order contributions agree ,E n v 0 G ( g 2 v H 18 , = 0 = discussed above to the superconfor- g 41 0 v ] recognised that the one-string elliptic , I 1) 8 , ( =0 0 ∞ +235331998114532 7 ,k, , , 37 X n (1 + denoted by b 0 expansion reads 6 , 0 12 6 0 92 , / , -string elliptic genera , t k τ v ) 1 0 0 I ,E , k 59 v t Q , 0 4 , +10844316461 2 − τ 0 to extract the BPS invariants / , 4 1 0 ,F – 51 – Q , v 2 4 − 0 , (1 + 59 E ,k 1), (0 (0 , 0 v ,D , 1 118 2 N 0 ) E , A v = 0 ) = +161349436368883950 i , +82360817736515085 − 1) 0 = 28 , +5249113972780491 , 0 22 v , m , 0 (1 0 16 v τ can be obtained by geometric engineering on non-compact G +316301853 v +32482207865920 which indeed agrees with the Hilbert series of two ,k, 2 ]. 0 , k, , 8 v 10 (1) 0 G 0 , 42 2 v ,E 0 , / , (0) 2 H -instanton moduli space in [ 0 1 0 v, x, Q , 8 g , ( = 1) = − 0 = 2 4d SCFTs of rank , 0 E 8 i ,k ) are rational functions. Turning off flavor fugacities and SU(2) , (0 (2) E 1) are 0 i m h , N N m 0 E , = 0 = (0 , 1) 0 , β v, x, Q 0 , ( v, x, Q 0 8 , ( , k, ) 0 8 , ,E n ) 0 0) 6d SCFTs with 0 ( ). 2 ,E +3262175735364 +161349436368883950 +1273365718136904 +41781447040327605 , n , ]. ( ] to the higher rank cases. To be precise [ 2 g ) = 31374+4996185 ,k, 0 2 0 g v , / , 37 ( 1 0 37 D.9 0 , 8 We have computed We also use the expressions of The two-string reduced elliptic genus in − 0 , , = (1 = 2 SCFTs namely 0 ,k ,E (0 (1) 0 , 2 N P mal indices of the N del Pezzo geometriessult and is contains an the extensioncase Minahan-Nemeschansky of [ theories. a surprising The conjecture main by re- Del Zotto-Lockhart from the rank one 5 On the relationThe with purpose 4d of SCFTs this of sectionN is type to connect the (0 which in fact can be provedand from ( the one-string formula ( purpose, we need tocone use generators. the The results K¨ahlermoduli areboard tabulated in pattern, appendix and reproducebase the degree known one, genus we zerocurve notice Gopakumar-Vafa classes invariants a [ pattern that the only non-vanishing BPS invariants for the reduced moduli space in [ where where the ellipses are completedseries by of palindome. reduced one Note with [ JHEP12(2019)039 3 G , 2 H and , . As ) 8 , k 7 ( G , theories theories = 1 theories, 6 e H ) ) f ) k k ( ( k G ,E G N ( G 4 e H H e H -string elliptic theories is the ,D k ) gauge theories 2 k G for the low energy ,A H 1 G — and ], and explain in some ,A ) ∅ theories. Here we follow the k 37 ( G 2 theories. For example, the , = 1 H , G 0 G H H type theory is in S-duality with SU(3), 6 k E are sometimes more natural than specialises to the Hall-Littlewood index ] and denote these theories as G v 37 e 1 are equivalent to USp(2 H . We also extend the analysis to some rank we analysed, the surprising relation between G and are also known as the Minahan-Nemeschansky k > – 52 – G τ 8 , given by the strings stretched between D3-branes 7 , Q 6 G E here. theory is well known to be the SU(2) gauge theory with H 4 1 there is one more SU(2) symmetry coming from the , which give the gauge symmetries (1) G D that are determined from the elliptic blowup equations. ∗ is usually called the rank of L theories. With the two string elliptic genera computed in ) H are well known to exist for k II k ( G ) SU(2). k > h (1) G k ( E G × H In type IIB superstring theory, they are realized as the world- H , and G theory construction, and then review the superconformal indices ∗ ] becomes 34 This function is on the one hand the ingredient of multiple D3-branes probing a stack of exotic seven-branes. Such S ]. 37 ], for higher rank cases, theories type theories are also traditionally denoted as k 33 66 ,III theories appear as certain limit of SU(2) gauge theory with 2 . 45 ]. ∗ – G ) ) can be decomposed in terms of a seemingly more fundamental function 2 k ,A ( G 67 62 ]. The rank one 1 ,A ], where the simplest example rank one , v [ L 1 ,IV theory can be coupled with a free hypermultiplet associated to the center τ ]. k H 62 ,A ∗ 0 ,A ) 66 = 2 SCFTs ∅ Q ∅ , 37 ( k ... function in [ ( G = H 3 ), which for special choices of 65 (1) G G N , H h called G L 2 , v E , τ k All We first review some basic properties of 4d rank Q = 4, while the higher rank cases with = 6 theory [ The The ( = 1 33 34 f f G transverse space in theshare seven-brane. flavour symmetry By coupling a free hypermultiplet, all notations in [ of mass motion ofwas the observed instantons. in Wetheories. [ follow [ Oneflavour major symmetry. difference Besides the between flavour and rank exotic one seven-brane, and for higher rank with four fundmental hypermultiplets and oneLagrangian antisymmetric theories. hypermultiplet, which The are rank all theories one [ N seven-branes in F-theory areII,III,IV,I defined as codimension8d one SYM singularities theories. with The number Kodairerank type one respectively [ N 5.1 Rank The 4d k volume theory for rank genus, on the other handat gives special the Hall-Littelwood choices index ofindices and parameters. efficiently Schur index from This of the general structure allows us to calculate the latter including their class of 4d SCFTs inthem. various Next physically we motivated statedetail limits the the as conjectural new well relation relationsthree at at as cases. rank rank the two one methods For for fromelliptic all to all [ genera choices compute and of superconformal rank indices and exists. We define an intermediate function at L or the Schur indexour of previous the sections, weprinciple are at arbitrary able rank, to and study find this indeed that conjectural similar relation striking at relations exist. rank two and in genus JHEP12(2019)039 ) 1 S k − ( G We ] to N H (5.3) (5.1) (5.2) A 69 35 ). to obtain theories and one instanton } 2  ), thus can 4.7 2 A 7 k N 7371 ... E { + + k H 4 ) and ( v 133 ,
... . To be precise, the as they are directly ) 4.6 3 τ G 8 2 + ], i.e. they are class , v 7 Q 8 7 = 1 + , ), (
( 70 6 v 573440 E O χ 7 ... 4.5 ,E 133 E 8645 4 + 2 + χ (2,0) SCFT on a sphere with 7 ) +  E 5248750 ,D 1 + th 2 χ k ... 7 A . 1 A 7 238602 E +
+ = 8 and appropriate Higgsing as ) E 7331 − partition function of the 4d SCFTs: χ 6 = k χ f ( G v 2 7 + G N S H + E 7371 ] to preserve half of the supersymmetries ,..., χ 7 × 7 β 7 1 ) 8645 E 2 68 E 133 E 238602 2 + − χ χ T S χ , × 7 + th 2 + E 1539 k ) makes it sometimes possible to compute the T 4 1 ( – 53 – χ 7 . The 6d construction for rank 1 + 2 v Z 7371 E 5.1 theories is that they all admit 6d construction. It + 7 χ ) 12 = + -twisted E 7371 7 k ,..., ( . See a good description of such twist in for example ) G β E 133 χ 7 theories can be realized by compactifying a 6d ]. For example, the elliptic genus of one k 2 + 2 ( G χ E 133 H S h 8 + , 48 7 χ 7 E 2 , – E 133 × 6 v diag(1 χ 2 in table ,E 3 + 2 7 3 1 + 46 4 1 T , E 133  z 8  , D + , where the agreement holds to all known orders. χ 7 τ 2 τ , 6 37 ] via SU(4) gauge theory [ 0) SCFTs with corresponding gauge group 6 Q Q E , , i.e. the residual flavour symmetry is SU(2). Thus the resulting 7 48 , one also need to use the Joseph relation Sym k H Φ = , } ,E + + 2 τ partition function. This was indeed achieved for one string elliptic 6 2 4 Q 1 = 1 + 2 and D { ,E S
4 4 7 ]. The identification ( = theory is obtained by compactifying 6d D × (1) D E 37 2 ) decomposes and what is the residual flavour symmetry. For example, the H theories of interest in this paper are = G T ) N β k G ) SO(8) ( G 2 -twist was introduced by Kapustin in [ theory on a sphere with one regular puncture with Young diagram H S H β × 1 2 − T k ( 2 Another important feature of The Z In the coefficients of A 35 irregular puncture of form the identification. 4d theory has gaugeabove. symmetry We summarize SU(2) the and gaugetheories four algebras and with fundamentals, punctures as for the washowever 6d involves mentioned construction irregular already of punctures. all For6d example, they can be realized by compactifying theories. The regular singularities arebe classified by denoted embeddings as of SU(2) Young in diagrams.how SU( the Such SU( punctures withrank associated one Young diagram represent four full punctures which completely agrees with ouralso universal checked expansion for formula ( is well known all(2,0) rank SCFT on some punctured sphere with regular singularities [ compute the genus with string was obtained in [ Here the on the backgrounds such as section 3.2 of [ elliptic genus from 4dinvoked setting, and in one which can cases use the certain S-duality analogy with of a Spiridonov-Warnaar Lagrangian inverse theory formula is [ related to 6d minimal (1 RR elliptic genus is identified as the Adding the “tildes”, one can also obtain the equality with the free hypermultiplet coupled. JHEP12(2019)039 2 R , 1 j with ], we (5.4) which ]. See 44 i 10), and 73 f with (8) affine , SO(8) 2 so (5 theory with H and SU(2) theories, the C + g r 3 in [ ) k − ( G ]. In particular, construction and H = 71 S ]. Besides, the rank d 2 79 } , k – 6 i ] } f i k 75 4 { a k systems on cones over the 82 for rank one owing to the , { i i theory. 7 Y } 81 and βγ } denote the U(1) ,E 3 G R 2 6 k } t k E 3 { R { ) 34 k H H j k q and two theories are identified as (2 and 12 } { four 6 j , three 2 ]. E p r ) } punctures Λ ]. 2 k r ]. H ) ) Argyres-Douglas theory. See also the 6d 48 80 4 k ,  (2 72 { t (3 47 pq ,D and { 1  4 – 54 – denote the rotation generators in A D F 1 1 have the same type of matrix [ j − H 1) 1 N (6) affine Lie algebra at level − are the fugacities for the flavour generators − − e 1 1 1 1 = 2 SCFT is defined as [ A i z 2 − − − − j a k k k k N 0) 6 4 3 2 theory with regular punctures. For a genus , respectively in [ . 6d construction for rank = A A A A S and 4), the complex orthogonal Grassmannian OG 2 2 and ) = Tr ( , 2 34 − j − OP 6d (2 z 0. Superconformal index when taking the Macdonald limit is = p, q, t ( = 1 Lagrangian flow [ 4 8 7 6 d Table 12 and → 2 I G E E E D ]. This correspondence relies directly on the class k N 1 p ) are difficult to compute. For example, among all j theory coincides with ( 74 , 2 + A 73 theories are also connected with the curved 2 p, q, t j H 7 4d SCFTs are also known to be connected to 2d vertex operator algebra, i.e. ,E = 6 S ,E 12 4 j D (Macdonald) computable for all class H Certain limits of superconformal index are particularly interesting due to symmetry en- Class • hancement. The name of limitcorresponding comes from symmetric the polynomial observation that known thelist in resulting indices three mathematics involve of literature. them Following here: [ indices with ( full superconformal indices toarbitrary our rank knowledge owing are to onlyexistence of their computable certain so Lagrangian far nature for and where representing each SU(2) Lorentzgenerators symmetry, respectively. and Besides, sometimes are set to be zero for simplicity. For generic 4d SCFT, the full superconformal the complex Cayley plane 5.2 Hall-Littlewood andThe Schur superconformal indices index of 4d chiral algebras associated toLie rank algebra one at level some recent works trying to explainone VOA/SCFT correspondence [ complex Grassmannian Gr(2 chiral algebra [ can be understood fromsurface. certain This generalized relation sometimes TQFT givesby structure a realizing on new them approach the as to punctured the compute Riemann the vacuum indices character of of 4d associated SCFT chiral algebra. For example, the where the coefficients of the rank one construction involving irregular punctures in [ JHEP12(2019)039 . s ) is b 2+ (5.9) (5.5) (5.6) (5.7) (5.8) ; (5.12) (5.10) (5.11) − ) a g ). The 2 i q, t N ) | ) i q, t | , (Λ i k ) 2 − a j ) 1 ( ( j σ ( λ , σ x P 2 x 2 ) v − i , . . . , t s − a , ) 3 − ( ) )] , i i i v 2 − ) ( | v 2 i k Λ , − | ) theory is σ ( − i , ˆ σ K x N , t s ,q 1 +1 i ) x 2 1 − k = 0. For example, the Hall- a − (Λ j / 2 − 1 i =1 λ 2  1 k s i N g a v v t q − i k ivector space. With our normalization, the bilinear form satisfies The Cartan matrix is then defined by where Here phic to each other via the natural inner product the paper. Given alattices of simple importance, Lie the algebar rootP,P and coroot lattices mechanics, geometry and strings”). A Lie algebraic convention We collect some definitions in (affine) Lie algebras and fix our convention used throughout We thank Babak HaghighatChristopher for Beem, the Giulio Bonelli, earlyGuglielmo Lockhart, Michele participation Joonho Kim, Del of Seok Zotto, Kim, thisdro Hiraku Lothar Nakajima, project. Tanzini, Min-xin Leonardo G¨ottsche, Rastelli, Wenbin Huang, Alessan- Yan, Wesupported also Don by thank Zagier the and Fonds Rui-Dong National Zhu Suisse, for subsidiary useful 200020-175539 discussion. (project “Quantum JG is Acknowledgments JHEP12(2019)039 . τττ κ (A.8) (A.9) (A.12) (A.13) (A.14) (A.15) (A.16) (A.10) (A.11) , which δ in the affine . We add an ∨ I ˆ g ) b ω , which involves by respectively. The A.7 g . τττ ˆ g 0 κ I, δ . · δ rk . + , . rk and coweights JI ,..., ˆ I . A and the imaginary root ) = 1 b J ω ) = 2 0 θ b = 0 ω ∨ 0 b ω ,..., = 0 1 δ, , α , =0 , rk ∨ J ( X J ∨ 0 are related those in , ∨ a 0 α 0 b ω ˆ g = = 0 IJ , b ω ∨ i θ . ˆ ,I,J i ∨ I A a a − IJ ) = ( + = 0 =0 rk δ δ 0 + X J i ∨ i i , α ∨ i = = , α ω ω – 77 – ,I,J 0 = 0 i 0 i ), we find the affine version of ( I, , α α = = ∨ J ∨ J α δ is irrelevant for the blowup equations in any case. 0 ( IJ i · ∨ ˆ b i ω A , α b , α ω A.7 h δ b ω I I ∨ I , τττ a α b we do not expect all the triple intersection numbers to ω = + ∨ 0 κ h h i α ∨ J =0 rk 0 can determine all the triple intersection numbers in the and has a vanishing norm square, by X I X = b = ω b ω = ∗ i , IJ h i 0 IJ associated to a minimal 6d SCFT except for the number J 2.2 ˆ ˆ α A α b ω A h satisfying or , X I 0 =0 rk h X α J α h = I α is imaginary, and it satisfies rk the fundamental (co-)weights in by can be written in terms of the longest root are the marks and the comarks of the affine Lie algebra ∨ 0 ˆ g 0 b ω ∨ J α ,..., = , a . I 0 4 a b ω = 1 Similarly we can also define the fundamental weights Most of these definitions can be generalized to the affine Lie algebra i Given the non-compactness of be computable, and theNevertheless number we propose herea a reasonable local normalisation version schemesection of for mirror symmetry. We use this normalisation scheme in the example B Mirror symmetry forThe elliptic prescription non-compact in Calabi-Yau three-folds section non-compact Calabi-Yau Using these relations together with ( while Lie algebra For Note that annihilates anything in where affine Cartan matrix satisfies additional simple root The affine Cartan matrix is defined to be JHEP12(2019)039 . , : J ). n 45 up F b τ (1) i 0 n M 0 ˆ = 1, Π ( (B.2) (B.1) , which ω 2 1 1) , ω . Note 4 0 c I b − t ω 0) ( , B 0 23 R n (0 = , ω F × P 0 0 3 is different τ b ω . ∂ . = 4 30 0 0 > F · ω ω 43 χ n ) can also be ignored 2 = 7 in table = 2 , remain well-defined − G τ ) by one. Hence the I F = 3 0 L M ) = -cycles ω ( n n in table 2 1 c , S comp B ) R k X ( n ( l ) = 60. ) χ j ( n l M = SU(3) is special as the reduction ) ( i . ( n χ l of compact for the model with G 2 n ijk b κ ) = 0 = 3, 0) , is embedded, one can define for every toric ijk n X (0 2 the elliptic singularity is constant over the 0, the Picard-Fuchs (PF) operator associated from the toric charge of the elliptic fiber 1 3 X F τ > – 78 – τ → ∂ L n = 3 case, the resolved geometry can be described 0 elliptic fibre type ) = = 1 curve decreases de ω n which annihilates the homogeneous periods ˆ ( z 8 X i τ where n ( E ˆ 3 L L − c ˆ X X . The case of = Z 4 = 1 we normalise the Euler characteristic using the formula b 2 ] with n = C + 96 χ 2 b It annihilates all the finite homogeneous periods, but not the is contained in the homogeneous B-period 44 ). = 2 has more supersymmetry and odd Betti numbers τττ ) = 1 + n κ ,... X 1 ( χ = 2. In the case 3 a Picard-Fuchs operator ,..., b 0 ) does not become a constant but remains a non-trivial holomorphic function i , ( = 0 l 3 ω 2 , b 2 The Euler characteristics thus computed for the theories with a pure gauge bulk agree There are several ways to understand this normalisation scheme. Once all the triple The number For the compact Calabi-Yau . In the decompactification limit , curve in the base. Except for the For these elliptic Calabi-Yau threefolds decompactified inThe charge the 1 horizontal correspondsHere direction, to it the the is zero fundamental understood section of that the we elliptic omit fibration. the toric charge of the (0)-curve in the summation. If one wishes 6 = 43 44 45 n (2) i 1 − ˆ period ˆ the charge entry associated toas the it pullback is of only the nonzero base for curve the (denoted (0)-curve. by See for instance table for the compact threefoldThe [ effect of blowingcontribution of up the a non-compact geometry should be with the naive definition in terms of the numbers as configuration of Hirzebruch surfacesnumbers inside and the compact CY-3-foldfrom with the only even compact Betti geometryThe to geometry the of non-compact geometryb also involves a flop operation. we list the results of allthat the minimal the 6CFTs calculation except for offrom the the cases that of normalised for Euler the characteristic− first for three the cases. case For to a scaling factor, which may have some openintersection string numbers interpretation. are known, the normalised Euler characteristic can be computed by condition Note that the resulting homogeneous B-periodWhen is it not is a solution acted to upon the by complete PF the system. other PF operators it does not vanish but produces system, in the sensewe that define they in have addition( extra the independent PF solutions. operator other To superfluous cure solutions, thus this making problem, the PF system complete. corresponds to the zero section. We find that it is completely fixed by the normalisation charge Π to the (0)-curve inand the non-trivial, base and vanishes, theyof while the annihilate the resulting all local other the Calabi-Yau. operators finite These homogeneous operators, periods however, do not form a PF complete JHEP12(2019)039 (B.3) (B.4) (B.5) , ) 3 θ 2 − 8 , 2 E θ , 2 , , , (3) − + 5) − =0 7 2 l t ρ 1 1 | =0 E =0 0 θ θ ) ρ ρ | ω | 1 1 1 ρ ) ) ( (2) − − 6 ρ l ρ 1) ( 0 = ˆ ( ( ˆ , E ω 0 0 2 − ) ˆ ˆ ω 0 1 0 0 0 1 0 1 0 1 0 3 0 0 2 0 0 6 0 0 ˆ ω  + 1) (6 2 3 X (1) 2 θ θ − 4 l we can compute the genus 0 GW invari-  1  2 θ 3 F 2 3 − , ∂ρ 0) , ∂ ∂ 1 − 1 (6 3 t ∂ρ ∂ρ 1 1 (0 ∂ θ 1 2 0 1 1 2 z ∂ρ θ − − F ω )( (8) τ ∂ρ 3 12 − + – 79 – ∂ − = ˆ θ 1 1 − 3 2 2 1 SO − 1 θ − ∗ ∂ ( ˆ i ) 1 2 X − ∂ρ ∂ρ 3 3 ∂ ∂ 2 θ 2 z 1 0 0 1 2 ∂ρ θ 2 , − ∂ρ − − − 0 − − ) ω    46 1 1 6 6 8 9 10 1 2 4 5 6 8 12 2 2 − − 2 3 1 0 2 3 0 1 2 3 2 3 0 0 2 3 0 0 1 0 0 0 0 0 0 0 10 5 5 7 8 9 θ θ θ = = ˆ = = the compact genus zero free energy, we have the periods: ( − 2 3 0 4 5 − − 0 ) 60 0 12 12 16 18 20 z ˆ ˆ ˆ ˆ ˆ 1 3 F X X X X 0 2 1 0 4 2 θ θ X n S G − F F b b D ν ( ( K S ( D D D 2 1 2 3 χ 6 models. θ θ θ , 5 = = = , 2 1 2 3 , ˆ ˆ ˆ . Denote L L L i ∂ ∂z = 1 direction, which should be the same as the Euler characteristics. We checked i z n τ := . The normalised Euler characteristics of the non-compact elliptic Calabi-Yau threefolds i θ In the following, we illustrate this idea with two examples. For the remaining models the first non-vanishing invariants appear at very high degree and we fail to = 1. 46 where obtain them within a reasonable period of time. The Picard-Fuchs operators of the compact geometry are this for the n Table 23 associated to minimal 6d SCFTs. Furthermore, by integrating the B-period ants in the JHEP12(2019)039 , 3 10, x 3 z − 2 (B.6) (B.7) (B.8) (B.9) x 2 (B.10) (B.11) z 1 = x 1 z derivative 1 2 1 J t 2 c which satisfies +1) = 2 , R 0) x 0 τ , → (0 3 F z | 3 ∂ . There are one period +1)Γ( is a special period, we ∂τ ˆ 0 3 0 X , 4 x ) is the deformed funda- → ω ρ 3 X = z ( | = 0 0 ) + 5) ω 4 F ρ , +1)Γ( 1 ( 2 0 X 3 0 θ ∂ , which are also known as toric , the limit happens to take the x ω → ∂t ) , . i 3 0 0 + ( z = 60 as well as 1 l 2 | ω ω = 1 is kind of trivial. Let us now 2.2 J 2 x ) = ˆ 2 2 ˆ χ − ρ = X n + 1) (6 J ( vanishes while the other two become: = 2 1 -direction. If 0 = 4 3 θ +1) − =0 τ ω ˆ 1 ρ 2 L 2 | 1 X (6 . x ) +1)Γ( 2 J 2 1 3 ρ 2 θ Γ( L z ( x J = 5. Notice in the compact cases, we embed ) 0 2 2 ,X 12 ω n 0 θ – 80 – − − , 2 − 3  → 1 x − 3 2 ) J ) becomes z = 0 1 2 − | ρ − θ θ 1 1 ( 4 ∂ρ ( terms is because we do not know the ∂ x ˆ 0 1 ). X 2 − X = ω τ z 1 1 ∂ρ = θ L R + 2.11 ( 1 − +1)Γ( 2 1 2 1 X θ θ 2 1 x ∂ and fix the full triple intersection ring = = ∂ρ 4 1 2 2 1 ) and ( L L − +1)Γ( ,X 1 3 B.2  x without singularity, two periods with logarithmic singularities X indeed descends from a period of the compact geometry. In the non- = Γ( 0 4 0 3 → ≥ i 3 X X Z ρ z is precisely the PF operator in the | ∈ + 3 i 0 0. Then the Picard-Fuchs operator (0). The non-compact geometry is related to the compact geometry by setting 1 n ,n ω 0 X 12 using ( 2 = L The computation for the geometry with i ω ,n → x − = ˆ 1 3 n = ˆ 0 = z ω 0 1 ) = ω Note that The reason we cannot fix the J := ρ = 2 ( = 5. c 0 0 dc ˆ ω our elliptic Calabi-Yau 3-foldits into star a triangulation. toriccharges, Then variety are the described related Mori by to conepoint a the generators of star reflexive the triangulation polytope polytope directly.in is and In missing, principle leaving the have a de-compactification infinite non-reflexive limit, size. polytope. a The As dual depicted polytope in now section compact geometry it ismirror the Calabi-Yau. properly normalised integral over an non-compact cycle in the consider a more complicated model with can integrate both and thenR proceed to compute the Euler number of the free energy, however, there(up is to an all interesting the “period” orders we have checked) Note that and no solution with triple logarithmic singularities. The deformed fundamental period ˆ X one with double logarithmic singularities z mental period and ˆ and another period with triple logarithmic singularity. Here ˆ JHEP12(2019)039 ) 4 i ( F l (C.3) (C.1) (C.2) of the (B.12) ) i direction ( . l , τ . , ) (7) 6 l θ (9) (9) l l = − 3 + + 4 , θ , . Σ ) + (8) (7) (7) l l l 5 , 2 (7) θ l θ , we say the associated = = 2 (6) + l + 2 2 2 6 4 − + 2 θ 1 Σ Σ = (6) 2 θ l 3 (2) l − , , Σ + 3 1)( . ) θ + (6) , , , l 6 − (1) ) θ 6 l 7 in table (1) 6 (9) (1) ). There are five B-periods solved = . The charges of the new basis θ l , l +5) − l θ i 1)(2 , 6 5 3 4 z − , − = 2 = θ θ − = 3 Σ 4 3 + 2 5 θ , 1 4 B.12 annihilated by all the operators except 2 + , θ θ , , ) +1) 3 3 Σ Σ + Σ 5 + (7) θ , 2 + ) 0) 2 , , l θ 2 θ (8) 6 2 2 4 θ , l 2 θ +3)(2 θ , (0 θ 2 (6) − − 2 3 2 so that the solutions do not change, and the l F − − (7) + 2 + 1 2 θ − l i 4 − θ 4 θ − ∂ 1 = 1 + ∂τ z θ 1 , θ 3 θ – 81 – (6) (7) 0 θ θ ( l l , i + + 2 )(2 1)( (6) (3) 2 ) X 1)( 3 )( 2 l l (2 i z + + ( 5 − θ θ +1)(2 4 l (6) − θ 6 2 3 − = = z l 5 − θ θ ) (2) (1) + θ 0 3 − 1 l l 6 − + 4 − ]. θ 2 2 (2 ) θ Σ Σ θ 4 8 θ 6 2 = = 5 , − (2 . The Euler number can be predicted from the 2 θ − θ (2) z 4 0 )( 1 4 l 1 4 4 4 − + θ 6 (6) − z θ ω z Σ Σ l ( 3 3 θ 4 = 3 3 θ θ 5 − + θ ∼ z − + ) 1 z 2 3 , 2 2 3 6 + ) θ , Σ z − − θ θ , 3 2 (8) (1) 1 0) ) given in [ 2 1)(2 θ l , l , which correspond exactly to nodes in the Dynkin diagram. Then z + 5 (4) , − θ − 2 (4) l (0 θ 8 2 3 − l ˆ 2 2 + X θ θ 5 − (3) F θ + − l = = 2 2 θ 2 = is the direction of decompactification. 4 5 + ∂ )( θ (7) b ∂τ 2 θ θ − 2 0 l + +1)( = 2 0 Σ θ 2 1 4 Σ 3 )( θ Σ (5) 0 ω θ 2 1) θ θ 5 is the direction of decompactifiction. l is the direction of decompactification. − ( ( − 2 θ + 2 Σ b , 3 − − (2 6 1 2 θ 1 5 − z θ L 3 (3) to 0. This is equivalent to deleting one Mori generator, and keeping the others. (5) , l θ θ (5) (6) 3 − l z l l ( (2 ( − (2 θ 4 2 (4) dc 2 1 5 6 1 θ l . z = = θ θ θ − θ . − 6 4 b 3 ======(2 = ( = , with E F Σ Σ b b b 4 1 2 3 0 τ For now, we try to triangulate the non-reflexive polytope, it has 16 star triangulations. L L Σ L L L L L L associated to the minimal 6d SCFTs in terms of the Mori cone generators = = = SO(8). G G We express here the Mori coneX generators of the elliptic non-compactcompact Calabi-Yau threefolds Calabi-Yau G C Geometric data from these operators, and anfor extra one genus zero invariant as 12. curves form the toric basisFuchs of equations, compact and curves. then One try to canvariables find in of solutions. principle complex write structure For down this parameters mirror the model, maps Picard- it have is expansions possible withcan to positive change be the powers found of inthe table complete Picard-Fuchs operators are given in ( From the polytope pointtriangulation on of the view, remaining we part.symmetry delete for We may a a assume compact sub-polytope that hyper-surface from the embedded standard in it, method a and of compactFor keep one mirror toric of variety the them, still same the holds. toric charges are variable JHEP12(2019)039 .
, , 1 − 2 (C.4) (C.5) (D.7) (D.2) (D.3) (D.4) (D.5) (D.6) (D.1) (10) q (11) l 1 l − 1 + +2 (7) Qq l , . − ) ) + (10) 2 2 l 1 z z (1) + ) l − − 2 q (2) , = 1 1 l 1 3  + , Σ τ, z τ, z z n τ Qq a (1) Q Q (2 (2 − l 0, the total contribution , 2+ ] ] / 2 2 (1 = − a a ≥ (9) Q l 2 2 +1 − − 4  1 , 1 1 k k z 2 . + ) Σ a a /  . [ [ 1+ i i Q 1 . )
θ θ 2 , (8) (9) − 2 , 1 ) ) / l l , a τ, z q a 2 2 2 − ( ) τ, z (11) z z + + n τ 1 a l − 2+ (7) ( k z +2 θ l / 1 2 Q + + 1 / (8) θ 2+ Q z l 1 = 2 +2 / − z z 1 1 +1 (7) 2 , l q k z / Q 3 ) , = 1 for all Q Q 2 +1 ) + +2 a 2 2 Σ Q k ) τ, z τ, z (10) − a / / 2  ( τ l + 1 1 ,k / τ, z 2 + k z 1 (2) (7) (2 (2 2 Q ( / ] ] + k l l − τ − τ k , (0 ) ( τ 1 1 a Q 2 2 a θ Q Q a a 2 − N = = 1 + (6) (1) Q / 2 2 =1 + + k ∞ l l q a 2 7 2+ − − − 2 Y – 82 – 1 1 n ) / a a + − Σ Σ 1) ) = = a [ [ i i k 2 τ θ θ 2 8 +1 + − / ( ) = ) = k k ( 1) , , 3 4 1 1 Σ Σ ( τ ( τ , , η τ τ Z q − 1 2 ∈ Q Q (6) (9)  ( =1 =2 X X + 1) = l l − + k X , i i − z Z Z Z i , = ∈ ∈ ∈ Q = = k k k X X X − (5) (11) 1 6 1 τ, z  τ, z τ, z 12 l l ) = ) = τ ( ( ( 2 Σ Σ 2 2 can be defined as triple products 1 / 1 1 Q = = 1 i θ ) = ) = ) = ) = θ θ 1 1 7
, − 2 τ, z τ, z θ 2 q ( ( Σ Σ / ] ] τ, z τ, z τ, z τ, z ) = 1 2 2 ( ( ( ( − (10) ) ] ] ] ] a a l 2 [ [ , 4 3 2 a a a a / [ [ [ [ 2 3 4 1 , θ θ q + τ, z 1 , 2 θ θ θ θ ) ) 1 ( q 1 1 q (4) (10) 1 l l (8) (5) ( θ l l  = = τ, z τ, z 2 = = ( ( is the direction fo decompactification. is the direction of decompactification. 0 6 / ] ] +1 0 5 1 1 1 k Σ Σ a a 2 − Σ Σ 1 [ [ 4 3 (3) (4) q l l χ θ θ , , k , , − Q 2 . . (9) (8) / (3) (7) l l 8 7 l l 1 1 q E E = = = =  For a cluster of refined BPS invariants Jacobi theta function b 5 b 4 = = Σ Σ Σ Σ =0 ∞ k X to BPS partition function is which satisfies the quasi-periodicity which satisfy the well-known addition formulas D Useful identities Jacobi theta functions with characteristics are defined as G G JHEP12(2019)039 . . = )
2 1  2) − 2 / (D.8) (D.9) q 1 (D.15) (D.11) (D.12) (D.14) (D.10) (D.13) +1) 1 − n − 1 ,k +( k (0 1 Qq , 0, the total  ) η N − 2  ≥ 1 2 . +1) / , q 1. In such case, ) . 1 k 1 ) m
q − − 2 2 1 Q  ( q q +( 2 ) − 2

 1 / z r = 1 q 2 ( 1 τ − ,j / 1 1 − 2 l − 1 θ 2) Qq 2 j . Q − 1 q ) ) ( / . / 2 1 1 2 2 1  1 f 2 − ) , − − q q − 1 2 1 for all 0 q Qq τ 1 0 1 / R q (0 1 ≥ (1 ( − q 1 − q ≤ ) − η ( N  Y − 2 + n   R 2 1 2 2 q 1  2 m,n = q +( + 2 / / k ) − z , z τ τ, z 2 1 q m 1 2 ) 2 − ( / ) 1 2) ) − Q 2 n τ Q 1 Q − 1 χ 1 2 R / q 2 2 2 q ) ) , q q ) q θ k q 1 / / 12 2 2 τ 2 ( 2 1 1 n 1  2 +1 − q Q r − − q QQ + − Q 1 j 2 1 ,k ) /q , q i 1 / 2 q Qq 1 1 2 χ k +1) (0 1 2 η / ( − ) +1 q q   n 1 q 1 = m Q ( ( n 2 − N L 1 2 q ) ) q + q / r  +(  R + ( 1 z  1 (1 ,j +1 l (  =0  l − 2 ∞ − j 1 1 , j Y m 1 τ n  / Q θ q ( 2 =1 χ q ∞ 1 2 / f k X z 1 Q / +1) 1 − = 1 − , 1 1 − Q 0 q ( m + + − 2 R − – 83 –  2 (0 ≥ η (1 + / q − ≤ τ 1 R 1 1 +( 1  12  Y 2 n Bl q z 2 q / ) can be written as 2 / Q ( − m,n ) + 1 τ / / 1 1  1  1 1 Q 1 2 + Q θ m q i − z / τ − 2 − z  1 /q 2 2 q ) 1 Q Q D.12 2 q − 2 Q ) = 0 +1)  = q ) − 2 R , q ≥ R 1 2 + ( ≤ 1 2    q , q Y z n / 6 q 2 2 R PE , q 1 1 ( 2 / m,n / + 1 q ) Q 1) 1 1 q q ( r + ( m 
− − − ) ( 1 2 ,j 2 2 1 2 ) r  q l q R / / /   j ,j ( 1 ( 1 1 ,R l 2 ) − − j ) ) − 2 f / − 2 ( 2 ) 2 / 2 ) PE 1 2 / f n 1 2 2 q q n / , 1 − 2 1 q q 1 1 1 1 q + q 1 (0 q q ) = m 1 q 1 q ( ( q 2 (  − z  Bl 2  m , q Q 2 2 2, the expression ( R  ( 1 / + / +2
0 are combined with a “zero” degree invariants = η 1 q 1 z 1 k − 2 ≥ ( ( q 2 / 12 − 1 ) 1 ≥  1 /  θ q χ 1 τ R − z ,R k k r Q − i Q ,j Q PE 2 1 l / 12 j 1 − 1 ( / 0 q 1 τ R ≥  In the computation of vector multiplets, we often encounter the following expressions: We also often encounter the case where a cluster of refined BPS invariants ≤ Bl Q i Y n m,n + 1 for all =0 ∞ k = X m Supposing Here and In counting the totaloften index encounter quadratic the form following of expression: the contribution from vector multiplets, we and − the total contribution is contribution to BPS partition function is Similiarly, for a cluster of refined BPS invariants JHEP12(2019)039 (E.2) (E.3) (E.4) (E.5) (E.6) (E.1) are (D.16) -gauge α , c ) 2 2 k G  47 ]. + 2 25  , 1 )  α + m 2 1 functions in the above  , + ( ) η − G,k red G,k ,  ˜ ˜ , ) D D + 1) 4 ` τ, m ) ) ) ) and ( 2 2 2 R α α ,F ( 1 ,α i 2 i has actually fewer poles and as − θ c ( ( , m , m 12 R α k 2 2 1 2 1 1 2 with a fixed norm square in the can be computed as , , 1) + 1) E ,  ,  1 1 ˘ k/c 1 θ − 1 ∨ ]). It can be written as SU(2) − − b − i N α ϕ ϕ D τ,  R τ,  59 ) ) ( ( ( s : 1 1 + ( τ ∆ i i − G,k G,k ∈ + SO(2 Y ( ( Q )  D α N , 1 2 ), i.e. those 1 2 2 ) , , )  1 1 N , − − + + 1) ) = ] claims that 2 ϕ ϕ D.12 – 84 – τ, m 1 α i (  G 97 k ( k (( =1 =1 i Y i Y z , m 1 2 = Sp( = , 2 ∨ G ∨ G 1 -string elliptic genus satifies the following ansatz SU(2) k,α G G ,  − k kh kh + 1) 1 D 4 4 ϕ . See more details in the appendix A in [ ) l ) R 1 Z ( τ τ ∆ τ,  =0 − ( ( 3 i Y ( ` R ∈ Y ∈ ] the η η ) α k 1) ( G k =1 = 2 if = 3 if R = = h i e Y = − 37 , α α E ∆ R c c ( ∈ 24 red G,k G,k . The index quadratic form of Y red G,k α ˘ θ ˜ D D − D = 2 z 2 2 G,k R ˜ D − ] and the correct leading order of are the set of long roots and short roots respectively, the constants s 43 ) = ∆ , l z, R ( ˘ θ It was proposed in [ Here “red” means “reduced”, i.e. the number of poles reduces. The “tilde” and “reduced” in this section 47 Ind should not be confuseddecoupled. with them in the main text where them mean a free hypermultiplet is coupled or with the gauge group related factor where ∆ multiplying over the set ofa roots. consequence Later the [ denominator is smaller (see also [ with the gauge group related factor thermore the denominator hasform the which following reproduces unique the structureinstantons poles [ as a of Weyl the invariant Hilbert Jacobi series of the moduli space of recursion formula. where both the numerator and the denominator are Weyl invariant Jacobi forms. Fur- E Relation with modularIn ansatz this appendix, wethe show how elliptic the genus modular emergesof ansatz, from the or its our modular denominator exact ansatz to formulas. comes be from specific, Simply suming for speaking, over the all denominator expression together as which actually holds for all We normally denote the modular part of ( JHEP12(2019)039 ) ) ), E.5 3.32 D.2 (E.7) (E.8) (E.9) (E.15) (E.12) (E.13) (E.14) (E.10) (E.11) = 5 with , . ) and ( n ) ) ) = 0 of order , α α α ) ) E.4 . Using ( 2 , m m m  α α ! + + m m + − ) < n ) 2 2 2 2 2 1 . According to the | + +       n 2 2 2 ij 2 /   s | + 2 + 2 + 3 − . ij − s 1 1 ) 1 1 1 < + 2 + 2     α  ( ( ( 1 1 = 1 2 1 2 2) 1 2   m 2) , , , ( ( . j 1 1 1 − 1 2 1 2 + − a ∨ . − − − , , 1 1 2 α n ) ϕ ϕ n ϕ ( − − − ) ) ( ) α A b α α i α ϕ ϕ } m 0 ) ) a , ) are consistent with ( + nτ, m m m 0 1 nτ, α α , ( 1 + contain the zero ( ∨ ] 1 ] D α + + 1 { m m + 2 2 1 a a  a 3.32 [ 2 2 3 2 D ( [ 3 + +    θ D θ 1 2 ∨ = 1 where the recursion formulas ( , 2 2 , we spell out explicitly the components of α 1 + + + 3 |   ) ) k + 2 2 2 − 1 1 1 are intergers and 0 1) G,k   + +    and 1 ϕ (  − 1 1 D 0 k 2) 2) ij – 85 – (2 (2 0 1 2 (   , , s ≤ (2 1 2 1 2 0 ∨ 1 , , , − − (2 (2 1 2 Y =2 α 1 1 1 − , a,b> a,b 2 1 2 1 2 . To be concrete, we take the model of n n 1 − − , , || ϕ D k 1 1 , − ) ∨ X ϕ × ϕ × − − ) ) ) ) ) α α ϕ + ( + ( ) = || , ) α α α α α ϕ ϕ 1 1 m ) ) ) α   m m m m m = α α α 2 2 + m τ, m 1 + + + + + m m m ( 2 − − E +  2 2 2 2 2 + + +      2 +  2 2 2 nτ, nτ, + + + + + ( (    SU(2) k,α 1 ] ] +  1 1 1 1 1 1 2 D ). + + +      a a 1 [ [ 3 3 ( (3 ( ( ( ( , we know that all  1 1 1 θ θ 3 1 2 1 2 1 2 1 2 1 2 1 2    , , , , , , , E.3 ( ( ( (3 2 1 1 1 1 1 1 , which differ from each other by

, 2 1 2 1 2 1 2 1 is actually smaller than 3 1 ). For example, the minor − − − − − − , , , , , | ), ( a 1 1 1 1 2 ϕ ϕ ϕ ϕ ϕ ϕ , ij l l l s s − − − − 1 s red G,k ∆ ∆ ∆ a ∆ ∆ ϕ ϕ ϕ ϕ E.2 = det ∈ ∈ ∈ ∈ ∈ D Y Y Y Y Y ∆ ∆ ∆ − | α α α for some small values of α α 2 ∈ ∈ ∈ , Y Y Y α α α 1 = = = , n | ∆ red G,k . On the one hand, = = = ij 4 ˜ red 1 D 3 1 2 red red 3 2 s F , | ˜ ˜ ˜ ˜ ˜ ˜ Let us first take a look at the case of D D D D D D = fields with G,k ˜ requirement for it is not difficultmin( to show that both We consider the polesr contributed by each component. Suppose we choose three unity We will demonstrate that our recursionrather formulas than ( ( simply read On the other hand, To see that D G and that JHEP12(2019)039 ), = − α 2 z m 3.32 (E.19) (E.20) (E.16) (E.17) (E.18) ( − . Thus 1 1 1 = 0 for θ have the z 2 1 ) by point- /D  0 , D 0 ∨ , − E.4 α 1 naively contain , 1 ,     ) D k and ]. The remaining ) ) ) β )) 3 3 3 E 0 2 − m , η z z z 59  ( 0 ∨ n n n , = 5 model. 1 α α 1 θ − n τ, τ, τ, m ) for D 1 n n n  =1 ( ( ( ∆ ] ] ] ( ∨ 1 2 3 ). Now by the universality ∈ | Y α 3.32 a a a . The components β · [ [ [ i i i ij β ∨ θ θ θ in recursion formulas. Let us s ) ), which can either be seen in α nτ, n ) ) ) 2 ∆ and 3 . Both , 2 2 2  −| 2 n (2 z z z , ] , / − ∈ 1 n n n 2 ) E.14 1 a , n . a 2 |  τ, τ, τ,  β ) 2 − ij = 1 n n n 2 − 4 always has simple zeros at  ( ( ( − s 1 ] ] ] , | 1 α a 1 2 3 α − a [ i a a a m [ [ [ i i i , which are consistent with the slimmer θ 2 m α ) ( l  θ θ θ = 3 2 − 1 )( 1 m  θ − ∆ i 1  ) ) ) of course. ( ( 2 )  ) 1 1 1 1 1 2 a = 0 for 2 τ θ θ  )) z z z ∈  ) ) − has zeros/poles 1 is n n n 1 1 − =  β α   − – 86 – 1 α 1 ( α τ, τ, τ, ) are not present in the recursion formulas ( 4 1 2 m 1 m − −  E η n n n θ a ( m )( ( ( ( /D 2 α ( ] ] ] 2 0  1 E.2 48 1 2 3  , − m ( θ 0 a a a ∨ ( ) , [ [ [ i i i 3 − 1 α 1 1 θ θ θ ) for ). a θ  = 0 of order min( 1  D = 0 for    ( 2 − ). In the following, we will argue in favor of ( nτ, n  E.8 2 α E.14  (2 − ] m E.2 ( 1 2 − 1  a θ 1 ) chosen as above and 2 ) = det −  α in the root associated to the coroot 3 3 1 or be argued formally from the Hilbert series of gauge instanton , , we can choose arbitrary three m a 2 2 − ( , z , [ i || 1 1 2 ) from ( θ 49 α θ ∨ a 3.2 3 = 0. In fact, more is true, the determinant , , z α m 1 E.4 2 = 0. Therefore || z =2 X  ] as well as from W-algebra via AGT correspondence [ ) rather than ( i / ) = ( 1 2 z − 43 D · = should contribute poles ) cancel with corresponding components in the numerator of my 1 . The former poles, however, get canceled with some factors in the numerator ∨ − 2 2 (  E.11 , 1  α 3 ∨ ∨ 1 E z α α − ∆ = A A α = to α l 3 m ( z ∆ The discussion above indicates that the recursion formulas ( The other component in ( 1 There are also polesThis at cancellation positions can shifted be by checked 1 in or all the minimal models, not only in the θ − ∈ ) 48 49 2 1 Nevertheless, all true poles should alreadyus be to visible distinguish in ( the recursioning formulas, out which that allows extra poles indicated in ( moduli space [ genuine poles are expression ( both genuine poles and spurious poles, while the cancellation of the latter is not obvious. where  naively α after performing the summationexplicit over short calculations, coroots in ( which can be boosted to the modular object with characteristics z choose three successive onessimple with zero which clearly contain zeros argument in section can be rewritten as JHEP12(2019)039 ) E.4 power -string k v ) or ( l l l E.2 ∆ ∆ ∆ . structure similar ) i d ∈ ∈ ∈ m /D is enough high, we } , α , α , α 2 2 2 2 n ,d    1 1) (E.21) poles, the poles which 2 2 2 , ,d v, x, Q 0 ( 1 ∨ − − − d ) , l l α { n 1 1 1 ( k,G    ∆ ∆ D (0 g , ∈ ∈ − − − n τ 1) α α α , relevant Q 0 there are five sectors on the right , α , , α , ). Usually, the leading 2 2 2 =0 , m , m , m i ∞ 3    X n 2 2 2 (2 m E  2  2 3  6 / goes up. When / − − − − − − 0) 1) n , 1 1 1 1 1 1 can be written as − = 5 models, we find the following relevant 1       l ∨ G , 2 2 2 τ v, x, Q n ( − − − ∆ ∆; ∆; ∆ kh (2 Q ) − − − ( , α α α n ∈ ∈ ∈ ∈ order − τ ( k,G α α α 0) g τ Q – 87 – , m , m , m , 1 has three sectors with 2 2 2 , α , m , α , m , α , m , α 0 Q    − 2 2 2 2 2 2 2 , 2        ∨ G ). Likewise, for E − − − kh − − − − − − − 1 1 1 v    1 1 1 1 1 1 1 ).    2   2 3   ) = (4 E.12 = 0, which are guaranteed by the 50 2 ) for 2 − − − − − − − − − − ). ) =  E.4 i , d α α α α α α α α α α 1 = m 3.32 m m − m m m m m m m m -expansion of 1 , d E.13  v 2 ,Q ) or ( τ || ) relevant poles ) relevant poles order results for the reduced elliptic genus here. Recall the ∨ 2 2 τ α E.2 , d , d || Q 1 1 ( v, x, Q , d , d ( 2 2 ) ), and we find the relevant poles as follows k || || ( G (4,0,0) (0,1,1) (6,0,0) (2,1,1) (0,2,1) ∨ ∨ h α α E 3.32 || || (2,1,0)/(2,0,1) (4,1,0)/(4,0,1) (2,2,0)/(2,0,2) ( ( from the recursion formulas, and consider only 3 E , 2 The right hand side of ( We also suppress the poles E 50 respectively. With an explicit calculation inpoles the in each of the three sectors cancel automatically. With thisof comment in mind,appear let also us in now either ( look at the denominators and assume along the way that the spurious poles not consistent with either ( They are clearly consistent withhand ( sie of ( We are interestedbecomes in more the and more negative when to the previous discussion. We record some high elliptic genus when expanded with respect to which are consistent with ( F Elliptic genera JHEP12(2019)039 8 25 v v 22 19 v 34 v v 10 9 12 v v 28 v 17 v +98441 (F.3) (F.2) (F.1) v 10 7 15 .... 18 , 131 v v v +9777494
v + − 34874 +198723 +5587892 2 , 24 48 +17019 increases.
− 39 v 11 v v 9 18 21 8 v v
v . n 279728 v v 3785597548 36 + v +5014
11102 v 170936 − − 9 . ··· ... 823 84 − + v . − ) 16 33 v +718317837 6 +1510639
+ − v 17 + v v .... v ··· ( 2, + +2582 27 v 26644 14 ) 24 10 2 8 v + v ) 4 v +1 v +4) +7044615 − ··· v +136089 ≥ 69861741 n ,A n 7 n − 19 +4316 ( 23 2 ) + v 17 8 v − χ i n 6( v 347 v v P θ 222250 43 v +62988 + 20 116300 5803 − v n +23808502207 4) v − 9 15 × − − 5 3( − v 38 v 2 7 10671 4250592736 5 v n 16 v − v ) +5994478 A ( +1930514 − v +27125 − +1716 2 v 6 χ +75426 +940214412 7 13 ( v 23 18 v 32 v v +220 v +4189517 16 26 v v 6735 + 8 v v − 22 89646012 v 206886 v − 419 v v − O +241846 6 118109 − −
− v
4 14 +2483 − 6 19 v , v 5 +35302136340 v 2) v (1 + +428 ...
15 +28966 v = SU(3) model, recall 7 6 − 42 v +27073 3110 +6633691 ) 54 +1816145 v + n 1 17 v +16075828305 963 ( v v 15 v − G n 12 22 3545719630 − 37 v − 5 + v v i χ +1729760 168868 , v 5 v +868425362 − θ +7553
68637 χ v +256 21 ··· − + palindrome up to +260250 (1 + 4 25 31 θ 3) 6 v 77137569 3 42 − v 2128 v v + v 13 10 v v − 2 399 − ) 2)) 14 v +27125 − n +577 28 + v v A ( − − 4 18 v 16 14 n 4 v χ v +41 − · · · − – 88 – v v ··· ) becomes more and more negative as v +5410138 5 +1497688 2( +5510 4 v v + +30457 3 (1 120018 +36965592032 11 ( − 21 − -expansion behavior. For two-strings elliptic genus, i 11462 v 2 v 1) 20 22 v ( 33 1 − ) 41 v +8659446500 v v − +185964 2 12583 v +138 +1695 n χ − ,A − 2275980208 ( v − 36 3 3 2 +641219584 n 4 13 12 − +21826 v v v − 5 v 52091127 v v P χ n 24 +11727151 13 15 − ( θ +1432 30 v − ) 10 v v 2 27 i v = 1) = in 2) 815649 72335 v 17 v − − + i +198 +880 − v +3612563 v 3 +244336 +1054375 − 2 n − 2 2 n v m v A ( 4 10 v v 810 3( 21 20 19 +25783 +90316 10701 v 5 v v v − χ − − +9 12 11 ,Q − +14419 n 2, − v v 2 − v v 3 2), +50 v +35302136340 v 12 14 − − 31129 v +2569710910 ≥ 951170969 v v 40 874 +377669387 = 1 v − 27314880 − +6 35 ) = v 28 223 +12428414 − n 963287 − i 23 v n v − − − 26 29 v ) = +260628 +605425 − m 4110 − i +1959712 16 v v +38096 +16620 2( order we have reached, this is true. It is nice to see how to explain this ) = v, x 6561 16 81 1 256 9 20 19 18 v − +7721 m ( v 11 10 v v v v τ 2 ( − − − − − ≥ 11 13 v v ) 2 Q v v i ) , we also observe the following general expansion: for ,A n ( 2 13 10 34 16 19 n ,A x v,x,Q ( 2 g − − − − − ( v v v v v 2 P 9064153 1612094548 23600 692866 v, x, Q ) ( ,A ... n +11727151 +773013 − +154737705 +93293293 − +30625221095 +2174 +244336 +31735 +231810 − − +2652 ( 2 ) = ) = ) = ) = ) = (2) g v v v v v i,G ( ( ( ( g )( 2 2 2 2 For the reduced two-string elliptic genus for +4 v ( ,A ,A ,A ,A n (4) (5) (2) (3) 2 2 2 2 2 3 . P P P P v ,A 2 (10) 2 With generic Note that the leading power of In fact, we notice for P A We have we observe for As high as the phenomenon. observe some patterns for the leading JHEP12(2019)039 , ) 74 . v 41 ) 31 v + 32 v 90 23 v 9 17 v ··· v v v , + 37 35 + ) 24 (F.5) (F.4) v v v 12 18 38 10 ··· 82 26 9 v v v v v v + v 19 29 25 + v v 16 27 46 136950773 v 20 +274511 v 69062433 v v 11 v ··· − 8 +2189163 . v 33 23440 +222042686 − v +4488 . ) + +25397471432 30 700938 v 16 − ) v 22 31 v 11 +38030 42 40 v − 9 ( v 1857927117 v v +19133352 v 8 v v v 2 ( +4501676 v 17 ) +569846 +3111882801 − 4 23 1280918 v n +6130 +635332178 ,A ) 25 v +3310035 ( 18 36 34 3 +417362 n ,D v +89188 − 10 37 v v v ( +622477442 2 24 7 P +12524957 v 19 v 12817 15 +4336 v v P 28 v 26 72005488 v − × v +60785174 10 22497646 v +44712 +1862626 +48157593258 8 448583 . − × v 703145 7 6 v 32 − ) 15 45 − v 29 v − +126228382 )) 2 v v +2718 19288 11 21 v v 16 v ) 9 +290609 +5892000612 22 30 v +4318298 − ( 2 v v +16332876522 1500 v v 6 + 17 v 41 3347473 +337366 24 +1740 1140297436 +6254586 O v 39 v v − v 9 v − 18 + v 7 +661625872 +2019390168 − 23 v +6368124 +21793 v v + v 14 v +880 6 36 35 33 +339731228 25 v +927411 θ 8 v 112582 v v v v 183 (1 + 46930 v 27 +7529926 +58351834 3) 14 − 5 v +9184210 − v 10030208 − ) (1 + − 28 31 8 15 2 5 +133960 +2876 n 20 v v − +57171170 v v v v ( +50110832268 12 6 1 v +3848955 16 ) +1292 +267 +199630 v 21 29 χ 44 2963132 v v 5 7 23 v v +6461123 622 1 17 v v v +6133530828 2) v − v +8636915664 (1 + − 22 241280313 +2662413 − 40 31191 7 13 v +101910 n 38 v +635332178 − (1 + 10 v 24 v − +84 v +2242 ) 5771 +1103016208 +91413273 13 v 2( +103012 4 6 22 35 32 5 v v +64404 ) − v v 34 26 v v χ v +63444248 +51803451 367 14 v 4 ( v v +22001562 15 10727974 v +77007 – 89 – v 30 27 5 − v − 19 v v 6 − +3145582 +16177917 (1 + − 16 +74 v v 5 v 1614464 n +487 10596 (1 20 22 28 16 +4741886 4 309898 v 4 4518 ) v v v − +839446 48 − +48157593258 v v − 21 − − 3 − 12 23 v v 3 43 v +5892000612 − 5 12 v 65395112 v +150494 +35306 +30 v v +2994640544 426 v v +4576 4 39 − +463211226 +563129124 13 14 (1 v − 37 v +438669558 15 v v ) = 3 25 31 34 v +42159674 +83251991 = 1) = 2195270 v 6887108 +766 33 v 1295 v v v +44 +20686636 2 i 29 26 v 4 − − +2376984 +11 374722 − v 18 v v v v, x 3 341004 m , 2 27 19 21 339 v − +2411835 +265962 ( ) v , v v v v − 2 ) 4 17424 − 11 20 22 66 +97493 +24774 ,Q = 1) = 2 +12 11 v v v 52 − v − v 3 12 13 v i v 2 (3) n,A +3044 + v 14 v v 398271211 v g +276 v m +42690290417 84 + 9 124631077 v = 1 5 v +5220665928 − ... − 42 − +43752233 − +813952151 +457167512 − 5447913 2210183 ··· 38 36 2 v v ,Q + +31405824 +72876216 v 32 24 30 33 v v v +12688484 − − +1628376 + 2 192767 v, x 25 34 28 v v v v +288140 +512476 +255946 12504 ( 17 26 18 20 27 v v v 10 − +25630 − +17594 = 1 4 10 19 21 v v v v v − 1 − ) 11 10 12 v v v − ,D 13 n v v v (512+1690 (32+40 (162+412 ( (2 ( 2 v g v, x 29 21 25 10 17 ( − − − − − 2 v v v v v ) 1829759686 113553734 38688 124639134 16831 495362 1672684 1116 3, we notice the following universal leading behavior ,A n +828007584 − +34642293136 +419176 +4247141 − − +1001340 +46719280 − − − − +337620393 − +382080 +21034734 +58351834 +11512 +1020364 +4318298 +4234930428 ( 3 ) = ) = ) = ) = ) = For the reduced two-string elliptic genus of SO(8) 6d SCFT, recall ≥ g v v v v v ( ( ( ( ( For the reduced elliptic genus of three-strings, recall n 2 2 2 2 2 . 4 ,A ,A ,A ,A ,A (6) (5) (4) (3) (2) 3 3 3 3 3 D For P P P P We obtain P JHEP12(2019)039 42 v 16 18 37 v 17 27 v v 11 8 v v , v v 28 ) 38 (F.6) (F.7) 23 v 15 18 v 8 21 24 v 54 v v v v v v 33 9 v 65210 + 1974439 v 32 34 − +7677132 v v − 1467224 22 ··· 10 24 v 10 15 v 17 − 118023 +39847612 + . 1483241349 v v v v 9612992092064 7 − 19726713 ) 16 28 29 59506626 v − 5445277331 − 7 v 3786292890317 v v v − 415787266 +57106 − v 26 ( − 41 1107115384 8 − 526950799964 v 4 − +1885916954 14 17 ) v v 13965 27 − v 36 v +12149 n − ,F 23 22 v ( − v 9 2 20 3067485 188200853472 v v 9 37 v +15592 55230 . v 79826800694 P +8955862 v 228511951 v − − 6 ) 144463390006 +211444789 − − v v − 14 16 32 6 × − 21 ( v +22410 33 v v v +14488449 23 v 7 3305 31 v O v 13251946148 17 32918254 +1877 v 15 31673046 v +3566407238 ) − 8 v − 2 − 8 − v 27 +3046056345 v 2306012154 369460130 , v 384 28 v 13 ) + ) 16 661966959 25 +410682 9274316155458 v − − + v v − x +12844 1770622 5 v 72 − +1274332193 6 ( − 5 v 1930 26 22 2450540661913 v v +2514527 v 2 − 1762 v 19 21 v 506517773112 v 40 74227039 − − + v v − 13 − 15 v 7 − n +114419603 − v 7 35 v v +4473554 (1 + 116221363135 65951082224 ··· v 36 v 5794 χ 20 22 59027415291 6d SCFT, recall 9090084 14 v θ − − + 22 v v − 1 15280924 v − ) − 4 5 4) 861 +16874 31 32 37 1411 +159328 v 7028420238 − v 4 30 F v v v +3718703248 4 − 12 4 − − v 364906 v − v v n 15 303095099 6 6 1085075 26 D ( +4109758618 1148224939 107579194 v v − v 27 v − − χ (1 + 3786 24 v − − +782158341 14 3166625 11 21 v 34 12 − − v 10767498 184 +911821 25 18 20 ) v − 31436 v − 4 +2027 8322655072044 v v v v − − 13 5 v , 3 – 90 – − 19 5 ) v v − v 3 21 v 449506711376 1433909622002 − v ) = 6645978 v +8314028 v 50 i 39 − − 64179900027 50298462814 89424037262 v − (1 1338 v 11 m 28634625983 747606 35 34 − − − 2495 + v 14 +29 +226932 − +329 82859 v v − − 4 3376185023 v 3 4 − 31 36 30 +3566407238 40296 13 229273231 ··· v − v v 2 29 v v v − 11 542911427 v v 25 v − +2672286699 − + v 12 58843606 . 26 v 2 +183703487 − +434067674 ) v , 281 v, x, Q 23 20 26 v v − ) +88 = 1) = ( 831 17 24 19 v v v 84 +10304550 +108 − 3 4 18 v v v 76 v 3 i 2 v 2589008 − ) +8007211 20 v v v v v + m ,D n v 6724 − 42 ( 10 2 + +240220 23 148110 v − g +306961 13 6932828077068 ··· − ,Q +47 12 v 749797256633 367411722852 ··· v − 2 − − + 10 2 35281600040 92856436862 31372682848 v v v +7 − − + v v 11 16826056321 n 38 43 − − − v 1479273049 98377407 159566189 415787266 22 3 = 1 33 34 39 v v − +3144719815 48153754 30 35 29 v v v v +557841530 − − − − − )(84+146 28 24 v v v +22 +186559606 − +217119419 2 v 884324 22 25 23 19 25 v v )(2+3 +10272490 v v 8878 +1088019 16 17 18 2 v v v v v − v, x 9 19 v v v v ( − +113904 +412658 12 v v 4 ) 9 10 11 v (1200+3374 (1+ (2+2 (1+3 ,F n v v v (1+ ( 25 22 19 8 2 g − − − − 11 v v v v − − − v − − 9274316155458 832956121 8623040702 347331281609 5341633935638 13585439053 275432340276 506517773112 2279805 13638464 601571798 22656311766 89424037262 260932 101871752 432409780 4, we observe the following general leading order behavior − +67719233 − − − − − − − − +45251 − +164548302 +38843 +3262797 − − − +65200 +97908991 +2545465687 − − − ) = ) = ) = ) = ) = ≥ For the reduced two-string elliptic genus of v v v v v ( ( ( ( ( n 4 4 4 4 4 . ,D ,D ,D ,D ,D 4 (6) (5) (3) (4) (2) 2 2 2 2 2 F For P P P P P We have JHEP12(2019)039 23 v 13 42 46 v 22 v v 27 27 v v v 21 38 39 43 v v 16 18 v v 26 v v 9 v 34 v (F.8) 10 9 39 35 v 17 7 v v v v v v 15 31 35 v 22 30 v v v 31 193165328741 v +201171570880 177701238 135552 683123 22028614537 v +40026745086453 − 27 12 − − 44757889196479 − − 22 v v +23995511 . 9 21 8 15 v 8 − +23893104753132 20 v v v v ) v v 26 26 +546538072 +5813458948881 v +354378137 +374497086121044331 +550330680323572096 v v ( 6 16 17 1141697391 23647746652092629 34949875241183086 +11730377038184 6 v 41 45 ) v v 57418 v v − − − +115501696318370322 . 25 n ,E 786424 ( ) 7182956156002759 v 2 − 14 38 42 37 v 8 − v v v v P − +15128265661342010 +1792334859311106 ( 81039308 +317784470985 v 7 +8097680 921916213048446 v 34 33 38 − 7 O × 21 1369097445 48437902734 v v v − +987634558444181 +84638789902 v v 14 +748700764659463 − − 9266 23 30 34 v 11 +239466759024769 +90874761 ) 29 v v v +73260270949890 ) + − 19 20 5 2 30 v +111764209 +30278882934513 7 v v x v v 411322 26 v v ( 17700027945928 15 21 990656365 v 3 − + v v − +11543089455336 6 − − v +3348446417048 , v 25 25 7574716 n +1875127 ) 13 16 v v 6 +3391440788051 v χ − 6d SCFT, recall v +5832 80 v θ 24 6 v (1 + 13 6 v v 6) , v +301112810847585121 +537304885313273980 + 18955600691329352 34110133094328607 +107176864396 +20938534 32 E 146242 − 1 56 ) 4 4 40 44 10239254139 − − +75560202097716654 n +33173318084 20 4676542813241321 ··· v +9932344114 v − v F ( v v +21374265 v − 5 37 41 36 10 − +7864675977003847 +1664270696390466 + + χ 18 476759190688969 v 14 v v v v +256583 +5964 19 v 33 32 37 41 441768092 v − +749891181508407 − 5 5 ··· v (1 + v v v v +297084435895112 v v − 29 33 +67951300040637 + +146800695692191 46 28 v v – 91 – +21734938030680 ) 12 6491341536008 +19396808 37612 25 29 29 v ) = v v 20 v v v i − +5224968453408 − 12 +4293754 +388 v +1818950197662 − 4 v 3 m 24 24 +2976 4 +853983234794 v v . 15 v v 4 v , ) (1 +4373263 23 v v ) v 92 +33193123730 13 1704624732 86 v +9413229443 v 6740 +12068000241 19 8588 v − 77639509 9 + 17 v − v, x, Q − + v 18 v − 3 ( +1071 3 ··· v +770928 v 4 3 v ) 11 ··· 14445439874041454 31707630617443838 +17299644 2 = 1) = + +230299469534789175 +500006416441373208 v ,F v 2881832101851776 n v + +46797420093685421 − − ( 2 11 231557373968430 i 729 47 39 43 +3839341467548316 +1470501793161916 +1792334859311106 − g v +957484 44 35 36 40 v v v − +540578567158590 m 1942 − 31 40 32 36 v v v v +109311773773871 +59931545438647 +73260270949890 +85029824340612 2 12 2198459381202 − +277 28 32 v v v v +14791974953829 v v +10 2 2 27 24 28 28 v v − +930177353330 +2209544249477 9 n 19 v v v v v v +143994570705 328644969 3 23 23 14 v − +9374157248 +118305 +4057153817 22 v v v v +4322862537 − v v v, x, Q 18 8 187 v +45403031 +58 ( 16 17 v v +7863641 − 6 v 10 v v +55432 ) v 10 v ,E n 11 v ( 2 (2+11 v g (1+ (1+7 15 − 18 12 v − − v − v 15689824140 1678228390978831 10456822688468996 28066464498494385 34110133094328607 228893241 683657580163 105345430184255 169110 6, we observe the following general leading order behavior +27394204368470677 +167393226212698984 +443382800033202476 +537304885313273980 +47743204 +314912356 − +37232002426385 +1754971776388633 − − − − +1519991 − − − +46455227531026 +369631201685833 +1235929372818009 +1837134386523548 +75118982210308 − +2392703794 +870042414425 +1254181600 +446852450528 +9534395062259 +50246339190488 ) = ) = ) = ) = 1653+14307 ≥ For the reduced two-string elliptic genus of v v v v ( ( ( ( n 4 4 4 4 . ,F ,F ,F ,F 6 (5) (4) (3) (2) 2 2 2 2 P E For P P P We have JHEP12(2019)039 24 17 7 v v v 29 8 21 v v v 9 20 v 25 v 12 v v 13 28 v . 7 v ) v 39 42 17 21 16 76 421607 36 v v v v v v v − 39 33 +4894721995 + 8 +992270036 v v 6 v 15 12 7 30 v v ··· v v 20 36 30 33 v +15241390482586 27 v v v v + 24 34 v 16 v 38 27 +88082527 v v 24 v v 6 +162200225646883046 364431 31 35 v v +25305594210396759 +4768406721146 32 23 v v − +32099832942977106 +3538777989264 28 v v 7 11 38487665 20 v v v +4367001323365453 +11088020153427871 12 v − v +966630727 24 19 +4857570355921361 +136827854 5 14 v v 6 v 27 v +554884515982156 +354773291170080 +376267731853770 +82957003626 v v +28747116555 16 15 20 11 177594 +17567385 v v v v 19 5 − v 6 v +277900332002367574858 +365533661213463473501 v +23475175955326 +6017588149603 +151556399981844604507 38 41 +11476648364287371362 +58842809938209124305 23 15 35 v v v v 38 32 v +11206753 +9717463741648588196 +16067738341715188425 +2537602212267590587 +5887558970641741644 +15949836 +172731506 v v +3032155653955325045 +1392335683050 64197 +831787432544 5 13 35 29 29 32 4 +2096960972906450647 +123925354694394472 +770246305708025175 v − 26 v v v v v 10 v +386049290828201197 11 +3176394 5 33 37 26 v v +130358984643118795 +61367940071565626 +104007721490984500 +88666687652950697 v 4 23 v v v v +17365401325615558 30 31 34 22 v v +12653182331604747 +24365673656 +4014329887 27 v v v v +2530053857442778 +5052883393549785 19 v 10 18 23 18 v v +1428966813600884 v 18313 v v +173571288630315 +181712020504595 +131729726893973 26 – 92 – +1465146 − 15 19 14 v 4 . +18191652 4 +27606333 v v v ) v +515477 3 v +5061754204737 +2249848989493 . 12 3 v 86 v v 22 14 78 v v v v 4214 +379809898432 + +179602878296 9 + − 10 v ··· +204232919 +97959 3 +6687039606 v ··· 3 v +235652392733234663776 +346060446910493617699 + 9 17 v +73825 +39727277739048609244 +114875661886259702901 +11266596547576742504 + v v 2 +3873323 37 40 43 +9612446692952872376 +1774271549538256254 +4620696016056616197 +8535984817703595260 v 2 769 31 34 37 40 v v v +875864441137943176 +1595815215747637914 +477678248920949928 +121537998101131452 +10737460 769 28 28 31 34 v v v v v +46247090390273044 +177003542946746952 +46391596457503471 +47551931562200552 +90737018750916024 − 32 25 25 36 v v v v +4724849885190467 +11433257908228549 − 11 2 29 22 21 30 33 v v v v +2193217132886674 +1402570753853399 2 v v 18 26 v v v v v +392058843196059 +50873087148945 +83623554563863 +46244599549903 v 17 22 v v +9129 25 14 18 13 v v +994624050267 +794925309293 112 v v v v v 112 235567050 +96385144324 21 13 − +464913 +35347238184 8 v v − − v v +1707996910 9 v v 16 8 v 11 v v +3722444 11 − − 10 1 v 1 )(82+896 − 2 ( − v ( 16 4 − − v v 147892027 − +177469181418 +99852192764195 +15462381593811917 +347206472133377093 +192548720900049944088 +315856987228959670753 +372262200765577638648 +420751 +77298235563217848 +803129445851193549 +5515192004170661557 +25779182779041415519 +83817103245449704330 +11266596547576742504 +6276463714 +13918341585911 +1667591219224745 +284262935556020849 +1192619227946678339 +3490792045588793343 +7223464858978994614 +10658977913348459838 +22642609833 +15311706805952 +905322588772629 +23207949295452654 +743340720549339 +7218179887542376 +35398201343930359 +76109936363599780 +114645634265369518 +404122599 +264812209428 +36623148751459 ) = ) = ) = 3486+44488 ) = (1+ v v v v ( ( ( ( 6 6 6 6 ,E ,E ,E ,E (4) (3) (2) (1) 2 2 2 2 P P P P We have JHEP12(2019)039 . ) 110 7 v v + ··· 41 (F.9) 25 + 49 52 v 20 v 38 v v v 46 49 52 55 46 v v v v v 43 35 v v v 43 8 v 40 32 v 21 12 40 +1887255497 v v v v v 6 37 . 37 v v v 29 ) 16 34 v v v v ( 26 7 13 31 v v v . +74152233 (2) 17 n,E 23 28 7 v v v P v 122 +268811177 v +921861932483495244 × 5 +576354715341079126 v + 24 35 19 v +10556715862964 ) ··· v 2 11 v + v +12390621413785440 +2135260459421424616916 +4857787518331415856898 +7920702637212573264754 +9316044470623822160548 56 58 60 62 + +15465256039202958794051688 +3665936157860425562998541 +8053206553397859455383003 +11476871 20 45 48 51 54 +665245395666408218767 +305215505275 54 v v v v +3107832107172475492688890 6 v v +3615374924636545 v v v v 48 51 40 v +482772800850075541856889 +378216261606533139497461 +1324728639658651633703727 v 42 12 +144526191934600175231 37 v v v 15 +34901534 v v +21407072068973867721 42 45 34 v +84802326792798968389732 +57361948999125457686102 6d SCFT, recall v 4 39 v v v +88338636956104 36 v v (1 + 39 31 7 +14820947289312246349740 v v v +5146221002718346735055 16 E 52 36 1 +2000571291562232590513 v ) 28 v +343131359707453293583 v 33 v +206242719899419463791 25 +1638666 v 30 5 v +16674709176092326437 +16018745367471142680 v v +2140990474296 (1 + 22 27 +4077137 v v 10 +331836472263902521 3 70 – 93 – v +173164585658174276 ) v +65121712327 23 v 18 v 11 v +3832341391241046 v − +213819 19 +907396069059573 (1 4 v v 14 v +419921 +23021851542594 2 +3834519574899765264582 +6979905103120498497197 +9149752299658818747593 +1505001425501016329934 15 v +416666541458353995971 v 47 50 53 44 +79898824171145741885 41 v v v v +10350471755632365352 +16154939755233169917249815 +19698620890606501833935055 +21750009714684524653667914 +21750009714684524653667914 38 v +25193 = 1) = +9316106764452824593797657 +2679565476854052143878502 +6354435158683924634396271 +11992251412402642586454948 +408614442098 35 v 3 55 57 59 61 +244657061843538883914723 +1719956874446161848789295 +236104043071240448693797 +895677339869346647226824 9 i v 39 50 53 47 v v v v v +48764087264312469202730 +26512271515436823962474 v +13133928036 33 36 41 44 v v v v +7828073649219480743672 m +37024 38 30 v v v v +2158650362542725175948 +967171231109021529977 10 v 35 v v +129836595656234291790 +90876821066275028695 v 27 32 v +5649018624246617909 +6399961693050532054 29 24 v v +2628 21 26 v v +49857926256318138 +115175603408208175 2 v v v v, x, Q 17 22 +1138031848052668 ( v v +216730904533865 7 18 +5725284334978 ) 13 v ,E n 14 v +237 ( 2 )(137+2597 +73094300214 v v g 2 8 v +2495671432 v 9 v +8668249061573197611391 +42290120178103689822 +250551336753503603832 +1020471334680763094257 +2915910779649516764294 +5930740363027659407709 +4773710145181408321 ) = (1+ ) = 1+17 For the reduced two-string elliptic genus of v v ( ( 7 7 . ,E ,E 7 (2) (2) 1 0 +875542039936399623515 +11870970192394860758359 +120296712068566252009120 +924703269912581018952608 +5457334614794588632799143 +12196657853 +49151597538306 +13736293007916068 +1840835604225541174 +47394303096706259811 +14093734406768042617860546 +18062065264884658609927825 +20956986683280640928389866 +22020920210850484561094012 +143449590902653729399624 +589822792928957883073617 +1908697079658876873038411 +4888414479465062757831170 +9951646269406905770095206 +2471530433876763846 +38781560068496818142 +453576963793872584712 +4016126507767354504238 +27267076918737091016348 +445070980 +1356033968529 +324035139906700 +38509222288582663 P P E We have JHEP12(2019)039 50 53 47 v v ]. v 44 31 v (G.1) (G.2) do not is easily τ r d is the first ≤ τ 1-loop d , 12 c Z ) 1, we observe an τ ≤ ), d b ), the curve-divisor ≥ d b cls 2.36 d + (1 + Z d 14 d/ ]. b -fields, the unity ≤ 1 , of divisors, the intersection r 42 b symmetry, in which case the 57 59 b +439563294320255738140535927 +838003231433873171234645238 d 55 v v we considered in this paper. +182784785431420765743008945 d ], the Weyl symmetry of gauge 8 δ 49 52 v +60003949644181883287996554 . ijk 46 v v , E + 24 G 43 v κ 1) v , as well as the one-loop partition d 2 + b − / r + 1) b 1) b d d , ( − 2 1)( d b / , we observed for d 8 , − + 1) ( 7 b , b is the degree of the base, and d , for complete lists, one can find them at [ 6 d − d Hilbert series in [ b E b 31 7 − d – 94 – . d – ( E ) ) = ) = ( b 1) b b d 24 , we observed for 120 − 4 . v − b d, d d, d F ) (these are two ingredients of for all the models. + field with nonzero ( ( . On the other hand, as seen in ( d max l,r X r ··· j ( 1)( + 1) max max +1387339824356009883032251758 +1534199301083129786878770830 + 2 r l +1134218396741161783456978908 +336728327510605097378060508 +693350791574808124298361559 j j c b . For 1-loop 56 58 60 1-loop +129479490500199449940430040 +39238673033178891558314265 − appears. For d 54 48 51 v v v Z ( Z 42 45 d v v v τ 12 models, there is no vanishing d v v max l,r , j , we compute all the BPS invariants up to total degree 12, with 8 , ) = ) = ( 7 , b b = 5 is strictly speaking also not necessary, as the bounds can be generated , one unity 6 E C n d, d d, d max l,r ( ( j and the bounds max max r l of divisors with j j agree with the two-instanton 1-loop GV i b is not manifest in the refined BPS invariants. This is simply because the Weyl 7 Z is the total degree of the fibers, ,E model, we compute all the refined BPS invariants up to total degree 14, with 4777 d (2) G 0 4 g In the case of The input of F is a global symmetry other than the gauge symmetry 8 +1477400106011059794784293700 +1553610262702054425407310320 +89325800153382434388949763 +251347720581234682951528991 +559192119429259737303598283 +987373409206497489976018270 +1270346407634715071774123344 +24973370386295921380753761 group symmetry will change the sign ofis some always K¨ahlerparameters, while the in refined positive BPSthe degrees. expansion refined Note BPS invariants this ofE should E-strings not do be have confused manifest with the situation where For non-vanishing. For 10383, 10491, 10068 non-vanishing respectively.in We the list affine part Lie of algebraIt the bases is refined in worthwhile BPS tables to point invariants out that unlike the elliptic genus [ where total degree of fibers when the program makes theexperimental computation formula much for faster. In any case, for have enough constraints on computed, we simply input from the blowup equations with the other input data, but the inclusion of the bounds in The refined BPSfollowing invariants are initial solved input: fromnumbers the the triple generalised intersection blowupintersection numbers matrix equations with the function Note G Refined BPS invariants JHEP12(2019)039 ⊕ ⊕ ⊕ 2) 3) 3) / 3) , , , 7 2 2 , 2 / / / (0 (1 (1 (1 ⊕ ⊕ ⊕ ⊕ 2) 2) 2) / 2) ) / / 5 / r , 2) 7 7 7 , , , ,j / l 5 j 2(0 , ( 2(0 2(0 2(0 r 2) ⊕ (0 , 2) 2) 3) 2) 2) 2) 3) 2) ,j ⊕ ⊕ ⊕ l , , , , , , , , 2) ⊕ β j (0 2) 2) / 2) (0 (0 (0 (0 (0 (0 (0 (0 continued on next page N 2) 2) / 2) / 2) 3 / 2) ⊕ , / / 5 / 5 / 5 / ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , , , 1) 5 3 5 5 5 , , , , , , 1) 1) 2) 1) 1) 1) 2) 1) 2(0 , , , , , , , , 1) 1) 2) 3) 1) 1) 1) 2) 2) 3) 1) 1) 2) 3) (0 (0 3(0 (0 3(0 (0 3(0 (0 , , , , , , , , , , , , , , ⊕ 3(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (0 (0 (0 2(0 (0 (0 (0 (0 2(0 2(0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 2(0 2(0 2) ⊕ 2) 2) 2) 4) 2) 2) 4) 2) / 2) 4) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , , , 0) / / / / / / 1 / / , 2 2 , 2 1) 0) 0) 2) 1) 0) 3) 2) 4) 5) 0) 0) 0) 0) 1) 2) 1) 1) 0) 3) 2) 4) 5) 0) 0) 1) 2) 0) 0) 1) 1) 0) 2) 3) 3 1 3 3 3 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , / , , / , , / , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (1 (0 2(0 (0 (1 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 2 0 1 2 0 1 0 0 0 1 2 1 2 1 0 1 2 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 1 1 1 2 2 2 3 3 4 5 1 1 1 2 2 3 0 1 1 2 0 1 0 0 1 1 2 3 1 2 1 0 1 0 0 2 2 3 2 3 1 2 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 3 3 4 5 1 1 1 1 2 2 3 1 2 1 1 0 0 0 1 1 2 2 3 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (1 (1 (1 (2 (2 (3 (4 (0 (0 (0 (0 (0 (0 (0 (0 (1 – 95 – ⊕ ⊕ 4) 4) , , 2 2 / / (1 (1 ⊕ ⊕ ) 2) 2) 2) r 4) / / 4) / , , ,j 9 5 9 l 2 , , 2 , j / / ( (0 r (1 2(0 (1 2(0 2) 2) 2) ,j ⊕ l , , , 2) 3) 2) 3) 3) 2) 3) ⊕ ⊕ ⊕ ⊕ β j , , , , , , , 2) 2) 2) (0 (0 (0 N / 2) 2) 2) / 2) 2) 2) / (0 (0 (0 (0 (0 (0 (0 / / / 3 / / / ⊕ ⊕ ⊕ ⊕ , 5 7 7 11 5 7 7 11 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , , , , , , , , 1) 1) 1) , , , 1) 2) 1) 2) 3) 2) 4) 1) 1) 2) 3) 4) 2) 2) 1) 2) 3) 2) 4) 1) 1) 1) 2) 3) 4) 1) 2) 2) (0 (0 (0 (1 2(0 (0 (0 (0 (1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 2) 2) 2) 2) 5) 2) 2) 2) 2) 2) 5) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , , / / / / / / / / / 2 2 0) 0) 1) 1) 1) 0) 1) 2) 1) 3) 1) 0) 0) 1) 0) 2) 3) 1) 1) 0) 1) 2) 1) 3) 0) 0) 0) 0) 1) 0) 2) 3) 0) 1) 1) 5 3 5 5 1 5 3 5 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , / , , , , , / (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 (1 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 2 2 1 2 3 1 2 1 0 1 2 0 1 0 0 0 1 0 0 0 0 0 0 1 2 1 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 1 1 1 2 2 2 3 3 4 0 1 1 2 2 3 4 1 1 2 3 1 2 1 0 1 1 2 0 1 0 0 0 0 0 2 2 3 4 2 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2 2 2 2 3 3 4 1 1 1 1 2 2 3 4 2 3 2 0 0 0 0 1 2 2 3 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (1 (1 (1 (1 (2 (2 (3 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 2) 2) 2) 2) / 2) / / ) / 2) 5 / 5 2) 5 r , , , 5 / 5 / ,j , , ) 5 5 l 2) (0 (0 (0 r , , j / 2) (0 (0 ( ⊕ ⊕ ⊕ ,j 5 / (0 (0 r l ⊕ ⊕ , 5 j ,j 2) 2) 2) ⊕ ⊕ , ( l (0 2) 2) / 2) / 2) 2) 2) 2) / β j r (0 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) 3 / 3 2) 2) 2) 2) 2) / 2) / / 2) / 2) 3 ,j ⊕ , , , N / / / / / / / / / 3 / 3 / / 3 / / / / / 3 / 3 3 / 3 / l ⊕ β continued on next page , , , , , , , j 3 5 3 5 3 3 3 3 3 5 3 5 3 7 7 7 9 5 3 5 ⊕ 2) , , , , , , , , , , , , , , , , , , , , N / 2) 2) (0 2(0 2(0 (0 (0 (0 (0 2(0 3 / / (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , 3 9 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ , , 2) 2) 2) 2) 2) 2) 2) 2) 2 2(0 (0 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / 2) / 2) 2) 2) 2) 2) 2) 2) / 2) / / 2) / 2) / / / / / / / / / / / / / / / / / / 1 / / / / / 1 / 1 / / / / / / / 1 / 1 1 / 1 / 1 ⊕ ⊕ , , , , , , , , (1 1 5 9 1 1 3 3 7 9 1 3 1 1 1 1 1 3 1 5 1 3 1 1 5 5 5 9 7 1 3 1 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2) 2) ⊕ / / (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 3(0 2(0 (0 2(0 (0 2(0 1 1 3) , , , 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) minimal SCFT. (0 (0 (0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 1 3 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , F 2) 2) 3) 0 0 0 1 2 2 3 4 5 0 0 1 1 1 1 2 2 3 3 0 0 1 1 1 2 3 0 1 1 0 1 0 0 0 1 1 2 2 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 4 4 0 1 1 1 1 1 1 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 1 3 5 0 0 0 0 0 0 1 3 0 2 0 2 1 1 0 0 0 2 0 2 1 1 0 1 0 0 0 0 0 1 3 1 0 0 0 0 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (2 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ – 96 – 3) , 2 / (1 ⊕ 2) 2) 2) 2) ) 2) 2) 2) / 2) / 2) / 2) / r ) / / / 5 / 5 / 7 / 5 r ,j , , , , 7 5 5 5 7 7 l , ,j , , , , , j l (0 (0 (0 (0 ( j (0 (0 (0 (0 (0 ( r ⊕ ⊕ ⊕ ⊕ 2(0 r ,j ⊕ ⊕ ⊕ ⊕ ⊕ . Refined BPS invariants of 6d l ,j ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) β j l β 2) 2) 2) 2) 2) / 2) / 2) / 2) 2) 2) 2) 2) / 2) 2) / 2) / 2) 2) 2) 2) / 2) / 2) 2) / j 2) N / / / / / 3 / 3 / 3 / / / / / 3 / / 3 / 5 / / / / 3 / 3 / / 3 N , , , , , , , , , / 2) 2) 3 3 7 3 7 5 3 3 5 7 5 3 5 3 5 5 9 3 3 3 3 5 ⊕ , , , , , , , , , , , , , , , , , , , , , , / 5 / ⊕ (0 (0 (0 (0 (0 , 5 9 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 , , ⊕ ⊕ ⊕ ⊕ ⊕ 2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (0 3(0 / 2) 2) 2) 2) 2) 2) Table 24 ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) / 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) / 2) / 2) 2) 2) (1 / / / / / / / / / / / / 1 / 1 / 1 / / / / / / / / / / / / / / / / 1 / 1 / / / , , , , , 2) 2) 4) ⊕ 1 3 7 1 3 1 5 1 5 3 1 5 3 1 1 3 5 3 1 1 3 1 1 7 3 3 3 7 11 1 1 1 1 3 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , / / 2 3 3 3) (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 3(0 (0 (0 (0 , , / , (0 (0 (1 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 2 0 2 1 0 0 0 0 0 1 0 1 2 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2) 2) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 2 2 3 3 4 0 0 0 1 1 1 2 2 2 3 4 0 0 1 1 2 2 0 0 1 2 0 0 0 0 1 1 1 2 3 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 5 1 1 1 1 1 1 1 2 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 2 4 0 0 0 0 0 0 0 2 4 1 3 1 0 0 0 1 0 1 3 1 0 0 0 0 2 1 0 1 0 0 2 0 2 1 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 ⊕ 2) 2) 2) 2) / / 2) / 2) 2) / 2) 2) ) 5 5 / 5 2) / / 4) 5 / 2) / 4) r , , , , , , 7 / 5 5 5 / 5 ,j , , , 2 , , 2 7 5 l (0 (0 (0 , / , / j (0 (0 (0 2(0 (0 (0 ( ⊕ ⊕ ⊕ (0 (1 (0 (1 r ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ,j 2) 2) 2) ⊕ ⊕ ⊕ ⊕ l / / 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) β j 3 3 / 2) 2) 2) 2) 3 2) 2) 2) 2) 2) / 2) / 2) / / 2) 2) 2) 2) 2) / / / 2) / / / 2) 2) 2) 2) 2) / / 2) 2) / 2) , , , N 5 / / / / / / / / / 3 / 3 / 3 3 / / / / / 3 3 3 / 3 3 3 / / / / / 3 3 / / 3 / continued on next page , , , , , , , , , , , , , , 9 3 3 3 5 5 5 3 3 3 7 5 7 3 3 7 5 3 3 3 7 5 5 3 7 ⊕ , , , , , , , , , , , , , , , , , , , , , , , , , 4(0 2(0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 (0 (0 2(0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / / 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) / / 2) 2) 2) 2) 2) 2) / / / 2) 2) / / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 1 1 / / / / / 1 / / / / / / / 1 / / / 1 1 / / / / / / 1 1 1 / / 1 1 / / / / / / / / / / / / / / , , , , , , , , , , , 3 7 1 1 1 3 3 3 7 9 1 1 1 1 5 3 5 1 3 1 5 3 1 1 1 1 5 5 5 3 1 7 1 3 1 1 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4(0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 3(0 (0 (0 (0 2(0 2(0 (0 (0 (0 (0 (0 (0 2(0 2(0 2(0 (0 (0 2(0 3(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 0 0 0 1 1 0 0 0 1 0 0 0 0 1 2 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 3 4 0 0 1 1 2 2 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 4 1 1 1 2 3 1 1 0 0 1 1 1 1 2 3 1 1 0 0 1 0 0 0 1 1 1 2 2 1 1 1 0 1 0 0 0 1 1 1 1 1 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 1 1 1 1 2 2 2 2 2 3 3 4 5 1 1 1 1 1 1 2 2 3 4 2 2 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 1 0 0 0 0 1 0 0 0 0 0 0 2 1 0 1 0 0 0 0 0 0 0 2 4 2 1 1 1 1 1 1 1 2 2 3 0 0 1 3 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (2 (2 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 – 97 – ⊕ 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) / 2) / / ) / / / 2) 5 2) / / / 2) / / / 2) 4) 2) 5 / 5 5 r , , , , , 5 5 7 / / 5 5 5 / 5 5 5 / / 5 ,j , , , , , , , , , 2 , 5 5 5 5 5 l (0 (0 (0 , , , , / , j (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 ( ⊕ ⊕ ⊕ (0 (0 (0 (0 (1 (0 r ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ,j 2) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) / 2) β j / / 2) / 2) 2) 2) 3 2) 2) 2) / / / 2) / 2) / / / 2) 2) 2) 2) / / 2) / 2) / / / 2) 2) 2) 2) 2) 3 / 3 / , , , N 3 3 / 5 / / / / / / 3 3 3 / 3 / 3 3 3 / / / / 3 3 / 3 / 3 3 3 / / / / / 3 3 , , , , , , , , , , , , , , , , , , 7 3 3 3 5 3 7 5 3 3 3 5 3 3 7 3 7 5 3 5 ⊕ , , , , , , , , , , , , , , , , , , , , 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 2(0 2(0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 2(0 (0 (0 (0 (0 2(0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / / / 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) / 2) / 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) / / / / / / / / 1 / / / / / 1 1 1 / / / 1 / / / / / / / / 1 / / / 1 / 1 / / / / / / / 1 / 1 / / , , , , , , , , , , 1 1 5 3 3 1 1 1 5 3 1 5 1 3 1 1 1 1 1 1 1 3 1 1 1 1 5 1 1 3 5 3 5 1 3 1 1 7 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 2(0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 3 0 0 1 1 1 2 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 3 0 1 1 2 2 0 1 2 1 0 1 1 1 2 2 0 1 2 1 0 0 0 0 1 1 1 1 2 3 1 2 1 0 0 0 0 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 1 1 1 1 2 2 2 2 2 3 3 3 4 1 1 1 1 1 1 2 2 2 3 2 3 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 3 1 3 2 1 2 1 2 1 1 3 2 4 0 0 2 0 2 1 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (2 (3 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 4) 4) , , 2 2 / / ⊕ (1 (1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ ⊕ 2) 2) 3) 2) 2) 4) 2) 5) / 2) 2) 3) 2) 2) 2) 4) 2) , , , , , / 3) / / 3) / / 2) 2) 2) 2) / / / 2) / 2) 2) 2) 2) / / ) , 2 , 2 2 2 2 5 5 5 7 / 13 / / / 7 5 2) 5 4) / 7 / / / / 7 r , 2 , / , 2 , / / , , , / , , , , / 11 9 5 5 5 / 5 5 5 9 11 ,j / / , , , , , 2 , , , , , 5 11 l (1 (1 (0 , / , j 4(0 (1 7(0 2(1 4(0 (1 5(0 3(1 (1 (0 (0 (0 2(0 7(0 2(1 (0 2(0 (0 (0 5(0 3(1 (1 ( ⊕ ⊕ ⊕ 2(0 (0 (1 2(0 (1 r ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ,j 4) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ l , 2) 2) 2) 2) 2) 2) 2) 3) 2) / 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 3) 2) β j , , , , 2 , , / / / / / 2) / / / / 4) / 2) 3 2) / / 4) 2) 2) / / 2) 2) 2) 2) 5) / / 2 2 2 2 / , 2 , , , 2 N 3 3 3 5 9 / 11 3 3 3 5 3 / / 3 5 / / 3 3 / / / / 5 9 continued on next page , / , / , / , / , , , , , , 2 , / , , 2 , , 2 , / , 7 3 7 5 5 7 7 7 7 ⊕ (1 , / , , / , , , , , , / (1 (1 (1 (1 (1 (1 (1 2(0 (1 (1 ⊕ 3(0 5(0 3(0 3(0 (0 2(0 2(0 2(0 3(0 (1 5(0 (0 (0 2(0 3(0 (1 (0 (0 2(0 2(0 (0 (0 (0 (0 (1 3(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 5) / 6) 2) 2) 2) 5) , , , 2) / 2) / 4) 2) / 2) / 2) 7) 2) 2) 2) 2) 3) 2) / 4) 2) / 2) 2) 2) 2) 3) 2) 2) 2) 2) 2) 2) 2) 2) 4) 2) / , 2 2 , , , , , 2 / 7 / 7 / 7 / 9 / 11 / / / / / 7 / 1 / / / / / / / / / / / / / 9 , , 2 , , / , / 2 2 , 2 , 2 2 , / 1 1 1 3 5 1 1 1 3 1 1 5 5 1 3 3 3 1 1 5 5 5 5 3 , , / , , , / , , , , / , / , , , , , / , , , , , , , , / , (0 2(0 (0 3(0 (1 (0 2(0 (0 3(0 2(1 (0 2(0 2(1 (3 (0 (0 (0 (0 (1 (0 3(0 (1 (0 2(0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 3(0 2(1 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 3) 3) 3) 3) , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 2 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 3 4 1 1 1 2 2 1 1 0 0 1 2 0 1 1 1 1 1 1 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 0 0 0 1 1 1 1 1 2 2 3 4 1 1 3 0 0 1 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (0 (0 (0 (0 (0 – 98 – 4) , 2 / (1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 4) 2) 2) 2) 2) 2) 3) 2) 4) 2) 2) 2) 2) 2) 2) 4) 2) , , , , / / / 2) 2) / / / / / / / / / / 3) / 2) 2) 2) / 2) / / ) 2 2 2 , 2 7 7 7 / / 7 5 5 5 5 7 5 5 5 / / / 7 2) / 2) 7 r , , , , / , , , , / , / , , 2 , , , / 9 11 11 5 5 5 / 7 / 11 ,j , , , / , , , , , 11 7 7 l (0 (0 (0 (0 , , , j 2(0 2(0 2(0 5(0 3(1 (1 7(0 2(1 5(0 3(1 (1 4(0 (1 2(0 (0 (0 (0 2(0 (0 5(0 3(1 (1 ( ⊕ ⊕ ⊕ ⊕ 2(0 (1 (0 (0 r ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ,j 2) 2) 2) 2) ⊕ ⊕ ⊕ ⊕ l 2) 2) 2) 2) 3) 2) / / / 2) 2) 2) 3) 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 2) β j , , , , , / 4) 4) / / 4) 2) 5) / / 2) 3 3 3 / / / 3 / / 2) / / 2) / / 4) 2) / 2) / / , , , , 2 , , , 2 2 , 2 , 2 N 5 5 5 / 5 9 / 3 5 9 3 3 / 3 3 / 3 5 / 5 / 5 9 , 2 2 , , 2 2 , / , , / , / , , / , , , , , 2 , , / , 7 5 5 5 5 5 ⊕ / / / , / , , , / , , (1 (1 2(0 3(0 2(0 (1 (1 (1 2(0 (1 (1 (1 3(0 (1 (1 3(0 3(0 (1 (0 (1 3(0 (0 5(0 3(0 3(0 2(0 (0 2(0 2(0 (0 2(0 3(0 (1 (0 2(0 (0 3(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 5) 2) 2) 2) 2) 2) 5) 2) 2) 2) 5) , , , 2) 3) 3) 2) 2) 3) 2) 4) 2) / 2) / / / 2) / 4) 2) / / 2) / 2) 2) 2) 2) 2) 2) 2) 2) 3) 2) 2) 2) 2) 2) / , , , , 2 , 2 , 2 / / / / / 9 / 1 1 1 / 7 / 9 1 / 7 / / / / / / / / / / / / / 9 2 2 2 2 , / , , , , 2 , / , , 2 , / 3 3 3 5 3 3 1 3 1 7 1 3 1 1 3 1 3 7 3 3 3 3 , / / , , / , / , , , / , , , , , , , , , , / , , , , , (0 (1 (1 (0 (0 (1 (0 (1 (0 3(0 2(1 (0 2(0 3(0 2(0 (0 3(0 (1 (0 3(0 2(1 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 3(0 2(1 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 3) 3) 3) 3) , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 0 1 0 0 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 3 3 1 1 1 1 2 3 1 2 1 1 1 1 0 1 2 1 1 1 1 2 , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 0 0 0 1 1 1 1 1 1 2 2 3 1 1 2 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 0 0 0 2 1 0 1 0 1 0 0 0 0 0 3 2 1 0 2 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (0 (0 (0 (0 (0 JHEP12(2019)039 ⊕ ⊕ 2) 2) ⊕ / 2) 5) / , / 5 2) 2) ) , 2 5 / 2) / 13 r ) , / , 7 / 9 r (0 ,j , , 7 l (0 (1 (1 ,j , ⊕ j l (0 ( ⊕ j ⊕ ⊕ (0 2(0 r ( 2) 2) 2) 2) ⊕ r 2) / 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) 2) 2) 2) / 2) 2) ,j 4) 2) ⊕ ⊕ l , / 3 / / / / / / / 3 / 3 / / / / / 3 / / ,j 2) / β j l , , , , 2 3 5 3 5 3 3 3 3 3 3 5 7 5 3 5 5 2) / 2) 2) 2) 2) β j , , , , , , , , , , , , , , , , / N / 5 / / / / 11 (0 (0 (0 , , N 7 5 7 9 7 ⊕ (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (1 , , , , , ⊕ ⊕ ⊕ ⊕ (1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (0 2(0 (0 (0 (0 (0 2) 2) 2) ⊕ continued on next page 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ / / / / / / 1 / / / / / / / / / / 1 / 1 / / / / / / / / / 6) 2) , , , , 1 5 9 1 3 1 5 3 7 9 1 3 1 1 1 1 1 1 3 5 3 1 1 3 3 2) 2) 2) 2) 2) 2) 7) / , , , , , , , , , , , , , , , , , , , , , , , , , 2 , / / / / / / 11 , / 2 5 3 3 5 7 5 11 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 , , , , , , / , (0 (0 (0 (0 (0 (0 2(0 2(1 (3 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 3 5 0 0 0 0 0 0 0 0 1 3 0 2 0 2 1 1 1 0 1 0 1 3 1 0 0 3) 3) 3) 3) 4) 4) 5) minimal SCFT. , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 1 2 0 1 0 0 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 1 1 E 0 1 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 2 2 2 3 3 4 5 0 0 1 1 1 1 1 2 2 2 3 4 0 0 1 1 1 β 1 1 1 1 1 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 2 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 (0 – 99 – 2) 2) / 2) 2) / 2) 5 / , ⊕ ⊕ / 5 / 5 , , ⊕ ⊕ ⊕ ⊕ ) 5 5 (0 , , 3) 6) r ⊕ (0 (0 , , 2) 7) 2) 4) 2) ⊕ ,j (0 (0 , 2 2 , / 2) 2) / / ) l ⊕ ⊕ 2 / / 2 / / 7 j r 2) ⊕ ⊕ / , / ( 11 7 7 11 2) 2) 2) 2) / 2) 2) (1 ,j , , , , r 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) / / 2) / 2) 2) 3 / l (3 3(1 ,j , j ⊕ / / / / / / / 3 / 3 / / 3 3 / 3 / / 3 (0 5(0 3(1 (1 l ( , , , , , , β ⊕ ⊕ 3 3 5 3 7 3 7 5 3 3 5 3 5 j 2(1 3(0 r , , , , , , , , , , , , , 2) ⊕ ⊕ ⊕ ⊕ (0 (0 (0 (0 2(0 N ,j 6) / 5) ⊕ ⊕ l . Refined BPS invariants of 6d (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 2(0 , , 2) 2) 3) 2) β j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2 2 , 2) 2) 11 / / / 2) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ / , / 2 N / / 5 5 9 / / / 2) 2) 2) 2) 2) , , / , 9 5 7 9 9 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) / 2) 2) / / 2) 2) 2) 2) 2) / 2) ⊕ (3 , , , , , / / / / / / / / / / / / / / 1 / 1 / / 1 1 / / / / / 1 / 4(0 3(1 (1 (1 , , , , , ⊕ 1 3 7 1 1 1 1 3 3 1 5 3 1 5 3 1 1 3 1 5 1 3 1 (1 (0 2(0 3(0 (0 (0 (0 , , , , , , , , , , , , , , , , , , , , , , , ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 (0 2(0 2(0 (0 (0 (0 (0 (0 2(0 (0 / 2) 2) 4) 2) 5) Table 25 , , 7) / 2) / 2) 2) / 2) 2) 2) 2) , 2 2 13 / 9 / / 9 / / / / 2 , , / , / 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 15 3 3 3 5 9 7 7 , , , , , , , , , , , , , , , , , , , , , , , , , , , , / , , , , , , , , 0 2 4 0 0 0 0 0 0 0 0 0 2 4 1 3 1 0 0 0 0 0 0 0 2 0 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 3(1 (2 (0 3(0 2(1 (0 (0 3(0 2(1 (0 (0 (0 (0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 1 2 3 1 2 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 3) 3) 3) 3) 4) 4) 4) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 1 2 2 2 3 3 4 0 0 0 1 1 1 1 2 2 2 3 3 0 0 1 1 1 β 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 0 0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 , , , , , , , 0 1 0 0 0 0 0 , , , , , , , (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 2) 2) / 2) 2) 2) 2) / 2) 2) 2) / / / / / / / 5 2) 2) 5 2) 2) 4) , , , / 5 5 5 7 / / 5 5 5 / , , , , , , , 2 5 5 5 5 ) (0 (0 , , , , / r (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ (0 (0 (0 (0 (1 ,j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ j ( / 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) r 3 2) 2) 2) 2) 2) / 2) / / / 2) 2) / / / 2) / 2) 2) 2) 3 2) 2) 2) / / / 2) / 2) / / / 2) 2) / 2) / / 2) 2) , , / / / / / 3 / 3 3 3 / / 3 3 3 / 5 / / / / / / 3 3 3 / 3 / 3 3 3 / / 3 / 3 3 / / ,j l , , , , , , , , , , , , , , , , , , 3 7 7 7 9 5 3 5 7 3 3 3 5 3 7 5 3 3 3 5 7 5 β j , , , , , , , , , , , , , , , , , , , , , , 2(0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 N (0 (0 (0 (0 (0 (0 (0 (0 2(0 2(0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 2(0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) continued on next page / 2) 2) 2) 2) 2) 2) 2) / 2) / / / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / / / 2) 2) 2) / 2) 2) 2) 2) 2) / 2) / / 2) 2) 2) 1 / / / / / / / 1 / 1 1 1 / / / / / / / / / / / 1 / / / / / 1 1 1 / / / 1 / / / / / 1 / 1 1 / / / , , , , , , , , , , , , , 1 5 5 5 9 7 1 3 1 3 1 1 1 5 3 3 1 1 1 5 3 1 5 1 3 1 1 1 1 1 1 1 3 5 3 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 3(0 2(0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 2(0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 (0 2(0 2(0 (0 (0 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 1 0 0 0 0 0 1 3 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 4 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 3 0 1 1 0 1 0 0 0 1 1 1 2 3 0 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 0 β , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 3 3 4 4 0 1 1 1 1 1 1 1 2 2 2 3 3 0 1 1 2 2 0 1 2 1 0 1 1 1 2 2 0 1 2 1 0 0 0 1 2 1 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 1 1 1 1 1 1 2 2 2 3 2 3 1 1 1 2 2 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (1 (0 (0 (0 – 100 – 2) 2) 2) 2) 2) 2) / / / 2) / 2) 2) / 2) / 2) 5 5 5 / 5 2) / 4) , , , , , / 5 / 7 / 7 / 5 , , , , 2 ) 5 7 7 7 (0 (0 (0 (0 , , , , / r (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ,j (0 (0 (0 (0 (1 l ⊕ ⊕ ⊕ ⊕ j 2) 2) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ( 2) 2) 2) / / / / 2) 2) 2) 2) 2) 2) 2) 2) r 2) / 2) / 2) 2) 2) 2) / 2) / 2) / 2) 3 3 3 / 2) 2) 2) 2) 3 2) 2) 2) 2) 2) / 2) / 2) / / 2) 2) 2) 2) / 2) 2) 2) ,j , , , , / 3 / 5 / / / / 3 / 3 / 3 / 5 / / / / / / / / / 3 / 3 / 3 3 / / / / 3 / / / l , , , , , , , , , , , β 3 5 5 9 3 3 3 3 5 9 3 3 3 5 5 5 3 3 3 7 5 7 3 3 3 5 7 j , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 2(0 4(0 2(0 2(0 (0 (0 (0 (0 N (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) / 2) / 2) / 2) / / / 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) / / 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / / / / / / / / / 1 / 1 / 1 / 1 1 1 / / / / / 1 / / / / / / / 1 / / / 1 1 / / / / / 1 / / / / / , , , , , , , , , , , 1 1 7 3 3 3 7 11 1 1 1 1 3 3 7 1 1 1 3 3 3 7 9 1 1 1 1 5 3 5 1 3 1 1 3 5 5 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 3(0 (0 2(0 (0 2(0 4(0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 3(0 (0 (0 (0 2(0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 2 1 0 1 0 0 2 0 2 1 0 0 0 2 1 0 1 0 0 0 0 1 0 0 0 0 0 0 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 3 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 0 0 1 2 0 0 0 0 1 1 1 2 2 0 0 1 2 0 0 0 1 1 0 0 0 1 0 0 0 0 1 2 0 0 0 1 0 0 0 0 0 1 0 0 0 3 4 β , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 3 3 3 3 4 5 1 1 1 1 1 1 1 2 2 2 2 3 4 1 1 1 2 3 1 1 0 0 1 1 1 1 2 3 1 1 0 0 1 0 1 1 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 4 5 1 1 1 1 1 1 2 2 3 4 2 2 1 1 2 3 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 (1 (1 (0 (0 (0 JHEP12(2019)039 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 3) 2) 2) 4) 2) 2) 2) 2) 3) 2) 2) , , , / 2) 2) / 3) / / 3) / / / 2) 2) 2) 2) / / / 2) / 2) , 2 , 2 2 / / 5 5 5 7 / / / / 7 5 / 7 / 2) 5 2) 4) 2) 5 4) 2) , , , 2 , / , 2 , / , , / , , , / 5 / 5 11 11 9 5 5 5 / 5 / 7 , , 2 / / , , , , , , 2 , , 5 5 5 7 ) (0 (0 , , / , / , r (0 (0 4(0 (1 7(0 2(1 4(0 (1 5(0 3(1 (1 (1 (0 (0 (0 2(0 7(0 2(1 (0 2(0 (0 ⊕ ⊕ ⊕ (0 (0 (1 2(0 (0 (1 (0 ,j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) 4) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ j , ( / 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 2) 6) 7) 2) 2) 2) 2) 2) 2) / 2) 2) 2) , , , , 2 , , , r 2) 2) 3 / 2) / 2) / / / / / 2) / / / / 4) / 2) 3 2) / / 4) 2) 2) / , 2 2 2 2 / 2 2 , 2 , , / / 3 / 3 / 3 3 3 5 9 / 3 3 3 5 3 / / 3 5 / / 5 ,j l , , , / , / , / , / , / / , , , , 2 , / , , 2 , 3 5 3 7 7 3 7 5 5 β j (1 , , , , , / , , / , , 2(0 (1 (1 (1 (1 (1 (3 (1 2(0 N ⊕ (0 (0 2(0 (0 2(0 (0 3(0 5(0 3(0 3(0 2(1 (0 2(0 2(0 2(0 3(0 (1 5(0 (0 (0 2(0 3(0 (1 (0 (0 2(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 5) / 2) 2) 2) , continued on next page 2) 2) / 2) 2) 2) 2) 2) / 4) 2) / 4) 2) / 4) 2) / 5) / 2) 2) 2) 2) 2) 3) 2) / 4) 2) / 2) 2) 2) 2) 3) 2) 2) 2) 2) , , , 2 , , , , / / 1 / / / / / 7 / 7 / 7 / 9 11 / / / / / / 7 / 1 / / / / / / / / , , 2 , 2 , 2 , / , 2 2 , 2 , 2 1 3 1 1 1 5 1 1 1 3 13 5 1 1 1 3 1 1 5 5 1 3 3 7 3 3 , , , , , , , / , / , / , / , , , , , , / , / , , , , , / , , , , (0 (0 2(0 (0 (0 (0 (0 (0 2(0 (1 (0 3(0 (1 (0 2(0 (1 (0 3(0 2(1 2(0 (1 (1 (0 (0 (0 (0 (0 (1 (0 3(0 (1 (0 2(0 (0 (0 (0 (0 (1 (0 (0 (0 (0 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 3) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 0 2 1 0 0 0 2 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 2 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 2 2 0 0 1 2 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 β , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 2 2 2 2 3 4 1 1 1 2 2 1 1 0 0 1 2 0 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 3 4 1 1 3 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ – 101 – ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 4) 2) 2) 2) 2) 2) 3) 2) 4) 2) 2) 2) 2) , , , / 2) / / / / / 2) 2) / / / / / / / / / / 3) / 2) 2) 2 2 2 , 5 / 5 5 7 7 7 / / 7 5 5 5 5 7 5 5 5 / / , , , , , , , / , , , , / , / , , 2 , 5 9 11 11 5 5 , , , , / , , ) 11 (0 (0 (0 (0 (0 , r 2(0 (0 2(0 2(0 2(0 2(0 5(0 3(1 (1 7(0 2(1 5(0 3(1 (1 4(0 (1 2(0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ,j 2(0 (1 l ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ j 2) 2) 2) 2) 2) ⊕ ⊕ ( / / / / / 2) 2) 2) 2) 2) 2) 2) 3) 2) 2) 2) 2) 3) 2) 2) 2) 2) 2) 2) r , , , , 2) / / 2) 2) 3 / / 4) / 4) 4) / 2) 5) / / 2) 3 3 3 / / / 3 / / 2) / / 2) 2) 2) ,j , , , , , 2 , , , 2 2 , 2 / 3 3 / / 3 5 5 5 / 5 9 / 3 5 9 3 3 / 3 3 / / / l , , , , 2 , 2 2 , 2 , / , , / , / , , / , , , β 5 5 5 7 5 5 5 7 7 j , , , / / / , / , , , , , 2(0 (1 (1 2(0 3(0 2(0 (1 (1 (1 2(0 (1 N (0 2(0 2(0 (0 (0 2(0 3(0 (1 3(0 (1 (1 3(0 (0 (1 3(0 (0 5(0 3(0 3(0 2(0 (0 2(0 2(0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 5) 2) 2) 2) 2) 2) 5) 2) 2) , , 2) 2) 2) 2) 2) 2) / 2) 2) 2) 3) 2) 3) 3) 2) 2) 4) 2) / 2) / / / 2) / 4) 2) / / 2) / 4) 2) 2) 2) 2) 2) 2) 2) 2) , , , , 2 , 2 , / / / / / / 1 / / / / / / / 9 / 1 1 1 / 7 / 9 1 / 7 / / / / / / / / , 2 2 2 2 , / , , , , 2 , / , , 2 3 1 7 1 3 3 1 7 3 3 3 5 3 3 1 3 1 7 1 3 1 1 3 5 5 , , , , , , , , , / , / / , , / , , , / , , / , , , , , , , , (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (1 (0 (1 (1 (0 (0 (1 (0 3(0 2(1 (0 2(0 3(0 2(0 (0 3(0 (1 (0 3(0 2(1 2(0 (0 2(0 (1 (0 (0 (0 (0 (0 (0 (0 (0 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 3) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 3 1 0 0 1 0 1 0 0 0 0 0 2 1 0 1 0 1 0 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 1 2 3 0 1 1 0 1 0 0 1 2 0 0 0 0 0 0 1 0 0 0 1 β , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 2 2 2 3 3 1 1 1 1 2 3 1 2 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 3 1 1 2 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (0 JHEP12(2019)039 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ 2) / 2) 2) 4) 2) 5) 2) 4) 2) 2) / 5 2) / , , , / / 2) 2) 2) 2) / / 2) / , 5 / 5 2 2 2 7 / / / / 7 / 2) , , 5 , / / , / ) / 11 7 9 7 11 9 11 (0 , r , , , , , , , 7 15 (0 (0 ⊕ ) ⊕ , , (0 r ⊕ ,j 5(0 3(1 (1 3(1 (1 (0 5(0 3(1 (1 (1 ⊕ ⊕ l 2) ⊕ (0 3(0 (2 2(0 2) ⊕ j ,j 2) / ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l ( 2) / 2) 2) / 4) ⊕ ⊕ ⊕ ⊕ j r / 3 2) 2) / 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) , ( 2) 3) 2) 4) 7) 13 2) 2) 3) 2) 6) 7) , 3 / / 3 / 3 / / / / / / / / / ,j , , , , , 2 , , r 2) / / 2) 11 7) / / / 2) 2) 2) l , , , 5 3 5 3 3 3 3 3 5 3 5 3 β 2 , 2 2 , 2 / 2 2 j / 5 9 / 5 5 9 / / / ,j , , , , , , , , , , , , l , / , / / 2 , , / , / / 5 5 7 9 7 2(0 (0 β j N 3(1 (1 , , / , , , 2(0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (1 4(0 (1 (1 (1 (3 ⊕ ⊕ N ⊕ ⊕ ⊕ (0 3(0 (0 2(1 (3 2(0 3(0 (0 (0 (0 2(1 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ continued on next page 2) / 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 5) 2) 6) / 2) 5) / 2) 2) / 1 / / / / / 1 / / / / / / / / / / / / / / / / , , , 2) 2) / 2) / 3) 6) 2) 2) / 2) 2) 2) 5) / / , , 1 3 1 5 1 3 1 1 1 1 1 3 1 9 7 3 3 1 1 9 5 1 2 , 2 , 2 , / / 9 / 9 11 / / 9 / / / 11 , , , , , , , , , , , , , , , , , , , , , , , / , 2 / , 2 , / , 2 3 3 3 3 3 5 7 5 13 11 , , , / / , , , , , / , , (0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 3(0 2(1 (0 3(0 (1 3(1 2(1 (3 (0 (0 3(0 2(1 (0 (0 (0 2(0 (1 (1 (0 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , 3) 3) 3) 3) 3) 3) 4) 4) 5) 1 2 0 2 0 0 0 1 1 2 0 2 0 3 1 0 0 0 0 0 0 5 3 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1 3 1 1 0 0 0 minimal SCFT. , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 1 1 1 0 0 3 3 2 2 1 1 1 1 0 0 5 4 3 2 2 1 0 0 0 7 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 E 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 2 0 0 0 0 0 0 0 0 β 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 2 3 1 2 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ – 102 – ⊕ 2) 4) 2) 2) , 2) 2) / / 2) / 2) 2) 2 / / 7 / 2) 5 / / , / 9 11 7 / , 5 5 , , , ) ) 11 7 , , , , r r (0 5(0 3(1 (1 (0 (0 (0 ,j ,j 2(0 (1 (0 ⊕ l l ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ j j ⊕ ⊕ ⊕ ( ( 2) 2) 2) 2) 2) 3) 2) 2) r r 2) / 2) 2) 2) 2) 2) / 2) / 2) / 2) 2) 2) 2) 2) , 2) 2) 5) / / 2) / 2) 2) 2) / 3 / / / / / 3 / 3 / 3 / / / / / ,j ,j , 2 / / 5 9 / 5 / / / l l , , , , 5 3 5 7 5 3 3 5 7 3 7 3 3 2 , / , , β β 7 7 7 5 9 9 j j , , , , , , , , , , , , , , , / , , , , (0 (0 (0 (1 (1 N N (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 . Refined BPS invariants of 6d (0 (0 (1 3(0 (0 2(0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 5) / / / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) , 2) 2) 4) 2) / 2) 2) 2) 2) 2) 2) / / / / / / / 1 / 1 / 1 / / / / / / / / / / / / , 2 / / / 9 / / / / / / , , , 3 1 1 3 5 3 1 1 3 5 1 3 5 1 5 1 3 1 7 3 1 2 , / 5 5 3 5 3 3 9 7 7 , , , , , , , , , , , , , , , , , , , , , , , / , , , , , , , (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 3(0 2(1 (0 (0 (0 (0 (0 (0 Table 26 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 3) 3) 3) 3) 3) 3) 4) 4) 4) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 3 1 0 1 0 0 0 1 3 1 4 2 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 0 0 0 2 1 0 1 0 0 0 0 0 1 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 0 0 4 3 2 2 2 1 1 1 0 0 0 4 3 3 2 2 1 0 0 0 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 β , , , , , , , , , , , , , , , , , , , , , , , , β , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 2 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 2) 2) 2) 2) 2) 2) / 2) / / / / / 2) 5 / 5 5 5 2) 5 , , , , , 5 / 7 / , , 7 5 ) (0 (0 (0 (0 (0 , , r (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ (0 (0 ,j ⊕ ⊕ l 2) 2) 2) 2) 2) ⊕ ⊕ j ( 2) 2) 2) 2) 2) / 2) / / / 2) 2) 2) 2) / r 2) 2) 2) / 2) 2) 2) 2) / / 2) / 2) / 2) 2) 2) 2) 2) 3 2) 2) 2) 2) / 3 3 3 2) / 2) / / 2) / 2) 2) 2) 2) 2) 3 , , , , , / / / 3 / / / / 3 3 / 3 / 3 / / / / / / / / / 5 / 3 / 3 3 / 3 / / / / / ,j l , , , , , , , , , , 3 5 3 3 3 7 5 7 3 3 3 5 5 5 3 3 3 9 5 3 5 9 7 7 7 3 β j , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 2(0 2(0 4(0 2(0 (0 (0 (0 (0 2(0 N (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) continued on next page 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / / 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) / / / 2) / 2) / / 2) / 2) 2) 2) 2) 2) 2) 2) / / / / / / / 1 / / / / / 1 1 / / / 1 / / / / / / / 1 / / / / / 1 1 1 / 1 / 1 1 / 1 / / / / / / / 1 , , , , , , , , , , , , , 5 5 1 1 3 1 1 3 1 5 3 5 1 1 1 1 9 7 3 3 3 1 1 1 7 3 3 1 3 1 7 9 5 5 5 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 2(0 (0 (0 (0 3(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 2(0 4(0 2(0 (0 2(0 (0 2(0 3(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 2) 2) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 0 0 0 0 0 0 1 0 0 0 0 1 0 1 2 0 0 0 0 1 3 1 0 0 0 0 0 1 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 1 0 0 0 0 1 0 0 0 1 1 0 0 0 2 1 0 0 2 2 1 1 0 0 0 1 0 1 1 0 3 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 1 1 0 1 0 0 1 1 3 2 1 1 1 1 0 0 1 1 3 2 1 1 1 4 3 2 2 2 2 1 1 1 1 1 1 0 4 4 3 3 3 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 2 1 3 2 1 1 2 2 4 3 2 2 1 1 1 1 1 1 5 4 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 1 1 1 1 3 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 – 103 – 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 4) 2) / / / 2) 5 2) / / / / 2) / 2) / 2) , , / 5 5 5 / / 7 5 5 5 / 7 / 5 / 2 , , , , , , , , , 5 5 5 7 7 5 ) (0 / , , , , , , r (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ (1 (0 (0 (0 (0 (0 (0 ,j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ j ( 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) 2) r 2) 2) / / 2) / 2) 2) / / / 2) / 2) / / / 2) 2) 2) 3 2) 2) 2) / 2) / / / 2) 2) / 2) / 2) 2) 2) 2) / 2) / 2) , / / 3 3 / 3 / / 3 3 3 / 3 / 3 3 3 / / / / / / 5 / 3 3 3 / / 3 / 3 / / / / 5 / 3 / ,j l , , , , , , , , , , , , , , , , , , 7 3 5 3 3 3 5 7 3 5 3 3 3 7 5 3 3 3 3 9 5 5 3 β j , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 2(0 (0 (0 N (0 (0 (0 (0 (0 2(0 2(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 2(0 2(0 (0 (0 (0 (0 (0 (0 (0 2(0 (0 2(0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / / / / / / / / / / 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2) 2) 2) 2) / / / 1 1 / 1 / / / / / 1 / / / 1 1 1 / / / / / 1 / / / / / / / / / / / 1 / 1 / / / / / / / / / , , , , , , , , , , 5 1 1 3 1 1 1 1 1 1 1 3 1 5 1 3 5 1 1 1 3 3 5 1 1 1 3 1 1 1 1 11 7 3 3 3 7 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 2(0 2(0 (0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 2(0 2(0 2(0 (0 (0 (0 (0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 3(0 (0 2(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 2) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 2 0 2 0 0 1 0 1 2 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 3 2 1 1 1 0 0 0 0 2 1 0 0 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 0 1 2 1 0 0 0 1 2 1 0 2 2 1 1 1 0 1 2 1 0 2 2 1 1 0 3 3 2 2 2 1 1 1 1 1 1 1 5 4 3 3 3 3 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 1 1 2 2 1 1 1 3 2 3 2 2 2 1 1 1 1 1 1 4 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 2 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 3) 2) 2) 2) 4) 2) 2) 2) 3) 2) 2) , , , / 2) / / / 2) 2) 2) 2) / / / / 3) / / 3) 2) 2) / 2 2 , 2 , 2) 7 / 4) 5 2) 5 7 / / / / 7 5 5 5 4) / 2) / 5 4) , , , , / , , / , 2 , / , 2 , , , / 5 / 5 5 5 9 11 11 5 / 5 , 2 , , , , , , / / 2 , , 2 7 5 5 ) (0 , / , / , / r 2(0 (0 7(0 2(1 2(0 (0 (0 (0 (1 5(0 3(1 (1 4(0 (1 7(0 2(1 4(0 (1 (0 (0 2(0 ⊕ ⊕ (0 (1 (0 2(0 (1 (0 (1 ,j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) 4) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ j , ( 2) 2) / 2) 2) 2) 2) 2) 2) 6) 7) 2) 3) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) , 2 , , , , , , r 2) 2) / 4) / 2) 3 2) / / 4) / / / 2) / / / / / 2) / 2) 2) / / 2) 2) , , , 2 / 2 2 2 2 2 2 / / 5 3 / / 3 5 3 3 3 / 5 9 3 3 3 / 3 / / 3 3 / / ,j l , 2 , , / , 2 , , , / / , / , , / , / , / , , , 5 5 7 3 7 7 3 5 5 7 β j (1 , , / , , / , , , , , , 2(0 (1 (3 (1 (1 (1 (1 (1 N ⊕ (0 (0 3(0 (1 2(0 (0 (0 5(0 3(0 (1 2(0 2(0 2(0 (0 2(1 3(0 3(0 5(0 3(0 (0 2(0 (0 (0 2(0 2(0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) / 2) 2) 5) 2) 2) 2) , continued on next page 2) 2) 2) 2) 3) 2) 2) 2) / 2) 2) / 4) 2) 3) 2) 2) 2) 2) 5) / 2) / 2) / 4) 2) / 4) 2) / 4) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , 2 , , , / / / / / / / 1 / / 7 / / / / / 11 / 9 / 7 / 7 / 7 / / / / / / / / / 2 , , 2 2 , 2 , / , 2 , 2 , 2 3 7 3 3 1 5 5 1 1 3 1 1 1 5 13 3 1 1 1 5 1 1 3 1 1 7 3 5 , , , , / , , , , , / , / , , , , / , , , / , / , / , , , , , , , , , (0 (0 (0 (0 (1 (0 (0 (0 2(0 (0 (0 3(0 (1 (0 (1 (0 (0 (0 (0 2(0 (1 (1 (0 3(0 2(1 (0 2(0 (1 (0 3(0 (1 (0 2(0 (1 (0 (0 (0 (0 (0 (0 (0 (0 (0 3) 3) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 2 0 0 0 0 1 3 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 2 1 0 0 2 2 1 1 0 0 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 0 2 1 0 0 1 1 2 2 1 1 1 4 3 2 2 2 2 1 1 1 1 1 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 0 3 1 1 4 3 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ – 104 – 2) 2) 2) 2) 4) 2) 2) 3) 2) 2) 2) 2) 4) 2) 2) 2) 2) 2) 2) 2) , , , 2) 2) / / 3) / / / / / / / / / 2) 2) / / / / / 2) / , 2 2 2 / / 5 5 5 7 5 5 5 5 7 / / 7 7 7 5 5 / 5 2) , , 2 , , / , / , , , , / , , , , , , 5 5 11 11 9 5 / , , / , , , , 11 5 ) (0 (0 (0 (0 (0 (0 , , r (0 (0 2(0 4(0 (1 5(0 3(1 (1 7(0 2(1 5(0 3(1 (1 2(0 2(0 2(0 2(0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2(0 (1 (0 ,j ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 2) 2) 2) 2) 2) 2) ⊕ ⊕ ⊕ j ( 2) 2) 2) 2) 2) / 2) 3) 2) 2) 2) / / / 2) 3) 2) 2) 2) 2) 2) / 2) / , , , , r 2) 2) / / 2) / / 3 / / / 3 3 3 2) / / 2) 5) / 4) / 4) / 4) / 3 / 3 2) 2) 2) 2 , 2 2 , , , 2 , , , , , , / / 3 3 / 3 3 5 9 3 / 5 9 / 5 5 5 3 3 / / / ,j l , , , , / , / , , / , / , 2 , , , , , 2 2 2 7 5 5 5 7 5 3 5 β j , , , , , / / / / , , , (1 2(0 (1 (1 (1 2(0 3(0 2(0 (1 (1 2(0 2(0 N (0 (0 2(0 2(0 (0 2(0 3(0 3(0 5(0 (0 3(0 (0 (1 3(0 (1 3(0 (1 3(0 (1 2(0 2(0 (0 (0 (0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 2) 5) 2) 2) 2) 2) 2) 5) 2) 2) , , / / / / 4) / / 2) 2) 2) 2) 2) 2) 2) 2) / 4) 2) / 2) / 4) 2) 2) / 2) 2) 3) 2) 3) 2) 3) 2) 2) 2) 2) 2) 2) 2) , 2 , 2 , , , , / / / / / / / / 7 1 / 9 / 7 1 1 1 / / 9 / / / / / / 1 / 1 / / / / , 2 , , / , 2 , , , , / 2 2 2 2 , , 5 3 1 1 3 1 7 1 3 1 3 3 5 3 3 3 1 7 1 3 1 5 3 , , , , , , , , / , , / , , , / , / , / , / , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 2(0 (1 2(0 (0 3(0 2(1 (0 3(0 (1 2(0 3(0 2(0 (0 (0 3(0 2(1 (0 (1 (0 (1 (0 (1 (0 (1 (0 (0 2(0 (0 2(0 (0 (0 (0 (0 3) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) 2) , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 0 0 0 0 0 0 1 0 1 0 1 2 0 0 0 0 0 1 0 1 1 2 0 2 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 1 0 0 0 0 0 0 2 1 0 0 1 0 1 1 0 3 2 1 1 1 0 0 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 1 1 1 1 2 1 3 2 1 1 1 1 3 3 2 2 2 1 1 1 1 1 1 1 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , β 0 1 2 1 1 3 2 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 JHEP12(2019)039 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) 4) 2) 5) 2) 4) 2) , , , 2) / / / 2) 2) 2) 2) / / 2) 2 2 2 / 7 / / / / 7 2) / , / / , / 9 11 11 7 7 9 11 / 7 , , , , , , , , 15 7 ⊕ ) , , r ⊕ (1 5(0 3(1 (1 (0 3(1 (1 5(0 3(1 (1 (0 2) ⊕ 2(0 3(0 (2 (0 ,j 2) / ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ l 4) / ⊕ ⊕ ⊕ ⊕ j , ( 6) 7) 2) 3) 2) 2) 4) 7) 13 2) 3) 2) 2) 2 , , , , , , , r 2) 2) 2) / / / 2) 11 7) / / 2) / / 2 2 2 , 2 2 , 2 / / / 5 9 5 / 5 9 / 5 ,j l / / , / , , / / 2 , / , , 7 9 7 5 5 β j (1 3(1 , , , , / , (3 (1 (1 4(0 (1 (1 (1 N ⊕ ⊕ (0 2(1 (0 (0 3(0 2(0 (0 2(1 (3 3(0 (0 2(0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 2) ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) / 2) 2) 5) 2) 6) / 2) 5) , , , / 2) 5) / 2) 2) 2) / 2) 2) / 3) 6) 2) / 2) 2) , 2 , 2 , 2 / 11 / / / 9 / / 9 11 / 9 / / , 2 , / , 2 / , 2 , / 11 5 13 7 5 3 3 3 3 3 3 , , / , , , , , , / / , , , (0 (0 2(0 (1 (1 (0 (0 (0 3(0 2(1 (0 (0 3(0 (1 3(1 2(1 (3 (0 3(0 2(1 (0 (0 5) 4) 4) 3) 3) 3) 3) 3) 3) 3) , , , , , , , , , , 0 0 1 0 0 1 0 1 0 1 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 minimal SCFT. , , , , , , , , , , 0 0 0 0 0 0 0 0 2 1 8 , , , , , , , , , , E 1 2 1 1 2 1 3 2 1 1 , , , , , , , , , , β 0 0 0 1 1 1 0 0 0 0 , , , , , , , , , , 0 0 0 1 0 0 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ⊕ ⊕ ⊕ – 105 – 2) 4) 2) , 2) / / 2) 2) 2 2) / 7 / / , / / 7 11 9 , , , 7 11 ) , , r (0 5(0 3(1 (1 (0 (1 2(0 ,j ⊕ ⊕ ⊕ ⊕ l ⊕ ⊕ ⊕ j ( 2) 2) 3) 2) , r 2) 2) 2) / 2) / / 5) 2) 2) 2) 2 , / / / 5 / 5 9 / / / ,j l , , / , 2 9 9 5 7 7 7 7 β j , , , , / , , , (1 (1 N (0 (0 (0 2(0 (0 3(0 (1 (0 (0 (0 . Refined BPS invariants of 6d ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2) 5) , 4) 2) 2) 2) 2) 2) 2) 2) / 2) 2) 2) 2 , / / / / / / / 9 / / / , / 2 7 7 9 3 3 5 3 5 5 5 , , , , , , , / , , , (0 (0 (0 (0 (0 (0 (0 3(0 2(1 (1 (0 (0 (0 Table 27 4) 4) 4) 3) 3) 3) 3) 3) 3) 3) , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 1 0 , , , , , , , , , , 0 1 0 0 1 0 1 0 1 1 , , , , , , , , , , 1 1 1 1 1 1 2 2 1 1 , , , , , , , , , , β 1 0 0 2 1 1 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , (0 (0 (0 (0 (0 (0 (0 (0 (0 (0

JHEP12(2019)039

4 iia SCFT. minimal F 6d of degrees selected for invariants BPS Refined . 28 Table

4) , 0 , 0 , 1 , 7 , (2 = β

12 1 2 2 1

11 3 8 9 5 1

03 59 25 31 20 6 1 10

9 46 66 31 23 66 86 61 24 6 1

57 5 0 5 34 53 150 205 159 75 25 6 1 8

57 9 6 3 1 1 12 112 311 438 366 198 79 25 6 1 7

58 1 5 5 4 8 2 21 32 221 586 843 752 457 212 80 25 6 1 6

58 1 9 4 4618 02427 4 71 402 1022 1489 1406 943 495 216 80 25 6 1 5

58 1 0 0613 3529 63681412 144 678 1633 2390 2365 1730 1016 504 216 80 25 6 4

97 1 0 0415 8732 5820 08283 1 31 258 1048 2402 3518 3625 2857 1854 1034 504 216 79 19 3

7 6 7 7327 0443 5333 45446 4 62 414 1455 3137 4553 4835 4064 2878 1783 975 466 177 2

2 4 4621 5848 3348 4510 5 0 9 107 556 1701 3425 4886 5303 4687 3568 2415 1456 743 222 1

3 2721 7339 8832 5219 2 2 18 124 528 1394 2572 3528 3828 3492 2793 2015 1287 630 0

R L 01 21 41 61 81 02 22 42 62 829 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

4) , 0 , 0 , 1 , 8 , (1 = β

15 1 2 1

– 106 – 14 3 7 5 1

13 72 8 20 17 6 1

12 14 92 1 20 49 46 21 6 1

11 25 0 0 44 44 105 107 59 22 6 1

10 26 4 2 0 911 89 207 224 141 63 22 6 1

9 26 5 0 3 8 7 81 28 172 384 433 302 155 64 22 6 1

26 5 3 8 7 6 0 24 62 308 660 772 588 338 159 64 22 6 1 8

26 6 5 7 0819 09521312 123 522 1079 1295 1058 670 352 160 64 22 6 1 7

26 6 5 0 2015 0816 3 2 01 30 229 839 1660 2028 1757 1210 702 355 160 64 22 6 1 6

26 6 5 1 2621 7829 4816 9 64 66 390 1266 2418 2999 2728 2016 1276 712 356 160 64 22 6 1 5

16 6 5 1 2520 0931 1630 82671512 125 617 1802 3306 4136 3919 3079 2105 1285 710 355 160 64 21 4

916316815 0534 2357 2849 3992282 1 28 218 902 2379 4197 5278 5179 4293 3140 2075 1252 688 341 146 39 3

4 9 1717 8645 2264 9748 8118 3 63 56 332 1185 2841 4782 5997 6043 5232 4050 2866 1871 1117 593 248 2

4 7 4220 1544 2154 7549 8412 3 88 88 431 1326 2874 4599 5705 5848 5241 4247 3175 2209 1412 773 244 1

3 0614 2122 4532 7638 0515 9 0 21 12 102 398 1051 2055 3081 3726 3828 3495 2920 2271 1649 1076 534 0

R L 01 21 41 61 81 02 22 42 62 82 03 23 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

4 F

JHEP12(2019)039

6 al 29 Table iia SCFT. minimal E 6d of degrees selected for invariants BPS Refined .

0 , 5 , 1 , 0 , 1 , (0 = β 5) , 0 ,

16 1 2 1

15 3 7 5 1

14 72 8 20 17 6 1

13 14 92 1 20 49 46 21 6 1

12 26 1 1 74 47 112 112 60 22 6 1

11 26 4 4 2 712 97 225 241 148 64 22 6 1

10 26 6 2 8 2 9 01 30 191 429 480 328 163 65 22 6 1

26 6 6 6 8 6 5 94 69 356 768 889 661 368 167 65 22 6 1 9

26 6 8 5 2114 27651313 143 625 1297 1541 1231 754 382 168 65 22 6 1 8

26 6 8 9 4023 5827 05223 1 34 272 1035 2074 2518 2135 1420 791 386 168 65 22 6 1 7

26 6 8 0 5528 4239 1413 8 74 77 485 1638 3164 3896 3472 2481 1505 805 387 168 65 22 6 1 6

26 6 8 0 5122 0356 6744 44891512 155 809 2444 4543 5657 5262 4003 2624 1531 807 387 168 65 22 5 5

916327712 6340 9674 7465 4015 8 11 31 285 1251 3430 6157 7724 7449 5986 4203 2653 1522 797 382 166 59 4

6327515 5841 0783 6092 7849 71406 4 69 470 1791 4493 7748 9729 9660 8136 6067 4112 2558 1458 755 342 96 3

2 2725 6653 6797 15 07 7550 39651512 125 695 2319 5308 8765 10971 11154 9776 7677 5532 3686 2253 1227 527 2

7 4329 1859 8495 09 09 3753 5182132 1 24 193 872 2551 5332 8357 10299 10592 9553 7804 5899 4158 2690 1483 470 1

9 9030 1054 2962 6455 7720 8 0 72 37 208 783 2005 3777 5557 6654 6824 6249 5241 4100 3001 1980 992 0

R L 01 21 41 61 81 02 22 42 62 82 03 23 435 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

6) , 0 , 0 , 4 , 1 , 0 , 1 , (0 = β – 107 –

15 1 2 1

14 3 7 5 1

13 72 8 20 17 6 1

12 14 92 1 20 49 46 21 6 1

11 25 0 0 44 44 105 107 59 22 6 1

10 26 4 2 0 011 90 209 225 141 63 22 6 1

9 26 5 0 3 8 7 81 28 173 386 434 302 155 64 22 6 1

26 5 3 9 8 7 1 14 61 311 670 780 591 339 159 64 22 6 1 8

26 6 5 7 0110 00571312 123 527 1090 1303 1061 671 352 160 64 22 6 1 7

26 6 5 0 2917 0518 4 2 11 31 227 845 1686 2055 1774 1219 705 356 160 64 22 6 1 6

26 6 5 1 2922 7632 4118 9 64 66 390 1280 2451 3029 2746 2025 1279 713 356 160 64 22 6 1 5

16 6 5 1 2422 1336 1835 87681811 128 618 1817 3359 4198 3969 3113 2123 1294 713 356 160 64 21 4

916316915 0435 3953 3746 4594292 1 28 219 904 2405 4260 5347 5234 4329 3159 2084 1255 689 341 146 39 3

4 9 1618 9040 3063 0545 8819 3 34 53 336 1194 2868 4859 6095 6137 5310 4104 2900 1889 1126 596 249 2

4 7 4022 2449 3652 7846 9613 3 88 88 435 1335 2906 4668 5788 5927 5306 4293 3204 2225 1420 776 245 1

4 1118 3729 5831 8734 0716 0 915 99 400 1063 2077 3142 3807 3913 3578 2991 2327 1688 1101 546 0

R L 01 21 41 61 81 02 22 42 62 82 03 23 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

6 E

JHEP12(2019)039

7 al 30 Table iia SCFT. minimal E 6d of degrees selected for invariants BPS Refined .

, 0 , 1 , 5 , 1 , 0 , (0 = β 5) , 0 , 0

16 1 2 1

15 3 7 5 1

14 72 8 20 17 6 1

13 14 92 1 20 49 46 21 6 1

12 26 1 1 74 47 112 112 60 22 6 1

11 26 4 4 2 712 97 225 241 148 64 22 6 1

10 26 6 2 8 2 9 01 30 191 429 480 328 163 65 22 6 1

26 6 6 6 8 6 5 94 69 356 768 889 661 368 167 65 22 6 1 9

26 6 8 5 2114 27651313 143 625 1297 1541 1231 754 382 168 65 22 6 1 8

26 6 8 9 4023 5827 05223 1 34 272 1035 2074 2518 2135 1420 791 386 168 65 22 6 1 7

26 6 8 0 5528 4239 1413 8 74 77 485 1638 3164 3896 3472 2481 1505 805 387 168 65 22 6 1 6

26 6 8 0 5122 0356 6744 44891512 155 809 2444 4543 5657 5262 4003 2624 1531 807 387 168 65 22 5 5

916327712 6340 9674 7465 4015 8 11 31 285 1251 3430 6157 7724 7449 5986 4203 2653 1522 797 382 166 59 4

6327515 5841 0783 6092 7849 71406 4 69 470 1791 4493 7748 9729 9660 8136 6067 4112 2558 1458 755 342 96 3

2 2725 6653 6797 15 07 7550 39651512 125 695 2319 5308 8765 10971 11154 9776 7677 5532 3686 2253 1227 527 2

7 4329 1859 8495 09 09 3753 5182132 1 24 193 872 2551 5332 8357 10299 10592 9553 7804 5899 4158 2690 1483 470 1

9 9030 1054 2962 6455 7720 8 0 72 37 208 783 2005 3777 5557 6654 6824 6249 5241 4100 3001 1980 992 0

R L 01 21 41 61 81 02 22 42 62 82 03 23 435 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

– 108 –

4) , 2 , 0 , 0 , 1 , 5 , 0 , 0 , (0 = β

12 1 2 2 1

11 3 8 9 5 1

10 03 59 25 31 20 6 1

9 46 66 31 23 66 86 61 24 6 1

8 57 6 0 5 44 54 152 207 160 75 25 6 1

57 9 6 4 1 1 12 113 313 440 367 198 79 25 6 1 7

58 1 6 6 5 9 2 11 31 225 597 857 762 460 213 80 25 6 1 6

58 1 9 4 4610 05487 4 71 408 1035 1504 1416 946 496 216 80 25 6 1 5

58 1 0 0715 4823 65681113 141 688 1665 2438 2408 1752 1027 507 217 80 25 6 4

97 1 0 0716 8037 5324 06283 1 31 258 1066 2444 3573 3670 2880 1865 1037 505 216 79 19 3

7 6 8 8622 1144 6639 46426 3 66 412 1476 3199 4656 4945 4141 2928 1806 986 469 178 2

2 4 4623 6145 3648 4912 5 0 9 108 558 1728 3489 4980 5396 4754 3611 2436 1466 746 223 1

4 3727 8130 9732 6711 3 3 41 14 131 531 1417 2627 3627 3957 3608 2891 2079 1327 647 0

R L 01 21 41 61 81 02 22 42 62 829 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 j 2 \ j 2

7 E JHEP12(2019)039 , Nucl. , SCFTs Nucl. , D 6 Nucl. Phys. 1 1 , (1978) 149 minimal SCFT. (2015) 017] B 135 8 1 3 2 E 06 ]. 4) 3) Atomic Classification of , , Nucl. Phys. ]. 0 2 1 2 2 1 , , , SPIRE 1 4 8 5 ]. ]. 0 0 IN , , Erratum ibid. 0 0 ][ [ SPIRE , , 3 3 IN , , SPIRE SPIRE 1 1 ][ IN IN , , On the Classification of 6D SCFTs and 1 1 ][ ][ , , 1 5 9 8 3 – 109 – 1 1 1 4 10 15 9 1 , , (2014) 028 instantons and tensionless noncritical strings 1 1 , , 8 05 E BPS states of exceptional noncritical strings = (1 = (0 arXiv:1502.05405 β β [ ]. 1 6 20 31 25 9 Compactifications of F-theory on Calabi-Yau threefolds. 2. JHEP Small hep-th/9607139 , [ 1 4 11 20 25 15 2 hep-th/9603161 hep-th/9602120 [ [ SPIRE IN 17 44 79 91 63 23 2 ][ (2015) 468 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 6 22 51 68 50 17 1 1 4 10 19 28 31 19 4 (1997) 177 63 . 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