Symmetry Enhancements Via 5D Instantons, Q 시-Algebrae and (1, 0) Superconformal Index

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JHEP09(2016)053 Springer algebrae. July 11, 2016 : W q September 1, 2016 September 9, 2016 , : : Received 10.1007/JHEP09(2016)053 Accepted Published doi: [email protected] Published for SISSA by , superconformal index singularity. Upon circle compactiﬁcation to 5 dimen- 0) k , 0) superconformal six-dimensional theories arising from A , [email protected] , (1 = (1 N . 3 0) index, which we compute by letter counting. We ﬁnally show that , 1606.03036 The Authors. Field Theories in Higher Dimensions, Solitons Monopoles and Instantons, c

We explore , [email protected] -algebrae and W SISSA/ISAS and INFN —via Sezione Bonomea di 265, Trieste, 34136, Trieste,E-mail: Italy [email protected] Open Access Article funded by SCOAP ArXiv ePrint: matches the 6d (1 S-duality of the pq-web implies new relations among vertexKeywords: correlators of Supersymmetry and Duality, Topological Strings Abstract: M5 branes probing asions, transverse we describe this systemof with basic a dual five-dimensional pq-web instantona of operators five-branes single driving and M5 propose global brane, the symmetry we spectrum find enhancement. that For the exact partition function of the 5d quiver gauge theory q Sergio Benvenuti, Giulio Bonelli, Massimiliano Ronzani and Alessandro Tanzini Symmetry enhancements via 5d instantons, JHEP09(2016)053 14 ]. On one hand 3 2 ]. On the other it has 4 11 7 5 ]. S 6 , 5 – 1 – 5 Abelian linear quiver Abelian circular quiver d d 21 16 1 : 6d and 5d gauge theory descriptions k Z algebrae ]. In this context, the study of supersymmetric gauge theories via / 2 20 2 , C W 1 q 0) superconformal index , 0) superconformal theories. We calculate those indices for a single M5 via letter , In this paper we address the problem of circle compactification of M5 brane sys- 5.1 Modularity and partition function on = (1 counting and show that theyquiver coincide gauge theories. with partition The functions five-dimensionalerties gauge of which, theories suitable after we five-dimensional T-dualizing consider to fulfilof type modular IIB, prop- a reveal pq-web to be five-branement encoded system. of in the global These S-duality symmetries properties latter induced are crucial by in instanton order operators. to expose More the precisely, we enhance- identify of strongly coupled super-conformalin field this theories case has in been sixbranes used in dimensions. to terms capture BPS of informations supersymmetric state about indices counting the [ circle compactificationtems of M5 transverse to anN ALE orbifold singularity, which encodes indices of six-dimensional time backgrounds [ localization and BPS state counting hasthis revealed to has be produced a new very powerful correspondences tooland among [ quantum two-dimensional field conformal theories, topologicalstimulated field the theories theories, study of as higher dimensional for supersymmetric instance gauge theories [ as deformations 1 Introduction and results The description of systems ofunderstanding multiple of M5 branes M-theory. isin still Nonetheless, an the elusive many BPS problem progresses in protected our have sector current been by obtained studying recently M5 brane compactifications on various space- A Nekrasov partition function 6 pq-webs and 7 Open questions 3 6d (1 4 Exact partition functions: 5 5 Exact partition functions: 5 Contents 1 Introduction and results 2 M5 branes on JHEP09(2016)053 - k q Z . / (2.1) 2 3 parti- C d ], which partition 12 5 NS5 branes S NS5 branes. N N -algebrae. The D6 ]: terms, associated W k q 11 -algebrae, which we – 9 singularity. This is an W , with the q . k 3 . 0) superconformal index ]. Using the modularity Z 1 , 7 k / − spurious 2 N C d 6 = (1 k,A T D6 branes. ). This generalizes the result of k N ] appeared. Their expressions for k 8 5.17 k special functions [ 2 – 2 – G ······ NS5 0) supermultiplets agree with ours in section , N abelian nodes ( k . k NS5 branes on top of of the form 6 1 N − computation is the presence of . N 0) field theory that we call k d ) , : 6d and 5d gauge theory descriptions D6’s. hypermultiplets, computed in section k k partition function. k 2 = 1. Z 5 k / k S 2 k : the M5’s become NS5 branes and the orbifold geometry Figure 1 k C tensor multiplets parameterizing the positions of the Z ] for / N 5 2 M5 branes sitting at the tip of the orbifold C while this paper was being finalized, [ N D6 branes. So we end up with the following brane setup [ k There are also If we separate the NS5 branes we go on the nowadays called tensor branch [ We can gain some knowledge about this class of theories reducing to Type IIA on a These results have also a nice interpretation in terms of representation theory of One crucial issue in the 5 theory is a linear quiver SU( Both from the Type IIA brane setup and from the quiver, it is easy to see the global circle inside the becomes sitting on top of a stack of gives a Lagrangian IR description of the deformed SCFT. On the tensor branch the field 2 M5 branes on We start from interacting superconformal (1 pq-web S-duality suggests relationscheck among in correlators some of examples different in section Note added: the superconformal index of the free (1 are infinite towers oftion spurious function terms. can Once beproperties, we written we remove build in these the terms contributions, of the 5 deformed infinite-dimensional Lie algebrae.tion Five-dimensional functions quiver can gauge be theories interpreted parti- as correlators of vertex operators of for a tensor multiplet and function of a necklaceLockart quiver and with Vafa [ to parallel external legs in the pq-web: for a pq-web on a cylinder we show that there a set of basicperturbative five-dimensional corrections instanton and operators allows whichplethystic to generates exponential write the form. full the By tower supersymmetricplethystic going of to partition formulae non- the functions in S-dual in termsThe frame a we of main are result characters then we of able obtain to the is write expected the these enhanced comparison global of symmetries. the 6d JHEP09(2016)053 , , ) ). 1 2 2 F S the M5 R ( and F (2.3) (2.4) (2.2) ( 2 k × tr N Z R tr R / ∗ 2 C surrounding gauge theory, singularity, so 4 k S k ) Reducing Z / N = 2 quivers. Their 2 C N . d . k N || ]. ]. One alternative description 18 13 ) on the sphere N 2 k . If this pq-web was on N 1 F ( S tr 5d gauge theories contain non pertur- N k N ]. The gauge theory is a linear quiver = 1 the theory is free, we will compute its = 1 circular quiver SU( 15 linear quiver, while putting the pq-web on , then the results of the two computations , partition functions. Notice that it is not the 1 N Let us first analyze the brane setup on N 1 singularity on a circle transverse to d 14 S − k k – 3 – ······ ) Z × ······ ······ / 4 N 2 S C 0) free supermultiplet. N , k NS5’s [ N k NS5’s. The gauge theory in this case is the linear quiver NS5 branes along the N N k N N ] for a recent discussion in the case of 3 || : S-duality for the rectangular pq-web. D4 branes sitting at the tip of the orbifold 20 d k , which are clearly perturbatively different. However, if we are able ]. N N and and 5d S-duality. 1 − 2 19 R k , ) 1 flavors at both ends: R , charged under the topological symmetries whose currents are flavors at both ends: − . There is also an additional global U(1) symmetry, that acts on all the N I 17 2 N D5’s intersecting N , k ) d k 5 16 k,A N D5’s intersecting T with with or SU( k . In a balanced linear quiver, there is a special set of ‘minimal’ monopole or 1 1 1 2 − − − S k N N ) ) ) This is analogous to the 3d case, where monopole operators carry a flux for The right way to think about this 5d “duality” is that there is a strongly coupled It is also possible to perform a type IIB S-duality on the brane setup, getting the k k N bifundamentals hypers with charge 1. For D5 branes along the point measures the instanton charges of the operatorson [ an instanton operators, see [ topological charges are 0 or 1 and are defined by the property that the non vanishing charges SCFT, like the partitionshould function agree on [ Instanton operators andbative global operators symmetry. When inserted at a point in space-time, the flux of 5d SCFT correspondingtop to of the each completely other,This unresolved UV emanating pq-web, SCFT from admits where relevant aSU( all deformations single that branes can point are lead andto to on perform respecting either computations IR a in Lagrangian the rescaling QFT: IR symmetry. QFT’s that can be uplifted to the strongly coupled UV pq-web of SU( a pq-web of SU( the cilinder the fieldthat theory we becomes call the 5d The pq-web on along the M5’s, we get the gauge theory can beis understood given using in the terms methods of ofk a [ pq-web of 5gauge branes theory in would type have IIB: been the the branes are SU( on a cilinder N superconformal index and compare itusual with gauge 5 theory orbifold of the (2 Compactification to 5 branes sitting at the tip of the orbifold symmetry SU( JHEP09(2016)053 ], th th N i i 21 (2.6) (2.5) ). k . Here . N − i, and the under each ) node with , each gauge , putting to- . The corre- 1 1 N th I − − − k k 1) i, ) ) − N N is charged precisely i we can organize the j 2 anti-monopoles and / N 1) − N k bifundamentals. This is the global symmetry. This global ( to node i      1 on the ( k 2 i th 1 1 1 1 ) − , , , , bar; N 1 1 1 0 ] it is shown how topological and , , , , N N 1 1 0 0 , , , , 21 1 0 0 0 = 1 5d theories. Let us define + 1) I I I I U(1) 1 currents of the enhanced SU( N ], j SU( and are charged under U(1) 0 0 0 N , , , − × 21 1 1 1 + and neutral under U(1) , , , i 2 i, 1 1 0 , , , N k k bar 1 0 0 + I I I top; i, 0 0 , , N and the ( – 4 – U(1) 0 0 , , ] and in 1 1 U(1) , , × th 1 0 i 1 I I 25 − 1 2 , 0 k top , 0 20 ≡ , 0 , 1 i, I to act with charge +1 and      i U(1) bar; 2 basic monopoles, the correponding / = 1 supersymmetry the story should be similar, but it is not much 1) = 4 theories it is known that these non perturbative operators form a = 2 3d theories [ − N k ]. N ( N k 24 – ) gives a U(1) “topological” or “instantonic” global symmetry, whose current is N 22 1 topological U(1) currents, we get the ). Each bifundamental hypermultiplet is charged under a standard U(1) “baryonic” 2 − Armed with these results we can study the global symmetries that can be inferred One important feature of these instanton operators is their baryonic charge spectrum: In 5d with In 3d for F k ( tr symmetry is actually enhancedbaryonic in symmetries the can UV enhance SCFT. toquiver) a In where non [ Abelian every group: nodesquare if is of we balanced, have the a then group IR thezero whose quiver Chern-Simon global (or Dynkin coupling a symmetry diagram is sub of is balanced the the if UV the quiver total SCFT in number question. is of the flavors A is U( precisely 2 sponding anti-instantons are neutral under U(1) from the low energygroup Lagrangian U( description. In∗ the gauge theory SU( symmetry. The theory enjoys a U(1) the basic instantons are charged under U(1) instanton state whose topological chargesunder are the 1 symmetry from node thatcase rotates both the in baryonic symmetry U(1) bifundamental, respectively. Then thetimes basic the instantons topological have charges. baryonic charges So, equal denoting as to in [ see also [ the instanton operators arewhich charged are under usually the calledstudying Abelian baryonic fermionic symmetries. factors zero of modes The around the charges the can global instanton in symmetries, background. principle It be turns computed out that a basic contain conserved currents withgether scaling these dimension 2.the For a quiver U( discussed in the literature. Symmetry enhancements of this type have been studied in [ where the superscriptsgauge group. denote the topological charges ofsupermultiplet whose the primary operator component has scaling dimension 1. Such supermultiplets 4 + 3 + 2 + 1 basic instanton states as are contiguous. For instance for a quiver JHEP09(2016)053 . † Q , as 1 (3.2) (2.9) (3.1) (2.7) − . SU(2). k and N A ' d ,A × 6 Q 1 1 ). \ T − k k N A A 0) supermultiplets, , ]. Here we only need ) symmetries together, 28 0) type N , 1. However thare are also , it is possible to define a − free (1 † = 1, a well known conjec- ) 0) superconformal index. − Q , d k . k (1 R + SO(6). R. d SU( 4 56 , . . . , k × J × 2 2 − q 2) with R-symmetry Sp(2) , R , we can repeat the same arguments. + | U(1) (2.8) to the 6d (2 R ) 1 2 56 + k , − J × N = 1 are the angular momenta on the three 34 , the affine extension of SU( N d 2 (6 l J ,A 1 = 1, wrapping instantons corresponds to ) 5 1 − ) In the case 1 q 1) SU( k − T ) Dyinkin diagram so the global symmetry +1 R N k 34 N osp , and it is natural to conjecture a relation k ,..., i i + [ J A → k 1 , J and its conjugate 1) basic instanton operators with the property 12 1 Z – 5 – − , with SU( 4) of SU(2) J 0 spin. (1 l / − , − q Q I 2 k − 12 × R F k C J = SU( ( 2 1) ) k 1 − k − U(1) − ( k = 0 contribute to the superconformal index, where × tr A δ SU( 1 = ∆ = − = = 2 field theory δ k + I } is the SU(2) † N . R 0) superconformal indices are discussed in [ U(1) , , N Q, Q 6 d transform in the (2 { 6 R k,A i α ) global symmetries are lost, but we gain one additional topological T symmetry is enhanced to the infinite dimensional group Q N 1 − and 0) and (2 k bar , N ]. There is also a U(1) symmetry acting on all bifundamentals with charge +1. 0) superconformal index we will show explicitly how, for d ] relates the 5d 5 , 21 k,A U(1) 5 T 27 , × 0) case: the superconformal algebra is , ‘non-wrapping’ instantons of length 26 k top Only states with Picking an appropriate supercharge The 6d (1 For the circular quiver there are Here we are adding the orbifold Compactifying the pq-web on a circle we are gauging theThe SU( two SU( Starting from the S-dual gauge theory SU( k hortogonal planes in ∆ is the scaling dimension of the states, the (1 The supercharges Witten index, with fugacities associatedThe to index the reads symmetries that commute with 3 6d (1 In this section weusing derive the letter counting. superconformal indices for the 6 ‘wrapping’ instantons of the form In section Kaluza-Klein modes, summing all of them reproduces a 6 U(1) Also, the sum ofU(1) all baryonic symmetries actsargued trivially in on [ the theory. The remaining that at least one topological chargeare is zero. We call these ‘non-wrapping’ instantons. There between so the quiver is now the Dynkin diagram of This is a well known first checkpq-web of the on pq-web the S-duality. ture cilinder: [ 6d/5d duality. enhancement in the UV is In both models one concludes that the total global symmetry in the UV is the quiver has the shape of the JHEP09(2016)053 - } R = 0 , ψ and Φ δ 2 } 2 { q 1 2 2 q 1 q q q 1 0 1 2 q 0 0 1 , ψ 0 0 0 0 0 0 0 0 0 0 0 0 q q q q 0 q q q there is a 0 q Φ ]. The full Index − − − q { , we obtain η √ i α 30 , Q R 29 4 − 2) [ J | 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 4 2 0 2 2 2 , P , we obtain a SU(2) i α 2 2 2 2 4 3 3 3 1 1 1 − (6 Q =∆ osp δ 1 4 4 4 4 1 4 4 4 4 6 6 6 10 10 10 10 10 10 10 SO(6) irrep SU(4) Dynkin labels of the superconformal 10) in the 6 of SO(6) and the self-dual tensor 2 2 2 2 2 ' / / / / / 2, transforming in the 4 of SO(6), while the . Acting with the supercharges 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 – 6 – ⊕ 1 1 1 1 1 R / R . Only the elementary fields (and derivatives) with 2 2 2 2 , whose scaling dimensions and spins are shown in 4 = 6 2 2 2 2 } / / / / 1 1 1 } 56 0 / / 0 / / 1 1 1 0 0 0 0 1 1 1 1 − ⊗ J 1 1 1 1 − − − − , η, φ 2, transforming in the 4 of SO(6). Acting on , η, φ + / + 2 2 2 2 with ∆ = 5 2 2 2 2 H / / / / 1 1 1 34 0 / / 0 / / 1 1 1 0 0 0 0 H { 1 1 1 1 − J ψ 1 1 1 1 { − − − − minimal susy superconformal algebra -singlet. Acting with the supercharges R d 2 2 2 2 2 2 2 2 / / / / 1 1 1 12 0 / / 0 / / 1 1 1 0 0 0 0 1 1 1 1 − J 1 1 1 1 with ∆ = 5 − − − − η 2 2 2 2 2 2 2 2 2 / / / / 2 / / / / 3 3 3 3 3 3 3 1 1 1 ∆ 5 5 5 5 5 5 5 5 , an SU(2) φ singlet fermion 2 4 6 R , , , + + + + + + + . Single-particle contributions to the index of the (1,0) massless hypermultiplet -triplet is a null state. 1 3 5 η η η η φ ψ ψ ψ ψ Φ . H H H H H H H ∂ ∂ ∂ R 1 Letter In the case of the self-dual tensor multiplet, the superconformal primary is the ∆ = 2 In the case of the half hypermultiplet the superconformal primary is the ∆ = 2 complex Let us study explicitly the free superconformal multiplets: hypermultiplet The two free supermultiplets are simple cases in the list of all possible short unitary real scalar doublet fermion null state (recall that forin SO(6), the 4 10 of SO(6). classification is given inprimary terms of of the the entire SO(6) superconformal multiplet. scalar Φ, transforming ina the SU(2) 2 ofSU(2) SU(2) self-dual tensor multiplet table representations of the 6 Table 1 and self-dual tensor multiplet contribute to the single-particle index. JHEP09(2016)053 (3.4) (4.1) (3.3) ], here we 34 – 32 , 2 basic instantons 4 1 / ] function explicitly q . 19 1) , 31 # ) − 16 4 4 n K k m ( m 2 orbifold, that on the tensor q k ) 0 2 k q ) q 2 Z , . . . , t n 3 3 q − / n 2 1 − q q 2 2 , t − q C n 1 1 t )(1 . q ( 1 3 2 )(1 0) 3 q f 1 q − m 1 q m ,..., − 1 q 0 =1 q ∞ , 0 − X n 0 1 q q " )(1 √ 0 2 2 ,..., = 4 NS5 on the plane. )(1 1 − q 1 q q , 0 1 nodes, there are k 1 2 q , – 7 – − q 0 − Abelian linear quiver 1 − q 2 (1 k 2 ,..., 0 d 0 )] = exp q (1 , m m K (0 = = I . 1 D5 and 1 1 1 q q , . . . , t 2 2 hyper , t / 1 1 1 SD tensor 1 . This is done by using the so called Plethystic Exponential t I I ( m m Figure 2 . We first consider the simpler case of the linear quiver and we 2 f [ A free hypers plus 1 self-dual tensor. PE 1 2 , the two indices are computed k it will be easy, using these results, to write down the superconformal index 1 5 ]: 0) SCFT corresponding to 1 M5 brane at the , 35 = 1 it is possible to compute the Nekrasov instanton partition [ ][ For the Abelian linear quiver with We need to take the multi-particle partition function generated by the basic instantons In section Using table f N [ with topological charges inferred from to theperform topological a direct string gauge amplitudes theory computed calculation. in [ we described inPE section For to all orders in thefunction instanton fugacities. in We appendix review thepostpone definition the of discussion the Nekrasov onthe partition the next circular section. quiver, Although which the is 5d the Nekrasov main partition result function of of the this linear paper, quiver to can be of the (1 branch is simply 4 Exact partition functions: 5 JHEP09(2016)053 , i /t = 1 1 (4.2) (4.3) (4.4) k → i t # ) 2 t # hypers: . These terms 1 ) t . These terms 2 l k t ) factors. ) (4.5) + 1 i k ,... 1 t 2 ,... − t m i I − 1 1 t ) factor. + m . − 1 − i k − +1 3 +1 I ) can also be checked to m i i 1 )(1 + q − i 1 m + 1 4.2 − 2 − m i +1 i t . The spurious contributions # i 1 )( ) m q t m 1 i 2 i i 1 t − q − ) i − m m i 2 i − − t q m ], see figure , m )(1 =1 1 ( − k i − r spurious − 38 , − )(1 i ! – 2 1 m ) P s t t 2 2 terms: m )(1 1 1 2 )(1 t i perturbative terms. We will show shortly t m 1 / t 36 s − − i q i t s , l 1 − k t q 1) = m + − m i r − (1 1 s i 19 i q − q − , " Q )(1 = I 1 (1 k − 1 s Y – 8 – + 17 )( ( − t i i s I k with Mathematica against the Nekrasov partition PE q = + −

i i s i l q = = 1 + free hypers, a expected from S-duality. We construct (1 Q i s =1 − ( 2 k X ), after removing the spurious contributions, reproduces i,I Q k 1 pert 1 ( =1 , 1 S − + 4.2 ): k i,I − × =0 2 k 4 k l t 1) instanton plus Nekrasov partition functions. We analyze the cases R P linear 1 of a sum of t P − i 1 Z + in the Nekrasov partition function. ( 1 SU( k S m 2 − ( =1 t PE × k i k = 2 first. 1 +1 × k i ) t pert+inst 1 =1 , i q ) k P 4 X k S i i " q R × 2 instantonic terms decomposes in two positive terms and two negatives 4 m / R linear 1) 1 is the length of the basic instantons, i.e. the number of non zero topo- =1 PE 1) negative terms, beside being negative, are not invariant under Z k i 2 negative terms of the form − 2 negative terms of the form / − − = k ) must be supplemented by the perturbative contribution of P / partition functions multiplying the contribution from the North and South and to ( k ( 1) ( i k 1) 1 " k q 4.2 inst 1 − , S − S positive terms, k k × , . . . , k × ( ( 2 PE 4 1 transform like ‘half’ adjoints (positive roots) of the other SU( k transform like ‘half’ adjoints (positive roots) ofk one of the two SU( k that the positivesymmetries terms SU( transform like the bifundamental of the enhanced global 4 R linear , We now want to show that ( We get, for It is easy to check that this formula correctly reproduces the terms proportional to S Z • • • = 0 precisely the partition function of the poles, that are the (which is trivial) and The latter they must be removed. Wecome call from these D1 negative branes terms slide stretching off between to parallel infinity, external as was NS5’s understood in in the [ pq-web, that can Each of the terms, so in the double sum in the numerator we can collect one single high orders in the instantonfunction. fugacities ( l logical charges. The formula is thus the JHEP09(2016)053 . ) 2 = 2 t (4.9) (4.6) (4.7) (4.8) 1 (4.13) (4.10) (4.11) (4.12) k t x, A, y ) . So we 2 2 y 2 t ) 1 2 2 t + t 2 2 √ − 2 ) t 1 /x 1 ) − 0 t )(1 2 2 t 1 m y t 1 + (1 t + ˜ 1 − 2 − 2 0 2 . t 1 m /y 1 , with fugacities x (1 m t 3 ( ˜ − ) A qm − 2 √ . + 1 t 2 √ ) 2 ) t 2 )(1 2 xy 2 ) 2 1 − t 2 0 x t t t 2 = t PE 1 ( ) t . t 1 = m 2 y √ − t 0 − 1 t )(1 = − ) 1 √ q − 1 − amounts to multiply the partition m 0 2 t   − t )(1 )(1 ˜ 1 = ) m 1 qm (1 1 )(1 1 − t 1 xyA S t q t 2 1 t )(1 − 2 + t 1 1 √ 2 t 1 + (1 1 × − + ) − t − 2 0 t t − t 1 4 1 2 − t 1 t /y (1 (1 − R yA 1 qm − 1 m 1 (1 √ t t 0 ) PE x )(1 (1 xy/A + 1 1 1 m q – 9 – = 1 y − 1 + = m 1 √ t − 0 PE . Changing variables to )( 1 + 1 ˜ − = m m = 1 pert S 0 , /A S × ,inst (1 Ay/x m 4 × =1 we get ( 4 R k =2 + 1 + + R k 1 Z 2 ) 1 A Z S t 2 spurious S q A 1 , = t )( 0 t × PE 1 × 2 4 xA pert S =2 1 − t xA/y , 4 √ m /x R k = full S ( 1 × y , S t 4 √ × Z =2 U(1) global symmetry, is actually a trifundamental R 4 k )(1 =1 + 1 √ = 1 R = k 0 inst Z t × x 0 + Z ˜ x PE ( 2 m . On = − m 2 5d index of a free hyper with mass ˜ 1 = t S ,pert 1 1 (1 t × S 4   PE =2 √ inst R k 0 × + = 1 Z 4 pert+inst m S PE we get , 1 S × S 1 ,pert 4 = inst = =2 × In this case the linear quiver is In this case we just have the perturbative contribution of a free hyper R S , with SU(2) k + 4 0 =2 1 R Z k × full S m 2 1 , Z S 4 × ,pert 4 Removing the spurious negative terms on =1 × S S 4 k =2 = 2. = 1. S k Z 2 Z recover the partition function of 4 free hypers, asfunction expected by from a S-duality. factor In the 8 positiveThe terms we 4 recognise negative the terms trifundamental are of spurious SU(2) and must be removed. Recall that in the case On with inverse we get Where k This is the where ˜ k JHEP09(2016)053 i 2 2 q k / . 1) 2 (4.14) (4.17) (4.19) (4.18) (4.15) (4.16) − 2 t spurious t 1 k 1 Z t ( t k √ 1 ) # inst I − 1 2 I + , Z t − j # 1 couplings + 2 1 y j ) t t A 2 − -charge ‘+1’, y 1 ) pert ) symmetry, and t t 1 √ j 1 k k A 1 Z ) I − t j √ − 1 I /y + y j j − = + y − j SU( 1 x +1 and A 1 j ] x − -charge ‘+1’, + i i × − y full 1 ) y I A A 1 1 [ j m − Z k 1 I + ) I − j P − i x k − + I + ) groups, while they are not x 2 j I A )( j m t + k 1 terms with x + j 1 x fund SU( ) j )( 1 t = − +1 x 2 i 2 i 1 χ x − t ] 1 k y √ 1 j j − i masses i i j − ) : y y /x − x y s − i 2 j +1 [ x k i i A m t x x x + we need to sum, inside the PE, the m x s ) − − )(1 2 terms with 1 q k A P 1 / j = ) 1 t S 1 I 2 − =1 i j − )(1 − )(1 + t antifund SU( k I X q + ( I − j I = s × 1 χ Y 1 s t +1) y + + − 4 A j m 1 − j =1 k (1 partition function ) s = 1) and + j k S X − ( y − ) from the q j j 1 k k )(1 +1 A 1 2 x x (1 y S ] 1 j – 10 – t 4.2 i − t ) = 1 i I y A t 2 = × /y [ j i = t + − j i s 4 y 1 k 1 √ − =1 y t ) , inverted, so we get I k I X x − S (1 Q k 2 I +1 +1 + 1 ( i t i + P j + = − =1 y x , y j 1 1 j i k X )( SU( antifund =1 1 0 y i − − t x y χ I 1 k i,I ] + 1 i j /x + − 2 = = − j x y j P t positive terms, [ i y 0 1 +1 ) m i t 2 k x + 1 m x k = √ 2 − t 1’. The full j 1, 1’. SU( fund s A 1 P − t χ j − m m − ) ( (( ( y s i +1 s j " q y m +1 1 m j j j s − x x q I =1 = PE PE k i Y s 1 , . . . , k + Let us change variables in ( will be the chemical potentials of the enhanced SU( j j k -charge ‘ − =1 -charge ‘ = = P j i I X = ( A A Y s y + . = 1 " j full 1 k i , j S ( − =1 × i and k I X 4 PE y i S 1 linear , x When we consider the gauge theory on We are then left with The negative terms are associated to strings stretching between external D5 branes, i − =1 Z j k X x is the chemical potential of the U(1) symmetry. Recalling that our formula for generic is contribution from the Northall pole the chemical and potentials, the and terms South with pole. Thecan South be pole simplified to contribution has so they must bethe removed. pq-web, and transform Their like flavourcharged a under fugacities ‘half-adjoint’ the of are U(1). the precisely two SU( the ones expectedterms from with while the negative terms become k under the above change of variables, the positive terms in the numerator become The A to which implies Generic JHEP09(2016)053 . ) ). k A k (4.21) k ]. 40 , 39 partition function and 5 S ) (4.20) 2 A t 1 ] it is shown how in circular quivers 1 ,A t − j power of the basic one. i 20 y √ 3 , y i th i y n m +1 x i i ( 2 x ] x q k [ 3 − 2 ] = 2 m q k ) the partition function (see appendix [ i q q 5 2 Z = 2 case. In [ free hypers in the bifundamental . It turns out that, in order to reproduce m 1) spurious contributions. These are D1 branes 2 N 2 2 1) and q k ) = − m d are simply the j 1 2 1 4 ,..., k q – 11 – t 1 q = 3. ( n Abelian circular quiver 2 , 3 1 2 k q A t , m k (1 i 1 i 1 m d 1 m q q I y √ q 2 q ( 2 1 m +1 +1 [1] i i m m − 1 1 y x 1 i − 1) non wrapping instantons, but we also need to consider k m m x − (1) 0) index. = , − k ( i [1] k m Z From the index computation point of view the instanton quantum 1 1. ] is the character of the (anti)fundamental representation of SU( i ≥ x [ n ) k for all ) is the same partition function of . Graphical representation of the SU( (anti)fund ) χ 4.19 The topological string partition forFor the this abelian necklace pq-web quiver has (figures been studied in [ Notice that this aspect looks different from the 3 1 n,n,...,n ( the Nekrasov partition function,I we need to sum over a full tower ofwith wrapping flavors at instantons each node there+1. is The only one wrapping wrapping monopoles monopole with in all the chiral charges ring, + with all topological charges necklace quiver, we study thewe compare spurious it terms, with we the construct 6d the (1 receives contribution from the wrapping instantons of the form 5 Exact partition functions:In this 5 section we write down the exact Nekrasov instanton partition function for the abelian where the two sets of variables are related by that can slide off to infinity. In the picture where Eq. ( Summarizing, we proved that Figure 3 JHEP09(2016)053 (5.4) (5.2) (5.3) (5.1) a ) 2 t 1 # t ) ) ) 1 2 2 − t t − i 1 1 ) 1 i t t − m I 1 1 − − + , m − i − i 1 1 I q 1 − i − k m ( + I q − i ]. We are thus led to 1 m )( + N k − i 1 m 41 ) )( − i 2 m + ) m i t − 2 m ) )( t 2 m , 1 − t kQ Q − 1 # − − 3 − 1 t i ) q − − i 2 )(1 (1 2 m (1 t hypermultiplets − 2 1 )(1 t m t t 3 2 k 1 1 ! =1 1 k − − t t t i X s i + m 1 − t − 2 m Q m 1 t )(1 s )(1 kQt s 1 − q 1 )(1 i t t 2 =1 1 i )(1 − m k i ) + Q i q m − s 2 ) − Q m = I NS5 on the cylinder. q t 2 P − ( s Y t + + 2 1 k − i (1 − 1 k − =1 (1 m − i " i I X −

– 12 – (1 = I + 1 1 i s )(1 + + − 1 =1 )(1 − i PE t I k X 1 Q 1 t ( m q kQ ( − = k =1 − =1 i X ! 1 1 − (1 . 1 D5 and s S k i,I (1 2 + m × Q t m 4 Q 1 s P R pert q + 1 + Z k i ! − kQt q 2 = I Q t PE s Y Figure 4 + − 1 i t Q − ) = i 2 is the lenght of the circle the pq-web lives on. We checked this result

1 t m 1 i " 1 − =1 − m S k I k X =1 i i X × q 4 PE k )(1 =1

R pert+inst i 1 X ) =1 t = k i Z Q + 1 − Q S − × (1 = 4 (1 R Q inst Q Z We see that Let us focus on the numerator inside the PE and split the instanton sum into wrapping There is also the perturbative contribution of the Which can be rewritten as instantons and non wrapping instantons: where to high orders with Mathematica. number corresponds to thepropose the Kaluza-Klein following charge formula: on the circle [ JHEP09(2016)053 2 q ). ). On the right 2 5.3 5.3 m Q Q 1 q 2 2 Q Q Q Q = 2. On the left the terms 3 1 k q m 2 3 1 m m 2 4 2 q q q U(1) nodes. 1 1 k 1 4 m m – 13 – 1 5 1 1 q q k 2 Q 2 . Circular quiver with m q 2 q q 2 2 m , associated to wrapping instantons, ﬁrst line of ( 1 2 m Q m k Q 1 q 1 Figure 5 q q 2 2 2 , associated to non-wrapping instantons, second line of ( 1 i Q 1 m q Q 1 m q 1 m q 1 q 1 1 m 1 m m and m i 2 q q 1 2 m . Spurious contributions on the cylinder. Here we display q Figure 6 dependent on the terms dependent only on JHEP09(2016)053 . , ], ), ). 1 ) 1 q k 2 ) q 1 t 2 ,... 1 (5.8) (5.6) (5.7) (5.5) t . We − ) 1 m 2 1 t m SU( t 1 q )(1 ( 1 kQ t 1 1 − × partition i t ). Notice − 1 ) 2 Q m − 5 Q/ m t 6 k − ( i 1 − (1 S / 1 + [ 4 m 1 1 kQt − − I − # ,Q 1 PE I + ) ) 1 − i + 1 2 ]. First we rewrite − i q m 1 45 m )( ) πiω s – 2 2 m − ω e ( m 1) negative terms. s / + 2 42 q 3 1 − t , 1 − 1 ω t − 6 0 + i k ,Q , I 0 )(1 ( ) = 5 + − 2 ω 1 i s ) [ 1 > k t 2 q 2 + t 1 5 ) Q Q t 1 z i t ( πiω S 1 1 − − 2 t ω m − ( − =1 e Z 1 + . We will see in the next subsection ( k I 1 πi − / )(1 i 2 − Q 2 1 /x P t e 5 m Q − =1 -nodes circular quiver is − − k i )(1 S +1 + k 0 or = x P 1 )(1 to form 2 πiz − t πiω 2 Q 1 I 1 2 e 1 ) + ,... S t e + − 3 − i ) contains the negative terms Q − × 1 4 + – 14 – − (1 for the Q m − i R 1 ], where for Im( ( 5.4 2 q t positive terms and 2 m 7 (1 Z [ ! 1 1 instead of s 2 " + Q 2 k 2 Qt ) m , m t G − s 1 2 Q t q PE + 1 i Q/x ) in the denominator, we are really claiming that there is 1 , there are also D1 branes going around the circle more Q 1 1 m i − i Q + Q − ( 6 q 1 = I x 1 ) = m s Y − S =1 + partition function which matters for comparison with the M5 2 − k i i × = 1. + 5 4 P , ω

2 R S pert+inst+spurious k 1 1) towers of spurious states. t ) displays 2 Q, m 1 + i 1 1 Z term, which is just a McMahon function, that is such contributions, one for every NS5 brane. It looks like these − =1 t 2 − q , ω i I t k X 2 1 0 1 , m k t k t ] for i m ω ( 1 m k ( q =1 5 ; t k − i X ( z k ( =1 Q N i 2 + X − 1) negative terms are all spurious terms to be factored out (figure G − + positive terms transform in the bifundamental representation of SU( k ) = i ( 2 Q k k Q , m in terms of modular forms − . There are i q 1 1 ( 6 S The final result for The first part of the numerator in ( The 2 The 2 × N 4 R PE We now patch together threeZ copies of 5.1 Modularity and partition function on We also added the in agreement with [ interpret these as towers ofNS5 spurious to contributions associated itself to andfigure D1 wrapping branes going the from circle one spurious an terms integer are number not ofpartition independent, function times. so a to term See avoid overcounting the we right have part to add of back to the for each pair ofbut semi-infinite we NS5 branes, can we alsoAs can strech shown strech a graphically a D1 D1 inthan going brane, figure once, around of of length the length, circle say, This the explains other the 2 way, of length check that it is reallybrane the index. that, beacuse of the factoran (1 infinite tower of spurious terms. This fact has a natural interpretetation in the pq-web: The formula for except that they are paired as that the correct character of thefunction, bifundamental is with reproduced once a we proper build the analytic continuation. Let us underline that this is a non-trivial with JHEP09(2016)053 1 = S ) 1 × ) = 4 2 S (5.9) 3 πi R full (5.12) (5.13) (5.11) (5.15) × 2 β 4 3 , σ 2 , Z 2 R full 1 πi Z , σ 2 β 1 , σ Ω 3 πi β 2 (0; 3 and we defined we have 1 ) for ; 0 2 , σ σ ) 2 i − l G Ω − + / 5.7 βτ i ) , σ s , e 3 µ 3 1 2 Ω ] − ω σ σ σ ) πiµ ) (5.14) = , s 2 3 2 − 46 τ − i µ 2 [ Ω; ω ω , − , Ω πi + − ω 2 / e 1 , , q 5 2 s ( ) i , τ σ σ 1 3 l ( S 1 i = ) (5.10) l ω ω + − βµ Ω i = + ω iτη , s , i s e | −

µ 1 3 ˜ √ m P σ Ω z l ( = Ω ω ) , − partition function we get β q ~m , ~q,

+ i 2 Ω i ) 1 2 3 s partition function are of the form ( πi 3 5 ` / ) 2 | − 2 2 z β ) = µ ( µ ) s 1 ω 3 S functions as ` 1 s , , m ( G G (0 σ τ S 2 1 − − , β 2 πiτ : harmless terms which do not play any role 2 0 2 β πi i ) l πi/ω πi/ω πi/ω 2 τ 2 Ω) ) − . 2 β β ` , − × G 24 = ( 2 2 2 =0 s i G + ( 2 , − 3 k l , G / i s − ) 3 2 4 µ 1 e q e 1 ,ω , Ω 2 πi σ Q − R ) 2 q β P 1 is used. Using the modular properties of 2 − 3 q ` 1 2 1 + ) = ( = − ( 1 q ) ,ω σ 2 3 1 2 ’s in the ω ω ` 2 s , 1 s ≡ πi ( 2 2 2 ( τ 1 − , − ˜ ω ω ω (0; ω q 2 − β ) , η 3 | ( 1 0 2 3 ( , 3 2 σ q , t i , G ) z l q S 2 ( η – 15 – 1 =1 1 2 Ω q ω ω G = ~τ − 0 i + 4 ) × πi / , i s + ; , i β 2 4 1 β G 4 3 | − ) )(1 z q Q 2 1 2 3 1 R , ` 2 − full − 2 B 1 ]. 1 ( . It’s non trivial that our result ( P ( 1 q Ω q ω e ω ω ω (0 r πiω σ Ω − Z πi Ω − πi 12 =0

+ 0 2 2 β 40 k l 2 2 β G − 1 2 q Ω = − − ; )(1 e G 3 1 z ( =1 1 e ) Q e q ω is obtained taking the product of three copies of Y ` 1 q i η + t 2 ` ) 1 2 3 µ − 3 5 =: = = 1 q ) = =1 = − πi 2 (1 G k i S − ) 2 ~τ − 1 i 2 5 i 1 G (Ω q q σ µ Q S − β − − =0 ; ) + k l ( 1 − Z i q 3 1 3 z 2 ( τ − ( Q σ ω ω η G − h ( can be written in terms of , ( =1 η r k i 2 1 =1 ) k i 1 PE ω ω 2 G P =1 S , k i σ Q β × ( 1 e 4 Q η Ω ). They satisfy the modularity property ω Ω and the convention = R ) spurious [1] = ) 2

/ 1 = πi 5 i Z Ω 1 πi 2 β σ i S z 1 β 2 ω ( PE ω , S ( ) η m Z 1 − 3 i η × πi 4 q 2 2 β , σ = = R ] inst , 2 = G 7 5 i =1 [ Z Ω 1 i k πi , σ S σ 1 β 2 1 S (0) contains a zero mode that can be regularized replacing it with r S 2 σ Q Z ; × The partition function on Using this equation for every triple of G × 4 z This is computed up to Bernoulli polynomials and G 4 ( 2 3 R = (0; full R 2 pert 2 Z These parameters take the following values in the 3 patches of Q Z The exponentiated variables are in our analysis. G the and exponentiated variables and the double elliptic gammaG functions in the where The modular properties ofmasses the equal, topological have string been partition studied in function, [ in the case of all the JHEP09(2016)053 ) 1 S 3.4 × (5.16) (5.17) 4 -flavors ]. More S !# !# . 4 + 2-point N # + 2-point i +1 y i k N i y 1 − k =1 i X A ] = 1. Let us now theories [ i y Ω S [ ], our final result is ! . The brane system ) i ], generalising to five 1 Ω k x 3 l 7 [ , the effective field the- +1 πi q l x 2 48 2 one expects the SU( fund 2 , x algebrae and − q χ A 7 1 e ) ] 47 i N k q k =1 linear quiver with l x X = [ √ W 1

SU( (anti)fund q 1 s corresponds to a linear quiver − 1 ) χ µ 2 − k k i ) − 7 + − y i s ), 3 3 A τ 1 y N q k q SU( antifund i − 2 algebrae [ 2 i ) as x + χ =1 q A q k s x , are described by ﬁve-dimensional gauge -punctures, corresponding to full vertex 1 ! + W P − − 2 simple punctures corresponding to semi- − = q N 5.13 1 3 Ω i 3 A k +1 R i q y i ] k q ˜ t i πi 1 1 y 2 y q q [ − k e =1 − − i X ) – 16 – , . . . , y 2 2 k 3 2 = q q q 1 1 , y 2 s ! SU( antifund q 1 q q ˜ A y χ 1 m parallel D4 branes (horizontal black lines) suspended l ] − s +1 − q + 2-point correlators of , i l x ˜ q ) (second line). 1 3 x 0) superconformal index of a free self-dual tensor ( 3 k x √ [ N , − q q ) i 1 k 2 =1 -Toda, and 2 full 2 k 3.3 y k s (1 k q q − q =1 1 i 0, we can rewrite ( l 1 X 1 y Q d ] and references therein. − SU( fund q q i i

> χ x x

51 algebrae h ) ) A , . . . , x ) i i 3 2 i " = σ q q q 3 i 2 , x W q satisfy − q ˜ 1 − 2 m q 1 x q s 1 1 algebra on the sphere with q algebrae. Indeed pq-webs on free hypers ( (1 1 (1 ] the known AGT relation for four-dimensional class ˜ flavors at both the ends. As depicted in figure m q √ k 2 N , =1 =1 k 3 i k s √ 3 i 50 + W q , W q Q Q q + 49 " " NS5 branes (vertical red lines). As described in section with 1 k PE PE − = = N ) 5 5 The most interesting consequence of this relation relies in the fact the S-duality in k S S Z Z degenerate vertex operators of ory living on theat D4 both system ends. is aSU( The five-dimensional S-dual SU( system onsupersymmetric the partition right function of ofcorrelator figure of the first linear quiver to compute the superstring theory, dubbed fiber-basetudes, duality predicts in a duality the between subclasscorrelators of of topological stringtheories ampli- associated to linearon quivers the of left-hand the sidebetween kind consists in depicted in figure recently five dimensional AGTdimensional duality algebra, has known been as understanding Ding-Iohara-Miki in the algebra the integrable or study elliptic properties offunction, Hall some of for algebra, infinite example the describ- see refined [ topological string and 5d Nekrasov partition 6 pq-webs and Five dimensional gauge theories describing theto dynamics have of the a above relation pq-websdimensions with are [ expected representation theory of It is easy to(first recognize line) plus the 6 Using the charachters of the (anti)fundamental of SU( change variables to and notice that, for Im( The variables ˜ JHEP09(2016)053 - q algebra with k -full insertions k . W q N momenta. On the 3 parameters. The linear quiver theory k − 1 and ) k − -Heisenberg algebra. A k k k q + 2 W + 2-point correlator of = 1 by making use of the N - + q k N N N k ]. Studies of the non Abelian cases 1 momenta and one position. Overall, 56 simple punctures on the sphere count − – semi-degenerate and two N k 53 N – 17 – partition function of the S-dual theory is expected to 1 S algebra with × 4 N N ) symmetry, this amounts to 2( S k C + 2 W W - , q -punctures count each k q that the partition function for the U(1) N 4 ]. According to the duality stated above, the ]. N 52 57 . Linear quiver: a cross is a simple puncture, a cross with a circle is a full puncture. ). 7 -full and one semi-degenerate insertion [ k Let us now proceed to theWe comparison saw of in the section two dual correlators. A more explicit check can be made in the simplest case A first check of this duality is the matching of the dimensions of the space of parameters positions of the vertex operator insertions and the corresponding Figure 7 two appeared in [ has three contributions: perturbative (1-loop), instanton and spurious (due to semi-infinite explicit computations of the supersymmetricsections. partition In functions displayed this incorrespondence case the between the previous this left-hand vertex sidediscussed algebra of in and the [ five-dimensional story gaugeHeisenberg reduces theories vertex has to operators been should capture the three-point correlator of k other hand, the two full taking into account PSL(2 counting for the dual correlators is obtained by simply swapping compute the correlator of (figure the correlation functions depend on. Indeed, operators. Analogously, the JHEP09(2016)053 k 1 W can q − (6.3) (6.4) (6.5) (6.6) (6.7) (6.1) (6.2) β !#)# − e 1 4 i − y i = th entry is 1 y i , . . . , k q j − . y j ) k =1 i y i X = 2 ) 1 ) i j 2 j − t ˜ . j α, e is the Weyl vector x ! x 1 2 , whose ρ − i i − t 1 ) − , l Q, h 2 y i 2 u − j t l ρ t y x − Q , n 1 u ) )(1 h − x t ) 1 1 2 ( 1 i − t x i ˜ t q 2 α − k , =1 − i t x k − h j − =1 X function, with ) n + 1 l x q X − k t )Υ )(1 )( 1 q 1 + k i 1 −

− ) gauge group it corresponds to the i t i (1 + +1 2 1 i i + )( t x N ˜ − α − 1 q i α, e = ( t +1 − 2 1 u A i i − n ( 1 k − =1 =1 2 x , h q i Y i k y X t i ( 1)( Q deﬁned by 2 Υ 2 u – 18 – i < j gauge group, 1 t − = = ) 1 + − 1 1 κ i − 1 t 1 ! are the weights of the fundamental representation , n ≤ =1 k 2 k − i 1 − ( i k )(1 1 1 l ˜ − k i,j ) α − − 1 h j l (0) q x , − A t 1 − i j 0 q Q x x x α − − − Υ + 1 ) k , =1 l 2 k X (1 j , y t ( ) u ) = i 1

i − ]. k − , y − =1 κ − ) = (1 i A i x X n ) 2 58 i x " k [ PE )( ˜ α, 2 1 1 u 2 )(1 , − 1 is needed to remove a zero mode, for SU( t i 1 1 +1 2 = 1 t 2 S i α, = = t )( t t | ( α 1 1 × e ρ − t x PE +1 − t √ 2 i ( ) in a form which is more suitable for the comparison with the 4 i t 1 1 q α α S 1 (1 + + S ( S t − Υ 6.1 β i " × × 4 α 4 − ( 2 S e pert+inst S pert,vector k β PE 1 k/ Z κ Z − ) β − =1 e 2 i = Y n − t the above can be written in term of 1 denotes the scalar product on the root space. The Υ e 1 1 term in the n 1 are the positive roots of the 1 t − =1 − ( i Y 1 x k ·i − e , S k = = = × h· A 4 The 1 k n 4 S pert+inst x x A Z Haar measure on the For one and all others zero, as where (half the sum ofand all positive roots), be defined as follows we can rewrite ( correlator By introducing the new variables where we included therole 1-loop in contribution the of identification the of the partition function with the M5 brane SCFT index parallel branes). The 1-loop and the instanton part can be written as JHEP09(2016)053 ) 1 =4 − = 1 N k (6.8) T k h = 2 it κ ) in the N ˜ α, κ , that has ( i represent D7 q α, ( ⊗ q 2, for C Q, Q h NS5’s. The vertical should correspond N > N 5 1 + 12 ) (if − . The difference between the N ] corresponds to the gener- k k 55 A . = 1 it has indeed been shown SU( = ) = U(1). The symbols c × N +1 i i × ) ) m u i N N − Q Hanany-Witten transitions, that, starting i SU( u SU( ( , while ours corresponds to the standard N i × × 8 φ ) 1 -Heisenberg on the torus computes the N . The results of section − =1 q 1 i k X i U(1)) S – 19 – = m can be expanded on the base of the simple roots × × i i ˆ e 4 φ i Q R φ 1 (U(1) − =1 S i k X symmetry). On the right the standard pq-web, one D5 intersect- = 3 φ 1 D5’s and also bend the upper part of the ]. The two formulae are different by a factor Υ − NS5-branes keeping track of the Hanany-Witten brane creation 56 k N , Toda theory with central charge ). algebrae on a torus. For the case q κ 55 - ( k q W q ] diagram on left side of figure ]. From the viewpoint of the index computation, on which we focus in ) can be compared to the three-point correlation function ] = Υ 59 61 2 6.4 ) – t 1 2 1 t t − ) of the 59 k this is described in terms of circular quivers. In this case one expects a relation − k 1 A 2 − . On the left a generalized pq-web which can be obtained from Higgling the ], the global symmetry is A )(1 − k 1 k 1)-dimensional vector of fields t 59 h − − ] that the one point chiral correlator of A − NS5’s, the global symmetry is the same, SU( (1 Let us now make some comments on the case of pq-webs on the cylinder. As explained Formula ( − k k [ 52 of the i ˆ in section with correlators of in [ circular quiver partition function on all the way through the effect, see [ this paper, this extra factorbetween the is two a sets spurious of oneside parallel due brane horizontal diagram. to branes, the From which the contributionmore are viewpoint of natural present of strings and only q-deformed stretching has in CFT, also the the the left-hand left-hand correct side four-dimensional diagram limit. is tation been conjectured innumerator, [ which is due toalized the pq-web fact [ that the computationpq-web in on [ the right hand side. The two diagrams are related by moving one D7 brane with one degenerate insertion (parallel to the highest weight of antifundamental represen- two partition functions is justPE that the left diagram has one more spurious term, which is precisely A( e is a standard pq-web withing SU(2) branes. The two pq-webs arefrom related the by right, a create sequencedisplacement the of between the two D7equals branes the attached to product the of semi-infinite all D5’s the does masses not in change, and the it linear quiver Figure 8 SCFT [ JHEP09(2016)053 . -algebrae -algebrae 9 W W N . It would be interesting k W 9 - q -algebrae, the analysis of the S-duality W – 20 – ]. From the viewpoint of the index computation this set- ]. 62 . Circular quiver: a cross is a simple puncture. on a circle, and has a link to topological string amplitudes on 65 – 1 k 63 Figure 9 0) SCFT [ , = (1 N N -point chiral correlator of the same vertices, see figure k in the supersymmetric partition functions and their brane realization. their interpretation in terms ofnot dualities clear of to q-deformed us correlators. what In is particular thethe it S-dual is interpretation of of the circular ouranalysed. quiver In results displayed this in in context, figure it terms would be of useful to quantum study integrable the insertion systems of is defect operators to be abelian Nekrasov partition functionsinvestigating this in problem. five and six dimensions.the study of We S-duality of are linearhas and currently circular to pq-webs be and further its investigated, relation informulae with order q for to the have supersymmetric an independent partition proof functions of derived the in plethystic this paper as well as of pq-web diagrams onas the considered torus in should [ be useful toit investigate would elliptic be very interestingprovide to information about analyze interacting the M5 case braneover of systems. the non-abelian This Coulomb theories, implies branch the which parameters integration would and a full control of the polar structure of non- elliptically fibered Calabi-Yau’s whichof are relevant forup the would classification also and bestretched study useful between to semi-infinite branes, analysetion which the limit. should issue From appear the of viewpoint in spurious of the factors deformed decompactifica- due to the strings a natural extension ofpactify our the work, pq-web on which braneD6 we branes diagram hope in on to figure the report torus. soon, is This to amounts fully to com- compactify the • • • • There are some openbriefly mention questions them: which in our view deserve further investigation. Let us to the to check this relation explicitly. 7 Open questions JHEP09(2016)053 ]. (A.5) (A.2) (A.4) 67 , µ ) (A.3) 66 i µ , − 1 ) − , µ − 31 t . 1 ( 1 +1 − γ i nodes is − ~ ) Y ) W k 2 , s i A ) + 1) ( ) (A.1) ) + 1) 2 ~ s i Y n α s ) ( , Y ( 2 α + , µ α A − Y +1 Y 1 2 i n +1 A A i ( ( ,~a + m i 2 − 2 ,~a 1 ~a i ( + ) + 1) ~a t ( m ( µ bif ) quiver with ) + α ) + z nodes is s + Y s − ) + 1 ) ( N ( i k s L γ γ γ γ 1-loop bif ( ~ ( Y a b Y W z γ , 1 ) i Y L L i − − ~a 1 1 L ~a ( α α + ( a a 1 γ − ( ( vec − b z 2 2 γ β β | γ i a b + − 1-loop vec ~ Y | i z α α − − q – 21 – a a ) quiver with sinh sinh k α α =1 i Y k 1 1 =1 a a − . And N i Y 2 β 1 γ } µ k a 2 2 Y Y β β ) = ~ Y m,n> m,n> } ≡ i sinh 2 ,... β X µ 1 =1 =1 γ ~ Y sinh sinh { N N +1 Y Y { W k , α α α,γ α,γ Y ∈ } µ Y Y sinh t i ∈ ∈ , Y Y ~a s s ) = 1 × × { ) = ) = } ( q i ~a =1 =1 k ( ) µ N N ≡ Y Y b, µ { ~ N α,γ α,γ , } 1-loop ~a, U( +1 i ( k 1-loop vec Z q z q ) = ) = { , , ~ Y 1 } 1-loop bif i ~a z ~a, ~ ~a W , µ ( { ≡ ( vec ~ Y, K z ) +1 b, k ~ N ~a ~a, U( ( The instanton partition function for the circular U( The 1-loop part for the circular U( Z bif z where where A Nekrasov partition function In this appendix we recall the definition of the Nekrasov partition function [ and the Theoretical Physics Departmentcollaborative of environment. the M.R. University thanks ofthe the Oviedo hospitality Simons for during Center hosting the for in Simons Geometry a Summer and very Workshop. Physics for We thank Can Kozcaz, ConstantinosGomez Papageorgakis, Sara for Pasquetti and useful DiegoST&FI. discussions. Rodriguez- The research The of researchThis S.B., of work M.R. was G.B. supported and isG.B. by A.T. and supported M.R. National is would Group by like supported to of the thank by the Mathematical INFN the COST Physics action INFN project “The (GNFM-INdAM). String project Theory Universe” GAST. MP1210 Acknowledgments JHEP09(2016)053 , 2 0. , ) β γ → − a (A.7) (A.8) (A.6) e − ] = YM g 2 massive . 2 B 500 ) t − 1 nodes it is α ~ , ]. W , µ 1 a 1 k m − − ) b, ~ β t − m k ( 1 ( − (2009) 63. 4 with Mathematica 0 when γ − e , SPIRE W in the fundamental )+1 4 )+1 s antif ) s A IN → ~ = ( Y , µ s k ( z = 3 ( − 2 α ) α ][ 1 Nucl. Phys. t Y α k ~a, Y t , =1 Y N ( -th gauge group, indeed A 2 A 2 Y arXiv:0906.3219 γ t 2 A YM t k )+1 ) [ ) t − 2 s s ( t ( /g ( antif α γ 1 ) = γ z Y Y ) quiver with W − ) gauge groups, )+1 L 1 − L L s t N ( ) − 1 N − 1 Jpn. J. Math. γ γ ~ t t W , µ b , ) Y ) ] = , exp( γ γ L 1 5 − (2010) 167 µ a b t ~ α ∅ ) 1 U( ∼ − − a α − ( α α b, ~ 91 k arXiv:0904.2715 a β − a a 2 q ( ( [ − − ~a, k β β γ ) with j e ( a − − ( e e + − bif β . So A.3 1 − 1 − − e Liouville correlation functions from πi i 2 YM 1 1 4 , z γ g – 22 – − ) + (2012) 034 W γ 1 α α = 2. We have checked it also for = Y α ∈ b Y Y t ]. in the antifundamental at the other endpoint. This a k ∈ ∈ τ Y Y 08 [ s s + × × k 2 β Lett. Math. Phys. =1 =1 , N N and Y Y SPIRE ~ 0. This correspond to freeze the Y , µ ), which permits any use, distribution and reproduction in α,γ α,γ 1 JHEP sinh IN S , ~a, → α ( ][ . The 5D exponentiated variables are Y k ) = ) = ∈ πiR q ) +1 ~ Y Y i fund i,j z ( m . ~a, massive flavour at the endpoints of the quiver: = 2 i i ( ~ dualities W , m =1 N k =1 β CC-BY 4.0 Y N βτ m γ Y α 0 vec e ~ Y, √ z = 2 This article is distributed under the terms of the Creative Commons / = Instanton partition functions and M-theory b, i ) = ~ ) = i N q Solutions of four-dimensional field theories via M-theory q ~a, , µ ( hep-th/9703166 = ) where ~ ]. ∅ [ ~ k Y , µ 0 bif i q z and ~ Y, βτ ~a, i ( b, ~ SPIRE βµ e ~a, IN fund four-dimensional gauge theories [ (1997) 3 ( To consider the instanton contribution of the linear U( We obtain a linear quiver gauge theory with These expressions can be substituted in ( z L.F. Alday, D. Gaiotto and Y. Tachikawa, E. Witten, D. Gaiotto, N. Nekrasov, It is easy to check this analytically when = = exp( 5 bif i z [4] [1] [2] [3] k up to instanton number 10. References Open Access. 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