Gauge Theories and Beyond Rui-Dong Zhu (朱睿东) Soochow University Institute for Advanced Study (苏州⼤学⾼等研究院）

Topological Vertex for SO(N) Gauge Theories and Beyond Rui-Dong Zhu (朱睿东) Soochow University Institute for Advanced Study (苏州⼤学⾼等研究院）

based on arXiv:2012.13303 (JHEP04(2021)292) in collaboration with Hirotaka Hayashi (Tokai Univ.) and work in progress with Satoshi Nawata (Fudan Univ.) What we study:

5d N=1 supersymmetric gauge theories (on C2q,t x S1)

gauge group: U(N) → SO(N) → G2 → … Why? • Exactly solvable (with localization method) • non-perturbative physics • dualities (S-duality, ﬁber-base duality, AGT (dual with CFTs) ) • quantum integrability • 5d N=1 → 4d N=2

… Current situation:

many interesting properties discovered in the case of U(N) • 4d/2d duality (with chiral algebra or non-unitary CFT) • modular tensor category • quantum integrability (~Calogero-Sutherland system)

… but not so many results known in SO(N) or Sp(N) theories

because of technical diﬃculties • in the calculation of Nekrasov partition function, instanton counting is very diﬃcult.

what we try to solve • in the study of chiral algebra, BCD-type Macdonald symmetric polynomial is diﬃcult. A sketch of what we did:

5d N=1 supersymmetric gauge theory

gauge group: gauge group: A-type BCD-type topological vertex computes the partition function

topological string extended topological string with new topological vertex Plan of Talk:

1. Review 1: SU(N) gauge theories and localization

2. Review 2: brane construction and topological string

3. Review 3: AGT duality and DIM algebra

4. proposal: topological vertex formalism for SO(N) theory

5. consistency checks

6. Kim-Yagi’s prescription and physical interpretation ⇢ µ0⇢v Ex = ,By = (156) ✏02⇡x 2⇡x

1 ⇢ v2 2 E0 = cosh ⌘E sinh ⌘cB = 1 (157) x x y ✏ 2⇡x c2 0 ✓ ◆

⇢ µ0⇢v Ex = ,By = (156) B0 = cosh ⌘B sinh ⌘E =0 (158) y ✏02⇡x y 2⇡xx

1 ⇢ v2 2 2 Ex0 = cosh ⌘Ex sinh ⌘cBy = v 1 2 (157) ✏02⇡x c `0 = `0 1 2 ✓ ◆ (159) r c

B0 = cosh ⌘B sinh ⌘E =0 (158) y y x t0 t0 = (160) 1 v2 v2 c2 `0 = `0 q1 2 (159) r c (161) 't0 t0 = (160) 1 v2 c2 n q an q an, for n Z (162) ! 3 2 (161) ' Cµ⌫

n an q an, for n Z (162) ! 3 2

Cµ⌫

Uq,t(gl1) (29)

b Uq,t(gl1) q 1,tb= q (29) (30) ! b q 1,tb= q (30) ! 1+ [µ] (31) W 1 2 1 5d N=1 gauge theory on C q,t x S 1+ [µ] (31) W 1 R✏1 R✏2 1 q1 = e = t, q2 = e = q . (32)

R✏1 R✏2 1 Ω-background with two deformation parametersq1 = e = t, q2 = e = q . (32)

R: radius of S1 (33) Its partition function can be found via localization method. 8 < (33) 8 : ( 1) Z = ZclZone loopZinstanton. < (163) : ( 1) perturbative non-perturbative part part Z = ZclZone loopZinstanton. (163) perturbative part is completely determined by the root system ( 1) 16 ⇤ of the gauge group G. ( 1) ⇤ G G Zone loop = ZCartanZroot, (164)

16 D A A (34) 2 ' 1 ⇥ 1 D A A (34) 2 ' 1 ⇥ 1 C⌫t t (q, t) (35) ;

C⌫t t (q, t) (35) ; v, v, (36) | i⌦| i X v, v, (36) | i⌦| i X = t/q (37) p = t/q (37) 2 p 2 ⇢ µ0⇢v Ex = ,By = (156) ✏02⇡x 2⇡x

1 ⇢ v2 2 E0 = cosh ⌘E sinh ⌘cB = 1 (157) x x y ✏ 2⇡x c2 0 ✓ ◆

B0 = cosh ⌘B sinh ⌘E =0 (158) y y x

v2 `0 = `0 1 2 (159) r c

t0 t0 = (160) 1 v2 c2 q (161) '

n an q an, for n Z (162) B Some useful formulae ! 3 2

In this appendix,C weµ summarize⌫ some useful formulae that we make use of in the computations in this paper.

B.1 Nekrasov partition functions λ 2 ThereforeThe partition the factor functionsgλ in of ( certain2.13) turns 5d out=1 tosupersymmetric be gλ =( 1) gauge| |fλ and theories this may is the be framing com- N − factorputed assigned by the localization to the horizontal method. legs It whenconsists we of use two the factors, diagram the perturbative of (2.10). part and the instantonIt is now part. possible The perturbative to compute part the partitionof a pure gauge function theory for with the diagram a gauge group (2.10)G whichis in is equivalentgeneral given to (2.9 by). The partition function is given by proofs JHEP_213P_0121

Inserting the explicitG expressions (2.1q) and (2.3) givest α a Z = P.E. + e− · , (B.2) root ⎛ (1 q)(1 t) (1 q)(1 t) ⎞ # $ α ∆ 2 proofs JHEP_213P_0121 − q − − −λ N%∈ t+(Q ,q) ⎝ 2 2 | | 3 λλ ⎠ Z(Q)=P.E. 2 Q Q fλ , (2.17) and −(1 q) Nλλ(1,q) $rank(−G) % qλ " # t ZG = P.E. ! + . (B.3) Cartan 2 (1 q)(1 t) (1 q)(1 t) where the Nekrasov factor in the# unrefined# limit− is given− by (−B.35)− and$$P.E. represents the ∆ is the set of positive roots of the Lie algebra g of G and a =(a , ,a ) is the Plethystic+ where exponential Δ+ is the defined set byof all positive roots, 1 ··· rank(G) Coulomb branch moduli inZ the= CartanZclZone subalgebra. loopZinstantonq, t are related. to the Ω-deformation (163) ϵ1 ϵ2 parameters ϵ1, ϵ2 by q = e− ,t= e . The unrefined∞ 1 casek correspondsk k to q = t. P.E. (f(x1,x2,...,xn)) := exp f(x1,x2,...,xn) . (2.18) On the other hand, the partition function& of thek instanton part is more' involved. The k!=1 instanton part of the pure SU(N) gauge theoryG with theG zero CS level is given by For obtainingnon-perturbative (2.17) we used part:Z theone Cauchy loop = identityZCartan (B.34Zroot) to, sum over the Young diagram (164) k µ and SU(N) ∞ 1 dφi SU(N) Z qk Z , instantonloc, counting inst =1 λ ρ ρ k (B.4) N ,q W (SU(k))s q ) s 2t πqi+ . λλ− k(1=0 )=(| 1)| | |λ( − i)=1λ ( − ) (2.19) in% 4d −F = F. * (codimension 4 object) (165) withIn this article, we compute such partition⇤ functions as series expansions by Kähler parameters using the mathematicain 5d:package particle-like developed in [19]. Since the diagram (2.10) n! G = SU(n) or (2.9) gives the pure “Sp(0)” gaugein string theory,n 1+theory:δ the partition D(p-4) functionbranes (on2.17 Dp) should brane. be trivial, W (G) = ⎧ 2 − n! G = O(2n + δ) , (B.5) namely | | ⎨⎪ 2nn! 16G =Sp(n) ⎪ and ⎩ Z(Q)=1. (2.20)

k k N k 2 Indeed, one canSU( checkN) this[2ϵ+ statement] with mathematica1 to[φij ﬁnd] [φij 2ϵ+] Zk = k [φi aj ϵ+]− ± , (B.6) [ϵ1,2] − ± [φij ϵ1][φij ϵ2] i=1 j=1 11i,j=1 ± ± * *Z(Q)=1+o(Q i

This diagram (2.22) is obtained by sending Q 0 with P ﬁxed for the diagram, – 43′ –→ ′ ν 2 Q′ P ′ + O5− O5 O5− (2.23)

–6– B Some useful formulae B Some useful formulae In this appendix, we summarize some useful formulae that we make use of in the compu- B Some useful formulae In thistations appendix, in this we paper.summarize some useful formulae that we make use of in the computations inIn this this paper. appendix, we summarize some useful formulae that we make use of in the compu- B SomeB.1tations useful Nekrasov in this formulae paper. partition functions B.1BThe Nekrasov Some partition useful partition functions formulae functions of certain 5d =1supersymmetric gauge theories may be com- In this appendix,B.1 Nekrasov we summarize partition some functions useful formulaeN that we make use of in the compu- The partitionputed by functions the localization of certain method. 5d =1 It consistssupersymmetric of two factors, gauge thetheories perturbative may be com- part and the tationsIn this inThe this appendix, partition paper. we functions summarize of certain someN useful 5d formulae=1supersymmetric that we make gauge use of theories in the compu- may be com- putedinstanton by the localization part. The method. perturbative It consists partN of of two a pure factors, gauge the theory perturbative with a part gauge and group the G is in tationsputed in this by paper. the localization method. It consists of two factors, the perturbative part and the proofs JHEP_213P_0121 B.1instanton Nekrasovgeneralinstanton part. given The partition part. by perturbative The perturbative functions part of part a pure of a gauge pure gauge theory theory with with a gauge a gauge group groupG Gisis in in

B.1 Nekrasov partition functions proofs JHEP_213P_0121 general given by proofs JHEP_213P_0121 The partitiongeneral functions given by of certain 5d =1GsupersymmetricG G gauge theories may be com- The partition functions of certain 5dN =1Zpertsupersymmetric= ZCartanZroot gauge, theories may be com- (B.1) puted by the localization method. ItG consistsN G of twoG factors, the perturbative part and the puted by the localization method.Z It consists=ZGZ of= twoZZG factors,,ZG the, perturbative part and the(B.1)(B.1) instanton part. The perturbative partpert of apert pureCartan gaugeCartanroot theoryroot with a gauge group G is in instantonwhere part. The perturbative part of a pure gauge theory with a gauge group G is in proofs JHEP_213P_0121 general given by proofs JHEP_213P_0121 wheregeneralwhere given by G q t α a Z = P.E. + e− · , (B.2) rootG q q t t α a G Z P.E. ⎛G (1 q)(1G t)G (1 q)(1 t) αea , ⎞ Z = P.E.root = #G G+ +G e$−α· −∆+·, (B.2)(B.2) root ⎛ZZpert(1 =−=qZ)(1ZCartan−t)ZZroot(1,, q−)(1 t)− %∈ ⎞ (B.1)(B.1) ⎛ (1 q#)(1pert t) Cartan(1 rootq)(1 t) $ α ∆+ ⎞ # ⎝ − − − $−α ∆+%∈ ⎠ and −⎝ − − − %∈ ⎠ where and ⎝ rank(G) q t⎠ andwhere G rank(G) q t ZCartanG = rankP.E.(G) q +t . (B.3) GG G ZCartan = P.E. q q 2 (1 t q)(1 +t) (1 α αa qa)(1 t.) (B.3) ZZCartanZ = P.E.==P.E.P.E. # 2 ++ #(1 q)(1 + t) (1 eq−e)(1−· · ,t),. $$(B.2)(B.2)(B.3) rootroot ⎛ 2 #q (1t #q)(1−q−t) −t(1− q)(1− −t)⎞− $$− ⎛ (1(1 q)(1)(1 t)) (1(1 q)(1 t)) α ⎞ ∆ is the set of positive### roots− of#− the− Lie− algebra− − g $of$−αG∆∆+and+ − a $$=(a , ,a ) is the +∆+ is the set of positive− roots of− the Lie algebra− g−of G and%∈%a =(a1, 1,arank(G)rank) is( theG) ⎝ g G a∈ a , ⎠ ···,a ··· ∆+ andisCoulomb theCoulomb set of branch positive branch moduli moduli roots⎝ of in the the the Lie Cartan Cartan algebra subalgebra. subalgebra.of andq, tq,are t=(are related1 related to⎠ therank toΩ the(-deformationG)) Ωis-deformation the and rank(G) q t ··· Coulomb branch moduliG in therank Cartanϵϵ1(1G) subalgebra.ϵ2ϵ2 q q, t are relatedt to the Ω-deformation parametersparametersGZCartanϵ1ϵ,1ϵ,2ϵ2=bybyP.E.qq== ee− ,t,t==ee .. The The unrefined unrefined+ case case corresponds corresponds. to q = tot.(B.3)q = t. ZCartan = P.E.ϵ1 ϵ22 (1 q)(1 t) + (1 q)(1 t) . (B.3) parameters ϵ1On, ϵ2 theby otherq = e hand,− ,t# the= 2e partition. The#(1 unrefined function−q)(1 − oft case) the(1 instantoncorresponds− q)(1− partt$$) to isq more= t. involved. The On the other hand,# the partition# − function− of the− instanton− $$ part is more involved. The ∆On+ is theinstanton the other set of hand, part positive of the the roots partition pure of SU( theN function Lie) gauge algebra of theory theg of instanton withG and thea zero=( parta CS1, is level more,arank is involved. given(G)) is by the The ∆+ is theinstanton set of positive part of the roots pure of the SU( LieN) algebra gauge theoryg of G withand thea =( zeroa1···, CS,a levelrank( isG) given) is the by instantonCoulomb part branch of the moduli pure SU( in theN) Cartan gauge subalgebra. theory withq, the t are zero related CS level to the··· isΩ given-deformation by Coulomb branch moduli in theϵ1 Cartanϵ2 subalgebra. q, t arek related to the Ω-deformation parametersinstantonϵ1, ϵ2 countingby q = e− isSU(,t notN=) eeasy.∞ Thek unrefined1 case correspondsdkφi SU( to Nq )= t. ϵ1 ϵ2 parameters ϵ1, ϵ2 by q = e− Z,tSU(loc,=N inst)e =. The∞ q unrefinedk 1 casek correspondsdφZik to SU(q, =Nt). (B.4) On the other hand,SU(N the)Z partition∞ = functionq1 W (SU( of thek)) instanton)dφi 2 partπiSU(+ isN more) Zk involved., The (B.4) ADHM constructionloc, inst → kNekrasovk%=0 | W partition(SU(k|))( functioni*=1 2πi Oninstanton the other part hand, ofZ theloc, purethe inst partition SU(= N)q gauge functionk=0 theory of with the instanton the zero) CSi=1Z partk level is+ is, more given involved. by (B.4) The W%(SU(|k)) ) | (2πi+* instantonwith partFor of U(N) the or pure SU(N) SU( Ntheory:k%=0) gauge| theory with| ( thei*=1 zero CS level is given by with k with SU(N) ∞ k 1 dφi SU(N) Zloc, inst = q n! Gk = SU(nZ)k , (B.4) SU(N) ∞ k W (SU(1 nk))1+δ )i 2dπφii+ SU(N) Z = kW%=0q(G)| = ⎧ 2 − |n(n! ! *G=1=G O=(2 SU(nZ+nδ)) , , (B.4)(B.5) loc, inst | W| (SU(n! k)) G = SU(2nπ)i k k=0 ⎨⎪ nnn1+δ)iG=1 +n with %W (G| )n=1+⎧δ 22−| (! n*! =Sp(G = O)(2n + δ) , (B.5) W (G)| = ⎧ 2| − n! G = O(2n + δ) , (B.5) where | | ⎪⎨⎪ 2nn! G =Sp(n) with and ⎨⎪ 2n⎩nn! ! GG==Sp( SU(nn)) W (G) = n 1+⎪δn G O n δ , (B.5) and k⎪⎧k2 −Nn!⎩ ! G == SU((2 n)+k ) 2 and SU(N|) [2ϵ|+]⎩⎪ n 1 [φij] [φij 2ϵ+] Zk = k⎨n 1+2 [nδφ!i aGj =Sp(ϵ+]−n) ± , (B.6) W (G)[ϵ=1,2]⎧ 2k− N n!−G =± O(2n + δ)[k,φij ϵ1][φij ϵ2] (B.5) k Nk i=1 j=1 k i,j=1 ± 2 ± SU(|N) k [2| ϵ+] ⎪* *n i

– 43 – Jeﬀrey-Kirwan (JK) residueF =in U(N)F. case (165) ⇤

poles are labeled by a set of N partitions (Young diagram) =( , ,..., ), . (166) 1 2 n 1 2 ··· n

F = F. α1 (165) ⇤ F = F.α (165) ⇤ 2 α3

β3

=(1, 2,...,n=(), 1,12,...,2n), 1 n2. n. (166) (166) β2 ··· ··· β1 pole Figure 4. The Frobenius coordinates = a +( canr be1) read✏ +( as thes length1)✏ . of each row and column(167) to the i = aj +(r 1)i✏1 +(j s 1)✏2.1 box2 in Young diagram(167) diagonal line for a Young diagram λ. As an example, we show the Young diagram λ =(5, 4, 4, 2, 2, 1) proofs JHEP_213P_0121 here. Then the Frobenius coordinates are given by (α , α , α β , β , β )= 9 , 5 , 3 11 , 7 , 1 . * Frobenius basis of Young diagram 1 2 3| 1 2 3 2 2 2 2 2 2 ! " # " =(↵1, ↵2,... 1, 2,...). (168) Jn and ψn, ψn∗ satisfy the following commutation| relations,

ψ , ψ = ψ∗, ψ∗ =0, ψ , ψ∗ = δ , (B.21) { n m} { n m} { n m} n+m,0 [J , ψ ]=ψ , [J , ψ∗]= ψ∗ , [J ,J ]=nδ . (B.22) n k n+k n k − n+k n m n+m,0 λ is a fermion basis with the label of the Frobenius coordinate (see ﬁgure 4) of a Young dia- | ⟩ gram λ. When the Frobenius coordinate of a Young diagram λ is λ =(α , α ,... β , β ...), 1 2 | 1 2 then λ is given by | ⟩ s β1+β2+ +βs+ 2 λ =( 1) ··· ψ∗ β1 ψ∗ β2 ...ψ∗ βs ψ αs ψ α(s 1) ...ψ α1 0 , (B.23) | ⟩ − − − − − − − − | ⟩ where s is the number of boxes on the diagonal line in λ and the vacuum state 0 satisﬁes | ⟩ ψ 0 = ψ 0 =0for any α > 0, β > 0. α | ⟩ β∗ | ⟩ There are two Cauchy identities known for the skew Schur functions.

1 sλ/µ(x)sλ/ν(y)= (1 xiyj)− sν/η(x)sµ/η(y), (B.24) − η $λ %i,j $ sλ/µt (x)sλt/ν(y)= (1 + xiyj) sνt/η(x)sµ/ηt (y). (B.25) η $λ %i,j $ To see the ﬁrst identify (B.24) we ﬁrst note that the lefthand side of (B.24) can be written as

µ V+(⃗x) λ λ V (⃗y) ν = µ V+(⃗x)V (⃗y) ν . (B.26) ⟨ | | ⟩⟨ | − | ⟩ ⟨ | − | ⟩ $λ Then by using the commutation relation 1 V+(⃗x)V (⃗y)= V (⃗y)V+(⃗x), (B.27) − 1 xiyj − %i,j − Eq. (B.26) can be further written as

1 µ V+(⃗x)V (⃗y) ν = (1 xiyj)− µ V (⃗y)V+(⃗x) ν ⟨ | − | ⟩ − ⟨ | − | ⟩ %i,j 1 = 17(1 xiyj)− µ V (⃗y) η η V+(⃗x) ν , (B.28) − η ⟨ | − | ⟩⟨ | | ⟩ %i,j $

– 46 –

17 0 1 2 3 4 5 6 7 8 9 D5 • • • • • • NS5 • • • • • • 7-brane • • • • • • • • Table 1: Conﬁguration of branes in the brane web construction. Bar represents the direction branes stretch along, and dot means the point-like direction for Appendixbranes. A F = F. (165) ⇤ F = F. (165) Preliminaries⇤ F = F. (165) =( , ,..., ), ⇤ . (166) 1 2 n 1 2 ··· n =( , ,..., ), . (166) Analytic expression1 2 n 1 2 ··· n =( , ,..., ), . (166) A.1 Nekrasov Factor1 2 n 1 2 ··· n the partitioni = afunctionj +(r can1)✏ 1be+( writtens 1) in✏2 .terms of Nekrasov factors(167) = a +(r 1)✏ +(s 1)✏ . (167) The Nekrasov factori hasj various equivalent1 expressions,2 which all look so di[Nekrasov↵erently. (2002)] Its deﬁnition i = aj +(r 1)✏1 +(s 1)✏2. (167) is given by

t t ⌫ +j 1 j i j ⌫j +i 1 N (Q;q =(,q ):=↵1, ↵2,...1 1Qq, 2,...i )q. 1 Qq i q (168) , (A.1.1) ⌫ 1 2 | 2 1 2 1 =(↵1,(↵i,j2),... 1, 2,...). (i,j) ⌫ (168) Y2 =(⇣ | ↵1, ↵2,... 1, 2⌘,...Y2). ⇣ ⌘ (168) | and two of the equivalent expressions were derived in [90]: for example, pure U(N) gauge theories can be expressed as N ⌫ i t+j 1 1 j i N 1 Qq2 q1 i 1 NIKV 1 1 1 Z = q| |N (1)N⌫ (Q;Nq1,q2)=(Q /Q )N (Qi j /Q1 ),N (Q /Q ). (A.1.2) i i1i i11j i j 11ij i j 11j i j i Z = q| |N (1) Ni (Qi/Qj)N1 Qq(Q2iq/Q1 j)N (Qj/Qi). Z = ii q| |Nij (1)i,j=1 Nij (Qi/Qj)Nji(Qj/Qi). 1,2,...,N i=1 i

17 function (B.4) of the pure SU(N) gauge theory with the CS level zero,

N SU(N) ∞ k 1 Z = q N (a a ; ϵ , ϵ )− , (B.8) loc, inst λiλj i − j 1 2 k=0 λ1,2,...,N : partitions i,j=1 ! N! # λ =k i=1 | i| function (B.4) of the pure SU(N) gauge theory with the CS level zero, " where the Nekrasov factor Nλν(a; ϵ1, ϵ2) is givenN by [8, 76, 77] function (B.4SU()N of) the∞ purek SU(N) gauge theory with the CS level zero,1 Zloc, inst = q Nλiλj (ai aj; ϵ1, ϵ2)− , (B.8) t − Nλν(a; ϵ1, ϵ2k):==0 λ1,2,...,N[:a partitions+ ϵ1(λi,j=1 jN)+ϵ2( νj + i 1)] ! ∞ N! i# SU(N) k λ =k − − − 1 Z = q i=1 i N (a a ; ϵ , ϵ )− , (B.8) (i,j) λ | | λiλj i j 1 2 proofs JHEP_213P_0121 loc, inst #∈ − k=0 λ1,2,...,N : partitions i,j=1 ! " N! #t where the Nekrasov factor Nλν(a; ϵ1, ϵ2)[ais+ givenλi ϵ=1k( byνi [8+, 76j , 771)] + ϵ2(λj i)]. (B.9) × i=1 | | − − − (i,j) ν #∈" t where the NekrasovN (a; ϵ , factorϵ ):=Nλν(a; ϵ[a,+ϵ ϵ) is(λ givenj)+ byϵ [8(, 76ν, 77+ ]i 1)] λν 1 2 1 2 1 i − 2 − j − (i,j) λ N δ proofs JHEP_213P_0121 The instanton part of the partition#∈ function oft the pure SO(2 + ) gauge theory for Nλν(a; ϵ1, ϵ2):= [a + ϵ1(λit j)+ϵ2( νj + i 1)] δ =0, 1 is also given by a contour integral.[a The+ ϵ1 Losev-Moore-Nekrasov-Shatashvili( νi −+ j 1) +−ϵ2(λj −i)]. (B.9) (LMNS)

× (i,j) λ − − − proofs JHEP_213P_0121 (i,j)#ν∈ #∈ integrand of the integral for the k-instanton partition[a + ϵ ( ν functiont + j 1) of + ϵ the(λ purei)]. SO(2N +(B.9)δ) gauge × 1 − i − 2 j − The instanton part of the partition(i,j function) ν of the pure SO(2N + δ) gauge theory for theory is [78–82] #∈ δ =0, 1 is also given by a contour integral. The Losev-Moore-Nekrasov-Shatashvili (LMNS) The instanton part of the partition function of the pure SO(2N + δ) gauge theory for ZSO(2integrandN+δ) of the integral for the k-instanton partition function of the pure SO(2N +δ) gauge k δ =0, 1 is also given by a contour integral. The Losev-Moore-Nekrasov-Shatashvili (LMNS) theory is [78–82] integrandk of the integral for the k-instantonk N partition function of thek pure SO(2N +δ) gauge k [2c.f.ϵ+ SO(N)] instanton partition function1 1 [ 2φi][ 2φi +2ϵ+] =( 1)SO(2N+δ) ( φi φj ϵ )− [ φi aj ϵ ]− ± ± , − Z theory[ϵ ]k is [78S–82±] ± − + ± ± − + [ φ ϵ ]δ k 1,2 i

– 44 – – 44 – – 44 – brane construction in string theory [Aharony, Hanany, Kol (1997)] Web of 5-branes We can build a large family of 5d =0 1 gauge1 2 theories3 4 in the5 type6 7 8 9 D5 N IIB string theory. [Aharony-Hanany, 1997]• • • • • • NS5 • • • • • • 0 1 7-brane2 3 4 5 6 7 8 9 D5 • • • • • • • • • • • • NS5Table 1: Conﬁguration of branes inall the non-trivial brane web information construction. Bar represents • • • • 7-branethe direction branes stretch along,contained and dot meansin this 2d the plane point-like direction for branes. • • The balance of tensionWe draw requires a web various diagram types on of this (p, q plane.) 5-branes (balance to of tension ⇒ appear in the construction,various kind and of they (p,q) form 5-branes a web of stretching 5-branes. along the vector (p,q) ) Example: pure SU(2) gauge theory.

6 axio-dilaton charge F = F. (165) 5 ⇤ N D5 branes ⇒ U(N) gauge theory 7-brane =( , ,..., ), . (166) 1 2 n 1 2 ··· n

= a +(r 1)✏ +(s 1)✏ . (167) i j 1 2

=(↵ , ↵ ,... , ,...). (168) 1 2 | 1 2

N 1 1 1 1 Z = q| i|N (1) N (Q /Q )N (Q /Q )N (Q /Q ). ii ij i j ij i j j i j i , ,..., i=1 i

N (mf ) (169) ⇠ ;

i j q t . (170) ⇠ (i,j) Y2

Z = ( Q )| |( Q )| | ...C C ... (171) top 1 2 µ⌫ µ⌧ ,µ,X⌫,,⌧,...

G Ztop = ZrootZinstanton. (172)

17 1 1 (192)(192) 2 2 ✓ ✓ ◆ ◆

() (3)()3() instantoninstanton + + 2 2 2 2 2 2 1 1 1 1 ZSOZ(4)SO(4) = = Q| |Q| |Q|0 ||Q|0q | |q NN(1,q(1)N,q)N(1,q(1) ,q) X, X, 2 2 2 22 22 22 2 2 2 N tN(Qt 0,q(Q)0M,q)M(P (,PP ,P,P ,PQ0 )Q.0 ) . ⇥ ⇥ , ,

In U(N) theory

instanton solutions labeled by N Young diagrams 1 (192) 2 ✓ ◆

D1 () 3() instanton + 2 2 2 1 1 ZSO(4) = Q| | | |Q0 | |q N (1,q)N (1,q) X, 2 2 2 2 2 2 N t (Q0,q)M (P ,P ,P Q0 ) . ⇥ , ( , , ) (193)(193) ; ; D5 Our results suggest that in the unreﬁned limit of SO(N) theories, the above picture still holds. D1

20 20 string duality with topological string theory

topological string (A-model) on toric Calabi-Yau [Leung, Vafa (1997)]

captures the BPS spectrum [Aganagic, Klemm, Marino, Vafa (2003)]

M-theory on toric Calabi-Yau type IIB on Taub-NUT =

(p,q) brane web

on S1

The topological string is a convenient way to compute the index (partition function on S1) or the instanton partition function of 5d N=1 gauge theories. Topological String?

*The concrete deﬁnition etc. are not useful in this talk.

• It is a topologically twisted N=(2,2) sigma model. [Witten, (1988)] [Vafa, (1991)] …

• Due to diﬀerent ways of topological twist, we have A- and B-models. They are connected through the mirror symmetry.

[Candelas et al., (1985), Dixon, (1987), Lerche et al., (1989) ….]

• There are certainly the open and closed version of the string theory, and there is a open/closed duality. The open theory is deeply related to Chern-Simons theories.

[Witten, (1992)] … under the transformation

I I¯ I¯ I¯ J¯ ,x =0, ,x = ⌘ , , ⌘¯ =0, , ✓ = G ¯F , (2.29) Q Q {Q } {Q I } IJ general,I the ( q, p)circleisgeneratedbyI I pr + qrI, and for genericI ( Iq, p)thecircledegeneratesatL¯ J K I¯ I¯ J¯ K¯ ⇥ ⇤ h i ↵ ¯ , ⇢µ = @µx , ,F = Dz⇢z¯ Dz¯⇢z + R JLK ⌘ ⇢z ⇢z¯ , 2 2,F = J¯K¯ ⌘ F (2.30). {Qone point}z1 = z2 = z3, but{Q there is one} exception: (1, 1) circle, generated by r↵ r = z1 {Qz2 , } There are equivalent ways to draw the toric diagram | by| choosing| | di↵erent patch to take the canonical degenerates at z1 =0=z2, which is described by r↵ r =0andr = 0. The degenerating 3 The stringloci always field lie on formulation a two-dimensionaldirection can of planeC be ofand theobtained all base, di with↵erent inr =0,andthereforewecandrawa a choices similar are way. related As by discussed SL(2,Z) transformation. before, can Let be us give an one-dimensional diagramequivalent on the r↵-r figure plane, that which will is usually be frequently referred to as used the toric later. diagram. The patches In are taken toQ be the same, but we identifiedthe as current@¯, which case, the has toric not diagram thing is given to by do with ✓ ’s in the field, and the selection rule is imposed set I separately on the holomorphic and anti-holomorphic part, we thus can factorize the string field r = x 2 x 2,r= z 2 x 2. (3.8) Lagrangian into two parts, ↵ | 1| | 2| | 1| | 2| The toric1 diagram in this choice is given2 by More general variety canS be= obtained by⌦ gluingtr severalA patches,@¯A + amongA whichA theA transformation, (2.31) g ^ ^ 3 ^ ^ rule is given by the defining relations Z ofX the toric Calabi-Yau.✓ ◆ 1 Let us see some examples. The first example is ( 1) ( 1) CP with Q~ =( 1, 1, 1, 1) where this time A is an anti-holomorphic (0O, 1)-form O and! ⌦ is a (d =3, 0)-form. and thus the indentification of the coordinates (x ,x ,z ,z )reads 1 2 1 2 t 1 1 (x ,x ,z ,z ) ( x , x , z , z ). (3.4) 1 2 1 2 ! 1 2 1 2 3 3 ToricWe choose Variety (x1,x2,z1)todescribethefirstThis geometry and MirrorC is alsopatch. known The second Curve as the patch resolved is related conifold.to the first one via 2 2 2 2 x1 x2 + z1 + z2 = t. 1(3.5) 1 1 1 Another| | frequently| | | | considered| | example is local CP CP ,or ( 2, 2) CP CP , with 2 2 2 ⇥2 O ! ⇥ Among variousThe (0, 1) 3-dimensional circle is generatedtwo U(1) by Calabi-Yaur↵ identifications,= x1 z1 manifolds,and (1, 0) circle there generated is by ar class = x2 of specialz1 . simplicity for topo- general, the ( q,There p)circleisgeneratedby are equivalent ways topr draw+ qr the, toric and| diagram for| generic| | by choosing ( q, p)thecircledegeneratesat di↵erent patch to| take| the| | canonical Switching to the (x1,x2,z↵2)patch,wehave 2 3 logical strings3 of A-model, the toric Calabi-Yau’s. The geometry is always T fibered over , one point z = directionz = z , but of C thereand is all one di exception:↵erent choices (1, 1) are circle, related generated by SL(2, byZ)r transformation.r = 2z 22 Letz 2, us give anR R 1 2 3 In the Calabi-Yau language,2 the web2 diagram(x, y1 corresponds,y22,z1,z↵22) to( µ1 x, y21, y2, µz⇥1, µz2). (3.9) r↵ = t + z2 x2 ,r3 = t + z2 x1 ,⇠ | | | |(3.6) degenerates atwhichequivalentz =0= can figure bezthe, locally whichtoric that diagram, will is embedded described be frequently in which by into| |eachr used a| C|line later.r patch. =0anddenotes The patches The |degenerater| = toric| are 0.| taken The variety locus degenerating to be is the given same, by but the we following defining 1 2 ↵ set whichof the lead torus to the fiber transformationEquations (of toric Calabi-Yau). relating di↵erent patches read general,loci the always( q, p)circleisgeneratedby lieequation, on a two-dimensionalpr↵ + planeqr, of and the for base, generic with ( rq, p=0,andthereforewecandrawa)thecircledegeneratesat 2 2 one point z1 = z2 = z3, but there is one exception: (1, 1) circle, generatedi i↵ by ri(↵↵+) r = z1 z2 , one-dimensional diagram on the r↵-r plane, which(x1,x22,z2 is) usuallyd+(2er Tx12,e x referred Rx2,e2 toz2 as). 2 the2 toric diagram.2 2 (3.7) In r↵ = x1 !x2 ,r = z1 x2 .2 x| |+y|1 |+ y2 = t1, (3.8) (3.10) degenerates at z =0=z , which is described by r | | r |=0and| r |=| 0. | The| degeneratingr the current1 case, the2 toric diagramToric isCalabi-Yau given by↵ a 2 a 2 | | | | | | (0, 1) circle degenerates at x2 =0= z2, with rQ↵ =i zti, r == tt x/1 U(1),and(12 ,, 0) circle2 degenerates2 (3.1) | | 2 x + z1 + z2 = t2. (3.11) loci always lie on a two-dimensionalThe toric diagram plane in this of the choice base,2 is with givenr by =0,andthereforewecandrawa|3 | at r = t, r↵ = t x2 . The rest one( i dimensional=1 R degenerating) loci is| for| r↵ | r|circle| stretching| | | X 2 2 one-dimensional diagram on the fromr↵-r (r↵plane,,r)=( whicht, t)as( isr usually↵,r)=( referredt + z2 , 2 tot + asz2 the2 ). The toric toric diagram. diagram2 can In2 therefore be | | | | a a By choosing r↵ = y1 x and r = z1 x , we obtain the following toriciQ diagram.↵a the current case, the toricwhere diagramt ’sdepicted areSimplest is given Kähler as examples: by parameters, and r| U(1)’s| | | are identified| | as| | the transformation zi e i zi ! with transformation parameter ↵a. The Calabi-Yau condition is equivalent to t a t Q =0, for 8a. 1 (3.2) More general variety can be obtained by gluing several patches,i t among which the transformation rule is given by the defining relation of the toric Calabi-Yau.i This geometry is also known as the resolvedX conifold. t2 3 1 11 ~ 1 1 Let us see someLocallyAnother examples. we frequently can The take first considered a exampleC patch, example is say( 1) is with local( coordinatesCP1) CPCP ,orwith (z1(Q,z2=(2,,z2)3)todescribethegeometry.There1, CP1, 1, 1)CP , with More general variety can be obtained by gluing several patches,O among O12 which!⇥ the transformationO ! ⇥ and thus the indentificationtwo U(1) identifications, of the coordinatesIt ( isx1 known,x2,z1,z that2)reads there is a mirror symmetry (see [3] for a review) between A-model and B-model, rule is given by the definingare three relation U(1) of actions the toric can Calabi-Yau. be considered in this patch, but thanks to one U(1) identification, only under which the2 Calabi-Yau2 manifold used in the A-model is mapped to a mirror Calabi-Yau. For two are left and they(x, form y ,y ,z the1,z ) torus1( fiber.µ x, 1y We, y are, µz free, µz ) to. choose two circles(3.9) of the torus to be Let us see some examples. The( firstx1,x example2,z1,z2) is1atoricvarietydescribedby(2( 11)x12, ⇠(x21), z1, CPz2).1with2 Q~ =(1 21, 1, 1, 1) (3.4) 2 2 !O 2 O 2 ! r↵ = z1 z3 and r = z2 z3 , which via the standard symplectic form ! = i i dzi d¯zi and thus the indentification of the coordinates (x1,x2,z1,z2)reads Equations| | relating | | di↵erent patches3 | | read | | d+r ^ We choose (x1,x2,z1)todescribethefirstC patch. The second patch is related to the first one via generate the U(1) transformations a 2 a 1 1 Qi zi = t , P (3.12) (x ,x ,z ,z ) ( x , x , z2 , z ).2 2 (3.4)| | 1 2 1 2 2 1 2 22 x2 1+ y212 + y2 = t1, i=1 (3.10) !x1 x2 +z1| |+ z|2 | = t.| i|↵ i i(X↵+) (3.5) | | | | (|z1|,z2,z| 3|) 2 (e 2z1,e z2,e z3). (3.3) 3 the mirror2 x variety+ z1 ! is+ givenz2 = byt2. (3.11) We choose (x1,x2,z1)todescribethefirstC patch.2 The2 second| | patch| | is| related| to the first one via2 2 The (0, 1) circle is generated by r↵ = x1 z1 and (1, 0) circle generated by r = x2 z1 . 3 2 | |2 | | 2 2 | | | | TheBy choosing base isrR↵2 = andy1 2 spannedx and2 r by =2 r↵z,1 r andx , wer obtain:= Im( thez1 followingz2z3). We toric want diagram. to know the location where Switching to the (x1,x2,z2x)patch,wehave1 |x2| +| z|1 + z2 =| t.| | | xx˜ = H(3.5)(u, v, xa), (3.13) the torus| degenerates.| | | | The| (0| ,|1) circle r↵ degenerates at z1 =0=z3, for which r↵ = r =0and r = t + 2z 2 2x 2,r= t + z 2 x 2, 2 213 (3.6) The (0, 1) circle is generated 0. by↵ Ther↵ = (1x1, 0)2 circlez1 and2 r (1degenerates,0) circle generated2 at z 1=0= by r =zx,2 for whichz1 . r = r =0andr 0. In | || | ||| | | | |2 | |3 | | | ↵ Switching to the (x1,x2,z2)patch,wehave t1 which lead to the transformation 11 2 2 2 t22 r↵ = t + z2 x2 ,r =i t + iz↵2 i(x↵+1 ), (3.6) | (x| ,x| ,z| ) (e x ,e| x| ,e | | z ). (3.7) 1 2 2 ! 1 2 2 which lead to the transformationIt is known that there is a mirror symmetry (see [3] for2 a review) between A-model and B-model, (0, 1) circle degenerates at x2 =0=z2, with r↵ = t, r = t x1 ,and(1, 0) circle degenerates under which the Calabi-Yau manifold used in the A-model | | is mapped to a mirror Calabi-Yau. For at r = t, r = t x 2. The rest onei dimensionali↵ i(↵+ degenerating) loci is for r r circle stretching ↵ atoricvarietydescribedby(x1,x2 2,z2) (e x1,e x2,e z2). ↵ (3.7) | | ! 2 2 from (r↵,r)=( t, t)as(r↵,r)=( t + z2 , t + z2 ). The toric diagram can therefore be | | d+r | | 2 (0, 1) circledepicted degenerates as at x2 =0=z2, with r↵ = t, r = t x1 ,and(1, 0) circle degenerates |a | 2 a 2 Qi zi = t , (3.12) at r = t, r↵ = t x2 . The rest one dimensional degenerating| loci| is for r↵ r circle stretching | | i=1 2 2X from (r↵,r)=( t, t)as(r↵,r)=( t + z2 , t + z2 ). The toric diagram can therefore be the mirror variety is given| | by | | depicted as

xx˜ = H(u, v, xa), (3.13) t 13

t 12

12 Elliptic Brane Web µ, t Cµ⌫(t, q) 1 Introduction ⌫,q The A-model partition function can be computed Figure 2: Thewith reﬁned the topologicaltopological vertex vertex. with labels. We use two short parallel lines to denote the [Aganagic, Klemm, Marino, Vafa, (2003)] preferred direction. Cµ⌫

We can easily generalize this computation to conclude

(µ) ⇢ t ⇢ C q 2 s t (q )s (q ), (4.36) µ,⌫, / µ /⌘ ⌫/⌘ ⌘ X where the proportional prefactor only depends on . By using the symmetric property, C , , = ; ; It can be expressed in terms of (skew) Schur functions. C ,, ,wefinallyarriveat 2 Partition Function; ; of M-string (µ) ⇢ ⇢ t ⇢ Cµ,⌫, = q 2 s(q ) sµt/⌘(q )s⌫/⌘(q ). (4.37) ⌘ The partition function of M-strings [1] can be computed from the refinedX topological string on a Calabi-Yau geometry specified by a toric diagram with its top and bottom external legs identified together. It can also5 be interpreted Refinement as the instanton partition function of the corresponding 6d =(1, 0) theory on R4 T 2 with omega background twisting. This partition function is an elliptic N ⇥ version of the NekrasovThe partition refinement function, of the which topological is a natural vertex consequence given in [10 from] does the not fact have that worldsheet the description, but in refined topological vertex on toric Calabi-Yau is dual to 5d =1theoryconstructedfrom(p, q)- stead, it is established in the meltingN crystal picture. The modification of (4.29)isgivenby brane web, and the compactification (i.e. identification of external legs) of this brane web along the 1 L0 L0 L0 L0 L0 L0 L0 t i j 1 1 NS5 direction lifts the theory to⌫ 6d.q V We+(1) giveq a...q briefV review+(1)t onV it(1) int this...t section,V (1) rederivingt µ it by(1 t q ) . (5.1) h | | i/ identifying external legs of toric Calabi-Yau, with the Awata-Feigin-Shiraishi (AFS) version ofi,j the=1 Y refined topological vertex [2] (see also [3] for a review and generalization). The building block forL0 a L0 The proper prescription is that whenever we switch V (1) to V+(1), t has to be replaced by q . compactified brane web is With this prescription, we can immediately see that the refined topological vertex is proportional to

⌘ + µ ⌫ | | | || | q 2 ⇢ t ⇢ Cµ,⌫,(t, q) sµt/⌘(q t )s⌫/⌘(t q ). (5.2) ⇤/[v1] t ⌘ X ⇣ ⌘ The prefactor depends on our[v2] precise definition of the vertex. We have lost the symmetry among three legs, so the only obvious guiding principle is that Cµ,⌫,(t, q)reducesbacktoCµ,⌫, when we set t = q. The leg corresponding to is called the preferred direction of the refined topological vertex, which is also the slicing direction of the melting crystal. For convenience, let us name the We note that by replacinglegs of⇤ and the vertexwith the with generalized Young diagram AFS verticesµ, ⌫,and introduced respectively in [3], it is by possiblet-direction, q-direction and the to build the compactified brane web with arbitrary rank of gauge group. What we need to do is to preferred direction as in Figure 2. The t-andq-direction exchange when we replace the parameter take the trace over the Fock space of the horizontal representation of the DIM algebra. There are t q in the refined topological vertex. $ We set the gluing rule1 that only pairs of legs that are both in the preferred direction, or one is in the t-direction and another is in the q-direction can be glued together.

22 FF==F.F. (165) (165) ⇤⇤

=( , ,..., ), . (166) =( 1, 2,...,n ), 1 2 ··· n . (166) 1 2 n 1 2 ··· n

= a +(r 1)✏ +(s 1)✏ . (167) i j 1 2 = a +(r 1)✏ +(s 1)✏ . (167) i j 1 2

=(↵ , ↵ ,... , ,...). (168) 1 2 | 1 2 =(↵ , ↵ ,... , ,...). (168) 1 2 | 1 2 N 1 1 1 1 Z = q| i|N (1) N (Q /Q )N (Q /Q )N (Q /Q ). ii ij i j ij i j j i j i , ,..., Ni=1 i

N (mf ) (169) ; i j ⇠ q t . (170) ⇠ It is similar to the Feynman (diagrami,j) to compute the Y2 partition function of topological vertex, i j q t . (170) Z = ( ⇠Q )| |( Q )| | ...C C ... (171) top (1i,j) 2 µ⌫ µ⌧ ,µ,⌫,,⌧,... Y2 X

Qi: Kahler parameters Z = ( Q )| |( Q )| | ...C C ... (171) top 1 2 µ⌫ µ⌧ ,µ,⌫,,⌧,... It reproduces part ofX the full Nekrasov partition function of the corresponding gauge theory.

G Ztop = ZrootZinstanton. (172)

*Remark: following from the pole cancellation (or blow-up equation), one can determine the classical piece and the Cartan part of the full partition function. [Grassi, Hatsuda, Marino (2014)]

17 ⌃out (1) ⌃ (2) (3) N (4) A() S ()= + S (5) gen 4G¯h out S = tr⇢ log ⇢ (6) out out out ⌃ = ⌃in + ⌃out (7)

⇢out =tr⌃in ⇢ (8) (9) y (10) = V (y) (11)

4G¯h Sgen ⇥[V (y); y1]= (12) V g(y1) V (y1)

V dpA = g(y1)dy (13) q G¯h 0 (14) ! (1) (15) O 1 dA ✓ =lim (16) A 0 A d ! 8y1,y2 (17) q ⇥[V (y); y1] 0 (18) V (y2)  t Sgen 0 (19) q A, Sout const (20) t ! QFC GSL (21) ) A(@L) Its partition function is given by S(L) (22) 1 1 n ⇢ ⇢ i 1/2 j 1/2  4G¯h (Q t/q) ( Q)| |s (q )s t (t )= (1 Qq t )=exp . (5.10) A(@Vn)(1 q n)(1 tn) i,j=1 n=1 p ! X Y S(V ) X (23) This is exactly the Cauchy identity of the Schur function, and a natural4G¯h candidate for refinement is to realize the same factor withQFC MacdonaldQuantum function. Bousso That is to bound!say, we would like to further impose (24) ) the constraint that the partition function does not depend on the choiceL of the preferred direction (25) in the toric diagram. 2 L +2✏⇢L (26) From (D.7), we see that the natural candidate for Z˜(q, t)isP(t ,q,t), with which the calculation of p Q t 1t Z(✏ , ✏ )=exp . (27) 1 2 ✏ ✏ F The original topological stringq is dual✓ toq the1 2 “self-dual”◆ point, with two omega-background parameters can be carried out as ✏1 + ✏2 =0. (28) 1 ⇢ ⇢ i 1/2 j 1/2 ( Q)| |C , ,(t, q)C , ,t (q, t)= ( Q)| |P(t ,q,t)Pt (q ,t,q)= (1 Qq t ). We call; ; it an unrefined; ; setup, and we mainly focus on this special Ulimitq,t(gl in1) this talk. (29) i,j=1 X X 1 Y The refined version corresponding to a general omega-background b (5.11) q 1,t= q (30) was soon proposed. ˜ ! There is some ambiguity in the particular choice of[Awata,Z(q, Kanno, t), but (2005)] it is only [Iqbal, conventional. Kozcaz, Vafa, Finally(2007)] we obtained the full expression for the refined topological vertex as

1+ [µ] (31) t 2 2 ⌘ + µ ⌫ 1 µ µ W t || || || || ⇢ q | | |2|| | ⇢ ⇢ 2 2 t Cµ,⌫,(t, q)=q t P(t ,q,t) ⌘ t sµ /⌘(q t )s⌫/⌘(t q ).

P R✏1 1 R✏2 1 So far, we have only considered the toric geometry of ( 1)q1 = (e 1)= t,CP q2 ,= lete us= examineq . (32) 1 O O ! the exampleIt is again(0) expressed( 2) CPin termsfor the of (skew) independence Schur function. of choice of preferred direction. O O ! *The reﬁned topological string has qno world sheet description, and is based on the melting crystal model picture. There is a special leg, usually named thet preferred direction of the vertex. q

2 It is straightforward to derive

loc N(+s)⌫(Q; q1,q2) 1 q1xs/x0 loc = , (A.5) N⌫ (Q; q1,q2) 1 xs/x0 x ⌫ Y02 N loc (Q; q ,q ) It is straightforward to derive(⌫+s) 1 2 1 q2x0/xs loc = , (A.6) N⌫ (Q; q1,q2) 1 q1q2x0/xs loc x0 ⌫ N(+s)⌫(Q; q1,q2) Y2 1 q1xs/x0 loc = , (A.5) +1 N⌫ (Q; q1,q2) 1 xs/x0 k 1 k x ⌫ where xs = v1q1 q2 for some s =(k, k +1) Y0A2 ()intheﬁrstrecursiverelationandasimilar N loc (Q; q ,q ) 2 quantity in the second one for some(⌫+s box) s1 in2 A(⌫). 1 q2x0/xs loc = , (A.6) N⌫ (Q; q1,q2) 1 q1q2x0/xs x0 ⌫ From (A.1), we can easily derive a useful identity,Y2 k 1 k+1 where xs = v1q1 q2 for some s =(k, k +1) A()intheﬁrstrecursiverelationandasimilar + ⌫ n() n(⌫) ⌫ 2 n(⌫t)+n(t) ⌫ 1 quantityN (Q in; q the,q second)=( oneQ) for| | some| |q box s in|A|(q⌫). | |N (Qq q ) ; q ,q . (A.7) ⌫ 1 2 1 2 ⌫ 1 2 1 2 From (A.1), we can easily derive a useful identity,

+ ⌫ n() n(⌫) ⌫ n(⌫t)+n(t) ⌫ 1 N (Q; q ,q )=( Q)| | | |q | |q | |N (Qq q ) ; q ,q . (A.7) B Schur⌫ functions1 2 1 2 ⌫ 1 2 1 2 The deﬁnition of a Schur function s ( x n )is B Schur functions { i}i=1

j +n j n det(xi ) The definition of a Schur function s( xi i=1)is s(x1,...,x{ } n):= n j , (B.1) det(x ) j +in j det(xi ) s(x1,...,xn):= n j , (B.1) for some Young diagram = . It is a symmetricdet( polynomialx ) and the set of all Schur functions { j} i forms afor complete some Young basis diagram of symmetric = . Itpolynomials. is a symmetric Therefore, polynomial theand the product set of all of Schur two Schur functions functions { j} can againforms be a expressed complete basis in Schur of symmetric functions, polynomials. Therefore, the product of two Schur functions can again be expressed in Schur functions, sµs⌫ = cµ⌫s. (B.2) sµs⌫ = cµ⌫s. (B.2) X X With this fusion coecient cµ⌫ of Schur functions, we define the skew Schur function, With this fusion coecient cµ⌫ of Schur functions, we define the skew Schur function, s := c s . (B.3) s/µ/µ := cµµ⌫⌫s⌫⌫. (B.3) ⌫⌫ XX withThe theLet most cmmutationThe us importanthave most importanta look relation, fact at factthe about aboutdetails the the skewof skew Schur Schur Schur functions. function function we we use use in this in this article article is that is it that can be it can be expressed as a fermion correlation. expressedIt can as abe fermion expressed correlation. as an expectation value of a vertex operator. n, m = n⇤, m⇤ =0, n, m⇤ = n+m,0,Jn := j j⇤+n, (B.6) { } { } { } s/µ(~x)= µ V+(~x) = V (~x) µ , j +1/2 (B.4) h | | i h | | i Z s/µ(~x)= µ V+(~x) = V (~x) µ , 2X (B.4) [J , ]= h | , [J |, i⇤]=h | ⇤ , | [iJ ,J ]=n . (B.7) where n k n+k n k n+k n m n+m,0 where where 1 is a fermion basis with the label of Frobenius coordinates1 n =(↵1, ↵2,... 1, 2 ...)oftheYoung | i V (~x)=exp xi J n , | (B.5) ± n ± diagram , n1=1 1 i ! X X n V (~x)=exp xi J n , (B.5) ± n ± s n=1 i ! 1+2+ +s+ 2 Equivalence =( 1) between··· Two ⇤ 1 ⇤ TypesX2 ... ⇤Xs of ↵ Refineds ↵(s 1) ... Topological↵1 vac , (B.8) with the cmmutation relation, | i Vertex 29 | i where s is the number of diagonal boxes in . n, m = n⇤, m⇤ =0, n, m⇤ = n+m,0,Jn := j j⇤+n, (B.6) { } { } { }29 j Z+1/2 There are two Cauchy identities known for the skew Schur functions.2X 1 Melting Crystal[J , Picture]= , [J , ⇤]= ⇤ , [J ,J ]=n . (B.7) n k n+k n k n+k n m n+m,0 1 Then we have s/µ(x)s/⌫(y)= (1 xiyj) s⌫/⌘(x)sµ/⌘(y), (B.9) is a fermion basis with the label of Frobenius coordinates =(↵1, ↵2,... 1, 2 ...)oftheYoung | i i,j ⇢+ 1/2 ⌘ t ⇢ 1/2 | XC (t, q) s t (tY q { })s (Xq t { }) diagram , µ,⌫, / µ /⌘ ⌫/⌘ s t (x)s ⌘t (y)= (1 + x y ) s t (x)s t (y). (B.10) /µ X /⌫ i j ⌫ /⌘ µ/⌘ t s ⇢+ 1/2 t ⇢ 1/2 1+2+ +s+ i,j ⌘ =( 1) = ···µ V 2( t⇤ q ⇤ { ...}) V+⇤ (q t { }) ...⌫ . vac , (1.1)(B.8) X 1 Y2 s X↵s ↵(s 1) ↵1 | i h | | i | i The first identity can simply be derived from where s is the number of diagonal boxes in . 2AFSVertex 1 There areµ twoV+( Cauchy~x) identitiesV (~y) ⌫ known= for(1 thexi skewyj) Schurµ functions.V (~y) ⌘ ⌘ V+(~x) ⌫ , (B.11) h | | ih | | i h | | ih | | i i,j ⌘ X Y 1 X s/µ(x)s/⌫(y)= (1 xiyj) s⌫/⌘(x)sµ/⌘(y), (B.9) which follows from the commutation[u, ti]=tn(,u,ti): (ti) ⌘(x):, (2.1) i,j ; ⌘ X Y x X Y2 s/µt (x)st/⌫(y)= (1 + x1iyj) s⌫t/⌘(x)sµ/⌘t (y). (B.10) V⇤ [+u,(~x ti)]=V t(n⇤~y()=,u,ti):⇤(ti) V⇠((~yx)):V+, (~x). (2.2) (B.12) i,j 1 ; x y ⌘ X Yi,j i xjX Y Y2 whereThe first identity can simply be derived from There exists an automorphism of the fermionic algebra, f : ⇤, under which transforms n 1 n $ t µ V+(~x) V (~y) ⌫1 =z (1 xiyj) 1 µzV (~y) ⌘ ⌘ V+(~x) ⌫ , (B.11) to as f exchangesh | (z)=exp↵| i ihand| i |.i In particular,a n exp we haveh | an| ih, | | i (2.3) ; { } { } n 1 qi,jn ⌘ n 1 q n X n>0 Y ! n>X0 ! X X t n n n which follows from the commutation1 (z) =( 1)| |f( ). 1 z (B.13) ⇤(z)=exp | i a n exp | i an , (2.4) ; n 1 qn n 1 q n n>0 ! 1 n>0 ! The second Cauchy identity followsV+X(~x)V directly(~y)= from the factVX( that~y)V+( the~x). transformation does not(B.12) change 1 x y f i,j i j the expectation value of a correlator and fY2 = id . i.e. n n 1 t n 1 t n ⌘(z)=exp z a n exp z an , tThere exists an automorphism t ofn the fermionic algebra,⌫ t f :n ⇤,t under which1 transformst V (~y) ⌫ =( 1)| | ( n)V1 (~y) ⌫ =(!1)| | | | n 1 (V ($~y)) ⌫! = V ( ~y) ⌫ ,(B.14) t f f toh | as exchanges| i ↵i andh X| i . In| particular,i we havehX| | i h | | i f n n { } { } 1 t n n 1 t n n ⇠(z)=exp z a n exp z an . where we used that f(Jn)= Jn, andn thereforet n n 1 =( 1)| |f!( ). n 1 ! (B.13) X | i | i X t t t t µ V+(~x) V (~y) ⌫ = (1 + xiyj) µ V (~y) ⌘ ⌘ V+(~x) ⌫ . (B.15) TheWe second changeh Cauchy the| normalization identity| ih follows| to (for directly| in>0) from the fact thath the| transformation| ih | f does| i not change i,j ⌘ X Y2 X the expectation value of a correlator andn f = id. i.e. 1 q n an = aˆn,an = q aˆ n, (2.5) t t 1 tn ⌫ t t 1 t V (~y) ⌫ =( 1)| |f( )V (~y) ⌫ =( 1)| || | f(V (~y)) ⌫ = V ( ~y) ⌫ ,(B.14) h | | i h | | i h | | i h | | i so that where we used that f(Jn)= Jn, and therefore 30 [ân, aˆm]=nn+m,0. (2.6) t t t t µ V+(~x) V (~y) ⌫ = (1 + xiyj) µ V (~y) ⌘ ⌘ V+(~x) ⌫ . (B.15) h | | ih | | i h | | ih | | i i,j ⌘ X Y X

30 α1 α2 α3

β3 α1 α2 β2 α3 β1 β3 α1 Figure 4. The Frobenius coordinates can be read as the length of each row and column to the α2

λ β λ , , , , , proofs JHEP_213P_0121 diagonal line for a Young diagram . As an example, we showα the21 Young diagram =(5 4 4 2 2 1) α3 9 5 3 11 7 1 here. Then the Frobenius coordinates are given by (αα11, α2, α3 β1, β2, β3)= , , , , . βα12 2 2 2 2 2 2 β3 | α2 ! " # α3 " Jn and ψn, ψ∗Figuresatisfy 4 the. Thefollowing Frobenius commutation coordinatesα3 relations, can be read as the length of each row and column to the n β2 β3 diagonal line for a Youngβ diagram3β λ. As an example, we show the Young diagram λ =(5, 4, 4, 2, 2, 1) proofs JHEP_213P_0121 ψ , ψ ψ , ψ , ψ ,1ψ δ , 9 5 3 11 7 1 n m here.= Thenn∗ m∗ the=0 Frobeniusn coordinatesm∗ = n+m, are0 given by (α1, α2, α3 β1, β2, β3)=(B.21), , , , . { } { } { β}2 | 2 2 2 2 2 2 [J , ψ ]=ψ , [J , ψ∗]= ψ∗ , [J ,J ]=nδ . (B.22) Figuren 4.k The Frobeniusn+k coordinatesβ2 n cank be readα n as+k the lengthn ofm each rown+m, and0 column to! the " # β1 −1 " diagonal line for a Young diagramβ λ. As an example,α we show the Young diagram λ =(5, 4, 4, 2, 2, 1) proofs JHEP_213P_0121 λ is a fermionJn basisand withψn, ψ then∗ satisfy label1 of the the Frobenius following2 coordinate commutation (see relations,ﬁgure 4) of a Young dia- α , α , α β , β , β 9 , 5 , 3 11 , 7 , 1 | here.⟩ Then the Frobenius coordinates are givenα by ( 1 2 3 1 2 3)= 2 2 2 2 2 2 . gramFigureλ. When 4. The the Frobenius Frobenius coordinates coordinate ofcan a Young be3 read diagram as the length|λ is λ =( of eachα1, α row2,... andβ1, columnβ2 ...), to the

Figure 4diagonal. The Frobenius line for a Youngψ coordinatesn, ψm diagram= canψλn∗., be Asψm∗ read an=0 example, as, the length weψn show, ψm∗ of the each= Youngδn row+m, diagram0 and,! columnλ"| =(5 to, 4 the#, 4, 2, 2, 1) (B.21)proofs JHEP_213P_0121 then λ is given by β3 { } { } { } 9 5 " 3 11 7 1 diagonal linehere. for| Then⟩ a Young the Frobenius diagram λ. coordinates As an example, are given we show by (α the, α Young, α β diagram, β , β )=λ =(5, , 4, , 4, 2,,2,,1) . proofs JHEP_213P_0121 Jn and ψn, ψn∗ satisfy[Jn, theψk]= followingψn+k, commutation relations,1[Jn2, ψk∗3]=1 2ψn∗3+k, 2 2[Jn2 ,J2m2]=2 nδn+m,0. (B.22) β +β + +β + s | 9 5 3 11 7 1 here. Then the Frobenius coordinates1 2 ares given2 by (α , α , α β , β , β )=− , , , , . λ =( 1) ··· ψ∗ β1 ψ∗ β2 1...ψ2 ∗ β3s ψ 1αs ψ2 α3(s 1) ...2 ψ2 α21 20 ,2 2 (B.23) | ⟩ − β2 − − − | − − − − !| ⟩ " # ψn, ψm = ψ∗, ψ∗ =0, ψn, ψ∗ = δn m, , " (B.21) { λ} is{ a fermionn m} basis with{ them label} of+ the0 Frobenius! coordinate" (see# figure 4) of a Young dia- whereJn ands isψn the, ψ| n∗ number⟩satisfy of the boxes followingβ1 on the commutation diagonal line in relations,λ and the vacuum state" 0 satisfies [Jn, ψgramk]=ψλn.+ Whenk, the Frobenius[Jn, ψ coordinatek∗]= ψn∗+k of, a Young[Jn,Jm diagram]=nδn+λm,|is0⟩.λ =((B.22)α1, α2,... β1, β2 ...), Jn and ψn, ψn∗ satisfy the following commutation relations,− ψα 0 = ψβ∗ 0 =0for any α > 0, β > 0. | | ⟩ ψn, ψ|mthen⟩ = λψ∗,isψ given∗ =0 by, ψn, ψ∗ = δn m, , (B.21) Figure 4.λ TheThereis{ a Frobenius fermion are} two basis{| coordinates Cauchy⟩n withm} identities the can label be of known{ read the Frobenius as form} the the length skew+ coordinate0 Schur of each functions. (see row figure and4 column) of a Young to the dia- ψn, ψ| m⟩ = ψn∗, ψm∗ =0, ψn, ψm∗ = δn+m,0s, (B.21) diagonal line for a[J Youngn, ψk]= diagramψn+k,λ. As an example,β1+β[2J+n we,+ψβk∗ shows]=+ theψn∗+ Youngk, diagram[Jn,Jm]=λ =(5nδn, +4,m,4,02. , 2, 1)(B.22) proofs JHEP_213P_0121 { gram} λ.{ When the} Frobeniusλ { coordinate} of··· a Young2 ψ∗ diagramψ∗ ...λ isψλ∗ =(ψα1, αψ2,... β1..., β2ψ...), , =( 1) − 1β1 β2 βs αs α(s |1) α1 0 (B.23) J , ψ ψ , sλ/µ| (⟩x)sλ/−νJ(y,)=ψ (1 ψ xiyj,)−− J−sν,J/η(x)sµ/−n9η(δy5)−, 3 11.− 7 −1 (B.24)− | ⟩ here. Then[ n then thek]= Frobeniusλ isn+ givenk coordinates by are[ givenn k∗]= by (α−1,n∗α+2k, α3 β1,[β2n, β3m)=]= , n+, m,0 , , (B.22). λ is a| fermion⟩ basisλ with the label ofi,j the− Frobenius| coordinateη (see2 2 figure2 2 42) of2 a Young dia- | ⟩ where$s is the number of% boxes on the$ diagonal line in λ and the vacuum state 0 satisfies β +β + +β + s ! " # λ is a fermiongram λ basis. When with the thes Frobenius label1t (x2)s oft coordinatethe(sy)= Frobenius2 (1 of + a coordinatex Youngiyj) diagrams t (see(x) figuresλ istλ(y=(4).) ofα1 a, α Young2,...(B.25) dia-β1, β2 ...), | ⟩ λ =( 1)λ/µ λ···/ν ψ∗ β1 ψ∗ β2 ...ψ∗ βs ψν α/sηψ α(µ/s η1) ...ψ "α1 0 , (B.23) ψ 0 = ψ 0 =0for any α > 0, β > 0.− − − − | J| ⟩and ψ , ψ satisfy| ⟩ theα λ following− β∗ commutation−i,j − relations,−η | ⟩ gramn λ.n Whenthenn∗ λ theis Frobenius given|$⟩ by coordinate| ⟩ of a Young% diagram$ λ is λ =(α1, α2,... β1, β2 ...), where| ⟩s is the numberThere areof boxes two Cauchy on the diagonal identities line known in λ and for the the vacuum skew Schur state| functions.0 satisfies then λ Tois see given the by firstCauchy identify identitiesβ (B.24+β +)of we Schur+β first+ s notefunctions that therevisited lefthand side of (B.24) can be written| ⟩ as ψ| n⟩, ψm = ψ∗λ, ψ∗ =0,1 2 ψn,sψ2∗ψ = ψδn m,..., ψ ψ ψ ...ψ ,(B.21) ψα 0 = ψnβ∗ 0=(m=01)for any α··· > 0, mβ >∗ β01. ∗+β2 0 ∗ βs αs α(s 1) 1 α1 0 (B.23) { }| ⟩ { | ⟩| ⟩ }− s { s }− (x)−s (y)=− −(1 −xiyj−)− −s |(x⟩)s (y), (B.24) βe.g.1+β2+ +βs+ λ/µ λ/ν ν/η µ/η [J , ψλ]==(Thereψ 1) are, two··· Cauchyµ V2+ψ( identities⃗x∗[)Jλψ, ψ∗λ∗]=V known...(⃗yψ)ψ∗ ν∗ forψ= the, αµψ skewV+[αJ(⃗x) Schur,JV−...(⃗y]=) functions.ψνnα.δ 0 , . (B.23)(B.22)(B.26) n k n+k β1nλ βk2 − βns+k s i,j (ns 1)−m 1n+ηm,0 where| ⟩ s is− the numberλ ⟨ of| boxes− $| on⟩⟨− the| diagonal−−| ⟩ − line⟨ | −% in λ−and the| −⟩ vacuum$| ⟩ state 0 satisfies $ 1 | ⟩ ψ 0 = ψ 0 =0forsλ any/µ(xα)s>λ/ν0s(,λy/µβ)=t>(x0).s(1λt/ν(xyi)=yj)− (1s +ν/ηx(ixy)js)µ/η(ys)ν,t/η(x)sµ/ηt(B.24)(y). (B.25) λwhereis a fermions isα the basis numberβ∗ with of the boxes label on of the the diagonal Frobenius line coordinate− in λ and the (see vacuum figure 4 state) of a0 Youngsatisfies dia- Then| ⟩ by using| the⟩ commutationλ relation i,j i,j η η | ⟩ | ⟩ There are two$ Cauchy identities$λ known% for the skew%$ Schur functions.$ gramψα 0λ.= Whenψβ∗ 0 the=0 Frobeniusfor any α coordinate> 0, β > 0 of. a Young diagram λ is λ =(α1, α2,... β1, β2 ...), | ⟩ | ⟩ can be derivedsλ/µt (x as)sλt/ν(y)= (1 +1 xiyj) sνt/η(x)sµ/ηt (y). | (B.25) There are two CauchyTo see the identities firstV+( identify⃗x) knownV (⃗y)= (B.24 forcomplete the) we skew first basisV Schurnote(⃗y)1 (ofV that+ functions. (Fock⃗x), the space) lefthand side of(B.27) (B.24) can be written as then λ is given by λ − i,j x y − η $ sλ/µ(x)sλ/ν(y)=i,j%1 (1 i xjiyj)$− sν/η(x)sµ/η(y), (B.24) | ⟩ % − − λ s i,j η β +β +$+βs+ % 1 $ Toλ see=( the1) first1sλ identify/µ2 (x···)sλ/ (νB.24(2yψ)=∗) weψ first∗(1µ note...Vx+ψiy that(∗j⃗x))−ψλ theλ lefthandψsνV/η(⃗yx))s sideνµ/...η=( ofψy) (µ,B.24V+0)(⃗x can, )V be((B.24)⃗y written(B.23)) ν . as (B.26) Eq. (B.26) can be further writtenβ1 as β⟨2 −| βs| ⟩⟨αs | −α(s 1)| ⟩ ⟨ α|1 − | ⟩ | ⟩ − λ sλ/µt (x)−sλt/i,jνλ(−y)= −(1 + −xηiyj)− s−νt/η(x)s−µ/ηt |(y⟩). (B.25) $ %$ $ η $λ µ V+(⃗x) λ λ V%i,j(⃗y)1 ν = µ$V+(⃗x)V (⃗y) ν . (B.26) where s is the numberµsλ ofV/µ+t boxes((⃗xx))Vsλt(/⃗y on⟨ν)(|yν the)== diagonal| (1(1⟩⟨ +|xxiy− lineijy)j−) in| µ⟩λsVνand⟨t/(η⃗y|()xV the+)s(µ/⃗x vacuum)ηt−ν(y).| ⟩ state 0 satisfies(B.25) ⟨Then| by−λ using| the⟩ commutation− ⟨ relation| − | ⟩ | ⟩ λ $ i,ji,j η ψα 0 = ψToβ∗ 0 see=0 the$ firstfor any identifyα > (0B.24, β >) we0%%. first note that$ the lefthand side of (B.24) can be written as | ⟩ | ⟩ 1 1 Then by using the commutation= relation(1 Vxiyj⃗x)−V ⃗y µ V (⃗y) η η V+V(⃗x)⃗y νV , ⃗x , (B.28) ToThere see the are first two identify Cauchy (B.24 identities) we first known note that− for+ the the( ) lefthand skew( ⟨)= Schur| side− of functions.| (⟩⟨B.24| ) can( |) be⟩+ written( ) as (B.27) µ V+(⃗xi,j) λ λ V (⃗y)−νη = µ V+1(⃗x)xViy(j⃗y)−ν . (B.26) ⟨ | %| ⟩⟨ | − $|1⟩ ⟨ %i,j| − − | ⟩ λ V+(⃗x)V (⃗y)= 1 V (⃗y)V+(⃗x), (B.27) sλ/µ(x)$sλ/ν(y)= − (1 xiyj1)− x y sν−/η(x)sµ/η(y), (B.24) Eq.µ (B.26V+()⃗x can) λ beλ furtherV (⃗y)−ν writteni,j= −µ asVi+j(⃗x)V (⃗y) ν . (B.26) Then by usingλ ⟨ the| commutation| ⟩⟨ | i,j− relation| ⟩% ⟨ | η − | ⟩ $$λ % $ Eq. (B.26with) can the be commutation further written relation as (Baker-Campbell-Hausdorff formula) s t (x)s t (y)= (1 + x y ) s t (x)s 1t (y). (B.25) Then by using the commutationλ/µ λ /ν relationµ V+(⃗x)V (⃗y) νi j= 1(1ν /ηxiyj)µ/−η µ V (⃗y)V+(⃗x) ν λ ⟨ V|+(⃗x)Vi,j (−⃗y)=–| 46⟩ – η −V (⃗y)V+(⟨⃗x)|, − | ⟩ (B.27) $ %− $i,j1 − µ V+(⃗x)V (⃗y) ν = (1 xi,jiyj1)−%xµiyVj (⃗y)V+(⃗x) ν ⟨ | − | ⟩ −1% − ⟨ | − 1 | ⟩ V+(⃗x)V (⃗y)= %i,j =V (⃗y(1)V+(x⃗xiy),j)− µ V (⃗y) η (B.27)η V+(⃗x) ν , (B.28) To see the first identify (B.24) we first− note that the lefthand− − side of (B.24⟨) can| − be written| ⟩⟨ | as | ⟩ Eq. (B.26) can be further writteni,j as1 xiyj i,j1 η = (1− xiyj)−% µ V (⃗y) η$η V+(⃗x) ν , (B.28) % − i,j − η ⟨ | | ⟩⟨ | | ⟩ µ V+(⃗x) λ λ V %(⃗y) ν = µ 1V$+(⃗x)V (⃗y) ν . (B.26) Eq. (B.26) can be furtherµ⟨V|+ written(⃗x)V (|⃗y as)⟩⟨ν |=− (1| ⟩xiy⟨j)−| µ V (−⃗y)V+|(⃗x⟩) ν λ − − $⟨ | | ⟩ i,j − ⟨ | | ⟩ % 1 Then by usingµ theV+( commutation⃗x)V (⃗y) ν = relation(1 xiyj)− µ V (1⃗y)V+(⃗x) ν ⟨ | − | ⟩ −= (1 x⟨ iy|j)−− µ V| (⟩⃗y) η η V+(⃗x) ν , (B.28) i,j − ⟨ | − | ⟩⟨ | | ⟩ % i,j η – 46 – % 1 – 46 – $ V+(⃗x)=V (⃗y(1)= xiyj)− Vµ (V⃗y)V(⃗y+)(⃗xη), η V+(⃗x) ν , (B.28)(B.27) − − − i,j − 1 xiηyj ⟨ | | ⟩⟨ | | ⟩ % %i,j − $ Eq. (B.26) can be further written as – 46 – 1 µ V+(⃗x)V (⃗y) ν = (1 xiyj)− µ V (⃗y)V+(⃗x) ν ⟨ | − | ⟩ − ⟨ | − | ⟩ %i,j – 46 – 1 = (1 xiyj)− µ V (⃗y) η η V+(⃗x) ν , (B.28) − η ⟨ | − | ⟩⟨ | | ⟩ %i,j $

– 46 – and then the righthand side of (B.28) precisely reproduces the righthand side of (B.24). The second identity can be derived using the automorphism of the fermionic algebra, t ! : ψ ψ∗, under which λ transforms to λ as ! exchanges αi and βi . In particular, ↔ { } { } we have

t λ λ =( 1)| |!( λ ). (B.29) | ⟩ − | ⟩ The second Cauchy identity then follows directly from the fact that the transformation ! does not change the expectation value of a correlator and !2 = id, i.e.

t λ proofs JHEP_213P_0121 λ V (⃗y) ν =( 1)| |!( λ )V (⃗y) ν ⟨ | − | ⟩ − ⟨ | − | ⟩ λ 2 =( 1)| |! ( λ )!(V (⃗y))!( ν ) − ⟨ | − | ⟩ λ ν t =( 1)| |−| | λ !(V (⃗y)) ν − ⟨ | − | ⟩ 1 t = λ V − ( ⃗y) ν , (B.30) ⟨ | − − | ⟩ where we used !(Jn)= Jn. Therefore − t t t 1 t µ V+(⃗x) λ λ V (⃗y) ν = µ V+(⃗x) λ λ V − ( ⃗y) ν ⟨ | | ⟩⟨ | − | ⟩ ⟨ | | ⟩⟨ | − − | ⟩ !λ !λ t 1 t = µ V+(⃗x)V − ( ⃗y) ν ⟨ | − − | ⟩ t 1 t = (1 + xiyj) µ V − ( ⃗y)V+(⃗x) ν ⟨ | − − | ⟩ "i,j t 1 t = (1 + xiyj) µ V − ( ⃗y) η η V+(⃗x) ν η ⟨ | − − | ⟩⟨ | | ⟩ "i,j ! t t = (1 + xiyj) µ V (⃗y) η η V+(⃗x) ν , (B.31) η ⟨ | − | ⟩⟨ | | ⟩ "i,j ! where we used 1 1 V+(⃗x)V − ( ⃗y)= (1 + xiyj)V − ( ⃗y)V+(⃗x). (B.32) − − − − In the topological vertex formalism, "i,jthese Cauchy Weidentities can see thatlead (toB.31 the) yieldsappearance (B.25). of Nekrasov factors in theThe partition specification function. of the variables, x = q ρ σ and y = q ρ τ , is very useful in { } { − − } { } { − − } this article, and in this case (B.24) and (B.25) become a typical summation in the computation: λ ρ σ ρ τ Q| |sλ/µ(q− − )sλ/ν(q− − ) !λ q 1 Elliptic Braneµ + ν η Web ρ σ ρ τ = P.E. 2 Q Nσ−tτ (Q, q) Q| | | |−| |sν/η(q− − )sµ/η(q− − ), (B.33) (1 q) η # − $ ! λ ρ σ ρ τ (1one-loopQ) Introduction| |s factort (q− − )Nekrasovs t (q− factor− ) − λ/µ λ /ν !λ q µ + ν η ρ σ ρ τ Remark:= P.E. all summationsQ overN tYoung(Q, q )diagrams( Q) | | C| |−| |s t (q− − )s t (q− − ), (B.34) q 2 σ τ µ⌫ ν /η µ/η in non-horizontal#−(1 directions) $ can be takenη in− this way. − ! where the unrefined Nekrasov factor is defined by Then we can see that the exact matching with t t Nekrasov’s formula for U(N) theories,λ i +ν i j+1 νi λ +i+j 1 N (Q, q):= (1 Qq j − − ) (1 Qq− − j − ). (B.35) λν − − i.e. eqn. ( * ). (i,j) λ (i,j) ν "∈ "∈

2 Partition Function of M-string – 47 – The partition function of M-strings [1] can be computed from the refined topological string on a Calabi-Yau geometry specified by a toric diagram with its top and bottom external legs identified together. It can also be interpreted as the instanton partition function of the corresponding 6d =(1, 0) theory on R4 T 2 with omega background twisting. This partition function is an elliptic N ⇥ version of the Nekrasov partition function, which is a natural consequence from the fact that the refined topological vertex on toric Calabi-Yau is dual to 5d =1theoryconstructedfrom(p, q)- N brane web, and the compactification (i.e. identification of external legs) of this brane web along the NS5 direction lifts the theory to 6d. We give a brief review on it in this section, rederiving it by identifying external legs of toric Calabi-Yau, with the Awata-Feigin-Shiraishi (AFS) version of the refined topological vertex [2] (see also [3] for a review and generalization). The building block for a compactified brane web is

⇤[v1]

[v2]

We note that by replacing ⇤ and with the generalized AFS vertices introduced in [3], it is possible to build the compactiﬁed brane web with arbitrary rank of gauge group. What we need to do is to take the trace over the Fock space of the horizontal representation of the DIM algebra. There are

1 (q, 1)u (q +1, 1) uv

(q) ⇤[u, v] (1, 0)v (1, 0)v (q)[u, v]

(q +1, 1) uv (q, 1)u

(q) (q) Figure 3.1: Correspondence between [u, v], ⇤[u, v] and graphic vertices. fig_vertex 3.5 Topological Vertex as Awata-Feigin-Shiraishi Intertwiner

Now we would like to define the refined topological vertex in terms of the DIM algebra as a three-leg object. As we want to identify the weight parameter v in the (0, 1) representation as the position of D5-branes, we decompose the Kähler parameter as Q = u1/u2 and absorb ui’s into two vertices sandwiching the degenerate locus line whose Kähler parameter is Q. We have already defined the shift-vert-ele-1shift-vert-ele-2shift-vert-ele-3shift-vert-ele-4 shifted vertex in (3.1.18), (3.1.19), (3.1.20) and (3.1.21), and roughly speaking, the three-leg vertex we would like to define takes the form,

v, [v]:= [v] ⌘(x), (3.5.1) | i⇠ ; x Y2 v, ⇤ ⇤ [v]:=⇤[v] ⇠(x). (3.5.2) h | ⇠ ; x Y2 To complete the prescription in terms of the DIM algebra, we need to introduce the framing de- (n) (n) pendence in the preferred direction. We add an integer label (n)tothevertex, and ⇤ . See fig_vertex Figure 3.1 for the way to assign vertices with respect to (p, q)braneandwenotethata(p, q)brane is mapped to a (`1 = q, `2 = p)representationoftheDIMalgebra. Taking the convenient normalization we chose in the previous section for the (0, 1) representation,

1 a =(v)| | N (1; q ,q ), (3.5.3) 0 (i,j)1 1 2 (i,j) Y2 the following definitionZ of= the topological0 ...@... vertices0 0 ...A... 0 ... 0 ...... 0 . (102) h | | ih | µ | i h | ⌫ | i (n) (n),µ,X⌫,... n+1 [u, v]:= [u, v] v, := ( uv)| | (/x) : [v] ⌘(x):, (3.5.4) | i ; x x µ + ⌫ + Y2 Y2 (n) Z = ( Q(n1))| | | |( Q2)| | | |...C↵µCµtnC⌫t C⌫tt ... (103) ⇤ [u⇤,v]:= v, ⇤ [u⇤,v]:=( u⇤)| | (x/) : ⇤[v] ⇠(x):, (3.5.5) ,µ,h⌫,, | ; ↵,X,,,... x x Y2 Y2 reproduce theRewriting calculation the of vertex refined operators topological string partition[Awata, function. Feigin, Shiraishi (2011)] C = ⌫ AFSµ . (104) Interestingly, Awata, Feigin and Shiraishiµ,⌫, h in| [31| i⌦] proved| iq,t that the refined topological vertices defined above behavewhere as intertwiners of representation in the DIM algebra.

(n) [ ±(z), ±(w)][u, = v]: 0, (1,n)u (0, 1)v (1,n+1) uv, (3.5.6) i.e. they satisfy ⌦ ! + (n) g (ˆz/w) + is( thez) intertwiner(w)=⇤[u, v in]: the (1 Ding-Iohara-Miki,n+1) uv(w) (1 (algebra,nz))u (quantum(0, 1)v, (3.5.7) 1 (1,n) (0,1) g (ˆ z/w(n)) ! (n) ⌦(1,n+1) toroidal(⇢u algebra⇢v ) of (agﬃ(zne)) gl⇤1), [u, v]=⇤ [u, v]⇢ uv (g(z)), (3.5.8) +⌦ 1 + ±(z) x(n) (w)=(0,1)g ˆ±(1,n2 z/w) x56(w) (1±,n(z+1)) (n) [u, v](⇢v ⇢u )(g(z)) = ⇢ uv (g(z)) [u, v], (3.5.9) ⌦ 1 ⇣ 1 ⌘ What (isz )ax quantum(w)= toroidalg AFS_coproductˆ algebra?2 z/w x (w) (z) where the coproduct± is given by (3.1.42⌥ ). We refer to this± relation as the Awata-Feigin-Shiraishi ⇣ 1 ⌘ (AFS) propertyx in±( thisz)Liex± article. algebra(w)=g (z/w)± x±(w) x±(za)ﬃne Lie algebra

quantization (1 q1)(1 q2) 1 1 + + 2 1 2 The abovex property[Drinfeld(z),x (1985)](w can) be= checked by explicit1 computation(ˆw/z) asˆ follows.w ˆ We w/z need to computeˆ w , [Jimbo (1985)] (1 q ) the contraction of vertex operators,a andﬃnization the3 result⇣ is listed below.⇣ ⌘ ⇣ ⌘⌘ ⇥ quantum group⇤ quantum aﬃne algebra [Ding, Iohara (1997)]

[Drinfeld (1988)] [Beck1 (1994)] quantum toroidal [ ±(z), ±(w⌘)](z) =[u, 0, v]= : ⌘(z)[u, v]:, (3.5.10) (z) aﬃnization algebra + g (ˆz/wY ) + (z) (w)= 1 vq3 (w1 ) (z) [u, v]⌘g(z(ˆ)= z/w) 1 : [u, v]⌘(z):, (3.5.11) z (zq3 ) + 1 + ±(z) x (w)=g ˆ± 2 z/w Yx (w) ±(z) 1 ⇠(z)[u, v]= ( z):⇠(z)[u, v]:, (3.5.12) 1 ⇣ 1Y ⌘ ±(z) x(w)=g ˆ⌥ 2 z/w1 z x(w1) ±(z) [u, v]⇠(z)= (z ):[u, v]⇠(z):, (3.5.13) ⇣ 1 ⌘v Y x±(z) x±(w)=g (z/w)± x±(w) x±(z) + '(z)[u,1/ v2]=:'1/(2z)[u, v]:, + 1 1 (3.5.14)1 x (z),x(w) =(q q ) (ˆw/z) ˆ 2 w ˆ w/z ˆ 2 w 1/2 2 (z ) [u, v]'(z)= Y : [u, v]'(z):, (3.5.15) ⇥ ⇤ ⇣ 1/2 1 ⇣ ⌘ ⇣ ⌘⌘ (z q3 ) Y 1/2 1 (z 1q qz) '+(z) [u, v]=Yg(z)= 3 :.'+(z) [u, v]:, (105)(3.5.16) (z1q/2) z Y + + [u, v]' (z)=:[u, v]' (z):, (3.5.17) ˆ (106)

1 ⌘(z)⇤ [u, v]= (z ):⌘(z)⇤ [u, v]:, (3.5.18) Y + 1 / 1 (107) ⇤ [u, v]⌘(z)= z/v0 0(z ):⇤ [u, v]⌘(z):, (3.5.19) Y 1 ⇠(z)⇤ [u, v]= 1 : ⇠(z)⇤ [u, v]:, (3.5.20) (zq ) 11 Y 3 v 1 ⇤ [u, v]⇠(z)= : ⇤ [u, v]⇠(z):, (3.5.21) z (z) Y

'(z)⇤ [u, v]=:'(z)⇤ [u, v]:, (3.5.22) 3/2 2 (z ) ⇤ ⇤ [u, v]' (z)= Y 1/2 : [u, v]' (z):, (3.5.23) (z ) Y 1/2 + (z ) + ' (z)⇤ [u, v]=Y : ' (z)⇤ [u, v]:, (3.5.24) (z 5/2) Y + + ⇤ [u, v]' (z)=:⇤ [u, v]' (z):. (3.5.25)

57 Z = 0 ...... 0 0 ...... 0 ... 0 ...... 0 . (102) h | | ih | µ | i h | ⌫ | i ,µ,X⌫,... Z = 0 ... ... 0 0 ...µ ... 0 ... 0 ...⌫ ... 0 . (102) h | | ih | |fixi theseh | parameters| i for all the vertices and decompose all Kähler parameters in a consistent way. ,µ,⌫,... X µ + ⌫ + Z = ( Q )| | | |( Q )| | | | ...C UnderC t thisC decomposition,t C t t ... the(103) position parameter v can be completely absorbed into the vertex 1 2 ↵µ µ ⌫ ⌫ ,µ,⌫,, operator, and we define the vertex and dual vertex after this shift as µ + ⌫ + Z↵=,X,,,... ( Q1)| | | |( Q2)| | | | ...C↵µCµtC⌫t C⌫tt ... (103) n n ,µ,⌫,, 1 v 1 1 1 v ↵,X,,,... [v]:=exp J n exp Jn , (3.1.18) ; n 1 t n n 1 qn n=1 ! n=1 ! X X Cµ,⌫, = µ q,t ⌫ . (104) n n n n h | ⌦ h | | i 1 1 v 1 1 v C = µ ⌫ . ⇤[v] := exp (104) J n exp Jn , (3.1.19) µ,⌫, q,t ; n 1 t n n 1 qn h | ⌦ h | | i n=1 ! n=1 ! X X

[ ±(z), ±(w)] = 0, 1 1 [ ±(z), ±(w)] = 0, 1 n n n 1 n n ⌘(x)=exp t (1 q )xJ n exp (1 t )x Jn , (3.1.20) + g (ˆz/w) + n n g (ˆz/w) n=1 ! n=1 ! (z+) (w)= 1 (w) (+z) (z) (w)=g (ˆ z/w) (w) (z) X X 1 1 1 g (ˆ z/w) 1 n n n n 1 n n n 1 ⇠(x)=exp t (1 q ) J n exp (1 t ) Jn , (3.1.21) + 2 + 1 x x ±(z) x (w+)=g ˆ± z/w1 x (+w) ±(z) +(z)=n +z n. (108)n ±(z) x (w)=g ˆ± 2 z/w x (w) ±(z) n=1 n ! n=1 ! X n=0 X ⇣ 1 ⌘ 1 X ⇣ 1 ⌘ 1 where we redeﬁned ±(z) x(w)=g ˆ⌥ 2 z/w x(w) ±(z) ±(z) x(w)=g ˆ⌥ 2 z/w x(w) ±(z) + `1/2 `2 (ˆ, / ) (q ,q ). (109) ⇣ 1 ⌘ 0 0 7! 3 i 31 j+1 x (z) x (w)=g (z/w⇣ )± x1 (⌘w) x (z) x := vt q , for x =(i, j) . (3.1.22) ± x±(z±) x±(w)=g (z/w)± ±x±(w)±x±(z) 2 Z = 0 ... ... 0 0 ...µ ... 0 ... 01...⌫ ... 0 . (102) 1 + (1 q1)(1h | q2) | ih | | +i h | 1 | i 1 1 + (1,µ,⌫q,...1)(1 q2) + 2 1 2 x (z),x(w) = (ˆw/z) Multiplyingˆ w2 these ˆ vertex `w/z1 =0 operators, `2 Nˆ. together 2w and, taking the(110) expectation value of the correspond- x (z),x(w) = X 1 1 (ˆw/z) ˆ w ˆ w/z 2 ˆ w , (1 (1q3q) ) 3 ⇣ ⇣ ing correlator,⇣⇣ ⌘⌘ we obtain the factors⇣⇣ in the⌘⌘⌘⌘ unpreferred direction of the reﬁned topological string ⇥ ⇥ ⇤ ⇤ µ + ⌫ + Z = ( Q )| | | |( Q )| | | | ...C C t C t C t t ... (103) 1 2 partition↵µ µ function.⌫ ⌫ We will give the prescription for preferred directions in a later section, and ,µ,⌫,, `1 =1, `2 N. (111) [ ±(z), ±(w)] =↵,X, 0,,... before that we want to know the algebraic2 structure of the above vertex operators, especially that [ ±(z), ±(w)] = 0, of ⌘(z)and⇠(z). + g (ˆgz/w(ˆz/w) ) + + (z) quantum(w)= toroidal algebra (of( glw)1+) (z) (z) (w)= 1 (w) (zU) q,t(gl )(`1, `2)v (29) (112) g (ˆ1 z/wC)µ,⌫, = µ q,t From⌫ 1. explicit computation,(104) we have g (ˆ z/w) h | ⌦ h | | i [Ding, Iohara (1997)] (Ding-Iohara-Miki algebra)1 + + [Miki (2007)] ±(z+) x (w)=g ˆ1 ± 2 z/w +x (w) ±(z) 1 ±(z) x (w)=g ˆ± 2 z/w x (w) ±(z) b (1 w/z)(1 q3 w/z) ⌘(z)z⌘(w)=zv : (113)⌘(z)⌘(w):, (3.1.23) [ ±(z), ±(w)]⇣ =1 0, ⌘ 1 ! (1 q1w/z)(1 q2w/z) 1 ±(z) x(w)=⇣g ˆ1 ⌥ 2 z/w⌘g (ˆz/w)x(w) ±(z) ±(z) x(w)= +(z)g (ˆw⌥)=2 z/w x( w)(w )± (+z(z)) (1 q3w/z)(1 w/z) 1 ⇠(z)⇠(w)= : ⌘(z)⌘(w): g1(ˆ z/w) 1 1 ⇣ ⌘ (1 q1 w/z)(1 q2 w/z) x±(z) x±(w)=g (z/w)± x±1 (w) x±(z) (114) ⇣+ 1 ⌘ + x±(z) x±(w)= ±(z)gx(z/w(w)=)± gx±ˆ(±w2 z/w) x±x(z)(w) ±(z) 1 1 (1 1 q3 z/w)(1 z/w) + 1/2 1/2 + 2 1 = 2 : ⌘(z)⌘(w):, (3.1.24) x (z),x(w) =(q q⇣ 1 ) ⌘(ˆ1 w/z) 1ˆ w ˆ w/z ˆ1 w + 1/2 1/2 + 1 (1 q1z/w)(1 q2z/w) x (z),x(w) =(±(z) xq(w)=qg ˆ)⌥ 2 z/w (ˆw/zx)(w ) ±(zˆ) 2 w ˆ w/z ˆ 2w v1 (1 q w/z)(1 q w/z) (115) ⇥ ⇤ ⇣ ⇣1 ⌘ ⇣ ⌘ ⇣ 1 ⌘⌘ 2 x±(z) x±(w)=g (z/w)± x±(w) x±(z) ⌘(z)⇠(w)= : ⌘(z)⇠(w):, (3.1.25) ⇣ ⇣ ⌘ v2 ⇣ ⌘⌘ 1 (116) ⇥ ⇤ (1 w/z)(1 q3 w/z) + (1 q1)(1 1 q2)qz + 1 1 1 x (z),x (w) = (ˆw/z) ˆ 2 w ˆ w/z ˆ 2 w , g(z)=1 . (105)(1 q1w/z)(1 q2w/z) (1 q3 ) ⇠(z)⌘(w)= : ⌘(z)⇠(w):, (3.1.26) with 1 qqz ⇣z ⇣ ⌘ ⇣ ⌘⌘ 1 ⇥ ⇤ (1 w/z)(1 q3 w/z) where g(z)= . Q1 =(105)v1/v2 (117) q (1z q1z)(1 q2z)(1 q3z) g(z)= 1 where 1 we repeatedly1 , used q1q2q3 =1.Wethushavethefollowingalgebraicrelations,(3.1.29) (1 q z)(1 q z)(1 q z) ˆ 1 2 3 (106) 1 1 1 satisfying g(z )=g(z) . ⌘(z)⇠(w)and⇠(w)⌘(z)arealmostthesame,butwehavetobecareful(q1 = t, q2 =⌘q(z))⌘(w)=g(z/w)⌘(w)⌘(z),(118) (3.1.27) because ˆ :center of the group (106) 1 ⇠(z)⇠(w)=g(z/w) ⇠(w)⇠(z), (3.1.28) + 1 i(1 aiz) i j i(z ai) i(1 ai/bk) |{ }||{/}| is also a center (107) z 0 0 1 = (bkz). (3.1.30) j(1 bjz) j(z bj) j=k(1 bj/bk) Q Q Xk Q6 41 As a consequence,Q Q Q 11 (1 11q1)(1 q2) 1 ⌘(z)⇠(w) ⇠(w)⌘(z)= 1 (w/z) ( w/z) : ⌘(z)⇠(w): (1 q3 ) (1 q1)(1 q2) + 1/2 1 1/2 = 1 (w/z)' ( w) ( w/z)'( w) ,12 (3.1.31) (1 q3 ) where we introduced two new vertex operators,

+ 1/2 1/2 1/2 1/2 ' (z)=:⌘(z )⇠(z ):, '(z)=:⌘(z )⇠(z ):. (3.1.32)

We can compute their algebraic relations to ﬁnd

1/2 ⌘(z)'±(w)=g11(⌥ w/z)'±(w)⌘(z), (3.1.33) 1/2 ⇠(z)'±(w)=g(⌥ z/w)'±(w)⇠(z). (3.1.34) DI Interestingly, the above algebraic relations are exactly those of Ding-Iohara type [27] with g(z) g-DI speciﬁed by (3.1.29). This algebra of Ding-Iohara type more generally is given by

[ ±(z), ±(w)] = 0, (3.1.35)

+ g (ˆz/w) + (z) (w)= 1 (w) (z) (3.1.36) g (ˆ z/w) + 1 + ±(z) x (w)=g ˆ± 2 z/w x (w) ±(z) (3.1.37) 1 ⇣ 1 ⌘ ±(z) x(w)=g ˆ⌥ 2 z/w x(w) ±(z) (3.1.38) 1 x±(z) x±(w)=g (⇣z/w)± x⌘±(w) x±(z) (3.1.39)

(1 q1)(1 q2) 1 1 + + 2 1 2 x (z),x(w) = 1 (ˆw/z) ˆ w ˆ w/z ˆ w (3.1.40), (1 q3 ) ⇥ ⇤ ⇣ ⇣ ⌘ ⇣ ⌘⌘ where ˆ is a central element of the algebra. We can see that the realization discussed above,

+ x (z) ⌘(z),x(z) ⇠(z), ±(z) '±(z), ˆ , (3.1.41) 7! 7! 7! 7!

2 A unifying framework of dualities integrable Nakajima quiver system Bethe/Gauge Maulik-Okounkov’s correspondence R-matrix symmetric polynomials quantum group q-Virasoro vertex F-theory operator quantum toroidal algebra algebra W-algebra

algebraic topological geometry SL(2,Z) duality vertex

brane construction topological string (type IIB) string Calabi-Yau M-theory theory Uq,t(gl1) (29)

b q 1,t= q (30) ! Uq,t(gl1) (29) Relation with W-algebras b Uq,t(gl1) (29) b U (gl )q-deformedq 1+1 (29),t[µ=]q (31)(30) q,t 1 W! 1 b q 1,t= q (30) !(4d limit)b

Uq,t(gl1)Yangian/SHc algebra (A-type) (29) 1+ [µ] (31) 1 … [Prochazka,W 2015] [Shiﬀmann, Vasserot,b 2012] [Maulik, Okounkov,b 2012] • Ding-Iohara-Miki algebra on the tensor product of N Fock spaces contains a U(1)x (q-deformed) WN algebra.

[Feigin, Hoshino, Shibahara, Shiraishi, Yanagida, 2010]

2 2

Uq,t(gl1) (29) b b

(30) q (32) 1,t= q (33) (30) !

1+ [µ] (31) W 1

R✏1 R✏2 1

q1 = e = t, q2 = e = q . (32)

. S-duality :

q 1 < symmetry

= 8 - W dual another N

= e R

µ3✏ ,Q3 1,Qb

✏ 1

-symmetry of AGT of -symmetry W N µ1,Q1

R 1 ,q t, 4,Qf Q4

e (33) 1 1 q 3,Qf Q1 Q2 ]8 (31) ) (29) µ5,Q5

= µ

2 µ4,Q4 = µ2,Q2

[

< S-duality W S-duality 2 = b b gl 1 1

,Q Q Q Q 1 2 b 1 2 4

(

,t e 1+

1 1 2016] Pestun, [Kimura, 8 < : 1+

q,t : W) (quiver

Figure 14: Toric✏ diagram of T theory.

2 3

-symmetry of Kimura-Pestun Kimura-Pestun of -symmetry W N

t, q q ! U [

! µ

q (31) ]

= q

// 1 1 ,t = .

˜1 || Then ZT2 /(ZU(1)ZU(1)ZU(1))coincideswiththepartitionfunctionoftheT2-theory. According

✏ // = U

R || b to the argument in section 3, the factors Z ,Z and Z are identiﬁed withb the con-

U(1) U(1) U(1) q,t

13년 9월 8일 일요일 q

tribution from decoupled M2-branes on two-cycles associated with Q2Q3, Q3Q1( and Q1Q2,

respectively. All the two-cycles can be continuously moved to inﬁnity, and1 therefore it is

1 (29) ) (33) ˜ = reasonableq to eliminate them. Now let us consider ZT2 /ZU(1). From (4.25), it follows that ˜ = ZT2 /ZU(1) coincides with the Nekrasov partition function of the U(1) gauge theory with two

fundamental matters. This strongly suggests that the TN -theory partition function is related to the partition(32) function of some simpler gauge theory with Lagrangian description.

4.1.2.2 T3 theory

3 We now consider the T3 theory engineered by C /(Z3 Z3). The relevant toric web diagram ⇥ is shown in Figure 14. The theory has a global symmetry SU(3)3, which is visible from the

diagram, and in fact the symmetry is enhanced to E6 [7]. The Coulomb branch of this theory is one dimensional corresponding to(30) the size of the hexagon in the center of the diagram. The theory also has a Higgs branch of complex dimension 22, which is the same dimension as the

dimension of the one instanton moduli space of E6 gauge theory. These properties of the T3

theory allow us to relate it with SU(2) gauge theory with Nf =5 fundamentalhypermultiplets in 5d.16 We will compare the partition functions of two theories in the next section, which provides a strong evidence for the relation.

The diagram leads to the T3 partition function

˜ 1/2 ZT3 =(M(t, q)M(q, t)) Z(t, q, Q) ,

µ1 µ2 µ3 µ4 µ5 ⌫1 1 1 ⌫2 Z(t, q, Q)= ( Q )| |( Q )| |( Q )| |( Q )| |( Q )| |( Q )| |( Q Q Q Q )| | 1 2 3 4 5 b b 1 2 4 X~⌫ ,µ~ 16As mentioned already, for a 5d Sp(1) gauge theory with N 5, the relevant Calabi-Yau threefold is not f toric with a generic complex structure. However, in the case of Nf = 5, there is a special choice of complex structure with which the Calabi-Yau threefold becomes toric.

2 After all, what we want to study is SO(N) theory. T 1 ∅ → (a) (b)

Figure 3. (a): the brane web by sending T 1 from the diagram in figure 1(b). (b): the diagram → to be used for the computations, where we shall set the leg attached to the D7-brane to be empty. proofs JHEP_213P_0121 the line with the Kähler parameter T , it turns out that a finite sum gives the correct answer, which we will see in section 3.2. We can use this technique to compute the partition functions of SO(2N +1) gauge theories and also G2 gauge theories utilizing the construction in [18]. Furthermore it is also possible to compute the pure SU(3) gauge theory with the CS level κcs =9by applying the brane web proposed in [33], which may be obtained by a twisted compactification of 6d pure SU(3) gauge theory with a tensor multiplet [38].

3 Examples

In this section, we give several examples of calculations that involve the usage of the O- vertex defined in section 2. More concretely, we first check the validity of our formalism in section 3.1, 3.2 and 3.3 by comparing the partition functions of the SO(2N), SO(2N +1) and G2 gauge theories obtained in the topological vertex formalism with the O-vertex with the known one-instanton and two-instanton partition functions in the literature. In section 3.4, we will further apply the O-vertex to the calculation of the partition function of the 5d pure SU(3) gaugeAs theory we mentioned with the before, CS level theκ integralcs =9, byis di usingfficult the to perform brane diagram for SO(N) proposed for it in [33]. and Sp(N) theories.

3.1 SO(2N) gauge theoriesHowever, the brane construction is simple. 0 1 2 3 4 5 6 7 8 9 N We start from the computation of the partitionD5 function of the pure SO(2 ) theory. The • • • • • • brane web diagram is obtained by puttingNS5 a stack of N D5-branes on the top of an O5−- • • • • • • plane. Hence we can directly apply the7-brane O-vertex proposed in section 2.2. • • • • • • • • O5± • • • • • • 3.1.1 Pure SO(4) gauge theory TableAccording 2: Conﬁguration to SO or Sp of gauge branes group, in the we brane add webO5+ construction.or O5- orientifold Bar represents The ﬁrst example is the simplestthe(O-plane). direction case pure branesN =2 SO(8), stretch i.e. gauge SO(4) along, theory gauge and is dot theory. given means by The the 5-brane point-like web direction for branes. diagram for the pure SO(4)(Well-known theory is given story by in string theory text book) Q Q T R Q ′ Q′ P P + + (3.43) O5 O5− O5 (3.1) TheSO(4) Coulomb theory branch moduli a1,a2,a3SO(8),a4 are theory the height of the color D5-brane. Hence a Ai = e− i , (i =1, 2, 3, 4) are parametrized by proofs JHEP_213P_0121 A1 = TRQ′P, A2 = RQ′P, A3 = Q′P, A4 = P. (3.44) Extrapolating– 11 – the external (2, 1) and (2, 1) 5-brane webs on the O5-plane, the instaton − fugacity is related to the length between the 5-branes on the orientifold. Namely it is given by 2 4 2 4 4 4 q = QT A1− = QT − Q′− P − R− . (3.45) From the web diagram in (3.43) it is possible to compute the partition function using the topological vertex and the O-vertex and the partition function becomes SO(8) Ztop 18 λ + σ + τ + υ α + β +2 σ µ + ν γ + δ ι + π +2 υ = ( Q)| | | | | | | |( Q′)| | | | | |( P )| | | |( R)| | | |( T )| | | | | | − − − − − µ,ν,α,β,λ!,σ,γ,δ,τ,υ,ι,π 1 3 1 1 1 3 fλfα− fβfσ− fγ− fδfτ− fι− fπfυ− CαtγλCδβtλt Cνtασt CβµtσCγtιτ Cπδtτ t Cιt υC πtυt VνWµ. ∅ ∅ (3.46) Since there is no parallel external lines, (3.46) does not contain an extra factor. The perturbative part is obtained by the limit Q 0 and the partition functin splits → into two parts, SO(8) SO(8) SO(8) Zpert = Zleft Zright , (3.47) where SO(8) α ν γ ι 1 1 1 Zleft = ( Q′)| |( P )| |( R)| |( T )| |fα− fγ− fι− Cαtγ Cνtα Cγtι Cιt Vν, (3.48) ν,α,γ,ι − − − − ∅ ∅ ∅ ∅∅ ! SO(8) β µ δ π Zright = ( Q′)| |( P )| |( R)| |( T )| |fβfδfπCδβt Cβµt Cπδt C πt Wµ. (3.49) − − − − ∅ ∅ ∅ ∅ ∅ µ,!β,δ,π The computing the summation of the left part yields

SO(8) q 2 2 2 2 Z = P.E. Q′ + R + T + Q′R + RT + Q′RT + P Q′ + RQ′P + RQ′ P left (1 q)2 " − # 2 2 2 2 2 2 4 3 3 3 + Q′RT P + Q′ RT P + Q′ R TP + o(P ,Q′ ,R ,T ) (3.50) $% q 1 1 1 1 1 1 = P.E. A A− + A A− + A A− + A A− + A A− + A A− + A A (1 q)2 3 4 2 3 1 2 2 3 1 3 1 4 3 4 " − # 4 3 3 3 + A2A4 + A2A3 + A1A4 + A1A3 + A1A2 + o(P ,Q′ ,R ,T ). (3.51) $%

– 18 – where we also introduced the parameter P . As explained in section 2.1, the partition function of the diagram (2.23) can be computed by deforming the diagram into the following one, In fact one can infer a candidate for another form of the vertex operator O(P, q) which satisfies (4.1). In order to see that, let us computeν the perturbative part of the pure Q2 SO(2N +2) gauge theory from the O-vertex. We focusP on the contribution from the left half of the 5-brane web diagram of the pure SO(2N +2) gauge theory as it contains the O5 O-vertex Vν. The partition function from the− left half of the diagram contains the(2.24) square root of the perturbative part of the partition function of a pure SU(N +1) gauge theory. 1 Q Q P PQ proofs JHEP_213P_0121 Fromwhere the topological′ = − and vertex′ = computation,and taking it is the possible fundamental to factor region out such as in a ( part2.10). by The using partition function for the diagram (2.24) is then the identity (B.33) and the remaining contribution is given by 2 µ 2 λ 2 ν ZνN(P, Q)= ( Q )| |Q | |f ( P )| |CνµλC t . (2.25) λ µ λ proofs JHEP_213P_0121 ν ηi − ρ −ρ ∅ ρ ρ | | µ,λ Vν( P )| | Qi !sν/η1 (q− )sη1/η2 (q− ) ...sηN 1/ηN (q− )sηN (q− ) − − ν,η1,η2,...ηN "i=1 $ Therefore! the function# associated to the diagram (2.22) is obtained by applying the limit(4.5) ν ρ ˜ ρ ˜ ρ ˜ ρ = Vν( P )| |sν/η1 (q− )sη1/η2 (Q1q− ) ...sηN 1/ηN (QN 1q− )sηN (QN q− ), Q with PQ−fixed in (2.25). Then it seems that (2.25− ) diverges− from the contribution ν→∞,η1,η2,...ηN coming! from Young diagrams satisfying 2 µ +2 λ > ν . However we observe that there is | | | | | | wherenon-trivial we defined cancellation for higher orders of Q. When ν = , then (2.25) reduces to (2.17) i ∅ and it is just 1 as we checked in (2.21). Even when ν is non-trivial, interestingly, the result Q˜i = Qj. (4.6) as an expansion over Q2 stops at a finite order. For example, for ν =(5, 1),wehave j#=1 WeIt remarkis possible that to write the instaton (4.5) in terms part of ofq the the13P expectation6 partition function value of is vertex expected operators.q9Q2 toP 6 be First symmetric note Z (P, Q)= underthat the(5, skew1) Weyl Schur group(1 function ofq)6 SU(3).(1+ mayq)3(1+ Until beq2 written)(1 theq order+ byq2)(1+ weq computed+ q2)2 − (1 weq can)5(1+ forq)2 example(1+q2) see a − 6 4 6 − 3 6 6 − part of the symmetry fromq Q theP part q Q P 11 + 4 s2λ+/µ(⃗x)=3 λ V (⃗x) µ , 2 + o(Q ). (2.26)(4.7) (1 q) 2(1+qq q) (1 q) ⟨(1+| q−)(1+|q+⟩ q ) ˆSU(3)9 − − 2 2 2 2 Ztop = P.E. + 2 A1/A2 + A1/A2 + A2/A1 + A2/A1 + ... , (3.134) where V (⃗x) is defined··· in(1 (B.19q)). Then using (4.1) and (4.7), (4.5) can be written as ν For various− choices! of ν we− have checked" the function (2.25) terminates# at the order$ Q| |. 5 ν in (3.132Hence), we which conjecture is symmetricν that the expansion underρ the˜ by exchangeQρof (2.25A)1 terminatesA2˜. at theρ order˜ Q| | ρfor any Vν( P )| |sν/η1 (q− )sη1/η2 (Q1q− ) ...sηN 1↔/ηN (QN 1q− )sηN (QN q− ) νν.,η Then,η ,...η it is possible− to take the limit Q with PQ− fixed and− (2.25) becomes 1 !2 N →∞ ρ ρ ρ 4 O-vertex= 0 (P, q as) ν a vertexν V (q− operator) η η V (Q˜2 qµ− )2 λη 2 ην V Q˜ q− 0 O lim Zν(P, Q)= 1 1 ( Q 1)| |Q | |2fλ( P )| N|CνµλC µtNλ. (2.27) ⟨ | Q| ⟩ − − ··· − ∅ − − proofs JHEP_213P_0121 PQ→∞fixed% & N & '!µ,(λ & & ) ( & * + & ) The topological vertex computationρ& &2 µ may+2ρ λ = be&ν rephrased& by expectation& values of some& vertex = 0 O(P, q)V (q− ) V (Q˜i|q−| )| |0 &| | & & & ⟨ | − − | ⟩ operators [5, 34]. In this sectioni=1 we propose a vertex operator corresponding to the O-vertex Therefore we define a vertex# function Vν by introduced in section 2 along the lineN ofñ then vertex operator formalism. Some technical ∞ 1 (1 + i=1 Qi )q 2 = 0 O(P, q) exp J µn 2 0 , (4.8) details of the vertex operatornVν formalism:= 1 qn are( summarized1)−| |fλCνµλC inµt appendixλ, B.2. (2.28) , & "n=1 - − $ & . ∅ & ! −µ,λ & & ! & [Hayashi, RZ (2020)] & Proposal 2 µ +2 λ = ν & 4.1where O-vertex we& used the operator completeness of| | the| | Frobenius| | basis& to sum over[Nawata, the Young RZ (wip)] diagrams fromwhich the is second associatedThe topological line to the the vertex diagram third seems line. to be applicable to brane webs with O-plane, We consider reformulatingexcept for the intersection the computation point of 5-brane using and the orientifold. O-vertex obtained in section 2 by Since (4.8) gives a part of the perturbative part of the partition function of the pure using operators and their expectation values. This mayν be regarded as a first step to develop SO(2N +2) gauge theory except for the pureVν SU(N +1) part, (4.8) should be equal to a purely analytic method to compute SO(N) and G2 partition functions in a closed form + at each order of the instanton2 number.O5 MoreN detailsO5− will be presentedN 2 in a future work(2.29) [54]. P q ˜2 ˜ Let us try toWe express proposeP.E.V aas new an topological expectation vertex1+ value (O-vertex)Qi of+ a vertexfor1+ this intersection operatorQi .point.(q), (4.9) µ"2(1 q)2 "− $$O V − / i=1 0 / i=1 0 We will call ν O-vertex as it is associatedν ! to a line intersecting! with the O5-plane. Vν( P )| | = 0 O(P, q) ν . (4.1) Then comparing (4.8) with (4.9),− we find that⟨ | a candidate| ⟩ for another form of the vertex Theoperator ket stateO(ExpectationP,ν q) herein the is value bosonic the of fermion a vertex-operator, basis basis may be labeled given by by the Frobenius coordinates of ν (see | ⟩ appendix B.2 for more details). It is straightforwardn n to obtainn the operator O(P, q) as ∞ P 2–7–(1 + q ) P 2 a form in the expansionO(P, q)=exp of the fermion basis. SinceJ2n the+ fermionJnJn basis. is orthonormal(4.10) n qn n "n=1 "− 2 (1 ) 2 $$ ν′ ν = δν ν, one way to express! the operator−O(P, q) is given by ⟨ | ⟩ ′ It directly follows that Vν vanishes for odd-size Young diagram. µ O(P, q)= ( P )| |Vµψµ∗, (4.2) µ − % with – 36 –

s β +β + +βs+ ψ∗ =( 1) 1 2 ··· 2 ψ∗ ψ∗ ψ∗ ψ ψ ψ , (4.3) µ − α1 α2 ··· αs βs ··· β2 β1 where (α , α , , α β , β , , β ) is the Frobenius coordinate of a Young diagram µ (see 1 2 ··· s| 1 2 ··· s ﬁgure 4). For example, the expression of the operator O(P, q) until the order P 4 is given by qP 2 P 2 q3P 4 O(P, q)=1+ ψ∗ ψ1/2 + ψ∗ ψ3/2 ψ∗ ψ1/2 1 q 3/2 1 q 1/2 − (1 q)(1 q2) 7/2 − − − − qP 4 (1 + q3)P 4 ψ∗ ψ ψ∗ ψ∗ ψ ψ (4.4) − (1 q)(1 q2) 5/2 3/2 − (1 q)(1 q2) 3/2 1/2 1/2 3/2 − 2 4− − 4 − q P P 4 + ψ∗ ψ + ψ∗ ψ + o(P ), (1 q)(1 q2) 3/2 5/2 (1 q)(1 q2) 1/2 7/2 − − − − from the explicit form of Vν, which can be computed by (2.28). 5We need to compute higher order terms for seeing the paired part under the exchange A A for the 1 ↔ 2 other terms in the instanton part in (3.132).

– 35 – After a little computation, we observe that Vν takes the form of

Pν(q) Vν = , (2.30) We can compute the explicit expression(q; qof) ν the/2 O-vertex. After a little computation, we observe that V| ν| takes the form of where we defined Pnν(1q) V = − , (2.30) qν q qk+1 . ( ; )n (=q; q)(1ν /2 ) (2.31) | | − A Explicit form of O-vertex k!=0 where we defined n(n+1) ν APν in Explicit (2.30) is form a polynomial of O-vertex of q of degreen 1 at most m( ν )= 2 for n = |2| and can and it seems that Pν is a polynomial.− | | In thisonly appendix, be non-zero we provide when ν theis even. explicit Some form of the of the explicitk O-vertices+1 expressionsVν,W ofν Pdefinedν are listed in (2.28 below,) | | (q; q)n = (1 q ). (2.31) proofs JHEP_213P_0121 In this appendix, we provide the explicit form of− the O-vertices Vν,Wν defined in (2.28) and (2.37) respectively. As discussed in sectionk=0 2, we observed that the O-vertex Vν may and (P2.37(2) )= respectively.q, P(1,1) As=1 discussed, in section! 2, we observed that the O-vertex Vν may(2.32) be written as (2.30− ). We here write down the explicit form of Pnν(nq) in (2.30) andν see 3 3 2 ( +1) Pν inbe (2.30 writtenP) is= a asq polynomial,P (2.30). We= here of q,q writeof P degree down=1+ at theq most explicit,Pm( formν=)= ofqP,Pν(q) infor (2.30=1n)= and, | | see(2.33)and can an interesting(4) pattern. The(3,1) expression− (2, of2) Pν(q) for ν (2=2,1,1)| ,|3 −has been2 (1, written1,1,1) in (22.32) only bean non-zero interesting when pattern.ν is The even. expression Some of ofP theν(q) explicitfor| ν| =2 expressions, 3 has been of writtenPν are in listed (2.32) below, and (2.33) and the explicit form for 6 ν 10 is | | proofs JHEP_213P_0121 andand (2.33 more) and can the be found explicit| | in form appendix for ≤6 A| .| Weν≤ will10 is also give a candidate for the refined version of this vertex in6 section 4.2. ≤ | |3≤ 5 6 P(2) = P(6)q,= Pq (1,P,1)6 =1, (5,1) = q ,P3 (4,2) = (q+q5 +q6 ), (2.32) −In theP(6) same−= way,q ,P we can compute(5,1) another= q ,P O-vertex which is(4, associated2) = −(q+q to+ theq ) diagram, P 3 = q4 +−q5,P=1+q43+q5,P2 =0−, P(4) =(4q,1P,1),P=(3q4,1)+q=5,Pq, P(2(3,2),3)=1+=1+qq4,P+q5,P(2,1,1) = (3q,2,1),P=0(1, ,1,1,1) =1, (2.33) proofs JHEP_213P_0121 given by(4,1,1) (3,3) (3,2,1) proofs JHEP_213P_0121 2 − 2 6 − 5 P(3,1,1,1) = (q+q ),P2 (2,2,2) = (q+q +2 q 6),P(2,2,1,1) =1+q+q5 , and more canP(3,1 be,1,1)− found= (q in+q appendix),PA(2.,2 We,2)ν=− will(q+ alsoq + giveq ),P a candidate(2,2,1,1) =1+ forq+ theq , refined version 3− −W P(2,1P,1,1,1) = =q ,Pq3,P(1,1,1,1,1,1) =1=1, , ν (A.1)(A.1) of this vertex(2,1,1 in,1,1) section− − 4.2. (1,1,1,1,1,1) O5 O5+ In the same way,10 10 we can compute6 6 another− O-vertex which is associated7 88 99 to the(2.34) diagram WeP do(8) Pnot=(8)q have=,Pq ,Pa closed-form(7,1)(7,=1) =q ,Pq formula,P for them.(6(6,1,1,1),1) == (q ++qq ++qq )),, given by − − −− ForP computingP = q3=+qq this38++qq8 type9++q9q+10 ofq,P10 O-vertex,P we start from the diagram== (q+ inqq66 (++2.24qq77++)qq and88),), take the (6,2) (6,2) (5(5,3),3) −− fundamental3 region34 as4 5 5 P(5,1P,1(5,1),1,=1,1)q=+qq ++qq+,Pq ,Pν (5(5,2,2,1),1) == PP(3(3,2,,11,,11,,1)1) =0=0,, Wν P = (q2+q2 +3 q3 +7 q7 +8 q8 +ν9q9),P= (q6 +q88), P(4,2,1(4,1),2,=1,1) (q−+q +q +q +q +q ),P(4(4,3,3,1),1) = −(q +q ), − 5 6 7 + − 5 6 7 8 10 P 5= q 6+q 7+q ,PO5− O5 =1+q5+q 6+q 7+q 8+q 10, P(4,1,1(4,1,11),1,=1,1)q +q +q ,Pλ (4(4,4),4) =1+q +q +q +q +q , (2.34) 2 3 4 5 7 8 9 10 P(4,2,2) 2= q 3+q 4+q +5 q +7 q +8q +9q +10q , P(4,2,2) = q +q +q +q +q +q +q +q , For computing this type of2 O-vertex3 7 8 we9 startλ fromO5− the diagram2 in4 (2.24) and take the P(3,3,2) = (q2+q +3 q +7 q +8 q +9q ),P(3,2,2,1) = (q2+q 4), P(3,3,2) = (q−+q +q +q +q +q ),P(3,2,2,1) = −(q +q ), (2.35) fundamental region− as 2 3 − P(3,1,1,1,1,1) = (q2+q +3 q ), P , , , , , = (q−+q +q ), (3The1 1 1 associated1 1) − partition2 function3 5 is6 given7 by8 2 3 4 5 10 P(3,3,1,1) =1+q2+q +3 q +5 q +6νq +7q +8q ,P(2,2,2,2) =1+q2+q 3+q 4+q 5+q 10, P(3,3,1,1) =1+q+q +q +q +q +q +q ,P(2,2,2,2) =1+q +q +q +q +q , P = (q2 +q3 +q4 +q9) 2 µ 2 λ P4 ν =1+q+q2 +q7, (2,2,2,1,1) 2 Z3ν(P,4 Q)=9 ( Q )| |Q | |fλ− (2(,2,P1,1),|1,1)| CµνλCµt λ.2 7 (2.36) P q− q q q λ P q ∅ q q , (2,2,2,1,1) = ( +4 + + ) µ,λ − (2,2−,1,1,1,1) =1+ + + P(2,1,1,1,1,1,1)−= q ,P# (1,1,1,1,1,1,1,1) =1, (A.2) 4− P , , , , , , = q ,P" , , , , , , , =1, (A.2) (2 1 1 1 1 1 1) (1O51 1−1 1 1 1 1) ν Again we observe− 15 that (2.36) terminatesλ at the finite order Q| | for various choices10 of ν. P(10) = q ,P(9,1) = νq , Hence we conjecture15 that the expansion of (2.36)byQ terminates at the order Q−| |.10 Then(2.35) P(10) = q ,P6 12 13 14 15 (9,1) = q , we canP take(8,2) the= q limit+q Q+q +qwith+qPQ, fixed to obtain another O-vertex W assoicated− the 6 12 13→∞14 15 ν The associatedP(8P,2) = partitionq=+q(q11+ function+q q12++qq13++ isqq given14,), by diagram(8, of1,1) (2.34−). Namely we define P = (q11 +3 q129+q1310+q1411), 12 (8,1,1)P(7,3) = q q q q 2 qµ ,P2 λ µ4 4 ν (7,2,1) =0, − − − −W :=− − ( 1)| |f − C C t . t (2.37) Z3ν(6P,9 Q7 )=810ν 9 (11 Q )12| |Q | |fλ− (λ Pµ)νλ| | Cµ µλνλCµ λ. (2.36) PP(7,3), ,=, =q q +qq +qq +q ,q − q ,P− − ∅ ∅ (7,2,1) =0, (7 1 1 1)− − − µ,−λ −#µ,λ 6 7 7 8 8 9#9 102 µ +211λ = ν12 13 14 15 P(7,1,1,1)P(6=,4)q="+qq++q q+q++q q, +q | +| q | +| q| | +q +q +q , ν Again we observeWe also observe that7 9 (2.368 that10)9 terminates1110 1211 1312 at the13 finite14 order15 Q| | for various choices of ν. P P(6,=3,1)q=+q (+q q++q q++qq ++q q ++q q), +q +q +q , (6,4) − ν Hence we conjecture that8 9 the10 expansion11 12 of (2.36˜ )byQ terminates at the order Q| |. Then P(6,1,1,1,1) = q9 +q10+2q11 +q12+q 13, Pν(q) P(6,3,1) = (q +q +q +q +q W)ν, = , (2.38) we can take the limit− Q 3 4 with5 6PQ10fixed11 ( toq; q12 obtain) ν /213 another14 O-vertex Wν assoicated the P(6,2,1,1) 8= (9q +q10+q 11+q +12q +q +2q +q +q ), P(6,1,1,1,1) = q +−q +2→∞q +q +q , | | diagram of (2.34). Namely6 we7 define8 9 10 11 12 13 14 P(5,5) = 3 (1+4q +5q +6q +10q +q11+q 12+q 13+q +14q ),P(5,4,1) =0, P(6,2,1,1) = (q−+q +q +q +q +q +2q +q +q ), P − q q2 q3 q4 q5 q6 q7 q8 µ q9 4 q10 q11 q12,P , (5,3,1,1) = + 6W+ 7 :=+ 8+ 9+ 10+ (+2111)|+212|f −+213C +14C t + . (5,2,1,1,1) =0 (2.37) P(5,5) = (1+q +qν +q +q +q +q +q +λ q +µνλq )µ,Pλ (5,4,1) =0, − 3 4 5 6 7 − ∅ P(5,1,1,1,1,1) = 2(q +3 q +24 q 5+q µ,+6λq )7, –8–8 9 10 11 12 P(5,3,1,1) = q+q−+q +q +q2 µ++2q#λ+=q ν+2q +2q +2q +q +q ,P(5,2,1,1,1) =0, P q q2 q3 q4 | q|5 |q6| |q|7 q8 q9 q10 q11 q12 q13 q14 q15. (4,4,2) = 3+ 4+ +5 +6 +7 + +2 +2 +2 +2 +2 + + + P(5,1,1,1,1,1) = (q +q +2q +q +q ), We also observe− that (A.3) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P(4,4,2) = q+q +q +q +q +q +q +2q +2q +2q +2q +2q +q +q +q . P˜ν(q) Wν = , (A.3)(2.38) (q; q) ν /2 | | – 41 –

– 41 – –8– In fact one can infer a candidate for another form of the vertex operator O(P, q) which T 1 satisﬁes (4.1). In order to see that, let us compute∅ the perturbative→ part of the pure SO(2N +2(a)) gauge theory from the O-vertex. We(b) focus on the contribution from the left

Figure 3half. (a): of the the brane 5-brane web by sending web diagramT 1 from of the the diagram pure in figure SO(21(b)N .+2 (b):) the gauge diagram theory as it contains the → to be usedO-vertex for the computations,Vν. The where partition we shall function set the leg from attached the to the left D7-brane half of to the be empty. diagram contains the square root of the perturbative part of the partition function of a pure SU(N +1proofs JHEP_213P_0121 ) gauge theory. the lineFrom with the the Kähler topological parameter vertexT , it computation, turns out that a it finite is possible sum gives to the factor correct out such a part by using answer, which we will see in section 3.2. the identity (B.33) and the remaining contribution is given by We can use this technique to compute the partition functions of SO(2N +1) gauge theories and also G2 gauge theories utilizingN the construction in [18]. Furthermore it is also proofs JHEP_213P_0121 ν ηi ρ ρ ρ ρ possible to compute the pure SU(3) gauge| theory| with the CS level κcs =9by applying Vν( P )| | Qi sν/η1 (q− )sη1/η2 (q− ) ...sηN 1/ηN (q− )sηN (q− ) the braneν web,η ,η proposed,...η in− [33], which" may be$ obtained by a twisted compactification− of 1 !2 N i#=1 (4.5) 6d pure SU(3) gauge theory with a tensorν multipletρ [38]. ˜ ρ ˜ ρ ˜ ρ = Vν( P )| |sν/η1 (q− )sη1/η2 (Q1q− ) ...sηN 1/ηN (QN 1q− )sηN (QN q− ), − − − ν,η1,η2,...ηN 3 Examples ! In fact one can infer a candidate for another form of the vertex operator O(P, q) which where we defined satisfies (4.1). In order to see that, let us compute the perturbative part of the pure In this section, we give several examples of calculations that involvei the usage of the O- SO(2N˜ +2) gauge theory from the O-vertex. We focus on the contribution from the left vertex defined in section 2. More concretely, we first checkQi the= validityQj. of our formalism in (4.6) half of the 5-brane web diagram of the pure SO(2N +2) gauge theory as it contains the section 3.1, 3.2 and 3.3 by comparing the partition functions ofj#=1 the SO(2N), SO(2N +1) O-vertex Vν. The partition function from the left half of the diagram contains the square and G2 Itgauge is possible theories toobtained write in (4.5 the) in topological terms of vertex the expectation formalism with value the O-vertex of vertex operators. First note root of the perturbative part of the partition function of a pure SU(N +1) gauge theory. with thethat known the one-instanton skew Schur and function two-instanton may partition be written functions by in the literature. In section 3.4, we will further apply the O-vertex toFrom the calculation the topological of the partition vertex computation,function it is possible to factor out such a part by using of the 5d pure SU(3) gauge theory with the CS leveltheκ identity=9, by (B.33 using) theand brane the remaining diagram contribution is given by s (cs⃗x)= λ V (⃗x) µ , (4.7) proposed for it in [33]. λ/µ ⟨ | − | ⟩N proofs JHEP_213P_0121 ν ηi ρ ρ ρ ρ V P Q| | s q s q ...s q s q where V (⃗x) is defined in (B.19). Then usingν ((4.1))| and| (4.7i), (4.5ν/)η1 can( − be) η written1/η2 ( − ) as ηN 1/ηN ( − ) ηN ( − ) 3.1 SO(2N) gauge− theories ν,η ,η ,...η − " $ − 1 !2 N i#=1 (4.5) We start from the computation ofν the partitionρ function of˜ theρ pureV SO(P2Nν )s theory.˜q ρ Thes ρ Q˜ q ˜ρ ...sρ Q˜ q ρ s Q˜ q ρ , Vν( P )| |sν/η (q− )sη =/η (Q1q− ) ...sν( η )| |/ην/η(1Q( N− )1qη1−/η2)(sη 1 (−QN) q− η)N 1/ηN ( N 1 − ) ηN ( N − ) 1 1 2 − N 1 N N − − brane web diagram is obtained− by putting a stack of Nν,ηD5-branes1,η2,...ηN on the− top of an O5− - ν,η1,η2,...ηN ! − plane. Hence we! can directly apply the O-vertex proposedSO(4) in section 2.2. ρ Zleft is the partitionρ function associted with the diagramρ = 0 O(P, q) ν ν V (q− ) ηwhere1 η1 weV defined(Q˜1q− ) η2 ηN V QÑ q− 0 ⟨ | | ⟩ − − ··· − i ˜ 3.1.1 Pure SO(4) gaugeReasoning theory% & N behind& this' (proposal:& & ) ( & Q*i = Q+j.& ) (4.6) ρ& & ρ & & & & The first example= 0 isO the(P, simplest q)V (q− case) N =2V ,(Q i.e.˜iq SO(4)− ) 0 gauge& theory. The& 5-brane web& j#=1 & ⟨ | − − | ⟩ (3.10) diagram for the pure SO(4)e.g. theory in SO(4) is giveni=1 theory by It is possible to write (4.5) in terms of the expectation value of vertex operators. First note # n SO(4)N SO(4) and Zñ is2 associated with ∞ 1 (1 +Zthatleft i=1is theQ therighti skew) partitionq Schur function function associted may with be the written diagram by = 0 O(P, q) exp Q J n 0 , (4.8) n qn − , & "n=1 -1 $ & . & Q′ ! − & sλ/µ(⃗x)= λ V (⃗x) µ , (4.7)

& & ⟨ | − | ⟩ proofs JHEP_213P_0121 & P & where we& used the completeness of the Frobenius basis& to sum over the Young diagrams (3.10) (3.11) + where V (+⃗x) is defined in (B.19). Then using (4.1) and (4.7), (4.5) can be written as from the secondO5 line to theO5 third− line. O5− (3.1) UsingZSO(4) the O-vertex (2.28) for (3.8) we checked that full brane diagramand right is associated withν perturbativeρ ˜ partρ ˜ ρ ˜ ρ Since (4.8) gives a part of the perturbativeVν( partP )| of|sν the/η1 (q partition− )sη1/η2 ( functionQ1q− ) ...s ofηN the1/η pureN (QN 1q− )sηN (QN q− ) − SO(4) q 2 −6 6 − ν,η1,η2,...ηN Z = P.E. (Q′ + P Q′) + o(P ,Q′ ), (3.12) SO(2N +2) gauge theory except for the! pure SU(Nleft+1) part,(1 (4.8q))2 should be equal to To reproduce the one-loop part, we need to require!ρ " ρ ρ − proofs JHEP_213P_0121 = 0 O(P, q) ν ν V (q− ) η1 η1 V (Q˜1q− ) η2 ηN V QÑ q− 0 k1 k2 − − −l1 l2 – 11 – where⟨ o|(X1N,X2 ,| ⟩) means thatN the computation2 was done until··· the order(3.11)X1 X2 2 ··· N ( & & ) ( & * ··· + & ) P q with l k ,l 2k , % .& Similarly applying& ' the& O-vertex (2.37& )to(3.9) yields& & P.E. 1+1 1 2Q˜ 2 + ρ&1+ Q˜& i ρ . (4.9) 2Using= the0 O-vertex≤O(P, q (≤)2.28iV )(···q for− ()3.8) weV checked(Q˜iq− that) 0 & & & & "2(1 q) "− ⟨ | − − $$| ⟩ − / i=1 0 SO(4)/ i=1 i=1 q0 2 6 6 ! SO(4)Zright =#P.E.!q 2 (Q2′ + P Q′) +6 o(P6 ,Q′ ). (3.13) Z = P.E. (1 (Qq′N)+ P Qn′) n+ o(P ,Q′ ), (3.12) left ∞ 1(1(1!q +)2− Q˜ )q 2 " Then comparing (4.8) with (4.9), we find that a candidate! for− anotheri=1 i form" of the vertex Hence= 0 weO obtained(P, q) exp n J n 0 , (4.8) ,k1 & k2 " n 1 q − $ & . l1 l2 operator (P, q) in the bosonic basiswhere mayo(X1 ,X& be2 given, ) means by n that=1 the computation-− was done until& the order X1 X2 O ···SO(4) SO(4)! 2q 1 6 6 ··· with l k ,l& Zk , Z. Similarly= P.E. applying the O-vertex(A1A− + (2.37A1A)to(2)& +3.9o) yieldsP ,Q′ . (3.14) 1 ≤ 1 &2 ≤ left2 ··· right (1 q)2 2 & this determines thewhere O-vertex we&2 usedn (up the ton terms completeness annihilated2!n of− the by Frobenius basis"& to# sum$ over the Young diagrams ∞ P (1 +SO(4)q ) P q 2 6 6 fromSince the the second topologicalZ line= vertex toP.E. the reproduces third line.(Q only′ + P theQ′) root+ o part(P ,Q of′ ) the. perturbative(3.13) partition theO( P,vacuum q)=exp state). right J2n + (1 q)J2nJn . (4.10) − 2n(1 qn) ! 2n " "n=1function"Since we (4.8 compare) gives (3.14 a part) with of (−B.2 the). The perturbative$$ root contribution part to of the the perturbative partition partition function of the pure Hence! we obtained− SO(funciton2N +2 of) the gauge pure SO(4)theory gauge except theory for from the ( pureB.2) is SU( givenN by+1) part, (4.8) should be equal to 2q SO(4) SO(4) q 1 6 6 Zleft Zright = P.E.SO(4) 2 (A12A2− + A1A21) + o P ,Q′ . (3.14) Z =(1P.E.q) (A1A− + A1A2) . (3.15) pert !P 2−q (1 q)2 N 2 " # $N 2 ! − ˜2 " ˜ Since the topological vertexP.E. reproduces only the1+ root partQ ofi the+ perturbative1+ Q partitioni . (4.9) Hence we see the agreement2(1 betweenq)2 (3.14− ) and (3.15) until the orders we computed. function we compare– 36 – (3.14) with" (B.2−). The root" / contributioni=1 to0 the perturbative/ i=1 partition0 $$ Namely the O-vertices (2.28) and (2.37) reproduced! the correct result! until the orders we funciton of the pure SO(4) gauge theory from (B.2) is given by Thencomputed. comparing (4.8) with (4.9), we find that a candidate for another form of the vertex SO(4) SO(4) 2q 1 ˆ operatorLet us( moveP, q) oninZ to the the= bosonic comparisonP.E. basis of the( mayA instantonA− be+ A givenA part) by.Ztop in (3.3), namely,(3.15) O pert q 2 1 2 1 2 !(1 ) " SO(4)− SO(4) SO(4) Zˆ = Zˆ /Zn . n n (3.16) Hence we see the agreement betweentop, (3.14 inst∞) andtop (P3.152 )(1pert until + q the) ordersP we2 computed. O(P, q)=exp J2n + JnJn . (4.10) NamelyFrom the the O-vertices identification (2.28) in and (3.6 (2.37), the) reproduced one-instanton− 2 then(1 correct partitionqn result) function until2 then can orders be extracted we "n=1 " − $$ computed.from (3.3) as the coefficient of Q1 in (!3.16) and it is given by ˆSO(4) Let us move on to the comparison of the instanton part Ztop in (3.3), namely, SO(4) 2q 4q 2 6q 3 8q 4 10q 5 Zˆ = Q′ + Q′ + Q′ + Q′ + Q′ top, 1-inst (1 q)2 (1ZˆSO(4)q)2 = ZˆSO(4)(1 /Zq)SO(4)2 . (1 q)2 (1 q)2 (3.16) − top,− inst top − pert − − 12q 6 2q 2 4q 4 2 6q 6 3 6 6 + Q′ + P Q′ + – 36P –Q′ + P Q′ + o(Q′ ,P ) . From the identification(1 q in)2 (3.6),(1 the one-instantonq)2 (1 partitionq)2 function(1 q can)2 be extracted from (3.3) as the coeffi−cient of Q1 in− (3.16) and it is− given by − (3.17) SO(4) 2q 4q 2 6q 3 8q 4 10q 5 Zˆ = Q′ + Q′ + Q′ + Q′ + Q′ top, 1-inst (1 q)2 (1 q)2 (1 q)2 (1 q)2 (1 q)2 − − − − − 12q 6 2q 2 4q 4 2 6q 6 3 6 6 + Q′ + P Q′ + P Q′ + P Q′ + o(Q′ ,P ) . (1 q)2 (1 q)2 (1 q)2 (1 q)2 − − –− 13 – − (3.17)

– 13 – (4.16) where λ 2 = λ2 for λ = λ and P is the Macdonald function with the normalization || || i i { i} λ such that ! λ 2 t ρ || || λ i+1 λi j 1 P t (t− ,q,t)=t 2 (1 t j − q − )− . (4.17) λ − (i,j) λ "∈ To use the reﬁned topological vertex, we need to choose a preferred direction and we assign the reﬁned topological vertex (4.16) so that λ is the preferred direction. The framing factor

is also reﬁned [5, 55] and we assign proofs JHEP_213P_0121

λ | | λt 2 λ 2 2 λ || || || || t f˜ (t, q)=( 1)| |q− 2 t 2 , (4.18) λ q − # $ for non-preferred directions and

λt 2 λ 2 λ || || || || f (t, q)=( 1)| |t− 2 q 2 , (4.19) λ − However, in the computation of partition function, the vertex for preferred directions. operator is enough. Then, suppose that the refined version of the O-vertex Vν(q) is given by Vν(t, q),the partition function corresponding to (4.5) will become6 The part thati involves O-vertex in the calculation: where Ai := j=1 Qj. This simplifies an expression of the 5d partition function N ν ηi ρ ρ ρ ρ Vr t, q P | | Q| | s q− s q− ...s q− s q− ν( )( ) iN ν/η1 ( ) ηN1/η2 ( ) s ηN 1/ηN ( ) ηN ( ) N (1) (5 2s)Ÿ(⁄( )) − ν,η ,η ,...η −⁄(s) 2⁄ r ⁄(r) ≠ fl fl 1 2 N s # i=1 $ 2( 3) 2 ZSO(2N%) = Q =1 | |Q2| " | Qr| ≠ | q s⁄(s) (q≠ )s(⁄(s)) (q≠ ) N ñ n ‚ ˛⁄ q ∞ 1 (1 +r=3 i=1 Qi )q 2 s=1 = 0 Oÿ(P, t, q) exp Ÿ J Ÿn 0 , 2 n 1 qn − 1 2 & ' (˛n=1 ! (t) (u) ) ' * ' M˛⁄(A%) N−⁄ ⁄ (AuAt≠ ,q' )≠ , ' ◊ 1 t

X2 Y Y X [Y,X2 ]+ 1 [Y,[Y,X]] 1 [Y,[Y,[2Y,X]]]+4 ... qPe e = e e ≠ P 2 ≠ 3! q tP , (2.6) O(P, t, q)=1+ ψ∗ ψ1/2 + ψ∗ ψ3/2 ψ∗ ψ1/2 1 t 3/2 1 t 1/2 − (1 t)(1 t2) 7/2 − − − − the M-factor can be(q writtenq2 + qt as)P 4 (1 + q2t)P 4 − ψ∗ ψ ψ∗ ψ∗ ψ ψ (4.23) − (1 t)(1 t2) 5/2 3/2 − (1 t)(1 t2) 3/2 1/2 1/2 3/2 − − −qX − M4 A˛ 4 (q t + qt)P ˛⁄( )=PE P 2 5 (2.7) + − ψ∗ ψ + 2(1 q) ψ∗ ψ + o(P ). (1 t)(1 t2) 3/2 5/2 (15 t)(1≠ t62) 1/2 7/2 − − − − ν where | | 6 t 2 The direct applicationN of the reﬁned topological¸ vertex(⁄(s)) gives a factor N and we deﬁned Vν (t, q) s 2 q 2 ¸(⁄(s)) i 1 ⁄( ) with the factorX included.:= A q +(1 q) q ≠ ≠ i A s ≠ ≠+ , s 5sÿ=1 1 ÿi=1 26 5sÿ=1 6 (2.8) N ¸(⁄(s)) s 2 2¸(⁄(s)) 2 2(i 1 ⁄( )) A q 1+(1 q ) q ≠ ≠ i ≠ s ≠ ≠ 5sÿ=1 1 – 38 – ÿi=1 26 One can show that this can be written as

˛ M˛⁄(A) N s 2 a +l +2(i ⁄( )) = (1 A q i,j i,j ≠ i ) (2.9) ≠ s s=1 (i,j) ⁄(s) Ÿ Ÿœ (u) (t) i 1 ai,j ⁄ 1 l j+1+(⁄ ) 1 (1 A A q ≠ ≠ ≠ j )≠ (1 A A q i,j ≠ i‚ )≠ . ≠ t u ≠ t u 1 t

M⁄(1),...,⁄(N) (A1,...,AN )=M⁄(‡(1)),...,⁄(‡(N)) (A‡(1),...,A‡(N)) for ‡ S . ’ œ N

–3– proofs JHEP_213P_0121 - ) − +1 (3.1) 1 N ) gauge → 2 T +1 by applying ) theory. The N ), SO( 2 N N =9 . (b): the diagram 2 . 2 cs κ 2.2 1(b) ∅ (b) ]. Furthermore it is also 18 (192) + , by using the brane diagram . O5 ) =9 D5-branes on the top of an O5 ‚ m ]. ) cs (3.7) (3.8) (3.9) ) N κ u 38 (3.11) (3.10) ( -factor ⁄ . ) M 2 +( ) ‚ n ,q ) 2 , i.e. SO(4) gauge theory. The 5-brane web ) 0 − t from the diagram in figure ( (1 ) Q 1 ⁄ ) Q 2 1 O5 =2 – 11 – 6 2 N → +( ( s 6 . , it turns out that a finite sum gives the correct proofs JHEP_213P_0121 N n ‡ 2 T T A ) ≠ N ,P 3.2 ) ) s ) gauge theory with the CS level m 2 ( i 3 =1 N ≠ ⁄ ÿ s , 1 ,q 5 ≠ q ′ ... ], which may be obtained by a twisted compactification of ,P 1 - ) + P u ,...,A . Note that the Q 2 (1 ≠ ≠ − 33 i 1 A ) ]]]+ O5 C 2 (1) t P ) 2( . More concretely, we first check the validity of our formalism in +1 6 (3.1) ) ‡ 1 ( q 2 s A 2 by comparing the partition functions of the SO( ( i gauge group receives contri- ) Y,X ) A [ N ) ⁄ , N s ) gauge ( s ) → ( i 2 ≠ ]. ( ≠ Y, ) gauge theories utilizing the construction in [ . 6 3.3 [ =1 ⁄ )) ) ) gauge theory with the CS level (a) ⁄ | i ÿ s 2 2 ( 3 33 N ) T N ≠ ( j ( Y, ¸ 2 (1 ) s ( +1 [ ) M 1 G ) (  by applying ‡ q ) ‚ ) 1 u ( 3! and ≠ 3 gauge theories ⁄ 2 ( i ⁄ ) theory. The ⁄ | N ⁄ ≠ q ), SO( ) q ) gauge theory with a tensor multiplet [ ≠ 2 ,q ]] œ +( 3 qX SO(2 ) 3.2 N 0 ) ) ,..., N j ) N =9 =1 Ÿ . (b): the diagram , , we will further apply the O-vertex to the calculation of the partition function ≠ 2 s formula N s . ≠ ( 2 2 (2 0 ◆ ( Q i . (a): the brane web by sending =1 Y,X gauge theories obtained in the topological vertex formalism with the O-vertex 2(1 ⁄ (1)) [ i ÿ cs (  3.1 m,n 3.4 ( q 20 q ‡ Q ( 5 ¸ 2 ( 2 s κ 2.2 Y, 1 2 1(b) 2 q [ ) ⁄ ) G ∅ t | We can use this technique to compute the partition functions of SO( 1 2 ) A =

instanton part of SO(4) theory part of SO(4) instanton q u PE (b) –7– ( j ]. Furthermore it is also M | 1+(1 ]+ k ⁄ ≠ 2 ≠ N 3.1 SO We start from the computationbrane of web the diagram partition is functionplane. obtained of by Hence the putting we pure a can SO( stack directly of apply the O-vertex3.1.1 proposed in section Pure SO(4)The gauge first theory example isdiagram the for simplest the case pure SO(4) theory is given by section and with the known one-instantonsection and two-instanton partition functionsof in the the 5d literature. pure SU( proposed In for it in [ Figure 3 to be used for the computations, where we shall set the legthe attached line to the with D7-brane toanswer, the be which Kähler empty. we will parameter see in section theories and also possible to compute thethe pure brane SU( web proposed6d in pure [ SU( 3 Examples In this section, wevertex give defined several in examples section of calculations that involve the usage of the O- 0 18 ≠ ✓ ≠ ) ) is given in Appendix t Y,X (1 ⇥ )= ) Q of this kind of string [ ( i )= ) ) | s + ˛ ⁄ s ≠ , by using the brane diagram ÿ representing the effect representing A ( N ( with Q ( +(1 ≠ ⁄ | ⁄ X 1 3.10 ( ⁄ ˛ ) ) œ O5 e ¸ ) ) ) + Ÿ ≠ =9 2 s | j Y D5-branes on the top of an O5 ( M N q i,j ]. e ( ⁄ + ( | cs i ( 1 ⁄ ¸ N 2 s q κ 38 ,...,A = q Q =1 u N 1 Ÿ A ) and ( s 1 Y s A A t , e ( =1 N A 3.8 ) ,..., ÿ s X A X 5 N e ( (1) =1 N , i.e. SO(4) gauge theory. The 5-brane web ⁄ ≠ ÿ s = − instantons ≠ ⁄ from the diagram in figure 5 Q ,..., 1 (1 O5 ) =2 – 11 – := t =( (1) ( → ⁄ ⁄ . +1) , it turns out that a finite sum gives the correct N ⁄ ˛ X œ T ) . Ÿ M T 3 2 1 -instanton partition function with (4) N N 3.2 N i,j k Q Q Q ( ) gauge theory with the CS level Q S 3 instanton SO N )= Æ œ Z -factor can be written as ˛ SO(2 ′ A ], which may be obtained by a twisted compactification of + ( P ‡ Ÿ Q t

1 M with . More concretely, we ﬁrst check the validity of our formalism in

2 3.2 is invariant under the permutation group for = The equality between ( One can show that this can be written as the where butions from Thus, the by comparing the partition functions of the SO( More precisely More

gauge theories utilizing the construction in [

3.3 ) gauge theory with the CS level (a)

2 3

and gauge theories )

3) gauge theory with a tensor multiplet [ 3.2

, N , we will further apply the O-vertex to the calculation of the partition function (2 . (a): the brane web by sending gauge theories obtained in the topological vertex formalism with the O-vertex 3.1 3.4 2 G We can use this technique to compute the partition functions of SO( plane. Hence we can directly apply the O-vertex3.1.1 proposed in section Pure SO(4)The gauge first theory example isdiagram the for simplest the case pure SO(4) theory is given by proposed for it in [ 3.1 SO We start from the computationbrane of web the diagram partition is function obtained of by the putting pure a SO( stack of section and with the known one-instantonsection and two-instanton partition functionsof in the the 5d literature. pure SU( In 3 Examples In this section, wevertex give defined several in examples section of calculations that involve the usage of the O- theories and also possible to compute thethe pure brane SU( web proposed6d in pure [ SU( Figure 3 to be used for the computations, where we shall set the legthe attached line to the with D7-brane toanswer, the be which Kähler empty. we will parameter see in section 0 1 2 3 4 5 6 7 8 9 D5 • • • • • • NS5 • • • • • • 7-brane • • • • • • • • Table 1: Configuration of branes in the brane web construction. Bar represents 0 1 2 3 4 5 6 7 8 9 the direction branes stretch along, and dot means the point-like direction for D5 branes.• • • • • • NS5 • • • • • • 7-brane • • • • • • • • Table 1: Configuration of branes in the brane web construction. Bar represents the direction branes stretch along, and dot means the point-like direction for F = F. (165) branes. ⇤

=( , ,..., ), . (166) 1 2 n 1 2 ··· n F = F. (165) ⇤

i = aj +(r 1)✏1 +(T s 1 1)✏2. (167) ∅ → (a) (b) =(1, 2,...,n), 1 2 n. (166) Figure 3. (a): the brane web by sending T 1 from the diagram in figure 1(b). (b): the diagram → ··· to be used for the computations, where we shall set the leg attached to the D7-brane to be empty.0 1 2 3 4 5 6 7 8 9 =(↵ , ↵ ,... , ,...). (168) 1 2 D51 2 proofs JHEP_213P_0121 | • • • • • • the line with the Kähler parameter T , it turns out that a finite sumNS5 gives the correct answer, which we will see in section 3.2. • • • • • • i = aj +(r 1)✏1 +(s 1)✏2. 7-brane (167) We can use this technique to compute the partition functions of SO(2N +1)• gauge• • • • • • • N O5± theories and also G2 gauge theories utilizing the construction in [18]. Furthermore it is• also• • • • • possible to compute the pure SU(3) gauge theoryi with1 the CS level κ =91by applying 1 1 Z = q| |N (1) csN (Qi/Qj)N (Qi/Qj)N (Qj/Qi). the brane web proposed in [33], which may be obtainedTablei byi 2: a Configuration twisted compactificationij of branes of in theij brane web construction.j i Bar represents 1,2,...,N i=1 0 1 2 3i

18 18

18 0 1 2 3 4 5 6 7 8 9 D5 • • • • • • NS5 • • • • • • 7-brane • • • • • • • • Table 1: Conﬁguration of branes in the brane web construction. Bar represents the direction branes stretch along, and dot means the point-like direction for branes.

F = F. (165) ⇤

=( , ,..., ), . (166) 1 2 n 1 2 ··· n

Note that (3.23)i is= theaj 1-instanton+(r 1) part✏Note1 of+(Z thatsloc, extra (1)3.23.✏2) Therefore. is the 1-instanton we expect part that of theZloc,(167) full extra. Therefore we expect that the full extra factor for the partition function forextra the pure factor SO(4) for the gauge partition theory function computed for by the (B.12 pure) SO(4) gauge theory computed by (B.12) is given by (3.2). Subtracting (3.23) fromis given (3.20) by one (3.2 ﬁnds). Subtracting (3.23) from (3.20) one ﬁnds

1 1 2q A A− A A 2q A A− A A ZSO(4) ZSO(4) Z1 SO(4)2 ZSO(4)1 2 , 1 2 1 2 , loc, 1-inst =(loc,↵ extra, ↵= ,... 2 , ,...loc,) 1-inst.1 + loc, extra =2 2 (3.24)(168) 1 + 2 (3.24) − 1 2(1 q) 1!(1 2 A A− )2 − (1 A1A2) "(1 q) !(1 A A− )2 (1 A1A2) " − | − 1 2 − − − 1 2 − which agrees with the 1-instanton partitionwhich function agrees with (3.19 the) computed 1-instanton by thepartition topological function (3.19) computed by the topological vertex formalism. One can also see thatvertex (3.24 formalism.) or (3.19) is One given can by also the see sum that of ( the3.24 one-) or (3.19) is given by the sum of the one- instantonN partition function of the pureinstanton SU(2) gauge partition theory function with discrete of the theta pure angle SU(2) zero, gauge theory with discrete theta angle zero, proofs JHEP_213P_0121 i 1 1 1 1 proofs JHEP_213P_0121 Z = q| |NˆSO(4)(1) SU(2)N (1Qi/QSU(2)j)NˆSO(4)(Qi/QjSU(2))N (Q1j/QiSU(2)). Zi 1-insti = Z1-inst Ai1Aj2− + Z1-inst (Ai1AZj21-inst) , = Z1-inst jA1iA(3.25)2− + Z1-inst (A1A2) , (3.25) 1,2,...,N i=1 ⇣ ⌘ i

3.1.2 Pure SO(6) and SO(8) theories3.1.2 Pure SO(6) and SO(8) theories G Let us then considerZ thetop partition= Zroot functionsZLetinstanton us of then the. pure consider SO(6) the and partition SO(8) functions gauge theories. of(172) the pure SO(6) and SO(8) gauge theories. The method of the computation is essentallyThe method parallel of to the that computation in the case ofis essentally the pure SO(4) parallel to that in the case of the pure SO(4) gauge theory discussed in section 3.1.1. gauge theory discussed in section 3.1.1.

Pure SO(6) gauge theory. We startPure from SO(6) the pure gauge SO(6) theory. gauge theory.We start The from pure the pure SO(6) gauge theory. The pure SO(6) SU(4) SO(6) gauge theory can be• realized= on theSO(6) following gauge theorybrane diagram, can be realized on the following(173) brane diagram, ⇠ hint: SU(3) subgroup Q Q R 17 R Q′ Q′ P P (3.28) (3.28)

The relation between the Kähler parametersThe relationQ, Q′,P,R betweenand the the Kähler gauge parameters theory pa- Q, Q′,P,R and the gauge theory parameters may be also read oﬀ from therameters diagram may in (3.28 be also). The read Coulomb oﬀ from branch the diagram moduli in (3.28). The Coulomb branch moduli pure SO(8)a1 gauge theorya2 isSO(8) givena3 by a1 a2 a3 A1 = e− ,A2 = e− ,A3 •= e− are relatedA1 = bye− ,A2 = e− ,A3 = e− are related by We checked that our results match with those known in the literature. A1 = PQT ′R, A2Q= PQ′,A3 = P.A1 = PQ′R, A2(3.29)= PQ′,A3 = P. (3.29) R Q ′ one instanton ☑ P (3.43)two instantons ☑ – 15 – –three 15 – instantons ☑ The Coulomb branch moduli a1,a2,a3,a4 are the height of the color D5-brane. Hence a Ai = e− i , (i =1, 2, 3, 4) are parametrized by proofs JHEP_213P_0121 A1 = TRQ′P, A2 = RQ′P, A3 = Q′P, A4 = P. (3.44) Extrapolating the external (2, 1) and (2, 1) 5-brane webs on the O5-plane, the instaton − fugacity is related to the length between the 5-branes on the orientifold. Namely it is given by 2 4 2 4 4 4 q = QT A1− = QT − Q′− P − R− . (3.45) From the web diagram in (3.43) it is possible to compute the partition function using the topological vertex and the O-vertex and the partition function becomes SO(8) Ztop λ + σ + τ + υ α + β +2 σ µ + ν γ + δ ι + π +2 υ = ( Q)| | | | | | | |( Q′)| | | | | |( P )| | | |( R)| | | |( T )| | | | | | − − − − − µ,ν,α,β,λ!,σ,γ,δ,τ,υ,ι,π 1 3 1 1 1 3 fλfα− fβfσ− fγ− fδfτ− fι− fπfυ− CαtγλCδβtλt Cνtασt CβµtσCγtιτ Cπδtτ t Cιt υC πtυt VνWµ. ∅ ∅ (3.46) Since there is no parallel external lines, (3.46) does not contain an extra factor. The perturbative part is obtained by the limit Q 0 and the partition functin splits → into two parts, SO(8) SO(8) SO(8) Zpert = Zleft Zright , (3.47) where SO(8) α ν γ ι 1 1 1 Zleft = ( Q′)| |( P )| |( R)| |( T )| |fα− fγ− fι− Cαtγ Cνtα Cγtι Cιt Vν, (3.48) ν,α,γ,ι − − − − ∅ ∅ ∅ ∅∅ ! SO(8) β µ δ π Zright = ( Q′)| |( P )| |( R)| |( T )| |fβfδfπCδβt Cβµt Cπδt C πt Wµ. (3.49) − − − − ∅ ∅ ∅ ∅ ∅ µ,!β,δ,π The computing the summation of the left part yields

– 18 – D7 D7

D7 D7 T + + + + O5 O5− O5 O5 O5− O5T + + + + O5 O5− O5 O5 O5− O5 (a) (b) (a) (b)

FigureD7 1. (a): the brane web for the SO(8) gauge theory with one flavor in the vector represen- Figure 1. (a): the brane web for the SO(8) gauge theory with one flavor in the vector represen- proofs JHEP_213P_0121 tation. The red dotted line represents the branch cut for the D7-brane. (b): the brane web after proofs JHEP_213P_0121 tation. The red dotted line represents the branch cut for the D7-brane. (b): the brane web after setting the height of the flavor D5-brane equal to the height of the bottom color D5-brane. setting the height of the flavor D5-braneD7 equal to the height of the bottom color D5-brane. T + + + + O5 O5− O5 O5 O5− O5 (a) (b)

Figure 1. (a): the brane web for the SO(8) gauge theory with one ﬂavor in the vector represen- SO(2N+1) theories proofs JHEP_213P_0121 tation. The red dotted line represents the branch cut for the D7-brane. (b): the brane web after + + + + setting the heightthe of brane the ﬂavor construction D5-brane equal involvesO5 to the O5 height and of the O5 bottom − .O5 color− O5 D5-brane.O5

! ! ! Figure 2Figure. The brane 2. The web brane for web the! for pure the SO( pure7) SO( gauge!7) gauge theory. theory.! The The green green dotted dotted line line represents represents the the branch cutbranch for a cut half for D7-brane. a half D7-brane. SO(7)

+ + O5 O5− O5 figure 1(b). Then we take T 1 and the D7-brane is on top of the O5-plane. When a figure 1(b)there. Then is a Higgsing we take prescriptionT 1 and →to write the down D7-brane an effective is on top of the O5-plane. When a Figure 2.D7-brane Theweb brane diagram web is for on that the top! pure can of SO( be an→ 7used) O5-plane gauge! to theory. do computations. it! The can green be dotted split into line represents two half the D7-branes [37]. A piece of D7-brane is on top of an O5-plane it can be split into two half D7-branes [37]. A piece of branch cutD5-brane for a half D7-brane. between the two half D7-branes can move in the direction which the D7-branes D5-braneextend, between which the represents two half D7-branesthe Higgs branch. can move After in removing the direction the piece which of the the D5-brane, D7-branes we figure 1(b). Then we take T 1 and the D7-brane is on top of the O5-plane. When a extend, which represents→ the Higgs branch. After removing the piece of the D5-brane, we D7-brane ispull on top the of two an O5-plane half D7-branes it can be split in into the two opposite half D7-branes directions [37]. A into piece the of infinity. This operation pullD5-brane the betweenleaves two half a the configuratioin two D7-branes half D7-branes inwith canthe a move branch opposite in the cut direction directions of one which of the theinto half D7-branes the D7-branes infinity. on This top of operation the whole extend, which represents the Higgs branch. After removingT the piece1 of the D5-brane, we leaves aO5-plane configuratioin and also with a half a branch D5-brane∅ cut on of top one→ of of the the O5 half−-plane, D7-branes which on is depicted top of the in figure whole2. pull the two half D7-branes in the opposite directions into the infinity. This operation (a) O5-planeThen and the also resulting a half D5-brane theory(b) should on top be of interpreted the O5−-plane, as the purewhich SO( is7 depicted) gauge theory. in figure Since2. leaves a configuratioinWe checked with SO(5) a branchand SO(7) cut oftheories one of the matching half D7-branes with the on known top of the whole results in the literature up to three instantons. ThenO5-plane the andthe resulting also SO( a7 half) gauge D5-brane theory group on should top can of be the be realized O5 interpreted-plane, on which an asO5-plane, is depicted the pure in we figure SO( may2.7 interpret) gauge the theory. orientifold Since in Figure 3. (a): the brane web by sending T 1 from the diagram in figure 1(b). (b):− the diagram theThen SO( the→7figure resulting) gauge2 theoryas group an shouldO5-plane. can be be interpreted realized Namely as on an theO5-plane an pureO5-plane, SO(7) is gauge effectively we theory. may realizedSince interpret as the an O5-plane orientifold with in a to be used for the computations, where we shall set the leg attached to the D7-brane to be empty.! figurethe SO(27)asbranch gauge an groupO5-plane. cut can of be a half realized Namely D7-brane on an anO5-plane, onO5-plane it (and we may is a interpret half effectively D5-brane the orientifold realized for an in asO5− an-plane). O5-plane with a

figure 2 as an O5-plane. Namely! an O5-plane is effectively! realized as an O5-planeproofs JHEP_213P_0121 with a ! ! branchbranch cut cut of a ofSince half a D7-branehalf it is D7-brane not on it clear (and on howa half it to D5-brane(and apply a half for the an D5-brane topologicalO5−-plane). for vertex an O5 for− a-plane). configuration with the the line with the Kähler parameter T , it turns! out that a finite! sum gives the correct ! Sincebranch it is not! clear cut, how we to find apply it the convenient topological! to vertex use for the a configuration diagram before with the moving the half D7-branes Since it is not clear how to apply the topological! vertex for a configuration with the answer, which we will see in sectionbranch3.2. cut,into we infinity. find it convenient Moving to the use D7-branes the diagram into before the moving left the or the half right D7-branes direction! in the diagram does branch cut, we find it convenient to use the diagram before moving the half D7-branes We can use this technique tointo compute infinity.not the Moving change partition the the D7-branes theory functions into and the of weleft SO( or can the2N equally right+1 direction) gauge use the in the diagram diagram before does the moving depicted in not change the theory and we can equally use the diagram before the moving depicted in theories and also G2 gauge theoriesinto utilizing infinity. the Moving construction the D7-branes in [18]. Furthermore into the left it is or also the right direction in the diagram does figure 3(a)figure. For practically3(a). For applying practically the topological applying vertex the to topological the diagram in vertex figure 3(a) to the, diagram in figure 3(a), possible to compute the pure SU(not3) gauge change theory the theory with the and CS we level canκ equally=9by use applying the diagram before the moving depicted in we can usewe the can diagram use in the figure diagram3(b) instead in figure ofcs the3(b) one ininstead figure 3(a) of, theThen one the partition in figure 3(a), Then the partition the brane web proposed in [33], whichfigurefunction may3(a) forfunction the. be For diagram obtained practicallyfor in the figure by diagram a3(b) twisted applyingcan in be figure compactification computed the3(b) topological usingcan the be topological of computed vertex vertex to using the and diagramthe topological in figure vertex3(a) and, 6d pure SU(3) gauge theory withwe athe tensor can O-vertex. usethe multiplet theThe O-vertex. Young diagram [38 diagram]. The in Young assigned figure diagram3(b) to theinstead bottom assigned D5-brane of the to isthe one empty bottom in as figure reviewed D5-brane3(a), Then is empty the as partition reviewed functionin section 2.1 forfor the the diagram application in of the figure topological3(b) vertexcan be to certain computed non-toric using diagrams. the topological vertex and The Kählerin parameter section T2.1is reintroducedfor the application and it is set of to the1 at topological the end of the vertex computation. to certain non-toric diagrams. 3 Examples theAlthough O-vertex.The it seems Kähler The that we Young parameter need to diagram sumT overis reintroduced infinitely assigned many to Young and the it bottom diagrams is set to assigned D5-brane1 at the to end is empty of the as computation. reviewed in sectionAlthough2.1 for it the seems application that we need of the to topological sum over infinitely vertex many to certain Young non-toric diagrams diagrams. assigned to In this section, we give several examplesThe Kähler of calculations parameter thatT is involve reintroduced the usage and of it the is O- set to 1 at the end of the computation. vertex defined in section 2. MoreAlthough concretely, it we seems first check that the we validity need– to 10 of – sum our over formalism infinitely in many Young diagrams assigned to section 3.1, 3.2 and 3.3 by comparing the partition functions of the SO(2N), SO(2N +1) and G2 gauge theories obtained in the topological vertex formalism with the O-vertex– 10 – with the known one-instanton and two-instanton partition functions in the literature. In section 3.4, we will further apply the O-vertex to the calculation of the partition function – 10 – of the 5d pure SU(3) gauge theory with the CS level κcs =9, by using the brane diagram proposed for it in [33].

3.1 SO(2N) gauge theories We start from the computation of the partition function of the pure SO(2N) theory. The brane web diagram is obtained by putting a stack of N D5-branes on the top of an O5−- plane. Hence we can directly apply the O-vertex proposed in section 2.2.

3.1.1 Pure SO(4) gauge theory The ﬁrst example is the simplest case N =2, i.e. SO(4) gauge theory. The 5-brane web diagram for the pure SO(4) theory is given by

Q′ P + + O5 O5− O5 (3.1)

– 11 – (176) (174) (175) proofs JHEP_213P_0121 9 • - ) − 8 • +1 (3.1) 1 7 • N ) gauge → 2 6 • T +1 by applying ) theory. The N 5 • • ), SO( 1 at the end. It appears 1 at the end. It 2 N N =9 → . (b): the diagram 2 . Q 2 4 • • • • T Q ... Q cs | 18 κ | 2.2 1(b) 3 • • • • ⇥ ∅ T 0 (b) 0 ]. Furthermore it is also Q 2 • • • • 2 Q X 18 P 1 • • • • + 0 , by using the brane diagram Q 0 • • • • O5 =9 D5-branes on the top of an O5 ]. cs ± N κ 38 D5 NS5 O5 7-brane , i.e. SO(4) gauge theory. The 5-brane web cult to do, since the computation involves to be difficult to do, since We need to take a special limit need to We and furthermore, in the vertex-operator formalism we proposed, formalism we proposed, in the vertex-operator and furthermore, and the limit is we have a closed-form expression to take. straightforward but the summation converges with finite non-zero contributions, non-zero with finite but the summation converges − from the diagram in figure Q 1 O5 =2 – 11 – Remark:

→

. , it turns out that a ﬁnite sum gives the correct N T Table 2: Conﬁguration ofthe branes in direction the branes branebranes. stretch web construction. along, Bar and represents dot means the point-like direction for

3.2 ) gauge theory with the CS level 3

′

], which may be obtained by a twisted compactiﬁcation of

P Q

33 O5 . More concretely, we first check the validity of our formalism in 2 by comparing the partition functions of the SO( ]. gauge theories utilizing the construction in [ 3.3 ) gauge theory with the CS level (a) 2 3 33 G and gauge theories ) ) gauge theory with a tensor multiplet [ 3 3.2 N , , we will further apply the O-vertex to the calculation of the partition function (2 . (a): the brane web by sending gauge theories obtained in the topological vertex formalism with the O-vertex 3.1 3.4 2 G We can use this technique to compute the partition functions of SO( 3.1.1 Pure SO(4)The gauge first theory example isdiagram the for simplest the case pure SO(4) theory is given by We start from the computationbrane of web the diagram partition is functionplane. obtained of by Hence the putting we pure a can SO( stack directly of apply the O-vertex proposed in section In this section, wevertex give defined several in examples section ofsection calculations that involve theand usage of thewith O- the known one-instantonsection and two-instanton partition functionsof in the the 5d literature. pure SU( proposed In for it in [ 3.1 SO the brane web proposed6d in pure [ SU( 3 Examples Figure 3 to be used for the computations, where we shall set the legthe attached line to the with D7-brane toanswer, the be which Kähler empty. we will parameter see in section theories and also possible to compute the pure SU( The presence of two parallel external (2, 1) 5-branes implies an SU(2) flavor symmetry associated to the spinor matter. To perform the Higgsing which breaks the SU(2) one needs to put the (2, 1) 5-branes on top of each other. For that we perform a generalized in addition to α1, α2. Associating the Kähler parameters Q1 and Q2 to the simple roots flop transition [30] which yields α1 and α2 respectively, we can see that (3.90) or (3.91) reproduces the root contribution to the perturbative part of the partition function. Let us move on to the instanton part. The instanton part is given by removing the perturbative contribution from the full partition function

G2 G2 G2 Ztop, inst = Ztop/Zpert. (3.93) (3.81) proofs JHEP_213P_0121 It is possible to compare (3.93) with the result in the literature. For the one-instanton part,From the (3.81 universal), we perform formula the (B.14 usual) gives ﬂop transition twice and arrive at proofs JHEP_213P_0121 2q Q3Q4(1 + Q + Q Q +3Q Q2 + Q Q3 + Q2Q3 + Q2Q4) ZG2 = 1 2 2 1 2 1 2 1 2 1 2 1 2 , (3.94) 1-inst (1 q)2 (1 Q )2(1 Q Q3)2(1 Q2Q3)2 − − 1 − 1 2 − 1 2 where we can see that the denominator is completely determined by the set of positive long roots, α , α +3α , 2α +3α . The two-instanton part may be computed by using { 1 1 2 1 2} (3.82) the blow up formula in [42]. Then we found a perfect match between (3.93) and the two- instantonThen we can part tune obtained the length from of the 5-branes blow up to formula put the as two well parallel as the external one-instanton(2, 1) 5-branes part (3.94 on) 5 5 untiltop of the each order other. of Q Supposing1Q2. that a 7-brane ends on each external 5-brane, we can remove a pieceIn the of a diagram(2, 1) 5-brane in (3.80 between), a (2, 1) two5-brane(2, 1) 7-branes. was introduced This corresponds on the right to side. the Higgsing It is also possibleand sending to realize the piece spinor of matter the 5-brane by having far away a (1, giving1) 5-brane rises on to the a diagram left side of of the the pure diagram.G2 Whengauge theory the diagram given byis reﬂected along a vertical axis, the following diagram

G2 gauge theory (3.95) (3.83) also realizes the SO(7) gauge theory with a hypermultiplet in the spinor representation. No ADHM construction known, but there is brane construction where the 7-brane on which the (2, 1) 5-branes end is depicted as a black dot. One can performproposed, the Higgsingand there to is the also diagram, a proposal which givesfor its a diagramblow-up of equation the pure G 2 gauge We then consider applying the topological vertex as well as the O-vertex to the diagram theory [33to] determine the Nekrasov partition function recursively. in (3.83). At the practical level, we use the following diagram for computation. Two proposed brane web: [Hayashi, Kim, Lee, Yagi (2018)] ① ∅

Q′ Q 2 ∅ (3.96) Q1 It is also possible to apply the topological vertex and the O-vertex to the diagram in (3.96). For that we use the followingQ diagram,2 ∅ (3.84)

The constraint that the two (2, 1) 5-branes with empty Young∅ diagrams assigned coincide ② at the inﬁnity forces the Kähler parameters to satisfy Q2′ = Q2. The Coulomb branch Q2 moduli of the pure G2 gauge theory can be parametrized by ∅ Q 1 1 Q1 = A1A2− ,QQ 2 = A2. (3.85)

Q one instanton ☑ 2 ∅P two instantons(3.97) ☑ – 24 –

– 26 – Combining (3.101) withCombining (3.103), ( we3.101 obtain) with (3.103), we obtain

G 2q G 2q 2 3 2 3 2 3 2 3 Z′ 2 = P.E. Z′ 2Q=+P.E.Q + Q Q + QQQ++QQ+QQ +QQ+QQ Q , + Q Q(3.104)+ Q Q , (3.104) pert (1 q)2pert 1 2 (11 q2)2 11 2 2 1 21 2 1 21 2 1 2 1 2 ! − " ! − " #$ #$

which reproduces thewhich perturbative reproduces part the of perturbative the pure G2 partition part of the function pure G (23.90partition). function (3.90). The instanton part ofThe the instanton pure G2 partition part of the function pure G is2 partition given by removing function is the given pertur- by removing the perturbative part (3.104) frombative (3.99 part), (3.104) from (3.99),

G2 G2 G2 G2 G2 G2 Ztop,′ inst = Ztop′ /Zpert′ Z.top,′ inst = Ztop′ /Zpert′ . (3.105) (3.105) proofs JHEP_213P_0121 proofs JHEP_213P_0121 We checked that bothWe the checked one-instanton that both partition the one-instanton function and partition the two-instanton function and partition the two-instanton partition function extracted outfunction from the extracted above expression out from the agree above with expression the result agree (3.94) with and the the result blow (3.94) and the blow 2 2 2 2 up result until the orderup resultQ1Q2 until. the order Q1Q2.

3.4 Pure SU(3) gauge3.4 Pure theory SU(3) with gauge the Chern-Simons theory with the level Chern-Simons 9 level 9 Combining (3.101) with (3.103), we obtain

LastlyG we compute2q theLastly partition we compute function2 the of the partition3 5d2 pure3 function SU(3) gaugeof the 5d theory pure with SU(3) the gauge Chern- theory with the Chern- Z′ 2 = P.E. Q + Q + Q Q + Q Q + Q Q + Q Q , (3.104) pert (1 q)2 1 2 1 2 1 2 1 2 1 2 Simons level!9 found− "Simons in [31, 32 level]. This9 found theory in [31 may, 32 be].#$ obtained This theory by maya circle be compactiﬁcation obtained by a circle compactiﬁcation whichwith reproduces a twist the of perturbative the 6dwith pure part a of twist SU(3) the pure of gaugeG the2 partition 6d theory pure function SU(3) with (3.90 a). gauge tensor theory multiplet with [38 a]. tensor A 5-brane multiplet [38]. A 5-brane webThe instanton diagram part for of the the pureweb theoryG diagram2 partition has been function for the also is theory given found by inremovinghas [33 been] and the also pertur- it is found given in by [33] and it is given by bative part (3.104) from (3.99),

G2 G2 G2 Ztop,′ inst = Ztop′ /Zpert′ . (3.105) proofs JHEP_213P_0121 We checked that both the one-instanton partition function and the two-instanton partition function extracted out from the above expression agree with the result (3.94) and the blow 2 2 up result until the order Q1Q2.

3.4 Pure SU(3) gauge theory with the Chern-Simons level 9 Lastly we compute the partition function of the 5d pure SU(3) gauge theory with the Chern- PotentialSimons application: level 9 found in [31, 32]. This theory may be obtained by a circle compactiﬁcation with a twist of the 6d pure SU(3) gauge theory with a tensor multiplet [38]. A 5-brane proposed brane diagram for 5d SU(3) theory at Chern-Simons web diagram for the theory has been also found in [33] and it is given by level 9.

In (3.107) the dotted lines represent O5-planes which is the S-dual of the ON-planes (3.106) (3.106) in (3.106). Here the vertical direction becomes periodic and it implies that this theory ! " is a marginalwhere theory we whose have UV an completionON-planewhere we is on related have the an left toON-plane a side 6d theory.and another on the leftON-plane side and on another the rightON-plane side. The on the right side. The Using the O-vertex, it is also possible to compute the partition function of the pure three color D5-branesthree imply color the D5-branes presence of imply the SU(3) the presence gauge group. of the TheSU(3) S-duality gauge group. of type The S-duality of type SU(3) gauge theory with the CS level% 9 from the diagram% (3.107). For that,% ﬁrst note that % IIB string theory exchangesIIB string a D5-brane theory exchanges with an a NS5-brane D5-brane and with it an amounts NS5-brane to rotating and it amounts a to rotating a the upper half or the lower half of the diagram (3.107) is the same as the lower half of the brane web diagram bybrane 90 degrees. web diagram Hence by the 90 S-dual degrees. version Hence of the the S-dual diagram version (3.106 of) becomes the diagram (3.106) becomes diagram (3.96) for the pure G2 gauge theory. Hence we can compute the partition function “non-traditional”(3.106) theory in the same way as we have done in the latter part of section 3.3. Namely for applying the topologicalwhere vertex we andhave an O-vertexON-plane we on use the left the side following and another diagram,ON-plane on the right side. The anthree S-dual color D5-braneseﬀective imply web the diagram presence of can the SU(3) be computed gauge group. Thewith S-duality our O-vertex of type proofs JHEP_213P_0121 % % IIB string theory exchanges a D5-braneP with an NS5-brane and it amounts to rotating a Q2 brane web diagram by 90 degrees. Hence∅ the S-dual version of the diagram (3.106) becomes

Q Q1 one-loop ☑ (3.107) (3.107) Q2 Gopakumar-Vafa ∅ P (3.108) (3.107) where we need again to take the P 1 limit at the end. Q and Q are related to the → 1 2 Coulomb branch moduli A1,A2,A3 (A1A2A3 = 1) of SU(3)– by 28 – – 28 –

1 – 28 – 1 Q2 = A1A2− ,Q1 = A2A3− , (3.109) and the length between the O5-planes is the instanton fugacity of the SU(3) gauge theory and it is ! 2 q = QQ2. (3.110) In this case, the diagram also involves the vertex of

O5−

ν (3.111) which has not appeared before. We can also determine the function associated to the vertex (3.111) by redoing the argument in section 2.2. For that we consider

P Q2 ν (3.112)

Taking the limit Q with PQ ﬁxed for the diagram (3.112) reproduces the dia- →∞ gram (3.111). The partition function for the diagram (3.112) can be computed by the method in [1], which was reviewed in section 2.1, and it gives

˜ 2 µ 2 λ 4 ν Zν(P, Q)= ( Q )| |Q | |fλ− ( P )| |CµνλCµt λ. (3.113) − − ∅ #µ,λ

– 29 – 0 1 2 3 4 5 6 7 8 9 D5 • • • • • • NS5 • • • • • • 7-brane • • • • • • • • O5± • • • • • • Table 2: Conﬁguration of branes in the brane web construction. Bar represents the direction branes stretch along, and dot means the point-like direction for branes.

Q0 (174)

2 P Q0

(175) ⇥ Q

Comment:

As far as we computed, our prescription literally reproduces each contribution in the localization integral.

In the unrefined limit, all JK polesT | are| ... labeled by Young diagrams. (176) X For example, we can effectively choose ✏ + ✏ = a +(r 1)✏ +(s 1)✏ + 1 2 . (177) i ± j 1 2 2 So the refinement is not straightforward to do in this framework.

On the other hand, Sp(N) theories are more complicated.

18 then the framing factor associated to the internal line in the above diagram is

(ad bc) fλ− − . (2.5)

The information above is enough to compute the topological string partition functions for toric Calabi-Yau threefolds. In fact one can apply the topological vertex to certain non- toric diagrams. Higgsing a 5d theory realized on a 5-brane web dual to a toric diagram may yield a non-toric diagram where a line in the diagram jumps over other lines [36]. A typical conﬁguration in such a case is given by proofs JHEP_213P_0121 then the framing factor associated to the internal line in the above diagram is

(ad bc) fλ− − . (2.5) (2.6)

The information above is enough to computewhere the topologicalthe lines with string the partition slope 1 are functions external lines. For applying the topological vertex to for toric Calabi-Yau threefolds. In fact one canthe apply diagram the topological (2.6) we vertex deform to the certain diagram non- in the following way [12, 14] toric diagrams. Higgsing a 5d theory realized on a 5-brane web dual to a toric diagram may yield a non-toric diagram where a line in the diagram jumps over other lines [36]. A ∅ typical conﬁguration in such a case is given by proofs JHEP_213P_0121 ∅ (2.7)

Then we apply the topological vertex to the diagram (2.7) with the trivial Young diagrams assigned to both the lines with the slope(2.6)1. Kähler parameters we assign in a deformed diagram need to respect the conﬁguration before the deformation. For example, the legnth where the lines with the slope 1 are external lines.between For theapplying vertical the lines topological in (2.7) vertex should to be equal to the length between the horizontal the diagram (2.6) we deform the diagram in thelines following in the way diagram. [12, 14 We] will make use of this technique in section 3. In [1] the topological vertex formalism has been further extended to 5-brane webs with O5-planes.∅ We review the formalism by using an example which we will utilize later. We start from the following diagram, ∅ (2.7) 2 Q′ Then we apply the topological vertex to the diagram (2.7) with the trivial Young diagrams + assigned to both the lines with the slope 1. Kähler parameters we assign in a deformedO5− O5 O5− (2.8) diagram need to respect the conﬁguration before the deformation. For example, the legnth between the vertical lines in (2.7) should be equalwith the to the Kähler length paramter betweenQ the′. This horizontal diagram may be thought of as the one for the pure lines in the diagram. We will make use of this“ techniqueSp(0)” gauge in section theory,3. which is trivial. Instead of directly applying the topological vertex In [1] the topological vertex formalism hasto been the further diagram extended of (2.8 to), we 5-brane may webs change with the Kähler parameter Q and deform the diagram O5-planes. We review the formalismKim-Yagi’s by prescription usinginto an example [30] which we will utilize later. We start from the following diagram, [Kim, Yagi (2017)] Sp(0) theory “microscopic” picture Q2 2 (geometric transition) Q′

+ O5− O5 O5− (2.8)O5− (2.9) 1 where Q′ = Q− . Then we use the mirror image for the half of the diagram of (2.9), which with the Kähler paramteryields Q′.way This to diagramcalculate may be thought of as the one for the pure “Sp(0)” gauge theory, which is trivial. Instead of directly applying the topological vertex to the diagram of (2.8), we may change the Kähler parameter Q and deform the diagram λ –4– into [30] O5− λ Q2 (2.10) Namely we take a diﬀerentuse fundamental a mirror image region and compared sum over to (λ2.9. ). Since this is a strip

diagram it is possible to apply the topological vertex to (2.10). The only unusual thing proofs JHEP_213P_0121 O5− (2.9) compared to an ordinary strip diagram is that we need to sum over all possible Young diagram conﬁgurations λ assigned to the horizontal legs. A subtlety is that the Young diagram λ assigned to the horizontal leg of the mirror image is transposed compared to the original diagram and we need to be careful of the framing factor for the glued–4– horizontal legs. Assuming that we can deﬁne a framing factor labeled only by the Young diagram for the gluing legs, we can determine the framing factor by requiring that the application of the topological vertex to the following local diagrams, which should represent the equivalent diagrams due to the orientifold, gives the same partition functions,

∅ Q2 ∅ ∅ λ λ ∅ ∅ ∅ O5− O5− ∅ λ

∅ (2.11)

where we assign trivial Young diagrams to all the external legs in (2.11). The diagram on the left-hand side of (2.11) is an ordinary 5-brane web and the application of the topological vertex to the diagram yields

2 λ 3 2 λ 3 ρ ρ ( Q )| |fλC λC λt =( Q )| |fλsλ(q− )sλt (q− ). (2.12) − ∅∅ ∅∅ − On the other hand, the contribution from the diagram on the right-hand side of (2.11)is given by 2 λ 2 λ 2 ρ ρ ( Q )| |gλC λC λ =( Q )| |gλfλsλ(q− )sλ(q− ), (2.13) − ∅∅ ∅∅ − with some factor gλ, which is assigned to the glued legs with the Young diagram λ. Since they are the contributions from the equivalent diagrams (2.12) should be equal to (2.13). In fact by using the identity

ρ λ ρ s t (q− )=( 1)| |f s (q− ), (2.14) λ − λ λ it is possible to rewrite (2.12) as

2 λ λ 4 ρ ρ ( Q )| |( 1)| |f s (q− )s (q− ). (2.15) − − λ λ λ

–5– Remark: We mainly set the gauge group in each node of the quiver diagram to be U(1), but (n) (n ) we can easily generate it to any A-type gauge group by replacing and ⇤ 0 with generalized

(n,m) (n0,m0) Remark: Weintertwiners mainly set theand gauge⇤ groupintroduced in each in [4]. node For of more the details, quiver refer diagram to [4]. to be U(1), but (n) (n ) we can easily generate it to any A-type gauge group by replacing and ⇤ 0 with generalized (n,m) (n ,m ) intertwiners 3and A Twisted⇤ 0 0 introduced Construction in [4]. For more details, refer to [4].

In this section, we make our main claim of this paper, which will be checked in the remaining sections. We propose one way to assign topological vertices, or equivalently DIM operators, to the 3 A Twistedorbifold-construction Construction diagram for D-type quiver gauge theories.

0 The brane constructionS-dual description with ON for D-type quiver gauge theories was originally reported in In this section, we[5, make 6], but instead our main of that claim ”macroscopic” of this paper, picture, we which adopt[Bourgine, will the ”resolved”Fukuda, be checked Matsuo, picture RZ in(2017)] of the this remaining orbifold sections. We proposeconstruction one way proposedWe to again assign in [7].use topological a resolved brane vertices, diagram. or equivalently DIM operators, to the orbifold-construction diagram for D-type quiver gauge theories. The brane construction with ON0 for D-type quiver gauge theories was originally reported in 0 [5, 6], but instead of that ”macroscopic” picture,ON we adopt the ”resolved”ON picture of this orbifold construction proposed in [7]. [Hayashi, Kim, Lee, Taki, Yagi, 2015]

The above diagramprescription: is a D -quiver construction for the ”SU(1)”forcing”SU(1)” two legs gauge to be labeled theory 1,butwe 2 ⌦ can generalize it to arbitrary D-type gauge theories by addingby morethe same branes Young on diagram the left. We only give a description in this speciﬁc0 case of D2, as the generalization is straightforward (see Figure3). ON ON

The above diagram is a D -quiver construction for the ”SU(1)” ”SU(1)” gauge theory1,butwe 2 ⌦ can generalize it to arbitrary D-type gauge theories by adding more branes on the left. We only give a description in this speciﬁcFigure case 3: Example of D2 of, as generalization the generalization to D3 quiver is ”SU(1)” straightforward gauge theory. (see Figure3).

We ﬁrst describe our proposal in the language of intertwiners of the DIM algebra, as it looks much clearer in this framework. 1The tensor product here means that the two gauge groups are completely decoupled.

Figure 3: Example of generalization to D3 quiver ”SU(1)” gauge theory.

5 Uq,t(gl1) (29)

b q 1,t= q (30) !

Uq,t(gl1) (29)

1+ [µ] ✓p(q1)✓p(q2) (31) b 1 b (55) W 2 q 1,t= q (30) (p; p) ✓p(q1q2) 1 !

R✏1 R✏2 1 q1 = e = t, q2 = e = q k. Z (32) 1+ [µ] (56) (31) 2 W 1

R✏ R✏ 1 q = e 1 = t, q = e 2 = q . (32) ✓p(qaz) 1 2 g(z)= 1 . (57) a=1,2,3 ✓p(qa z) (33) 8 Y < ✓p(q1)✓p(q2) 2 (55) (p; p) ✓p(q1q2) (33) : (1 x1) ✓ (x) 8 (58) ! p < ( 1) : k Z ( (56)1) 2 n 1 1 t n exp a nz (59) n (1 qn)(1 pn) n>0 ✓p(qaz) ! Xg(z)= 1 . (57) a=1,2,3 ✓p(qa z) Y dx 1 1 Y (xq1q2)+ =0 (60) C 2⇡ x z(1 x) ✓ (x) Y (x) (58) ⌧I ✓ ! p ◆

1 dx 1 1 dx 1 1 n Y (zq1q2)+ = 1 1 t Y (xq1q2)+n + Y (xq1q2)+ (61) Y (z) exp x 0 2⇡ x z a nz Y (x) x(59)2⇡ x z Y (x) ⌧ ⌧I ⇠n (1 qn)(1 ✓ pn) ◆ ⌧I ⇠1 ✓ ◆ n>0 ! D2 A1 A1 (34) X ' ⇥

D2 A1 A1 n (34) n n ' ⇥1 dx 1 z 1 1 (1 t )z (z)=exp Y (xq qa)+n exp =0 (60)an (62) ; n (1 qn)(1 qn1) 2 n (1 qCn⌫t)(1t (q,p t)n) (35) n>0 C 2⇡ x 2 z !Y (x) n>0 1 ; ! X⌧I ✓ ◆X

C⌫t 1t (q, t) dx 1 1 (35) dx 1 1 ; n n n n v, n v, (36) ReflectionY (zq1 qstate2)+ in quantumq3 z= 1 toroidalq1 algebraY (xq1q2)+1 (1 q2 )(1|+ ti⌦) | n i Y (xq1q2)+ (61) ⌘(z)=exp Y (z) x 0 2a⇡ nx expz Y (x) x z 2a⇡nx (63)z Y (x) be found in [23]).⌧ Using these contractions,n ⌧1 I ⇠qn the ✓ instanton n partition1◆Xpn function⌧I ⇠1 of linear✓ quivers is ◆ n>0 ! n>0 ! reproduced by the (Fock) vacuumX expectation value ofX the product of intertwiners following the n n n v, v1, z 1 (1(36)t)=z t/q (37) prescription of the (p,(z)=exp q)-web diagram [23]. a n exp an (62)⌦, ↵ ; | i⌦| n (1 i qn)(1 qn) n (1 qn)(1 p$n) | ii n>0 ⌦ 2:= v, ! v, n>0 1 p ! (64) X X | ii | i⌦| iX X We can use a reflection state in the quantum toroidal algebra 2 n n n n n q3 z 1 q1 1 (1 q2 )(1 t ) n ⌘(z)=exp a n exp z an (63) 3 Reflection states (x ofn±(z)1 the1+1qn Ding-Iohara-Mikix⌥(z)) ⌦ n=0 1 pn algebra(65) n>0 Figure 4: Graphical! representationn>0 of the boundary! state in the (p, q)-web diagram. X ⌦ ⌦ |Xii where we showed the property in the unrefined limit (for simplicity) Our proposal for the reflection states in DIM algebra is inspired by the boundary states of Virasoro where2 ✏ is the⌦ co-unit,:= v,✏(xk±)=0,v, ✏( U±k)=1.¯(gl )⇢ provides(64) a representation of (29)DIM, it is the transposed | ii x±|(z)i⌦| i ±q,t 1 (66) algebra [24] that play an importantof the contragredient role in 2dX boundary representation conformal⇢ ˆ defined field in theories (2.14). More [33]. explicitly, These states the characterization (3.2) • One intriguing observation here is that the reflection10 bstate above live in the tensor product of twoof the Verma boundary modules, state in and terms satisfy of the action of the Drinfeld currents (2.2) reads reduces to the boundary state in 7the 4d limit q 1 ,t = q . (30) (x±(z) 1+1 x⌥(z)) ⌦ =0! (65) ⌦ ⌦ (0,m| )ii + (0,m) (⇢ (x (z)) 1) ⌦ = (1 ⇢ (x(z))) ⌦ , (Ln 1 1 L n) ⌦ =0. ⌦ | ii ⌦ | (3.1)ii (0,m) (0,m) + ⌦ ⌦ (⇢ | i(x(z)) 1) ⌦ = (1 ⇢ (x (z))) ⌦ , (3.4) “2d” picture ofx this±(z) construction? ⌦ | ii (66)⌦ | ii (0,m) (0,m) In the second term, L n =(Ln)† corresponds to the(⇢ contragredient( ±(z)) 1) ⌦ action=(1 of Virasoro⇢ ( ±(z generators.))) ⌦ . ⌦ | ii ⌦ | ii Note the presence of the morphism Ln = L n that7 sends the Virasoro algebra with central charge These constraints· are satisfied by a coherent state given by c to a Virasoro algebra with central charge c. ⌦ = a~ ~v,~ ~v,~ . (3.5) Compared to the Virasoro algebra, the DIM algebra has| ii a richer (auto)morphism| ii ⌦ | ii structure, X~ and we expect a large class of boundary states. In this paper, we will not attempt to a general Indeed, this state is obtained from the identity operator classification of boundary states but present only the simplest constructions associated with the id = a ~v,~ ~v,~ , (3.6) horizontal and vertical representations. We hope to come back to the classification~ | iihh issue| in the near future. X~ with the second state has been transposed. The identities (3.2) are the equivalent of ⇢(0,m)(e) id = The action of DIM algebra on instanton partition functions is very di↵erent from the action · id ⇢ˆ(0,m)(e)withˆ⇢ replaced by its transpose⇢ ˆt =¯⇢. of Virasoro algebra on 2d conformal· field theory. In our context, the interpretation of the states satisfying (3.1) as boundary statesIn the seems definition misleading, (3.5) of the and boundary we will state, use instead we have theassumed terminology that the two vertical modules have the same weights. In the following, we will need a more general definition that relaxes this reflection state. condition, ⌦, ↵ = a ~v,~ ↵~v,~ . (3.7) | ii ~ | ii ⌦ | ii 3.1 Vertical reflection state X~ From the scaling property of the vertical representation with respect to the automorphism ⌧↵,it By analogy with Virasoro boundarycan be shown states, that we this are state looking satisfies for the states property in the tensor product of two vertical modules and satisfying the constraint ⇢(0,m)(e) 1 ⌦, ↵ = 1 ⇢¯(0,m)(⌧ e) ⌦, ↵ ,eDIM. (3.8) ⌦ | ii ⌦ ↵ · | ii 2 ⇢(0,m)(e) 1 ⌦ = 1 ⇢¯(0,m)(e) ⌦ ,eDIM. (3.2) In⌦ the| correspondenceii ⌦ with (p,| qii)-web diagrams,2 the boundary state will be associated to an orientifold plane that realizes the vertical reflection V . Graphically, it will be represented as a The RHS involves the dual actiondashed that line has in the been brane expressed web diagram in terms (see Figure of the 4). reflection symmetry V .

(0,m) ✏(e) 1 (0,m) We⇢¯ would(e)=( like to) conclude ⇢ this(V sectione), with two important remarks.(3.3) Firstly, the vertical representation of rank m =1canberephrasedasanactionoftheDIMalgebraonMacdonaldsymmetric· 10This condition is not restrictive enough to deﬁne these states, and one should rely on the Sugawara construction a a 2 and impose the condition (Jn 1+1 J n) B = 0 [24]. ⌦ ⌦ | i 12

11 1 (192) 2 ✓ ◆

*Sp(N) theory in ADHM basis (193) In the ADHM construction, we; have O(k) theory on D1 branes. The Nekrasov partition function diﬀers in the pieces of O(k)+ and O(k)-.

Z + Z Z(k) = O(k) ± O(k) . (194) Sp(N), ✓=0,⇡ 2 The integral expression of the partition function diﬀers for instanton number k odd and even.

In total, we have four diﬀerent contributions.

We gave an analytic expression for each piece of the contribution in terms of the M-factor. [Nawata, RZ (wip)]

But what is the corresponding brane-web realization?

20 2

Future directions:

• adding matter fundamental matter ☑ (straightforward)

spinor?:

• topological vertex for 8 (33) Sp(N) in ADHM basis? [work in progress with S. Nawata] • qq-characters

• meaning in the context of topological string?

2 O7-plane 1

(32) . q = e = q t, = e =

• q

2 1 1 ✏ R ✏ R

• algebraic description of 3d N=2* web?

AGT for SO/Sp gauge theories1 (algebraic structure, integrability) 1+ (31) ] µ • [

…

(30) q = ,t 1 q

1 q,t gl (29) ) ( U Summary

• beautiful results known for U(N) theories, but not so many things known for other gauge groups.

• The main reason is the diﬃculty to perform the localization integral.

• We used an extended version of the topological vertex formalism to write down an analytic expression of SO(N) (and some other) gauge theories.

• We expect our new formula to be useful in the analysis of properties, e.g. algebraic structure, in gauge theories beyond U(N).