JHEP05(2021)155 Springer May 18, 2021 April 19, 2021 : : February 2, 2021 : , Published Accepted Received Published for SISSA by https://doi.org/10.1007/JHEP05(2021)155 [email protected] superconformal indices for 3d/1d coupled systems, , = 2 N . -Virasoro construction for the superconformal indices. 3 q 2101.05689 The Authors. c Duality in Gauge Field Theories, Supersymmetry and Duality We compute the 3d

, gauge theories. We generalize some known 3d dualities, including non-Abelian 3d [email protected] = 1 N School of Physics, Sun Yat-Sen University, E-mail: [email protected] Guangzhou 510275, China Open Access Finally we propose a Keywords: 5d mirror symmetry and 3d/3d correspondence, to some of the simple 3d/1d coupled systems. indices Intersecting surface defects and 3d superconformal Junfeng Liu, Yiwen Pan and Hong-Hao Zhang Article funded by SCOAP which arise as the worldvolume theories of intersecting surface defects engineered by Higgsing ArXiv ePrint: Abstract: JHEP05(2021)155 6 ], which are closely 6 – 2 16 ]. Among these BPS insertions are the 39 1 – S 34 10 × – 1 – 4 S ], which can be further decorated with local BPS 23 ], numerous exact results have been derived using the 3 – 29 1 10 [ 31 12 1 21 24 14 13 ] of partition functions and indices. 9 – 7 28 theory 2 1 ]. A second approach is to place a supersymmetric theory on the locus 41 1 6 13 ] or non-local BPS defects [ U(1) 33 – ]. 24 40 4.1.1 4.1.2 3d mirror symmetry from fiber-base duality 3.2.1 Reduction to 2d -Virasoro construction The field-theoretic construction of surface defects usually falls into three, probably See also some localization computations performed on manifolds with boundaries [ q 4.1 3d mirror symmetry 4.2 3d/3d correspondence 3.1 Higgsing 3.2 Index of intersecting gauge theory 1 overlapping if properlypreserving identified, singular categories. behavior of thethe One fundamental defect fields is resides in [ by the theory prescribing near some the locus symmetry- where related to the factorization [ operators [ particularly interesting codimension-two defects, whichsurface are defects. usually They referred are totheories considered simply [ important as tools to identify phases of quantum field 1 Introduction Since the seminal work bytechnique of Pestun supersymmetric [ localization for supersymmetric theories in different dimensions. and sphere partition functions C Factorization of 3d index 5 A Special functions B Instanton partition function 4 3d dualities 3 Intersecting surface defects on Contents 1 Introduction 2 Higgsing and surface defects Two simplest quantities that admit localization computations are superconformal indices JHEP05(2021)155 1 3 R S S ]. In ]. A × ]. In ∪ ]. For 2 R 43 3 L 53 69 , S – S ]. When 60 , partition ∪ , 42 50 54 1 68 1 , 59 S S d part in the 45 gauge theories ]. × × 4) ) 2 2 L 49 d coupled system. , − S – 1 L,R 4  4) n n 44 S ( version [ − mirror symmetry that d/ n ( 2) construction [ = 4 d/ = 2 U( = 2 − S 2) N n SQED, it is well-known that it N ( N theory, a class of (intersecting) − superconformal indices of 3d/1d n theory. Starting with the 5d index ( theory with one fundamental chiral = 1 U(1) = 2 d/ 1 2 SCFTs engineered by coupling a class N n = 1 N N U(1) = 2 ]. The list goes on. N d part (or the 57 – 2) – 2 – can be engineered by a twisted compactification 55 − ], similar to the class- , ] n 67 47 ( M – [ T 61 [ M theories ], which is a generalization to the which further interact with some 1d bifundamental chiral multiplets SQCD, we perform the Higgsing procedure to extract the indices 58 -partition function and to 5d index were also discussed [ . These 3d/1d systems if circle-reduced to 2d/0d would correspond 1 5 , ) = 2 1 S S S N 56 × N 4 U( S gauge theories (possibly with Chern-Simons term) are known to enjoy 3d ⊂ 1 SQCDA with one flavor are shown to enjoy 3d = 2 mirror symmetry between 3d/1d coupled systems, generalizing the usual duality S × N = 2 = 2 2 L,R A class of 3d 3d In this paper, we continue to investigate the 3d The sphere partition functions in the presence of a wide variety of surface defects, N N S from 6d on a three manifold between SQED and thepartition XYZ functions, model. and in This thissuperconformal has paper indices been we for checked provide such by further 3d/1d comparing evidence coupled the by systems. also computing the example, a standard mirror symmetryand is the between free theory withand one couple chiral. them When to wesymmetry a place generalizes. bifundamental two chiral Similarly, such on if theories the oneenjoys on starts intersection, a with we fiber-base a show duality 5d that with the some mirror 5d free theory. The duality is expected to descend to on the intersection to the general degenerate Liouville/Todarepresentations. vertex operators labeled by a pair of symmetric mirror symmetry [ coupled systems, viewed as surfaceof defects the in standard a 5d of the resulting 3d/1dfunction, coupled the systems. 3d/1d Viewing coupled the systemson 5d are built index out as of a two 3d intersecting case) of the fullFor instance, theory as can defect be worldvolume isolated theories and in one a can 5d studydescends dualities from they the enjoy. fiber-base duality [ including intersecting surface defects, aredualities. computed For and example, shown surface to defectsof participate 2d in in gauged a different linear 4d sigma models to the bulksimilar are 5d shown uplift to to bethe dual bulk to theory general is degenerate a free theory, the submanifolds, and some furthercodimensions, while supersymmetric all theories these onbulk lower-dimensional theories theory the are in intersection further a of coupled supersymmetric to higher manner, the forming original a is preserved. A thirdexpectation way value is to to some operator trigger and ansimplest then cases, RG-flow look the by at locus the giving of resulting a thebe theory defect position in is the the dependent a union IR submanifold vacuum [ of in the multipleabove bulk intersecting would space, submanifolds. require while generally one In it to the can place latter two cases, supersymmetric the theories second on approach the two codimension-two that will couple to the bulk theory in such a way that a certain amount of supersymmetry 3d 3d vertex operators in the Liouville/Toda theory through the AGT-duality [ We shall refer to these general defects as intersecting surface defects [ JHEP05(2021)155 = 1 = 2 , we U(1) U(1) 5 N N × ) N ( ]. Another 72 SU factor from the × ) ) N N ( ( . One can bring in an ) SU SU N ( factor and trigger an RG- and has an additional we will review the Higgsing SU ˜ T 2 partition function of a simplest U(1) and one 1 T S × , we apply the procedure to 5d theories can be constructed using the ], to accommodate partition functions 2 3 from 73 D ) = 2 are known to compute the partition functions N was proposed and was shown to be the only ] ( ) N – 3 – ], and therefore they sit in the kernel of some M SU [ 3 at real level 0, which is one simple entry in the (3) 70 T S ∪ M ( ) supersymmetric gauge theories, referred to as the . One can then turn on the vacuum expectation value indices of the worldvolume theories as 3d/1d coupled 3 (2) T S = 2 ]. ∪ = 2 42 N 3 (1) N mirror symmetry and 3d/3d correspondence. In section S with a flavor symmetry subgroup T = 2 (or 4d , we investigate some possible dualities that these types of theories N -Virasoro algebras [ 4 ]. Generalization to higher dimension is also possible [ q -Virasoro stress tensors. In this paper we propose a similar construction q = 1 71 , free hypermultiplets with flavor symmetry subgroup N 70 2 -Virasoro construction of the superconformal indices for the 3d/1d coupled q N , while for non-trivial dependence, one recovers the original theory coupled to a T Consider a theory The organization of the paper is as follows. In section Partition functions and indices of 3d expectation vanishes. If the positiontheory dependence is trivial, in thesurface IR defect one of recovers the codimension-two original sittinga at submanifold, the but core. in generalcase, Note could we that will be the refer core the to is union the not of surface necessarily two defects intersecting as submanifolds. an In intersecting such surface defect. additional and gauge the diagonalfree subgroup hypermultiplets. of the The resultingflavor symmetry, theory compared will with beof called a baryonic Hiiggsflow. branch In operator particular, associated the expectation to value the can be position dependent with a core where the In this section wedefects review in an 5d approachHiggsing to procedure constructing following a [ class of codimension-two BPS propose a systems and argue its uniqueness. 2 Higgsing and surface defects procedure that engineers athe type corresponding of brane construction. surfaceSQCD In defects and section that extract we the willsystems. 3d be In studying, section asenjoy, well including 3d as of intersecting theories on solutions under the requirement that allthe the participating participating screening charges commute withfor all intersecting superconformal indices, and also argue the uniqueness of the construction. future study. screening charges differential operators if properand formal variables indices are [ included intogeneralization, the refereed partition to functions as a modular triple [ of complex Chern-Simons theory on theories. We report antheory equality of at intersecting the SQEDs level andknown of a duality. matrix However, the integral, precise which physical could interpretation be remain a unclear generalization and to is the left for particular, the superconformal indices of 3d/3d correspondence. It is therefore natural to ask if it generalizes to intersecting 3d JHEP05(2021)155 . 1 ˜ T ) q − , (2.3) (2.4) (2.5) (2.1) (2.2) flavor p N ; i satisfies up front. ˜ µ i 1 ! µ 1 U(1) − A , related to N . We choose 4 z , 2 S 2 pq | ) πi , √ 2 q e anti-fundamental , N 1 )( p S ). The residue can q ; ≡ ι , N  free hypermultiplets, ∈ p q ˜ µ ; 2 , , , 1  1 ) N − i q ˜ rotations of µ , πi flavor fugacities as 2 z, µ A p ) , ι ; e ; z . T N Q pq N pq ≡ ( ( U(1) 1 L,R p √ ∈ − B n . × ( SU inst z independent, and one simply needs 1 2 R ) = A Z q | − , which effectively gauges A theory. By a redefinition of variables, z , ) , n b z R A p ) fundamental and z )( U(1) L n ; L,R A ( q i . The index has poles at special values of − N , n b N q in 4d by p bµ 1 2 ; U( from the 1 2 =1 1 N , − X − 1 A A b − B L A  , n z z n variables 1 2 A 1-loop-HM – 4 – − pq gauge theory. One can compute the index of z + p Z ( N √ R ) , A n ι z B )( , while the degeneracy then removes the 1 2 =1 q ( µ such that the integration contour is pinched by poles q 1 = 1 6= N 1 2 − A Y , SQCD with b A R p = + A,B flavor fugacity ) n ; = N L theory into a 1 1 A n − N A − (on top of the sum over Higgs vacua ) z − i p q ( ι flavor fugacity 1 2 µ N N R ) 1 U(1) 1-loop-VM ( − → SU n q − L A Z , b ’s making all b n p SU z A imposed implicitly everywhere) A U(1) ; − A z , q ! p and ( A A πiz dz 1 pq ι N z L 2 = 1 − are fugacites associated to the n √ 1 bµ ( -deformation parameters N A q − ) =1 z Ω Y q = A N N =1 , Y i =1 p A and ; =1 z N A | =1 p N i Y p ( z A Q | . I as usual. ≡ ≡ 1 = = 1 will be the standard I i < ˜ µ T | The baryonic simple poles that we will be interested in are The above construction can be further visualized in detail at the level of sphere partition q | =1 , N i | 1-loop-HM 1-loop-VM can be absorbed into the , which correspond to the values of p Z Z to collect the residues of the integrand at trivial, and one may simply put and as a result, theinteger contour partitions integral of picks upalso residues be at extracted these by poles, performingsymmetry which and a is making contour a integral the sum of b over one-loop contributions, of theindependent variables integrand pinch the contour since there are only | Here we haveQ separate the the exponentiated then hypermultiplets. the index reads functions or superconformal indices.the For superconformal the index purpose of of a this 5d paper, we considerb the case of of the integrand therein. More concretely, if one starts with where (with which will be a function of the A symmetry observation implies that the dependence of the residue on the Higgs vacua is At these values, the poles of the following form, coming from the fundamental hypermultiplet JHEP05(2021)155 ) 4 L 1 N ( D3 L SU n denotes the · flavor symmetry. SQCD, which can ]. ) 74 (2) N ( SU SU quiver gauge theory with ) × ) N ( N ··· implies that a (1,1)-brane extends ( SU / SU the spacetime directions spanned 1 // web, by aligning the D5/NS5 branes on ) p, q ( – 5 – -web diagram that engineers a ) -web leaves the partition function invariant, which is ) p, q ( p, q ( 0 1 2 3 4 5 6 7 8 9 − −− − −− − − − −− − − − − · · · · · · − · · − · · · − · · − · · · − − − − − · − · · · hypermultiplets to one side corresponds to gluing in an additional L R 2 D5 engineered by a five-brane web in type IIB string theory; see figure D3 NS5 D3 N (1,1) slope within the 56 plane. T 1 web diagram can also be understood in terms of refined topological vertex, ) branes which extend in different directions as shown in the above table. These R p, q ( . The figure on the left is a D3 . Directions along which a brane extends are denoted by a bar (–), while R n The One may look at the brane realization of such procedure. Consider a linear 5d theory. particular, the S-duality of the associated to identities involving the (skew-) Macdonald polynomials [ one additional gauge node.the One root can of then Higgs tunethe branch, the or, side. adjacent in Coulomb One terms branch can of parameterand further the to pull away the NS5D3-branes brane support from the the worldvolume web theories while of stretching the codimension-two defects in the bulk quiver gauge theory for an simplest example.by the We branes. also tabulate Thefundamental horizontal in hypermultiplets external table associated legs to on the theGauging manifest in two sides additional represents fundamental/anti- strip of D5-NS5 geometry to that side of the web, making a right is pulled away of the web while suspendingfundamentals some in D3-branes this (represented case. by the dashed lines) Figure 1 be constructed by gluinghas two strip been geometry. tuned In to thebranes special middle and values corresponding figure, D5-branes to the are the Coulomb aligned branch root reconnected. parameter of In the the Higgs figure branch, where on some the NS5 right, the NS5-brane on the directions along which a branealong takes a some line fixed with values. a – Finally, which engineer codimension-two defects in the remaining 5d theory, which is a free theory with which straightforwardly produces the Nekrasov partition function of the 5d theory. In Table 1 JHEP05(2021)155 1 − ) q q anti- (3.5) (3.6) (3.7) (3.8) (3.1) (3.2) (3.3) (3.4) , ≡ p t ; N i ˜ , µ 1 A and -deformation . z − A Ω , p ) z 1 2 | ≡ − , ) pq q , q ) q , √  , R 1 ˜ p µ n 1 )( − ; , q   p , = ˜ ; ], µ p 1 fundamental, , ; R A  − z, µ 1 76 n ( ˜ , µ − i N , ˜ µ z, µ A 1 -tuples of Young diagrams HM ; X A 75 − , z N Z Q ) ( , µ 43 z pq . 1 encodes the Yang-Mills coupling ( − 2 √ , inst pq ( z Q L Z VM ; ) | n √ 1 ) q Z µ outside the unit circles of , z − ) are related to four-dimensional = , ( p 1 z } Q q ) = ˜ ; ( , and ( L A i S , q  ] p f n , µ CS ˜ L,R A µ ) B 1 p together with all other fugacities, 77 × q n Z ; − A A | to be the corresponding contributions in the , { X 4 z ~ Y p Q pq | ; 1-loop-HM S 1 – 6 – pq ˜ Q , µ Z − B ) √ z . The index reads [ pq z ~ , Y 1 A 1 )( ( X . 1 2 z √ q S z, µ, , µ − ; )( . In some literature, one uses the notation p × R A 2 q ) = ; Q = n , gauge theories coupled with 4 ( q 1 πi p  − , S f − i 2 . ; ) q p µ e 1 µ 1-loop-fund/afund ; 1 2 1-loop-VM pq = N  − A B Z = Z − denotes the shift [ ˜ z z 2 µ = 1-loop-afund | L A q , A  A ) 1 n  Z pq A , z q − − 1 ( , √ p πiz dz qt p ( πi z, µ B 2 =1 A ; 2 ; ≡ = 1 U( 6= N ˜ e µ µ Y N =1 p A Q Y =1 i N A,B ( Y = = A ! N 1-loop-fund p =1 1 N A z, µ, Y inst N I A Z z ; Z and we define by are flavor fugacities for the fundamental and anti-fundamental hypermultiplets Q = 2 ( 3 ≡ ≡ ≡ , ˜ µ I 1 f |  and µ To access the partition functions of intersecting surface defects, we focus on the following The instanton partition function is given by a sum over Here we adopt the convention in which the fugacities The distinguishing of anti-fundamental from fundamental hypermultiplets is merely for 1-loop-HM 1-loop-VM 2 3 Z Z parameters instead. Consequently, bookkeeping purpose. set of simple poles labeled by two partitions constant. The superscript Here respectively, above factor. For convenience, we reproduce the one-loop factors here, 3.1 Higgsing Let us consider fundamental hypermultiplet on 3 Intersecting surface defects on The modular square is defined by inverting where relevant factors are collected in appendix JHEP05(2021)155 =1 N A (3.9) } (3.12) (3.13) (3.10) (3.11) A Y { , 1 rectangle of − j , ˜ µ i R A i ). The factor 2 n µ  . ], it is crucial to 1) × C.6 45 L A , − . n , R A 44 ) n q VF-extra , Z − p ; s J ˜ µ + ( 1 afund-extra R − I 1 Z ~n fund+adj  does not contain the “forbidden µ Z -Pochhammer symbol. The anti- 1) A which soon participate in partial pq R q Y ~n afund − ) √ 2 Z 1 L A )( L,R A s q L n n , ~n L fund+adj p + ~n − afund Z ; 1 r Z 1 1 = − J r S ˜ µ – 7 – × I + where each 4 ) + ( µ pole  I I S free

index of the world-volume gauge theory living on ˆ ˆ µ µ =1 Z pq N A ) of the double − − √ } = ( = 2 A A A z A.8 z Y -free hypermultiplets with flavor fugacity (ˆ ) associated with the fundamental hypermultiplets, while (ˆ =1 contains that box. As explained in [ N { pole 2 )( N h

Y 1-loop-fund A I,J 3.3 N Z πiβ Y A.7 captures the contributions from the fundamental and adjoint πiβ = in ( inside. Tuples that are not large are then referred as small. It ) evaluated at the pole gives 1 Res S . This is because the fundamental hypermultiplets contribute a × 1) 2 sinh sinh 3.3 4 , A A 1-loop-afund S free Y Y + 1) (1 ). The residue of vector multiplet and fundamental hypermultiplets Z ~n Z fund+adj ∈ ∈ denote the contributions from anti-fundamental chirals to the Higgs- ) ) R A Z are fixed natural numbers. These poles come from the perturbative Y Y 1-loop-HM , C.6 r,s r,s 1-loop-VM , n ( ( Z N L,R Z ~n afund =1 =1 N N ∈ + 1 are simple factors that depend on Y Y ~n Z afund A A is the 5d index of Z L A . See ( =1 =1 n N N 1 Y Y I I ( L, R free S n Z → × Next we turn to the instanton partition function which is a sum over tuples The residue of the one-loop factors can be computed straightforwardly and simplified matter-extra 2 L,R the former type of tuples,boxes every with member the Young box diagramis contains easiest the to deal withlarge the Yong diagram contributions can from be theleft large alternatively and described Young upper diagrams. by right two First corner, subdiagrams of sandwiching located all, the at each rectangle. the The lower lower left subdiagram will be divide these tuples of hook Young diagrams into two classes, called “large” and “small”. In box” at factor at the pole, chirals to the Higgs-branched-localizedZ form of thecancellation index, with defined similar in factors ( from the evaluation of theof instanton Young partition diagrams. function. It cantuples be of hook seen immediately Young diagrams that the sum is truncated to a sum over factors read Similar to branch-localized form of the 3d S one-loop factor the remaining perturbative and non-perturbative factors are regular atusing these the values. shiftfundamental properties factor ( in ( where which vanishes whenever one The factors where JHEP05(2021)155 , 1 (3.15) (3.16) (3.18) (3.20) (3.17) (3.19) (3.14) − , R A R . ,  q p , , , . . . , n 1 = = appears with 3d R − . L R   1 β 2 = 0 ˜ t, v, τ, q R A S / n 1 t, R L A − ; n L,R ,~n q q L , ν τ 5d , q ,Q , q CS ~n i VF-extra × k 1 1 ˜ t ν Z 2 5d −  − CS − − p q  k S R A ; = = = m L +1) ∨ A,n on R  L  R A  R ,~n 2 n R ( L / 1 1 2 ~n afund-extra Y L,R R − −  Z ~n vortex ~n vortex q q  Z τ Z = i , v , v is depicted with a red “forbidden box”, where ˜ t L R +1) . From them, one can further define two 2 2 appearing in the factorization cancel those   / / Y L A ,~n R Aν R 1 1 1 A n L ( q p m Y − ~n cl-extra 1 2 1 1 ) Z − i − i – 8 – − , p  µ µ R ˜ t, v, τ, q µ R R R , ,~n = ˜ = ˜   Y m t, 1 L 2 , ; L R A R ~n VF-extra /
L
n 1 − Z τ τ L A − ,Q m i i L A n ˜ t q ˜ t  as the shaded parts. Apparently, the large diagram contains 5d τ CS Q i k t R ;  =1 L,R N ,~n Y m A Y R L , . . . , n =  ~n afund-extra , , L ,~n = L L Y 2 2 Z / / = 0  ~n ( intersection 1 1 2 L R Z / ~n − − vortex ,~n 1 q p L Z × − height)-rectangle of boxes colored in white. On the right, we depict a small 1 1 ~n cl-extra q , µ − − µ Z τ × → p q i i i − t µ µ L A  = = L A,n while the one at the upper right L Y R (width ) for the detail definition. It is easy to observe the fugacity relations ) ) . In the left figure, a large Young diagram denotes transposition of a Young diagram. The contributions from such a tuple of L τ R A τ ≡ i i n Y t t ∨ ( ( C.11 L Aµ × L m large diagram n multiplets. The factors from the one-loop factors when computing the residue. The classical extra factor reads See ( fugacities large Young diagram at the pole factorizes into a called non-decreasing sequences of natural numbers Figure 2 we identify the subdiagrams Young diagram. which arise from superpotentials coupling the free 5d hypermultiplets and the 3d chiral where (summands of) vortex partition functions where JHEP05(2021)155 . . 2 1

# − . R ) by (3.24) (3.25) (3.21) (3.22) (3.23)   L  R A R 2 m  m  rectangle  , 1  L + R R A . 1 m n m − , . . . , n  ˜ t, v, τ, q 1 ,  1
, and since the L − × t,  µ  ; Q m
= 0 L A  µ + n ,Q is a product of the R R + ,~n ,~n R Bν 5d CS L , ν L k ~n m intersection ~n R R Bν VF-extra ν ; Z − ,~n Z m R m L R A  −  ,~n ~n intersection R 2 L − Z  ∨ A,n intersection ~n m 0 ) 2 R     R R Z R ν ,~n ~n vortex 1  L Y m Z ν ~n + afund-extra 1 R  L Z ~n vortex + = (   L Aµ Z ) R m L Aµ R R Aν L ~n vortex  m ,~n m m Z L  ( +  ~n cl-extra ˜ t, v, τ, q L + Z AB t, m  , and can be relocated outside of the sum over ; , ˆ µ – 9 –  AB R ~  Y 1 ˆ µ ,Q m L  , − ~n L semi-vortex 5d L CS πiβ ~n vortex L A Z k m Z ; πiβ  m X small  2 sinh X large , . . . , n + 1   2 sinh R − 1 =0 ,~n R B Y = 0 ν L − n =0 intersection ~n B R Y R 0 1 ν L n − ,~n ~n semi-vortex Z =0 1 L L A Y µ , µ Z − ~n n fund-extra × =0 L A A L Y µ Z n n → N Y A,B − N Y A,B × µ R − = ,~n L A L ~n afund-extra A,n Z is a monomial of these fugacity, the classical extra factor actually cancel within the Y R R . As advertise before, the extra factors from the anti-fundamental and fundamental ≡ ,~n ,~n 2 L Putting both the large and small Young diagram contributions together, we have the The contributions from the small Young diagram tuples are less trivial. Nonetheless, | L small diagram ~n intersection ~n cl-extra Aµ L Z cl-extra Z ... m

that the modular-square isZ defined by inverting all fugacities| including hypermultiplets will annihilate with those from the residue computation of the perturbative rectangle region. instanton partition function evaluated at the pole Here, as indicated bysame the factors prime, as the in the intersection large factor Young diagram case, but now only over the boxes inside the diagram reads region, which of course precisely fill that entire region sinceone we can are still dealing define with non-decreasing integers large (which however could be negative for One may view this factor as a product over all the boxes inside the hook Young diagrams. Finally thediscussion, intersection which factor reads is the most crucial one in the following Obviously, such factor is independent of To avoid clutter, we have omitted the fugacities from the expression. Immediately we recall The contribution from a tuple of small (or a tuple containing both small and large) Young Young diagrams. JHEP05(2021)155 , 1 . , 2 S   L and )  2 πiβ )  2 (3.27) (3.28) (3.26) ,B 2 } × N e  R + N = 1 ,z + {  L L 1 ( . theory with  q 1 2 2 ( SQCDA with ,B

, where 1 2 L ) ± chiral multiplet   1 z  = 1   S ± fundamental and L,R 1 R   n = 2 N m 1 N R a } × ,  U( L B N R S a =  1 2 { m B R 1 af  1 2 captures the contribution S intersection − m n , ) Z R b + and L = R  ˆ z 1 R f b m R  1 R ˆ , z z ’s. We propose the index to be S n  gauge theory on an intersecting 2 L  ,~n − 1 ,B z L ) S R (  ~n − intersection } × R z 2 Z    n R N 2 L a 1  {  ,~n S U( R B L L a R R 1 2 ~n q intersection ~n m fund+adj × B × 1  Z . 1 2 Z ) 2 L − S intersection  L S 3 L . The quiver diagram of the theory associated a Z + n R Z ˆ 1 z R L a  m S ~n vortex ˆ z U( L   – 10 – L Z ~n fund+adj -deformation parameters are given by  }× ,B  Z Ω L L S πiβ R z { ~n vortex m  πiβ R  Z 1 ~n afund S  L Z L . The q 2 sinh ) × m 1 2 sinh 2 R L L  =1 S S n Y b ~n afund R L Z =1 R n Z Y L b q ~n semi-vortex =1 R a n denotes the usual integrand of the index of a L R Y a R a × L Z ~n , and additionally a pair of bifundamental 1d vortex =1 ) ,~n n Y 1 a 2 ), while the last factor × R L πiz Z dz X S ~n 2 S L,R X = ( 1 small R z R C.1 q S =1 ( X  n large ∪ Y a 1 + × × R   ) 4 S L a 1 2

R S free L a , z given by S S L,R Z L × , see ( q πiz dz L z . The gauge theory is coupled to two copies of 2 q × =  N 1 L,R × and L ~n =1 2 L,R n Y = 2 a 1 L S πiβ = S S 2 af Z I ( e R L n R q ,n ~ B 1 = , × L = refer to the common north and south poles of the two S L X intersection n 2 L f R ~ I B Z Here the first line comesand the from second the lines degrees at ofto the south freedom this pole localized superconformal at index the is north given in pole figure n from the one-dimensional bifundamental chiral multiplet, S given simply by a sum of contour integrals space q ant-fundamental chiral multiplets and oneS adjoint chiral multiplets livingliving on at two each different of the two common circle intersections coupled system. 3.2 Index of intersecting gauge theory the index of a free 5d theory coupled to a partition one-loop factor. Altogether, we have the sum of residues of all the simple poles labeled by where We are now ready to identify the above result from Higgsing the 5d We will see shortly that this expression can be reorganized into the index of a 5d/3d/1d JHEP05(2021)155 . } 1 +1 − ], all i,µ (3.32) (3.29) (3.30) (3.31) -poles m R 78 R z , ≤ . . , 0 45 0 iµ 1 0 , , ≥ m , . . . , n ≥ − hypermultiplets L,R 0 that one would 0 ≥ 0 44 R i L i , [  = ≤ 2 i 1 L,R 1 i m m iµ S 0  N m × ≥ anti-fundamental and ≥ , ν iµ q m µ 2 L,R q N ∈ {− ...... S µ will be reorganized into τv i , ν i +1) Z ≥ ˆ ≥ ν type NS t separately. Concretely, 0 i R τv 1 1  ν i and the intersection factor. 1 = 0 t (ˆ ≥ − − /n S  1 0 R L = i R L − labeling the isolated massive and S × R i,n L i,n = m × = 2 L L,R m m =1 = S ( N i  2 L,R fundamental, } L,R 0 Z a L i B 1 S , . . . , n  ... , − N m S type S a B L,R i q 1 = , n − × 0 ≥ = 1 R i iµ { − q ˆ ν 1 n z 2 R i R L i ≥ iµ S  = i m m z a/b i  ≥ ≥ , type N , ν 1 N N take values in some different range rather than -poles from , S ... L ... – 11 – is coupled to L,R L z ) , , 1 1 , ν ≥ ≥ m +1) L iµ S S . , 0 1 i 1 n m } L ν − − L,R × × = 0 (ˆ q 1 ≥ ) R i L i 2 m µ type old 0 2 L R i − S iµ − S R i,n L i,n m m τv = i m m q t µ = L,R i ( 0 L i , τv , = ( ... i 0 m 0 t = ≥ ≥ ≥ L,R i , . . . , n = ( ˆ ν 0 1 0  R i R i L i L i a B : m m m m = 0 1 q L,R ≥ ≥ ≥  ≥ − iµ are non-decreasing sequences of natural numbers, e.g., z a B  = ...... q ... i iµ L,R iµ ˆ ≥ ≥ ν ≥ ≥ z , and then a combination of 1 1 m 1 1 1  : , : − − − − S , . . . , N, µ ˆ ν ˆ . ν i R L i R L × 1 . The worldvolume theory of the surface defect is described by the quiver diagram. Each of L,R iµ 2 R S R i,n L i,n R i,n L i,n = 1 S m m m m m i × Z | 4 The poles of type old are simply the familiar poles of the factors To identify the index with the result from the previous section, it is crucial to correctly ) type old: Type S Type N S iµ ( • • natural numbers, determined for each type by a set of integers from Concretely, they are poles of the form Higgs branch vacua. Doing so,{ the color indices pick up to perform factorization of the usual index on specify the integration contour.four We require types that of the poles, contourof integral which which picks we depend up call on residues a of choice of integer partitions Figure 3 the two unitary gauge theories onone each adjoint chiral multiplets, andintersecting, as further colored to in the purple.on 1d bifundamental The chiral red multiplet line supported above indicates on the the free theory of where the non-decreasing sequences where not all where The other three types of poles arise from first taking a subset of the standard JHEP05(2021)155 , (3.36) (3.33) (3.34) (3.35) ], which to zero = 0 1 79 0 R i S , m 1) by hand. In = − ) τ q ... +2ˆ i − ˆ ˜ t = + i (1 S ˆ t i effectively shrinks the gauge theory given by ˆ ν R i 2( . 0 ) − or in more general cases, m n ) 4 +1 q → of the temporal ≥ v U( 2ˆ β − β − . ) ... , . (1 q -partition function 2d gauge theories, 1 f ) ) 1 ≥ to zero. In this section the reduction 2 0 n =1 b 0 a − whose length is controlled by the − Y i S 1 − β ≥ 1 − Γ( Γ( = ≥ (1 n = a = R i Y S S i S N i i =1 = N n i R i,n = Y b a,b , ν m − , ν a  = )+1 )  τ q -partition function, +ˆ +1 2 i + 1 , , ν − ˆ ˜ t , ν v + 1 S S 2ˆ i – 12 – 0 2( 0 ˆ ν N (1 i − . Therefore, sending − b ˆ ν u ) = 0 ≥ ) a z ≥ q z q   2ˆ 0 L,R 2ˆ ; ; R i − q b a − − − a m q q ) z , and similarly for other fugacities, one has the needed ( ( q 2ˆ ˆ z = = ] to reduce the 3d index, the matter contributions therein ) = 1 q ( ( − 11 → πiβ 0 0 lim q 2 i L i L , − ... shares the same circle (1 e one would recover poles of type old which we have avoided in m m 1 10 = (1 1 = S )+1 ≥ ≥ N i τ − ˆ =1 z ˆ n ν 1 1 × q +ˆ R i Y i L i L i a,b = ˆ t m m m = i 2( 2 L,R ˆ ν and also for the adjoint chiral contributions, such factor are harmless. ≥ z − ≥ ≥ -partition function by sending the radius S a 2 , af z 2ˆ S n ...... : ) πiβ S q 2 ˆ = ν ≥ ≥ ≥ e entering into fugacities N f − 1 1 1 ˆ ν n = − β − − ]. (1 L i L i R i q f 10 -partition function from the index computed in the previous discussions. n =1 i,n R i,n L i,n L Y i 2 R m m m = S n a The reduction of indices is due to the limit It is straightforward, though somewhat tedious, to check that summing the residues Y Note that there maybe more than one limits one can take in such dimensional reduction [ and in the end one expects to find the ∪ Type NS 4 2 1 L are associated to differentof scaling the of index of parametersfollowing the when [ 3d sending gauge theories is chosen to reproduce the and for the adjoint in particular, to the case of Defining factors for the fundamental and anti-fundamental chirals in a S 3.2.1 Reduction toNote 2d that the two parameter S from all four types ofare poles in one-to-one recovers the correspondence resultin with from the the Higgsing. instanton double part, sum Indeed, which overof the is tuples type poles a of N of product large then type of Young correspondnorth old diagrams the and to north south the and pole double south respectively, sumFinally, while pole poles of poles of contributions. tuples of type of Poles type NS large S correspond and correspond to small to the diagrams the double in other sum the way over around. tuples of small Young diagrams. Obviously, when all the definition. • while rescaling the fugacities properly and inserting the factors of This limit was exploited in [ JHEP05(2021)155 . - 1 2 L S S = 2 × (3.37) (3.38) (3.39) (3.40) ’s and N mirror theory 2 R z S 1 2 = ∪ k = 2 . 1 -deformation R a S N B Ω  2 × U(1) ) 2 2 L  , S πiβ  e + through the Higgsing ) 1 2 = 1   ( S R a . The contribution from 1 2 + 1 B × 0 1 1 2 S R  are taken as an evidence of 2 R ± ( 3 S R 1 2  , S 0 R a , q ∪ 1 2 mirror symmetry, at the level of B 1 ± ∪ R a ˆ z S 3  L 1 ± πiβ S R 2 a × 2 = 2 R a e B 2 ˆ ]. Such duality can usually be realized L z πiβ 2  S = N ± e 59 1 R 1  R a = ˆ z − 1  R which originates from the 5d a 1  2  theories enjoy an IR duality called 3d mirror L a 2 – 13 – , q ,  B − 2 1   ± , z L = 4 a 2 πiβ L a L a B 2 ˆ z B e  N 2 theories admit deformations to theories with only ± 2 =  L a ˆ πiβ  z L e q  = 4 2 =  encodes the radius of the common πiβ N  L a β B 1 2 L πiβ ]. 3d 2 sinh 2 0 0 81 where they intersect. , q ± ± Y Y , L a 2 ˆ R z 2 ’s. 80 S ± ± Y Y B πiβ 2 0 and e → 2 L β = −−−→ S -partition functions. The remaining factors to reduce are the contributions from the L a 2 R z S This procedure straightforwardly carries over to the intersecting case, where the chiral coupled to a fundamentalexample chiral is and to the reducesymmetry theory the between of S-duality, a a i.e., class free of theprocedure. simple chiral fiber-base The 3d multiplet. duality, identical theories to indices on Themirror the of duality. second these 3d theories on supersymmetry, such as turning on complex masses, FI parametersmirror and/or symmetry. superpotentials. However it remainsthe an following, interesting we yet will challenging discusssuperconformal arena two indices, to examples generalized explore. of to 3d In the case with intersecting space symmetry, where two UV supersymmetricthe gauge Higgs theories branch flow of in onemany the other SCFT IR identifications identified to of with two physical theby SCFTs quantities with Coulomb the [ branch S-duality of in thegauge type other, theories IIB among [ string theory acting on brane systems that engineer these 4 3d dualities 4.1 3d mirrorIt symmetry is well-known that various 3d pole of In this parametrization, the 1d chiral multiplet then reads parameters, and define the hatted fugacities, for example, the sum over multiplets and the vector multipletand contributions in the index reduce to those in the The first example is the generalization of the basic duality between the which is simply the contributions from the 0d chiral multiplets living at the north and south These factors are simple constant factors that can be taken out of the integral over 1d matters. Let us parametrize, using With less supersymmetry, one typically has less control of various dualities, including 3d JHEP05(2021)155 , (4.3) (4.4) (4.5) (4.1) (4.2)   q q ; ]. To see ; ˜ 2  B ˜ 2 q B 82  − ; , 1 − q 2 B ; q 1 58 2 − B wq − 1 and the integral . − 1 2 q w ˜ 1 − B 1 2 .  , − = 1 tτq  ) 1 2 1 3 1 2 iπ  − ) − tτ 5  q − z ( q +1 − ;  k k ; z ˜ 2 ST tτq 2 B 2 ˜ 2 (  tτq B  e −  1 −   +1 q  1 2 2 , k 1 with unit Cherns-Simons level ) 2 wq − ˜ 2 B − − 1 2 w ˜  B w ( ˜ − 1 2 w tτq − . The sum over 1 2 1  ) z 1 −  1 2 − ˜ +2 − fund B 1 2 k k z . − ( + 4 2 tτq − ˜ k − i 1 theory coupled to one fundamental chiral  1 2 theory coupled to one fundamental chiral tτq − fund tτq 1 2 ˜ − z  chiral B 1 2 U(1) +  ) q I I =  − 1 2 = 3 k  w k )  = ˜ B ˜ − B tτ ) = ( – 14 – U(1) − )  ( ) I h U(1)  B q w U(1) w ( 1 2 ; q tτ ; q − = − 1 ( ) by the above relation. Even though the expression for w ˜ − fund ˜ w z = ( B theory coupled to a fundamental chiral [ 1 2 ) + k w − ˜  k − k w ( − 1 2 ˜  ) B  ˜ k, 1 2 ˜ theory coupled to one fundamental chiral. In particular, the z  ( , 1 2 U(1) − πiz dz − 1 2 I U(1) 3 − 2 ( → ˜ fund − + 2 z tτq ) I ˜ z tτq +  k k U(1) πi d  1 2 2 4  2 B with additional bifundamental 1d chiral multiplets. The index can  X k, w = 1 I U(1) = = :( S I ˜ k ˜ B theory coupled to a fundamental chiral is × X ST theory fund theory, and the above equality is refined to k 2 R action on the = + S 1 2 1 2 ) 1 2 Z ∪ , U(1) fund 1 U(1) + (2 S U(1) I U(1) k can be easily performed, giving × SL ˜ z 2 L U(1) Let us now consider the intersecting ∈ I S looks complicated, it is straightforward to check that indeed ˜ Define the map w on over effectively replaced by a index of a meaning that any chiral multiplet contribution in the index of an interacting theory can be Note that this basicST duality between the two theoriesthis, can we be introduce a used background tointo Chern-Simons the induce term the order-3 Let us first recall themultiplet (fund) duality between and a the theory of a free chiral multiplet. At the level of indices, one has 4.1.1 where where the last factor can be replaced by JHEP05(2021)155 ,     , ) (4.7) (4.8) (4.9) (4.6)  R 1  m − f , R µ L q     1 2 L ) m q − ( R w − m , 1 − L − R f q m µ ( 1 1 2 − L ) intersection w q α Z ( 1) L )     R − m f R R 1 m  ( ) µ 1 − 0 R 0 m − 1 2 m − q intersection − − equals the index of the theory ( 1 1 q . w Z → L → | m − L , ( R q m m → m L → − R − ) fund  q w 1 1d B B L → − | − R → + I → − f 2 m B ) m 1 2 B L R m µ ( L α q − ) ( B 1 2 q L  B vortex R chiral − ; q B 1 2 U(1) Z I q  I w − ( L vortex  . )  Z 2 α 1 L tτq L ( chiral L L = / . Under this duality, the 1d bifundamental  −  I )  R R  ) 4 q ) q L | /   m | 1d 1) m ) ) 1
= ≡ – 15 – = 1 m R 1 − m m m + − α 1+ ) ) ( (  ( ) − m k m 1 2 1 2 q − ( 2 ( | ( tτ 1 2 / 1 2 fund ( , 1 2 1 m tτq 1 + ,I ( tτq tτq − 1 2 S vortex vortex tτq →   tτq tτq R  Z Z  ∪ →  × tτq R L,R ( L → → U(1) z L,R L 2  =0 =0 R vortex → I → ) ) ∞ ∞ z →  q R R S α α Z ) ; L,R q L X X + z z + ( ( L α ; q z ∪ = m m ( z 1 α z w 1 + + − 1 2 S Q 1 2 1 2 w L −  − L 1 2 1 2  1 2 ≡ S 2 R N   × ∪ NS 1 2 − old / R q f 2 1 1 2 q 1 2 / 2 − L L µ 1 2 1 − − q S − − L − tτq q tτq tτq 1 2   − tτq  − R  1 1 2  R q  ) − q 1 2 ) 1 2 L w = − L q w q − ( − theories using the one with both (     k L,R     chiral × I = chiral multiplets in themultiplets SQED on on the the right. leftU(1) It are would dual be to interesting to a generalize pair to of dualities free for 1d intersecting fermi and chiral It is straightforward to check that theof above free index chirals on be computed by summing over four types of poles, where the indices on the right are given by The full residue can be organized into a factorized form, with the fugacity where JHEP05(2021)155 , 1 1 . ] S S  74 × q × , (4.11) (4.12) (4.13) (4.14) (4.10) . The 2 R 2 p q ; S S , 1 2 p ∪ where the ) 5 SQED and , 1 . 2 . pq | S ( 1  ] (up to some  . 2 λ, µ S q . × r = 1 ) , × 74 2 r 2 Q − L p 4 / ) + ∨ s  S N 1 ; R SQED q s ∨ λ ∅ , − 2 λ q Z / Y p ) | − 1 ˜ 1 t q M ) − N 1 , 2 s (1; −  / p s pq + encodes the gauge cou- 1 ; q ( r + , r µ YY 0 2 p pq M µ − ; 2 N Q β Q − p 1 2 2 1 Q 2 YM x π ) xq 0 g 8 Q 0 − − pq − Q instead. . ( e Q 1 ) 2 (1 1 )( / -brane web, and their full partition  − µ -deformation parameters q 1 ) . Q ∈ q, t µ   , ) function is given by − Ω ;  1 = ∈ q p Y R x ) , S r,s ; ( p, q 0 Y ( n λµ Y 1 p ( × ∅ ) r,s − ; 4 λµ N ( Q q N R 2 free +1 N 0  2 | / actually equals the partition function [ r 1 / Z (including the perturbative contributions) is 1 − Q Y 1 ) 1 | − ( s ∨ , the 1 r − S  = µ 1 ,Q L t pq 0 S + × 1 − 1 – 16 – s ( n 4 ∨ s q − Q S − 0 × − R µ SQED r 1 2 × ≡ p λ − 4 Q Z 4 t − q  R zM R SQED ) xq s , M p Z −   − = pq r gauge theory on the intersecting space  ≡ ( → λ 2 (1 q  ) p q z λ , x R ∈ p ) Y n ; X Y − 2 1 function is defined for two Young diagrams by r,s (  ) U( 1 1 q  ≡ , λµ × pq λ ) p ˜ ( ) N 1 ; i M,Q ∈ ) L 1 2 1 − Q Y ) n can be effectively computed using the method of refined topological − 1 r,s  ( q, t z 1 ; pq U( ( x 2 S =1 ≡ = i ( Y i ) 1 Q ×   encode the flavor and gauge fugacities and λµ q  4 Q , ]. Here we mainly follow the formalism and convention in [ N 2 ). = ], the exact same function is denoted as , p R 1 1 2 ; =1 1 83 74 Y i x S − Q , ( × = 4 74 1 λµ R free S N Z × The full 5d index of the SQED is given by a contour integral of The fiber-base duality states that Here To begin with, the 4 Note that in some literature with R 5 SQED Z coupled to one pair of fundamental and anti-fundamental chiral multiplets on each However, in [ into the index of a modular square inverts allHiggsing fugacities procedure including on the both sides picks up residues at the poles Namely, pling constant given by Next we turn to thesome 3d free consequence theory. of They thefunction can fiber-base on be duality realized between by 5d simple powers of 4.1.2 3d mirror symmetry from fiber-base duality vertex [ The SQED full partition function on As we have discussed in the previous section, the residue on one side of the equality organizes JHEP05(2021)155 .  ν L Y I + from 1 2 (4.20) (4.16) (4.17) (4.18) (4.19) (4.15) L  X L q I  p )] µ ; L p Z 1 q ; + I  ; ν − 1 2 v q ν p = ( +1 1 2 − 2 ν / / L q ) ) q 1 L     q 1 2 −     pq ; ) +1 )  ( q µ 1  q 1 1 2  pq  p q ; − ( q − q ; − 1 2 ˜ ; ˜ ; v 1 2 1 2 M  q . M 1 2 1 2 1 1 [(  1 q q 1 1 L − − − 1 M # − − ( wq ˜ − w M M m ˜ w 1 2 w  2 1 w M

m − 1 2 p q M − − ) 1 2 2 ; 1 ;
− M q  1
2 )  2
˜ ; tτ 2   +1 2 p t 1 w q 1 ν ˜ tτ ; L ˜ tτ , ( w ˜ tτ − reduces to the index of a collection t  1 t ˜ − tτ t   2  =0 R =0  t pq

2 − R Y can be further rewritten in terms of ν n ν  | " 1 ν n  ˜  1 q  M 1
− − Q m L ; ˜ S     =0 1 q q R M − Y 1     q ν × =0  ( L ; − n L Y vortex , 4 − 1 µ 1 2 L 1 n    
R L =0 free Z −      2 − − M

 n µ 1 M
 q  Z  `  R  =0 2  | gives the index of the standard XYZ model, q ; ˜ tτ q 1 q  Q n ν − 1 q ; t 1 ; ; q ) 1 2 − ˜ tτ ; 1 2 − µ − Q 1 2 ; 1 t q 1 2 q 1-loop R µ p q 5d =0 free 1 2 2 − k – 17 – . 1 n ν − 1 2 Z  ν Z − wq − p ) ˜ tτ  wq ) − Q 1 2 wq t 1 2 1 = 1 2 q w q 1 2 p )   pq 1 2 ; − ) 1 − 2 1 2 ( q − q q

/ − w ; ;

1 2 1 2 2 pq 1 µ . The residue of the 5d SQED index gives the 3d index 2 2 v ( )  ) − − ( − +1 ˜ tτ 1 ˜ tτ q SQED 1 2 v ˜ ˜ tτ tτ  t p t µ ; ( I − ∞ t t =0 1  1 = 0 X − + 1   m ˜ − qp  −  M   R − µ ˜ ˜ (1 fugacities ). The two results on both sides are of course equal. = = M         M ˜ − 1 M , n M 1 . On the other side, p ( L L − ) → ( M −     R =0 w R M 1  3.17 n ν = 1    n − M − 1-loop vortex q q w L =0 Q 

L 1 ; ; w Z Z n µ  q q U( 1 1 n  q q  ; ; − − Q 1 1 × , and the 5d flavor fugacities R 1

L L =0 =0 =1 ) − − n 2 2 β Y µ n µ n ` −

=0 L 2 fugacities L Y 2 2 2 YM π µ ˜ ˜ tτ tτ L Q Q n  n g 8 =1 t t n Y k ˜ ˜ tτ tτ − × × × × t t U( e → adjoint chiral to be omitted),   2 ) |     L 1 ≡ − S n = = w × 4 Let us go through a few simple instances in the above dualities. We start with the U( R free L XYZ Z I | On the other hand, the residue from the simplest case where of the SQED theory inthe the factorized form (with a decoupled factor Here the 3d fugacities using ( gauge group of free chiral multiplets togetherat with the 1d north contributions and from south the circles, Fermi multiplets localized and additional pair of 1d chiral multiplets in the bifundamental representation under the where JHEP05(2021)155 . , R 2 2  / / 2 1 1 (4.27) (4.23) (4.24) (4.25) (4.26) (4.21) (4.22) / ) 1 ∓ ∓ R R α q q − ( 2 2 )  / / α q 1 1 ( ˜ tτq ; , ∓ ∓  L L  2 B q q q  , , . ; 2 = R R 1+ 2 0 0 B ), L   ˜ q tτ  + t 2 2 B B 1 ≥ ≥ SQCDA theory on 2 q  / . − − 1 = 0
L 4.24 R 1 zq
) = −  m zq m ˜ tτ α − ˜ tτ (   L Z 2 ) U(1)  1  / z L ˜ tτq 1 α 1 m − (  2  B  × − ) q 2 z B ; − R )  tτq q q q , α 1 1  ) L ( , , , m q α ) − − R U(1) v . ( 1 ( 1 α = z z  q R L (  − , 1 2   ) )   q   / − R ) α q w = ; 1 m m q ) ) ( r r q ; 2 ; 2 q − . One may encode the charges of the B  ; Z ) / 2 ; B tτ 1 1 − − 1 2 ± L α − B ( tτq tτq z ( m − 1 az q 2 2 − (   tτq − L v ( q q / / . As claimed in the previous discussions, Y   ) 1 1   = 1 1 = = zq 1 ) ± ± m R R = − − = tτ q α  − q q ) L ( ( = 1 q R X 2 2 L m 1 2  B  / / z  tτ R ) m 1 1 − 2 2 B tτq ( 2 B ) ˜ tτ n m z m – 18 – ± ± and / L L t  α − z ) ( ) − 1 ( q q ,  q q  = ± L − ) w ; ; zq R R zq ) ) α = q a L  2   (  α − 1 = ( ( ( ) ( n tτq 2 2 B B Y ˜ tτ ) m   − t q − ) ∞ α =0 2 1 ( B 2 q α X + m / ( − 1 tτq  , 1 = ( z 1 − 1 ) πiz dz  z − a − R L 2 , ˜ t = (  q q L , t ) L  L L L  w, m = q α ) 2   L,R B ( ( 2 2 B 1 , Y R 2 =  / / B 1 − − − 1 1 α 2 R B − R z q − ) zq zq   ± I    , v tτq zq 2 tτ B 1 1 Z w   coupled to some 1d chiral multiplets. Indeed, the index can be computed − r r 2 ∈ − R L,R zq q / 1 R q = ( = Y 1 2 =  ± ± S Y α Y ,B X − L = R L ˜ tτ ) L  t   × L × × × B 2  ˜ tτ 2 B 2 B B 2 R t S = = zq zq zq , v  ∪   X R = (( 1 w m S z R Next we consider the case with ∪ = × L SQCDA L I 2 L w Poles of type N are given by the left hand side reducesS to the intersecting indexby of a the JK-residue prescription applied to the contour integral mirror symmetry. More explicitly, the equalit is due to the identity It is well-known that the indices of the two theories are equal, as guaranteed by the 3d These poles obviously satisfy, thanks to the above fugacity relations ( where the fugacities satisfy the relations which satisfy the superpotential constraint The JK-residue prescription picks out four set of poles. The first set are of type old, given by with X, Y, and Z chiral multiplets in JHEP05(2021)155 . α ) Z (4.38) (4.34) (4.35) (4.36) (4.37) (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) I Y . I 1 2 ) , X # I R 2 ( q

m ) L
m q . q ) R , . ( ; L,R q . 1 2 0 0 ; Y m = − 1 q , − L α ˜ tτ 1 2 ( t L ≥ ≥  i − 1 ) = 1 m L R " R − ( − R 2 1 q m , m v ) ) L − =0 2 ) q Y tτq k m / ( tτ α → 1 , since  ) ( m # . 1 z −   2 0 z   = ( = S / , v  1 L − 1 1 2 q L
R L intersection −
∼ × − ; 1 ) −   q (1 1 2 Z q L 2 L − ) )  = 0) ; 2 m 2 / B B −  1 1 R q q S 1
q ) 1 ; , − R − − − z ˜ tτq 2 2 q q t
− → ( q m ; ( (1; R ˜ 2 tτ zq 2 zq ˜ tτ ) , t ) 1 ) t  m  q m 1 ˜ tτ q − L ˜ tτ − ( zτq ; h t tτ ) t ; −  (1 q  w     zq  ( ( − = = ( =  w
(  vortex 2 1 1 q L type NS L   " 1 2 R ) −−−−−→ ; Z − →  q  m  q
z L ( L ; 2 1 2 L B ) B R Y – 19 – ) R ) = v ) q − ) can be reorganized into the following (ignoring q v ( − q − q ; m v ; ; L ; L Y 2 zq v = 0) = m zq B 2 m B (    = Res ( R Z  4.34 R − m − , 1 2  q m m q ) L ) L,R , intersection 1 L Z 2  − Y 1 = / vortex − Z 1 vortex tτ , 1 m α ) 1 ( − Z − Z − 1 R ) tτ z ), = = − ( − 1 ) =0 ,   , z tτq L ∞ z tτ α − ( 1 2 1 ( (  X + L R ] R,L m 2   ) ) ) / m q = ( = m m R X ; 1-loop 1  )( ) semi-vortex q q 1 2 z + ; L Z R 1 Z m ( ) v

 −  ( tτq tτq ) ) L X R v 2 2 B B ( q q v [ ; ; m L = ( = ( q z zq  zq v 1-loop ( ( (   L L,R R Z intersection − 1   = − SQCDA Z 1 2 α → 2 B B = z 1  Q  zq zq       Res U(1) R ∪ = × L SQCDA Collecting the residues from all four type of poles, we have the index of the intersecting R I semi-vortex ∪ Z L XYZ ( I the factor and U(1) Note that although this polea is simple a zero in double the pole fundamental of chiral the contribution 1d on contribution alone, it also leads to poles of type S are given by Finally, theres is one pole of type NS, given by where They combine to produce a simple pole of the full integrand. More precisely, one uses Thanks to the S-duality, the index ( which corresponds a simple pole of the 1d contribution in the contour integral. Similarly, JHEP05(2021)155 . , , 1 1 1 w S B − ) > L a × L (4.43) (4.44) (4.45) (4.46) (4.39) (4.40) (4.41) (4.42) z 2 R n . S L,R by L =1 ˜ Q n a ∪ symmetry , reducing = ,...,n 1 Q 1 Q, S α ( , . , = 1 α × U(1) ) a = 0 2 L L 1 βγ 2 z I S a ˜ tτ t q Q gauge theory with a , µ L XYZ L ) . Here again we notice = ( with I ] = L  Z ) µ α Z µ and n ≡ L − q v [Φ   ; to the other side, we have U( 1 2 also reduces the contribution µ and , = 0 m µ q γ 1 m γ ( I µ L 1 2 ] a I /
adj 2 Q, , m of a µ ) − , q I B ,Y 1 − β L  µ =0 µ [ a L − Y q µ adj 2 µ − n ; and X w Φ v P ˜ tτ   L L 2 XYZ t ˜ µ q Q / I   β  1 1 µ I µ -SQCDA 1 − − α γ ] = ) = β 1 ) − L 2 L I m 1 n − w ( q, m =0 − L  L X µ L q U( =2 ,...,n n n v ≡ , compatible with an XYZ-type superpotential Y 2 have fugacities satisfying I µ L ] = ] = =2 + 2 = ˜ tτ q – 20 – µ µ µ -SQCDA t adj adj − ) models living on the intersecting I β Y I L =0 = L [ [ adj [Φ µ adj n µ   n matter = ( . The denominator Φ topological symmetry on both side, namely, integrate } Z I m )   L µ µ U( Y α µ L m I γ ] β Y , m µ n XYZ { q I [˜ ( − tr L βγ µ 1 U(1) I  − m X L q =2 SU , m L I n and µ ] X µ n 1 q L gauge field that weakly gauge the topological − Y [ − v  =0 L =2 w, m 1 µ L m Y m µ n µ 2 , which simultaneous force n µ − − / , m } ] w v  1 X Z µ U(1) µ 2 L 1 2 I α − B ) can be further refined by including a background Chern-Simons term β γ . Rearranging the factors of m  ) q [   { ˜ tτ 1 1 1 2 t Z ) − − m =  − 4.44 µ L q   2 n − 2 1 v ˜ tτ ] = −  t ˜ tτ L XYZ SQCDA. As a result, the integrand on the left contains a term 2 L 2 t b I n −  ) ( ˜ tτ L = ( gauge symmetry to collectively denotes the contributions from L t Y w n  ) µ α X L  and sum over ,...,n . On one side, we obtain the index X U( n m ] = 1 XYZ =0 w µ ] = I − The equality ( Next we consider the non-abelian but non-intersecting case of the duality with µ U( = 0 captured by the fraction in front. The fugacities are defined naturally for L =0 Z γ X [ [ [ n µ 1 R m m m at level 1 coupled toin a the over the coming from the superpotential constraint the fugacity relations P and n pair of fundamental/anti-fundamental, and oneone adjoint has chiral multiplets. On the other side, S Here we recognize the index of two where we denotes the fundamental and anti-fundamental chiral multiplets by where we have the free chiral indices where the free fields with additional contributions from 1d free chiral and free Fermi multiplets on the intersection At this point, one can gauge the JHEP05(2021)155 ) . b L to  1) n 1 ( )] − k, -type (4.53) (4.49) (4.50) (4.51) (4.52) (4.47) (4.48) ( theory N SU L ) fug U( L , n 3 XYZ → ( S [ SU T fug . As mentioned ) 1  0) q gauge theory with at (complex) level ; . This is equal to × components of the a ) > ) ]. This construction z 1 R  ]. The M R N 1 2 n 63 − Θ( on Lens spaces  84 , n , ) U( with the moduli space of 1 L U( ] 0 =1 N − Y R n 1 61 Y a q [ × flavors with an > S 1 ) ) ↔ 1 L , − L L L . × ) M,G can be engineered by compact- q M q [ n n 1 n Z ] ; ] X ( q C I 2 T , ( R k,σ ν ; q 3 U( b γ b  [ S 1d CS XY 1 2 . /z I /z Z M,G m  a [ ] a ] z z k,σ R ν = T ( R ν ( R , βγ b β U(1)+ 1 β R I [ XYZ 1 Argyres-Douglas theory. The 3d mirror [ I 6= -CS S =1 I S a of the intersecting unitary gauge theories m M defined on G theory is given by ) × adjoint chiral to m ] × = ] 2 N a,b 1 ] Q 2 Z L µ ) )] R − L ν D βγ free -partition function in [ γ D Q L free L [ L I XYZ 3 = Z N X R n a n Z I [ ] – 21 – S 2 m b a ∪ ] M,G = theory coupled to L U( SQCDA m [ = U( 1) L = µ ] πiz dz , I ], ,A on a three manifold -SQCDA )] k, M,G β L T 3 ) 2 µ ) 1 [ ( [ 1 L N β 67 S L T G A q [ [ n m is the IR theory of the 3d reduction of a 4d 1 ( =1 ; N 1 R U( Z U(1) Y , S  a ` T 2 m ∪ 3 ] q × L SQCDA S L ( 2 µ − [ I I +SU( β, γ D ! T 1 X [ βγ =1 N 1 , the Z  Y ` N I 3 m  ] 1 S ≡ = R − R ν , while the partition function of q = β )] [ 1 M N − L ]. m theory of type M 1 q ] U( supersymmetric gauge theories , S  L µ 3 86 × 2) , β S 2 , [ [ − = 2 D T 85 1 m (0 Z  N − Cartan components of the 1 L  n 1 adjoint chiral multiplet. In the end, the equality of indices now reads − -connection on =0 ) R C Y ν L , n ) In the case with Finally we are ready to conclude the general case with G n 1 − =0 L Y µ k, σ n an adjoint chiral, and thea duality collection has of been free studied chiral via multiplets another [ duality from Concretely, one has flat 4.2 3d/3d correspondence ifying from 6d identifies the supersymmetric vacua of coupled to 3d fundamental/anti-fundamental/adjoint chiralchiral multiplets, multiplets and in additional the 1d bifundamental representation of dual of that 4dsuperpotential SQCD and goes flows through intosuperpotential a the [ sequential confinement after adding a monopole above, on one side we have the index SU( coupled with the “flipSQCD fields” which flows (in 4d) to the from the with the partition function of the complex Chern-Simons theory on where we have the contributions from the one dimensional matters given by This duality was studied in the context of A class of 3d ( JHEP05(2021)155 . 2 . a free σ 2 a Z σ =1 (4.57) (4.58) (4.59) (4.54) (4.55) (4.56) . The N a 1 =1 N a P CP π P k 4 π k − , 4 , giving → e ) Chern-Simons. − N 2 with A-branes ,  e q k 1) b )
) 3 z, q σ 1 1 ) − S 2 → 1 q q − ( N ∗ a q ; ; Θ( 1 σ 2 ; Q T n U( b q e B` 1 z b σ ` 1 q ⊕ O σ q − z dz a − − 1) a σ =1 N σ e a Y − I ` ( σ e ( is also the well-known result 2 ( , capturing the contribution − b = = O n b σ z ` 6= 2 N e free Y ) a 2 − 2 x  n Z a π . N b 4 q  , ) 6= (log dσ Z . The geometric transition can be Y n a 1 , the partition function of the refined k 1 1 ∈ q q 1 − N − n ; 1 =1 N − ) e =0 Y ` 2 a B` Y 1 2 ` CP B ∈ n q P q reads ( a x ∞ , q B 1 − ≡ + B x q dσ −∞ =1 N dx 1 Y ) ; ` Z z  =1 N Q ∞ Y 1 = a N – 22 – with + ) − Θ( =0 0 1 Y Z n = ( B B 1 Z q ] top q ;
N 2 1 partition function of the refined Z 2 1 ) 87 top π k q = q 1 3 4 ; q Z 1 !( 2 S ; B Res 1 1 q 2 s . N q q π k 2 !( = N 1 e N 2 −   =0 π Y ` 4 N 2 N = kσ   π k 1 − 2 4 = . From the perspective of the brane web that engineers the pure q e π k R s 4 n and the topological string theory on nσ − 1   3 s ]. The latter agrees with the refined Chern-Simons partition function q S L   dσe = ∗ 87 n = ], T − 2 ∞ refined CS q + k 67 −∞ CS ⊂ ) Z k → N 3 ) 3 2 S S U( encodes the Kahler parameter of the N Q π k refined-CS , and applying the identity [ Z 4 U( k a Q ) x s N 3 Based on the above discussion, it is natural to consider a more general residue of The relation between the refined Chern-Simons and The resulting contour integral picks up the constant term of the integrand, and gives = S U( a refined Z σ Z the form implemented in the partition functionas by the taking result the [ residue at a pole of the open-closed duality between the openlatter topological partition string function on is given by under the identification e For example, at the specialChern-Simons value theory with integer refinement from the boundary degreesof of the freedom, integrand and [ the contour integral picks up the constant term which is expected to be dual to the where warpping This partition function can be rewritten into a contour integral by a change of variable where the Theta function is defined by JHEP05(2021)155 (4.66) (4.63) (4.64) (4.65) (4.60) (4.61) (4.62) .  2 1 z z 2 1 2 . / 1 2 , one extracts its − 2 | q )] 2 2 n . z / +1) | − 1 ) 0 n ` 2 2 m − 1 . + ( q < q q ` m 1 2 ; while leaving the more − | ( . 2 2 − 1 1 1 2 − − 1 z q q z 1) )] | 1 n )+ − . 1 − − = 1 n n − 2 )Θ( − + 1 1   m q 1 R ( 1 2 m m 1 q ( z z n − [( R ; = − 1 1 2 =1 2 2 1 2 0 1 n 1 Y q / ` 1 2 − z = q 1 2 ) L − +1) q 1 − 2 L =1 n 2 n Θ( )] Y ≥ ` 1 / n n + n /q , 1 ) X ) − 1 m in the region 2 m 2 − 1 q m [( q q q ( ( 2 1 2 ; ; 1 1 = 0 . − q 1 ` 1 − 2 =0 , z ∞ q 1 2 1 2 1 2 n q 1 , ( 1   X + ( z − − − 0 m )  +1) )] )] )] 1 n < ) ) R n n n =1 q n – 23 – 2 1 + 0 , n Y X − − − ; with additional 1d chiral multiplets, 1 ` q q 1 2 m m 2 m m m which keeps the sum invariant. As a result, we ; ; 1 ) ) ( ( [( q − 1 2 1 1 ( 1 2 + ) − +( − q q n q 2 2 2 2 0 S q ( ; 1 )( 0 = ≥ ` ) 2 2 1 /q n q 2 q ↔ ≥ , +1) +1) +1) q 1 X ( q n n n n ; m q , × X ; 1 top ( 2 + + + m m 1 L 2   -series q Z =1 m m m q n Y ( q ` ( D [( [( [( 1 2 ≡ ( 1 1 2 1 2 2 2 − = ) q q q 2 Res ∪ q 0 0 0 = ( ) = ) πiz dz < ≥ < top 1 q 2 2 n n n G ( , , , X X X Z S 1 m m m G 1 1 q = ≡ = πiz dz Res × ) 2 2 q SQED(A) ( D I Z ( G = 2 SQCDA In the following we focus on the simplest case Z ) that In the last step,may write we swapped Moreover, we have Let us consider the doubleq sum above. First of all, one can verify (to very high order in constant terms, and obtain a Next we look for thethe corresponding dual gauge to theory be dual the of intersecting this SQED(A) free whose theory, partition and function we is propose given by general situations for futuremultiplets study. on The above residue then takes the form of free chiral D5 and NS five branes in figure (). The result is simply super-Yang-Mills, the residue corresponds to suspending two set of D3 branes between the Treating the integrand as a Laurent series in JHEP05(2021)155 . i m , m T z ) (5.2) (5.1) , one | (4.67) (4.68) (4.69) m 2 − ] = 2 z t theories | m p − Q k , < . of the false . 1+ )(1 | 0 P ) 1 . [  m n, ) z q | , q U(1) 2 1) + − 2  − 2 q m (1 n ( δ + n − , 0 with commutation
c , 0 1  1 m + } n, ( m> , − − + Q 1 2 mn . The algebra admits a ( p P b , m 1 top h , m δ + 2 2 ± − 1 P , Z − σ ;  1) 1 2 z 2 2 π in the region m − g σ m 2 k 4 exp p m m 1 2 1 2 a ( − m/  -Viraosro algebra to construct { m T 2 = 1
− ] and then generalize to theories , z q a σ 1 − 1 = Res p k m q 1 π z 70 − z k 4 1 n 2 t k + P y σ q − f 2 2 1 m e − p  ≡ − x 1 ∞  1 1 =0 m/ n q ) ] using − + k p z + ) πi 1 ( − 73 m q P superconformal index of intersecting gauge -partition function of two , 1   T 3 1) 2 − ) S 72 2 − sinh ), and show that ( q = 2 m/ , (1 t ; ± – 24 – 1   − 70 N q Q 0 − 4.51 , 2 < 1 2 )( ). n 1 , X q 46 ) = dσ q m m/ 1 k − 1 ; 1 − 2 + + t q is generated by a set of infinitely many generators dσ m 0 ( ≥   T ∞ n q,t 2 , k = X + m ], − V −∞ 2 m/ n Z   in the stress tensor 88 − T q ∝ ≡ m − ) − T SQCDA k is defined by the relation top . Then we argue the uniqueness of the algebraic construction. 2 Z + 1 Z k n m/ S f q T x, y, q ×  k ( − 2 R 1 and m = Res m S a,b,c 1 2 − g T ∪ − ( -Virasoro algebra 1 k qt q f S ] = 0 n ≡ ≥ × a SQCDA -Virasoro construction X p k , 2 L Z The To summarize, the partition function evaluates to q m S a [ free field realization inrelations terms of the Heisenberg operators Here One can also pack the and study the algebraic propertiestheories. of We the will begin byon reviewing the construction in [ satisfying the commutation relations somewhat trivial rational function 5 In this section we follow the idea of [ coupled through some 1d bifundamentalinterpretation chiral of multiplets. the However, expression thenon-abelian and precise cases the physical remains generalization to ofbe be the interesting explored, above to computation which understand we tointerpretation leave if the (although to the in future appearance the work. of case Also considered the it above false-theta the would function false-theta have function physical evaluates to a can retrace the steps which proves the ( reproducing the free theorytreating index the coming integrand from of the the refined contour geometric integral transition. as Finally, series in Such double sum turns out to coincide with the special value The expression on the right looks like the Theta function defined in [ JHEP05(2021)155 ] i ˜ t β − to . , 70 2 i t : 0 n/ (5.7) (5.8) (5.3) (5.4) (5.5) (5.6) since t < 2 ( / 1 a 1 n 1 − , p t ) 2 P depending by P β ↔ 6  n in certain way, β ) ˜ z a √ q ˜ x q ˜ z q p (   , ˜ z, → 2 : , and ˜ z n − O z, P sharing the same m/ τ ( 1 )  − m f 2 1 − x p − − : β and 1   n/ ˜ t z − − ˜ + √ q, P Q t q m qz β − 2 V β  a z − will also affect , √ ) p 2 m/ → O 1   z z n p −  n/ q z   − Q q n i 1 1 x q,t ( − ˜ t 2 6=0 Q − V X β m (1 by β m/   n p m n − )] = p ˜ q − − af x =1 + a ˜ n z i ( − X − is actually invariant under 2 − m : exp 2 − 2 a S ≡ ) m/ q,t , m/ n m/ m ) t − m V q m ˜ q z q − n − unitary gauge theories coupled to one adjoint ( − T − − i x x [ t − 6=0 2 – 25 – Y m m (1 X 2 m : to denote the normal ordering pushing a n a = 2 m/ , m/ − f − ... )) q t =1 n -Virasoro algebras N i 2 X , : x q ( 1 + 6=0 6=0 m/ − X X n m m m − τ  q and − z − O − − , 7 z 2 ) is. → ) β     − / m 1 q n 2 τ a qx p 5.1 (  = m/ )] = 0 O q : exp : exp t ( Y ’s from two x ≡ ≡ ( + 1 x 6=0 + ) ) ± S . After taking normal ordered product of the screening currents, one may shift [ X via are not independent, and therefore the shift in m x x satisfies certain periodic condition under  S n ( ( τ z ˜ z ) β − ∂ 2 + − dx ) )] = /   P 2 S S 1 x β partition functions of ( I and − n/ ], two , + p 1 − √ z exp p  m S 73 S ˜ z, , , + ≡ T Y [ × m 2 z, ) 70 2 ( ˜ z T n/ [ ⇒ f S p ) = z, )( z In [ The algebra contains special operators refereed to the screening currents given by ( 2 Note that To do this, one first represents the Heisenberg operators ( or S 6 7 n/ T 3 − to generate additional factorsencode associated the to flavor fundamental fugacities. andchiral anti-fundamental Alternatively contributions. chiral one multiplets, may where consider additional vertex operator insertions to include construction are implemented. depending on the specific modular double construction. t on the geometry. TheS (sum of) normal orderedchiral products multiplet, of and modular fundamental/anti-fundamental doubles chiral then multiplets produce if further shifts in the for an appropriate contour. are fused into a modular double screening current, the commutation relation ( the left, and, by defining where which satisfy An immediate observation is that the algebra JHEP05(2021)155 , , ]. , ˜ S Q 9   R q from (5.9) ; (5.15) (5.12) (5.13) (5.14) (5.10) (5.11) and ,B j ± B R  ˜ S S 2 − , q . . In other ,z a ; P B b + R β B S 2 with − 1+ √ ,B a q P B L 1 B ± β z − 1d chiral − S t  √ tq t Z 1 a    B : z q − a 1 Q z R    b − β b ) 1 z b z Q √ 1 1d chiral β − t ,B − Z     b p =   q m q z ˜ t ) , ,  i ; by merging ( ; + − 1 ( b b b P ) −  which merge into − B m 2 ± B B β m q z 2 S B 2 − ) 2 − ˜ − t 2 ( S − = a √ a − 2  ) a 2 ˜ q m R B − t i ) B =1 B B 2 ( n adj 1 B Y q b − V 2 2 q m B ˜ 1 q − q Z 1 tq L 1 q z − 1+ ) − , except that they now share the 1 ) − ˜ t i − a 1 − q  ( z a ˜ − S z b 2 z 1 ) m − ( and b m z  ˜ z − z q z + ˜ a a ,B t ( ( m z ˜ b S a q,t P ) ˜ a − z z i and  6=0 β V 1 ( ( VM 1 X ) m − a + 6=0 √ Z S ) z and X S − m – 26 – q, t 2    B ) m L L,R implicit) − ) b 2 =1 Y B b n − Y = B a zq ) i m 2 m B  2 i −   − t ( zq 2 a 2 m − B : (  a B − = ( − − R 2 + B B = : q q 2 ) m ( zt 1 P S a − − − zq R β  − b t ) ) 2  m z , t b m √ ,B q a 2 R a m + a B / z ) a z S 1 ,B q a ( b 6=0 -Virasoro algebras − − z − 6=0 X q q m ( ,B 1 ( X S m factor in a −  − z R S ( − =1 b    Y a

P, Q  n/ n/ 2 < B 2 + 3 V B 2 − 2 2 t | . q t q q 1 1 but not | 1 − by − − − − n 2 z z 2 1 3    1 a n/ n/ 2 > − n n commute with − 3 q , 2 3 t q | 3 ) ; a a q m ,t k | and 1 2 3 q − n zq T 2 X X V zq a – 28 – −  Q − −

, and (1 2 n n P , thus reproducing the construction discussed above. 2 n − − m ∞ ) = n/ =0 1 1 n/ Y q   + k + 2 2 a − 1 ; T t 2 2 B B ,t 1 1 q z 2 ≡ ( q − ) zq zq − 2 V q with   2 ; n/ n n z n

n/ 1

2 1 (

− 2 3 Q q , t

a a a

and also identify the zero modes P t 1 -Virasoro algebras would fail the commutativity requirement, unless the X X ,t q 1 = q − − 1           V t = 3 , since it commutes only with t , 23 ) = exp ) = exp q S . Gluing three = z, B z, B 1 ( ( -Pochhammer symbol is defined by the (regularized) infinite q q 23 12 S S = 3 One can extend the definition to regions with A Special functions suggestions. This work isChina supported (NSFC) in under part Grant by Nos.Research the 11905301 National Funds (YP) Natural for and Science the 11875327Foundation. Foundation (HHZ), of Central the Universities, Fundamental and the Sun Yat-Sen University Science mute with make is to further identify Acknowledgments It is straightforward to observe that Figure 4 third algebra is identified with the first, as shown on theq right. construct modular doubles The The authors would like to thank Yongchao Lü, Fabrizio Nieri and Wolfger Peelaers for JHEP05(2021)155 (B.3) (B.1) (B.2) (A.7) (A.8) (A.3) (A.4) (A.5) (A.6) , ,  N ] ˜ µ . ∈ denotes the 92 . –  ∨ ` ,  z, µ, 89 − Y 1 ; m, n q  − k ~ Y k . −  − , q p n 1 ) z zq − , and n q HM p 2 , . ; z -deformation paramters r − captues the contribution − ; Z q ) 1 1 Y Ω 1 ( 1 p  , −   ;  CS z < 2 ` q ` =1 ∞ ; || =0 | 1 q r Z q n X =1 + ∨ n n q =1 k ~ A − Y Y z ` | Y with k p Y P ( ,  | || = z . zp 1 1 =1 1 2 m = p ) Y  | ) ≡ − k S =0 − r − VM ) q n `  2 p 2 ; q 1 1 = × Z p ) 2 Q ; z ||  ; 4 || ) q ( −  ) n ` 1 1 ; Y A q , R − − z − Y − || ; , z =0 (1 ( q || ` n k  r zq 2 z k ` p 5d CS ( Y Q q  -tuples of Young diagrams [ p, q z − , k 1 k 1 ; ( q ; n 1 =1 | N n − 1 =1 X r =0 z ~ Y zp Y A ` − − k m Y =0 – 29 –  |  − 1) , m k − zq Q q − ( 2 ) 5d CS ; 1 n CS Q = k n k ( . It can be shown that k  ) n Z 2 − = A − | q Y zq = q z || p ; ~ Y ) =0 ∞ n | ∨ z ) q − Y + A ( q  , q region, one uses Q k,` 1) Y Y 1 ], , 1 ) || (1 p − − q ≡ ~ =1 = Y m ( ; 91 1 Y , p X k ) 1 p, q n − ; p =0 q ∞ n k 1 ; , we use the symbol CS =0 ≡ = -Pochhammer symbol is defined by X z + n − m Z p, q Y q Q ( )  p ; q q zp z z , , ( = ( ( ) = p p ) q ) is given by a sum over ; ; q ; q ) n ˜ µ 2 , ; z q ) = − p ( ; n q ; πi z 2 z ( m zq p, q ( z, µ, e ( ; − ; z p ≡ ( z 5d CS ( -Pochhammer symbols enjoy well-known shift properties. For SQCD instanton partition function on q -Pochhammer symbol has simple and useful series expansions given by q , ) denotes the exponentiated Yang-Mills coupling, and q 1 Q, k N  Q πi 2 U( Both Similarly, the double The e inst ≡ Z For any Young diagram transposition of the Young diagram from the Chern-Simons term [ p B Instanton partition function and also To extend beyond the above where The JHEP05(2021)155 . is = 1 − into . z |  (B.7) (B.8) (B.9) (B.4) (B.5) (B.6) 2 ~ ) Y ~ (B.11) (B.10) Y | Ar Q , R Y ( ~n L Bs Ar =1 ~n VF-extra sinh Y sinh Y . , r ,   Z q p ) 2 2 P  2   ) = = , each summand s 2 5d CS R Bs Bs 2 1 k L R Y Y m − + − , A R 1 − − L n Ar ) ) , we can encode Y r m r r − ( q 1 =1 − − =1 + 2 1 , q , q − r s s N A 1 1 Ar  I − ( (  } sinh Y ˆ − − 1 1 µ P A L ) .   R p q is a tuple of only large Young n 2 ) − 5d CS  ,~n − − − 2 k = = ~ + 1) L Y p ) A ~n R + intersection 2 1 A k L z R R AB AB ~n (ˆ r Z µ + ˆ ˆ L z z 1 + , n 1 1 ~n A R R afund-extra  ) n = − 2 − 2 x Z ( πiβ   ( + q  ,q + 1 A   2  I z ) R m , v , v L A ) µ ( πiβ are factors in the vortex partition function ∨ 2 2 Ar Bs n As / / sinh Y sinh Y ( ,q Y − 2 sinh 1 1 ) ) ( { q p 2 2 m A   1 1 ( Ar =1 =1 z s Y Y s − i − i (ˆ 2 sinh – 30 – Bs Bs µ µ 1 P ∞ ∞ Y Y =1 =1 − Y Y r r =0 + + = ˜ = ˜ Y 2 − − k 5d CS m k R L R =1 =1 N N ~n vortex-fund-adj ) Y Y ) − I I ≡ R vortex-fund-adj . The latter requires a bit of straightforward compu- τ τ Z +1) +1) ∨ i ~n vortex-afund i As Z R A ˜ L t r r ˜ t =1 =1 , and we define the (regularized) infinite product Y N N ) Z n ( ( Y Y A A sinh m B − − L L A ) ˆ ,q =1 z ) n s s s x = and ( ( m ( ,q 1 1 − ( ) = factorizes. In particular, when =1 P    I N A m A , , ( − − ~ 5d Y CS z 2 2 P k / / , the Chern-Simons factor reads ) z, µ 1 1 = ˆ z Q 1 2 AB AB ; | , and we have − − z z + R A ~ } q (ˆ (ˆ p Y L A ( 1 m 1 AB | n vortex-afund ˆ z − − ( =1 L,R Aµ L L p q Z ∞ , p =1 Y ) that we review in the next appendix. The parameters in the 5d and the i i | ˆ ~n ~n m z vortex-afund vortex-fund-adj r,s N A A µ µ are specified to the special values { Z Z Y factor and the Chern-Simons term also happily factorize when (a)fund | P =1 πiβ z A = = 2 N C.11 → → Q z 5d CS Y e | L R k Q   A,B ) L A ) A − ≡ τ m τ ~ ~ Y Y | i = µ i t z   t ( ( =1 =1 N The When The vector multiplet contribution is given by N A Y vect VF A z Z P afund Z tation. At the special specialized.Q It is easy to see that the former indeed factorizes into Here the factors summand ( the sequences corresponding to a tuple diagrams with respect to the forbidden boxes however, it is crucial tobetween the keep 5d it and general 3d in theories order related to by completely Higgsing. fix the fugacities relation 3d theories are identified by The fundamental and anti-fundamental hypermultiplets contribute where Although most of the discussions in the main text focus on vanishing Chern-Simons level, JHEP05(2021)155 ) , for | b (C.1) . B (B.12) (B.13) (B.14) R Aµ 2 − m a R Aν B m | + q 5d CS . In the end, gauge theory 1 L A k R A − ) b n − z n .  antifundamental a = z   R Aν af ν q q m  , − n − ; ; ], while towards the 1 2 q , . . . , n  ) | | R A 9 ; p a a s q = 2 U( | (1 n  ν ; B B 2 | a 2 2 ) | | q | 2 b = 1 − B ; ∨ a  | , A B N 2 − q 2+ 2+ | q B s 4 b q − ) | 1 Y q q ; ) , a ( B r | q − 1 1 (1 τ B b − 2 ) L A , q | i ∨ s − − 2 a B 2 τ 1 t ) ) ) − i 2 B / − ( − | τ τ ˜ t R 1 q a 1 A i i − ( b ˜ t t B − 2+ − Y a a (1 | ( ( p L A ( 6= z z q 1 a Y a , . . . , n L 1 A Ar R A z   vq − a =1 a b − n µ Y fundamental and X z s  z B v  ). f | = 1 )  1 1 L A a − 1 =1 n | 2 n − a − r X w b r B , . . . , n a − | 2 z =0 z = B R A Y − | ν a − − B.14  ( n 2 z for − a = 0 )) || =  ) B τ )) L Aµ ∨ µ s i | 2 ) τ m t b CS − i || ∨ – 31 – k ]. B ˜ t R A )( ) 5d CS ∨ 2 − R A − 1 n k ) )( R ( a ) 44 n A , for L − a a − a A B R A L A Y | z z  z Y n L Aµ m − ( (( − − − L Aµ ( ( 5d  CS || || m ( = k 1 1 m v 2 2 1 2 − − 1 − n + =1 = = 2 Y R q a − q q As ! 2 s 2 R A ( ( µ q ) ) a Y n −  a +1 ∨ coupled to an adjoint, , n n =1 =1 2 A p ) Ar Y Y R A a a 1 r Y = πiz dz n Y =1 ( n f CS − 2 − ( af ) Y − n =1 =1 p k n µ L A Y Y a,b i i q +1 2 +1 n − n =1 / ( L A ∞ Y L A × × × ∞ a 1 L A +1 n L A 2 X + n n X + − L A = ( = m n q r =1 r A | A − a , and Y = z p µ 1 |  A I L Ar − 1 µ n Y Z −

R A =0 ∈ L A Y µ X ~ n B =1 N , between flavor fugacities, and therefore the identification of the vortex partition ! Y A 1 × 1 n − j t = , . . . , n i = The index can be computed in terms of a contour integral t I = 0 chiral multiplets. Part of theend following the computation result follows is that reorganized in following [ [ C Factorization of 3dHere index we collect some detail on the factorization of the index of a 3d Let us pause and comment on the role of the Chern-Simonse.g. factor in parameter identification. function at vanishing Chern-Simons levelratios only and 3d fixes fugacity the ratios. relations The between Chern-Simons the term 5d however provide fugacity additional constraints the Chern-Simons factor factorizes ν One can rewrite with Chern-Simons level which lead to the final complete identification ( where again Without the Chern-Simons factor, the vortex partition function depends only on the ratios, JHEP05(2021)155 i of − } z f (C.6) (C.2) (C.3) (C.4) (C.5) → . Using 0 + . ] z , while the ≥ , . . . , n 2 h / 1   m , 2 , − = 1 }
/ ) ~n 1-loop  1 m q  ,  B q , i ; Z q − i q ; q ; − iµ ; n ! µ q µ m , v { 1 − zq q + + + , b a  j 1 ( j  −  µ z z ) q n n − q q ) v − v ;

B ;  q − ); τv 2  1 1 2 q q ; i v / + τ a  1 t − ij − − b i 1 1 ± B {z ij t q t ) − 1 1 2 t 2 − /z (  ) ~n = /z fund+adj q / ) B ); ± + 1 b 1 τ 1 Z − τ a τ − 1 i z i i − − iµ ˜ t z ! 2 t − ˜ t xq  =0 (  B i x ( zq ( ( Y + + µ b a ( n 1  +  a − + z z a − a ][( ) z − q z b z f 2 )  =1

B . Explicitly, they are given by  / B n  iµ − Y . a 1 1    2 − z i,j | |∓ z a − ( } B iµ a − )  | B 2 ≡  ( B Y i aν ) n µ q µ,ν 1 + µ +1+ iµ 2 x  i ) ˆ − − 1 )) m f ( − ˆ t q ν b k =1 n x ν f ( − i Y − v j v m ) −  1 − 1 Also, ν µ 2 − ij j,n +1)ˆ − − ij jν − πiβ / t v 8 t a µ 1 k )) m 2 m v ( x . Some more massaging turns b − m − τ ˆ − x ij 1 ( µ j )( − − 1 t 1 ˜ t 2 q  ( i iµ  v / − iµ 2sinh f t 1 πiβ ) ˆ 1 ) a m 2 iµ 1 2 1 j − m =1 ) )+ ˆ − e µ  ν m k x b − n − ( − jν =0 µ =0 2 q − +( iµ ν Y − Q µ = ) − k m k ( v v m 2 − − , while the second factor will be the one that is µ +ˆ jν − )ˆ x / ) ˆ 1 Q  a ( n 1 v m ν µ 1 − ab 1 µ x  =0   i − − ˆ t ( Y ( ) − ij ) − µ − =0 n ( − q t i ) ; ν Y iµ ij − ( µ j q µ n f 1 ˆ t ; j,ν m Y f q ( n n =1 +ˆ − ( f 1 ( Y = i 1 − z ) πiβ 1 4 n =1 ( +( Y − i ab )  =0 – 33 – af j ˆ +( − t 1 i,µ Y =1 ij ν Y ) n n ( ( Y q af j ˆ t − ij iµ =1 1 ) n Res Y ˆ x  t j k sinh +1 m πiβ iµ − k =0  q i µ jν ) − Y − m µ πiβ − . The former reads − 2 iµ n µ 4 − m )  w q m j ν 1 v πiβ m − n v µ 2 w )(1 − − 1 ≥ iµ ( τ − x q i − sinh 2sinh − ij j πiβ m v ( t n ˜ t ( 1 | Y iµ − ( 1 i iµ j − 2 1 2 m − t f iµ − ij m 6= 2sinh (1  i 1 t m iµ 1  =0 − 2 ) CS − Y =0 m k a µ i n − Y Y × ) ; =0 f µ n i 1 n f f Y =1 =1 µ f 1 n n n m − " Y Y =0 i,µ n =1 − i ( (0) Y =0 i ( i,j i,j Y f i µ f n Y µ f n =1 n n Y i,j i × × × × Y The remaining factors go into the vortex partition function which factorizes into those f To obtain this factor, it is convenient to apply the following identity with = f n =1 =1 Y 8 i = × n ~n Y vortex i,j = Here the first factor isgrouped related into to the the Higgs-branch-localized residues note one-loop. that The remaining factors will be sent into the vortex piece; this expression into a more recognizable form Z second factor comes from the fundamental and adjoint chirals. containing where we define the hatted variables uniformly by where the first factor obviously comes from the anti-fundamental chiral one-loop, and the JHEP05(2021)155 , , and 12 . 2  JHEP JHEP 04 (C.12) (C.13) and R ,  , 1 2 × − (2012) 120 JHEP 2 q (2015) 155 ]. , ; S 2 1 04 JHEP ]. ]. T − 11 , . × , τ
SPIRE 1 2 q of the full vortex − IN D ; SPIRE JHEP SPIRE ) , JHEP ][ IN , v Supersymmetric Gauge IN q ]. supersymmetric theories 1 , ; ][ ][ − ˜ t ˜ t, v, τ ]. , = 2 = 2 1 t, , elliptic genera and dualities ; SPIRE − D ˜ Gauge Theories on S t, v, τ 2 gauge theories on t N IN T ; t, theories on , w SPIRE 2) 1 ; ]. ][ , × − IN = 1 CS I , w k = 1 ][ , w ; N = (2 CS m N SPIRE CS k ; N k IN arXiv:2007.07664 ].  [ arXiv:1206.2356 m ][ arXiv:1812.11188 ( [ [ ~n vortex Exact Results in Z SPIRE ~n vortex ~n vortex ]. ]. IN Z m Z arXiv:1409.6713
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(2015) 1483 Theories on the Squashed 3-Sphere Notes on SUSY Gauge Theories on Three-Sphere (2020) 115168 SUSY Gauge Theories on Squashed Three-Spheres q Localization of 4d IN IN ; SPIRE SPIRE ˜ t, v, τ ][ ][ Partition functions of (2020) 2050207 IN IN = 2 334 arXiv:1206.2606 960 t, ][ ][ [ ; Supersymmetric indices on ]. N ), which permits any use, distribution and reproduction in Localization of three-dimensional 35 ˜ t, v, τ Partition Functions of ]. (2012) 71 , w t, Factorisation and holomorphic blocks in 4d (2020) 113B02 ; CS Factorization of the 3d superconformal index with an adjoint matter SPIRE k 313 , w SPIRE arXiv:1506.03951 IN [ IN CS 2020 (2013) 093 ][ k ][ CC-BY 4.0 Nucl. Phys. 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