PHYSICAL REVIEW D 98, 126002 (2018)

3D mirror symmetry from S-duality

† ‡ Fabrizio Nieri,1,* Yiwen Pan,2, and Maxim Zabzine1, 1Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 2School of Physics, Sun Yat-Sen University, 510275 Guangzhou, Guangdong, China

(Received 27 September 2018; published 5 December 2018)

We consider type IIB SLð2; ZÞ symmetry to relate the partition functions of different 5d supersymmetric Abelian linear quiver Yang-Mills theories in the Ω-background and squashed S5 background. By Higgsing S-dual theories, we extract new and old 3d mirror pairs. Generically, the Higgsing procedure yields 3d defects on intersecting spaces, and we derive new hyperbolic integral identities expressing the equivalence of the squashed S3 partition functions with additional degrees of freedom on the S1 intersection.

DOI: 10.1103/PhysRevD.98.126002

I. INTRODUCTION partition function [23,24]. In fact, over the past few years, the results of supersymmetric localization (see, e.g., [25] One of the most beautiful features in the family of 3d for a review) have been systematically exploited to predict gauge theories with N ¼ 4 supersymmetry is the existence and test dual pairs. of mirror symmetry [1]. When 3d supersymmetric gauge In this paper, we continue the study of 3d dualities theories admit brane constructions through D3 branes inherited from the SLð2; ZÞ symmetry of type IIB string suspended between ðp; qÞ branes [2–6], mirror symmetry theory. Our strategy is to consider first 5d N ¼ 1 SYM can be understood from the SLð2; ZÞ symmetry of type IIB theories with unitary gauge groups engineered by ðp; qÞ- string theory. From the QFT perspective, mirror symmetry webs in type IIB string theory in which the SLð2; ZÞ action is deeply related to S-duality of the boundary conditions in can be manifestly realized, for instance, through the 4d N 4 supersymmetric Yang-Mills theory (SYM) [7], ¼ exchange of D5 and NS5 branes (a.k.a. the fiber-base or and for Abelian theories it can also be traced back to the S-duality [26,27]). Second, we engineer codimension 2 existence of a natural SL 2; Z action on path integrals ð Þ defects of the parent 5d theories by the Higgsing procedure (functional Fourier transform) [8,9]. For non-Abelian [28,29], and in simple configurations we can identify theories, this action can be implemented at the level of candidate 3d mirror pairs (this is the perspective also localized partition functions [10,11]. Moreover, the class of – N 4 adopted in [30 32]). In order to be able to explicitly test 3d ¼ theories can be deformed in many interesting their IR equivalence through the exact evaluation and N 2 ways to ¼ , such as the inclusion of masses, Fayet- comparison of the partition functions, we focus on 5d Iliopoulos (FI) parameters for Abelian factors in the gauge Abelian linear quivers in which the instanton corrections group, or superpotential terms. While the reduced super- can be easily resummed [33]. In fact, the fiber-base dual symmetry implies a weaker control over the dynamics, picture of such theories provides a very simple duality mirror-like dualities are known to exist for a long time frame for the resulting 3d theories, which look free. Our – [12 14]. Lately, this has been a very active research field, reference example is 5d SQED with one fundamental and and significant progress is made possible thanks to the one anti-fundamental flavors and its fiber-base dual. From – careful analysis of (monopole) superpotentials [15 22].In this very simple example, we can already extract nontrivial many cases, the IR equivalence of proposed dual pairs has dualities for 3d non-Abelian theories. One of our main been tested using the exact evaluation of supersymmetric results is indeed a non-Abelian version of the basic SQED/ S3 observables through localization, such as the (squashed) XYZ duality. Remarkably, this duality has implicitly appeared in [34] (at the level of the squashed S3 partition *[email protected] function) as an intermediate step to test the mirror dual of † [email protected] ðA1;A2n−1Þ Argyres-Douglas (AD) theories reduced to 3d, ‡ [email protected] which has been shown to follow from an involved cascade of sequential confinement and mirror symmetry [20,21] Published by the American Physical Society under the terms of starting from the 3d reduction of the 4d “Lagrangian” the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to description [35,37]. Here, we provide a first principle the author(s) and the published article’s title, journal citation, derivation of this crucial bridge from the 5d physics and DOI. Funded by SCOAP3. viewpoint.

2470-0010=2018=98(12)=126002(21) 126002-1 Published by the American Physical Society FABRIZIO NIERI, YIWEN PAN, and MAXIM ZABZINE PHYS. REV. D 98, 126002 (2018)

Another motivation for this paper comes from the recent research. In Appendix A, we collect the definitions of the studies of supersymmetric gauge theories on intersecting special functions which we use throughout the paper. In spaces [38–44]. In our case, we are interested in pairs of 3d Appendix B and C, we present few technical definitions theories supported on two codimension 2 orthogonal and derivations. In Appendix D, we collect useful infor- spaces in the ambient 5d space (which we take to be either mation and notation of the refined topological vertex. Ω C2 S1 S5 – the -background q t−1 × or the squashed [45 55]), ; 1 interacting along a common codimension 4 locus (S ) II. 5D INSTANTON PARTITION FUNCTIONS where additional degrees of freedom live. A natural question is whether 3d mirror symmetry survives in these In this section, we review the instanton partition func- more complicated configurations. Remarkably, we are able tions of 5d Abelian linear quiver theories with unitary Ω to generalize known dualities to this more refined setup too gauge groups in the -background, usually denoted by C2 S1 by studying the relevant compact and noncompact space q;t−1 × . The geometric engineering of these theories partition functions using the integral identities descending through ðp; qÞ-webs in type IIB string theory or M-theory from the fiber-base duality, up to some subtleties inherent on toric Calabi-Yau 3-folds [4–6,56–58] allows us to to intersecting theories [36]. perform the various computations using the topological The rest of the paper is organized as follows. In Sec. II, vertex formalism [59–62]. In this paper, we mainly follow we review instanton partition functions of 5d Abelian linear the conventions of [63], summarized in Appendix D.Ina C2 S1 quiver theories on q;t−1 × through the refined topo- nutshell, in any toric diagram there is a frame in which one logical vertex, exploiting their ðp; qÞ-web realization in associates internal white arrows which point in the same type IIB string theory or M-theory on toric Calabi-Yau (preferred/instanton) direction and correspond to unitary 3-folds. In particular, the slicing invariance of the refined gauge groups, with the ranks determined by their number in topological vertex implies the equivalence of supersym- each segment (one in this paper); consecutive gauge groups metric partition functions of different looking field theories are coupled through bi-fundamental hypers, while non- (duality frames) associated to the same string geometry. compact white arrows correspond to (anti)fundamental Two of the duality frames are exactly related by S-duality in hypers. type IIB, but we also discuss another one. In Sec. III, Our reference examples are the diagrams listed in Fig. 1. we extract candidate 3d mirror pairs by following the By explicit computation, it is easy to verify that the Higgsings of the parent 5d theories across different duality associated topological amplitudes correspond respectively frames, and compare the resulting partition functions. For to the instanton partition functions of: (i) the U(1) theory special Higgsings, the 3d theories live on a single compo- with one fundamental and one antifundamental hypers nent codimension 2 subspace in the 5d ambient space, in (SQED); (ii) the theory of four free hypers and “resummed which case we reproduce known results and propose a new instantons,” which will be simply referred to as the “free mirror pair which is a non-Abelian version of the basic theory”; (iii) the Uð1Þ ×Uð1Þ theory with one bifunda- SQED/XYZ. However, we show that generic Higgsings mental hyper. Note however that, although the case (i) and produce 3d/1d coupled theories which live on distinct (iii) can be precisely identified with conventional 5d gauge codimension 2 subspaces mutually intersecting along theory partition functions, the case (ii) involves contribu- codimension 4 loci, and we generalize and test the dualities tions from nonconventional matter. in these cases too. In Sec. IV, we discuss further our results The first diagram, corresponding to the U(1) theory, has and outline possible applications and extensions for future amplitude

FIG. 1. From left to right, the diagrams correspond to 5d U(1) gauge theories with 2 hypers, a free theory, and Uð1Þ2 quiver theory with one bifundamental hyper multiplet.

126002-2 3D MIRROR SYMMETRY FROM S-DUALITY PHYS. REV. D 98, 126002 (2018) Y2 1 X p1=2; q t−1 p1=2; q t−1 −1=2 jλj N∅λðQ1 ; ÞNλ∅ðQ2 ; Þ Z1 ≡ ðp Q0Þ ; ð2:1Þ p1=2; q t−1 1; q t−1 i¼1 ðQi ; Þ∞ λ Nλλð ; Þ where p ≡ qt−1. The prefactor in front of the instanton sum can be identified with the perturbative or 1-loop contribution from the fundamental hypermultiplets. We refer to Appendix A for the definition of q-Pochhammer symbols and Nekrasov’s function. The second diagram, if read naively corresponding to the four free hypermultiplets, has amplitude given by

ZResummedðQ0;Q1;Q2Þ Z2 ≡ Q : ð2:2Þ 2 p1=2; q t−1 p1=2; q t−1 p1=2; q t−1 ½ i¼1ðQi ; Þ∞ðQ0 ; Þ∞ðQ0Q1Q2 ; Þ∞ We note that the factors in the bracket captures the contribution from four free hypermultiplets, however, the resummation of instantons has also produced a factor ≡ ; q t−1 p; q t−1 ZResummedðQ0;Q1;Q2Þ ðQ0Q1 ; Þ∞ðQ0Q2 ; Þ∞; ð2:3Þ in the numerator, which does not have straightforward interpretation in terms of conventional 5d supersymmetric matter content. Here, we simply take the expression (2.2) as a computational result. We refer the readers to [64,65] for more detail on these nonconventional matter, which are termed “non-full spin content.” Finally, the third diagram, corresponding to the Uð1Þ ×Uð1Þ theory, has amplitude X 1=2 −1 1 Nλ λ ðQ0p ; q; t Þ ≡ p−1=2 jλ1j p−1=2 jλ2j 1 2 Z3 1=2 −1 ð Q1Þ ð Q2Þ −1 −1 : ð2:4Þ Q0p ; q; t Nλ λ ð1; q; t ÞNλ λ ð1; q; t Þ ð Þ∞ λ1;λ2 1 1 2 2

The prefactor in front of the instanton sum can be identified triality relation among distinct gauge theories has been with the perturbative contribution of the bi-fundamental recently obtained also in 6d [66,67]. hyper. The above computation can be generalized to more A. Duality frames complicated toric diagrams. For instance, a strip of 2N The three configurations in Fig. 1 share the same toric vertices can be associated to three QFT frames, corre- diagram. In fact, they all give equivalent amplitudes. Let us −1 sponding respectively to: (i) the Uð1ÞN theory coupled start by focusing on the first two diagrams in Fig. 1. They to N − 2 bifundamentals, one fundamental at first node can be understood as two different ðp; qÞ-webs related by and one anti-fundamental at last node; (ii) the theory of S-duality in type IIB string theory, under which D5 and 2N free hypers and “resummed instantons”; (iii) the NS5 branes are exchanged. Upon a clockwise rotation Uð1ÞN theory coupled to N − 1 bifundamentals. A similar by 90 degrees, the S-duality is represented by Figure 2.

FIG. 2. S-duality between the two ðp; qÞ-webs.

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Since D5s correspond to horizontal (1,0) branes, NS5s ðp; qÞ charge vectors. In this particularly simple example, correspond to vertical (0,1) branes and diagonal segments we can explicitly check the invariance of the amplitude. We correspond to (1,1) branes, the duality map is indeed can expand Z1 and Z2 in series of Q0, and both Z1 and represented by the S element in SLð2; ZÞ acting on the Z2 equal

Y2 1 Q0 1=2 Z1 ¼ 1 þ ðqQ2 þ tQ1 − ðqtÞ ð1 þ Q1Q2ÞÞþ p1=2; q t−1 q − 1 t − 1 i¼1 ðQi ; Þ ð Þð Þ Y2 1=2 1=2 1=2 1=2 1 ðq Q2 − t Þðt Q1 − q Þ ¼ 1 − Q0 þ ¼ Z2; ð2:5Þ p1=2; q t−1 1 − q 1 − t i¼1 ðQi ; Þ ð Þð Þ confirming one of the predictions. More generally, ðp; qÞ- B. S5 partition functions webs constructed by gluing vertically N copies of the left In this section, we use the refined topological string/ diagram in Fig. 2 or constructed by gluing horizontally N Nekrasov partition functions in the various duality frames copies of the right diagram are S-dual to each other and to write S5 partition functions related by type IIB SLð2; ZÞ hence give equivalent amplitudes: the former is nothing but transformations. The study of compact space partition 2 N−1 the UðNÞ SQCD, while the latter is a linear Uð Þ quiver functions is useful because one can get rid of subtleties with bifundamental hypers between gauge nodes and one related to boundary conditions, at the price of introducing fundamental and one antifundamental hypers at each ends. an integration over some modulus. The round S5 ≡ The third duality frame is related to the first one by a 3 2 2 2 fðz1;z2;z3Þ ∈ C jjz1j þjz2j þjz3j ¼ 1g admits a toric clockwise rotation by 45 degree of the (1,1) branes, which 3 iφα −1 Uð1Þ action given by zα → e zα. Denoting by eα the acts as the STS element in SLð2; ZÞ on the ðp; qÞ charge corresponding vector fields, the vector R ¼ e1 þ e2 þ e3 is vector. One can verify that Z1 ¼ Z2 ¼ Z3 using the the so-called Reeb vector, and it describes the Hopf identities of [63]. 1 → S5 → CP2 The map between the Kähler parameters of the string fibration Uð Þ . A useful generalization is ω geometry and the physical masses and coupling constants obtained by replacing the Reeb vector with R ¼ 1e1 þ ω ω ω ∈ R of the gauge theory depends on the duality frame. First of 2e2 þ 3e3 ( i >0), and the resulting manifold is S5 ω’ all, it is convenient to introduce exponential variables referred to as the squashed and the s as squashing (or equivariant) parameters. We refer to [68] for further details

2π βϵ −2π βϵ 2π β 2π β of this geometry. q ≡ i 1 t ≡ i 2 ≡ i a1;2 ≡ i a0 e ; e ;Q1;2 e ;Q0 e ; The partition functions of 5d N ¼ 1 gauge theories on 5 ð2:6Þ the (squashed) S can be computed via localization. In the Coulomb branch localization scheme [51–54] (as opposed to the Higgs branch scheme [38,44,69]), the result is given where β measures the S1 radius. In the frame corresponding in terms of a matrix-like integral over the constant vector to the U(1) theory, we can identify multiplet scalar in the Cartan subalgebra of the gauge group. It is known that the integrand can be constructed by −2πβ8π2 ˜ 1=2 2 gluing three Nekrasov partition functions [51,54,68,70,71], a1 ≡ −iðΣ − MÞ;a2 ≡ iðΣ − MÞ;Q0ðQ1Q2Þ ≡ e g ; one for each fixed point of the toric action on CP2, with ð2:7Þ equivariant parameters ϵ1;2 and radius β of the Ω-back- ground related to (complexified) squashing parameters. For where M; M˜ are the 5d fundamental and antifundamental each of the fixed points labeled by α ¼ 1, 2, 3, where the 2 1 Σ C −1 S masses, is the v.e.v. of the vector multiplet scalar and g is space looks like a copy of q;t × β, we can choose the YM coupling. Similarly, on the Uð1Þ ×Uð1Þ side we can identify 123

ϵ1 ω1 þ ω2 ω2 þ ω3 ω1 þ ω3 8π2 −2πβ 2 : ð2:9Þ −1 2 g ≡ Σ − p = ≡ 1;2 ϵ2 ω3 ω1 ω2 a0 ið 12 MbifÞ; Q1;2 e ; ð2:8Þ β 1=ω1 1=ω2 1=ω3 Σ ≡ Σ − Σ where Mbif is the 5d bi-fundamental mass, 12 1 2 and Σ1;2 are the v.e.v.’s of the vector multiplet scalars and On the U(1) theory side (frame 1), the product of g1;2 are the YM couplings. Nekrasov partition functions yields

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−iπ − Σ− ˜ ω Σ− ω S5 S5 6 ðB33ð ið MÞþ2ÞþB33ðið MÞþ2ÞÞ Z Z ; : 3 ≡ e SQED ¼ free ð2 12Þ jZ1j ˜ ω ω S3ð−iðΣ − MÞþ2ÞS3ðiðΣ − MÞþ2Þ 5 C2×S1 ˜ 3 where we defined the squashed S partition function of the × jZ 1 ðg; Σ;M;MÞj ; ð2:10Þ instjUð Þ SQED by 3 where ω ≡ ω1 þ ω2 þ ω3 and j · j denotes the product Z 5 1 of three objects with parameters related by Table (2.9). ZS ≡ dΣZcl ðg; ΣÞZ -loopðΣ;M;M˜ Þ Notice that the 1-loop contributions have fused into triple SQED SQED SQED 2 1 Sine functions (and exponential factors) by using the × jZC ×S ðg; Σ;M;M˜ Þj3; ð2:13Þ definition (A13). instjUð1Þ On the free theory side (frame 2), the product of q- Pochhammer symbols yields 8π3 Σ2 −ω ω ω 2 cl ; Σ ≡ 1 2 3 g ZSQEDðg Þ e ; 2 2 Q −iπ 8π i i ˜ − ω − 8π i i ˜ − Σ 6 ðB33ð 2 þ2ðM MÞþ2Þ B33ð 2 þ2ðMþMÞ i ÞÞ 1 1 3 e g g -loop Σ ˜ ≡ ≡ ZSQEDð ;M;MÞ ω ω ; jZ2j iπ ˜ ω ω ˜ 33 − Σ− 33 Σ− S3ð−iðΣ − MÞþ ÞS3ðiðΣ − MÞþ Þ e 6 ðB ð ið MÞþ2ÞþB ðið MÞþ2ÞÞ 2 2 1 ð2:14Þ × ˜ ω ω S3ð−iðΣ − MÞþ2ÞS3ðiðΣ − MÞþ2Þ 5 2 “ ” S Y 8π i i ˜ − Σ and the Fourier-like transform of the squashed S3 g2 þ 2 ðM þ MÞ i × : ð2:11Þ partition function of free theory by 8π2i i ˜ ω S3 2 þ 2 ðM − MÞþ2 g Z 3 8π iΣðiMþiM˜ −ωÞ S5 2 ≡ Σ g ω1ω2ω3 Σ ˜ Zfree d e ZResummedð ;M;M; gÞ Using Z1 ¼ Z2 (type IIB S-duality), after removing common exponential factors on both sides and integrating 1-loop × Z ðΣ;M;M;˜ gÞ; ð2:15Þ with the classical action, we can obtain the identity1 free

8π3 −ω ˜ −ω ðiM 2ÞðiM 2ÞY 2 − 2 8π i i Σ ˜ ≡ g ω1ω2ω3 − Σ ˜ ZResummedð ;M;M; gÞ e S3 i 2 þ 2 ðM þ MÞ ; g 1 1-loop Σ ˜ ≡ Zfree ð ;M;M; gÞ ˜ ω ω S3ð−iðΣ − MÞþ2ÞS3ðiðΣ − MÞþ2Þ Y 1 × 2 : ð2:16Þ 8π i i ˜ − ω S3ð g2 þ 2 ðM MÞþ2Þ

Z As for the Ω-background case, we simply take this result as S5 ≡ Σ Σ cl ; Σ Σ 1-loop Σ Z 1 2 d 1d 2Z 1 2 ðg1;g2 1; 2ÞZ 1 2 ð 12;MbifÞ a computational fact and we do not attempt to give here a Uð Þ Uð Þ Uð Þ gauge theory interpretation, which is not needed for the C2×S1 ; Σ 3 × jZ 1 2 ðg1;g2 12;MbifÞj ; ð2:18Þ purposes of this paper. instjUð Þ Σ2 Σ2 On the Uð1Þ ×Uð1Þ side (frame 3), we can write − 8π3 1 2 ω1ω2ω3ð 2 þ 2 Þ cl g1 g2 Z 2 g1;g2; Σ1; Σ2 ≡ ; Uð1Þ ð Þ e

iπ ω − B33ðiΣ12−iM þ Þ 1 1 e 6 bif 2 2 1 -loop Σ ≡ 3 C ×S 3 Z 2 ð 12;MbifÞ ω : ð2:19Þ jZ3j ≡ jZ 2 ðg1;g2;Σ12;M Þj ; Uð1Þ Σ − Σ − ω instjUð1Þ bif S3ðið 12 MbifÞþ2Þ S3ðið 12 MbifÞþ2Þ ð2:17Þ Substituting Z3 ¼ Z2 ¼ Z1 and using the dictionary (2.6)–(2.8), one can obtain two more identities. In the following, we are going to focus on first one, namely 5 and in order to reproduce the squashed S partition function type IIB S-duality in relation to 3d mirror symmetry. we need to bring the exponential factor on the other side and integrate with the classical action, namely III. MIRROR SYMMETRY

1The identity is actually stronger because it is really an identity In this section, we will follow type IIB S-duality acting even before taking the integral. on 5d gauge theories, and extract mirror dual partition

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FIG. 3. The brane moves of a simple type of Higgsing applied to a 5d linear quiver gauge theory. The NS5s fill the 012348 directions, while the D5s fill the 012347 directions, hence 78 represents the ðp; qÞ-plane (here we are slightly simplifying the picture). The D3s are stretched along the 6 direction and fill also the 012 and/or 034 directions, hence they all share a common direction and are supported on two orthogonal planes inside the 5-brane world volumes. functions of 3d gauge theories defined on the squashed S3 integrand at a collection of poles of the perturbative S3 ∪ S3 ⊂ S5 determinant as a function of the v.e.v.’s of the scalars in or on the intersecting space ð1Þ ð2Þ . The spheres 3 the vector multiplet(s). S are submanifolds associated to the equations zα ¼ 0, ðαÞ Let us consider the partition function of the SQED on the α 1 S3 S3 5 ¼ , 2, 3. We will focus on ð1Þ and ð2Þ, which clearly squashed S expressed as an integral as in (2.13). In the fol- 2 1-loop intersect transversally along the circle z3 1. We will Σ j j¼ lowing we will focus on the poles of ZSQEDð Þ of the form S3 S3 denote the squashing parameters of ð1Þ and ð2Þ by bð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ ð2Þ ð3Þ ω ω ω ω iðΣ − MÞþω=2 ¼ −n ω1 − n ω2 − n ω3; 2= 3 and bð2Þ ¼ 1= 3 respectively, and we will set ≡ −1 ðαÞ ∈ Z QðαÞ bðαÞ þ bðαÞ as usual. We will review few aspects of n ≥0: ð3:1Þ gauge theories on this type of geometries in the following, 3 while for further details we refer to [39,40]. It is sufficient to study the cases with nð Þ ¼ 0 as they already demonstrate many core features of more general nð3Þ ≠ 0 A. Higgsing, residues, and mirror symmetry cases. The cases are a straightforward generali- zation. As was extensively discussed in [40], the residue of Higgsing [28,29] a higher dimensional bulk theory is an the integrand can be organized into the partition function of effective procedure for accessing lower dimensional super- a 5d/3d/1d coupled system. Indeed, upon taking the symmetric theories that preserve half (or fewer) the super- residue, a few things happen which we now summarize charges that the bulk theory enjoys. More precisely, the (we refer to [40] for a full account, and to Appendix B for resulting lower dimensional supersymmetric field theories the sketch of a slightly different derivation). The non- are worldvolume theories of codimension 2 Bogomol’nyi- perturbative factors and the classical factor are simply ð1Þ ð2Þ Prasad-Sommerfield defects inserted into the bulk theory. evaluated at the pole Σ ¼ M þ iω=2 þ in ω1 þ in ω2. The procedure can be more easily described when there is a Because of the different S1 periodicities at the fixed points (flat space) brane construction. If the 5d theory T admits a ð1Þ jzαj¼1 labeled by α ¼ 1, 2, 3 as in Table 2.9, the n construction in terms of an array of D5s suspended between dependence in the first instanton partition function drops parallel NS5s, for example when T is a unitary linear 2 out, and it only depends on nð Þ, and similarly the second quiver gauge theory, then one type of Higgsing amounts to depends only on nð1Þ, while the third depends on both. aligning the outermost flavor D5 with the adjacent gauge Therefore, among the three instanton partition functions D5, and subsequently pulling the in-between NS5 away associated to the three fixed points, two simply reduce to from the array while stretching a number of D3s. See Fig. 3 the vortex partition functions of two SQCDAs with gauge for an example. At the level of the compact space partition 2 1 1 ð Þ ð Þ C −1 S function, Higgsing T implies taking the residues at certain groups Uðn Þ and Uðn Þ supported on ð t × Þð1Þ and C S1 poles of the partition function as a meromorphic function of ð q × Þð2Þ respectively, while the remaining one encodes mass parameters. In practice, when the compact space the vortex partition functions of the two SQCDA,3 now 1 1 C S C −1 S partition function is written as a Coulomb branch integral, supported on ð q × Þð3Þ and ð t × Þð3Þ respectively, this is often equivalent to computing the residues of the and their intricate interaction along the common S1 at the origin. Schematically, we have the reduction 2The intersection is transversal from the perspective of the two C’s in the two individual tubular neighborhoods C × S1 ⊂ S3 . Put differently, the two complex planes intersect only 3 ð1Þorð2Þ We refer to the UðnÞ SYM theory coupled to nf fundamental, at the origin. antifundamental and 1 adjoint chiral multiplets with SQCDA.

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Σ Σ cl Σ C2×S1 Σ 3⟶¼ cl Σ ZSQEDð ÞjZinstjSQEDð Þj ZSQEDð Þ C S1 1 1 C S1 1 t−1 × Cq×S S ðþ“extra”Þ t−1 × Cq×S × ðZ 2 Þ 1 ðZ 2 Z 2 1 Z 1 Þ 3 ðZ 1 Þ 2 ; ð3:2Þ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}vortexjUðnð ÞÞ ð Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}vortexjUðnð ÞÞ intjUðnð ÞÞ×Uðnð ÞÞ vortexjUðnð ÞÞ ð Þ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}vortexjUðnð ÞÞ ð Þ 2 1 2 1 2 1 ZC ×S Σ ZC ×S Σ ZC ×S Σ ð instjUð1Þð ÞÞð1Þ ð instjUð1Þð ÞÞð3Þ ð instjUð1Þð ÞÞð2Þ where the “extra” factors are remnants that will eventually The proof of this equality relies on formal manipulations cancel out in the final result. Also, the residue of the 1-loop of Nekrasov’s functions and brute force computational factors can be simplified to checks, as briefly explained in Appendix B. To summarize, the result of the residue computation can 1-loop Σ S5 ˜ − be naturally interpreted as the partition function of a free Res ZSQEDð Þ¼ZHMðM MÞ × ; ð3:3Þ Σ→Σ hypermultiplet on the squashed S5 in the presence of two ’ S5 −1 Bogomol nyi-Prasad-Sommerfield codimension 2 defects ≡ ω 2 3 3 where ZHMðMÞ S3ðiM þ = Þ denotes the 1-loop S S supported respectively on ð1Þ and ð2Þ which intersect determinant of a free hyper of mass M on the squashed 1 3 3 5 along a common S ¼ S ∩ S . Each defect is charac- S , while the dots denote 1-loop determinant factors similar ð1Þ ð2Þ to those which would arise in a Higgs branch localization terized by its worldvolume theory, being 3d N ¼ 2 α computation of SQCDAs on each S3 [72,73], plus inter- Uðnð ÞÞ-SQCDA with α ¼ 1, 2 respectively. It is crucial action terms. Because of the form of the q, t parameters at to emphasize that the two defect worldvolume theories 1 each fixed point and the 3d holomorphic block factorization interact at an S , which harbors a pair of additional 1d of S3 partition functions [74–76], one can readily under- N ¼ 2 chiral multiplets transforming in the bifundamental stand that the above reduction describes the partition representation of the two 3d gauge groups. Figure 4 function of the combined system of two SQCDA on summarizes the quiver structure of the 5d/3d/1d coupled 3 3 3 S S system. Each SQCDA on S α contains one fundamental, ð1Þ and ð2Þ, interacting through additional degrees of ð Þ freedom at the common S1.4 one antifundamental and one adjoint chiral multiplet, of ðαÞ ˜ ðαÞ ðαÞ To make our life easier when dealing with the defect masses m , m , and madj respectively. The FI term is ζ theories, it is convenient to recast the above Higgs branch-like turned on with coefficient ðαÞ. These parameters can be representation of the partition function sketched above, into a identified with the 5d hyper multiplet masses and gauge Coulomb branch-like integral, making the structure of the coupling according to the dictionary world volume theories manifest. This is possible thanks to the following nontrivial observation: one can reorganize all ðαÞ i m ¼ λα M þ ðω þ ωαÞ ; the (intricated) factors into an elegant matrix integral, namely 2 Proposition 1 (residues). i ˜ ðαÞ λ ˜ ω −ω m ¼ α M þ 2ð αÞ ; cl Σ 1-loop Σ C2×S1 Σ 3 Res ZSQEDð ÞZSQEDð ÞjZinstjSQEDð Þj 2 Σ→Σ ðαÞ 8π λα 5 m ¼ iωαλα; ζ α ¼ ; ð3:5Þ cl ω 2 S ˜ − adj ð Þ 2 ZSQEDðM þ i = ÞZHMðM MÞ g Z 2 ðαÞ α Y Yna σð Þ S3 d a ð1Þ ð1Þ ¼ Z 1 ðσ Þ 2π ðαÞ! Uðnð ÞÞ-SQCDA α¼1 a¼1 in 3 1 S S ð1Þ ð2Þ ð2Þ ð2Þ × Z ðσ ; σ ÞZ 2 ðσ Þ≡ 1d chiral Uðnð ÞÞ-SQCDA S3 ∪S3 1 2 ≡ Z ð1Þ ð2Þ Uðnð ÞÞ-SQCDA ∪ Uðnð ÞÞ-SQCDA: ð3:4Þ

The explicit expression of the integrand of the matrix model on the right-hand side (r.h.s.) can be found in Appendix C, and the definition of the integral is given by the Jeffrey-Kirwan prescription discussed in [40]. FIG. 4. The quiver structure of the 5d/3d/1d theory describing 4 Notice that a generalization to the three-component subspace the 5d theory in the presence of intersecting codimension 2 S3 ∪ S3 ∪ S3 ⊂ S5 ð1Þ ð2Þ ð3Þ features in the Higgs branch localization defects. The hyper or chiral multiplets supported on spheres of on S5 [44]. different dimensions are indicated by their colors.

126002-7 FABRIZIO NIERI, YIWEN PAN, and MAXIM ZABZINE PHYS. REV. D 98, 126002 (2018) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ζ ζ where λα ≡ ωα=ω1ω2ω3. The 3d chiral multiplets q; q˜ theories are also related by ðb Þð1Þ ¼ðb Þð2Þ, indicating 5 couple to the bulk hyper multiplet qbulk via cubic super- that the two theories also share the same U(1) topological ˜ ˜ potentials qð1Þqð1Þqbulk and qð2Þqð2Þqbulk, leading to the mass symmetry. relations Now we are ready to extract candidate 3d mirror pairs. The two sides of the fiber/base duality between ð1Þ ð2Þ i 2 2 b 1 m − b 2 m b − b ; frame 1 and 2 share the same poles in the integrand. In ð Þ ð Þ ¼ 2 ð ð2Þ ð1ÞÞ fact, the integral equality trivially follows from the i ˜ ð1Þ − ˜ ð2Þ − 2 − 2 equality of the integrand, and therefore by taking the bð1Þm bð2Þm ¼ ðb 2 b 1 Þ; 2 ð Þ ð Þ residue at the same pole Σ → Σ on both sides (and ð1Þ − ð2Þ 2 − 2 dropping the common factors), we extract a family of bð1Þmadj bð2Þmadj ¼ iðb 2 b 1 Þ: ð3:6Þ ð Þ ð Þ nontrivial integral identities labeled by non-negative 3 3 S S ð1Þ ð2Þ In other words, the theories on ð1Þ and ð1Þ share the integers n and n , namely same U(1) flavor group. The FI parameters in the two

Proposition 2 (master identity).

S3 ∪S3 1 2 Z ð1Þ ð2Þ Uðnð ÞÞ-SQCDA ∪ Uðnð ÞÞ-SQCDA 8π3 16π3 − ˜ ω ð1Þω ð2Þω − ð1Þ ð2Þ ¼ exp 2 ð iðM þ MÞþ Þðn 1 þ n 2Þ 2 n n g ω1ω2ω3 g ω3 Y2 nYðαÞ−1 ˜ S2ðiðM − MÞþω þ kωαjω3; ωγÞ × Q − 1 ω ω ω 8π2i i ˜ − ω ω ω ω α¼1 k¼0 S2ð ðk þ Þ αÞj 3; γÞ S2ð g2 þ 2 ðM MÞþ2 þ k αj 3; γÞ Q 2 Ynð1Þ Ynð2Þ 8π i i ˜ − ω − 1 ω l − 1 ω ω S1ð 2 þ 2 ðM MÞþ2 þðk Þ 1 þð Þ 2j 3Þ × g ; 3:7 − ω − lω ω ˜ − ω − 1 ω l − 1 ω ω ð Þ k¼1 l¼1 S1ð k 1 2j 3ÞS1ðiðM MÞþ þðk Þ 1 þð Þ 2j 3Þ

S3 where γ ¼ 1, 2 when α ¼ 2, 1. We refer to Appendix A Z ð1Þ Uð1Þ-SQCDA for the definitions of the double Sine and single Sine iQ sbð 2 þ m Þ 1 functions. Notice that this mathematical identity, which adj ð Þ iQ m − m˜ we will refer to as the master identity,isnewand ¼ e−πiζðmþm˜ Þs þ m − m˜ s −ζ − provides a huge generalization of the hyperbolic iden- b 2 b 2 tity in Theorem 5.6.8 of [77]. The proof relies on − ˜ ζ − m m formal manipulations of Nekrasov’s functions and × sb þ 2 : ð3:8Þ brute force computational checks. We will shortly see ð1Þ that these integral identities, derived from type IIB We refer to Appendix A for the definition of the double sine S-duality, capture 3d N ¼ 2 mirror symmetry on function. This integral equality is nothing but the mirror 3 intersecting S ’s. symmetry relation between 3d N ¼ 2 SQED and the XYZ S3 model at the level of ð1Þ partition functions. As expected, B. Warming up: SQED/XYZ duality the complexified masses of the three free chiral multiplets 1 We begin with a warm-up exercise to see that the well- in the XYZ model, namely (suppressing the label ð Þ) known Abelian mirror symmetry between 3d N ¼ 2 − ˜ ≡ ζ − m m − iQ ≡ − ˜ SQED and the XYZ model arises from the integral mX;Y 2 2 ;mZ m m; ð3:9Þ identities discussed above. For this, we consider nð1Þ ¼ 1 2 satisfy and nð Þ ¼ 0. Upon substituting in (3.5), the master equality

(3.7) implies mX þ mY þ mZ ¼ −iQ; ð3:10Þ signaling the presence of the superpotential XYZ. On the 5 ˜ SQED side, the additional 1-loop factor signals the pres- The 5d hyper multiplet has two scalars qbulk; qbulk. It can be decomposed into two 3d N ¼ 2 chiral multiplets, into which ence of a decoupled chiral multiplet β1 interacting with the ˜ qbulk and qbulk enter separately. adjoint chiral Φ through the superpotential β1Φ.

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C. Generalization: Intersecting SQED/XYZ duality Now we are ready to generalize the mirror symmetry relationQ between the SQED and XYZ models to intersecting 2 2 spheres. Dropping from both sides the common 1-loop factors like α¼1 sbðiQ= þ madjÞðαÞ, the master equality (3.7) with nð1Þ ¼ nð2Þ ¼ 1 implies a more involved integral identity, namely

S3 ∪S3 ð1Þ ð2Þ ∪ ZSQED SQED iπ ð1Þ ð2Þ iπ ð1Þ ð2Þ sin 2 ðb 1 m þ b 2 m Þ sin 2 ðb 1 m þ b 2 m Þ ¼ ð Þ X ð Þ X ð Þ Y ð Þ Y π ð1Þ ð2Þ iπ ð1Þ ð2Þ sin i ðbð1Þmadj þ bð2ÞmadjÞ sin 2 ðbð1ÞðmZ þ madjÞ þ bð2ÞðmZ þ madjÞ Þ Y2 −πiζðmþm˜ Þ iQ iQ iQ × e sb 2 þ mZ sb 2 þ mX sb 2 þ mY ; ð3:11Þ α¼1 ðαÞ

S3 1 where the masses in the XYZ models are defined as usual Z ð1Þ Uðnð ÞÞ-SQCDA ðαÞ by (we suppress the label ) Yn−1 −πinζðmþm˜ Þ iQ iQ ¼ e s þ mΦμ s þ m μ m − m˜ iQ b 2 b 2 Z ≡ ζ − − ≡ − ˜ μ¼0 mX;Y 2 2 ;mZ m m: ð3:12Þ iQ iQ × sb 2 þ mXμ sb 2 þ mYμ ; ð3:13Þ The l.h.s. of the above identity is the partition function of ð1Þ S3 S3 two SQED on ð1Þ and ð2Þ, coupled through a pair of 1d bi-fundamental chiral multiplets along the common S1 where we used the shorthand notations (suppressing again ð1Þ intersection. The r.h.s. can be naturally interpreted as the the label ) S3 S3 partition function of two XYZ models on 1 and 2 , − ˜ ð Þ ð Þ ≡ ζ − m m − iQ − μ coupled to a pair of 1d free Fermi multiplets and another mXμ 2 2 madj; pair of 1d chiral multiplets on S1. The fact that the masses − ˜ ≡ −ζ − m m − iQ − − 1 − μ of the 1d multiplets are combinations of those of the 3d mYμ 2 2 ðn Þmadj; multiplets indicates the presence of a certain 1d super- ≡ − ˜ μ ≡ μ 1 potential that involves both the 3d and 1d chiral multiplets. mZμ m m þ madj;mΦμ ð þ Þmadj: ð3:14Þ As a result, the 1d multiplets are charged under the 3d For convenience, we can reorganize the following products global symmetries, in particular, the Fermi multiplets are −1 charged under the 3d topological U(1) symmetry. Yn iQ 1 Some remarks follow. In the previous subsection, the s þ mΦμ ¼ Q ; b 2 n s ðiQ=2 − μm − iQÞ integral equality reproduces the well-known 3d mirror μ¼0 μ¼1 b adj symmetry between SQED and XYZ model, confirming ð3:15Þ at the level of partition function that the SQED flows to the free XYZ model as the IR limit. In this subsection, our Yn−1 2 − ˜ − 1 iQ sbðiQ= þ m m þðn ÞmadjÞ derived integral identity shall be viewed as a piece of s þ m μ ¼ Q ; b 2 Z n−2 s ðiQ=2 − m þ m˜ − μm −iQÞ mathematical evidence from which we identify a possible μ¼0 μ¼0 b adj 3 ∪ flow from the gauge theory on the intersecting space S 1 3:16 3 ð Þ ð Þ Sð2Þ to a free theory on the same space. However, one should also note that surprises arise in intersection theories and move the denominators to the l.h.s. of (3.13). Defining [36], which might bring subtleties to such naive RG flow. the leftover mass on the r.h.s. We leave detailed investigation of these subtleties to ≡ − ˜ − 1 future study. mZ mZðn−1Þ ¼ m m þðn Þmadj; ð3:17Þ

D. Generalization: Non-Abelian SQCDA/XYZ duality one easily finds the masses satisfy

We can now move to discuss more interesting mXμ þ mYμ þ mZ ¼ −iQ; μ ¼ 0; …;n− 1; ð3:18Þ examples, generalizing the previous Abelian examples to P n−1 non-Abelian gauge groups. Let us start by considering which is compatible with the superpotential μ¼0 XμYμZ. nð1Þ > 0, nð2Þ ¼ 0, in which case the master equality On the l.h.s., the additional 1-loop factors are compatible specializes to with free chiral multiplets βμ and γμ interacting with the

126002-9 FABRIZIO NIERI, YIWEN PAN, and MAXIM ZABZINE PHYS. REV. D 98, 126002 (2018) adjoint chiralP Φ and the quarksP q; q˜ through the super- n−2 γ ˜Φμ n β Φμ potential μ¼0 μq q þ μ¼1 μ . The mathematical relation (3.13) has implicitly appeared in [34] as an intermediate step to test another duality, involving the SUðnÞ theory coupled to one fundamental, FIG. 5. Another duality that can be obtained by integrating over one anti-fundamental and one adjoint chiral on the one the FI parameter, viewed as the “mass” for the topological U(1) hand, and the U(1) theory coupled to n hypers on the other symmetry. hand as shown in Fig. 5, which was motivated by the study of the mirror dual of ðA1;A2n−1Þ AD theories reduced to 3d [20,21]. This duality is simply related to ours by gauging E. Generalization: Intersecting non-Abelian the topological U(1). Hence, we have physically interpreted SQCDA/XYZ duality and derived both dualities as 3d N ¼ 2 mirror symmetry It is now straightforward to take the further generaliza- descending from type IIB S-duality. tion nð1Þ;nð2Þ > 0. In this case, the master identity yields

S3 ∪S3 1 2 Z ð1Þ ð2Þ Uðnð ÞÞ-SQCDA ∪ Uðnð ÞÞ-SQCDA Y2 Yn−1 −πinζðmþm˜ Þ iQ iQ iQ iQ ¼ e sb 2 þ mΦμ sb 2 þ mZμ sb 2 þ mXμ sb 2 þ mYμ α¼1 μ¼0 ðαÞ nYð1Þ−1 nYð2Þ−1 πi − ð1Þ − ð2Þ 2 2 → sin 2 ðbð1ÞðmXμ mΦμÞ þ bð2ÞðmXμ mΦνÞ þ ib 2 þ ib 1 ÞðX YÞ × ð Þ ð Þ ; ð3:19Þ π ð1Þ ð2Þ πi ð1Þ ð2Þ μ¼0 ν¼0 sin iðbð1ÞmΦμ þ bð2ÞmΦνÞ sin 2 ðbð1ÞðmZμ þ mΦμÞ þ bð2ÞðmZν þ mΦνÞ Þ

P nð1Þ −1 where we used the same shorthand notations as before. We implies integration with the constraint δð ðb σ Þ 1 þ P a¼1 a ð Þ can reorganize the factors as we did in the previous ð2Þ n b−1σ , whose field theory interpretation subsection, and the difference compared to the previous a¼1ð aÞð2ÞÞ result (besides the doubling of all factors) is the presence of remains unclear to us at the moment. the additional 1-loop contributions from the 1d matter living on the Sð1Þ intersection, represented by the last line. F. Quiver gauge theories This picture provides the generalization of the non-Abelian It is possible to generalize the above computations to SQCDA/XYZ duality to the more complicated geometry quiver gauge theories. As shown in Fig. 3, one could start involving 1d degrees of freedom, and we have shown that it from a 5d linear quiver gauge theory and engineer also descends from type IIB S-duality. intersecting codimension 2 defects with quiver world It is worth noting that one can further integrate over the volume theories by multiple Higgsings. For example, it ζ FI parameters ðiÞ to obtain the intersecting space version of is not hard to convince oneself that by Higgsing twice the the SUðnÞ-SQCDA/U(1) duality mentioned at the end of 5d linear quiver gauge theory with two U(1) gauge nodes, the last subsection. However, the fact that the FI parameters one will obtain 3d quiver theories of the form depicted in ζ ζ on each component space are related by ðb Þð1Þ ¼ðb Þð2Þ Figure 6. It is possible to apply the Higgsing procedure by taking the residues of the resulting partition functions and their fiber/base dual, and repeat the computations in the previous discussions. However, the technical computations are more involved and we do not consider them here explicitly.

IV. DISCUSSION AND OUTLOOK In this paper, we have studied a class of 3d N ¼ 2 non- Abelian gauge theories which can be realized as codimen- sion 2 defects in the parent 5d N ¼ 1 Abelian gauge theories, which in turn can be realized in type IIB string FIG. 6. Quiver world volume theories of intersecting codimen- theory. Generically, the defect theories are not supported on sion 2 defects following from Higgsing twice. The purple arrows a single component subspace, instead, they live on mutually denote bi-fundamental 1d chiral multiplets, while the blue dotted orthogonal submanifolds intersecting at codimension lines denote 1d Fermi multiplets. 4 loci where additional degrees of freedom live. We have

126002-10 3D MIRROR SYMMETRY FROM S-DUALITY PHYS. REV. D 98, 126002 (2018) considered some implications of type IIB SLð2; ZÞ sym- and P. Koroteev for discussions, and UC Berkeley and UC metry for these systems, and we have generalized to this Davis for hospitality. The research of F. N. and M. Z. is class of more complicated geometries the known fact that supported in part by Vetenskapsrådet under Grant type IIB S-duality reduces to 3d mirror symmetry. Using No. 2014-5517, by the STINT grant and by the grant the refined topological vertex, we have been able to test this “Geometry and Physics” from the Knut and Alice idea in simple cases where the parent 5d gauge theory is Wallenberg foundation. Y. P. is supported by the 100 simply the SQED with two flavors, while the dual 3d Talents Program of Sun Yat-sen University under Grant theories are SQCDA with two chirals and a generalized No. 74130-18831116. XYZ model. Interestingly enough, the QFT/string theory methods have also allowed us to physically explain existing APPENDIX A: SPECIAL FUNCTIONS integral identities in the math literature, and moreover, to In this Appendix, we recall the definitions of several derive new ones and interpret them as the equivalence of special functions which we use in the main body. Below, r partition functions of mirror dual theories on (intersecting) is a positive integer, and ω⃗≡ ðω1; …; ω Þ is a collection of squashed spheres. One should bare in mind, however, r nonzero complex parameters. We frequently take r ¼ 1,2, that the field-theoretical interpretation of the intersecting 3 for concreteness. We refer to [103] for further details. mirror symmetry requires further investigation due to The multiple Bernoulli polynomials B ðXjω⃗Þ are surprises/inconsistences hidden within theories on inter- rn defined by the generating function secting spaces. Along the lines of this paper, one should also be able to r Xt X n study more complicated 5d theories and hence derive new Q t e t ω ≡ BrnðXjω⃗Þ : ðA1Þ r it − 1 ! or generalized 3d mirror pairs. As byproduct, one may also i¼1 e m≥0 n obtain new mathematical identities expressing the equiv- alence of dual partition functions. Moreover, what we In particular, we use B22ðXjω⃗Þ and B33ðXjω⃗Þ in this paper, have discussed in this paper is expected to have a higher and they are given explicitly by dimensional lift [78] by considering 6d theories engineered by periodic ðp; qÞ-webs [58,79,80] and the resulting 4d/2d 2 2 2 X ω1 þ ω2 ω þ ω þ 3ω1ω2 defect theories. ω⃗ ≡ − 1 2 B22ðXj Þ ω ω ω ω X þ 6ω ω ; Finally, it is worth noting that the type of 3d/1d defects 1 2 1 2 1 2 that we have considered in this paper appear in the Higgs ðA2Þ branch localization approach to SQCD on S5 [44], whose partition functions are identified with correlators in the 3 X 3ðω1 þ ω2 þ ω3Þ 2 q-Virasoro modular triple [81]. Therefore, another inte- B33ðXjω⃗Þ ≡ B33ðXÞ¼ − X resting route of investigation would be the study of type IIB ω1ω2ω3 2ω1ω2ω3 2 Z 2 2 2 SLð ; Þ symmetry from the viewpoint of the ω1 þ ω2 þ ω3 þ 3ω1ω2 þ 3ω2ω3 þ 3ω3ω1 ’ þ X Bogomol nyi-Prasad-Sommerfield/CFT and 5d AGT cor- 2ω1ω2ω3 respondences [82–96] and the DIM algebra [97,98], whose ðω1 þ ω2 þ ω3Þðω1ω2 þ ω2ω3 þ ω3ω1Þ representation theory is known to govern the topological − þ 4ω ω ω : amplitudes associated to toric CY 3-folds or ðp; qÞ-webs 1 2 3 [99–102]. From this perspective, the SLð2; ZÞ symmetry ðA3Þ group is identified with the automorphism group of the DIM algebra, and it would be interesting to systematically The q-Pochhammer symbols are defined as study how different q-deformed correlators are related to each other. In turn, this perspective may give powerful tools Y∞ ; … ≡ 1 − n1 … nr for handling 3d mirror symmetry very efficiently. This ðx q1; ;qrÞ∞ ð xq1 qr Þ is a topic which deserves further investigations, and in n1;…;nr¼0 Appendix D we have collected few preliminary comments when all jq j < 1: ðA4Þ and background material for the interested readers. i

ACKNOWLEDGMENTS Other regions in the q-planes are defined through the replacements We thank S. Pasquetti for valuable comments and discussions. We also thank the Simons Center for 1 x; q ; …;q → : ð 1 rÞ∞ −1 ; … −1 … ðA5Þ Geometry and Physics (Stony Brook University) for ðqi x q1; ;qi ; ;qrÞ∞ hospitality during the program “Localization Techniques in Quantum Field Theories,” at which some of the research The multiple Sine functions SrðXjω⃗Þ can be defined by for this paper was performed. F. N. also thanks N. Haouzi the ζ-regularized product

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Y Xr ð−1Þrþ1 The reflection property of s ðzÞ is simply SrðXjω⃗Þ ≃ ðX þ miωiÞ b m1;…;m ∈N i¼1 r Xr sbðXÞsbð−XÞ¼1: ðA12Þ × −X þ ðmi þ 1Þωi : ðA6Þ i¼1 The triple Sine function S3ðXjω⃗Þ ≡ S3ðXÞ also has a S ðXjω⃗Þ is symmetric in all ω , has the reflection property r i useful factorization property. When Imðωi=ωjÞ ≠ 0 for all ω⃗ ω − ω⃗ ð−1Þrþ1 ω ≡ ω ω SrðXj Þ¼Srð Xj Þ for 1 þþ r, the i ≠ j, then × homogeneity property SrðλXjλω⃗Þ¼SrðXjω⃗Þ for λ ∈ C , and the shift property Y 2π ω ωj iπ i 2π i 2π − B33ðXÞ ω X iω iω S3ðXÞ¼e 6 ðe k ; e k ; e k Þ∞: ðA13Þ SrðXjω⃗Þ 1≤i≠j≠k≤3 SrðX þ ωijω⃗Þ¼ ; Sr−1ðXjωˆ Þ ωˆ ≡ ω ω ω … ω The Nekrasov function is defined as ð 1; i−1; iþ1; ; rÞ: ðA7Þ Y ∨ The single Sine function S1ðXjω⃗Þ is simply defined as −1 λ −j μ −iþ1 Nλμðx; q; t Þ ≡ ð1 − xq i t j Þ ði;jÞ∈λ S1ðXjω⃗Þ ≡ 2 sinðπX=ω1Þ: ðA8Þ Y −μ −1 −λ∨þi × ð1 − xq iþj t j Þ; ðA14Þ The double Sine function S2ðxjω⃗Þ enjoys a factorization ði;jÞ∈μ property when Imðω1=ω2Þ ≠ 0, namely ∨ iπ where denotes transposition of the Young diagrams. B22ðXjω⃗Þ 2πiX=ω1 2πiω2=ω1 S2ðXjω⃗Þ¼e 2 ðe ; e Þ∞

2πiX=ω2 2πiω1=ω2 × ðe ; e Þ∞: ðA9Þ S3 ∪ S3 APPENDIX B: DERIVATION OF THE ð1Þ ð2Þ MATRIX MODEL There is also a shifted version of the double Sinep functionffiffiffiffiffiffiffiffiffiffiffiffiffi ≡ ω ω which is often denoted by sbðXÞ where b 1= 2, Here we sketch how to derive the matrix model (3.4) related to S2ðXjω⃗Þ by following the argument given above (3.3). The exact equality between the residue of the S5 integrand at the selected poles iQ iX ð3Þ 0 S3 ∪ S3 ω⃗ ≡ − ffiffiffiffiffiffiffiffiffiffiffi (3.1) (with n ¼ ) and the 1 2 matrix model is S2ðXj Þ sb 2 þ pω ω ; ðA10Þ ð Þ ð Þ 1 2 established in the next section in the notation used in the main body. See also [40] for another derivation. ≡ −1 where Q b þ b . In terms of the double sine sbðxÞ, the We start by rewriting the instanton sum (2.1) using the factorization is rewritten as manipulations considered in [43]. Shown in Fig. 7 is a large Hook Young diagram λ decomposed into an upper-left full iQ iπ − −1 2π 2π 2 − 2 B22ð iXjb;b Þ bX; ib rectangle with exactly r rows and c columns, an upper-right sb 2 þ X ¼ e ðe e Þ∞ subdiagram YR with at most r rows and a lower-left sub- 2πb−1X 2πib−2 L × ðe ; e Þ∞: ðA11Þ diagram Y with at most c rows. For such a diagram, we can write the corresponding summand in the instanton partition function (2.1) as

1=2 1=2 Yr Yc 1=2 j−1 1−i 1=2 i −j N∅λðQ1p ; q; tÞNλ∅ðQ2p ; q; tÞ 1 ð1 − p Q2q t Þð1 − p Q1t q Þ ¼ i j−1 1−i −j Nλλ 1; q; t N ∅∅ 1 − t q 1 − t q ð Þ i¼1 j¼1 ð Þð Þ Y −1 r −c −1 r −c −1 ðη t q z R =x; qÞ∞ ðη t q z L =x; t Þ∞ −1 R Yi L Yi Δ ; q Δ −1 ; t × tðzYR Þ q ðzYL Þ −r c −1 −r c −1 ðtη t q x=z R ; qÞ∞ ðq η t q x=z L ; t Þ∞ i≥1 R Yi L Yi Y 1 × −1=2 −1=2 ð1 − p z L =z R Þð1 − p z R =z L Þ i;j≥1 Yj Yi Yi Yj Y 1=2 −1 1=2 −1 ðtη p Q1x=z R ; qÞ∞ ðq η p Q1x=z L ; t Þ∞ R Yi L Yi × −1 1=2 −1 1=2 −1 ; ðB1Þ ðη p Q2z R =x; qÞ∞ ðη p Q2z L =x; t Þ∞ i≥1 R Yi L Yi

126002-12 3D MIRROR SYMMETRY FROM S-DUALITY PHYS. REV. D 98, 126002 (2018) ffiffiffiffiffi η η η pqt where L;R are free parameters such that L= R ¼ ,we defined

−r i−1 −YL c 1−i YR z L ≡ η xt q t i ;zR ≡ η xq t q i ; ðB2Þ Yi L Yi R and N ∅∅ denotes the whole factor beginning in the second line and evaluated for empty diagrams. The nonperturbative instanton partition function is obtained as the weightedP sum λ p−1=2 jλj λ ≡ λ over with weight ð Q0Þ , where j j i i implies the total number of boxes in λ. The sum can be further decomposed into a form respecting the hook Young FIG. 7. A large Young diagram with subdiagrams YL and YR. PdiagramP decompositionP as shown in Fig. 7, namely − n λ ¼ r;c≥0 YL;YR , such that r c ¼ is a fixed λ ≤ λ∨ ≤ arbitrary integer expressing a linear relation between r diagrams with n2þ1 n1, n1þ1 n2: they include all large 1=2 −n1 n2 and c. Note that if we tune p Q2 ¼ q t , the first factor hook Young diagrams with an upper-left rectangle of the in (2.1) vanishes, and therefore the instanton sum only shape r ¼ n2;c¼ n1, and infinitely many diagrams that we receives non-vanishing contributions from diagrams λ call small hook diagrams. Let us focus on the large Hook which do not contain the box ðn2 þ 1;n1 þ 1Þ, i.e., Hook diagrams. In this case we get the simplification

1=2 1=2 Yr Yc −j i 1=2 i −j N∅λðQ1p ; q; tÞNλ∅ðQ2p ; q; tÞ ð1 − q t Þð1 − p Q1t q Þ ¼ i j−1 1−i −j Nλλ 1; q; t 1 − t q 1 − t q ð Þ i¼1 j¼1 ð Þð Þ −1 Y 1=2 −1 1=2 −1 Δt z R ; q Δ −1 z L ; t ðtη p Q1x=z R ; qÞ∞ ðq η p Q1x=z L ; t Þ∞ ð Y Þ q ð Y Þ R Yi L Yi × −r c −1 −r c −1 N ∅∅ ðtη t q x=z R ; qÞ∞ ðq η t q x=z L ; t Þ∞ i≥1 R Yi L Yi Y 1 × −1=2 −1=2 : ðB3Þ ð1 − p z L =z R Þð1 − p z R =z L Þ i;j≥1 Yj Yi Yi Yj

1=2 −n1 n2 Also, the residue of the perturbative factor in (2.1) at a pole p Q2 ¼ q t reads

Y −1 −1 Yr Yc Yc Yr 1 Res 1ðz; q; t Þ∞ 1 1 1 → z¼ : ðB4Þ p1=2 ; q t−1 p1=2 ; q t−1 1 − q−jti q−j; t−1 ti; q i¼1;2 ð Qi ; Þ∞ ð Q1 ; Þ∞ i¼1 j¼1 j¼1 ð Þ∞ i¼1 ð Þ∞

Notice that the second factor will cancel against the first factor in the numerator of (B3). We can also set

c −r 1=2 Q1 ¼ q t p w=x; ðB5Þ and redefine

−c r −1 −cþi −YL −c r r−i YR z L q t ¼ η q xq t i → z L ;zR q t ¼ η txt q i → z R ; ðB6Þ Yi L Yi Yi R Yi so that

1=2 1=2 Yr Yc −j i i−r c−j N∅λðQ1p ; q; tÞNλ∅ðQ2p ; q; tÞ ð1 − q t Þð1 − pt q w=xÞ ¼ i j−1 1−i −j Nλλ 1; q; t 1 − t q 1 − t q ð Þ i¼1 j¼1 ð Þð Þ −1 Y −1 −1 Δt z R ; q Δ −1 z L ; t ðtη pw=z R ; qÞ∞ ðq η pw=z L ; t Þ∞ ð Y Þ q ð Y Þ R Yi L Yi × −1 −1 N ∅∅ ðtη x=z R ; qÞ∞ ðq η x=z L ; t Þ∞ i≥1 R Yi L Yi Y 1 × −1=2 −1=2 : ðB7Þ ð1 − p z L =z R Þð1 − p z R =z L Þ i;j≥1 Yj Yi Yi Yj

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For convenience, we can also set

tη p ≡ q−1η p ≡ tη ≡ q−1η ≡ R w wR; L w wL; Rx xR; Lx xL; ðB8Þ so that

1=2 1=2 Yr Yc −j i i−r c−j N∅λðQ1p ; q; tÞNλ∅ðQ2p ; q; tÞ ð1 − q t Þð1 − pt q w=xÞ ¼ i j−1 1−i −j Nλλ 1; q; t 1 − t q 1 − t q ð Þ i¼1 j¼1 ð Þð Þ −1 Y −1 Δt z R ; q Δ −1 z L ; t ðw =z R ; qÞ∞ ðw =z L ; t Þ∞ ð Y Þ q ð Y Þ R Yi L Yi × −1 N ∅∅ ðx =z R ; qÞ∞ ðx =z L ; t Þ∞ i≥1 R Yi L Yi Y 1 × −1=2 −1=2 : ðB9Þ ð1 − p z L =z R Þð1 − p z R =z L Þ i;j≥1 Yj Yi Yi Yj

Notice that

−1=2 −1=2 Θðξp Q0=z R ; qÞΘðξ; qÞ Θðξp Q0=z∅R ; qÞΘðξ; qÞ Yi i −1 2 R p = jYi j −1=2 ¼ −1=2 ð Q0Þ ; ðB10Þ Θðξ=z R ; qÞΘðξp Q0; qÞ Θðξ=z∅R ; qÞΘðξp Q0; qÞ Yi i

−1=2 −1 −1 −1=2 −1 −1 Θðξp Q0=z L ; t ÞΘðξ; t Þ Θðξp Q0=z∅L ; t ÞΘðξ; t Þ Yi i −1 2 L p = jYi j −1 −1=2 −1 ¼ −1 −1=2 −1 ð Q0Þ ; ðB11Þ Θðξ=z L ; t ÞΘðξp Q0; t Þ Θðξ=z∅L ; t ÞΘðξp Q0; t Þ Yi i where ξ is arbitrary. Since

−1=2 jλj −1=2 rc −1=2 jYLj −1=2 jYRj ðp Q0Þ ¼ðp Q0Þ ðp Q0Þ ðp Q0Þ ; ðB12Þ we can recognize the weighted sum over the left and right diagrams [second and third line of (B9)] as the vortex part of the partition function I Yr dz Yc dz B ≡ Ri Lj ϒ ðz Þϒ ðz ;z Þϒ ðz Þ¼ ðB13Þ LR 2πiz 2πiz L L int L R R R i¼1 Ri j¼1 Lj X ϒ ϒ ϒ LðzYL Þ intðzYL ;zYR Þ RðzYR Þ z ¼z ϒ ϒ ϒ ¼ Res Lj ∅L LðzLÞ intðzL;zRÞ RðzRÞ ; ðB14Þ j ϒ L ϒ L R ϒ R L R Lðz∅Þ intðz∅;z∅Þ Rðz∅Þ zRi¼z∅R Y ;Y i where

Yr Θ ξp−1=2 ; q Θ ξ; q Yr ; q ð Q0=zRi Þ ð Þ ðwR=zRi Þ∞ ϒ ðz Þ ≡ Δtðz ; qÞ ; ðB15Þ R R Θ ξ ; q Θ ξp−1=2 ; q R x =z ; q i¼1 ð =zRi Þ ð Q0 Þ i¼1 ð R Ri Þ∞

Yc Θ ξp−1=2 ; t−1 Θ ξ; t−1 Yc ; t−1 ð Q0=zLj Þ ð Þ −1 ðwL=zLj Þ∞ ϒ ðz Þ ≡ Δq−1 ðz ; t Þ ; ðB16Þ L L Θ ξ ; t−1 Θ ξp−1=2 ; t−1 L ; t−1 j¼1 ð =zLj Þ ð Q0 Þ j¼1 ðxL=zLj Þ∞

Yr Yc 1 ϒ ðz ;z Þ ≡ ; ðB17Þ int L R 1 − p−1=2 1 − p−1=2 i¼1 j¼1 ð zLj=zRiÞð zRi=zLjÞ and the contour is chosen to encircle the poles6

6 ’ We simply integrate the z s one after the other, starting from zR;i¼r around xR and zL;j¼c around xL.

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− −YL − R S3 Y − ω ω q cþjt j tr iqYi 1 S2ðZ Z j 1; 3Þ zLj ¼ z L ¼ xL ;zRi ¼ z R ¼ xR : ð Þ Ri Rj Yj Yi Z Z ≡ 1-loopð RÞ −ω − ω ω S2ð 2 þ ZRi ZRjj 1; 3Þ ðB18Þ 1≤i≠j≤r Yr − ω ω S2ðWR ZRij 1; 3Þ × − ω ω ; ðB24Þ This corresponds to the block integral [75] of the 1 S2ðXR ZRij 1; 3Þ 1 i¼ SQCDA-UðrÞ ∪ SQCDA-UðcÞ theory on ½Cq × S ∪ 1 ½Ct−1 × S , interacting through a pair of 1d chiral multip- 1 S3 Y − ω ω S 2 S2ðZ i Z jj 2; 3Þ lets in the bifundamental of UðrÞ ×UðcÞ at the common Z ð Þ Z ≡ L L 1-loopð LÞ −ω − ω ω intersection at the origin (plus superpotential terms). The 3d 1≤i≠j≤c S2ð 1 þ ZLi ZLjj 2; 3Þ ζ ζ FI/vortex counting parameters L, R are identified with Yc S2ðW − Z jω2; ω3Þ × L Lj ; ðB25Þ −1=2 ζ −ζ S2ðX − Z jω2; ω3Þ p Q0 ¼ q R ¼ t L : ðB19Þ j¼1 L Lj

3 S iπω2 2 1 1 ð1Þ 2ω ω ðr −1Þðω1þω2þω3Þ C S C −1 S ≡ 1 3 Now let us think of q × and t × as two halves Zcl ðZRÞ e 3 S ’ iπ of two squashed s, namely − rðW −X ÞðW þX −ω1−ω3Þ ×e 2ω1ω3 R R R R Yr Yr 3 1 1 iπ − 2πi ζ S ≃ ½Cq × S # ½Cq˜ × S ; ω ω ðWR XRÞZRi ω ω ZRi ð1Þ S × e 1 3 × e 1 3 ; ðB26Þ 3 1 1 1 1 S ≃ C −1 S # C˜−1 S i¼ i¼ ð2Þ ½ t × S½ t × ; ðB20Þ

3 ˜ S iπω1 2 q˜ t q t ð2Þ ðc −1Þðω1þω2þω3Þ where and are related to and by the S element in ≡ 2ω2ω3 Z ðZLÞ e SLð2; ZÞ performing the boundary homeomorphism [75], cl iπ − cðW −X ÞðW þX −ω2−ω3Þ and form the partition function on the intersecting space ×e 2ω2ω3 L L L L 3 3 S 1 ∪ S 2 . In order to do that, it is convenient to para- Yc Yc ð Þ ð Þ iπ ðW −X ÞZ 2πi ζZ metrize the variables as × eω2ω3 L L Lj × eω2ω3 Lj ; ðB27Þ j¼1 j¼1 ω ω ρ 2πi 1 −1 2πi 2 2πi q ≡ e ω3 ; t ≡ e ω3 ; p ≡ e ω3 ; iπ ρ 2π 2π 2π Yr Yr Y ω iZ iZ iX 1 e 3 z ≡ eω3 Lj ;z≡ eω3 Ri ;x≡ eω3 L;R ; ZS Z ;Z ≡ : Lj Ri L;R intð L RÞ 4 π − ρ ðB28Þ 2π 2π 2π 1 1 sin ω ½ZRi ZLj 2 iW −1 2 iζ iΞ i¼ j¼ 3 ≡ ω3 L;R p = ≡ ω3 ξ ≡ ω3 wL;R e ; Q0 e ; e : − 2 ðB21Þ Notice the renormalization of the FI by ðWL XLÞ= ¼ − 2 ðWR XRÞ= (we impose this equality), as usual when Then we can multiply (B13) with another left block integral going from K-theoretic to field-theoretic notation. The with ω3 ↔ ω2 and another right block integral with vortex part of the above matrix model captures the Hook 5 ω3 ↔ ω1. This will convert truncation of the S integrand at the poles specified in (3.1) with nð3Þ ¼ 0. In order to obtain the exact equality between iπ 5 − B22ðjω1;ω3Þ S ð ; qÞ∞ → S2ðjω1; ω3Þe 2 ; the matrix model and the residue of the integrand at −1 −iπ ω ω these poles, one needs to carefully study the extra factors in ; t → ω ω 2 B22ðj 2; 3Þ ð Þ∞ S2ðj 2; 3Þe ; the first line of (B9), their combination with the 5d Θð ; qÞ → e−iπB22ðjω1;ω3Þ; perturbative contributions (B4) as well as the residue of the matrix model at the trivial poles (perturbative part). Θ ; t−1 → −iπB22ðjω2;ω3Þ ð Þ e : ðB22Þ Also, in order to fully specify the matrix model, one needs to choose an integration contour. The right choice turns out Then the matrix model we are interested in becomes to be a Jeffrey-Kirwan prescription as studied in [40]. 3 Z Intuitively, the poles coming from the S ’s integrands will 3 3 S3 S3 S ∪S ð2Þ ð2Þ capture the contribution from large Hook diagrams (namely Z ð1Þ ð2Þ ≡ dcZ drZ Z ðZ ÞZ ðZ Þ L R cl L 1-loop L those constructed over a rectangle of size r × c and 3 3 1 S S considered in this appendix), while the contribution from S ð1Þ ð1Þ × ZintðZL;ZRÞZcl ðZRÞZ1-loopðZRÞ; ðB23Þ small Hook diagrams (namely those which do not contain the box ðr; cÞ) are accounted by additional poles coming where from the S1 piece.

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S3 S3 ∪ S3 APPENDIX C: AND ð1Þ ð2Þ PARTITION recalling useful definitions of partition functions on a FUNCTIONS squashed spheres or their intersections. The squashed S3 partition function of a UðnÞ gauge In this Appendix, we establish the exact equality 5 theory coupled to n ¼ n fundamental and antifundamen- between the residue of the S integrand at the selected f af tal chirals and one adjoint, which we will refer to as poles (3.1) (with nð3Þ ¼ 0) and the S3 ∪ S3 matrix model ð1Þ ð2Þ UðnÞ-SQCDA, is given by (3.4) in the notation used in the main body. We start by

Z nσ P Y S3 d −2πiζ σ −1 Z ≡ e a a 2 sinh πbðσ − σ Þ2 sinh πb ðσ − σ Þ UðnÞ-SQCDA ð2πiÞnn! a b a b Q a>b Ynf n Yn 1 s ðþiQ=2 þ σ − m˜ Þ iQ × Qa¼ b a i s − σ þ σ þ m : ðC1Þ n s −iQ=2 σ − m b 2 a b adj i¼1 a¼1 bð þ a iÞ a;b¼1

As usual, b denotes the squashing parameter, Q ≡ b þ b−1, 5d/3d superpotentials, which is indeed the case throughout ˜ while mi, mi, and madj denote the complexified masses our paper. For example, we have mass relations of fundamental, antifundamental, and adjoint chiral multiplets ð1Þ ð2Þ i 2 2 b 1 m − b 2 m b − b : C5 ð Þ i ð Þ i ¼ 2 ð ð2Þ ð1ÞÞ ð Þ ≡ R − iQ ˜ ≡ ˜ R ˜ iQ m m q 2 ; m m þ q 2 ; The matrix model (C3) should be understood as a contour integral with a Jeffrey-Kirwan residue prescription. iQ 1 2 m ≡ mR − q ; Take nð Þ ¼ 1;nð Þ ¼ 1 as an example. There are two sets adj adj adj 2 ðC2Þ of poles, the first of which is given by ζ and is the FI parameter. Let us denote the integrand ð1Þ ð1Þ ð1Þ ð1Þ −1 3 σ ¼ m − im b 1 − in b ; simply as ZS ðσÞ. Then the partition function of a ð Þ ð1Þ UðnÞ-SQCDA 2 2 2 2 −1 α 3 3 σð Þ ð Þ − mð Þ − nð Þ ð Þ S ∪ S ¼ m i bð2Þ i bð2Þ; ðC6Þ pair of Uðn Þ-SQCDA on ð1Þ ð2Þ, interacting through a pair of 1d bi-fundamental chiral multiplets at the α α S1 S3 ∩ S3 for all mð Þ; nð Þ ≥ 0, while the second intersection ¼ ð1Þ ð2Þ,isgivenby σð1Þ ð1Þ − −1 − nð1Þ −1 S3 ∪S3 ¼ m ið Þbð1Þ i bð1Þ; ð1Þ ð2Þ Z 1 2 2 2 2 −1 Uðnð ÞÞ-SQCDA∪Uðnð ÞÞ-SQCDA σð Þ ð Þ − nð Þ ¼ m i bð2Þ; ðC7Þ Z 2 ðαÞ α Y Yn σð Þ S3 d a ð1Þ ð1Þ α ≡ Z 1 ðσ Þ nð Þ ≥ 0 ðαÞ Uðnð ÞÞ;n ;n for all . Clearly, the second set come from the poles α 1 1 2πin ! f af 1 ¼ a¼ of ZS , since this set of poles satisfies 3 1d chiral 1 S S σð1Þ σð2Þ ð2Þ σð2Þ × Z1 ð ; ÞZ ð2Þ ð Þ; ðC3Þ d chiral Uðn Þ;nf ;naf ð1Þ ð2Þ i 2 2 sinh π b 1 σ − b 2 σ − b b 0; C8 ð Þ ð Þ 2 ð ð2Þ þ ð1ÞÞ ¼ ð Þ where the contribution from the 1d chiral multiplets is captured by thanks to the mass relations mentioned above. With these definitions, the equality (3.4) and the master identities (3.7) S1 σð1Þ σð2Þ Z1d chiralð ; Þ can be explicitly verified (e.g., by using Mathematica). YYnð1Þ Ynð2Þ 1 ¼ : ð1Þ ð2Þ i 2 2 1 1 2 π σ − σ APPENDIX D: THE REFINED TOPOLOGICAL a¼ b¼ isinh ðbð1Þ a bð2Þ a 2 ðbð1Þ þ bð2ÞÞÞ VERTEX AND DIM ALGEBRA ðC4Þ 1. The refined topological vertex In general, the parameters in the two SQCDA are The topological vertex formalism [59] and its refinement independent, however, when they are the world volume [62,63] are powerful tools to study 5d instanton partition theories of intersecting codimension 2 defects in a bulk 5d functions and their properties. In this paper we will mainly N ¼ 1 theory, the masses are likely to be related due to follow the conventions of [63], which we now review.

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The relevant vertices7 are graphically represented in Fig. 8. Note that at each vertex there are two black and one white arrows (the preferred/instanton direction), each labeled by a Young diagram. The three arrows are ordered in a clockwise manner, keeping the white arrow in the middle. For example, in the two diagrams in the Fig. 8, the white arrows are labeled with λ2, and is also the second index of the vertex. Lowered/raised indices of the vertex correspond to incoming/outgoing arrows. These graphical FIG. 8. Refined topological vertices. vertices represent the following contributions to the full amplitude,

X λ − λ N 1=2 λ ρ j j j 3j −1 where we defined Q ≡ −qð−yÞ =t x. The state j{Pμi 3 t ; q t p 2 q t Nðx;yÞ Cλ1λ2 ¼ Pλ2 ð ; Þ fλ3 ð ; Þ λ and its dual h{Qμj give a Fock basis, and the labels ðn; kÞx λ∨ ρ λ ρ are DIM representations specified by the integer values of × {Pλ∨ λ∨ ð−t q ; t; qÞPλ λðq t ; q; tÞ; ðD1Þ 1 = 3= the two central charges and the complex spectral parameter. 0 1 1 X λ − λ In particular, ð ; Þx is called vertical, while ð ;NÞx is λ λ ρ j 3j j j 1 2 ∨ −q ; t q p 2 q t C λ3 ¼ Pλ2 ð ; Þ fλ3 ð ; Þ called horizontal. They are isomorphic and related by the λ so-called spectral duality [101,105,106], a manifestation of λ ρ λ∨ ρ q ; q t ∨ ∨ −t q ; q t the SLð2; ZÞ group of automorphism of the DIM algebra. × {Pλ1=λð t ; ÞPλ3 =λ ð ; Þ: ðD2Þ In the web diagram, the choice of preferred/white direction

The Pλ=μðx; q; tÞ is the skew Macdonald function of the correspond to the choice of vertical representation, to which Φ Φ sequence of variables x ¼ðx1;x2; …Þ with Young dia- or are attached. See Fig. 9 for an illustration. As the basic example, let us consider the resolved grams λ ¼ðλP1; λ2; …Þ and μ ¼ðμ1; μ2; …Þ as parameters, λ ≡ λ conifold amplitude with preferred direction or (0,1) repre- while j j i i denotes the total number of boxes in λ − sentation along the vertical direction the diagram and { is the involutionP {ðpnÞ¼ pn acting ≡ n on the power sums pn ixi . The other parameters q ≡ 2πiϵ1 t ≡ −2πiϵ2 p ≡ qt−1 1;N 0; 1 1;N 1 e ; e and are complex numbers. ð Þb ð Þa ð þ Þ−uv h∅jΦ∅ Φ∅ j∅i The vertices can be joined together to form web diagrams 1 1 0 1 1 ð ;Nþ Þ−ab ð ; Þv ð ;NÞu corresponding to CY or ðp; qÞ-webs engineering 5d super- X jλj ∅ q t λ∅ q t symmetric gauge theories. In doing so, each internal line ¼ ðv=aÞ Cλ∅ ð ; ÞC ∅ð ; Þ; ðD4Þ λ is further associated to a complex parameter Qjλj and a n framing factor fλðq; tÞ (for us n ¼ 0), and the correspond- ing Young diagrams are summed over. where uv ¼ ab. Alternatively, we could have put the preferred direction or (0,1) representation along the hori- zontal direction 2. DIM intertwiners The topological vertex can be interpreted as matrix ð1;N− 1Þ− 0 0 ð0; 1Þ 0 v =u u elements of DIM intertwining operators in the h∅jΦ∅ ð1;NÞ 0 Macdonald basis [104], namely v 1 ð ;NÞa0 N × Φ∅ j∅i μλ jλj −1 2 ν − μ fλ ðq; tÞfνðq; tÞ q t = j j j j ð0; 1Þ 0 ð1;N− 1Þ− 0 0 C νð ; Þ¼QN ðt vÞ b a =b ðu;vÞ hPλjPλi X 0 0 jλj ∅ q t λ∅ q t 1 1 ¼ ðb =u Þ Cλ∅ ð ; ÞC ∅ð ; Þ; ðD5Þ ð ;Nþ Þ−uv λ × h{PμjΦλ j{Qνi 0 1 1 ð ; Þv ð ;NÞu 1 0 0 0 ν −jλj −1=2 jμj−jνj where a =b ¼ v =u . The two results should agree Cμλ ðq; tÞ¼Q ðt uÞ N v;u N ð Þ fλ ðq; tÞfνðq; tÞ because of slicing invariance of the topological vertex, 0 0 and they do provided v=a ¼ b =u ≡ Q0, which is the ratio 1;N 0; 1 ð Þv ð Þu × h{PνjΦ j{Qμi; ðD3Þ of the outgoing/incoming spectral parameters associated to λ 1 1 ð ;Nþ Þ−uv the (0,1) representations. From the DIM perspective, this should descend from the SLð2; ZÞ automorphism of the 7There are two more vertices with different directions of the algebra, see Fig. 10 for an illustration. A more complicated white arrows. However, we choose to build the web diagrams choice is to assign the preferred direction or (0,1) repre- with just the two in Fig. 8. sentation to the diagonal direction. Now the composition of

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FIG. 9. DIM intertwining operators. the intertwiners acts on the tensor product of two Fock Z2 ¼h∅jΦ∅Φ∅Φ∅Φ∅j∅i; ðD8Þ spaces, and the corresponding amplitude is X ∅ ⊗ ∅ ⊗ ∅ 1 ð1; 1 − MÞ 00 Z3 ¼h j h j h j ∅ ⊗ ∅ Φ b h j h j λ 0 1 1 − X 1 ⊗ Φλ ⊗ Φ Φλ ⊗ Φ ⊗ 1 λ hPλjPλi ð ; Þ−b00=a00 ð ; MÞa00 ð 1 λ1 Þð 2 λ2 Þ × hPλ jPλ ihPλ jPλ i ð1; −MÞ 00 ð0; 1Þ− 00 00 λ1;λ2 1 1 2 2 v u =v ⊗ Φλ j∅i ⊗ j∅i 1; 1 − M 00 × j∅i ⊗ j∅i ⊗ j∅i: ðD9Þ X ð Þu 00 00 jλj ∅λ ∅ ¼ ða =v Þ C ∅ðq; tÞC∅λ ðq; tÞ; ðD6Þ λ Of course, we need suitable identifications between param- eters. Anyhow, from the form of the matrix elements it is 00 00 00 00 where b =a ¼ u =v . This corresponds to the Nekrasov immediate that Z1 should correspond to a U(1) theory, Z2 to partition function of the 5d pure U(1) SYM theory with a free theory and Z3 to a Uð1Þ ×Uð1Þ theory. Also, since 00 00 −1 q instanton counting parameter a =v . This expansion coin- the Wq;t ðA1Þ or -Virasoro algebra can be represented on cides with the previous ones provided we identify a00=v00 ¼ the tensor product of two horizontal DIM representations, Q0. See Fig. 11 for an illustration. while Wq;t−1 ðA2Þ can be represented on the tensor product For the next level of complication, we can consider the of three horizontal DIM representations, the resulting 5d geometries considered in the main text. As we discussed, N ¼ 1 quiver gauge theories match with Kimura-Pestun there is a frame corresponding to a U(1) theory with two construction of quiver Wq;t−1 algebras [94]. In their con- flavors (Fig. 1 left), a frame corresponding to four free struction, the basic object is the Z operator, which is an hypers (Fig. 1 center) and a frame corresponding to a infinite product of the Wq t−1 screening charges. From the Uð1Þ ×Uð1Þ theory with one bifundamental hyper (Fig. 1 ; DIM perspective, we can identify right). It is now easy to recognize the various topological amplitudes as (vacuum) matrix elements of intertwining X Φλ ⊗ Φλ operators between various representations, and the fact that Z½A1¼ ; they should agree is expected from the SLð2; ZÞ auto- λ hPλjPλi morphism of DIM. In particular, we can identify (we X 1 ⊗ Φ ⊗ Φ Φ ⊗ Φ ⊗ 1 ð λ1 λ1 Þð λ2 λ2 Þ neglect the unnecessary labels in order to avoid cluttering) Z½A2¼ : ðD10Þ hPλ jPλ ihPλ jPλ i λ1;λ2 1 1 2 2 X ðΦ∅ ⊗ 1ÞðΦλ ⊗ ΦλÞð1 ⊗ Φ∅Þ Z1 ¼h∅j ⊗ h∅j j∅i ⊗ j∅i; λ hPλjPλi On the other hand, it is known that Kimura-Pestun construction as an analogous for 3d N ¼ 2 quiver ðD7Þ gauge theories, which involves a finite number of Wq;t−1

FIG. 10. Action of the S element in SLð2; ZÞ (for N ¼ 0).

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FIG. 11. The third triality frame for the resolved conifold. screening charges [90,107–109]. An efficient control on the the 5d theories) would imply an elegant description of transformation relations between the DIM operators in some 3d dualities. The peculiar example of the self-mirror different duality frames and at specific points in the T½UðNÞ theory [7] has been recently considered in [30] parameter space (corresponding to complete Higgsing of from the Wq;t−1 perspective.

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