JHEP05(2015)095 ) gauge Springer N May 4, 2015 May 19, 2015 : : February 11, 2015 : . We explain how 1 S Accepted × Published 2 10.1007/JHEP05(2015)095 , Received R doi: b Published for SISSA by in the presence of codimension two defects. 1 S × 4 R and Peter Koroteev b [email protected] , . 3 Hee-Cheol Kim a 1412.6081 The Authors. Supersymmetric gauge theory, Duality in Gauge Field Theories, Integrable c ⃝ We study the Nekrasov partition function of the five dimensional U( , [email protected] Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, OntarioE-mail: N2L2Y5, Canada [email protected] Institute for Advanced Study, Einstein Dr., Princeton, NJ 08540, U.S.A. b a Open Access Article funded by SCOAP Equations in Physics, Differential and Algebraic Geometry ArXiv ePrint: We check, as anfunctions expansion are in eigenfunctions of thethe an instanton Seiberg-Witten elliptic number, geometry integrable of that many-body the the five-dimensional system, gauge aforementioned which theory. partition quantizes Keywords: Abstract: theory with maximal supersymmetry on The codimension two defects can beto described a either certain as class monodromythese of defects, or three computations by dimensional coupling are quiver connected gauge theories with on both classical and quantum integrable systems. Mathew Bullimore, Defects and quantum Seiberg-Witten geometry JHEP05(2015)095 9 6 4 3 1 30 39 20 40 15 15 12 43 27 13 11 30 14 44 21 48 24 27 30 31 26 39 43 38 40 41 34 36 38 18 18 32 = 4 SYM –i– N [U(3)] [U(2)] T T )] theory N ) theory N [U(2)] theory [U(3)] theory [U(2)] theory [U( T T T T )] N partition function [U( 3 4.5.2 U(3)4.5.3 theory Comments4.5.4 on normalizability of wavefunctions Connection to 4d index of class S 4.3.1 U(2)4.3.2 theory U( 4.5.1 U(2) theory 3.6.2 3.5.1 3.5.2 3.5.3 Givental J-functions 3.6.1 S T 2.5.1 Higgsing 2.5.2 Higgsing 2.2.1 Electric2.2.2 frame Magnetic frame 4.6 5d theory4.7 coupled to 3d S-transformation hypermultiplets 4.4 Wilson loops4.5 5d/3d coupled system 4.1 Instanton counting 4.2 Ramified instantons 4.3 3d decoupling limit 3.6 Difference equations in electric frame 3.1 3.2 ’t Hooft3.3 operators Interfaces 3.4 3.5 Holomorphic blocks 2.5 More genetic quiver varieties 2.1 The Nekrasov-Shatashvili2.2 correspondence Quantum/classical duality 2.3 Line operators2.4 and interfaces in Quantum equivariant K-theory 4 5d/3d partition functions 3 3d partition functions Contents 1 Introduction 2 Twisted chiral rings JHEP05(2015)095 63 54 61 63 58 57 66 58 53 54 61 64 60 61 56 66 66 51 52 69 66 61 67 51 ]. We focus on a class 2 , ] or by coupling to a class 1 4 , 3 –1– = 4 supersymmetry in three dimensions. They = 2 supersymmetric gauge theories with codimen- N N ) N [U(2)] T partition functions ρ T 6.1.3 Gaudin-spin chain family 6.1.1 Calogero-Ruijsenaars-Dell6.1.2 family Toda family 5.2.1 Open5.2.2 Toda from decoupling limit Closed Toda chain 5.4.1 Equivariant5.4.2 quantum cohomology Coupled 4d/2d systems 5.1.1 Chiral5.1.2 limit of Effective5.1.3 twisted superpotential Open Toda Lax matrix In this work, we study 5d U( A.2 Double sineA.3 functions Theta functions 6.2 Open problems A.1 Q-hypergeometric functions 6.1 Integrable systems 5.3 Connections to5.4 Kapustin-Willett results 4d/2d construction 5.1 Complete flags and open Toda chains 5.2 Pure super Yang Mills theories with defects and closed Toda chains sion two and codimension fourthe integrable defects system and associated how to theyof its are codimension Seiberg-Witten connected geometry two to [ defects thecan preserving be quantization described of either as Gukov-Witten monodromy defects [ Supersymmetric gauge theories provide amathematics. rich source From of a inspirationniques practical for to various viewpoint, branches solve they of challenging cansupersymmetric mathematical also gauge problems provide theories using and a physics.preserve mathematics powerful some The is set amount interplay enhanced of of between by supersymmetry. tech- introducing defects that 1 Introduction B Factorization of C Perturbative 5d partition functions A Conventions of special functions 6 Summary and outlook 5 Gauge theories with chiral matter and 4d reduction JHEP05(2015)095 =2 -body -body ]. The N N N 0 where partition partition 11 ) , 3 3 → N S S 10 Q in the presence of surface defects ]. We will show that the twisted 2 6 ,ϵ )] theory as a surface defect in 5d )] theory in the limit 1 4 ϵ N N R . × [U( [U( Q 1 T )] theory, we check that this computation T S N ]. The codimension four defects are described 5 –2– [U( . This computation is performed by treating the )] [ T 1 ]. This is indeed a deformation of the trigonometric 2 ϵ N R 15 . The Seiberg-Witten curve of the 5d U( × [U( 4 1 T R S can then be obtained by factorization of the × 2 ϵ 1 R , building on the work of [ S 2 × R , otherwise known as the three-dimensional Omega background partition function of 2 1 1 × S 2 ϵ R = 4 theory , which is related to the 5d holomorphic gauge coupling. According 1 R ) flavor symmetries. The twisted chiral ring relations, or equivalently S Q N × )] theory using results from supersymmetric localization [ N 1 N S ]. We show that these supersymmetric partition functions are eigenfunctions . The corresponding eigenvalues are given by supersymmetric Wilson loops [U( 13 ϵ T ) supersymmetric gauge theory on , ]. As it is technically simpler, we will first consider the squashed ]. We will also explore the connection with quantum K-theory for more general 0. N ]. In order to check that the Gukov-Witten monodromy defect is reproducing 9 8 12 = 2 gauge theory, the twisted chiral ring relations are deformed by an additional [ = 4 supersymmetric gauge theories. We will concentrate on the surface defect ob- , 17 2 ϵ → )] theory on 7 for rotations in N , ] it is expected that this deformation provides a presentation of the Seiberg-Witten N R ϵ N ϵ ) In order to test this relationship, we will compute the 5d Nekrasov partition function of In coupling the three-dimensional theory This classical integrable system can be quantized by turning on an equivariant param- As a preliminary step towards understanding and computing with surface defects, we 16 14 × = 2 U( N [U( 1 reproduces the the coupling to theGukov-Witten monodromy 5d parameters degrees are of identifiedof freedom with the the is 3d Fayet-Iliopoulos turned parameters gauge theory off. supported In on the particular, defect. we note that the N wrapping one of thesurface two-planes defect as ain monodromy [ defect and applyingthe the same orbifolding surface procedure defect introduced as coupling to to [ curve of the 5dsupersymmetric theory gauge theory on is knownelliptic Ruijsenaars-Schneider to system correspond [ to theRS spectral system curve by of an the additional complex parameter the spectral curve oflimit the classical integrable system, are reproduced inU( the semi-classical complex parameter partition functions on function [ of the quantized trigonometric Ruijsenaars-Schneider systemportional with to the Planck constant pro- for background U( linear quiver gauge theories. eter S function of T chiral ring relations are equivalentintegrable to system, the known spectral as curvetem. of the an In complex associated addition, trigonometric classical the this Ruijsenaars-Schneider cotangent provides (RS) bundle a sys- todence reformulation a [ of complete the flag equivariant variety, quantum via K-theory the of Nekrasov-Shatashvili correspon- tained by introducing the mostcoupling generic to monodromy the for 3d theby gauge supersymmetric field, Wilson or loops. alternatively by will first find a reformulation of the twisted chiral ring of a canonical deformation of the of 3d JHEP05(2015)095 )]. We N = 2 gauge [U( N , deformed by T we show that 2 R 3 × mass deformation. 1 ∗ we present a cursory S and demonstrate that 5 =2 4 R N × 1 S ] that the equations determining the we will relate the twisted chiral ring of . In the the Nekrasov-Shatashvili limit 8 are eigenfunctions of the corresponding 2 , Q 2 7 we explain how to compute the expectation R 4 elliptic Ruijsenaars-Schneider system. Indeed, × , or the twisted chiral ring relations, can be ] relates this partition function to the partition 2 1 –3– R 20 S ) gauge theory on dual × N 1 and S 3 = 4 linear quiver gauge theories on S N = 2 U( N ]. We focus on the simplest example where two free hypermultiplets 19 we summarize the connections of this work to integrable systems and , 6 18 as an expansion in the parameter 2 ,ϵ -body integrable system known as the complexified trigonometric Ruijsenaars- 1 4 ϵ N R )] to the spectral curve of the classical trigonometric Ruijsenaars-Schneider system × 0, we will show that the expectation value of the most generic surface defect is formally N 1 We will explain how the corresponding statements for more general linear quivers can We will focus for the most part on the triangular quiver gauge theory: The paper is organized as follows. In section We will also study another type of codimension two defect by coupling directly to 3d After performing this preliminary check, we study the full Nekrasov partition function in the 5d gauge theory. This computation provides a quantization of the Seiberg- S → [U( 1 2 be obtained by aa combination simple of example. Higgsing and Weliterature mirror will on symmetry, also and the briefly demonstrate discuss equivariantmanifolds. this connections quantum in to K-theory results of in the the cotangent mathematical bundles to partial flag will reformulate the statement ofclassical its twisted chiral ringSchneider in system. terms of Alternatively, thediagonalizes it spectral this curve can of classical be a integrablewe viewed system. come as to a This couple reformulation Lagrangian this will theory correspondence as be that a important codimension when two defect in five-dimensions. In this section, we will studyhypermultiplet 3d masses, FI parameters,It and with was the observed canonical bysupersymmetric Nekrasov massive and vacua Shatashvili on identified [ with Bethe ansatz equations for a quantum XXZ integrable spin chain. in section section discuss areas for further research. 2 Twisted chiral rings partition functions on squashed quantized integrable system. Then invalues section of surface defectsthey in are eigenfunctions ofdiscussion the of elliptic various limits RS and system. degenerations of Later the results in presented section in this paper. Finally, the S-transformation of the 3dfunction theory of in the [ U(2) gauge theory with a monodromyT defect. and discuss connections to equivariant quantum K-theory. In section Witten geometry. hypermultiplets [ of U(2) flavor symmetrytheory. are We coupled will to show that the thetion bulk partition equation function gauge of of the field this so-called of coupled two-body system the solves 5d an eigenfunc- U(2) on ϵ an eigenfunction of theeigenvalues elliptic are RS given by system. theS expectation Furthermore, values we of find supersymmetric that Wilson the loops corresponding wrapping JHEP05(2015)095 C G (2.1) ) and × L and FI ). This × ×··· H N ) 1 G m U( M is the diago- R-symmetry m, t, ϵ ( R . Similarly, we . The physical C ) i } } ( j ) j × ×··· +1 ( M . Furthermore, we W ) 2) theory obtained ϵ L 1 , SU(2) N × H = (2 ,...,t ,...,m ) 1 j t N ( 1 { L L m supersymmetry. We refer the N M { ∗ . The anti-diagonal combination C = 2 vectormultiplets for =2 , N N . This symmetry can be enhanced by =1 SU(2) . It is convenient to introduce a U(1) j with gauge group U( t 1 " ...... × L i + 1) in the IR. − H W A with real mass parameter ∂s ∂ L ϵ +1 j –4– t πR 2 2 2 . In this case, the real deformation parameters are ! N M R = 4 supersymmetry have SU(2) exp ) for the dynamical vectormultiplets. The supersymmet- -th gauge node. i acting on the fields parametrizing the Higgs and Coulomb N = 2 supersymmetry. The corresponding U(1) with corresponding parameters C 1 1 G N +1 N 2) supersymmetry. M L × m, t, ϵ, s , ( . We refer to this setup as H t -th gauge node is W G j = (2 N each with an associated effective twisted superpotential = 4 quiver gauge theory of type 2 R N × ] for a more complete description of this setup. 1 6 S . 3d ). The corresponding mass parameters are denoted by L For generic deformation parameters there is a discrete set of massive supersymmetric We will focus on he twisted chiral ring of the effective 2d Here, we focus on theories with a UV description as a linear quiver with unitary gauge fundamental hypermultiplets at M i the UV theory one cantwisted integrate superpotential out three-dimensional chiral multiplets toric find vacua an are effective then solutions to complexified by background Wilson linesin wrapping the the circle language and of behave as twisted masses vacua on is a holomorphic function that is independent of superpotential and gauge couplings. In the UV description is U(1) FI parameter at the monopole operators up to a maximum of U( by compactifying on a circle of radius groups. Our notationsymmetry acting is trivially summarized such that inU( the figure Higgs branch symmetryintroduce is an given additional by topological U( U(1) so that the Coulomb branch symmetry manifest in turn on real deformationand parameters by giving coupling a to vacuumformation expectation parameters value enter to as theparameters real real denoted hypermultiplet scalar. by masses In denotedreader a typically to UV by [ description, these de- Three-dimensional theories with and flavor symmetries branches respectively. For the purposedeformation of preserving this only paper, it isnal important combination to of turn Cartan on generatorsbecomes a of an SU(2) canonical additional flavor symmetry U(1) Figure 1 M 2.1 The Nekrasov-Shatashvili correspondence JHEP05(2015)095 . i , is i δ 1 (2.3) (2.4) (2.5) (2.6) (2.2) 1) − ,...,N =( , which is a ) =1 . ) i i ( a m ( α µ ) α σ ϵ πR − − e ) ) i for i ( a = ( α ) . s, m, ϵ ] that these equations i ( ηµ ( α ) ησ 8 ). We refer the reader ) σ i , i ( a 7 =1 µ M & a = 1-loop ,η · πRm − ) , i ) u W ( α i u ( 1 α ( +1) + σ i i − ( β ) πRs =1 i M ]. − 2 ’s. For this purpose, we introduce & σ a ( α = ) e 6 s i − ( α ) +1) i = σ i i =1 +1 )= ( ( α β N − i ) # +1 α i u + i . ( α ( Q ησ ησ i % a Q ) = log(2 sinh j ]. R m ++ i +1 6 =1 i iδ m 2 −− i i ) can be expressed in terms of these polyno- & Q to be the 1-loop contribution to the scaling ( β N =1 1 ℓ Q M · i a + # ,σ 1 − 2.4 –5– m − i ,P i ) ) − N i i + i ) ( ( β α +1 +1 2 Q ) i The second term includes the 1-loop contributions i i Q t t πRt − ( α i 2 ησ ησ − 1 + πR∂ i σ P e − j 1 =1 1. − − P i i # i ) ) − − +1 t i i = i τ i − ( ( α β i u τ $ L =1 σ σ N ( i j # 1 1 +1 i i ∆ L − − =1 + =1 N i + # − & η η α N η α +1 i + ,τ i ̸= and monic polynomials )= i N ) & )= i β N ( a N u u · ) from a three-dimensional chiral multiplet of mass ( + i ) + m i 1) i ( α ( 1 πRm Q − ℓ 2 σ i − M ( β e i . It was observed by Nekrasov and Shatashvili [ s, m, t, ϵ − σ ( N = = 1) − ) W + ) − i i i i i ( a ( ( α β µ ,...,L M ησ ησ 1 = =1 − i =1 i i δ & β N ] for an explicit expression. The final term is included to ensure that mirror symmetry In order to write down the twisted chiral ring it is necessary to introduce a generating As the imaginary components of the twisted mass parameters are periodic, it is con- The effective twisted superpotential of the generic linear quiver shown in figure Sign conventions are slightly altered compared to [ 6 i +1 1 τ i τ where the polynomialsWe are defined understood ∆ to be evaluated at The equations for supersymmetricmials vacua as ( can be identified withcomplete the dictionary for Bethe linear equations quivers of can a be quantum found in integrablefunction [ for XXZ the spin gauge chain. invariantan combinations auxiliary of A parameter the for all venient to introduce exponentiated parameters With this notation, the equations for the supersymmetric vacua are from the KK tower of chiralis multiplets. the The basic contribution building blocksolution of the of 1-loop the contributions differentialto equation [ 2 acts straightforwardly in the presence of the spurious U(1) symmetries. The first term includesphase contributions from the FI parameters at each node together with dimensional theory. given by which can be identified with the twisted chiral ring relations of the effective two- JHEP05(2015)095 ]. ]. ). 6 6 2.4 (2.8) (2.7) . 1 − η to absorb − j ∆ , for example . 1 3 η u =1 − i j ! η . To recover equa- ) in i τ +1 i , 2.7 = N . i +1 i u was mapped onto ' τ + Q η i 1 N − ) in N i − Q i 2.6 1 P − ], where ) i 6 i 1 ) cancel out and we reproduce ( ' τ N hypermultiplets at the final node. − u ( η -th gauge group. The superscripts on i − + i N N-1 ' i Q ↔ +1 i M η ' τ )] theory consists of a sequence of gauge groups N =( –6– ). Dividing one equation by the other, the com- [U( a + i u τ T ( ' Q i − i ↔ Q Q 2 a ). The twisted chiral ring relations for gauge invariant i of ' µ τ u ) 1 mass deformation. Schematically, we have. i − ( α − ∗ σ η − i ( ) of the linear quiver, which is spelled out in reference [ i ' Q i ) — this will be our main example throughout this paper. =2 1 Q 2 We first rescale the FI parameters by + i has a di fferent sign compared to [ ,M Q 2 N η i )= +1 N . Then we introduce the polynomial equations i u j ' τ ( = 4 theories have a remarkable duality known as mirror symme- ’s are given by expanding equations ( − i ) 1) with bifundamental matter and i ( α ) and the auxiliary polynomials Q i N σ − ' τ ), N − )] (see figure ηu ( N U( ) are auxiliary polynomials of rank +1 i i ]. This can be understood from brane constructions in Type IIB String the- u ' τ Q ) for the supersymmetric vacua we shift the argument this polynomial equation by ( . A Lagrangian description of the [U( i 22 ' ], where it is realized as the S-duality. Mirror symmetry acts in quite a non-trivial T , Q 2.6 )= × ×··· 23 21 This correspondence is most straightforward to understand in the case of the triangular Finally, 3d In what follows, it will be useful to introduce another slightly less familiar reformulation u We thank Davide GaiottoHere for the exhibiting mirror us map this for calculation in a sample example. and evaluate both at roots ( 2 3 + i ± ticles in one dimension — the complexified trigonometric Ruijsenaars-Schneider system [ quiver 2.2 Quantum/classical duality As mentioned above, theBethe equations equations for for supersymmetric aequations vacua are quantum can related integrable be to a spin identified second, chain. with classical the integrable Remarkably, system of the interacting same relativistic par- system of acts non-trivially on the where the check symbol designates parameters of the dual theory. try [ ory [ manner on the dataMirror symmetry ( interchanges mass parameters and FI parameters of the quivers and also where tions ( η bination ( The twisted chiral ring relations are given by expanding equations ( of the twisted chiral ring. the dependence on ∆ dimensions of monopole operatorsthe charged polynomials are under shorthand forQ multiplicative shifts of the argumentscombinations by of Figure 2 U(1) JHEP05(2015)095 L⊂ (2.9) (2.12) (2.13) (2.14) (2.10) (2.11) [U(2)] T for )) L . ϵ 2 1 . This theory is − + η ) and we have corre- ↔ +1) ′ , j . N ( α ,...,N " ,η s j τ j j τ j τ . p + W dp =1 ∂t of the cotangent bundle to 1 1 ) ∂ j ↔ σ ∧ ( α j τ ησ s j j πR − j . = 0 to simplify notation. For µ − M 2 τ − 2 dτ ( ! , and generating function given by 2 with ℓ j is consistent with the above conventions +1 =1 2 ηµ µ ). τ τ =1 N j j 1 2 N µ j and # + ) ) τ δ ,ϵ t Ω τ 1 σ 1 1 j µ ) µ = σ = exp − = ,t ϵ and 2 ησ i τ j τ M − µ = 2 µ j and Ω + m − 2 µ 2 τ ( α 2 , ) ) of this theory is given by µ ηµ W j +1) m ′ ) ( α j ϵ ( α –7– )( )( s ,p , ,p 2.2 1 1 s = " − j µ 1 1 ,p σ j =1 j µ j ) − ) ησ σ 1 ′ # α p j dp − ) N ησ W ( α σ j ( α − − s % 1 ( α ∂ ∂m s ∧ − s j 1 1 = R − j 1 ( µ ηµ and FI parameters ) sweep out a complex Lagrangian in the product iδ j 2 1 τ ) ( ( πR ℓ ηµ µ p j µ j 1 2 2 dµ ( α 2 ( , so the this definition of + τ τ µ s τ ! j 2.10 ( =1 =1 N +1 = ′ +1 = ℓ j # j # j α ′ j t 1 µ α = N p j =1 ̸= − = exp # ) µ # α α j j µ 1 1 t Ω 2.4 p and we define − − $ =1 =1 j j # # N N 4 =1 N 1, + + j # − = j with holomorphic symplectic form Ω W = τ j with the following holomorphic symplectic forms δ ×M N It is straightforward to find an explicit description of the Lagrangian By introducing the conjugate momenta to The effective twisted superpotential ( ) In the notations of ( µ 4 ∗ C and for the masses modulo two. The conjugate momenta for the FI terms are as follows M the on-shell twisted effective superpotential theory. The supersymmetric vacua equations read as follows This is the phase spacefor of the our conjugate complex classical momenta integrable ( system. The defining equations we provide canonical coordinates( on two copies this theory, the Higgs andsponding Coulomb exponentiated branch symmetries mass are bothinvariant under U( mirror symmetry with the transformation where JHEP05(2015)095 are L (2.19) (2.15) (2.17) (2.18) (2.16) . The 2 τ ,p 1 τ p ). Therefore, ) are true it is . , 2.8 ) for each node 2 2 τ u µ 1 ( ) can be presented 1 2.18 τ j µ ' Q 2 is rather non-trivial = ) evaluated on super- 2.12 = 2 µ α ), 2 τ p u p 1 µ ( and momenta ) and ( N> 1 τ . , j ,ϵ, s 2 ) , j α a Q j ) p ,τ τ 2.17 j ,t ) 1 a µ 1 τ − ,p − ,p m 2 ) so the Lagrangian correspondence − from the supersymmetric vacuum u ( β 2 τ k ( u 2 µ µ k α α ( W + =1 in the case N σ α ,p & j + 1 − j τ =1 N 1 µ τ & j 1 − i p p = , and to the latter relation as presented µ α β polynomials j = + ( or i i ) µ = 2 µ j 1 ̸= α j µ N )) = , p ( k 2 τ − p j p η ,η . ̸= 1 –8– N ,η 1 , 1 τ k 1 µ µ − − µ means this Lagrangian also diagonalizes the same 1 η η τ − τ,p 1 1 ( τ − µ, p )= 1 − ( → − L 2 − η − L 2 ,β µ η η τ − α 2 η − 2 ↔ µ u . τ j u η α, p τ * + ( + 1 µ -body complex trigonometric RS system. One can clearly see det ( ij 1 τ p det p L and N η 1 2 j are related to masses − 2 τ µ 2 η µ τ L 2 − ) are related to each other by mirror symmetry map ( τ ↔ − − 1 j − 1 − 1 µ τ µ η 2.18 η 1 1 τ µ of the quiver. Note that we treat the matter polynomial in a uniform manner, ) and ( ) the combinations appearing on the left are the Hamiltonians of the complex 2.17 ,...,N As we mentioned above, in order to understand why ( To eliminate dynamical vectormultiplet scalars 2.15 diagonalizes the system. From this perspective, =1 identified with FI parameters convenient to re-formulate the supersymmetricsuperpotential. vacuum equations arising from We this introducej twisted monic degree- is the Lax matrixthat for ( the somewhat artificially, we can referwhere to eigenvalues the of former relationin as the written in ‘magnetic’ the frame, ‘electric’ in frame, which, using mirror frame variables, the eigenvalues of and, second, as In both relations above equations in favor of the conjugatetask. momenta Below we demonstrated that one can do this in two equivalent ways. First as L symmetric vacua are solutionssymmetry of under the relativistic Hamilton-Jacobisystem with equation. the coupling The inverted evident In ( trigonometric RS system forright two hand particles sides with are positions independent of the momenta in two equivalent ways. First, as and, second, as Given the above definitions of conjugate momenta vacua equation ( JHEP05(2015)095 ) ' ' Q Q =2 Q Q = 1, 2.23 j (2.21) (2.25) (2.23) (2.24) (2.20) (2.22) ) it is j u ( 1 ) from the Q u , evaluating ) with ) ( ± ) with η +1 ). They can be j 2.20 2.10 u 2.20 Q ( j ' Q unknown coefficients ) and hence find . j u ( +1 . The original supersym- 2 j j ' τ Q Q 1 . Then from equation ( − i τ . j p , Q +1 . . ) 1 − ) using equation ( j τ j p u τ + + we can determine − 2 3 u ( 1 ). With this definition, we have q q 3 j u − − = − − 1 1 = j τ j τ ( ) as follows Q q q = 0 we can immediately solve for the ≤ i 1 1 1 1 N u (0) +1 q i τ τ τ τ +1 +1 ( j 1 (0) + j − j u Q j τ j ...p ...p − ˜ − − − − j Q Q Q 1 τ 1 τ Q in favor of the momenta conjugate to the FI 3 2 + 1 in the same equation and evaluating it on p p Q − − 2 3 =( τ τ )= ) for ) j j j q q −− j ++ j j − u –9– u ( α + j + + ). Then, from equation from ( 1 1 = 2 on the root of the polynomial 1) 1) ( ( Q Q i σ q q ' u 1 1 → Q = − − ) and P 2 3 ( j ' − − Q − j τ τ j 2 u j τ + j − j q ( p Q j Q Q = = j equation at τ Q 2 2 j +1 )= ' ) and (0) = ( (0) = ( τ − j ' Q Q Q u j j ) with τ u ( . ( − j ' Q ) Q Q 1 i j ' Q ' and hence ( α Q Q 2.20 + j σ ) = 0 and there is no need to redefine N Q = ( a ) and eliminating the auxiliary polynomials j σ u +1 u ) we have just enough data to determine the ( j ) and u j τ = ( u ( j Q a 1 ). Then, by shifting Q q m of ). For this purpose, we will set up an inductive procedure to solve the of ) j 2.20 ) ( α ) )= j σ ( 2.17 α u ( σ 2.7 1 ). Q u ( roots +1 j equation ( To illustrate this process, let us perform the first interation explicitly. It is convenient j ' Q ˜ Q straightfoward to compute the coefficient of the linear term in We can now immediately compute the polynomial we have we find Now, evaluating equation ( Q the in to introduce the monic degree one polynomials Now, given the polynomials and hence byconstant evaluating terms the in the polynomials 2.2.1 Electric frame Firstly, we explain how toparameters eliminate ( the equations recursively node by node. We first note that by the definition ( expressed uniformly as evaluated on the roots equations ( Note that in thismetric case vacuum ∆ equations are obtained byat shifting arguments the in roots the above by that is we define JHEP05(2015)095 ), j we µ u (2.32) (2.26) (2.27) (2.29) (2.30) (2.31) (2.28) − u ( ) in matrices , =1 N , j j ⎤ ⎥ ⎥ ⎦ 2.17 1 1 × − . . . − 1 j )= j j j i i u . τ τ ( . ) ) , P . k . ··· ··· +1 +1 µ ,j ,j j 1 i i . . . 1 1 , τ τ ∈I − − and found experimentally & k r . . . 1 1 } u j ) ,...,j ,...,j ⎡ ⎢ ⎢ ⎣ (0) 1 1 r ,η = = ,...,N , i τ M M , 1 # j r ( ( j τ |I| ,p u p , i ) τ + I⊂{ ,...,i i ) det det ( 1 1 (0) µ i r L − = ( ) T r η k k τ − )= χ τ k p =0 N τ u u r # ( − ( =0 N j ∈I − r i # & k = ' ). By expanding both sides of ( ,M τ Q η ( 1 + i i ) ⎤ ⎥ ⎥ ⎦ − τ – 10 – )= ̸= j N j η * j , j 2.17 k ,η − τ j − j 1 ) = det µ j i τ 1 i 1 , i τ ̸= N u q , q − . . . k ( − ,p 1 ) ) 1 − i i ) becomes the matter polynomial N − τ − u τ 1 j = j η j u i ( ( i Q ( i τ ,...,j ,...,j τ (0) L τ ij 1 1 =1 N N , the ratio of determinants can be reexpressed as a single . & j L . Then the solution is given by a ratio of Vandermonde-like . − M M Q . η ∈I ∈I ··· ··· i ( ( & j/ u are monic of degree one the above ratios can be simplified and * } ,...,N 3 3 -body complex trigonometric RS system. At the final stage of and we remind the reader that superscripts are not exponentials (0) i 1 − − 1 j q det det j j i τ i i r N . . . M q q p det = ,...,N j 1 i i 1 # − |I| τ τ )= . u u 1 1 ), or, more explicitly as ( I⊂{ i j − = − . . . 1 ]). j j i µ j i i Q ( q q q 24 r ⎢ ⎣ ⎡ ⎢ χ ,...,N 1 = , j )= =0 ,η ,...,i i τ 1 r i ,p Since all polynomials We have implemented this procedure to many orders in i M τ ( r T where and their eigenvalues Thus we can explicitly write the full set of conserved charges for trigonometric RS system providing us with thefind required explicitly relation the ( Hamiltonians is the Lax matrix of the the recursion, the polynomial by inverting the matrix spectral determinant where Solutions of this formliterature. for similar We expect functional thatfound equations these (see have e.g. techniques appeared [ could in be the used integrability to prove the solution we have where we define but shifts of the argumentdeterminants by that the solution can be expressed as follows. We introduce the following JHEP05(2015)095 , ∞ ) as ) ) u ). u u = (2.37) (2.33) (2.34) (2.36) (2.35) ( ( ( u j 2 2 ( u 1 ) ) − − Q − u u + N − N ( ( N 1 1 Q Q ) − − Q ) ) n , u u ++ N N ) u ( ( ) ) . Q Q 1 1 n = 0 and u u . Our proposition − u ) ) − − ) ) ( ( α , ). Now we want to u u 1 1 u −− N N m m m . In this case, it will ( ( − µ k − − j u u u Q Q τ = 1. The momentum ), − N + N ( ( µ 2 N + N − m 1 1 p u 2.18 r − Q Q u u ( Q Q ∈I − − du ( η 1 & k ) n ( + N − N 1 ) u − } du u ̸= − u , ( , Q Q =1 − N ( 1 N m proceeds in a similar manner. r 1 r ) − , ) Q C , τ N = r −− N ,...,N m N 5 m,n ) ) m 1 u Q # u C ) ) + Q |I| r of u 2 ( r 5 ( N N η ( ( u α α 1 2 u I⊂{ 2 du N ... du . σ σ −− N η − η − ( ( 2 Q = surrounds only roots of N ) i.e. the masses N N α + Q 1 1 1 η τ ... − u C σ k µ 1 − − 1 1 ( 1 5 1 p − − − =1 1 r + + N − N N & 2 u N N N N m . Q Q du ∈I ) using the supersymmetric vacuum equa- η τ τ τ Q 1 C & 2 1 ··· k r N 1) − N η − τ − − + + η η − C . It is now straightforward to see that the first + N 1 j 5 α × N ) N N N 1 – 11 – j ( α µ 1 τ τ τ τ µ ↔ N µ σ = τ − ( η − = = = = η k µ − ( 1 N = ) p ) ) i − N Q u u α µ N µ η τ ( α ( ( and p ∈I i 1 1 & σ k 4 j = µ − − µ η η ) can be expressed in terms of the polynomials + N − N α µ j β ∈I p ∈I j Q Q i & µ j/ ↔ β µ µ ) } j µ ) 2.10 − u τ − u ( − r 1 − ( 1 i = − ,...,N surrounds the roots of − α N µ 1 −− N η # η µ |I| i Q N Q α µ I⊂{ C µ u du α ∈I such that it surrounds the roots ∈I i & j/ ̸= N & } β in favor of the momentum conjugate to the masses, N C 5 ) C 3 r j ( 2 α = ,...,N ), and then express the result once again as contour integral. Performing these η 1 σ =1 N # 2 |I| # α η − I⊂{ 2.21 1 The argument for the Hamiltonian appearing at order We prove this proposition by induction. To perform the inductive step, we contract Let us first consider the first independent Hamiltonian with N τ as required. In the second line above contour We first express the Hamiltonian as a contour integral the contour then eliminate the dependencetions ( on steps, we find where the contour is that this contour integral evaluates to follows where we remind the readerHamiltonian that can be expressed as a contour integral related to those above by conjugate to the masses ( Let us now lookeliminate at the other presentationnot of be possible twisted to chiral provideof ring the an ( complex argument trigonometric that RS lands model. directly Instead, on we the attempt Lax to verify matrix the formulation mirror equations 2.2.2 Magnetic frame JHEP05(2015)095 ) N (2.38) (2.39) ]. 6 of U( ) ) ). Note the m m M m ) ) u u u ( ( m m ( 2 2 u u − − N ( ( 2 2 + N − N Q − − Q Q + N − N ) ) Q Q m m ) ) ) ) u u and complex coordinates coupled to the 3d gauge ( ( m m m m 1 1 u u N u u 1 ( ( − − ( ( S 1 1 1 1 −− N N − − − − Q Q + N − N −− N N 1 Q Q Q Q . =1 − ) r & r ) m i m =1 r τ 1 m 1 & u m − u ( − ( SYM 1 r ··· N − τ N ) can be interpreted as partition functions 1 −− N i ) r N ) in ensuring that the non-zero residues are τ Q n Q τ n ) can be interpreted as an identity between =4 r ) ) u ) u 2.35 1 n n =1 r N − − − u ) u & <...i # 2.39 m n – 12 – 1 N m i theory. m − − u r N u u ∗ τ ( 2 m m − n ) − u )= u n ̸= ( 2 η , m τ =2 τ ,..., ) u n ( m ( − u 1 n r ̸= ( τ η , − 1 u =1 N ( n ( χ m − r , ̸= ) m r − , =1 χ m,n m u r N , 2 m 1 1 . This contour integral is the path integral of a supersymmetric − m,n u − r )+ ( − . There is always a region at infinity in the moduli space where η r r corresponding to complexified electric and magnetic Wilson lines 1 r n 3 ( r − u u ̸= R du ,i j and giving a vacuum expectation [U( is isomorphic to I m , T / G ) N ib∂ 7 N e 1 P , interacting relativistic particles called τ ∗ ± i µ = T p − N j ,µ j µ 1 p µ p ( ± ij δ ,η q 1 ± j – 18 – = ····· ) i j ) τ 1 ). µ πbm ( τ i µ ). we obtain that the ideal given in ( 2 , e p 1 N a − ± p 2.56 . p = p a are promoted to difference operators ( 6 N j is trivial in this case; which confirms the known fact that − i = 2 vectormultiplet for 1 P µ C η µ η ,...,m N ≃ 1 → (and, as we explain below, also ) and m a N b , as computed using supersymmetric localization in [ p i P 3 τ ∗ partition function, we will obtain the partition function of =( S T 3 ( retracts to ⃗m S • T -equivariant quantum K-ring of N P T , where the quantization parameter is identified with squashing parameter ∗ QK 3 is given by relation ( T S , we explained how twisted chiral ring of by factorization. I 2 2 partition function R 3 × S 1 As in the previous section, it is convenient to introduce exponentiated mass parameters. In this section, we will quantize this classical integrable system by studying Note that in the limit when K¨ahlerparameter vanishes S symmetry we define in this section They obey the following relation by coupling to a background value to the realshorthand scalar. notation For the most part we considerThe U( conjugate momenta now become elementary difference operators. For a U( 3.1 Let us begin bytheories summarizing on the squashed supersymmetricThis partition partition function function of depends on 3d valued a in the squashing Cartan parameter subalgebra of the global symmetry group conjugate momenta to spectral curve becomesFrom an the squashed operator equation,on which annihilates the partition function. the trigonometric RS system. theories on a curved three-dimensional background.on For most squashed of this section, wetraditionally concentrate denoted by equivariantly 3 3d partition functions In section spectral curve of a classical integrable system of which is in therings agreement with of the complex known projectivegesting result that spaces. about the classical action Note K-rings of also and U(1) that cohomology the dependence on where ideal Therefore, by redefining Corollary 2.2 JHEP05(2015)095 =2 (3.5) (3.6) (3.7) (3.3) (3.4) N for the % m (2) j ) j im m and momenta ) gauge groups. + − ' τ i , N (1) i τ m ( 1 partition function of im − 3 ) − j supersymmetry in three S πb ∗ ∗ 2 ε im $ = 2 vectormultiplet measure − =2 i S , N ) 2 sinh N % ) im , j im im ) (2) j m − + − N ε + − − im m ( b b i ( + ) S − j m = ··· πib ( , the contributions from the adjoint + 2 ∗ =1 (1) i 1 N im e + & πb ) gauge group is a real mass parameter for breaking to i,j b m ( − im 1 = N C i − → πit + ) – 19 – 2 . They are defined in exactly the same fashion j ε im ' ∗ q πibε 2 sinh − 2 2 ε e U(1) im + − im , ε $ N × ε ij ( k " " − µ µ k ' 1) ' =1 q ! ) of the form & ( ( − n − N i + − s ij ij → k 0 ( N for ( N k k ) can have poles only from the vortex and anti-vortex series. Fur- 2) ) ) ) # 1) s = 0. Here we have multiplied the prefactor }≥ U( 1 − 1 q q ( Nt − | ( 1) ; ; − − Z N ) =1 1) ∗ i i ( N j j 2 in the similar manner. Upon the redefinition of the vortex numbers i − N j N µ µ − µ µ k ' N 1. ( U( ( ! πiε ( 2 { − N ). This is the Higgs branch representation of the imb k ' k ⃗ 2 , − | Z # − − τ, µ, η, q − e U( = + ··· =1 )] theory. We note that the partition function is factorized by the vortex N n , N i qη = (( = 1 in Z 0 and they are also cancelled by zeros of the vector multiplet contribution at k ' 3.36 ( N | ! (1) 2 V j σ ) # ∈W 1) gauge group as k ⃗ inb − ̸= n V 1) We still need to check the our assumption that the vector multiplet and the function Finally we combine all integrated subpartition functions and find N AV N − Z = U( Z N U( ( i i Z is completely cancelled by the vectorthe multiplet function contribution thermore, we find that possiblewith poles from the vortex seriess are perform the contour integral startingcontribution in with and anti-vortex partition functions, contributions. Z to first fix where omitted in ( of the for U( one can see thatis the reduced partition to function takes the same form as We can consecutively apply thisfor procedure lower and evaluate the partition functions k in the third line can be further simplified using the equations in ( Here we have redefined the vortex numbers of the U( JHEP05(2015)095 )= ) at ) the s obeys (3.39) (3.40) (3.41) (3.42) ( [U(3)] )). We 1 a, q T ( V − ). 3.35 θ Z , a 3.30 p N,N A.1 % z Q 2 ) by prefactors , − 1] , . Then qη [U(2)] and 2.15 [1 V ; q T Z ; A.3 ) 1 2 2 µ µ µ q ) for . According to ( ; + 1 ) is correctly derived. 1 2 P 1 ∗ µ µ µ 3.37 , 2 T 1] 3.37 , ,η =( [1 2 V ]. η 1] Z , , ) $ [1 ) V 40 . This proves that the relevant poles 1 1] 1 , ,q Z , ,q F 2 − 2 [1 V 2 % 31 τ N [U(2)] theory in one of the vacua (when, 2 τ . Note that the operator in the left hand Z 2 ητ = p i τ Z ( T µ p 1 θ ( = k j ) τ θ τ ) 1 ) 1] 1 [U(2)] theory is , k ,q 1 2 ij − τ ) τ τ 1 for trigonometric RS model ( δ [1 ,q – 30 – V T η q τ 2 q 1 − 1 ; Z quiver theories can be identified with the cotan- τ 1 − − q 2 τ T 2 1 = − ( ∗ τ p 2 µ qη η j ( 1 τ ( ( ), which are also cancelled by zeros of τ ητ θ is one solution of the difference equation p θ s τ i · =2 ) satisfies the following difference relation ( 2 p 1 η k ∞ = µ ) − ) N V q 3.39 N . Therefore the integral contour involves no pole from the . We can remove those factors by redefining + ,q ; 1 ν Z η 1 1 τ 1 2 k − − µ µ p ) in 2 2 q qa τ ; η and ( 1 imb 1 2 2 ( µ µ 2 − τ ∞ k + ) q η + ( − ,µ q i>j 1 − ; 1 a, q µ 2 τ inb 1 theory η * ητ + η ∗ 2 ε 0 and =0 )=( ∞ 1 k i # ). Our theta function conventions are given in appendix µ ≥ [U(2)] − = $ a, q T ( j a, q 1] θ ( , = 1) reads m p, q θ [1 V i 1 The corresponding quiver variety for Z − = a i where − and where momenta operators actside as of the above equationwhich differs depend from on We can explicitly check that ( It is instructive to explicitly demonstrateabove, that holomorphic satisfy blocks, difference which we relations have computed shall in demonstrate the this electric fact frame fortheories. (i.e. 3d second vortex partition relation functions in ( ( 3.6.1 where we used the definition of the Q-hypergeometric3.6 function (see appendix Difference equations in electric frame corresponding vortex partition functionsay, of the Since the Higgs branchgent bundle of of 3d thethereof, corresponding Nakajima which quiver is varieties, theGivental the J-function vortex generating of partition function the function corresponding of variety the [ BPS states on Higgs branch, gives the s vector multiplet contribution and thecan come function only from thethe contributions Higgs of branch the representation bifundamental of hypermultiplets, the and partition therefore function3.5.3 ( Givental J-functions with JHEP05(2015)095 ) 1 1 ), = τ µ k 3.39 − 3.42 (3.44) (3.43) q and 2 µ ). [U(3)] theory, . [U(2)] theory. T = 2) we would T 1] 3.43 , 2 τ i , 1 , p ) one puts V 3 [1 V )( τ Z Z 1 3 , ) ) 3.42 − τ 3 2 V η 3.35 µ µ − ) (again, with Z − 2 2 + + τ 2 µ 2 1 between each other. Thus in 1 3.42 µ µ ητ 3 µ 1 µ + τ =( = 1 1 1 in this case. In the classical limit µ − τ V V η 1 and Z Z − m =( 2 − 2 τ 2 1 % p τ symmetric Macdonald polynomials in 2 2 τ 1] πt 1 τ ,µ , p 1 k 1 p ) that obeys the same difference equation. ητ , 1 [U(2)] theory in ( +2 µ η τ [1 1 V 2 T 1 t 1 1 µ τ ∗ Z − τ 1 η A + µ – 31 – πiε − 3 τ ( e 1 τ ] it was shown that Macdonald polynomials with − p 2 p τ 2 /θ 2 3 ) 41 τ τ 1 1 ητ 1 1 τ 3 − τ [U(3)] theory are slightly cumbersome, so we shall only ( ) is not unique. For example, one may replace the factor − τ η + θ η T − − . In [ 1 τ − 3 − p η 1 3.42 τ 3 τ 2 1 τ ητ 1 ητ 2 1 and − τ 2 τ η τ 1 q 1 − 1 . After the redefinition similar to ( 2 − τ ) are infinite series in FI parameters. Note, however, that for some − 1 1 − τ η τ 1 η µ − 2 the series become degree- − − , 3 ητ 3.37 − 1 1 τ 3 τ 1 , $ ), this prefactor reproduces the classical FI terms up to proper rescaling and theory ) by another one ητ ητ 1 2 τ 3 2 =0 2 1 µ η µ η k µ →∞ 1 ( ) provide the desired quantization of the twisted chiral ring of + b µ [U(3)] /θ for @ T ) with parameters 2 1 3.43 τ 2 0 (or 1 /µ τ At this step let us make the following observation. Vortex partition functions which Note that the prefactor in ( Had we chosen another vacuum of the 1 − µ η → 2 ( The vortex partition functions of specify difference operators which they satisfy.meaning There are that 3! there = 6 are vacuaeach in as other the many by different interchanging vortex massone partition of parameters functions the vacua which the are partition related function to satisfies the followingff di erence equation and proper normalization are eigenfunctions of difference operators of tRS type. 3.6.2 values of the coefficientswe combine these them series with the truncatewe corresponding and can prefactors containing reproduce become theta-functions, the polynomials. like so-calleddifferent ( Macdonald contexts Remarkably, polynomials, if in which recentlyη mathematical appeared physics. in many For example, if in ( theory living on theb surface defect, i.e. of the parameters. were computed in ( by replacing interchanged) the partition function in the second vacuum alsoθ satisfies ( They correspond to factorizations of mixed Chern-Simons terms (or FI terms) for the 3d Hamiltonians. The eigenvalues on rightmetric hand Wilson sides loops are in the the expectationThus fundamental values ( and of skew supersym- symmetric tensor representation of U(2). have obtained the second vortex partition function which can be obtained from ( which are nothing but conservation conditions for quantum two-body trigonometric RS the following set of eigenfunction difference equations JHEP05(2015)095 ) ], N 42 ) are (4.1) (4.2) , (3.45) µ ( 14 r χ ) with the ) symmetry N 3.44 . The 5d gauge 4 , we expect that deformation. The R 4 , ∗ ) R × in the anti-symmetric 1 × ,Q =1 S 1 1 N . S S N ). The functions 1] , 1 , ⊂ )] is deformed as follows A.3 [1 V 2 ,...,µ N Z 1 R ) µ ) ( [U( . Interestingly, one can reproduce × r ,q ,q 2 T 3 3 χ 1 of the three-dimensional theory are T τ τ S 3 j 2 = µ µ . η ). Furthermore, the three-dimensional ( i τ ( 1 θ R p θ )] is gauged in coupling to five-dimensional ) 2 2 S ) g π 8 N ∈I ,q ,q i & 2 − 2 τ ) τ e 2 [U( ) ( by the formula Q – 32 – µ θ T = ; Q ( ) 1 ). j ; θ τ j S ) Q ,q 2 1 /τ ,q τ of 3.44 i /η 1 τ 2 i τ ( τ R − 1 1 ( η )] is deformed by an additional parameter that we denote µ θ 1 ( ( θ θ N θ )] theory can be coupled to maximally supersymmetric U( ∈I ∈I N = i & j/ [U( ) symmetry of } T V N [U( r Z T can be identified with the five-dimensional = ,...,N 1 # η |I| and the radius are additional parameters associated to the remaining U( I⊂{ 2 j g τ ) is the first Jacobi theta function (see appendix x, q ( deformation 1 θ ∗ . This parameter is related to the dimensionless combination of the five-dimensional The Higgs branch U( We have checked in the series expansion in the FI parameters that ( Q =2 where the expectation values of supersymmetric Wilson loops wrapping scalar (complexified byN the holonomy around FI parameters of the defect. We claim that the twisted chiral ring of we can turn on agauge background coupling. holonomy in the vector multiplet and itmaximal complexifies SYM. the 5d In particular,identified the with twisted the masses vacuum expectation value of the five-dimensional real vectormultiplet This deformed twisted chiral ringtion of of the the defect Seiberg-Witten theory curvecoupling is of can expected the be to five-dimensional provide considered theoryto a as on the representa- the topological scalar instanton field charge. in When the the background 5d vector gauge theory multiplet is coupled compactified on a circle, if the surface defectthe twisted is chiral supported ring of onby the subspace gauge coupling 4 5d/3d partition functions The three-dimensional Yang-Mills theory in five dimensions to form a surface defect. Following references [ above redefinition is theit true is eigenfunction an of eigenfunctions thethe of tRS entire the model. vortex quadratic partition tRS Wethe function operator have difference using also relations the verified of perturbation that the theory form in ( FI parameters given three-particle trigonometric RS model we redefine the vortex partition function as The prefactors again correspond to factorization of the FI terms. Again, in order to convert the above equation into the momentum conservation of the JHEP05(2015)095 . . 8 1 2 2 ϵ ϵ R (4.4) (4.3) × and 1 0 where 1 S which are ϵ → N Q with the surface ) defines a family 5 τ ,..., S 1 4.2 0 of the normalized τ , ) → . Q ) ( 2 , ϵ Q O ( ) O + Q ( r i O µ )+ 1 + ··· j t, ϵ ⃗ , including instanton corrections. It is 1 i ητ j µ m, ⃗ τ r ( − )] is reproduced in the limit ) , over which equation ( ]. − i i 4 N ,ϵ ρ,k by 1 of the standard instanton moduli space 1 with M λ λ ⃗ ⃗ − a, ϵ ⃗ Let us now compute the equivariant character of the tangent space to the ramified For example, let us consider U(2) theory in the two-instanton sector Let us now explain how to compute the answer. The first statement is that each fixed ( j λ ⃗ N,k ρ, + In addition, for claim that this mass parameterinstanton is partition invariant under function the is orbifold given action. by Finally, the ramified Analogously to the unramified case we define over the orbifold action,always write therefore using the a summation answer over is weights automatically invariant, which we can χ the decomposition of space parameters Then we compute the character of the ramified moduli space of instantons via averaging instanton moduli space at a fixed point point 2, so we needgenerated to by the describe following three tuples spaces of Young tableaux total number of boxes in the Young tableaux Then the boxes ini the M point one ramified instanton moduli space identify which sector space JHEP05(2015)095 ) in 2 ϵ 4.24 (4.30) (4.25) (4.26) (4.27) (4.28) (4.29) , , ) 3 3 ) z z ( ( O O 1], which is expected , + + 2 2 z z = [1 + + . , 1 2 ρ [U(2)]. The additional label µ µ % % ) ) 1 2 z z T − − 2 2 µ µ , q q − − . 2 1 − − 2 1 z Q τ τ z. µ µ 2 1 2 2 µ µ = = η η = 2 2 η ,q,q η ,q,q 2 2 z q q 1 2 1 2 +* +* Q, 1 2 µ µ µ µ )( )( ). µ µ = 1 2 ,q ,q µ µ s 1 1 2 2 − − 4.18 Q ,Q µ µ µ µ 2 1 − − 2 2 µ µ 2 1 z Q 2 2 z, Q – 38 – z z ,η ,η η η can be identified with the holomorphic scale the 2 2 qµ qµ + + = = ····· ) ) η η 1 2 +* +* 1 2 1 1 1 Q µ µ 1 1 $ $ µ µ can be understood as a ratio of the FI parameters of 1) ( 1) ( Q Q Q 1 1 [U(2)]. − − − − z − − − − F F 2 1 T 2 2 of the monodromy parameters, corresponding to the two q q 2 1 2 2 ) µ 2 µ 2 q q 2 2 η η − = = ± (+) ( qµ qµ η η ) 1)( 1)( 1] 1] +* +* , , − +* +* 1 1 ( (+) [1 [1 1) ( 1) ( − − 1 1 . Note that the dependence on the equivariant parameter Z Z − − theory with gauge group U(2) in great detail. In this case, there is q q − − 1 − − ( ( ϵ 2 2 ∗ ). We shall immediately see physical meaning of both choices. As 4 4 q q e 2 2 η η ( ( η η η η * * 2 2 * * = =1 2 2 η η 4.25 q q q q . It is straightforward to recognize them as q-hypergeometric series q 2 N + + µ 0 in order to decouple the five-dimensional degrees of freedom. The first ↔ =1+ =1+ → 1 ) µ 1] 1] , , Q − ( (+) [1 [1 Z Z Let us compute the instanton contributions to the Nekrasov partition function ( It is convenient to make the following replacements corresponds to permutations related by where we identify the plane orthogonal to the defect has dropped out in this limit. The two expressions are the 3d defect theory coupled to the bulk 5dand theory send few terms are in order to satisfywe ( have already mentioned,five-dimensional variable bulk theory while massive supersymmetric vacua of 4.3.1 U(2) theory We shall consider only one non-trivial monodromy defect labelledto by the correspond partition to theσ surface defect obtained by coupling to instanton parameter which is also implied by the last equality4.3 in ( 3d decoupling limit Note that the product of instanton parameters for each sector is equal to the unramified JHEP05(2015)095 1] , (+) [1 and Z . (4.34) (4.31) (4.32) (4.33) 2 Q µ and 1 µ theory with maximal ) gauge group, which ∗ 0. Thus the expectation N . α ) from in favor of → . n which can be reproduced in =1 [U(2)] theory in one of the α 2 − ] n ϵ T ∞ N 4.24 − ++ ) N [1 i α . 2 ) gauge group is then given by ++ ∞ ) w j Z ) α τ in ( 2 * N ∂ N w ⟩ ⟩ j ) i . * τ . Using the language of equivariant 1 ⟩ N 1 Q/q, Q Q . It turns out that the numerator has ϵ 2 1 S e U(1) (1) U( (1) η (1) Q/q, Q 2 sinh S ( ( * = W W W 2 sinh ∞ ⟨ ⟨ ⟨ α j ∞ ) * 0 ) in order to insert the fundamental Wilson τ ) p α , = → ,Q ) 2 ⟩ , 2 ϵ E ) 4.14 – 39 – ( λ | ⃗ Q, Q λ ⃗ [U(2)] example, they can be mapped onto vortex N ( χ = lim | | q T λ ⃗ Q/η | ( λ ⃗ SU( (1) (1) q λ E ⃗ ) in the numerator represents the contributions from a ? W = ⟨ ⟩ ? 4.10 = ( ) gauge group by the overall U(1) contribution. The overall U(1) (1) ⟩ ) coincides with the the vortex partition function in the other different partition functions N E ) the defect partition function satisfies trigonometric RS differ- W ). ( λ ⃗ ) (1) ⟨ 1] , χ N − ( [1 W ⟨ ] we can eliminate each of 3.42 3.37 Z N ), where momenta act as ]. The U(1) contribution is given by 3.43 theory 47 ) ), whereas N ( . Thus there are i 3.39 Q After redefinition ( The fundamental Wilson loop expectation value of SU( differs from that ofU(1) the contribution SU( can be interpretedone as can a compute contribution from it agauge using heavy theory the free [ Wilson BPS loop particle expectation and value of unit charge in the abelian value is finite and we can denote it by Remember that this is the Wilson loop expectation value of U( The additional character heavy charged BPS particlethe propagating same around universal divergence as the denominator in the limit An essential ingredient of ourvalues construction of is supersymmetric the Wilson computationcharacters loops of we can the wrapping easily vacuum expectation modifyloop formula operator ( inside other the decoupling limit. Similarlypartition to functions the in ( 4.4 Wilson loops ence relations ( 4.3.2 U When we consider themonodromy decoupling defect limit [1 of the 5d U( evaluated in each of the supersymmetriccontributions vacua. Alternatively to they are the the non-perturbative exactly two independent matches with holomorphic thevacua blocks. ( vortex partition We can functionvacuum. of now Recall that see the both that expressions are related merely by interchanging These are indeed the equivariant vortex partition function of U(1) theory with two flavors, JHEP05(2015)095 . ). 4 " 2 ) ) 2 2 ,τ 2 (4.38) (4.39) (4.35) (4.36) (4.37) Q Q ( ( µ ( O O and hence ↔ to be some + + 2 ) 2 , . 1 Q Q Q µ ) q ,τ 3 2 1 . It appears that ··· ··· η 1 and µ qµ 2 and + + ) 1 /τ − 2 1 2 . Q 1 z z µ Q Q τ µ 2 ) µ ) 1 + + µ + 2 3 ) ) 2 1 = µ 1 η 2 1 µ µ z µ µ µ qµ )( − ( 2 and − − q − − − − 2 1 2 1 qµ 1 q q µ µ µ µ ++ 1 2 µ 2 2 − q µ µ )( η η 1 )( 2 * * + , µ 2 theory with a fundamental Wilson 2 µ 2 q q 1 . 1) ( 1) ( η η ∗ + + qµ − − ) ⟩ 1 1 1] + − − )( , q ± − 3 4 ( q q [1 Z 1 =1 − − η 1 µ ( ( µ U(1) * Z 2 2 2 2 2 (1) µ ( η η η η η q 0 N q )( W * * 4 3 − ⟨ → + 2 0 1 η 2 µ . ϵ + + -th skew symmetry power of the fundamental µ η → – 40 – z z r * 2 − lim = lim )( ϵ + + 2 1 2 ) ) 1 2 ) 0. We introduce the notation 1] µ µ 1 2 µ , ≡ µ µ 1 ± 2 µ µ ( [1 )( µ η → 2 − − (1) ) D − − 2 µ − 2 2 1 ϵ N 1 q q η µ µ 2 1 − µ 2 2 − µ µ 1 η η )( q * * µ 2 ( )( q q η , the partition function in the presence of the defect is no longer 1) ( 1) ( q 2 + + 6 Q − η 1 1 − − expansion there are negative powers of η q − 0. The divergence exponentiates and the anomalous dimension is q q − − = 1. For later convenience we define )( ( ( Q cyclic permutations) ) we must modify the computation. It will be discussed elsewhere. 2 (1 i 2 2 2 2 → 3 η η η η η µ N − * * i 2 − 1 ,µ ϵ 3 2 , (1 ) we find the following expectation value 2 ,µ are related by interchanging the mass and FI parameters ( − 1 µ 1 ) 1 =1+ =1+ S µ − ! + ( 1 ) 1] 1] 0. It is very important that we have a regular expansion in 1 , , µ − D +( (+) ( µ [1 [1 → = D D =( Q and As the divergence is universal, the normalized expectation value of the monodromy Let us consider some examples. For U(2) U(3) (1) U(2) (1) (+) E E One may check that thelimit above expressions reduce toat the higher orders vortex in partition the functionsD in the The first few terms of the expansion are expectation value of the surface defect with unconstrained defect is well defined in the limit Turning on the parameter finite in the limit universal, that is, it isarising the from same the divergence without anomalous thediscrete dimension defect. will solutions. The fix saddle the For point mass now equations parameters we will ignore this issue and concentrate on the normalized In order to insertrepresentation of a U( Wilson loop in the 4.5 5d/3d coupled4.5.1 system U(2) theory given by Similarly, the Wilson loop expectation value in the fundamental representation of U(3) is which will be used in the difference equations below. loop around with a condition JHEP05(2015)095 ) and (4.42) (4.40) (4.41) ) with 2 n 2 4.36 Q 1 n 1 , , , Q ) ) ( 1] 1] , , ± − ( O (+) ( [1 [1 . into U(3) labelled ) D D D 3 ± ( (1) (1) (1) D ). Therefore if we im- E E E 2 = 1 in what follows. µ (1) (1) (1) 1 = U(1) σ 4.35 N N N µ L , , = = = = ) 1] 1] , , ) theory in the presence of full ) ) − (+) ( [1 [1 − ± ± (+) ( ∗ ( ( D D D D D D ) ) ) ) 2 τ A A ,q ,q A =1 ,q ,q p 2 τ 2 τ 2 2 2 τ 2 2 1 τ p p τ p τ τ p + + 1 1 2 + ητ N ) ) ( ) − µ µ θ ( ( ) η ,Q ,Q ,Q ) θ θ ( 1 1 ,Q ,Q 1 ,Q ) ) N θ τ τ 1 1 τ 1 ,q ) 2 2 2 we have checked up to order 1 ,q ,q /τ /τ τ /τ 1 1 – 41 – ,q 2 2 2 /η /η 2 /η τ τ 1 1 τ τ 2 2 τ 2 0 which we have computed earlier in ( 2 1 Q ( ( − τ τ ( τ µ µ * * ητ η 1 1 * 1 ( ( ( ( θ θ 1 1 θ → 1 θ θ θ θ θ θ θ and = 1, the eigenvalues of the difference equations are the 2 1 2 ϵ = = i 1 + ) ηµ ηµ µ τ 1 Q i − p (+) ( + + + , D D 1 τ 1 τ ) p p ,Q + + 2 ,Q ) ) 0. As discussed in the previous sections, this limit is well defined τ 2 2 ,Q ,Q /τ 2 2 ,Q ,Q → 1 τ τ /η 2 2 ] monodromy defect. In this section we will present the results for U(3) τ 2 2 2 1 . We shall focus on a particular embedding ( ϵ τ /τ /τ N σ * 1 1 1 /η /η τ τ θ 1 1 1 ( ( τ τ θ * * 1 1 θ θ @ 1 1 θ θ is the normalized expectation value of a Wilson loop in the fundamental rep- 1] := [1 1 2 η η µ µ = 5 that the above normalized expectation values obey the following difference (1) @ @ 2 E ,..., n is the inverse of the overall U(1) contribution defined in ( We are interested in the normalized partition function in the presence of a monodromy As before, we can define new partition functions As a regular expansion in + (1) = [1 1 ρ theory. There are 3! different embeddingsby of a the Levi permutation subgroup defect in the limit system, at least as a series expansion. 4.5.2 U(3) theory The generalization toramified many-body instanton system partition is function straightforward of — U( one needs to compute The same non-uniqueness caveatsthe as perturbative contributions before obtained holds fromfrom orbifolding also factorize the here. nicely defect into and One contributions and the should bulk. eigenvalues also of At check this the that point two-body we elliptic claim to (complexified) have Ruijsenaars-Schneider found integrable formal eigenfunctions which obey the same difference equations N pose the traceless condition fundamental Wilson loop expectation value of SU(2) gauge group. where resentation of U(2) in the limit n equations JHEP05(2015)095 2 1 z ] ) , ) 3 2 ] 2 3 [1 (4.43) (4.44) (4.46) (4.45) µ µ 2 [1 2 D q 1 in the D 1) , qη 0 one can − 3 < (1) (1 z − 1 2 | 1 E → E µ z 3 µ z ) )( (1) | (1) 3 Q )( 2 , 1 τ ) | 2 N N 3 p 2 qµ µ qµ 3 τ µ z 2 = | = p − η , τ 2 ] − ) ] , − | 3 . 3 2 p ) ) 1 1 − 2 2 [1 ) [1 µ ] z 1 , ,Q µ µ | µ ) 3 2 1 ,Q D 1 D µ )( η [1 )( τ 1 Z ,Q 2 ,µ 3 2 )( % 2 ,Q ··· % Z )( 1 q 2 τ /τ 3 τ 2 µ 2 τ 3 µ 0 2 2 /η p p 3 + ) in one of the vacua, after a µ − τ 2 qη /τ ) − 1 τ → ( ,µ τ " 3 /η ) 2 p lim 3 1 1 ( − − ϵ τ 3 (1 ) µ θ 1 ( ,Q τ 2 1 ) 3.37 2 ) ) θ 1 3 (1 ( ,Q qµ η µ µ τ θ 1 3 ) ( ,Q ,q 2 ,q 2 )( θ 2 ) ,Q 3 3 η + q → /τ τ 2 ) τ ) τ ,Q 2 /η 2 3 ) 3 3 )( 3 ,Q τ 2 2 − z /τ 2 µ τ ( 3 ,µ τ ,Q 2 3 − /η ( η 2 1 2 ( 2 ,Q τ z 3 η (1 θ θ 1 τ /τ 2 ( ,µ τ ( ) 2 ) − θ ) 2 2 /η 1 1 ( 3 θ η 3 τ = 5. 2 θ /τ 1 µ ) ) ,q µ ( ( τ 1 θ /η (1 ) 3 is the inverse of the overall U(1) Wilson loop 2 J J 2 1 ( τ 1 qµ 0, this partition function precisely reproduces ,q ) τ θ 1 n η ( ,Q τ – 42 – 2 2 ) + θ (1) 1 3 ( ,Q using the fundamental Wilson loop expectation τ − (1) − µ τ θ 1 3 + 1 ( ,Q → E ( 2 2 2 θ 1 z + θ ,Q N 2 1) cyclic permutations) [U(3)] theory in ( θ /τ µ , ) τ µ 1 ) ) n ) 1 3 τ /η 2 Q 2 T 3 + (1 )( 2 τ )( 1 p ,q )= /τ µ ,q 3 ( 3 + τ 1 1 τ 2 τ 3 E 3 ,z . We have also assumed that /η 2 1 1 ( τ qµ µ τ p p 2 3 1 τ η θ 1 ( Q qµ τ ) ) 2 ,µ 1 n θ 1 ( 1 3 ,z − − ) ) 2 η . Here µ − τ τ − θ 1 1 ). ] ( ( 1 2 2 ,Q ,Q θ 3 1 z 1 θ + ,µ θ 1 3 µ ,Q ,Q µ µ [1 − µ 1 i = ( τ τ 1 3 2 ( + )( 2 2 µ D = η ,τ 3 2 ( ! 3 /τ /τ ) with ] and 1 ) ) 3 1 µ )( /η /η 3 3 µ 1) ,z 2 − i , τ τ q 2 3 1 3 n 2 3 [1 3 2 ( ( τ τ η µ − z µ (1 τ τ µ 1 1 ( ( ( − 2 D ,µ 1 1 θ θ 1 1 E − − n 2 2 θ θ µ 1 = z (1 → qµ ) ) 1 2 µ ,µ (1 ( ) ) 2 ) n = 1 2 1 q η 2 i z is the expectation value of a Wilson loop in the fundamental representation η ] ,Q ,Q is the Wilson loop expectation value in the rank-two antisymmetric tensor µ η ,z ( ( 3 1 2 ,τ ,Q ,Q 2 1 i τ τ 1 2 [1 τ τ O 1) (1) , 2 2 µ + +( ). D /τ /τ E (1 3 τ 2 1 = /η /η =1+ p E τ τ 1 2 ] ( ( 2 τ 1 τ τ 3 3.44 1 1 ( ( z p [1 Z θ θ 1 1 1 τ θ θ Z p ’s up to The difference equations are the eigenvalue equations of the three-particle eRS inte- Let us define a new function as i 0 $ $ z → 2 lim ϵ grable system. Thereforeeigenfunctions of we the found integrable thatsymmetric Hamiltonians the representations and are the monodromy Wilson their defectalso loops eigenvalues. partition notice in In functions rank that the 1 are theseans decoupling and in equations limit 2 ( reduce anti- to the equations for three-body tRS Hamiltoni- which holds only for U(3).in The difference equations have been checked in a series expansion and factor and of U(3). representation. We canvalue compute by a simple replacement of the gauge fugacities such as It follows that this function satisfies three difference equations: relabelling ( where expansion. In thethe decoupling vortex limit partition function of the and we end up with a finite expression. The partition function is expanded as JHEP05(2015)095 2 ], 48 CP of the . In the 2 ] realizes wrapping W 47 CP λ (1) we will fix ) with respect → together with a O 5 . The parameters 5 S 4.38 + 1 S S W 2 1 on ϵ λ with a monodromy defect → 4 Z R . The 6d superconformal index 5 × S 1 S partition function, the integrand fac- 5 ) maximal SYM on S . N λ partition function with a monodromy defect , the contributions from 2 of the fixed points 5 3 S S – 43 – are Hopf-linked inside 1 S (putting the solution on shell) will resolve the normalizability to discrete set of values. It is possibly the case that for these i and a Wilson loop in the representation 2 µ in the presence of a codimension 2 defect of ‘maximal’ type wrap- µ 3 5 and the S S 3 ] two of the author’s have conjectured that the eigenfunctions appro- . The point is that we have found only formal eigenfunctions in that and S × 2 . The identification of parameters given in equation (5.2) of [ 1 47 β 1 µ 2 S µ R and a codimension 4 defect labelled by the dominant weight × and ]. The hamiltonians act on the superconformal index by introducing a surface 3 1 S 1 48 S µ × where the ]. This conjecture was checked explicitly in the ‘Schur’ limit where only a single 1 1 β is an eigenfunction of a pair of RS systems, one associated to each fixed point on 47 S S 3 The results of this paper imply that the Now, in the localization computation of the In reference [ The quantum elliptic Ruijsenaars-Schneider model in connection with supersymmet- 0) theory on × S , 1 β wrapping the NS-limit for theseThus Nekrasov we partition arrive functions at in the the partition planes functions transverse studied to in theon this defect. paper. parameter of the superconformal index is turned on. torizes into contributions from 3presence fixed of points a in monodromy the defect baseare wrapping of two the copies fibration of the Nekrasov partition function on ping S is computed by themonodromy partition defect function on of 5dof SU( the 4d andence 6d [ superconformal index are identified according to equation (5.2) of refer- appropriate for the superconformal indexand are are eigenfunctions labelled of by a a pair dominant of integral elliptic weight RS sytems priate for the 4d(2 superconformal index are equivalent to the superconformal index of the 4.5.4 Connection to 4dThe elliptic index RS of system classof also S class arises S in [ thedefect study of and the the superconformal eigenfunctionspuncture index in of are the 4d the TQFT theories that wavefunctions computes associated the superconformal to index. a In so-called fact the ‘maximal’ wavefunctions some approximate solutions to theparameter. elliptic RS We eigenvalue therefore problem are expectto labeled that the by extremization mass a of parameters discrete issues. eigenfunctions We ( plan to discuss it elsewhere. defect theory coupled tothe the parameters 5d theory invalues the we divergence obtain log normalizable wavefunctions and hence obtain aric discrete spectrum. gauge theories has been discussed in the the literature before. For instance, in [ The complexification plays a crucial role here.system has Indeed, a normally discrete one spectrum, may whereas expectvariables our that eigenfunctions elliptic are parametrized by parameters they may not bearising normalizable from wavefunctions. the In saddle solving point the analysis additional of Bethe equations the effective twisted superpotential 4.5.3 Comments on normalizability of wavefunctions JHEP05(2015)095 (4.47) , instantons. ]. Here we ) symmetry 4 i k 51 N φ ) and we set – − N e 49 , k =1 ) are the same un- i # 4 2 i N ϵ φ ) gauge theory. The 3d for the auxiliary gauge + e 1 i N ϵ φ k e =1 e ) i # 2 ) ϵ 2 − ϵ e − ) and SU( e − N − (1 (1 − is incorporated by adding a Wilson loop − I ) a 1 of the universal bundle at ) λ ϵ I − 1 a ) ϵ − e e E e − ( e =1 χ − =1 2 I 2 I − – 44 – ). Naively U( -tuple of Young tableaux. We note that the path (1 ? ? N (1 N 3 3 x x + − 1 ) flavor symmetry and we couple this SU( ϵ ]. The defect introduces an additional three dimensional 1 ϵ 2 − N 2 − 19 m m e e SU( = 1 as the diagonal U(1) part decouples in the field theory = = duality. × ) i ) ∗ µ is the equivariant parameter for the U(1) flavor symmetry. These ) E E ) 1 ( k ( k 1 ϵ x are the equivariant characters of the 3d chiral and anti-chiral multi- ϵ χ χ =1 − N i x x − . The additional parameter instanton moduli space. Due to the 3d contribution the saddle point e e 3 − + , anti k 1 1 S − ϵ ϵ − χ 2 2 − − bispectral (1 m m (1 e e and ]by = = is not fully classified by the 2 ) of the i k = 1 to get the results for SU( d φ chiral 3 anti d partition function and checking the result takes the form of a superconformal index = [1 i χ 3 chiral χ ]. However, it should be emphasized that evaluating the Coulomb branch integral of µ 5 ρ χ In order to compute the partition function in the presence of this 3d defect we shall In this section, however, we carry out the computation using U( S 47 =1 N i characters are expressed ingroup term U( of the equivariantvalue parameter of integral on the instantonthan moduli the space previous has saddle extra points saddle labeled points by Young from tableaux. the 3d factors other where plets respectively, and computed using the equivariantWe get character follow the prescription given invector [ bundle on thedefect instanton moduli theory space are of inbulk the the 5d gauge fundamental gauge group and theory. andconjugation, anti-fundamental The representations thus respectively. 3d of the fields the The of corresponding SU(2) the equivariant bundles characters are of the the universal 3d bundle hypermultiplets and can its be limit. However, thisrelated is to no the longer U(1)assume true part that in the in U(1) the the partto instanton 5d does the computation. partition not partition affect functions function The the can are gauge subtle be discussed theory subtracted issues in dynamics by and [ hand. its contribution of defect turns outtype to be related to the, U(2) gauge theory with ader monodromy the defect constraint of 4.6 5d theory coupledWe to now 3d consider hypermultiplets a differentcouple 3d type hypermultiplets of to co-dimension the 5d twohypermultiplets maximal defect have supersymmetric U(1) in SU( five dimensions.to the We bulk shall gauge symmetry. Our motivation is that the U(2) gauge theory with this type at the remaining fixed point.in [ This provides somethe further evidence of theremains conjecture a made formidable challenge. intersected by the JHEP05(2015)095 (4.49) (4.50) (4.48) give the per- . d k 3 k Z d 5 k ) and the appropriate Z , i i ) with specified boundary . 1 4.47 ) 2 πiρ dρ − i 1 2 ∞ D − i ) τρ ∞ q 1 k =1 × ) τρ ; i & − q 1 1 1 ; − − I S 1 qη 5 − I τµ − = pqη 1 k − − ητµ )(1 ( i qη ]. We will discuss the latter in the next ρ ( 1 )(1 ,Z i 53 theory in the presence of the 3d hypermul- − k 2 ρ =1 . , τ & 1 ∗ I 2 Z 1 ϵ ) − for detailed 1-loop computations. The 1-loop 52 − e x k τ – 45 – η 1 − =1 Q ] using localization. We will compute the 1-loop 1 = C − πi − ( η 54 N p =0 ∞ 1 = 4 superpotential. For being supersymmetric with instanton partition function of the original 5d theory m k ϵ # ,p − − x − k e N e (1 loop (1 = 4 vector multiplet, we impose the Dirichlet boundary = p d = − 5 1 k Z =1 N i ,τ & loop i φ d − = loop 3 e 1 d − d Z 3 1 3 k = Z due to the presence of the Omega background. The superpotential i Z 2 = ,ρ T d 1 ϵ 5 are the 1-loop and ∼ = + d/ d which amounts to the mixed Chern-Simons term between the background 3 m ) 5 k 1 − Z 2 ϵ Z 1 e ϵ mx R e = × and 2 1 η instanton partition function in the integral expression. It is given by S ( The full partition function of the The partition function of 3d theories on solid torus Let us focus on the standard There are subtleties in the perturbative contribution related to the boundary condition We find from the above character formulae the extra contribution of the 3d fields to k loop ∂ d − 5 1 Z prefactor U(1) flavor gauge field and R-symmetry current. tiplets is given by The prefactor comes from regularization. Note that this 1-loop contribution includes the conditions was recently computeddeterminant in for [ the 3d hypermultipletsboundary using character conditions. formula ( Theturbative terms contributions. independent See ofdeterminant appendix of the the instanton 3d number hypermultiplets is given by condition. The boundaryboundary condition conditions on for theanti-chiral the chiral multiplets, multiplets hypermultiplets, or and vice we the versa.of impose Dirichlet This the boundary the choice superpotential condition guarantees in Neumann the on the supersymmetry the presence invariance of the boundary. symmetry in the presencesubsection of when a we gauge boundary the [ 3d flavor symmetry. boundary we should imposejoint the chiral relevant multiplet boundary in conditions the for the fields. For the ad- where on and the Chern-Simons term in the 3d theory are in general not invariant under the super- the JHEP05(2015)095 n ) k i 2 / of the (4.54) (4.52) (4.53) (4.51) ,Q 1 " k R a set of ∞ ··· ) , in negative } ∈ 1 . i i i ∞ ∗ ) ) p, q Q H ) should not be j j φ Q ; J ∈ η /ρ /ρ i i p, q φ ; ρ ρ /µ J J 2 2 I J i Cone( ] is: , − − µ Q ) 2 η /µ ∈ 56 j 1 I qη − η − /ρ ≡{ − qη i if ( qp ) pqµ ρ ( ∗ ∞ 1 )(1 − φ − ) ∞ j . − ) | . This vector , } ) (1 /ρ k )(1 j p, q k i ) i j ; k p, q ̸= ∗ R ρ i & ; J 2 Q =0 φ /ρ J ) that there would be extra degrees ) i − i z /µ ρ − η 0 otherwise ··· I /µ 2 1 /ρ φ I 1 µ − − 4.48 ) gauge group, the Witten index of the i I independent charge vectors. The result ( )+ 2 µ ) η p k µ i ( Q φ i − N k 2 ( − − . The JK prescription can be applied when i plane for each charge vector Q − /ρ pη η I ( Q φ det( (1 (1 1 qη J J | and denote by Q( ! ··· ]. The JK prescription has also been applied ) becomes singular ) k – 46 – ) arbitrarily in j J − ∗ pqµ K can be evaluated using the Jeffrey-Kirwan (JK) ∗ 2 C 57 ̸= φ & η φ I i /ρ − = , i 4.50 ∈ ρ )(1 ( = − ) I ) ∞ 56 qρ j φ ) k )(1 φ φ φ { I ( /µ ( Z − /ρ i k ) 1 ∞ i i i p, q = ρ theory with U( ) φ /µ ; 2 i ( i ). ) over Q 2 )(1 ∗ ∗ ρ k − dQ pqρ j i H − p, q η φ ; 1 − Q − /ρ 4.50 q qη =1 i − is Laurent expanded around the singular point ( ( p (1 (1 ··· pρ ∞ ∞ k N ) ) ) − mass parameter. The JK residue defined in [ ∧ ∧··· Z φ − ) ( ∗ (1 ). Among others, the JK residue yields nonzero result only at simple ] using localization. When we couple the 3d fields, the supersymmetry 1 p, q p, q φ 2) and one can deduce from ( are in Q( i (1 ∗ ( ; ; , 1 k 2 55 j φ k p Q =1 i =1 i i & ( k − = i & − 2 N dQ ,Q =1 ] for more detailed explanations. pη = (2 φ × & I ( ) ( i 59 = ∼ N ,η ··· , Q ∗ , d 1 5 k i 58 4) ADHM gauged quantum mechanics gives the instanton partition function, which denotes other chemical potentials (or log of fugacities). Let us then consider when , loop Z , Q hyperplanes meet at a point d − z 5 1 51 k We will choose a reference vector The integrand of We can first define a hyperplane in the The contour integral ( One can obtain the instanton part using the quantum mechanics on the instanton Z = (4 ≥ JK-Res(Q where ’Cone’ denotes theturns cone out to formed be by independent the of the choice of where confused with charge vectors at the point.JK prescription. The residue at the singular point canpowers be of computed using the poles of the form Here n residue prescription introducedto in the [ localization ofSee the [ index in quantum mechanics, which we willmultiplets briefly where the review integrand now. in moduli space. ForN the was computed in [ reduces to of freedom, a chiral and an anti-chiral multiplets, added to the quantum mechanics. without the 3d fields, JHEP05(2015)095 } is ) 1 ρ (1) pητ 2 . The (4.59) (4.57) (4.56) (4.58) E k so that pη = Z τ 1 = , 2 ,ρ 0. From the d 2 qτp ) and 5 µ , > in the presence d/ = d i 3 5 = 4.35 pητ,ρ τ Q =2 D d/ 1 τ k 3 = p Z ,ρ (1) D 1 1 E to cancel the induced ρ ) (4.55) µ with ( (1) (1) 1 1 , E = ϵ φ mx ) φ, z 2 N i 1 ( 1 ) − (1) Q d 1 ρ ρ e 3 is given in ( k 1 { N ,Q − Z τ sinh ) q . = 1). However, after coupling the (1) p , 2 d d = N 5 5 − φ, z 2 ( d/ d/ ··· Z d 3 3 qη , 5 k ( 1 D Z θ , Z 0 % ) pητ,ρ = ) → 2 ) d lim ,η = ϵ = (1 5 ∗ ,Q 1 1 d/ 1 ,Q η Q ϵ ρ mx 3 1 ). We have checked this relation by expanding . The factor − τ ( – 47 – − − τ p τ D , 2 e p + ) ( η ) ]. This turns out to be the case with our 2 4.36 ) gauge group. 1 ( = µ k θ 1 56 d the difference operator satisfying θ ,Q 5 JK-Res( = τ 1 τ τ p d/ ∗ 2 p ( 3 φ + θ # D 1 | ) ) − 1 τ W pητ,ρ ,Q ,Q | 1 τ τ − = p p = η 4 2 1 k − − ρ . η η ( 2 Z )+ ) gauge theory without coupling the 3d fields, the JK prescription 2 2 , 0 q q k> ) N ( ( ,Q 1 given above one can see that the poles at 1 1 Z -tuple of Young tableaux, whereas the last pole is the new pole arising τ → θ θ µ p N 2 τ 4 =1 ϵ = k − $ 2 η computation. Z 2 q is the Weyl group of the U( ( =1 k pητ,ρ W Z ητθ * = One can also recast the abovefference di equation as an eigenfunction equation Let us now define the normalized partition function of the 3d-5d coupled system and At two instanton sector, we have nontrivial poles due to the 3d factors at For instance, at the single instanton sector, we should pick up all the poles from the For the 5d U( becomes the conjugate momentum of 1 ρ τ ( the Wilson loop expectationboth value sides in up ( to two instanton order. where, as before, wep denoted by take the limit Here we turned onChern-Simons term. the This classical normalized mixed partition Chern-Simons function obeys term the difference equation apart from the poles inthe the these Young poles, tableaux one classification. can Summing computeof the over the two all instanton 3d JK partition residues matter functions at instanton fields. sectors We expect that the Jeffrey-Kirwan method also works for higher from the 3d factors.in The the JK residue at the last pole is nontrivial and should be involved fields of positive charge which giveformula the for factors of theare form inside the contour.labeled Note by that the the first two poles are from the 5d factors which were reproduces the Young tableaux sum rulewe of need the to instanton choose partition3d function. the fields, reference To check the vector this, residue like the prescription extra implies poles that developed there by would the be 3d factors nontrivial as contributions well at as the Young tableaux summation. partition function now takes the form of where the projective condition is satisfied [ JHEP05(2015)095 ) 1 ϵ Z )), mx , e 4.59 . We have for the new 1 η B ), the prefactor theory coupled to two . Moreover, the mixed 4.49 x . Secondly we introduce ) and the eigenfunction, ∗ and later in transformation should be A Q S =1 4.36 for the U(1) symmetry. The Hamiltonian (l.h.s. of ( A N transformation simply shifts by one transformation also breaks the gauge T S ]. This action maps a theory to other theo- 20 which is an extension of the S-duality in three – 48 – ) and show that the result agrees with the partition as a dynamical gauge field. The theory then has a new ) action. The at unit level with background gauge field Z 4.50 A . The Chern-Simons coupling induces the terms violating , transformation on the 3d partition function of the defect πi of ( S ). AdB +2 5d / a dual elliptic Ruijsenaars-Schneider 4.57 S-transformation 3d ∼ transformation acts on the U(1) flavor symmetry with parameter Z a S in the SL(2 S and T , the naive gauging procedure fails to work due to the gauge anomaly. The Chern- A Let us first consider the In the partition function, the gauge invariance is associated to the periodicity of the When the theory is on the manifold with boundary, the Let us first review the S transformation in three dimension. There is a natural SL(2 transformation is more complicated. Firstly we gauge the U(1) global symmetry and . We turn off all the classical background Chern-Simons terms. However, the 1-loop x effect generates the nontrivialffective e symmetry mixed and Chern-Simons the U(1) term R-symmetry. betweeninduced In the from the U(1) the 3d flavor 1-loop partition function determinantcoupling. ( amounts to This the prefactor dynamically violates generated the Chern-Simons periodicity of the parameter holonomy parameter, this periodicity. To cancel thesegenus gauge of non-invariant terms, the we proper should 2d multiply theory. the We elliptic will nowafter see decoupling this with the ourSU(2) 5d example. doublet. theory. The The 3d theory consists of free hypermultiplets of the invariance one need toanomaly couple of relevant the boundary 2d degreesthe theory of mixed compensates freedom Chern-Simons the such term gaugeinvariance. that introduced anomaly The the by on 2d the flavor the theory boundary. should be In chosen addition, to cancel this anomaly as well. global symmetry. carefully taken. If therefield exists the effectiveSimons Chern-Simons term term makes of the the gauge background invariance gauge anomalous on the boundary. To keep the gauge unit the Chern-Simons level ofS the background gauge field make the background gauge field global symmetry whose conserved current isa the mixed magnetic Chern-Simons flux term of function of the U(2) gauge theory in the presenceaction of on a 3d monodromy CFTs defect. ries with which U(1) could symmetry begenerators [ inequivalent to the original theory. It is sufficient to understand two 4.7 S-transformation We will now showdimensions that can relatedefect two and different a types defectthe of of partition 3d defects function hypermultiplets. in More five precisely, we dimensions: will act a the monodromy S operation on checked for somespectrum lowest of orders the two-body inwhose these eigenvalue expansions. is the expectationup Therefore value to of we the a have Wilson normalization,free loop computed is 3d ( the the hypermultiplets partition ( function of the 5d To verify this equation we should expand both sides first in JHEP05(2015)095 ≥ %% 1 πi ,k ϵ is the 2 (4.61) (4.62) (4.63) (4.60) I , µ y 1 1 | , − y % η 2) theories. , k − πi u 2 2 − m q , − where . Applying the 1 − . = (0 1 = , µ ∞ ϵ x yx ) 1 of the monodromy 0 e ∞ τ q,η q q − N − $ ] ) ; ; e 2 2 η q = 1 2 1 (+) k [1 ; − I µ µ 1 τ − Z πiζ q − I ,q τµ 1 2 1 = +2 and − µ µ 2) multiplets and cancel the . ητµ τ % 2 ( , 1 . We introduce the boundary . This result shows that the ) qη 1 1 πi q ( ϵ C ,η /ϵ 2 ; . Then the S transformation on 2 + /τ ) , u τ 2 η = (0 2 =1 q 2 1 & I τ 12 | ; $ η / 1 ) 1 N 1 2 q πi − 1 x = − ϵ F ; 2 ) ) q 2 u τ ( q u uµ − − 2 ( ; θ 3 , × 3 ) 1 η consists of two fermi and one chiral multi- 0 η 1 q / − 1 ] $ 2 2 ; ∞ loop − 2 − ) 1 d π q − T 13 q uτ q 3 ( − 1 1 ( 2 ; θ πiζ θ Z η 2 – 49 – ∞ ) ηµ 1 ) ) 2+ q ( q ; − / /µ +2 θ ; 1 1 q 1 2 ( % u, τ − µ θ ( 2 τ −F transformation adds a term like 1 πiϵ d πi /µ ] for details of elliptic genera of the ( ϵ e − 2 1 2 θ S − − 60 µ , Z qη ( 1 , ( 1 | ϵ yx = −F ∞ y ∞ 52 e e ) 0 ) πim q − q * Z ) τ τ ; dτ dτ ; x in the q πi q is precisely the vortex partition function 2 ; ( = 2 ) + 1 − ) upon the identification u q 8 8 transformed partition function reproduces the holomorphic block 2 F ; F m 2 qη 1 η = S ( 1 AdB − − 4.30 )= , − 0 × u q uτ ( ( 2) theory on the boundary $ 0 2 2 d θ , 3 η S Z ) , this 1 ] in ( q 0 Z 2 πiζ ; − Z 2 1 q . One can check that the prefactor precisely cancels the gauge non-invariant = (0 )= ( y − − u θ e τ $ ( = [1 N ( . In this section we couple instead boundary θ = ρ 1 (+) · ϵ mx u d, The integral can be evaluated using the residue theorem with an appropriate pole We now gauge the U(1) symmetry parametrized by The 3 S = exp − . These gauge non-invariant terms must be canceled. In the previous section, we cancel e d x Z 2 Z Apart from of the 3d U(1)hypergeometric gauge series theory with 2defect flavors in one3d of theories the on supersymmetric the defects vacua. of The twoff di erent q- types are related by the S transformation. The where prescription. We propose that the0 integral contour which encloses come the from poles the at be 1-loop possible determinant to of justify the thisnot 3d pole attempt fundamental prescription to chiral by do multiplet. the it.residue JK-like It theorem, Let residue may us we prescription, focus obtain but we on will the residues at poles of where the prefactor is the 3d partition function can be written as [ with terms of the 3d theory on the boundary. plets. We refer the readerThe to regularized [ elliptic genustheory is whose summarized the in elliptic appendix genus is given by holonomy parameter for the newof U(1) flavor symmetry, which alsothe violates former the anomalous periodicity factor byto turning on the classicalanomalous Chern-Simons term boundary corresponding terms. Chern-Simons term JHEP05(2015)095 S (4.70) (4.66) (4.67) (4.68) (4.69) (4.64) (4.65) and the . 2 0 and the µ 1 ··· Q/u − + η . k< k ) ] − exchanged which and q 19 Q/u 2 u . u, Q ) = # µ . We note that the ) ( 2 ) 4 2 Z τ n µ . µ x (+) ) ( , ∈ − and f − p 1 inst S 1 =1 , τ,Q µ pq ,k ∞ n Z ( µ 1 I d " 2 µ µ " 5 and find ) × η ] ! 1 ( in section q 2 d/ 1 1) − q 3 (+) 1)( ] , this instanton partition function [1 ∞ µ η − 2 1 ) u. 1 − We have confirmed this relation up Z 1) Z k for negative integer (+) q − [1 2 p − ) − p ; /τ η 4 ' (+) I Z q − η 2 2 2 q N ∞ 10 1 η ( k )] = exp τ d, µ ) η 2 2 − 5 1 x q u, τ − Z η /µ η = q ( ; ( ( − q = 1)( 1 d/ f )= 1)( 2 d 3 η S µ τ 2 u k → − = + 2 Z − /µ q Z ,Q − 2 τ 1 u τ 0 q 2 1 1 η ϵ ) ( yx µ τ τ ) ( = qη → 1 ( e 1 ( 2 – 50 – ϵ µ τ Q = µ ∞ ,p transformation is expanded in τ ∞ ) dτ ) 1 − = lim Q u q − − S q ( − u ; 2 ; . One can also obtain the second ramified partition 1 q 8 p q 2 µ 2 ( ! (+) S 2 (+) − , µ yields the same result with η → D ( )= 2 qη q τ inst ( 1)( S µ 1 Z 1) = PE − × − u, Q ( q η − 0 ' ( d N k 2 Z 5 2 q η η d/ ( 3 = S bispectral dual )= Z τ u, Q ( )=1+ from the sum over residues at the second set of poles (+) d, ] ) order. This result provides a strong evidence that the two types of defects, the 5 2 u, Q − 2 ( transformation exchanges the mass parameter and the corresponding momentum ( [1 d/ ) 3 S Z S (+) , Z Q/u ( inst S The We sum over all residues of the poles at The S transformation can be promoted to five dimensions. It acts on the U(1) flavor 2 PE denotes the Plethystic exponential defined as PE[ Z u transformation: i.e. 10 Therefore the normalized partition function defined as S function as follows: which is independent of theto mass and FI parameters. monodromy defect and the defect of 3d hypermultiplets, are dual to each other under the reproduces the ramified instanton partition function up to the factor first few terms are given by Remarkably, having identified the parameter as The instanton partition function after The contour is choseninstanton to contribution enclose also has thecontributions poles poles from at at the residuesfunction at of the these dual poles theory. are crucial to compare with the partition corresponds to the holomorphic block in the second vacuum. symmetry of the 3dtransformation theory on on the the surface defect defect as partition we function did is above defined in as the in decoupling [ limit. The residues at the poles JHEP05(2015)095 , j δ 1) (5.3) (5.1) (5.2) (4.71) − = 4 R- =( N ) ) j j ( a ( n µ εσ , − ) − j ) ( n j (+) S ( a D ηεσ ηµ . We start with the (1) . In physics literature = 2 by this axial mass, j N =1 E M C & a 2.4 N · (1) ) j N ( n , +1) N ′ σ j ) = ( n j ( a − 1) theory with fundamental matter µ εσ is formulated using complete N- − 1 (+) S ⊂ ⊂··· +1) − ′ 2 − j ) N D j ( n , 2 j ε ( n % C U( u N-1 ησ ηεσ → p ⊂ ) ) ) as follows →∞ j +1 =1 ( a ) j ′ C & ,Q n N 2.4 × ×··· 1 · ,Q − ) ) 1 ⊂ ′ ,η j j – 51 – ,µ u ( ( n n − 0 2 ) u j − ( i ( ησ ησ η 1 :0 → σ ( θ j − − 2 1 N ε ) ) ε ′ θ j j F ( ( n n σ σ → + 1 1 ) 1 − − j ( i − u η η p σ n ) 1 j ̸= ′ ) N & n ). · u, Q ) 2 j 1) u, Q ( n − ( − ′ 4.42 η 1 j ( n ( εσ θ 1 σ − θ − 1) $ ) j − ( ′ n j ( ). n 4 )] theories let us rewrite equation ( ηεσ ησ . A Lagrangian description of the U(1) 1 N − chiral multiplets at the final node. =1 j ′ & n N In order to see how the vacua equations for chiral quivers arise form vacua equations [U( N T j +1 τ j τ where we scaled and then take the limit of dimensional complex flagconnection variety of equivariant quantum K-theorythe of corresponding complete 3d flags gaugecomplete and N-flags theories twisted arise along chiral as rings the target(see spaces of lines figure of of supersymmetric sigma section models with chiral matter shall study chiral limitsthe of lines three-dimensional of theories the and previous sections. their 5d/3d completions5.1 along Complete flags andThe open chiral Toda version chains of the quiver theory from figure dimensions which we have considered so farwhose included masses both were chiral and split antichiral viasymmetry. matter fields the Since twisted the mass supersymmetry ofnothing is the already prevents axial broken us U(1) down from to subgroupsome examining of of the the the mass possibility parameters of become more large. extreme This is mass the splitting goal when of the current section where we jsenaars system in ( 5 Gauge theories with chiralIn this matter section and we 4d discuss reduction chiral limits of our 5d/3d construction. Gauge theories in three obeys the difference equation which is in perfect agreement with the eigenfunction equation of the 2-body elliptic Rui- Figure 4 and JHEP05(2015)095 σ ). )we 5.3 (5.7) (5.8) (5.9) (5.4) (5.5) (5.6) (5.10) (5.11) (5.12) 5.7 , where , . B j j δ δ 1) 1) − − . =( =( 2 ) τ ) ) 1 ) j τ j ( a , ( a µ µ 2 (see appendix = τ , − σ ρ 2 τ 1 − ) p τ )] ℓ, ) j 1 τ j ( n ( n =0 N = = σ σ 2 2 ( 2 τ . ( Λ τ ) , ,p [U( j 2 j 1 =1 2 2 πR T =1 τ M & . Let us see how to describe its a M 2 & µ µ a 5.7 =1 1 · e · + P 2 ) + + σ, p ) 2 j = µ εσ j ( 1 1 n τ ( n = ] we get 1 σ µ µ 2 1 , σ − − τ ) 6 τ τ 1 τ j ) j 2 − 2 ( n = = 2 τ − 1 µ j µ 2 τ 2 τ 2 1 ηεσ ηµ ησ p p ϵ ,p +1) µ 1 ′ +1) + 1 j ′ j ( n τ + µ εσ 1 ( n ) we get for ( → – 52 – 1 σ 1 . From [ 1 τ µ σ =1 − − τ j )= − ( 1 p τ 2 5.5 η 1 p +1 1 ℓ P 2 =1 σ j =1 − ′ µ − ∗ ′ & µ & j n − N − 2 n − 2 ηεσ ηµ T − − · τ τ 2 · ) − 2 2 2 1 ℓ ) ) ) σ µ ′ ′ − τ , so, after additional scaling j j j ητ 1 ( ( ( n n ε n j ) and ( − )( ησ ηµ = 2). 1 2 µ σ σ σ . In order to derive other chiral theories corresponding to 1 + 2 1 τ τ σ = n µ n N 5.2 µ N 1 τ µ j j j ̸= 1 ̸= F p ′ − ′ N − & ∗ & [U(2)] µ − n 2 N n 1 ( τ · T σ T · 2 1 ( 2 ) for p ησ ηµ − τ 1) 1) 1 2 η − − ′ τ τ ′ − j j 1) ( . One can recognize in the above equation equivariant quantum K-ring n ( 2.56 n 1) − 1 p − σ σ − τ ′ 1 j ′ j ( n ( n = − − ητ σ ) (cf. ( ) 1 τ ησ j j ( p n ( n 1 = 1 to get the following σ σ P 1 εη =1 − =1 +1 ′ j ′ & j & n n N j +1 j +1 τ j τ j τ τ which can be identified withcan the derive trace of two-body open Toda Lax matrix. Fromwhere ( we put relation for where Λ is thein dynamically the generated l.h.s. scale. of the By above expanding expression the and quadratic using expression the in definition of the momenta we arrive at Performing scaling limits in ( or The set of the above three equations can be rewritten as 5.1.1 Chiral limitThe of simplest examplequantum of cohomology a starting complete from flag variety is We have just described howthe to theory obtain supported a on chiralnon theory complete which flags lives one on needswe start a complete from the complete theories corresponding flag of vortex from partition type functions) and take a limit similar to ( we get For flag manifold we have such that JHEP05(2015)095 , ) } ). ) i j N (1) 1 ( j µ σ with (5.13) (5.14) (5.15) ) − i ( j = u s ( 1 τ ,...,s ) p N =1 j , i ( 1 s { )= )] case they may u ( ) and anti-chiral fields N ...,j, , ) M 2.9 ). [U( u . ( =1 T j ⎤ ⎦ 5.13 ' ) )= Q I u ( j ) . reads ( s } u ( ) and read as N 4 N j ,i are mass parameters for U( +1) Q Q I 4 which is generated in one loop, and ) } ( i . The parameters j / 5.13 +1) ( i s j 2 = N j s s ( k ,...,t m =1 σ 1 ) N j +1 t =1 ) # j τ i i,j = # I { =1 +1 ( j -flag. As in the & ) j k ) s % − σp ) ) ,...,m N N j I 2 ) ( j ) 1 R − ( j j s iδ u ( 2 i s A m ( ) we introduce real mass parameters ) 1) σ 1 { I ( − + − ( i − 1 I ) s 2.9 j and ( i – 53 – I − s ( i j +1 Q I j s =1 ( t i Q # ( ℓ j j +1 δ ℓ τ @ j − i )= p 3 τ ) ) 1 τ u u p i ( u p u 3 2 δ ( ( 1 . τ τ 1 2 )+( + ) " Q 1) 2 2 τ ' ' 3 τ u δ Q Q 2 τ p % ( − p ) p ) = 0 we find that ( 2 τ 1) ' 3 2 2 1 σ u Q − p τ τ u τ τ ( ( − − ) 2 1 2 1 2 u ( u u δ τ τ = 1 ( Q Q )( 1 = 2 example. Thus we have δ − 1 1) δ 1 τ 3 τ p = = 1) Q +1 − 1) N p ). Thus we have constructed the limit ( 2 τ 3 τ − i,i 2 τ − − = p ( − and the phase space of the (complexified) u 2 τ 1 2 up up ( 2 τ 2.17 τ τ − 2 1 3 2 p )+( N $ 1 2 τ τ τ τ τ σp = 2 case so that δ 2 1 2 τ ) p ) 1 τ τ p 2 1 3 τ ,L δ 1) – 54 – u 1 τ τ N p i δ ( + − 1 1) − p 1 δ ( 1 τ − 1) u − Q p − ( 1) u − 2 − )( $ δ ( u 1 τ − 1 τ = 3 case. We have two equations − ( u p 1) p +( − ) = 3 τ = 0 it is straightforward to see that N − u − 3 τ − 3 τ − p ( ( p p ) i,i u 2 2 u u + u = 2 for simplicity. There is a single equation − 3 2 ( + Q τ τ 3 τ ) 2 τ = ) )) analogously to ( p N u M p 1 τ 2 τ u ( )=( ( p p u ,L + ( L M − + 1 τ p M 2 τ u =1 ! ( p 1 τ u 2 ,i 2 p u ) is the matter polynomial of the δ ! and therefore +1 u i − 1) ) = det( ( u 2 τ L 2 3 u p − ( + u Q − M = u )=( u ( . Furthermore, by evaluating the second equation at )= 2 τ Now we can graduate on to Again, let us consider M u p ( 1 − ' using gauge theories. 5.2 Pure superThus Yang far Mills we have theories identified parameter with spacechiral of defects supersymmetric matter and vacua with of closed quiver global theories Toda with symmetry chains of rank components and then when trigonometric Ruijsenaars-Schneider model is reduced to relativistic Toda lattice which are Hamiltonians of the three-body relativistic5.1.3 Toda system. Open TodaNow Lax we matrix systematize thematrix above of the computation open in relativistic the Toda chain. general We case should find the matrix We can recycle the informationu from the Recall that Thus we have recovered the Q-Toda Hamiltonians. By evaluating the equationQ at JHEP05(2015)095 ) =1 =1 4.24 " (5.23) (5.24) (5.25) (5.22) 2 τ N N p 1 τ p 1 3 is described τ τ 3 4 δ 1 τ " are included to 1) 1 τ relativistic Toda 2 up p − ) µ 3 1 ( ] for nonrelativistic 1 u τ τ ) (5.21) ( , we shall investigate µ 3 − u 1 14 δ 4 ( 1 τ √ − closed j 1) , p N ' Q 3 τ − z p Q ) Q ( z. 3 2 u 1 N 2 2 τ τ ( − τ τ j µ µ 2 3 τ δ 1 1 N 1 1 Q p δ µ µ 1) 2 3 = τ τ 1) √ √ − 2 ( − δ +1 ( = = j − 1) τ 2 2 − 3 τ − p ( up N 2 τ τ ) p − u 2 1 ( 2 τ up τ τ ,Q 1 p ) 1 − δ 1 2 u z j Q τ τ z, Q ( – 55 – 1) 1 2 Q 2 2 δ − − µ µ in the formulae for the Nekrasov partition function ( j 1) +1 1 1 N 1 1 τ j . 2 µ µ − τ − Q ) . The normalization factors of . Now, along the lines of section µ ( j 1 ) = 1, whereas at the final node there is a modification √ √ 2 1 1 τ u 2 δ τ τ ( u − p − N 1 ( 3 τ 1) τ = = N 1]. We make the replacement 0 3 τ − p = and τ , − p 1 1 Q 1 N ( 1 z + Q Q − ' + µ Q 3 τ − N = [1 ) 2 τ δ p ) ) p u 2 τ ρ particles. In other words, we have described the ‘chiral’ version of u ( 1) p ( − 1 + ( (+) − − N ( + 1 τ +1 , j N p 2 τ = 3 we would find 3 τ − ! p Q Q 2 and with p ) 1 τ 2 2 τ N u p u = = 1 SYM in five dimensions. We expect a deformation of the twisted chiral − p ( ! 1 τ − u p N M 3 ). Then we introduce the monodromy defect according to the prescription of ) + − u . For U(2) theory the instanton partition function will have the form ( N ,...,N 4.11 )= 4.2 u ( =1 We now discuss the computation of ramified instanton partition functions of We therefore claim that the twisted chiral ring of quiver theory in figure j M Again, later we shallremove replace fractional powers of in the presencenormalization of will the be used defect, formove which to the the difference we center equations. shall of Note mass address that frame they later i.e. would consider in cancel gauge if this group we SU(2) section. instead of The U(2). same labelled by the partition SYM theories in five dimensions.to For compute simplicity the we start correspondingtheory with ( instanton gauge partition group U(2). functionsection we In order use character for which are nothing buttheory to closed five Toda dimensions transforms spectral openchain relations. relativistic — Toda chain a to Thus natural weToda generalization systems. claim of the that similar coupling statement the found in [ For example, for for of the equation to radius of the circle. by the functional equations the XXZ/tRS duality from section how the above construction is modifiedin pure when U( the same theoryring is coupled depending as on a the surface dynamicaluse defect scale a of dimensionless the five quantity dimensional Λ theory.obtained In by what follows, multiplying we by the dynamical scale by the open Toda chain with JHEP05(2015)095 . 1 /τ 2 ]. At τ 0 and (5.27) (5.28) (5.29) (5.26) 31 → Q from [ , . ) 1 1] 1] , , , − P (+) ( [1 [1 ) l Z Z q sigma model we can = = − 1 ) P 1] 1] , , , − ) (+) ( , , ) (1 [1 [1 1] 1] , , l − n n Z Z (+) ( q [1 [1 k z z 2 τ 2 τ 1 ) Z Z ) ) p p µ ) ) 1 2 qz 1 τ 1 τ ( µ µ ) obey the followingfference di − ,q ,q j j ) ) 2 1 2 q q ) of holomorphic blocks and vortex µ µ µ ,q ,q ( ( ( 2 1 5.27 − − In this limit we expect to find con- θ θ µ µ 2 2 5.5 2 1 ,p ,p ) ) / / 2 2 =1 µ µ k l τ τ 1] 1] ,q ,q 12 , , ( ( ), ( )( )( 2 2 on the left we must include contributions +1) +1) , (+) (+) θ θ [1 [1 and perform the chiral limit in order to j j τ τ n n ( ( q q 2 ( ( Z Z 1 5.3 . The above solutions represent the non- θ θ n n =0 µ ) ) ∞ 2 P − − k q q # ) ) 2 2 µ ∗ , namely study 3d partition functions of chiral quivers µ µ ,q ,q = 3 T (1 (1 ) ) – 56 – 1 2 and + + ]. ↔ µ µ ,q ,q % =1 =1 1 1 1 ( ( j n j n 1 2 z 61 1 µ θ θ µ µ ; µ µ ) ) , , µ q 1 1 ) for ; τ τ ,q ,q ( ( =( =( 1 2 1 1 =1 =1 . The above result can be understood as hypergeometric ∞ ∞ θ θ ) we get τ τ µ µ # # ) n n 2 1] 1] 3.39 ( ( , , q − θ θ (+) ( [1 [1 = = 0; 5.25 z/µ Z Z ) , = = 1] 1] , , we discuss the decoupling limit first, so we send − 0 ) % % (+) ( [1 [1 → 2 1 2 1 $ − (+) ( Z Z τ τ τ τ z 1 4.3 Z Z F − − 2 2 τ 2 τ p p → 2 1 ). We could choose, for example . V µ µ 1 Z F + + 3.42 0 1 τ 1 τ p p 1 2 µ µ $ $ 0 limit using notations ( It is straightforward to prove that expansions ( Equivalently, from the computation of the ramified instanton partition functions in the For example, in order to find the vortex partition function for One can also study ‘chiral’ version of section → 12 and prove that they satisfytheories Q-Toda in di fference three equations. dimensions) This was approach pursued (albeit in without [ reference to gauge The second equations in eachIn line order impose to that remove the theto answer dependence the depends on classical only action on from theaddressed FI ratio parameters. in ( These are exactly the same ambiguities we have which are relatedperturbative by contributions interchanging to the twotwo independent chiral holomorphic multiplets. blocks of U(1) theory with equations Q which leads us tothe the last formula step for we rescaled equivariantseries K-theoretic of J-function type for be the case. start with Givental J-functionsee that ( Analogously to section decouple the five-dimensional degreestributions of that freedom. can beanswer accounted should for be by consistent with degreespartition chiral of limits functions freedom ( which supported we have on computed the earlier defect. in the The paper. Indeed, this turns out to 5.2.1 Open Toda from decoupling limit JHEP05(2015)095 ) ± ( with Z ] the (5.34) (5.30) (5.31) (5.35) (5.32) (5.33) 2 63 /τ 1 τ = , , ) , z ) − ( (+) ± ··· ··· ( ) we arrive to the R R Z + + ) obey the difference = = 2 5.1 z z ) µ Q Q − 1 ) ) ( (+) 5.32 q q µ 1 2 R R µ µ = τ 2 τ 2 ]. , p p ) − − τ 1 τ 1 ± 13 ( 2 1 , q q =1 µ µ ). Z 2 τ 2 = 5 that ( )( )( p . µ ). Now both of the functions q q 2 Q 1 ] 1 τ 5.32 ,p ,p 2 n µ − − ) ,ηϵ ) a, q − + ( 4 =[1 ± Z ( (+) (1 (1 → ( θ ρ 1 1 R R n Qϵ Z + + ,p − 0 ) a (1) (1) z z → ± ) ) → E E − expansion. This fact is not surprising since one ( ,Q 2 q q ϵ 2 1 – 57 – 1 2 Z = = Q ) with τ τ ) µ µ = lim ) 2 1 2 )= n 2 ,Q ) 1 − µ − − µ ( (+) ± 1 2 Q 1 2 ( + τ τ 1 R R a, q q q µ µ is also rescaled as follows ( n 1 2 → 1 R ϵ θ % % )( µ Q 2 1 Q a ( q 1) ( Q Q τ τ p → 1 2 1 2 O =( − − τ τ τ τ 1 2 , the partition function the presence of the defect diverges in ) and taking the limit described after ( ) q τ τ (1 ( − − ± Q ( 2 1 2 1 4.39 Z τ τ τ τ ) obeys % − − 1 2 =1+ =1+ 2 τ 2 τ τ τ a, q at higher order in the ) p p ( θ − 2 2 1 − as we obtain the above expressions via taking the limit from the elliptic (+) ( µ µ µ 2 τ 0 since we have degrees of freedom propagating in this plane. However, R R p Q + + ↔ + → 1 τ 1 τ 1 p p 1 τ 2 µ ] the K-theoretic J-function of the flag variety was shown to be the (universal) 1 2 p ϵ µ µ $ 62 0. In addition to the above scaling one can redefine the FI parameters as $ $ . Starting from ( 1 → We have checked to order In [ /τ ϵ 2 equations as then the elliptic RS eigenfunctions take the form of ( RS eigenfunctions. Yet, theQτ expressions are symmetric ifabove one expressions interchanges provided that We can see that,related unlike by the elliptic RSneeds eigenfunctions, to these scale two functions are no longer The first few terms in the expansion are Turning on the parameter the limit the divergence exponentiates anddefect. is Thus universal the i.e. expectation it value is of independent the of defect the is presence finite of in the the limit eigenfunction of the relativisticanalysis Toda was system extended to for non-simply any laced simply groups. laced5.2.2 group and Closed in Toda [ chain obey the same difference equations which are the Hamiltoniansmorphic of blocks the in this two-body example open were Q-Toda discussed integrable in system. detail in The [ holo- where, we recall, JHEP05(2015)095 . In k (5.38) (5.39) (5.37) (5.40) (5.36) by the same . ) ) ± ± ( ( R R % flavors 2 ) µ 2 , 1 M Q 2 ( µ )). For simplicity let us copies of the tautological O ϵµ = . ) + , → M ± 2 ) ]. Here we discuss chiral limit ( 2 q M,N µ =1 µ 2 R 13 2 1 65 i τ µ j 2 τ µ 1 p (Gr( 1 τ − ησ ησ 1 τ ∗ µ ) theory with 1 2 − T − = µ qQ in the difference equation controls the ϵ N i j 2 )( σ σ µ Q U( 2 1 1 → 1 ,p µ − − ,q ∗ 3 µ ) η η − ± i ( q =2 ̸= N =1 1). . Again, as in the non chiral case, they should j & 1 R – 58 – , 2 i 1 µ a N 1 2 µ ( (1) σ , µ − ϵσ , µ − E σ − (3 1 − 1 → 1+ a − i = T and µ theory in two dimensions. σ ) $ 1 1 ηµ ησ ∗ − ± ) ) over the Grassmannian. µ ( 2 η fundamental twisted chirals and Chern-Simons level 2) ηµ µ =1 R , M 2 & a η,σ k, N + 1 % M 2 1 qσ − τ τ 1 Q − ϵ µ 1 2 ] computed quantum K-ring of a family of line bundles of Grass- τ τ = 3 first. The variables in the above equation then can be rescaled M → 64 =( − for convenience. η ), which naturally appear in the study of twisted chiral rings of 3d M 2 1 k Gr( (1) ). From a three-dimensional perspective they were contributions to τ τ − E → − to 5.29 2 τ k M p M,N,k ) theories with ( 1) = 1 and + T − N 1 τ ( N p O $ Let us write Bethe equations for 3d We changed = 2 U( 13 For completeness we exhibit 2d analoguesthe of main some text. properties Quantum of cohomology 3din of theories connection complete we flag with have varieties monodromy used and in defects theirof cotangent were the bundles discussed twisted in chiral [ ring of (2 which is the chiral ring relation for 5.4 4d/2d construction As we know theseconsider are quantum ringas follows relations for to get Kapustin and Willett [ manninas N other words, the authorsbundle computed the quantum K-ring for we have continuous parameters be fixed by Betherelated equations to coming normalizability from of the the divergent wavefunctions. prefactor. It5.3 could be that Connections this to is Kapustin-Willett results Thus we can seefive-dimensional how instanton turning corrections. on thebody We parameter have closed found Q-Toda formal Hamiltonians. eigenfunctions of Note the that two- the spectrum should be discrete, whereas is the vacuum expectationfactors value of as the in Wilsonclassical ( loop. action from We now FI multiply parameters. The final result is that both functions obey where JHEP05(2015)095 , ) j j δ ( ϵ ϵ 2 2 a ) ) 1) j j M (5.41) (5.42) (5.44) (5.45) (5.46) (5.47) (5.48) (5.49) (5.43) − − − =( = ϵ 2 +(1 +(1+ ϵ 2 ϵ 2 ) ) , ) j j + j + ( a ( n ) ) ) s j j − m ( n ( a ] we rewrite the − s +1) ′ ) − j m ( n j ) 14 − ( a j s +(1 − ( n ) ) and the coordinates ) s m j j − j ( a ( a i ( n j ) ' s j m m Q =1 ( n vanish and vacua equa- M & a s − j · i ( ) and read as follows =1 M & a ϵ M · +1 ) =1 j ′ j 2.4 ϵ + 2 & ϵ 2 ( n , n N s 1 + + +1) +1 − ′ . ) − j j , ( n j ( n i,j = 1. Following [ N i such that s +1) ′ , δ s +1 j ' − ( n j Q +1) − 1 ′ N i j s − ) − ( n − +1 N j i j Q Q s ( n − N s ) M, →∞ − Q +1) +1 ′ j i = 1 j + ( n L. ) 1 ( n j i, i,j j D i s +1 s =1 − ( a δ j δ ′ sigma model. We can also write the above M j j +1 & 2 i n i N m +1 = N · =1 j N =Λ Q ′ & = det j 2 i ϵ ϵ n – 59 – N i is nonzero, the rest + +Λ · ϵ 2 Q Λ )=Λ CP − − j ϵ ϵ L 0 M j ) ij M ) ) ′ j 1 i,j j j δ − − − ( M Q ( n ( n a i ) ) − − L ′ 1 s s i j j ' ) Q ( n ( n M = − = j Q i − − s s ( n i ) ) = ′ s − 1 j j Q − − ( n ( n D ) ) ) ′ s s j j j M for each gauge group, so we have the following chiral ring i,j ( n → n ( n ( . It is assumed that n combined with i s s s L j ) j ( ̸= j ' ′ Q n ( n N j & j n s ̸= =1 ′ M N · & a & n at each gauge group. · · ϵ →∞ i ) ϵ ϵ 2 2 ϵ + 1) 1) ) + + − j ′ − ) ( n j ′ j j ( n 1) s ( n ( n s − s s ′ − j ( n − − − ) s 1) j ) − ( n 1) ′ − j j s ( n ) − ( n ′ j j s s ) can be written as n ( n ( , ( j s s 1 1 δ 1 − − =1 =1 j j ′ ′ − 5.44 =1 1) & & j ′ n n N N & n N − Supersymmetric vacua equations can be obtained from ( j +1 τ j j +1 =( τ j τ τ and it also follows that Interestingly the momenta inare Toda dynamical are scales the Λ auxiliary polynomials where tions ( using auxiliary polynomial above equation using Toda Lax matrix where For a complete flag case only which is the quiverequation generalization in of Baxter form a We now take the limit is kept fixed.thereby In generating order a to scalerelations keep Λ the whole expression finite the FI couplings have to run Now we make some shifts JHEP05(2015)095 ). ], in (5.50) (5.52) (5.51) 68 3.37 gauge the- we need to ∗ 2.1 =2 and polynomials ] for details). i N 65 Toda quiver theories in two I / ∗ ] ] (see [ 2) N , = 2 and , 67 and (exponential of) relative ) i . N i µ p ,... µ theories were studied in [ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ 1 − N ∗ 0 0 0 u ,µ p ( 1 2) 2) and (2 , , 1 − =1 N & j N − ...... ] one can derive quantum J-function for 3 2 with dynamical scales Λ 0 = (2 τ 66 p )) = 0, and, second, take the chiral limit in mass τ ,... 1 +1 2 1 i 1 N t τ p ) were examined using cluster algebra methods. − → − ,τ p, τ i – 60 – 1 ( t 1 The equivariant quantum cohomology ring of the N 0 000 2.7 e R p ··· − = ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ is isomorphic to Toda i L τ = N ,...,p − 1 C p u [ Toda C L ≃ det ( ) N ). This theorem can now be viewed as an important limiting case ]. In particular, two-dimensional vortex partition functions for the F ( 65 • T (Lax matrix of open Toda chain) 5.48 is generated by thecientsffi coe of the following polynomial QH ). Toda -type relations, similar to ( L ' Toda Q 5.3 I Q Further connections to integrability in Note that currently in mathematical literature all necessary tools for computing quan- A detailed analysis of twisted chiral rings of (2 In the Lax matrix above we have identified momenta respectively in ( i ' tum cohomology of vector bundles overis complex already projective available. spaces and Thus theirthe using generalizations cotangent the bundle results to of the [ flag variety in theparticular, framework of [ mology was done in [ theories on two-dimensional defects insideories four-dimensional were computed, and thecan identification also with obtain Givental these J-functions expressions was by presented. taking One the 2d limit of 3d holomorphic blocks ( of our construction. Infirst, order shrink to the compactification obtainparameters radius the ( above result from Proposition dimensions in the context of ramified instanton counting and equivariant quantum coho- coordinates between Toda particles Q where matrix where ideal One of the motivatingwhich results we for list this below for note completeness. is the famous theoremTheorem by 5.1 Givental and (Givental, Kim Kim, complete 1993) manifold of flags inside 5.4.1 Equivariant quantum cohomology JHEP05(2015)095 ], and 0 bulk 72 ] the five → 46 Q theory. ∗ =2 gauge theory by taking N ∗ ) . When we send N =2 Q N ], Ruijsenaars-Schneider [ ] and references therein). These are 71 74 – 69 vanishes. Provided that an appropriate ϵ – 61 – = 2 gauge theory with a 3d surface defect), which has elliptic N ], in particular, the 4d analogue of the deformed twisted chiral ring ]) and Seiberg-Witten curves of supersymmetric gauge theories with ] this correspondence was extended and systematized to a large class 14 76 – 77 73 the first row contains Calogero-Moser-Sutherland (CMS) family, the second 5 ) was constructed in the cited paper for the SU( exhibits a family on many-body classical integrable systems and our proposal 4.2 5 In figure Recently, in [ There exists a correspondence between the spectral curves of a family of many-body arrows. Thus the right columna describes role of elliptic the models, bulk whosedegrees (4,5, ellipticity of or parameter freedom 6-dimensional) plays decouple instantonare fugacity and located we are in left theone with middle the may column. corresponding take These defect a models theories limit which are when trigonometric the in coordinates. adjoint mass Next, row describes Ruijsenaars-Schneider (RS)that are family, whereas bispectrally the dualBlue double to last arrows eCMS, row describe eRS, contains bispectralthe and, dualities models properties between finally, of the the the modelsDell double Dell in and system the elliptic itself table. are (Dell) is largely Because designated system. unexplored, by the the dotted anticipated blue duality double between arrow. Various limits are shown by periodicity properties of the coordinates andand momenta: elliptic. each may be rational,(corresponding The trigonometric to most 5d generaldependence system on studied positions in andhow this trigonometric some dependence paper of the on is remaining momenta. the members of We elliptic have the RS indicated family can system be obtained by various limits. 6.1.1 Calogero-Ruijsenaars-Dell family Figure for how theirgauge eigenfunctions theories are in various realized dimensions. by The integrable the models partition are characterized functions by of how the supersymmetric the examples that we have focussed on in thisfour paper. dimensional quiver gauge theoriesand of six-dimensional finite and versions affine ofdeformation ADE the are type. also Seiberg-Witten Later discussed. in geometry [ together with its quantum many-body systems. integrable systems (Calogero-Moser-SutherlandDouble-elliptic [ [ adjoint matter in four, five and six dimensions (see [ 6 Summary and outlook 6.1 Integrable systems In this section we outline how the results of this paper fit into the wider context of integrable Using the results ofmensional the quiver previous theories section coupledthe we to radius can four of compute dimensional twisted thebeen chiral conducted compact in rings circle [ of inrelation two ( the di- 5d computation to zero. Classical analysis has 5.4.2 Coupled 4d/2d systems JHEP05(2015)095 . ) )] 4.3 N 4.71 [U( 0 limit. T → ). ϵ 0 descends us N → R ). By applying the S- ) gauge theories in the ) as a partition function 4.59 4.45 3.28 . The eigenvalues in this case 4.5 ) and U(3) ( 0 thereby shrinking one of the extra compact 4.42 → – 62 – ' (rows). R ). U(2) ( p N ∗ )] theory, then in terms of vortex partition functions N =1 [U( N T ) and correspond to VEVs of Wilson loops in the skew powers of ]), one obtains a family of rational models from the 6 4.31 ) we have shown that the eigenfunctions of the elliptic RS model ( 4.64 (columns) and momenta ), where the eigenfunctions are written in ( q ). After quantization these relations become operator equations which we 3.30 . Classification of integrable many-body systems according to their periodicity properties 2.32 ), and finally as a decoupling limit of the 5d/3d partition function in section Finally, we were able to find the spectrum of the two-body dual eRS model using Next, we have found the eigenfunctions of elliptic RS system for two and three particles Recall that the classical tRS model was derived from the twisted chiral ring of 3.37 the fundamental representation of U( the U(2) 5d gaugetransformation theory ( coupled tocan 3d be free reproduced hypermultiplets from ( the eigenfunctions of the dual model. The eigenvalues are characters of antisymmetric tensor representation of U( as partition functions ofpresence of 5d monodromy defectscan of be maximal computed type in section ( theory ( solved in ( on the Coulomb branchin of ( descendants in the last rowdirection take of a the limit 6d theory.the This other limit circle moves (again, usfurther with to to a the the CMS properly RS family. chosen family.the One scaling Further can diagram on, of use starting shrinkage the parameters) from of arrows Dell. we described above to navigate through Figure 5 in coordinates scaling is chosen (seeEquivalently, one [ can take the limits row-wise. First, starting from the Dell model and its JHEP05(2015)095 can 7 . In the last row of the ]. and can be compared to 6 0 should be supplemented 78 7 Toda → Toda ϵ (??) ] for the details about XXX and affine 6d(1,0) 6 2ddefect 3ddefect 4ddefect closed 4dN=2 SYM 5dN=1 SYM ) Toda chains. In the limit when the closedelliptic remain constant. (see [ closed relativistic Toda closedrelativistic i ϵ 5.37 ϵm

– 63 – ] and references therein for the discussion about 79 Toda ) and closed ( finite 3dN=2 4dN=1 open 2dN=(2,2) 5.30 open elliptic Toda chiralquiver theory chiralquiver theory chiralquiver theory

open relativistic Toda openrelativistic ). The classification is presented in figure

t r e p 5.4 0) on codimension two surface defects in the four, five and six-dimensional → we have discussed 3d quiver gauge theories with chiral matter and pure 5d Q 5 . Classification of rational, trigonometric (relativistic) and elliptic Toda integrable sys- . 5 The spin chain family (XXX, XXZ, and XYZ) in the middle column of figure be obtained from theconstruction spin is chain family represented by byXXZ turning the off models). adjointthe mass We ‘Planck referXYZ constant’, the spin which chain reader in and to XYZ our (elliptic) [ by Gaudin scaling model. of The anisotropy limit parameters such that theories we have discussedfigure above. This is illustrated in figure be described using the Nekrasov-Shatashvilidimensions correspondence two, using three, quiver gauge and theories four in respectively. The Gaudin family in the first column can 6.1.3 Gaudin-spin chain family For completeness we alsoreproduce discuss the the twisted quantum chiral integrabledecoupling rings models of whose the Bethe supersymmetric equations gauge theories appearing (after SYM theories with defectsspectra which support of chiral relativistic quiver opencompactification theories. ( radius We have of found thechains quantum (see 3d section theory becomestable one small can we find recover elliptic nonrelativistic Toda systems Toda which were studied in [ tems. Closed Toda chainsunder exhibit the affine action Lie of algebra the symmetry, Lie whereas algebra open itself. chain are symmetric 6.1.2 Toda family In section Figure 6 JHEP05(2015)095 . ]. 7 84 14 ]. 82 – 80 (rows). Blue double arrows describe m – 64 – (columns) and anisotropies t these conjectures. We understand that work in this direction is ]. 83 prove . Classification of integrable models of Gaudin and spin chain type according to the The Hamiltonians of trigonometric Ruijsenaars-Schneider modelsubalgebra form inside a double commutative affine HeckeFurther algebras understanding (DAHA) of and the Cherednik role algebras [ of Cherednik algebras in gauge theories is due. Ruijsenaars-Schneider model and (inchecked the our 2-body conjectures case) by itstions expanding bispectral of the dual. 5d eigenfunctions theories (instanton Webe with have partition possible codimension func- to two defects)being to pursued first in several [ orders. It should In this paper we have conjectured the structure of formal eigenfunctions of the elliptic To the best of our knowledge there are no known quantum models with elliptic de- Some work understanding these integrable models is being done in [ • • 14 6.2 Open problems Let us now articulate whatpresent we work. believe We to list be them some below important in questions the unanswered random by order. the pendence on twists whichMoreover, our could analysis be involving defects consistentlymatter in fields placed higher does not dimensional in immediately gauge suggest theis theories any candidates certainly with right for an adjoint such column models. intriguing question of Their on existence, figure its own and should be pursued independently. Figure 7 periodicity properties in twists bispectral dualities between the models in the table. JHEP05(2015)095 ] 93 – 90 , 81 , 80 ] for a related study of q-Toda 89 1) theory with a four dimensional , – 86 – 65 – ]. 96 – 94 ] correspondence is modified to conformal blocks of affine algebras. 85 a legitimate question toelliptic many-body ask system. how the correspondence is generalized to the related in the limit ofrespondence large number to of the particles. Ruijsenaars-Schneideralready One family. been may done Some study in generalizations [ work of in theIn this cor- this direction paper has weRuijsenaars-Schneider system discussed and the Bethe relationship ansatz between of the quantum many-body XXZ spin trigonometric chains. It is functions in the presence of suchin monodromy the defects. case One of may wonder irregular what singularities. happens The Calogero-Moser-Sutherland family of integrablerteweg — models de is Vries/Benjamin-Ono/Intermediate connected Long to Wave hierarchy the [ Ko- Understanding of five-dimensional gaugenew theories ideas with in surface quantum defects affinedegenerate might algebras conformal lead (see, blocks). to e.g. [ There is a notionthis of paper regular we and have only irregular addressed monodromy the defects regular case in and gauge computed theories. instanton partition In generalization would be to study sixdefect dimensional and (1 use it towould find be formal interesting eigenfunctions to of understand the the double-elliptic actionIn (Dell) of model. bispectral the duality It in presenceTachikawa this [ case. of monodromy defects, it is known that the Alday-Gaiotto- We have analyzed in this paper five-dimensional theories with defects. An immediate MB gratefully acknowledges support from the Perimeter Institute for Theoretical • • • • • Dimensions” at CERN andResearch Foundation also (Korea) CERN-Korea for the Theory hospitality andnumber Collaboration support. of funded conferences This and by work seminars was National metric presented including at Correspondences” “IV a and SISSA KITP Workshop program onField “New Dualities Methods Theory”. and in Geo- We Nonperturbative are Quantum grateful to the organizes for invitations and kind hospitality. Physics and IAS Princeton throughthanks W. the Fine Martin Institute for A. Theoretical andfor Physics at Helen Theoretical University Physics Choolijan of at Minnesota, Membership. Kavli Universityat Institute PK of Berkeley, and California Simons Santa Center Barbara, forfor University Geometry of kind and California Physics hospitality. where HC partTheoretical gratefully of his Physics, acknowledges work the support was organizers from done, of the “Exact Perimeter Results Institute in for SUSY Gauge Theories in Various We would like toMina thank Aganagic, Davide Shamil Gaiotto, Shakirov, Jaumedrey Nathan Zayakin, Gomis, Mikhail Haouzi, Bershtein, Nikita Satoshi Aleksey Morozov, Nekrasov, Nawata, AntonioGiulio Amihay Sciarappa, Vasily Bonelli, Alessandro Hanany, Pestun, Tanzini, Heeyeon An- Kim for fruitful discussions. Acknowledgments JHEP05(2015)095 (A.7) (A.3) (A.4) (A.5) (A.6) (A.1) (A.2) 0 defined for > 2 , ,ω t % dt 1 ω +1 1) a m − q . t , , . k 1 − ) 0 0 , ω ) with k q 1 2 e ) ; z =0 zt $ q . e +1 ,ω ( ) 1)( n> n n< ) 1 m k m ω q ) − q t for a, q aq ; k 1 − ( =( ) b 1 ω ( − q e − ) for . (1 k ; ⃗ω ( i ) 0 ) KW c i ) (1 θ q ( ≥ 0 0 − ⃗ω xq i ; 1 2 & | m ≥ ) 8 a + 1 a · & xq − ( ⃗ω m R 1 8 ω | ) 1 for z − − (1 = ( =1 + ) where ∞ as 1 )+ k 2 # iq – 66 – a, q ⃗ω z (1 − ⃗ω 1 | ∞ ( S =0 ( | − n i ) ˆ n z θ 2 A z =1 ( . For instance − i ( )= S , 2 ,q q 2 1 , S , )= 2 )= − q, z B ; ⎧ ⎪ ⎨ ⎪ ⎩ qa a, q c a, q 2 ( ( ; iπ ( ≡ 1 ∞ θ n ) ) a, b KW ( q )= θ 1 ; a, q ⃗ω | F x ( 2 z ( 2 )=( S a, q log ( θ ) is the multiple Bernoulli polynomial. The most important property for ⃗ω | z ( 2 , 2 B Our research was partly supported by the Perimeter Institute for Theoretical Physics. which in turn iscurve related constructed to on complex conventional vector elliptic theta functions associated to the elliptic It is related to Kac-Weyl character for We have used basic theta-function in the computations in the main text where, we recall, the q-Pochhammer symbol is defined as follows this paper is the functional relations A.3 Theta functions has an integral representation where A.2 Double sineHere functions we use the double sineexample function in the appendix of. Let us summarize the properties written there. This function We summarize our conventions on various functions weA.1 have used in the Q-hypergeometric main functions We text. use the following definition for Q-hypergeometric functions Research at Perimeter Institute is supportedtry Canada by and the by Government of the Province Canadaand of through Innovation. Ontario Indus- through the Ministry of Economic Development A Conventions of special functions JHEP05(2015)095 , = 0. (B.2) (B.3) (B.1) ) % n ×···× ) ( i ) s ). 1 8 +1) N n ( j . x i N − ) (B.4) ) Then the contour − ,t n ) ( i 1 j x ( +1 15 − i ( σ i b N ) 1 n − ± ( i = b s, m k ' N ) ∗ ( i 2 T i n ε ( i ρ Z + $ k ' i b ) b ) n S ( i + n πi 0. It is convenient to define new ( i +1 2 b ds =1 n ) L & ik j ≥ n N n ( i ) =1 + n N N i & n ik =1 ( i N i & ∗ k ' ε ) =1 L + theory where , ) & n ) ρ n ∗ ) n . This effectively shifts all poles to ( ij ε ( i ) )] 8 +1 + 1) k n ix ( i n 2 N . N t n x S + − + 1) – 67 – ∈ 3.5 n [U( # 2 − mass parameters for the flavor symmetry. ε t σ n ( T ( ) ! L b i n ( N − ( i ρ S i x 2 L n 1 n ( =1 + =1 =1 i L i N N i & = 4 quiver gauge theory with gauge group U( j N instead of i,j , the i theory. The partition function is given by ! ) , ) + m ) N n j ρ m ( i n =1 ( ij + s )] L n partition functions m x ) )= = n ! N ± n ρ ( i = π s 1 T )] theory in section ,t 2 ) πb +1) i e [U( n N L ( i = ) N ( i such as m ! T x n ) ( x n ( ) ) [U( n . Quiver diagram for ( i n N T and non-negative integers ( i N dx x 2 sinh( s ρ U( and n =1 L Z ,N & n N theory is the 3d fundamental hypermultiplets for the last gauge group (see figure i t = = = ··· ,w ) vec loop , , 1 ? hyper q loop w | − 1 , 5d 1 vec loop z − , w ( 5d 1 hyper Z | loop − , 5d 1 z Z − +1 ( 5d 1 Z N These partition functions are divergent infinite products which need to be properly We find the following 1-loop determinants N Z Γ Γ Assuming The 1-loop determinants regularized using Barnes’ gamma functions can be written as When Im( where Barnes’ zeta function is defined by the series where we assumed that regularized. We willgamma regularize functions them are defined using as Barnes’ regularized multiple infinite gamma products [ functions. Barnes’ the factor follows we set potentials. charge conjugation. We also remind the reader that the equivariant indices implicitly have JHEP05(2015)095 1 − 4 ] and (C.8) (C.9) ) (C.10) (C.11) ) for the , 13 1 1 1 πi " πi ϵ ϵ ϵ 2 2 ) − e − 2 − πi , ϵ , 2 1 1 J J , J J 1 1 . 1 ϵ ϵ πi ) ϵ 1 2 2 ∆ 2 − ∆ , ) 1 n + J J . − q ) + 2 m n ϵ / πi q πi 2 ∆ = 2 field theories in three − 2 +(1 2 q / ) πi m m = 2 chiral multiplet, we can ∆ a 2 N )+2 ( 1 − − a ) ) ρ 1 ( − )+ a . N q ) ( m − α a ) ρ ( m a , m ( − ρ 0 − ρ e )+ )+ ( e a 1 − a 3 ( )+ − ( 1 ( ρ ζ ρ has been recently computed in [ a 2 ρ e ( . ( # (1 2 Γ 2 + ρ 0 ) + − 1 Γ ) 1 ϵ ) D ≥ ϵ 1 ) + 2 & 2 n 1 πi ∆ ϵ − (1 πi ϵ 2 ϵ × 1 ) πi 0 2 e 1 ϵ 2 − πi ϵ 2 ϵ 1 , 2 ≥ e ) πi , 2 ϵ & , 2 − n S – 71 – 1 n , 1 m + 1) J J ) πi − 1 ϵ 1 1 e , πi 1 2 J J 1 n ϵ 1 πi ϵ ϵ 2 | , 2 1 2+ , = 1 ϵ , 2 1 ∆ ϵ 1 ) J J / 1 | 2 J J 2+ ∆ 1 2 ∆ + ϵ ) / m + ) (∆ a − 2 m 3d chiral πi πi ∆ ∆ ( m 2 πi − − 2 πi χ − α ) − )+ 2 2 a − , a ( πi πi +(1 ρ ( 0 +(1 2 ) 2 πit +( − ρ a ( m , m ( , 3 (2 )+ ρ ζ (0 0 0 a 2 πit ! ( ] implies that we have to expand it in power series of ≥ )+ ( − ρ & n 3 a , (2 πiζ ( ζ ( 54 α Z 0 (0 2 − ρ # ∈ 2 ≥ & Γ e t ρ & ( n # 2 2 πi πiζ Z ρ ρ ρ Γ − & & & ∈ πi − & t 3 e = = = ρ ρ = = ρ & & & N , d 5 vec = = = d F 5 hyper d D , 3 chiral for the chiral multiplet with the Dirichlet boundary condition. If we impose the F ] using localization. Z 1 We now turn to the perturbative partition functions of The equivariant index of the chiral multiplet with R-charge ∆is ϵ d 54 3 e chiral Z The prefactors involve thematter 1-loop fields. corrections to the (mixed-) Chern-Simons terms by the has the following 1-loop determinant We regularized the infinitetheories. products using On the the Barnes’ other gamma hand, functions the as chiral we multiplet did with in the 5d Dirichlet boundary condition choose either theboundary Neumann condition or determines how theThe to Dirichlet computation expand in boundary the [ denominator condition.chiral of multiplet the with It equivariant index. the appearsof Neumann that boundary the condition,Neumann while boundary expand condition, it the 1-loop in determinant power of series the chiral multiplet becomes In the 1-loopin computation, supersymmetric we and need gauge to invariant take fashion. into For account the a proper boundary condition These prefactors encode the 1-loop corrections to the effective action of thedimensions. 5d The gauge theory. 3d partitionin function on [ where the prefactors induced after the regularization are written as JHEP05(2015)095 ) ] for 1 πi Prog. ϵ 2 (C.12) (C.13) 99 , , − , =2 60 1 ]. J J , (2010) 87 ) 1 N Nucl. Phys. ϵ (1994) 19 1 57 ]. ) πi , ϵ 14 1 2 ϵ SPIRE ) . − IN 2 ) , ∆ Super Yang-Mills q 1 ][ SPIRE ) J J ; B 426 − 2 IN 1 m / ϵ =4 ][ ∆ πi 2 ∆ q 2 ]. N )+ + (1 ]. m + a ( − . m ) ]. ρ m a 1 πi ( SPIRE 2 − ρ + − Nucl. Phys. ]. ) )+ SPIRE IN − 1 1 , ) q a e ϵ 2) multiplets. See [ IN ; − a ( ][ SPIRE ( , hep-th/9408099 ) ( 2 ρ [ θ 2) ][ / IN ρ / m ) ∆ SPIRE − ][ − Adv. Theor. Math. Phys. q 1 πi ϵ IN , 1 2 arXiv:0807.3720 [ m [ , − )+ ( +∆ 1 − | a 2 1 ) 2 ( 1 a Γ ( ϵ n ( ρ 2 ) ( ρ 2 ∆ (1994) 484 Γ − + + 1 − πi e ) ϵ – 72 – 1 m πi ( 2 1 ϵ 2 − θ 1 , ) πi ϵ a (2009) 721 1 2) arXiv:0901.4748 ) 2 Quantization of Integrable Systems and Four Quantum integrability and supersymmetric vacua Supersymmetric vacua and Bethe ansatz ( [ J J ]. / ρ πin , arXiv:1304.0779 1 πi B 431 1 [ ϵ − 1 2 ϵ 13 , (2 J J , arXiv:0901.4744 ) 1 [ Z 1 (0 | 2 +∆ ∆ ϵ 2 ∈ 1 arXiv:0908.4052 2 ϵ ), which permits any use, distribution and reproduction in 2 SPIRE 2 ∆ On Three Dimensional Quiver Gauge Theories and , ]. 2 − ∆ πiζ ,n n & IN ]. 1 2 ( S-duality of Boundary Conditions In [ Monopoles, duality and chiral symmetry breaking in Electric-magnetic duality, monopole condensation and + + n e Gauge Theory, Ramification, And The Geometric Langlands Rigid Surface Operators m πi (2009) 105 − 2 − πi m ρ ρ (2013) 126 ) 1 + (1 πi Nucl. Phys. 2 supersymmetric Yang-Mills theory & & a SPIRE (2009) 91 2 ( , SPIRE − ρ IN 177 m 05 = = ) − IN πin , ][ a =2 CC-BY 4.0 ( ][ (2 (0 )+ ρ 2 d Z N a 2 fermi ∈ This article is distributed under the terms of the Creative Commons − ( 2 πiζ JHEP Z 192-193 ρ 2 ( ,n & , 2 ( 1 − 2 n Γ e hep-th/0612073 Γ , Adv. Theor. Math. Phys. ρ ρ ρ , & & & × = = = arXiv:0804.1561 hep-th/9407087 Proc. Suppl. Dimensional Gauge Theories Theor. Phys. Suppl. Theory Integrability supersymmetric QCD Program [ confinement in [ N.A. Nekrasov and S.L. Shatashvili, N.A. Nekrasov and S.L. Shatashvili, D. Gaiotto and E. Witten, D. Gaiotto and P. Koroteev, N.A. Nekrasov and S.L. Shatashvili, S. Gukov and E. Witten, S. Gukov and E. Witten, N. Seiberg and E. Witten, N. Seiberg and E. Witten, Lastly, let us compute the elliptic genera of 2d (0 d 2 chiral Z [8] [9] [5] [6] [7] [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. 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