Intersecting Surface Defects and Instanton Partition Functions
UUUITP-31/16
Intersecting Surface Defects and Instanton Partition Functions
Yiwen Pan and Wolfger Peelaers Department of Physics and Astronomy, Uppsala University,Box 516, SE-75120 Uppsala, Sweden New High Energy Theory Center, Rutgers University,Piscataway, NJ 08854, USA
Abstract
We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 super-symmetric quantum field theories. We employ the realization of a class of such systems as theinfrared fixed points of renormalization group flows from larger theories, triggered by perturbedSeiberg-Witten monopole-like configurations, to compute their partition functions. These results arecast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provideconcrete expressions for the instanton partition function in the presence of intersecting defects andwe study the corresponding ADHM model.October 2, 2018 a r X i v : . [ h e p - t h ] M a y ontents N free hypermultiplets 11 S (cid:126)ω . . . . . . . . . . . . . . . . . . . . . . . 113.2 Intersecting surface defects on S b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.1 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2 Double- and triple-sine functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3 Υ b functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B The S and S b SQCDA partition function 41
B.1 The S SQCDA partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41B.2 The S b SQCDA partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.3 Forest-tree representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
C Factorization of instanton partition function 45
C.1 The instanton partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46C.2 Reduction to vortex partition function of SQCD instanton partition function . . . . 47C.3 Factorization of instanton partition function for large N -tuples of Young diagrams . 48C.4 Factorization for small N -tuples of Young diagrams . . . . . . . . . . . . . . . . . . . 50 D Poles and Young diagrams in 3d 51
D.1 Poles of type-ˆ ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52D.2 Constructing Young diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53D.3 Residues and instanton partition function . . . . . . . . . . . . . . . . . . . . . . . . 54D.4 Extra poles and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 E Poles and Young diagrams in 2d 60
E.1 Four types of poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60E.2 Extra poles and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
Introduction
Half-BPS codimension two defect operators form a rich class of observables in supersymmetricquantum field theories. Their vacuum expectation values, as those of all defect operators, arediagnostic tools to identify the phase of the quantum field theory [1–3]. Various quantum fieldtheoretic constructions of codimension two defects have been proposed and explored in the literature,see for example the review [4]. First, one can engineer a defect by defining a prescribed singularityfor the gauge fields (and additional vector multiplet scalars) along the codimension two surface, asin [5]. Second, a defect operator can be constructed by coupling a quantum field theory supportedon its worldvolume to the bulk quantum field theory. The coupling can be achieved by gauginglower-dimensional flavor symmetries with higher-dimensional gauge fields and/or by turning onsuperpotential couplings. Third, a codimension two defect in a theory T can be designed in terms ofa renormalization group flow from a larger theory (cid:101) T triggered by a position-dependent, vortex-likeHiggs branch vacuum expectation value [6, 7]. Naturally, some defects can be constructed inmultiple ways. Nevertheless, it is of importance to study all constructions separately, as theircomputational difficulties and conceptual merits vary. Such study is helped tremendously by thefact that when placing the theory on a compact Euclidean manifold, all three descriptions are,in principle, amenable to an exact analysis using localization techniques. See [17] for a recentcomprehensive review on localization techniques.The M-theory construction of four-dimensional N = 2 supersymmetric theories of class S (oftype A N − ) [18] allows one to identify the class of concrete defects of interest to this paper: addingadditional stacks of M2-branes ending on the main stack of N M5-branes can introduce surfacedefects in the four-dimensional theory. The thus obtained M2-brane defects are known to be labeledby a representation R of SU ( N ). In [19], the two-dimensional quiver gauge theory residing onthe support of the defect and its coupling to the bulk four-dimensional theory were identified indetail. In fact, for the case of defects labeled by symmetric representations two different coupledsystems were proposed. For the purposes of this paper, it is important to remark that one of thesedescriptions can alternatively be obtained from the third construction described in the previousparagraph. Allowing for simultaneous insertions of multiple half-BPS defects, intersecting each other alongcodimension four loci, while preserving one quarter of the supersymmetry, enlarges the collection ofdefects considerably and is very well-motivated. Indeed, in [21] it was conjectured and overwhelmingevidence was found in favor of the statement that the squashed four-sphere partition function oftheories of class S in the presence of intersecting M2-brane defects, wrapping two intersectingtwo-spheres, is the translation of the insertion of a generic degenerate vertex operator in the The gauged perspective of [6] is equivalent to considering sectors with fixed winding in a ‘Higgs branch localization’computation. See [8–16] for such computations in various dimensions. The fact the application of this Higgsins prescription introduces M2-brane defects labeled by symmetric represen-tations was understood in the original paper [6], see for example also [20]. R (cid:48) , R ), which is precisely the defining information of a generic degenerate vertex operator inLiouville/Toda theory. In [21], the insertion of intersecting defects was engineered by considering a coupled 4d/2d/0dsystem. In this description, the defect is engineered by coupling quantum field theories supportedon the respective codimension two worldvolumes as well as additional degrees of freedom residing attheir intersection to each other and to the bulk quantum field theory. The precise 4d/2d/0d coupledsystems describing intersecting M2-brane defects were conjectured. As was also the case for a singledefect, intersecting defects labeled by symmetric representations can be described by two differentcoupled systems.A localization computation, performed explicitly in [21], allows one to calculate the squashedfour-sphere partition function of such system. Let T denote the four-dimensional theory and let τ L/R denote two-dimensional theories residing on the defects wrapping the two-spheres S and S ,which intersect each other at the north pole and south pole. The full partition function then takesthe schematic form Z ( T ,S ∪ S ⊂ S b ) = (cid:88)(cid:90) Z ( T ,S b )pert Z ( τ L ,S )pert Z ( τ R ,S )pert Z +intersection Z − intersection (cid:12)(cid:12)(cid:12) Z ( T , R ∪ R ⊂ R )inst (cid:12)(cid:12)(cid:12) , (1.1)where the factors Z ( T, M )pert denote the product of the classical action and one-loop determinant ofthe theory T placed on the manifold M (in their Coulomb branch localized form). Furthermore, Z ± intersection are the one-loop determinants of the degrees of freedom at the two intersection pointsrespectively, and | Z ( T , R ∪ R ⊂ R )inst | are two copies of the instanton partition function, one for thenorth pole and one for the south pole, describing instantons in the presence of the intersectingsurface defects spanning the local coordinate planes R ∪ R in R . In [21] the focus was onthe already very rich dynamics of 4d/2d/0d systems without four-dimensional gauge fields, thusavoiding the intricacies of the instanton partition functions. In this paper we aim at consideringintersecting defects in interacting four-dimensional field theories and addressing the problem ofinstanton counting in the presence of such defects. Our approach will be, alternative to that in [21], to construct theories T in the presence ofintersecting M2-brane defects labeled by symmetric representations using the aforementioned thirdstrategy, i.e. , by considering a renormalization group flow from a larger theory (cid:101) T triggered by A generic degenerate momentum reads α = − b Ω R − b − Ω R (cid:48) , in terms of the highest weight vectors Ω R , Ω R (cid:48) ofirreducible representations R and R (cid:48) respectively, and b parametrizes the Virasoro central charge. See also [25] for a localization computation in the presence of a single defect. By taking one of the intersecting defects to be trivial, one can always simplify our results to the case of a singledefect. In [19] an extensive study was performed of the squashed four-sphere partition function of theories of freehypermultiplets in the presence of a single defect. Whenthe theory (cid:101) T is a Lagrangian theory on S b , this Higgsing prescription offers a straightforwardcomputational tool to calculate the partition function Z ( T ,S ∪ S ⊂ S b ) of T in the presence of saidintersecting defects. In more detail, it instructs one to consider the residue of a certain pole of thepartition function Z ( ˜ T ,S b ) , which can be calculated by considering pinching poles of the integrandof the matrix integral computing Z ( (cid:101) T ,S b ) . The result involves intricate sums over a restricted setof Young diagrams, which we subsequently cast in the form of a coupled 4d/2d/0d system as in(1.1), by reorganizing the sums over the restricted diagrams into the integrals over gauge equivariantparameters and sums over magnetic fluxes of the partition functions of the two-dimensional theories τ L/R . This step heavily relies on factorization properties of the summand of instanton partitionfunctions, which we derive in appendix C, when evaluated at special values of their gauge equivariantparameter. More importantly, we obtain concrete expressions for the instanton partition function,computing the equivariant volume of the instanton moduli space in the presence of intersectingcodimension two singularities, and their corresponding ADHM matrix model.The main result of the paper, thus obtained, is the S b -partition function of a four-dimensional N = 2 SU ( N ) gauge theory with N fundamental and N antifundamental hypermultiplets, i.e. ,SQCD, in the presence of intersecting M2-brane surface defects, labeled by n R and n L -fold symmetricrepresentations respectively. It takes the form (1.1) and can be found explicitly in (4.13). To bemore precise, the coupled system we obtain involves chiral multiplets as zero-dimensional degrees offreedom, i.e. , it coincides with the one described in conjecture 4 of [21] with four-dimensional N = 2SQCD. The left subfigure in figure 1 depicts the 4d/2d/0d coupled system under consideration. Wederive the instanton partition function Z ( T , R ∪ R ⊂ R )inst in the presence of intersecting planar surfacedefects and find it to take the form Z ( T , R ∪ R ⊂ R )inst = (cid:88) (cid:126)Y q | (cid:126)Y | z R vect ( (cid:126)Y ) z R afund ( (cid:126)Y ) z R fund ( (cid:126)Y ) z R defect ( (cid:126)Y ) z R defect ( (cid:126)Y ) , (1.2)where we omitted all gauge and flavor equivariant parameters. It is expressed as the usual sumover N -tuples (cid:126)Y of Young diagrams. The summand contains the new fctors z R defect , which can befound explicitly in (4.17), capturing the contributions to the instanton counting of the additionalzero-modes in the presence of intersecting surface defects, in addition to the standard factors z R vect , z R fund and z R afund describing the contributions from the vector multiplet and N + N hypermultiplets.The coefficient of q k of the above result can be derived from the ADHM model for k -instantonsdepicted in the right subfigure of figure 1. We have confirmed this ADHM model by analyzing To be more precise, the configuration that triggers the renormalization group flow is a solution to the (perturbed)Seiberg-Witten monopole equations [26], see [16]. While there is no distinction between a fundamental and antifundamental hypermultiplet, it is a useful terminologyto keep track of the respective quiver gauge theory nodes. We choose to call the right/upper node of each link thefundamental one. L n R NNN
N S
0d chiral2d chiral4d hyper ADHMquiver
NNN kn L n R
0d Fermi0d chiral
Figure 1:
On the left, the coupled 4d/2d/0d quiver gauge theory realizing the insertion, in four-dimensional N = 2 SQCD, of intersecting M2-brane surface defects labeled by symmetric representationsof rank n R and n L respectively is depicted. The zero-dimensional multiplets are denoted using two-dimensional N = (0 ,
2) quiver notation reduced to zero dimensions. Various superpotential couplingsare turned on, in direct analogy to the ones given in detail in [21]. On the right, the ADHM modelfor k -instantons of the left theory is shown. The model preserves the dimensional reduction to zerodimensions of two-dimensional N = (0 ,
2) supersymmetry. We used the corresponding quiver conventions.A J-type superpotential equal to the sum of the U ( k ) adjoint bilinears formed out of the two pairs ofchiral multiplets is turned on for the adjoint Fermi multiplet. The flavor charges carried by the variousmultiplets are also compatible with a quadratic J- or E-type superpotential for the Fermi multipletscharged under U ( n L/R ). the brane construction of said instantons, see section 5 for all the details. In section 6 we presentconjectural generalizations of the instanton counting in the case of generic intersecting M2-branedefects.The paper is organized as follows. We start in section 2 by briefly recalling the Higgsingprescription to compute squashed sphere partition functions in the presence of (intersecting) M2-brane defects labeled by symmetric representations. We also present its brane realization. In section3 we implement the prescription for the case where T is a four- or five-dimensional theory of N freehypermultiplets placed on a squashed sphere. The vacuum expectation value in T of intersectingM2-brane defects on the sphere has been computed in [21] from the point of view of the 4d/2d/0dor 5d/3d/1d coupled system and takes the form (1.1) (without the instanton contributions). For thecase of symmetric representations, we reproduce this expression directly, and provide a derivation ofa few details that were not addressed in [21]. We notice that the superpotential constraints of thecoupled system on the parameters appearing in the partition function are reproduced effortlesslyin the Higgsing computation thanks to the fact that they have a common origin in the theory (cid:101) T ,which in this case is SQCD. These relatively simple examples allow us to show in some detail theinterplay of the various ingredients of the Higgsed partition function of theory (cid:101) T , and how to cast itin the form (1.1). In section 4 we turn our attention to inserting defects in four-dimensional N = 2SQCD. We apply the Higgsing prescription to an SU ( N ) × SU ( N ) gauge theory with bifundamentalhypermultiplets and for each gauge group an additional N fundamental hypermultiplets, and castthe resulting partition function in the form (1.1). As a result we obtain a sharp prediction for theinstanton partition function in the presence of intersecting surface defects. This expression provides The partition function is insensitive to the presence of superpotential couplings.
In this section we start by briefly recalling the Higgsing prescription to compute the partitionfunction of a theory T in the presence of (intersecting) defects placed on the squashed four/five-sphere [6, 7]. We also consider the brane realization of this prescription, which provides a naturalbridge to the description of intersecting surface defects in terms of a 4d/2d/0d (or 5d/3d/1d) coupledsystem as in [21]. We will be interested in four/five-dimensional quantum field theories with eight supercharges. Let us for concreteness start by considering four-dimensional N = 2 supersymmetric theories.Consider a theory T whose flavor symmetry contains an SU ( N ) factor, and consider the theoryof N free hypermultiplets, which has flavor symmetry U Sp (2 N ) ⊃ SU ( N ) × SU ( N ) × U (1). Bygauging the diagonal subgroup of the SU ( N ) flavor symmetry factor of the former theory withone of the SU ( N ) factors of the latter theory, we obtain a new theory (cid:101) T . As compared to T , thetheory (cid:101) T has an extra U (1) factor in its flavor symmetry group. We denote the corresponding massparameter as ˇ M .
The theory (cid:101) T can be placed on the squashed four-sphere S b , and its partition function can becomputed using localization techniques [27, 28]. Let us denote the supercharge used to localize thetheory as Q . Its square is given by Q = b − M R + b M L − ( b + b − ) R + i (cid:88) J M J F J + gauge transformation , (2.1)where M R/L are generators of the U (1) R/L isometries of S b (see footnote 10), R is the SU (2) R Cartan generator and F J are the Cartan generators of the flavor symmetry algebra. The coefficients The localization computations we will employ throughout this paper rely on a Lagrangian description, but theHiggsing prescription is applicable outside the realm of Lagrangian theories. We will restrict attention to (Lagrangian)four-dimensional N = 2 supersymmetric quantum field theories of class S and their five-dimensional uplift. We consider S b defined through the embedding equation in five-dimensional Euclidean space R = R × C withcoordinates x, z , z x r + | z | (cid:96) + | z | ˜ (cid:96) = 1 , in terms of parameters r, (cid:96), ˜ (cid:96) with dimension of length. The squashing parameter b is defined as b = (cid:96) ˜ (cid:96) . The isometriesof S b are given by U (1) R × U (1) L , which act by rotating the z and z plane respectively. The fixed locus of U (1) R isa squashed two-spheres: S = S b (cid:12)(cid:12) z =0 and, similarly, the fixed locus of U (1) L is S = S b (cid:12)(cid:12) z =0 . The two-spheres S and S intersect at their north pole and south pole, i.e. , the points with coordinates z = z = 0 and x = ± r . J are mass parameters rescaled by (cid:112) (cid:96) ˜ (cid:96) , where (cid:96) and ˜ (cid:96) are two radii of the squashed sphere (seefootnote 10), to make them dimensionless. Localization techniques simplify the computation of the S b partition function to the calculation of one-loop determinants of quadratic fluctuations aroundthe localization locus given by arbitrary constant values for Σ (cid:101) T , the imaginary part of the vectormultiplet scalar of the total gauge group. The final result for the S b partition function of thetheory (cid:101) T is then Z ( (cid:101) T ,S b ) ( M ) = (cid:90) dΣ (cid:101) T Z ( (cid:101) T ,S b )cl (Σ (cid:101) T ) Z ( (cid:101) T ,S b )1-loop (Σ (cid:101) T , M ) | Z ( (cid:101) T , R )inst ( q, Σ , M (cid:15) ) | , (2.2)where Z ( (cid:101) T ,S b )cl denotes the classical action evaluated on the localization locus, Z ( (cid:101) T ,S b )1-loop is the one-loop determinant and | Z ( (cid:101) T , R )inst ( q, Σ , M (cid:15) ) | are two copies of the Nekrasov instanton partitionfunction [29, 30], capturing the contribution to the localized path integral of instantons residing atthe north and south pole of S b .In [6,7], it was argued, by considering the physics at the infrared fixed point of the renormalizationgroup flow triggered by a position dependent Higgs branch vacuum expectation value for the baryonconstructed out of the hypermultiplet scalars, which carries charges M L = − n L , M R = − n R , R = N/ F = N , that the partition function Z ( (cid:101) T ,S b ) ( M ) necessarily has a pole when i ˇ M = b + b − b − n R N + b n L N . (2.3)Moreover, the residue of the pole precisely captures the partition function of the theory T in thepresence of M2-brane surface defects labeled by n R -fold and n L -fold symmetric representationsrespectively up to the left-over contribution of the hypermultiplet that captures the fluctuationsaround the Higgs branch vacuum. These defects wrap two intersecting two-spheres S , the fixedloci of U (1) R/L .The pole at (2.3) of Z ( (cid:101) T ,S b ) ( M ) finds its origin in the matrix integral (2.2) because of poles ofthe integrand pinching the integration contour. To see this, let us separate out the SU ( N ) gaugegroup that gauges the free hypermultiplet to T , and split Σ (cid:101) T accordingly: Σ (cid:101) T = (Σ T , Σ) , where Σ T is the vector multiplet scalar of the full gauge group of theory T , and Σ the SU ( N ) vector multipletscalar. We can then rewrite (2.2) as Z ( (cid:101) T ,S b ) ( M ) = (cid:90) dΣ T (cid:90) dΣ Z ( (cid:101) T ,S b )cl (Σ T , Σ) Z ( T ,S b )1-loop (Σ T , Σ , M ) | Z ( (cid:101) T , R )inst ( q, Σ T , Σ , M (cid:15) ) | × N (cid:89) A,B =1 A (cid:54) = B Υ b ( i (Σ A − Σ B )) N (cid:89) A =1 N (cid:89) I =1 Υ b (cid:18) i (Σ A − M I − ˇ M ) + Q (cid:19) − . (2.4) More precisely, this is the “Coulomb branch localization” locus. Alternatively, one can perform a “Higgs branchlocalization” computation, see [15, 16]. SU ( N ) vector multiplet, whilethe second factor is the contribution of the N extra hypermultiplets, organized into N SU ( N )fundamental hypermultipets. Here M I , I = 1 , . . . , N denote the mass parameters associated tothe SU ( N ) flavor symmetry (with (cid:80) I M I = 0). The integrand of the Σ-integral has poles (amongmany others) located at i Σ A = iM σ ( A ) + i ˇ M − n R A b − − n L A b − b + b − n R/L A ≥ , A = 1 , . . . , N , (2.5)where σ denotes a permutation of N variables. These poles arise from the one-loop determinant ofthe extra hypermultiplets. When the U (1) mass parameter ˇ M takes the value of (2.3), they pinchthe integration contour if n R = N (cid:88) A =1 n R A , n L = N (cid:88) A =1 n L A , (2.6)since we only have N − SU ( N ) integration variables. Note that the residue of thepole of Z ( (cid:101) T ,S b ) at (2.3) is equal to the sum over all partitions of n R , n L in (2.6) of the residue of theΣ-integrand of Z ( (cid:101) T ,S b ) at the pole position (2.5) when treating the Σ A as N independent variables. A similar analysis can be performed for five-dimensional N = 1 theories. The theory (cid:101) T canbe put on the squashed five-sphere S (cid:126)ω , and its partition function can again be computed usinglocalization techniques [31–36]. The localizing supercharge Q squares to Q = (cid:88) α =1 ω α M ( α ) − ( ω + ω + ω ) R + i (cid:88) J M J F J + gauge transformation , (2.8)where M ( α ) are the generators of the U (1) (1) × U (1) (2) × U (1) (3) isometry of the squashed five-sphere S (cid:126)ω (see footnote 14). The localization locus consists of arbitrary constant values for the vectormultiplet scalar Σ (cid:101) T , hence the partition function reads Z ( (cid:101) T ,S (cid:126)ω ) ( M ) = (cid:90) dΣ (cid:101) T Z ( (cid:101) T ,S (cid:126)ω )cl (Σ (cid:101) T ) Z ( (cid:101) T ,S (cid:126)ω )1-loop (Σ (cid:101) T , M ) | Z ( (cid:101) T , R × S )inst ( q, Σ (cid:101) T , M ω ) | . (2.9) See appendix A for the definition and some useful properties of the various special functions that are usedthroughout the paper. Upon gauging the additional U (1) flavor symmetry and turning on a Fayet-Iliopoulos parameter, which coincideswith the gauged setup of [6,7], the residues of precisely these poles were given meaning in the “Higgs branch localization”computation of [16] in terms of Seiberg-Witten monopoles. The squashed five-sphere S (cid:126)ω =( ω ,ω ,ω ) is given by the locus in C satisfying ω | z | + ω | z | + ω | z | = 1 . (2.7)Its isometries are U (1) (1) × U (1) (2) × U (1) (3) , which act by rotations on the three complex planes respectively. Thefixed locus of U (1) ( α ) is the squashed three-sphere S α ) = S (cid:126)ω (cid:12)(cid:12) z α =0 , while the fixed locus of U (1) ( α ) × U (1) ( β (cid:54) = α ) is thecircle S α ∩ β ) = S (cid:126)ω (cid:12)(cid:12) z α = z β =0 . The notation indicates that it appears as the intersection of the three-spheres S α ) and S β ) . A convenient visualization of the five-sphere and its fixed loci under one or two of the U (1) isometries is as a T -fibration over a solid triangle, where on the edges one of the cycles shrinks and at the corners two cycles shrinksimultanously. Z ( (cid:101) T ,S (cid:126)ω ) ( M ) has a pole at i ˇ M = ω + ω + ω (cid:88) i =1 ω α n ( α ) N , (2.10)whose residue computes the S (cid:126)ω partition function of T in the presence of codimension two defectslabeled by n ( α ) -fold symmetric representations and wrapping the three-spheres S α ) obtained asthe fixed loci of the U (1) ( α ) isometries (see footnote 14), respectively. These three-spheres intersecteach other in pairs along a circle. Again, this pole arises from pinching the integration contour bypoles of the one-loop determinant of the N hypermultiplets located at i Σ A = iM σ ( A ) + i ˇ M − (cid:88) α =1 n ( α ) A ω α − ω + ω + ω n ( α ) A ≥ , A = 1 , . . . , N , (2.11)if n ( α ) = (cid:80) NA =1 n ( α ) A . The residue of Z ( (cid:101) T ,S (cid:126)ω ) ( M ) at the pole given in (2.10) equals the sum overpartitions of the integers n ( α ) of the residue of the integrand at the pole position (2.11) with theΣ A treated as independent variables. To sharpen one’s intuition of the Higgsing prescription outlined in the previous subsection, onemay look at its brane realization [7]. Consider a four-dimensional N = 2 gauge theory T describedby the linear quiver and corresponding type IIA brane configuration N N · · ·
N N ←→ NS5 NS5 · · ·
NS5 NS5 N D4 Gauging in a theory of N hypermultiplets amounts to adding an additional NS5-brane on theright end of the brane array. The Higgsing prescription of the previous subsection is then triviallyimplemented by pulling away this additional NS5-brane (in the 10-direction of footnote 16), whilesuspending n R D2 R and n L D2 L -branes between the displaced NS5-brane and the right stack ofD4-branes, see figure 2. In [13], these residues were interpreted as the contribution to the partition function of K-theoretic Seiberg-Wittenmonopoles. The branes in this figure as well as those in figure 2 and the following discussion span the following dimensions:1 2 3 4 5 6 7 8 9 10NS5 — — — — — —D4 — — — — —D2 L — — —D2 R — — —D0 — S5 NS5 NS5 N D . . . N D N D N D NS5 NS5 NS5 N D . . . N D N D n L D2 L n R D2 R NS5 NS5 NS5 N D . . . N D N D n L D2 L n R D2 R Figure 2:
Gauging the diagonal subgroup of the SU ( N ) flavor symmetry carried by the right-hand stackof D4-branes and an SU ( N ) subgroup of the flavor symmetry of an additional N free hypermultipletsamounts to adding an additional NS5-brane on the right end of the brane array. This leads to the figureon the left. Higgsing the system as in subsection 2.1 corresponds to pulling away this NS5-brane fromthe main stack, while stretching n R D2 R and n L D2 L -branes in between it and the D4-branes, producingthe middle figure. The final figure represents the system in the Coulomb phase. R L NN · · · NN n R n L Figure 3:
Coupled 4d/2d/0d quiver gauge theory realizing intersecting M2-brane surface defectslabeled by symmetric representations, of rank n R and n L respectively, in a four-dimensional N = 2linear quiver gauge theory. The two-dimensional degrees of freedom, depicted in N = (2 ,
2) quivernotation, are coupled to the four-dimensional ones through cubic and quartic superpotential couplings.The explicit superpotentials can be found in [21]. The zero-dimensional degrees of freedom, denotedusing two-dimensional N = (0 ,
2) quiver notations dimensionally reduced to zero dimensions, with solidlines representing chiral multiplets, participate in E- and J-type superpotentials.
Various observations should be made. First of all, the brane picture in figure 2 was alsoconsidered in [21] to describe intersecting M2-brane surface defects labeled by n R and n L -foldsymmetric representations respectively. Its field theory realization is described by a coupled4d/2d/0d system, described by the quiver in figure 3 (see [21]). Note that the two-dimensionaltheories, residing on the D2 R and D2 L -branes, are in their Higgs phase, with equal Fayet-Iliopoulosparameter ξ FI proportional to the distance (in the 7-direction) between the displaced NS5-braneand the next right-most NS5-brane. Before Higgsing, this distance was proportional to the inversesquare of the gauge coupling of the extra SU ( N ) gauge node: ξ FI = 4 πg . (2.12)In particular, the Higgsing prescription will produce gauge theory results in the regime where ξ FI ispositive, and where the defect is inserted at the right-most end of the quiver. In this paper we will10estrict attention to this regime. Note however that sliding the displaced NS5-brane along the branearray in figure 2 implements hopping dualities [19, 37] (see also [38, 39]), which in the quiver gaugetheory description of figure 3 translate to coupling the defect world volume theory to a differentpair of neighboring nodes of the four-dimensional quiver, while not changing the resulting partitionfunction.In [21], a first-principles localization computation was performed to calculate the partitionfunction of the coupled 4d/2d/0d system when placed on a squashed four-sphere, with the defectswrapping two intersecting two-spheres S , the fixed loci of U (1) R/L , in the case of non-interactingfour-dimensional theories. Our aim in the next section will be to reproduce these results from theHiggsing point of view. When the four-dimensional theory contains gauge fields, the localizationcomputation needs as input the Nekrasov instanton partition function in the presence of intersectingplanar surface defects, which modify non-trivially the ADHM data. The Higgsing prescription doesnot require such input, and in section 4 we will apply it to N = 2 SQCD. This computation willallow us to extract the modified ADHM integral.The brane realization of figure 2 already provides compelling hints about how the ADHM datashould be modified. In this setup, instantons are described by D0-branes stretching between theNS5-branes. Their worldvolume theory is enriched by massless modes (in the Coulomb phase, i.e. ,when ξ FI = 0), if any, arising from open strings stretching between the D0-branes and the D2 R and D2 L -branes. These give rise to the dimensional reduction of a two-dimensional N = (2 , N = (0 ,
2) chiral multiplet and Fermi multiplet. We will provide more details about the instantoncounting in the presence of defects in section 5. Our Higgsing computation of section 4 will providean independent verification of these arguments. N free hypermultiplets In this section we work out in some detail the Higgsing computation for the case where T is atheory of free hypermultiplets. We will find perfect agreement with the description of intersecting M2-brane defects labeled by symmetric representations in terms of a 4d/2d/0d (or 5d/3d/1d) system [21].Our computation also provides a derivation of the Jeffrey-Kirwan-like residue prescription used toevaluate the partition function of the coupled 4d/2d/0d (or 5d/3d/1d) system, and of the flavorcharges of the degrees of freedom living on the intersection. In the next section we will consider thecase of interacting theories T . S (cid:126)ω As a first application of the Higgsing prescription of the previous section, we consider thepartition function of a theory of N free hypermultiplets on S (cid:126)ω in the presence of intersecting11 ∩ S S ∩ S S ∩ S S ~ω b (1) b (2) b − b − b (3) b − b (1) = p ω /ω b (2) = p ω /ω b (3) = p ω /ω radii S ∩ : 1 /ω S ∩ : 1 /ω S ∩ : 1 /ω Figure 4:
A convenient representation of S (cid:126)ω in terms of a T -fibration over a triangle. Each edgerepresents a three-sphere invariant point-wise under one of the U (1) isometries, and each vertex representsan S , where two S ’s intersect, invariant point-wise under two U (1) isometries. Each S has two tubularneighborhoods of the form S × R in the two intersecting S ’s, with omega-deformation parametersgiven in terms of b ± α ) , as shown in the figure. codimension two defects wrapping two of the three-spheres S α ) fixed by the U (1) ( α ) isometry (seefootnote 14, and also figure 4), say S and S . Our aim will be to cast the result in the manifestform of the partition function of a 5d/3d/1d coupled system, as in [21]. We consider this casefirst since the fact that the intersection S ∩ S = S ∩ has a single connected component is asimplifying feature that will be absent in the example of S b in the next subsection. S (cid:126)ω partition function of (cid:101) T Our starting point, the theory (cid:101) T , is described by the quiver N N N .That is, it is an SU ( N ) gauge theory with N fundamental and N anti-fundamental hypermultiplets, i.e. , N = 2 SQCD. The S (cid:126)ω -partition function of (cid:101) T is computed by the matrix integral (2.9)[31–36, 40, 41] Z ( (cid:101) T ,S (cid:126)ω ) ( M, ˜ M ) = (cid:90) dΣ Z ( (cid:101) T ,S (cid:126)ω )cl (Σ) Z ( (cid:101) T ,S (cid:126)ω )1-loop (Σ , M, ˜ M ) | Z ( (cid:101) T , R × S )inst ( q, Σ , M (cid:15) , ˜ M (cid:15) ) | . (3.1)The explicit expression for the classical action is given by Z ( (cid:101) T ,S (cid:126)ω )cl (Σ) = exp (cid:20) − π ω ω ω g Tr Σ (cid:21) , (3.2) Recall our terminology of footnote 7. Z ( (cid:101) T ,S (cid:126)ω )1-loop is the product of the one-loop determinants of the SU ( N )vector multiplet, the N fundamental hypermultiplets and the N antifundamental hypermultiplets: Z ( (cid:101) T ,S (cid:126)ω )1-loop (Σ , M, ˜ M ) = Z S (cid:126)ω vect (Σ) Z S (cid:126)ω fund (Σ , M ) Z S (cid:126)ω afund (Σ , ˜ M ) (3.3)= (cid:81) NA,B =1 A (cid:54) = B S ( i (Σ A − Σ B ) | (cid:126)ω ) (cid:81) NA =1 (cid:81) NI =1 S ( i (Σ A − M I ) + | (cid:126)ω | / | (cid:126)ω ) (cid:81) NA =1 (cid:81) NJ =1 S ( i ( − Σ A + ˜ M J ) + | (cid:126)ω | / | (cid:126)ω ) , (3.4)written in terms of the triple sine function. Here we used the notation | (cid:126)ω | = ω + ω + ω . Notethat we did not explicitly separate the masses for the SU ( N ) × U (1) flavor symmetry, but insteadconsidered U ( N ) masses. Finally, there are three copies of the K-theoretic instanton partitionfunction, capturing contributions of instantons residing at the circles kept fixed by two out of three U (1) isometries. Concretely, one has | Z ( (cid:101) T , R × S )inst ( q, Σ , M ω , ˜ M ω ) | ≡ Z ( (cid:101) T , R × S ∩ )inst (cid:16) q , Σ ω , M ω ω , ˜ M ω ω , πω , ω ω , ω ω (cid:17) × Z ( (cid:101) T , R × S ∩ )inst (cid:16) q , Σ ω , M ω ω , ˜ M ω ω , πω , ω ω , ω ω (cid:17) Z ( (cid:101) T , R × S ∩ )inst (cid:16) q , Σ ω , M ω ω , ˜ M ω ω , πω , ω ω , ω ω (cid:17) , (3.5)where q α = exp (cid:16) − π g Y M πω α (cid:17) . Each factor can be written as a sum over an N -tuple of Youngdiagrams [29, 30] (cid:126)Y = ( Y , Y , . . . , Y N ) , with Y A = ( Y A ≥ Y A ≥ . . . ≥ Y AW YA ≥ Y A ( W YA +1) = . . . = 0) (3.6)of a product over the contributions of vector and matter multiplets: Z ( (cid:101) T , R × S )inst (cid:16) q, β π Σ , β π M ω , β π ˜ M ω , β, (cid:15) , (cid:15) (cid:17) = (cid:88) (cid:126)Y q | (cid:126)Y | z R × S vect (cid:18) (cid:126)Y , β π Σ (cid:19) z R × S fund (cid:18) (cid:126)Y , β π Σ , β π M ω (cid:19) z R × S afund (cid:18) (cid:126)Y , β π Σ , β π ˜ M ω (cid:19) . (3.7)Here we have omitted the explicit dependence on (cid:15) , (cid:15) in all factors z R × S . The instanton countingparameter q is given by q = exp (cid:16) − π βg Y M (cid:17) , and | (cid:126)Y | denotes the total number of boxes in the N -tupleof Young diagrams. The expression for z fund reads z R × S fund (cid:18) (cid:126)Y , β π Σ , β π M ω (cid:19) = N (cid:89) A =1 N (cid:89) I =1 ∞ (cid:89) r =1 Y Ar (cid:89) s =1 i sinh πi (cid:18) β π ( i Σ A − iM ωI ) + r(cid:15) + s(cid:15) (cid:19) , (3.8)13hile those of z R × S vect and z R × S afund are given in (C.2)-(C.3) in appendix C. Note that the massesthat enter in (3.7) are slightly shifted (see [42]): M ω ≡ M − i ω + ω + ω ) , ˜ M ω ≡ ˜ M − i ω + ω + ω ) . (3.9) As outlined in the previous section, to introduce intersecting codimension two defects wrappingthe three-spheres S and S and labeled by the n (1) -fold and n (2) -fold symmetric representationrespectively, we should consider the residue at the pole position (2.11) with n (3) = 0 (and hence n (3) A = 0 for all A = 1 , . . . , N ) i Σ A = iM σ ( A ) − n (1) A ω − n (2) A ω − ω + ω + ω A = 1 , . . . , N , (3.10)while treating Σ A as N independent variables, and sum over all partitions (cid:126)n (1) of n (1) and (cid:126)n (2) of n (2) . As before, σ ( A ) is a permutation of A = 1 , ..., N which we take to be, without loss of generality, σ ( A ) = A . At this point let us introduce the notation that “ → ” means evaluating the residue at thepole (3.10) and removing some spurious factors. As we aim to cast the result in the form of a matrixintegral describing the coupled 5d/3d/1d system, we try to factorize all contributions accordinglyin pieces depending only on information of either three-sphere S or S . As we will see, thenon-factorizable pieces nicely cancel against each other, except for a factor that will ultimatelydescribe the one-dimensional degrees of freedom residing on the intersection.It is straightforward to work out the residue at the pole position (3.10). The classical action (3.2)and the one-loop determinant (3.3) become, using recursion relations for the triple sine functions(see (A.8)), Z ( (cid:101) T ,S (cid:126)ω )cl Z ( (cid:101) T ,S (cid:126)ω )1-loop → Z ( T ,S (cid:126)ω )1-loop (cid:16) Z S cl | (cid:126)n (1) Z S | (cid:126)n (1) (cid:17) (cid:16) Z S cl | (cid:126)n (2) Z S | (cid:126)n (2) (cid:17)(cid:16) Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) cl,extra Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) (cid:17) . (3.11)Let us unpack this expression a bit. First, Z ( T ,S (cid:126)ω )1-loop is the one-loop determinant of N free hypermul-tiplets, which constitute the infrared theory T . It reads Z ( T ,S (cid:126)ω )1-loop = N (cid:89) A =1 N (cid:89) J =1 S ( − iM A + i ˜ M J + | (cid:126)ω | | (cid:126)ω ) = N (cid:89) A =1 N (cid:89) J =1 S ( iM A − i ˜ M J | (cid:126)ω ) . (3.12) In appendix C we have simultaneously performed manipulations of four-dimensional and five-dimensional instantonpartition functions, which is possible after introducing the generalized factorial with respect to a function f ( x ), definedin appendix A.1, with f ( x ) in four and five dimensions given in (C.1). Recall that we have regrouped the mass for the U (1) flavor symmetry and those for the SU ( N ) flavor symmetryinto U ( N ) masses. Here we omitted on the right-hand side the left-over hypermultiplet contributions mentioned in the previoussection as well as the classical action evaluated on the Higgs branch vacuum at infinity, i.e. , on the position-independentHiggs branch vacuum. N free hypermultiplets, represented by a two-flavor-node quiver, are M AJ = M A − ˜ M J + i | (cid:126)ω | . Recall that N (cid:80) NJ =1 i ˜ M J = i ˇ˜ M, while N (cid:80) NA =1 iM A = | (cid:126)ω | + n (1) N ω + n (2) N ω .Second, we find the classical action and one-loop determinant of squashed three-sphere partitionfunctions of a three-dimensional N = 2 supersymmetric U ( n ( α ) ) gauge theory with N fundamentaland N antifundamental chiral multiplets and one adjoint chiral multiplet, i.e. , the quiver gaugetheory NN n ( α ) We will henceforth call this theory ‘SQCDA.’ These quantities are in their Higgs branch localizedform, hence the additional subscript indicating the Higgs branch vacuum, i.e. , the partition (cid:126)n ( α ) .Their explicit expressions can be found in appendix B.2. The Fayet-Iliopoulos parameter ξ ( α )FI ,the adjoint mass m ( α ) X , and the fundamental and antifundamental masses m ( α ) I , ˜ m ( α ) I entering thethree-dimensional partition function on S α ) are identified with the five-dimensional parameters asfollows, with λ ( α ) ≡ (cid:113) ω ( α ) / ( ω ω ω ), ξ ( α )FI = 8 π λ ( α ) g , m ( α ) X = iω ( α ) λ ( α ) , (3.13) m ( α ) I = λ ( α ) (cid:18) M I + i | (cid:126)ω | + ω ( α ) ) (cid:19) , ˜ m ( α ) J = − iω ( α ) λ ( α ) + λ ( α ) (cid:18) ˜ M J + i | (cid:126)ω | + ω ( α ) ) (cid:19) . (3.14)Note that the relation on the U (1) mass N (cid:80) NI =1 iM I = | (cid:126)ω | + n (1) N ω + n (2) N ω translates into arelation on the U (1) mass of the fundamental chiral multiplets. Finally, both the classical action andthe one-loop determinant produce extra factors which cannot be factorized in terms of information Note that the rank of the gauge group is the rank of one of the symmetric representations labeling the defectssupported on the codimension two surfaces, or in other words, it can be inferred from the precise coefficients of thepole of the (cid:101) T partition function, see (2.10). The squashed three-sphere partition function of a theory τ can be computed using two different localizationschemes. The usual “Coulomb branch localization” computes it as a matrix integral of the schematic form [43–46] Z ( τ,S b ) = (cid:90) d σ Z ( τ,S b )cl ( σ ) Z ( τ,S b )1-loop ( σ ) , while a “Higgs branch localization” computation brings it into the form [10, 11] Z ( τ,S b ) = (cid:88) HV Z ( τ,S b )cl | HV Z ( τ,S b )1-loop | HV Z ( τ, R × S )vortex | HV ( b ) Z ( τ, R × S )vortex | HV ( b − ) . Here the sum runs over all Higgs vacua HV and the subscript | HV denotes that the quantity is evaluated in the Higgsvacuum HV. Furthermore, one needs to include two copies of the K-theoretic vortex partition function Z R × S vortex . Thetwo expressions for Z are related by closing the integration contours in the former and summing over the residues ofthe enclosed poles. In the main text the theory τ will always be SQCDA and hence we omit the superscripted label.Note that for SQCDA, the sum over vacua is a sum over partitions of the rank of the gauge group. See appendix Bfor all the details. (cid:126)n (1) or (cid:126)n (2) , Z ˜ T ; (cid:126)n (1) ,(cid:126)n (2) = Z (cid:126)n (1) ,(cid:126)n (2) vf,extra ( M ) Z (cid:126)n (1) ,(cid:126)n (2) afund,extra ( ˜ M ) , Z (cid:126)n (1) ,(cid:126)n (2) cl,extra = ( q ¯ q ) − (cid:80) NA =1 n (1) A n (2) A , (3.15)where Z (cid:126)n L ,(cid:126)n R afund,extra captures the non-factorizable factors from the antifundamental one-loop deter-minant, while Z (cid:126)n L ,(cid:126)n R vf,extra captures those from the vector multiplet and fundamental hypermultipletone-loop determinants, which can be found in (C.21)-(C.22). These factors will cancel againstfactors produced by the instanton partition functions, which we consider next.When employing the Higgsing prescription to compute the partition function in the presence ofdefects, the most interesting part of the computation is the result of the analysis and massagingof the instanton partition functions (3.5) evaluated at the value (3.10) for their gauge equivariantparameter. We find that each term in the sum over Young diagrams can be brought into an almostfactorized form. As mentioned before, certain non-factorizable factors cancel against the extrafactors in (3.11), but a simple non-factorizable factor remains. When recasting the final expression inthe form of a 5d/3d/1d coupled system, it is precisely this latter factor that captures the contributionof the degrees of freedom living on the intersection S ∩ of the three-spheres on which the defectslive.Let us start by analyzing the instanton partition functions Z ( (cid:101) T , R × S ∩ )inst and Z ( (cid:101) T , R × S ∩ )inst . Itis clear from (3.8) that upon plugging in the gauge equivariant parameter (3.10) in Z ( (cid:101) T , R × S ∩ )inst ,the N -tuple of Young diagrams (cid:126)Y has zero contribution if any of the Young diagrams Y A hasmore than n (2) A rows. Similarly, Z ( (cid:101) T , R × S ∩ )inst does not receive contributions from (cid:126)Y if any of itsmembers Y A has more than n (1) A rows. Hence the sum over Young diagrams simplifies to a sum overall possible sequences of n ( α ) non-decreasing integers. The summands of the instanton partitionfunctions undergo many simplifications at the special value for the gauge equivariant parameter, andin fact one finds that they become precisely the K-theoretic vortex partition function for SQCDAupon using the parameter identifications (3.13) (see appendix C.2 for more details): Z ( (cid:101) T , R × S ∩ )inst → Z R × S ∩ vortex | (cid:126)n (2) ( b − ) , Z ( (cid:101) T , R × S ∩ )inst → Z R × S ∩ vortex | (cid:126)n (1) ( b − ) , (3.16)with the three dimensional squashing parameters defined as b (1) ≡ (cid:112) ω /ω , b (2) ≡ (cid:112) ω /ω , b (3) ≡ (cid:112) ω /ω . (3.17)The third instanton partition function, Z ( (cid:101) T , R × S ∩ )inst , behaves more intricately when substitutingthe gauge covariant parameter of (3.5). From (3.8) one immediately finds that N -tuples of Young This fact has for example also been observed in [47–50], and can also be read off from the brane picture in figure2. Before Higgsing, the instantons of the extra SU ( N ) gauge node are realized by D0-branes spanning in betweenthe NS5-branes. After Higgsing, the D0-branes can still be present if they end on the D2 R and D2 L -branes. If, say, n L = 0, they precisely turn into vortices of the two-dimensional theory living on the D2-branes. R ν ... = ν Y R m L µ Y L = µ Large Y Y L Y R Figure 5:
A constituent Y of a large N -tuple of Young diagrams (cid:126)Y for n (1) = 4 , n (2) = 8. The red boxdenotes the “forbidden box” with coordinates ( n (1) + 1 , n (2) + 1) . The green and blue areas denote Y L and Y R respectively. The definitions of m L µ and m R ν , see (3.19), are also indicated. diagrams (cid:126)Y have zero contribution if any of its constituting diagrams Y A contain the “forbidden box”with coordinates (column,row) = ( n (1) A + 1 , n (2) A + 1). We split the remaining sum over N -tuples ofYoung diagrams into two, by defining the notion of large N -tuples, as those N -tuples satisfying therequirement that all of its members Y A contain the box with coordinates ( n (1) A , n (2) A ), and calling allother N -tuples small . Let us focus on the former sum first.Given a large N -tuple (cid:126)Y , we define (cid:126)Y L and (cid:126)Y R as the Young diagrams Y L Ar = Y Ar − n (2) A for 1 ≤ r ≤ n (1) A , and Y L Ar = 0 for n (1) A < rY R Ar = Y A ( n (1) A + r ) for 1 ≤ r . (3.18)Furthermore, we define the non-decreasing sequences of integers m L Aµ ≡ Y L A ( n (1) A − µ ) , µ = 0 , ..., n (1) A − , m R Aν ≡ ˜ Y R A ( n (2) A − ν ) , ν = 0 , ..., n (2) A − , (3.19)where ˜ Y R A denotes the transposed diagram of Y R A . Figure 5 clarifies these definitions. With thesedefinitions in place, one can show (see appendix C.3) the following factorization of the summand ofthe instanton partition function for large tuples of Young diagrams (cid:126)Yq | (cid:126)Y large | Z ( (cid:101) T , R × S ∩ )inst (cid:16) (cid:126)Y large (cid:17) → q | m L | + | m R | Z R × S ∩ vortex | (cid:126)n (1) ( m L | b (1) ) Z R × S ∩ vortex | (cid:126)n ( m R | b (2) ) × Z large | (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) (cid:16) Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) cl,extra Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) (cid:17) − . (3.20)Here we used Z R × S vortex | (cid:126)n ( m | b ) to denote the summand of the U ( n ) SQCDA K-theoretic vortex partition17unction, i.e. , Z R × S vortex | (cid:126)n ( b ) = (cid:88) m Aµ ≥ m Aµ ≤ m A ( µ +1) z | m | b Z R × S vortex | (cid:126)n ( m | b ) , (3.21)where | m | = (cid:80) A (cid:80) µ m Aµ . (See appendix B.2 for concrete expressions.) The factor Z large | (cid:126)n (1) ,(cid:126)n (2) intersection isgiven by Z large | (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) ≡ N (cid:89) A,B =1 n (1) A − (cid:89) µ =0 n (2) B − (cid:89) ν =0 (cid:16) i sinh πi (cid:104) i β π ( M A − M B ) + (cid:15) ( m L Aµ + ν ) − (cid:15) ( m R Bν + µ ) − (cid:15) (cid:105)(cid:17) − × (cid:16) i sinh πi (cid:104) i β π ( M A − M B ) + (cid:15) ( m L Aµ + ν ) − (cid:15) ( m R Bν + µ ) + (cid:15) (cid:105)(cid:17) − . (3.22)As announced, the extra factors in the second line of (3.20) cancel against those in (3.11).For small diagrams, we can still define (cid:126)Y R as in the second line of (3.18), but (cid:126)Y L is not a proper N -tuple of Young diagrams due to the presence of negative entries. Nevertheless, we can define setsof non-decreasing integers as m L Aµ ≡ Y A ( n (1) A − µ ) − n (2) A , for 0 ≤ µ ≤ n (1) A − , m R Aν ≡ ˜ Y R A ( n (2) A − ν ) , for 0 ≤ ν ≤ n (2) A − . (3.23)It is clear that m L Aµ can take negative values. Then one can show (see appendix C.4) that q | (cid:126)Y large | Z ( (cid:101) T , R × S ∩ )inst (cid:16) (cid:126)Y small (cid:17) → q | m L | + | m R | Z R × S ∩ (semi-)vortex | (cid:126)n (1) ( m L | b (1) ) Z R × S ∩ vortex | (cid:126)n ( m R | b (2) ) × Z (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) (cid:16) Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) cl,extra Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) (cid:17) − . (3.24)The intersection factor for generic (small) N -tuples of Young diagrams is a generalization of(3.22) that can be found explicitly in (C.25). The factor Z R × S ∩ (semi-)vortex | (cid:126)n (1) ( m L | b (1) ) is a somewhatcomplicated expression generalizing Z R × S ∩ vortex | (cid:126)n (1) , which we present in (C.26).Putting everything together, and noting that summing over all N -tuples of Young diagramsavoiding the forbidden box is equivalent to summing over all possible values of m L/R Aµ , we find thefollowing result for the Higgsed partition function Z ( (cid:101) T ,S (cid:126)ω ) → Z ( T ,S (cid:126)ω )1-loop (cid:32) (cid:88) (cid:48) large (cid:126)Y Z (cid:126)n (1) ( m L | b (1) ) Z large | (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) Z (cid:126)n (2) ( m R | b (2) )+ (cid:88) (cid:48) small (cid:126)Y ˆ Z (cid:126)n (1) ( m L | b (1) ) Z (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) Z (cid:126)n (2) ( m R | b (2) ) (cid:33) (3.25)18here Z (cid:126)n (1) ( m L | b (1) ) = Z S cl | (cid:126)n (1) Z S | (cid:126)n (1) q | m L | Z R × S ∩ vortex | (cid:126)n (1) ( m L | b (1) ) Z R × S ∩ vortex | (cid:126)n (1) ( b − ) , (3.26)and similarly for Z (cid:126)n (2) ( m R | b (2) ). The expression for ˆ Z n ( m L | b (1) ) is obtained by replacing Z R × S ∩ vortex | n with Z R × S ∩ (semi-)vortex | n . The prime on the sums over Young diagrams in (3.25) indicates that only N -tuples of Young diagrams avoiding the “forbidden box” are included. To obtain the final resultof the Higgsed partition function, we need to sum the right-hand side of (3.25) over all partitions (cid:126)n (1) of n (1) and (cid:126)n (2) of n (2) . Our next goal is to write down a matrix model integral that reproduces the S (cid:126)ω -partition functionof the theory T of N free hypermultiplets in the presence of intersecting codimension two defects, i.e. , a matrix integral that upon closing the integration contours appropriately reproduces theexpression on the right-hand side of (3.25), summed over all partitions of n (1) and n (2) , as its sumover residues of encircled poles.A candidate matrix model is obtained relatively easily by analyzing the contribution of the largetuples of Young diagrams in (3.25). It reads Z ( T ,S ∪ S ⊂ S (cid:126)ω ) = Z ( T ,S (cid:126)ω )1-loop n (1) ! n (2) ! (cid:90) JK n (1) (cid:89) a =1 d σ (1) a n (2) (cid:89) b =1 d σ (2) b Z S ( σ (1) ) Z intersection ( σ (1) , σ (2) ) Z S ( σ (2) ) , (3.27)where Z S ( σ (1) ) denotes the classical action times the one-loop determinant of the S partitionfunction of SQCDA, that is, of a three-dimensional N = 2 gauge theory with gauge group U ( n (1) ), and N fundamental, N antifundamental and one adjoint chiral multiplet, and similarly for Z S ( σ (2) ). The contribution from the intersection S ∩ reads Z intersection ( σ (1) , σ (2) ) = n (1) (cid:89) a =1 n (2) (cid:89) b =1 (cid:89) ± (cid:20) i sinh πi (cid:16) ∆ ab ± (cid:0) b + b (cid:1)(cid:17)(cid:21) − , (3.28)with ∆ ab = − ib (2) σ (2) b + ib (1) σ (1) a . Note that from (3.13) we deduce that the Fayet-Iliopoulosparameters ξ (1)FI and ξ (2)FI are both positive. The mass and other parameters on both three-spheres See appendix B.2 for concrete expressions for the integrand of the three-sphere partition function. b (1) ξ (1)FI = b (2) ξ (2)FI , b (1) (cid:18) m (1) I + i b (1) (cid:19) = b (2) (cid:18) m (2) I + i b (2) (cid:19) , m (1) X = i b b (1) ,b (1) (cid:18) ˜ m (1) J − i b (1) (cid:19) = b (2) (cid:18) ˜ m (2) J − i b (2) (cid:19) , m (2) X = i b b (2) , (3.29)where m ( α ) I , ˜ m ( α ) J and m ( α ) X are the fundamental, antifundamental and adjoint masses on the respectivespheres. Moreover, the differences of the relations in (3.14), for fixed α , relate the three-dimensionalmass parameters on S α ) to the five-dimensional mass parameters of the N free hypermultiplets, i.e. , to M IJ = M I − ˜ M J + i | (cid:126)ω | : M IJ = λ − α ) (cid:16) m ( α ) I − ˜ m ( α ) J (cid:17) − iω α + i | (cid:126)ω | . (3.30)The matrix integral (3.27) is evaluated using a Jeffrey-Kirwan-like residue prescription [51].We have derived it explicitly by demanding that the integral (3.27) reproduces the result of theHiggsing computation (see below). The prescription is fully specified by the following chargeassignments: the matter fields that contribute to Z S ( σ (1) ) and Z S ( σ (2) ) are assigned theirstandard charges under the maximal torus U (1) n (1) × U (1) n (2) of the total gauge group U ( n (1) ) × U ( n (2) ), while all factors contributing to Z intersection ( σ (1) , σ (2) ) are assigned charges of the form(0 , . . . , , + b (1) , . . . , , . . . , , − b (2) , . . . , η = ( ξ (1)FI ; ξ (2)FI ),where we treat the Fayet-Iliopoulos parameters as an n (1) -vector and n (2) -vector respectively. Recallfrom (3.13) that both are positive.Before verifying that the matrix model (3.27), with the pole prescription just described, indeedfaithfully reproduces the expression (3.25) summed over all partitions (cid:126)n (1) , (cid:126)n (2) , we remark thatit takes precisely the form of the partition function of the 5d/3d/1d coupled system of figure 6,which is the trivial dimensional uplift of figure 3 specialized to the case of N free hypermultipletsdescribed by a two-flavor-node quiver. This statement can be verified by dimensionally upliftingthe localization computation of [21]. In some detail, Z ( T ,S (cid:126)ω )1-loop captures the contributions to thepartition function of the five-dimensional degrees of freedom, i.e. , of the theory T consisting of N free hypermultiplets, while Z S α ) encodes those of the degrees of freedom living on S α ) , described by U ( n ( α ) ) SQCDA, for α = 1 ,
2, and the factor Z intersection precisely equals the one-loop determinantof the one-dimensional bifundamental chiral multiplets living on the intersection S ∩ S = S ∩ .Moreover, the mass relations (3.30), which we find straightforwardly from the Higgsing prescription,are the consequences of cubic superpotential couplings in the 5d/3d/1d coupled system, which wereanalyzed in detail in [21]. The mass relations among the (anti)fundamental chiral multiplet massesin (3.29) are in fact a solution of (3.30) obtained by subtracting the equation for α = 1 and α = 2and subsequently performing a separation of the indices I, J . The separation constants appearing20 d3d (1) (2) NN n (1) n (2) Figure 6:
Coupled 5d/3d/1d quiver gauge theory realizing intersecting M2-brane surface defects labeledby n R - and n L -fold symmetric representations in the five-dimensional theory of N free hypermultiplets.The three-dimensional degrees of freedom are depicted in N = 2 quiver gauge notation, while the one-dimensional ones are denoted using one-dimensional N = 2 quiver notations, with solid lines representingchiral multiplets. Various superpotential couplings are turned on, as in figure 3 (see [21]). Applying theHiggsing prescription to SQCD precisely results in the partition function of this quiver gauge theory. in the resulting solutions can be shifted to arbitrary values by performing a change of variablesin the three-dimensional integrals, up to constant prefactors stemming from the classical actions.The Higgsing prescription also fixes the classical actions and hence we find specific values for theseparation constants. The adjoint masses in (3.29) are the consequence of a quartic superpotential.Also observe that our computation fixes the flavor charge of the one-dimensional chiral multiplets,which enter explicitly in Z intersection , and for which no first-principles argument was provided in [21].The integrand of (3.27) has poles in each of the three factors; the Jeffrey-Kirwan-like residueprescription is such that, among others, it picks out classes of poles, which we refer to as poles oftype-ˆ ν . They read, for partitions (cid:126)n (1) and (cid:126)n (2) of n (1) and n (2) respectively, over all of which wesum, and for sequences of integers { ˆ ν A } where ˆ ν A ∈ {− , , . . . , n (2) A − } ,poles of type-ˆ ν : σ (1) Aµ = m (1) A + µm (1) X − i m L Aµ b (1) − i n L Aµ b − , µ = 0 , . . . , n (1) A − σ (2) Bν = m (2) B + νm (2) X − i m R Bν b (2) − i n R Bν b − , ν = 0 , . . . , n (2) B − . (3.31)where m L µ , m R ν , n L µ , n R ν are non-decreasing sequences of integers, such that n L Aµ , n R Bν (cid:62) ν enters) m L Aµ ≥ ≥ ν A = − m L Aµ ≥ ≥ m L A = − ˆ ν A − ν A ≥ , m R0 (cid:54) ν (cid:54) ˆ ν A = 0 , m R ν ≥ ˆ ν A +1 ≥ . (3.32)Note that if all ˆ ν A = − σ (1) a pole position of Z S and to σ (2) a pole position of Z S , whose residues precisely reproduce the sum over large diagramsin (3.25). Precisely this fact motivated the candidate matrix model in (3.27). In appendix D.2, wedescribe a simple algorithm to construct Young diagrams avoiding the “forbidden box” associatedwith poles of type-ˆ ν . Furthermore, we show in appendix D.3 that the sum over the corresponding21esidues precisely reproduce the sum over Young diagrams in (3.25). Finally, we show in appendixD.4 that the residues of poles not of type-ˆ ν , but contained in the Jeffrey-Kirwan-like prescription,cancel among themselves by studying a simplified example. We thus conclude that the integral(3.27) indeed faithfully reproduces the sum over Young diagrams in (3.25). S b Let us next study the partition function of N free hypermultiplets on S b in the presence ofintersecting codimension two defects wrapping the two-spheres S , the fixed loci of the U (1) L/R isometries (see footnote 10). The intersection of S with S consists of two points. The analysislargely parallels the one in the previous subsection, so we will be more brief. S b partition function of (cid:101) T The theory (cid:101) T is an N = 2 supersymmetric gauge theory with gauge group SU ( N ) and N fundamental and N antifundamental hypermultiplets. Its squashed four-sphere partition function iscomputed by the matrix integral (2.2) (or (2.4)), Z ( (cid:101) T ,S b ) ( M, ˜ M ) = (cid:90) dΣ Z ( (cid:101) T ,S b )cl (Σ) Z ( (cid:101) T ,S b )1-loop (Σ , M, ˜ M ) | Z ( (cid:101) T , R )inst. ( q, Σ , M (cid:15) , ˜ M (cid:15) ) | . (3.33)The classical action is given by Z ( (cid:101) T ,S b )cl (Σ) = exp (cid:20) − π g Tr Σ (cid:21) (3.34)and the one-loop factor reads Z ( (cid:101) T ,S b )1-loop (Σ , M, ˜ M ) = Z S b vect (Σ) Z S b fund (Σ , M ) Z S b afund (Σ , ˜ M ) , (3.35)where Z S b fund (Σ , M ) = N (cid:89) I =1 N (cid:89) A =1 b ( i Σ A − iM I + Q/ , Z S b vect (Σ) = N (cid:89) A,B =1 A (cid:54) = B Υ b ( i Σ A − i Σ B ) ,Z S b afund (Σ , ˜ M ) = N (cid:89) J =1 N (cid:89) A =1 b ( − i Σ A + i ˜ M J + Q/ . (3.36)We have denoted the masses associated with the U ( N ) flavor symmetry of the N fundamentalhypermultiplets as M I and those of the N antifundamental hypermultiplets as ˜ M J . We also denote Q = b + b − . 22he instanton partition functions can be written as a sum over N -tuples of Young diagrams as Z ( (cid:101) T , R )inst. ( q, Σ , M (cid:15) , ˜ M (cid:15) ) = (cid:88) (cid:126)Y q | (cid:126)Y | z R vect ( (cid:126)Y , Σ , (cid:15) , (cid:15) ) z R fund ( (cid:126)Y , Σ , M (cid:15) , (cid:15) , (cid:15) ) z R afund ( (cid:126)Y , Σ , ˜ M (cid:15) , (cid:15) , (cid:15) ) . (3.37)The various factors in the summand are defined in (C.2) and (C.3) in appendix C. The Ω-deformationparameters are identified as (cid:15) = b and (cid:15) = b − , the superscript (cid:15) denotes the usual shift ofhypermultiplet masses [42] M (cid:15) ≡ M − i (cid:15) + (cid:15) ) , ˜ M (cid:15) ≡ ˜ M − i (cid:15) + (cid:15) ) , (3.38)and the modulus squared simply entails sending q = exp(2 πiτ ) → ¯ q , with τ = ϑ π + πig . The Higgsing prescription instructs us to consider the poles of the fundamental one-loop factorgiven by i Σ A = iM σ ( A ) − n L A b − n R A b − − b + b − A = 1 , . . . , N , (3.39)with σ a permutation of N elements, which we choose to be the identity. At the end of thecomputation, we should sum over all partitions (cid:126)n L/R of n L/R , i.e. , n L/R = (cid:80) A n L/R A .The fact that the two two-spheres intersect at two disjoint points, namely their north poles andsouth poles, adds another layer of complication compared to the analysis in the previous subsection.Even so, when evaluating the residue at (3.39), the analysis of the classical action and one-loopdeterminants is straightforward. Both can be brought into a factorized form in terms of piecesdepending only on information on either two-sphere, using the shift formula (A.11) for the latter, upto extra factors which will cancel against certain non-factorizable factors coming from the instantonpartition functions. Explicitly, Z ( (cid:101) T ,S b )cl Z ( (cid:101) T ,S b )1-loop → Z ( T ,S b )1-loop (cid:16) Z S cl | (cid:126)n L Z S | (cid:126)n L (cid:17) (cid:16) Z S cl | (cid:126)n R Z S | (cid:126)n R (cid:17)(cid:16) Z (cid:101) T ; (cid:126)n L ,(cid:126)n R cl,extra Z (cid:101) T ; (cid:126)n L ,(cid:126)n R (cid:17) , (3.40)where Z ( T ,S b )1-loop is the one-loop determinant of N hypermultiplets, which constitute the infraredtheory T , and have masses M IJ = M I − ˜ M J + i Q . Furthermore, Z S . . . | (cid:126)n L/R denote factors in theHiggs branch localized two-dimensional N = (2 ,
2) SQCDA two-sphere partition function (seefootnote 22 for the equivalent three-sphere discussion, and appendix B.1 for explicit expressions).The two-dimensional FI-parameter ξ FI , fundamental masses m I , antifundamental masses ˜ m J and23djoint masses m X are related to the four-dimensional parameters as ξ LFI = ξ RFI = 4 πg , m L I = bM I + i ib , ˜ m L J = b ˜ M J + i , m L X = ib (3.41) ϑ L/R = ϑ , m R I = b − M I + i ib − , ˜ m R J = b − ˜ M J + i , m R X = ib − . (3.42)Finally the extra factors are Z (cid:126)n L ,(cid:126)n R = Z (cid:126)n L ,(cid:126)n R vf,extra ( M ) Z (cid:126)n L ,(cid:126)n R afund,extra ( ˜ M ) , Z cl,extra = ( q ¯ q ) − (cid:80) NA =1 n L A n R A , (3.43)where Z (cid:126)n L ,(cid:126)n R vf,extra and Z (cid:126)n L ,(cid:126)n R afund,extra are as before the non-factorizable pieces produced by applying theshift formulae to the vector and (anti)fundamental one-loop determinant and can be found in(C.21)-(C.22).The massaging of each of the two instanton partition functions, which now both describeinstantons located at intersection points, is completely similar to the one we performed above.First, the sum over N -tuples of Young diagrams (cid:126)Y can be restricted to a sum over tuples whoseconstituents Y A all avoid the “forbidden” box at ( n L A + 1 , n R A + 1). Second, the left-over sum can bedecomposed into sums over large and small diagrams, and moreover their summands can almostbe factorized in terms of the summands of vortex partition functions, after canceling some overallfactors with the extra factors from the classical action and one-loop determinants in (3.40). Theremaining non-factorizable factor is an intersection factor, Z large | (cid:126)n (1) ,(cid:126)n (2) intersection ( m L , m R ) = N (cid:89) A,B =1 n (1) A − (cid:89) µ =0 n (2) B − (cid:89) ν =0 (cid:16) i ( M A − M B ) + (cid:15) ( m L Aµ + ν ) − (cid:15) ( m R Bν + µ ) − (cid:15) (cid:17) − × (cid:16) i ( M A − M B ) + (cid:15) ( m L Aµ + ν ) − (cid:15) ( m R Bν + µ ) + (cid:15) (cid:17) − . (3.44)for large diagrams, and (C.25) for generic diagrams. The full expression for the residue at the polelocation (3.39) thus involves the product of the two massaged instanton partition functions, togetherwith the leftover classical action and one-loop determinant factors, Z ( (cid:101) T ,S b ) → Z ( T ,S b )1-loop (cid:16) Z S cl | (cid:126)n L Z S | (cid:126)n L (cid:17) (cid:16) Z S cl | (cid:126)n R Z S | (cid:126)n R (cid:17) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:48) large (cid:126)Y q | m L | + | m R | Z R vortex | (cid:126)n L ( m L ) Z large | (cid:126)n L ,(cid:126)n R intersection ( m L , m R ) Z R vortex | (cid:126)n R ( m R ) (3.45)+ (cid:88) (cid:48) small (cid:126)Y q | m L | + | m R | Z R semi-vortex | (cid:126)n L ( m L ) Z (cid:126)n L ,(cid:126)n R intersection ( m L , m R ) Z R vortex | (cid:126)n R ( m R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The final result for the Higgsed partition function is obtained by summing the right-hand side ofthis expression over all partitions (cid:126)n
L/R of n L/R . 24 .2.3 Matrix model description and 4d/2d/0d coupled system
As in the previous subsection, the contribution of large tuples in both instanton partitionfunctions suggests the following matrix integral Z ( T ,S ∪ S ⊂ S b ) = Z ( T ,S b )1-loop n L ! n R ! (cid:88) B R ∈ Z n R (cid:88) B L ∈ Z n L (cid:90) JK n R (cid:89) a =1 dσ R a π n L (cid:89) c =1 dσ L c π Z S ( σ R , B R ) Z S ( σ L , B L ) × (cid:89) ± Z ± intersection ( σ L , B L , σ R , B R ) , (3.46)where Z S ( σ R , B R ) denotes the summand/integrand of the S partition function for SQCDA withgauge group U ( n R ), and similarly for Z S ( σ L , B L ). The intersection factors read Z ± intersection ( σ L , B L , σ R , B R ) = n R (cid:89) a =1 n L (cid:89) c =1 (cid:20)(cid:18) ∆ ± ac + b + b − (cid:19) (cid:18) ∆ ± ac − b + b − (cid:19)(cid:21) − , (3.47)with ∆ ± ac = b − (cid:16) iσ R a ± B R a (cid:17) − b (cid:16) iσ L c ± B L c (cid:17) and where b is the four-sphere squashing parameter.The factor labeled by the plus sign arises from the intersection point at the north pole, and the otherfactor from the south pole. The mass and other parameters on both two-spheres satisfy relations,which can be derived from (3.41)-(3.42), ξ LFI = ξ RFI , b − (cid:18) m L I + i (cid:19) = b (cid:18) m R I + i (cid:19) , m L X = ib ,ϑ L = ϑ R , b − (cid:18) ˜ m L J − i (cid:19) = b (cid:18) ˜ m R J − i (cid:19) , m R X = ib − , (3.48)while the hypermultiplet masses M IJ = M I − ˜ M J + i Q are related to the two-dimensional massparameters as ib − = (cid:20) M IJ + i b + b − ) (cid:21) − b − ( m L I − ˜ m L J ) , ib = (cid:20) M IJ + i b + b − ) (cid:21) − b ( m R I − ˜ m R J ) . (3.49)The residue prescription used to evaluate the integrals in (3.46) is completely similar to Jeffrey-Kirwan-like prescription introduced in the previous subsection: the matter fields contributingto Z S are assigned their natural charges under the Cartan subgroup U (1) n R × U (1) n L of thetotal gauge group, while all factors of the intersection factors are assigned charges of the form(0 , . . . , , b − , . . . , , . . . , , − b, . . . , η = ( ξ RFI , ξ
LFI ). We have derived this prescription by demanding that the matrix integralreproduces the result of the Higgsing computation.It was shown in [21] that the partition function of the 4d/2d/0d coupled system of figure 3 for Concrete expressions can be found in appendix B.1. d2d L R NN n L n R Figure 7:
Coupled 4d/2d/0d quiver gauge theory realizing intersecting M2-brane surface defects labeledby n R - and n L -fold symmetric representations in the four-dimensional theory of N free hypermultiplets.Various superpotential couplings are turned on and are given in detail in [21]. The Higgsing prescriptionapplied to SQCD precisely reproduces the partition function of this coupled system. the case of N free hypermultiplets described by a two-flavor-node quiver, reproduced in figure 7 forconvenience, precisely equals the matrix integral (3.46). In particular, (cid:81) ± Z ± intersection computes theone-loop determinant of the zero-dimensional bifundamental chiral multiplets at the two intersectionpoints of the two-spheres S and S . In the first-principles localization computation of [21], therelations (3.49) are consequences of cubic superpotential couplings. Up to separation constants,their solutions can be found to be the mass relations in (3.48). As explained in the previoussubsection, the Higgsing computation fixes the separation constants to specific values. Note thatour computations fixes the flavor symmetry charges of the zero-dimensional fields and provides aderivation of the residue prescription.The proof that the matrix integral reproduces the result of the Higgsing computation follows thesame logic as the one in the previous subsection, but is substantially more involved due to the factthat two copies of the intersection factor are present. We present some of the details in appendix E. In the previous section, we have computed the expectation value of intersecting surface defectsin four-/five-dimensional theories T of free hypermultiplets placed on the four-/five-sphere. In thissection, we consider intersecting surface defects inserted in interacting theories. More precisely, wefocus on T being an N = 2 supersymmetric theory with gauge group SU ( N ) and N fundamentaland N anti-fundamental hypermultiplets, i.e. , N = 2 SQCD.The partition function of SQCD on the four-sphere has appeared in our earlier computations, see(3.33). In particular, it involves the contribution of instantons located at the north pole and southpole of the four-sphere. When decorating the computation with intersecting surface defects, whichprecisely have these points as their intersection locus, we should expect the instanton counting tobe modified non-trivially. By performing the Higgsing procedure on a theory (cid:101) T described by the N = 2 quiver 26 N N N ,we will be able to derive a precise description of the modified ADHM integral by casting both theHiggsed partition function as well as its instanton contributions in a matrix integral form.
The S b -partition function of (cid:101) T is given by Z ( (cid:101) T ,S b ) ( M, ˜ M , ˆ M ) = (cid:90) N (cid:89) A,B =1 d Σ A d Σ (cid:48) B Z ( (cid:101) T ,S b )cl (Σ , Σ (cid:48) ) Z ( (cid:101) T ,S b )1-loop (Σ , Σ (cid:48) , M, ˜ M , ˆ M ) × (cid:12)(cid:12)(cid:12) Z ( (cid:101) T , R )inst. ( q, q (cid:48) , Σ , Σ (cid:48) , M (cid:15) , ˜ M (cid:15) , ˆ M (cid:15) ) (cid:12)(cid:12)(cid:12) , (4.1)where M I and ˜ M J denote the masses associated to the U ( N ) flavor symmetry of the N fundamentaland antifundamental hypermultiplets respectively, while ˆ M is the mass associated to the U (1) flavorsymmetry of the bifundamental hypermultiplet. The classical action reads Z ( (cid:101) T ,S b )cl (Σ , Σ (cid:48) ) = exp (cid:20) − π g Tr Σ − π g (cid:48) Tr Σ (cid:48) (cid:21) , (4.2)while the one-loop determinant is given by Z ( (cid:101) T ,S b )1-loop (Σ , Σ (cid:48) , M, ˜ M , ˆ M ) = Z S b vect (Σ) Z S b vect (Σ (cid:48) ) Z S b fund (Σ , M ) Z S b afund (Σ (cid:48) , ˜ M ) Z S b bifund (Σ , Σ (cid:48) , ˆ M ) , (4.3)where all factors were defined in (3.36) but Z S b bifund (Σ , Σ (cid:48) , ˆ M ) = N (cid:89) A =1 N (cid:89) B =1 b ( i Σ (cid:48) B − i Σ A + i ˆ M + Q/ . (4.4)The instanton partition function is given by a double sum over N -tuples of Young diagrams Z ( (cid:101) T , R )inst. ( q, q (cid:48) , Σ , Σ (cid:48) , M (cid:15) , ˜ M (cid:15) , ˆ M (cid:15) ) = (cid:88) (cid:126)Y ,(cid:126)Y (cid:48) q | (cid:126)Y | q (cid:48)| (cid:126)Y (cid:48) | z R vect ( (cid:126)Y , Σ) z R vect ( (cid:126)Y (cid:48) , Σ (cid:48) ) z R fund ( (cid:126)Y , Σ , M (cid:15) ) × z R afund ( (cid:126)Y (cid:48) , Σ (cid:48) , ˜ M (cid:15) ) z R bifund ( (cid:126)Y , (cid:126)Y (cid:48) , Σ , Σ (cid:48) , ˆ M (cid:15) ) . (4.5)The contributions of the various multiplets can be found in appendix C. The superscripts (cid:15) againdenote the usual shift [42] M (cid:15) = M − i (cid:15) + (cid:15) ) , ˜ M (cid:15) = ˜ M − i (cid:15) + (cid:15) ) , ˆ M (cid:15) = ˆ M − i (cid:15) + (cid:15) ) . (4.6)27 mplementing the Higgsing prescription once again amounts to considering the poles of thefundamental one-loop factor given by i Σ A = iM σ ( A ) − n L A b − n R A b − − b + b − A = 1 , . . . , N , (4.7)with σ a permutation of N elements, which we choose to be the identity. Here (cid:126)n L/R is a partition of n L/R , and we will sum over all.It is straightforward to compute the residues of the one-loop determinant at (4.7): Z ( (cid:101) T ,S b )cl Z ( (cid:101) T ,S b )1-loop → Z ( T ,S b )cl Z ( T ,S b )1-loop (cid:16) Z S cl | (cid:126)n L Z S | (cid:126)n L (cid:17) (cid:16) Z S cl | (cid:126)n R Z S | (cid:126)n R (cid:17)(cid:16) Z (cid:101) T ; (cid:126)n L ,(cid:126)n R cl,extra Z (cid:101) T ; (cid:126)n L ,(cid:126)n R (cid:17) , (4.8)Here Z ( T ,S b )cl Z ( T ,S b )1-loop are the classical action and one-loop determinant of the theory T , i.e. , offour-dimensional N = 2 supersymmetric SQCD. Their expression can be found in (3.34) and (3.35)respectively. The antifundamental masses of T are simply given by ˜ M J , but the fundamentalmasses take the values M (cid:48) A = M A − ˆ M + iQ/ (cid:101) T . As before, Z S . . . | (cid:126)n L/R denote factors in the Higgs branch localized SQCDA two-sphere partition function. The two-dimensional FI-parameters ξ L/RFI , fundamental masses m L/R I , antifundamental masses ˜ m L/R J andadjoint masses m L/R X are now related to the four-dimensional parameters of theory (cid:101) T as ξ LFI = 4 πg , m L I = b ( M I + iQ/
2) + i b , ˜ m L J = b (Σ (cid:48) J + ˆ M ) + i , m L X = ib (4.10) ξ RFI = 4 πg , m R I = b − ( M I + iQ/
2) + i b − , ˜ m R J = b − (Σ (cid:48) J + ˆ M ) + i , m R X = ib − , (4.11)together with ϑ L/R = θ . Note that the two-dimensional masses depend on the four-dimensional gaugeparameter. The explicit expressions for the extra one-loop factors, which now receives contributionsfrom the fundamental hypermultiplet, vector multiplet and bifundamental hypermultiplet one-loopdeterminant, can be found in (C.22)-(C.23). Again, Z (cid:126)n L ,(cid:126)n R cl,extra = ( q ¯ q ) − (cid:80) A n L A n R A .When substituting the gauge equivariant parameter (4.7) in the instanton partition functions(4.5), the only non-vanishing contributions arise from N -tuples (cid:126)Y avoiding the “forbidden box” andarbitrary N -tuples (cid:126)Y (cid:48) . As before, we can split the sum over the former into one over large and oneover small tuples. As we have learned in the previous section, the analysis of the large tuples issufficient to derive the matrix model integral describing the infrared system, i.e. , the theory T with In the previous section the theory (cid:101) T was SQCD. q | (cid:126)Y large | q (cid:48)| (cid:126)Y (cid:48) | Z ( (cid:101) T , R × S ∩ )inst ( (cid:126)Y (cid:48) , (cid:126)Y large ) → q (cid:48)| (cid:126)Y (cid:48) | z R vect ( (cid:126)Y (cid:48) , Σ (cid:48) ) z R afund ( (cid:126)Y (cid:48) , Σ (cid:48) , ˜ M (cid:15) ) z R fund ( (cid:126)Y (cid:48) , Σ (cid:48) , M (cid:48) (cid:15) ) z R defect ( (cid:126)Y (cid:48) , Σ (cid:48) , m L ) z R defect ( (cid:126) ˜ Y (cid:48) , Σ (cid:48) , m R ) × q | m L | + | m R | Z R vortex | (cid:126)n L ( m L ) Z large | (cid:126)n L ,(cid:126)n R intersection ( m L , m R ) Z R vortex | (cid:126)n R ( m R ) (cid:16) Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) cl,extra Z (cid:101) T ; (cid:126)n (1) ,(cid:126)n (2) (cid:17) − , (4.12)The intersection factor was already given in (3.44). The expression for the factors z R defect can befound in (C.20). They clearly correspond to new ingredients in the instanton partition functionof T , arising due to the presence of the defects on the local R . Momentarily, we will study themodified ADHM data and its corresponding ADHM integral computing this modified instantonpartition function. Recall that to obtain the final expression for the Higgsed partition function, weneed to sum over all partitions of n L/R . A matrix model integral describing the S b partition function of SU ( N ) SQCD in the presenceof intersecting surface defects supported on S can be inferred from (4.8) and (4.12) to be Z ( T ,S ∪ S ⊂ S b ) = 1 n L ! n R ! (cid:88) B R ∈ Z n R (cid:88) B L ∈ Z n L (cid:90) d Σ (cid:48) (cid:90) JK n R (cid:89) a =1 dσ R a π n L (cid:89) b =1 dσ L b π Z ( T ,S b )cl (Σ (cid:48) ) Z ( T ,S b )1-loop (Σ (cid:48) , M (cid:48) , ˜ M ) × Z S ( σ R , B R ; Σ (cid:48) ) Z S ( σ L , B L ; Σ (cid:48) ) Z +intersection ( σ L , B L , σ R , B R ) Z − intersection ( σ L , B L , σ R , B R ) (4.13) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) (cid:126)Y (cid:48) q (cid:48)| (cid:126)Y (cid:48) | z R vect ( (cid:126)Y (cid:48) , Σ (cid:48) ) z R afund ( (cid:126)Y (cid:48) , Σ (cid:48) , ˜ M (cid:15) ) z R fund ( (cid:126)Y (cid:48) , Σ (cid:48) , M (cid:48) (cid:15) ) z R defect ( (cid:126)Y (cid:48) , Σ (cid:48) , σ L , B L ) z R defect ( (cid:126)Y (cid:48) , Σ (cid:48) , σ R , B R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Here the factors in the first lines are the classical action and one-loop determinant of T , i.e. ,four-dimensional SQCD, and the factors in the second line are the S partition functions forSQCDA as well as the intersection factors (3.47). The last line contains two copies of the instantonpartition function, computed in the presence of the locally planar intersecting surface defects.The mass parameters on the two two-spheres are related as in (3.48), while the parameters ofthe four-dimensional theory T are related to the two-dimensional ones as ib − = (cid:20) M (cid:48) I − Σ (cid:48) J + i b + b − ) (cid:21) − b − ( m L I − ˜ m L J ) , ib = (cid:20) M (cid:48) I − Σ (cid:48) J + i b + b − ) (cid:21) − b ( m R I − ˜ m R J ) . (4.14)Note that when performing the integral over the four-dimensional gauge parameter Σ (cid:48) , one shoulduse ˜ m L J = b Σ (cid:48) J + ˜ m L U (1) , ˜ m R J = b − Σ (cid:48) J + ˜ m R U (1) , (4.15)where ˜ m L/R U (1) = N (cid:80) NK =1 ˜ m L/R K . These follow directly from (4.14). In the two-dimensional one-loopdeterminants we have made explicit this Σ (cid:48) -dependence. Note that by performing the change of29 d2d L R NNN n L n R Figure 8:
Coupled 4d/2d/0d quiver gauge theory realizing the insertion, in four-dimensional N = 2SQCD, of intersecting M2-brane surface defects labeled by symmetric representations of rank n R and n L respectively . Various superpotential couplings are turned, in direct analogy to the ones given indetail in [21]. The Higgsing prescription applied to a linear quiver gauge theory with two gauge nodesreproduces the partition function of this coupled system. variables σ L/R I → σ L/R I + ˜ m L/R U (1) in the two-dimensional integrals, one effectively changes the U (1)masses as ˜ m L/R U (1) → m L/R U (1) → m L/R U (1) − ˜ m L/R U (1) in the matrix integral, up to an overall constantfactor originating from the two-dimensional classical actions. Henceforth, we choose to work withthis effective new integral.The contribution of the locally planar surface defect, supported on the local R , to the northpole copy of the instanton partition function is given by z R defect ( (cid:126)Y (cid:48) , Σ (cid:48) , σ L , B L ) = n L (cid:89) a =1 N (cid:89) B =1 W Y (cid:48) B (cid:89) r =1 Y (cid:48) Br (cid:89) s =1 − (cid:15) ( iσ a + B a / − i ˜ m B ) + r(cid:15) + s(cid:15) − (cid:15) ( iσ a + B a / − i ˜ m B ) + ( r − (cid:15) + s(cid:15) , (4.17)where W Y (cid:48) B denotes the width of the Young diagram Y (cid:48) B . Similarly, z R defect is obtained by swappingL ↔ R, (cid:15) ↔ (cid:15) and Y B ↔ ˜ Y B . The combination iσ + B is valid for the north pole contributions;to get the south pole counterpart one replaces it with iσ − B .One can verify that if we perform the integrations over σ L/R and the sums over B L/R using thesame Jeffrey-Kirwan-like residue prescription as discussed in the previous section, the matrix model(4.13) reproduces the result obtained from the Higgsing prescription.
The 4d/2d/0d coupled system whose partition function is computed by (4.13) is depictedin figure 8. The first line of (4.13) captures the classical action and one-loop determinant of thefour-dimensional theory, while the second line captures the contributions of the two-dimensionaldegrees of freedom residing on the intersecting two-spheres as well as the one-loop determinants of the Note that in terms of the effective variables, the relation (4.14) remains unaffected, but (3.48) is modified as b − (cid:18) m L I + i (cid:19) = b (cid:18) m R I + i (cid:19) + c , b − (cid:18) ˜ m L J − i (cid:19) = b (cid:18) ˜ m R J − i (cid:19) + c , (4.16)with c = b − ˜ m L U (1) − b ˜ m R U (1) + i ( b − b − ). S5 NS5 NS5 N D N D k D N D n L D2 L n R D2 R NNN kn L n R Figure 9:
The left part of the figure depicts the brane configuration realizing k -instantons in N = 2SQCD, in the presence of intersecting surface defects, of M2-type and labeled by symmetric representations,represented by the gray branes. The right part of the figure shows the quiver description of theworldvolume theory of the D0-branes. As the system preserves two-dimensional N = (0 ,
2) supersymmetrydimensionally reduced to zero dimensions, the quiver is drawn using N = (0 ,
2) notations, with fulllines representing chiral multiplets and dashed lines Fermi multiplets. In the absence of the defects, thepreserved supersymmetry is N = (0 , J Λ for the adjoint Fermi multiplet Λ consisting of the sum of the adjoint bilinears of the scalars of thetwo pairs of chiral multiplets. The charges in table 1 are also compatible with quadratic E- or J-typesuperpotentials for the Fermi multiplets charged under U ( n L/R ). zero-dimensional bifundamental chiral multiplets at their intersection points. The most salient newfeature of this coupled system is the fact that part of the two-dimensional flavor symmetry is gaugedby the four-dimensional gauge symmetry. This fact is reflected in the relations in (4.14), relatingthe two-dimensional mass parameters ˜ m to the gauge parameter Σ (cid:48) , which are the consequenceof the usual cubic superpotential couplings. As mentioned above, when computing the squashedfour-sphere partition function of the coupled system, the instanton counting is modified non-triviallydue to the presence of the intersecting surface defects. The argument of the modulus squared in thelast line of (4.13) provides a concrete expression for the modified instanton partition function. Inthe next section, we turn to a more detailed analysis of the degrees of freedom which give rise tothis instanton partition function. Let us start by considering the familiar brane realization of a k -instanton in SU ( N ) N = 2SQCD as an additional stack of k D0-branes as depicted in the the left part of figure 9, ignoringthe gray branes for the time being. The supersymmetry preserved by the worldvolume theoryof the D0-branes is the dimensional reduction to zero dimensions of two-dimensional N = (0 , The brane directions are as in footnote 16. k -instanton partition function,which we denote as Z R k .strings D0-D4 D0-D4 D0-D4 D0-D0 D0-D2 R D0-D2 L N = (0 ,
4) FM HM FM VM HM (not preserved) (not preserved) N = (0 ,
2) FM CM CM FM VM FM CM CM CM FM CM FM J
12 12
12 12 J l −
12 12 − Table 1:
Massless excitations of strings stretching between the branes indicated in the first row organizedin multiplets of the dimensional reduction of two-dimensional N = (0 ,
4) and N = (0 ,
2) supersymmetryto zero dimensions in the second and third row respectively. Here VM denotes vectormultiplet, HMhypermultiplet, FM Fermi multiplet and CM chiral multiplet. Note that the system including theD2-branes only preserves N = (0 , N = (0 ,
4) entries corresponding to D0-D2strings open. The last two rows list the charges of the mutliplets under the flavor symmetry charges J and J l . In some more detail, the instanton partition function of a four-dimensional N = 2 supersymmetrictheory is computed by a localization computation on R (cid:15) ,(cid:15) , i.e. , in the Ω-background parametrizedby (cid:15) , (cid:15) [29, 30]. The localizing supercharge Q squares to Q = ( (cid:15) + (cid:15) )( J r + R ) + ( (cid:15) − (cid:15) ) J l + i Σ · G + iM · F , (5.1)where J l , J r are the Cartan generators of the SU (2) l × SU (2) r (cid:39) SO (4) rotational symmetriesof R , while R is the SU (2) R generator. We define J = J r + R . Furthermore, G denotesthe collection of Cartan generators of the gauge symmetry and F those of the flavor symmetry; φ and M are their respective equivariant parameters. The localization locus consists of point-instantons located at the origin. The integration over the k -instanton moduli space is capturedby the non-perturbative k -instanton partition function, which equals the partition function of thezero-dimensional ADHM model, read off as the worldvolume theory on the D0-branes, computedby localization with respect to the induced supercharge, that is, with respect to the superchargein its N = (0 ,
2) supersymmetry (sub)algebra satisfying the same square as in (5.1) (up to gaugetransformations). From the N = (0 ,
2) zero-dimensional point of view, the charges J = J r + R and J l appear as flavor charges, as do both G and F . Q additionally includes φ · G d .Dimensionally reducing the localization results of [54] and in particular [55], it is now straight- The Ω-deformation breaks the rotational symmetry to SO (2) × SO (2) . In terms of their Cartan generators J , J one has J l = ( J − J ) and J r = ( J + J ). Z R k as Z R k = (cid:90) JK k (cid:89) I =1 dφ I Z D0-D0 ( φ ) Z D0-D4 ( φ, ˜ M ) Z D0-D4 ( φ, Σ (cid:48) ) Z D0-D4 ( φ, M ) , (5.2)where Z D0-D0 ( φ ) = k (cid:89) I,J =1 ( φ IJ ) (cid:48) ( φ IJ + (cid:15) + (cid:15) )( φ IJ + (cid:15) )( φ IJ + (cid:15) ) , (5.3) Z D0-D4 ( φ, ˜ M ) = k (cid:89) I =1 N (cid:89) A =1 ( φ I − i ˜ M A ) , Z D0-D4 ( φ, M ) = k (cid:89) I =1 N (cid:89) A =1 ( φ I − iM A ) , (5.4) Z D0-D4 ( φ, Σ (cid:48) ) = k (cid:89) I =1 N (cid:89) A =1 φ I − i Σ (cid:48) A + ( (cid:15) + (cid:15) ))( − φ I + i Σ (cid:48) A + ( (cid:15) + (cid:15) )) , (5.5)where φ IJ = φ I − φ J , and the prime on ( φ IJ ) (cid:48) indicates to omit the factors with I = J . Here wedenoted the equivariant parameters for the various SU ( N ) symmetries as in the previous section.The integral (5.2) is computed using the Jeffrey-Kirwan residue prescription. We choose to selectthe contributions of negatively charged fields, and thus collect the residues of the poles defined bysolving the equations φ I = i Σ (cid:48) A + 12 ( (cid:15) + (cid:15) ) , φ I = φ J + (cid:15) , φ I = φ J + (cid:15) . (5.6)The contributing poles are labeled by N -tuples of Young diagrams (cid:126)Y = { Y A } such that (cid:80) A | Y A | = k , φ I = i Σ (cid:48) A −
12 ( (cid:15) + (cid:15) ) + r(cid:15) + s(cid:15) , ( r, s ) ∈ Y A . (5.7)It is easy to convince oneself that summing over the residues precisely reproduces the q k term ofthe SQCD instanton partition function given in (3.37).Let us now re-introduce the intersecting surface defects in the setup. The brane configurationwas already depicted in the left part of figure 9, now also considering the gray branes. Uponinserting the defects, the N = (0 ,
4) symmetry, dimensionally reduced to zero dimensions, carriedby the D0-branes is broken to the dimensional reduction of N = (0 , L andD2 R -branes. They each contribute an additional N = (0 ,
2) Fermi and chiral multiplet, and the See also [7] for an analysis of the equivariant integral of a five-dimensional theory in the presence of three-dimensional chiral multiplets. The brane system consisting of a stack of D0-branes and one stack of D2-branes, each ending on an NS5-brane,preserves on the D0-brane the dimensional reduction to zero dimensions of N = (2 ,
2) supersymmetry. The openstring modes thus organize themselves in an N = (2 ,
2) chiral multiplet. J and J l as in table 1. It is then straightforward to include their contributions to theADHM matrix model. It is given by Z D0-D2 ( φ, Σ (cid:48) , σ, B ) ≡ K (cid:89) I =1 N (cid:89) A =1 (cid:34) n L A − (cid:89) a =0 φ I − (cid:15) ( iσ L a + B L a ) + ( (cid:15) + (cid:15) ) φ I − (cid:15) ( iσ L a + B L a ) − ( (cid:15) − (cid:15) ) n R A − (cid:89) a =0 φ I − (cid:15) ( iσ R a + B R a ) + ( (cid:15) + (cid:15) ) φ I − (cid:15) ( iσ R a + B R a ) + ( (cid:15) − (cid:15) ) (cid:35) . (5.8)where we used σ L/R a − i B L/R a as the gauge equivariant parameter of the U ( n L/R ) symmetry, asthis is the combination that enters in our computations on S b at the north pole. (The south polecontribution would be obtained by changing the sign in front of B .) Noting that our JK-prescription does not select the poles of the above factor, it is straightforwardto see that the matrix integral Z R ∪ R ⊂ R k = (cid:90) JK k (cid:89) I =1 dφ I Z D0-D0 ( φ ) Z D0-D4 ( φ, ˜ M ) Z D0-D4 ( φ, Σ (cid:48) ) Z D0-D4 ( φ, M ) Z D0-D2 ( φ, Σ (cid:48) , σ, B )(5.9)precisely reproduces the modified instanton partition function as it appeared in the last line of(4.13). In this paper, we have extended the study of intersecting codimension two defects, initiatedin [21], to interacting four-dimensional theories. We have employed the Higgsing prescriptionof [6, 7] to compute the vacuum expectation value of intersecting M2-brane defects, labeled by n L and n R -fold symmetric representations respectively, inserted in four-dimensional N = 2 SQCD.Subsequently we cast the result in the form of a partition function of a coupled 4d/2d/0d system,see (4.13), which takes the schematic form Z ( T ,S ∪ S ⊂ S b ) = (cid:88)(cid:90) Z ( T ,S b )pert Z ( τ L ,S )pert Z ( τ R ,S )pert Z +intersection Z − intersection (cid:12)(cid:12)(cid:12) Z ( T , R ∪ R ⊂ R )inst (cid:12)(cid:12)(cid:12) . (6.1)The leftmost subfigure of figure 10 depicts the 4d/2d/0d coupled system under consideration. Thetheory T is four-dimensional N = 2 SQCD and τ L/R are identified as two-dimensional N = (2 , U ( n L/R ) SQCDA. Our computation provides an explicit formula for the instanton partition functionin the presence of the above-mentioned intersecting defects, Z ( T , R ∪ R ⊂ R )inst , appearing in (6.1), see Note that we are using the effective description obtained by performing a change of variables in the two-dimensionalintegrals and omitting some irrelevant constant prefactor as explained below equation (4.13). Before tackling this computation, we also considered the theory of N free hypermultiplets. Also for this case, wecast the result in the form of a partition function of a coupled system. L n R NNN
N S
0d chiral2d chiral4d hyperD4,D2worldvolume
NS5 NS5 NS5 N D4 N D4 k D0 N D4 n L D2 L n R D2 R D0worldvolume
NNN kn L n R
0d Fermi0d chiral
Figure 10:
The type IIA brane-configuration in the middle describes the 4d/2d/0d coupled systemon the left as the worldvolume theory of the D4/D2 L /D2 R -branes, see also figure 8. The worldvolumetheory of the k D0-branes is shown on the right, see figure 9 for more details. Its partition functioncomputes the k -instanton partition function of the 4d/2d/0d coupled system. NS5 NS5 NS5 N D4 N D4 N D4 N D4 NS5 NS5 NS5 N D4 N D4 n L2 D2 L n R2 D2 R n L1 D2 L n R1 D2 R NN n L2 n L1 n R2 n R1 Figure 11:
One starts with the theory T of N free hypermultiplets and successively gauges in twomore theories of N free hypermultiplets. The brane realization of the resulting theory (cid:101) T is shown inthe leftmost figure. One can then apply the Higgsing prescription twice, corresponding to pulling thetwo rightmost NS5-branes away from the main stack, and stretching ( n L2 − n L1 , n R2 − n R1 ) (D2 L , D2 R )branes and ( n L1 , n R1 ) (D2 L , D2 R ) branes respectively in between them and the flavor D4- branes. Thetwo-dimensional part of the system is in its Higgs phase, and can be brought into its Coulomb phaseby aligning the two displaced NS5-branes, as shown in the middle figure. The corresponding 4d/2d/0dcoupled system can be read off easily, and is shown in the last figure. This system was also consideredin [21]. equation (4.13). We also found the ADHM model whose partition function computes the k -instantoncontribution to Z ( T , R ∪ R ⊂ R )inst , see the rightmost subfigure in figure 10. This model can be read offfrom a D-brane construction, as also indicated in the figure.Starting from a theory T whose flavor symmetry contains an SU ( N ) factor, one can gaugein successively multiple, say p , theories of N free hypermultiplets. The resulting theory (cid:101) T has p additional U (1) factors in its flavor symmetry group compared to the original theory T . It isclear that one can apply the Higgsing prescription consecutively to each of these starting from theoutermost one along the quiver. The associated type IIA brane-realization is a simple generalizationof the one we have discussed in section 2.2. We depict the case p = 2 for T the theory of N free hypermultiplets in figure 11. The corresponding 4d/2d/0d coupled system can be read offfrom the brane picture and is given in the rightmost subfigure in figure 11. We conjecture thatthe M-theory interpretation of this procedure corresponds to the insertion of multiple intersectingM2-branes ending on the main stack of M5-branes, describing theory T , all labeled by symmetric35epresentations.General intersecting M2-brane defects labeled by two generic irreducible representations ( R L , R R )of SU ( N ) can also be described by 4d/2d/0d coupled systems [21]; when the four-dimensional theoryis N = 2 SQCD, we have depicted an example in the bottom left of figure 12. The two-dimensionaldegrees of freedom are described by quiver gauge theories which encode the representation R throughtheir gauge group ranks [19]. The coupled system involves zero-dimensional Fermi multiplets,transforming in the bifundamental representation of the innermost two-dimensional gauge groups,as degrees of freedom living at the intersection points. Such 4d/2d/0d coupled system can beengineered as the worldvolume theory of the D4/D2 L /D2 R -branes in the type IIA system shown infigure 12.When attempting to use this coupled system to compute the vacuum expectation value of generalintersecting M2-brane defects, one needs as an input the instanton partition function in the presenceof the defects. We propose that the structure of its k -instanton ADHM model can also in this casebe read off from the D0-brane worldvolume theory in the type IIA system. The resulting quivertheory, which has two-dimensional N = (0 ,
2) supersymmetry reduced to zero dimensions, is alsoincluded in figure 12. It is almost the same as the one in figure 10, up to the orientation of anarrow. This new ADHM model however leads to a dramatically different ADHM integration: extrapoles coming from the factor Z D0-D2 will be selected by the JK-prescription, and the result of theADHM integral is a double sum over N -tuples of Young diagrams and, separately, n L -tuples ofYoung diagrams, together having k boxes in total. It would be very interesting to study in moredetail these new ADHM integrals.When R L/R are both symmetric representations, the descriptions of figures 12 and 10 areboth valid. In [21], the equality of the resulting squashed four-sphere partition functions wasverified for four-dimensional theories without gauge fields, and for defects labeled by fundamentalrepresentations. It would be of interest to study the duality between the two descriptions ininteracting theories.A construction for general intersecting M2-brane defects in terms of a renormalization groupflow from a larger theory (cid:101) T triggered by some position-dependent Higgs branch vacuum expectationvalue is currently unknown. Presumably it requires (cid:101) T to be a non-Lagrangian theory of class S .Reversing the logic, one might hope to recover information about the partition function of the UVnon-Lagrangian theory (cid:101) T by investigating the partition function of all intersecting M2-brane defects,for which we have nice quiver description, and to which the UV theory can flow. Note that Higgsbranch localized expressions of partition functions are a simple example of this aspiration [16].It was conjectured in [21] that the partition function of SU ( N ) SQCD in the presence ofgeneral intersecting M2-brane defects can be identified with Liouville/Toda five-point functions. Inparticular, identifying x (cid:48) = qz , x = z , with | q | , | z | <
1, along with other parameter identifications,36 d Fermi
NNNn L W L n L W L − · · · n R W R n R W R − · · ·
0d Fermi2d chiral4d hyper D , D w o r l d v o l u m e NS5NS5 123456 · · · ·
NS5 R NS5 R NS5 R . . . · · · · NS5 L NS5 L NS5 L . . . · · · · N D4 N D N D k D n R W R D2 R · · · · · · · D2 R n R W R − D2 R n R W R − n L W L D2 L · · · · · · · D2 L n L W L − D2 L n L W L − D w o r l d v o l u m e NNN kn L W L n R W R
0d Fermi0d chiral
Figure 12:
The type IIA brane realization of general intersecting M2-brane defects labeled by twoirreducible representation ( R L , R R ) inserted in SQCD, as well as its corresponding 4d/2d/0d coupledsystem and ADHM model are depicted. one expects Z ( T ,S ∪ S ⊂ S b ) ( q, z, M (cid:48) , ˜ M ) = A ( x, x (cid:48) ) (cid:68) ˆ V α (0) ˆ V − b Ω R L − b − Ω R R ( x (cid:48) ) ˆ V β ( x ) ˆ V α (1) ˆ V α ∞ ( ∞ ) (cid:69) , (6.2)where A ( x, x (cid:48) ) ≡ A | x (cid:48) | γ | − x (cid:48) | γ | x | γ | − x | γ | x − x (cid:48) | γ , for some γ i . Furthermore α , α ∞ aregeneric, while β , α are semi-degenerate momenta determined in terms of the masses of the gaugetheory. Let us perform a few checks of this statement, leaving a more thorough analysis for thefuture.Consider the simple case with R L = and R R = symm n R (cid:50) . In the OPE limit 1 > | q | > | z | → z read, up to the factor A , (cid:88) t | z | α − b − t ) − α ) − − n R b − h ) ˆ C α − b − t α , − n R b − h (cid:68) ˆ V α − b − t (0) ˆ V β ( x ) ˆ V α (1) ˆ V α ∞ ( ∞ ) (cid:69) . (6.3)Here t = (cid:80) NA =1 n R A h A and h A are the weights of the fundamental representation of SU ( N ). Theset of natural numbers (cid:126)n R is any partition of n R , and the sum over t means summing over allsuch partitions. On the gauge theory side, in the z → σ R in the partition function Z ( T ,S ⊂ S b ) , as in (4.13) with n L = 0, and obtainthe Higgs branch localized expression as a sum over the two-dimensional Higgs vacua labeled bypartitions (cid:126)n R . This sum is mapped to the sum over t in (6.3). The leading terms in z simply come37 S b N N N N ! (cid:10) V α V β V β V α V α ∞ (cid:11) AGT Z S ∪ S ⊂ S b NNN n L n R ! h V α V − n L bh − n R b − h V β V α V α ∞ i Higgsing ˇ M → ˇ M n L ,n R Degeneration β → − n L bh − n R b − h Figure 13:
A commuting diagram showing the relation between the Higgsing prescription anddegenerating semi-degenerate momentum. from the zero-vortex sector. The four-dimensional matrix integral in each leading term, which nowdepends on the Higgs vacuum (cid:126)n R , is simply an S b -partition function with shifted fundamentalmasses M n R A ≡ M (cid:48) A − ˆ M + i ( b + b − ) / in R A b − , thanks to z R fund ( (cid:126)Y (cid:48) , M (cid:48) ε ) z R ⊂ R defect ( (cid:126)Y (cid:48) , Σ (cid:48) , iσ R , B R ) → z R fund ( (cid:126)Y (cid:48) , ( M n R ) ε ) , (6.4)where → indicates the evaluation at the Higgs vacuum (cid:126)n R . This S b -partition function is mapped tothe four-point function in (6.3). In particular, the four-dimensional one-loop determinant togetherwith the two-dimensional one-loop determinant evaluated at the Higgs vacuum (cid:126)n R is precisely equalto the structure constants ˆ C α − b − t α , − n R b − h ˆ C αα − b − t ,β ˆ C α,α ,α ∞ , up to some uninteresting constants.In fact, the statement (6.2) in the case of R L/R being both symmetric, can be viewed as adegeneration of the well-established AGT conjecture without surface defects, by considering thecommuting diagram in figure 13 [47–49]. We remind ourselves that i ˇ M n L ,n R = ( bn L + b − n R ) /N +( b + b − ) /
2, and in the AGT correspondence β (cid:48) = (cid:104) N ( b + b − ) / − (cid:80) NA =1 iM A (cid:105) h . The Higgsingprescription sends the U (1) mass ˇ M → ˇ M n L ,n R , which is equivalent to degenerating the semi-degenerate momentum β (cid:48) → − n L bh − n R b − h .The correspondence (6.2) is a great tool to discover and understand new dualities of the 4d/2d/0dcoupled system. The Liouville/Toda correlation functions enjoy various symmetries, including, butnot limited to, the invariance under conjugation and Weyl reflection of the momenta, and conformaland crossing symmetries. It would be interesting to translate these CFT symmetries to dualitieson the gauge theory side, especially when the intersecting defects are coupled to four-dimensionalinteracting theories. 38 cknowledgments The authors would like to thank Giulio Bonelli, Jaume Gomis, Bruno Le Floch, Fabrizio Nieri,Daniel Park, Massimiliano Ronzani, Alessandro Tanzini and Maxim Zabzine for helpful conversationsand useful suggestions. We are also grateful to Bruno Le Floch for comments on a draft of thispaper. Y.P. is supported in part by Vetenskapsr˚adet under grant
A Special functions
In this appendix we briefly recall the definitions and some useful properties of the specialfunctions which play an essential role in this paper.
A.1 Factorials
In analyzing vortex and instanton partition functions, one often encounters products of the form (cid:81) m − k =0 f (cid:15) ,(cid:15) ( z + k ) , for some function f (cid:15) ,(cid:15) . Due to its frequent occurrence and to streamline thediscussion we introduce the factorial( z ) fm ≡ m − (cid:89) k =0 f (cid:15) ,(cid:15) ( z + k ) . (A.1)As a trivial example, it includes the standard Pochhammer symbol ( x ) m ≡ (cid:81) m − k =0 ( x + k ), for f (cid:15) ,(cid:15) = id. We often abbreviate f (cid:15) ,(cid:15) as f when no confusion is expected. A.2 Double- and triple-sine functions
Double-sine function s b ( z ) : the double-sine function s b ( z ) is defined as the regularized versionof the product s b ( z ) ≡ (cid:89) m,n (cid:62) ( m + 1 / b + ( n + 1 / b − − iz ( m + 1 / b + ( n + 1 / b − + iz . (A.2)It is a common practice to define Q ≡ b + b − . The function s b ( z + iQ/
2) has poles at z =+ im ≥ b + in ≥ b − and zeros at z = − im ≥ b − in ≥ b − . The double sine function satisfies thefollowing recursion relation s b (cid:18) z + iQ imb + inb − (cid:19) = ( − mn s b (cid:16) z + iQ (cid:17)(cid:81) mk =1 i sinh πb ( z + ikb ) (cid:81) nk =1 i sinh πb − ( z + ikb − ) , (A.3)39s well as the symmetry properties s b ( z ) s b ( − z ) = 1 , s b (cid:18) z + iQ (cid:19) s b (cid:18) − z + iQ (cid:19) = 12 sinh( πbz ) 2 sinh( πb − z ) . (A.4)An alternative definition of the double-sine function is given by S ( z | ω , ω ) ≡ (cid:89) m,n ≥ mω + nω + z ( m + 1) ω + ( n + 1) ω − z . (A.5)The definitions of s b ( z ) and S ( z | ω , ω ) are related by b = (cid:112) ω /ω and S ( z | ω , ω ) = s b (cid:18) − i Q i z √ ω ω (cid:19) , S (cid:18) ω + ω − iz (cid:12)(cid:12)(cid:12) ω , ω (cid:19) = s b (cid:18) z √ ω ω (cid:19) . (A.6) Triple-sine function S ( z | (cid:126)ω ) : the triple-sine function S ( z | (cid:126)ω ) for any triplet (cid:126)ω = ( ω , ω , ω ) isdefined as the regularization of the product S ( z | (cid:126)ω ) = + ∞ (cid:89) n ,n ,n =0 ( z + (cid:126)n · (cid:126)ω ) (( (cid:126)n + 1) · (cid:126)ω − z ) . (A.7)It has no poles, and its zeros are located at z = ( (cid:126)n + 1) · (cid:126)ω or z = − (cid:126)n · (cid:126)ω for n α ≥
0. It satisfies aconvenient recursion relation (where S ( z | ω ) ≡ πz/ω )) S (cid:32) z + (cid:88) α =1 n α ω α | ω , ω , ω (cid:33) = ( − n n n S ( z | ω , ω , ω ) n − (cid:89) k =0 S ( z + k ω | ω , ω ) n − (cid:89) k =0 S ( z + k ω | ω , ω ) n − (cid:89) k =0 S ( z + k ω | ω , ω ) × n − (cid:89) k =0 n − (cid:89) k =0 S ( z + k ω + k ω | ω ) n − (cid:89) k =0 n − (cid:89) k =0 S ( z + k ω + k ω | ω ) n − (cid:89) k =0 n − (cid:89) k =0 S ( z + k ω + k ω | ω ) , (A.8) and has the following symmetry property S ( z | (cid:126)ω ) = S ( ω + ω + ω − z | (cid:126)ω ) . (A.9) A.3 Υ b functions The function Υ b ( z ) is defined asΥ b ( z ) = (cid:89) m,n ≥ ( mb + nb − + z )(( m + 1) b + ( n + 1) b − − z ) . (A.10)40t has no poles, but zeros located at z = − mb − nb − and z = ( m + 1) b + ( n + 1) b − , for m, n ≥ m, n ∈ Z Υ b ( z − mb − nb − ) = ( − | mn | Υ b ( z ) (cid:81) − m − r =0 (cid:81) − n − s =0 ( z + rb + sb − ) (cid:81) mr =1 (cid:81) ns =1 ( z − rb − sb − ) (cid:81) − m − r =0 (cid:81) ns =1 ( z + rb − sb − ) (cid:81) mr =1 (cid:81) − n − s =0 ( z − rb + sb − ) × (cid:81) − m − r =0 γ ( b ( z + rb )) (cid:81) − n − s =0 γ ( b − ( z + sb − )) (cid:81) mr =1 γ ( b ( z − rb )) (cid:81) ns =1 γ ( b − ( z − sb − )) (cid:81) mr =1 b − z − rb ) b (cid:81) ns =1 b − z − sb − ) b − (cid:81) − m − r =0 b − z + rb ) b (cid:81) − n − s =0 b − z + sb − ) b − . (A.11)Here we defined Q ≡ b + b − , and for each product, when the lower limit is strictly larger than theupper limit, the product reduces to 1. Some other useful properties include Υ b ( Q − z ) = Υ b ( z ) andΥ b ( Q/
2) = 1.
B The S and S b SQCDA partition function
In this appendix we briefly recall the two-dimensional N = (2 ,
2) supersymmetric and three-dimensional N = 2 supersymmetric sphere partition function of a U ( n c ) gauge theory with n f fundamental, n af antifundamental, and one adjoint chiral multiplet, which we henceforth callSQCDA, and present its Higgs branch localized form.Here and in the next appendices, we will use the notation ( i, µ ) in substitution of the originalcolor index a ∈ { , ..., n c } , where i = 1 , ..., n f , µ = 0 , ..., k i −
1, with (cid:80) i k i = n c . Each partition (cid:126)k = { k i } corresponds to a Higgs vacuum of the theory. Therefore, for arbitrary functions Φ ofarbitrary sequences of n c variables x a , we have n c (cid:89) a =1 Φ( x a ) → n f (cid:89) i =1 k i − (cid:89) µ =0 Φ( x iµ ) ≡ (cid:89) ( i,µ ) Φ( x iµ ) , n c (cid:88) a =1 Φ( x a ) → n f (cid:88) i =1 k i − (cid:88) µ =0 Φ( x iµ ) ≡ (cid:88) ( i,µ ) Φ( x iµ ) . (B.1) B.1 The S SQCDA partition function
The two-sphere partition function of an N = (2 ,
2) supersymmetric gauge theory with gaugegroup U ( n c ) and with n f fundamental chiral multiplets with masses m j , n af antifundamental chiralmultiplets with masses ˜ m t and an adjoint chiral multiplet with mass m X is computed by [8, 9] Z S SQCDA = 1 n c ! (cid:88) B ∈ Z n c (cid:90) n c (cid:89) a =1 (cid:20) dσ a π e − πiξ FI σ a − i ( ϑ − n c − B a (cid:21) (cid:89) a>b (cid:34) ( σ a − σ b ) + ( B a − B b ) (cid:35) × n f (cid:89) j =1 n c (cid:89) a =1 Γ( − iσ a − B a + im j )Γ(1 + iσ a − B a − im j ) n af (cid:89) t =1 n c (cid:89) a =1 Γ(+ iσ a + B a − i ˜ m t )Γ(1 − iσ a + B a + i ˜ m t ) × n c (cid:89) a,b =1 Γ( − iσ a + iσ b − B a − B b + im X )Γ(1 + iσ a − iσ b − B a − B b − im X ) . (B.2)41ere we have set the r -charges to zero. They can be reinstated by analytically continuing themasses. In comparison to higher-dimensional sphere partition functions, the two-sphere partitionfunction also involves a sum over magnetic fluxes B. We have also included the two-dimensional ϑ -angle in the classical action. The one-loop determinants for the chiral multiplets are expressed interms of the standard Gamma-function. Finally, note that we have turned off background fluxes forthe flavor symmetries, as they will play no role in this paper.Assuming that n f > n af or n f = n af with ξ FI > , we can close the integration contours of (B.2)in the lower half-plane. The integrand of the σ -integrations in (B.2) is an infinite sum over B ∈ Z n c .One can show that the sum of the residues of different permutations of the color label give the sameresult, canceling the n c ! in (B.2). For each summand with a given B a , the relevant poles are thengiven by iσ jµ = im j + µm X + N jµ + | B jµ | , for all 0 ≤ N j ≤ N j ≤ ... ≤ N j ( k j − ∈ N , (B.3)where µ = 0 , ..., k j −
1, and (cid:126)k is a partition of n c : (cid:80) n f j =1 k j = n c , labeling the Higgs vacua.These poles can be rewritten into a more useful form, by introducing m jµ = N jµ + B jµ + | B jµ | , n jµ = N jµ + | B jµ | − B jµ . We collectively denote these poles as σ m , n ,poles of type σ m , n : iσ jµ + B jµ im j + µim X + m jµ , iσ jµ − B jµ im j + µim X + n jµ . (B.4)Note that m , n are sequences of non-decreasing natural numbers, such that 0 ≤ m j ≤ m j ≤ ... ≤ m j ( k j − , 0 ≤ n j ≤ n j ≤ ... ≤ n j ( k j − , ∀ j = 1 , ..., n f . Alternatively, one can solve N and B interms of m , n by definition.The sum over residues can be brought into the form [19] Z S SQCDA = (cid:88) k Z S cl | (cid:126)k Z S | (cid:126)k (cid:88) m ˆ z | m | Z R vortex | (cid:126)k ( m ) (cid:88) n ¯ˆ z | n | Z R vortex | (cid:126)k ( n ) , (B.5)where ˆ z is defined as ˆ z = e − πξ FI + iϑ (cid:48) , with ϑ (cid:48) = ϑ + π ( n f − n af ) . Furthermore, we introduced | m | ≡ (cid:80) jµ m jµ , and similarly for | n | . The classical and one-loop factors read, with m jl ≡ m j − m l and ˜ m jt ≡ m j − ˜ m t , Z cl | (cid:126)k = exp − πiξ FI n f (cid:88) j =1 (cid:16) k j m j + m X k j − k j (cid:17) (B.6) Z S | (cid:126)k = (cid:81) n f j =1 (cid:81) n f l =1 (cid:81) k j − µ =0 γ ( − im jl + n l im X − µim X ) (cid:81) n f j =1 (cid:81) n af t =1 (cid:81) k j − µ =0 γ (1 − i ˜ m jt − µim X ) , (B.7)42here γ ( x ) = Γ( x )Γ(1 − x ) . The summand Z R vortex | (cid:126)k ( m ) of the vortex partition function is given by Z R vortex | (cid:126)k ( m ) = n af (cid:89) t =1 n f (cid:89) j =1 k j − (cid:89) µ =0 (1 − i ( m j − ˜ m t ) − µim X − m jµ ) m jµ × n f (cid:89) j,l =1 (cid:34) k j − (cid:89) µ =0 k l − (cid:89) ν =0 − im jl − ( µ − ν ) im X + m lν − m jµ ) m jµ − m jµ − × k j − (cid:89) µ =0 (1 + im jl − ( k l − µ ) im X + m jµ − m l ( k l − ) m l ( kl − (1 + im jl − ( k l − µ ) im X ) m jµ (cid:35) , (B.8)and (cid:80) m denotes a sum over all possible non-decreasing sequences of natural numbers m j . B.2 The S b SQCDA partition function
The squashed three-sphere is defined by its embedding in C as ω | z | + ω | z | = 1 , (B.9)and the parameter b is given by b = (cid:112) ω /ω . The partition function of an N = 2 supersymmetricgauge theory with gauge group U ( n c ) and with n f fundamental chiral multiplets with masses m j , n af antifundamental chiral multiplets with masses ˜ m t and an adjoint chiral multiplet with mass m X is computed by [43–46] Z S b = 1 n c ! (cid:90) (cid:32) n c (cid:89) a =1 dσ a (cid:33) e − πiξ FI Tr σ (cid:89) a>b πb ( σ a − σ b )) 2 sinh( πb − ( σ a − σ b )) × (cid:81) n af t =1 (cid:81) n c a =1 s b (cid:16) iQ + σ a − ˜ m t (cid:17)(cid:81) n f j =1 (cid:81) n c a =1 s b (cid:16) − iQ + σ a − m j (cid:17) n c (cid:89) a,b =1 s b (cid:18) iQ − σ a + σ b + m X (cid:19) , (B.10)where Q = b + b − and the matter one-loop determinants are expressed in terms of the double-sinefunction s b . We have taken the Chern-Simons level to be zero.When n f > n af or n f = n af and the FI-parameter ξ FI > we again consider poles of type σ m , n in the lower half-plane, labeled by ascending sequences of natural numbers 0 ≤ m j ≤ m j ≤ ... ≤ m j ( k j − , ∀ j = 1 , ..., n f ,poles of type σ m , n : σ iµ = m j + µm X − i m jµ b − i n jµ b − , µ = 0 , ..., k j − , m jµ , n jµ ∈ N . (B.11)Summing over the residues, one obtains the Higgs branch localized S b -partition function [10, 11], Note that if we had turned on a Chern-Simons level, the convergence criterion would have been slightly moresubtle than in two dimensions, as was explained in [11]. (cid:126)k , Z S b SQCDA = (cid:88) k Z S b cl | (cid:126)k Z S b | (cid:126)k (cid:34)(cid:88) m z | m | b Z R × S vortex | (cid:126)k ( m | b ) (cid:35) (cid:34)(cid:88) n z | n | b − Z R × S vortex | (cid:126)k (cid:0) n | b − (cid:1)(cid:35) , (B.12)where the classical and 1-loop factors, with m ij ≡ m i − m j , are given by Z S b cl | (cid:126)k ≡ exp (cid:104) πiξ FI (cid:16)(cid:88) n f j =1 m j k j + m X (cid:88) n f j =1 ( k j − k j (cid:17)(cid:105) (B.13) Z S b | (cid:126)k ≡ n f (cid:89) j =1 (cid:89) ( l,µ ) s b (cid:18) iQ − m lj + ( k j − µ ) m X (cid:19) n af (cid:89) t =1 (cid:89) ( j,µ ) s b ( iQ m j − ˜ m t + µm X ) . (B.14)The summand Z R × S vortex | (cid:126)k ( m | b ) of the vortex partition function is given by Z R × S vortex | (cid:126)k ( m | b ) ≡ n af (cid:89) t =1 n f (cid:89) j =1 k j − (cid:89) µ =0 (1 − ib − ( m j − ˜ m t ) − iµb − m X − m jµ ) f m jµ × n f (cid:89) j,l =1 (cid:34) k j − (cid:89) µ =0 k l − (cid:89) ν =0 − ib − m jl − i ( µ − ν ) b − m X + m lν − m jµ ) f m jµ − m j,µ − × k j − (cid:89) µ =0 (1 + ib − m jl − i ( k l − µ ) b − m X + m jµ − m l,k l − ) f m l,kl − (1 + ib − m jl − i ( k l − µ ) b − m X ) f m jµ (cid:35) . (B.15)Here we used the function ( x ) fm ≡ (cid:81) m − k =0 f ( x + k ) with f ( x ) = 2 i sinh πib x . See appendixA. Expression (B.15) is summed over all possible sequences of non-decreasing natural numbers0 ≤ m j ≤ m j ≤ ... ≤ m j ( k j − , with weighting factor given in terms of z b ± ≡ e − πξ FI b ± | m | ≡ (cid:88) n f j =1 (cid:88) k j − µ =0 m jµ . (B.16) B.3 Forest-tree representation
The poles (B.4) and (B.11) admit a useful graphical representation in terms of forests of trees.Such representation will turn out to be useful in later appendices, so we introduce it here alreadyfor the simple case of SQCDA [56]. We will consider the example of S b ; the case of S is completelysimilar.When n f ≥ n af and the FI-parameter ξ FI >
0, the Jeffrey-Kirwan residue prescription, mentionedbelow (3.30), selects as poles the solutions to the equations σ a = m i a − i m a b − i n a b − m a , n a (cid:62) σ a = σ b + m X − i ∆ m ab b − i ∆ n ab b − ∆ m ab , ∆ n ab (cid:62) , a (cid:54) = b . (B.17)where for each label a the component σ a appears exactly once on the left-hand side, and i a ∈ i σ i ...σ iµ ...σ i ( k i − m j σ j ...σ jµ ...σ i ( k j − m i σ i σ i σ .. σ i m i σ i ... σ i ... Figure 14:
Examples of forests of trees. The figure on the left shows two branch-less trees, associatedwith masses m i and m j ( (cid:54) = i ) respectively. Forests consisting of such branch-less trees will give non-zerocontributions to the SQCDA partition function. The figure in the middle and on the right show treeswith branches, or two trees associated to the same mass m i ; a forest that contains such trees does notcontribute to the partition function by symmetry arguments. { , ..., n f } . Note that (B.17) contains more poles than those described by (B.11).The poles constructed by solving n c of the equations in (B.17) for the n c components σ a canbe represented by forests of trees by drawing nodes for all components σ a and all masses m i andconnecting the nodes associated with the first symbol on the right-hand side of (B.17) ( i.e. , acomponent of σ or a mass m ) to that associated with the component on the left-hand side witha line, for all n c equations that were used. Note that trees consisting of a single mass node, canbe omitted from the forest. As a result, each component σ a is linked to a fundamental mass m i a (which occurs as the root node of the tree containing the node of σ a ), and the interrelations betweencomponents σ a form the structure of the forest of trees. Figure 14 demonstrates a few examples.When no confusion is expected, we will sometimes omit the mass node at the root of the tree.Using the symmetries of the one-loop determinants, one can show that, after summing overall possible poles, namely over all possible forest diagrams, only those forests whose trees are all branch-less and where each fundamental mass is only linked to (at most) one branch-less tree, willcontribute. The rest of the diagrams cancel among themselves.In the residue computation, we encountered partitions (cid:126)k of the rank n c of the gauge group. Eachentry k j is precisely the length of the length of the tree (or number of descendant nodes under mass m j ) C Factorization of instanton partition function
In this appendix, we analyze the factorization of the summand of the instanton partition function,evaluated at special values of its gauge equivariant parameter, into the product of the summandsof two (semi-)vortex partition functions. We can simultaneously consider the four-dimensionaland five-dimensional instanton partition function by using the notation ( x ) fm (see appendix A),45here f ( x ) = f (cid:15) ,(cid:15) ( x ) is some odd function that might depend on the Ω-deformation parameters.Replacing f by 4d : f ( x ) = (cid:15) x,
5d : f ( x ) = 2 i sinh( πi(cid:15) x ) , (C.1)the following results apply to the familiar instanton partition function respectively on R (cid:15) ,(cid:15) and R (cid:15) ,(cid:15) × S β . In appendix D, E, we will discuss the relation between the factorization results inthis appendix, and the poles and residues of the matrix models that describe gauge theories in thepresence of intersecting defects. C.1 The instanton partition function
We start with a four-/five-dimensional supersymmetric quiver gauge theory with gauge group SU ( N ) × SU ( N ) , with N fundamental hypermultiplets, N anti-fundamental hypermultiplets andone bi-fundamental hypermultiplet, with masses M I , ˜ M J and ˆ M respectively. Let Σ and Σ (cid:48) denotethe Cartan-valued constant scalars of the two vector multiplets. The instanton partition functioncan be written as a sum over N -tuples of Young diagrams (cid:126)Y , (cid:126)Y (cid:48) and the individual contributions toeach summand read z vect ( (cid:126)Y ; Σ) ≡ N C (cid:89) A,B =1 ∞ (cid:89) r,s =1 ( i(cid:15) − Σ AB − b ( s − r + 1) − Y Bs ) fY Ar ( i(cid:15) − Σ AB − b ( s − r + 1) − Y Bs ) fY Bs ( i(cid:15) − Σ AB − b ( s − r ) − Y Bs ) fY Bs ( i(cid:15) − Σ AB − b ( s − r ) − Y Bs ) fY Ar , (C.2) z (a)fund ( (cid:126)Y , Σ , µ (cid:15) ) ≡ N (cid:89) A =1 N (cid:89) I =1 ∞ (cid:89) r =1 ( i(cid:15) − (Σ A − µ (cid:15)I ) + b r + 1) fY Ar , (C.3) z bifund ( (cid:126)Y , (cid:126)Y (cid:48) , Σ , Σ (cid:48) , ˆ M (cid:15) ) ≡ N (cid:89) A,B =1 ∞ (cid:89) r,s =1 ( − i(cid:15) − (Σ (cid:48) B − Σ A + ˆ M (cid:15) ) − b ( s − r + 1) − Y (cid:48) Bs ) fY (cid:48) Bs ( − i(cid:15) − (Σ (cid:48) B − Σ A + ˆ M (cid:15) ) − b ( s − r + 1) − Y (cid:48) Bs ) fY Ar × ( − i(cid:15) − (Σ (cid:48) B − Σ A + ˆ M (cid:15) ) − b ( s − r ) − Y (cid:48) Bs ) fY Ar ( − i(cid:15) − (Σ (cid:48) B − Σ A + ˆ M (cid:15) ) − b ( s − r ) − Y (cid:48) Bs ) fY (cid:48) Bs . (C.4)Here Σ AB = Σ A − Σ B and b ≡ (cid:15) /(cid:15) . The full instanton partition function is thus Z inst ≡ (cid:88) (cid:126)Y ,(cid:126)Y (cid:48) q | (cid:126)Y | q (cid:48)| (cid:126)Y (cid:48) | z vect ( (cid:126)Y , Σ) z vect ( (cid:126)Y (cid:48) , Σ (cid:48) ) z afund ( (cid:126)Y (cid:48) , Σ (cid:48) ) z bifund ( (cid:126)Y , (cid:126)Y (cid:48) , Σ , Σ (cid:48) ) z fund ( (cid:126)Y , Σ) , (C.5)where we omitted the mass dependence.We are interested in the instanton partition function evaluated at special values for its gauge Instanton counting is typically performed for U ( N ) gauge groups. We will not be careful about the distinction. Infact, removing the U (1) factors is expected to just amount to some overall factor (1 − q ) , as in [22]. On the one hand, the simpler case of SU ( N ) SQCD, which we used in section 3, can be easily extractedfrom this expression, by setting all Y (cid:48) A to empty Young diagrams and identifying the antifundamental mass as˜ M A = Σ (cid:48) A + ˆ M − i(cid:15) − i(cid:15) . On the other hand, it can also easily be generalized to linear SU ( N ) quivers. A → Σ (cid:126)n L ,(cid:126)n R A ≡ M (cid:15)A + i ( n L A + 1) (cid:15) + i ( n R A + 1) (cid:15) , (C.6)for integers n L / R A ≥
0. Here M denotes the mass of the fundamental hypermultiplets. We denotethe collection of natural numbers simply by (cid:126)n L/R ≡ { n L/R A } , and their sums as n L/R ≡ (cid:80) NA =1 n L/R A .Remarkably, when evaluated at these special values, the instanton partition function simplifies andexhibits useful factorization properties.The most significant simplification comes from the evaluation of z fund : if any Young diagram Y A of the N -tuple (cid:126)Y contains a box (the “forbidden box”) at position ( n L A + 1 , n R A + 1), then z fund ( (cid:126)Y , Σ)evaluates to zero. Hence, the sum over all (cid:126)Y is effectively restricted to those tuples all of whosemembers avoid the “forbidden box”. C.2 Reduction to vortex partition function of SQCD instanton partition func-tion
Let us consider the SQCD instanton partition function and look at the case where n R = 0. Theforbidden boxes sit at ( n L A + 1 , Y A in a contributing tuple (cid:126)Y must have width W Y A ≤ n L A .Let z (cid:126)n L ,(cid:126)n R vf ( (cid:126)Y ) denote the product z vect ( (cid:126)Y , Σ (cid:126)n L ,(cid:126)n R ) z fund ( (cid:126)Y , Σ (cid:126)n L ,(cid:126)n R , M (cid:15) ). It simplifies in the case (cid:126)n R = (cid:126) z (cid:126)n L ,(cid:126) ( (cid:126)Y ) = z vect ( (cid:126)Y , Σ (cid:126)n L ,(cid:126) ) z fund ( (cid:126)Y , Σ (cid:126)n L ,(cid:126) , M (cid:15) )= ( − N | (cid:126)Y | N (cid:89) A,B =1 (cid:34) (cid:81) n L A r =1 (cid:81) n L B s =1 (1 − i(cid:15) − M AB + b ( s − r + n L A − n L B ) − Y Ar + Y Bs ) fY Ar − Y Ar +1 × (cid:81) n L B s =1 (1 + b s − i(cid:15) − M AB + b ( n L A − n L B ) + Y Bs − Y A ) fY A (cid:81) n L B s =1 (1 + b s − i(cid:15) − M AB + b ( n L A − n L B )) fY Bs (cid:35) . (C.7)Multiplying in also z afund ( (cid:126)Y , Σ (cid:126)n L ,(cid:126) , ˜ M (cid:15) ), we can identify the resulting product with a summand of atwo-/three-dimensional SQCDA vortex partition function. We identify the number of colors andflavors as n c = n L , n f = N , and n af = N . The integer partitions are identified as { n L A } ↔ { k i } , andfinally m Aµ = Y A ( n L A − µ ) . Then we recover (B.8), (B.15) in an obvious way, if one also sets2d: f ( x ) ≡ (cid:15) x, (C.8)3d: f ( x ) ≡ i sinh πi(cid:15) ( x ) b ≡ √ (cid:15) , (C.9) Such diagrams are sometimes referred to as hook Young diagrams. R ν ... = ν Y R m L µ Y L = µ Y R n L A +1 r i such that Y Ari − Y A ( ri +1) > s = n R A ν = n R A − Y A ( ri +1) − ν = n R A − Y Ari ν =0 ) m R Aν = r i − n L A Figure 15:
The left figure demonstrates the decomposition of a large Young diagram Y A into Y L A and Y R A . The latter are filled in gray, while the “forbidden box” is colored red. The figure on the rightdemonstrates some convenient relations between m R Aµ and Y R Ar . and identifies the masses as2d ξ FI > m X ≡ i(cid:15) /(cid:15) , m AB ≡ (cid:15) − M AB , m A − ˜ m J ≡ (cid:15) − ˜ M A,J + m X , (C.10)3d ξ FI > m X ≡ i(cid:15) / √ (cid:15) , m AB ≡ (cid:15) − / M AB , m A − ˜ m J ≡ (cid:15) − / ˜ M A,J + m X , (C.11)where M AB = M A − M B and ˜ M AB = M A − ˜ M B . C.3 Factorization of instanton partition function for large N -tuples of Youngdiagrams Given the set of natural numbers (cid:126)n L , (cid:126)n R , we have defined in the main text the notion of large N -tuples of Young diagrams, see above equation (3.18). For such large N -tuples we introducedsubdiagrams Y L A and Y R A in (3.18), and finally sequences of non-decreasing integers m L Aµ and m R Aν in (3.19). In figure 15 we remind the reader of these definitions.Now we are ready to state the factorization of the various factors in the (summand of) thetwo-gauge-node instanton partition function of (C.5), associated to large Young diagrams, whenevaluated on Σ (cid:126)n L ,(cid:126)n R A defined in (C.6). Introducing the shorthand notations z (cid:126)n L ,(cid:126)n R afund ( (cid:126)Y , ˜ M ε ) = z afund ( (cid:126)Y , Σ (cid:126)n L ,(cid:126)n R , ˜ M (cid:15) ), z (cid:126)n L ,(cid:126)n R vf ( (cid:126)Y ) = z vect ( (cid:126)Y ; Σ (cid:126)n L ,(cid:126)n R ) z fund ( (cid:126)Y , Σ (cid:126)n L ,(cid:126)n R , M (cid:15) ) and z (cid:126)n L ,(cid:126)n R bifund ( (cid:126)Y , (cid:126)Y (cid:48) , ˆ M (cid:15) ) = z bifund ( (cid:126)Y , (cid:126)Y (cid:48) , Σ (cid:126)n L ,(cid:126)n R , Σ (cid:48) , ˆ M (cid:15) ), it is straightforward to show that z (cid:126)n L ,(cid:126)n R afund ( (cid:126)Y , ˜ M ε ) = z (cid:126)n L ,(cid:126) ( (cid:126)Y L , ˜ M ε ) z (cid:126) ,(cid:126)n R afund ( (cid:126)Y R , ˜ M ε ) ( Z (cid:126)n L ,(cid:126)n R afund,extra ( ˜ M )) − (C.12) z (cid:126)n L ,(cid:126)n R vf ( (cid:126)Y ) = ( − N(cid:126)n L · (cid:126)n R z (cid:126)n L ,(cid:126) ( (cid:126)Y L ) z (cid:126) ,(cid:126)n R vf ( (cid:126)Y R ) Z large | (cid:126)n L ,(cid:126)n R vf,intersection ( m L , m R ) ( Z (cid:126)n L ,(cid:126)n R vf,extra ) − (C.13) z (cid:126)n L ,(cid:126)n R bifund ( (cid:126)Y , (cid:126)Y (cid:48) , ˆ M (cid:15) ) = ( − N(cid:126)n L · (cid:126)n R z (cid:126)n L ,(cid:126) ( (cid:126)Y L , (cid:126)Y (cid:48) , ˆ M (cid:15) ) z (cid:126) ,(cid:126)n R bifund ( (cid:126)Y R , (cid:126)Y (cid:48) , ˆ M (cid:15) ) × Z bifund,intersection ( (cid:126)Y (cid:48) ) ( Z (cid:126)n L ,(cid:126)n R bifund,extra ) − . (C.14)48he product of the latter two can be simplified further to z (cid:126)n L ,(cid:126)n R vf ( (cid:126)Y ) z (cid:126)n L ,(cid:126)n R bifund ( (cid:126)Y , (cid:126)Y (cid:48) , ˆ M (cid:15) ) = Z vortex | (cid:126)n L ( m L ) Z vortex | (cid:126)n R ( m R ) z fund ( (cid:126)Y (cid:48) , Σ (cid:48) , ( M (cid:48) ) (cid:15) ) × ( Z (cid:126)n L ,(cid:126)n R vf,extra Z (cid:126)n L ,(cid:126)n R bifund,extra ) − Z large | (cid:126)n L ,(cid:126)n R vf,intersection ( m L , m R ) z Ldefect ( Y (cid:48) , m L ) z Rdefect ( ˜ Y (cid:48) , m R ) . (C.15)Let us spell out in detail the various factors and quantities appearing in these factorization results.First of all, new masses of fundamental hypermultiplets, which we denoted as M (cid:48) , appear. They aregiven by M (cid:48) I = M I − ˆ M + i ( (cid:15) + (cid:15) ) /
2, and their shifted versions are as usual ( M (cid:48) ) (cid:15) = M (cid:48) − i ( (cid:15) + (cid:15) ) / (cid:126)n L · (cid:126)n R ≡ (cid:80) NA =1 n L A n R A . Next, as in the previous appendix, Z vortex | (cid:126)n L/R denotes the vortex partition function of U ( (cid:80) n L/R A ) SQCDA with n f = n af = N , whose explicitexpressions on R and R × S can be found in appendix B.1 and B.2. The fundamental and adjointmasses are identified as in (C.10)-(C.11), while the antifundamental masses are given by2d ξ FI > m A − ˜ m J = ε − ( M A − Σ (cid:48) J − ˆ M ) + m X (C.16)3d ξ FI > m A − ˜ m J = ε − / ( M A − Σ (cid:48) J − ˆ M ) + m X . (C.17)The factors labeled with ‘intersection’ are given by Z large | (cid:126)n L ,(cid:126)n R vf,intersection ( m L , m R ) ≡ N (cid:89) A,b =1 n L A − (cid:89) µ =0 n R B − (cid:89) ν =0 f (∆ C ( m ) − b ) f (∆ C ( m ) + 1) , (C.18) Z bifund,intersection ( (cid:126)Y (cid:48) ) ≡ N (cid:89) A,B =1 W Y (cid:48) B (cid:89) r =1 Y (cid:48) Br (cid:89) s =1 f ( − i(cid:15) − (Σ (cid:48) B − M A + ˆ M ) − b r − s ) , (C.19)with ∆ C ( m ) ≡ i(cid:15) − ( M A − M B ) + ( m L Aµ + ν ) − b ( m R Bν + µ ). The factor z Ldefect is defined as z Ldefect ( Y (cid:48) , m L ) = N (cid:89) A,B =1 n L A − (cid:89) µ =0 W Y (cid:48) B (cid:89) s =1 ( − iε − (Σ (cid:48) B − ( M (cid:48) A ) (cid:15) ) + ( µ + 1 + s ) b + m L Aµ − Y (cid:48) Bs ) fY (cid:48) Bs ( − iε − (Σ (cid:48) B − ( M (cid:48) A ) (cid:15) ) + ( µ + s ) b + m L Aµ − Y (cid:48) Bs ) fY (cid:48) Bs , (C.20)and z Rdefect ( ˜ Y (cid:48) , m R ) is the same expression but with ( n L , m L , (cid:15) , Y (cid:48) ) ↔ ( n R , m R , (cid:15) , ˜ Y (cid:48) ).49inally, the factors labeled by ‘extra’ read Z (cid:126)n L ,(cid:126)n R afund,extra ( ˜ M ) ≡ N (cid:89) A,B =1 n L A (cid:89) r =1 n R A (cid:89) s =1 f ( iε − ( M A − ˜ M B ) + b ( r − n L A −
1) + ( s − n R A − Z (cid:126)n L ,(cid:126)n R vf,extra ≡ N (cid:89) A,B =1 (cid:81) n L A − n L B − r =0 (cid:81) n R A − n R B − s =0 f (∆ − ( r, s )) f (∆ − ( r, s ) − b − (cid:81) n L B − n L A r =1 (cid:81) n R A − n R B s =1 f (∆ + ( r, s ) − b ) f (∆ + ( r, s ) + 1) (C.22) × N (cid:89) A,B =1 n L A (cid:89) r =1 n R A (cid:89) s =1 f ( iε − ( M A − M B ) + b ( r − n L A −
1) + ( s − n R A − Z (cid:126)n L ,(cid:126)n R bifund,extra ≡ N (cid:89) A,B =1 n L A (cid:89) r =1 n R A (cid:89) s =1 f ( i(cid:15) − (Σ (cid:48) B − M A + ˆ M ) + b r + s ) , (C.23)where ∆ ± ( r, s ) ≡ i(cid:15) − ( M A − M B ) ± b r − s . C.4 Factorization for small N -tuples of Young diagrams For N -tuples of Young diagrams that are not large, which we refer to as small , a similarfactorization of the summand of the instanton partition function occurs, but is more involved. A(tuple of) small Young diagram (cid:126)Y , namely Y An L A < n R A for some A , again defines two non-decreasingsequences of integers as in (3.23). In particular, m L Aµ can be negative: for each A , we define ¯ µ suchthat m L¯ µ A (cid:62) m L¯ µ A − <
0. For simplicity, we show the results for the SQCD instanton partitionfunction. The summand of this instanton partition function evaluated at (C.6), i.e. , z (cid:126)n L ,(cid:126)n R vf z (cid:126)n L ,(cid:126)n R afund ,factorizes into, for small N -tuple of Young diagrams (cid:126)Y , z (cid:126)n L ,(cid:126)n R vf ( (cid:126)Y small ) z (cid:126)n L ,(cid:126)n R afund ( (cid:126)Y small , ˜ M ε )= Z semi-vortex | (cid:126)n L ( m L ) Z vortex | (cid:126)n R ( m R ) Z (cid:126)n L ,(cid:126)n R vf,intersection ( m L , m R ) (cid:0) Z (cid:126)n L ,(cid:126)n R afund,extra ( ˜ M ) Z (cid:126)n L ,(cid:126)n R vf,extra. (cid:1) − (C.24)where the ‘extra’ are as before, and the intersection factor reads, again with ∆ C = i(cid:15) − ( M A − M B ) +( ν + m L Aµ ) − b ( µ + m R Bν ), Z (cid:126)n L ,(cid:126)n R vf,intersection ( m L , m R ) = N (cid:89) A,B =1 n L A − (cid:89) µ =0 n R B − (cid:89) ν =0 f (∆ C − b ) N (cid:89) A,B =1 A (cid:54) = B or ¯ µ A =0 n L A − (cid:89) µ =0 n R B − (cid:89) ν =0 f (∆ C + 1) × N (cid:89) A (= B )=1 | ¯ µ A > ¯ µ A − (cid:89) µ =0 n R A − (cid:89) ν = − m L Aµ f (∆ C + 1) n L A − (cid:89) µ =¯ µ A n R B − (cid:89) ν =0 f (∆ C + 1) , (C.25)50 igure 16: An example of a small Young diagram, with the “forbidden box” shown in red. Therectangular region, enclosed by the dashed-lines, is partially filled. In general, the intersection factor Z (cid:126)n L ,(cid:126)n R vf,intersection involves a product over those filled boxes. and we defined Z SQCDA(semi-)vortex | (cid:126)n ( m )= N (cid:89) A>B =1 n A − (cid:89) µ =0 n B − (cid:89) ν =0 f ( − i(cid:15) − M AB + ( µ − ν ) b − ( m Aµ − m Bν )) N (cid:89) A =1 n A − (cid:89) µ>ν =0 f (+( µ − ν ) b − ( m Aµ − m Aν )) × N (cid:89) A,B =1 ¯ µ − (cid:89) µ =0 ¯ ν − (cid:89) ν =0 ( − i(cid:15) − M AB + ( µ − ν − b + m Bν ) f − m Aµ ( i(cid:15) − M AB + ( ν − µ + 1) b ) f − m Bν × N (cid:89) A,B =1 ¯ µ − (cid:89) µ =0 ¯ ν − (cid:89) ν =0 ( i(cid:15) − M AB + ( ν − µ − b + m Aµ ) f − m Bν ( i(cid:15) − M AB + ( ν − µ + 1) b + 1) f m Aµ ( i(cid:15) − M AB + ( ν − µ − b ) f m Aµ ( i(cid:15) − M AB + ( ν − µ + 1) b + 1) f − m Bν × N (cid:89) A,B =1 n A − (cid:89) µ =¯ µ n B − (cid:89) ν =¯ ν ( − i(cid:15) − M AB + ( µ − ν − b ) f m Bν ( i(cid:15) − M AB + ( ν − µ + 1) b − m Bν + 1) f m Aµ × N (cid:89) A =1 N (cid:89) B ( (cid:54) = A )=1 n A − (cid:89) µ =0 i(cid:15) − M AB − µ b + 1) f m Aµ N (cid:89) A =1 n A − (cid:89) µ =¯ µ − µ b + 1) f m Aµ × (cid:81) NB =1 (cid:81) A,µ | m Aµ ≥ (1 − iε − ( M A − ˜ M B ) + ( µ + 1) b − m Aµ ) f m Aµ (cid:81) NB =1 (cid:81) A,µ | m Aµ < (1 − iε − ( M A + ˜ M B ) + ( µ + 1) b ) f − m Aµ . (C.26) We remark that in Z (cid:126)n L ,(cid:126)n R vf,intersection , the second line is in fact a product over the boxes filled insidethe n L A × n R A rectangle, namely the gray boxes inside the region enclosed by the dashed lines in figure16. Also note that when all ¯ µ A = 0, (C.25) turns into (C.18), and the expression for Z (semi-)vortex | (cid:126)n L in (C.26) reduces to the usual vortex partition function, since the small Young diagram has deformedinto a large Young diagram. D Poles and Young diagrams in 3d
In this appendix we analyze the correspondence between poles in the three-dimensional Coulombbranch matrix model describing the worldvolume theory of intersecting codimension two defects,and (Young) diagrams. We will show that one can construct generic Young diagrams using a classof poles of the matrix model, which we call poles of type-ˆ ν , and the sum over the corresponding51esidues is precisely the instanton partition function evaluated at (C.6). All other classes of polesare spurious and their residues should cancel among themselves: we will indeed argue that this isthe case by showing that they give rise to certain diagrams, consisting of boxes and anti-boxes, andthat these diagrams pair up and the corresponding residues cancel each other. We will first considergeneric intersecting defect theories on S ∩ S with gauge groups U ( n (1) ) and U ( n (2) ), sharing n f = n af = N . D.1 Poles of type- ˆ ν We recall from subsection 3.1.3 that the proposed matrix model that computes the partitionfunction of the worldvolume theory of intersecting defects has an integrand of the form, see (3.27), Z ( T ,S ∪ S ⊂ S (cid:126)ω ) ( σ (1) , σ (2) ) = Z ( T ,S (cid:126)ω )1-loop n (1) ! n (2) ! Z S ( σ (1) ) Z intersection ( σ (1) , σ (2) ) Z S ( σ (2) ) , (D.1)where Z S ( σ (1) ) denotes the integrand of U ( n (1) ) SQCDA on S with ξ (1)FI >
0, and similarly for Z S ( σ (2) ). Recall that the parameters entering the two three-sphere integrands satisfy variousrelations, see (3.29). The intersection factor reads Z intersection ( σ (1) , σ (2) ) = n (1) (cid:89) a =1 n (2) (cid:89) b =1 (cid:89) ± (cid:20) i sinh π ( − b (1) σ (1) a + b (2) σ (2) b ± i b + b )) (cid:21) − . (D.2)The Jeffrey-Kirwan-like prescription selects a large number of poles in the combined meromorphicintegrand (D.1). We now focus on the subclass of poles, defined in (3.31)-(3.32), and referred toas poles of type- ˆ ν . It is useful to observe that n L and n R can be decoupled from the followingdiscussion. Using the recursion relations of the double-sine function s b ( iQ/ z ) and the factthat sinh πi ( x + n ) = ( − n sinh πx , they can be seen to give rise to Z R × S ∩ vortex | (cid:126)n and Z R × S ∩ vortex | (cid:126)n ,independent of the values of m L and m R . Therefore, without loss of generality, we will ignore thedetails of n L , n R .It may be helpful to remark that the poles of type-ˆ ν , as defined in (3.31)-(3.32), can be obtainedby solving the component equations Z S : σ (2) → σ (2) m , n Z intersection : b (1) σ (1) A = b (2) σ (2) A ˆ ν A + i b + i b Z S : σ (1) A − m (1) A = − i m L A b (1) − i n L A b − σ (1) A ( µ (cid:62) = σ (1) A ( µ − + m (1) X − i ∆ m L Aµ b (1) − i ∆ n L Aµ b − , (D.3)with the requirement that m R A ˆ ν A = 0 (which automatically implies that for all µ ≤ ˆ ν A also m R Aµ = 0since the m R Aµ are a non-decreasing sequence). One should also bear in mind the parameter relations52 + + + ++ + + ++ + + ++ + + ++ + + + n (2) An (1) An (2) A − ... n (1) A − · · · + + + ++ + + ++ + + ++ + + ++ + + + −−−−− + +++ + + ++ + m L A − ˆ νA − m L Aµ> ≥− ˆ νA − | m L Aµ | ( m L Aµ< m L Aµ ( m L Aµ ≥ m R Aν m R A (ˆ νA +1) ≥ m R A ˆ νA =0 ... m R A + + + + + + ++ + + + + ++ + ++ ++ ++ +++ Figure 17:
The three steps in constructing a Young diagram Y using the combinatoric data from a poleof type-ˆ ν . Boxes with a black + are normal boxes, while boxes with a red − are anti-boxes. Coincidentboxes and anti-boxes, i.e. , the ones with red edges in the middle figure, annihilate to create vacant spots. b (1) m (1) A − b (2) m (2) A = i ( b − b ). Here we assigned to σ (2) the poles locations defined in (B.11).As usual, for each A , σ (1) A should be solved either with the equation in the second or third line.The class with all ˆ ν A = − σ (1) A via the equation in the third line,since σ (2) A (ˆ ν A = − does not exist. The resulting poles are simply (the union of) the poles σ (1) m , n and σ (2) m , n of Z S and Z S respectively, which were discussed in detail in appendix B.2. Their residuesare equal to the product of the summand of two SQCDA vortex partition functions times theintersection factor evaluated at the pole position. The remaining classes with at least one ˆ ν A ≥ { ˆ ν A = − } are associated with large Young diagrams, while the remaining poles of type-ˆ ν are associated withsmall Young diagrams. The poles of type-ˆ ν are special cases of the more general poles that will bediscussed in later appendices. D.2 Constructing Young diagrams
We now construct Young diagrams associated with poles of type-ˆ ν, labeled by the integers m L/R ,through the following steps. We only present the construction of Y A for a given A , which can berepeated to construct the full N -tuple of Young diagrams { Y A } . The procedure is also depicted infigure 17.1. Start with a rectangular Young diagram Y (cid:50) with n (1) A columns and n (2) A rows of boxes. Thecolumns can be relabeled by µ , starting as µ = n (1) A − n (1) A -th column has label 0, and columns tothe right of it are negatively-labeled. Similarly, the rows can be labeled by ν , starting as ν = n (2) A − σ (2) Aν = m (2) A + νm (2) X − i m R Aν b (2) − i n R Aν b − . For each ν = 0 , ..., n (2) − S ( σ (1) ) Z (1 , ( σ (1) , σ (2) ) Z S ( σ (2) ) Z R × S vortex | ~n (1) ( n L | b − ) h Z R × S (semi-)vortex | ~n (1) ( m L | b (1) ) Z ~n (1) ,~n (2) intersection ( m L , m R ) ˜ Z R × S vortex | ~n (2) ( m R | b (2) ) i ˜ Z R × S vortex | ~n (2) ( n R | b − ) Z C × S ∩ inst ( ~Y (1 ∩ , Σ , M, ˜ M ) Z C × S ∩ inst ( ~Y (1 ∩ , Σ , M, ˜ M ) Z C × S ∩ inst ( ~Y (2 ∩ , Σ , M, ˜ M ) Figure 18:
Schematic representation of how taking the sum over residues of D.4, drawn by thedownward arrows, reproduces the result as obtained from the five-dimensional Higgsing analysis, depictedby the upward arrows. In the bold-face upward pointing arrow, we have omitted the ‘extra’ factors, seeappendix C.3. In the bold-face downward arrows, we have omitted the classical action and one-loopfactors (at the Higgs branch vacuum (cid:126)n (1) or (cid:126)n (2) respectively). attach an extra segment of m R Aν boxes to the ν -th row at the right edge of Y (cid:50) extendingtowards the right.2b. Consider each component σ (1) Aµ = m (1) + µm (1) X − i m L Aµ b (1) − i n L Aµ b − . For each µ = 0 , ..., n (1) A − m L Aµ >
0, attach an extra segment of m L Aµ boxes to the µ -th column at the bottom edgeof Y (cid:50) hanging downwards, or, if m L Aµ <
0, attach a segment of m L Aµ anti-boxes to the µ -thcolumn at the bottom edge standing upwards.3. An anti-box annihilates a box at the same location, creating a vacant spot.It is now obvious that, poles of type- { all ˆ ν A = − } generate large Young diagrams, since all m L/R are non-negative. When there is at least one ˆ ν A ≥
0, the poles of type-ˆ ν generate small Youngdiagrams whose n (1) A -th column (labeled as µ = 0) has length n (2) A − ˆ ν A − < n (2) A . D.3 Residues and instanton partition function
The correspondence between poles and Young diagrams in the previous subappendix does notstop at the combinatoric level: it also leads to an equality between residues of the matrix modeland the summand of the instanton partition function evaluated at Σ (cid:126)n L ,(cid:126)n R A as in (C.6). Namely, wewill show that (cid:88) σ pole ∈{ poles of type-ˆ ν } Res σ → σ pole Z ( T ,S ∪ S ⊂ S (cid:126)ω ) ( σ (1) , σ (2) ) = right-hand side of equation (3.25) . (D.4)Figure 18 indicates schematically how the various factors in the integrand (D.1) reorganize themselvesupon taking the residues of the poles of type ˆ ν (ignoring the classical and overall one-loop factors,which are trivial to recover).Let us present some more details. We start with the poles of type-( − Z S and Z S arejust the summand of the relevant vortex partition functions multiplied by the classical action and54ne-loop determinant at the Higgs branch vacuum. The remaining factor Z intersection ( σ (1) , σ (2) ) canbe trivially evaluated at the pole σ pole , giving, with ∆ C ( m ) as defined below (C.19) and using themass relation (C.11), Z intersection ( σ pole ) = N (cid:89) A,B =1 n (1) A − (cid:89) µ =0 n (2) A − (cid:89) ν =0 i sinh iπ ( b ∆ C ( m ) + b ) 2 i sinh iπ ( b ∆ C ( m ) − b ) . (D.5)This is precisely the intersection factor appearing in (C.15), see (C.18), with f ( x ) = 2 i sinh πib x .Summing the product of all factors just described over m L , m R , n L , n R reproduces the sum overlarge diagrams in the right-hand side of (3.25). Note that we have inserted a trivial factor of onewritten as the ratio of the extra factors appearing in (C.15). One factor of this ratio completes theHiggsed instanton partition function (of the large N -tuples of diagrams), and the other one mergeswith the three-dimensional one-loop determinants at the Higgs branch vacuum to form the Higgsedfive-dimensional one-loop determinant. Of course this should come as no surprise, since the matrixmodel integrand (D.1) was designed to reproduce the instanton partition functions for large Youngdiagrams, when evaluated at these poles.Next we consider the poles of type-ˆ ν with some ˆ ν A > −
1, which we claim will reproduce thesmall Young diagram contributions to the instanton partition function. Define ¯ µ A as the smallestinteger for which m L A ¯ µ A ≥ , i.e. , m L A (¯ µ A − < ≤ m L¯ µ A . Notice that ˆ ν A (cid:62) ⇔ ¯ µ A >
0. At this point,we will suppress the details about n L and n R , as their computational details are similar to the onesjust presented for the large diagrams. We first compute the reside of the fundamental one-loopfactor in Z S . It reads Res σ → σ pole N (cid:89) A =1 n (1) (cid:89) a =1 s b (1) ( iQ (1) − σ (1) a + m (1) A ) = (cid:104) Res z → s b (1) ( iQ (1) − z ) (cid:105) N (cid:80) A =1 δ ¯ µA × N (cid:89) A,B =1 A (cid:54) = B n (1) A − (cid:89) µ =0 s b (1) ( iQ (1) − m (1) AB − µm (1) X ) N (cid:89) A =1 n (1) A − (cid:89) µ =1 s b (1) ( iQ (1) − µm (1) X ) N (cid:89) A =1¯ µ A > s b (1) ( iQ (1) − ib (1) ) × N (cid:89) A =1 n (1) A − (cid:89) µ =¯ µ A m L Aµ (cid:89) k =1 i sinh πb (1) ( − µm (1) X + ikb (1) ) N (cid:89) A,B =1 A (cid:54) = B n (1) A − (cid:89) µ =0 m L Aµ (cid:89) k =1 i sinh πb (1) ( − m (1) AB − µm (1) X + ikb (1) ) × N (cid:89) A =1¯ µ A > ¯ µ A − (cid:89) µ =1 − m L Aµ − (cid:89) k =0 i sinh πb (1) ( − µm (1) X − ikb (1) ) ˆ ν A − (cid:89) k =0 i sinh πb (1) ( − ib (1) − ikb (1) ) . (D.6) Since we are considering SQCD, we should set all Young diagrams (cid:126)Y (cid:48) to be empty. In particular, z fund → z L/Rdefect →
1. See footnote 36.
55e note that s b (1) ( iQ (1) / − ib (1) ) = b (1) . Next we take the residue of one of the factors of (D.2) Z intersection,1 ≡ n (1) (cid:89) a =1 n (2) (cid:89) b =1 (cid:16) i sinh π (cid:16) b (2) σ (2) b − b (1) σ (1) a + i b + i b (cid:17)(cid:17) − . (D.7)We also denote the other factor in (D.2) as Z intersection,2 . We use again b ∆ C ( m ) = ib (1) m (1) AB + b ( m L Aµ + ν ) − b ( m R Bν + µ ), and observe that N (cid:89) A (= B )=1 − m L A − (cid:89) ν =0 i sinh iπ (∆ C ( m ) + b ) ¯ µ A − (cid:89) µ A =1 − m L Aµ − (cid:89) ν =0 i sinh iπ (∆ C ( m ) + b ) = N (cid:89) A (= B )=1 ˆ ν A − (cid:89) k =0 i sinh iπ ( − ( k + 1) b ) ¯ µ A − (cid:89) µ =1 − m L Aµ − (cid:89) k =0 i sinh πb (1) ( − µm (1) X − ikb (1) ) . The residue of Z intersection,1 can then be written asRes σ → σ pole Z intersection,1 = b − (cid:80) A δ ¯ µA (1) N (cid:89) A,B =1 A (cid:54) = B or ¯ µ A =0 n L A − (cid:89) µ =0 n R B − (cid:89) ν =0 i sinh iπ (∆ C + b ) × N (cid:89) A (= B )=1¯ µ A > ¯ µ A − (cid:89) µ =0 n R A − (cid:89) ν = − m L Aµ i sinh iπ (∆ C + b ) n L A − (cid:89) µ =¯ µ A n R B − (cid:89) ν =0 i sinh iπ (∆ C + b ) (D.8) × N (cid:89) A (= B )=1 ˆ ν A − (cid:89) k =0 i sinh iπ ( − ( k + 1) b ) ¯ µ A − (cid:89) µ =1 − m L Aµ − (cid:89) k =0 i sinh πb (1) ( − µm (1) X − ikb (1) ) − . Observe that the last line and b cancel against the last line in (D.6) and the products of s b (1) ( iQ (1) − ib (1) ). The factors in the second line are precisely a product over the filled boxesinside the n (1) A × n (2) A rectangular region, and, together with the leftover factor in the first line and Z intersection,2 ( σ pole ), reproduce the intersection factor in the factorization result for small diagrams(C.24). The leftover factors of (D.6) together with the residues of other one-loop factors combineinto the “(semi-) vortex” partition function factor in (C.24). D.4 Extra poles and diagrams
The matrix model (D.1) has more simple poles, which are selected by the JK prescription, thanjust those of type-ˆ ν . All of them assign to σ (2) poles of type σ (2) m , n ,σ (2) jν = m (2) j + νm (2) X − i m R jν b (2) − i n R jν b − , m R jν , n R jν (cid:62) , (D.9)56hile σ (1) are solutions to the component equations σ (1) a = m (1) i a − i m L a b (1) − i n L a b − , m L a , n L a (cid:62) σ (1) a = σ (1) b + m (1) X − i ∆ m ab b (1) − i ∆ n ab b − , ∆ m ab , ∆ n ab (cid:62) b (1) σ (1) a = b (2) σ (2) b + i b + i b − i n L a , n L a (cid:62) b (1) σ (1) a = b (2) σ (2) b − i b − i b − i n L a , n L a (cid:62) . (D.10)where at least one of the component σ (1) a should be solved by a type I or type II equation (otherwiseone just gets back the poles σ (1) m , n , σ (2) m , n , which are already discussed). Similar to those of the SQCDApartition functions on S b , the poles specified by (D.9)-(D.10) can be characterized by forest-treediagrams. However, there are now three possible types of links between two nodes, correspondingto the equations of type-adj., type I and type II. Note that the poles of type-ˆ ν discussed in theprevious section, which gave rise to large and small Young diagrams, can be recovered as specialcases of (D.9)-(D.10).For simplicity and clarity of the presentation, we consider the cases of n f = n af = 1, gaugegroups U ( n (1) = 2) on S and U ( n (2) ) on S . The flavor index A is spurious in this case and willbe omitted. For more general unitary gauge groups, the poles can be analyzed following exactlythe same logic. We will show that the general simple poles not of type-ˆ ν cancel among themselves.Again, we decouple the n L , n R in the following discussions.First of all, there are many families of poles. The components σ (1) µ , σ (2) ν , given by solving (D.10),can be written universally as σ (1) µ = m (1) + h µ m (1) X − i m L µ b (1) , σ (2) ν = m (2) + νm (2) X − i m R ν b (2) , (D.11)where h µ (which later determines the horizontal position of the appended boxes) is closely relatedto the tree structure that describes the pole, and can be negative. In figure 19, we list all classes ofcontributing forests that describe the above poles, and we tabulate the corresponding values of h µ and m L µ in table 2. Note that poles of type-ˆ ν with ˆ ν ≥ S ∪ S is Z cl. = exp (cid:104) − πiξ (1)FI (cid:80) µ σ (1) µ − πiξ (2)FI (cid:80) ν σ (2) ν (cid:105) .Substituting in (D.11), and using the fact that ξ (1)FI b (1) = ξ (2)FI b (2) ≡ ξ b , one has Z cl. = N exp − πiξ (1)FI m (1) X ( h + h ) + 2 πξ b m L0 + m L1 + n (2) − (cid:88) ν =0 m R ν , (D.12)where N denotes some common factors shared across all families of poles. Clearly, one necessarycondition for two poles to potentially cancel is that they have equal classical contributions, and57 (2) ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 A.1 m (2) ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 A.2 m (2) ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 A.3 m (2) ... σ (2)ˆ ν ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 B.1 m (2) ... σ (2)ˆ ν ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 B.2 m (2) ... σ (2)ˆ ν ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 C.1 m (2) ... σ (2)ˆ ν ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 C.2 m (2) ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 D.1 m (2) ... σ (2)ˆ ν ... m (1) σ (1)0 σ (1)1 D.2
Figure 19:
Classes of forests that describe the extra poles. Note that we have omitted some otherclasses that are obviously not contributing due to symmetry reason. Green and red lines correspond totype I and type II equations, which are used to solve σ (1)0 , in terms of component(s) of σ (2) . Poles oftype-ˆ ν form a subclass of class A.1. The residues of poles corresponding to non-type-ˆ ν diagrams enclosedwithin a dashed rectangle cancel each other. hence equal h + h and m L0 + m L1 + (cid:80) n (2) − ν =0 m R ν .An excellent tool to pinpoint the canceling pairs of poles is again given by diagrams associatedwith the poles (D.11). These diagrams consist of boxes and anti-boxes, and it is possible thatanti-boxes survive after annihilation. The construction is a simple generalization of that in appendixD.2, and is illustrated in figure 20: step 1. and 2a. are identical. When it comes to appendingvertical boxes or anti-boxes corresponding to σ (1) µ , one should, generalizing 2b., append to the h µ -thcolumn. Now that h µ can be negative, these vertical segments of boxes can sit to the right of Y (cid:50) ,and can have annihilation with the horizontal segments of boxes corresponding to σ (2) ν . Figure 21demonstrates a few examples of such diagrams, constructed from several poles.It can be shown that if two poles contribute opposite residues, then their corresponding diagrams(after annihilation) must be the same. Moreover, given a pole not of type-ˆ ν with associated diagram,58 m L0 h m L1 A.1 − m Rˆ ν − (ˆ ν + 1) 1 − m Rˆ ν − (ˆ ν + 1 − ∆ m L1 ) , ∀ ∆ m L10 ∈ N A.2 − m Rˆ ν − (ˆ ν + 1) − m Rˆ ν − − ˆ ν A.3 − m Rˆ ν − − ˆ ν − m Rˆ ν − (ˆ ν − ∆ m L10 ) , ∀ ∆ m L10 ∈ N B.1 (ˆ ν (cid:54) = ˆ ν ) − m Rˆ ν − (ˆ ν + 1) − m Rˆ ν − (ˆ ν + 1)B.2 (ˆ ν (cid:54) = ˆ ν ) − m Rˆ ν − − ˆ ν − m Rˆ ν − − ˆ ν C.1 (ˆ ν (cid:54) = ˆ ν ) − m Rˆ ν − − ˆ ν − m Rˆ ν − (ˆ ν + 1)C.2 (ˆ ν (cid:54) = ˆ ν ) − m Rˆ ν − (ˆ ν + 1) − m Rˆ ν − − ˆ ν D.1 − m Rˆ ν − (ˆ ν + 1) 0 ∀ m L10 ∈ N D.2 − m Rˆ ν − − ˆ ν ∀ m L1 ∈ N Table 2:
Values of of h µ and m L µ in the pole equation (D.11) corresponding to the tree-diagrams infigure 19. + + + ++ + + ++ + + + +++ −− + + + + m (2) ν column 3 2 1 0 − − · · · h µ m (1) µ > · · · h µ m (1) µ < Figure 20:
Construction of a general diagram associated with the poles (D.11). + + + ++ ++ ++ + −−− + ++ ++ + −− A.1: n (1) = n (2) =2 , ˆ ν =1 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+0 , m (2)1 =+3 A.3: n (1) = n (2) =2 , ˆ ν =1 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+0 , m (2)1 =+2 + + + + ++ ++ ++ + −−−− + ++ ++ + −−− A.1: n (1) = n (2) =2 , ˆ ν =1 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+0 , m (2)1 =+3 A.2: n (1) = n (2) =2 , ˆ ν =1 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+0 , m (2)1 =+2 + + + + ++ + ++ ++ + −−− + ++ ++ + − B.1: n (1) = n (2) =2 , ˆ ν =1 , ˆ ν =0 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+1 , m (2)1 =+3 B.2: n (1) = n (2) =2 , ˆ ν =1 , ˆ ν =0 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+0 , m (2)1 =+2 + + + + ++ + + + ++ ++ + −−− + ++ ++ ++ + − A.1: n (1) = n (2) =2 , ˆ ν =1 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+3 , m (2)1 =+3 A.3: n (1) = n (2) =2 , ˆ ν =0 h = − , h = − m (1)0 = − , m (1)1 = − m (2)0 =+2 , m (2)1 =+2 Figure 21:
Some examples of diagrams constructed from the indicated poles. The double arrowsindicate that the residues from the related poles, which generate the same diagrams, are opposite. one can always find another pole within the same class (A,B,C, or D) with the same diagram; hencethey cancel . See figure 21 for some examples. In all these examples, the pairs of poles have indeedequal h + h and m L0 + m L1 + (cid:80) n (2) − ν =0 m R ν . E Poles and Young diagrams in 2d
In this appendix we study the poles and their residues of the matrix model computing thepartition function of intersecting surface defects supported on S ∪ S ⊂ S b . Throughout theappendix we will use (sub-)superscripts L, R for quantities on S , and N, S for quantities associatedto the north- or south-pole contributions. The main idea is very similar to the discussion in appendixD, but slightly more involved, due to the fact that the intersection between S and S have twoconnected components, namely the north and south poles. We will need to bring the contributionsfrom both poles together to reproduce the square of the instanton partition function. E.1 Four types of poles
Recall that for a theory T of N free hypermultiplets in the presence of intersecting defects with U ( n L ) SQCDA on S and U ( n R ) SQCDA on S respectively, the partition function Z ( T ,S ∪ S ⊂ S b ) Note that poles of type-ˆ ν with ˆ ν ≥ Z ( T ,S ∪ S ⊂ S b ) ( σ L , σ R ) = Z ( T ,S b )1-loop n L ! n R ! (cid:88) B L ,B R Z S ( σ L , B L ) (cid:89) ± Z ± intersection ( σ L , B L , σ R , B R ) Z S ( σ R , B R ) , (E.1)where the intersection factor is defined in (3.47).The combined meromorphic integrand (E.1) has many poles. Recall that m R X = ib − . It is easyto check that all poles take the form iσ L Aν + B L Aν im L A + h L ν im L X + m L Aν iσ L Aν − B L Aν im L A + h L ν im L X + n L Aν iσ R Aν + B R Aν im R A + h R µ im R X + m R Aν iσ R Aν − B R Aν im R A + h R µ im R X + n R Aν . (E.2)First of all, we define type-old poles by simply taking the (union of) poles of Z S and Z S discussed in appendix B.1. Additionally, we introduce three special classes of poles, which we refer toas type-N +ˆ ν , S +ˆ ν and NS +ˆ ν poles. Their definition goes as follows. We start by selecting partitions (cid:126)n L , (cid:126)n R of the ranks n L , n R : this corresponds to choosing a Higgs branch vacuum of the SQCDA theoryliving on S and S respectively. Next we select a set of integers { ˆ ν N/S A , A = 1 , . . . , N } , where eachˆ ν N/S A ∈ {− , , . . . , n R A − } and (cid:80) NA =1 ˆ ν N/S A > − N . In the end we will sum over all such partitions (cid:126)n L , (cid:126)n R and sets { ˆ ν N/S A } to obtain all relevant poles. Then the three special types of poles are givenby the abstract equations (E.2) with h R ν = ν , h L µ = µ , together with the following conditions: • Poles of type-N +ˆ ν N : m R A ( n R A − (cid:62) ... (cid:62) m R A (ˆ ν N A +1) (cid:62) m R A ˆ ν N A = m R A (ˆ ν N A − = ... = m R A = 0 , n R Aν (cid:62) m L A ( n L A − (cid:62) ... (cid:62) m L A (cid:62) m L A , n L Aµ (cid:62) m L A = − (ˆ ν N A + 1) if ˆ ν N A ≥ , or m L A ≥ ν N A = − . (E.3) • Poles of type-S +ˆ ν S : n R A ( n R A − (cid:62) ... (cid:62) n R A (ˆ ν S A +1) (cid:62) n R A ˆ ν S A = n R A (ˆ ν S A − = ... = n R A = 0 , m R Aν (cid:62) n L A ( n L A − (cid:62) ... (cid:62) n L A (cid:62) n L A , m L Aµ (cid:62) n L A = − (ˆ ν S A + 1) if ˆ ν S A ≥ , or n L A ≥ ν S A = − . (E.4)61 Poles of type-NS +ˆ ν N ˆ ν S : m R A ( n R A − (cid:62) ... (cid:62) m R A (ˆ ν N A +1) (cid:62) m Rˆ ν N A = m Rˆ ν N A − = ... = m R0 = 0 , n R A ( n R A − (cid:62) ... (cid:62) n R A (ˆ ν S A +1) (cid:62) n R A ˆ ν S A = n R A (ˆ ν S − = ... = n R A = 0 , m L A ( n L A − (cid:62) ... (cid:62) m L A (cid:62) m L A , n L A ( n L A − (cid:62) ... (cid:62) n L A (cid:62) n L A , m L A = − (ˆ ν N A + 1) if ˆ ν N A ≥ , or m L A ≥ ν N A = − n L A = − (ˆ ν S A + 1) if ˆ ν S A ≥ , or n L A ≥ ν S A = − . (E.5)A few remarks are in order. Poles of type-N +ˆ ν N come from solving the equations iσ R Cν + 12 B R Cν = im R C + νim R X + m R Cν , with m R C ˆ ν N C = 0 , C = 1 , . . . , Niσ L A + 12 B L A − im L A = + m A ( ≥ b − (cid:18) iσ L A + 12 B L A (cid:19) − b (cid:18) iσ R A ˆ ν N A + 12 B Rˆ ν N A (cid:19) + b + b − iσ L A ( µ +1) + 12 B L A ( µ +1) = iσ (L) Aµ + 12 B L Aµ + im L X + ∆ m A ( µ +1) µ ( (cid:62) , µ = 0 , . . . n L A − . (E.6)If ˆ ν N A = − A , one should use the equation in the second line to obtain σ L A + B L A ( σ R A ( ν = − does not exist anyway), otherwise the equation in the third line. If ˆ ν N A = − A , onesimply recovers the poles of type-old which we define separately, and therefore we exclude such casewhen defining poles of type-N +ˆ ν N . Among the solutions, most of those with n L A < Z S . Similarly for poles of type-S − ˆ ν N . However, thereare survivors from the cancellation, which involve simultaneous solutions to the set of equations iσ R Cν + 12 B R Cν = im R C + νim R X + m R Cν , m R B ˆ ν N C = 0 iσ R Cν − B R Cν = im R C + νim R X + n R Cν , n R B ˆ ν S C = 0 b − (cid:18) iσ L A + 12 B L A (cid:19) − b (cid:18) iσ Rˆ ν N A + 12 B Rˆ ν N A (cid:19) + b + b − b − (cid:18) iσ L A − B L A (cid:19) − b (cid:18) iσ Rˆ ν S A − B Rˆ ν S A (cid:19) + b + b − . (E.7)Naively, simultaneous solutions to the last two equations seem to correspond to double poles ofthe integrand, since two separate intersection factors develop a pole. However, they are actuallysimple poles after canceling with the zeros of Z S . These poles are called type-NS +ˆ ν N ˆ ν S in the aboveclassification: they have negative m L / R0 , n L / R0 controlled by ˆ ν N/S . The presence of these delicatepoles forbids us to decouple n from the discussion of m as we did in the previous appendix.It is clear that one can construct all pairs of N -tuples ( (cid:126)Y N , (cid:126)Y S ) from the four types of poles.62he construction is essentially the same as outlined in appendix D.2, where m L/R will now takecare of (cid:126)Y N , and n L/R will take care of (cid:126)Y S . More precisely, one has the correspondencetype-old type-N +ˆ ν N type-S +ˆ ν S type-NS +ˆ ν N ˆ ν S (large, large) (small, large) (large, small) (small, small) Y N An L A , Y S An L A ≥ Y N An L A = n R A − ˆ ν N A − Y S An L A = n R A − ˆ ν S A − Y N An L A = n R A − ˆ ν N A − Y S An L A ≥ Y N An L A ≥ Y S An L A = n R A − ˆ ν S A − N -tuples of Young diagrams.Again, the residues of the four types of poles sum up to the modulus squared | Z inst | of the instantonpartition function, evaluated at the specific value of its gauge equivariant parameter, which appearsin the full S b partition function. E.2 Extra poles and diagrams
There are many extra poles in the integrand E.1 selected by the JK prescription, besides thefour types of poles discussed above. For simplicity, here we only present the cancellation in thesimplest case of n L = n f = n af = 1. The main idea is very similar to the discussion in appendix D.4and techniques to analyze more general cases can be found there as well.There are four types of extra poles selected by the JK prescription (we recycle the namingappearing in the previous subsection): h L0 m L0 n L0 h R ν m R ν n R ν type-N +ˆ ν N − m Rˆ ν N − (ˆ ν N + 1) ∈ Z ν ≥ ≥ − ˆ ν N − ( m Rˆ ν N + 1) − ˆ ν N ∈ Z ν ≥ ≥ +ˆ ν S − n Rˆ ν S ∈ Z − (ˆ ν S + 1) ν ≥ ≥ − ˆ ν S − ( n Rˆ ν S + 1) ∈ Z − ˆ ν S ν ≥ ≥ +ˆ ν N cancel those of type-N − ˆ ν N , andsimilarly between type-S +ˆ ν S and -S − ˆ ν S . Again, from the four types of poles one can construct pairsof general diagrams consisting of boxes and anti-boxes. Two poles contribute opposite residueswhen their corresponding pairs of diagrams coincide (taking into account of annihilation betweencoincident boxes/anti-boxes). 63 eferences [1] K. G. Wilson, “Confinement of Quarks,” Phys. Rev.
D10 (1974) 2445–2459. [,45(1974)].[2] G. ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,”
Nucl. Phys.
B138 (1978) 1–25.[3] S. Gukov and A. Kapustin, “Topological Quantum Field Theory, Nonlocal Operators, andGapped Phases of Gauge Theories,” arXiv:1307.4793 [hep-th] .[4] S. Gukov, “Surface Operators,” in
New Dualities of Supersymmetric Gauge Theories ,J. Teschner, ed., pp. 223–259. 2016. arXiv:1412.7127 [hep-th] .[5] S. Gukov and E. Witten, “Gauge Theory, Ramification, And The Geometric LanglandsProgram,” arXiv:hep-th/0612073 [hep-th] .[6] D. Gaiotto, L. Rastelli, and S. S. Razamat, “Bootstrapping the superconformal index withsurface defects,”
JHEP (2013) 022, arXiv:1207.3577 [hep-th] .[7] D. Gaiotto and H.-C. Kim, “Surface defects and instanton partition functions,” JHEP (2016) 012, arXiv:1412.2781 [hep-th] .[8] F. Benini and S. Cremonesi, “Partition Functions of N = (2 ,
2) Gauge Theories on S andVortices,” Commun. Math. Phys. (2015) no. 3, 1483–1527, arXiv:1206.2356 [hep-th] .[9] N. Doroud, J. Gomis, B. Le Floch, and S. Lee, “Exact Results in D=2 Supersymmetric GaugeTheories,”
JHEP (2013) 093, arXiv:1206.2606 [hep-th] .[10] M. Fujitsuka, M. Honda, and Y. Yoshida, “Higgs branch localization of 3d N = 2 theories,” PTEP (2014) no. 12, 123B02, arXiv:1312.3627 [hep-th] .[11] F. Benini and W. Peelaers, “Higgs branch localization in three dimensions,”
JHEP (2014)030, arXiv:1312.6078 [hep-th] .[12] W. Peelaers, “Higgs branch localization of N = 1 theories on S x S ,” JHEP (2014) 060, arXiv:1403.2711 [hep-th] .[13] Y. Pan, “5d Higgs Branch Localization, Seiberg-Witten Equations and Contact Geometry,” JHEP (2015) 145, arXiv:1406.5236 [hep-th] .[14] C. Closset, S. Cremonesi, and D. S. Park, “The equivariant A-twist and gauged linear sigmamodels on the two-sphere,” JHEP (2015) 076, arXiv:1504.06308 [hep-th] .[15] H.-Y. Chen and T.-H. Tsai, “On Higgs branch localization of Seiberg–Witten theories on anellipsoid,” PTEP (2016) no. 1, 013B09, arXiv:1506.04390 [hep-th] .[16] Y. Pan and W. Peelaers, “Ellipsoid partition function from Seiberg-Witten monopoles,”
JHEP (2015) 183, arXiv:1508.07329 [hep-th] .[17] V. Pestun et al. , “Localization techniques in quantum field theories,” arXiv:1608.02952[hep-th] .[18] D. Gaiotto, “N=2 dualities,” JHEP (2012) 034, arXiv:0904.2715 [hep-th] .[19] J. Gomis and B. Le Floch, “M2-brane surface operators and gauge theory dualities in Toda,” JHEP (2016) 183, arXiv:1407.1852 [hep-th] .6420] L. F. Alday, M. Bullimore, M. Fluder, and L. Hollands, “Surface defects, the superconformalindex and q-deformed Yang-Mills,” JHEP (2013) 018, arXiv:1303.4460 [hep-th] .[21] J. Gomis, B. Le Floch, Y. Pan, and W. Peelaers, “Intersecting Surface Defects andTwo-Dimensional CFT,” arXiv:1610.03501 [hep-th] .[22] L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions fromFour-dimensional Gauge Theories,” Lett. Math. Phys. (2010) 167–197, arXiv:0906.3219[hep-th] .[23] N. Wyllard, “A(N-1) conformal Toda field theory correlation functions from conformal N = 2SU(N) quiver gauge theories,” JHEP (2009) 002, arXiv:0907.2189 [hep-th] .[24] L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, and H. Verlinde, “Loop and surfaceoperators in N=2 gauge theory and Liouville modular geometry,” JHEP (2010) 113, arXiv:0909.0945 [hep-th] .[25] J. Lamy-Poirier, “Localization of a supersymmetric gauge theory in the presence of a surfacedefect,” arXiv:1412.0530 [hep-th] .[26] E. Witten, “Monopoles and four manifolds,” Math. Res. Lett. (1994) 769–796, arXiv:hep-th/9411102 [hep-th] .[27] V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,” Commun. Math. Phys. (2012) 71–129, arXiv:0712.2824 [hep-th] .[28] N. Hama and K. Hosomichi, “Seiberg-Witten Theories on Ellipsoids,”
JHEP (2012) 033, arXiv:1206.6359 [hep-th] . [Addendum: JHEP10,051(2012)].[29] N. A. Nekrasov, “Seiberg-Witten prepotential from instanton counting,” Adv. Theor. Math.Phys. (2004) 831–864, arXiv:hep-th/0206161 [hep-th] .[30] N. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions,” arXiv:hep-th/0306238 [hep-th] .[31] K. Hosomichi, R.-K. Seong, and S. Terashima, “Supersymmetric Gauge Theories on theFive-Sphere,” Nucl. Phys.
B865 (2012) 376–396, arXiv:1203.0371 [hep-th] .[32] J. K¨all´en, J. Qiu, and M. Zabzine, “The perturbative partition function of supersymmetric 5DYang-Mills theory with matter on the five-sphere,”
JHEP (2012) 157, arXiv:1206.6008[hep-th] .[33] H.-C. Kim and S. Kim, “M5-branes from gauge theories on the 5-sphere,” JHEP (2013)144, arXiv:1206.6339 [hep-th] .[34] Y. Imamura, “Perturbative partition function for squashed S ,” arXiv:1210.6308 [hep-th] .[35] G. Lockhart and C. Vafa, “Superconformal Partition Functions and Non-perturbativeTopological Strings,” arXiv:1210.5909 [hep-th] .[36] H.-C. Kim, J. Kim, and S. Kim, “Instantons on the 5-sphere and M5-branes,” arXiv:1211.0144 [hep-th] .[37] A. Gadde and S. Gukov, “2d Index and Surface operators,” JHEP (2014) 080, arXiv:1305.0266 [hep-th] . 6538] B. Assel and J. Gomis, “Mirror Symmetry And Loop Operators,” JHEP (2015) 055, arXiv:1506.01718 [hep-th] .[39] C. Closset and H. Kim, “Comments on twisted indices in 3d supersymmetric gauge theories,” JHEP (2016) 059, arXiv:1605.06531 [hep-th] .[40] J. Qiu and M. Zabzine, “Factorization of 5D super Yang-Mills theory on Y p,q spaces,” Phys.Rev.
D89 (2014) no. 6, 065040, arXiv:1312.3475 [hep-th] .[41] J. Qiu, L. Tizzano, J. Winding, and M. Zabzine, “Gluing Nekrasov partition functions,”
Commun. Math. Phys. (2015) no. 2, 785–816, arXiv:1403.2945 [hep-th] .[42] T. Okuda and V. Pestun, “On the instantons and the hypermultiplet mass of N=2* superYang-Mills on S ,” JHEP (2012) 017, arXiv:1004.1222 [hep-th] .[43] A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in SuperconformalChern-Simons Theories with Matter,” JHEP (2010) 089, arXiv:0909.4559 [hep-th] .[44] D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z,” JHEP (2012) 159, arXiv:1012.3210 [hep-th] .[45] N. Hama, K. Hosomichi, and S. Lee, “Notes on SUSY Gauge Theories on Three-Sphere,” JHEP (2011) 127, arXiv:1012.3512 [hep-th] .[46] N. Hama, K. Hosomichi, and S. Lee, “SUSY Gauge Theories on Squashed Three-Spheres,” JHEP (2011) 014, arXiv:1102.4716 [hep-th] .[47] G. Bonelli, A. Tanzini, and J. Zhao, “The Liouville side of the Vortex,” JHEP (2011) 096, arXiv:1107.2787 [hep-th] .[48] G. Bonelli, A. Tanzini, and J. Zhao, “Vertices, Vortices and Interacting Surface Operators,” JHEP (2012) 178, arXiv:1102.0184 [hep-th] .[49] F. Nieri, S. Pasquetti, F. Passerini, and A. Torrielli, “5D partition functions, q-Virasorosystems and integrable spin-chains,” JHEP (2014) 040, arXiv:1312.1294 [hep-th] .[50] M. Aganagic, N. Haouzi, C. Kozcaz, and S. Shakirov, “Gauge/Liouville Triality,” arXiv:1309.1687 [hep-th] .[51] L. C. Jeffrey and F. C. Kirwan, “Localization for nonabelian group actions,” in eprintarXiv:alg-geom/9307001 . July, 1993.[52] M. R. Douglas, “Branes within branes,” in Strings, branes and dualities. Proceedings, NATOAdvanced Study Institute, Cargese, France, May 26-June 14, 1997 , pp. 267–275. 1995. arXiv:hep-th/9512077 [hep-th] .[53] M. R. Douglas, “Gauge fields and D-branes,”
J. Geom. Phys. (1998) 255–262, arXiv:hep-th/9604198 [hep-th] .[54] K. Hori, H. Kim, and P. Yi, “Witten Index and Wall Crossing,” JHEP (2015) 124, arXiv:1407.2567 [hep-th] .[55] C. Hwang, J. Kim, S. Kim, and J. Park, “General instanton counting and 5d SCFT,” JHEP (2015) 063, arXiv:1406.6793 [hep-th] . [Addendum: JHEP04,094(2016)].[56] C. Hwang and J. Park, “Factorization of the 3d superconformal index with an adjoint matter,” JHEP (2015) 028, arXiv:1506.03951 [hep-th]arXiv:1506.03951 [hep-th]