Intersecting surface defects and instanton partition functions

Journal of High Energy Physics, Jul 2017

We analyze intersecting surface defects inserted in interacting four-dimensional \( \mathcal{N}=2 \) supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.

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Intersecting surface defects and instanton partition functions

JHE Intersecting surface defects and instanton partition Yiwen Pan 0 1 3 Wolfger Peelaers 0 1 2 Instantons, Supersymmetric Gauge Theory 0 Piscataway , NJ 08854 , U.S.A 1 Box 516 , SE-75120 Uppsala , Sweden 2 New High Energy Theory Center, Rutgers University 3 Department of Physics and Astronomy, Uppsala University We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 supersymmetric quantum eld theories. We employ the realization of a class of such systems as the infrared xed points of renormalization group triggered by perturbed Seiberg-Witten monopole-like con gurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model. Extended Supersymmetry; Nonperturbative E ects; Solitons Monopoles and - HJEP07(21)3 3 Intersecting defects in theory of N 2 free hypermultiplets HJEP07(21)3 C.1 The instanton partition function C.2 Reduction to vortex partition function of SQCD instanton partition function 43 C.3 Factorization of instanton partition function for large N -tuples of Young 1 Introduction 2 Higgsing and codimension two defects The Higgsing prescription Brane realization 2.1 2.2 3.1 3.2 Intersecting surface defects on S4 b Intersecting codimension two defects on S5 !~ 4 Intersecting surface defects in interacting theories 5 Instanton partition function and intersecting surface defects 6 Discussion A Special functions A.1 Factorials A.2 Double- and triple-sine functions A.3 b functions B The S2 and Sb3 SQCDA partition function B.1 The S2 SQCDA partition function B.2 The Sb3 SQCDA partition function B.3 Forest-tree representation C Factorization of instanton partition function diagrams C.4 Factorization for small N -tuples of Young diagrams D Poles and Young diagrams in 3d D.1 Poles of type-^ D.2 Constructing Young diagrams D.3 Residues and instanton partition function D.4 Extra poles and diagrams E Poles and Young diagrams in 2d E.1 Four types of poles E.2 Extra poles and diagrams { i { Introduction Half-BPS codimension two defect operators form a rich class of observables in supersymmetric quantum eld theories. Their vacuum expectation values, as those of all defect operators, are diagnostic tools to identify the phase of the quantum eld theory [1{3]. Various quantum eld theoretic 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 de ning a prescribed singularity for the gauge elds (and additional vector multiplet scalars) along the codimension two surface, as in [5]. Second, a defect operator can be constructed by coupling a quantum eld theory supported on its worldvolume to the of a renormalization group ow from a larger theory Te triggered by a position-dependent, vortex-like Higgs branch vacuum expectation value [6, 7].1 Naturally, some defects can be constructed in multiple ways. Nevertheless, it is of importance to study all constructions separately, as their computational di culties and conceptual merits vary. Such study is helped tremendously by the fact 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 recent comprehensive review on localization techniques. The M-theory construction of four-dimensional N = 2 supersymmetric theories of class S (of type AN 1) [18] allows one to identify the class of concrete defects of interest to this paper: adding additional stacks of M2-branes ending on the main stack of N M5branes can introduce surface defects in the four-dimensional theory. The thus obtained M2-brane defects are known to be labeled by a representation R of SU(N ). In [19], the two-dimensional quiver gauge theory residing on the support of the defect and its coupling to the bulk four-dimensional theory were identi ed in detail. In fact, for the case of defects labeled by symmetric representations two di erent coupled systems were proposed. For the purposes of this paper, it is important to remark that one of these descriptions can alternatively be obtained from the third construction described in the previous paragraph.2 Allowing for simultaneous insertions of multiple half-BPS defects, intersecting each other along codimension four loci, while preserving one quarter of the supersymmetry, enlarges the collection of defects considerably and is very well-motivated. Indeed, in [21] it was conjectured and overwhelming evidence was found in favor of the statement that the squashed four-sphere partition function of theories of class S in the presence of intersecting M2-brane defects, wrapping two intersecting two-spheres, is the translation of the insertion of a generic degenerate vertex operator in the corresponding Liouville/Toda conformal eld theory correlator through the AGT dictionary [22, 23], extending and completing [19, 24]. 1The gauged perspective of [6] is equivalent to considering sectors with xed winding in a `Higgs branch localization' computation. See [8{16] for such computations in various dimensions. 2The fact the application of this Higgsins prescription introduces M2-brane defects labeled by symmetric representations was understood in the original paper [6], see for example also [20]. { 1 { Note that such defects are labeled by a pair of representations (R0; R), which is precisely the de ning information of a generic degenerate vertex operator in Liouville/Toda theory.3 In [21], the insertion of intersecting defects was engineered by considering a coupled 4d/2d/0d system. In this description, the defect is engineered by coupling quantum eld theories supported on the respective codimension two worldvolumes as well as additional degrees of freedom residing at their intersection to each other and to the bulk quantum eld theory. The precise 4d/2d/0d coupled systems describing intersecting M2-brane defects were conjectured. As was also the case for a single defect, intersecting defects labeled by symmetric representations can be described by two di erent coupled systems. A localization computation, performed explicitly in [21], allows one to calculate the squashed four-sphere partition function of such system.4 Let T denote the four-dimensional theory and let L/R denote two-dimensional theories residing on the defects wrapping the two-spheres SL2 and SR2, which intersect each other at the north pole and south pole. The full partition function then takes the schematic form Z(T ;SL2[SR2 Sb4) = Z X Zp(Ter;tSb4) Zpert ( L;SL2) Zp(erRt;SR2) Zi+ntersection Zintersection Zi(nTst;R2L[R2R R4) 2 ; (1.1) where the factors Zp(Ter;tM) denote the product of the classical action and one-loop determinant of the theory T placed on the manifold M (in their Coulomb branch localized form). Furthermore, Zintersection are the one-loop determinants of the degrees of freedom at the two intersection points respectively, and jZi(nTst;R2L[RR R4)j2 are two copies of the instanton 2 partition function, one for the north pole and one for the south pole, describing instantons in the presence of the intersecting surface defects spanning the local coordinate planes R2L [ R2R in R4. In [21] the focus was on the already very rich dynamics of 4d/2d/0d systems without four-dimensional gauge elds, thus avoiding the intricacies of the instanton partition functions. In this paper we aim at considering intersecting defects in interacting four-dimensional eld theories and addressing the problem of instanton counting in the presence of such defects.5 Our approach will be, alternative to that in [21], to construct theories T in the presence of intersecting M2-brane defects labeled by symmetric representations using the aforementioned third strategy, i.e., by considering a renormalization group ow from a larger theory Te triggered by a position-dependent vacuum expectation value with an intersecting vortex-like pro le.6 When the theory Te is a Lagrangian theory on S4, this Higgsing b prescription o ers a straightforward computational tool to calculate the partition function Z(T ;SL2[SR2 Sb4) of T in the presence of said intersecting defects. In more detail, it instructs one to consider the residue of a certain pole of the partition function Z(T~ ;Sb4), which can be 3A generic degenerate momentum reads = b R b 1 R0 , in terms of the highest weight vectors 4See also [25] for a localization computation in the presence of a single defect. R ; R0 of irreducible representations R and R0 respectively, and b parametrizes the Virasoro central charge. 5By taking one of the intersecting defects to be trivial, one can always simplify our results to the case 6To be more precise, the con guration that triggers the renormalization group ow is a solution to the (perturbed) Seiberg-Witten monopole equations [26], see [16]. { 2 { calculated by considering pinching poles of the integrand of the matrix integral computing Z(Te;Sb4). The result involves intricate sums over a restricted set of 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 equivariant parameters and sums over magnetic uxes of the partition functions of the two-dimensional theories L/R. This step heavily relies on factorization properties of the summand of instanton partition functions, which we derive in appendix C, when evaluated at special values of their gauge equivariant parameter. More importantly, we obtain concrete expressions for the instanton partition function, computing the equivariant volume of the instanton moduli space in the presence of intersecting codimension two singularities, and their corresponding ADHM matrix model. dimensional N The main result of the paper, thus obtained, is the Sb4-partition function of a four= 2 SU(N ) gauge theory with N fundamental and N antifundamental hypermultiplets,7 i.e., SQCD, in the presence of intersecting M2-brane surface defects, labeled by nR and nL-fold symmetric representations respectively. It takes the form (1.1) and can be found explicitly in (4.13). To be more precise, the coupled system we obtain involves chiral multiplets as zero-dimensional degrees of freedom, i.e., it coincides with the one described in conjecture 4 of [21] with four-dimensional N = 2 SQCD. The left sub gure in gure 1 depicts the 4d/2d/0d coupled system under consideration. We derive the instanton partition function Zi(nTst;R2L[R2R R4) in the presence of intersecting planar surface defects and nd it to take the form Zi(nTst;R2L[R2R R4) = 4 4 2 2 X qjY~ j zvRect(Y~ ) zaRfund(Y~ ) zfRu4nd(Y~ ) zdReLfect(Y~ ) zdReRfect(Y~ ) ; (1.2) ~ Y where we omitted all gauge and avor equivariant parameters. It is expressed as the usual which can be found explicitly in (4.17), capturing the contributions to the instanton counting of the additional zero-modes in the presence of intersecting surface defects, in addition to the standard factors zvRe4ct, zfRund and zaRf4und describing the contributions from the vector 4 multiplet and N + N hypermultiplets. The coe cient of qk of the above result can be derived from the ADHM model for k-instantons depicted in the right sub gure of gure 1. We have con rmed this ADHM model by analyzing the brane construction of said instantons, see section 5 for all the details. In section 6 we present conjectural generalizations of the instanton counting in the case of generic intersecting M2-brane defects. The paper is organized as follows. We start in section 2 by brie y recalling the Higgsing prescription 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 section 3 we implement the prescription for the case where T is a fouror ve-dimensional theory of N 2 free hypermultiplets placed on a squashed sphere. The 7While there is no distinction between a fundamental and antifundamental hypermultiplet, it is a useful node of each link the fundamental one. 0d chiral 2d chiral 4d hyper ADHM quiver nL N 0d Fermi 0d chiral N nR tial couplings are turned on, in direct analogy to the ones given in detail in [21]. On the right, the ADHM model for k-instantons of the left theory is shown. The model preserves the dimensional reduction to zero dimensions 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 of chiral multiplets is turned on for the adjoint Fermi multiplet. The avor charges carried by the various multiplets are also compatible with a quadratic J- or E-type superpotential for the Fermi multiplets charged under U(nL/R).8 vacuum expectation value in T of intersecting M2-brane defects on the sphere has been computed in [21] from the point of view of the 4d/2d/0d or 5d/3d/1d coupled system and takes the form (1.1) (without the instanton contributions). For the case of symmetric representations, we reproduce this expression directly, and provide a derivation of a few details that were not addressed in [21]. We notice that the superpotential constraints of the coupled system on the parameters appearing in the partition function are reproduced e ortlessly in the Higgsing computation thanks to the fact that they have a common origin in the theory Te , which in this case is SQCD. These relatively simple examples allow us to show in some detail the interplay of the various ingredients of the Higgsed partition function of theory Te , and how to cast it in the form (1.1). In section 4 we turn our attention to inserting defects in four-dimensional N = 2 SQCD. We apply the Higgsing prescription to an SU(N ) SU(N ) gauge theory with bifundamental hypermultiplets and for each gauge group an additional N fundamental hypermultiplets, and cast the resulting partition function in the form (1.1). As a result we obtain a sharp prediction for the instanton partition function in the presence of intersecting surface defects. This expression provides concrete support for the ADHM matrix model that we obtain in section 5 from a brane construction. We present our conclusions and some future directions in section 6. Five appendices contain various technical details and computations. 2 Higgsing and codimension two defects In this section we start by brie y recalling the Higgsing prescription to compute the partition function of a theory T in the presence of (intersecting) defects placed on the squashed 8The partition function is insensitive to the presence of superpotential couplings. { 4 { four/ ve-sphere [6, 7]. We also consider the brane realization of this prescription, which provides a natural bridge to the description of intersecting surface defects in terms of a 4d/2d/0d (or 5d/3d/1d) coupled system as in [21]. factor, and consider the theory of N 2 free hypermultiplets, which has avor symmetry HJEP07(21)3 SU(N ) SU(N ) U(1). By gauging the diagonal subgroup of the SU(N ) avor symmetry factor of the former theory with one of the SU(N ) factors of the latter theory, we obtain a new theory Te . As compared to T , the theory Te has an extra U( 1 ) factor in its avor symmetry group. We denote the corresponding mass parameter as M . The theory Te can be placed on the squashed four-sphere Sb4,10 and its partition function can be computed using localization techniques [27, 28]. Let us denote the supercharge used to localize the theory as Q. Its square is given by bra. The coe cients MJ are mass parameters rescaled by p``~, where ` and `~ are two radii of the squashed sphere (see footnote 10), to make them dimensionless. Localization techniques simplify the computation of the Sb4 partition function to the calculation of oneloop determinants of quadratic uctuations around the localization locus given by arbitrary constant values for Te , the imaginary part of the vector multiplet scalar of the total gauge group.11 The nal result for the Sb4 partition function of the theory Te is then 9The localization computations we will employ throughout this paper rely on a Lagrangian description, but the Higgsing prescription is applicable outside the realm of Lagrangian theories. We will restrict attention to (Lagrangian) four-dimensional N = 2 supersymmetric quantum eld theories of class S and 10We consider S b4 de ned through the embedding equation in ve-dimensional Euclidean space R5 = r2 + jz`12j2 + jz2j2 = 1 ; x2 `~2 The isometries of S b4 are given by U( 1 )R The xed locus of U( 1 )R is a squashed two-spheres: SR2 = S in terms of parameters r; `; `~ with dimension of length. The squashing parameter b is de ned as b2 = ``~ . U( 1 )L, which act by rotating the z1 and z2 plane respectively. b z1=0 and, similarly, the xed locus of U( 1 )L 4 with coordinates z1 = z2 = 0 and x0 = r. is SL2 = S b z2=0. The two-spheres SR2 and SL2 intersect at their north pole and south pole, i.e., the points 4 11More precisely, this is the \Coulomb branch localization" locus. Alternatively, one can perform a \Higgs branch localization" computation, see [15, 16]. { 5 { where Zc(lTe;Sb4) denotes the classical action evaluated on the localization locus, Z1(T-elo;Sopb4) is the one-loop determinant and jZi(nTest;R4)(q; ; M )j2 are two copies of the Nekrasov instanton partition function [29, 30], capturing the contribution to the localized path integral of instantons residing at the north and south pole of Sb4. In [6, 7], it was argued, by considering the physics at the infrared xed point of the renormalization group ow triggered by a position dependent Higgs branch vacuum expectation value for the baryon constructed out of the hypermultiplet scalars, which carries charges ML = Z(Te;Sb4)(M ) necessarily has a pole when nL; MR = nR; R = N=2 and F = N , that the partition function iM = Moreover, the residue of the pole precisely captures the partition function of the theory T in the presence of M2-brane surface defects labeled by nR-fold and nL-fold symmetric representations respectively up to the left-over contribution of the hypermultiplet that captures the uctuations around the Higgs branch vacuum. These defects wrap two intersecting two-spheres SR2/L, the xed loci of U( 1 )R/L. The pole at (2.3) of Z(Te;Sb4)(M ) nds its origin in the matrix integral (2.2) because of poles of the integrand pinching the integration contour. To see this, let us separate out the SU(N ) gauge group that gauges the free hypermultiplet to T , and split Te accordingly: Te = ( T ; ), where T is the vector multiplet scalar of the full gauge group of theory T , and the SU(N ) vector multiplet scalar. We can then rewrite (2.2) as while the second factor is the contribution of the N 2 extra hypermultiplets, organized into N SU(N ) fundamental hypermultipets.12 parameters associated to the SU(N ) avor symmetry (with P the -integral has poles (among many others) located at Here MI ; I = 1; : : : ; N denote the mass I MI = 0). The integrand of i A = iM (A) + iM nAb 1 R nLAb of (2.3), they pinch the integration contour if n R = N X nRA ; A=1 nL = N X nLA ; A=1 (2.6) N Y N Y A=1 I=1 { 6 { 12See appendix A for the de nition and some useful properties of the various special functions that are used throughout the paper. (2.3) 1 independent SU(N ) integration variables. Note that the residue of the pole of Z(Te;Sb4) at (2.3) is equal to the sum over all partitions of nR; nL in (2.6) of -integrand of Z(Te;Sb4) at the pole position (2.5) when treating the A the residue of the as N independent variables.13 Te can be put on the squashed A similar analysis can be performed for ve-dimensional N = 1 theories. The theory ve-sphere S5 ,14 and its partition function can again be computed using localization techniques [31{36]. The localizing supercharge Q squares to 3 X n =1 ( ) A ! !1 +!2 +!3 2 with n ( ) A 0 ; A = 1; : : : ; N; (2.11) where M( ) are the generators of the U( 1 )( 1 ) U( 1 )(2) U(1)(3) isometry of the squashed ve-sphere S!~5 (see footnote 14). The localization locus consists of arbitrary constant values for the vector multiplet scalar Te , hence the partition function reads Z(Te;S!~5)(M ) = Z d Te Zc(lTe;S!~5)( Te ) Z1(T-elo;Sop!~5)( Te ; M ) jZi(nTest;R4 S1)(q; Te ; M !)j3 : (2.9) One can argue that Z(Te;S!~5)(M ) has a pole at iM = !1 + !2 + !3 + 2 3 X ! i=1 n( ) N ; (2.10) whose residue computes the S!~5 partition function of T in the presence of codimension two defects labeled by n( )-fold symmetric representations and wrapping the three-spheres S(3 ) obtained as the xed loci of the U( 1 )( ) isometries (see footnote 14), respectively. These three-spheres intersect each other in pairs along a circle. Again, this pole arises from pinching the integration contour by poles of the one-loop determinant of the N 2 hypermultiplets located at i A = iM (A) +iM 13Upon gauging the additional U( 1 ) avor symmetry and turning on a Fayet-Iliopoulos parameter, which coincides with 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. 14The squashed ve-sphere S!~=(!1;!2;!3) is given by the locus in C3 satisfying 5 !12jz1j2 + !22jz2j2 + !32jz3j2 = 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. The xed locus of U( 1 )( ) is the squashed three-sphere S ( ) = S!~ z =0, while the xed locus of U( 1 )( ) 3 5 U( 1 )( 6= ) is the circle S(1 \ ) = S!~ z =z =0. The notation indicates that it appears as the intersection of 5 the three-spheres S ( ) and S(3 ). A convenient visualization of the ve-sphere and its xed loci under one 3 or two of the U( 1 ) isometries is as a T 3- bration over a solid triangle, where on the edges one of the cycles shrinks and at the corners two cycles shrink simultanously. if n( ) = PN A=1 n(A ). The residue of Z(Te;S!~5)(M ) at the pole given in (2.10) equals the sum over partitions of the integers n( ) of the residue of the integrand at the pole position (2.11) with the A treated as independent variables.15 T described by the linear quiver and corresponding type IIA brane con guration16 one may look at its brane realization [7]. Consider a four-dimensional N = 2 gauge theory N N · · · N N ←→ N D4 · · · NS5 NS5 NS5 NS5 Gauging in a theory of N 2 hypermultiplets amounts to adding an additional NS5-brane on the right end of the brane array. The Higgsing prescription of the previous subsection is then trivially implemented by pulling away this additional NS5-brane (in the 10-direction of footnote 16), while suspending nR D2R and nL D2L-branes between the displaced NS5brane and the right stack of D4-branes, see gure 2. Various observations should be made. First of all, the brane picture in gure 2 was also considered in [21] to describe intersecting M2-brane surface defects labeled by nR and nL-fold symmetric representations respectively. Its eld theory realization is described by a coupled 4d/2d/0d system, described by the quiver in gure 3 (see [21]). Note that the two-dimensional theories, residing on the D2R and D2L-branes, are in their Higgs phase, with equal Fayet-Iliopoulos parameter FI proportional to the distance (in the 7-direction) between the displaced NS5-brane and the next right-most NS5-brane. Before Higgsing, this distance was proportional to the inverse square of the gauge coupling of the extra SU(N ) gauge node: FI = 4 2 gYM : (2.12) In particular, the Higgsing prescription will produce gauge theory results in the regime where FI is positive, and where the defect is inserted at the right-most end of the quiver. In this paper we will restrict attention to this regime. Note however that sliding the displaced NS5-brane along the brane array in gure 2 implements hopping dualities [19, 37] 15In [13], these residues were interpreted as the contribution to the partition function of K-theoretic Seiberg-Witten monopoles. 16The branes in this dimensions: gure as well as those in gure 2 and the following discussion span the following NS5 D4 D2L D2R D0 1 | | | 2 | | | 3 | | | 4 | | | 5 | 6 | 7 | | N D 4 N D 4 nL D2L nR D2R NS5 NS5 NS5 nL D2L nR D2R NS5 NS5 NS5 hypermultiplets amounts to adding an additional NS5-brane on the right end of the brane array. This leads to the gure on the left. Higgsing the system as in subsection 2.1 corresponds to pulling away this NS5-brane from the main stack, while stretching nR D2R and nL D2L-branes in between it and the D4-branes, producing the middle gure. The nal gure represents the system in the Coulomb phase. N N · · · N N 4d nL 0d nR linear quiver gauge theory. The two-dimensional degrees of freedom, depicted in N = (2; 2) quiver notation, 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, denoted using two-dimensional N = (0; 2) quiver notations dimensionally reduced to zero dimensions, with solid lines representing chiral multiplets, participate in E- and J-type superpotentials. (see also [38, 39]), which in the quiver gauge theory description of gure 3 translate to coupling the defect world volume theory to a di erent pair of neighboring nodes of the four-dimensional quiver, while not changing the resulting partition function. In [21], a rst-principles localization computation was performed to calculate the partition function of the coupled 4d/2d/0d system when placed on a squashed four-sphere, with the defects wrapping two intersecting two-spheres SR2/L, the xed loci of U( 1 )R/L, in the case of non-interacting four-dimensional theories. Our aim in the next section will be to reproduce these results from the Higgsing point of view. When the four-dimensional theory contains gauge elds, the localization computation needs as input the Nekrasov instanton partition function in the presence of intersecting planar surface defects, which modify non-trivially the ADHM data. The Higgsing prescription does not require such input, and in section 4 we will apply it to N = 2 SQCD. This computation will allow us to extract the modi ed ADHM integral. { 9 { S(31) b(−11) S(33) S(32) b(−31) b(−21) radii edge represents a three-sphere invariant point-wise under one of the U( 1 ) isometries, and each vertex represents an S1, where two S3's intersect, invariant point-wise under two U( 1 ) isometries. Each S1 has two tubular neighborhoods of the form S1 R2 in the two intersecting S3's, with omega-deformation parameters given in terms of b( 1 ), as shown in the gure. The brane realization of gure 2 already provides compelling hints about how the ADHM data should be modi ed. In this setup, instantons are described by D0-branes stretching between the NS5-branes. Their worldvolume theory is enriched by massless modes (in the Coulomb phase, i.e., when FI = 0), if any, arising from open strings stretching between the D0-branes and the D2R and D2L-branes. These give rise to the dimensional reduction of a two-dimensional N = (2; 2) chiral multiplet to zero dimensions, or equivalently, the dimensional reduction of a two-dimensional N = (0; 2) chiral multiplet and Fermi multiplet. We will provide more details about the instanton counting in the presence of defects in section 5. Our Higgsing computation of section 4 will provide an independent veri cation of these arguments. 3 Intersecting defects in theory of N 2 free hypermultiplets In this section we work out in some detail the Higgsing computation for the case where T is a theory of free hypermultiplets. We will nd 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 Je reyKirwan-like residue prescription used to evaluate the partition function of the coupled 4d/2d/0d (or 5d/3d/1d) system, and of the avor charges of the degrees of freedom living on the intersection. In the next section we will consider the case of interacting theories T . 3.1 Intersecting codimension two defects on S5 !~ As a rst application of the Higgsing prescription of the previous section, we consider intersecting codimension two defects wrapping two of the three-spheres S(3 ) the partition function of a theory of N 2 free hypermultiplets on S!~5 in the presence of xed by the U( 1 )( ) isometry (see footnote 14, and also gure 4), say S(31) and S(32). Our aim will be to cast the result in the manifest form of the partition function of a 5d/3d/1d coupled system, as in [21]. We consider this case rst since the fact that the intersection S(31) \ S(32) = S(11\2) has a single connected component is a simplifying feature that will be absent in the example of Sb4 in the next subsection. Our starting point, the theory Te , is described by the quiver The explicit expression for the classical action is given by d Zc(lTe;S!~5)( )Z1(T-elo;Sop!~5)( ; M; M~ )jZinst (Te;R4 S1)(q; ; M ; M~ )j3 : Zc(lTe;S!~5)( ) = exp while the one-loop determinant Z1(T-elo;Sop!~5) is the product of the one-loop determinants of the SU(N ) vector multiplet, the N fundamental hypermultiplets and the N antifundamental hypermultiplets: Z1(T-elo;Sop!~5)( ; M; M~ ) = Zve!~ct( ) Zfu!~nd( ; M ) Zaf!~und( ; M~ ) S5 S5 S5 QAN;B=1 S3(i( A A6=B B) j !~) = QN A=1QIN=1 S3(i( A MI )+j!~j=2 j !~) QN QN A=1 J=1 S3(i( A +M~ J )+j!~j=2 j !~) written in terms of the triple sine function. Here we used the notation j!~j = !1 + !2 + !3. Note that we did not explicitly separate the masses for the SU(N ) U( 1 ) avor symmetry, but instead considered U(N ) masses. Finally, there are three copies of the K-theoretic instanton partition function, capturing contributions of instantons residing at the circles kept xed by two out of three U( 1 ) isometries. Concretely, one has (3.1) (3.2) (3.3) ; (3.4) ; (3.5) (3.6) jZi(nTest;R4 S1)(q; ; M !; M~ !)j3 Zi(nTest;R4 S(11\3)) Zi(nTest;R4 S(11\2)) q1; !1 ; . Each factor can be written as a sum over an N -tuple of Young where q = exp diagrams [29, 30] 17Recall our terminology of footnote 7. Y~ = (Y1; Y2; : : : ; YN ) ; with YA = (YA1 YA2 : : : YAWYA YA(WYA +1) = : : : = 0) ; 2 Here we have omitted the explicit dependence on 1; 2 in all factors zR4 S1 . The instanton counting parameter q is given by q = exp boxes in the N -tuple of Young diagrams. The expression for zfund reads , and jY~ j denotes the total number of of a product over the contributions of vector and matter multiplets: Zi(nTest;R4 S1) q; 2 ; 2 M !; 2 = 4 S1 and zaRfund 4 S1 are given in (C.2){(C.3) in appendix C.18 Note that the masses that enter in (3.7) are slightly shifted (see [42]): (3.8) (3.9) As outlined in the previous section, to introduce intersecting codimension two defects wrapping the three-spheres S(31) and S(32) and labeled by the n( 1 )-fold and n(2)-fold symmetric representation respectively, we should consider the residue at the pole position (2.11) with n(3) = 0 (and hence n(3) = 0 for all A = 1; : : : ; N )19 i 2 i A = iM (A) n(A2)!2 !1 + !2 + !3 2 for A as N independent variables, and sum over all partitions ~n( 1 ) of n( 1 ) and ~n(2) of n(2). As before, (A) is a permutation of A = 1; : : : ; N which we take to be, without loss of generality, (A) = A. At this point let us introduce the notation that \!" means evaluating the residue at the pole (3.10) and removing some spurious factors. As we aim to cast the result in the form of a matrix integral describing the coupled 5d/3d/1d system, we try to factorize all contributions accordingly in pieces depending only on information of either three-sphere S(31) or S(32). As we will see, the non-factorizable pieces nicely cancel against each other, except for a factor that will ultimately describe 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 18In appendix C we have simultaneously performed manipulations of four-dimensional and vedimensional instanton partition functions, which is possible after introducing the generalized factorial with respect to a function f (x), de ned in appendix A.1, with f (x) in four and ve dimensions given in (C.1). 19Recall that we have regrouped the mass for the U( 1 ) avor symmetry and those for the SU(N ) avor symmetry into U(N ) masses. triple sine functions (see (A.8)),20 Zc(lTe;S!~5) Z1(T-elo;Sop!~5) ! Z1(T-lo;Sop!~5) Let us unpack this expression a bit. First, Z1(T-lo;Sop!~5) is the one-loop determinant of N 2 free hypermultiplets, which constitute the infrared theory T . It reads (3.11) Z1(T-lo;Sop!~5) = N Y N Y A=1 J=1 S3( iMA + iM~ J + j!~j j !~) A=1 J=1 S3(iMA 1 = N Y N Y 1 iM~ J j !~) : (3.12) Note that the masses of the N 2 free hypermultiplets, represented by a two- avor-node quiver, are MAJ = MA 2 j!~j + n( 1 ) N !1 + n(2) N MJ + i j!~2j . Recall that N1 PN ~ J=1 iM~ J = iM~ , while N1 PN A=1 iMA = !2. Second, we nd the classical action and one-loop determinant of squashed three-sphere partition functions of a three-dimensional N U(n( )) gauge theory with N fundamental and N antifundamental chiral multiplets and = 2 supersymmetric one adjoint chiral multiplet, i.e., the quiver gauge theory N N n(α) We will henceforth call this theory `SQCDA'.21 These quantities are in their Higgs branch localized form,22 hence the additional subscript indicating the Higgs branch vacuum, i.e., the partition ~n( ). Their explicit expressions can be found in appendix B.2. The FayetIliopoulos parameter F(I), the adjoint mass m(X ), and the fundamental and antifundamental 20Here we omitted on the right-hand side the left-over hypermultiplet contributions mentioned in the previous section as well as the classical action evaluated on the Higgs branch vacuum at in nity, i.e., on the position-independent Higgs branch vacuum. 21Note that the rank of the gauge group is the rank of one of the symmetric representations labeling the defects supported on the codimension two surfaces, or in other words, it can be inferred from the precise coe cients of the pole of the Te partition function, see (2.10). 22The squashed three-sphere partition function of a theory can be computed using two di erent localization schemes. The usual \Coulomb branch localization" computes it as a matrix integral of the schematic form [43{46] Z( ;Sb3) = Z d Zc(l ;Sb3)( ) Z1-loop ( ;Sb3)( ) ; while a \Higgs branch localization" computation brings it into the form [10, 11] HV Z( ;Sb3) = X ZcljHV ( ;Sb3) Z1-loopjHV Zv(o;rRte2xjHSV1)(b) Zv(o;rRte2xjHSV1)(b 1) : ( ;Sb3) Here the sum runs over all Higgs vacua HV and the subscript jHV denotes that the quantity is evaluated in the Higgs vacuum HV. Furthermore, one needs to include two copies of the K-theoretic vortex partition function ZvRo2rteSx1 . The two expressions for Z are related by closing the integration contours in the former and summing over the residues of the enclosed poles. In the main text the theory 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 B for all the details. with the ve-dimensional parameters as follows, with ( ) p!( )=(!1!2!3), masses m ( ) entering the three-dimensional partition function on S(3 ) are identi ed I ( ) FI = 8 2 ( ) ; 2 gYM ( ) ( ) J mX = i!( ) ( ) ; i m ( ) I = ( ) MI + 2 (j!~j+!( )) ; m~ = i!( ) ( ) + ( ) M~ J + 2 (j!~j+!( )) : (3.14) i Note that the relation on the U( 1 ) mass N1 PN I=1 iMI = j!~2j + nN( 1 ) !1 + nN(2) !2 translates into a relation on the U( 1 ) mass of the fundamental chiral multiplets. Finally, both the classical action and the one-loop determinant produce extra factors which cannot be factorized in terms of information depending only on ~n( 1 ) or ~n(2), Z1T~-1;~no(o1p),;e~nx(t2r)a = Zvf,extra (M ) Zafund,extra(M~ ) ; ~n( 1 );~n(2) ~n( 1 );~n(2) Zcl,extra ~n( 1 );~n(2) = (q3q3) PAN=1 n(A1)n(2) A ; (3.15) where Zafund,extra captures the non-factorizable factors from the antifundamental one-loop determinant, while Zvf,extra captures those from the vector multiplet and fundamental hypermultiplet one-loop determinants, which can be found in (C.21){(C.22). These factors will cancel against factors produced by the instanton partition functions, which we (3.13) consider next. When employing the Higgsing prescription to compute the partition function in the presence of defects, the most interesting part of the computation is the result of the analysis and massaging of the instanton partition functions (3.5) evaluated at the value (3.10) for their gauge equivariant parameter. We nd that each term in the sum over Young diagrams can be brought into an almost factorized form. As mentioned before, certain nonfactorizable factors cancel against the extra factors in (3.11), but a simple non-factorizable factor remains. When recasting the nal expression in the form of a 5d/3d/1d coupled system, it is precisely this latter factor that captures the contribution of the degrees of freedom living on the intersection S(11\2) of the three-spheres on which the defects live. Let us start by analyzing the instanton partition functions Zi(nTest;R4 S(12\3)) and Zi(nTest;R4 S(11\3)). It is clear from (3.8) that upon plugging in the gauge equivariant parameter (3.10) in Zi(nTest;R4 S(12\3)), the N -tuple of Young diagrams Y~ has zero contribution if any of the Young diagrams YA has more than n(A2) rows. Similarly, Zi(nTest;R4 S(11\3)) does not receive contributions from Y~ if any of its members YA has more than n(A1) rows. Hence the sum over Young diagrams simpli es to a sum over all possible sequences of n( ) non-decreasing integers. The summands of the instanton partition functions undergo many simpli cations at the special value for the gauge equivariant parameter, and in fact one nds that they become precisely the K-theoretic vortex partition function for SQCDA upon using the parameter identi cations (3.13) (see appendix C.2 for more details):23 23This fact has for example also been observed in [47{50], and can also be read o from the brane picture in gure 2. Before Higgsing, the instantons of the extra SU(N ) gauge node are realized by D0-branes spanning in between the NS5-branes. After Higgsing, the D0-branes can still be present if they end on the D2R and D2L-branes. If, say, n L = 0, they precisely turn into vortices of the two-dimensional theory living on the D2-branes. with the three dimensional squashing parameters de ned as p!1=!2 : (3.17) The third instanton partition function, Zi(nTest;R4 S(11\2)), behaves more intricately when substituting the gauge covariant parameter of (3.5). From (3.8) one immediately nds that N -tuples of Young diagrams Y~ have zero contribution if any of its constituting diagrams YA contain the \forbidden box" with coordinates (column,row) = (n(A1) + 1; n(A2) + 1). We split the remaining sum over N -tuples of Young diagrams into two, by de ning the notion of large N -tuples, as those N -tuples satisfying the requirement that all of its members YA contain the box with coordinates (n(A1); n (A2)), and calling all other N -tuples small. Let us focus on the former sum rst. Given a large N -tuple Y~ , we de ne Y~ L and Y~ R as the Young diagrams Here we used ZvRo2rteSx1j~n(mjb) to denote the summand of the U(n) SQCDA K-theoretic vortex (3.18) 1 : (3.20) (3.21) 1 i i 1 1 : (3.22) As announced, the extra factors in the second line of (3.20) cancel against those in (3.11). YALr = YAr YARr = YA(n(A1)+r) n(2) A mLA Furthermore, we de ne the non-decreasing sequences of integers where Y~AR denotes the transposed diagram of YAR. Figure 5 clari es these de nitions. With these de nitions in place, one can show (see appendix C.3) the following factorization of the summand of the instanton partition function for large tuples of Young diagrams Y~ for for 1 1 r r : n(A1) ; and YALr = 0 for n( 1 ) < r A mRA Y~ R ; = 0; : : : ; n(A2) mA . (See appendix B.2 for concrete expressions.) The factor 2i sinh i i 2i sinh i i h h 2 2 (MA (MA MB) + 2(mLA + ) 1(mRB + ) MB) + 2(mLA + ) 1(mRB + ) + 2 mLμ Y L . 3 2 1 0 ν = Large Y Y L Y R HJEP07(21)3 box denotes the \forbidden box" with coordinates (n( 1 ) + 1; n(2) + 1). The green and blue areas denote Y L and Y R respectively. The de nitions of mL and mR, see (3.19), are also indicated. For small diagrams, we can still de ne Y~ R as in the second line of (3.18), but Y~ L is not a proper N -tuple of Young diagrams due to the presence of negative entries. Nevertheless, we can de ne sets of non-decreasing integers as mLA mRA YA(n(A1) ) Y~ R ; n(A2) ; for 0 for 0 n( 1 ) A n(2) A 1 ; It is clear that mLA can take negative values. Then one can show (see appendix C.4) that qjY~largejZi(nTest;R4 S(11\2)) Y~small 3 ! q3 jmLj+jmRjZR 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(semi-)vortexj~n( 1 ) (mLjb( 1 )) is a somewhat complicated expression generalizing Zvo2rteSx(1j1~n\(12)) , which we present in (C.26). R Putting everything together, and noting that summing over all N -tuples of Young diagrams avoiding the forbidden box is equivalent to summing over all possible values of mLA/R, we nd the following result for the Higgsed partition function R where Z~n( 1 ) (mLjb( 1 )) = ZS(31) and similarly for Z~n(2) (mRjb(2)). The expression for Z^n1 (mLjb( 1 )) is obtained by replacing 2 S(11\2) with Z(semi-)vortexjn1 . The prime on the sums over Young diagrams in (3.25) R2 S(11\2) indicates that only N -tuples of Young diagrams avoiding the \forbidden box" are included. To obtain the nal result of the Higgsed partition function, we need to sum the right-hand side of (3.25) over all partitions ~n( 1 ) of n( 1 ) and ~n(2) of n(2). Matrix model description and 5d/3d/1d coupled system Our next goal is to write down a matrix model integral that reproduces the S!~5 -partition function of the theory T of N 2 free hypermultiplets in the presence of intersecting codimension two defects, i.e., a matrix integral that upon closing the integration contours appropriately reproduces the expression on the right-hand side of (3.25), summed over all partitions of n( 1 ) and n(2), as its sum over residues of encircled poles. A candidate matrix model is obtained relatively easily by analyzing the contribution of the large tuples of Young diagrams in (3.25). It reads HJEP07(21)3 (3.27) n(2) b=1 where ZS(31) ( ( 1 )) denotes the classical action times the one-loop determinant of the S(31) partition function 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 ZS(32) ( (2)).24 The contribution from the intersection S(11\2) reads Z(T ;S(31)[S(32) S!~5) = Z1(T-lo;Sop!~5) Z n( 1 )!n(2)! JK a=1 n( 1 ) with ab = ib(2) b (2) +ib( 1 ) a( 1 ). Note that from (3.13) we deduce that the Fayet-Iliopoulos parameters F(1I) and F(2I) are both positive. The mass and other parameters on both threespheres satisfy relations which follow from the identi cations in (3.13){(3.14). Concretely, we nd b( 1 ) F(1I) = b(2) F(2I) ; b( 1 ) m( 1 ) + I b( 1 ) m~ ( 1 ) J i i 24See appendix B.2 for concrete expressions for the integrand of the three-sphere partition function. where m ( ) and m(X ) are the fundamental, antifundamental and adjoint masses on the J respective spheres. Moreover, the di erences of the relations in (3.14), for xed , relate the three-dimensional mass parameters on S(3 ) to the ve-dimensional mass parameters of the N 2 free hypermultiplets, i.e., to MIJ = MI MJ + i j!~2j : ~ MIJ = 1 ( ) m ( ) I m~ ( ) J i! + i j!~j : 2 (3.30) The matrix integral (3.27) is evaluated using a Je rey-Kirwan-like residue prescription [51]. We have derived it explicitly by demanding that the integral (3.27) reproduces the result of the Higgsing computation (see below). The prescription is fully speci ed by the following charge assignments: the matter elds that contribute to ZS(31) ( ( 1 )) and ZS(32) ( (2)) are assigned their standard 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 Zintersection( ( 1 ); (2)) are assigned charges of the form (0; : : : ; 0; +b( 1 ); 0 : : : ; 0 ; 0; : : : ; 0; b(2); 0 : : : ; 0). Furthermore, we pick the JK-vector = ( F(1I); F(2I)), where we treat the Fayet-Iliopoulos parameters as an n( 1 )-vector and n(2)-vector respectively. Recall from (3.13) that both are positive. Before verifying that the matrix model (3.27), with the pole prescription just described, indeed faithfully reproduces the expression (3.25) summed over all partitions ~n( 1 ); ~n(2), we remark that it takes precisely the form of the partition function of the 5d/3d/1d coupled system of gure 6, which is the trivial dimensional uplift of gure 3 specialized to the case of N 2 free hypermultiplets described by a two- avor-node quiver. This statement can be veri ed by dimensionally uplifting the localization computation of [21]. In some detail, Z1(T-lo;Sop!~5) captures the contributions to the partition function of the ve-dimensional degrees of freedom, i.e., of the theory T consisting of N 2 free hypermultiplets, while ZS(3 ) encodes those of the degrees of freedom living on S(3 ), described by U(n( )) SQCDA, for = 1; 2, and the factor Zintersection precisely equals the one-loop determinant of the one-dimensional bifundamental chiral multiplets living on the intersection S(31) \ S(32) = S(11\2) the mass relations (3.30), which we nd straightforwardly from the Higgsing prescription, . Moreover, are the consequences of cubic superpotential couplings in the 5d/3d/1d coupled system, which were analyzed in detail in [21]. The mass relations among the (anti)fundamental chiral multiplet masses in (3.29) are in fact a solution of (3.30) obtained by subtracting the equation for = 1 and = 2 and subsequently performing a separation of the indices I; J . The separation constants appearing in the resulting solutions can be shifted to arbitrary values by performing a change of variables in the three-dimensional integrals, up to constant prefactors stemming from the classical actions. The Higgsing prescription also xes the classical actions and hence we nd speci c values for the separation constants. The adjoint masses in (3.29) are the consequence of a quartic superpotential. Also observe that our computation xes the avor charge of the one-dimensional chiral multiplets, which enter explicitly in Zintersection, and for which no rst-principles argument was provided in [21]. The integrand of (3.27) has poles in each of the three factors; the Je rey-Kirwan-like residue prescription is such that, among others, it picks out classes of poles, which we refer to as poles of type-^. They read, for partitions ~n( 1 ) and ~n(2) of n( 1 ) and n(2) respectively, where the classical and 1-loop factors, with mij mi mj , are given by S3 Z b clj~k S3 Z1-bloopj~k h exp 2 i FI nf Y j=1 (l; ) Y sb iQ 2 Xnf j=1 mj kj + mX Xnf (kj 2 )mX j=1 naf Y t=1 (j; ) Y sb 1)kj iQ 2 i The summand ZR2 S1 vortexj~k (mjb) of the vortex partition function is given by Y Y naf nf kj 1 Y (1 t=1 j=1 =0 Y j;l=1 nf " kj 1 kl 1 Y Y where Q = b + b 1 and the matter one-loop determinants are expressed in terms of the double-sine function sb. We have taken the Chern-Simons level to be zero. When nf > naf or nf = naf and the FI-parameter FI > 0,34 we again consider poles m;n in the lower half-plane, labeled by ascending sequences of natural numbers poles of type m;n : i = mj + mX imj b inj b 1 ; = 0; : : : ; kj 1; mj ; nj 2 N : tion [10, 11], with Higgs vacua speci ed by a partition ~k, Summing over the residues, one obtains the Higgs branch localized Sb3-partition funcm X zbjmjZR2 S1 vortexj~k (mjb) n X zbjnj1 ZR2 S1 vortexj~k njb 1 # ; (B.12) Here we used the function (x)fm Qkm=01 f (x + k) with f (x) = 2i sinh ib2x. See appendix A. Expression (B.15) is summed over all possible sequences of non-decreasing natural numbers 0 mj0 mj1 : : : mj(kj 1), with weighting factor given in terms of zb 1 e 2 FIb 1 jmj Xnf Xkj 1 j=1 =0 mj : 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 34Note that if we had turned on a Chern-Simons level, the convergence criterion would have been slightly more subtle than in two dimensions, as was explained in [11]. mlj +(kj +mj m~ t + mX : (B.14) (B.11) (B.13) (B.15) (B.16) Y =0 =0 =0 (1 ib 1mjl kj 1 (1 + ib 1mjl i( i(kl (1 + ib 1mjl i(kl f )b 1mX )mj 1 )b 1mX + ml )b 1mX + mj mj )mj f mj; 1 f ml;kl 1)ml;kl 1 : # σi0 ... σiμ ... ... σjμ ... σi1 σ.. σi0 σi2 σi0 ... mi σi1 ... gure on the left shows two branch-less trees, associated with masses mi and mj(6=i) respectively. Forests consisting of such branch-less trees will give non-zero contributions to the SQCDA partition function. The gure in the middle and on the right show trees with branches, or two trees associated to the same mass mi; a forest that contains such trees does not contribute to the partition function by symmetry arguments. case of S2 is completely similar. here already for the simple case of SQCDA [56]. We will consider the example of Sb3; the When nf naf and the FI-parameter FI > 0, the Je rey-Kirwan residue prescription, mentioned below (3.30), selects as poles the solutions to the equations a = mia imab a = b + mX i mabb inab 1 ma; na > 0 i nabb 1 mab; nab > 0; a 6= b : (B.17) where for each label a the component a appears exactly once on the left-hand side, and ia 2 f1; : : : ; nfg. Note that (B.17) contains more poles than those described by (B.11). The poles constructed by solving nc of the equations in (B.17) for the nc components a can be represented by forests of trees by drawing nodes for all components a and all masses mi and connecting the nodes associated with the rst symbol on the right-hand side of (B.17) (i.e., a component of or a mass m) to that associated with the component on the left-hand side with a line, for all nc equations that were used. Note that trees consisting of a single mass node, can be omitted from the forest. As a result, each component a is linked to a fundamental mass mia (which occurs as the root node of the tree containing the node of a), and the interrelations between components 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 over all 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, will contribute. The rest of the diagrams cancel among themselves. In the residue computation, we encountered partitions ~k of the rank nc of the gauge group. Each entry kj is precisely the length of the length of the tree (or number of descendant nodes under mass mj ) 4d : f (x) = 2x; 5d : f (x) = 2i sinh( i 2x) ; (C.1) HJEP07(21)3 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 summands of two (semi-)vortex partition functions. We can simultaneously consider the four-dimensional and ve-dimensional instanton partition function by using the notation (x)fm (see appendix A), where f (x) = f 1; 2 (x) is some odd function that might depend on the -deformation parameters. Replacing f by and R41; 2 the following results apply to the familiar instanton partition function respectively on R41; 2 S1. In appendix D, E, we will discuss the relation between the factorization results in this appendix, and the poles and residues of the matrix models that describe gauge theories in the presence of intersecting defects. C.1 The instanton partition function We start with a four-/ ve-dimensional supersymmetric quiver gauge theory with gauge group SU(N ) SU(N ),35 with N fundamental hypermultiplets, N anti-fundamental hypermultiplets and one bi-fundamental hypermultiplet, with masses MI , M~ J and M^ respectively. Let and 0 denote the Cartan-valued constant scalars of the two vector multiplets. The instanton partition function can be written as a sum over N -tuples of Young diagrams Y~ ; Y~ 0 and the individual contributions to each summand read zvect(Y~ ; ) NC Y Y 1 (i 2 1 A;B=1 r;s=1 (i 2 1 AB AB f b2(s r+1) YBs)YAr (i 2 1 f b2(s r+1) YBs)YBs (i 2 1 AB AB f b2(s r) YBs)YBs ; f b2(s r) YBs)YAr z(a)fund(Y~ ; ; ) zbifund(Y~ ; Y~ 0; ; 0; M^ ) N Y N Y A=1 I=1 r=1 N Y 1 Y 2 4 A;B=1 r;s=1 ( i 2 1( 0B ( i 2 1( 0B ( i 2 1( 0B ( i 2 1( 0B 1 Y (i 2 1( A I )+b2r+1)fYAr ; A +M^ ) b2(s r+1) YB0 s)fYB0 s A +M^ ) b2(s r+1) YB0 s)fYAr A +M^ ) b2(s r) YB0 s)YB0 s A +M^ ) b2(s r) YB0 s)fYAr 5 : (C.4) f 3 (C.2) (C.3) 35Instanton counting is typically performed for U(N ) gauge groups. We will not be careful about the distinction. In fact, removing the U( 1 ) factors is expected to just amount to some overall factor (1 q)#, as in [22]. Here AB = B and b2 1= 2. The full instanton partition function is thus36 Zinst X qjY jq0jY~ 0jzvect(Y~ ; )zvect(Y~ 0; 0)zafund(Y~ 0; 0)zbifund(Y~ ; Y~ 0; ; 0)zfund(Y~ ; ) ; ~ Y~ ;Y~ 0 (C.5) (C.6) where we omitted the mass dependence. gauge equivariant parameter, We are interested in the instanton partition function evaluated at special values for its A ! ~nL;~nR A MA + i(nLA + 1) 1 + i(nRA + 1) 2 ; for integers nL=R A denote the collection of natural numbers simply by ~nL/R A=1 nLA/R. Remarkably, when evaluated at these special values, the instanton partition function simpli es and exhibits useful factorization properties. The most signi cant simpli cation comes from the evaluation of zfund: if any Young diagram YA of the N -tuple Y~ contains a box (the \forbidden box") at position (nLA +1, nRA +1), then zfund(Y~ ; ) evaluates to zero. Hence, the sum over all Y~ is e ectively restricted to those tuples all of whose members avoid the \forbidden box".37 fnLA/Rg, and their sums as C.2 Reduction to vortex partition function of SQCD instanton partition function Let us consider the SQCD instanton partition function and look at the case where nR = 0. The forbidden boxes sit at (nLA + 1; 1), implying that each YA in a contributing tuple Y~ Let zv~nfL;~nR (Y~ ) denote the product zvect(Y~ ; ~nL;~nR ) zfund(Y~ ; ~nL;~nR ; M ). It simpli es must have width WYA nL . A in the case ~nR = ~0 to zvf ~nL;~0(Y~ ) = zvect(Y~ ; ~nL;~0) zfund(Y~ ; ~nL;~0; M ) = ( 1 )NjY~ j N Y A;B=1 2 4 1 QnLA r=1 Qsn=LB1 (1 i 2 1MAB +b2(s r+nLA f nL ) YAr +YBs)YAr YAr+1 B Qsn=LB1 (1+b2s i 2 1MAB +b2(nLA Qsn=LB1 (1+b2s i 2 1MAB +b2(nLA f nLB)+YBs YA1)YA1 5 : 3 nLB))fYBs (C.7) Multiplying in also zafund(Y~ ; ~nL;~0; M~ ), we can identify the resulting product with a summand of a two-/three-dimensional SQCDA vortex partition function. We identify the number of colors and avors as nc = nL, nf = N , and naf = N . The integer partitions are 37Such diagrams are sometimes referred to as hook Young diagrams. 36On the one hand, the simpler case of SU(N ) SQCD, which we used in section 3, can be easily extracted from this expression, by setting all YA0 to empty Young diagrams and identifying the antifundamental mass as M~A = 0A + M^ i 1 i 2. On the other hand, it can also easily be generalized to linear SU(N ) quivers. ri such that YAri −YA(ri+1)>0 ν=nRA−YA(ri+1)−1 ) ν=nRA−YAri ν=0 mRAν= ri−nLA HJEP07(21)3 mLμ Y L . right demonstrates some convenient relations between mRA and YARr. and YAR. The latter are lled in gray, while the \forbidden box" is colored red. The gure on the identi ed as fnLAg $ fkig, and nally mA an obvious way, if one also sets = YA(nLA ). Then we recover (B.8), (B.15) in 2d: 3d: f (x) f (x) 2x; 2i sinh i 2(x) b3d p 2 ; (C.8) (C.9) and identi es the masses as mA m~ J m~ J 2 1M~ A;J + mX ; 2 1=2M~ A;J +mX ; (C.10) (C.11) where MAB = MA MB and M~ AB = MA M~ B. C.3 Factorization of instanton partition function for large N -tuples of Young diagrams Given the set of natural numbers ~nL, ~nR, we have de ned in the main text the notion of large N -tuples of Young diagrams, see above equation (3.18). For such large N -tuples we introduced subdiagrams YAL and YAR in (3.18), and integers mLA and mRA in (3.19). In gure 15 we remind the reader of these de nitions. nally sequences of non-decreasing Now we are ready to state the factorization of the various factors in the (summand A of) the two-gauge-node instanton partition function of (C.5), associated to large Young diagrams, when evaluated on de ned in (C.6). Introducing the shorthand notations za~nfLu;n~ndR (Y~ ; M~ ") = zafund(Y~ ; ~nL;~nR ; M~ ), zv~nfL;~nR (Y~ ) = zvect(Y~ ; ~nL;~nR ) zfund(Y~ ; ~nL;~nR ; M ) and zb~niLfu;~nnRd(Y~ ; Y~ 0; M^ ) = zbifund(Y~ ; Y~ 0; ~nL;~nR; 0; M^ ), it is straightforward to show that za~nfLu;n~ndR(Y~ ; M~ ") = za~nfLu;n~0d(Y~ L; M~ ") zafund(Y~ R; M~ ") (Za~nfLu;n~ndR,extra(M~ )) 1 ~0;~nR zvf ~nL;~nR(Y~ ) = ( 1 )N~nL ~nR ~nL;~0(Y~ L) zvf zvf ~0;~nR(Y~ R) Zvlaf,rignetje~nrLse;~nctRion(mL; mR) (Zv~nfL,e;~xntRra) 1 Zbifund,intersection(Y~ 0) (Zb~niLfu;~nnRd,extra) 1 : The product of the latter two can be simpli ed further to zvf ~nL;~nR(Y~ ) zb~niLfu;~nnRd(Y~ ; Y~ 0; M^ ) = Zvortexj~nL(mL) Zvortexj~nR(mR) zfund(Y~ 0; 0; (M 0) ) (Zv~nfL,e;~xntRraZbifund,extra) 1Zvlaf,rignetje~nrLse;~nctRion(mL; mR)zdLefect(Y 0; mL)zdRefect(Y~ 0; mR) : (C.15) ~nL;~nR 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 0, appear. They are given by MI0 = MI as usual (M 0) = M 0 M^ + i( 1 + 2)=2, and their shifted versions are i( 1 + 2)=2. We also used the dot product ~nL ~nR PN A=1 nLAnR. A U(P nL/R) SQCDA with nf = naf = N , whose explicit expressions on R2 and R2 A Next, as in the previous appendix, Zvortexj~nL/R denotes the vortex partition function of S1 can be found in appendix B.1 and B.2. The fundamental and adjoint masses are identi ed as in (C.10){(C.11), while the antifundamental masses are given by 2d FI > 0 : 3d FI > 0 : mA mA m~ J = "2 1(MA m~ J = "2 1=2(MA M^ ) + mX M^ ) + mX : The factors labeled with `intersection' are given by Zvlaf,rignetje~nrLse;~nctRion(mL; mR) Zbifund,intersection(Y~ 0) N nLA 1 nRB 1 Y Y Y N Y WY B0 YB0 r Y Y A;b=1 =0 =0 f ( C(m) b2)f ( C(m) + 1) A;B=1 r=1 s=1 f ( i 2 1( 0B MA + M^ ) b2r ; s) ; 0 J 0 J 1 1 (C.12) (C.13) (C.14) (C.16) (C.17) (C.18) (C.19) HJEP07(21)3 with C(m) i 2 1(MA MB) + (mLA + ) b2(mRB + ). The factor zdLefect is de ned as zdefect(Y 0; mL) = L Y Y Y N nLA 1 WY B0 ( i"2 1( 0B (M A0) ) + ( + 1 + s)b2 + mLA A;B=1 =0 s=1 ( i"2 1( 0B (M A0) ) + ( + s)b2 + mLA YB0 s)fYB0 s ; YB0 s)fYB0 s (C.20) and zdRefect(Y~ 0; mR) is the same expression but with (nL; mL; 2; Y 0) $ (nR; mR; 1; Y~ 0). A;B=1 r=1 s=1 f (i"2 1(MA M~ B)+b2(r nLA 1)+(s nRA 1)) Qrn=LA0 nLB 1 Qsn=RA0 nRB 1 f ( (r; s)) f ( (r; s) b 2 1) Qrn=LB1 nLA Qsn=RA1 nRB f ( +(r; s) b2) f ( +(r; s) + 1) (C.21) (C.22) 1 1 A;B=1 r=1 s=1 f (i"2 1(MA MB) + b2(r nL A 1) + (s nR A 1)) Finally, the factors labeled by `extra' read 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 similar factorization of the summand of the instanton partition function occurs, but is more involved. A (tuple of) small Young diagram Y~ , namely YAnLA < nRA for some A, again de nes two nondecreasing sequences of integers as in (3.23). In particular, mLA can be negative: for each such that mLA > 0, mLA 1 < 0. For simplicity, we show the results for the SQCD instanton partition function. The summand of this instanton partition function evaluated at (C.6), i.e., zvf zafund ~nL;~nR, factorizes into, for small N -tuple of Young diagrams Y~ , zvf ~nL;~nR(Y~ small) za~nfLu;n~ndR(Y~ small; M~ ") = Zsemi-vortexj~nL(mL) Zvortexj~nR(mR) Zv~nfL,i;n~ntRersection(mL; mR) Zafund,extra(M~ ) Zvf,extra. ~nL;~nR ~nL;~nR 1 where the `extra' are as before, and the intersection factor reads, again with N Y A;B=1 A6=B or A=0 Y Y 1 =0 =0 f ( C + 1) N Y A(=B)=1j A>0 2 A 1 nRA 1 Y =0 = mLA f ( C + 1) = A =0 f ( C + 1) 5 1 Y Y 1 3 7 ; C = (C.25) Zvf,intersection involves a product over those lled boxes. rectangular region, enclosed by the dashed-lines, is partially lled. In general, the intersection factor N Y Y nA 1 nB 1 A>B=1 =0 =0 Y f ( i 2 1MAB +( )b2 (mA mB )) Y N nA 1 Y A=1 > =0 f (+( )b2 (mA mA )) N Y N Y N Y A;B=1 =0 =0 A;B=1 =0 =0 Y1 Y1 ( i 2 1MAB + ( Y1 Y1 (i 2 1MAB + ( (i 2 1MAB + ( (i 2 1MAB + ( A;B=1 = N Y N Y nA 1 Y A=1 B(6=A)=1 =0 (i 2 1MAB QN B=1 QA; jmA 0 (1 i"2 1(MA 1 nA 1 nB 1 Y Y ( i 2 1MAB + ( = (i 2 1MAB + ( + 1)b2 1)b2)fmB mB + 1)fmA N nA 1 Y Y b2 + 1)fmA A=1 = ( b2 + 1)fmA M~ B) + ( + 1)b2 1 f mA )mA : 1)b2 + mB )f mA + 1)b2)f mB 1)b2 + mA )f mB + 1)b2 + 1)fmA (i 2 1MAB + ( (i 2 1MAB + ( 1 )b2)fmA + 1)b2 + 1)f mB QN B=1 QA; jmA <0 (1 i"2 1(MA + M~ B) + ( + 1)b2)f mA (C.26) We remark that in Zvf,intersection, the second line is in fact a product over the boxes lled inside the nL A nRA rectangle, namely the gray boxes inside the region enclosed by the dashed lines in gure 16. Also note that when all A = 0, (C.25) turns into (C.18), and since the small Young diagram has deformed into a large Young diagram. the expression for Z(semi-)vortexj~nL in (C.26) reduces to the usual vortex partition function, D Poles and Young diagrams in 3d In this appendix we analyze the correspondence between poles in the three-dimensional Coulomb branch 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 class of poles of the matrix model, which we call poles of type-^, and the sum over the corresponding residues is precisely the instanton partition function evaluated at (C.6). All other classes of poles are spurious and their residues should cancel among themselves: we will indeed argue that this is the case by showing that they give rise to certain diagrams, consisting of boxes and anti-boxes, and that these diagrams pair up and the corresponding residues cancel each other. We will rst consider generic intersecting defect theories on S(31) \ S(32) with gauge groups U(n( 1 )) and U(n(2)), sharing nf = naf = N . D.1 Poles of type-^ We recall from subsection 3.1.3 that the proposed matrix model that computes the partition function of the worldvolume theory of intersecting defects has an integrand of the form, HJEP07(21)3 Z1(T-lo;Sop!~5) Z(T ;S(31)[S(32) S!~5)( ( 1 ); (2)) = ZS(31) ( ( 1 )) Zintersection( ( 1 ); (2)) ZS(32) ( (2)) ; where ZS(31) ( ( 1 )) denotes the integrand of U(n( 1 )) SQCDA on S(31) with F(1I) > 0, and similarly for ZS(32) ( (2)). Recall that the parameters entering the two three-sphere integrands satisfy various relations, see (3.29). The intersection factor reads (2) 2i (b(21) + b(22))) The Je rey-Kirwan-like prescription selects a large number of poles in the combined meromorphic integrand (D.1). We now focus on the subclass of poles, de ned in (3.31){ (3.32), and referred to as poles of type-^. It is useful to observe that nL and nR can be decoupled from the following discussion. Using the recursion relations of the double-sine function sb(iQ=2 + z) and the fact that sinh i(x + n) = ( 1 )n sinh x, they can be seen to R give rise to Zvortexj~n1 2 S(11\3) and Zvortexj~n2 R without loss of generality, we will ignore the details of nL; nR. 2 S(12\3) , independent of the values of mL and mR. Therefore, It may be helpful to remark that the poles of type-^, as de ned in (3.31){(3.32), can be obtained by solving the component equations (D.1) 1 : (D.2) (D.3) Zintersection : ZS(32) : ZS(31) : (2) (2) m;n b( 1 ) A(10) = b(2) A(2^)A + A0 m(A1) = ( 1 ) A( >1) = imLA0b( 1 ) A( 1 ) + m(X1) ( 1 ) i 2 with the requirement that mRA^A = 0 (which automatically implies that for all = 0 since the mRA are a non-decreasing sequence). One should also bear in mind the ^A also parameter relations b( 1 )m(A1) b(2)m(A2) = 2i (b(22) locations de ned in (B.11). As usual, for each A, b(21)). Here we assigned to (2) the poles A(10) should be solved either with the equation in the second or third line. The class with all ^A = (2) 1 is obtained from solving all A(10) via the equation in the third line, since the poles m(1;)n and A(^A= 1) does not exist. The resulting poles are simply (the union of) m(2;)n of ZS(31) and ZS(32) respectively, which were discussed in detail in appendix B.2. Their residues are equal to the product of the summand of two SQCDA vortex partition functions times the intersection factor evaluated at the pole position. The remaining classes with at least one ^A 0 are then obtained by solving all four equations. Shortly, we will see that poles of type-f^A = 1g are associated with large Young diagrams, while the remaining poles of type-^ are associated with small Young diagrams. The poles of type-^ are special cases of the more general poles that will be discussed in later appendices. We now construct Young diagrams associated with poles of type-^, labeled by the integers mL/R, through the following steps. We only present the construction of YA for a given A, which can be repeated to construct the full N -tuple of Young diagrams fYAg. The procedure is also depicted in gure 17. 1. Start with a rectangular Young diagram Y with n( 1 ) columns and n(A2) rows of boxes. A The columns can be relabeled by , starting as 1 for the rst column and decreasing towards the right in unit steps. Note that the n(A1)-th column has label 0, and columns to the right of it are negatively-labeled. Similarly, the rows can be = n( 1 ) A labeled by , starting as downwards. = n(2) A 1 for the rst row and decreasing in unit steps 2a. Consider each component (2) = m(A2) + A 1, attach an extra segment of mRA right edge of Y extending towards the right. 2b. Consider each component A ( 1 ) = m( 1 ) + imLA b( 1 ) imRA b(2) inRA b(21). For each boxes to the -th row at the > 0, attach an extra segment of mLA column at the bottom edge of Y hanging downwards, or, if mLA segment of mLA anti-boxes to the -th column at the bottom edge standing upwards. inLA b(11). For each boxes to the -th < 0, attach a A D.3 1 < n(A2). Residues and instanton partition function 3. An anti-box annihilates a box at the same location, creating a vacant spot. It is now obvious that, poles of type-fall ^A = 1g generate large Young diagrams, since all mL/R are non-negative. When there is at least one ^A ^ generate small Young diagrams whose n(A1)-th column (labeled as 0, the poles of type= 0) has length The correspondence between poles and Young diagrams in the previous subappendix does not stop at the combinatoric level: it also leads to an equality between residues of the n(A2)−1 + + + + . . . 1 0 mRAνˆA =0 . . . mRA0=0 + + + + + + + + + + + + + mRA(νˆA+1)≥0 + + + + + + + + + + + + + + + + + + a pole of type-^. Boxes with a black + are normal boxes, while boxes with a red are anti-boxes. Coincident boxes and anti-boxes, i.e., the ones with red edges in the middle gure, annihilate to create vacant spots. ZS(31)(σ( 1 )) Zi(n1t,e2r)section(σ( 1 ), σ(2)) ZS(32)(σ(2)) ZvRo2r×teSx1|~n( 1 )(nL|b(−11)) hZ(Rsemi-)vortex|~n( 1 )(mL|b( 1 )) Zintersection(mL, mR) Z˜vRo2rtex|~n(2)(mR|b(2)) 2×S1 ~n( 1 ),~n(2) ×S1 i Z˜vRo2rtex|~n(2)(nR|b(−21)) ×S1 Zinst C2×S(11∩3)(Y~ (1∩3), Σ, M, M˜ ) Zinst C2×S(11∩2)(Y~ (1∩2), Σ, M, M˜ ) Zinst C2×S(12∩3)(Y~ (2∩3), Σ, M, M˜ ) downward arrows, reproduces the result as obtained from the ve-dimensional Higgsing analysis, depicted by the upward arrows. In the bold-face upward pointing arrow, we have omitted the `extra' factors, see appendix C.3. In the bold-face downward arrows, we have omitted the classical action and one-loop factors (at the Higgs branch vacuum ~n( 1 ) or ~n(2) respectively). matrix model and the summand of the instanton partition function evaluated at in (C.6). Namely, we will show that X pole2fpoles of type-^g ! pole Res Z(T ;S(31)[S(32) S!~5)( ( 1 ); (2)) = right-hand side of equation (3.25) : themselves upon taking the residues of the poles of type ^ (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-( 1 ). These poles are simply the familiar SQCDA poles (B.11), and the corresponding residues of ZS(31) and ZS(32) are just the summand of the relevant vortex partition functions multiplied by the classical action and one-loop determinant at the Higgs branch vacuum. The remaining factor Zintersection( ( 1 ); (2)) can be trivially evaluated at the pole pole, giving, with C(m) ~nL;~nR A as Res ! pole A=1 a=1 Y N n( 1 ) Y sb( 1 ) Y A6=B N n(A1) 1 Y A;B=1 =0 Y Y N n(A1) 1 mLA Y Y AA=>10 4 Y Y k=0 We note that sb( 1 ) (iQ( 1 )=2 Y Y as de ned below (C.19) and using the mass relation (C.11), Zintersection( pole) N Y A;B=1 n( 1 ) 1 n(A2) 1 A Y Y =0 1 : (D.5) =0 2i sinh i b( 1 ) C(m) + b(21) 2 2 b( 1 ) C(m) This is precisely the intersection factor appearing in (C.15),38 see (C.18), with f (x) = 2i sinh ib(21)x. Summing the product of all factors just described over mL; mR; nL; nR reproduces the sum over large diagrams in the right-hand side of (3.25). Note that we have inserted a trivial factor of one written as the ratio of the extra factors appearing in (C.15). One factor of this ratio completes the Higgsed instanton partition function (of the large N -tuples of diagrams), and the other one merges with the threedimensional one-loop determinants at the Higgs branch vacuum to form the Higgsed vedimensional one-loop determinant. Of course this should come as no surprise, since the matrix model integrand (D.1) was designed to reproduce the instanton partition functions for large Young diagrams, when evaluated at these poles. Next we consider the poles of type-^ with some ^A > 1, which we claim will reproduce ^A > 0 , the small Young diagram contributions to the instanton partition function. De ne as the smallest integer for which mLA A 0, i.e., mLA( A 1) < 0 mLA . Notice that A A > 0. At this point, we will suppress the details about nL and nR, as their rst compute the reside of the fundamental one-loop factor in ZS(31) . It reads computational details are similar to the ones just presented for the large diagrams. We 2 a( 1 ) + m(A1) = zR!es0 sb( 1 ) N n(A1) 1 Y Y A=1 =1 P A=1 A0 z m(X1) N Y sb( 1 ) AA=>10 1 2 A=1 = A k=1 2i sinh b( 1 )( m(X1) +ikb( 1 )) A;B=1 =0 k=1 2i sinh b( 1 )( m(A1B) m(X1) +ikb( 1 )) zfund ! 1 and zdLe/fRect ! 1. See footnote 36. 2i sinh b( 1 )( m(X1) ikb( 1 )) Y 2i sinh b( 1 )( ib( 1 ) ikb( 1 ))5 : (D.6) ib( 1 )) = b( 1 ). Next we take the residue of one of the factors 2i sinh b(2) b (2) b( 1 ) a ( 1 ) + i 2 i 2 N Y A6=B n(A1) 1 mLA Y Y ^A 1 k=0 3 1 : N Y A(=B)=1 N Y 2 mLA0 2 4 Y =0 2^A 1 Y 4 A(=B)=1 k=0 The residue of Zintersection,1 can then be written as Res P A A0 A(=B)=1 A>0 N Y N Y 2 2^A 1 Y 4 =0 Y Y Y =1 A 1 mLA 1 Y k=0 3 3 3 7 (D.8) 3 1 HJEP07(21)3 =0 2i sinh i ( C + b(21)) 1 1 2i sinh i ( C + b(21)) = A =0 2i sinh i ( C + b(21)) 5 2i sinh i ( (k + 1)b(21)) 2i sinh b( 1 )( m(X1) We also denote the other factor in (D.2) as Zintersection,2. ib( 1 )m(A1B) + b(21)(mLA + ) b(22)(mRB + ), and observe that We use again b(21) C(m) = 2i sinh i ( C(m) + b(21)) 2i sinh i ( C(m) + b(21))5 2i sinh i ( (k + 1)b(21)) m(X1) ikb( 1 ))5 : Observe that the last line and b(#1) cancel against the last line in (D.6) and the products of sb( 1 ) ( iQ( 1 ) ib( 1 )). The factors in the second line are precisely a product over the lled boxes 2 inside the n( 1 ) A A n(2) rectangular region, and, together with the leftover factor in the rst line and Zintersection,2( 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 combine into 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, than just those of type-^. All of them assign to (2) poles of type j imjR b(2) injR b(21) ; mjR ; njR > 0 ; (D.9) while ( 1 ) are solutions to the component equations ( 1 ) = mi(a1) a ( 1 ) = a ( 1 ) + m(X1) b i mab b( 1 ) i nab b(11) ; maL; naL > 0 mab; nab > 0 type I : b( 1 ) a ( 1 ) = b(2) b (2) + type II : b( 1 ) a ( 1 ) = b(2) b (2) i 2 2 b( 1 ) 2 b(2) i 2 inaL ; inaL ; naL > 0 naL > 0 : (D.10) A 1 mLA Y A=1 Y =0 A 1 mLA 1 1 Y =1 Y k=0 D.1 D.2 B.1 (^0 6= ^1) B.2 (^0 6= ^1) C.1 (^0 6= ^1) C.2 (^0 6= ^1) gure 19. mR^m0 R^0 mR^m0 R^0 mR^m0 R^0 1 1 1 1 (^0 + 1) (^0 + 1) (^0 + 1) (^0 + 1) mR^m1 R^1 mR^m1 R^1 0 0 1 1 1 (^0 + 1 m1L); 8 m1L0 2 N m1L0); 8 m1L0 2 N ^1 ^1 (^1 + 1) (^1 + 1) 8m1L0 2 N 8m1L 2 N where at least one of the component a( 1 ) should be solved by a type I or type II equation (otherwise one just gets back the poles m(2;)n, which are already discussed). Similar to those of the SQCDA partition functions on Sb3, the poles speci ed by (D.9){(D.10) can be characterized by forest-tree diagrams. However, there are now three possible types of links between two nodes, corresponding to the equations of type-adj., type I and type II. Note that the poles of type-^ discussed in the previous section, which gave rise to large and small Young diagrams, can be recovered as special cases of (D.9){(D.10). For simplicity and clarity of the presentation, we consider the cases of nf = naf = 1, gauge groups U(n( 1 ) = 2) on S(31) and U(n(2)) on S(32). The avor index A is spurious in this case and will be omitted. For more general unitary gauge groups, the poles can be analyzed following exactly the same logic. We will show that the general simple poles not of type-^ cancel among themselves. Again, we decouple the nL; nR in the following discussions. First of all, there are many families of poles. The components ( 1 ); (2), given by solving (D.10), can be written universally as ( 1 ) = m( 1 ) + h m(X1) (2) = m(2) + m(X2) imRb(2) ; (D.11) where h (which later determines the horizontal position of the appended boxes) is closely related to the tree structure that describes the pole, and can be negative. In gure 19, we list all classes of contributing forests that describe the above poles, and we tabulate the corresponding values of h and mL in table 2. Note that poles of type-^ with ^ 0 all lie in class A.1. It is easiest to look for potential cancellations by rst inspecting the classical factor. The Coulomb branch classical factor on S(31)[S(32) is Zcl. = exp h 2 i F(1I) P Substituting in (D.11), and using the fact that F(1I)b( 1 ) = F(2I)b(2) 2 0 Zcl. = N exp 4 2 i F(1I)m(X1)(h0 + h1) + 2 b @m0L + m1L + n(2) 1 X =0 b, one has mR 13 A5 ; ( 1 ) 2 i F(2I) P (D.12) .. B.2 . .. C.1 . .. C.2 . σ(2) νˆ1 . . . . . . σ( 1 ) 1 σ(2) νˆ1 . . . . . . .. A.2 . σ( 1 ) 0 σ( 1 ) 1 .. D.1 . .. D.2 . σ( 1 ) 1 σ(2) νˆ1 .. A.3 . σ(2) νˆ0 . . . σ( 1 ) 1 σ( 1 ) 0 σ( 1 ) 1 σ(2) νˆ1 . . . . . . .. B.1 . .. A.1 . classes that are obviously not contributing due to symmetry reason. Green and red lines correspond to type I and type II equations, which are used to solve 0(1;1) in terms of component(s) of (2). Poles of type-^ form a subclass of class A.1. The residues of poles corresponding to non-type-^ diagrams enclosed within a dashed rectangle cancel each other. contributions, and hence equal h0 + h1 and m0L + m1L + Pn(2) 1 mR. =0 where N denotes some common factors shared across all families of poles. Clearly, one necessary condition for two poles to potentially cancel is that they have equal classical An excellent tool to pinpoint the canceling pairs of poles is again given by diagrams associated with the poles (D.11). These diagrams consist of boxes and anti-boxes, and it is possible that anti-boxes survive after annihilation. The construction is a simple generalization of that in appendix D.2, and is illustrated in gure 20: step 1. and 2a. are identical. When it comes to appending vertical boxes or anti-boxes corresponding to ( 1 ), one should, generalizing 2b., append to the h -th column. Now that h can be negative, these vertical segments of boxes can sit to the right of Y , and can have annihilation with the horizontal (2). Figure 21 demonstrates a few examples of such segments of boxes corresponding to 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-^ with + + h0=−3, h1=−2 m(01)=−2, m(11)=−1 m(02)=+0, m(12)=+3 + + + + +− + + + h0=−3, h1=−1 m(01)=−2, m(11)=−1 m(02)=+1, m(12)=+3 + + − − h0=−3, h1=−2 m(01)=−1, m(11)=−1 m(02)=+0, m(12)=+2 + + m(01)=−1, m(11)=−0 m(02)=+0, m(12)=+2 A.3: n( 1 )=n(2)=2, νˆ0=1 A.1: n( 1 )=n(2)=2, νˆ0=1 A.2: n( 1 )=n(2)=2, νˆ0=1 hμ0 + + + +− +− h0=−3, h1=−2 m(01)=−2, m(11)=−2 m(02)=+0, m(12)=+3 + + + + +− + + + +− + − h0=−3, h1=−2 m(01)=−2, m(11)=−1 m(02)=+3, m(12)=+3 + + + +− + + − − h0=−2, h1=−3 m(01)=−2, m(11)=−1 m(02)=+0, m(12)=+2 + + + +− h0=−3, h1=−2 m(01)=−0, m(11)=−1 m(02)=+2, m(12)=+2 m(ν2) + + + + + + + + A.1: n( 1 )=n(2)=2, νˆ0=1 A.3: n( 1 )=n(2)=2, νˆ0=0 indicate that the residues from the related poles, which generate the same diagrams, are opposite. associated diagram, one can always nd another pole within the same class (A,B,C, or D) examples, the pairs of poles have indeed equal h0 + h1 and m0L + m1L + Pn(2) 1 mR. =0 with the same diagram; hence they cancel.39 See gure 21 for some examples. In all these E Poles and Young diagrams in 2d In this appendix we study the poles and their residues of the matrix model computing the partition function of intersecting surface defects supported on S2 L [ S2 R b S4. Throughout the appendix we will use (sub-)superscripts L, R for quantities on SL2/R, and N, S for quantities associated to the north- or south-pole contributions. The main idea is very similar to the discussion in appendix D, but slightly more involved, due to the fact that the intersection between SL2 and SR2 have two connected components, namely the north and 39Note that poles of type-^ with ^ 0 although being special case of A.1, do not have such canceling siblings, therefore they have non-zero contributions in the end. south poles. We will need to bring the contributions from both poles together to reproduce the square of the instanton partition function. E.1 Four types of poles Z(T ;SL2[SR2 Sb4) has integrand Recall that for a theory T of N 2 free hypermultiplets in the presence of intersecting defects with U(nL) SQCDA on SL2 and U(nR) SQCDA on SR2 respectively, the partition function Z(T ;SL2[SR2 Sb4)( L; R ZSL2 ( L; BL) Y Zintersection( L; BL; R; BR) ZSR2 ( R; BR) ; where the intersection factor is de ned in (3.47). The combined meromorphic integrand (E.1) has many poles. Recall that mRX = ib 2. (E.2) >> i AL + >>: i AL 8 >> i AR + >>: i AR BAL 2 BAL BAR BAR 2 2 2 = imLA + hLimLX + mLA = imLA + hLimLX + nLA = imRA + hRimRX + mRA = imRA + hRimRX + nA R : First of all, we de ne type-old poles by simply taking the (union of) poles of ZSL2 and ZSR2 discussed in appendix B.1. Additionally, we introduce three special classes of poles, which we refer to as type-N+^, S +^ and NS+^ poles. Their de nition goes as follows. We start by selecting partitions ~nL, ~nR of the ranks nL, nR: this corresponds to choosing a select a set of integers f^AN/S; A = 1; : : : ; N g, where each ^N/S A 2 f 1; 0; : : : ; nRA Higgs branch vacuum of the SQCDA theory living on SL2 and SR2 respectively. Next we 1g and PN A=1 ^N/S > A N . In the end we will sum over all such partitions ~nL, ~nR and sets f^AN/S g to obtain all relevant poles. Then the three special types of poles are given by the abstract equations (E.2) with hR = , hL = , together with the following conditions: Poles of type-N+^N : mRA(nRA 1) > : : : > mRA(^AN+1) > mRA^AN = mRA(^AN 1) = : : : = mRA0 = 0; nA > 0 R mLA(nLA 1) > : : : > mLA1 > mLA0; mLA0 = (^AN + 1) if ^AN 0; or mLA0 0 if ^AN = 1 : L nA > 0 (E.3) Poles of type-S+^S : nA(nLA 1) > : : : > nA1 > nLA0; L L L nA0 = (^AS + 1) if ^AS 0; Poles of type-NS+^N ^S : nA(nRA 1) > : : : > nA(^AS+1) > nA^AS = nA(^AS 1) = : : : = nRA0 = 0; mRA > 0 R R R R or L nA0 0 if ^AS = 1 : mLA > 0 (E.5) 1 : (E.6) 1 for (E.7) >> i CR + >>> i AL0 + 1 2 >> b 1 i AL0 + >>: i AL( +1) + mRA(nRA 1) > : : : > mRA(^AN+1) > mR^N = mR^AN 1 = : : : = m0R = 0 ; A nA(nRA 1) > : : : > nA(^AS+1) > nA^AS = nA(^S 1) = : : : = nRA0 = 0 ; R R R R mLA(nLA 1) > : : : > mLA1 > mLA0 ; mLA0 = L nA0 = (^AN + 1) if ^AN (^AS + 1) if ^AS 0; 0; nA(nLA 1) > : : : > nA1 > nLA0; L L or or mLA0 L nA0 0 if ^AS = 1 : A few remarks are in order. Poles of type-N+^N come from solving the equations 2 BCR = imCR + imRX + mCR ; with mCR^CN = 0; C = 1; : : : ; N imLA = +mA( 0) 2 b i AR^AN + 1 b + b 1 2 = 0 12 BAL( +1) = i A(L) + 2 1 BAL + imLX + mA( +1) (> 0) ; = 0; : : : nLA ( AR( = 1) does not exist anyway), otherwise the equation in the third line. If ^AN = If ^AN = 1 for a given A, one should use the equation in the second line to obtain AL0+ 12 BAL0 all A, one simply recovers the poles of type-old which we de ne separately, and therefore we exclude such case when de ning poles of type-N+^N . Among the solutions, most of those with nLA0 < 0 are canceled by zeros of the fundamental one-loop determinant ZSL2 . Similarly for poles of type-S^N . However, there are survivors from the cancellation, which involve simultaneous solutions to the set of equations 8 > i CR + >>: b 1 i AL0 1 1 2 BCR = imCR + imRX + mCR ; 2 BCR = imCR + imRX + nCR ; 2 2 b i R^N + A b i R^S A 1 1 mRB^CN = 0 R nB^CS = 0 b + b 1 b + b 1 2 2 = 0 = 0 : Naively, simultaneous solutions to the last two equations seem to correspond to double poles of the integrand, since two separate intersection factors develop a pole. However, they are actually simple poles after canceling with the zeros of ZSL2 . These poles are called The presence of these delicate poles 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 (Y~ N; Y~ S) from the four types of poles. The construction is essentially the same as outlined in appendix D.2, where mL/R will now take care of Y~ N, and nL/R will take care of Y~ S. More precisely, one has the correspondence type-old (large, large) YANnLA ; YASnLA type-N+^N (small, large) type-S+^S (large, small) YANnLA = nR A YASnLA 1 YASnLA = nR A YANnLA 1 type-NS+^N ^S (small, small) YANnLA = nR A YASnLA = nR A A 1 1 Exhausting all four types of poles, one recovers all possible pairs of N -tuples of Young diagrams. Again, the residues of the four types of poles sum up to the modulus squared equivariant parameter, which appears in the full Sb4 partition function. jZinstj2 of the instanton partition function, evaluated at the speci c value of its gauge E.2 Extra poles and diagrams There are many extra poles in the integrand (E.1) selected by the JK prescription, besides the four types of poles discussed above. For simplicity, here we only present the cancellation in the simplest case of nL = nf = naf = 1. The main idea is very similar to the discussion in appendix D.4 and techniques to analyze more general cases can be found there as well. 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Yiwen Pan, Wolfger Peelaers. Intersecting surface defects and instanton partition functions, Journal of High Energy Physics, 2017, 73, DOI: 10.1007/JHEP07(2017)073