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 , SE75120 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 fourdimensional 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 SeibergWitten monopolelike 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 triplesine 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 Foresttree 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
HalfBPS 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 positiondependent,
vortexlike 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 Mtheory construction of fourdimensional 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 M2branes ending on the main stack of N
M5branes can introduce surface defects in the fourdimensional theory. The thus obtained
M2brane defects are known to be labeled by a representation R of SU(N ). In [19], the
twodimensional quiver gauge theory residing on the support of the defect and its
coupling to the bulk fourdimensional 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 halfBPS defects, intersecting each
other along codimension four loci, while preserving one quarter of the supersymmetry,
enlarges the collection of defects considerably and is very wellmotivated. Indeed, in [21] it
was conjectured and overwhelming evidence was found in favor of the statement that the
squashed foursphere partition function of theories of class S in the presence of intersecting
M2brane defects, wrapping two intersecting twospheres, 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 M2brane 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 M2brane 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 foursphere partition function of such system.4 Let T denote the fourdimensional
theory and let L/R denote twodimensional theories residing on the defects wrapping the
twospheres 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 oneloop determinant
of the theory T placed on the manifold M (in their Coulomb branch localized form).
Furthermore, Zintersection are the oneloop 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 fourdimensional gauge elds, thus avoiding the intricacies of the instanton
partition functions. In this paper we aim at considering intersecting defects in interacting
fourdimensional 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 M2brane 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 positiondependent vacuum expectation value with an
intersecting vortexlike 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) SeibergWitten 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 twodimensional 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 Sb4partition 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 M2brane surface defects,
labeled by nR and nLfold 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 zerodimensional degrees of freedom, i.e., it coincides with
the one described in conjecture 4 of [21] with fourdimensional 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 zeromodes 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 kinstantons 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 M2brane 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) M2brane 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 vedimensional 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 kinstantons of the left theory is shown. The model preserves the dimensional
reduction to zero dimensions of twodimensional N = (0; 2) supersymmetry. We used the
corresponding quiver conventions. A Jtype 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 Etype
superpotential for the Fermi multiplets charged under U(nL/R).8
vacuum expectation value in T of intersecting M2brane 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 fourdimensional 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/ vesphere [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 foursphere 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) fourdimensional N = 2 supersymmetric quantum eld theories of class S and
10We consider S
b4 de ned through the embedding equation in vedimensional 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 twospheres: 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 twospheres 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(Telo;Sopb4) is
the oneloop 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 M2brane surface defects labeled by nRfold and nLfold symmetric
representations respectively up to the leftover contribution of the hypermultiplet that captures
the
uctuations around the Higgs branch vacuum. These defects wrap two intersecting
twospheres 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 vedimensional N = 1 theories. The theory
vesphere 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
vesphere 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(Telo;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 threespheres
S(3 ) obtained as the
xed loci of the U(
1
)( ) isometries (see footnote 14), respectively.
These threespheres intersect each other in pairs along a circle. Again, this pole arises
from pinching the integration contour by poles of the oneloop 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 FayetIliopoulos 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 SeibergWitten monopoles.
14The squashed
vesphere 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 threesphere 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 threespheres S
( ) and S(3 ). A convenient visualization of the vesphere 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 fourdimensional 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 NS5brane on
the right end of the brane array. The Higgsing prescription of the previous subsection is
then trivially implemented by pulling away this additional NS5brane (in the 10direction
of footnote 16), while suspending nR D2R and nL D2Lbranes between the displaced
NS5brane and the right stack of D4branes, 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 M2brane surface defects labeled by nR and
nLfold 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
twodimensional theories, residing on the D2R and D2Lbranes, are in their Higgs phase,
with equal FayetIliopoulos parameter FI proportional to the distance (in the 7direction)
between the displaced NS5brane and the next rightmost NS5brane. 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 rightmost end of the quiver.
In this paper we will restrict attention to this regime. Note however that sliding the
displaced NS5brane 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 Ktheoretic
SeibergWitten 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 NS5brane 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 NS5brane from the main stack, while stretching nR D2R and nL D2Lbranes in between
it and the D4branes, 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 twodimensional degrees of freedom, depicted in N = (2; 2) quiver
notation, are coupled to the fourdimensional ones through cubic and quartic superpotential
couplings. The explicit superpotentials can be found in [21]. The zerodimensional degrees of freedom,
denoted using twodimensional N = (0; 2) quiver notations dimensionally reduced to zero
dimensions, with solid lines representing chiral multiplets, participate in E and Jtype 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
fourdimensional quiver, while not changing the resulting partition function.
In [21], a rstprinciples localization computation was performed to calculate the
partition function of the coupled 4d/2d/0d system when placed on a squashed foursphere,
with the defects wrapping two intersecting twospheres SR2/L, the xed loci of U(
1
)R/L, in
the case of noninteracting fourdimensional theories. Our aim in the next section will be
to reproduce these results from the Higgsing point of view. When the fourdimensional
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 nontrivially 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 threesphere invariant pointwise under one of the U(
1
) isometries, and each
vertex represents an S1, where two S3's intersect, invariant pointwise under two U(
1
) isometries.
Each S1 has two tubular neighborhoods of the form S1
R2 in the two intersecting S3's, with
omegadeformation 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 D0branes
stretching between the NS5branes. 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 D0branes and the D2R and D2Lbranes. These give rise to the dimensional
reduction of a twodimensional N = (2; 2) chiral multiplet to zero dimensions, or
equivalently, the dimensional reduction of a twodimensional 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 M2brane 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
reyKirwanlike 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 threespheres 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(Telo;Sop!~5)( ; M; M~ )jZinst
(Te;R4 S1)(q; ; M ; M~ )j3 :
Zc(lTe;S!~5)( ) = exp
while the oneloop determinant Z1(Telo;Sop!~5) is the product of the oneloop determinants of the
SU(N ) vector multiplet, the N fundamental hypermultiplets and the N antifundamental
hypermultiplets:
Z1(Telo;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 Ktheoretic
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 threespheres 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 threesphere S(31) or S(32). As we will see, the nonfactorizable pieces nicely cancel
against each other, except for a factor that will ultimately describe the onedimensional
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 oneloop determinant (3.3) become, using recursion relations for the
18In appendix C we have simultaneously performed manipulations of fourdimensional 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(Telo;Sop!~5) ! Z1(Tlo;Sop!~5)
Let us unpack this expression a bit. First, Z1(Tlo;Sop!~5) is the oneloop determinant of N 2 free
hypermultiplets, which constitute the infrared theory T . It reads
(3.11)
Z1(Tlo;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 avornode
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 oneloop determinant of
squashed threesphere partition functions of a threedimensional 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 righthand side the leftover 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 positionindependent 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 threesphere 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)( ) Z1loop
( ;Sb3)( ) ;
while a \Higgs branch localization" computation brings it into the form [10, 11]
HV
Z( ;Sb3) = X ZcljHV
( ;Sb3) Z1loopjHV 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 Ktheoretic 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 vedimensional parameters as follows, with ( )
p!( )=(!1!2!3),
masses m
( ) entering the threedimensional 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 oneloop 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 nonfactorizable factors from the antifundamental oneloop
determinant, while Zvf,extra captures those from the vector multiplet and fundamental
hypermultiplet oneloop 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 nonfactorizable
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 threespheres 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( ) nondecreasing
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 Ktheoretic 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 D0branes
spanning in between the NS5branes. After Higgsing, the D0branes can still be present if they end on the
D2R and D2Lbranes. If, say, n
L = 0, they precisely turn into vortices of the twodimensional theory living
on the D2branes.
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 Ktheoretic 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 nondecreasing 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 nondecreasing 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 righthand
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 righthand 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 oneloop determinant of the S(31)
partition function of SQCDA, that is, of a threedimensional 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(Tlo;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 FayetIliopoulos
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 threesphere 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 threedimensional mass parameters on S(3 ) to the vedimensional 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 reyKirwanlike 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 JKvector
= ( F(1I); F(2I)), where we treat the FayetIliopoulos 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 avornode quiver. This statement can be
veri ed by dimensionally uplifting the localization computation of [21]. In some detail,
Z1(Tlo;Sop!~5) captures the contributions to the partition function of the vedimensional 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 oneloop determinant of the onedimensional
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 threedimensional 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 onedimensional chiral multiplets, which enter
explicitly in Zintersection, and for which no rstprinciples argument was provided in [21].
The integrand of (3.27) has poles in each of the three factors; the Je reyKirwanlike
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 1loop factors, with mij
mi
mj , are given by
S3
Z b
clj~k
S3
Z1bloopj~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 oneloop determinants are expressed in terms of the
doublesine function sb. We have taken the ChernSimons level to be zero.
When nf > naf or nf = naf and the FIparameter FI > 0,34 we again consider poles
m;n in the lower halfplane, 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 Sb3partition
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 nondecreasing
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
Foresttree 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 ChernSimons 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 branchless trees,
associated with masses mi and mj(6=i) respectively. Forests consisting of such branchless trees will
give nonzero 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 FIparameter FI > 0, the Je reyKirwan 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 lefthand 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 righthand side
of (B.17) (i.e., a component of
or a mass m) to that associated with the component on
the lefthand 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 oneloop determinants, one can show that, after summing
over all possible poles, namely over all possible forest diagrams, only those forests whose
trees are all branchless and where each fundamental mass is only linked to (at most) one
branchless 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
fourdimensional and
vedimensional 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/ vedimensional supersymmetric quiver gauge theory with gauge
group SU(N )
SU(N ),35 with N fundamental hypermultiplets, N antifundamental
hypermultiplets and one bifundamental hypermultiplet, with masses MI , M~ J and M^
respectively. Let
and 0 denote the Cartanvalued 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/threedimensional 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 nondecreasing
Now we are ready to state the factorization of the various factors in the (summand
A
of) the twogaugenode 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~ ")
= Zsemivortexj~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 dashedlines, 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 threedimensional
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 antiboxes, 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(Tlo;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 threesphere integrands
satisfy various relations, see (3.29). The intersection factor reads
(2)
2i (b(21) + b(22)))
The Je reyKirwanlike 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 doublesine
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 nondecreasing 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 typef^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 negativelylabeled. 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 antiboxes 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 antibox annihilates a box at the same location, creating a vacant spot.
It is now obvious that, poles of typefall ^A =
1g generate large Young diagrams,
since all mL/R are nonnegative. 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 antiboxes.
Coincident boxes and antiboxes, 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
)(nLb(−11)) hZ(Rsemi)vortex~n(
1
)(mLb(
1
)) Zintersection(mL, mR) Z˜vRo2rtex~n(2)(mRb(2))
2×S1 ~n(
1
),~n(2) ×S1
i Z˜vRo2rtex~n(2)(nRb(−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
vedimensional Higgsing analysis,
depicted by the upward arrows. In the boldface upward pointing arrow, we have omitted the `extra'
factors, see appendix C.3. In the boldface downward arrows, we have omitted the classical action
and oneloop 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)) = righthand side of equation (3.25) :
themselves upon taking the residues of the poles of type ^ (ignoring the classical and overall
oneloop 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 oneloop 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 righthand 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 oneloop determinants at the Higgs branch vacuum to form the Higgsed
vedimensional oneloop 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 oneloop 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
oneloop 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 foresttree diagrams. However, there are now three possible types of
links between two nodes, corresponding to the equations of typeadj., 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 nontype^ 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 antiboxes, and it is
possible that antiboxes 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 antiboxes 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 southpole 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 nonzero 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 typeold 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 typeN+^, 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 typeN+^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 typeS+^S :
nA(nLA 1) > : : : > nA1 > nLA0;
L L
L
nA0 =
(^AS + 1) if ^AS
0;
Poles of typeNS+^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 typeN+^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 typeold which we de ne separately, and therefore
we exclude such case when de ning poles of typeN+^N . Among the solutions, most of those
with nLA0 < 0 are canceled by zeros of the fundamental oneloop determinant ZSL2 . Similarly
for poles of typeS^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
typeold
(large, large)
YANnLA ; YASnLA
typeN+^N
(small, large)
typeS+^S
(large, small)
YANnLA = nR
A
YASnLA
1
YASnLA = nR
A
YANnLA
1
typeNS+^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.
There are four types of extra poles selected by the JK prescription (we recycle the
naming appearing in the previous subsection):
hL
0
(mR^N + 1)
(nR^S + 1)
(^N + 1)
m0L
(^S + 1)
It is straightforward to verify that the residues of poles of typeN+^N cancel those of
typeN^N , and similarly between typeS+^S and S^S . Again, from the four types of poles one
can construct pairs of general diagrams consisting of boxes and antiboxes. Two poles
contribute opposite residues when their corresponding pairs of diagrams coincide (taking
into account of annihilation between coincident boxes/antiboxes).
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