Modular and duality properties of surface operators in \( \mathcal{N}={2}^{\star } \) gauge theories
HJE
Modular and duality properties of surface operators in
S.K. Ashok 0 1 2 3 4
M. Bill`o 0 1 2
E. Dell'Aquila 0 1 2 3 4
M. Frau 0 1 2
R.R. John 0 1 2 3 4
A. Lerda 0 1 2
0 Via P. Giuria 1, I10125 Torino , Italy
1 Training School Complex , Anushakti Nagar, Mumbai, 400085 India
2 C.I.T. Campus , Taramani, Chennai, 600113 India
3 Homi Bhabha National Institute
4 Institute of Mathematical Sciences
We calculate the instanton partition function of the fourdimensional N = 2⋆ SU(N ) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from Sduality, we show that the effective twisted superpotential, which governs the infrared dynamics of the twodimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasimodular forms which resum all nonperturbative corrections. We also show that our results, derived for monodromy defects in the fourdimensional theory, match the effective twisted superpotential describing the infrared properties of certain twodimensional sigma models coupled either to pure N = 2 or to N = 2⋆ gauge theories.
Duality in Gauge Field Theories; Extended Supersymmetry; Supersymmetry

N
⋆
1 Introduction
2 Instantons and surface operators in N = 2⋆ SU(N ) gauge theories
3 Partition functions for ramified instantons
3.1
3.2
Summing over fixed points and characters
Map between parameters 3.3 Extracting the prepotential and the twisted superpotential
4
Modular anomaly equation for the twisted superpotential
4.1 Sduality constraints
5.1 Relation to CFT results
5 Surface operators in N = 2⋆ SU(2) theory
6 Surface operators in N = 2⋆ SU(N ) theories
6.1
6.2
Simple surface operators
Surface operators of type {p, N − p}
6.3 Surface operators of general type
7 Duality between surface operators
7.1
7.2
The pure N = 2 SU(N ) theory
The N = 2⋆ SU(N ) theory
7.3 Some remarks on the results
8 Conclusions
A Useful formulas for modular forms and elliptic functions
B Generalized instanton number in the presence of fluxes
C Ramified instanton moduli and their properties
C.1 SU(2)
D Prepotential coefficients for the SU(N ) gauge theory
– 1 –
Introduction
The study of how a quantum field theory responds to the presence of defects is a very
important subject, which has received much attention in recent years especially in the
context of supersymmetric gauge theories. In this paper we study a class of twodimensional
defects, also known as surface operators, on the Coulomb branch of the N = 2⋆ SU(N )
gauge theory in four dimensions.1 Such surface operators can be introduced and analyzed
in different ways. They can be defined by the transverse singularities they induce in the
fourdimensional fields [2, 3], or can be characterized by the twodimensional theory they
support on their worldvolume [4, 5].
A convenient way to describe fourdimensional gauge theories with N = 2
supersymmetry is to consider M5 branes wrapped on a punctured Riemann surface [6, 7]. From the
point of view of the sixdimensional (2, 0) theory on the M5 branes, surface operators can be
realized by means of either M5′ or M2 branes giving rise, respectively, to codimension2 and
codimension4 defects. While a codimension2 operator extends over the Riemann surface
wrapped by the M5 brane realizing the gauge theory, a codimension4 operator intersects
the Riemann surface at a point. Codimension2 surface operators were systematically
studied in [8] where, in the context of the of the 4d/2d correspondence [9], the instanton
partition functions of N = 2 SU(2) superconformal quiver theories with surface operators
were mapped to the conformal blocks of a twodimensional conformal field theory with an
affine sl(2) symmetry. These studies were later extended to SU(N ) quiver theories whose
instanton partition functions in the presence of surface operators were related to
conformal field theories with an affine sl(N ) symmetry [10]. The study of codimension4 surface
operators was pioneered in [11] where the instanton partition function of the conformal
SU(2) theory with a surface operator was mapped to the Virasoro blocks of the Liouville
theory, augmented by the insertion of a degenerate primary field. Many generalizations
and extensions of this have been considered in the last few years [12–19].
Here we study N = 2⋆ theories in the presence of surface operators. The lowenergy
effective dynamics of the bulk fourdimensional theory is completely encoded in the
holomorphic prepotential which at the nonperturbative level can be very efficiently determined
using localization [20] along with the constraints that arise from Sduality. The latter turn
out to imply [21, 22] a modular anomaly equation [23] for the prepotential, which is
intimately related to the holomorphic anomaly equation occurring in topological string theories
on local CalabiYau manifolds [24–27].2 Working perturbatively in the mass of the adjoint
hypermultiplet, the modular anomaly equation allows one to resum all instanton
corrections to the prepotential into (quasi)modular forms, and to write the dependence on the
Coulomb branch parameters in terms of particular sums over the roots of the gauge group,
thus making it possible to treat any semisimple algebra [41, 42].
1For a review of surface operators see [1].
2Modular anomaly equations have been studied in various contexts, such as the Ωbackground [21, 22, 28–
34], the 4d/2d correspondence [35–37], SQCD theories with fundamental matter [21, 22, 38–40] and in
N = 2⋆ theories [21, 22, 41–44].
– 2 –
In this paper we apply the same approach to study the effective twisted superpotential
which governs the infrared dynamics on the worldvolume of the twodimensional surface
operator in the N = 2⋆ theory. For simplicity, we limit ourselves to SU(N ) gauge groups
and consider halfBPS surface defects that, from the sixdimensional point of view, are
codimension2 operators. These defects introduce singularities characterized by the pattern
of gauge symmetry breaking, i.e. by a Levi decomposition of SU(N ), and also by a set of
continuous (complex) parameters. In [45] it has been shown that the effect of these surface
operators on the instanton moduli action is equivalent to a suitable orbifold projection
which produces structures known as ramified instantons [
45–47
]. Actually, the moduli
spaces of these ramified instantons were already studied in [48] from a mathematical point of
view in terms of representations of a quiver that can be obtained by performing an orbifold
projection of the usual ADHM moduli space of the standard instantons. In section 2 we
explicitly implement such an orbifold procedure on the nonperturbative sectors of the
theory realized by means of systems of D3 and D(−1) branes [49, 50]. In section 3 we
carry out the integration on the ramified instanton moduli via equivariant localization.
The logarithm of the resulting partition function exhibits both a 4d and a 2d singularity in
the limit of vanishing Ω deformations.3 The corresponding residues are regular in this limit
and encode, respectively, the prepotential F and the twisted superpotential W. The latter
depends, in addition to the Coulomb vacuum expectation values and the adjoint mass, on
the continuous parameters of the defect.
In section 4 we show that, as it happens for the prepotential, the constraints arising
from Sduality lead to a modular anomaly equation for W. In section 5, we solve this
equation explicitly for the SU(2) theory and prove that the resulting W agrees with the
twisted superpotential obtained in [35] in the framework of the 4d/2d correspondence with
the insertion of a degenerate field in the Liouville theory. Since this procedure is appropriate
for codimension4 defects [11], the agreement we find supports the proposal of a duality
between the two classes of defects recently put forward in [52]. In section 6, we turn our
attention to generic surface operators in the SU(N ) theory and again, order by order in
the adjoint mass, solve the modular anomaly equations in terms of quasimodular elliptic
functions and sums over the root lattice.
We also consider the relation between our findings and what is known for surface
defects defined through the twodimensional theory they support on their worldvolume.
In [5] the coupling of the sigmamodels defined on such defects to a large class of
fourdimensional gauge theories was investigated and the twisted superpotential governing their
dynamics was obtained. Simple examples for pure N = 2 SU(N ) gauge theory include
the linear sigmamodel on CPN−1, that corresponds to the socalled simple defects with
Levi decomposition of type {1, N − 1}, and sigmamodels on Grassmannian manifolds
corresponding to defects of type {p, N −p}. The main result of [5] is that the SeibergWitten
geometry of the fourdimensional theory can be recovered by analyzing how the vacuum
structure of these sigmamodels is fibered over the Coulomb moduli space. Independent
3We actually calculate the effective superpotential in the NekrasovShatashvili limit [51] in which only
one of the Ωdeformation parameters is turned on.
– 3 –
analyses based on the 4d/2d correspondence also show that the twisted superpotential for
the simple surface operator is related to the line integral of the SeibergWitten differential
over the punctured Riemann surface [11]. In section 7, we test this claim in detail by
considering first the pure N = 2 gauge theory. Since this theory can be recovered upon
decoupling the massive adjoint hypermultiplet, we take the decoupling limit on our N = 2⋆
results for W and precisely reproduce those findings. Furthermore, we show that for simple
surface defects the relation between the twisted superpotential and the line integral of the
SeibergWitten differential holds prior to the decoupling limit, i.e. in the N = 2⋆ theory
itself. The agreement we find provides evidence for the proposed duality between the two
types of descriptions of the surface operators.
Finally, in section 8 we present our conclusions and discuss possible future perspectives.
Some useful technical details are provided in four appendices.
2
Instantons and surface operators in N = 2
⋆ SU(N ) gauge theories
The N = 2⋆ theory is a fourdimensional gauge theory with N = 2 supersymmetry that
describes the dynamics of a vector multiplet and a massive hypermultiplet in the adjoint
representation. It interpolates between the N = 4 super YangMills theory, to which it
reduces in the massless limit, and the pure N = 2 theory, which is recovered by decoupling
the matter hypermultiplet. In this paper, we will consider for simplicity only special unitary
gauge groups SU(N ). As is customary, we combine the YangMills coupling constant g and
the vacuum angle θ into the complex coupling
on which the modular group SL(2, Z) acts in the standard fashion:
with a, b, c, d ∈ Z and ad − bc = 1. In particular under Sduality we have
The Coulomb branch of the theory is parametrized by the vacuum expectation value
of the adjoint scalar field φ in the vector multiplet, which we take to be of the form
hφi = diag(a1, a2, · · · , aN )
with
N
u=1
X au = 0 .
The lowenergy effective dynamics on the Coulomb branch is entirely described by a single
holomorphic function F , called the prepotential, which contains a classical term, a
perturbative 1loop contribution and a tail of instanton corrections. The latter can be obtained
from the instanton partition function
τ =
θ
2π
4π
+ i g2
,
τ
→
S(τ ) = − τ
1
.
Zinst =
∞
X qk Zk
k=0
– 4 –
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
where
and Zk is the partition function in the kinstanton sector that can be explicitly computed
using localization methods.4 For later purposes, it is useful to recall that the weight qk
in (2.5) originates from the classical instanton action
strength F equals the instanton charge k. Hence, the weight qk is simply e−Sinst .
Let us now introduce a surface operator which we view as a nonlocal defect D
supported on a twodimensional plane inside the fourdimensional (Euclidean) spacetime (see
appendix B for more details). In particular, we parametrize R4 ≃ C2 by two complex
variables (z1, z2), and place D at z2 = 0, filling the z1plane. The presence of the surface
operator induces a singular behavior in the gauge connection A, which has the following
generic form [8, 45]:
as r → 0. Here (r, θ) denotes the set of polar coordinates in the z2plane, and the γI ’s are
constant parameters, where I = 1, · · · , M . The M integers nI satisfy
(2.6)
(2.7)
(2.9)
(2.10)
and define a vector ~n that identifies the type of the surface operator. This vector is
related to the breaking pattern of the gauge group (or Levi decomposition) felt on the
twodimensional defect D, namely
SU(N ) → S U(n1) × U(n2) × · · · × U(nM ) .
The type ~n = {1, 1, · · · , 1} corresponds to what are called full surface operators, originally
considered in [8]. The type ~n = {1, N − 1} corresponds to simple surface operators, while
the type ~n = {N } corresponds to no surface operators and hence will not be considered.
In the presence of a surface operator, one can turn on magnetic fluxes for each factor
of the gauge group (2.10) and thus the instanton action can receive contributions also from
the corresponding first Chern classes. This means that (2.7) is replaced by [2, 8, 11, 45]
Sinst[~n] = −2πiτ
1 Z
Tr F ∧ F
− 2πi X ηI
M
I=1
4Our conventions are such that Z0 = 1.
X nI = N
where ηI are constant parameters. As shown in detail in appendix B, given the
behavior (2.8) of the gauge connection near the surface operator, one has
with mI ∈ Z. As is clear from the second line in the above equation, each mI represents
the flux of the U(1) factor in each subgroup U(nI ) in the Levi decomposition (2.10);
furthermore, these fluxes satisfy the constraint
HJEP07(21)68
Using (2.12), we easily find
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
Using (2.14), we deduce that the weight of an instanton configuration in the presence
Sinst[~n] = −2πiτ k − 2πi X
ηI + τ γI mI = −2πiτ k − 2πi ~t · m~
where in the last step we have combined the electric and magnetic parameters (ηI , γI ) to
form the M dimensional vector
~t = {tI } = {ηI + τ γI } .
This combination has simple duality transformation properties under SL(2, Z). Indeed, as
shown in [2], given an element M of the modular group the electromagnetic parameters
transform as
to show that
γI , ηI
→
γI , ηI M
−1 = d γI − c ηI , a ηI − b γI .
Combining this with the modular transformation (2.2) of the coupling constant, it is easy
In particular under Sduality we have
of a surface operator of type ~n is
so that the instanton partition function can be written as
e−Sinst[~n] = qk e2πi ~t· m~ ,
Zinst[~n] =
X qk e2πi ~t· m~ Zk, m~[~n] .
k, m~
In the next section, we will describe the computation of Zk, m~[~n] using equivariant
localization.
– 6 –
Partition functions for ramified instantons
As discussed in [45], the N = 2∗ theory with a surface defect of type ~n = {n1, · · · , nM },
which has a sixdimensional representation as a codimension2 surface operator, can be
realized with a system of D3branes in the orbifold background
C × C2/ZM × C × C
with coordinates (z1, z2, z3, z4, v) on which the ZM orbifold acts as
(z2, z3) → (ω z2, ω−1 z3) ,
where ω = e M .
2πi
Like in the previous section, the complex coordinates z1 and z2 span the fourdimensional
spacetime where the gauge theory is defined (namely the worldvolume of the D3branes),
while the z1plane is the worldsheet of the surface operator D that sits at the orbifold fixed
point z2 = 0. The (massive) deformation which leads from the N = 4 to the N = 2∗ theory
takes place in the (z3, z4)directions. Finally, the vplane corresponds to the Coulomb
moduli space of the gauge theory.
Without the Z
M orbifold projection, the isometry group of the tendimensional
background is SO(4)×SO(4)×U(1), since the D3branes are extended in the first four directions
and are moved in the last two when the vacuum expectation values (2.4) are turned on. In
the presence of the surface operator and hence of the Z
M orbifold in the (z2, z3)directions,
this group is broken to
U(1) × U(1) × U(1) × U(1) × U(1) .
In the following we will focus only on the first four U(1) factors, since it is in the first four
complex directions that we will introduce equivariant deformations to apply localization
methods. We parameterize a transformation of this U(1)4 group by the vector
~ǫ =
ǫ1, Mǫ2 , ǫ3 , ǫ4
M
= {ǫ1, ǫˆ2, ǫˆ3, ǫ4}
~l = {l1, l2, l3, l4}
where the 1/M rescalings in the second and third entry, suggested by the orbifold
projection, are made for later convenience. If we denote by
with a phase given by e2πi ~l·~ǫ, while the ZM action produces a phase ωl2−l3 .
the weight vector of a given state of the theory, then under U(1)4 such a state transforms
On top of this, we also have to consider the action of the orbifold group on the
ChanPaton factors carried by the open string states stretching between the Dbranes. There are
different types of Dbranes depending on the irreducible representation of Z
M in which this
action takes place. Since there are M such representations, we have M types of Dbranes,
which we label with the index I already used before. On a Dbrane of type I, the generator
of ZM acts as ωI , and thus the ChanPaton factor of a string stretching between a Dbrane
– 7 –
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
{±1, 0, 0, 0}0 , {0, ±1, 0, 0}0 , {0, 0 ± 1, 0}0 , {0, 0, 0 ± 1}0 , {0, 0, 0, 0}±1 ,
(3.7)
where the subscripts denote the charge under the last U(1) factor of (3.3). They correspond
to spacetime vectors along the directions z1, z2, z3, z4 and v, respectively. In the fermionic
Ramond sector one finds states with weight vectors
1
1
1
± 2 , ± 2 , ± 2 , ± 2
1
with a total odd number of minus signs due to the GSO projection. They correspond to
antichiral spacetime spinors.5
It is clear from (3.7) and (3.8) that the orbifold phase ωl2−l3 takes the values ω0, ω+1 or
ω−1 and can be compensated only if one considers strings of type II, I(I + 1) or (I + 1)I,
respectively. Therefore, the Z
M invariant neutral moduli carry ChanPaton factors that
transform in the (dI , d¯I ), (dI , d¯I+1) or (dI+1, d¯I ) representations of the ADHM group (3.6).
Let us now consider the colored states, corresponding to strings stretched between a
Dinstanton and a D3brane or vice versa. Due to the twisted boundary conditions in the
first two complex spacetime directions, the weight vectors of the bosonic states in the
NeveuSchwarz sector are
1
1
± 2 , ± 2 , 0, 0
0, 0, ± 2 , ± 2
1
1
0
In the resulting D3/D(−1)brane systems there are many different sectors of open strings
depending on the different types of branes to which they are attached. Here we focus only
on the states of open strings with at least one endpoint on the Dinstantons, because they
represent the instanton moduli [49, 50] on which one eventually has to integrate in order
to obtain the instanton partition function.
Let us first consider the neutral states, corresponding to strings stretched between
two Dinstantons. In the bosonic NeveuSchwarz sector one finds states with U(1)4 weight
vectors
while those of the fermionic states in the Ramond sector are
Assigning a negative intrinsic parity to the twisted vacuum, both in (3.9) and in (3.10) the
GSOprojection selects only those vectors with an even number of minus signs. Moreover,
5Of course one could have chosen a GSO projection leading to chiral spinors, and the final results would
have been the same.
– 8 –
since the orbifold acts on two of the twisted directions, the vacuum carries also an intrinsic
ZM weight. We take this to be ω− 12 when the strings are stretched between a D3brane
and a Dinstanton, and ω+ 12 for strings with opposite orientation. Then, with this choice
we find Z
II, whose ChanPaton factors transform in the (nI , d¯I ) representation of the gauge and
M invariant bosonic and fermionic states either from the 3/(−1) strings of type
ADHM groups, or from the (−1)/3 strings of type I(I + 1), whose ChanPaton factors
transform in the (dI , n¯I+1) representation, plus of course the corresponding states arising
from the strings with opposite orientation.
In appendix C we provide a detailed account of all moduli, both neutral and colored,
and of their properties in the various sectors. It turns out that the moduli action, which
can be derived from the interactions of the moduli on disks with at least a part of their
boundary attached to the Dinstantons [50], is exact with respect to the supersymmetry
charge Q of weight
Therefore Q can be used as the equivariant BRSTcharge to localize the integral over the
moduli space provided one considers U(1)4 transformations under which it is invariant.
This corresponds to requiring that
1
2
+ , + , + , +
1
2
1
2
1
2
.
ǫ1 + ǫˆ2 + ǫˆ3 + ǫ4 = 0 .
M
dI
V = Y
Y (χI,σ − χI,τ + δστ ) .
– 9 –
Thus we are left with three equivariant parameters, say ǫ1, ǫˆ2 and ǫ4; as we will see, the
latter is related to the (equivariant) mass m of the adjoint hypermultiplet of N = 2∗ theory.
As shown in appendix C, all instanton moduli can be paired in Qdoublets of the type
(ϕα, ψα) such that
Q ϕα = ψα ,
Q ψα = Q2ϕα = λα ϕα
where λα are the eigenvalues of Q2, determined by the action of the Cartan subgroup of the
full symmetry group of the theory, namely the gauge group (2.10), the ADHM group (3.6),
and the residual isometry group U(1)4 with parameters satisfying (3.12) in such a way that
the invariant points in the moduli space are finite and isolated. The only exception to this
structure of Qdoublets is represented by the neutral bosonic moduli with weight
{0, 0, 0, 0}−1
transforming in the adjoint representation (dI , d¯I ) of the ADHM group U(dI ), which
remain unpaired. We denote them as χI , and in order to obtain the instanton partition
function we must integrate over them. In doing so, we can exploit the U(dI ) symmetry to rotate
χI into the maximal torus and write it in terms of the eigenvalues χI,σ, with σ = 1, · · · , dI ,
which represent the positions of the Dinstantons of type I in the vplane. In this way we
are left with the integration over all the χI,σ’s and a CauchyVandermonde determinant
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
More precisely, the instanton partition function in the presence of a surface operator of
type ~n is defined by
M
{dI } I=1
where z{dI } is the result of the integration over all Qdoublets which localizes on the fixed
points of Q2, and qI is the counting parameter associated to the Dinstantons of type I.
With the convention that z{dI =0} = 1, we find
z{dI } = V
Y
α
λα
(−)Fα+1
,
where the index α labels the Qdoublets and λα denotes the corresponding eigenvalue of Q2.
This contribution goes to the denominator or to the numerator depending upon the bosonic
or fermionic statistics (Fα = 0 or 1, respectively) of the first component of the doublet.
Explicitly, using the data in table 1 of appendix C and the determinant (3.15), we find
(3.16)
(3.17)
where dM+1 = d1, nM+1 = n1 and aM+1,t = a1,t. The integrations in (3.16) must be
suitably defined and regularized. The standard prescription [41, 42, 53] is to consider aI,s
to be real and close the contours in the upperhalf χI,σ planes with the choice
Im ǫ4 ≫ Im ǫˆ3 ≫ Im ǫˆ2 ≫ Im ǫ1 > 0 ,
and enforce (3.12) at the very end of the calculations.
In this way one finds that these integrals receive contributions from the poles of z{dI },
which are in fact the critical points of Q2. Such poles can be put in onetoone
correspondence with a set of N Young tableaux Y = {YI,s}, with I = 1, · · · , M and s = 1, · · · nI , in
the sense that the box in the ith row and jth column of the tableau YI,s represents one
component of the critical value:
χI+(j−1)modM,σ = aI,s +
(i − 1) +
ǫ1 +
(j − 1) +
ǫˆ2 .
(3.20)
1
2
1
2
Note that in this correspondence, a single tableau accounts for dI ! equivalent ways of
relabeling χI,σ.
where YI(,js) denotes the height of the jth column of the tableau YI,s, and the subscript
index I + 1 − J is understood modulo M .
The instanton partition function (3.16) can thus be rewritten as a sum over Young
tableaux as follows
M
Y I=1
Zinst[~n] = X Y qdI(Y ) Z(Y )
I
where Z(Y ) is the residue of z{dI} at the critical point Y . This is obtained by deleting
in (3.18) the denominator factors that yield the identifications (3.20), and performing these
identifications in the other factors. In other terms,
Z(Y ) = V(Y )
Y
α : λα(Y )6=0
[λα(Y )](−)Fα+1 ,
where V(Y ) and λα(Y ) are the Vandermonde determinant and the eigenvalues of Q2
evaluated on (3.20).
A more efficient way to encode the eigenvalues λα(Y ) is to employ the character of the
action of Q2, which is defined as follows
X{dI} = X(−)Fαeiλα .
α
Summing over the Young tableaux collections Y we get all the nontrivial critical points
corresponding to all possible values of {dI }. Eq. (3.20) tells us that we get a distinct χI,σ
for each box in the jth column of the tableau YI+1−j mod M,s. Relabeling the index j as
with J = 1, . . . M , we have
j → J + j M ,
dI (Y ) = X
X YI(+J1+−jMJ,s) ,
If we introduce and
dI
σ=1
nI
s=1
VI = X eiχI,σ− 2i (ǫ1+ǫˆ2) ,
WI = X eiaI,s
T1 = eiǫ1 , T2 = eiǫˆ2 , T4 = eiǫ4 ,
M
I=1
we can write the contributions to the character from the various Qdoublets as in the last
column of table 1 in appendix C. Then, by summing over all doublets and adding also the
contribution of the Vandermonde determinant, we obtain
X{dI} = (1 − T4) X
−(1 − T1)VI∗VI + (1 − T1)VI∗+1VI T2 + VI∗WI + WI∗+1VI T1T2 .
(3.21)
(3.22)
sions dI (Y ) and the eigenvalues λα(Y ). Thus, the character X(Y ) associated to a set of
Young tableaux is obtained from X{dI } by substituting (3.20) into the definitions of VI ,
namely
VI =
By analyzing X(Y ) obtained in this way we can extract the explicit expression for the
eigenvalues λs(Y ) and finally write the instanton partition function. This procedure is easily
implemented in a computer program, and yields the results we will use in the next sections. In
appendix (C.1), as an example, we illustrate these computations for the SU(2) gauge theory.
In our analysis we worked with the moduli action that describes Dbranes probing
the orbifold geometry. An alternative approach works with the resolution of the orbifold
geometry [54, 55]. This involves analyzing a gauged linear sigmamodel that describes a
system of D1 and D5branes in the background C × C/ZM × T ⋆S2
× R2. One then uses
the localization formulas for supersymmetric field theories defined on the 2sphere [56, 57]
to obtain exact results. This is a very powerful approach since it also includes inherently
stringy corrections to the partition function arising from worldsheet instantons [54]. The
results for the instanton partition function of the N = 2⋆ theory in the presence of surface
operators obtained in [55] are equivalent to our results in (3.18).
3.2
Map between parameters
One of the key points that needs to be clarified is the map between the microscopic counting
parameters qI which appear in (3.23), and the parameters (q, tI ) which were introduced in
section 2 in discussing SU(N ) gauge theories with surface operators. To describe this map,
we start by rewriting the partition function (3.16) in terms of the total instanton number k
and the magnetic fluxes mI of the gauge groups on the surface operator which are related
to the parameters dI as follows [8, 45]:
d1 = k ,
dI+1 = dI + mI+1 .
Therefore, instead of summing over {dI } we can sum over k and m~ and find
X (q1 · · · qM )k (q2 · · · qM )m2 (q3 · · · qM )m3 · · · (qM )mM Zk, m~[~n]
k, m~
Furthermore, if we set we easily get
qI = e2πi(tI −tI+1) for I ∈ {2, . . . M − 1} ,
qM = e2πi(tM −t1)
and q = Y qI ,
M
I=1
(3.30)
(3.31)
(3.32)
(3.33)
where in the last step we introduced m1 such that P
I mI = 0 (see (2.13)) in order to write
the result in a symmetric form. This is precisely the expected expression of the partition
function in the presence of a surface operator as shown in (2.20) and justifies the map (3.32)
between the parameters of the two descriptions. From (3.33) we see that only differences
of the parameters tI appear in the partition function so that it may be convenient to use
as independent parameters q and the (M − 1) variables
zJ = tJ − t1 for J ∈ {2, . . . M } .
(3.34)
This is indeed what we are going to see in the next sections where we will show how to
extract relevant information from the instanton partition functions described above.
Extracting the prepotential and the twisted superpotential
The effective dynamics on the Coulomb branch of the fourdimensional N = 2⋆ gauge
theory is described by the prepotential F , while the infrared physics of the twodimensional
theory defined on the worldsheet of the surface operator is governed by the twisted
superpotential W. The nonperturbative terms of both F and W can be derived from the
instanton partition function previously discussed, by considering its behavior for small
deformation parameters ǫ1 and ǫ2 and, in particular, in the socalled NekrasovShatashvili
(NS) limit [51].
To make precise contact with the gauge theory quantities, we set
hφi = diag(a, −a) ,
where m is the mass of the adjoint hypermultiplet, and then take the limit for small ǫ1 and
ǫ2. In this way we find [8]:
log Zinst[~n] ≃
−
Finst(ǫ1)
ǫ1ǫ2
+ Winst(ǫ1)
ǫ1
+ O(ǫ2) .
The two leading singular contributions arise, respectively, from the (regularized)
equivariant volume parts coming from the fourdimensional gauge theory and from the
twodimensional degrees of freedom supported on the surface defect D. This can be understood
from the fact that, in the Ωdeformed theory, the respective supervolumes are finite and
given by [1, 58]:
Z
Rǫ41,ǫ2
d4x d4θ −→ ǫ1ǫ2
1
and
Z
R2
ǫ1
d2x d2θ −→ ǫ1
.
1
The nontrivial result is that the functions Finst and Winst defined in this way are analytic
in the neighborhood of ǫ1 = 0. As an illustrative example, we now describe in some detail
the SU(2) theory.
SU(2).
When the gauge group is SU(2), the only surface operators are of type ~n = {1, 1},
the Coulomb branch is parameterized by
(3.35)
(3.36)
(3.37)
(3.38)
and the map (3.32) can be written as
,
where, for later convenience, we have defined z = (t2 − t1). Using the results presented in
appendix C.1 and their extension to higher orders, it is possible to check that the instanton
prepotential arising from (3.36), namely
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45a)
f2inℓ+st1 = 0 for ℓ = 0, 1, · · · ,
f inst = −
2
m2
f inst =
4
f inst =
6
1
2a2
1
16a4
ǫ
2
1
1
ǫ
1
One can check that this precisely agrees with the NS limit of the prepotential derived
for example in [21, 22]. This complete match is a strong and nontrivial check on the
correctness and consistency of the whole construction.
Let us now consider the nonperturbative superpotential, which according to (3.36) is
Winst = lim
ǫ2→0
ǫ1 log Zinst[1, 1] + Finst
.
ǫ2
Differently from the prepotential, Winst is, as expected, a function both of q and x. If we
expand it as
Finst = − ǫl2i→m0
ǫ1ǫ2 log Zinst[1, 1]
Finst =
∞
X f inst
ℓ
ℓ=1
is, as expected, a function only of the instanton counting parameter q and not of x.
Expanding in inverse powers of a, we have
HJEP07(21)68
where fℓ ∼ a2−ℓ. The first few coefficients of this expansion are
with winst ∼ a1−ℓ, using the results of appendix C.1 we find
ℓ
w1inst = −
m − 2
ǫ1
1
x
+
+ 2 + x + · · · q
+ 3 + · · · q2 + · · · ,
Winst =
∞
ℓ=1
X winst
ℓ
x +
+
x
2
2
1
2x2 +
x
3
3
1
x
+
+
x
4
4
2x
2x2 +
x
2
2
+
+
3x3
4
1
2x2 +
1
4x
+ · · · q2 + · · · ,
+ x4 + · · ·
+
+ · · · q2 + · · · ,
x
4x
x
4
+
+ · · · q
w1′ = −
m − 2
ǫ1
w2′ = − a
1
m2
ǫ
2
1
2
1
x + x2 + x3 + x4 + · · · −
1
x − x + · · · q
+ · · · q2 + · · · ,
−
x
2
+
−
x
4
1
x2 +
1
x
+ x2 +
3x3
2
9x3
4
1
x2 +
1
2x
+ x2 +
1
x2 +
1
4x
+ 2x4 + · · ·
+
+ · · · q2 + · · · ,
+ 4x4 + · · ·
−
+ · · · q2 + · · ·
x
2
+
1
2x
+ · · · q
1
4x − 4
x
+ · · · q
and so on. For later convenience we explicitly write down the logarithmic derivatives with
where wℓ′ := x ∂∂x winst . In the coming sections we will show that these expressions are
ℓ
the weakcoupling expansions of combinations of elliptic and quasimodular forms of the
modular group SL(2, Z).
4
Modular anomaly equation for the twisted superpotential
41, 42], and also to N
superpotential W.
In [21, 22] it has been shown for the N = 2⋆ SU(2) theory that the instanton expansions of
the prepotential coefficients (3.42) can be resummed in terms of (quasi) modular forms of
the duality group SL(2, Z) and that the behavior under Sduality severely constrains the
prepotential F which must satisfy a modular anomaly equation. This analysis has been
later extended to N = 2⋆ theories with arbitrary classical or exceptional gauge groups [34,
= 2 SQCD theories with fundamental matter [38, 39]. In this
section we use a similar approach to study how Sduality constrains the form of the twisted
For simplicity and without loss of generality, in the following we consider a full surface
operator of type ~n = {1, 1, · · · , 1} with electromagnetic parameters ~t = {t1, t2, · · · , tN }.
Indeed, surface operators of other type correspond to the case in which these parameters
are not all different from each other and form M distinct sets, namely
~t = n
t1, . . . , t1, t2, . . . , t2, · · · , tM , . . . , tM

{z
n1
} 
{z
n2
}

{z
nM
}
o
.
Thus they can be simply recovered from the full ones with suitable identifications.
Before analyzing the Sduality constraints it is necessary to take into account the
classical and the perturbative 1loop contributions to the prepotential and the superpotential.
The classical contribution. Introducing the notation ~a = {a1, a2, · · · , aN } for the
vacuum expectation values, the classical contributions to the prepotential and the
superpotential are given respectively by
and
Note that if we use the tracelessness condition (2.4), Wclass can be rewritten as
Fclass = πiτ ~a · ~a
Wclass = 2πi ~t · ~a .
Wclass = 2πi X zI aI
N
I=2
S Fclass = − Fclass ,
S Wclass = − Wclass .
where zI is as defined in (3.34).
These classical contributions have very simple behavior under Sduality. Indeed
,
ǫ2
(4.2)
(4.3)
(4.4)
(4.5a)
(4.5b)
(4.6)
(4.7)
(4.8)
HJEP07(21)68
where auv = au − av, and the ceiling function ⌈y⌉ denotes the smallest integer greater than
or equal to y. The first term in (4.8) represents the contribution of the vector multiplet,
while the second term is the contribution of the massive hypermultiplet. Expanding (4.8)
To show these relations one has to use the Sduality rules (2.3) and (2.18), and recall that
S ~a = ~aD :=
1 ∂F
which for the classical prepotential simply yield S(~a) = τ ~a.
The 1loop contribution. The 1loop contribution to the partition function of the
Ωdeformed gauge theory in the presence of a full surface operator of type {1, 1, · · · , 1} can
be written in terms of the function
γ(x) := log Γ2(xǫ1, ǫ2) =
d
ds
Λ
s Z ∞
Γ(s) 0
dt
ts−1e−tx
(e−ǫ1t − 1)(e−ǫ2t − 1)
s=0
where Γ2 is the Barnes double Γfunction and Λ an arbitrary scale. Indeed, as shown for
example in [55], the perturbative contribution is
log Zpert[1, 1, · · · , 1] =
γ auv +
N
X
u,v=1
u6=v
v − u
N
ǫ2 − γ auv + m + ǫ1 +
2
v − u
N
for small ǫ1,2 and using the same definitions (3.36) used for the instanton part, we obtain
the perturbative contributions to the prepotential and the superpotential in the NS limit:
Fpert = − ǫl2i→m0
Wpert = lim
ǫ2→0
ǫ1ǫ2 log Zpert[1, 1, · · · , 1] ,
ǫ1 log Zpert[1, 1, · · · , 1] + Fpert
.
ǫ2
Exploiting the series expansion of the γfunction, one can explicitly compute these
expressions and show that Fpert precisely matches the perturbative prepotential in the NS limit
obtained in [34, 41], while the contribution to the superpotential is novel. For example, in
the case of the SU(2) theory we obtain
Fpert =
1
2
ǫ
2
− 4
m2
Note that, unlike the prepotential, the twisted superpotential has no logarithmic term.6
Furthermore, it is interesting to observe that
4.1
Sduality constraints
Wpert = − 4 ∂a
1 ∂Fpert .
We are now in a position to discuss the constraints on the twisted superpotential arising
from Sduality. Adding the classical, the perturbative and the instanton terms described
in the previous sections, we write the complete prepotential and superpotential in the NS
limit as
F = Fclass + Fpert + Finst = πiτ ~a · ~a + X fℓ(τ, ~a) ,
W = Wclass + Wpert + Winst = 2πi X zI aI + X wℓ(τ, zI , ~a)
N
I=2
∞
ℓ=1
∞
ℓ=1
where for later convenience, we have kept the classical terms separate. The quantum
coefficients fℓ and wℓ scale as a2−ℓ and a1−ℓ, respectively, and account for the perturbative and
instanton contributions. While fℓ depend on the coupling constant τ , the superpotential
coefficients wℓ are also functions of the surface operator variables zI , as we have explicitly
seen in the SU(2) theory considered in the previous section.
6This fact is due to the superconformal invariance, and is no longer true in the pure N = 2 SU(2) gauge
theory, for which we find
2
Wpert = − 2 − 2 log 2a a + ǫ1 −
ǫ
4
Λ
24a
ǫ
6
1
2880a3 +
1
40320a5 + · · · .
The coefficients fℓ have been explicitly calculated in terms of quasimodular forms
in [34, 41] and we list the first few of them in appendix D. Their relevant properties can
be summarized as follows:
satisfy the scaling relation7
• All fℓ with ℓ odd vanish, while those with ℓ even are homogeneous functions of ~a and
f2ℓ(τ, λ ~a) = λ2−2ℓ f2ℓ(τ, ~a) .
Since the prepotential has massdimension two, the f2ℓ are homogeneous polynomials
of degree 2ℓ, in m and ǫ1.
• The coefficients f2ℓ depend on the coupling constant τ only through the Eisenstein
series E2(τ ), E4(τ ) and E6(τ ), and are quasimodular forms of SL(2,Z) of weight
2ℓ − 2, such that
1
f2ℓ − τ , ~a
= τ 2ℓ−2 f2ℓ(τ, ~a)
E2→E2+δ
where δ = π6iτ . The shift δ in E2 is due to the fact that the second Eisenstein series
is a quasimodular form with an anomalous modular transformation (see (A.4)).
• The coefficients f2ℓ satisfy a modular anomaly equation
∂f2ℓ +
∂E2
24
1 ℓ−1
X ∂f2n
n=1 ∂~a ·
∂f2ℓ−2n = 0
∂~a
which can be solved iteratively.
Using the above properties, it is possible to show that Sduality acts on the prepotential
F in the NS limit as a Legendre transform [41, 42].
Let us now turn to the twisted superpotential W. As we have seen in (4.5), Sduality
acts very simply at the classical level but some subtleties arise in the quantum theory. We
now make a few important points, anticipating some results of the next sections. It turns
out that W receives contributions so that the coefficients wℓ do not have a welldefined
modular weight. However, these anomalous terms depend only on the coupling constant τ
and the vacuum expectation values ~a. In particular, they are independent of the continuous
parameters zI that characterize the surface operator. For this reason it is convenient to
consider the zI derivatives of the superpotential:
W
(I) :=
2πi ∂zI
1 ∂W = aI + X wℓ(I)(τ, zI , ~a)
∞
ℓ=1
where, of course, wℓ(I) := 21πi ∂∂wzIℓ .
Combining intuition from the classical Sduality transformation (4.5b) with the fact
that the zI derivative increases the modular weight by one, and introduces an extra factor
of (−τ ) under Sduality, we are naturally led to propose that
7To be precise, one should also scale Λ → λΛ in the logarithmic term of f2.
S W
(I) = τ W
(I) .
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
This constraint can be solved if we assume that the coefficients w(I) satisfy the following
properties (which are simple generalizations of those satisfied by fℓ):
• They are homogeneous functions of ~a and satisfy the scaling relation
wℓ(I)(τ, zI , λ ~a) = λ1−ℓ wℓ(I)(τ, zI , ~a) .
Given that the twisted superpotential has massdimension one, it follows that w(I)
must be homogeneous polynomials of degree ℓ in m and ǫ1.
• The dependence of w(I) on τ and zI is only through linear combinations of
quasiℓ
modular forms made up with the Eisenstein series and elliptic functions with total
weight ℓ, such that
We are now ready to discuss how Sduality acts on the superpotential coefficients wℓ(I).
Recalling that
S(~a) = ~aD :=
where f = Fpert + Finst, we have
S w(I)
ℓ
= w(I)
ℓ
1
− τ , − τ
zI , ~aD
= τ ℓ wℓ(I)(τ, zI , ~aD)
where in the last step we exploited the scaling behavior (4.18) together with (4.20). Using
this result in (4.16) and formally expanding in δ, we obtain
w(I)
ℓ
1
− τ , − τ
zI , ~a
= τ ℓ wℓ(I)(τ, zI , ~a)
.
(4.18)
ℓ
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
(4.24)
1
τ
S W
(I) = W
(I) τ, zI , ~a +
= W
(I) + δ
∂W
∂E2
(I)
δ ∂f
12 ∂~a
+
E2→E2+δ
1 ∂W
12 ∂~a
(I) ∂f
· ∂~a
+ O(δ2) .
The constraint (4.17) is satisfied if
which also implies the vanishing of all terms of higher order in δ. This modular anomaly
equation can be equivalently written as
where we have defined w(I) = aI .
0
∂W
∂E2
(I)
+
1 ∂W
12 ∂~a
(I) ∂f
· ∂~a
= 0 ,
∂wℓ(I)
∂E2
+
1 ℓ−1
X ∂fℓ−n
12
n=0
∂~a
∂wn(I)
· ∂~a
= 0
(see for example [34])
m → ∞
and q → 0 such that q m2N = (−1)N Λ2N
is finite,
(7.1)
only one parameter x = e2πi z, this scaling is
where Λ is the strong coupling scale of the pure N = 2 theory. In presence of a surface
operator, this limit must be combined with a scaling prescription for the continuous
variables that characterize the defect. For surface operators of type {p, N − p}, which possess
m → ∞
and
x → 0 such that x mN = (−1)p−1x0 ΛN
is finite.
(7.2)
Here x0 = e2πi z0 is the parameter that labels the surface operator in the pure theory `a la
Performing the limits (7.1) and (7.2) on the localization results described in the
prex0+ x0
1
We have explicitly verified this expression in all cases up to SU(7), and for the low rank
groups we have also computed the higher instanton corrections.11 With some simple algebra
one can check that, up to the order we have worked, W
both i and j are ≤ p or > p. Furthermore, one can verify that
′ is not singular for ai → aj when
as a consequence of the tracelessness condition on the vacuum expectation values.
We now show that this result is completely consistent with the exact twisted chiral
ring relation obtained in [5]. For the pure N = 2 SU(N ) theory with a surface operator
parameterized by x0, the twisted chiral ring relation takes the form [5]
1
x0
N
Y
Λ2N
2
1
x
∂
N
Y
1
2 ∂ai j6=i aij
0 2
+ O Λ3N . (7.4)
(7.3)
(7.5)
(7.6)
(7.7)
GukovWitten [2–5]. vious sections, we obtain where
Wi′ = −ai − ΛN
x0 +
with
′
W1 = −a− 2a
Λ
2
where a = a1.
W
′ =
Wi
′
p
X
i=1
N
X
i=1
Wi′ = 0
PN (y) − ΛN
x0 +
1
x0
PN (y) = Y
y − ei
where ei are the quantum corrected expectation values of the adjoint scalar. They reduce
to ai in the classical limit Λ → 0 and parameterize the quantum moduli space of the theory.
The ei, which satisfy the tracelessness condition
N
were explicitly computed long ago in the 1instanton approximation in [64, 65] by evaluating
the period integrals of the SeibergWitten differential and read
ei = ai − Λ2N ∂
Y
1
+
1
x
2
0
∂
N
Y
1
for i = 1, · · · , N . Comparing with (7.4), we see that, up to an overall sign, yi coincide with
the derivatives of the superpotential Wi′ we obtained from localization. Therefore, we can
rewrite the left hand side of (7.6) in a factorized form and get
y + Wi ) − PN (y) + ΛN
′
x0 +
1
x0
The higher instanton corrections can be efficiently computed using localization
methods [66–69], but their expressions will not be needed in the following.
Inserting (7.9) into (7.7) and systematically working order by order in ΛN , it is possible
to show that the N roots of the chiral ring equation (7.6) are
1
x0
N
Y
i=1
X1 =
1 ∂W
This shows a perfect match between our localization results and the semiclassical expansion
of the chiral ring relation of [5], and provides further nontrivial evidence for the equivalence
of the two descriptions. Let us elaborate a bit more on this. According to [5], a surface
operator of type {p, N −p} has a dual description as a Grassmannian sigmamodel coupled to the
SU(N ) gauge theory, and all information about the twisted chiral ring of the sigmamodel is
contained in two monic polynomials, Q and Qe of degree p and (N −p) respectively, given by
Q(y) =
p
ℓ=0
X yℓ Xp−ℓ ,
N−p
k=0
with X0 = Xe0 = 1. Here, Xℓ are the twisted chiral ring elements of the Grassmannian
sigmamodel, and in particular
where W is the superpotential of the surface operator of type {p, N − p}. The
polynomial Qe encodes the auxiliary information about the “dual” surface operator obtained by
(7.8)
(7.9)
(7.11)
(7.12)
(7.13)
HJEP07(21)68
sending p → (N − p). The crucial point is that, according to the proposal of [5], the two
polynomials Q and Qe satisfy the relation
Q(y) Qe(y) − PN (y) + ΛN
x0 +
= 0 .
Comparing with (7.11), we are immediately led to the following identifications12
1
x0
N
Y
j=p+1
Thus, using (7.13) and (7.3), we find
p
i=1
Q(y) = Y
y + Wi′ ,
X
i=1
Wi′ = W ′ .
This equality shows that our localization results for the superpotential of the surface
operator of type {p, N − p} in the pure SU(N ) theory perfectly consistent with the proposal
of [5], thus proving the duality between the two descriptions. All this is also a remarkable
consistency check of the way in which we have extracted the semiclassical results for the
twisted chiral ring of the Grassmannian sigmamodel and of the twisted superpotential we
have computed.
Inspired by the previous outcome, we now analyze the twisted chiral ring relation for simple
operators in N
= 2⋆ theories using the SeibergWitten curve and compare it with our
localization results for the undeformed theory. To this aim, let us first recall from section 6.1
(see in particular (6.3) with ǫ1 = 0) that for a simple surface operator corresponding to
the following partition of the Coulomb parameters
(7.14)
(7.15)
(7.16)
(7.17)
(7.18)
HJEP07(21)68
the zderivative of the superpotential is
′
Wi′ = −ai + m2 X h
1 +
m3
X
j6=i aij
2 j6=k6=i aij aik
n
ai, {aj with j 6= i}

N{−z 1
o
,
}
h′′
1
+
m4
6
3
aij
Let us now see how this information can be retrieved from the SeibergWitten curve of
the N = 2⋆ theories. As is well known, in this case there are two possible descriptions
(see [44] for a review). The first one, which we call the DonagiWitten curve [70], is written
12We have chosen a specific ordering in which the first p factors correspond to the first p vacuum
expectation values ai; of course one could as well choose a different ordering by permuting the factors.
naturally in terms of the modular covariant coordinates on moduli space, while the second,
which we call the d’HokerPhong curve [71], is written naturally in terms of the quantum
corrected coordinates on moduli space. As shown in [44], these two descriptions are linearly
related to each other with coefficients depending on the second Eisenstein series E2.
Since our semiclassical results have been resummed into elliptic and quasimodular
forms, we use the DonagiWitten curve, which for the SU(N ) gauge theory is an N fold
cover of an elliptic curve. It is described by the pair of equations:
Y 2 = X3
E4
− 48
X +
E6 ,
864
FN (y, X, Y ) = 0 .
(7.19)
The first equation describes an elliptic curve and thus we can identify (X, Y ) with the
Weierstraß function and its derivative (see (A.11)). More precisely we have
X = −℘e = −h′1 +
Y =
1 ℘ ′ =
2 e
1 ′′
2 1
h
1
12
E2 ,
The second equation in (7.19) contains a polynomial in y of degree N which encodes the
modular covariant coordinates Ak on the Coulomb moduli space of the gauge theory:
where Pk are the modified DonagiWitten polynomials introduced in [44]. The first few of
them are:13
N
k=0
FN (y, X, Y ) =
X(−1)kAk PN−k(y, X, Y )
P0 = 1 ,
P2 = y2 − m2 X ,
P1 = y ,
P3 = y3 − 3 y m2 X + 2 m3 Y ,
P4 = y4 − 6 m2 y2 X + 8 y m3 Y − m4 3 X2
On the other hand, the first few modular covariant coordinates Ak are (see [44]):
A2 =
X aiaj +
A3 =
A4 =
i<j
X
i<j<k
X
i<j<k<ℓ
+
m4
288
m2
12
N
2
m4
aiaj ak − 144
aiaj akaℓ +
E22 − E4
E2 +
m4
288
E22 − E4
m2
12
N − 2
2
E22 − E4
X
X
2
i j6=i aij
ai
X
2
i6=j aij
1
+ O(m6) ,
E2
X aiaj +
i<j
2
i<j k6=ℓ akℓ
X
X aiaj + 3 X
X
2
i j6=i aij
a
2
i
m4
48
−
E22
N
2
1
− 24
E4 .
+ O(m6) ,
+ O(m6) ,
(7.20)
(7.21)
(7.22)
(7.23)
and so on.
also here.
13The E4 term in P4 is one of the modifications which in [44] were found to be necessary and is crucial
We now have all the necessary ingredients to proceed. First of all, using the above
expressions and performing the decoupling limits (7.1) and (7.2), one can check that the
DonagiWitten equation FN = 0 reduces to the twisted chiral ring relation (7.6) of the pure
theory. Of course this is not a mere coincidence; on the contrary it supports the idea that
the DonagiWitten equation actually encodes also the twisted chiral ring relation of the
simple codimension4 surface operators of the N = 2⋆ theories. Secondly, working order by
order in the hypermultiplet mass m, one can verify that the N roots of the DonagiWitten
equation are given by
′
yi = ai − m2 X h
1
m3
X
where z∗ is an arbitrary reference point. Indeed, in the DonagiWitten variables, the
differential is simply λSW (z) = y(z) dz. Given that the DonagiWitten curve is an N fold
W(z) =
λSW
Z z
z∗
Remarkably, this precisely matches, up to an overall sign, the answer (7.18) for the simple
codimension2 surface operator we have obtained using localization. Once again, we have
exhibited the equivalence of twisted chiral rings calculated for the two kinds of surface
operators. Furthermore, we can rewrite the DonagiWitten equation in a factorized form
as follows
N
Y
i=1
y + Wi′ − FN (y, X, Y ) = 0
which is the N = 2⋆ equivalent of the pure theory relation (7.11).
At this point one is tempted to proceed as in the pure theory and try to deduce also
the superpotential for surface operators of type {p, N − p}. However, from our explicit
localization results we know that in this case W
tentials of type {1, N − 1}, differently from what happens in the pure theory (see (7.3)).
Thus, a naive extension to the N = 2⋆ of the proposal of [5] to describe the coupling of a
two dimensional Grassmannian sigmamodel to the four dimensional gauge theory can not
work in this case. This problem as well as the coupling of a flag variety to the N = 2
theory, which is relevant for surface operators of general type, remains an open question
⋆
′ is not simply the sum of the
superpowhich we leave to future investigations.
7.3
Some remarks on the results
The result we obtained from the twisted superpotential in the case of simple operators
is totally consistent with the proposal given in the literature for simple codimension4
surface operators labeled by a single continuous parameter z, whose superpotential has been
identified with the line integral of the SeibergWitten differential of the fourdimensional
gauge theory along an open path [11]:
(7.24)
(7.25)
(7.26)
cover of the torus, the twisted superpotential with the classical contribution proportional
to ai can be obtained by solving for y(z) and writing out the solution on the ith branch.
As we have seen in the previous subsection, the general identification in (7.26) works
also in the pure N = 2 theory, once the parameters in the SeibergWitten differential are
rescaled by a factor of ΛN [5]. This rescaling can be interpreted as a renormalization of
the continuous parameter that labels the surface operator [72].
The agreement we find gives further evidence of the duality between defects realized as
codimension2 and codimension4 operators that we have already discussed in section 5.1,
where we showed the equality of the twisted effective superpotential computed in the two
approaches for simple defects in the SU(2) theory.
We have extended these checks to
defects of type {p, N − p} in pure N = 2 theories, and to simple defects in N = 2⋆ theories
with higher rank gauge groups. All these checks support the proposal of [52] based on a
“separation of variables” relation.
8
Conclusions
In this paper we have studied the properties of surface operators on the Coulomb branch of
the four dimensional N = 2⋆ theory with gauge group SU(N ) focusing on the superpotential
W. This superpotential, describing the effective twodimensional dynamics on the defect
worldsheet, receives nonperturbative contributions, which we calculated using equivariant
localization. Furthermore, exploiting the constraints arising from the nonperturbative
SL(2, Z) symmetry, we showed that in a semiclassical regime in which the mass of the
adjoint hypermultiplet is much smaller than the classical Coulomb branch parameters, the
twisted superpotential satisfies a modular anomaly equation that we solved order by order
in the mass expansion.
We would like to remark some interesting properties of our results. If we focus on the
derivatives of the superpotential, the coefficients of the various terms in the mass expansion
are linear combination of elliptic and quasimodular forms with a given weight. The explicit
expression for the twisted superpotential can be written in a very general and compact form
in terms of suitable restricted sums over the root lattice of the gauge algebra.
The match of our localization results with the ones obtained in [5] by studying the
coupling with twodimensional sigma models is a nontrivial check of our methods and
provides evidence for the duality between the codimension2 and codimension4 surface
operators proposed in [52]. Further evidence is given by the match of the twisted
superpotentials in the N = 2⋆ theory, which we proved for the simple surface operators using
the DonagiWitten curve of the model. A key input for this match is the exact quantum
expression of the chiral ring elements calculated using localization [44, 69]. It would be
really important to extend the discussion of this duality to more general surface operators
described by a generic Levi decomposition.
There are several possible extensions of our work. A very direct one would be to check
that the general expression given for the twisted superpotential is actually valid for all
simply laced groups, in analogy to what happens for the fourdimensional prepotential.
A technically more challenging extension would be to study surface operators for theories
HJEP07(21)68
with nonsimply laced gauge groups. The prepotential in these cases has been calculated
in [42] using localization methods and expressed in terms of modular forms of suitable
congruence subgroups of SL(2,Z), and it would be very interesting to similarly calculate
the twisted superpotential in a semiclassical expansion.
Another interesting direction would be to study surface operators in SQCD theories.
For SU(N ) gauge groups, the prepotential as well as the action of Sduality on the infrared
variables have been calculated in a special locus of the Coulomb moduli space that has
a Z
N symmetry [38, 39]. Of special importance was the generalized Hecke groups acting
on the period integrals and the period matrix of the SeibergWitten curve. It would be
worthwhile to explore if such groups continue to play a role in determining the twisted
A related development would be to analyze the higher order terms in the ǫ2 expansion
of the partition function (see (3.36)) and check whether or not they also obey a modular
anomaly equation like the prepotential and the superpotential do. This would help us in
clarifying the properties of the partition function in the presence of a surface operator in
a general Ω background.
There has been a lot of progress in understanding M2 brane surface operators via the
4d/2d correspondence. For higher rank theories, explicit results for such surface defects
have been obtained in various works including [73–77]. In particular in [75], the partition
functions of theories with Nf2 free hypermultiplets on the deformed 4sphere in the presence
of surface defects have been related to specific conformal blocks in Toda conformal field
theories. This has been extended in [76, 77] to study gauge theory partition functions in the
presence of intersecting surface defects. It would be interesting to study such configurations
directly using localization methods.
Acknowledgments
We would like to thank Dileep Jatkar, Madhusudhan Raman and especially Jan Troost
for useful discussions and Matteo Beccaria for comments on the manuscript. The work of
M.B. and M.F. is partially supported by the Compagnia di San Paolo contract “MAST:
Modern Applications of String Theory” TOCall320120088. The work of M.B., M.F. and
A.L. is partially supported by the MIUR PRIN Contract 2015MP2CX4 “Nonperturbative
Aspects Of Gauge Theories And Strings”.
A
Useful formulas for modular forms and elliptic functions
In this appendix we collect some formulas about quasimodular forms and elliptic functions
that are useful to check the statements of the main text.
Eisenstein series.
We begin with the Eisenstein series E2n, which admit a Fourier
expansion in terms of q = e2πiτ of the form
E2n = 1 +
2
ζ(1 − 2n)
∞
k=1
X σ2n−1(k)qk ,
(A.1)
where σp(k) is the sum of the pth powers of the divisors of k. More explicitly we have
(A.2)
E6 = 1 − 504 X σ5(k)qk = 1 − 504q − 16632q2 − 122976q3 − 532728q4 + · · · .
Under a modular transformation τ → acττ++db , with a, b, c, d ∈ Z and ad − bc = 1, the
Eisenstein series transform as
HJEP07(21)68
6
πi
E2 → (cτ + d)2 E2 +
c (cτ + d) , E4 → (cτ + d)4 E4 , E6 → (cτ + d)6 E6 . (A.3)
In particular, under Sduality we have
E2(τ ) → E2 − τ
E4(τ ) → E4 − τ
E6(τ ) → E6 − τ
1
1
1
= τ 2 E2(τ ) + δ ,
= τ 4E4(τ ) ,
= τ 6 E6(τ )
where δ = π6iτ .
Elliptic functions.
The elliptic functions that are relevant for this paper can all be
obtained from the Jacobi θfunction
where x = e2πiz. From θ1, we first define the function
and the Weierstraß ℘function
θ1(zτ ) =
q 12 (n− 21 )2 (−x)(n− 12 )
1 ∂
2πi ∂z
h1(zτ ) =
log θ1(zτ ) = x
log θ1(zτ ) ,
℘(zτ ) = − ∂z2 log θ1(zτ ) − 3
E2(τ ) .
∂
In most of our formulas the following rescaled ℘function appears:
℘e(zτ ) :=
℘(z, τ )
4π2
= x
∂
log θ1(zτ ) − 12 E2(τ ) ,
1
which we can write also as
℘e(zτ ) = h′1(zτ ) − 12 E2(τ ) .
1
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
Another relevant elliptic function is the derivative of the Weierstraß function, namely
The Weierstraß function and its derivative satisfy the equation of an elliptic curve, given by
℘e′(zτ ) :=
1 ∂
2πi ∂z ℘e(zτ ) = x
∂
∂x ℘e(zτ ) = h′1′(zτ ) .
℘e′(zτ )2 + 4 ℘e(zτ )3 − E124 ℘e(zτ ) − 216
By differentiating this equation, we obtain
which, using (A.9) and (A.10), we can rewrite as
℘e′′(zτ ) = −6 ℘e(zτ )2 +
E4
24
h′1′′(zτ ) = −6 h′1(zτ )
2 + E2 h′1(zτ ) −
E2
2 − E4
24
.
The function h1, ℘ and ℘e′ have wellknown expansions near the point z = 0. However,
e
a different expansion is needed for our purposes, namely the expansion for small q and x.
To find such an expansion we observe that q and x variables must be rescaled differently,
as is clear from the map (3.32) between the gauge theory parameters and the microscopic
counting parameters. In particular for M = 2 this map reads (see also (3.39))
q = q1q2 ,
x = q2 ,
q → λ2q
x → λx .
so that if the microscopic parameters are all scaled equally as qi −→ λqi, then the gauge
theory parameters scale as
With this in mind, we now expand the elliptic functions for small λ and set λ = 1 in the
end, since this is the relevant expansion needed to compare with the instanton calculations.
Proceeding in this way, we find14
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
HJEP07(21)68
h1(xq) = h1(λxλ2q)
1
− 2
+ λ
λ=1
x − x
+ λ2 q
2
x2 − x
2
+ λ3 q
3
x3 +
q
2
x − qx − x
3
− λ4 x4 + λ5
q
3
x − q2x − x
5
− λ
6 q2x2 + x6 + · · ·
λ=1
1
= − 2 −
x − x − x
2 q2 +
1
x3 +
1
x
+ · · · q3 + · · · ,
1
x − x q
14Depending on the context, we denote the arguments of the elliptic functions by either (zτ ) as we did
so far, or by their exponentials (xq) when the expansions are being used.
− λ
1
x3 +
2
℘e′(xq) = ℘e′(λxλ2q)
λ=1
= − 12 − x + 2x2 + 3x3 + 4x4 + · · · −
2
x2 +
1
x − 6 + · · · q2 − x3 + · · · ,
3q3
x − x + λ2 4q2
q
x2 − 4x2
+ λ3 9q3
x3 +
2
x − qx − 9x3
= − x + 4x2 + 9x3 + 16x4 + · · · +
+ x + λ2
2q2
− x2 + 2q − 2x2
+ qx + 3x3
+ λ4 6q2 − 4x4 + · · ·
1
x − 2 + x q
λ=1
− 16λ4x4 + · · ·
1
x − x q+
λ=1
4
x2 +
1
+ · · · q2 + 9q3
x3 + · · · .
As a consistency check it is possible to verify that, using these expansions and those of the
Eisenstein series in (A.2), the elliptic curve equation (A.11) is identically satisfied order by
order in λ.
As we have seen in section 2, the modular group acts on (zτ ) as follows:
(zτ ) →
z
cτ + d cτ + d
with a, b, c, d ∈ Z and ad − bc = 1. Under such transformations the Weierstraß function
and its derivative have, respectively, weight 2 and 3, namely
℘(zτ ) → ℘
℘ ′(zτ ) → ℘
cτ + d cτ + d
cτ + d cτ + d
= (cτ + d)3 ℘ ′(zτ ) .
℘e(zτ ) → ℘e − τ
℘e′(zτ ) → ℘e
z
z
− τ
1
− τ
1
− τ
= τ 2 ℘e(zτ ) ,
= −τ 3 ℘e′(zτ ) .
Of course, similar relations hold for the rescaled functions ℘e and ℘e′. In particular, under
B
Generalized instanton number in the presence of fluxes
In this appendix we calculate the second Chern class of the gauge field in the presence of
a surface operator for a generic Lie algebra g.
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
From this expression we obtain
Tr F ∧ F = Tr Fb ∧ Fb + 2 Tr d g(r) dθ ∧ Fb
− 2 i Tr dθ ∧ g(r), Ab ∧ Fb
= Tr Fb ∧ Fb + 2 Tr d g(r) dθ ∧ Fb
+ 2 Tr g(r)dθ ∧ dFb − i Ab ∧ Fb − i Fb ∧ Ab
.
The last term vanishes due to the Bianchi identity, and thus we are left with
Tr F ∧ F = Tr Fb ∧ Fb + 2 Tr d g(r) dθ ∧ Fb
We now assume that the function g(r) has components only along the Cartan directions
of g, labeled by an index i, such that
Surface operator Ansatz.
A surface operator creates a singularity in the gauge field
A. As discussed in the main text, we parametrize the spacetime R4
≃ C2 by two complex
variables (z1 = ρ eiφ , z2 = r eiθ), and consider a twodimensional defect D located at z2 = 0
and filling the z1plane. In this setup, we make the following Ansatz [8]:
where Ab is regular all over R4 and g(r) is a gvalued function regular when r → 0. The
corresponding field strength is then
F := dA − i A ∧ A = Fb + d g(r) dθ − i dθ ∧ g(r), Ab .
HJEP07(21)68
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
This means that near the defect the gauge connection behaves as
rli→m0 gi(r) = −γi and
rl→im∞ gi(r) = 0 .
≃ − diag γ1, · · · , γrank(g) dθ
for r → 0. Using this in (B.4), we have
Tr F ∧ F = Tr Fb ∧ Fb + 2 X d gi(r) dθ ∧ Fbi .
mi =
2π
D
Fi .
i
Notice that in the last term we can replace Fbi with Fi because the difference lies entirely in
the transverse directions of the surface operator and thus does not contribute in the wedge
product with dθ. Since the defect D effectively acts as a boundary in R4 located at r = 0,
integrating (B.7) over R4 we have
Tr F ∧ F =
Tr Fb ∧ Fb + X γi Z
2π
D
Fi = k + X γi mi .
i
Here we have denoted by k the instanton number of the smooth connection Ab and taken
into account a factor of 2π originating from the integration over θ. Finally, we have defined
These quantities, which we call fluxes, must satisfy a quantization condition that can be
understood as follows. All fields of the gauge theory are organized in representations15
of g and, in particular, can be chosen to be eigenstates of the Cartan generators Hi with
eigenvalues λi. These eigenvalues define a vector ~λ = {λi}, which is an element of the weight
lattice ΛW of g. Let us now consider a gauge transformation in the Cartan subgroup with
parameters ω~ = {ωi}. On a field with weight ~λ, this transformation simply acts by a phase
factor exp i ω~ · ~λ . From the point of view of the twodimensional theory on the defect,
the Cartan gauge fields Ai must approach a puregauge configuration at infinity so that
with ωi being a function of φ, the polar angle in the z1plane. In this situation, for the
corresponding gauge transformation to be singlevalued, one finds
Ai ∼ dωi for ρ → ∞ ,
(B.10)
with integer n. In other words ω~ · ~λ must be a map from the circle at infinity S1∞ into S1
with integer winding number n. Given this, we have
Z
D
I
S1∞
2πmi =
Fi =
dωi = ωi(φ + 2π) − ωi(φ) .
Then, using (B.11), we immediately deduce that
m~ · ~λ ∈ Z .
m~ ∈ (ΛW )∗ .
For the group SU(N ) this condition amounts to say that m~ must belong to the dual of the
weight lattice:
The SU(N ) case. For U(N ) the Cartan generators Hi can be taken as the diagonal
(N ×N ) matrices with just a single nonzero entry equal to 1 in the ith place (i = 1, · · · , N ).
The restriction to SU(N ) can be obtained by choosing a basis of (N −1) traceless generators,
for instance (Hi − Hi+1)/√2. In terms of the standard orthonormal basis {~ei} of RN , the
(N − 1) simple roots of SU(N ) are then {(~e1 − ~e2), (~e2 − ~e3), · · · } and the root lattice ΛR is
the Zspan of these simple roots. Note that ΛR lies in a codimension1 subspace orthogonal
to Pi ~ei, and that the integrality condition for the weights is simply α~ · ~λ ∈ Z for any root
α~. This shows that the weight lattice is the dual of the root lattice, or equivalently that the
dual of the weight lattice is the root lattice: (ΛW )∗ = ΛR. Therefore, the condition (B.14)
implies that the flux vector m~ must be of the form
m~ = n1(~e1 − ~e2) + n2(~e2 − ~e3) + · · · + nN−1(~eN−1 − ~eN )
with ni ∈ Z .
This simply corresponds to
m~ =
X mi ~ei
with mi ∈ Z
and
X mi = 0 .
The fact that the fluxes mi are integers (adding up to zero) has been used in the main text.
15Here for simplicity we consider the gauge group G to be the universal covering group of g; in particular
for g = AN−1, we take G = SU(N ).
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
Generic surface operator.
The case in which all the γi’s defined in (B.5) are distinct,
corresponds to the surface operator of type [1, 1, . . . , 1], also called full surface operator.
If instead some of the γi’s coincide, the surface operator has a more generic form. Let us
consider for example the case in which the SU(N ) gauge field at the defect takes the form
(see (2.8)):
for r → 0, which corresponds to splitting the gauge group according to
HJEP07(21)68
SU(N ) → S U(n1) × U(n2) × · · · × U(nM ) .
The calculation of the second Chern class (B.8) proceeds as before, but the result can be
(B.17)
(B.18)
(B.19)
(B.20)
written as follows
with
8π2
M
Tr F ∧ F = k + X γI mI
mI =
X mi =
nI
i=1
2π
X Fi =
D i=1
Tr F U(nI ) .
M
I=1
2π
D
Here we see that it is the magnetic flux associated with the U(1) factor in each subgroup
U(nI ) that appears in the expression for the generalized instanton number in the presence
of magnetic fluxes.
C
Ramified instanton moduli and their properties
In this appendix we describe the instanton moduli in the various sectors. Our results are
Let us first consider the neutral states of the strings stretching between two
Dsummarized in table 1.
instantons.
• (−1)/(−1) strings of type II: all moduli of this type transform in the adjoint
representation (dI , d¯I ) of U(dI ). A special role is played by the bosonic states created in the
NeveuSchwarz (NS) sector of such strings by the complex oscillator ψv in the last complex
spacetime direction, which is neutral with respect to the orbifold. We denote them by χI .
They are characterized by a U(1)4 weight {0, 0, 0, 0} and a charge (+1) with respect to the
last U(1). The complex conjugate moduli χ¯I , with weight {0, 0, 0, 0} and charge (−1), are
paired in a Qdoublet with the fermionic moduli η¯I coming from the ground state of the
Ramond (R) sector with weight
1 1 1 1
− 2 , − 2 , − 2 , − 2
and charge (− 21 ). All other moduli in
this sector are arranged in Qdoublets. One doublet is (AIz1 , MIz1 ), where AIz1 is from the
ψz1 oscillator in the NS sector with weight {+1, 0, 0, 0} and charge 0, and MIz1 is from the
R ground state + 12 , − 2 , − 2 , − 2
1 1 1
with charge (+ 21 ). Another doublet is (AIz4 , MIz4 ), where
AIz4 is from the ψz4 oscillator in the NS sector with weight {0, 0, 0, +1} and charge 0, and
MIz4 is from the R ground state with weight − 2 , − 2 , − 12 , + 1
1 1
2
and charge (+ 12 ). Also the
Doublet (−)Fα ChanPaton
U(1)4charge
(χ¯I , η¯I )
(AIz1 , MIz1 )
(AIz4 , MIz4 )
(λI , DI )
(λIz1 , DIz1 )
(AIz2 , MIz2 )
(λIz2 , DIz2 )
(dI , d¯I )
(dI , d¯I )
(dI , d¯I )
(dI , d¯I )
(dI , d¯I )
(nI , d¯I )
(nI , d¯I )
(dI , n¯I+1)
(dI , n¯I+1)
+1, 0, 0, 0
+ 21 ,+ 12 ,+ 12 ,+ 12
+ 21 ,− 12 ,− 12 ,+ 12
0, +1, 0, 0
− 21 ,+ 12 ,− 12 ,+ 12
0, 0, −1, 0
+ 21 ,+ 12 ,− 12 ,− 12
+ 12 ,+ 12 , 0, 0
0, 0,− 12 ,+ 21
+ 12 ,+ 12 , 0, 0
0, 0,− 12 ,+ 21
χI,σ − χI,τ
χI,σ − χI,τ + ǫ1
χI,σ − χI,τ + ǫ4
χI,σ − χI,τ
χI,σ − χI,τ + ǫ1 + ǫ4
χI,σ − χI+1,ρ + ǫˆ2
χI,σ − χI+1,ρ + ǫˆ2 + ǫ4
χI,σ − χI+1,ρ − ǫˆ3
χI,σ − χI+1,ρ + ǫ1 + ǫˆ2
VI VI T4
−VI∗VI T1T4
VI∗+1VI T2
−VI∗+1VI T2T4
VI+1VI T1T2T4
−VI∗+1VI T1T2
VI∗WI
−VI∗WI T4
WI∗+1VI T1T2
χI,σ − aI+1,t + 12 (ǫ1 + ǫˆ2) + ǫ4 −WI∗+1VI T1T2T4
For each of them, we display their statistics (−)Fα , the representation of the color and ADHM groups
in which they transform, their charge vector with respect to the U(1)4 symmetry, the eigenvalue
λα of Q
2 and the corresponding contribution to the character. The neutral moduli carrying a
superscript z1, z2, z3 or z4, and the colored moduli in this table are complex. The quantities
appearing in the last column, namely VI , WI , T1,T2 and T4 are defined in (3.26) and (3.27).
complex conjugate doublets are present. Finally, there is a (real) doublet (λI , DI ) where
λI is from the R ground state with weight + 12 , + 1 , + 21 , + 1
2 2
and charge (− 21 ), and DI is
an auxiliary field, and a complex doublet (λIz1 , DIz1 ) with λz1 associated to the R ground
I
state with weight + 21 , − 2 , − 12 , + 1
1
2
and charge (− 12 ), and DIz1 an auxiliary field.
• (−1)/(−1) strings of type I(I + 1): in this sector the moduli transform in the
bifundamental representation (dI , d¯I+1) of U(dI ) × U(dI+1). In order to cancel the phase
ω−1 due to the different representations on the ChanPaton indices at the two endpoints,
the weights under spacetime rotations of the operators creating the states in this sector
must be such that l2 − l3 = 1. In this way they can survive the Z
Applying this requirement, we find a doublet (AIz2 , MIz2 ), AIz2 is from the ψz2 oscillator in
the NS sector with weight {0, +1, 0, 0} and charge 0, and MIz2 is from the R ground state
M orbifold projection.
− 21 , + 1 , − 2 , − 2
1 1
2
with charge (+ 12 ). Another doublet is (A¯Iz3 , M¯ Iz3 ) where A¯Iz3 is from the
ψ¯z3 oscillator in the NS sector with weight {0, 0, −1, 0} and charge 0, and M¯ Iz3 is from the
R ground state
complex Qdoublets, (λIz2 , DIz2 ) and (λIz3 , DIz3 ) where λIz2 and λIz3 are associated to the R
+ 12 , + 1 , − 21 , + 1
2 2
with charge (+ 21 ).16 Furthermore, we find two other
ground states with weights
− 21 , + 21 , − 12 , + 1
2
and
+ 21 , + 21 , − 2 , − 2
1 1
and charges (− 21 ),
− 12 , − 21 , + 21 , − 21 .
16Notice that this last doublet is actually the complex conjugate of a Qdoublet of type (I + 1)I, which is
made of (AIz3 , MIz3 ) with AIz3 corresponding to the weight {0, 0, 1, 0} and MIz3 corresponding to the weight
HJEP07(21)68
the Z
M invariant spectrum, and arise from strings with the opposite orientation.
while DIz2 and DIz3 are auxiliary fields. Also the complex conjugate doublets are present in
• 3/(−1) strings of type II: these open strings have mixed NeumannDirichlet
boundary conditions along the (z1, z2)directions and thus the corresponding states are
characterized by the action of a twist operator Δ [50]. We assign an orbifold charge ω− 12
to this twist operator, so that the states which survive the Z
M projection are those with
weights such that l2 − l3 = 1/2. The moduli in this sector belong to the bifundamental
representation (nI × d¯I ) of the gauge and ADHM groups, and form two complex
doublets. One is (wI , µ I ) where the NS component wI has weight
is (µ ′I , h′I ) where µ ′I is associated to the R ground state with weight
1 1
0, and the R component µ I has weight 0, 0, − 2 , − 2
and charge (+ 12 ). The other doublet
0, 0, − 21 , + 1
2
and
charge (− 12 ), while h′I is an auxiliary field. Also the complex conjugate doublets, associated
to the (−1)/3 strings of type II, are present in the spectrum.
+ 21 , + 21 , 0, 0
and charge
• (−1)/3 strings of type I(I + 1): these open strings have mixed DirichletNeumann
boundary conditions along the (z1, z2)directions and transform in the bifundamental
representation (dI × n¯I+1) of the gauge and ADHM groups. As compared to the previous
case, the states in this sector are characterized by the action of an antitwist operator Δ¯
which carries an orbifold parity ω+ 12 . Thus the ZM invariant configurations must have
again weights with l2 − l3 = 1/2 in order to compensate for the ω−1 factor carried by
the ChanPaton indices. Taking this into account, we find two complex doublets: (wˆI , µˆI )
where the NS component wˆI has weight + 21 , + 21 , 0, 0 and charge 0, and the R component
1 1
µˆI has weight 0, 0, − 2 , − 2
and charge (+ 12 ), and (µˆ′I , hˆ′ ) where µˆ′I is associated to the
I
and charge (− 12 ), while hˆ′I is an auxiliary field.
Also the complex conjugate doublets, associated to the 3/(−1) strings of type (I + 1)I,
are present in the spectrum.
Notice that no states from the 3/(−1) strings of type I(I + 1) or from the (−1)/3
strings of type (I + 1)I survive the orbifold projection. Indeed, in the first case the
phases ω− 21 and ω−1 from the twist operator Δ and the ChanPaton factors cannot be
compensated by the NS or R weights; while in the second case the phases ω+ 12 and ω+1
from the antitwist operator Δ¯ and the ChanPaton factors cannot be canceled.
All the above results are summarized in table 1, which contains also other relevant
information about the moduli. As an illustrative example, we now consider in detail the
SU(2) theory.
C.1
SU(2)
In this case we have M = 2, and thus necessarily n1 = n2 = 1. Therefore, in the SU(2)
theory we have only simple surface operators. Furthermore, since the index s takes only
one value, we can simplify the notation and suppress this index in the following.
Each pair Y = (Y1, Y2) of Young tableaux contributes to the instanton partition
function with a weight q1d1 q2d2 where d1 and d2 are given by (3.22), which in this case take the
simple form [8]
d1 =
X
j
Y12j+1 + Y22j+1 ,
d2 =
X
j
Y12j+2 + Y22j+2 .
(C.1)
with YIk representing the length of the kth column of the tableau YI .
Let us begin by considering the case of pairs of Young tableaux with a single box.
There are two such pairs that can contribute. One is Y = ( , •) corresponding to d1 = 1
and d2 = 0. Using these values in (3.18), we find
z{1,0} =
(ǫ1 + ǫ4) a1 − χ1,1 + 21 (ǫ1 + ǫˆ2) + ǫ4
χ1,1 − a2 + 12 (ǫ1 + ǫˆ2) + ǫ4
ǫ1 ǫ4 a1 − χ1,1 + 12 (ǫ1 + ǫˆ2)
χ1,1 − a2 + 12 (ǫ1 + ǫˆ2)
Due to the prescription (3.19), only the pole at
(C.3)
(C.5)
HJEP07(21)68
χ1,1 = a1 +
(ǫ1 + ǫˆ2)
1
2
contributes to the contour integral over χ1,1, yielding
Z
(ǫ1 + ǫ4) (4a + 2ǫ1 + ǫ2 + 2ǫ4)
ǫ1 (a12 + ǫ1 + ǫˆ2)
ǫ1 (4a + 2ǫ1 + ǫ2)
where in the last step we used the notation a12 = a1 −a2 = 2a and reintroduced ǫ2 = 2ǫˆ2. A
similar analysis can be done for the second pair of tableaux with one box that contributes,
) corresponding to d1 = 0 and d2 = 1. In this case we find
Z(•, ) =
(ǫ1 + ǫ4) (−4a + 2ǫ1 + ǫ2 + 2ǫ4) .
ǫ1 (−4a + 2ǫ1 + ǫ2)
In the case of two boxes, we have five different pairs of tableaux that can contribute.
They are: Y = ( ,
), Y = (
, •), Y = (•,
), Y =
, •
and Y =
•
contributions of these five diagrams are listed below in table 2.
Multiplying all contributions with the appropriate weight factor and summing over
them, we obtain the instanton partition function for the SU(2) gauge theory in the presence
of the surface operator:
Zinst[1, 1] = 1 + q1
(ǫ1 + ǫ4) (4a + 2ǫ1 + ǫ2 + 2ǫ4)
ǫ1 (4a + 2ǫ1 + ǫ2)
+ q2
(ǫ1 + ǫ4) (−4a + 2ǫ1 + ǫ2 + 2ǫ4)
ǫ1 (−4a + 2ǫ1 + ǫ2)
+q1q2
+q12 (ǫ1 + ǫ4) (2ǫ1 + ǫ4) (4a + 2ǫ1 + ǫ2 + 2ǫ4) (4a + 4ǫ1 + ǫ2 + 2ǫ4)
2ǫ21 (4a + 2ǫ1 + ǫ2) (4a + 4ǫ1 + ǫ2)
+q22 (ǫ1 + ǫ4) (2ǫ1 + ǫ4) (−4a + 2ǫ1 + ǫ2 + 2ǫ4) (−4a + 4ǫ1 + ǫ2 + 2ǫ4)
2ǫ21 (−4a + 2ǫ1 + ǫ2) (−4a + 4ǫ1 + ǫ2)
(ǫ1 + ǫ4)(ǫ2 + ǫ4)(4a + ǫ2 − 2ǫ4)(4a + 2ǫ1 + ǫ2 + 2ǫ4)
ǫ1ǫ2(4a + ǫ2)(4a + 2ǫ1 + ǫ2)
+
+
(ǫ1 + ǫ4)(ǫ2 + ǫ4)(−4a + ǫ2 − 2ǫ4)(−4a + 2ǫ1 + ǫ2 + 2ǫ4)
(ǫ1 + ǫ4)2(4a + ǫ2 + 2ǫ4)(−4a + ǫ2 + 2ǫ4)
ǫ1ǫ2(−4a + ǫ2)(−4a + 2ǫ1 + ǫ2)
ǫ21(4a + ǫ2)(−4a + ǫ2)
(C.6)
weight
( ,
) q1q2
,
, •) q1q2
) q1q2
, •
q
2
1
q
2
2
poles
χ1,1 = a1 + 12 (ǫ1 + ǫˆ2)
χ2,1 = a2 + 12 (ǫ1 + ǫˆ2)
χ1,1 = a1 + 12 (ǫ1 + ǫˆ2)
χ2,1 = χ1,1 + ǫˆ2
χ2,1 = a2 + 12 (ǫ1 + ǫˆ2)
χ1,1 = χ2,1 + ǫˆ2
χ1,1 = a1 + 12 (ǫ1 + ǫˆ2)
χ1,2 = χ1,1 + ǫ1
χ2,1 = a2 + 12 (ǫ1 + ǫˆ2)
χ2,2 = χ2,1 + ǫ1
ǫ1ǫ2(4a+ǫ2)(4a+2ǫ1+ǫ2)
ǫ1ǫ2(−4a+ǫ2)(−4a+2ǫ1+ǫ2)
2ǫ21(4a+2ǫ1+ǫ2)(4a+4ǫ1+ǫ2)
2ǫ21(−4a+2ǫ1+ǫ2)(−4a+4ǫ1+ǫ2)
partition function in all five cases with two boxes for the SU(2) theory.
where the ellipses stand for the contributions originating from tableaux with higher number
of boxes, which can be easily generated with a computer program.
We have explicitly
computed these terms up six boxes, but we do not write them here since the raw expressions
are very long and not particularly illuminating. To the extent it is possible to make
comparisons, we observe that the above result agrees with the instanton partition function
reported in eq. (B.6) of [8] under the following change of notation
q1 → y ,
q2 → x , ǫ4 → −m ,
2a → 2a +
2
.
Note then that the mass m appearing in [8] is the equivariant mass of the
hypermultiplet [78], which differs by ǫcorrections from the mass we have used in this paper (see (3.35)).
D
Prepotential coefficients for the SU(N ) gauge theory
The prepotential F of the N = 2⋆ SU(N ) gauge theory has been determined in terms of
quasimodular forms in [34, 41]. Expanding F as in (4.12), the first few nonzero coefficients
fℓ in the NS limit turn out to be
f2 =
1
4
f4 = − 24
f6 = − 288
m2
1
1
2
1
m2
X log
2 2
1
E2 C2 ,
2 2( 2
1
(au − av)2
Λ2
+ N
m2
1 log η ,
1
2
m2
2
1
m2
ǫ
2
1
5E22 + E4 − 6 ǫ12 E4 C4
E22 − E4 C2;1,1 ,
)
(C.7)
(D.1)
(D.2)
(D.3)
f8 = − 1728
m2
1
2 2(
1
1
− 5
− 5
1
m2
m2
2
1
2
1
2 2
m2
24 ǫ2
1
m2
2
1
m2
1
1
175E23 + 84E2E4 + 11E6
7E2E4 + 3E6 +
E6 C6
24 ǫ4
5E23 − 3E2E4 − 2E6 − 6 ǫ2 E2E4 − E6
C4;2
5E23 − 3E2E4 − 2E6 − 3 ǫ2 E2E4 − E6
C3;3
E23 − 3E2E4 + 2E6 C2;1,1,1,1 .
Here E2, E4 and E6 are the Eisenstein series and
log η = −
∞
k=1
X σ1(k) qk = − 24
1
k
log q + log η
with η being the Dedekind ηfunction. Finally, the root lattice sums are defined by
Cn;m1,m2,··· ,mk =
X
X
α~∈Φ β~16=β~26=···6=β~k∈Φ(α~)
1
(α~ · ~a)n(β~1 · ~a)m1 (β~2 · ~a)m1 · · · (β~k · ~a)mk
where Φ is the root system of SU(N ) and
Φ(α~) = {β ∈ Φ
~
α~ · β~ = 1} .
(D.4)
(D.5)
(D.7)
We refer to [41] for the details and the derivation of these results. Notice, however, that
we have slightly changed our notation, since f2hℓere = f there. By expanding the modular
ℓ
functions in powers of q and selecting SU(2) as gauge group, it is easy to show that the
above formulas reproduce both the perturbative part and the instanton contributions,
reported respectively in (4.10a) and (3.42) of the main text.
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