Modular and duality properties of surface operators in \( \mathcal{N}={2}^{\star } \) gauge theories

Journal of High Energy Physics, Jul 2017

We calculate the instanton partition function of the four-dimensional \( \mathcal{N}={2}^{\star } \) SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure \( \mathcal{N}=2 \) or to \( \mathcal{N}={2}^{\star } \) gauge theories.

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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, I-10125 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 four-dimensional N = 2⋆ SU(N ) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional 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 S-duality 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 two-dimensional 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 four-dimensional fields [2, 3], or can be characterized by the two-dimensional theory they support on their world-volume [4, 5]. A convenient way to describe four-dimensional 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 six-dimensional (2, 0) theory on the M5 branes, surface operators can be realized by means of either M5′ or M2 branes giving rise, respectively, to codimension-2 and codimension-4 defects. While a codimension-2 operator extends over the Riemann surface wrapped by the M5 brane realizing the gauge theory, a codimension-4 operator intersects the Riemann surface at a point. Codimension-2 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) super-conformal quiver theories with surface operators were mapped to the conformal blocks of a two-dimensional 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 codimension-4 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 low-energy effective dynamics of the bulk four-dimensional theory is completely encoded in the holomorphic prepotential which at the non-perturbative level can be very efficiently determined using localization [20] along with the constraints that arise from S-duality. 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 Calabi-Yau 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 semi-simple algebra [41, 42]. 1For a review of surface operators see [1]. 2Modular anomaly equations have been studied in various contexts, such as the Ω-background [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 world-volume of the two-dimensional surface operator in the N = 2⋆ theory. For simplicity, we limit ourselves to SU(N ) gauge groups and consider half-BPS surface defects that, from the six-dimensional point of view, are codimension-2 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 non-perturbative 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 S-duality 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 codimension-4 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 quasi-modular 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 two-dimensional theory they support on their world-volume. In [5] the coupling of the sigma-models 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 sigma-model on CPN−1, that corresponds to the so-called simple defects with Levi decomposition of type {1, N − 1}, and sigma-models on Grassmannian manifolds corresponding to defects of type {p, N −p}. The main result of [5] is that the Seiberg-Witten geometry of the four-dimensional theory can be recovered by analyzing how the vacuum structure of these sigma-models is fibered over the Coulomb moduli space. Independent 3We actually calculate the effective superpotential in the Nekrasov-Shatashvili limit [51] in which only one of the Ω-deformation 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 Seiberg-Witten 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 Seiberg-Witten 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 four-dimensional 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 Yang-Mills 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 Yang-Mills 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 S-duality 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 low-energy 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 1-loop 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 k-instanton 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 non-local defect D supported on a two-dimensional plane inside the four-dimensional (Euclidean) space-time (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 z1-plane. 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 z2-plane, 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 two-dimensional 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 electro-magnetic 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 S-duality 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 six-dimensional representation as a codimension-2 surface operator, can be realized with a system of D3-branes 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 four-dimensional space-time where the gauge theory is defined (namely the world-volume of the D3-branes), while the z1-plane is the world-sheet 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 v-plane corresponds to the Coulomb moduli space of the gauge theory. Without the Z M -orbifold projection, the isometry group of the ten-dimensional background is SO(4)×SO(4)×U(1), since the D3-branes 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 D-branes. There are different types of D-branes 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 D-branes, which we label with the index I already used before. On a D-brane of type I, the generator of ZM acts as ωI , and thus the Chan-Paton factor of a string stretching between a D-brane – 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 space-time 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 anti-chiral space-time 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 I-I, I-(I + 1) or (I + 1)-I, respectively. Therefore, the Z M -invariant neutral moduli carry Chan-Paton 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 D-instanton and a D3-brane or vice versa. Due to the twisted boundary conditions in the first two complex space-time directions, the weight vectors of the bosonic states in the Neveu-Schwarz 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 end-point on the D-instantons, 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 D-instantons. In the bosonic Neveu-Schwarz 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 GSO-projection 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 D3-brane and a D-instanton, and ω+ 12 for strings with opposite orientation. Then, with this choice we find Z I-I, whose Chan-Paton 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 Chan-Paton 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 D-instantons [50], is exact with respect to the supersymmetry charge Q of weight Therefore Q can be used as the equivariant BRST-charge 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 Q-doublets 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 Q-doublets 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 D-instantons of type I in the v-plane. In this way we are left with the integration over all the χI,σ’s and a Cauchy-Vandermonde 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 Q-doublets which localizes on the fixed points of Q2, and qI is the counting parameter associated to the D-instantons 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 Q-doublets 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 upper-half 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 one-to-one 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 i-th row and j-th 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 j-th 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 non-trivial 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 j-th 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 Q-doublets 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 D-branes probing the orbifold geometry. An alternative approach works with the resolution of the orbifold geometry [54, 55]. This involves analyzing a gauged linear sigma-model that describes a system of D1 and D5-branes in the background C × C/ZM × T ⋆S2 × R2. One then uses the localization formulas for supersymmetric field theories defined on the 2-sphere [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 world-sheet 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 four-dimensional N = 2⋆ gauge theory is described by the prepotential F , while the infrared physics of the two-dimensional theory defined on the world-sheet of the surface operator is governed by the twisted superpotential W. The non-perturbative 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 so-called Nekrasov-Shatashvili (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 four-dimensional 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 Ω-deformed theory, the respective super-volumes are finite and given by [1, 58]: Z Rǫ41,ǫ2 d4x d4θ −→ ǫ1ǫ2 1 and Z R2 ǫ1 d2x d2θ −→ ǫ1 . 1 The non-trivial 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 non-trivial check on the correctness and consistency of the whole construction. Let us now consider the non-perturbative 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 weak-coupling expansions of combinations of elliptic and quasi-modular 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 S-duality 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 S-duality 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 electro-magnetic 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 S-duality constraints it is necessary to take into account the classical and the perturbative 1-loop 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 S-duality. 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 S-duality 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 1-loop contribution. The 1-loop 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(x|ǫ1, 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 Γ-function 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 γ-function, 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 S-duality constraints Wpert = − 4 ∂a 1 ∂Fpert . We are now in a position to discuss the constraints on the twisted superpotential arising from S-duality. 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 quasi-modular 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 mass-dimension 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 quasi-modular 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 quasi-modular 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 S-duality 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), S-duality 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 well-defined 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 S-duality transformation (4.5b) with the fact that the zI -derivative increases the modular weight by one, and introduces an extra factor of (−τ ) under S-duality, 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 mass-dimension 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 S-duality 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) Gukov-Witten [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 1-instanton approximation in [64, 65] by evaluating the period integrals of the Seiberg-Witten 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 semi-classical expansion of the chiral ring relation of [5], and provides further non-trivial 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 sigma-model coupled to the SU(N ) gauge theory, and all information about the twisted chiral ring of the sigma-model 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 sigma-model, 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 semi-classical results for the twisted chiral ring of the Grassmannian sigma-model 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 Seiberg-Witten 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 z-derivative 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 Seiberg-Witten 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 Donagi-Witten 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’Hoker-Phong 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 semi-classical results have been resummed into elliptic and quasi-modular forms, we use the Donagi-Witten 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 Donagi-Witten 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 Donagi-Witten 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 Donagi-Witten equation actually encodes also the twisted chiral ring relation of the simple codimension-4 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 Donagi-Witten equation are given by ′ yi = ai − m2 X h 1 m3 X where z∗ is an arbitrary reference point. Indeed, in the Donagi-Witten variables, the differential is simply λSW (z) = y(z) dz. Given that the Donagi-Witten 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 codimension-2 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 Donagi-Witten 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 sigma-model 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 codimension-4 surface operators labeled by a single continuous parameter z, whose superpotential has been identified with the line integral of the Seiberg-Witten differential of the four-dimensional 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 Seiberg-Witten 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 codimension-2 and codimension-4 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 two-dimensional dynamics on the defect world-sheet, receives non-perturbative contributions, which we calculated using equivariant localization. Furthermore, exploiting the constraints arising from the non-perturbative SL(2, Z) symmetry, we showed that in a semi-classical 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 quasi-modular 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 two-dimensional sigma models is a non-trivial check of our methods and provides evidence for the duality between the codimension-2 and codimension-4 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 Donagi-Witten 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 four-dimensional prepotential. A technically more challenging extension would be to study surface operators for theories HJEP07(21)68 with non-simply 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 semi-classical 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 S-duality 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 Seiberg-Witten 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 4-sphere 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” TO-Call3-2012-0088. The work of M.B., M.F. and A.L. is partially supported by the MIUR PRIN Contract 2015MP2CX4 “Non-perturbative Aspects Of Gauge Theories And Strings”. A Useful formulas for modular forms and elliptic functions In this appendix we collect some formulas about quasi-modular 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 p-th 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 S-duality 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 θ-function where x = e2πiz. From θ1, we first define the function and the Weierstraß ℘-function θ1(z|τ ) = q 12 (n− 21 )2 (−x)(n− 12 ) 1 ∂ 2πi ∂z h1(z|τ ) = log θ1(z|τ ) = x log θ1(z|τ ) , ℘(z|τ ) = − ∂z2 log θ1(z|τ ) − 3 E2(τ ) . ∂ In most of our formulas the following rescaled ℘-function appears: ℘e(z|τ ) := ℘(z, τ ) 4π2 = x ∂ log θ1(z|τ ) − 12 E2(τ ) , 1 which we can write also as ℘e(z|τ ) = h′1(z|τ ) − 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′(z|τ ) := 1 ∂ 2πi ∂z ℘e(z|τ ) = x ∂ ∂x ℘e(z|τ ) = h′1′(z|τ ) . ℘e′(z|τ )2 + 4 ℘e(z|τ )3 − E124 ℘e(z|τ ) − 216 By differentiating this equation, we obtain which, using (A.9) and (A.10), we can rewrite as ℘e′′(z|τ ) = −6 ℘e(z|τ )2 + E4 24 h′1′′(z|τ ) = −6 h′1(z|τ ) 2 + E2 h′1(z|τ ) − E2 2 − E4 24 . The function h1, ℘ and ℘e′ have well-known 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(x|q) = h1(λx|λ2q) 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 (z|τ ) as we did so far, or by their exponentials (x|q) when the expansions are being used. − λ 1 x3 + 2 ℘e′(x|q) = ℘e′(λx|λ2q) 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 (z|τ ) as follows: (z|τ ) → 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 ℘(z|τ ) → ℘ ℘ ′(z|τ ) → ℘ cτ + d cτ + d cτ + d cτ + d = (cτ + d)3 ℘ ′(z|τ ) . ℘e(z|τ ) → ℘e − τ ℘e′(z|τ ) → ℘e z z − τ 1 − τ 1 − τ = τ 2 ℘e(z|τ ) , = −τ 3 ℘e′(z|τ ) . 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 space-time R4 ≃ C2 by two complex variables (z1 = ρ eiφ , z2 = r eiθ), and consider a two-dimensional defect D located at z2 = 0 and filling the z1-plane. In this set-up, we make the following Ansatz [8]: where Ab is regular all over R4 and g(r) is a g-valued 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 two-dimensional theory on the defect, the Cartan gauge fields Ai must approach a pure-gauge configuration at infinity so that with ωi being a function of φ, the polar angle in the z1-plane. In this situation, for the corresponding gauge transformation to be single-valued, 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 non-zero entry equal to 1 in the i-th 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 Z-span of these simple roots. Note that ΛR lies in a codimension-1 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 I-I: 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 Neveu-Schwarz (NS) sector of such strings by the complex oscillator ψv in the last complex space-time 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 Q-doublet 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 Q-doublets. 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 Chan-Paton 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 bi-fundamental 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 Chan-Paton 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 Q-doublets, (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 Q-doublet 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 I-I: these open strings have mixed Neumann-Dirichlet 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 bi-fundamental 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 I-I, are present in the spectrum. + 21 , + 21 , 0, 0 and charge • (−1)/3 strings of type I-(I + 1): these open strings have mixed Dirichlet-Neumann boundary conditions along the (z1, z2)-directions and transform in the bi-fundamental 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 anti-twist 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 Chan-Paton 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 Chan-Paton factors cannot be compensated by the NS or R weights; while in the second case the phases ω+ 12 and ω+1 from the anti-twist operator Δ¯ and the Chan-Paton 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 ǫ-corrections 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 quasi-modular forms in [34, 41]. Expanding F as in (4.12), the first few non-zero 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 η-function. 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. 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S. K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, R. R. John, A. Lerda. Modular and duality properties of surface operators in \( \mathcal{N}={2}^{\star } \) gauge theories, Journal of High Energy Physics, 2017, 68, DOI: 10.1007/JHEP07(2017)068