Surface operators, chiral rings and localization in \( \mathcal{N} \) =2 gauge theories

Journal of High Energy Physics, Nov 2017

We study half-BPS surface operators in supersymmetric gauge theories in four and five dimensions following two different approaches. In the first approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and five-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and five-dimensional gauge theories, and find a perfect match with the results obtained from the chiral ring equations. In the five-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for ramified instantons. We also comment on the necessity of including Chern-Simons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator.

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Surface operators, chiral rings and localization in \( \mathcal{N} \) =2 gauge theories

HJE 2 gauge theories S.K. Ashok 0 1 3 4 6 8 M. Billo 0 1 2 3 5 8 E. Dell'Aquila 0 1 3 5 8 M. Frau 0 1 2 3 5 8 V. Gupta 0 1 3 4 6 8 R.R. John 0 1 3 4 6 8 A. Lerda 0 1 2 3 8 0 Via P. Giuria 1, 10125 Torino , Italy 1 Mumbai 400085 , India 2 Arnold-Regge Center and INFN 3 Taramani , Chennai 600113 , India 4 Homi Bhabha National Institute, Training School Complex , Anushakti Nagar 5 Universita di Torino , Dipartimento di Fisica 6 Institute of Mathematical Sciences , C.I.T. Campus 7 Sezione di Torino 8 Viale T. Michel 11, 15121 Alessandria , Italy We study half-BPS surface operators in supersymmetric gauge theories in four ve dimensions following two di erent approaches. In the rst approach we analyze the chiral ring equations for certain quiver theories in two and three dimensions, coupled respectively to four- and ve-dimensional gauge theories. The chiral ring equations, which arise from extremizing a twisted chiral superpotential, are solved as power series in the infrared scales of the quiver theories. In the second approach we use equivariant localization and obtain the twisted chiral superpotential as a function of the Coulomb moduli of the four- and ve-dimensional gauge theories, and nd a perfect match with the results obtained from the chiral ring equations. In the ve-dimensional case this match is achieved after solving a number of subtleties in the localization formulas which amounts to choosing a particular residue prescription in the integrals that yield the Nekrasov-like partition functions for rami ed instantons. We also comment on the necessity of including ChernSimons terms in order to match the superpotentials obtained from dual quiver descriptions of a given surface operator. ArXiv ePrint: 1707.08922 Supersymmetry and Duality; Duality in Gauge Field Theories; Extended - N and Supersymmetry 1 Introduction SU(2) 2.1 2.2 2.3 SU( 3 ) 2 Twisted superpotential for coupled 2d/4d theories Twisted chiral ring in quiver gauge theories 3 Twisted superpotential for coupled 3d/5d theories 3.1 Twisted chiral ring in quiver gauge theories 3.2 SU(2) and SU( 3 ) 4 Rami ed instantons in 4d and 5d 4.1 4.2 Localization in 4d Localization in 5d 5 Superpotentials for dual quivers 5.1 Adding Chern-Simons terms 6 Conclusions and perspectives A Chiral correlators in 5d B Chern-Simons terms in an SU(4) example 1 3 surface operators in theories with eight supercharges. We focus on pure SU(N ) theories in four dimensions and in ve dimensions compacti ed on a circle, and explore their Coulomb branch where the adjoint scalars acquire a vacuum expectation value (vev). In four dimensions, a surface defect supports on its world-volume a two-dimensional gauge theory that is coupled to the \bulk" four-dimensional theory, see [1] for a review. This combined 2d/4d system is described by two holomorphic functions: the prepotential F and the twisted superpotential W . The prepotential governs the dynamics of the bulk theory and depends on the Coulomb vev's and the infra-red (IR) scale of the gauge theory in four dimensions. The twisted superpotential controls the two-dimensional dynamics on the surface operator, and is a function of the continuous parameters labeling the defect, the two-dimensional IR scales, and also of the Coulomb vev's and the strong-coupling scale of the bulk gauge theory. The twisted superpotential thus describes the coupled 2d/4d system. { 1 { defects) in four dimensions along the lines discussed in [2, 3] and compute the corresponding twisted superpotential W using equivariant localization as shown for example in [4{7]. A second approach is to focus on the two-dimensional world-volume theory on the surface operator [8]. In the superconformal theories of class S, a microscopic description of a generic co-dimension 4 surface operator in terms of (2; 2) supersymmetric quiver gauge theories in two dimensions was realized in [9]. Here we focus on a generic co-dimension 2 surface operator in pure N = 2 SU(N ) gauge theory [10], which has a microscopic description as a quiver gauge theory of the type shown in gure 1. Here the round nodes, labeled by an index I, correspond to U(kI ) gauge theories in two dimensions whose eld strength is described by a twisted chiral eld rightmost node represents the four-dimensional N = 2 gauge theory whose SU(N ) gauge group acts as a avor group for the last two-dimensional node. The arrows correspond to (bi-)fundamental matter multiplets that are generically massive. Integrating out these elds leads to an e ective action for the twisted chiral elds which, because of the twodimensional (2; 2)-supersymmetry, is encoded in a twisted chiral superpotential W . The contribution to W coming from the massive elds attached to the last node depends on the four-dimensional dynamics of the SU(N ) theory and in particular on its resolvent [10]. In this approach a key role is played by the twisted chiral ring equations that follow from extremizing the twisted superpotential with respect to (I). The main idea is that by evaluating W on the solutions to the twisted chiral ring equations one should reproduce (I). The precisely the superpotential calculated using localization. In this work, we extend this analysis in a few directions. We show that there exists a precise correspondence between the choice of massive vacua in two dimensions and the Gukov-Witten defects of the SU(N ) gauge theory labeled by the partition [n1; : : : nM ] with n1+ +nM = N . We also describe the relation between the (M 1) dynamically generated scales I associated to the Fayet-Iliopoulos (FI) parameters for the two-dimensional nodes and the (M 1) dimensionful parameters that naturally occur in the rami ed instanton counting problem. Both the chiral ring equations and the localization methods can be extended to ve-dimensional theories compacti ed on a circle of circumference , i.e. to theories de ned on R 4 S1. In this case, surface operators correspond to codimension-2 defects wrapping S1 and supporting a three-dimensional gauge theory coupled to the bulk ve-dimensional theory. In the 3d/5d case, one again has a quiver theory, and as before its infrared dynamics is encoded in a twisted chiral superpotential. However, the form of the superpotential is modi ed to take into account the presence of a compacti ed direction. { 2 { The twisted chiral rings for purely three-dimensional quiver theories have been studied in great detail in [11]. Here we extend this analysis and propose that the coupling between the last three-dimensional gauge node and the compacti ed ve-dimensional theory is once again determined via the resolvent of the latter. With this assumption, the analysis of the modi ed twisted chiral ring equations as well as the choice of vacuum follow exactly the same pattern as in the 2d/4d case. An important and non-trivial check of this proposal is provided by the perfect agreement between the twisted superpotential obtained from solving the chiral ring equations and the one obtained from localization in ve dimensions, which we perform in several examples. In the 2d/4d case, the quiver theory on the defect can be mapped to other quiver solutions of the twisted chiral ring equations for such dual theories lead to the same twisted superpotential. We obtain strong indications that each such superpotential matches the result of a localization computation carried out with a di erent residue prescription. If we promote the quiver theories to the compacti ed 3d/5d set-up, the superpotentials still agree at the classical level but, in general, they di er when quantum corrections are taken into account. The 3d/5d quiver gauge theories can be extended to include Chern-Simons (CS) terms in their e ective action. Quite remarkably, we nd in a simple but signi cant example that the equivalence between the dual quiver realizations of the same defect is restored at the quantum level if suitably chosen CS terms are added to the superpotentials. The paper proceeds as follows. In section 2, we study the coupled 2d/4d system and solve the twisted chiral ring equations as power series in the IR scales of the theory. In section 3, we lift the discussion to coupled 3d/5d systems compacti ed on a circle. In section 4, we analyze the rami ed instanton counting in four and ve dimensions and show that the e ective twisted chiral superpotential calculated using localization methods exactly matches the one obtained from the solution of the chiral ring equations in the earlier sections. In section 5, we discuss the relation between di erent quiver realizations of the same surface defect, and show that the equivalence between two dual realizations, which is manifest in the 2d/4d case, is in general no longer true in the 3d/5d case. We also show in a speci c example that the duality is restored by adding suitable Chern-Simons terms. Finally, in section 6 we present our conclusions and discuss some possible extensions of our results. Some technical details are collected in the appendices. 2 Twisted superpotential for coupled 2d/4d theories In this section our focus is the calculation of the low-energy e ective action for surface operators in pure N = 2 SU(N ) supersymmetric gauge theories in four dimensions. As mentioned in the Introduction, surface operators can be e ciently described by means of a coupled 2d/4d system in which the two-dimensional part is a (2; 2)-supersymmetric quiver gauge theory with (bi-)fundamental matter, as shown in gure 1. Such coupled 2d/4d systems have an alternative description as Gukov-Witten monodromy defects [2, 3]. The discrete data that label these defects correspond to the partitions of N , and can be { 3 { (2.1) (2.2) (2.3) summarized in the notation SU(N )[n1; : : : ; nM ] where n1 + + nM = N . The M integers nI are related to the breaking pattern (or Levi decomposition) of the four-dimensional gauge group on the defect, namely SU(N ) ! S U(n1) : : : U(nM ) : They also determine the ranks kI of the two-dimensional gauge groups of the quiver in gure 1 according to The (bi-)fundamental elds connecting two nodes turn out to be massive. Integrating the gauge multiplet is described by a twisted chiral eld (I) and the low-energy e ective action is encoded in a twisted chiral superpotential W ( (I)). The vacuum structure can be determined by the twisted chiral ring equations, which take the form [14{16] kI = n1 + + nI : exp where (I) are the diagonal components, with s = 1; : : : ; k [17]. This exponentiated form s of the equations is a consequence of the electric uxes which can be added to minimize the potential energy and which lead to linear (in (I)) terms in the e ective superpotential. We extend this analysis in the following manner: rst of all, we show that in the classical limit there is a very speci c choice of solutions to the twisted chiral ring equations that allows us to make contact with the twisted chiral superpotential calculated using localization. We establish the correspondence between the continuous parameters labeling the monodromy defect and the dynamically generated scales of the two-dimensional quiver theory. We then show that quantum corrections in the quiver gauge theory are mapped directly to corrections in the twisted superpotential due to rami ed instantons of the fourdimensional theory. 2.1 SU(2) As an illustrative example, we consider the simple surface operator in the pure SU(2) theory which is represented by the partition [1; 1]. From the two-dimensional perspective, the e ective dynamics is described by a non-linear sigma model with target-space CP1, coupled to the four-dimensional SU(2) gauge theory in a particular way that we now describe. We use the gauged linear sigma model (GLSM) description of this theory [17, 18] in order to study its vacuum structure. We essentially follow the discussion in [10] although, as we shall see in detail, there are some di erences in our analysis. The GLSM is a U(1) gauge theory with two chiral multiplets in the fundamental representation, that can be associated to the quiver drawn in gure 2. Let us rst analyze the simple case in which the quantum e ects of the SU(2) theory are neglected. We consider a generic point in the Coulomb branch parameterized by the vev's a1 = a2 = a of the adjoint SU(2) scalar eld in the vector multiplet. These have the interpretation of twisted masses for the chiral multiplet of the two-dimensional { 4 { been studied in some detail in a number of works and here we merely present the resulting e ective action which takes the simple form:1 HJEP1(207)3 W = 2 i ( ) = 2 i ( ) ai) log ) log 2 X( i=1 Tr ( 0 is the ultra-violet (UV) cut-o which we eventually take to in nity, and ( ) is the bare FI parameter at the scale . We can rewrite this superpotential using another scale 0 and get W = 2 i ( ) 2 log Tr ( ) log 1 From the coe cient of the linear term in , we identify the running of the FI coupling:2 2 i ( 0) = 2 i ( ) 2 log W = Tr ( ) log W = Tr ( ) log In particular, we can choose to use the complexi ed IR scale 1 at which ( 1) = 0, so that In this way we trade the UV coupling ( ) for the dynamically generated scale 1 . Let us now turn on the dynamics of the four-dimensional SU(2) gauge theory. As pointed out in [10], this corresponds to considering the following superpotential: The angular brackets signify taking the quantum corrected vev of the chiral observable in the four-dimensional SU(2) theory. The twisted chiral ring equation is obtained by extremizing W and is given by 1For notational simplicity we denote the super eld by its lowest scalar component . 2Recall that is actually the complexi cation of the FI parameter r with the -angle: 2 i = i r. The sign of the coe cient of the logarithmic running (2.5) is such that r( 0) grows with the scale 0. The same is true in the other cases we consider. exp = 1 ; { 5 { 1 1 ai 0 : 1 1 1 1 0 : : from which we obtain the two solutions is the characteristic polynomial appearing in the Seiberg-Witten solution where u = Tr 2 = a2 + 4 2a2 + ring relation (2.10) becomes which, using the superpotential (2.8), is equivalent to exp As explained in [10], the left-hand side of (2.10) is simply the integral of the resolvent of the pure N = 2 SU(2) theory in four dimensions which takes the form [19]: 1 and , which are related respectively to the two-dimensional and the four-dimensional dynamics. Clearly, the purely two-dimensional result can be recovered by taking the ! 0 limit. We can now substitute either one of the solutions of the chiral ring equation into the twisted chiral superpotential and obtain a function W? . The proposal in [10] is that this should reproduce the twisted superpotential calculated using localization methods. We shall explicitly verify this in section 4, but here we would like to point out an important simpli cation that occurs in this calculation. chiral ring relations, namely Let us consider the twisted e ective superpotential evaluated on the ?+ solution of the While W?+ itself is complicated, its logarithmic derivative with respect to 1 takes a re markably simple form. In fact W?+ seems to depend on ? + = 0 and so we simply have 1 both explicitly and through the W?+ u; 1 W ?+(u; 1); 1 : 1 d 1 dW?+ = 1 = 2 ?+ : { 6 { where in the last step we used (2.8) and took into account the tracelessness of . Using the explicit form of the solution given in (2.15), and inserting in it the weakcoupling expansion (2.13) of u, we thus obtain 1 2 1 d 1 dW?+ = a + + 1 2a 1 16a5 Twisted chiral ring in quiver gauge theories We will now show that the procedure described above generalizes in a rather simple way to any surface operator in the SU(N ) gauge theory labeled by a partition of N . In this case, however, it will not be possible to solve exactly the twisted chiral ring equations as we did in the SU(2) theory. We will have to develop a systematic perturbative approach in order to obtain a semi-classical expansion for the twisted chiral superpotential around a particular classical vacuum. Proceeding in this way we again nd that the derivatives of the twisted superpotential with respect to the various scales have simple expressions in terms of combinations of the twisted chiral eld evaluated in the appropriately chosen vacuum. Following [10], we consider a quiver gauge theory of the form U(k1) U(k2) : : : U(kM 1) with (bi)-fundamental matter between successive nodes, coupled to a pure N = 2 theory in four dimensions with gauge group SU(N ) acting as a avor symmetry for the rightmost factor in (2.19). All this is represented in gure 1. We choose an ordering such that As we shall show later in section 4, this result precisely matches the derivative of the twisted e ective superpotential calculated using localization for the simple surface operator in the SU(2) gauge theory, provided we suitably relate the dynamically generated scale two-dimensional theory to the rami ed instanton counting parameter in presence of the monodromy defect. (2.18) (2.19) (2.20) (2.21) (I) where the kI 's are related to the entries of the partition of N labeling the surface operator as indicated in (2.2). Our rst goal is to obtain the twisted chiral ring of this 2d/4d system. Only the diagonal components of are relevant for this purpose [17], and thus for the I-th gauge group we take (I) = diag 1(I); 2(I); : : : ; k(II) : The (bi)-fundamental matter elds are massive and their (twisted) mass is proportional to the di erence in the expectation values of the 's in the two nodes connected by the matter multiplet. In order to minimize the potential energy, the twisted chiral eld gets a vev and this in turn leads to a non-vanishing mass for the (bi)-fundamental matter. k1 < k2 < k3 : : : < kM 1 < N ; { 7 { Integrating out these massive elds, we obtain the following e ective superpotential W = 2 i X X where, for compactness, we have introduced the function with being the UV cut-o scale. Similarly to the SU(2) example previously considered, also here we can trade the UV parameters I ( ) for the dynamically generated scales I for each of the gauge groups in the quiver. To this aim, we unpackage of the terms containing the $-function and rewrite them as follows: $ (I) s (I+1) t = s(I) log s + s(I) log bII = e2 i I ( ) bI bI = kI+1 kI 1 { 8 { (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) HJEP1(207)3 Tr $ s (M 1) = Tr ( s(M 1) ) log s (M 1) 1 # + N s (M 1) log M 1 : Considering the linear terms in the (I) elds we see that the FI couplings change with the scale and we can de ne the dynamically generated scales I to be such that I ( I ) = I ( ) 2 i kI+1 kI 1 log I = 0 3We assume that k0 = 0 and kM = N . denotes the coe cient of the -function for the running of the FI parameter of the Using these expressions, the twisted superpotential (2.22) can thus be rewritten as W = + 1 #+ 2) in the rst line and the (I + 1)-th term in the second line of this expression are obtained by integrating out the bifundamental elds between the nodes I and (I + 1), while the last line is the result of integrating out the fundamental elds attached to the last gauge node of the quiver. The angular brackets account for the fourdimensional dynamics of the SU(N ) theory. One can easily verify that for N = M = 2, the expression in (2.29) reduces to (2.8). The twisted chiral ring. The twisted chiral ring relations are given by In order to write the resulting equations in a compact form, we de ne a characteristic gauge polynomial for each of the SU(kI ) node of the quiver exp = 1 : kI s=1 QI (z) = Y(z s(I)) : For I = 1; : : : ; M 2, the equations are independent of the four-dimensional theory, and read QI+1(z) = ( 1)kI 1 bII QI 1(z) (I) for each s, and with the understanding that Q0 = 1 and k0 = 0. Note that the power of I , which is determined by the running of the FI coupling, makes the equation consistent from a dimensional point of view. For I = M 1, the presence of the four-dimensional SU(N ) gauge theory a ects the last two-dimensional node of the quiver, and the corresponding chiral ring equation is exp Tr log z with z = s(M 1) for each s. We now use the fact that the resolvent of the four-dimensional SU(N ) theory, which captures all information about the chiral correlators, is given by [19] T (z) := Tr z 1 = pPN (z)2 P N0 (z) 4 2N { 9 { (2.29) is its dynamically generated scale. Since we are primarily interested in the semi-classical solution of the chiral ring equations, we exploit the fact that PN (z) can be written as a perturbation of the classical gauge polynomial in the following way: where ei are the quantum vev's of the pure SU(N ) theory given by [20, 21] HJEP1(207)3 PN (z) = Y(z ei) N i=1 ei = ai Integrating the resolvent (2.34) with respect to z and exponentiating the resulting expression, one nds exp 4 2N : Using this, we can rewrite the twisted chiral ring relation (2.33) associated to the last node of the quiver in the following form: q PN (z) + PN (z)2 4 2N = 2 ( 1)kM 2 bMM 11 QM 2(z) ; (2.35) (2.36) (2.37) (2.38) (2.39) where z = s (M 1) . With further simple manipulations, we obtain PN (z) = ( 1)kM 2 bMM 11 QM 2(z) + 2N ( 1)kM 2 bMM 11 QM 2(z) ! 0 which corresponds to turning o the four-dimensional dynamics, we obtain the expected twisted chiral ring relation of the last two-dimensional node of the quiver. Equations (2.32) and (2.39) are the relevant chiral relations which we are going to solve order by order in the I 's to obtain the weak-coupling expansion of the twisted chiral superpotential. Solving the chiral ring equations. Our goal is to provide a systematic procedure to solve the twisted chiral ring equations we have just derived and to nd the e ective twisted superpotential of the 2d/4d theory. As illustrated in the case of the SU(2) theory in section 2.1, we shall do so by evaluating W on the solutions of the twisted chiral ring equations. Each choice of vacuum therefore corresponds to a di erent surface operator. In order to clarify this last point, we rst solve the classical chiral ring equations, which are obtained by setting I and theory at a scale much bigger than to zero keeping their ratio xed, i.e. by considering the I and . Thus, in this limit the right-hand sides Our proposal is to solve the chiral ring equations (2.32) and (2.39) as a simultaneous power series in all the qI 's, including qM , which ultimately will be identi ed with the Nekrasov-like counting parameters in the rami ed instanton computations described in section 4. We will explicitly illustrate these ideas in some examples in the next section, but rst we would like to show in full generality that the logarithmic derivatives with respect to I are directly related to the solution ? (I) of the twisted chiral ring equations (2.32) and (2.39). The argument is a straightforward generalization of what we have already seen in the SU(2) = 0, the twisted superpotential W ( ?) depends on I only explicitly. Using the expression of W given in (2.29), ? I dW? = d I I = bI tr ?(I) ; where in the last step we used the tracelessness of . This relation can be written in terms of the parameters qI de ned in (2.42), as follows (2.40) (2.41) This is equivalent to assuming that the classical expectation value of for the I-th node is We will see that this choice is the one appropriate to describe a surface defect that breaks the gauge group SU(N ) according to the Levi decomposition (2.1). Let us now turn to the quantum chiral ring equations. Here we make an ansatz for (I) as a power series in the various I 's around the chosen classical vacuum. From the explicit expressions (2.32) and (2.39) of the chiral ring equations, it is easy to realize that there is a natural set of parameters in terms of which these power series can be written; they are given by qI = ( 1)kI 1 bI I 1. If the four-dimensional theory were not dynamical, these (M 1) parameters would be su cient; however, from the chiral ring equations (2.39) of the last two-dimensional node of the quiver, we see that another parameter is needed. It is related to the four-dimensional scale and hence to the four-dimensional instanton action. It turns out that this remaining expansion parameter is qI dqI dW? = tr ?(I) : If we express the solution ? of the chiral ring equations as the classical solution (2.40) plus quantum corrections, we nd dW? = a1 + : : : ak1 + corr.ns = a1 + : : : + an1 + corr.ns ; dW? = a1 + : : : ak2 + corr.ns = a1 + : : : + an1+n2 + corr.ns and so on. This corresponds to a partition of the classical vev's of the SU(N ) theory given by an1 ; an1+1; an1+n2 ; ; aN nM +1; : : : aN {z n1 } | {z n2 } | {z nM } o ; which is interpreted as a breaking of the gauge group SU(N ) according to the Levi decomposition (2.1). In fact, by comparing with the results of [6] (see for instance, equation (4.1) of this reference), we see that the expressions (2.46) coincide with the derivatives of the classical superpotential describing the surface operator of the SU(N ) theory, labeled by the partition [n1; n2; : : : ; nM ], provided we relate the parameters qI to the variables tJ that label the monodromy defect according to We now illustrate these general ideas in a few examples. We consider the surface operators in the SU( 3 ) theory. There are two distinct partitions, namely [1; 2] and [1; 1; 1], which we now discuss in detail. SU( 3 )[1,2]. In this case the two-dimensional theory is a U(1) gauge theory with three avors, represented by the quiver in gure 3. Since M = 2, we have just one and one chiral ring equation, which is given by (see (2.39)) where the gauge polynomial is de ned in (2.35). We solve this equation order by order in 1 and , using the ansatz (2.49) (2.50) 1 3 P3( ) = 31 + 6 3 1 where the expansion parameters are de ned in (2.42) and (2.43), which in this case explicitly read gure 4. node, from (2.32) we nd Inserting (2.50) into (2.49), we can recursively determine the coe cients c`1`2 and, at the rst orders, nd the following result aj . According to (2.45), this solution coincides with the q1-logarithmic derivative of the twisted superpotential. We will verify this statement by comparing (2.52) against the result obtained via localization methods. SU( 3 )[1,1,1]. In this case the two-dimensional theory is represented by the quiver of Since M = 3, there are now two sets of twisted chiral ring equations. For the rst while for the second node, from (2.39) we get P3 s (2) = 2 Y s=1 2 2 (1) (2) s (2) = s 2 1 ; (1) 2 2 (2) s 6 (1) for s = 1; 2. From the classical solution to these equations (see (2.41)), we realize that this con guration corresponds to a surface operator speci ed by the partition of the Coulomb vev's ffa1g; fa2g; fa3gg, which is indeed associated to the partition [1,1,1] we are considering. Thus, the ansatz for solving the quantum equations (2.53) and (2.54) takes the following form: with (2.53) (2.54) (2.55) (2.56) Solving the coupled equations (2.53) and (2.54) order by order in qI , we nd the following result: According to (2.45) these expressions should be identi ed, respectively, with the q1- and q2-logarithmic derivatives of the twisted superpotential. We will verify this relation in section 4 using localization. We have analyzed in detail the SU( 3 ) theory in order to exhibit how explicit and systematic our methods are. We have thoroughly explored all surface defects in the SU(4) and SU(5) theories and also considered a few other examples with higher rank gauge groups. In all these cases our method of solving the twisted chiral ring equations proved to be very e cient and quickly led to very explicit results. One important feature of our approach is the choice of classical extrema of the twisted superpotential which will allow us to make direct contact with the localization calculations of the superpotential for Gukov-Witten defects in four-dimensional gauge theories. A further essential ingredient is the use of the quantum corrected resolvent in four dimensions, which plays a crucial role in obtaining the higher-order solutions of the twisted chiral ring equations of the twodimensional quiver theory. 3 Twisted superpotential for coupled 3d/5d theories Let us now consider the situation in which the 2d/4d theories described in the previous section are replaced by 3d/5d ones compacti ed on a circle S1 of length . The content of these theories is still described by quivers of the same form as in gure 1. We begin by considering the three-dimensional part. 3.1 Twisted chiral ring in quiver gauge theories To construct the e ective theory for the massless chiral twisted elds, which is encoded in the twisted superpotential W , we have to include the contributions of all Kaluza-Klein (KK) copies of the (bi)-fundamental matter multiplets. When the scalars (I) are gaugexed as in (2.21), the KK copies of the matter multiplets have masses5 (I) s (I+1) + 2 i n= ; t (3.1) 5This is consistent with the fact that gauge- xing the scalars (I) as in (2.21) leaves a residual invariance under which the eigenvalues shift by s (I) ! s (I) + 2 ins= . 2 (with an independent integer n for each multiplet). Similarly, the copies of the matter multiplet attached to the 5d node have masses s (M 1) ai + 2 i n= when the 5d theory is treated classically. All these chiral massive elds contribute to the one-loop part of W . As we saw in (2.22) and (2.23), in 2d a chiral eld of mass z contributes a term proportional to $(z). Summing over all its KK copies results therefore in a contribution proportional to where the integration constant has been xed in such a way that where the sum has to be suitably regularized. This function satis es the property n2Z z + 2 i n= = log 2 sinh : (3.4) z 2 Note that the scale , present in the de nition (2.23) of the function $(z), no longer appears after the sum over the KK copies. Integrating this relation, one gets `(z) X $ z + 2 i n= n2Z = X log n2Z 1 `(z) = Li2 e z + z 4 2 6 2 ; `(z) !0 z log( z) 1 = $(z) : Note that here $(z) is de ned taking the UV scale to be = 1= ; as is natural in this compacti ed situation. Therefore, the twisted superpotential of the three-dimensional theory is simply given by (2.22) with all occurrences of the function $(z) replaced by `(z), for any argument z, and with the UV scale being set to 1= . Just as in the two-dimensional case, we would like to replace the FI couplings at the UV scale, I (1= ), with the dynamically generated scales I . Since the renormalization of these FI couplings is determined only by the lightest KK multiplets, the running is the same as in two dimensions and thus we can simply use (2.26) with identi ed with 1= according to (3.7). Note however that, in contrast to the twodimensional case described in (2.29), this replacement does not eliminate completely the UV scale from the expression of W , and the dependence on remains in the functions `(z). Altogether we have W = The expectation value in the last term is taken with respect to the ve-dimensional gauge theory de ned on the last node of the quiver and compacti ed on the same circle of length as the three-dimensional sector. (3.2) (3.3) Our aim is to show that the twisted superpotential (3.8), evaluated on a suitably chosen vacuum ?, matches the twisted superpotential extracted via localization for a corresponding monodromy defect. Just as in the 2d/4d case, the vacuum ? minimizes W , namely solves the twisted chiral ring equation (2.30). Moreover, the logarithmic derivatives of W with respect to I , or with respect to the parameters qI in (2.42), evaluated on a solution ?, still satisfy (2.44) or (2.45) respectively. These derivatives are quite simple to compute and these are the quantities that we will compare with localization results. In close parallel to what we did in the 2d/4d case, the chiral ring equations (2.30) can be expressed in a compact form if we introduce the quantity for each of the SU(kI ) gauge groups in the quiver; note that QbI (z) is naturally written in terms of the exponential variables QbI (z) = Y which are invariant under the shifts described in footnote 5. Indeed, starting from (2.30) and taking into account (3.4), for I = 1; : : : ; M 2 we nd QbI+1(z) = ( 1)kI 1 I bI QbI 1(z) with z = s(I). For the node I = M 1 we obtain exp (z 2 ) = ( 1)kM 2 M 1 . To proceed further, we need to evaluate in the compacti ed 5d theory the expectation value appearing in the left hand side of (3.12). To do so, let us brie y recall a few facts about this compacti ed gauge theory. The resolvent in the compacti ed 5d gauge theory. vector multiplet consists of a gauge eld A , a real scalar The ve-dimensional N = 1 and a gluino . Upon circle compacti cation, the component At of the gauge eld along the circle and the scalar give rise to the complex adjoint scalar = At + i of the four-dimensional N = 2 theory. The Coulomb branch of this theory is classically speci ed by xing the gauge [22]: = At + i = diag (a1; a2; : : : aN ) : However, there is a residual gauge symmetry under which preserve the vanishing of Pi ai. with ni 2 Z; since we are considering a SU(N ) theory, we must ensure that these shifts (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) The low-energy e ective action can be determined in terms of an algebraic curve and a di erential, just as in the usual four-dimensional case. The Seiberg-Witten curve for this model was rst proposed in [22] and later derived from a saddle point analysis of the instanton partition function in [23, 24]; it takes the following form Here is the strong-coupling scale that is dynamically generated and y2 = PbN2 (z) 4 PbN (z) = Y where ei parametrize the quantum moduli space and reduce to ai in the classical regime, in analogy to the four-dimensional case. Like the latter, they also satisfy a tracelessness condition: Pi ei = 0. Note that PbN can be written purely in terms of the exponential variables Ei = e ei ; Z = e z ; and is thus invariant under the shift (3.14). Indeed, using (3.17) we nd where Ui is the symmetric polynomial N 1 i=1 PbN (z) = Z 2 N ZN + X ( 1)iZN i Ui + ( 1)N ; Ui = X j1<j2<:::ji Ej1 : : : Ejk : In (3.18) we have used the SU(N ) tracelessness condition, which implies UN = Qi Ei = 1. The resolvent of this ve-dimensional theory, de ned as [25] Tb(z) = Tr coth (z 2 (z 2 ) Tr log 2 sinh ; (3.20) contains the information about the chiral correlators through the expansion On the other hand, the Seiberg-Witten theory expresses this resolvent as6 so that, integrating (3.20), we have exp 6Using (3.18) we can expand this expression in inverse powers of Z; then, comparing to (3.21), we can relate the correlators hTr e` i, of which the rst (N 1) ones are independent, to the U`'s. With manipulations very similar to those described in section 2 for the 2d/4d case, we can now rewrite the twisted chiral ring relation (3.12) as follows PbN (z) = ( 1)kM 2 + ( 1)kM 2 M 1 where Ai = e ai . These variables Ai represent the classical limit of the variables Ei de ned in (3.17). The SU(N ) tracelessness condition implies that Qi Ai = 1. Our aim is to solve the chiral ring equations (3.11) and (3.24), and then compare the solutions to the localizations results, which naturally arise in a semi-classical expansion. Therefore, we propose an ansatz that takes the form of an expansion in powers of , namely S?(I) = diag A1 + X S1(I;`); : : : ; AkI + X Sk(II );` : ` ` (3.24) (3.25) (3.26) (3.27) (3.28) Notice that also the chiral ring equation (3.24) of the last node can be expanded in . Indeed, the quantity PbN contains the moduli space coordinates Ui, which as shown in appendix A, admit a natural expansion in powers of 2N . Putting everything together, we can solve all chiral ring equations iteratively, order by order in and determine the corrections Ss(;I`) and thus the solution S?(I). In this way, repeating the same steps of the 2d/4d theories, we obtain the expression of the logarithmic derivative of the twisted superpotential, namely qI dqI dW? = 1 XkI log S?(I;s) = Tr ?(I) : s=1 (M 1). It is easy to check that in the limit equation (2.39) for the 2d/4d theory. ! 0 we recover the corresponding Solving the chiral ring equations. At the classical level the solution to the chiral ring equations takes exactly the same form as in (2.41). In terms of the exponential variables introduced in (3.10) we can write it as HJEP1(207)3 SU(2) and SU( 3 ) low rank. We now show how this procedure works in a few simple examples with gauge groups of SU(2)[1,1]. In this case the quiver is the one drawn in gure 2. Since M = 2, there is a single variable for the U(1) node and a single FI parameter . The only chiral ring equation is given by (3.24) with z = , namely Pb2( ) = 2 21 + 4 2 1 : 21 + 4 2 1 According to (3.27), this solution corresponds to the logarithmic q1-derivative of the superpotential, namely We will verify in the next section that this is indeed the case, by comparing with the superpotential computed via localization and nding a perfect match. SU( 3 )[1,2]. This case is described by the quiver in gure 3. Again, we have M = 2 and thus a single variable and a single FI parameter . In this case, the chiral ring U1 = 2 cosh( a) 1 + 4 sinh2( a) Substituting this into (3.30), we nd nally (3.29) (3.30) (3.31) HJEP1(207)3 (3.33) (3.34) (3.35) (3.36) (3.37) Using (3.18) we can express Pb2 in terms of S = e , obtaining where U1 = E1 + E2. A solution of the twisted chiral ring equation is therefore given by ? = 1 1 log S? = arccosh U1 + 2 21 + : 4 2 1 In appendix A we derive the semi-classical expansion of U1. This is given in (A.4) which, rewritten in terms of a, reads + 6 3 1 6 3 1 2 2 4 6 3 1 equation (3.24) reads where Using the semi-classical expansions of U1 and U2 given in (A.5) and (A.6), and solving the chiral ring equation order by order in according to the ansatz (3.26), we obtain A1(A1 + A2) A312A213 + A1(A1 + A3) A212A133 61 + 12 6 1 Aj . Rewriting this solution in terms of the classical vev's ai, we have 1 ? = log S? = a1 + It is very easy to see that in the limit ! 0 this reduces to the solution of the corresponding 2d/4d theory given in (2.52). In the next section we will recover this same result by computing the q1-logarithmic derivative of the twisted superpotential using localization. SU( 3 )[1,1,1]. In this case the quiver is the one drawn in gure 4. Since M = 3, we have two FI parameters and two sets of chiral ring equations. For the rst node the equation is given by (3.11) which, in terms of the exponential variables, explicitly reads 2 s=1 Y(S(1) Ss(2)) = For the last node, instead, the chiral ring equations are given by (3.24), namely Pb3 Ss(2) = 2 0 2 2 Ss(2) q S(1)Ss(2) 1 S(1) A for s = 1; 2. Here Pb3 is as in (3.35) with U1 and U2 given in (A.5) and (A.6). Solving these equations by means of the ansatz (3.26), we obtain (1) = ? 1 log S?(1) = a1 + p A12 A1A2 21 + p A1A3 A13 6 In terms of the classical vev's ai these solutions become, respectively, Tr ?(2) = 1 log S?(2;1) + log S?(2;2) = a1 + a2 p A23 A2A3 22 + p A1A3 A13 ! 0 these expressions reproduce the rst few terms of the 2d/4d solutions (2.57) and (2.58) and, as we will see in the next section, they perfectly agree with the qI -logarithmic derivatives of the twisted superpotential calculated using localization, con rming (3.27). We have also computed and checked higher order terms in these SU( 3 ) examples, as well as in theories with gauge groups of higher rank (up to SU( 6 )). 4 Rami ed instantons in 4d and 5d In this section we treat the surface operators as monodromy defects D. We begin by considering the four-dimensional case and later we will discuss the extension to a vedimensional theory compacti ed on a circle of length . 4.1 We parametrize R 4 ' C2 by two complex variables (z1; z2) and place D at z2 = 0, lling the z1 plane. The presence of the surface operator induces a singular behavior in the gauge connection A, which acquires the following generic form [4, 26]: A = A dx diag 1 ; M ; ; M d (4.1) as r ! 0. Here (r; ) denote the polar coordinates in the z2-plane orthogonal to D, and I are constant parameters that label the surface operator. The M integers nI are a partition of N and identify a vector ~n associated to the symmetry breaking pattern of the Levi decomposition (2.1) of SU(N ). This vector also determines the split of the vev's ai according to (2.47). A detailed derivation of the localization results for a generic surface operator has been given in [4{6], following earlier mathematical work in [27{29]. Here, we follow the discussion in [6] to which we refer for details, and present merely those results that are relevant for the pure gauge theory. The instanton partition function for a surface operator described by ~n is given by7 X ZfdI g[~n] with ZfdI g[~n] = Y M I=1 ( qI )dI Z dI ! YdI d I; M Y dI Y I=1 ; =1 g ( I; g ( I; M dI nI Y Y Y I=1 =1 s=1 g aI;s I; + ; ) I; + 1) 1 Y M dI dI+1 g ( I; Y Y I=1 =1 =1 g ( I; nI+1 Y t=1 g I; + 12 ( 1 + ^2) I; Let us now explain the notation. The M positive integers dI count the numbers of rami ed instantons in the various sectors, with the convention that dM+1 = d1.8 When these numbers are all zero, we understand that ZfdI =0g = 1. The M variables qI are the rami ed instanton weights, which will be later identi ed with the quantities qI used in the previous sections (see in particular (2.42) and (2.43)). The parameters 1 and ^2 = 2=M specify the -background [23, 24] which is introduced to localize the integrals over the instanton moduli space; the rescaling by a factor of M in 2 is due to the ZM -orbifold that is used in the rami ed instanton case [4]. Finally, the function g is simply This seems an unnecessary redundancy but we have preferred to introduce it because, as we will see later, in the ve-dimensional theory the integrand of the rami ed instanton 7Here, di erently from [6], we have introduced a minus sign in front of qI in order to be consistent with the conventions chosen in the twisted chiral ring. 8Also in nI , I and aI , the index I is taken modulo M . (4.2) (4.3) : (4.4) partition function will have exactly the same form as in (4.3), with simply a di erent function g. The integrations over I in (4.2) have to be suitably de ned and regularized, and we will describe this in detail. But rst we discuss a few consequences of the integral expression itself and show how to extract the twisted chiral superpotential from Zinst. An immediate feature of (4.3) is that, unlike the case of the N = 2? theory studied in [6], the counting parameters qI have a mass dimension. In order to x it, let us consider the contribution to the partition function coming from the one-instanton sector. This is a sum over M terms, each of which has dI = 1 for I = 1; : : : ; M . Explicitly, we have HJEP1(207)3 Z1 inst = nI XM Z d I qI Y I=1 2 i 1 s=1 aI;s 1 I + 12 ( 1 + ^2) nI+1 Y t=1 1 I Since the partition function is dimensionless and I carries the dimension of a mass, we deduce that mass dimension of qI is qI = ( 1)kI 1 bI for I = 1; : : : ; M I 1 ; 9The sign in this formula is the one that, given our conventions, is consistent with the standard eld theory results. log Zinst = Finst + 1^2 Winst + : : : 1 where the ellipses refer to regular terms. The key point is that the prepotential extracted this way depends only on the product of all the qI . On the other hand, it is well-known that the instanton contributions to the prepotential are organized at weak coupling as a power series expansion in 2N where is the dynamically generated scale of the four-dimensional theory and 2N is the one-loop coe cient of the gauge coupling -function. Thus, we are naturally led to write9 M I=1 Y qI = ( 1)N 2N : Notice that the mass-dimensions (4.6) attributed to each of the qI are perfectly consistent with this relation, since the integers nI form a partition of N . We therefore nd that we can use exactly the same parametrization used in the e ective eld theory and given in (2.42) and (2.43), which we rewrite here for convenience qI = nI + nI+1 = bI where the last step follows from combining (2.2) and (2.28). Another important dimensional constraint follows once we extract the non-perturbative contributions to the prepotential F and to the twisted e ective superpotential W from Zinst. This is done by taking the limit in which the -deformation parameters i are set to zero according to [4, 26, 30] M 1 Y qI I=1 1 : : (4.5) (4.6) (4.7) (4.8) (4.9) 2 21 (2a + 1 + ^2) (2a + 2 1 + ^2) 2 21 ( 2a + 1 + ^2) ( 2a + 2 1 + ^2) and the contribution to the partition function in all cases up to two boxes for the SU(2) theory. Here we have set a1 = a2 = a. Residues and contour prescriptions. The last ingredient we have to specify is how to evaluate the integrals over I in (4.2). The standard prescription [6, 31{33] is to consider aI;s to be real and then close the integration contours in the upper-half I; -planes with the choice Im ^2 Im 1 > 0 : (4.10) It is by now well-established that with this prescription the multi-dimensional integrals receive contributions from a subset of poles of zfdI g, which are in one-to-one correspondence with a set of Young diagrams Y = fYI;sg, with I = 1; ; M and s = 1; nI . This fact can be exploited to organize the result in a systematic way (see for example [6] for details). Let us brie y illustrate this for SU(2), for which there is only one allowed partition, namely [1; 1], and hence one single surface operator to consider [34]. In table 1 we list the explicit results for this case, including the location of the poles and the contribution due to all the relevant Young tableaux con gurations up to two boxes. Combining these results, we nd that the instanton partition function takes the following form Zinst[1; 1] = 1 + q1 q 2 1 1 (2a + 1 + ^2) 1 ( 2a + 1 + ^2) + 1 + ^2 + q1q2 21^2 ( 2a + 1 + ^2) (2a + 1 + ^2) q 2 2 (4.11) The prepotential and the twisted e ective superpotential are extracted according to (4.7) and using the map (4.9). Let us focus on the twisted superpotential Winst, or better on its q1-derivative. We nd This precisely matches, up to two instantons, the non-perturbative part of the result (2.18) obtained by solving the twisted chiral ring equations for the quiver theory representing the surface defect in SU(2). We have also checked the agreement at higher instanton orders (up to six boxes), which we have not reported here for brevity. The speci c prescription (4.10) we have chosen to compute the instanton partition function is particularly nice due to the correspondence of the residues with Young tableaux. However, there are many other possible choices of contours that one can make. One way to classify these distinct contours is using the Je rey-Kirwan (JK) prescription [35]. In this terminology, the set of poles chosen to compute the residues is described by a JK parameter , which is a particular linear combination of the I;s; the prescription chooses a set of factors D from the denominator of zfdI g such that, if we only consider the I;s-dependent terms of these chosen factors, then, can be written as a positive linear combination of these. For instance, our prescription in (4.10) corresponds to choosing10 the form M 1 X I=1 = I + M For a detailed discussion of this method in the context of rami ed instantons we refer to [7] where it is also shown that di erent JK prescriptions can be mapped to di erent quiver realizations of the surface operator. Let us consider for example the prescription corresponding to a JK parameter of where is a large positive number. In our notation this corresponds to closing the integration contours in the upper half-plane as before for the rst (M 1) variables, and in the lower half plane for M . Applying this new prescription to the SU(2) theory, we nd a di erent set of poles that contribute. They are explicitly listed in table 2. Comparing with table 1, we see that, although the location of residues has changed, for most cases the residues are unchanged. The only set of residues that give an apparently di erent answer is the one with d1 = d2 = 1 with weight q1q2. As opposed to the earlier case, where there were three contributions, now there are only two terms proportional to q1q2. However, it is easy to see that if we sum these contributions, we nd an exact match between the two prescriptions. This fact should not come as surprise since it is a simple consequence of the residue theorem applied to the 2 integral. Therefore, all 10We understand the extra index s running from 1 to nI . = M X I=1 I (4.12) (4.13) (4.14) HJEP1(207)3 weight poles q1 q2 q1q2 q1q2 2 1 2 2 1 1 in all cases up to two boxes for the SU(2) theory using the contour prescription corresponding to the JK parameter (4.14). results that follow from the instanton partition function (and in particular the twisted superpotential) are the same in the two cases. Of course what we have just seen in the simple SU(2) case at the two instanton level, occurs also at higher instanton numbers and with higher rank gauge groups. The price one pays in changing the contour prescription or equivalently in changing the JK parameter from (4.13) to (4.14) is the loss of a simple one-to-one correspondence with the Young tableaux, but the gain is that, as shown in [7], the second prescription produces at each instanton order an instanton partition that is already organized in a factorized form in which the various factors account for the 2d, the 4d and the mixed 2d/4d contributions. This is a feature that will play a fundamental role in the 3d/5d extension. Let us now list our ndings obtained by using the second residue prescription for the SU( 3 ) theory, limiting ourselves to the one-instanton terms for brevity. In the case of the surface operator corresponding to the partition [1,2] we get 1 (a12 + 1 + ^2) (a13 + 1 + ^2) 1 (a21 + 1 + ^2) (a31 + 1 + ^2) while for the surface operator described by the partition [1,1,1] we obtain + + 1 (a12 + 1 + ^2) 1 (a23 + 1 + ^2) 1 (a31 + 1 + ^2) q2 q3 q1 q2 Applying (4.7) to extract Winst, we nd that the qI -logarithmic derivatives of the twisted superpotential for the two partitions perfectly match the non-perturbative pieces of the solutions (2.57) and (2.58) of the twisted chiral ring equations. We have checked that this agreement persists at the two-instanton level. We have also thoroughly explored all surface operators in the SU(4) theory and many cases in higher rank theories up to two instantons, always nding a perfect match between the qI -logarithmic derivatives of W and the solutions of the corresponding twisted chiral ring equations. We now turn to discuss the results for a gauge theory on R 4 S1 in the presence of a HJEP1(207)3 surface operator also wrapping the compacti cation circle. This case has been discussed by a number of recent works (see for instance [36, 37]). Here we observe that the rami ed instanton partition function is given by the same expressions (4.2) and (4.3) in which the function g(x) is [23, 24, 38] qI = ( 1)kI 1 qM = ( 1)N I 2N bI Another di erence with respect to the 2d/4d case is that the counting parameters qI are now dimensionless and are given by The nal result is obtained by summing the residues of zfdI g over the same set of poles selected by the JK prescription (4.14). Let us illustrate these ideas by calculating the twisted e ective superpotential that governs the infrared behavior of the [1; 1] operator in SU(2). Up to two instantons, the partition function using these rules is Zinst.[1; 1] = 1 + q1 q2 4 sinh 2 1 sinh 2 (2a+ 1 +^2) 4 sinh 2 1 sinh 2 ( 2a+ 1 +^2) + + + + + : : : 16 sinh 2 1 sinh 1 sinh 2 (2a + 1 + ^2) sinh 2 (2a + 2 1 + ^2) 16 sinh 2 1 sinh 1 sinh 2 ( 2a + 1 + ^2) sinh 2 ( 2a + 2 1 + ^2) q1q2 sinh 2 ( 1 + 2^2) 16 sinh2 2 1 sinh ^2 sinh 2 ( 2a + 1 + ^2) sinh 2 (2a + ^2) 16 sinh 2 1 sinh ^2 sinh 2 (2a + 1 + ^2) sinh 2 (2a + ^2) q1q2 q 2 1 q 2 2 (4.17) (4.18) (4.19) where a1 = a2 = a. From this instanton partition function we can extract the twisted chiral superpotential in the usual manner according to (4.7). The result is It is very easy to check that in the limit ! 0 this expression reduces to the 2d/4d result in (4.12). Most importantly it agrees with the non-perturbative part of the solution (3.32) of the chiral ring equation of the 3d/5d SU(2) theory, thus con rming the validity of (3.27). Similar calculations can be performed for the higher rank cases without much di culty, and indeed we have done these calculations for all surface operators of SU(4) and for many cases up to SU( 6 ). Here, for brevity, we simply report the results at the one-instanton level for the surface operators in the SU( 3 ) theory. In the case of the defect of type [1,2] the instanton partition function is 8 sinh 2 1 sinh 2 (a12 + 1 + ^2) sinh 2 (a13 + 1 + ^2) 8 sinh 2 1 sinh 2 ( a12 + 1 + ^2) sinh 2 ( a13 + 1 + ^2) q2 (4.21) (4.22) while for the defect of type [1,1,1] we nd q1 q3 4 sinh 2 1 sinh 2 (a31 + 1 +^2) + 4 sinh 2 1 sinh 2 (a12 + 1 +^2) 4 sinh 2 1 sinh 2 (a23 + 1 + ^2) where aij = a1 aj . These expressions are clear generalizations of the 2d/4d instanton partition functions (4.15) and (4.16). Moreover one can check that the twisted superpotentials that can be derived from them perfectly match the ones obtained by solving the chiral ring equations as we discussed in section 3. 5 Superpotentials for dual quivers The 2d/4d quiver theories considered in section 2 admit dual descriptions [7, 12, 13]. In particular, with repeated applications of Seiberg-like dualities, one can prove that the linear quiver of gure 1 is dual to the one represented in gure 5. Here the ranks of the U(rI ) gauge groups are given by rI = N kI = nK ; (5.1) where in the second step we have used (2.2) to express kI in terms of the entries of partition [n1; : : : ; nM ] labeling the surface defect. Notice the reversal of the arrows with respect to the quiver in gure 1, and thus the di erent assignment of massive chiral elds to fundamental or anti-fundamental representations. M X K=I+1 The new quiver provides an alternative realization of the same surface operator [7]. modi cation of (2.22), and reads11 Its corresponding twisted superpotential, which we denote by Wf, is given by the obvious HJEP1(207)3 r1 X s=1 Tr $ e Wf = 2 i X X (I 1) (I) by tilde variables. Thus, if the FI parameters in the two dual models are related to each other by 11For later convenience, we denote the twisted chiral scalars and the FI couplings of the dual gauge groups As in (2.22), the linear terms in I ( ) are the classical contributions, while the other terms are the one-loop part. The dual FI couplings I ( ) renormalize like the orginal couplings I ( ) but with kI replaced by rI . In view of (5.1), this implies that the one-loop -function e coe cient in the dual theory is opposite to that of the original theory, namely ebI = rI+1 rI 1 = kI+1 + kI 1 = bI : In turn, this implies that the dynamically generated scale in the I-th node of the dual theory is given by ebII = e 2 i eI bI ; to be compared with (2.27). As usual we can trade the couplings I ( ) for these scales eI , and thus rewrite the twisted superpotential (5.2) in a form that is the straightforward e modi cation of (2.29). If we make the following classical ansatz e (I) = diag(an1+:::+nI +1; an1+:::+nI +2; : : : ; aN ) ; which is dual to the one for (I) given in (2.41), then it is easy to check that Tr e(I) = Tr (I) : e e I = I ; (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) one has Wfclass = Wclass. Notice that using (5.8) in (5.4) and comparing with (2.27), we have eI = I : (5.9) HJEP1(207)3 The relation (5.6) remains true also at the quantum level. This statement can be veri ed by expanding e (I) as a power series in the various eI 's around the classical vacuum (5.5), and iteratively solving the corresponding chiral ring equations in a semi-classical approximation. Doing this and using (5.8) and (5.9), we have checked the validity of (5.6) in several examples. Furthermore, we have obtained the same relations also using the localization methods described in section 4. Therefore, we can conclude that the two quiver theories in gure 1 and 5, indeed provide equivalent descriptions of the 2d/4d defect SU(N )[n1; : : : ; nM ]. This conclusion changes drastically once we consider the 3d/5d quiver theories compacti ed on a circle. In this case, the dual superpotential corresponding to the quiver in gure 5, is obtained by upgrading (5.2) to a form analogous to (3.8), namely Wf = X ebI log( eI )es(I) X X X ` et (I 1) (I) r1 X s=1 Tr ` (1) es Here we have used the loop-function `(x) de ned in (3.3), and taken into account the renormalization of the FI couplings to introduce the scales eI . Using for the original quiver the ansatz (3.25), and for the dual theory the ansatz (5.5), which can be rewritten as S(I) = diag(An1+:::+nI+1 ; An1+:::+nI+2 ; : : : ; AN ) e in terms of the exponential variables Se(I) = e e the relation (5.6) still holds true. However, in general, this is no longer valid for the full solutions of the chiral ring equations. This happens whenever the ranks kI of the original quiver theory and the ranks rI of the dual model are di erent from each other for some I, which is the generic situation. Let us show this in a speci c example, namely the defect of type [1,2] in the SU( 3 ) theory. The original quiver theory was discussed in detail in section 3 where we have shown that the solution of the chiral ring equation is (see (3.36)) (I) and Ai = e ai , one can easily check that : (5.10) (5.11) (5.12) The dual quiver for this defect is depicted in gure 6. From (5.10), it follows that the corresponding twisted superpotential is Wf = 2 X s=1 3 2 i log( 1)es Tr `( s) : Using the function Pb3 de ned in (3.35), we see that the twisted chiral ring equations are e1 for s = 1; 2. Solving iteratively these equations around the classical vacuum (5.5), we nd HJEP1(207)3 Tr e? = a2 + a3 + 2 e13 + 6 3 A1=2 2 (5.15) of the solution By comparing (5.12) and (5.15), we see that at the classical level Tr e? is equal to negative ? in the original quiver; this simply follows from the SU( 3 ) tracelessness condition. However, the rst semi-classical correction of order 2 spoils this relation, even if we use the relation (5.9) between the dynamically generated scales. Therefore, as anticipated, the two descriptions are not any more dual to each other. It is interesting to observe that the twisted superpotential corresponding to the dual solution (5.15) can also be obtained using localization. Indeed, if one evaluates the instanton partition function Zinst[1; 2] for the compacti ed theory using the JK prescription with where is positive and large, and then extracts from it the corresponding twisted superpotential using (4.7), one obtains precisely the above result.12 Notice that the JK parameter (5.16) is opposite in sign with respect to the one in (4.14) that we have adopted in the original quiver realization. Actually, what we have seen in this particular example can be generalized to other cases and for any M , we nd that the JK parameter which has to be used in the localization computations for the dual quiver theory to match the solution of the chiral ring equations is = + 1 2 ; = M 1 X I=1 I M : (5.13) (5.14) (5.16) (5.17) This fact points towards the nice scenario in which the twisted superpotentials W and Wf for a pair of quiver theories related by a chain of Seiberg-like dualities can be obtained in localization using two di erent JK prescriptions associated to opposite parameters. While in the 2d/4d systems all di erent JK prescriptions are equivalent to each other and lead to the same superpotentials, in general this is no longer true in the 3d/5d theories because of the particular structure of the instanton partition functions. 12We have checked this up to the two-instanton level, namely up to order 5. Adding Chern-Simons terms We now investigate the possibility of restoring the duality between the two 3d/5d descriptions of the SU( 3 )[1,2] defect by considering the addition of Chern-Simons (CS) couplings. These can be written as a term in the twisted chiral superpotential that is quadratic in the twisted scalars and proportional to the compacti cation circle [14, 39]. For the Ith node, the CS term is of the form: WCS = Tr ( (I))2 : Let us start from the original theory and let us turn on a CS term on the U(1) node with coupling k. The resulting quiver is represented in gure 7 and the corresponding twisted superpotential is W = 3 log( 1) 2 Tr `( ) : Repeating the same steps described in section 3, we easily obtain the modi ed twisted chiral ring equation Pb3(S) = where, as before, Pb3 is given in (3.35) and S = e . Solving this equation with the usual ansatz leads to Of course, for k = 0 one recovers the solution (5.12) in the absence of the CS term. Let us now consider the dual quiver with a CS interaction with coupling ek turned on in the U(2) node. This is represented in gure 8 and the corresponding twisted superpotential is Wf = 2 X s=1 3 log( 1) es + e 2 k 2 es Tr `( e s) : 1k 3 k 2 k 2 e 6 Ak 3 (5.18) (5.19) (5.20) (5.21) (5.22) From this we can easily derive the twisted chiral ring equations, namely 3 e1 + es for s = 1; 2, which are a simple generalization of (5.14). Solving these equations with the Of course for ek = 0 we recover the solution (5.15) in the absence of the CS coupling. Our main observation is that if we take then, using the SU( 3 ) tracelessness condition and the relations (5.8) and (5.9), we have k = ek = 1 2 e?;1 + e?;2 = ? e1 de1 dWf = 1 dW d 1 2 3 A1=2 ek e1 (5.23) (5.24) (5.25) (5.26) (5.27) e?;1 + e?;2 = a2 + a3 + This implies that so that the duality between W and Wf is restored at the quantum level under the map (5.8) and (5.9). We have checked that this match holds true at the next-order in the -expansion of the solutions of the chiral ring equations. Therefore, thanks to the CS terms also in the 3d/5d case we can realize the same kind of relation which was manifest in the 2d/4d theories. In appendix B, we discuss a slightly more complicated example in which a similar phenomenon occurs. It is a surface operator of type [1,1,2] in the SU(4) gauge theory compacti ed on S1. In this case as well, the two dual quivers lead to the same twisted chiral superpotential provided suitable CS couplings are turned on. 6 Conclusions and perspectives In this paper we have studied surface operators in four- and ve-dimensional SU(N ) gauge theories, focusing on the e ective twisted chiral superpotential which governs their infrared dynamics. Our results are a natural generalization and extension of those in [10]. We have illustrated our ndings in the context of the simplest defects in SU(2) and SU( 3 ) theories, even though we have performed several checks in a number of theories with higher rank gauge groups. Already in the simplest SU( 3 ) case we could observe that di erent realizations of the same surface operator in terms of dual quivers lead, in the ve-dimensional case, to di erent twisted superpotentials. We have found that this feature is reproduced also in the localization approach where the di erent superpotentials arise from di erent choices of the Je rey-Kirwan residue prescription. In an interesting twist, we have shown that the duality between the coupled 3d/5d quivers can be restored by the addition of suitable three-dimensional Chern-Simons terms. Clearly, it would be desirable to do a systematic analysis of this phenomenon and thoroughly explore the e ects of the Chern-Simons couplings, but we leave this to future work. It would be very interesting to extend our results to surface operators in N theories. As shown in our earlier work [6], the non-perturbative S-duality group of the four-dimensional theory constrains the twisted superpotential of the monodromy defect, which can be written in terms of elliptic and modular forms in a semi-classical expansion. = 2 ? Therefore, it would be worthwhile to understand if it is possible to obtain these exact results directly from the chiral ring analysis of a coupled quiver gauge theories and also to extend them to ve dimensions. Surface operators in four- and ve-dimensional gauge theories have been studied also by exploiting their connections to integrable systems and in particular the relation between the twisted chiral ring equations and Bethe ansatz for integrable spin chains [14{16]. In this context, the wave-functions of the quantum systems can be related to the instanton partition function in the presence of surface operators [40, 41]. For the 3d/5d theories studied in this work, recently there has been interesting developments on the connection between the instanton partition function and the wave functions of relativistic Toda theories [42]. It would be worthwhile to explore this direction using our methods. Another interesting possibility is to use dualities between three-dimensional quiver gauge theories with avor to study bi-spectral dualities between quantum integrable systems [11]. It would be desirable to investigate the possible implications of our results for these integrable systems, especially in the presence of Chern-Simons couplings. Acknowledgments We would like to thank Sourav Ballav, Noppadol Mekareeya, Madhusudhan Raman and Jan Troost for many useful discussions. S.K.A. would especially like to thank for the hospitality the Physics Department of the University of Torino and the Torino section of INFN where this work was initiated. 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 Chiral correlators in 5d In this appendix we brie y review some well-known results about the way in which chiral correlators are computed using localization [43{45] that are useful for the calculations presented in the main text. For details we refer to [46] and references therein. In a four-dimensional theory SU(N ) the generating function of all chiral correlators of the form Tr ` is Tr ez N X ezai i=1 1 X1 qk Z Zinst k=1 I=1 Yk d2 Ii zk( I ) O(z; I ) (A.1) where zk( I ) is the k-instanton partition function and O is the following observable O(z; I ) = X ex I (1 ez 1 )(1 ez 2 ) : (A.2) Rather interestingly, the same formula (A.1) can also be exploited to compute the quantum corrected correlators in the ve-dimensional SU(N ) theory provided one uses the appropriate function g(x) as in (4.17) and sets z = ` for ` 2 Z and ` < N . With an explicit calculation, we nd the following universal formula V` DTr e` E X Ai` + `2( N i=1 )2N X N i=1 Qj6=i(Ai i AN 2+` Aj )2 + O ( )4N (A.3) HJEP1(207)3 where Ai = e ai . Once the V` are obtained, the U`'s which appear in the ve-dimensional Seiberg-Witten curve can be calculated by forming the symmetric polynomials in the usual manner. In particular, we have U1 = V1 and UN 1 = V 1 . The last relation follows by utilizing the special unitary condition PiN=1 ai = 0 which implies QiN=1 Ai = 1. For example, for SU(2) we have while for SU( 3 ) we have U1 = A1 + A2 + ( 4 A1 + A2 A2)2 + O(( )8) ; A2 1 A2)2(A1 A3)2 U1 = A1 + A2 + A3 + ( A2 2 A1)2(A2 1 A1)2(A2 U2 = A1A2 + A2A3 + A3A1 + ( A3)2 + A3)2 + A1)2(A3 A2)2 1 (A1 A2)2(A1 A3)2 A1)2(A3 A2)2 + O(( )12) ; + O(( )12) : (A.4) (A.5) (A.6) B Chern-Simons terms in an SU(4) example In this section, we provide more evidence towards the duality that was discussed in section 5. We consider the gauge group SU(4) and the surface operator described by the partition [1,1,2]. There are two dual descriptions for this defect in terms of quiver diagrams: one is represented in gure 9, and the other is represented in gure 10. In both cases we have added CS interactions. In particular, following [47{50], we have turned on a CS terms in those gauge nodes where the e ective number of fermions is odd, which for both quivers of our example are the U(2) nodes. One way to justify this is to start from a parity invariant theory and generate these non-integer CS terms by integrating out an odd number of fermions. 4 2~ Let us rst consider the quiver of gure 9. The corresponding twisted e ective superpotential with a CS coupling k is W = 2 log( 1) (1) + 3 log( 2 2 X( t(2))2 2 The twisted chiral ring relation at the rst node is while at the second node we have Qb2( (1)) 1)2 = 0 ; (2) s 2 X t=1 Tr `( t(2) ) : 2k 3 2 2) X s=1 Pb4( s(2)) = 3 32 Qb1( s(2)) (Ss(2))k + 5 8 32Qb1( s(2)) (Ss(2))k for s = 1; 2. Solving these equations order by order in by using the standard ansatz and the chiral correlators of the SU(4) theory, we obtain 1 kA1=2A1=2 3 4 A13A14 8 1 2 2 3 + : : : ; 2 A1+kA1=2A1=2 3 4 A23A24 3 2 1 A 1 kA1=2A1=2 3 4 A13A14 8 ! 2 3 1 2 + : : : : We now consider the dual quiver represented in gure 10. In this case the twisted superpotential with a CS coupling ek is Wf = 2 log( e1) X es 3 log( e2) X 3 s=1 3 s=1 X `(es(1) 3 X `(es(1) 2 t=1 et (2) + e 2 3 X s=1 2 t=1 X(et(2))2 Tr `( (B.1) (B.2) (B.3) (B.4) (B.5) The corresponding chiral ring equations are for s = 1; 2; 3, and Pb4(es(1)) = 2 Qb2(es(1)) e12 + 8 Qb2(es(1)) e12 for t = 1; 2. Solving these equations with the usual ansatz, we nd Tr e?(1) = a2 + a3 + a4 2 2 Qb1(et(2)) = 3 3 e2 (Set(2))ek 1 A12 2 A31A34 A ek 1=2A ek 1A ek 1=2 2 A41A43 A1=2A13+ekA14=2 1 A31A34 A 1=2 ekA ek 1A 1=2 ek A14A34 1 2 3 3 Tr e(2) = a3 + a4 2 A12=2A13 ekA14=2 A32A34 Aek 1A 1=2+ekA 1=2+ek 2 A42A43 3 8 2 3 e1 e2 8 2 3 e1 e2 3 e2 + : : : : (B.6) (B.7) (B.9) (B.10) These expressions look very di erent from the solution (B.4) of the chiral ring equations of the original quiver. However, if we impose the SU(4) tracelessness constraint Pi ai = 0 and use the following map we nd the following relations 1 2 k = ek = and I = eI ; Tr e?(1) = Tr e?(2) = ?(1) ; Tr ?(2) : This proves that, to leading order in the instanton expansion, the superpotentials of the dual pair match as expected. We have checked that this match continues to hold up to two instantons as well. Open Access. 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S. K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, V. Gupta, R. R. John, A. Lerda. Surface operators, chiral rings and localization in \( \mathcal{N} \) =2 gauge theories, Journal of High Energy Physics, 2017, 137, DOI: 10.1007/JHEP11(2017)137