# Surface operators, dual quivers and contours

## Abstract

We study half-BPS surface operators in four dimensional \({{{\mathcal {N}}}}=2\) SU(*N*) gauge theories, and analyze their low-energy effective action on the four dimensional Coulomb branch using equivariant localization. We also study surface operators as coupled 2d/4d quiver gauge theories with an SU(*N*) flavour symmetry. In this description, the same surface operator can be described by different quivers that are related to each other by two dimensional Seiberg duality. We argue that these dual quivers correspond, on the localization side, to distinct integration contours that can be determined by the Fayet-Iliopoulos parameters of the two dimensional gauge nodes. We verify the proposal by mapping the solutions of the twisted chiral ring equations of the 2d/4d quivers onto individual residues of the localization integrand.

## 1 Introduction

Surface operators in 4d gauge theories are natural two dimensional generalizations of Wilson and ’t Hooft loops which can provide valuable information about the phase structure of the gauge theories [1]. In this paper we study the low-energy effective action of surface operators in pure \({{\mathcal {N}}}=2\) 4d gauge theories from two distinct points of view, namely as monodromy defects [2, 3] and as coupled 2d/4d quiver gauge theories [4, 5]. In the first approach, one specifies how the 4d gauge fields are affected by the presence of the surface operator by imposing suitable boundary conditions in the path-integral. In this framework the non-perturbative effects are described in terms of ramified instantons [2] whose partition function can be computed using equivariant localization methods [5, 6, 7, 8, 9, 10]. From the ramified instanton partition function one can extract two holomorphic functions [11, 12]: one is the prepotential \(\mathcal {F}\) that governs the low-energy effective action of the 4d \({\mathcal {N}}=2\) gauge theory on the Coulomb branch; the other is the twisted chiral superpotential \({\mathcal {W}}\) that describes the 2d dynamics on the defect.

In the second description of the surface operators, one considers coupled 2d/4d theories that are (2, 2) supersymmetric sigma models with an ultraviolet description as a gauged linear sigma model (GLSM). The low-energy dynamics of such a GLSM is completely determined by a twisted chiral superpotential \({{{\mathcal {W}}}}(\sigma )\) that depends on the twisted chiral superfields \(\sigma \) containing the 2d vector fields [13]. By giving a vacuum expectation value (vev) to the adjoint scalar of the 4d \({\mathcal {N}}=2\) gauge theory, one introduces twisted masses in the 2d quiver theory [14, 15]. At a generic point on the 4d Coulomb branch, the 2d theory is therefore massive in the infrared and the 2d/4d coupling mechanism is determined via the resolvent of the 4d gauge theory [5]. The resulting massive vacua of the GLSM are solutions to the twisted chiral ring equations, which are obtained by extremizing \({{\mathcal {W}}}(\sigma )\) with respect to the twisted chiral superfields.

The main goal of this work is to clarify the precise relationship between the above two descriptions of the surface operators. In our previous works [9, 10] the first steps in this direction were already taken by showing that there is a precise correspondence between the massive vacua of the 2d/4d gauge theory and the monodromy defects in the \({{{\mathcal {N}}}}=2\) gauge theory. In fact, the effective twisted chiral superpotential of the 2d/4d quiver gauge theory evaluated in a given massive vacuum exactly coincides with the one computed from the 4d ramified instanton partition function [9, 10]. This equality was shown in a specific class of models that are described by oriented quiver diagrams. Recently, this result has been proven in full generality in [16, 17].

An important feature of the (2, 2) quiver theories that was not fully discussed in our previous papers is Seiberg duality [18, 19]. This is an infrared equivalence between two gauge theories that have different ultraviolet realizations. In this work we fill this gap and consider all possible quivers obtained from the oriented ones by applying 2d Seiberg duality. While all such quivers have different gauge groups and matter content, once the 4d Coulomb vev’s are turned on, it is possible to find a one-to-one map between their massive vacua. Therefore it becomes clear that they must describe the same surface operator from the point of view of the 4d gauge theory; indeed, the different twisted chiral superpotentials, evaluated in the respective vacua, all give the same result. This equality of superpotentials gives a strong hint that the choice of a Seiberg duality frame might have an interpretation as distinct contours of integration on the localization side: the equality of the superpotentials would then be a simple consequence of multi-dimensional residue theorems.

In this work we show that this expectation is correct and provide a detailed map between a given quiver realization of the surface operator and a particular choice of contour in the localization integrals. This contour prescription can be conveniently encoded in a Jeffrey-Kirwan (JK) reference vector [20], whose coefficients turn out to be related to the Fayet-Iliopoulos (FI) parameters of the 2d/4d quiver. While the twisted superpotentials are equal irrespective of the choice of contour, the map relates the individual residues on the localization side to the individual terms in the solutions to the twisted chiral ring equations, thereby allowing us to identify in an unambiguous way which quiver arises from a given contour prescription and vice-versa.

This paper is organized as follows. In Sect. 2 we review and extend our earlier work [8, 9, 10], and in particular we show how to map the oriented quiver to a particular contour by studying the solution of the chiral ring equations and the precise correspondence to the residues of the localization integrand. In Sect. 3 we discuss the basics of 2d Seiberg duality and how it acts on the quiver theories we consider. In Sect. 4 we apply the duality moves to the oriented quiver of interest and show in detail (for the 4-node quiver), how it is possible to map each quiver to a particular integration contour on the localization side without explicitly solving the chiral ring equations. We also discuss how this integration contour can be specified in terms of a JK reference vector. In Sect. 5 we give a simple solution for the JK vector associated to a generic linear 2d/4d quiver with arbitrary number of nodes. Finally, we summarize our main results in Sect. 6 and collect the more technical material in the appendices.

## 2 Review of earlier work

To set the stage for the discussion in the next sections and also to introduce our notation, we briefly review the results obtained in our earlier work [10] where we studied surface operators both as monodromy defects in 4d and as coupled 2d/4d gauge theories.

### 2.1 Surface operators as monodromy defects

*N*) theory is specified by a partition of

*N*, denoted by \(\varvec{n}=(n_1, n_2, \ldots n_M)\), which corresponds to the breaking of the gauge group to a Levi subgroup

*N*) theory as follows:

*I*th partition in (2) is of length \(n_I\). Introducing the following set of numbers with cardinality \(n_I\):

*M*positive integers \(d_I\) count the numbers of ramified instantons in the various sectors, the variables \(q_I\) are the ramified instanton weights, and the parameters \(\epsilon _1\) and \({\hat{\epsilon }}_2=\epsilon _2/M\) specify the \(\varOmega \)-background [21, 22] which is introduced to localize the integrals over the instanton moduli space.

^{1}

*a*as real variables and assign an imaginary part to the \(\varOmega \)-deformation parameters according to

### 2.2 Surface operators as coupled 2d/4d quivers

*G*, the relevant sigma model is defined on the target space \(G/{\mathbb {L}}\) [2, 3]. Such a space is, in general, a flag variety which can be realized as the low-energy limit of a GLSM [13, 15], whose gauge and matter content can be summarized in the quiver diagram of Fig. 1.

Each circular node represents a 2d gauge group \(\mathrm {U}(r_I)\) where the ranks \(r_I\) are as in (3), whereas the last node on the right hand side represents the 4d gauge group SU(*N*) which acts as a flavour symmetry group for the \((M-1)^{\mathrm {th}}\) 2d node. The arrows correspond to matter multiplets which are rendered massive by non-zero v.e.v’s of the twisted scalars \(\sigma ^{(I)}\) of the \(I^{\mathrm {th}}\) node and of the 4d adjoint scalar \(\varPhi \). The orientation of the arrows specifies whether the matter is in the fundamental (out-going) or in the anti-fundamental (in-going) representation.

*I*th node at the scale \(\mu \), namely

*I*th gauge node. Finally, the angular brackets in the last term of (12) correspond to a chiral correlator in the 4d SU(

*N*) theory. This correlator implies that the coupling between the 2d and 4d theory is via the resolvent of the SU(

*N*) gauge theory [5], which in turn depends on the 4d dynamically generated scale \(\varLambda _{\text {4d}}\).

*i.e.*they are solutions of the twisted chiral ring equations [23, 24]

### 2.3 A contour from the twisted chiral ring

*N*) node, namely

^{2}Indeed, the dimensional argument allows us to express the ramified instanton counting parameters in terms of the 2d effective scales as follows [10]

^{3}:

## 3 2d Seiberg duality

The notion of Seiberg duality in 4d gauge theories [18] can be generalized to two dimensions (see for example [19]). Thus, by applying 2d Seiberg duality it is possible to obtain distinct quiver theories in the UV that have the same IR behaviour.

*r*) gauge theory with \(N_F\) fundamental flavours and \(N_A\) anti-fundamental flavours. For definiteness we take \(N_F > N_A\), and call this system “theory A”. Its classical twisted superpotential is simply

^{4}

## 4 Relating quivers and contours

In this section we discuss different 2d/4d theories related by Seiberg duality to the oriented quiver represented in Fig. 2. To any of these theories we can associate a system of twisted chiral ring equations that are distinct from the ones we have discussed in Sect. 2.3. However, being related by Seiberg duality, there is a simple one-to-one map among them and their solutions. Then, a natural question arises: how is this duality map reflected on the localization side?

To answer this question, consider again the oriented quiver of Fig. 2, which we now denote by \(Q_0\). From it we can generate equivalent quivers by dualizing any of the 2d nodes. We first carry out a very specific sequence of dualities that are shown in Fig. 8: at each step of the duality chain, the node being dualized has only fundamental matter. Therefore, Seiberg duality always acts as in (40).^{5}

For each quiver in the chain, we can proceed as we did in Sect. 2.3 for \(Q_0\). We integrate out the matter multiplets to obtain the effective twisted chiral superpotential, derive from it the twisted chiral ring equations, solve them about a particular massive vacuum order by order in the strong coupling scales, evaluate the superpotential on the corresponding vacuum and finally compare the result with the ramified instanton calculation with a specific integration contour for the \(\chi _I\) variables. In this program, the choice of the classical vacuum is the first important piece of information which we have to provide.

### 4.1 Classical vacuum

^{6}:

Quiver | \(\sigma ^{(1)}_{\text {cl}}\) | \(\sigma ^{(2)}_{\text {cl}}\) | \(\sigma ^{(3)}_{\text {cl}}\) |
---|---|---|---|

\(\,\,Q_0 \) | \({{\mathcal {A}}}_1\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2 \oplus {{\mathcal {A}}}_3\) |

\(\,\,Q_1 \) | \({{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2 \oplus {{\mathcal {A}}}_3\) |

\(\,\,Q_2 \) | \({{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2 \oplus {{\mathcal {A}}}_3\) |

\(\,\,Q_4 \) | \({{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3 \oplus {{\mathcal {A}}}_4\) |

\(\,\,Q_5 \) | \({{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3 \oplus {{\mathcal {A}}}_4\) |

\(\,\,Q_6 \) | \({{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_3\oplus {{\mathcal {A}}}_4\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3 \oplus {{\mathcal {A}}}_4\) |

\(\,\,Q_7 \) | \({{\mathcal {A}}}_4\) | \({{\mathcal {A}}}_3\oplus {{\mathcal {A}}}_4\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3 \oplus {{\mathcal {A}}}_4\) |

### 4.2 The *q* vs \(\varLambda \) map

The next necessary ingredient is the relation between the ramified instanton parameters \(q_I\) and the strong coupling scales \(\varLambda _I^{Q_i}\) of a given quiver.

*q*vs \(\varLambda \) map was already derived and written in (32). If we now consider the second quiver \(Q_1\), from the running of the FI parameters we find

*q*vs \(\varLambda \) map in this case, namely

For each quiver of Fig. 8, we list the *q* vs \(\varLambda \) map (up to sign factors, which can be found in Appendix B). The exponent of each strong coupling scale is determined by the number of effective flavours at that node in the quiver and is related to the \(\beta \)-function coefficient of the corresponding FI parameter

Quiver | \(q_1\) | \(q_2\) | \(q_3\) |
---|---|---|---|

\(\,Q_0\) | \(\varLambda _1^{n_1+n_2}\) | \(\varLambda _2^{n_2+n_3}\) | \(\varLambda _3^{n_3+n_4}\) |

\(\,Q_1\) | \(\big (\varLambda ^{Q_1}_1\big )^{n_1+n_2}\) | \(\frac{\big (\varLambda ^{Q_1}_2\big )^{n_1+2n_2+n_3}}{\big (\varLambda ^{Q_1}_1\big )^{n_1+n_2}}\) | \(\big (\varLambda ^{Q_1}_3\big )^{n_3+n_4}\) |

\(\,Q_2\) | \(\frac{\big (\varLambda ^{Q_2}_2\big )^{n_1+2n_2+n_3}}{\big (\varLambda ^{Q_2}_1\big )^{n_2+n_3}}\) | \(\big (\varLambda ^{Q_2}_1\big )^{n_2+n_3}\) | \(\frac{\big (\varLambda ^{Q_2}_3\big )^{N+n_2+n_3}}{\big (\varLambda ^{Q_2}_2\big )^{n1+2n_2+n_3}}\) |

\(\,Q_4\) | \(\frac{\big (\varLambda ^{Q_4}_3\big )^{N+n_2+n_3}}{\big (\varLambda ^{Q_4}_1\big )^{n_2+n_3} \big (\varLambda ^{Q_4}_2\big )^{n_3+n_4}}\) | \(\big (\varLambda ^{Q_4}_1\big )^{n_2+n_3}\) | \(\big (\varLambda ^{Q_4}_2\big )^{n_3+n_4}\) |

\(\,Q_5\) | \(\frac{\big (\varLambda ^{Q_5}_3\big )^{N+n_2+n_3}}{\big (\varLambda ^{Q_5}_2\big )^{n_2+2n_3+n_4}}\) | \(\big (\varLambda ^{Q_5}_1\big )^{n_2+n_3}\) | \(\frac{\big (\varLambda ^{Q_5}_2\big )^{n_2+2n_3+n_4}}{\big (\varLambda ^{Q_5}_1\big )^{n_2+n_3}}\) |

\(\,Q_6\) | \(\big (\varLambda ^{Q_6}_3\big )^{n_1+n_2}\) | \(\frac{\big (\varLambda ^{Q_6}_2\big )^{n_2+2n_3+n_4}}{\big (\varLambda ^{Q_6}_1\big )^{n_3+n_4}}\) | \(\big (\varLambda ^{Q_6}_1\big )^{n_3+n_4}\) |

\(\,Q_7\) | \(\big (\varLambda ^{Q_7}_3\big )^{n_1+n_2}\) | \(\big (\varLambda ^{Q_7}_2\big )^{n_2+n_3}\) | \(\big (\varLambda ^{Q_7}_1\big )^{n_3+n_4}\) |

*q*vs \(\varLambda \) map to be consistent with the power series expansion of the ramified instanton partition function. For instance for the quiver \(Q_1\), we see from Table 2 that if we want that both \(q_1\) and \(q_2\) be “small”, it is necessary to have

We can repeat this analysis for all linear quivers of the sequence, and always find the same pattern: when a hierarchy of scales is needed in order to have a meaningful ramified instanton expansion, this is automatically guaranteed by the duality relations among the real FI parameters of the various quivers. Moreover, the 4d low-energy scale \(\varLambda _{\text {4d}}\) is always the smallest scale in view of (57).

### 4.3 Contour prescriptions for dual quivers

We now address the question of how the non-perturbative superpotential associated to each quiver can be obtained from the ramified instanton partition function (8) using a suitable contour prescription for the \(\chi _I\)-integrals. In Sect. 2.3 we answered this question for the oriented quiver \(Q_0\) by comparing each term of the solution of the chiral ring equations with the localization results. Here we provide a general argument that allows one to derive the appropriate contour prescription for any quiver of the duality chain, without explicitly solving the twisted chiral ring equations and integrating them in. We perform a detailed analysis at the one-instanton level, but our conclusions are valid also at higher instantons.

Comparing \({\mathcal {W}}_{\text {cl}}^{\,Q_0}\) and \({\mathcal {W}}_{\text {cl}}^{\,Q_1}\) given in (42) and (43), we notice that an indication for the flipping of the \(\chi _1\) integration contour between \(Q_0\) and \(Q_1\) can be traced to the change in sign of the term containing \(\mathrm {Tr}\,\sigma ^{(1)}\), or equivalently to the change in sign of the \(\beta \)-function coefficient and of the FI parameter of the first node under the duality map from \(Q_0\) to \(Q_1\). We propose that this is in fact the rule, and that it is the sign of the \(\beta \)-function coefficient for a given node (or of its FI parameter) that determines whether the contour of integration for the corresponding \(\chi \) variable has to be closed in the upper or in the lower half-plane.

*q*vs \(\varLambda \) map of Table 2 into (64), we find

*i.e.*for \(Q_0\), \(Q_1\) and \(Q_2\), and then it flips to the upper half-plane \((+)\), remaining unchanged for the rest of the duality chain,

*i.e.*for \(Q_4\), \(Q_5\), \(Q_6\) and \(Q_7\). We have verified the validity of this proposal by explicitly solving the twisted chiral ring equations for all seven quivers to obtain the corresponding twisted superpotentials, and checking that these agree term by term with what the ramified instanton partition function yields with the proposed integration prescriptions (see Appendix B for details). Our results on the contour assignments for the various quivers are summarized in Table 3.

^{7}

For each quiver \(Q_i\) in Fig. 8, we list the signs of the \(\beta \)-function coefficients \(b_I^{Q_i}\) for the three 2d nodes, which are also the signs of the corresponding FI parameters \(\zeta _I^{Q_i}\). These signs determine whether the integration contour for the corresponding \(\chi \)-variable has to be closed in the upper (\(+\)) or lower (−) half-plane. The last column displays the contour prescription from which we can also read which \(\chi \)-variable is associated to which node of the quiver. The variable \(\chi _4\) is always the last one to be integrated

Quiver | \(\text {sgn}(b_1^{Q_i})\) | \(\text {sgn}(b_2^{Q_i})\) | \(\text {sgn}(b_3^{Q_i})\) | Contour prescription |
---|---|---|---|---|

\(\,\,Q_0\) | \(+\) | \(+\) | \(+\) | \(\big (\chi _1|_+,\chi _2|_+,\chi _3|_+,\chi _4|_-\big )\) |

\(\,\,Q_1\) | − | \(+\) | \(+\) | \(\big (\chi _1|_-,\chi _2|_+,\chi _3|_+,\chi _4|_-\big )\) |

\(\,\,Q_2\) | \(+\) | − | \(+\) | \(\big (\chi _2|_+,\chi _1|_-,\chi _3|_+,\chi _4|_-\big )\) |

\(\,\,Q_4\) | \(+\) | \(+\) | − | \(\big (\chi _2|_+,\chi _3|_+,\chi _1|_-,\chi _4|_+\big )\) |

\(\,\,Q_5\) | − | \(+\) | − | \(\big (\chi _2|_-,\chi _3|_+,\chi _1|_-,\chi _4|_+\big )\) |

\(\,\,Q_6\) | \(+\) | − | − | \(\big (\chi _3|_+,\chi _2|_-,\chi _1|_-,\chi _4|_+\big )\) |

\(\,\,Q_7\) | − | − | − | \(\big (\chi _3|_-,\chi _2|_-,\chi _1|_-,\chi _4|_+\big )\) |

### 4.4 The Jeffrey-Kirwan prescription for dual quivers

At one-instanton it is sufficient to specify whether the contours of integration for \(\chi _{I}\) are closed in the upper or lower half-planes to completely specify the prescription. However, at higher instantons this may be no longer sufficient since also the order in which the integrations are performed may become relevant to have a one-to-one correspondence between the terms appearing in the superpotential derived from the twisted chiral ring equations and the residues contributing in the localization integrals.

An elegant way to fully specify the contour of integration for all variables (including the order in which they are integrated) is using the Jeffrey-Kirwan (JK) residue prescription [20] (see also, for example, [7, 31, 32] for recent applications to gauge theories). The essential point of this prescription is that the set of poles chosen by a contour is completely specified by the so-called JK reference vector \(\eta \).

^{8}

For each quiver we list the JK reference vector that picks the appropriate contour on the localization side. The parameter \(\zeta _4\) is always positive and bigger in magnitude than any of the FI parameters. If \(\zeta _I^{Q_i}>0\) the associated \(\chi \)-variable is integrated along a contour in the upper-half plane, while if \(\zeta _I^{Q_i}<0\) it is integrated in the lower-half plane, in agreement with the prescription in the last column of Table 3

Quiver | JK vector |
---|---|

\(\,\,Q_0\) | \(~-\zeta _1\,\chi _1-\zeta _2\,\chi _2-\zeta _3\,\chi _3 + \zeta _4\,\chi _4~\) |

\(\,\,Q_1\) | \(~-\zeta _1^{Q_1}\,\chi _1-\zeta _2^{Q_1}\,\chi _2 -\zeta _3^{Q_1}\,\chi _3 + \zeta _4\,\chi _4~\) |

\(\,\,Q_2\) | \(~-\zeta _1^{Q_2}\,\chi _2-\zeta _2^{Q_2}\,\chi _1 -\zeta _3^{Q_2}\,\chi _3 + \zeta _4\,\chi _4~\) |

\(\,\,Q_4\) | \(~-\zeta _1^{Q_4}\,\chi _2-\zeta _2^{Q_4}\,\chi _3 -\zeta _3^{Q_4}\,\chi _1 - \zeta _4\,\chi _4 ~\) |

\(\,\,Q_5\) | \(~-\zeta _1^{Q_5}\,\chi _2-\zeta _2^{Q_5}\,\chi _3 - \zeta _3^{Q_5}\,\chi _1-\zeta _4\,\chi _4~\) |

\(\,\,Q_6\) | \(~-\zeta _1^{Q_6}\,\chi _3-\zeta _2^{Q_6}\,\chi _2 -\zeta _3^{Q_6}\,\chi _1 - \zeta _4\,\chi _4~\) |

\(\,\,Q_7\) | \(~-\zeta _1^{Q_7}\,\chi _3-\zeta _2^{Q_7}\,\chi _2- \zeta _3^{Q_7}\,\chi _1 -\zeta _4\,\chi _4~\) |

### 4.5 New quivers and the corresponding contours

For the quivers \({\widehat{Q}}_{1}\), \(Q_{3}\) and \({\widehat{Q}}_{5}\) drawn in Fig. 9, we list the classical expectation values of the twisted chiral fields in each of the 2d nodes, about which one finds the solution to the twisted chiral ring. Using this vacuum, along with the FI couplings in the classical twisted chiral superpotentials for each quiver, one finds identical expressions at leading order. The vacuum for the other quivers of the duality chain, namely \(Q_0\), \(Q_2\), \(Q_4\) and \(Q_7\), can be read from Table 1

Quiver | \(\sigma ^{(1)}_{\text {cl}}\) | \(\sigma ^{(2)}_{\text {cl}}\) | \(\sigma ^{(3)}_{\text {cl}}\) |
---|---|---|---|

\({\widehat{Q}}_{1} \) | \({{\mathcal {A}}}_1\) | \({{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2 \oplus {{\mathcal {A}}}_3\) |

\(Q_{3} \) | \({{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3\) | \({{\mathcal {A}}}_1\oplus {{\mathcal {A}}}_2 \oplus {{\mathcal {A}}}_3\) |

\({\widehat{Q}}_{5} \) | \({{\mathcal {A}}}_4\) | \({{\mathcal {A}}}_2\) | \({{\mathcal {A}}}_2\oplus {{\mathcal {A}}}_3 \oplus {{\mathcal {A}}}_4\) |

For the quivers \({\widehat{Q}}_{1}\), \(Q_{3}\) and \({\widehat{Q}}_{5}\) drawn in Fig. 9, we list the relations (up to signs) between the ramified instanton counting parameters \(q_I\) and the strong coupling scales \(\varLambda _I\), and also the JK reference vector that selects the contour prescription needed to compute the ramified instanton partition function using the localization formula

Quiver | \(q_1\) | \(q_2\) | \(q_3\) | JK vector |
---|---|---|---|---|

\({\widehat{Q}}_{1}\) | \(\big (\varLambda _1^{{\widehat{Q}}_{1}}\big )^{n_1+n_2}\) | \(\big (\varLambda _2^{{\widehat{Q}}_{1}}\big )^{n_2+n_3}\) | \(\frac{\big (\varLambda _3^{{\widehat{Q}}_{1}}\big )^{n_2+2n_3+n_4}}{\big (\varLambda _2^{{\widehat{Q}}_{1}} \big )^{n_2+n_3}}\) | \(-\zeta _1^{{\widehat{Q}}_{1}}\chi _1-\zeta _2^{{\widehat{Q}}_{1}}\chi _2 -\zeta _3^{{\widehat{Q}}_{1}}\chi _3 +\zeta _4\,\chi _4\) |

\(Q_{3}\) | \(\big (\varLambda _2^{Q_3}\big )^{n_1+n_2}\) | \(\big (\varLambda _1^{Q_3}\big )^{n_2+n_3}\) | \(\frac{\big (\varLambda _3^{Q_3}\big )^{N+n_2+n_3}}{\big (\varLambda _1^{Q_3}\big )^{n_2+n_3}\big (\varLambda _2^{Q_3}\big )^{n_1+n_2}}\) | \(-\zeta _1^{Q_3}\chi _2-\zeta _2^{Q_3}\chi _1-\zeta _3^{Q_3}\chi _3 + \zeta _4\chi _4\) |

\({\widehat{Q}}_{5}\) | \(\frac{\big (\varLambda _3^{{\widehat{Q}}_5}\big )^{n_1+2n_2+n_3}}{\big (\varLambda _2^{{\widehat{Q}}_5}\big )^{n_2+n_3}}\) | \(\big (\varLambda _2^{{\widehat{Q}}_5}\big )^{n_2+n_3}\) | \(\big (\varLambda _1^{{\widehat{Q}}_5}\big )^{n_3+n_4}\) | \(-\zeta _1^{{\widehat{Q}}_5}\chi _3-\zeta _2^{{\widehat{Q}}_5}\chi _2 - \zeta _3^{{\widehat{Q}}_5}\chi _1-\zeta _4\,\chi _4\) |

Next, we determine the classical vacuum for the quivers in this duality chain by equating the classical twisted chiral superpotentials for each dual pairs. In Table 5 we report the results for the three new quivers \({\widehat{Q}}_{1}\), \(Q_{3}\) and \({\widehat{Q}}_{5}\) of this sequence.

Using this information and following the same procedure described above, we can find the *q* vs \(\varLambda \) map and the contour prescription that has to be used in the localization formula in order to match term-by-term the superpotential with the one obtained from solving the twisted chiral ring equations. Of course, we do not repeat the derivation of these results since the calculations are a straightforward generalization of what we did for the other duality chain, and we simply collect our findings for the three new quivers \({\widehat{Q}}_{1}\), \(Q_{3}\) and \({\widehat{Q}}_{5}\) in Table 6. We have checked the validity of our proposal up to two instantons, while some details on the results at the one-instanton level can be found in Appendix B.

## 5 Proposal for generic linear quivers

*k*elements out of 4. In matrix form, we have

*i*written in binary notation.

*q*vs \(\varLambda \) map. Moreover, we see that two quivers whose vectors \(\mathbf {s}\) only differ by the value of \(s_3\) have the same permutation and that this permutation involves only cyclic rearrangements of the first two or the first three variables described by \(P_2\) and \(P_3\). In particular, introducing the vector \(\mathbf {\chi }=(\chi _1,\chi _2,\chi _3,\chi _4)\), we can check that

*M*nodes in a straightforward manner. In this case we have \((M-1)\) binary choices corresponding to \(2^{M-1}\) linear quivers that are related to each other by Seiberg duality. Therefore, they can be labelled by a vector \(\mathbf {s}=(s_1,s_2,\ldots ,s_{M-1})\) with \(s_i=0,1\). For each choice, the ranks of the

*M*nodes are determined by the permutation

*k*numbers, while the FI parameters of the \((M-1)\) nodes are obtained using \({\widehat{P}}\,[\mathbf {s}\,]\), which represents the permutation \(P[\mathbf {s}]\) in an irreducible way in an \((M-1)\)-dimensional space. Generalizing (85) to the

*M*-node case in an obvious way, and defining

## 6 Summary of results

The dictionary between the various features of surface operators in the two descriptions, as monodromy defects and as coupled 2d/4d quivers

Monodromy defect | 2d/4d quiver models |
---|---|

Partition of | Ranks of 2d gauge nodes |

4d Coulomb vev’s | 2d twisted masses |

Partition of Coulomb vev’s | Classical (massive) vacuum |

Ramified instanton counting parameters | 2d/4d strong coupling scales |

\(q_I\), \(q_M\) | \(\varLambda _I\), \(\varLambda _{\text {4d}}\) |

\({{{\mathcal {W}}}}_{\text {inst}}(a, q)\) | \(\left. {{{\mathcal {W}}}}(\sigma , a, \varLambda _I, \varLambda _{\text {4d}}) \right| _{\sigma _{\star }}\) |

Contour prescription | 2d Seiberg duality frame |

Establishing a precise correspondence between different integration contour prescriptions in the ramified instanton partition function for a monodromy defect and different quiver theories related to each other by a Seiberg duality has been the main focus of our present work. Dual quivers have different ultraviolet realizations but share the same infrared physics and thus the (massive) vacua of their low-energy theories can be mapped onto each other. These massive vacua are obtained by extremizing the effective twisted chiral superpotential of the 2d/4d quiver. The evaluation of the effective superpotential in a particular vacuum is in turn mapped to the twisted superpotential which is extracted from the ramified instanton partition function with a specific contour of integration.

For surface operators in pure \({{{\mathcal {N}}}}=2\) gauge theories, like the ones we have considered in this paper, residue theorem ensures that one always obtains the same superpotential irrespective of the contour of integration chosen. Nevertheless, by a careful study of the individual residues that contribute to the superpotential, we have been able to map distinct contours on the localization side to distinct Seiberg-dual 2d quivers coupled to the same 4d SU(*N*) flavour group. The duality frame one chooses affects the details of the other entries in the table above, such as the choice of the classical vacuum and the map between the ramified instanton counting parameters \(q_I\) and the strong coupling scales \(\varLambda _I\). We initially restricted ourselves to systems with four nodes to exhibit our explicit results, but in the end we have generalized our analysis to linear quivers with an arbitrary number of nodes providing the map between the data of the quiver and the corresponding JK prescription, which takes a universal form.

There is one caveat to our analysis. All quivers we have studied so far, have only a single 2d node that is connected to the flavour node that is gauged in 4d. It is only for such cases that the coupling of the 2d degrees of freedom to the 4d theory via its resolvent gives results that are consistent with those obtained using localization methods in the monodromy defect approach. It would be very interesting to understand whether quivers with more 2d nodes connected to the 4d node also have an interpretation as surface operators in a 4d gauge theory. Furthermore, there are many worthwhile but yet unexplored directions to pursue, such as the extension of our analysis to (conformal) SQCD models for which the integrands of ramified instanton partition function may have non-vanishing residues at infinity, or the lift of our techniques to five dimensions to study surface operators from the point of view of 3d/5d coupledsystems, with possible Chern-Simons interactions. We leave these extensions and generalizations to future work.

## Footnotes

- 1.
- 2.
In a purely 2d context, a relation between the solution of chiral ring equations for certain quiver theories and contour integrals has been noticed in [30].

- 3.
The signs have been chosen to match the two superpotentials exactly.

- 4.
In addition, an ordinary superpotential term is generated, but it plays no role in our discussion.

- 5.
The same sequence of dualities has also been mentioned in [7].

- 6.
For ease of notation we use the same symbol \(\sigma ^{(I)}\) to denote the chiral superfield before and after the duality.

- 7.
We remark that the results for the last quiver \(Q_7\) coincide with those derived in Ref. [10], once the nodes are numbered in the opposite order.

- 8.
Similar JK prescriptions have been considered in [33] for quiver theories in a 3d context.

- 9.

## Notes

### Acknowledgements

We would like to thank Stefano Cremonesi and Amihay Hanany for many useful discussions. S.K.A. would especially like to thank the Physics Department of the University of Torino and the Torino Section of INFN for hospitality during the final stages of this work. M.B, M.F., R.R.J. and A.L. would like to thank the “Galileo Galilei Institute for Theoretical Physics” in Florence for hospitality. The work of M.B., M.F., R.R.J. and A.L. is partially supported by the MIUR PRIN Contract 2015MP2CX4 “Non-perturbative aspects of gauge theories and strings”. The work of A.L. is partially supported by the “Fondi Ricerca Locale dell’Università del Piemonte Orientale”.

## Supplementary material

## References

- 1.S. Gukov, Surface Operators. arXiv:1412.7127
- 2.S. Gukov, E. Witten, Gauge Theory, Ramification, and the Geometric Langlands Program. arXiv:hep-th/0612073
- 3.S. Gukov, E. Witten, Rigid surface operators. Adv. Theor. Math. Phys.
**14**(1), 87–178 (2010). arXiv:0804.1561 MathSciNetCrossRefGoogle Scholar - 4.D. Gaiotto, Surface operators in
*N*= 2 4d Gauge theories. JHEP**11**, 090 (2012). arXiv:0911.1316 ADSMathSciNetCrossRefGoogle Scholar - 5.D. Gaiotto, S. Gukov, N. Seiberg, Surface defects and resolvents. JHEP
**09**, 070 (2013). arXiv:1307.2578 ADSCrossRefGoogle Scholar - 6.H. Kanno, Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver. JHEP
**06**, 119 (2011). arXiv:1105.0357 ADSMathSciNetCrossRefGoogle Scholar - 7.A. Gorsky, B. Le Floch, A. Milekhin, N. Sopenko, Surface defects and instanton? Vortex interaction. Nucl. Phys. B
**920**, 122–156 (2017). arXiv:1702.03330 ADSMathSciNetCrossRefGoogle Scholar - 8.S.K. Ashok, M. Billo, E. Dell’Aquila, M. Frau, R.R. John, A. Lerda, Modular and duality properties of surface operators in
*N*= 2* gauge theories. JHEP**07**, 068 (2017). arXiv:1702.02833 ADSMathSciNetCrossRefGoogle Scholar - 9.S.K. Ashok, M. Billo, E. Dell’Aquila, M. Frau, V. Gupta, R.R. John, A. Lerda, Surface operators in 5d gauge theories and duality relations. JHEP
**05**, 046 (2018). arXiv:1712.06946 ADSMathSciNetCrossRefGoogle Scholar - 10.S.K. Ashok, M. Billo, E. Dell’Aquila, M. Frau, V. Gupta, R.R. John, A. Lerda, Surface operators, chiral rings and localization in
*N*=2 gauge theories. JHEP**11**, 137 (2017). arXiv:1707.08922 ADSMathSciNetCrossRefGoogle Scholar - 11.L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, H. Verlinde, Loop and surface operators in
*N*= 2 gauge theory and Liouville modular geometry. JHEP**1001**, 113 (2010). arXiv:0909.0945 ADSMathSciNetCrossRefGoogle Scholar - 12.L.F. Alday, Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys.
**94**, 87–114 (2010). arXiv:1005.4469 ADSMathSciNetCrossRefGoogle Scholar - 13.E. Witten, Phases of
*N*= 2 theories in two-dimensions. Nucl. Phys. B**403**, 159–222 (1993). arXiv:hep-th/9301042 ADSMathSciNetCrossRefGoogle Scholar - 14.O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg, M.J. Strassler, Aspects of
*N*= 2 supersymmetric gauge theories in three-dimensions. Nucl. Phys. B**499**, 67–99 (1997). arXiv:hep-th/9703110 ADSMathSciNetCrossRefGoogle Scholar - 15.A. Hanany, K. Hori, Branes and
*N*= 2 theories in two-dimensions. Nucl. Phys. B**513**, 119–174 (1998). arXiv:hep-th/9707192 ADSMathSciNetCrossRefGoogle Scholar - 16.N. Nekrasov, BPS/CFT correspondence IV: sigma models and defects in gauge theory. arXiv:1711.11011
- 17.S. Jeong, N. Nekrasov, Opers, Surface Defects, and Yang-Yang Functional. arXiv:1806.08270
- 18.N. Seiberg, Electric-magnetic duality in supersymmetric non Abelian gauge theories. Nucl. Phys. B
**435**, 129–146 (1995). arXiv:hep-th/9411149 ADSMathSciNetCrossRefGoogle Scholar - 19.F. Benini, D.S. Park, P. Zhao, Cluster Algebras from Dualities of 2d
*N*= (2, 2) Quiver Gauge Theories. Commun. Math. Phys.**340**, 47–104 (2015). arXiv:1406.2699 ADSMathSciNetCrossRefGoogle Scholar - 20.J.C. Jeffrey, F.C. Kirwan, Localization for non-abelian actions. Topology
**34**, 291–327 (1995). arXiv:alg-geom/9307001 - 21.N. Nekrasov, Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys.
**7**, 831–864 (2004). arXiv:hep-th/0206161 MathSciNetCrossRefGoogle Scholar - 22.N. Nekrasov, A. Okounkov, Seiberg-Witten theory and random partitions. Prog. Math.
**244**, 525–596 (2006). arXiv:hep-th/0306238 MathSciNetCrossRefGoogle Scholar - 23.N.A. Nekrasov, S.L. Shatashvili, Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl.
**177**, 105–119 (2009). arXiv:0901.4748 ADSCrossRefGoogle Scholar - 24.N. Nekrasov, S. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories. arXiv:0908.4052
- 25.U. Bruzzo, F. Fucito, J.F. Morales, A. Tanzini, Multi-instanton calculus and equivariant cohomology. JHEP
**05**, 054 (2003). arXiv:hep-th/0211108 ADSMathSciNetCrossRefGoogle Scholar - 26.A.S. Losev, A. Marshakov, N.A. Nekrasov, Small Instantons, Little Strings and Free Fermions. arXiv:hep-th/0302191
- 27.R. Flume, F. Fucito, J.F. Morales, R. Poghossian, Matone’s relation in the presence of gravitational couplings. JHEP
**04**, 008 (2004). arXiv:hep-th/0403057 ADSMathSciNetCrossRefGoogle Scholar - 28.M. Billo, M. Frau, F. Fucito, L. Giacone, A. Lerda, J.F. Morales, D. Ricci-Pacifici, Non-perturbative gauge/gravity correspondence in
*N*= 2 theories. JHEP**1208**, 166 (2012). arXiv:1206.3914 ADSMathSciNetCrossRefGoogle Scholar - 29.S.K. Ashok, M. Billo, E. Dell’Aquila, M. Frau, A. Lerda, M. Moskovic, M. Raman, Chiral observables and S-duality in
*N*= 2* U(N) gauge theories. JHEP**11**, 020 (2016). arXiv:1607.08327 ADSMathSciNetCrossRefGoogle Scholar - 30.D. Orlando, S. Reffert, Relating Gauge theories via Gauge/Bethe correspondence. JHEP
**1010**, 071 (2010). arXiv:1005.4445 ADSMathSciNetCrossRefGoogle Scholar - 31.F. Benini, R. Eager, K. Hori, Y. Tachikawa, Elliptic genera of two-dimensional
*N*= 2 gauge theories with rank-one gauge groups. Lett. Math. Phys.**104**, 465–493 (2014). arXiv:1305.0533 ADSMathSciNetCrossRefGoogle Scholar - 32.K. Hori, H. Kim, P. Yi, Witten index and wall crossing. JHEP
**01**, 124 (2015). arXiv:1407.2567 ADSMathSciNetCrossRefGoogle Scholar - 33.C. Hwang, P. Yi, Y. Yoshida, Fundamental vortices. Wall-crossing, and particle-vortex duality. JHEP
**05**, 099 (2017). arXiv:1703.00213 ADSMathSciNetCrossRefGoogle Scholar

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