Vortex-strings in N=2 quiver X U(1) theories

We study half-BPS vortex-strings in four dimensional N=2 supersymmetric quiver theories with gauge group SU(N)^n X U(1). The matter content of the quiver can be represented by what we call a tetris diagram, which simplifies the analysis of the Higgs vacua and the corresponding strings. We classify the vacua of these theories in the presence of a Fayet-Iliopoulos term, and study strings above fully-Higgsed vacua. The strings are studied using classical zero modes analysis, supersymmetric localization and, in some cases, also S-duality. We analyze the conditions for bulk-string decoupling at low energies. When the conditions are satisfied, the low energy theory living on the string's worldsheet is some 2d N=(2,2) supersymmetric non-linear sigma model. We analyze the conditions for weak to weak 2d-4d map of parameters, and identify the worldsheet theory in all the cases where the map is weak to weak. For some SU(2) quivers, S-duality can be used to map weakly coupled worldsheet theories to strongly coupled ones. In these cases, we are able to identify the worldsheet theories also when the 2d-4d map of parameters is weak to strong.


Introduction and Summary
In this work we study the worldsheet theories of 1 2 -BPS vortex strings (strings from now on) configurations in four-dimensional N = 2 supersymmetric SU (N ) n × U (1) gauge theories. These theories are related to the well-studied SU (N ) n quiver theories, by gauging some U (1) flavour symmetry and adding to it a Fayet-Iliopoulos (FI) term. The study of strings in these theories allows us to understand better interesting physical phenomena such as bulk-string decoupling at low energies and weak→ strong mapping of parameters from the four-dimensional theory to the two-dimensional worldsheet theory. In addition, for SU (2) quivers these strings have interesting transformation rules under S-duality that relates strings in theories with different U (1) gauged, and strings in linear quivers to strings in generalized quivers. Some of these properties already appeared in [1,2] for the simpler case where the gauge group is SU (N ) × U (1).
The matter content consists of N fundamental hypermultiplets of SU (N ) 1 , N fundamental hypermultiplets of SU (N ) n and n − 1 bi-fundamental hypermultiplets of SU (N ) i × SU (N ) i+1 for i = 1, ..., n−1. Under the U (1), every hypermultiplet is assigned with a charge c i ∈ Z. We will denote the two scalars of the hypermultiplets by q andq. When adding an FI term associated with this U (1), some of the hypermultiplets scalars must get non-trivial vacuum expectation value (VEV) q a = v where the index a here labels the scalars that get VEV. The vacuum equations are solved by giving VEV to an SU (N ) n invariant operator, charged under the U (1). The sign of its U (1) charge should be the same as the sign of the FI parameter. This theory supports stable strings. The way to construct a string is to change the boundary conditions for the scalars getting VEV to lim r→∞ q a = ve ikaφ where r, φ are the polar coordinates on the plane transverse to the string, and {k a } a set of non-negative or non-positive integers. The string is labelled by the choice of vacuum and the total winding number K = a k a . The minimal tension configurations within a topological sector K, are 1 2 -BPS and preserve N = (2, 2) supersymmetry on the string worldsheet. These strings are generalizations of the well studied strings in U (N ) theories. For a partial list of references, see [3][4][5][6][7][8][9][10][11][12][13][14], and the reviews [15][16][17][18].
An important tool we will use in order to study the worldsheet theories of these strings is supersymmetric localization. We can write the partition function of the four-dimensional theory on a squashed sphere using the results of [19,20]. For some range of parameters, we can rewrite the partition function as a sum over Higgs (and mixed) branch contributions. Out of this sum, one can identify the string contributions [2,[21][22][23]. As was explained in [1], in some cases the low energy theory factorizes into a product of the four dimensional vacuum theory and the two dimensional worldsheet theory. Correspondingly, the string contribution factorizes into a product of the four-dimensional vacuum partition function and the worldsheet partition function on a two-sphere. In these cases, we can compare the worldsheet partition function to known results of S 2 N = (2, 2) supersymmetric partition functions [24,25]. This comparison of the partition functions allows a highly non-trivial check of any suggestion for the worldsheet theory. See also [26,27] where similar methods were used in the context of surface defects.
In [2], a condition on the U(1) charges was given such that the map of parameters from the 4d theory to the 2d theory is weak to weak. The classical analysis of the zero modes is useful only when the worldsheet theory is weakly coupled. Supersymmetric localization gives exact results for the partition function. However, when the map of parameters is weak to strong, the expression for the partition function we derive is expanded around some strongly coupled point. On the other hand, the expressions available in the literature for S 2 partition functions are expanded around the weakly coupled points. This makes the task of identifying the theory very hard in these cases. In this work, we identify the worldsheet theories in all the cases where the map of parameters is weak→weak. For SU (2) quivers, we identify strongly coupled worldsheet theories that are related to weakly coupled ones via S-duality.
The outline of this paper is as follows. In section 2 we analyze the vacua of SU (N ) n ×U (1) theories in the presence of an FI term. Section 3 is devoted to general properties of the string and its zero modes. We also present the conditions on U(1) charges {c i } that lead to bulkstring decoupling at low energies, and to weak→weak map of parameters. In sections 4 and 5 we study strings in SU (N ) 2 × U (1) theories and SU (2) M × U (1) theories respectively, and give ansatzes for the worldsheet theories based on semiclassical analysis. In section 6 we use S-duality properties of the four-dimensional SU (2) M × U (1) theories in order to study strongly coupled strings. In particular, S-duality relates linear SU(2) quivers to generalized SU(2) quivers, which allows us to study also strongly coupled strings on generalized quivers. In section 7 we go back to all the strings studied in the previous sections and study their worldsheet theory using supersymmetric localization. We extract their worldsheet partition function from the four-dimensional partition function and show agreement with our ansatzes. Some technical computations appear in the appendix.

Vacua analysis and tetris diagrams
In this section we describe what we call a tetris diagram which is a picturial way to represent the matter content of a quiver theory, and use it to classify the vacua in the presence of an FI parameter. Our starting point is the four dimensional N = 2 superconformal quiver theory with gauge group G = SU (N ) n ≡ SU (N ) 1 × SU (N ) 2 × ... × SU (N ) n . The matter content consists of N fundamental hypermultiplets of SU (N ) 1 , N fundamental hypermultiplets of SU (N ) n and n−1 bi-fundamental hypermultiplets of SU (N ) i ×SU (N ) i+1 for i = 1, ..., n−1.
The hypermultiplets can be represented by a diagram made out of n + 1 blocks, each one contains N 2 boxes arranged in an N ×N matrix. See figure 1 for an example. We modify the theory first by introducing small and generic masses for the hypermultiplets. In addition, Figure 1: This figure shows the tetris diagram for SU (2) 3 theory. Every box (one square) represents one component of a hypermultiplet. The diagram should be read from left-up to right-down. The first two columns represent two fundamentals of SU (2) 1 . The 2 × 2 block to their right represents one bifundamental of SU (2) 1 × SU (2) 2 . The 2 × 2 block below it, represents one bifundamental of SU (2) 2 × SU (2) 3 . Finally, the last two columns represent two fundamentals of SU (2) 3 .
we gauge some global U (1) and add an FI term associated with this U (1). This U (1) can be labelled by asigning an independent U(1) charge c i ∈ Z to every hypermultiplet. Due to the FI parameter, some of the hypermultiplets scalars must get non-trivial vacuum expectation value. The vacuum equations are solved by giving VEV to an SU (N ) n invariant operator, charged under the U (1). The sign of its U (1) charge should be the same as the sign of the FI parameter. These vacua can be represented nicely on the tetris diagram. We can denote a q getting VEV by a black dot and aq getting VEV by a white (empty) dot. We need to give VEV to several scalars such that for every SU (N ) factor we have a baryon, a meson or nothing getting VEV. See figure 2 for some examples.
In a given vacuum, the gauge symmetry is broken down to H ⊂ G × U (1). H can be read easily from the tetris diagram using the following simple rules: • The number of dots equals the reduction in the rank.
Rank (G × U (1)) − Rank(H) = # of dots . (2.1) • If one draws a line (lines) from every dot in the directions of the color indices, all the boxes with line on them represent hypermultiplets which are combined with gauge multiplets into massive W-boson multiplets. Therefore, the number of broken generators equals the number of boxes with line on them.
• H always contains a discrete Z |C| factor, where C is the U (1) charge of the operator getting VEV.
These rules are correct only for non-seperable vacua. By seperable we mean that the operator getting VEV can be written as a product of more than one G-invariant operators. The seperable vacua are excluded from the following reason. The mass term of a hypermultiplet scalar q looks schematically like (µ − a) 2 |q| 2 where µ is the bare mass of q and the  sum over a is the sum over the Cartan components of the gauge multiplet scalars in the relevant representations (The off-diagonal elements of a are taken to be zero). If q gets a VEV, this term must vanish and therefore we must also introduce a VEV for the gauge multiplet scalars such that a = µ in the vacuum. In a seperable vacuum, the VEV of a u(1) , the U (1) scalar, can be extracted from every G-invariant operator independently. For generic masses, these values will not coincide and therefore this solution is forbidden. As a simple example, consider the G = SU (3) 2 theory with the 2-baryonic vacuum illustrated in figure 3. In this example, the operator getting VEV is a product of two G-invariant operators. The vanishing of the mass terms gives six equations. By summing over the first three equations, one can extract the value of the U(1) scalar where µ i , c i with i = 1, 2, 3 are the masses and U(1) charges of the first three hypermultiplets. Similarly, by summing over the last three equations, one can find where now µ i , c i with i = 5, 6, 7 are the masses and U(1) charges of the last three hypermultiplets. For generic masses µ i , these values don't coincide and therefore this configuration doesn't solve the vacuum equations. On the same way, every seperable vacuum is excluded once generic values for the hypermultiplets masses µ i are turned on. Consider G = SU (N ) n with general N, n. We will list all the possible types of nonseperable vacua and the residual gauge symmetry in each of these vacua.
The rest of the vacua contain dots in all the blocks.
• Mesonic chain: These vacua are similar to the one presented in 2 (c). In these vacua there is a meson for every SU (N ) factor. There is one dot in every block which reduces the rank of the gauge group by n + 1. There are 2nN − n + 1 broken generators, which means that the residual gauge symmetry has rank n(N − 2) and dimension n((N − 1) 2 − 1). The residual gauge group is • (anti-) Baryonic chain: These vacua are similar to the one presented in 2 (d). In these vacua there is a baryon for every SU (N ) factor. We can classify the vacua according to the number of dots in the first block, denoted by m with 1 ≤ m ≤ N − 1. With this choice, the number of dots in all the odd blocks is m while the number of dots in all the even blocks is N − m. We will divide the analysis to two cases: 1. odd n: If n, the number of SU (N ) factors, is odd, there is an even number of blocks. The number of dots is always N (n+1)

2
, regardless of m. The number of broken generators is N 2 + n−1 2 (N 2 − 2m 2 + 2mN ). This implies that the residual gauge symmetry has rank n−1 2 (N − 2) and dimension n−1 2 (N 2 + (N − m) 2 − 2). The residual gauge symmetry is 2. even n: In this case, there is an odd number of blocks. The number of dots is N n 2 + m and the number of broken generators is m 2 + n 2 (N 2 − 2m 2 + 2N m). This implies that the residual gauge symmetry has rank nN 2 − n − m + 1 and dimension . The residual gauge symmetry in this case is  symmetry is to characterize the mixed vacua by the number of blocks with one dot, denoted by l. The residual gauge symmetry is then This can be simply understood in the following way. Start from a mesonic chain which is a special case of the mixed vacua with l = n + 1. One can contaminate the chain with baryons by replacing a block with 1 dot by a block with N − 1 dots, where two "contaminated" blocks cannot be one next to the other. It is easy to see that every contamination of this type, breaks one SU (N − 1) factor. See figure 4. Notice that the minimal value for l is n 2 + 1 for even n and n+1 2 for odd n which are barynoic chains.
An immediate result of this analysis is that fully Higgsed vacua exist only for N = 2 with arbitrary n, or n ≤ 2 with arbitrary N . In the next sections we will study vortex-strings above the fully Higgsed vacua of G = SU (2) n and G = SU (N ) 2 .

Strings: Generalities and classical analysis
In the next sections we will generalize the analysis made in [2] and study the low energy theories living on the strings worldsheet in cases where the original gauge group is SU (N ) n × U (1). We will do it only for the cases where the vacuum is fully Higgsed. In this section we will go over some of the main steps in the way. Our starting point will be a baryonic chain vacuum of an SU (N ) n × U (1) theory illustrated by some tetris diagram. The diagram contains nN + 1 − n dots describing nN + 1 − n hypermultiplets scalars getting VEV q i = v , i = 1, ..., nN +1−n. One also need to give VEV to the nN +1−n Cartan gauge multiplet scalars a I in order to eliminate the mass terms of q i . At energies much smaller than the mass of the W-bosons m W , the vacuum excitations consists of N 2 + n − 1 light hypermultiplets with masses which are some linear combinations of the hypermultiplets bare masses (The exact values will be specified later). These masses are taken to be much smaller than m W . Excitations with these masses are considered as light and dynamical, while excitations with masses ∼ m W are considered as heavy and frozen. In order to construct strings, one needs to modify the VEV of the scalars by changing the boundary conditions to lim r→∞ q i = ve ik i φ where r, φ are the polar coordinates on the plane transverse to the string and {k i } is a set of non-negative integers. 2 The VEVs of a I are left untouched, but one needs to introduce VEVs to the Cartan gauge fields A φ such that the kinetic terms |D µ q i | 2 vanish at r → ∞. The U (1) flux carried by such a configuration is where K = k i , and C = c i is the charge of the operator getting VEV. Minimal tension configurations within a topological sector labelled by K satisfy a set of BPS equations and preserve N = (2, 2) supersymmetry on the string worldsheet. Configurations with the same K but different {k i } are connected by magnetic monopoles [28] and are part of the same worldsheet theory. Different {k i } correspond to different vacua of the worldsheet sigma model, and the monopoles correspond to worldsheet kinks that connect the different vacua [12]. The worldsheet theory also inherits two vector-like U (1) R symmetries. The first one is a combination of the four-dimensional U (1) R ⊂ SU (2) R and gauge transformations preserved by the string. The second is a combination of rotation on the plane transverse to the string and gauge transformations. The exact massless zero modes of the string are only the positions of the cores of the string and their superpartners. However, we will treat light modes with masses of order of the hypermultiplets masses as approximate zero modes and include them in our zero modes analysis. There are three types of light modes. We will focus on the bosonic modes where their fermionic partners can be found using supersymmetry.
1. Size modes: These are modes that come from excitations of the light scalars q and q. They can be found in the following way. Among the BPS equations there are the equations These equations together with the boundary conditions lim r→∞ q i = ve ik i φ implies that q i has k i zeros at positions r l i . Close to the zeros it behaves as q i ∼ z − z l i , where z = re iφ . Given the boundary conditions, one can ask whether equations (3.2) allow non-trivial solutions for the light scalars. These solutions are reffered to as size modes. The number of modes equals the number of independent solutions. The number and functional behaviour of the size modes depend highly on the U(1) charges.
2. Off-diagonal modes: Off-diagonal modes are related to gapless excitations of the massive W-bosons. Their number is independent of the U(1) charges. Every pair of swallawed scalars q ij and q ji give rise to k i + k j complex modes. Evidences from localization and S-duality for the existance of these modes were given in [2].
3. Center of mass modes: These modes parametrize the positions of the string cores on the (x 1 , x 2 ) plane. There are K cores and therefore 2K real zero modes. These are the only exact zero modes of the string once the hypermultiplets masses are included.
As was already emphasized in [1,2], the size modes are responsible for two major properties of the low energy effective theory in the string background. We will summarize these two properties here for convenivence.

Bulk-string decoupling
The first property is related to the question whether the string light modes and the bulk light modes decouple at low energies such that the effective action can be written as a sum of two decoupled actions S ef f = S bulk + S string . (3. 3) The answer to this question depends on the asymptotic r-dependence of the size modes. Modes that decay like 1 r β with β ≥ 1 decouple from the bulk modes at low energies while long range modes that decay like 1 r β with 0 < β < 1 stay coupled to the bulk modes even at low energies. The demand that there are no long range modes coincides with the condition for no non-trivial Aharonov-Bohm phases for particles in the spectrum encircling the string.

Weak to weak mapping
Starting from a weakly coupled four-dimensional theory, the two-dimensional worldsheet theory can be either weakly coupled or strongly coupled. In other words, the 2d-4d map of parameters can map the weakly coupled regime of the 4d theory to weakly coupled or to strongly coupled regimes of the worldsheet theory. This property depends on the F-term constraints of the four-dimensional theory, that take the form where the sum over i is the sum over flavors, T α R i is an SU(N) generator in the representation of q i and the color index is suppressed. If for every flavor, there are only q zero modes orq zero modes but not both, all the F-terms vanish identically without imposing any constraints on the worldsheet. In this case the map of parameters will be weak to weak. If, on the other hand, there exists a flavor for which there are both q andq zero modes, the F-term constraints act non-trivially on the worldsheet and as a result, the worldsheet theory will be strongly coupled.

Strings in
In this section we will study the SU (N ) 2 quiver in which we gauge some U (1) flavor symmetry. The matter content of this theory consists N fundamentals of the first SU (N ), N fundamentals of the second SU (N ) and one bi-fundamental. Therefore, this theory is parameterized by 2N + 1 charges and masses. We will label the rows of the tetris diagram by a, b = 1, ..., 2N and the columns by a, b = 0, ..., 2N − 1. In the fully Higgsed vacua, 2N − 1 scalars get VEV. We will take them to be q aa with q aa = v for a = 1, ..., 2N − 1. We will also denote the masses and U(1) charges by µ a , c a with a = 0, ..., 2N . As mentioned above, the gauge symmetry is broken in this vacuum to Z C where C = 2N −1 a=1 c a is the U(1) charge of the operator getting VEV. 3 The spectrum of strings above this vacuum is given by where k a are non-negative integers. For finite tension configurations, |D µ q| 2 must decay at r → ∞ faster than r −2 . This implies that the Cartan components of the gauge fields A φ must be turned on. In particular, it is straight forward to show that the U(1) magnetic flux carried by the string is where A is the U(1) gauge field, in agreement with the allowed spectrum. We will be interested in computing the mass and the two U(1) R-charges of every hypermultiplet around the string solution. These three quantities are computed in a similar way.
• Mass: Taking all the off-diagonal elements of the adjoint scalars to zero, the mass of 3 Without loss of generality, we take the FI parameter and the charge C to be positive. the scalars q ab is M ab with where a , a 1,2 a denote the Cartan elements of the adjoint scalars of the U (1), SU (N ) 1,2 gauge multiplets with N a=1 a 1,2 a = 0.
• R (R) -charge: This is a two dimensional vectorlike U(1) R-charge which is a combination of the four-dimensional U (1) R ⊂ SU (2) R and gauge transformations preserved by the string solution. Under a general combination of U (1) R ⊂ SU (2) R and Cartan gauge transformation, the scalar q ab tranform as q ab → e iω ab q ab with where ω R , ω are the U (1) R and U(1) gauge symmetry parameters respectively, and ω 1,2 a are the Cartan parameters of SU (N ) 1,2 transformations satisfying N a=1 ω 1,2 a = 0.
• R (J) -charge: This is a two dimensional vectorlike U(1) R-charge which is a combination of rotation and gauge transformations. Rotation is broken by the string due to the explicit φ dependence of the string solution. If all the cores of the string coincide at the same point, there is a combination of rotation and gauge transformation that leaves the string solution invariant. Under a general combination of rotation δφ = 2φ 0 4 and Cartan gauge transformation, the scalars q ab transform as q ab → e iφ ab q ab with In the string vacuum, we need to demand that These are 2N − 1 equations for the 2N − 1 Cartan parameters. Plugging the values of the Cartan generators back gives us the mass and R-charges of the scalars q ab . It will be convenient to package the three quantities intô Notice that 2N −1 I=1Î = 0 is satisfied. Every box labelled by the row and column indices 1 witĥ where we emphasize the similarity to equations (4.3)(4.4)(4.5). Solutions to these equations are given by and on the same waỹ (4.14) The functions f (z),f (z) are general functions independent ofz that come from integrating overz in (4.12). Given the boundary conditions lim r→∞ q aa = ve ikaφ , q aa has k a zeros on the x 1 − x 2 plane. The functions f (z),f (z) should be restricted such that the scalars q ab ,q ab vanish at r → ∞ and are regular everywhere. For simplicity, the solution is written here for the case where all the zeros of q aa coincide at r = 0.
15) and (4.16) The parameters ρ,ρ are arbitrary complex numbers which parametrize the size modes. For general zeros, the solution is modified as in equation (4.28) of [1]. However, the number of zero modes and the asymptotic behaviour will not be affected. The conditions for bulkstring decoupling and for the weak to weak mapping can be read directly from equations (4.15), (4.16).
Bulk-string decoupling: Decoupling happens if there are no long range size modes that decay slower than 1 r . The conditions for this are for every i, i = 1, ..., N −1. This condition also coincides with the condition for no non-trivial Aharonov-Bohm phases.
Weak to weak mapping: The mapping of parameters from the four dimensional theory to the two dimensional worldsheet theory is weak to weak if all the F-term constraints are satisfied trivially. This happens if for every flavor, there are only q or onlyq modes, but not both. in addition to the size modes, there are also the off-diagonal modes as explained above. In particular, the scalars q iN , q N j with i = 1, ..., N − 1 and j = N + 1, ..., 2N − 1 give rise to the off-diagonal modes and therefore we must forbidq modes for the entire bi-fundamental. It means thatq ij = 0 which leads to the condition From the first column and the last row, we get four possibilities for weak to weak mapping: In the cases where the two conditions are satisfied, i.e. the bulk and the string decouple at low energies, and all the F-term constraints vanish identically, we give an ansatz for the low energy worldsheet theory with topological charge K = 1. We will show that the given ansatz is consistent both with the classical zero modes analysis and with results obtained from localization.

Noq case
In this section we will describe the worldsheet theory in the case where there are noq excitations. This happens when (4.19) In addition, we also assume that the other conditions required for bulk-string decoupling and weak to weak mapping (4.17) and (4.18) are satisfied. The size modes are given by the parameters ρ of equation (4.15). It is usefull to distinguish between two types of size modes. The size modes that exist for every choice of the partition {k a } give rise to decoupled chiral fields on the worldsheet. • ρ Stripping off the decoupled modes and ignoring the center of mass modes, we are left with the interacting size modes and off-diagonal modes as can be seen in the tetris diagram 6. There are 2N K complex interacting modes. The fact that the number of modes is independent of the partition {k a } is a sign for weak→weak mapping. Our ansatz is that the K = 1 worldsheet theory in this case is given by the low energy limit of a two-dimensional is the 4d complexified gauge coupling, and t a = θ (2d) a 2π + iξ a is the complexified 2d FI parameter. In addition, the theory contains the following chiral fields • X, parametrizing the center of mass modes.
The η fields parametrize the size modes that exist for every partition {k a }. The quantum numbers of these fields are given in table 1.

Comparison with the classical spectrum
In this section we will show that the ansatz for the worldsheet theory agrees with classical zero modes analysis. We will start from the decoupled sector. It is straight forward to see that the quantum numbers of the η fields coincide with the quantum numbers of the decoupled size modes. These are the ρ (n) s of equation (4.15) with n = n max + 1 − r. Now we will move on to the charged sector. The worldsheet theory has 2N − 1 vacua that correspond to the 2N − 1 choices of {k a } with K = 1. Due to the twisted masses, only two chiral fields can get non-trivial VEV. This is allowed thanks to the two gauge multiplets scalars σ 1,2 as the mass terms for the charged fields are (4.21) The chiral fields that get VEV must satisfy the D-term equations The 2N − 1 vacua and the corresponding {k a } partitions are given by (4.23) Lets focus for example on the vacua appearing on the first line of (4.23). By plugging in the VEV for σ 1,2 , we find that the masses of the dynamical fields around the vacuum are given by Similarly, the R-symmetries preserved by the vacuum are linear combinations of the original R-symmetries with some gauge transformations. This leads to a shift in the charges of the fields. The shift for a field with U (1) × U (1) charges (q 1 , q 2 ) is (4.25) The equations for these three quantum numbers can be summarized as where M can be mass or any of the two R-charges. Straight forward computation shows agreement between the spectrum of the charged fields in the vacuum to the off-diagonal and interacting size modes around the string vacuum k a = δ aJ , with the identifiaction • ψ + 0 with the off-diagonal mode of the pair q N,J , q J,N .
• ψ + I =J with the off-diagonal mode of the pair q I,J , q J,I .
• ψ − I with the size modes ρ The quantum numbers of the charged fields in the vacuum are exactly given by theÎ AB of equations (4.10) as summarized in table 2.

Field
Twisted Mass

Includingq excitations
In this section we will describe the worldsheet theory in the cases where some of the hypermultiplets admitq excitations but still the F-term constraints are satisfied trivially. This happens in one of the following three cases It is easy to see that almost all the analysis will be the same as in the previous section. The only difference comes from the size modes analysis of the first column or/and the last row. In case 1 in the above list, ρ N,0 of (4.16). In case 3, the two replacements should be made. From the worldsheet point of view, these replacements include two changes. One is a trivial change in the spectrum of the decoupled fields η →η. The second change includes adding neutral fields and couple them to the charged fields via a superpotential. Consider for example the first case in the list. The spectrum is the same as in table 1 with the following changes: Replace η 2N,j,r , η 2N,N,r with the decoupled fieldsη 2N,j,r with j = N +1, ..., 2N −1 , r = 1, ..., j=N +1 c j . Their quantum numbers appear in table 3. These fields represent the size modesρ

Field Twisted Mass
2N,N with n = n max + 1 − r. The second change invloves adding neutral chiral fields χ 2N and χ i with i = N + 1, ..., 2N − 1 together with the superpotential where α 2N , α J are some non-zero coefficients that cannot be fixed by our analysis. The quantum numbers of the χ fields are fixed from the superpotential. We will show that due to the superpotential, the field ψ − N is fixed to be zero on the target space. From the superpotential, we get (among others) the constraints In order to satisfy the D-term equations (4.22), we must have ψ + N = 0 or ψ + 0 ψ − J = 0 for some 1 ≤ J ≤ N − 1, which means that ψ − N = 0 on every point on the targetspace. The other non-trivial equation we get from the superpotential is On the vacua (4.23), one of the χs vanishes and we are left with N −1 χs. They represent the interacting size modesρ There are N − 1 such modes at any vacuum (4.23), and there is an exact agreement between the quantum numbers of the χs and theρs. As an example consider the vacuum described on the first line of (4.23). At this vacuum ψ + N = 0 and therefore, χ 2N = 0. Similarly, in this vacuum k a = δ a,J for some J = 1, ..., N − 1. This implies that there are noρ 2N,j . Due to the obvious symmetry between the first column and the last row of the tetris diagram, the other cases in the list presented at the beginning of 4.2 will be exactly the same.

Strings in SU
In this section we will study the SU (2) M quiver theories in which we gauge some U(1) flavor symmetry. We will focus on the fully Higgsed vacua in which M + 1 scalars get VEV. The tetris diagram contains M + 1 2 × 2 blocks. Each one of the M − 1 inner blocks represent one bifundamental and therefore the entire block should be accompannied with one mass and one U(1) charge. Each of the two outer blocks represent two fundamentals and therefore it is accompannied with two masses and two U(1) charges. There are M + 3 masses {µ i } and charges {c i } which are labelled by the index i, j = 0, ..., M + 2. without loss of generality, we will give VEV to the boxes that sit on the same diagonal. We will denote the boxes by two indices, where the first row is denoted by 1 and the first column is denoted by 0, such that the dots are located in q aa . See figure 7. The gauge symmetry is broken in this vacuum to Z C where C = M +1 a=1 c a . The spectrum of strings above this vacuum is given by Repeating the same procedure as in the previous section, we construct the string by changing the boundary conditions to where k a are non-negative integers. The asymptotic value of the gauge field can be easily computed from the demand that the tension is finite. The U(1) flux carried by this string is in agreement with the allowed spectrum. As before, we will compute the mass and two U (1) R-charges of every hypermultiplet around the string solution.
• Mass: The VEV of the adjoint scalars should be chosen such that the mass terms of q aa vanish. Denoting by a the U(1) scalar and by a I the Cartan scalar of the I'th SU(2) gauge group, the following equations must be satisfied in the vacuum These equations are solved by • R (R) -charge: the vacuum preserves a U (1) R-charge which is the original U (1) R ∈ SU (2) R accompannied by a Cartan gauge transformation that keeps the vacuum invariant. This R-symmetry transformation should satisfy where ω , ω I are the gauge parameters related to the U(1) and the Cartan of the I'th SU(2) respectivly, and α is the U (1) R ∈ SU (2) R parameter.
These equations are solved by Using the previous computations, we can write the mass and R-charges of the hypermultiplets around the string. The results are represented on the tetris diagram 7. Our next step is to  (q M,M +2 , q M +1,M +2 ). The size modes analysis for these hypermultiplets results in 10) and exactly the same for the last row/column with the replacement of 0 → M +2 , 1 → M +1. The conditions for bulk-string decoupling and the for weak to weak mapping can be read directly from equations (5.9),(5.10).
Bulk-string decoupling: Decoupling happens if there are no long range size modes that decay slower than 1 r . The conditions for this are This condition also coincides with the condition for no non-trivial Aharonov-Bohm phases.
Weak to weak mapping: The mapping of parameters from the four dimensional theory to the two worldsheet theory is weak→ weak if all the F-term constraints are satisfied trivially. This happens if for every flavor, there are only q or onlyq modes, but not both. in addition to the size modes, there are also the off-diagonal modes as explained above. These forbidq modes for all bi-fundamental size modes (5.9). It means thatq a±1,a∓1 = 0 for every 2 ≤ a ≤ M , which leads to the condition 2c a ≥ C ∀ 2 ≤ a ≤ M . The other cases can be dealt similarly with q ↔q. 7 We will give an ansatz for the low energy worldsheet theory with topological charge K = 1, and show that the given ansatz is consistent both with the classical zero modes analysis and with results obtained from localization.

K = 1 worldsheet theory
In this section we will describe the worldsheet theory for the K = 1 string in the case where (5.11), (5.12) are satisfied and We will keep for now K and restrict to K = 1 later on. In this case, there are noq excitations. The size modes are given by the parameters ρ in (5.9) and (5.10). We can identify the size modes that exist for every choice of the partition. They give rise to decoupled chiral fields on the worldsheet. These size modes are Stripping off the decoupled modes and ignoring the center of mass modes, we are left with the interacting size modes and off-diagonal modes as can be seen in the tetris diagram 8.
Our ansatz for the worldsheet theory in the K = 1 case is the low energy limit of a two- 14) and the following chiral fields: • X, parametrizing the center of mass modes.
Again, the η fields represent the decoupled size modes. The quantum numbers of these fields are given in table 4. Field (0, , ..., 0,

Comparison with the classical spectrum
It is straight forward to check that the spectrum of η fields matches exactly the spectrum of decoupled size modes ρ with n = n max + 1 − r. We will show that the charged sector also agrees with the expectations. The GLSM has M + 1 vacua corresponding to the M + 1 partitions {k a } in the following way I . Exactly on the same way, the matching of the spectra holds also when expanded around the other vacua.

Field
Twisted Mass

S-duality for SU(2) quivers
S-duality properties of N = 2 superconformal SU(2) quivers were studied for example in [29][30][31]. In the case of one SU(2) gauge group with four fundamental hypermultiplets, the theory enjoys a classical SO (8) global symmetry that acts on the eight half hypermultiplets (or, if you like, the eight N = 1 chiral multiplets). SO(8) has an S 3 outer automorphism group. The theory is invariant under the outer automorphism of SO(8) accompanied by S 3 transformations of the gauge coupling q ≡ e 2πiτ . 8 The S-duality group S 3 has six elements which are generated by two generators. We will denote them by S and T . 9 They act on the four SO(8) Cartan generators as Turning on masses for the hypermultiplets and/or gauging some U(1), break explicitly the SO(8) global symmetry. Now, instead of relating the theory to itself, the S-duality transformations relate between two theories that differ by their masses and U(1) charges. The masses and U(1) charges transform on the same way as the Cartan generators. This fact was used in [2] to find the worldsheet theory of strings in some cases where the 4d-2d map is weak→strong. This is done in the following way. Consider two different U(1)s with charges {c i } and {c i } related by SO(8) outer automorphism. It means that there exists some mapping τ → τ (τ ) such that a theory with U(1) charges {c i } and SU(2) gauge coupling τ is equivalent to a theory with U(1) charges {c i } and SU(2) gauge coupling τ . The same story holds also for the worldsheet theories. Lets say that in the first theory, there is a string 8 We use here what is known in the literature as τ uv that transforms non-trivially only under an S 3 subgroup of the SL(2, Z) that acts on τ IR . For a discussion about the differences between the two, see for example section 9.2 in [32]. 9 T is the same as ST S in the conventions of [2]. We decided to specify here the action of T instead of T since we work with it explicitly in 6.1.
such that the FI parameter of its worldsheet theory is given by t = f (τ ) where f (τ ) is some function. Under S-duality, the string is mapped to a string in the second theory. The worldsheet theory of the dual string will be the same up to the map of parameters, which is t = f (τ (τ )). As an example, we can start from a known weakly coupled worldsheet theory for which t = τ and act with the S transformation that takes e 2πiτ → e 2πiτ = 1 − e 2πiτ . (6.4) The dual worldsheet theory will be the same theory but with which is of course strongly coupled. We would like to apply this method on SU(2) quivers. Every SU (2) factor is coupled to four fundamental hypermultiplets. However, there are some difficulties coming from the coupling of these hypermultiplets to other SU(2)s. The exact Sduality group involves a complicated transformation of all the gauge couplings simultanously [31] For simplicity, we will use an approximate S-duality that acts only on one gauge group, as in the case of one SU(2) coupled to four fundamental hypermultiplets. The approximate S-duality is broken due to the other gauge couplings, however, we still expect it to be applicable in the limit where the gauge couplings of the adjacent SU(2)s are much smaller than the gauge coupling of the discussed SU(2). In the next section, we will consider two weakly coupled worldsheet theories related by S-duality and show that the correct mapping between the two as predicted from the approximate S-duality is achieved only in the limit described above. Later on, in 6.2, 6.3, we will use the S-duality transformation (6.4) to find strongly coupled worldsheet theories of some (generalized) quivers.

T transformation
Consider two different SU (2) M ×U (1) theories with U(1) charges related by T tranformation (6.2) of one of the SU(2) factors. Lets consider first the case where the SU(2) is in the middle of the quiver, i.e. coupled to two bifundamentals. We can take it to be SU (2) L for some 2 ≤ L ≤ M − 1. In order to understand the action of S-duality, it will be usefull to follow the Cartans. The two bifundamentals charged under SU (2) L have masses µ L , µ L+1 and charges c L , c L+1 . In addition, they are charged under SU (2) L−1 and SU (2) L+1 respectively. Their Cartan generators, denoted by α L∓1 , act with an opposite phase on the two halves of every bifundamental. The SO(8) automorphism acts on the four charges Under T which takes τ L → −τ L , the two Cartan generators α L±1 are interchanged α L−1 ↔ α L+1 . This is equivalent to interchanging the two bifundamentals. The meaning is that in the limit of Im(τ L ) Im(τ L±1 ), the worldsheet theory should be invariant under Lets see that this is indeed true for the worldsheet theory described in 5.1. The conditions (5.11), (5.12), (5.13) are invariant under (6.8). The decoupled sector, which is made out of the η fields of table 4 is also invariant under (6.8). The non-trivial part comes from the charged sector of table 4. It will be useful to see what happens to the D-term constraints. Before the transformation, the relevant D-terms were The transformation (6.8) takes ξ L → −ξ L and exchanges the masses and R-charges of ψ ± L and ψ ± L+1 . We can relabel ψ ± L ↔ ψ ± L+1 and write the new D-term constraints (6.10) These three equations can be written as In the Im(τ L ) Im(τ L±1 ) ⇔ ξ L ξ L±1 limit, these equations are the same as equations (6.9) which is what we expect to get from S-duality. Now we will consider the case where the transformed gauge group is on one of the edges. We will take it to be SU (2) 1 . The case of SU (2) M is equivalent. The relevant hypermultiplets are two SU (2) 1 fundamentals whose masses and charges are denoted by µ 0,1 , c 0,1 and one SU (2) 1 × SU (2) 2 bifundamental, whose mass and charge are denoted by µ 2 , c 2 . T now acts on the four charges {c 0 , c 1 , c 2 + α 2 , c 2 − α 2 } . (6.12) This transformation takes and similarly for the masses. As before, the conditions (5.11), (5.12), (5.13) are invariant under (6.13). The decoupled sector, which is made out of the η fields of table 4 is also invariant under (6.13). The relevant charged part of the theory contains 6 fields and two U(1)s with the D-terms Under ξ 1 → −ξ 1 the D-terms become (6.15) and the masses and R-charges of the fields are given by the action of (6.13) on table 4. Recall that the R-charges and the masses themselves are not physical because they can be shifted by gauge transformations and redefinitions of the scalars σ i . If we shift σ 1 by 1 2 (µ 0 − µ 1 ) then the new masses are 16) and similarly for the R-charges. We can rename ψ − 1 ↔ ψ − 2 and ψ + 1 ↔ ψ + 2 and in the limit where ξ 2 ξ 1 we get exactly the same theory before the T transformation with the correct spectrum.

S transformation on the edge
In this section we consider the action of the S transformation (6.1) on SU (2) 1 , the first SU(2) factor. The S transformation acts on (6.12) as The first thing that we observe is that C = C, simply because c 1 + c 2 = c 1 + c 2 . This means that we are not comparing the correct strings. The string we need to examine is a string above the mixed mesonic-baryonic vacuum illustrated in figure 9. Instead of analysing this string from the beginning, we can use the fact that the fundamental representation of SU (2) is pseudoreal, and therefore we can exchange mesons with baryons, and take the vacuum described in figure 9 back to the baryonic vacuum. This transformation effectively takes c 2 → −c 2 = 1 2 (c 1 − c 0 ). The next thing that we observe is that unlike the T transformation, the conditions (5.12), (5.13) are not invariant under the S transformation. This is becaue 2c 2 < C and c 0 < c 1 (After we took c 2 → −c 2 ). The size modes counting (5.9), (5.10) now implies that there are size modes excitations ofq 13 andq 10 . Therefore there are non-trivial F-term constraints of the formq and as a result the worldsheet theory is strongly coupled. The approximate S-duality gives us the worldsheet theory only in the limit τ 1 τ 2 . The K = 1 worldsheet theory in this case is given by the low energy limit of an N = (2, 2) U (1) M GLSM with M complexified FI parameters e 2πit 1 = 1 − e 2πiτ 1 , t a = τ a ∀ 2 ≤ a ≤ M, (6.19) and the following chiral fields • X, parametrizing the center of mass modes.
The quantum numbers of the fields are summarized in table 6.

S transformation on the middle: Generalized quivers
In this section we will study the S transformation acting on an intermediate SU (2) L with 2 ≤ L ≤ M − 1. This SU(2) is coupled to 2 bifundamentals. As in section 6.1, the S transformation acts as (6.1) on (6.7), which results in (0, , ..., 0, An interesting consequence is that the S transformation takes the quiver to the so called generalized quiver represented in figure 10. Lets see exactly how it works. Under S, we see from equation (6.20) that the two gauge groups SU (2) L±1 act on the same block. This block is now a trifundamental of the three gauge groups SU (2) L,L±1 . Take as an example L = 2. In order to understand better how SU (2) 1 acts on the trifundamental hypermultiplet, it will be usefull to look at the scalars The third gauge group, SU (2) 1 , now acts on q σ 2q σ 2 T as doublets. Notice that the mixing between q andq implies that the trifundamental field must be massless and U(1) neutral. One of the difficulties in studying strings on generalized quivers is that the worldsheet theory is inherently strongly coupled. In the previous cases, the F-terms couple q withq, then one can find simple conditions on the U(1) charges such that the F-terms are satisfied trivially and as a result, the worldsheet theory is weakly coupled. On the other hand, trifundamental F-terms couple also q with q andq withq. For example, from the Cartan of SU (2) 1 we get the constraint q 43 q 32 −q 43q32 + q 33 q 42 −q 33q42 + ... = 0 . (6.23) The off-diagonal modes that come from q 43 , q 32 impose non-trivial constraints on the worldsheet. As a result, the worldsheet theory is strongly coupled, regardless of the U(1) charges. We will study generalized quiver strings in the case where they are S-dual to weakly coupled strings. We will write everything explicitly for the SU (2) 3 quiver. The generalization to longer quivers is straight forward. The masses and U(1) charges of the first two columns and the last two columns, parametrized by a = 0, 1, 4, 5, are invariant under the S transformation and are equal to µ a , c a . The two SU (2) 2 fundamentals have masses and U(1) charges µ 2 ± µ 3 , c 2 ± c 3 . The trifundamental field is massless and neutral, as explained above. The U(1) charges satisfy It will be convenient to denote the charges by 25) and similarly for the masses. Putting all the details together, our ansatz for the worldsheet theory is a U (1) 3 GLSM with complexified FI parameters t 1,3 = τ 1,3 , e 2πit 2 = 1 − e 2πiτ 2 , (6.26) and the following chiral fields • One neutral chiral field X.
• Neutral fields η − I,r with I = 1, 3 and r = 1, ..., The quantum numbers of these fields are summarized in table 7 One can easily check that at the limit g 2 g 1,3 , the spectrum is mapped to spectrum of

Worldsheet partition functions from supersymmetric localization
In this section we will derive the worldsheet S 2 partition functions for all the strings discussed in the previous sections. The ideas of this derivation were presented in [21,22] and elaborated in [2]. We will review this method briefly. As a start, we put our four dimensional quiver theory with gauge group SU (N ) M × U (1) on the four ellipsoid The partition function on this manifold was computed in [19,20]. The partition function in this representation is written as an (M N − M + 1)-dimensional integral over the Coulomb branch coordinates For a wide range of the parameters of the theory, one can close the contour of integration over the U(1) Coulomb branch parameter in the complex plane. The partition function is then written as a sum over the residues of the poles that lie inside the contour of integration. The poles come from the zeros of the Υ b (x) functions that appear in the denominator of (7.2). One can try and close the contour of integration also for the other M (N − 1) integrals. For some of the terms, it is possible to eliminate all the integrals in this way. Inspired by the analysis of [34], these terms are interpreted as Higgs branch saddle points. Indeed, among these terms we can identify the fully Higgsed vacua and strings above these vacua that were studied in the previous sections. For the rest of the terms, it is not possible to close the contour for all the integrals. These terms are identified with mixed Higgs-Coulomb saddle points. The leftover Coulomb branch integrals represent the residual gauge symmetry of these configurations. Between these terms one can find the non-fully Higgsed vacua that were described in section 2. From now on we will focus only on the fully Higgsed contributions. This sum can be written as = 1 respectively. We will focus on configurations for which K = 0 that correspond to the 1 2 -BPS strings studied above. Configurations for which both K and K are non-zero are 1 4 -BPS configurations of intersecting strings. The function Z K,{la} ≡ Z K,0,{la} contains the information about the string dynamics. If the factorization conditions hold and the string is decoupled from the bulk at low energies, Z K,{la} is identifed with the S 2 partition function of the string worldsheet theory. In these cases, one can use the identities (7.3) to transform all the Υ b (x) functions to Γ-matrices, which are the "building blocks" of S 2 partition functions. The weak→weak condition can also be seen from the form of Z K,{la} . When the weak→weak conditions are satisifed, Z K,{la} has a fixed number of Γ functions, independent of the partition {k a }. This is a property of S 2 partition functions expanded around the weakly coupled point. When the weak→weak conditions are not satisifed, different terms in Z K,{la} contain different number of Γ functions. This can happen for S 2 partition functions expanded around some strongly coupled point. 10 Once Z K,{la} is found, we want to identify it with the S 2 partition function of the string worldsheet theory under some 2d-4d map of parameters. In the next sections, we will compute Z K,{la} for all the strings studied in sections 4, 5 and show that the results agree with our suggestions for their worldsheet theories.
In this section we will extract the worldsheet S 2 partition functions for the strings studied in section 4 and show that the results are consistent with the worldsheet theories presented in 4. The four-ellipsoid partition function for the SU (N ) × SU (N ) × U (1) theory is (7.6) The fully Higgsed vacuum and strings we are interested in are given by collecting the residues from the following poles Summation over all the terms with fixed K = 2N −1 a=1 k a gives the contribution from the string with winding number K, and the term with K = 0 gives the vacuum contribution.
It will be usefull to define (7.8) In terms of these variables, the poles are given by The contribution from this choice of poles with a fixed K = 2N −1 a=1 k a is (7.10) First, let us evaluate the vacuum contribution by plugging in K = 0: (7.11) Z 0 describes the N 2 + 1 light hypermultiplets which are dynamical in the vacuum. Their masses and R (R) charges can be read from the imaginary and real parts of the arguments of the Υ b functions. It is straight forward to check that these are in agreement with the expectations from the classical analysis. Now we will move on to evaluating Z K . Using the identities , (7.13) the partition function can be written as (7.14) From this form of Z K we can easily derive the factorization condition .., 2N − 1 (7.15) When these conditions are satisfied, we can use (7.12) to write Z K Z 0 as a product of γ functions which can be interpreted as some S 2 partition function describing the string worldsheet theory. Indeed, these are the same conditions as the conditions for no fractional size modes as seen from equations (4.15), (4.16). Besides the factorization conditions, we also have the weak→weak conditions. The worldsheet theory is weakly coupled when the number of γ-functions doesn't depend on the partition {k i }, only on K. For this to be satisfied, we demand In addition, we need to demand one of the following four possibilities These are the same conditions found using the classical zero modes analysis, see equation (4.18) and the list right after. Assuming that the conditions are satisfied, the result for Z K that we get is where Z overall contains overall factors that are interpreted as regularization ambiguities, see the discussion at the end of section (4) in [2]. The factors Z (7.21) For case 2, it is the same, multiplied by an overall For case 3, it is the same, multiplied by an overall For case 4, it is the same, multiplied by an overall charged coincides with the S 2 partition function of the U (1) × U (1) GLSM whose matter content consists of the chiral fields X, ψ ± I and χ described in section 4.2. Cases 3,4 can be treated exactly the same. The computations of the relevant S 2 partition functions and the exact matchings are presented in appendix A.

SU (2) n × U (1)
In this section we will extract the worldsheet S 2 partition functions for the strings studied in section 5 and show that the results are consistent with the worlsheet theories presented in 5. The four-ellipsoid partition function for the SU (2) n × U (1) theory is (7.25) The poles that give the baryonic chain studied in section 5 are given by (7.26) We will denotê The residues of the poles (7.26) are given by (7.28) The vacuum of the theory is given by plugging in k I = 0. This results in 29) This is the partition function of the n + 3 light hypermultiplets with the correct masses and R-charges. The conditions for factorization can read easily from the arguments of the Υ b (x) functions, and result in These are the same conditions derived from classical size modes analysis (5.11).

Weakly coupled worldsheet theories
Assuming the conditions are satisfied, we can write the worldsheet theory partition function. We will start from strings satisfying the conditions These are the weak→weak conditions analysed in section 5. Ignoring the instantons and some overall factors, we get 33) and (7.34) For K = 1 it simplifies to (7.35) The expression derived here for Z (K) dec coincides with the S 2 partition function of the decoupled size modes. For K = 1, these are the η fields of table 4. Z (1) charged coincides with the S 2 partition function of the U (1) n GLSM whose matter content consists of the chiral fields X and ψ ± I described in table 4. The computations of the relevant S 2 partition functions and the exact matchings are presented in appendix A.

S-dual strings
In this section we will compute the partition function for strongly coupled wordlsheet theories which are S-dual to the strings studied in the previous section. We will start from the case described in section 6.2, in which the charges satisfy (7.36) For Simplicity we will derive the worldsheet partition function in the case of two SU (2) factors. The generalization to higher number of SU (2)s is straight forward. For K = 1, the worldsheet partition function is 38) and (7.39) Z dec coincides with the S 2 partition function of the decoupled modes X and η of table 6. Z charged coincides with the S 2 partition function of the U (1) 2 GLSM whose matter content consists of the ψ ± 1,2,3 of table 6. The computation of the relevant S 2 partition function and the exact mapping of parameters appear in appendix A.

Generalized quiver localization
In this section we will study the G = SU (2) 3 theory with two fundamentals for every SU (2) and one trifundamental. We will denote the masses and U(1) charges of the six fundamental fields by µ Is , c Is with I = 1, 2, 3 and s = ±. The trifundamental field is massless and U(1) invariant. The S 4 partition function reads (7.40) We are interested in the poles (7.43) The result is (7.44) Lets start from the vacuum described by taking K = 0. The result is For K = 1, the contribution is (7.46) Z 1 factorizes if c I+ +c I− C ∈ Z ∀I = 1, 2, 3. Using the identities (7.12), one can write the expression for Z 1 Z 0 and see that there is no choice of charges such that the number of γ functions is independent of the partition {k a }. This means that the generalized quiver string is inherently strongly coupled due to the trifundamental matter. In order to simplify the expressions, we will assume that the S-dual string is weakly coupled such that the dual charges satisfy 2c 2,3 ≥ C , c 0 ≥ c 1 , c 5 ≥ c 4 , c 0 + c 1 ≥ C , c 5 + c 4 ≥ C . For charges satisfying (7.49), we can write  In this section we will compute the S 2 partition function of U (1) × U (1) gauge theory with 4 chiral multiplets with charges (±1, 0), (0, ±1) and masses m ± 0,N and 2(N −1) chiral multiplets with charges (±1, ∓1) and masses m ± I , I = 1, ..., N − 1. We will show that this agrees with (7.21) under some 2d-4d map of parameters. Ignoring instantons, the partition function is Here z a = e 2πita where t a are the complexified FI parameters. We can close the contours of integrations over σ a in the complex plane and obtain the partition function in the Higgs branch representation, given by There is an agreement with (7.21) if From the definition of M, we can write the explicit expressions (A.7) Using the relation (A.1), we can extract the masses and R-charges of the different fields and check agreement with the table 1.

A.1.2 Includingq excitations
In the previous section we studied only the case withoutq excitations. We can have a weakly coupled theory withq excitations. These are cases 2,3,4 from the list right after (4.18). The worldsheet theory will be very similar to that of case (1), but with additional neutral fields coupled to the charged fields via a superpotential. The S 2 partition function is independent of the superpotential coefficients. This means that when adding these fields we just need to multiply the partition function by the partition function of the neutral fields. In case (2), this factor is (A.8) These are the neutral fields χ 2N and χ i with i = N +1, ..., 2N −1 that are described in 4.2 . In addition, we replace η 2N,j,r and η 2N,N,r withη 2N,j,r j = N +1, ..., 2N −1 r = 1, ..., (c i −c 2N )K C −K andη 2N,N,r r = 1, ..., − K C 2N j=N +1 c j as described in 4.2. These changes of the partition function are exactly captured by the change in the I 2N factor of equations (7.17), (7.20) when moving from case (1) to case (2).
(A.9) Again, upon closing the σ a integral in the complex plane, we get

A.3 S-dual strings A.3.1 Linear quiver
In this section we will compute the S 2 partition function for the worldsheet theories described in 6 and show agreement with the results obtained from S 4 b partition function. Due to the weak→ strong mapping of parameters z a ≡ e 2πita = 1 − e 2πiτa for some a, we want to expand the partition function around 1 − z a . In [2] it was shown that the partition function of a U(1) GLSM with 2 chirals with charge +1 and masses m + 1 , m − 2 and two chirals with charge −1 and masses m − 1 , m + 2 at leading order in 1 − z 1 is .