On supersymmetric Wilson loops in two dimensions

We classify bosonic $\mathcal{N}=(2,2)$ supersymmetric Wilson loops on arbitrary backgrounds with vector-like R-symmetry. These can be defined on any smooth contour and come in two forms which are universal across all backgrounds. We show that these Wilson loops, thanks to their cohomological properties, are all invariant under smooth deformations of their contour. At genus zero they can always be mapped to local operators and computed exactly with supersymmetric localisation.


I. INTRODUCTION
Wilson loops are an important tool to understand the dynamics of gauge theories: they encode how an (infinitely massive) elementary excitation localized along the loop responds to the presence of a dynamical gauge field. They provide an efficient and operative way to characterize the confinement or deconfinement phases of the theory according to whether their expectation value increases with the area (area law) or, respectively, with the perimeter of the loop (perimeter law) [1].
In supersymmetric theories, a special role is played by the so-called supersymmetric or BPS Wilson loops. These observables are annihilated by a subsector of the supercharges of theory, and they are often amenable to exact evaluation through localization techniques. The first and probably most famous example is the Maldacena-Wilson loop along a circular path in N ¼ 4 super Yang-Mills (SYM) in four dimensions. It preserves half of the original supercharges, and its expectation value is given by a Gaussian matrix model whose form was originally conjectured in [2] and then rigorously derived by Pestun in [3].
According to the AdS/CFT correspondence, the very same quantity can be computed at strong coupling by a semiclassical string computation, the actual matching at leading [2] and subleading [4][5][6] order being in striking support for the gauge/gravity paradigm.
This initial success has prompted an intensive search for other examples, both in N ¼ 4 SYM in four dimensions and in other theories with a different number of supersymmetries or dimensions. In [7], Zarembo constructed entire families of BPS Wilson loops in N ¼ 4 SYM that preserve 1 16 , 1 8 or 1 4 of the supersymmetry depending on the subspace spanned by the contour supporting the loop. The shape of the contour is not relevant for the existence of unbroken supercharges. Unfortunately, the expectation value of these observable does not receive quantum correction, and it is one to all orders in the coupling constant. Nonetheless, this triviality carries significant information about the dynamics of the theory, especially in the context of the AdS/CFT correspondence: the associated family of minimal surfaces in AdS 5 × S 5 must have zero regularized area independently of the shape of the boundary [8].
On the other hand, generic contours on an S 3 embedded in R 4 and preserving at least two supercharges were constructed in [9]. Restricting the loop on S 2 , the authors conjectured that this can be computed in terms of the zeroinstanton sector of similar observables in purely bosonic Yang-Mills in two dimensions on the sphere. This conjecture was first verified perturbatively up to two loops in [10,11], and later an (almost) complete proof using localization techniques was given in [12] (see also [13]). The study of this family of Wilson loops has led to exact results for correlators of Wilson loops [14][15][16][17], correlators of Wilson loops and primary operators and theories living on one-dimensional defects [18][19][20]. The defect approach was also crucial in relating circular Wilson loops to the energy emitted by an accelerating particle, captured by the so-called bremsstrahlung function [21].
In the N ¼ 2 case, the original analysis by Pestun carries over, and the 1 2 -BPS circular Wilson loop has been studied: the relevant matrix model is much more complicated, and the eigenvalue measure also depends on instanton contributions. The large-N limit, where instantons decouple, has been thoroughly studied and successfully compared with AdS/CFT predictions [22][23][24]. Furthermore, also in this case, an exact prediction for the bremsstrahlung function has been put forward [25,26].
In three dimensions the situation is more intricate, especially when considering superconformal theories. One can construct bosonic Wilson loops, whose structure mimics the behavior in four dimensions: the connection appearing in the holonomy is built by adding to the gauge field a suitable bilinear in the scalars of the theory. These loops are supersymmetric only for a suitable choice of the contour (for instance, circles and straight lines) [27,28]. The expectation value of the circular bosonic Wilson loop can again be evaluated with a localization procedure, developed by Kapustin, Willett and Yaakov [29], and can, in principle, be computed for a very general class of N ¼ 2 theories on S 3 (see [30] for extensions on more general three-dimensional manifolds) through complicated matrix models.
In the superconformal case of ABJ(M) theories, the above Wilson loop has been explicitly evaluated by a tractable matrix model, which is closely related to that describing topological Chern-Simons theory on lens space [31,32]. However, in the context of the AdS/CFT correspondence, these bosonic loops are not dual to the fundamental strings since, in general, they possess the wrong residual symmetries. The holographic dual for the case of the straight line and the circle in ABJ(M) theories was constructed by Drukker and Trancanelli in [33] and quite surprisingly couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bifundamental representation of the UðNÞ × UðMÞ gauge group. In other words, the usual connection is replaced by a superconnection built out of the fundamental fields and living in the superalgebra UðNjMÞ. The presence of fermionic couplings naturally allows for a larger number of preserved supersymmetries. These loops were extended to more general contours in [34] and to theories with less supersymmetries in [35][36][37][38][39]. They also allowed, as in four dimensions, for the exact computation of the bremsstrahlung function [40][41][42][43].
In two dimensions the situation is expected to be somewhat simpler, but it is still much less explored nonetheless. Interestingly, exact and very nontrivial results can be obtained even for Wilson loops in nonsupersymmetric models, when the theory is defined on compact two-dimensional manifolds. For instance, in pure Yang-Mills a generic Wilson loop can be evaluated on any Riemann surface by exploiting either lattice [44] or localization [45] techniques. At large N, they exhibit a stringy behavior [46,47]; different scalings chart the phase structure [48] and admit explicit solutions of the Migdal-Makeenko equations [49,50].
It is quite natural therefore to wonder if a similar variety of phenomena is shared by the supersymmetric analogue of the "canonical" Wilson loops of twodimensional Yang-Mills. More generally, one would like to classify and, hopefully, compute new gauge invariant observables in two-dimensional supersymmetric theories that could be useful in checking the AdS/CFT correspondence, testing nonperturbative dualities and constructing defect field theories. We focus, in particular, on N ¼ ð2; 2Þ gauge theories with vector R-symmetry. Backgrounds for such theories on arbitrary Riemann surfaces were recently studied in [51,52]. More specifically, we would like to classify all bosonic Wilson loops built from the gauge supermultiplet which preserve some supersymmetries independently of the shape of the closed contour, realizing a sort of two-dimensional version of the Zarembo's charting in four dimensions [7]. We find two families of BPS observables: iðA a þ f a ϵ σ þf a ϵσ Þ_ x a dt; where σ andσ are the scalar fields in the gauge supermultiplet and the couplings f a andf a are defined in (9) and (10). These loops are 1 4 -BPS, and the analysis of the preserved supersymmetries extends to the case of a general Riemann surface. For both classes of Wilson loops, we find that the associated field strength is Q-exact. This, in turn, implies that the quantum expectation value of a generic non-self-intersecting loop does not change under smooth deformations of its contour, thus signaling a topological character of the observable.
We then proceed to extract exact expectation values and correlators of these Wilson loops. For N ¼ ð2; 2Þ gauge theories, exact results on various backgrounds have been computed in recent years using localization. On the sphere [53,54], these can be represented both as a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group (the so-called Coulombbranch representation) and as the product of a vortex times an antivortex partition function, weighted by semiclassical factors and summed over isolated points on the Higgs branch. This dual representation is reminiscent of the nonsupersymmetric case, where the exact partition function can be expressed both as a sum over the irreducible representations of the gauge group [44] and as a weighted sum over instanton solutions [45]. Fayet-Iliopoulos and theta terms for the Abelian factors of the gauge group and twisted masses and background fluxes for the chiral multiplets can also be added, enriching the parametric dependence of the results. In this theory a one-parameter family of Wilson loops with contours lying on latitudes of the round two-sphere were already considered in [53]: on the maximal circle the dimensionally reduced 1 6 -BPS Wilson loop of three-dimensional N ¼ 2 theories is recovered, and the quantum expectation value of the whole family is independent from the latitude angle.
These are recovered in our construction as the limit case in which the two Wilson loops in (1) coincide and the preserved supersymmetry is enhanced to 1 2 -BPS. It is only in this case that the matrix model of [53,54] can be directly applied to perform exact computations. Our cohomological argument outlined earlier, however, ensures that the result can be extended to encompass Wilson loops on arbitrary contours. The final result can be expressed both in the Coulomb branch representation and in the Higgs branch representation: the nontriviality of the vacuum expectation value is ensured by the presence of twisted masses or background fluxes.
Two-dimensional gauge theories with N ¼ ð2; 2Þ supersymmetry admit dual descriptions in the infrared regime, first described by Hori and Tong [55,56], analogously to the case of four-dimensional N ¼ 1 Seiberg duality [57]. For unitary and special-unitary groups, the duality has been explicitly checked, at the level of the partition functions, using results from localization [53,54]. More recently, the duality has been tested against correlators of Coulombbranch operators for topologically twisted theories [58]. Here, we show that the dictionary for unitary groups can be extended to include correlators of the Wilson loop operators in (1), similarly to what has been done in [59] for three-dimensional N ¼ 2 theories. We find that a single operator insertion is mapped to a combination of Wilson loops in different representations.
The structure of the paper is the following: in Sec. II we discuss the general construction of supersymmetric Wilson lines for two-dimensional N ¼ ð2; 2Þ supersymmetric gauge theories with Uð1Þ R vector R-symmetry on orientable Riemann surfaces. We derive the explicit form of the scalar couplings in term of the relevant Killing spinors, and we are able to associate to a generic curve two 1 4 -BPS Wilson lines. By further constraining the contours we observe a collapse of the two solutions into a single 1 2 -BPS Wilson line. We then look at specific backgrounds and derive the explicit form of these 1 2 -BPS lines. Section III concerns Wilson loops, and the main result of the paper is presented: non-self-intersecting Wilson loops whose paths are homotopically equivalent are Q-cohomologous. In other words, the vacuum expectation value of our 1 4 -BPS observables does not change under smooth deformations of their contour. We will prove this property by showing that the effect of an infinitesimal deformation results in a Q-exact quantity. In Sec. IV we specialize to a squashed background on S 2 and find exact results for the value of any of these Wilson loops and their correlators, using supersymmetric localization. We then discuss the dependence on geometrical data and external parameters. Finally, in Sec. V we determine how correlators of Wilson loops are mapped under Seiber-like dualities and provide explicit examples.

II. WILSON LINES
A. N = (2;2) supersymmetry Let us consider two-dimensional N ¼ ð2; 2Þ supersymmetric gauge theories with Uð1Þ R vector R-symmetry on some orientable Riemann surface M. Any supersymmetric background for such theories can be understood as a background off-shell supergravity multiplet as in the approach of [60]. In addition to the metric g, the multiplet contains, as bosonic degrees of freedom, a gauge field B, which couples to the R-symmetry current, and graviphotons C andC, which couple to the central-charge currents. The Killing spinor equations for the generators of rigid supersymmetry are obtained by imposing the variation of the gravitini to be vanishing. This gives where H andH are the dual field strengths of the graviphotons The various backgrounds induced by the solutions of the above equation on an arbitrary M have been studied and classified in [51,52]. The only backgrounds that apply to any genus g are the topological A-andĀ-twist [61,62], which preserve up to two supercharges of opposite R-charge. All other possible backgrounds apply to the case of g < 2.
For g ¼ 1 one can adopt a flat background as on the complex plane C. Finally, the case of g ¼ 0 admits various rigid supersymmetry realizations preserving a different number of supercharges. We refer the reader to [51,52] for more detail. In this paper, we will pay particular attention to the maximally supersymmetric round sphere and its squashing.
While still keeping the supersymmetric background generic, let us look at N ¼ ð2;2Þ vector multiplets. Here we denote with G the gauge group and with g its associated Lie algebra. In the Wess-Zumino gauge the multiplet consists of g-valued fields ðA; λ;λ; σ;σ; DÞ, where A is the gauge vector field, λ andλ are gaugini, σ andσ are scalars, and D is an auxiliary scalar field. The Uð1Þ Rcharges for the component fields are given by In general, the axial R-symmetry is explicitly broken. The supersymmetric variations for the component fields are given in Appendix B.

B. Supersymmetric connections
We are now ready to introduce a generic Wilson line of the form where Γ∶ð0; 1Þ → M is some smooth path and A is a G-connection defined as some combination of fields from the vector multiplet. In coordinates, we denote Γ as the embedding xðtÞ.
Since we are interested in bosonic deformations of the nonsupersymmetric Wilson line defined with A ¼ A, we write for some one-forms f andf. We want to find the choices of f andf for which W is annihilated by some supercharge. For this, we need the explicit expression of the supersymmetric variation of the fields A, σ andσ. Interestingly, these fields are precisely those whose variations are insensitive to the particular realization of supersymmetry considered, as can be seen from (B1).
When acting with a supersymmetry variation on A ¼ A a e a , one finds where From this, one can immediately obtain the variation of the integrand A a _ x a , i.e., of the pullback of A on the path Γ.
The kernels of both M a _ x a andM a _ x a are nontrivial for any choice of xðtÞ if When these conditions are satisfied, one can find solutions for f andf such that the Wilson line is annihilated by either Q ¼ ϵQ for for any choice of the path Γ. Notice how the above are well defined: the ratios of spin components are single-valued on M and, under a frame change, transform with a phase that cancels the opposite phase coming from the (anti)holomorphic combinations of zweibein elements.
To summarize our discussion so far, given a generic path xðtÞ we have defined two 1 4 -BPS Wilson lines, namely annihilated, respectively, by Q andQ. We want to stress that the above construction is fully general and, following from our previous discussion, holds for any supersymmetric background, provided that this preserves the selected supercharge Q orQ. For certain choices of Γ, the two Wilson lines may coincide. This happens when xðtÞ obeys the differential equation On such paths, we obtain a unique Wilson line which is 1 2 -BPS, as it is annihilated both by Q andQ. From (13), a necessary condition on the supercharges is imposed, namely, The Killing spinor equations fix the action of the exterior derivative of f andf. From (2), in fact, follow When using the above with (9) and (10), the dependence on ω and B drops, and one is left with Starting from the supersymmetric connections A ϵ and Aε, we can introduce the associated field strengths The dual forms read Here we have used the fact that A, σ andσ are uncharged with respect to R-symmetry. One crucial property of F ϵ and Fε is that they are, respectively, Q-andQ-exact. In particular, The above can be obtained by using the identities This fact will play a prominent role in Sec. III.
In the remainder of this section we will look at some specific backgrounds.

C. Flat backgrounds
We begin by studying Wilson lines on manifolds that admit a flat background, i.e., the plane, the cylinder and the torus. On a flat background, Killing spinors are constant. This, in turn, implies that f a andf a are constant for both choices of Wilson lines W ϵ and Wε. The paths associated with 1 2 -BPS Wilson lines are straight lines, as it can be seen by solving (13) for constant ϵ andε.

D. The round sphere
Let us now consider the case of M ≃ S 2 equipped with the round metric. We denote the radius of the sphere with r and use spherical coordinates θ ∈ ½0; π and φ ∈ ½0; 2πÞ in terms of which the zweibein reads We follow [54] and consider the supersymmetry algebra generated by Killing spinors obeying (2) with A generic solution is of the form for some constant spinors ϵ 0 andε 0 . It will prove to be useful to also work in terms of the complex stereographic projection With this choice of variables, we find We see that f ϵ has a unique singular point at z ϵ;1 ¼ 2irϵ − 0 =ϵ þ 0 and a unique zero at z ϵ; The same is true forf ϵ , but the locations of the singularity and the zero are swapped. Moreover, the fact that z ϵ;1zϵ;2 ¼ −4r 2 implies that the two points on S 2 are antipodal. Notice how here the components of ϵ 0 play the role of homogeneous coordinates.
The paths zðtÞ over which the Wilson line becomes 1 2 -BPS satisfy, according to (13), Suppose that detðϵ 0ε0 Þ ≠ 0; then we can recast (27) as so the differential equation reduces to a rational one. In particular, given the condition on the determinant, we can always find a Möbius transformation that brings the above to an equation of the form When the coefficients allow for solutions of the above, the corresponding paths on the sphere are circles. The situation is analogous for detðϵ 0ε0 Þ ¼ 0. In that case one can recast which, again, reduces to a rational equation of the form (29). Here we are assuming that certain components of ϵ 0 andε 0 are nonvanishing; other corner cases do not introduce solutions which are qualitatively different from the one discussed above. So far we have concluded that (27) admits solutions only for certain choices of ϵ 0 andε 0 , and these solutions can only be circles. Now, instead of deriving the explicit form of a generic solution of (27), without loss of generality we will focus only on those circles which are centered in z ¼ 0, i.e., latitudes. Any other solution can be mapped into this restricted class by a suitable SUð2Þ transformation.
When substituting the ansatz zðtÞ ¼ ρe iφðtÞ , (27) is satisfied for either In both cases one finds ξ a ¼εγ a ϵ ∝ 0 i.e., that the Killing vector defined by the two spinors generates the U(1) isometry whose orbits on the sphere are the paths of the 1 2 -BPS Wilson line considered. At the same time, the fixed points of the action of such isometry are the north and the south poles, where f andf have their zeros and their singularities.
As anticipated, these conclusions are fully general. In fact, one can proceed the other way around and consider any 1 2 -BPS Wilson line on S 2 . This will run along the action of some U(1) isometry induced by the supercharges annihilating the Wilson line. The fixed points of the isometry will be precisely the antipodal points where f andf become singular. One can then identify the two points with north and south poles and choose appropriate spherical coordinates in terms of which the 1 2 -BPS paths have θ fixed. For any value of θ, one finds that there are actually two Wilson lines annihilated by two different pairs of supercharges Q ¼ ϵQ andQ ¼εQ. The first one takes the form with while the second reads with Notice how, since at the poles either f orf is singular, the integrals do not vanish in the limits θ → 0 and θ → π. In these limits the Wilson lines reduce to the local operators These limits will be important for our analysis in the next sections.

E. The squashed sphere
The round sphere can be seen as a special case of a more general supersymmetric background. This is the squashed sphere, whose geometry is given by for some smooth lðθÞ > 0 with lð0Þ ¼ lðπÞ ¼ 1.
This supersymmetric background, studied in [63], takes the form A generic lðθÞ preserves only a Uð1Þ isometry group and two supercharges associated with the Killing spinors in (34), which, in turn, generate the residual isometry.
Inverting the sign of B leads to a different background in which the preserved supercharges are the ones in (36).
Here the only 1 2 -BPS Wilson lines are the ones running along the action of the preserved Uð1Þ, and are of the form we denoted with W a (or W b , for a different sign in B).

III. WILSON LOOPS
In this section we will focus on Wilson loops; i.e., we will consider the gauge-invariant observables obtained by taking the trace, in some representation R of G, of a Wilson line defined on a closed path Γ, For simplicity, we will consider only non-self-intersecting Wilson loops. We will also mainly focus on Wilson loops annihilated by Q, although what we will say will extend straightforwardly to those annihilated byQ. The fact that the field strength F associated with the supersymmetric connection A is Q-exact has the important consequence that a Wilson loop is invariant under a smooth deformation of Γ. It is possible to show, in fact, that two Wilson loops whose paths are homotopically equivalent are Q-cohomologous. We will prove this by showing that their difference is a Q-exact quantity.
We start by considering the Wilson line defined on a homotopy of paths Γðs; tÞ, where s and t parametrize, respectively, the homotopy and the path. We also introduce the components Using the variation formula Notice that if we are dealing with a closed loop, i.e., if Γðs; 0Þ ¼ Γðs; 1Þ, the first term in the right-hand side of (44) becomes a commutator. Therefore It follows from (19) that Likewise, the variation of the Wilson loop annihilated bỹ Q isQ-exact and reads For a finite homotopy one can integrate both sides of (46) and show that the difference between two homotopic Wilson loops is indeed Q-exact. Crucially, in extending our argument to a finite deformation of Γ, one should be careful to avoid singularities of (46), at which our argument breaks down. These singularities come from the zeros of ϵ þ and ϵ − , which are also the sources of singularities for f ϵ and f ϵ . In general, one should consider not just the homotopy of M but rather the homotopy of M with punctures corresponding to the points in which the components of ϵ vanish.
This property is already important at genus zero. In fact, because of homotopic invariance, one might be tempted to conclude that, at genus zero, all Wilson loops are necessarily trivial. However, while this is true for the Euclidean plane, this turns out not to be the case for the squashed sphere, despite the fact that π 1 ðS 2 Þ ¼ 0. This is precisely because, as noted in Sec. II D, for our choice of supercharges, the Killing spinors are singular at the north and south poles. When these two points are removed, different contours, which are homotopically inequivalent, lead to inequivalent Wilson loops as depicted in Fig. 1.

IV. LOCALIZATION AT GENUS ZERO
The aim of this section is to find an exact expression for supersymmetric Wilson loops on the squashed sphere, and their correlators. Exact results for N ¼ ð2; 2Þ gauge theories have been derived in recent years through supersymmetric localization [53,54,63]. All these results are obtained with a choice of localizing supercharge Q which is a combination of Q andQ. Since a generic Wilson loop defined as in either (11) or (12) is only annihilated by one chiral supercharge, one cannot directly apply the recipe of [53,54,63] to compute the expectation value of such a Wilson loop. However, the conclusions of the previous section come to the rescue, as one can use the invariance under homotopy to map a generic Wilson loop to a local 1 2 -BPS operator. If we consider, for instance, a Wilson loop of the kind depicted in Fig. 1(a), one can compute its expectation value by simply considering the associated local operator inserted at the pole obtained by shrinking the contour as in (37). Specifically, for a Wilson loop annihilated by Q ¼ ϵ a Q and wrapping the north pole anticlockwise, one finds A change in orientation will result in a simple sign flip in the exponent. In a similar fashion, one can recast a correlator of n Wilson loops annihilated by the same Q as where [n] is the homotopy class of paths that wind n times around the north pole. What is crucial for the success of this approach is that these local operators are annihilated by two supercharges, as noted in Sec. II D. In particular, they are annihilated by Q, and as such, are amenable to localization. Here we get to the final result by effectively using two cohomological arguments. The first, with respect to the supercharge Q, uses the invariance under a variation of the homotopy parameter s to map any Wilson loop to a 1 2 -BPS local operator. The second, with respect to the supercharge Q, uses the invariance under a smooth variation of the gauge coupling (and possibly other parameters) to reduce the path integral to a Coulomb-branch matrix model as in [53,54,63]. We will now briefly summarize the setup for the computation.
Let us consider a theory of a vector multiplet and chiral multiplets, with gauge group G. The Lagrangian for the vector multiplet is given by For every generator in the center of g, we can add a topological and a Fayet-Iliopoulus term, namely In the following we will rely on the presence of such a term. The chiral multiplet has components ðϕ; ψ; FÞ, while the antichiral has components ðφ;ψ;FÞ. The Lagrangian for the matter content is This can be complemented by the introduction of superpotential interactions L pot . The choice of the superpotential will determine the R-charge assignments for the matter fields. Finally, if the theory has some flavor group G F , we can gauge it by introducing a nondynamical vector multiplet and turning on background values for its bosonic component fields along the Cartan [53,54,65]. These will introduce twisted masses τ i =r and background monopole charges n i , where i ¼ 1; …; rkG F . It is possible to localize this theory [53,54] with respect to a charge In Coulomb-branch localization one uses the fact that L vec , L mat and L pot are all Q-exact. This implies that the expectation value of Q-closed observables will not depend on the couplings appearing in the action for the vector and the chiral multiplet, or on parameters in the superpotential. The result will depend, instead, on the parameters ϑ and ξ in the Fayet-Iliopoulus action and on the background flavor gauge multiplet through τ i and n i . The BPS locus is spanned by the field configurations aligned with the Cartan. Here y ∈ R rkG , m ∈ Z rkG are monopole charges, and κ is chosen to be þ1 in a coordinate patch that covers the north pole, and −1 in one covering the south pole. The one-loop determinant associated with the vector multiplet reads where the product is over the roots α of G. Then consider a chiral multiplet with R-charge q in a representation R mat of G and in a representation of G F with charges h i . Its oneloop determinant is given by Z mat ðm; y; n; τÞ where the product is over the weight ρ of R. The contribution associated with the classical action comes from L FI . This, evaluated on the locus, gives When putting things together, Coulomb-branch localization gives rise to the matrix model dy r 2π Z cl ðm; yÞZ vec ðm; yÞZ mat ðm; y; n; τÞ: One can compute the expectation value of a Q-closed operator O through the above matrix model simply by considering the insertion of O evaluated on the BPS locus (54). The expectation value in (48), in particular, corresponds to the insertion of In the following we will consider theories with unitary gauge groups.

A. U(1) gauge theory with matter
We will start by considering a theory of gauge group U (1) with the N f chiral multiplets of charge þ1 and twisted masses τ f , N a chiral multiplets of charge −1 and twisted massesτ a , and vanishing background flavor fluxes. The matrix model resulting from Coulomb-branch localization reads Z Uð1Þ ðξ; ϑ; τ;τÞ ¼ The R-charge contributions can be reabsorbed by giving an imaginary part to the twisted masses, as can be seen in (56). Without loss of generality we will assume that N f > N a , or that N f ¼ N a and ξ > 0. The other cases can be obtained by charge conjugation.
The issue about the convergence of the above matrix model with the insertion of the Wilson loop operator can be simply addressed by using the fact that the partition function is analytic in ξ and ϑ (see [54]). In the Abelian case, then, the insertion of the local operator (59) corresponds to a shift in the parameters ϑ and m. In particular, where Λ is the weight of R, i.e., an integer labeling the Wilson-loop representation. The integral representation in (60) can be recast into the "Higgs-branch" formula [53,54] Z Uð1Þ ðξ; ϑ; τ;τÞ where we set z ¼ e −2πξ−iϑ andz ¼ e −2πξþiϑ , and the functional determinants can be written in terms of hypergeometric functions as where M l f ¼ τ f − τ l andM l a ¼τ f þτ l .

B. U(N) gauge theory with matter
We now want to generalize the result above to the case of a UðNÞ gauge group with matter in the fundamental and antifundamental representations. To avoid supersymmetry breaking we consider theories with N ≤ N f . Coulomb branch localization leads to the matrix model Z UðNÞ ðξ; ϑ; τ;τÞ We notice that it is possible to obtain Z UðNÞ by acting with a differential operator Δ on N copies of Z Uð1Þ . Namely, Similarly, the insertion of (59) is obtained by evaluating Following [54], we introduce new coordinates and their complex conjugatesw r . For the partition function, we have Z UðNÞ ðξ; ϑ; τ;τÞ ¼ where l ¼ ðl 1 ; …; l N Þ is a combination C of N elements out of N f , and the functional determinants are here we use ∂ r ¼ ∂=∂w r and∂ r ¼ ∂=∂w r . Notice that the vortex and the antivortex functional determinants can be written in a nicer form as where jkj ¼ P r k r , andz r ¼ z p with p such that l p ¼ r. The computation of the operator insertion is similar: with the new variables w andw, we have The last line of (77) suppresses the l's containing repeated indices. As before, this gives a sum over l ∈ CðN f ; NÞ. Keeping into account that e i P r w r τ l r j w r ¼w−2πiΛ r ¼ e 2πΛðτ l Þ e iw P r τ l r ; ð78Þ we finally arrive at the expression where with ⟪O⟫ we denote the insertion of O in the Higgsbranch formula (71). Here, we have defined x i ¼ e 2πτ i . We notice that an analogous conclusion could be obtained, perhaps more directly, by using Higgs branch localization as in [53,54]. In this case, the localization locus for the bottom components of the vector multiplet is given by Interestingly, in order to perform Higgs branch localization, one has to assume the presence of a Fayet-Iliopoulus term, which in our case is crucial for the convergence of the matrix model. When N ¼ N f there is only one l in the sum, and the result takes the simple form At this point we have everything in place to address the original aim of this section, namely, to give a quantitative description of the correlator in (49). Because of the isomorphism between the representation ring and the character ring of any compact G, a correlator can always be recast as a single Wilson loop insertion associated with a product of representations. These are the representations (or their conjugates, according to the path orientation) of homotopically nontrivial Γ's in the correlator. Namely, Again, for N ¼ N f , one simply obtains V. DUALITIES Two-dimensional N ¼ ð2; 2Þ gauge theories enjoy a set of dualities [55,56] which are reminiscent of Seiberg duality in four dimensions [57]. For models with unitary gauge groups, the ones we are interested in, the duality, suggested by the brane construction in [66], relates a UðNÞ theory with N f > N chiral multiplets in the fundamental representation with a UðN f − NÞ theory with the same matter content, under the following identification of parameters: In the absence of antifundamental multiplets, the flavor symmetry group G F is SUðN f Þ, which implies that P f τ f ¼ 0 and, in turn, that Q f x f ¼ 1. In this section we provide additional evidence for such dualities by extending their dictionary with the supersymmetric Wilson loops defined in the present work. What we find is that a single Wilson loop in a given representation of UðNÞ is mapped, under duality, to a linear combination of Wilson loops in different representations of UðN f − NÞ, similarly to what was found in [59] for theories in three dimensions.
Without loss of generality, we will write down the duality map for a single Wilson loop in an irreducible representation of UðNÞ. An irreducible representation R is uniquely determined by its highest weight λ, so in the remainder of this section we will use the two interchangeably. We refer the reader to Appendix C, where we provide details about UðNÞ characters along with relevant mathematical identities.
In the Higgs branch formula (71), one has to sum over combinations l. Every l ∈ CðN f ; NÞ has a natural dual l D ∈ CðN f ; N f − NÞ, such that l ∩ l D ¼ ∅. Indeed, as proven in [53,54], the duality is realized in (71) at the level of individual terms in the sum, where a term labeled by a certain l is equal to the term of the dual theory labeled by the dual l D , with the correct identification of parameters as in (84).
As it turns out, this property also holds when extending the duality to Wilson loops. In fact, we can construct the dictionary for such operators by matching, term by term in the sum, both sides of the duality. Explicitly, one starts from the identity and finds, after applying the map (92) that prescribes The coefficients c μ are symmetric in all x's. This, in particular, means that they do not carry a dependence on l and, as such, are taken out of the sum over l in the Higgsbranch formula. The duality dictionary is fully specified by the explicit expression of the coefficients c μ . The algorithm to determine the c μ 's consists of three main steps, whose technical details are given in Appendixes C 1, C 2 and C 3, respectively.
In the first step one decomposes the character χ UðNÞ λ as a linear combination of power sums. Particular care is needed when λ contains some negative entries, and one cannot straightforwardly apply the Frobenius formula. In the second step one manipulates the power sums so that they either depend on the x's of the dual combinations l D or on all x's. Finally, in the third step we decompose all the power sums as UðN f Þ and UðN f − NÞ characters. The former will recombine to form the coefficients c μ .
The discussion so far has been somewhat abstract, so we now look at a concrete example for a simple but nontrivial case. We consider a Wilson loop in the adjoint representation of Uð2Þ with N f ¼ 3. For ease of notation we fix l ¼ f1; 2g and l D ¼ f3g.
The dictionary for conjugate representations can be obtained by inverting all x's.
It is immediate to check that the above maps are indeed involutions.
One might be puzzled by the fact that the above are written in terms of UðN f Þ characters, while G F is actually SUðN f Þ. However, since, as mentioned above, the sum of all τ's is vanishing, every χ UðN f Þ is secretly a G F character as well.
The maps in (92) suggest that a more direct interpretation on the duality could be obtained by considering Wilson loops associated with supersymmetric connections including matter fields. Operators of this kind have been considered in relation to theories with boundaries [67,68]. Also, correspondence of such boundary supersymmetric connections under Seiberg-like dualities has been studied in [69,70].

ACKNOWLEDGMENTS
It is a pleasure to thank Lorenzo Bianchi, Giulio Bonelli, Matthew Buican, Kimyeong Lee, Michelangelo Preti, Diego Rodriguez-Gomez, Dario Rosa, Antonio Sciarappa, Alessandro Tanzini, and Itamar Yaakov for useful discussions. We thank, in particular, Luca Griguolo for reading the manuscript and providing suggestions, and Kentaro Hori for many useful comments. The initial stage of this work was carried out at the Galileo Galilei Institute in Florence during the workshop Supersymmetric Quantum Field Theories in the Non-perturbative Regime. R. P. is grateful for the hospitality of the Kavli Institute for the Physics and Mathematics of the Universe in Tokyo.
APPENDIX A: GEOMETRY

Flat-space conventions
In two-dimensional Euclidean space we introduce Dirac spinors In this representation, the Clifford algebra is generated by gamma matrices They obey the identity where ε 12 ¼ 1 and γ 3 is the chirality matrix given by The charge conjugation matrix defines the invariant Majorana product Since it follows that, for anticommuting spinors, ψγ i 1 …γ i n χ ¼ ð−1Þ n χγ i n …γ i 1 ψ; where the i's run from 1 to 3. We also define the chiral projectors 2. Curved-space conventions Frame indices (a ¼ 1, 2) are denoted with a sans serif font and always appear as raised indices. The metric tensor g is written in terms of a zweibein e a as g ¼ e a ⊗ e a ; ðA10Þ while the spin connection ω satisfies de a þ ω ab ∧ e b ¼ 0: The action of the covariant derivative on spinors is defined as APPENDIX B: SUPERSYMMETRY The Killing spinors ϵ andε generating rigid supersymmetry are defined as Grassmann even and satisfy the Killing spinor equations (2).
Since we are working in Euclidean signature, the supersymmetry algebra and all the fields are complexified. In quantizing a theory, one specifies reality conditions for all bosonic fields. Spinor fields are defined as complex Dirac spinors, and their product is taken as in (A6), where the complex-conjugate components do not appear.

APPENDIX C: CHARACTERS
Irreducible representations of UðNÞ are labeled by a set of N integers which form the highest weight λ of the representation.
The character of such a representation is given by When all λ's are non-negative the character reduces to a Schur polynomial s λ ðx 1 ; …; x N Þ.
Given an integer k we define the power sum Analogously, for any set of ordered integers λ, we define