Exact Results for 't Hooft Loops in Gauge Theories on S^4

The path integral of a general N=2 supersymmetric gauge theory on S^4 is exactly evaluated in the presence of a supersymmetric 't Hooft loop operator. The result we find - obtained using localization techniques - captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S^4. As a by-product, our gauge theory calculations successfully confirm the predictions made for 't Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.


Introduction
Supersymmetry -apart from being phenomenologically appealing for physics beyond the standard model -is a powerful symmetry which constraints the dynamics of gauge theories. Investigations of supersymmetric gauge theories have yielded important physical (and mathematical) insights and serve as calculable models for the rich dynamics of four dimensional gauge theories. For instance, the exact low energy effective action of N = 2 super Yang-Mills constructed by Seiberg and Witten [1] provides an elegant physical realization of quark confinement in terms of the dual Meissner effect, via the condensation of magnetic monopoles.
The correlation functions of gauge invariant operators in supersymmetric gauge theories -despite enjoying more controlled dynamics in comparison to QCD -are highly non-trivial to calculate. Even for supersymmetric observables, which preserve some of the symmetries of the theory, generic correlation functions have perturbative corrections to arbitrary loop order as well as non-perturbative instanton corrections. Only in the past few years, exact calculations for the correlation functions of some supersymmetric operators started to be available. An important early step in this recent development was the calculation of the exact partition function of physical N = 2 gauge theories on S 4 and of the expectation value of supersymmetric Wilson loop operators in these theories [2]. Likewise, the computation of certain supersymmetric domain walls in N = 2 gauge theories on S 4 -such as Janus and duality walls -were presented in [3] (see also [4]).
Some of the most basic observables of four dimensional gauge theories are loop operators. These operators can be classified according to whether the loop operator is electric or magnetic, giving rise to Wilson and 't Hooft operators respectively. Gauge theory loop operators -which are supported on curves in spacetime -are order parameters for the phases that a gauge theory can exhibit, and serve as probes of the quantum dynamics of gauge theories. Loop operators are also the most basic observables on which S-duality is conjectured to act in supersymmetric gauge theories (or certain nonsupersymmetric lattice models), and therefore are ideal probes of this remarkable symmetry exhibited by some supersymmetric gauge theories and M-theory. Calculating these observables exactly allows for a quantitative study of S-duality and serves as a theoretical playground for gaining a deeper understanding of the inner workings of dualities.
In this paper we evaluate the exact path integral which computes the expectation value of supersymmetric 't Hooft loop operators in an arbitrary N = 2 supersymmetric gauge theory on S 4 admitting a Lagrangian description. The expectation value of 't Hooft loop operators -originally introduced [5] to probe the phase structure of gauge theories -are calculated by explicit evaluation of the path integral using localization [6]. In the localization framework, the path integral is one-loop exact with respect to an effective -parameter, but nevertheless the computation yields the exact result with respect to the gauge theory coupling constant We find that for an N = 2 gauge theory in S 4 , the expectation value of a supersymmetric 't Hooft operator carrying magnetic charge labeled by a coweight 4 B of the gauge group G takes the form (1.1) The integral is over the Cartan subalgebra of the gauge group. The coweight B of G can be identified with the highest weight for a representation of the Langlands (or GNO [7]) dual group L G. 5 The sum is then over the coweights v of G such that their corresponding weights of L G appear in the representation specified by B.
The path integral in the localization computation receives contributions which localize to the north and south poles of S 4 as well as to the equator, where the 't Hooft operator is supported. Each factor has an elegant interpretation as arising from specific field configurations in the effective path integral arising in the localization computation. The magic of localization is that it restricts the integral over the space of all field configurations to the submanifold of field configurations invariant under a fermionic symmetry Q, which also preserves the supersymmetric 't Hooft operator. These field configurations are solutions to the localization saddle point equations. Integrating out the fluctuations around each of the saddle points and summing over them in the path integral yield the exact result for the expectation value of the 't Hooft loop operator.
The north pole factor captures the effects of point-like instantons while the south pole one incorporates the contributions of point-like anti-instantons. These configurations are the solutions to the localization saddle point equations at the north and south poles of S 4 , given by F + = 0 and F − = 0 respectively. The result of summing over these saddle points can be written in terms of Nekrasov's instanton partition function [8] of the corresponding N = 2 theory in R 4 (more precisely in the Ω-background), with arguments depending on the effective magnetic charge v with Z cl , Z 1-loop,pole and Z inst given in (4.10,6.32,5.2). The parameters m f are the masses of the hypermultiplets in the N = 2 gauge theory, r is the radius of S 4 and q = exp(2πiτ ), where τ is the gauge theory coupling constant 6 τ = θ 2π + 4πi g 2 .
A crucial new contribution to the 't Hooft loop expectation value arises from the equator of S 4 , where the localization saddle point equations are the Bogomolny equations DΦ = * F . Z equator (B, v) captures the contribution to the path integral of field configurations which are solutions to the Bogomolny equations in the presence of a singular monopole background labeled by the magnetic charge B, created by the 't Hooft loop operator insertion. The sum over v in (1.1) appears due to the physics of monopole screening, whereby smooth non-abelian monopole field configurations screen the charge B of the singular mononopole down to an effective magnetic charge v. In the path integral we must sum over all possible effective magnetic charges labeled by coweights v, which are attainable given a singular monopole of magnetic charge B. 7 The L G-weights corresponding to v precisely span the weights of the representation of L G for which B corresponds to the highest weight. 8 The equatorial contribution is where Z 1-loop,eq is given in (6.59) and Z mono in section 7. Combining all the various contributions produces the exact expectation value for the supersymmetric 't Hooft loop operator in N = 2 gauge theories on S 4 . Our gauge theory computations are in elegant agreement with the conjectures and calculations in [10][11][12] for 't Hooft operators in certain N = 2 gauge theories using topological defect operators in two dimensional nonrational conformal field theory. In these papers, gauge theory loop operators in N = 2 gauge theories were identified with loop operators (topological webs more generically) in two dimensional Liouville/Toda conformal field theory, and some correlation functions were explicitly calculated. The Liouville/Toda conformal field theory computations are shown to capture in detail all the features of our gauge theory computation, thereby establishing the proposal put forward in [10][11][12].
The localization calculation performed in this paper is the first example of an exact computation of a path integral in the presence of a genuine singularity due to a disorder operator -an operator characterized by the singularities induced on the fields -and of which a 't Hooft operator is a prime example. 9 In order to treat precisely the fluctuations around the singular field configuration, we employ the mathematical correspondence between singular monopoles in three dimensions and U(1)-invariant instantons in four dimensions [14]. This turns out to be a particularly clean way to carry out the relevant index calculations.
The plan of the rest of the paper is as follows. Section 2 briefly introduces the key ingredients that will be needed to perform the localization computation of 't Hooft operators in N = 2 gauge theories on S 4 . In section 3 we derive the localization saddle point equations relevant for the localization computation, demonstrate that these equations interpolate between the anti-self-duality, self-duality and Bogomolny equations at the north pole, south pole and equator respectively, and find the most general non-singular solution to these equations. This section also describes the singular field configuration produced by the supersymmetric 't Hooft operator as well as the symmetries of the theory used to carry out the localization computation. Section 4 contains the calculation of the classical contribution of the 't Hooft loop path integral, which includes a discussion of the relevant boundary terms. In this section we demonstrate that the classical result can be factored into a contribution arising from the north pole and one from the south pole. Section 5 computes the contribution due to the singular solutions to the saddle point equations arising at the north and south poles, described by pointlike instantons and anti-instantons. In section 6 we calculate the localization one-loop determinants arising from the north and south poles of S 4 as well as from the equator. Section 7 describes the effect of monopole screening in the study of the equatorial Bogomolny equations and explains how to calculate the contribution to the 't Hooft loop expectation value due to screening. In section 8 we compare our gauge theory results with the Liouville/Toda computations conjectured to capture 't Hooft operators in certain N = 2 gauge theores. We finish with conclusions in section 9. The Appendices contain some technical details and computations 2 N = 2 Gauge Theories in S 4 and Localization In this section we introduce the main ingredients of the localization analysis in [2] that we require to calculate the exact expectation value of supersymmetric 't Hooft operators in an arbitrary four dimensional N = 2 gauge theory on S 4 admitting a Lagrangian description. 10 Such a theory is completely characterized by the choice of a gauge group G and of a representation R of G under which the N = 2 hypermultiplet transforms, the N = 2 vectormultiplet transforming in the adjoint representation of G. This includes gauge theories with several gauge group factors and multiple matter representations by letting G be the product of several gauge groups and by taking R to be a reducible representation of G. It therefore applies to any gauge theory with a Lagrangian description.
In this notation, the usual complex scalar field of the N = 2 vectormultiplet is constructed out of the real fields Φ 0 and Φ 9 . One complication in the construction of the N = 2 Lagrangian in S 4 overcome in [2] was to turn on in a supersymmetric way mass parameters for the flavour symmetries associated to the hypermultiplet. These N = 2 gauge theories on S 4 are invariant under the superalgebra OSp(2|4), where Sp(4) ≃ SO(5) is the isometry group on S 4 and SO(2) R is a subgroup of the SU(2) R-symmetry of the corresponding N = 2 gauge theory in flat spacetime.
The key idea behind localization [6] exploits that the path integral -possibly enriched with any observables invariant under the action of a supercharge Q -is unchanged upon deforming the supersymmetric Lagrangian of the theory by a Q-exact term L → L + t Q · V . (2.1) The restriction on the choice of V is such that if Q 2 generates a symmetry and a gauge transformation, as will be the case in our analysis, then V must be gauge invariant and also invariant under the action of the symmetry. Also we require the path integral to be still convergent after the deformation, and that the contribution from the boundary in the space of fields vanishes. In order to localize the gauge fixed path integral, the supersymmetry generated by Q must be realized off-shell, and a gauge fixing procedure must be implemented. This was accomplished in [2] by introducing suitable auxiliary fields and a ghost multiplet, which plays a key role in precisely determining the measure of integration of the fluctuations.
Since the path integral is independent of t, we can study it in the t → ∞ limit. In this limit the saddle points of the path integral are the solutions to the localization equations, which are the saddle points of the deformed action Q · V . In this limit, the path integral becomes one-loop exact with respect to the effective = 1/t parameter and can be evaluated by summing over all saddle points. Therefore, it can be calculated by evaluating the original Lagrangian L on the saddle points and by integrating out the quadratic fluctuations of all the fields in the Lagrangian deformation Q · V expanded around the solutions to the saddle point equations. 11 Of course, even though the path integral is one-loop exact with respect to t, it yields results to all orders in perturbation theory with respect to the original gauge coupling constant τ of the theory. This underlies the power of localization. In favorable situations, for a judicious choice of V , the deformation freezes out most of the fields that must be integrated over in the path integral, thus yielding a path integral for a reduced model, with fewer degrees of freedom.
In the analysis in [2], as well as in our analysis, it suffices to single out a single supersymmetry generator Q of the OSp(2|4) symmetry algebra present in any N = 2 gauge theory on S 4 . This supercharge generates an SU(1|1) subalgebra of OSp(2|4), given explicitly by J is the generator of a U(1) J subgroup of the SO(5) isometry group of the S 4 while R is the SO(2) R ≃ U(1) R symmetry generator in OSp(2|4). If we represent the S 4 of radius r by the embedding equation then J acts as follows We note that the action of J has two antipodal fixed points on S 4 , which can be used to define the north and south pole of S 4 . The U(1) symmetry associated to J + R will be denoted by We conclude this section by mentioning a property of the localization equations that we will exploit in the following section when studying the N = 2 gauge theory path integral on S 4 in the presence of a supersymmetric 't Hooft loop operator. The deformation term Q · V that we add to the action naturally splits into two pieces, one giving rise to localization equations for the vectormultiplet and one for the hypermultiplet. In formulas where Ψ and χ are the fermions in the vectormultiplet and hypermultiplet respectively. We represent the fermion fields in the N = 2 gauge theory by sixteen component, ten dimensional Weyl spinors of Spin(10) subject to the projection conditions (see appendix A for spinor notations and conventions) Since the bosonic part of deformed action Q · V -given by Tr(|Q · Ψ| 2 ) + Tr(|Q · χ| 2 ) -is positive definite, the saddle point equations are As shown in [2], the only solution of the saddle point equations forces all the fields in the hypermultiplet to vanish. 12 Therefore, we are left to analyze the non-trivial saddle point equations for the vectormultiplet fields 13 where A m ≡ (A µ , Φ A ) = (A µ , Φ 9 , Φ 0 ) and K j ≡ (K 1 , K 2 , K 3 ) are the propagating bosonic fields and three auxiliary fields of the N = 2 vectormultiplet respectively. Therefore in our conventions m = 1, 2, 3, 4, 9, 0, while µ = 1, 2, 3, 4 and A = 9, 0. ǫ Q is the conformal Killing spinor that parametrizes the supersymmetry transformation generated by the supercharge Q.
The equations (2.9) are Weyl invariant. That is Q · Ψ = 0 is invariant under the Weyl transformation (2.10) We will use this symmetry to study (2.9) in a Weyl frame where the localization equations take a simpler form.

't Hooft Loop in S 4 and Localization Equations
In this section we initiate our study of the expectation value of a supersymmetric 't Hooft loop operator [5] in an arbitrary N = 2 gauge theory in S 4 . We start by constructing a supersymmetric 't Hooft loop operator which is annihilated by Q (and therefore by J + R). This implies that we can localize the 't Hooft loop path integral using the supercharge Q.
The derivation and interpretation of the localization saddle point equations Q · Ψ = 0 in (2.9) follow. We will then find the most general non-singular solution to the localization equations in the presence of a supersymmetric 't Hooft loop operator. A 't Hooft loop operator inserts a Dirac monopole into (an arbitrary) spacetime. The operator has support on the loop/curve spanned by the wordline of the monopole. In an arbitrary gauge theory, the operator is characterized by a boundary condition near the support of the loop operator that specifies the magnetic flux created by the monopole. Since the choice of 't Hooft operator depends on the embedding of the U(1) gauge group of a Dirac monopole into the gauge group G, these operators are labeled by a coweight or magnetic weight vector B, which takes values in the coweight lattice Λ cw of the gauge group G [7].
Locally, near the location of any point on the loop -where the loop is locally a straight line -the 't Hooft operator creates quantized magnetic flux [16] where x i for i = 1, 2, 3 denote the three local transverse coordinates to any point in the loop. Since B ≡ B a H a ∈ t takes values in the Cartan subalgebra t of the Lie algebra g of the gauge group G, the magnetic flux (3.1) is abelian. Locally, this operator inserts quantized flux through the S 2 that surrounds any point in the loop In order to be able to apply localization we must consider supersymmetric 't Hooft loop operators invariant under the action of Q. These operators create a local singularity on the scalar fields of the N = 2 vectormultiplet. The singularity which will be locally compatible with our choice of Q is 14 A 't Hooft loop operator which is globally annihilated by Q and J +R, can be constructed by choosing -without loss of generality -the support of the 't Hooft operator to be the maximal circle on S 4 4) which is located at the equator of S 4 and left invariant by the action of J (see (2.4)). We find it convenient to study the localization equations Q · Ψ = 0 in (2.9) in the presence of the circular 't Hooft loop by choosing the following coordinates on S 4 (see appendix C for various useful coordinate systems) The coordinates x i , where | x| 2 ≤ 4r 2 , define a three-ball B 3 . In these coordinates, the support of the circular 't Hooft loop (3.4) is the maximal circle parametrized by the coordinate τ located at x i = 0. In these coordinates the action of J, defined in (2.4), is (see equation (C.15)) Therefore, the north and south poles of the S 4 -the fixed points of the action of J -are located at x = (0, 0, 2r) and x = (0, 0, −2r) respectively. Now by using the invariance of the saddle point equations (2.9) under the Weyl transformation (2.10), the solutions to the saddle point equations on S 4 can be obtained from the solutions of the saddle point equations in They are related by the transformation (2.10) with Ω = 1 + | x| 2 4r 2 . One advantage of this choice of Weyl frame is that the exact singularity produced by the circular 't Hooft loop operator in B 3 × S 1 is identical to the one produced by inserting a static point-like monopole in flat spacetime. The exact circular 't Hooft loop background on B 3 × S 1 annihilated by Q when the topological angle vanishes -that is when θ = 0 -is given by 15 When the topological angle is non-trivial -that is when θ = 0 -then the particle inserted by the 't Hooft operator is a dyon, which acquires electric charge through the Witten effect [17]. If the 't Hooft operator is labeled by a magnetic weight B, the induced electric weight is g 2 θB/4π. Moreover, the scalar field Φ 0 also acquires a singularity near the loop. The exact background created by a supersymmetric 't Hooft loop on B 3 × S 1 is given by 15 The corresponding singularity created by the insertion of the circular 't Hooft loop in S 4 can then be simply obtained by performing the Weyl transformation (2.10) with Ω = 1 + | x| 2 4r 2 .

Symmetries and Fields
We now proceed to determining the partial differential equations for the bosonic fields in the N = 2 vectormultiplet on B 3 × S 1 whose solutions yield the saddle points of the localization path integral upon Weyl transforming them back to S 4 . 16 We first have to choose the supercharge Q with which to localize the 't Hooft loop path integral. The supersymmetry transformations of an N = 2 gauge theory on a four manifold with metric h µν is parametrized by a sixteen component Weyl spinor of Spin(10) which solves the conformal Killing spinor equation 17 subject to the projection Γ 5678 ǫ = −ǫ . (3.11) ǫ is determined in terms of ǫ byǫ = 1 4 Γ µ ∇ µ ǫ. 18 It satisfies where R is the scalar curvature derived from h µν . The equations on B 3 × S 1 that we need to analyze are 19 for an specific choice of ǫ = ǫ Q . Here ǫ is the (commuting) conformal Killing spinor of the N = 2 gauge theory on B 3 × S 1 which parametrizes the supersymmetry transformation generated by the supercharges of the OSp(2|4) symmetry of the N = 2 gauge theory. The 16 As we mentioned earlier, the saddle point equations for the hypermultiplets force the fields in the multiplet to vanish. 17 The theory has maximal number of supersymmetries when the metric is conformally flat. 18 The spinor ǫ is referred to as a conformal Killing spinor because its defining equation ∇ µ ǫ = 1 4Γ µ Γ ν ∇ ν ǫ is invariant under the Weyl transformation (2.10). 19 In our conventions Q acts on a field as a fermionic operator and therefore ǫ is a commuting spinor.
general conformal Killing spinor on B 3 × S 1 -with metric (3.7) -is given by (see appendix A for details and conventions) whereε s andε c are two constant ten dimensional Weyl spinors of opposite ten dimensional chirality obeying Γ 5678ε s = −ε s and Γ 5678ε c = −ε c . We now identify the spinor ǫ Q which parametrizes the supersymmetry transformations of the supercharge Q generating the SU(1|1) subgroup of the OSp(2|4) symmetry of an N = 2 theory in S 4 . This is the supercharge used in our localization analysis. We take the spinor ε s to beε and the spinorε cε Therefore, the conformal Killing spinor associated to Q is given by and has norm ǫ Q ǫ Q = 1 2 1 + | x| 2 4r 2 . 20 We note that the spinor ǫ Q generates one of the unbroken supersymmetries 21 preserved by the circular Wilson loop coupled to the scalar field Φ 0 in the N = 2 vectormultiplet Tr exp supported on the maximal circle at x = 0 in B 3 × S 1 . Therefore Q can be used to localize the path integral in the presence of this Wilson loop operator, as in [2]. 20 Using (2.10) the norm of the spinor on S 4 is therefore 1/2. 21 Which obey (1 + iΓ 4 Γ 0 )ε s = 0 and (1 + iΓ 0Γ4 )ε c = 0.
Given our choice of supercharge Q, we can now calculate Q 2 , that is the symmetries and gauge transformation that Q 2 generates when acting on the fields of the N = 2 gauge theory. Due to the addition of suitable auxiliary fields, this symmetry is realized off-shell, as required for localization.
The spacetime symmetry transformation induced by Q 2 is generated by the Killing vector where eˆi = e i = dx i for i = 1, 2, 3 and e4 = r 1 − | x| 2 4r 2 dτ is a vielbein basis for the metric on B 3 ×S 1 given in (3.7). Therefore, Q 2 yields the infinitesimal U(1) J spacetime transformation (3.6) generated by J.
The operator Q 2 also generates a U(1) R R-symmetry transformation. It acts on the fields of the theory as a U(1) R ⊂ SU(2) R subgroup of the SU(2) R symmetry present when the N = 2 theory is in flat spacetime. Therefore, it acts on the gauginos Ψ in the vectormultiplet and the scalars (q,q † ) in the hypermultiplet. The infinitesimal R-symmetry transformation generated by Q 2 is parametrized by the rotation parameter 22 As advertised, our choice of supercharge Q corresponding to the Killing spinor (3.17) generates an SU(1|1) subalgebra of OSp(2|4) In the presence of N F hypermultiplets transforming in a representation R of G, the N = 2 gauge theory has a flavour symmetry group G F , and the masses m f with f = 1, . . . N F of the hypermultiplets take values in the Cartan subalgebra of the flavour symmetry algebra, which has rank N F . The action of Q 2 on the hypermultiplets fields generates an infinitessimal flavour symmetry transformation with parameters m f , while the flavour symmetry action on vectomultiplet fields is trivial.
Finally, the operator Q 2 further generates a gauge transformation with gauge group G on all the fields in the theory. The gauge transformation is a function of the scalar fields Φ A = (Φ 9 , Φ 0 ) of the N = 2 vectormultiplet. The associated gauge parameter is given by where Explicit calculation using (3.17) and Weyl transforming to S 4 using (2.10) gives 23 This implies that the gauge transformation parameter at the north and south poles of S 4 -which are located at x = (0, 0, ±2r) -are Therefore the gauge transformation acts differently at the north and south poles of the S 4 , which are the fixed points of the action of the U(1) generator J. This observation will have far reaching consequences in our computation of the expectation value of 't Hooft operators in these theories. At the equator of S 4 , on the other hand, we have that In summary, Q 2 acting on the bosonic fields of the N = 2 vectormultiplet -whose localization equations we are after -generates a J + R and a G-gauge transformation that can be encoded in terms of the vector field More explicitly, the action of Q 2 on these fields is 13 Including the action on hypermultiplets, we conclude that the action of Q 2 on all fields in the N = 2 theory defined on B 3 × S 1 generates an U(1) J+R × G × G F transformation.

Localization Equations in
Given our choice of supercharge Q, we can now proceed to finding the saddle point equations (2.9) of the localization path integral where we have used that and ǫ Q is given in (3.17). The equations can be found by projecting (3.29) on a basis of spinors generated by 24 Γ m ǫ Q m = 1, 2, 3, 4, 9 (3.31) We note that the projection equations along Γ 0 ǫ Q can be obtained from a linear combination of the projection equations along (3.31) since the conformal Killing spinor ǫ Q satisfies the linear constraint v m Γ m ǫ Q = 0 , (3.33) with v m given in (3.27).
In order to develop intuition for the saddle point equations, we first study them in the point These equations have a simple interpretation. They describe the Q 2 -invariance equations of the bosonic fields in the N = 2 vectormultiplet (obtained by setting equations (3.28) to zero), which at x = τ = 0 are generated by the vector field v m = (0, 0, 0, 1 r , 0, i) (see equation (3.27)). This captures the combined action of a J and a G-gauge transformation with vector field v µ = (0, 0, 0, 1 r ) and gauge parameter Λ = iΦ 0 respectively. The invariance equation for the scalar field Φ 0 is a linear combination of (3.34), a fact which follows from (3.33).
Projection of (3.29) along Γ 8 j+4 ǫ Q gives three dimensional equations. They are the Bogomolny equations We can move to an arbitrary point τ = 0 at x = 0 by acting on the equations (3.34) (3.35) by the U(1) J transformation generated by J. The invariance equations (3.34) remain the same while the three dimensional equations take the same form (3.35) upon replacing the auxiliary scalar fields K i by rotated ones We can now consider the general equations with x = 0 and τ = 0. The Q 2 -invariance 24 Since the N = 2 theory has eight supercharges, there are eight independent equations. equations, obtained by projecting (3.29) along Γ m ǫ Q , are given by 25 (3.37) The three dimensional equations, which we call deformed monopole equations, are obtained by projecting (3.29) along Γ 8 j+4 ǫ Q . They are given by We note that the equations near the location of the 't Hooft loop -at x = 0 -reduce to the familiar Bogomolny equations in R 3 , thus justifying their name. These equations are a supersymmetric extension of well known equations, which interpolate between F + = 0 at the north pole, the Bogomolny equations at the equator and F − = 0 at the south pole. This concludes our derivation of the saddle point equations of the 't Hooft loop path integral.
In appendix D we explicitly show that the background created by the insertion of a circular 't Hooft loop -given in equations (3.8) (and (3.9) when θ = 0) -is a solution of the localization equations derived in this section. This confirms that we can study the expectation value of a supersymmetric circular 't Hooft loop operator in any N = 2 gauge theory on S 4 by localizing the path integral with our choice of supercharge Q. 25 As already mentioned, the invariance equation for Φ 0 is a linear combinations of these equations.
We can now anticipate some key features in the evaluation of the 't Hooft loop path integral of the N = 2 theory defined on S 4 . As explained earlier, the fields and conformal Killing spinor in B 3 × S 1 and S 4 are related by the Weyl transformation (2.10) with Ω = 1 + | x| 2 4r 2 . We note that the conformal Killing spinor in S 4 -which we denote by ǫ sphere Q has negative/positive four dimensional chirality at the fixed points of the U(1) action of J, denoted as north/south poles of S 4 respectively. In formulas 26 This implies that the path integral for a 't Hooft loop receives contributions from equivariant instantons at the north pole and equivariant anti-instantons at the south pole. These are singular solutions to the localization equations which must be included in the computation of the 't Hooft loop expectation value. The equivariant instanton/anti-instanton partition function in R 4 is captured by the so-called Nekrasov partition function [8], which will play a prominent role in our analysis.
From our expression for the action of Q 2 on the fields (see (3.27)), we find that the U(1) ε 1 × U(1) ε 2 equivariant rotation parameters (ε 1 , ε 2 ) in Nekrasov's partition function [8] at the north and south poles in S 4 are fixed to Here (X 1 , . . . , X 4 ) are the S 4 embedding coordinates (2.3) which parametrize the local R 4 near the north and south poles. As Nekrasov's partition function is for the N = 2 topologically twisted theory in R 4 -which mixes the SU(2) Lorentz with SU(2) R-symmetry generators -the U(1) J+R symmetry generated by Q 2 in the physical theory on S 4 gets identified at the north and south poles with the (U(1) ε 1 × U(1) ε 2 ) diag symmetry in Nekrasov's partition function. 26 The volume form is given by ǫ4123 = 1. 27 Also known as N = 2 gauge theory in the Ω-background [8].
Moreover, it follows from equation (3.27) that the equivariant parameterâ ∈ t for the action of constant G-gauge transformations in R 4 in Nekrasov's partition function is fixed at the north and and south poles of the S 4 to 28 a(N) = iΦ 0 (N) − Φ 9 (N)â(S) = iΦ 0 (S) + Φ 9 (S) (3.44) respectively. Since the 't Hooft loop induces a non-trivial background for the scalar field Φ 9 (3.9), which is non-vanishing at the north and south poles, the instanton/anti-instanton partition function contributions arising from the fixed points of J explicitly depend on the magnetic weight B labeling the 't Hooft operator. We will return to the instanton and anti-instanton contributions to the 't Hooft loop path integral in section 5. Likewise, there are singular solutions to the localization equations arising from the equator in S 4 , where the 't Hooft loop is inserted. As we have shown, near the equator we must consider solutions to the Bogomolny equations in the presence of the singular monopole configuration created by the 't Hooft loop operator. We will consider the contribution of these singular solutions to the saddle point equations in sections 6.3 and 7.
Our next task is to study the non-singular solutions of the localization equations.

Completeness of Solutions
In the evaluation of the 't Hooft loop path integral using localization we must sum over all the saddle points of the localization action Q · V which have a prescribed singularity, induced by insertion of the 't Hooft operator. Therefore, we wish to obtain the most general solution of the localization equations (3.37-3.40) satisfying the appropriate boundary conditions imposed by the presence of the circular 't Hooft loop operator. The boundary condition requires that the solutions to the localization equations approach the background (3.9) near the location of the 't Hooft loop, supported at the equator of S 4 . In this section we obtain the most general non-singular solution to these equations (besides the singularity due to the ' t Hooft operator). Singular solutions to the localization equations, however, will play a central role in our computations. We will discuss singular solutions supported at the north and south poles of the S 4 and their contribution to the expectation value of the 't Hooft loop operator in section 5, while the contribution of the singular solutions supported at the equator will be analyzed in section 7.
In appendix D we show that the field configuration (3.45) 28 Here we note that the value of scalar fields at the north and south poles of B 3 × S 1 and S 4 are related by Φ S 4 = 2Φ B3×S 1 through Weyl rescaling, while at the equator Φ S 4 = Φ B3×S 1 .
solves the saddle point equations Q · Ψ = 0. This field configuration is the 't Hooft loop background (3.9) deformed by a "zeromode" 29 of Φ 0 , which is labeled by a. The auxiliary field K 3 in the N = 2 vectormultiplet is also turned on. Therefore, evaluation of the path integral requires integrating over the "zeromode" a ∈ t, which takes values in the Cartan subalgebra t of the gauge group G.
We will now show that the only solutions to Q · Ψ = 0 which are smooth away from the loop are given by (3.45). For this it suffices to consider the deformed monopole equations, the differential equations (3.38-3.40). We find it more transparent, however, to take instead a projection of the localization equations Q · Ψ = 0 along Γ 9µ ǫ Q . This gives where we have used [Φ 9 , Φ 0 ] = 0, which follows from the imaginary part of the last equation in (3.37). We have also defined three real one-forms w The field strength F = F (r) + iF (i) has real and imaginary parts. The imaginary part is due to the presence of the 't Hooft operator background with θ = 0, while the fluctuating part of the field that we integrate over in the path integral must be real. The imaginary part of equations (3.38) and (3.39) imply that while the imaginary part of (3.37) requires that Therefore, these equations completely determine Φ 0 in terms of the electric field produced by the 't Hooft operator when θ = 0 up to a zeromode, which we parametrize by a in (3.45). Moreover, the imaginary part of (3.40) locks in the value of the auxiliary field K 3 in terms of the zeromode part of Φ 0 . Therefore, the most general solution to the localization equations for the electric field F j4 and the scalar fields Φ 0 , K 1 , K 2 and K 3 is given in (3.45). Now it remains to show that the most general solution to the localization equations for the magnetic field F ij and the scalar field Φ 9 is also given by (3.45). From the real part of (3.46) we obtain We also note that the real part of the Q 2 -invariance equations (3.37) implies that Let us define a 1-formṽ = dx µ v µ /(v ν v ν ) dual to the four-vector v, so that i vṽ = 1. Now, in terms of the redefined gauge fieldÂ the Q 2 -invariance equations (3.50) imply that The scalar field in the S 4 conformal frame is Φ = Φ 9 1 + | x| 2 4r 2 . In the S 4 metric 30 (see appendix C) In this metric, the 1-formṽ is given byṽ = r(dψ + ω), and the redefined gauge field (3.51) iŝ in terms of which equation (3.55) is 30 The orientation is such that the volume form is proportional to dτ dx 1 dx 2 dx 3 ∝ dϑdψvol(S 2 ).
Let us introduce a 1-form λ and a function h as quantities that appear in the background values ofÂ and Φ specified by (3.9): Since the background solves the equation (3.57), λ and h satisfy the relation In order to derive useful identities, we square the left-hand side of the equation (3.57) and integrate it with an appropriate measure: The measure is chosen so that the cross-term becomes a total derivative: The second term in the last line has to vanish because it is a 4-form annihilated by i v . Thus the cross term is a total derivative, and the only potential contribution to its integral is from the equatorial S 1 where the 't Hooft loop is inserted. Let us consider a tubular neighborhood S 1 × B 3 of thickness δ > 0 and denote its boundary by Σ 3 . Then Because the fields must obey the boundary conditions associated to a 't Hooft operator at the equator of S 4 , their values on Σ 3 must approach the background values (3.58), for which the integrand vanishes. Thus the squares in the first line of (3.61) must vanish separately, and we have in particular The boundary condition near the operator then requires that Φ = Bh/(2 sin 2 ϑ) up to a gauge transformation, corresponding to the original 't Hooft operator background we started with. In summary, the most general non-singular solution to the localization equations is the field configuration (3.45).

Classical Contribution
In this section we calculate the classical contribution to the localization path integral computing the expectation value of a supersymmetric 't Hooft loop in an arbitrary N = 2 gauge theory on S 4 . The classical contribution to the path integral is obtained by evaluating the N = 2 gauge theory action on S 4 -including suitable boundary terms -on the saddle point solutions of the localization equations.
Using that the localization equations set to zero the scalar fields in the N = 2 hypermultiplet, the classical contribution to the path integral arises from evaluating the bosonic action of the N = 2 vectormultiplet on S 4 on the Weyl transformed (2.10) saddle point solution (3.45). The relevant part of the N = 2 gauge theory action on S 4 of radius r is given by (4.1) where we have denoted by h µν the S 4 metric and R = 12/r 2 is the scalar curvature. The classical action (4.1) is invariant under the Weyl transformation (2.10). Therefore, we can calculate the classical contribution to the expectation value of the 't Hooft loop by computing the N = 2 gauge theory action on B 3 × S 1 (3.7) evaluated on the background (3.45). The non-topological part of the action is thus 31 .
x is the radial coordinate in B 3 32 The volume form is given by ǫ4 123 = 1.
Explicit computation using the saddle point configuration (3.45) gives The unregulated on-shell action is clearly divergent, as it measures the infinite self-energy of a point-like monopole. This divergence -which is proportional to the length of the curve on which the 't Hooft loop is supported -can be regulated by introducing a cutoff δ in the integration over x, and subtracting terms in the action proportional to 1/δ. This subtraction can be implemented by adding to the action (4.1) covariant boundary terms supported on the x = δ hypersurface Σ 3 . The relevant boundary terms are Evaluating them on the saddle point solution (3.45) we get and (4.7) The terms proportional to 1/δ in the boundary terms cancel the self-energy divergences in the bulk on-shell action in (4.4). Moreover, the on-shell boundary term (4.7) generates a finite contribution, which precisely cancels the corresponding one appearing in the bulk onshell action (4.4). Therefore, the leading classical action for the circular 't Hooft loop in the N = 2 gauge theory is given by The classical action (4.8) can be split into the sum of two terms, which are the complex conjugate of each other This observation leads to an illuminating interpretation. The classical result for the 't Hooft loop path integral on S 4 is captured by the classical contribution to Nekrasov's equivariant instanton and anti-instanton partition functions on R 4 [8] localized at the north and south poles of the S 4 respectively. As we shall see, the classical, one-loop and instanton factors in Nekrasov's equivariant instanton/anti-instanton partition function in R 4 [8] will enter in the computation of the 't Hooft loop on S 4 . We first recall that the classical contribution to the N = 2 equivariant instanton partition function in R 4 -or the partition function of the N = 2 theory in the Ω-background -is given by [8] The constant fieldâ ∈ t is the equivariant parameter for the action of G-gauge transformations on the moduli space of instantons in R 4 , while ε 1 and ε 2 are the equivariant parameters The parameter q = exp (2πiτ ) is the instanton fugacity while q is the fugacity for antiinstantons, where τ is the complexified coupling constant of the N = 2 gauge theory In section 3.2 we have already mentioned that the supercharge Q with which we localize the 't Hooft loop path integral becomes near the north and south poles of the S 4 the supercharge which localizes the equivariant instanton and anti-instanton partition function in R 4 [8] respectively, with the following equivariant parameters (4.14) 33 We have evaluated the scalar field Φ 0 and Φ 9 at the north and south poles of S 4 . From equation (3.45) we find that the value of the field Φ 9 (Φ 0 ) at the north and south poles of B 3 × S 1 , which are located at . Weyl transforming to S 4 using Φ S 4 = 2Φ B3×S 1 , we get the formula (4.14). 34 In section 6.3 we will need the value ofâ at the equator of S 4 . It is given byâ This implies that the classical equivariant instanton/anti-instanton partition functions arising from the north and south poles are given by Therefore, the classical expectation value (4.9) for the 't Hooft loop operator with magnetic weight B in any N = 2 gauge theory on S 4 factorizes into a classical contribution associated to the north and south poles respectively the south pole contribution being the complex conjugate of the north pole one The identification of the integrand of the 't Hooft loop path integral with contributions arising from the north and south poles of S 4 will be a recurrent theme in our computation of the 't Hooft loop expectation value. As we shall see, however, an important contribution also arises from the equator of S 4 .

Instanton Contribution
In the previous section we have calculated the classical contribution to the expectation value of a 't Hooft loop with magnetic weight B on S 4 due to the non-singular solutions of the localization equations (besides the obvious singularity created by the insertion of the 't Hooft operator), which are labeled by a ∈ t (3.45). As discussed earlier, however, there exist singular solutions to the localization equations supported at the north and south poles. In this section we determine their contribution to the 't Hooft loop expectation value.
The localization equations (3.38)-(3.40) at the north and south pole of the S 4 become, respectively, the instanton and anti-instanton equations These equations describe singular field configurations, corresponding to point-like instantons, which are localized at the poles of S 4 . The inclusion of these singular field configurations in the localization computation implies that we must enrich the result in section 4 with the contribution of point-like instantons and anti-instantons arising at the north and south pole respectively. We now identify these contributions and include their effect in the computation of the 't Hooft loop path integral.
Nekrasov's equivariant instanton (anti-instanton) partition function in R 4 [8] computes the contribution of instantons (anti-instantons) to the path integral of an N = 2 gauge theory in the so-called Ω-background. We denote it by [8] where (ε 1 , ε 2 ,â,m) are the equivariant parameters for the U(1) ε 1 ×U(1) ε 2 ×G×G F symmetries of the N = 2 gauge theory.m f with f = 1, . . . , N F denote the equivariant parameters for the flavour symmetry group G F associated to the hypermultiplet and q is the instanton fugacity.
Since the N = 2 gauge theory action on S 4 and Q-complex near the poles reduces to those of the N = 2 gauge theory in the Ω-background, the contribution of the singular field configurations in our localization computation due to point-like instantons and antiinstantons at the north and south poles respectively, are precisely captured by Nekrasov's instanton and anti-instanton partition function.
As we have already mentioned, the Q-complex of the N = 2 theory near the north (south) pole of S 4 reduces to that describing Nekrasov's equivariant instanton (anti-instanton) partition function on R 4 with U(1) ε 1 ×U(1) ε 2 equivariant parameters ε 1 = ε 2 = 1/r. Furthermore, the equivariant parameterâ ∈ t for the action of the gauge group G on the instanton moduli space is given respectively by equations (4.13, 4.14) Therefore, the contribution to the 't Hooft loop expectation arising from the solutions to the F + = 0 equations at the north pole is given by while that due to the solutions of the F − = 0 equations at the south pole is We have used the relationm derived in [18] between the physical mass m f of a hypermultiplet and the equivariant parameterm f in Nekrasov's instanton partition function.
Taking into account the following identity obeyed by the instanton partition function [18,19] Z we find that the anti-instanton south pole contribution is the complex conjugate of the one in the instanton north pole one We can now combine the results of this section with the ones found in the previous one and write down the "classical" contribution to the expectation value of a 't Hooft loop with magnetic weight B. Summing over all saddle points of the localization equations -including both non-singular and singular solutions at the north and south poles -which are labeled by a ∈ t, leads to 35 with Z cl and Z inst given in (4.10) and (5.2) respectively.

One-Loop Determinants
The calculation of a path integral using localization enjoys the drastic simplification of reducing the computation to one-loop order with respect to the deformation parameter t, while being exact with respect to the original gauge theory coupling constant. In this section we calculate the relevant determinants required for computing the expectation value of 't Hooft operators on S 4 . Computation of the one-loop determinants in the N = 2 gauge fixed action is performed by expanding to quadratic order in all field fluctuations -which include vectomultiplet, hypermultiplet and ghost multiplet fields -the deformation term Q ·V around the saddle point configuration background (3.45). In the gauge fixed theory, the supercharge Q combines with the BRST operator Q BRST asQ = Q + Q BRST , such that the deformed action Q · V (2.1) together with gauge fixing terms can be written asQ ·V , withV = V + V ghost [2]. As shown in [2], the saddle points ofQ ·V coincide with those of Q · V , and we can borrow the saddle point configuration in (3.45) for the calculation of the determinants. Direct evaluation of the determinants by diagonalization of the quadratic fluctuation operator in the saddle point background is rather complicated. Instead, we calculate the relevant one-loop determinants using an index theorem. More precisely we use the Atiyah-Singer index theorem for transversally elliptic operators [20], which was also used in [2] to compute the partition function of N = 2 gauge theories on S 4 . Even though we are considering the physical N = 2 gauge theory on S 4 (not a topologically twisted theory), the combined supersymmetry and BRST transformations generated byQ can be written in cohomological form [2]. Fields of opposite statistics are paired into doublets under the action ofQ. Schematically, denoting the fields of even and odd statistics with a subindex e and o respectively, we have that Here R is the generator of the U(1) J+R × G × G F symmetries discussed in section 3.1, corresponding to the group U(1) J+R combining the U(1) J rotation on S 4 (2.4) with an SO(2) R R-symmetry transformation, the G-gauge and the G F flavour symmetries respectively. Therefore,Q acts as an equivariant cohomological operator sincê andQ 2 is nilpotent on R-invariant field configurations. The invariance of the deformation termQ ·V under the action ofQ and the pairing of of the fields as in (6.1) leads to cancellations between bosonic and fermionic fluctuations. The remainder of this cancellation is the following ratio of determinants over non-zeromodes [2] det The differential operators D vm and D hm are obtained from the expansion of the deformation termQ ·V for the vectormultiplet and hypermultiplet fields respectively. Therefore, the one-loop determinants that appear in the localization computation of the partition function of an N = 2 gauge theory on S 4 are given by the product of weights for the group action R generated byQ 2 on the vectormultiplet and hypermultiplet fields. Furthermore, the weights appearing in the determinants (6.3) can be determined from the computation of the R-equivariant index for D = D vm and D = D hm . In order to convert the index (Chern character) ind D in (6.4) into a fluctuation determinant (Euler character), we read off the weights w α (ε 1 , ε 2 ,â, m f ) from the index and combine them to get the determinant according to the rule 36 The relevant R-equivariant indices can then be calculated from the equivariant Atiyah-Singer index theorem for transversally elliptic operators [20], to which we now turn.
The index theorem localizes contributions to the fixed points of the action of R, that is to the north and south poles of S 4 . Therefore, the relevant index corresponds to the equivariant index of the vectormultiplet and hypermultiplet complexes of the N = 2 theory in the Ω-background, to which the N = 2 gauge theory on S 4 reduces at the poles. The presence of a 't Hooft loop, however, introduces a further contribution, arising from the equator, where the operator is supported.

Review of the Atiyah-Singer Equivariant Index Theory
Consider a pair of vector bundles Let T = U(1) n be the maximal torus of a compact Lie group G acting on M and the bundles E i , and let D : V 0 → V 1 be an elliptic differential operator commuting with the G-action. In this situation we can define the G-equivariant index of the operator D as a formal character where H 0 = ker D, H 1 = coker D. If D is elliptic and M is compact, H 0 and H 1 are finite dimensional vector spaces.
The index does not depend on small deformations of the operator D and, therefore, is a topological invariant. If the action of G on M has a discrete set of fixed points, Atiyah and Singer represent the index as a sum over the set of fixed points F Each fixed point contribution to the Atiyah-Singer index formula (6.7) is a rational function in t. For an elliptic operator D on a compact manifold M the sum over all of the fixed point contributions to the index is a finite Laurent polynomial in t = (t 1 , . . . , t n ), since the spaces H i are finite dimensional.
The basic example is the equivariant index of the Dolbeault operator ∂ : Ω 0,0 (C) → Ω 0,1 (C) from the space of functions to the space of (0, 1)-forms on the complex plane M = C under the T = U(1) action z → tz. Computing the index of ind(∂)(t) directly using (6.6) we just need to evaluate the U(1) character on the space of holomorphic functions where the last equality should be understood formally since for |t| = 1 the series does not actually converge.
On the other hand, we can evaluate ind(∂)(t) using the Atiyah-Singer fixed point theorem (6.7). Since there is a single fixed point at z = 0 of the U(1) action, we get 37 thus reproducing the previous computation. The index theory for elliptic operators can be generalized to transversally elliptic operators [20]. Let T be the maximal torus of a Lie group G that acts on a manifold M. An operator D on M is called transversally elliptic with respect to the G action on M if it is elliptic in all directions transversal to the G-orbits on M. As in the elliptic case, the index of D possesses the excision property. Therefore the index can be computed as a sum of local contributions, a sum over the fixed points of the G action. The total index ind D(t) is an infinite formal Laurent series n c n t n with n ∈ Z, since the cohomology spaces H i can be infinite dimensional. However, for each c n , the multiplicity of the representation n in ⊕(−1) i H i , is finite. Atiyah-Singer theory allows us to find c n unambiguously since the theory specifies whether each fixed point contribution is to be expanded in powers of t or t −1 , after choosing a deformation of the symbol for D. 38 In the paper [2], the partition function and the Wilson loop expectation value were computed, with the one-loop contributions evaluated using an index theorem. In the set-up of [2] and the current paper, the manifold is M = S 4 and the spacetime part of the relevant group G = U(1) J+R × G × G F is generated by J (2.4). The differential operators that appear in the quadratic part ofQ ·V fail to be elliptic on the equatorial S 3 , but they are still transversally elliptic and the generalized index theorem can be applied. In [2] the index is a sum of local contributions from the north and south poles of S 4 , which are the fixed points of J.
When we turn on the singular monopole background (3.45), there is an extra complication since some of the fields are singular along the equatorial S 1 where the loop operator is inserted. This gives rise to an extra contribution to the one-loop determinant, associated with the equator of S 4 . We believe that the index theorem for transversally elliptic operators can be generalized to the situation where such singular monopoles are present. A similar index theorem was established in [21] using a relation between singular monopoles and U(1)invariant instantons [14]. Assuming the existence of such an index theorem, we will compute local contributions from the equatorial S 1 , for which there is a natural expansion. The 37 The fiber (E 0 ) z=0 transforms trivially, the fiber ( 38 After summing over fixed points, c n is independent of the choice of deformation.
specific choice of a deformation of the symbol made in [2] led to the expansion in positive and negative powers of t at the north and south poles, respectively. In the presence of a 't Hooft loop we will apply the same deformation, and therefore obtain the same rules for expansion at the north and south poles.

North and South Pole Contributions
We wish to compute the vectormultiplet and hypermultiplet one-loop determinant contributions from the north and south poles, which are the fixed point set of J. The relevant complex for the vectormultiplet calculation is the self-dual complex while for the hypermultiplet it is the Dirac complex. We now consider the associated equivariant indices and one-loop determinants.

Vectormultiplet Determinant
Near the north pole, we consider the complex 39 of vector bundles associated with linearization of the anti-self-dual equation where d is the de Rham differential and d + is the composition of the de Rham differential and self-dual projection operator. We want to compute the equivariant index of D SD with respect the For the moment we take t 1 and t 2 generic though we will set t 1 = t 2 in the end, as U(1) J+R corresponds to (U(1) ε 1 ×U(1) ε 2 ) diag in the self-dual/anti-self-dual complex at the north/south pole. The Atiyah-Singer formula (6.7) for the complexification of (6.11) gives 40 .
The index for the real complex (6.11) is the half of (6.12). Unless there is a further input from the transversally elliptic Atiyah-Singer theory, we can expand the function (6.12) in various ways depending on whether we take |t i | > 1 or |t i | < 1. For example, expanding in positive powers of t 1 , t 2 we get (1 + t 1 t 2 )t n 1 t n 2 , (6.13) 39 The complex (6.11) can be turned into the two-term complex in (6.6) by "folding" the complex as while expanding in negative powers of t 1 , t 2 we get and there are several other available expansions as well.
In order to calculate the one-loop determinant for the N = 2 vectormultiplet, we must consider the self-dual complex (6.11) tensored with the adjoint representation of the gauge group G, and study the U(1) where χ adj (g) is the character of G in the adjoint representation. More explicitly, let us denote t 1 = exp(iε 1 ), t 2 = exp(iε 2 ) and g = exp(iâ), where ε 1 , ε 2 andâ are the elements of the Lie algebra of U(1) ε 1 ×U(1) ε 2 and of the Cartan subalgebra t of G respectively. Denoting by w be the weights of the adjoint representation of G, the index (6.15) can be written as As mentioned earlier, the one-loop determinant in the localization computation of the 't Hooft loop path integral can be computed as the product over all the weights of the generator R of the U(1) J+R × G × G F action on the space of fields (see (6.1)). Mathematically, the product of weights computes the equivariant Euler class of the normal bundle to the fixed point set. The corresponding index or equivariant Chern character determines the one-loop determinant or equivariant Euler character by taking the weighted product of all weights extracted from the exponents in the Chern character (using (6.5)). Therefore, we will calculate the one-loop determinant of the N = 2 vectormultiplet by determining the weights under the action of U(1) J+R × G × G F from the index (6.16). We remove the terms with w = 0 because they are independent ofâ, so that we are only left with the sum over the roots α of g.
with multiplicities 1/2. The one-loop determinant contribution from the north pole of the N = 2 vectormultiplet labeled by a root α of the Lie algebra g is therefore In our localization calculation on S 4 , we must specialize to the values ε 1 = ε 2 = ε = 1/r, which correspond to the U(1) J+R symmetry. The expression is divergent, and we regularize it by identifying it with the Barnes G-function [22] (see for example section 5.17 in [23]). It is an analytic function that has a zero of order n at x = −n for all integers n > 0, and can be defined by the infinite product formula Therefore, the corresponding vectormultiplet one-loop determinant is given by 42 At the other fixed point -at the south pole -we need to consider the anti-self-dual complex and an expansion in negative powers of t 1 and t 2 . However, the index of the anti-self-dual complex at the south pole coincides with the index of the self-dual complex at the north pole. Relative to the north pole, the difference amounts to the sign change (ε 1 , ε 2 ) → (−ε 1 , −ε 2 ), which can be absorbed into the redefinition of roots α → −α, which just exchanges positive and negative roots, yielding once again (6.20).
Therefore, recalling that the equivariant parameters for the G-action at the north and south poles are fixed (4.13, 4.14) we obtain that the vectormultiplet one-loop determinant contributions from the north and south poles are with Z vm 1-loop,pole (â) given in (6.20). Furthermore, the south pole contribution is the complex conjugate of the north pole Z vm south,1-loop = Z vm north,1-loop , (6.23) precisely the same relation that we found earlier for the classical and instanton contributions. 42 For asymptotically free gauge theories see discussion after equation (6.32).
Let us now compare these results with the computation in [2]. In the absence of a 't Hooft loop we haveâ(N) =â(S) = ia, and is precisely the one-loop determinant for the vectormultiplet obtained in [2], up to the ghostsfor-ghosts contributions. The ghosts-for-ghosts were introduced to gauge-fix the constant gauge transformations on S 4 , and they had the effect of removing the Vandermonde α>0 α·â from the one-loop factor, while the square of the Vandermonde reappeared as the volume of the adjoint orbit {gâg −1 |g ∈ G}. In the approach of this paper, we do not introduce ghosts-for-ghosts, and the Vandermonde is included in the one-loop factor (6.20).

Hypermultiplet Determinant
The index of the complex for the Dirac operator D Dirac that maps the space of positivechirality spinors S + to the space of negative-chirality spinors S − in R 4 with a suitable inversion of the grading, computes the contribution of a hypermultiplet to the one-loop determinant [2]. By applying the fixed-point formula (6.7) to the Dirac complex, we obtain 43 The kinetic operator for a hypermultiplet in the adjoint representation of the gauge group and the one-loop factor were analyzed in [2] in detail. The corresponding index is given by tensoring the Dirac bundle with the adjoint bundle. We also need to remember that the G F = SU(2) flavour symmetry associated to an adjoint hypermultiplet acts on the bundle. Therefore the U(1) ε 1 × U(1) ε 2 × G × G F equivariant index for this complex, taking into account the inversion of the grading, is given by e im + e −im 2 w∈adj e iw·â . (6.27) 43 The index can also be obtained by noting that the Dirac complex in C 2 = R 4 is related to the Dolbeault operator ∂ : Ω 0,0 → Ω 0,1 → Ω 0,2 . The bundle S + is given by Ω 0,0 ⊕ Ω 0,2 twisted by K 1/2 while S − is given by Ω 0,1 twisted by K 1/2 , where K is canonical bundle. We want to compute the equivariant index of D Dirac with respect the T = U (1) ε1 × U (1) ε2 action (z 1 , z 2 ) → (t 1 z 1 , t 2 z 2 ). Hence up to the twist by K 1/2 , which contributes a factor of (t 1 t 2 ) 1/2 to the index, the Dirac complex (6.25) is isomorphic to standard Dolbeault complex in C 2 . The factor t We recall that the equivariant parameterm = im for the SU(2) flavour symmetry, which takes values in the SU(2) Cartan subalgebra, is interpreted as the mass m of the adjoint hypermultiplet.
Given the formula for the equivariant index for the hypermultiplet in the adjoint representation, group theory completely determines the corresponding index for an arbitrary representation R of the gauge group. To explain this claim, let us recall that the precise flavour symmetry depends on the type of matter representation, and that in general we need to consider half-hypermultiplets although in the end half-hypermultiplets pair up into full hypermultiplets. For a complex irreducible representation R, half-hypermultiplets always appear as copies of conjugate pairs N F · (R ⊕ R), and the flavour symmetry is U(N F ). Halfhypermultiplets in a real irreducible representation R can only arise in an even number 2N F , in which case the flavour symmetry is enhanced to Sp(2N F ). 44 If the irreducible representation R is pseudo-real, classically an arbitrary number n of half-hypermultiplets can appear with SO(n) as the flavour symmetry group, but for odd n an anomaly renders the theory inconsistent [24]. Thus n = 2N F has to be even and the flavour symmetry is enhanced to SO(2N F ). In every case, the flavour symmetry group acts in the defining representation and there are N F mass parametersm f = im f with f = 1, . . . N F parametrizing the Cartan subalgebra of G F . As shown in appendix E, the following expression for the index holds for an hypemultiplet in an arbitrary representation R of G: At the north and south poles, we expand the index (6.28) in positive and negative powers of (t 1 , t 2 ) respectively, from which we read the weights of the the U(1) ε 1 × U(1) ε 2 × G × G F action. Both expansions give rise to identical one-loop determinants, given in terms of the weights by (6.5).
The relevant hypermultiplet one-loop determinant of the theory on S 4 is obtained by setting ε 1 = ε 2 = ε = 1/r, the G-equivariant parameters at the north and south poles to (6.21) andm f = im f , where m f with f = 1, . . . , N F are the masses of the N F hypermultiplets. Therefore the one-loop determinants of N F massive hypermultiplets in a representation R of G arising from the north and south poles are given by where w are the weights of the representation R. We note that for an arbitrary representation R, the hypermultiplet one-loop determinant at the south pole is the complex conjugate of the determinant at the north pole 45 We can now start gathering the results obtained until now. Combining the vectormultiplet and hypermultiplet determinants given in (6.20) and (6.30), we conclude that the pole contribution to the one-loop determinant for an arbitrary N = 2 Lagrangian theory in S 4 in the presence of a 't Hooft operator can be written in terms of where we recall that ε = 1/r. Formula (6.32) holds for an arbitrary N = 2 gauge theory admitting a Lagrangian description, and can be explicitly calculated given the choice of gauge group G and of a representation R of G under which the hypermultiplet transforms. For asymptotically free gauge theories, the localization calculation is most accurately performed by embedding such a theory into one that is ultraviolet finite, which then flows to the asymptotically free theory upon taking the mass parameters of the finite theory to be very large. As a prototype of this construction, N = 2 pure super Yang-Mills with arbitrary gauge group G can be regulated by embedding it in the N = 2 * theory, consisting of a vectormultiplet and massive hypermultiplet in the adjoint representation of G, by then taking the mass of the hypermultiplet to be very large. This construction exists for an arbitrary asymptotically free four dimensional N = 2 gauge theory. Given an asymptotically free N = 2 gauge theory, the end result of this procedure in the localization computation is that the one-loop determinants are given by (6.32) for the field content of the asymptotically free theory, together with the replacement of the bare coupling constant τ of the theory with the familiar one-loop corrected running coupling constant τ ren .
The complete one-loop determinants in our localization computation arising at the north and south poles are thus given by N F massive hypermultiplets in a representation R of G is given by 46 with Z cl , Z 1-loop,pole and Z inst given in (4.10), (6.32) and (5.2) respectively. Therefore, the path integral completely factorizes into north and south pole contributions as which furthermore are complex conjugate to each other When the gauge theory is asymptotically free, we must replace the bare instanton fugacity q by the renormalized one q ren in Z north and Z south . The ≃ symbol is used in (6.34) and (6.35) since in the presence of a 't Hooft loop operator, an extra contribution supported on the loop must be included, to which we now turn. 47

Equator Contribution
In the absence of a 't Hooft loop, the index is a sum of local contributions from the north and south poles [2], which are the fixed points of J. In defining the 't Hooft loop path integral in gauge theory, we must impose boundary conditions along the loop compatible with the field configuration of a singular monopole. In this subsection we calculate the contributions to the vectormultiplet and hypermultiplet indices as well as one-loop determinants from the equatorial S 1 where the 't Hooft loop is located, which are functions of the weights for the group action U(1) J+R × G × G F generated by Q 2 (also byQ 2 ). Let us recall from section 3.1 that the isometry generator J in Q 2 acts on B 3 × S 1 as a spatial rotation along the x 3 -axis as well as a shift in the periodic coordinate τ . The 46 After trivially shifting the integration variable ia → ia + ig 2 θ B 16π 2 r . 47 In section 7 we will include yet another contribution due to monopole screening, which is non-perturbative in nature.
conformal killing spinor ǫ Q in (3.14) with which we localize the 't Hooft loop path integral can be written as Note that ǫ Q changes its sign when going around the S 1 , under τ → τ +2π. Therefore while all the bosons are periodic, all the fermions in the vielbein basis are antiperiodic. In particular, within each supermultiplet bosons and fermions obey different boundary conditions around S 1 .
Recall that Q 2 also includes the U(1) R transformation (see (3.20)), which is generated by J 56 + J 78 . When we apply the index theorem it is convenient to redefine fields of the theory and ǫ Q using U(1) R as 48 where we have normalized the ten-dimensional Lorentz generators in the vector representation as (J M N ) P Q = δ M P δ N Q − δ M Q δ N P and used that U(1) R is generated by Γ 56 + Γ 78 when acting on spinors. After the field redefinition, the whole vectormultiplet is periodic, and all fields in the hypermultiplet are antiperiodic. 49 This redefinition makes the spinor ǫ Q independent of τ . The shift in τ now induces an R-symmetry transformation in addition to the S 1 part of isometry J.

Vectormultiplet Determinant
As we saw in section 3.2, the localization equations near the location of the 't Hooft operator -which wraps the S 1 at the origin in B 3 -are approximately the Bogomolny equations * 3 F = DΦ (6.40) in B 3 ×S 1 . The differential operator that appears in the kinetic term for the vectormultiplet is obtained by linearizing the Bogomolny equations. Linearization of the gauge transformation and the Bogomolny equations is described by the complex 50 D Bogo : Ω 0 → Ω 1 ⊕ Ω 0 → Ω 2 (6.41) 48 Here we are using ten dimensional notation for the bosonic fields of the N = 2 theory, so that A M = {A µ , q,q, Φ 9 , Φ 0 } with M = 1, . . . , 9, 0. 49 The R-symmetry group U (1) R acts non-trivially on the fermions in the vectormultiplet and on the scalars in the hypermultiplet. 50 This complex can also be turned into a two-term complex as in (6.6) by folding the complex.
in R 3 − {0}. In appendix F, we explain Kronheimer's observation that the Bogomolny equations in R 3 with a monopole singularity at the origin -where the 't Hooft operator resides -is equivalent to the anti-self-duality equations for gauge fields in R 4 invariant under the action of a spacetime symmetry group U(1) K . Using Kronheimer's correspondence, we can obtain this complex by projecting the self-dual complex (6.11) to the U(1) K -invariant sections. We can compute the index of the complex (6.41) by averaging the index of the self-dual complex over the U(1) K action, picking up the contributions only from the U(1) K invariant sections. 51 In equation (6.9), the index for the Dolbeault operator ∂ on C was obtained as the U(1) character on the space of holomorphic functions. In this toy example the index is an infinite power series corresponding to infinitely many monomials. The same logic can be used to derive the index (6.12) for the complex (6.11) through an expansion in a basis of local sections. Among such sections, those which are invariant under U(1) K correspond to the ordinary spherical harmonics for the bundles in three dimensions. We can keep track of the original expansion by introducing an infinitesimal positive parameter δ > 0: .

(6.42)
We now parametrize the U(1) × U(1) weights as where ν is the parameter for the group U(1) K used in Kronheimer's construction: ( The parameter ε is the angle for a rotation along the x 3 -axis in R 3 , and the factors of 1/2 ensure that for ε = 2π this rotation acts as −1 on C 2 even though it acts as +1 on R 3 . In order to describe the singular monopole background due to the 't Hooft operator, we also need to twist by the adjoint gauge bundle on which the gauge group G and U(1) K act as eâ +Bν , with B being the magnetic weight labeling the operator. The four-dimensional sections invariant under U(1) K can be identified with the monopole harmonics [25] of the corresponding bundles over R 3 − {0}. The index for the self-dual complex twisted by the gauge bundle is given by (6.44) 51 A similar computation was done in [21], where more than one singular monopole was considered on a compact manifold. Though our integrand to be averaged is a rational function with poles on the integration contour, the integrand in [21] was a polynomial due to cancellations among singular monopoles.
By averaging over U(1) K , we get the desired index for the complex (6.41) where we have renamed w as w → −w. We can evaluate the integral by summing over residues for the poles inside the unit circle. For w · B > 0 a pole at z = 0 contributes In addition there are always two poles at z = e −δ e iε/2 , e −δ e −iε/2 . In the limit δ → 0, the contribution of the pole at z = e −δ e iε/2 is given by In the last line we replaced the sum over the adjoint weights satisfying w · B > 0 by the sum over positive roots α > 0. This is possible because by taking B to be in the Weyl chamber all such w's are positive roots. For the vectormultiplet one-loop determinant computation, we also need to tensor with the space of periodic functions on S 1 . Thus we need to compute n∈Z e inε ind(D Bogo ). A simplification arises because the parameter n is summed over, and can be shifted by an integer freely. Finally, the equatorial index for the vectormultiplet is Note that we can write the last sum as n∈Z e i(n+α·B/2)ε in both cases. 52 Setting ǫ = 1/r and using that at the equatorâ(E) = ia − ig 2 θ B 16π 2 r (3.26), we find that the equatorial one-loop determinant for the vectormultiplet is given by computed from the index (6.50) using equation (6.5),

Hypermultiplet Determinant
We deal with the hypermultiplet in a similar way. The relevant differential operator is the Dirac operator plus a coupling to the Higgs field Φ 9 . In Kronheimer's correspondence, this lifts simply to the Dirac operator on C 2 given in (6.26). We regularize the index (6.26) by specifying the expansion in a local basis as . (6.53) 52 Physically, half-odd integer coefficients appear in the exponential for odd α · B because the relation between the angular momentum and statistics is reversed when the monopole charge measures in the unit amount is odd [25]. 53 We regulate the product by identifying it with the product representation of the sin function.
We can twist the Dirac complex by a vector bundle whose sections transform in representation R of the gauge group. Including the action of the gauge and flavour groups G × G F as in (6.28), and then averaging over U(1) K , we obtain We noted above that the hypermultiplet fields are antiperiodic in τ . Thus we must tensor with the space of anti-periodic functions on S 1 , and change the sign for the index because we shift the degrees for physical fields in the complex (as we did already for the hypermultiplet contribution at the poles). The equatorial index for the hypermultiplet is thus Therefore, the one-loop determinant contribution from the equator of N F hypermultiplets in a representation R of the gauge group is Z hm equator = Z hm 1-loop,R,eq (ia, im f , B) , Combining the vectormultiplet (6.52) and hypermultiplet (6.57) determinants, the complete equator contribution is given by (6.59) 54 Up to a phase, this expression is valid even if B is not in the Weyl chamber.
We are now in the position of writing the exact expectation value of a 't Hooft loop in an N = 2 gauge theory on S 4 with magnetic weight B. Multiplying the contributions associated to the poles and the equator, we have that 55 with Z cl , Z 1-loop,pole , Z inst and Z 1-loop,eq given in (4.10), (6.32), (5.2) and (6.59). In section 7 we will identify further non-perturbative corrections to this result arising due to monopole screening.

Examples
The formulae we have found for the one-loop determinants in the localization computation is valid for an arbitrary N = 2 gauge theory on S 4 admitting a Lagrangian description. Combining the contributions from the north pole, south pole and equator we get for a 't Hooft operator of magnetic weight B The choice of gauge group G and representation R characterizing the N = 2 theory is encoded in the one-loop determinant formulae (6.32) and (6.59) in the choice of the root system {α}, which characterizes the gauge group, and of the weights {w} of R. Here we write explicitly these formulae for two simple N = 2 gauge theories with G = SU(N): N = 2 * and N = 2 conformal SQCD. We also consider N = 4 super Yang-Mills, which is a special case of N = 2 * . From now on we set ε = r = 1.
The N = 2 * SU(N) theory For this theory the hypermultiplet is in the adjoint representation and has mass m. We parametrize a = i diag(a 1 , . . . , a N ) , (6.63) with i a i = 0. The magnetic weight B of an arbitrary 't Hooft loop is Therefore, the pole one-loop contribution (6.32) is given by Up to a phase, we have for the equator one-loop contribution (6.59) If we further restrict to the special case of G = SU(2), so that we have that α · B ≡ − Tr(αB) = p. Here the new a is a real number, and p is a non-negative integer (it is twice the usual SU(2) spin). The pole contribution (6.59) is thus For N = 4 super Yang-Mills, the one-loop factors trivialize. This result was already demonstrated in the perturbative computation of the 't Hooft loop path integral in [9] (see also [26,27]).

Conformal SQCD
This theory has gauge group SU(N) and N F = 2N massive hypermultiplets in the fundamental representation of SU(N) with masses m f with f = 1, . . . , 2N. We are interested in the 't Hooft loop specified by the magnetic weight Dirac quantization requires that n i ∈ Z. The one-loop pole contribution (6.59) is given by Up to a phase, the equatorial one-loop contribution (6.59) is given by As in (6.69), each sinh becomes cosh when n j in the numerator or n i − n j in the denominator is odd. Specializing further to G = SU(2), we have N F = 4 fundamental hypermultiplets. With the same parametrization as in the N = 2 * case, p = 2n needs to be even for Dirac quantization. Up to a phase, the one-loop factor (6.32) is for the north and south poles, and for the equator. We note that for real values of a i and m f , one encounters no branch point upon integrating over a i in (6.60). When the exponent of a sinh is a half-odd integer, the sinh actually becomes a cosh and has no zero.

Physical Picture of Monopole Screening
In the absence of a 't Hooft loop, Q-invariance requires the curvature F to vanish everywhere on S 4 , except at the north and south poles. 56 If we allowed only smooth configurations, we would conclude that only trivial gauge field configurations contribute. As shown in [2], however, localization permits instanton corrections at the north and south poles, which are precisely captured by the Nekrasov partition function.
One can argue in two steps that such corrections are necessary [2]. First, Q-invariance requires that the field strength F vanish only away from the north and south poles. If singular configurations arise as a limit of smooth configurations, there can be contributions to the path integral localized at the poles. Second, the localization Lagrangian Q · V in the neighborhood of the poles is approximately that of the twisted N = 2 Lagrangian in the Ωbackground in R 4 with the specific values of the equivariant parameters ε 1 = ε 2 = 1/r. The approximation becomes exact at the poles. Building on the earlier work [28][29][30], Nekrasov showed that the path integral of such a theory computes the equivariant integral of certain differential forms defined on the instanton moduli space [8]. The integral can be computed by a localization formula as a sum over fixed points. The fixed points in the moduli space of instantons indeed correspond to gauge field configurations that are non-trivial only at the origin of R 4 . We studied these instanton corrections in the presence of a 't Hooft loop in section 5 and found that the instanton contributions are given by the Nekrasov partition function at the north and south poles with its arguments shifted due to the insertion of the 't Hooft operator at the equator.
In this section we study another type of non-perturbative corrections due to the screening of magnetic charge associated to a 't Hooft operator. We begin by explaining how such corrections arise in our localization framework.

Monopole Screening
As we showed in section 3.3 the only possible field configurations that can contribute to the path integral are those of the form (3.9) in the bulk of S 4 . They are only allowed to deviate from (3.9) in an infinitesimal neighborhood of either the poles or the equator. The deviations at the poles were considered in [2] and have been reviewed in section 5; they are the small instanton solutions of the anti-self-dual/self-dual equations that approximate the Q-invariance equations near the poles. In the neighborhood of the loop, we saw in section The monopole moduli space M mono relevant for us is the space of solutions on R 3 , up to gauge transformations, to the Bogomolny equations with a prescribed singularity at the origin corresponding to the insertion of a 't Hooft operator. Since we are only interested in the behavior at the origin, the boundary condition at the infinity of R 3 is irrelevant. It is simplest to consider the solutions that have a vanishing Higgs expectation value at infinity. The vanishing Higgs vev will allow us to use the ADHM construction of instantons to describe the monopole moduli space in section 7.2. 57 We will describe the moduli space explicitly in the case G = SU (2). For the moment we proceed assuming that G is a general Lie group.
The magnetic charge of the singular monopole configuration created by the 't Hooft operator is specified by a coweight w ≡ B. Generally, smooth monopoles that surround the singular monopole screen its magnetic charge so that the asymptotic behavior of the fields at infinity is that of the background configuration (3.9) with the coweight w replaced by a smaller coweight v. This is because the magnetic charge of a smooth monopole is labeled by a coroot of G, and can screen the charge of the singular monopole by that amount. The coweight v is such that its corresponding weight appears in the irreducible L G-representation specified by the highest weight corresponding to w. In the terminology of [31], such v is said to be associated to w. One can show that the magnetic charge v seen at infinity must have a smaller norm than w by applying a method similar to the one we used to prove completeness of solutions in section 3.3. 58 Denoting by M(w; v) the moduli space of solutions whose asymptotic magnetic charge is given by v, we have that the relevant moduli space to consider is where the union is over coweights v such that (if we identify coweights with weights using a metric) v is a weight that appears in the highest weight representation specified by w.
The spaces M mono (w) and M(w; v) are in general singular. To understand the nature of the singularities in these spaces, it is useful to recall the situation with instantons. The Uhlenbeck compactified instanton moduli space M inst [32] is singular due to instantons of zero size. For G = U(N) the moduli space M inst can be conveniently resolved by turning on a Fayet-Illiopolous parameter for the real ADHM equation. The resolved space M inst is known to be isomorphic to the moduli space of non-commutative instantons [33], or the Gieseker resolution [34] in terms of torsion free sheaves.
As explained in [31], a natural resolution M mono of the moduli space of monopoles with a monopole singularity labeled by w at the origin involves all the coweights w ′ associated to 57 When the gauge group G is a classical group the moduli space can be constructed using the ADHM construction. In this paper we focus on the case where G is U (N ) or SU (N ). 58 The difference w · w − v · v can be expressed in terms of the integral of the instanton density, upon lifting the field configuration to instantons in C 2 using Kronheimer's correspondence explained in appendix F. the coweight w. The coweights w ′ represent the magnetic charge at the origin reduced by the smooth monopoles that are attracted to the singular monopole there. This effect was called monopole bubbling in [31]. This means that a natural resolution M(w; v) of a component in (7.1) contains smaller moduli spaces in its boundary, with w ′ being the coweights such that w ′ is associated to w while v is associated to w ′ . In the case G = U(N), one can see this structure explicitly using the ADHM construction. Thus we have the resolution of the whole moduli space where the union is over the coweights v associated to w.
We only need to study the neighborhood of the monopole bubbling locus, where all the smooth monopoles are almost on top of the singular monopole, because only these would be the approximate solutions to our genuine Q-invariance equation. For example, for gauge group SU(2) and for a 't Hooft operator with w = (1, −1) (spin 1) and v = (0, 0) (spin 0) the bubbling locus is the zero-section P 1 in the resolved A 1 space M(w; v) = T * P 1 (see section 7.3 for details). Because Q-invariance implies in particular the invariance under the U(1) J+R generated by Q 2 we are only interested in the U(1) J+R fixed points. Such fixed points are necessarily in the bubbling locus because when lifted by one dimension so that monopoles become instantons, the fixed points of the U(1) J+R × U(1) K action sit in the small-instanton locus; see section 7.2. Thus these fixed points represent all the subleading saddle points of the original gauge theory path integral. Upon evaluating the path integral, we need to sum over the fixed points.
At each fixed point of M(w; v), we need to compute the fluctuation determinants. The common factor that appears for fixed magnetic weight v was computed in section 6.3, where it was called Z 1-loop,eq (ia, im f , v). Let us denote by Z mono (ia, im f ; w; v) the sum of contributions from fluctuations at the fixed points in a single component M(w; v) divided by Z 1-loop,eq (ia, im f , v). The function Z mono (ia, im f ; w; v) is the monopole analog of the Nekrasov instanton partition function, whose computation is reviewed in appendix G from a related point of view.
Therefore, given the decomposition of the moduli space in (7.3), the expectation value of the 't Hooft loop operator with magnetic charge B = w takes the form Except Z mono (ia, im f ; w; v), all the expressions in the integrand were already calculated in the previous sections. Our remaining task is to compute this factor.

ADHM Construction of the Monopole Moduli Space
To perform explicit calculations we need an efficient way to describe the monopole moduli space M mono . The connection between monopoles and instantons [14] reviewed in appendix F, combined with the ADHM construction of instantons [35], provides a useful method to manipulate the monopole moduli space. Let us briefly review the ADHM construction of instantons in C 2 . For simplicity we will take the gauge group G to be U(N). The basic data in the construction are encoded in the complex . The z-dependent maps α(z) and β(z) are given by and their cohomology E z = Ker β(z)/Im α(z) is identified with the fiber of the gauge bundle (in the fundamental representation). We are particularly interested in E z=0 since it encodes the singularity of the 't Hooft loop. The U(1) K action on a vector space V is specified by the character χ(V ), which is a Laurent polynomial of e iν ∈ U(1) K . The 't Hooft loop with charge w = idiag(p 1 , . . . , p N ) corresponds to the case The U(1) K action on (z 1 , z 2 ) implies that The characters of H and E ∞ take the form where K is a diagonal k × k matrix and M = diag(q 1 , . . . , q N ) is a diagonal N × N matrix related to the coweight v = i(q 1 , . . . , q N ) corresponding to the magnetic charge at infinity. Both K and M have integer entries that we choose to be in the descending order. The characters of various spaces are related as For given w = i(p 1 , . . . , p N ), the choice of K and M is not necessarily unique, but we have the non-trivial condition that the whole right hand side of (7.10) has only positive coefficients. The moduli space M(w; v) is given as a hyperKähler quotient of the space of U(1) Kinvariant ADHM data (B 1 , B 2 , I, J). The action of U(1) K on (B 1 , B 2 , I, J) can be read off from the complex (7.5) and the action on (z 1 , z 2 ). In the usual ADHM construction of the instanton moduli space, we take a quotient by a certain action of the U(k) group. This action of U(k) on the ADHM data is induced from its natural action on H ≃ C k . The choice of K breaks the U(k) symmetry into the commutant subgroup r U(k r ), where k = r k r and k r is the number of entries of the r-th largest integer in the diagonal of K. Thus the moduli space is given as the hyperKähler quotient The hyperKähler quotient denoted by "///" can be implemented by imposing the ADHM equations and then considering the solutions up to the action of r U(k r ). Or alternatively, if we are only interested in the complex structure, we can drop the real equation µ R = 0 and divide by the complexified group r U(k r ) C . A resolution M(w, v) of the moduli space can be achieved by setting µ R to a non-zero constant matrix instead of requiring it to vanish. The U(1) J+R -fixed points can be found by demanding that for any e iε ∈ U(1) J+R there exists e φ ∈ r U(k r ) such that 59 e iε/2 e φ B s e −φ = B s , s = 1, 2 , e iε/2 e φ I = I , (7.14) By construction, the fixed points of U(1) J+R in M(w; v) automatically correspond to the fixed points of U(1) K × U(1) J+R in the instanton moduli space. The fixed points in the instanton moduli space were classified in [36], and they were found to sit on the boundary components of the moduli space corresponding to small instantons. This in turn implies that the U(1) J+R -fixed points on the monopole moduli space sit on the bubbling locus. We also know from the experience with instantons that the fixed points of U(1) K ×U(1) J+R ×G ×G F coincide with the fixed points of U(1) K × U(1) J+R .
At each fixed point, the ratio Z 1-loop (w; v)/Z 1-loop (v; v) can be calculated from the weights of the equivariant group action on the tangent space and the Dirac zeromodes. The ADHM construction provides a concrete procedure to derive such weights.
The tangent space can be described by considering the linearization of the ADHM system. Namely, let us consider the complex determine φ as a function of ε andâ, i.e., they define a homomorphism U(1) J+R × G → r U(k r ) at each fixed point. Thus we have an action of U(1) J+R × G on the complex (7.15), and the character on the tangent space is given as where V 1 , V 2 , V 3 are the three vector spaces that appear in (7.15) and g = (e iε , e iâ ) ∈ U(1) J+R × G. There is another method, heuristic but efficient, which can be used to compute the weights on the tangent space based on the character on the space of holomorphic functions. It is best explained in the example we consider next.

Example: SU (2) N = 2 * Theory
For G = SU(2), we can label the coweights with integers (corresponding to twice the spin). Also we slightly modify the ADHM construction above and allow w, v and K to have half odd integers. We define the integers p ≥ 0 and q by 60 Since v is associated to w, p − q is non-negative and even. The constraint (7.10) then implies that k = p − 1 and also that As a character, χ(H) is a polynomial with positive coefficients for −p ≤ q ≤ p. For ease of writing we will assume q ≥ 0, and sum over q < 0 in the end remembering that the Weyl group acts as q → −q,â → −â. We can then write where in the last expression the coefficient of the exponential increases from 1 to p−q 2 monotonically, stays constant, and then decreases monotonically to 1.
We see that (i 1 , i 2 , µ) are essentially the variables for T * P 1 that appear in the gauged linear sigma model description [37].
We also get the extra contribution for the adjoint hypermultiplet. To understand this, note the relations among the indices of the Dirac, self-dual, and Dolbeault complexes in four dimensions ind(D SD ) = 1 + e iε 1 +iε 2 2 ind(D) , According to the rule c j e w j (â,m,ε) → w j (â,m, ε) c j , this leads to the one-loop determinant where we have used that ε = 1/r. The second fixed point p 2 contributes the same amount. Thus There is another method based on contour integrals as applied in [38,39] to instantons. Let us temporarily ignore the matter contribution. In this approach, 61 we compute the character of the space holomorphic functions on the moduli space M = M(p; q), identify it with the index of the Dolbeault operator on the resolved moduli space and read off the weights. The holomorphic functions depend on B 1 , B 2 , I, J, and we need to take into account the complex ADHM equation (7.12) and the quotient by the group r U(k r ). Schematically, the character is computed by averaging over h ∈ r U(k r ), where the determinants are taken in the spaces of equations and variables and Vol is the volume of r U(k r ). For M(2; 0), To evaluate the integral by residues we need to specify the precise contour. Following [38] we assume that Im ε > 0 and treat φ andâ as real variables. we find two poles in z = e iφ , and the character is given as .
Given the weights we found above, (7.41) is consistent with the identification of the character with the index where j runs over the holomorphic tangent directions. After this practice, let us now include the matter contribution. It is convenient to consider the so-called χ y -genus: 62 with y = e im . Each weight w j (P ) will be of the form where n j and l j are integers. (7.33) implies that the contribution to ind(D Bogo ) at the fixed point P is given by − 1+e −iε 2 j e w j . Then the contribution to (7.36) is given by Summing over the fixed points P , the contribution to the path integral is Z mono (â,m; w, v) = P : fixed points j n∈Z (nε +m + n jâ + l j ε/2) 1/2 (nε −m + n jâ + l j ε/2) 1/2 (nε + n jâ + l j ε/2) = P : fixed points j sin 1/2 [π(n j râ + rm + l j /2)] sin 1/2 [π(n j râ − rm + l j /2)] sin[π(n j râ + l j /2)] , (7.46) where we recall that ε = 1/r. On the other hand, the χ y genus in (7.43) can be written as The χ y -genus also appeared in the instanton calculus for N = 2 * theory [40].
Thus we find that . (7.48) This is why the χ y -genus is useful for us. We now calculate the χ y genus using the ADHM construction of the monopole moduli space. Locally at the origin of the space of ADHM data, the space of holomorphic sections is the tensor product of the space of holomorphic functions and the space of Dirac zeromodes. (7.43) corresponds to Tr[det(1 − yg)], where the trace is over the holomorphic functions and the determinant is over the zeromodes. Since the space of zeromodes is given by the cohomology of the complex (7.15), the determinant over the zeromodes is given by In the case of M(2; 0), We note that (7.38) is indeed obtained from (7.50) using the relation (7.48). The magnetic charge p can be screened by monopoles and get reduced to q, also an even integer. We set l := p − q. The moduli space M(p; q) can be described using the ADHM construction as follows. The action of U(1) K is specified by the matrix require that the ADHM matrices take the following form: , J = 0 · · · J 1,l/2 · · · 0 · · · 0 0 · · · 0 · · · J 2,p−l/2 · · · 0 . We thus obtain a contour integral expression for the χ y -genus of M(p; q): The first line on the right hand side represents the Haar measure on r U(k r ), which would be clearer if the integral is written in terms of φ r,i such that z r,i = e iφ r,i . We again choose to use the prescription where we integrate over each z r,i along the unit circle |z r,i | = 1, assuming thatâ ∈ R and Im ε > 0. The integral can be evaluated by residues, and the computation can be automated as a Mathematica code. Applying the rule (7.48), we find experimentally 63 for p odd, for p even. (7.65) Combined with (6.69), the dependence of Z mono (ia, im; p, q)Z 1-loop,eq (ia, im; q) on q is in fact only in the binomial coefficient. We now put everything together. Including the terms with q ≤ 0, we get for p odd, for p even.

(7.66)
This is the complete gauge theory result for 't Hooft loops in SU(2) N = 2 * theory. This analysis, with the philosophy described, can be extended to other gauge theories.

Gauge Theory Computation vs Toda CFT
In this section we compare the results of our gauge theory analysis for the expectation value of 't Hooft loop operators in N = 2 gauge theories on S 4 with formulae in [10][11][12], which were obtained from computations in two dimensional Liouville/Toda CFT. As we shall see, for the theories for which we explicitly carry out the comparison, we find beautiful agreement. In [10,11] a dictionary was put forward relating the exact expectation value of gauge theory loop operators in N = 2 gauge theories on S 4 and Liouville/Toda correlation functions in the presence of Liouville/Toda loop operators (topological defects). This enriches the AGT correspondence [19], which identifies the gauge theory partition function with a correlation function in Liouville/Toda (see also [41]), to encompass more general observables. The identification in [10,11] has yielded explicit predictions for the exact expectation value of 't Hooft loop operators in N = 2 gauge theories on S 4 .
We compare the Liouville/Toda results for 't Hooft operators in N = 2 * with the corresponding gauge theory computations for both the one-loop determinants as well as for the non-perturbative contributions due to monopole screening.

't Hooft Loop Determinants from Toda CFT
We now explicitly compare the results obtained for 't Hooft operators in the N = 2 * theory -corresponding to an N = 2 SU(N) vectormultiplet with a massive hypermultiplet in the adjoint representation -with loop operator computations in Toda CFT on the oncepunctured torus. For a 't Hooft loop labeled by a magnetic weight B = h 1 -corresponding to the fundamental representation of SU(N) -the Toda CFT calculation yields [12] da C(ia, im)Z cl (ia, q)Z inst (ia, 1 + im, q) while Z inst (ia, 1 + im, q) is the instanton contribution (5.2). 65 Finally C(ia, im) is the Toda CFT three-point function 66 relevant for the once-punctured torus description of N = 2 * (à la [42]) with α the roots of the SU(N) Lie agebra. Since Υ b=1 (x) = G(x)G(2 − x)/2π, with G(x) being the Barnes G-function (6.19) and because α = h i − h j for j > i if α > 0 we obtain 67 . (8.4) We note that C(ia, im) is precisely given by the square of the one-loop factor in Nekrasov's partition function of N = 2 * in R 4 (see (6.32) and (6.65))
66 This is the three-point function of two non-degenerate and one semi-degenerate primary operators in Toda CFT when the background charge b = 1.
67 In this section, in order to avoid cluttering formulas, we drop inessential overall numerical factors. with Thus we can write the Toda loop correlator as We note that the result is given by the sum of N terms, associated to the N weights of the fundamental representation of SU(N). Each of the N weights yields an identical contribution, and therefore we can focus on the contribution of the highest weight term, labeled by h 1 . It is important to remark at this point that genuine new contributions appear for loop operators labeled by a representation with highest weight B for which not all weights are in the Weyl orbit of B (non-minuscule representations). These contributions correspond precisely to the non-perturbative contributions due to monopole screening encountered in our gauge theory analysis! Section 8.2 demonstrates for 't Hooft loops with higher magnetic weight B that Liouville theory precisely reproduces the non-perturbative screening contributions discussed in section 7.2. Focusing on the highest weight vector contribution, we trivially rewrite the answer as Without encountering any residues, we now shift the contour of integration ia → ia + h 1 /2 to express the answer in a more symmetric form Our next goal is to rewrite the second line in (8.9) as a complete square of a function with the same shifted argument ia − h 1 /2 as in the first line times a remainder, which we denote by E(ia, im) da |Z cl (ia − h 1 /2, q)Z 1-loop,pole (ia − h 1 /2, im)Z inst (ia − h 1 /2, 1 + im, q)| 2 × E(ia, im) .

(8.10)
To anticipate where this path will leads us when comparing with our gauge theory analysis, the complete square contributions reproduce the classical, one-loop and instanton contributions that arise from the north and south poles of S 4 , while the remainder captures the contribution from the equator! In order to determine E(ia, im) in (8.10) we need to calculate The ratio of one-loop factors can be determined by recalling that a j = a · h j , so that the shifts ia ± h 1 /2 in the arguments in (8.11) are given by (since Therefore, only a 1j ≡ a 1 − a j shifts, by ia 1j → ia 1j ± 1/2. Since α = h i − h j for j > i if α > 0, we decompose the product over positive roots appearing in (8.6) comprising the splitting of positive roots into h 1 −h j and the rest. Therefore only the factors N j=2 · shift, the rest cancel between the numerator and denominator in (8.11). We find Using the explicit form of the monodromy operators T k (ia, m) in (8.1) we arrive at which by Euler's reflection formula Γ(z)Γ(1 − z) = π sin(πz) yields sin(π( 1 2 + ia 1j − im)) sin(π( 1 2 − ia 1j − im)) sin(π( 1 2 + ia 1j )) sin(π( 1 2 + ia 1j )) The result can be written in a more covariant form to arrive at the final answer This is precisely the gauge theory formula for the equatorial one-loop determinant (6.59). This shows that the Toda prediction for the expectation of the 't Hooft loop operator labeled by the fundamental representation in the N = 2 * theory with SU(N) gauge group precisely agrees with our gauge theory computation. We identify in the Toda correlator the factor |Z cl (ia − h 1 /2, q)Z 1-loop,pole (ia − h 1 /2, im)Z inst (ia − h 1 /2, 1 + im, q)| 2 in (8.10) with the gauge theory contributions arising from the north and south poles of S 4 (see (6.34)), while comparison of (8.18) with (6.59) demonstrates that indeed E(ia, m) precisely captures the gauge theory contribution from the equator, so that The Toda calculation of [12] can be extended to describe 't Hooft operators with higher magnetic weight in N = 2 * . We have checked that the Toda calculation for B = 2h 1 also exactly reproduces the gauge theory prediction.

Monopole Screening from Liouville Theory
We now specialize to the A 1 Toda theory, i.e., Liouville theory. We compare the results in [10,11], which are the special case of the general Toda calculations above, with the nonperturbative contributions from monopole screening in SU(2) N = 2 * theory computed in section 7 .3 In Liouville theory, the 't Hooft loop expectation value is given in terms of shifted conformal blocks. To simplify formulas we set r = 1 without loss of generality, also set b to 1 and adapt the normalization of [10] Z L (â, im, τ ) ≡ e −2πiτâ 2 F (1 +â, 1 + im, τ ) , (8.20) where F (α, α e , τ ) is the conformal block of the 1-punctured torus with modulus τ in the standard normalization [43], with internal and external Liouville momenta α and α e . 68 Up to a normalization constant, it was shown in [10,11] that the loop operator expectation value is given by 69 The Liouville loop operator L 1,0 acts as a difference operator. For any meromorphic function f (â), let us define the operatorsĥ ± as multiplication by the functions h ± (â): We also define the shift operator ∆: With these definitions the Liouville loop operator is defined by whereĥ p,q is multiplication by a function h p,q (â). This function can be determined by the recursion relation The solution is given by .
The complex conjugate of τ appears with a minus sign because τ enters into Z L through e 2πiτ = e −2πiτ . In this subsection we avoid using the symbol q to denote e 2πiτ , in order to avoid confusion with screened magnetic charge q.
Up toâ-independent factors, the Liouville three-point function is related to the gauge theory one-loop determinant as forâ ∈ iR, where Z 1-loop,pole (â,m) is given in (6.68). Thus the Liouville correlator (8.21) becomes Assuming that we can shift the contour without picking up residues, 70 we can write this as for p odd, cos p 2 (2πâ + πim) cos p 2 (2πâ − πim) cos p (2πâ) for p even, (8.31) Comparing this with (6.69) and (7.65), this is precisely Z mono (ia, im; p, q)Z 1-loop,eq (ia, im, q).

Using the relation [19]
we thus obtain After reintroducing the dimensionful parameter r, the gauge theory result (7.66) for T p and the Liouville theory expression (8.33) for (L 1,0 ) p precisely agree, including the monopole screening contributions! We note that the charge p 't Hooft loop T p corresponds to the Liouville operator (L 1,0 ) p . Thus our charge p 't Hooft loop T p equals the power (T 1 ) p of the 't Hooft loop that is S-dual to the spin 1/2 Wilson loop, and differs from the S-dual of the spin p/2 Wilson loop. The origin of the power is in the natural resolution of the Bogomolny moduli space. As explained in [31], the moduli space of solutions describing an array of p minimal 't Hooft loops T p=1 develops a singularity when two of the loop operators collide. In the limit that all of them are on top of each other, the magnetic charge of the 't Hooft loop is p. Said another way, the singularity of the moduli space can be resolved by replacing the charge p 't Hooft loop with a collection of slightly displaced minimal 't Hooft loops. 71

Conclusion
We performed an exact localization calculation for the expectation value of supersymmetric 't Hooft loop opertors in N = 2 supersymmetric gauge theories on S 4 . These results combined with the exact computation of Wilson loop expectation values [2] constitute a suite of exact calculations for the simplest loop operators in these gauge theories and allow for a quantitative study of S-duality for this rich class of gauge theory observables.
A 't Hooft loop was defined by specifying a boundary condition of the fields in the path integral. We integrated over the non-singular and singular solutions to the saddle point equations in the localization computation. We integrated over the fluctuations of the fields around the singular monopole background (3.9) that represents an infinitely heavy monopole with a circular worldline.
In the leading classical approximation the expectation value was obtained by evaluating the on-shell action in the non-singular background (3.9), and the only perturbative quantum corrections 72 in the localization path integral are the one-loop determinants computed using the Atiyah-Singer index theorem, arising from the north pole, south pole and equator. The 't Hooft loop expectation value receives two types of non-perturbative corrections. The first is from instantons and anti-instantons localized at the north and south poles as in [2], arising because our localization saddle point equations become F + = 0 and F − = 0 there. One new feature in our calculation is that the Nekrasov instanton partition functions at the poles have their argument shifted due to the 't Hooft loop background. The second type of 71 As noted in [2], the localization supercharge Q is indeed compatible with parallel loop operators each located at a fixed latitude.
72 The one-loop determinants are the unique perturbative corrections with respect to the localization action Q · V . All the perturbative corrections with respect to the physical action [9] are reproduced by integrating over the zero-mode a.
non-perturbative correction occurs as new saddle point field configurations, where smooth monopoles in the bulk of S 4 screen the charge of the singular monopole inserted along the loop. These arise from non-abelian solutions to the Bogomolny equations DΦ = * F , which describe the saddle point equations in the equator. The field configurations were identified as the fixed points of an equivariant group action on the moduli space of solutions of the Bogomolny equations.
In this paper we have focused on the computation of 't Hooft operators for which the magnetic charge and the electric charge vectors are parallel, where the electric charge is acquired by the Witten effect, due to the non-vanishing topological angle θ. The techniques introduced here, however, can be used to compute general dyonic Wilson-'t Hooft operators. The new ingredient for a dyonic operator is the insertion of a Wilson loop for the unbroken gauge group preserved by the singular monopole background.
We compared our gauge theory calculations with some of the predictions in [10][11][12] obtained from computations with topological defects in Liouville and Toda field theories, and found a perfect match for all comparisons we have performed. The physical observables in Liouville/Toda theory are known to be invariant under the modular transformations (or more generally under the Moore-Seiberg groupoid) that are identified with the S-duality transformations in gauge theory. Thus our results prove S-duality invariance of the N = 2 gauge theories, in the sector of physical observables involving Wilson and 't Hooft loop operators. In turn, the progress we made on gauge theory loop operators provides motivation to study in more depth the two-dimensional observables. In particular the computational techniques for topological webs, -the defects involving trivalent vertices -are to be developed in order to make a useful comparison with more complicated loop operators in higher-rank gauge theories.
In our study an important role was played by the equivariant index for the moduli space of solutions of the Bogomolny equations in the presence of a singular monopole background, created by the 't Hooft operator. The analysis was similar to that of the instanton moduli space that led to the Nekrasov partition function, and we defined the quantity Z mono which is an analogous physical quantity in the monopole case. It is possible to generalize and formalize the definition of Z mono by setting up a localization scheme on S 1 × R 3 [44].
The localization techniques we developed for 't Hooft operators should also admit generalizations to other supersymmetric disorder operators, such as monopole and vortex loop operators in three dimensions and surface operators in four dimensions. For example, it would be interesting to formulate a path integral framework that realizes the mathematical calculations [45][46][47][48] for instantons in the presence of singularities representing surface operators. Also, the localization framework for N = 2 theories on S 4 should apply to surface operators preserving two-dimensional N = (2, 2) supersymmetry. Localization calculations for such observables should help understand disorder operators in the broad duality web involving quantum field theories in diverse dimensions.

A Supersymmetry and Killing Spinors
The spinors in this paper transform in a representation of Spin(10), whose generators are constructed from the Clifford algebra Cl(10) We take the Euclidean metric η M N = δ M N . In the chiral representation whereΓ M ≡ (Γ 1 , . . . , Γ 9 , −Γ 0 ), and Γ M ,Γ M are 16 × 16 matrices which satisfỹ The matricesΓ M and Γ M act respectively on the negative and positive chirality spinors of Spin(10) since In Euclidean signature, which we use in this paper, ten dimensional spinors are complex. We choose a basis in which Γ 1 , . . . , Γ 9 are real and Γ 0 imaginary. To describe Γ M explicitly it is convenient to break SO(10) to SO(8) × SO(2) and use the octonionic construction of the Clifford algebra Cl (10). For the explicit expressions which are needed for explicit construction of the supersymmetry equations in components we use matrices as defined in appendix A of [2] with a certain permutation of spacetime indices. If Γ M are the matrices in [2], then the present Γ M are given by Γ M = Γ M +1 for M = 1, 2, 3, 5, 6, 7 The factor of i appears in the relation to Γ 0 because our present conventions use the Euclidean metric η M N = δ M N , while [2] used the Lorentz metric with η 00 = −1.
The conformal Killing spinor equation (3.10) in the B 3 × S 1 metric (3.7) is In the vielbein basis eˆi = e i = dx i and e4 = r 1 − | x| 2 4r 2 dτ , the non-zero components of the spin connection are Equation (A.8) implies thatǫ = ǫ c (τ ) while the first three equations in (A.7) imply that ǫ = ǫ s (τ ) + x iΓ i ǫ c (τ ). The solution to the equation is (3.14) withε s andε c two constant ten dimensional Weyl spinors of opposite chirality.

B Lie Algebra Conventions
Let G be a compact Lie group and g the Lie algebra of G. As a vector space g is isomorphic to R dim G . In our conventions, for a gauge theory with gauge group G, the fields A µ , F µν and Φ A (A = 0, 9) of the vectormultiplet take values in g. In particular, we write the covariant derivative as D ≡ D A = d + A and the curvature as F µν = [D µ , D ν ]. If G is U(N) or SU(N), the basis {T α } of the Lie algebra g can be represented by N × N antihermitian matrices. Given the basis, the real coordinates a α of an element a ∈ g are defined by the expansion a = a α T α . Let g C = g ⊗ C be the complexification of g. An element a = x + iy of g C , where x, y ∈ g, can be written as a = a α T α with a α being complex numbers. We say that an element a of g C is real if the coordinates a α are real. Complex conjugation acts by conjugating the coefficients: a = a α T α → a α T α . If G is a compact Lie group, then the Lie algebra g of G can be equipped with a positive definite bilinear form (• , •) : g × g → R invariant under the adjoint action of G. Such a bilinear form g × g → R is defined uniquely up to a scaling, and extends holomorphically to g C × g C → C. For g = u(N) and g = su(N), we choose (• , •), also donoted by • · •, to be given by minus the trace in the fundamental representation: (a, b) = − Tr ab.
The basis elements T α in the Cartan algebra t of u(N) can be represented by the diagonal N × N matrices T α = i diag(0, . . . , 0, 1, 0, . . . , 0) , where 1 is at the position α. Since in this basis the bilinear form is the identity matrix, − Tr T α T β = δ αβ , we do not distinguish between contravariant and covariant Lie algebra indices. For an element a = a α T α of t we refer to a using the following equivalent notations where a α are real. When dealing with complexification t C we allow a α to be complex. The notation (B.2) is also used for g = su(N). For example, the Nekrasov instanton partition function Z inst takes a complex elementâ of t C , i.e. the Coulomb parameter, as one of its arguments. We use equivalently the following forms referring to Z inst evaluated atâ It should be clear from the context whatâ refers to in the main text.

C Coordinates and Weyl Transformations on S 4
The SO(5) isometry of the round metric on S 4 is made manifest by the induced metric on the following hypersurface in R 5 In this paper, a certain U(1) J ⊂ SO(5) isometry of S 4 generated by the generator J plays a key role. It acts on the embedding coordinates as and its fixed points X 5 = ±r define the north and south pole of S 4 . The following coordinates are of use in the paper: Latitude Coordinates: The metric is given by where dΩ n is the metric on the unit S n and ϑ is the latitude angle on S 4 , with ϑ = 0, π/2 and π corresponding to the north pole, equator and south pole respectively. The embedding coordinates are X a = r sin ϑ n a a = 1, 2, 3, 4 where n a is a unit vector in R 4 parametrizing S 3 . The U(1) J action induced by J is realized by the Hopf fibration. Consider S 3 : |w 1 | 2 + |w 2 | 2 = 1, (w 1 , w 2 ) ∈ C 2 and the U(1) action (w 1 , w 2 ) → (e iε w 1 , e iε w 2 ). Introduce angular coordinates on C 2 : w 1 = ρ cos η 2 e iψ and w 2 = ρ sin η 2 e iψ+iϕ so that the U(1) J acts by shifts ψ → ψ + ε, and consider the map C 2 ⊗ C 2 → R 3 x = w σw = ρ 2 (sin η cos ϕ, sin η sin ϕ, cos η), (C.5) so that (ρ 2 , η, ϕ) are the spherical coordinates on R 3 . Rewriting the flat metric on C 2 in the (ρ, η, ϕ, ψ) coordinates we get The unit S 3 is at ρ = 1 with metric where dΩ 2 = dη 2 + sin 2 η dϕ 2 (C.8) and In these coordinates, the U(1) J vector field is v = 1 r ∂ ∂ψ , and the dual 1-form used in section The one-form ω satisfies dω = 1 2 vol(S 2 ). S 2 × S 1 Foliation Coordinates: The metric is given by where τ is the coordinate on S 1 and 0 ≤ ξ ≤ π/2. The embedding coordinates are given by X 1 + iX 2 = r cos ξe iτ X 3 + iX 4 = r sin ξ sin αe iφ X 5 = r sin ξ cos α , where dΩ 2 = dα 2 + sin 2 αdφ 2 . The U(1) J symmetry generator J acts by shifts In these coordinates the north and south pole are at (ξ = π/2, α = 0) and (ξ = π/2, α = π) respectively. B 3 × S 1 Foliation Coordinates: The metric is given by and | x| 2 ≤ 4r 2 defines the three-ball B 3 . The embedding coordinates are given by (C.15) The U(1) J symmetry generator J acts by 16) In these coordinates the north and south pole are at x = (0, 0, 2r) and x = (0, 0, −2r) respectively.

D Q-Invariance of the 't Hooft Loop Background
The background created by a circular 't Hooft loop with magnetic weight B located at x = 0 in the B 3 × S 1 metric (3.7) takes the same form as that of a static 't Hooft line in flat spacetime (3.8) (for θ = 0) Since B ∈ t takes values in the Cartan subalgebra of the gauge group G, the singularity is abelian in nature. We can verify that the the deformed monopole equations (3.38, 3.39, 3.40) are solved by the 't Hooft loop background (D.1). For example, let's consider the first spatial equation (3.38). In the background (D.1) F 14 , F 34 , K 1 , D4Φ 9 vanish. We group the remaining terms to make the structure of cancellation obvious using that (D.1) satisfies where the first equality is due to the abelian nature of the background (D.1). Evaluation yields for (3.38) The cancellation in the second deformed monopole equation (3.39) is exactly the same with replacement of indices 1 → 2. In the third equation (3.40), the relative signs are different, but again all terms cancel similarly. In analyzing the last equation (3.40) we also find that Φ 0 can be turned on as long as This observation plays an important role in finding the most general solution to the saddle point equations, as discussed in section 3.3. Similarly, it is very easy to show that the invariance equations (3.37) are satisfied by the background (D.1). For these only D i Φ 9 and F jk contribute and cancel elementarily due to formula (D.2). These equations also exhibit that Φ 0 has a zeromode, given by and therefore, due to (D.4) where a ∈ t is constant. In comparison with [2] the profile of Φ 0 is not constant in B 3 × S 1 . However, since the metric on B 3 × S 1 and S 4 are related by a Weyl transformation with Ω = 1 + | x| 2 4r 2 , it follows from (2.10) that the Weyl transformation makes Φ 0 constant in S 4 , as found in [2] .
It is straightforward to show that the background created by the 't Hooft loop when θ = 0 (3.9) solves the localization equations Q · Ψ = 0 As we have already demonstrated that the terms involving Φ 0 and F jk cancel in the invariance equations (3.37) and deformed monopole equations (3.38)(3.39)(3.40), we just have to exhibit cancellation of the terms involving Φ 0 and F i4 . Since the 't Hooft loop background is τ independent and abelian (i.e. [Φ 0 , Φ 9 ] = 0), we are just left to verify from the invariance equations (3.37) that (D.8) Using that F i4 = r 1 − | x| 2 4r 2 F i4 and D i 1 + | x| 2 This concludes the explicit check that the direct sum of the monopole background configuration (D.7) and the Φ 0 zeromode profile (D.5) with the associated auxiliary field K 3 (D.6) solve the localization equations Q · Ψ = 0.

E Hypermultiplets in General Representations
In this appendix we will derive the formula (6.28) of the one-loop index for hypermultiplets in an arbitrary representation.
We will do this by generalizing, and also applying in a suitable way, the formula (6.27) that is valid for the adjoint representation. Let us begin with N = 2 * theory in flat space which we regard as a dimensional reduction of the super Yang-Mills in ten dimensions. The group SO(4) that rotates the 5678 directions factorizes into the product of the R-symmetry group SU(2) R and the flavour symmetry group SU(2) F .
where diag(e iâ 1 , e iâ 2 ) and diag(e im , e −im ) parametrize the maximal tori of U(2) and SU(2) F that we denote by U(1) 1 × U(1) 2 and U(1) F respectively. Under U(1) 1 × U(1) 2 × U(1) F , offdiagonal fields in the hypermultiplet transform in representations with charges (+1, −1, ±1) and their complex conjugate. The trick is to consider a new N = 2 theory that is obtained by setting all the off-diagonal components of the U(2) adjoint fields in the vector multiplet to zero, regarding G ′ = U(1) 1 as a new gauge group. We also project the hypermultiplet fields onto those with charges (+1, −1, +1) and their conjugate, and regard U(1) ′ F ≡ [U(1) 2 × U(1) F ] diag as a new flavour group. The hypermultiplet index for the new theory is obtained from (E.1) by keeping the relevant terms: whereâ ′ ≡â 1 , and the Coulomb parameterâ 2 and the original mass parameterm have combined into a new mass parameterm ′ ≡â 2 −m. Thus we have derived the formula (6.28) for the spacial case of gauge group U(1) and a single charged hypermultiplet. Noting that a general complex irreducible representation of an arbitrary gauge group G can be thought of as embedding G into U(dim R) whose maximal torus is U(1) dim R , this U(1) result implies the formula (6.28) for any complex R.
Similarly any strictly real irreducible representation defines an embedding of G into SO(dim R) with maximal torus SO(2) [dim R/2] . Noting that the vector representation of SO(2) ≃ U(1) gives the minimal real irreducible representation, the U(1) formula (E.2) also generalizes to (6.28) for any real representation R.
where A = A i dx i and Φ are the connection and a scalar on R 3 . If we assume that A is independent of ψ, or equivalently invariant under the U(1) K action, the four-dimensional curvature F = dA + A ∧ A decomposes as To be more precise, we need to specify the boundary conditions we impose in three and four dimensions. In three dimensions, we require that the Higgs field Φ vanishes at infinity. As we see from (F.5) this is indeed necessary if A at infinity becomes pure gauge g −1 dg with g : S 3 → G depending only on the angular directions of C 2 .
To understand the appropriate boundary condition at the origin, let us consider the trivial background A = 0 on C 2 . Let w be a coweight of the gauge group G. We recall that a coweight is an element of the Lie algebra, and discretely quantized in such a way that the exponential e Bψ is invariant under ψ → ψ + 2π. A singular gauge transformation by e Bψ induces a non-trivial field i.e., This is precisely the 't Hooft operator background in the transverse directions to the loop. If we start with a general gauge field, after the singular gauge transformation by e Bψ , the group U(1) K acts as an isometry that shifts ψ as well as a linear transformation on the fibers of the gauge bundle. In general, a smooth gauge field on C 2 in variant under the U(1) K group action becomes a field configuration in three dimensions that obeys the boundary condition appropriate for the 't Hooft loop. The linear transformation on the fiber at the origin encodes the magnetic charge of the 't Hooft operator. In fact, one can reverse the logic and use this connection with instantons to define the precise boundary conditions for singular solutions of the Bogomolny equations, which is otherwise difficult to specify. See for example [21], where this definition of boundary conditions was concretely used to compute the dimension of the moduli space by suitably applying the index theorem.

G Instanton Partition Functions for U (N )
For G = U(N), the localization calculation represents the instanton partition function Z inst as a sum over the set of the U(1) ǫ 1 × U(1) ǫ 2 × U(1) N -fixed points on the moduli space of noncommutative instantons on C 2 . For each fixed point, we need to compute the equivariant Euler character of the self-dual complex → Ω 2+ ⊗ ad(g) . (G.1) Note that we can decompose the complexified spaces of differential forms as Ω 0 C ≃ Ω 0,0 , Ω 1 C ≃ Ω 1,0 ⊕Ω 0,1 , Ω 2+ C ≃ Ω 2,0 ⊕Ω 0,0 κ⊕Ω 0,2 , where κ is the Kähler form. Using Hodge duality, we also have the relations Ω 2,2 ≃ Ω 0,0 and Ω 2,1 ≃ Ω 1,0 . It follows that the complexification of the self-dual complex (G.1) is isomorphic to the Dolbeault complex D : Ω 0,0 ⊗ ad(g) D → Ω 0,1 ⊗ ad(g) D → Ω 0,2 ⊗ ad(g) (G.2) twisted by Ω 0,0 ⊕ Ω 2,0 . The index of the self-dual complex (G.1) differs from the index of the Dolbeault complex (G.2) by a factor accounting for complexification, and another computing the weights of the toric U(1) ǫ 1 × U(1) ǫ 2 action on the fiber of Ω 0,0 ⊕ Ω 2,0 at the origin: Mathematically it is sometimes more convenient to consider the torsion free sheaves, which are known to be in a one-to-one correspondence with non-commutative instantons. Deformations of the torsion free sheaves are captured by the Dolbeault complex (G.2). Each fixed point is labeled by an N-tuple of Young diagrams Y = (Y 1 , . . . , Y N ). Each partition Y α defines an ideal sheaf of rank one E Y in the standard way [36]. Let V Y be the space of holomorphic sections of E Y . For Y = (λ 1 ≥ λ 2 · · · ≥ λ λ ′ 1 ), where λ i and λ ′ i are the number of squares in the i-th column and row respectively, the basis of V Y is given by monomials z i−1 1 z j−1 2 for all (i, j) such that j > λ i . (The counting of squares in each Young diagram starts from (i, j) = (1, 1)). In other words the basis in the V Y is enumerated by the squares outside of the Young diagram Y . Each basis element z i−1 1 z j−1 2 generates an eigenspace of the torus T = U(1) ǫ 1 × U(1) ǫ 2 with eigenvalue t 1−i 1 t 1−j 2 , where (t 1 , t 2 ) = (e iε 1 , e iε 2 ). Therefore the character of V Y as a U(1) where s i = e iâ i . We can extract the common infinite part independent of Y To convert the index (Chern character) ind(D vm ) to the fluctuation determinant (Euler character), we need to expand in powers of (t 1 , t 2 ) and take the product of weights according to the rule α c α e wα(ε 1 ,ε 2 ,â) → α w α (ε 1 , ε 2 ,â) cα . (G.10) Notice that ind(D) and t −1 1 t −1 2 ind(D) in (G.3) are exchanged by (ε 1 , ε 2 ,â) → (−ε 1 , −ε 2 , −â). For the common one-loop factor Z 1-loop at the north pole, it is important to use ind(D vm ) 1-loop = 1 + t −1 1 t −1 2 2 ind(D) 1-loop , (G.11) rather than ind(D) 1-loop , before expanding in positive powers of t 1 , t 2 as we did in section 6. 76 For the finite instanton part Z inst computed by the rule (G.10), however, the result obtained from is identical to the result from ind(D) inst because the signs that appear from (ε 1 , ε 2 ,â) → (−ε 1 , −ε 2 , −â) cancel out in the product. Thus the instanton partition function can be computed either from the self-dual complex or the Dolbeault complex. 76 Recall that one applies the positive and negative expansions to the north and south poles, respectively. Because of this, in the absence of a 't Hooft loop, the product of north and south pole contributions to the one-loop factor obtained from ind(D) is the same as the one from ind(D vm ).