A Localization Computation in Confining Phase

In this note we show that the gaugino condensation of 4d N=1 supersymmetric gauge theories in the confining phase can be computed by the localization technique with an appropriate choice of a supersymmetry generator.

In this paper, we will show that the gaugino condensation can actually be computed by the localization technique with an appropriate choice of a SUSY generator. For applying the localization technique, a SUSY theory on a curved compact manifold will be desirable because of the absence of the some infra-red divergence. One of the candidate of such theories is the 4d N = 1 SUSY gauge theory on S 1 × S 3 [7,28,29]. 2 We will show that for this theory a correlators of lowest components of chiral multiplets can be computed exactly in the weak coupling limit computation around the instantons(= anti-self-dual connections) on S 1 × S 3 , although we have not done it explicitly. Instead, we consider the theory on S 1 × R 3 which can be obtained from the theory on S 1 × S 3 by the flat space limit. For this, the explicit computation should be much easier because the explicit forms of the antiself-dual connections on it are known. Indeed, the actual computation is almost identical to the one performed by Davies et. al. [31,32], in which they consider the small radius limit of S 1 and argued that the gaugino condensation in the N = 1 SUSY Yang-Mills theory is independent of the radius of S 1 . Furthermore, it was shown in [32] that the value of the gaugino condensation agrees with the one obtained in [25,24] using the holomorphy. Therefore, we reproduce the gaugino condensation using the localization technique, 3 and we hope that the discussions in the paper can be generalized to broader classes of SUSY gauge theories with only minor changes.
The organization of this paper is as follows: In section 2 we first briefly review the 4d N = 1 SUSY gauge theory on S 1 × S 3 . Then, we show that by the localization technique the correlators of the lowest components of the chiral multiplets can be computed semiclassically around the anti-self-dual configurations. In section 3 we compute the gaugino condensation for the pure SUSY Yang-Mills on S 1 × R 3 using the localization technique. We comment on the inclusion of the chiral multiplets shortly in section 4. In the appendix, we construct the 4d N = 1 SUSY gauge theory on S 4 although the localization technique is not useful for it.

4d N = 1 SUSY Gauge Theory on S 1 × S 3 and Localization
In this section, we will briefly review the 4d N = 1 SUSY gauge theory on S 1 × S 3 [7,33] 4 and then show that the path-integral for the correlation function of the certain fields can be reduced to the semi-classical computations around the instantons by adding an appropriate regulator action. First, let us remind that the SUSY transformation of the vector multiplet for Euclidian 4d N = 1 SUSY gauge theory on R 4 is where m = 1, 2, 3, 4, all fields are in the adjoint representation of the gauge group G and ǫ,ǭ are constant spinors. We have introduced the chiral decomposed Gamma matrix σ m : σ a is the Pauli matrix for a = 1, 2, 3 and σ 4 = i, andσ m = (σ m ) † . We also defined σ mn = 1 2 (σ mσn − σ nσm ) andσ mn = 1 2 (σ m σ n −σ n σ m ). The spinor indices are raised or lowered by ǫ αβ which has ǫ 12 = −ǫ 12 = 1. 5 It is known [7,34] that the SUSY transformation (2.1) is consistent with the 4d N = 1 SUSY gauge theory on S 1 × S 3 if the derivative D m is defined as follows: where D m is the covariant derivative including the spin connection, for example D m = ∂ m + 1 4 w ab m γ ab for spinors. Here the V m is an appropriate background field and the q is the R-charge for which we assigned q(ǫ) = 1, q(ǭ) = −1, q(λ) = 1, q(λ) = −1. For simplicity, we will consider the unit 3-sphere and ds 2 = dt 2 + ds 2 S 3 where 0 ≤ t < 2πR is the periodic coordinate for S 1 . Then, the background field is fixed to be and the Killing spinors are given by the solutions of the following equations: It should be stressed that we regard λ andλ are independent holomorphic fermionic 2-components spinor fields in the 4d Euclidian spacetime. Indeed, these two spinors are fundamental representations of the former and the latter SU(2) of SU(2) × SU(2)(= Spin(4)), thus they can not be related by the complex conjugation. Because any reality condition can not be imposed on the fundamental representation of SU(2), we regard λ,λ as the formal holomorphic path-integral variables. The SUSY parameters ǫ andǭ are also independent fields.
Usual N = 1 SUSY Yang-Mills action with the theta term is SUSY invariant on S 1 × S 3 with the covariant derivative D m defined by (2.2). We will also use the complexified coupling constant τ ≡ θ 2π + 4πi g 2 . Now we are applying the localization technique to the theory. In this paper, the localization technique simply means the use of the following identity for the operators satisfying δO a = 0: where δ is a symmetry transformation of the theory and V should satisfies δ 2 S 3 ×S 1 V = 0. With this identity we can compute the correlator O 1 O 2 · · · O n with a sufficiently large t if the real part of δ S 3 ×S 1 V is non-negative.
In order to use the localization technique, we should choose a SUSY transformation δ and a regulator Lagrangian V . Here the choice of δ means the choice of ǫ andǭ. As in [2], we simply take the regulator Lagrangian as following: where we should define (δλ) † appropriately. First, let us assume both ǫ andǭ are nonzero. Then, the bosonic contributions from the first term is where we have used and F + (F − ) are the anti-self-dual (self-dual) part of the field strength F , respectively. Then, the saddle points for t → ∞ will satisfy F + = 0 and D = 0. Because the contribution from the other term in V , i.e. (δλ) † δλ ∼ (F − ) mn (F − ) mn + D 2 , will give F − = 0, we conclude that the saddle points satisfy F mn = 0 and D = 0 where the regulator action is essentially the Yang-Mills action. Thus the partition function can be calculated in the weak coupling limit. This is well-known result. Indeed, with the twist of the boundary condition along the S 1 , the partition function on S 1 × S 3 is the superconformal index [35,36,28,29,7]. In this paper, we consider another possibility for the localization computation on S 1 ×S 3 : We takeǭ = 0, but ǫ = 0, which implies δδ = 0. Thus the condition δδ V = 0 is trivially satisfied. 6 Explicitly, the SUSY transformation is With this δ, we have δλ = 0, thus a correlation function of any gauge invariant combinations of λ can be computed by the localization technique. The Killing spinor can be taken explicitly asǭ where we use the coordinate system with the metric ds 2 = cos 2 θdφ 2 + sin 2 θdχ 2 + dθ 2 . Taking V as in (2.19), we easily see that whereF is the dual of the field strength, and the saddle points equations are F − = 0 and D = 0, i.e. the instantons (anti-self-dual connections) on S 1 × S 3 . 7 Note that this includes the Yang-Mills action with a pure imaginary θ. This also means that by taking t → ∞ limit the theory is arbitrary weak coupling because only the Yang-Mills action affects the fluctuations around the instantons. The value of the Lagrangian (2.5) at the saddle points is 2πiτ × (instanton number) because F =F . It is interesting that there is a coupling constant dependence in the computation using the localization for the N = 1 SUSY gauge theory on S 1 × S 3 .
For the chiral multiplets on S 1 × S 3 , the SUSY transformation were given as where r is the R-charge of φ. Under the SUSY transformation, the following kinetic term is invariant: We can easily see that the usual superpotential terms are the SUSY invariant if the R-charge is 2.
For ǫ = 0, we have δφ = 0, δF = 0. Thus any gauge invariant combination of the lowest components of the chiral multiplets is invariant under the SUSY transformation generated by the δ with ǫ = 0. Now let us consider the following δ-exact Lagrangian: whereη is a Grassmann even spinor defined bȳ is an SU(2) matrix. Note thatǭ = U 1 0 and then we haveηǭ = 1. This does not seem to keep the rotational symmetry of S 3 because of the presence ofη, however, this keeps it. Indeed, we can see that this Lagrangian is the anti-chiral part of the superpotential W (φ). 8 Thus, the correlators we consider do not depend on parameters in the anti-chiral 8 For the fermionic part, we have where the last equation can be shown by taking ǫ 1 = 1, ǫ 2 = 0. Thus the LW is the anti-chiral part of the superpotential.
superpotential. This can be regarded as a derivation of the holomorphy without using the superfields. Now we consider the following regulator Lagrangian: where the Hermite conjugate ( †) is defined such that it is positive definite. We have schematically. For a field with r = 0, the saddle point is trivial, i.e. φ = 0 and F = 0. However, in the flat space limit ( 21) and the saddle points are D m φ = 0 and F = 0. In order to explicitly compute the correlation functions, we further need to construct the instantons on S 1 × S 3 and to find the 1-loop determinant around them. Although it is important and interesting to perform these explicitly, we leave these problems in future. Instead, we will consider the N = 1 SUSY gauge theory on S 1 × R 3 which is considered as a small curvature limit of S 1 × S 3 .

Gaugino Condensation in SUSY Yang-Mills by the Localization
In this section, for simplicity, we will consider the N = 1 SUSY Yang-Mills theory with gauge group G which has a simple Lie algebra of rank r on S 1 × R 3 where the radius of S 1 is R. Here we require the periodic boundary conditions along S 1 for all fields. The action is given by (2.5). For this theory, the important correlation function which can be computed using the localization is the gaugino condensation which is known to be non-vanishing due to the strong coupling effects, i.e. the confinement effects. Indeed, this has been computed exactly for the theory on R 4 for the classical groups in [26,37,38,39] which was also computed using the holomorphy of the superpotential. Furthermore, it was also computed by using the R → 0 limit for any simple gauge group in [31]. Although R itself is not holomorphic variable, in [31], it was argued that the gaugino condensation does not depend on R|Λ| because of the holomorphy for the SUSY Yang-Mills theory and it is only the dimensionless combination the correlator can depend on.
Here, we would like to compute the gaugino condensation by using the localization technique. 9 The computation we will see below is essentially same as the one in [31,32] and then the followings will be almost a brief review of [31,32]. However, we need to be careful to check that the validity of the computation in R → 0 can be applied to our computation in t → ∞ limit with R fixed finite. We hope that our localization computation can be generalized to more general ones. In this section, we will use the notation and convention used in [32].
First, we will add the regulator action, whose bosonic part is −t ((F − ) 2 + D 2 ), to the original SUSY Yang-Mills action in order to compute Tr (λλ) . Note that the path-integral measure is defined at t = 0. Then, taking t → ∞ limit, the path integral localized to the saddle point configurations which satisfy F − = 0 and D = 0. Here, the 1-loop determinant around the saddle points may be trivial because we are considering the flat space R 3 with the Euclidian time. 10 The saddle points (and every configurations) are characterized by the Wilson loop along S 1 , the instanton charge and the magnetic charges. The Wilson loop is defined by which should not be path-integrated and considered as a moduli of the vacua because we consider the theory on S 1 ×R 3 which is non-compact for the three directions. Note also that the instanton number k is not necessary integer because we consider non-compact manifold.
Assuming the generic non-zero Wilson loop, we have the decomposition of gauge group G to U(1) r which enable us to define the r different magnetic charges. Among the saddle point configurations, we need the configurations which have precisely two zero modes of the fermions λ under the anti-self-dual configurations in order to give a non-zero contribution to the gaugino condensation because it is a bi-linear of the fermions. These configurations are identified and are called the fundamental monopoles in [31,32]. These fundamental monopoles are usual BPS magnetic monopoles parameterized by i = 1, · · · , r which is related to the embedding of SU(2) into G, and the KK monopole of [42]. It is known that the 1-instanton without magnetic charges on S 1 × R 3 can be considered as a bound state of these r + 1 monopoles.
The BPS magnetic monopole has magnetic charge g given by α * i where α i is a simple root. The instanton charge k of it is 1 2π α * i · φ and then the classical action is Here α * ≡ 2α/(α · α). For the saddle points, the regularized action vanishes by definition, of course. For the KK monopole, we have g = α * 0 , k = 1 + 1 2π α * 0 · φ , and where α 0 is the lowest root which satisfies r i=0 k * i α * i = 0 with k * i is the Kac labels. The two fermionic zero modes are given by the SUSY transformation where the limit means |x µ | → ∞ and S F = σ µ x µ /(4π|x ν | 3 ) is the massless fermion propagator in 3d. Before computing the correlators, we need to find the quantum vacua of the theory. Here we will find the quantum vacua and compute the gaugino condensation at t → ∞ limit where the theory is weak coupling. Later, we will consider the original theory at t = 0.
The classical massless fields for the t → ∞ limit should be U(1) r Abelian multiplets with zero KK momentum of S 1 , which can be regarded as 3d fields. Because in the t → ∞ limit the zero modes and non-zero modes are decoupled each other, we can forget about the non-zero KK momentum modes. Thus the bosonic part of them are the Wilson loop scalars φ and dual photon scalars σ, which can be combined to r complex scalars z = i(τ φ + σ). (3.6) Thus the z are the classical moduli of vacua. By checking the SUSY transformations, we can see that this z forms the N = 1 chiral multiplets X with a rescaled massless fermions ψ = 2 5 2 π 2 R g 2 λ, where we integrated out the auxiliary fields D and have considered the on-shell multiplets. The kinetic terms are given by the original action as where no potential terms appears. Now we will compute the scalar potential semi-classically and determine the vacua. Note that our regularization term is the SUSY Yang-Mills term with the non-zero theta term, thus it is invariant under all SUSY transformations. Therefore, instead of the scalar potential, we will compute the fermion bi-linear terms which are related to the superpotential and the scalar potential by the SUSY transformation. The fermion bi-linear can be non-zero only for the fundamental monopole configurations, on which the original action is evaluated to the following value: for the j-th fundamental monopole. Note that the v.e.v. of the dual photon σ also contributes to it because the additional action i 4π d 3 xǫ µνρ σ∂ µ F νρ which forces the Bianchi identity gives the boundary term: The path-integral measure of the zero modes of the j-th monopole is given by where a µ is the position on R 3 , Ω is the U(1) phase and ξ is the Grassmann odd zero modes. Note that this includes the cut-off scale µ and the gauge coupling g. These two are defined by the original theory, especially g is defined by the original action because the path-integral measure is defined at the original action, i.e. t = 0. Using this measure and the asymptotic form of the fermionic zero modes, we have for |x| → ∞. The superpotential which gives the contributions of r + 1 fundamental monopoles is found to be (3.12) The vacua can be fixed by dW dX = 0 to and which has c 2 roots corresponding to c 2 vacua. Here c 2 = r i=0 k * i is the dual Coxeter number. We can compute the gaugino condensation by evaluating the (3.11) by the fermion zero modes without taking the asymptotic form. More conveniently, we can use the relation which can be derived by uplifting the τ to a superfield and Λ is the dynamical scale in the Pauli-Villars renormalization scheme at 2-loop order, and b 0 = 3c 2 . Using this relation, we finally find which does not depend on R. For example, Trλ 2 16π 2 = ±Λ 3 for G = SU(2). Now we will consider the original theory at t = 0. We will denote (· · · ) t as the correlator of the original theory, but choosing the vacuum (or the boundary condition at spatial infinity) as the one of the theory with the regulator action with t. Note that the vacua of the theory deformed by the regulator action is not need to be the vacua of the original theory in general. Then, we have where we have used δ(O)e −t δV t = 0. However, this will diverges if the vacuum at t + ∆t is not the vacuum at t. Now we assume the smoothness of changing the parameter t. 11 Furthermore, we have seen that the moduli space of vacua of the theory at t → ∞ is discrete. Thus, the vacuum is independent of t, which means that ∂ ∂t O 1 · · · O m e −t δV t = 0. Therefore, the gaugino condensation (3.17), which is correct value [32], is valid at t = 0, i.e. the original theory.

Including Chiral Multiplets
Let us consider the chiral multiplets. By the regulator action for the vector multiplets, the effective gauge coupling determined by t can be arbitrary weak. Thus, without introducing the regulator action for the chiral multiplets, the chiral multiplets can be integrated out first where the vector multiplets as the background fields. Indeed, for the (massive) SUSY QCD, the matters can be integrated out first trivially and the dynamical scale of the resulting SUSY Yang-Mills theory is computed by the usual way. Then, we can easily see that the resulting gaugino condensation is correct one.
If there is interaction terms in the superpotential, the effective superpotential after integrating out the chiral multiplets is non-trivial function of S = 1 32π 2 W α W α . This is the case for the Dijkgraaf-Vafa conjecture [43]. In order to evaluate the gaugino condensation, we need to take into account not only the fundamental monopoles, but general anti-self-dual configurations because of the interactions which are induced by the chiral multiplets. This would be rather difficult. It could be useful to introduce the regulator action for the chiral multiplets for this case. We hope to report further progress for this in near future.
A 4d N = 1 SUSY Gauge Theory on S 4 In this appendix, we will explicitly construct the SUSY transformations and the SUSY invariant actions for 4d N = 1 SUSY gauge theories on S 4 . 12 However, as we will see in later, it may be impossible to construct a SUSY exact regulator term with a (semi)-positive definite bosonic part because (δ ξ ) 2 can not be real nor pure imaginary. First, we will construct N = 1 SUSY theories from the N = 2 SUSY theories on S 4 , which are realized by a form given in [20,6], by a truncation of the fields. The notation in this section is the one used in [14,20]. Especially, the indices µ, ν, · · · runs from 1 to 4.
The metric of S 4 is taken to be where r 2 = 4 n=1 (x n ) 2 and we find e a = f δ a n dx n with f ≡ (1 + r 2 4ℓ 2 ) −1 . We can embed the S 4 in R 5 as Y 2 1 + · · · + Y 2 5 = l 2 . The relation between x n and Y n (n = 1, We assume the following Killing Spinor equation: Using the traceless 2 × 2 matrix t J I , which satisfies 12 For the chiral multiplets, they were already explicitly represented in [19].
the Killing Spinor equation is solved by where i, j = 1, . . . , 4 which are 4d flat indices and ǫ I , ǫ ′ I are constants, and Therefore, by the SU(2) R transformation, we will choose 13 For the N = 2 vector multiplets. the N = 2 SUSY variation of fields on S 4 was given by where m = 1, . . . , 5, A 5 is a scalar, F µ5 = D µ A 5 and D 5 ( * ) = −i[ * , A 5 ]. Note that the SU(2) R transformation including t J I acts covariantly on the SUSY transformation. It was shown that the commutator of the two SUSY generators is a sum of a translation (v m ), a gauge transformation (γ + iv m A m ), a dilation (ρ), an R-rotation (R IJ ) and a Lorentz rotation (Θ ab ): In [20,14], t ∼ σ 3 was chosen. Our choice here is more convenient for N = 1 SUSY case. Now let us consider the hypermultiplets, The system of r hypermultiplets consists of scalars q A I , fermions ψ A and auxiliary scalars F A I . Here, I = 1, 2 is the SU(2) R-symmetry index and A = 1, · · · , 2r. The fields obey the reality conditions where ǫ IJ , C αβ , Ω AB are antisymmetric invariant tensors of SU(2) ≃ Sp(1), Spin(5) ≃ Sp(2) and the "flavor symmetry" of r free hypermultiplets Sp(r). The coupling to vector multiplets can be introduced via gauging a subgroup of Sp(r). In the Euclidian signature, we regard the fields are holomorphic variables, and then we will forget these reality conditions. (Note that two complex fields with a reality condition have two real components and two holomorphic fields without conditions also have two components.) To introduce the coupling to gauge fields and other fields in the vector multiplet, we need first to introduce the covariant derivative Requiring Ω AB to be gauge-invariant, one finds (A m ) AB ≡ Ω AC (A m ) C B to be symmetric in the indices A, B.
The N = 2 SUSY transformation was given by (A.12) Here,ξ I ′ is a constant spinor which satisfies The square of δ is which is consistent with the one for the vectormultiplets. Now we take a SUSY generator with a Killing spinor which satisfies 14 16) which is equivalent to where P ≡ Γ 5 (σ 3 ) J I . Then, we find i.e. P ξ I = ξ I . Because η − ξ + = 0 for arbitrary chirality + and − spinors, these Killing spinors satisfy the followings: This implies that we can takeξ For this choice, R ′ I ′ J ′ = 0. For the scalars and vectors, we will define the action of P as This means that, schematically, Now, we consider only the P = 1 fields as an N = 1 SUSY fields. The SUSY transformation on the N = 1 SUSY fields are defined just by taking P = −1 fields vanish in the SUSY transformation of N = 2 SUSY. Thus, the SUSY algebra is obtained by taking P = −1 fields vanish in the one for N = 2 SUSY, which is consistent.
Explicitly, we find where λ + 1 = (1 + Γ 5 )/2 λ 1 and λ − 2 = (1 − Γ 5 )/2 λ 2 . If we define D ≡ D 12 , ξ ≡ ξ 1 + ξ 2 and λ ≡ λ + 1 + λ − 2 , we find which is same as the one in flat space except that the vierbein and the connections are the ones on S 4 . For the hypermultiplets, we can take We will take and we will change the notation for the index A, for example, . With this, we can consistently truncate the N = 2 fields to the P = 1 fields, i.e. N = 1 fields, and find where we have not summed over i and we defined q A,i ≡ q A,i i (no summation for i), This form is slightly different from the one in [19]. Forξ I ′ = iξ I , however, if we define we obtain the same form.
We will regard the fields as holomorphic and then forget the reality conditions. Indeed, we will not encounter any complex conjugate of the fields below.
Below we will try to construct SUSY invariant actions. We will drop the total divergent terms below for the notational convenience.
The N = 2 SUSY invariant action for vectormultiplets on S 4 is L vector where ∂ 5 = 0, which is the usual SUSY Yang-Mills Lagrangian of the vector multiplet used in [2] with some field redefinitions [20]. This action does not have any terms linear in fields with P = −1. Thus, the following truncated action is invariant under the N = 1 SUSY: which take the same form in the flat space. Now we will consider the hypermultiplets. We have the N = 2 SUSY invariant Lagrangian on S 4 : where, we have introduced the notationψ B ≡ ψ A Ω AB and suppress the indices A, B, · · · , such that Because this action does not have any term which is linear in the P = −1 fields, the following action is N = 1 SUSY invariant: where h AB ≡ Ω AB (−1) A = σ 1 ⊕ σ 1 ⊕ · · · ⊕ σ 1 .
(A. 35) We can easily see that the flat space superpotential terms written in F ′ = F + i l q is invariant under the N = 2 SUSY on S 4 because the superpotential is gauge invariant and the F ′ enters in the superpotential at most linearly. Note that the SUSY transformation is modified only for F ′ except the modification of the metric and connections. The SUSY transformationξ 2 Γ µ D µ ψ in F ′ gives an extra contribution D µξ2 Γ µ ψ, which indeed cancel with the one from the modified term in F ′ . The theta term is also N = 1 SUSY invariant because it is a topological term.
Note that q A,2 can be considered as (q A,1 ) † , thus the lowest components of the chiral superfields inserted at the north pole and the lowest components of the anti-chiral superfields inserted at the south pole are the SUSY invariants operators. (In the flat case, it is clear thatD 2 (Φ † ) ∼F is a chiral operator becauseD 3 = 0.) Now, we will try to apply the localization technique used in [2] to the N = 1 SUSY theory on S 4 . However, as we will see below, it may be impossible to construct a term S 4 δV which has a positive definite real part.
First we will try to construct S 4 δV on S 4 with a appropriate properties. We take ξ I as Grassmann-even spinors such that δ ξ is a fermionic transformation. We can easily see that ξ 1 ξ 1 = 0, which is followed form C T = −C and ξ 1 ξ 2 = 0 because Γ 5 ξ 1 = ξ 1 and Γ 5 ξ 2 = −ξ 2 . In order to use the localization technique, we need a regulator Lagrangian δ ξ V with S 4 (δ ξ ) 2 V = 0. The usual choice is a form like V = tr (δ ξ λ) † λ , where (δ ξ λ) † should be defined using the holomorphic fields and ξ I . However, if we assume the following form where M, N are arbitrary matrices, we find a contradiction. Indeed, the Γ 5 -chirality requires that M, N are diagonal, and we can assume (ξ I ) ⋆ = MCξ I . Then, we find ξ I = M * C * (ξ I ) * = M * C * MCξ I = −|M| 2 ξ I , which means ξ I = 0. We can use a tensor satisfying L v T = 0, however, as far as we have checked, there is no localization terms with a positive definite real part of bosonic terms. These difficulty will be originate from the nonzero complex value of the v µ = ξ I Γ µ ξ I even if we take ǫ 1 = 0.