A BRST gauge-fixing procedure for Yang-Mills theory on sphere

A gauge-fixing procedure for the Yang-Mills theory on an n-dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang-Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang-Mills field and associated fields are derived in manifestly O(n+1) covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.


I. INTRODUCTION
Manifestly O(n + 1)-covariant formulation of gauge theories on an n-dimensional sphere (or a hypersphere) has been discussed in several contexts [1,2,3,4,5,6,7,8]. Such a formulation was first developed by Adler to study massless Euclidean QED (quantum electrodynamics) on a hypersphere in 5-dimensional Euclidean space [1]. Infrared-finiteness of this theory was pointed out there as one of the advantages due to compactification of spacetime. Further study of massless Euclidean QED on a hypersphere was made in a manifestly O(n+1)-covariant way by using the dimensional regularization [2] and it was extended to the case of Yang-Mills theory [3]. The manifestly O(5)-covariant formulation was also applied to analyzing classical and semi-classical behaviors of the pseudoparticle solution in the SU (2) Yang-Mills theory [4,5,6]. The connection between the axial anomaly and the Atiyah-Singer index theorem was illustrated with the manifestly O(n + 1)-covariant formulation in the cases of n = 2 and n = 4 [7]. Recently, the manifestly O(n+1)-covariant formulation was reconsidered with the aid of conformal Killing vectors [8]. There, in addition to the Yang-Mills theory, a rank-2 antisymmetric tensor gauge theory and the case involving spinor fields were discussed by using formulas derived with the aid of conformal Killing vectors and spinors.
To study the quantum-theoretical properties of a gauge theory, we have to introduce a suitable gauge-fixing condition to this theory. In the study of massless Euclidean QED on a hypersphere, Adler adopted a gauge-fixing condition appropriate for manifestly O(n + 1)covariant analysis [1]. This condition makes the analysis quite simple and is applicable to higher-dimensions and to the case of Yang-Mills theory. However, Adler's condition is unusual in the sense that it has an extra free index. For this reason, Adler's condition is not favorable for the gauge-fixing procedure based on the Becchi-Rouet-Stora-Tyutin (BRST) invariance principle (or simply BRST gauge-fixing procedure) proposed by Kugo and Uehara [11,12]. In fact, Adler's condition has not been treated in connection with the first-order formalism of gauge-fixing [9] and with the BRST symmetry [10,12].
The purpose of this paper is to apply the BRST gauge-fixing procedure to the Yang-Mills theory on a hypersphere in such a way that manifestly O(n + 1)-covariance of the theory is maintained. To this end, we propose a gauge-fixing condition adapted for the BRST gaugefixing procedure. Our condition is equivalent to a generalization of Adler's condition, but has no extra free indices. A desirable gauge-fixing term is thus defined at the action level, and it is shown, with the aid of conformal Killing vectors, that this gauge-fixing term yields results compatible with those for the Yang-Mills theory on the flat space.
The paper is organized as follows: Section 2 is devoted to a brief review of manifestly O(n + 1)-covariant formulation of the Yang-Mills theory on a hypersphere. Section 3 dis-cusses the gauge-fixing conditions. Here the equivalence of a generalization of Adler's condition and our gauge-fixing condition without extra free indices is proved. Section 4 treats the BRST gauge-fixing procedure for the Yang-Mills theory on a hypersphere. In section 5, field equations for the Yang-Mills field and associated fields are derived in manifestly O(n + 1) covariant or invariant forms. They are shown to reproduce, in the large radius limit, the corresponding field equations defined on the n-dimensional flat space. Section 6 contains concluding remarks.

II. YANG-MILLS THEORY ON HYPERSPHERE
In this section, we briefly review a manifestly O(n + 1)-covariant formulation of the Yang-Mills theory on a hypersphere. Following the literature [1,4,5,7,8], we consider an n-dimensional unit sphere (or a unit hypersphere) S n 1 embedded in (n + 1)-dimensional Euclidean space R n+1 . The hypersphere S n 1 is characterized by the constraint r a r a = 1 imposed on Cartesian coordinates (r a ) (a = 1, 2, . . . , n + 1) on R n+1 . One may use (r µ ) (µ = 1, 2, . . . , n ; 0 ≤ r µ r µ ≤ 1) as local coordinates on S n 1 , treating r n+1 = ± 1 − r µ r µ as a dependent variable. In this treatment, the angular momentum operators L ab read or more concisely Noting that one can show that the operators in Eqs. (1) and (2) satisfy the O(n + 1) Lie algebra In terms of the stereographic coordinates (x µ ) which are related to (r a ) by the angular momentum operators in Eqs. (1) and (2) are written as [8] where K µ a are the conformal Killing vectors Equation (6) illustrates a mapping from the flat space (or hyperplane)R n ≡ R n ∪ {∞} to S n 1 . The inverse mapping from S n 1 toR n is then illustrated by x µ = r µ /(1 + r n+1 ). The Killing vectors K µ a satisfy the transversality condition LetÂ a be a Yang-Mills field on S n 1 which takes values in a Lie algebra g;Â a can be expanded asÂ a =Â i a T i in terms of the Hermitian basis {T i } of g which satisfy the algebra [T i , T j ] = if ijk T k with the structure constants f ijk . The normalization conditions Tr(T i T j ) = δ ij are also put for convenience. One may regardÂ a as a function of the independent variables (r µ ). The Yang-Mills fieldÂ a is assumed to live on the tangent space at a point P (r µ ) on S n 1 by imposing the transversality condition This implies that one component of (Â a ), for instanceÂ n+1 , depends on the other components, such asÂ n+1 = −(r µÂµ )/r n+1 . The differentiation of the condition (10) with respect to r µ is carried out to get Here Eq. (4) has been used. Equation (11) may be written as because, in the case a = µ, it reduces to Eq. (11) and, in the case a = n + 1, both sides of Eq. (12) vanish so that it gives rise to no additional conditions. (Note that ∂ a is understood The infinitesimal gauge transformation ofÂ a is given by [8] where λ is an infinitesimal gauge parameter taking values in g, L ab are covariantized angular momentum operators while P ab and D a are the tangential projection operator and the covariant derivative, respectively: The projection operator P ab in Eq. (13) guarantees that the Yang-Mills field transformed according to the rule (13), i.e.,Â a + δ λÂa , lives on the same tangent space.
The field strength ofÂ a can be written in a manifestly O(n + 1)-covariant form [4,5,8]: whereF ab is defined byF The gauge transformation ofF ab is found from Eq. (13) to be where Eqs. (4), (10) and (11) have been used. This is an inhomogeneous transformation involving terms that are not manifestly O(n + 1) covariant. Since field strengths should transform homogeneously, one cannot takeF ab as the field strength ofÂ a , even thoughF ab looks like the field strength as far as one sees only Eq. (18). Instead ofF ab , the rank-3 tensorF abc transforms homogeneously under the gauge transformation, HenceF abc has the property of field strength. With the field strengthF abc , the Yang-Mills action forÂ a is given by Here dΩ is an invariant measure on S n 1 which is written in terms of the coordinates (r µ ) as Obviously, the action S YM is gauge invariant. The variation of S YM can be calculated through integration by parts over (r µ ), which is performed by taking into account the factor |r n+1 | −1 contained in dΩ. Using Eq. (4) and noting that S n 1 has no boundaries, we obtain

III. GAUGE-FIXING CONDITIONS
For studying quantum-theoretical structure of the Yang-Mills theory on the hypersphere S n 1 , it is necessary to consider the gauge-fixing procedure in the theory. We here focus our attention on discussing gauge-fixing conditions, before setting a suitable gauge-fixing term.
In a study of massless Euclidean QED on S 4 1 , Adler proposed a gauge-fixing condition iL abÂb =Â a [1]. This is, of course, useful for making the analysis quite simple, and is also applicable to higher-dimensions as well as to the case of Yang-Mills theory. However, Adler's condition is unusual in the sense that it has an extra free index a in comparison with the well-known Lorentz condition ∂ µ A µ = 0 ; for this reason, Adler's condition is not favorable to the ordinary first-order formalism of gauge-fixing [9,12]. In this section, we shall prove that Adler's condition is equivalent to the condition ir a L abÂb = 0 with no free indices. This condition, which has a form more similar to ∂ µ A µ = 0 than Adler's condition, is essentially the same as the one used in Refs. [5,6] in a somewhat different context where the oneinstanton background is present. Although Adler's condition (or its quadratic equivalent) was compared with another condition ir a L abÂb = 0 in the literature [3], it seems that a complete proof of the equivalence has not been given yet.
To incorporate a gauge parameter α into the gauge-fixing conditions in a manifestly O(n + 1)-covariant manner, we introduce the Nakanishi-Lautrup (NL) fieldB on S n 1 [9]. Then the above-mentioned conditions are generalized as in Eqs. (24) and (25) given below.
The generalization of Adler's condition, Eq. (24), is also unusual in the sense that it has a free index a. We now show the following.
Proposition: The following conditions (a) and (b) are equivalent.
Proof: Consider the condition (a). Contracting Eq. (24) by r a yields Eq. (25), owing to the constraint r a r a = 1 and the condition (10). The condition (b) is thus derived from (a).
[ * ] Using Eqs. (10) and (12), we can readily show thatÂ a (iL abÂb −Â a ) = 0, which is compatible with the Next we shall derive (a) from (b). Using Eq. (12), r a r a = 1 and the condition (10), one can rewrite Eq. (25) as This is an alternative form of the condition (b). Substituting Eqs. (12) and (26) into From this formula, we readily see that the condition ir a L abÂb = 0 is equivalent to Adler's condition iL abÂb =Â a . Adding αr aB to the both sides of Eq. (28), we have iL abÂb −Â a + αr aB = r a (ir c L cbÂb + αB) .
This formula converts the conditions (a) and (b) into each other. Equation (29) will also be useful for simplifying field equations.

IV. BRST SYMMETRY AND A GAUGE-FIXING TERM
It is known that gauge-fixing is neatly performed by considering the BRST invariance as a first principle [11,12]. In the previous section, we have claimed that Adler's condition and its generalization, namely the condition (a), are not conventional owing to the presence of a free index. If one applies the BRST gauge-fixing procedure to the Yang-Mills theory on S n 1 , the condition (b) is much better to use than (a), though (a) and (b) are equivalent. The reason for choosing (b) is that it resembles the ordinary gauge-fixing condition ∂ µ A µ +αB = 0 which has often been adopted in the BRST gauge-fixing procedure [11,12]. With the condition condition (a). The equation derived here implies that the vector (iL abÂb −Â a ) is null or perpendicular to the vector (A a ) living in a tangent space of S n 1 . Whereas (iL abÂb −Â a ) can live in the same tangent space, the condition (a) requires that (iL abÂb −Â a ) is null or in the normal (or radial) direction, being perpendicular to S n 1 .
(b), it is easy to apply the conventional procedure to the present case since there is no free index.
The BRST transformation δ is defined forÂ a by replacing λ in Eq. (13) by the Faddeev-Popov (FP) ghost fieldĈ on S n 1 , By putting the transformation rule the nilpotency property δ 2 = 0 is guaranteed forÂ a andĈ. In particular, δ 2Â a = 0 is verified by using the property P ac P cb = P ab . In addition toĈ, the FP anti-ghost fieldĈ is introduced to satisfy Consequently the nilpotency property δ 2 = 0 is still held after incorporatingĈ andB.
Needless to say,Ĉ,Ĉ andB are treated as functions of (r µ ).
With the BRST transformation and the relevant fields in hand, we can discuss the BRST gauge-fixing procedure for the Yang-Mills theory on S n 1 . To deal with the condition (b) in a BRST symmetric manner, we now take the sum of gauge-fixing and FP ghost terms written in the following form: [ †] S GF = dΩ − iδTr Ĉ ir a L abÂb + α 2B .
Since this is a BRST-coboundary term, the BRST invariance of S GF is guaranteed due to the nilpotency of δ. In contrast, the BRST invariance of S YM is clear from its gauge invariance.
[ †] It is possible to deal with Adler's gauge-fixing condition by taking Eq. (33). HereB a andĈ a are a NL field and a FP anti-ghost field of the vector type, respectively, satisfying δĈ a = iB a , δB a = 0. The necessity of introducing the vector type of NL and FP anti-ghost fields is due to the fact that Adler's gauge-fixing condition has a free index. AlthoughS GF works well as a sum of gauge-fixing and FP ghost terms, it involves redundant degrees of freedom caused by the vectorial property ofB a andĈ a . As a result, the discussion based onS GF is fairly complicated. Also, in this case, the symmetry between a FP ghost field and a FP anti-ghost field is spoiled. This turns out to be a serious problem when one applies the superfield formalism [13] of the BRST and anti-BRST symmetries to the Yang-Mills theory on S n 1 [14].
Carrying out the BRST transformation contained in the right-hand side of Eq. (33) and using the formula we have Integrating by parts over (r µ ) and using Eqs. (4), (10) and r a r a = 1, we may rewrite Eq. (35) as Further integration by parts for the ghost term in Eq. (36) leads to with use of the formula Quantization of the fieldsÂ a ,Ĉ,Ĉ andB is performed based on the total action in a systematic way; thereby one may see quantum-theoretical structure of the Yang-Mills theory on S n 1 . Details of the quantization will be discussed elsewhere.

V. FIELD EQUATIONS
From the total action S, we can derive the Euler-Lagrange equation for each field, ir a L abÂb + αB = 0 , The field equation (40), which is directly found from Eqs. (23) after using Eq. (43) and r a r a = 1. In the Abelian case, using the commutation relations (5) and [L ab , r c ] = −i(r a δ bc − r b δ ac ), as well as Eqs. (29) and (41), we can rewrite Eq. (40) as They are the spherical analogues of the field equations on the flat space given in the literature [10,12]: where Comparing Eqs. (40)-(44) with Eqs. (46)-(50), we see that L ab and L ab correspond to ∂ µ and D µ , respectively. In the Yang-Mills theory on S n 1 , L ab and L ab are more fundamental than ∂ µ and D µ . This can be understood from the fact that translations on a plane are realized as rotations on a sphere, so that usual derivatives are replaced by angular derivations when one discusses on a sphere. It is possible to establish the correspondence between Eqs. (40)- (44) and Eqs. (46)-(50) through the following discussion.
The Yang-Mills fieldÂ a on the hypersphere S n 1 and the conventional one A µ on the flat spaceR n is related by the conformal Killing vectors K µ a [8]: The field strengthF abc can be expressed as [8] F with F µν defined by Eq. (52). Similarly to Eq. (53), the fieldsB,Ĉ andĈ on S n 1 are related to the corresponding fields B, C andC onR n bŷ These relations may be understood from a supersphere formulation of the Yang-Mills theory on S n 1 [14]. From Eq. (7), it follows that where r a r a = 1 and the condition (9) has been used. A covariantized version of Eq. (56) is also satisfied owing to Eq. (10), with D µ defined by Eq. (51). We now rewrite the field equations (40)-(44) in terms of the relevant fields onR n . To this end, the following formulas are particularly useful: Equation (62) may be regarded as a combination of the ordinary gauge-fixing condition ∂ µ A µ + αB = 0 and the Fock-Schwinger gauge condition x µ A µ = 0 [15], involving an extra factor (1 + x 2 ) −1 . In the case of α = 0, Eq. (62) is essentially a higher-dimensional analogue of the gauge-fixing condition proposed by Ore [5]. An analogue of Eq. (63) is also seen in Ref. [5]. In the present paper, however, Eqs. (62) and (63) have been derived systematically by considering the BRST symmetry and using the conformal Killing vectors.
We now make a replacement of r a by r a /R to explicitly incorporate the radius R of a hypersphere into the field equations. Thereby, the hypersphere S n 1 is scaled to be S n R characterized by r a r a = R 2 , and the Yang-Mills theory in question is formulated on S n R . By taking the limit R → ∞, the hypersphere S n R approximates to the flat space, i.e. S n ∞ = R n . To maintain the relation between two coordinates systems (r µ ) and (x µ ) even in the large radius limit, x µ has to be replaced by x µ /R. Correspondingly, the derivative gets replaced as ∂ µ → R∂ µ . Accordingly, the relevant fields on the flat space have to be modified appropriately. For instance, to maintain homogeneity of the changes, A µ → RA µ , leading to F µν → R 2 F µν . Any other transformation for A µ would lead to different transformations for the derivative piece and the commutator piece in F µν leading to a lack of homogeneity. We can also express Eq. (45) as an Abelian field equation for A µ and B written in terms of the stereographic coordinates. After making the replacement mentioned above in this equation, the term corresponding to the third term of Eq. (45) vanishes in the limit R → ∞. Hence, in the Feynman gauge α = 1, the NL field B is completely removed and the Abelian field equation turns out to be the Laplace equation ∂ µ ∂ µ A ν = 0.

VI. CONCLUSIONS
After briefly reviewing a manifestly O(n + 1)-covariant formulation of the Yang-Mills theory on the hypersphere S n 1 , we have discussed a gauge-fixing procedure for this theory by considering the BRST invariance as a first principle. It was stressed that although Adler's gauge-fixing condition is useful for concrete analyses, it is unfavorable for the BRST gaugefixing procedure owing to the presence of an extra free index. Instead of Adler's condition, we proposed a suitable gauge-fixing condition which is equivalent to a generalization of Adler's condition, but has no extra free indices. A complete proof of the equivalence was given by introducing the NL field.
Having obtained the suitable gauge-fixing condition, the BRST gauge-fixing procedure was applied to the Yang-Mills theory on S n 1 , toward investigating its quantum-theoretical properties. The gauge-fixing and FP ghost terms, as well as the Yang-Mills action, were written in manifestly O(n + 1)-invariant forms with the aid of the angular momentum operators and their gauge-covariantized versions. Consequently, the field equation for each of the Yang-Mills, NL, FP ghost, and FP anti-ghost fields was derived in a manifestly O(n + 1) covariant or invariant form. All the field equations were also written in terms of the stereographic coordinates using the conformal Killing vectors which act like a metric in translating formulas from the local coordinates (r µ ) to the stereographic coordinates (x µ ) and vice-versa.
Then it was shown that these equations reduce to corresponding field equations defined on the n-dimensional flat space, in the limit where the radius of the hypersphere is taken very large.
In quantizing the relevant fields in the Yang-Mills theory on the hypersphere S n R , one can expect that infrared divergences are automatically regularized, because S n R is a bounded space with a maximum length. For this reason, the Yang-Mills theory on S n R would be appropriate for studying properties of quantum chromodynamics (QCD) at a low-energy region, such as the gluon condensate of dimension 2 [16]. Also, the BRST symmetry discussed in this paper may be suitable for the geometric approach to non-Abelian chiral anomalies with the use of decent equations and BRST cohomology [17], because, there, space-time is assumed to be essentially the hypersphere S n R . In such an approach, non-Abelian chiral