Gauge and Scalar Fields on $\mathbb{CP}^2$: A Gauge-invariant Analysis II. The measure for gauge fields and a 4d WZW theory

We consider the volume of the gauge orbit space for gauge fields on four-dimensional complex projective space. The analysis uses a parametrization of gauge fields where gauge transformations act homogeneously on the fields, facilitating a manifestly gauge-invariant analysis. The volume element contains a four-dimensional Wess-Zumino-Witten (WZW) action for a hermitian matrix-valued field. There is also a mass-like term for certain components of the gauge field. We discuss how the mass term could be related to results from lattice simulations as well as Schwinger-Dyson equations. We argue for a kinematic regime where the Yang-Mills theory can be approximated by the 4d-WZW theory. The result is suggestive of the instanton liquid picture of QCD. Further it is also indicative of the mechanism for confinement being similar to what happens in two dimensions.


Introduction
In this paper, we continue our analysis of quantum fields on the manifold CP 2 focusing on the gauge-invariant volume element relevant to the functional integration over gauge fields [1]. As is well known, in a gauge theory, the physical degrees of freedom correspond to the space of all gauge potentials (A) modulo the set of all gauge transformations which are set to the identity at some chosen point of the spacetime manifold (G * ). The volume element on the gauge orbit space C = A/G * is what appears in the functional formulation of gauge theories, eliminating the redundant degrees of freedom corresponding to gauge transformations. Despite being a key foundational ingredient for the quantum description of gauge theories, there is no satisfactory expression for this volume element for four-dimensional nonabelian gauge theories [2]. While perturbation theory is well understood by use of gauge-fixing and Faddeev-Popov ghosts, or, equivalently, via the BRST (Becchi-Rouet-Stora-Tyutin) procedure, an analytic approach to nonperturbative questions such as confinement remains elusive.
The motivation to consider nonabelian gauge theories on the manifold CP 2 is from two and three dimensions. In two dimensions, it is possible to calculate the volume element for C exactly in terms of a Wess-Zumino-Witten (WZW) action [3]. The same calculation placed within a Hamiltonian formulation of (2+1)-dimensional gauge theories has led to an analytic formula for the string tension [4,5] which agrees very well, to within about 2%, with the results from numerical simulations [6]. Another prediction regarding the Casimir effect for nonabelian gauge theories also seems to agree, to within one percent or so, with numerical estimates [7]. The key feature which facilitated such calculations was the complex structure of the two-dimensional spaces and an associated parametrization of gauge potentials which allowed for factoring out gauge transformations and reduction to gauge-invariant degrees of freedom in a simple way.
The manifold CP 2 then emerges as the natural candidate for a similar scenario in four dimensions. This is a complex Kähler manifold with a standard choice of metric as the Fubini-Study metric which is given in local complex coordinates z a ,zā, a = 1, 2,ā = 1, 2, by ds 2 = dz · dz (1 + z ·z/r 2 ) −z · dz z · dz r 2 (1 + z ·z/r 2 ) 2 = g aā dz a dzā (1) Here r is a scale parameter defining the volume of the space as π 2 r 4 /2. As r → ∞, the metric becomes flat (albeit modulo some global issues), so that one can compare results with expectations in flat space. The main advantage is that CP 2 = SU(3)/U (2), so that, utilizing group theoretic techniques, one can obtain a parametrization of gauge fields similar to what was obtained in two dimensions. This was indicated in [8] and explained in detail in [1]. In the latter paper, we calculated the leading quantum corrections, i.e., monomials of fields and derivatives of the lowest dimensions generated by loops, due to a chiral scalar field on CP 2 coupled to gauge fields. The effective action from integrating out the scalar fields comprised of a quadratic divergence corresponding to a possible gauge-invariant mass term, standard logarithmic divergences corresponding to wave function and/or coupling constant renormalization and a finite WZW action, which is a dimensionally upgraded version of the 2d-WZW action. The natural next set of questions will be about contributions due to the gauge fields themselves, with the volume element on C being a key part of the one-loop results. This is the subject of the present paper.
The organization of the paper is as follows. The parametrization of the gauge fields and the factoring out of the gauge degrees of freedom are reviewed in section 2. The formal expression for the volume element for the gauge orbit space C = A/G * is given in section 3, where we also identify the relevant Jacobian determinant to be calculated.
In addition to the scalar propagator on CP 2 (with hypercharge Y = 0), which was calculated in [1], we will need the propagator for an antisymmetric rank-2 tensor (with Y = −2). This is calculated in section 4. A covariant point-splitting regularization, consistent with the isometries of CP 2 , is discussed in section 5. In section 6, we give the key results of our calculations. We have calculated the terms of the lowest scaling dimension (≤ 4), which are presumably the most relevant for the long wavelength modes of the fields. These include a four-dimensional WZW term with a finite coefficient, a masslike term with a quadratically divergent coefficient and a set of log-divergent terms of dimension 4. The physical implications of these results are discussed in section 7. There are two appendixes which give details of the calculations for the WZW term and the UV divergent terms.

Parametrization of gauge fields
We start by recalling the parametrization of the gauge fields introduced in [1]. The manifold CP 2 is taken to be the group coset space CP 2 = SU(3)/U (2), so that it can be coordinatized in terms of a group element g ∈ SU (3), with identification g ∼ gh, h ∈ U(2) ⊂ SU (3). Thus U(2) defines the local isotropy group. As a result, vectors, tensors, etc. transform as specific nontrivial representations of U(2).
Consider the group g ∈ SU(3) defined in its fundamental representation as a 3 × 3 unitary matrix g of unit determinant. It may be taken to be of the form g = exp(it a ϕ a ), where t a form a basis for traceless hermitian 3 × 3 matrices, with Tr (t a t b ) = 1 2 δ ab , and ϕ a are the coordinates for SU (3). The SU(2) ⊂ U(2) subgroup is the standard isospin subgroup, defined by the upper left 2 × 2 block, and corresponding to the generators t a , a = 1, 2, 3. The U(1) part of U(2) is defined by the hypercharge transformations, with the generator Y = 2 t 8 / √ 3. We also define right translation operators on g by In terms of the frame fields E a i for SU (3), we may write these as differential operators, Translations on CP 2 correspond to the coset directions t a , a = 4, 5, 6, 7 and we can define the complex translation operators as These will be denoted by R i , Rī, i,ī = 1, 2.
Functions on CP 2 are invariant under the U(2) subgroup. So they admit a mode expansion of the form Here D (s) AB (g) = s, A|ĝ|s, B are the finite-dimensional unitary representation matrices for SU(3) and in (4) the states on the right, namely, |s, w are invariant under the U(2) where T a are the matrix representatives of t a in the representation designated by s. The invariance condition for functions may therefore be stated as T a |s, w = 0, a = 1, 2, 3, 8.
In more detail, we can specify a state |s, A = |a 1 a 2 · · · a p , b 1 b 2 · · · b q corresponding to a finite dimensional SU(3) representation of the form T a 1 a 2 ···ap b 1 b 2 ···bq , a i , b j = 1, 2, 3. We will refer to this as a (p, q)-type representation. These are totally symmetric in all the upper indices a i 's and totally symmetric in all the lower indices b j 's with the trace (or any contraction between any choice of upper and lower indices) vanishing. Under the For functions, we need s, A|ĝ|s, w , with states of the (p, p)-type, with |s, A being general and |s, w = |333 · · · , 333 · · · (8) Vectors on CP 2 must transform the same way as R ±i , so they must be doublets of SU(2) ⊂ SU(3) and must carry hypercharge Y = ±1. Derivatives of functions of the form R ±i f obviously satisfy this requirement. Another possibility is given by representations of the type (p + 3, p) and (p, p + 3), so that a general parametrization of a vector takes the form The particular state |j33 · · · , 33 · · · (of the (p + 3, p)-type) can be obtained, by the application of Rj, from the SU(2) invariant states, with all indices equal to 3, as η iī ǫ¯ij|j33 · · · , 33 · · · = η iī ǫ¯ij Rj |33 · · · , 33 · · · (10) where η iī = δ iī is the metric for CP 2 in the tangent frame and ǫ¯ij is the Levi-Civita tensor.
With a similar result for the conjugate representation, the parametrization (9) can be written as In a coordinate basis, rather than the tangent frames we have used above, this becomes where χkm andχ km are antisymmetric rank-two tensors, which for dimensional reasons, are proportional to ǫkm and ǫ km and so can be reduced to χ andχ.
The nonabelian generalization of (12) leads to the following parametrization for the gauge potentials: In these expressions, M and M † are complex matrices which are elements of SL(N, C) if the gauge group is SU(N); i.e., they are N × N complex matrices with determinant equal to 1. (More generally, M, M † will be in the complexification of the gauge group.) Further, φ¯ij = ǫ¯ijφ, φ †ij = ǫ ij φ † are also complex N × N matrices in the Lie algebra of SL(N, C). We also use an antihermitian basis for the fields, so thatĀī = −(A i ) † . The derivatives D j andDj in (13) are defined by We have written these in terms of the action on a field Φ (like φ or φ † ) which transforms under the adjoint representation of the gauge group, i.e., as (∇ j ,∇j) are also taken to be Levi-Civita covariant, so that (13) behave correctly under gauge and coordinate transformations.
It is possible to write the parametrization (13) in terms of manifestly gauge-invariant fields by using the identities are defined using the connections H −1∇j H, −∇ j HH −1 ; i.e., By virtue of (15), the parametrization (13) can be written as Another equivalent version is given by Notice that a i ,āī obey the conditions so that, effectively, they have only one independent component each.  (19). Yet another choice, also equivalent to the above mentioned ones, would be From the point of view of carrying out the functional integration, these fields are the coordinates for the gauge-orbit space C. A polar decomposition of M as M = Uρ, where ρ is hermitian allows us to factor out the gauge degrees of freedom and define a volume measure on C = A/G * . This will be taken up in the next section.
An interesting feature which is worthy of comment is the holomorphic ambiguity or redundancy of the parametrization (13) or (18). Notice that (M, a i , M † ,āī) So there are no additional degrees of freedom associated with this. However, as mentioned in [1], this feature may be useful with locally defined V andV to express fields in a nonsingular way in various coordinate patches.
We close this section with another comment on the uniqueness of the parametrization of the fields. This will be important for the metric and volume element which we consider in the next section. We have argued, based on the group theoretic counting of functional degrees of freedom, that any vector can be parametrized as in (11); hence any gauge potential on CP 2 can be written in terms of M, M † , χ, χ † . The construction of A i ,Āī from the data (M, M † , χ, χ † ) is thus clear. Conversely, we can ask whether we can construct (M, M † , χ, χ † ) from (A i ,Āī). This can indeed be done, as explained in some detail in [1]. The fields (M, M † , χ, χ † ) will be nonlocal functions of A i ,Āī and their derivatives, consistent with the fact that there is no way to factor out the gauge degrees of freedom and obtain (unconstrained) gauge-invariant degrees of freedom in a local way in terms of A i ,Āī.

The metric and volume element for C
We now turn to the metric on the space of gauge potentials (A) and the reduction of the associated volume element to the gauge orbit space A/G * . The starting point is the standard Euclidean metric on the space of the fields A, given by Here dµ denotes the volume element for CP 2 . In terms of the parametrization of the fields (13), we find We have also used the definition of a i ,āī from (18). Upon using (21) in (20) and carrying out some integrations by parts, we find We can write this as a quadratic form We have written this in terms of components in the Lie algebra of the gauge group SU(N) by taking ξ = ξ α (−it α ) where t α form a basis for the Lie algebra. It is useful to compare this with the metric In the second line, we used the fact that From the structure of (23), we see that the volume element takes the form dV = √ det M dV 0 , where dV 0 is the volume element corresponding to (24). The first term in (24) is the integral over CP 2 of the Cartan-Killing metric for SL(N, C). The corresponding volume element is thus the product over all points of the Haar measure for SL(N, C). Using the polar decomposition M = Uρ, where ρ is hermitian, and writing out the differential form of the top rank, we can see that this gives the volume element where dU is the volume of SU(N), and dµ(H) is the Haar measure for SL(N, C)/SU(N). Further, since H αβ has unit determinant, we can write the volume Finally going back to (23), we get the corresponding volume element as We can now factor out the volume of gauge transformations, i.e., factor out x [dU] to obtain the volume element for C = A/G * as The differentials appearing in this expression are for gauge-invariant fields.
The next step is the calculation of the determinant of M. For this, it is simpler to write the determinant as a functional integral over a set of auxiliary fields B,B, C,C. Explicitly, Equation (28) expresses the required determinant in the form of a functional integral for a standard field theory. We have taken C andC to behave like θ, θ † , so they are scalar fields. B andB correspond to δφ ′ and δφ ′ † , so they are fields with Y = 2 and −2, respectively. The terms in S 1 involve powers of a,ā, while the differential operators in S 0 only depend on H via the covariant derivatives. Our strategy will be to evaluate the integral in (28) as a perturbation series by expanding e −S 1 in powers of S 1 . The integral over e −S 0 will define the lowest order result. Writing Γ = log √ det M, we find The evaluation of the determinant det(−D · D) Y =0 was considered in [1]. This was done by writing its variation as The free "propagator" G = (−∇ · ∇) −1 for scalar fields on CP 2 was calculated exactly.
The propagator G(x, y) could then be obtained by expansion in powers of ∇HH −1 , since D, as defined in (16), involves this connection. With regularization to take care of singularities as y → x in the expression for Ĵ (x) , we obtained the leading terms in the expansion of log det(−D ·D) Y =0 in terms of the monomials of H and its derivatives of increasing scaling dimension. The lowest order term was a WZW action for H, somewhat surprisingly, with a finite coefficient. The next set of terms of dimension four were the H-dependent part of Tr(F 2 ) and Tr(g iī F iī ) 2 , with a logarithmically divergent coefficient.
There are also terms of higher dimension which will have finite coefficients, these were not explicitly calculated in [1]. We do not display the leading terms of det(−D · D) Y =0 at this point, they will be given later, together with some of the other terms in (31) which will be calculated presently.
The calculation of the leading terms in det(−D · D) Y =−2 will proceed as in the case of det(−D · D) Y =0 . Since we will again expand in powers of ∇HH −1 , the key ingredient for this will be the free propagator (−∇ · ∇) −1 , now for Y = −2 fields. This, as well as the issue of regularization for such propagators, will be taken up in the next section.
Once we have the propagators, the calculation of terms resulting from S 1 will be straightforward. We will focus on the terms of scaling dimension ≤ 4, corresponding to possible quadratic and logarithmic divergences. For these terms, it will suffice to consider terms up to four powers of S 1 .

The propagator for Y = −2 fields
In this section, we calculate the propagator for fields with Y = −2, such as the fieldB in (29), (30). This field was identified as the variation of φ † in φ †ij = ǫ ij φ † , where φ †ij is a second rank antisymmetric tensor. The Laplace operator on a second rank antisymmetric tensor with holomorphic indices will lead to the equation for the propagator for B. Rather than using the tangent frame basis, if we use the coordinate basis for CP 2 , we can write Since CP 2 is a Kähler manifold with potential K = log(1 +z · z), the metric and Levi-Civita connections are easily worked out as The other components of the connection vanish. Using these results, we find This identifies the kinetic operator onB. The propagator thus obeys the equation whereG(z, y) is the free propagator for Y = −2 fields, to be distinguished from G(z, y) which denotes the free propagator for scalars with Y = 0. In the case of scalar fields, we could take the propagator G to be a function of the distance between the two points corresponding to z and y. However, in the present case, there is an additional phase factor due to the presence of the operator z · ∂ −z ·∂ in (37). This is ultimately due to Y = −2 for the field, which implies that it couples to the Y -component of the curvature and connection. To isolate the phase factor, we note that theB field has the mode As a result, the propagator for the Y = −2 fields takes the form where d p = 1 2 (p+1)(p+4)(2p+5) is the dimension of the (p, p+3)-representation. In the second line of this equation, we have indicated the result in terms of products of g † y g z and its conjugate, where g is the 3 × 3 matrix g ∈ SU(3) used to define local coordinates in the coordinate patch we are using. There are also lower powers of (g † y g z ) 33 (g † z g y ) 33 for each representation, as indicated by the ellipsis in the square brackets. The point is that there is a common term (g † y g z ) 33 3 which leads to a phase factor. To see this more explicitly, we note that g ∈ SU(3) is related to the local coordinates as The square of the distance, s, between two points with local coordinates z i , y i can be taken to be This is clearly translationally invariant (under left translations g → ug, u ∈ SU (3)) and agrees with s =z · z when y = 0, i.e., for g y = 1. Thus, we see from (39) thatG is a function of s, apart from the prefactor Combining this with (39) we see that a general ansatz for the propagator can be taken Having identified the phase factor, we can now use (37) to calculate F (s). For z = y, or s = 0, the δ-function has no support and we can check that the homogeneous equation is solved by where C 1 , C 2 are arbitrary constants. Considering the short distance behavior, we see that we need C 1 = 1 2 to reproduce the δ-function on the right hand side of (37). Further, there should be no singularity as s → ∞, this identifies C 2 = 0. Combining (43) and (44), we get the propagator for Y = −2 fields as G(z, y) = (1 + s) where s = σ 2 z,y is as given in (41). The last factor which is the phase can also be viewed as arising from the path-ordered integral of the Levi-Civita connection along a line joining y and z.

Regularization
The regularization of the propagator for Y = −2 fields will use the same point-splitting which was used for scalar fields in [1]. The key idea was to define the regularized version of G(z, y) as G(z, y ′ ) where y ′ specifies a point displaced from y by a small distance of order √ ǫ. Thus ǫ will serve as the regularization parameter, with ǫ → 0 recovering the unregularized results. The shift y → y ′ must be done in a way consistent with the isometries and gauge symmetries. It is useful to work this out in terms of the In terms of these coordinates, the distance between the points, given in (41), can be written as Notice that this expression has invariance under the scaling Z → λZ, Y → λ ′ Y . In a particular coordinate patch with Z 3 , Y 3 = 0, we can write where We then recover the expression given in (41). With s written in homogeneous coordinates as in (46), (45) gives a globally valid expression for the propagator.
For the point-splitting regularization, we shift the point Y to Y ′ , which is chosen to Here α is a small complex number, |α| ∼ √ ǫ, and W parametrizes the shift of coordinates. The added term is chosen so as to have the same scaling behavior as Y . We then find that In equation (45), we can replace all factors of s by σ 2 (Z, Y ′ ) given by (49). Under coordinate transformations, the propagator must transform as u † (z)G(z, y)u(y), where u † is the hypercharge phase transformation with Y = −2. (u will correspond to Y = 2.) To preserve this property, we add an extra phase factor joining y and y ′ . We can view this extra term as a path-ordered integral for the Levi-Civita connection along a line joining y and y ′ , analogous to the Wilson line for the gauge fields (see below) that is needed to maintain gauge covariance of the regulator. Thus the UV regularized form of the propagator is given bỹ where as in [1] we will do a suitable angular averaging over the displacement due to point-splitting. (It will turn out that, for some of the calculations we do, one of the arguments of the propagator can be shifted to the origin by virtue of translational invariance. The details of how this can be done in terms of the homogeneous coordinates are given in [1]; see also Appendix A of this paper.) Since σ 2 (Z, Y ) and σ 2 (Z, Y ′ ) are covariant quantities respecting the full isometry (namely, left SU (3) transformation on Z or Y ) of CP 2 , the procedure we have outlined provides a covariant point-splitting regularization. However, we will need to modify this slightly to take account of covariance with respect to gauge transformations as well. We have so far considered the free propagator. In the presence of gauge fields, the propagator isG(z, y) = ((−D · D) −1 z,y ) Y =−2 . We carry out the explicit calculations by expanding this in powers of the gauge field as G(z, y) =G(z, y) + y 1G (z, y 1 )V y 1G (y 1 , y) + y 1 ,y 2G (z, y 1 )V y 1G (y 1 , y 2 )V y 2G (y 2 , y) + · · · (51) where V =Ā · ∂ + A ·∂ + (∂ · A) +Ā · A. The propagatorG(z, y) transforms asG(z, y) → U(z)G(z, y) U † (y) under the gauge transformation M → UM, M † → M † U † . This should be respected in calculating currents such as Ĵ (z) = −D zG (z, y) y→z in (32), (33), with regularization. In other words, even though we shift y to y ′ , the gauge transformation must maintain the action of U † (y) at the second argument to be consistent with its role in the current Ĵ (x) . This means that a gauge-invariant point-splitting is given bỹ HereG Reg is as in (50) and y ′ = y + δy, with δy a δȳā → ǫ η aā in taking the small ǫ limit in a symmetric way. (This is for the case when y can be set to zero; which will cover the calculations for which we need this factor.) The path-ordered exponential helps to convert the U † (y ′ ) (due toG(z, y ′ )) back to U † (y). (Such a step was already included for the isometries since we added an extra phase factor inG Reg connecting y to y ′ .) Because it involves the integral of one-forms, it is adequate to use local coordinates y, y ′ in (52).
Turning to the infrared side of calculations, note that there are no infrared divergences since CP 2 is a compact space of finite volume. However, we are only calculating the leading terms in Γ, up to terms of scaling dimensions ≤ 4. Terms of higher dimension will be ultraviolet finite. While they are irrelevant for issues of renormalization, we need to identify a kinematic regime where such terms are suppressed to be able to make any conclusions with the terms we calculate. If we use an infrared cutoff λ, then the terms of higher scaling dimension will carry inverse powers of this cutoff and will be suppressed for modes of the fields with small momenta compared to λ. This is what we do here. (This rationale for the infrared cutoff is explained in more detail in [1].) The WZW action will be special because it is a term of lower dimension, yet appears with a finite coefficient. Just for that particular term, we do the calculations both with and without an infrared cutoff.
The details of the UV-and IR-regularized calculations will be given in the Appendix, but here we note that the IR regularization is easily incorporated by using a simple integral representation for F (s) in (45) We have introduced r 2 via the scaling of coordinates. The infrared cutoff λ appears as the lower limit of the integration over t. When λ is set to zero, we clearly reproduce F (s) in (45). This result, combined with (52), can be used for calculating the effective action.

Results
We are now ready to present the results regarding the volume element for A/G * . As mentioned at the end of section 3, we will consider the expansion of Γ as a series in terms of increasing scaling dimension, focusing on those with dimension ≤ 4. These are the terms we can expect to be relevant for the long wavelength modes of the fields; they are potentially UV-divergent terms, up to a possible logarithmic divergence. For scaling dimension 2, the possible terms correspond to a mass-like term for a,ā (with a coefficient of order 1/ǫ) and a WZW term S wzw (H), which, somewhat surprisingly, has a finite coefficient. There are also terms of dimension 4 which arise with a coeffcient of order log ǫ.

The WZW action
All purely H-dependent terms, such as the WZW action, come from Γ 0 as defined in (31). As mentioned above, the determinant det(−D · D) Y =0 for scalar fields was already found in [1]. Using the result from there Here S wzw (H) is the four-dimensional WZW action given by [9,10] S wzw (H) = 1 2π where ω is the Kähler two-form on CP 2 . This can be expressed in local coordinates as with g aā given by the Fubini-Study metric (1). The last term in (56) is, as usual, over a five-manifold which has CP 2 as the boundary. We have normalized the volume of CP 2 to 1, so there is an extra factor of π 2 /2 in (56) compared to the standard normalizations.
(Also, we use a slightly different convention for the normalization of ω, compared to [8].) It is straightforward to simplify the expression for C Y =0 in (55) to show that in the absence of an infrared cutoff (λ → 0) we get a finite result with no infrared divergence On the other hand, for λr 2 ≫ 1 Turning to the Y = −2 part of Γ 0 , i.e., Tr log(−D · D) Y =−2 , we can proceed analogously as in [1] by defining and calculating the current Ĵ (H, 1) . As for the scalar where the only difference is thatG(x, y) = (−∇ · D) −1 x,y is now defined as an expansion in free propagators for Y = −2 fields on CP 2 (with the covariant derivatives ∇ including the appropriate Levi-Civita connections).
Expanding Ĵ in powers of ∇HH −1 , and using the UV and IR regularized propagators as defined in (52) and (53), we can in principle find Ĵ as we did for the scalar case. However, as the result is much more complicated for Y = −2 fields, instead, here we present the results in the two relevant limits, one with no infrared cutoff (λ → 0) and the other for λr 2 ≫ 1: The details of this calculation are given in Appendix A. We may also note that the transition from the variation and the current in (60) As for the scalar case, other than the WZW action, Γ 0 will also include log ǫ terms (see below), and finite terms as well (if we include terms with scaling dimension > 4, involving more derivatives on H). We will not calculate the finite terms in this paper.

The mass term
Now we turn to terms with quadratic divergence. In Γ 0 the only term that could have 1/ǫ-divergence is the WZW action. However, as we have seen, for this term all divergences cancel out, giving an overall finite coefficient.
Among the next set of terms in ∆Γ in (31), there are a couple of possible candidates for quadratic divergence. However, the only term which survives is a mass term for a andā. This is expected as the only quadratically divergent term in ∆Γ that is invariant under the holomorphic transformation M † → V M † is a mass term Tr[āHaH −1 ].
Looking back at (31) and expanding the exponential in S 1 we can write where the sum is over connected diagrams. S 1 is defined in (30), and Since each term in the expansion of ∆Γ is a trace over the operators MaM The only quadratically divergent term is thus a mass term for a andā.

The log-divergent terms
Logarithmically divergent terms can arise from both Γ 0 and ∆Γ. Combining all such contributions, where the trace is in the adjoint representation. (Once again, the details are given in Appendix B.)

Discussion
The key results we have obtained are the following. We introduced a parametrization of the gauge potentials, see equation (18) in section 2, which allows for the explicit factoring out of gauge transformations and consequent reduction to gauge-invariant degrees of freedom. In particular, the volume element for the gauge orbit space C = A/G * can be written as Unlike the case of two dimensions, an exact calculation of Γ is not possible in four dimensions. We have calculated the first set of terms in Γ corresponding to monomials of the fields and their derivatives of scaling dimension ≤ 4. These are the terms which can be expected to be potentially ultraviolet divergent. The first of these terms is a four-dimensional WZW action S wzw (H) for H ∈ SL(N, C)/SU(N) (for an SU(N) gauge theory), given in (65). A priori, on dimensional grounds, one may expect this term to be quadratically divergent, but, somewhat surprisingly, it arises with a finite coefficient.
The sign of the coefficient is "correct" in the sense of ensuring convergence for functional integration over H. (In [1], we calculated a similar contribution due to chiral scalar fields on CP 2 . The coefficient of S wzw (H) was such that it tends to destabilize the theory for long wavelength modes of H. It is interesting that chiral scalars on CP 2 have this destabilizing effect. For the volume element for C, there is no such issue.) The second term in Γ is a mass term for the components a i ,āī of the potentials, which are related to χ ′ , χ ′ † as a i = −g iī ǫ¯ijH −1 (∇jχ ′ )H,āī = −gī i ǫ ij H(∇ j χ ′ † )H −1 . The coefficient of this term, say µ 2 div , is quadratically divergent. Since ultraviolet divergences are related to products of operators at the same point, i.e., to ultralocal geometry and hence not sensitive to global geometry, the divergence shows that this term should survive even if we take the large r limit of CP 2 . Further, since it is consistent with gauge invariance and all the isometries of CP 2 , there is no reason to reject this. This means that the functional measure has to be defined by introducing a similar counterterm (with a coefficient µ 2 counter ) from the beginning and then renormalizing by setting µ 2 div + µ 2 counter to a finite value µ 2 Ren . This finite renormalized value µ Ren has the dimensions of mass and will serve as the mass parameter defining the theory. This is in accordance with the fact that dimensional transmutation, with the introduction of an arbitrary scale factor, is needed to define four-dimensional gauge theories. In the usual perturbative approach, this would enter via loop corrections and the running coupling constant, but in our formulation, it appears as what is needed to make the volume element for C (or the measure of functional integration) well-defined. If loop calculations are carried out in our approach, the results will be functions of µ Ren ; there will be no need for additional dimensional parameters; what is conventionally considered as Λ QCD will be related to µ Ren .
There is also a set of terms which are of scaling dimension 4 with a logarithmically divergent coefficient, see (71). Unlike the case of flat space, these terms do not combine into Tr(F 2 ) since the reduced isometries of CP 2 allow for additional tensor structures.
Presumably, to make the volume element well-defined, similar counterterms have to be introduced a priori and renormalization has to be carried out. Finally there is also an infinity of terms of scaling dimension > 4, which are ultraviolet finite, which we have not calculated. They are presumably less relevant to the dynamics of long wavelength modes of the fields compared to the terms we have calculated.
Returning to the mass term (70), we note that the possibility of a soft gluon mass has been proposed already in the 1980s [11]. Lattice simulations of the gluon propagator in the Landau gauge also indicate its saturation to a finite value at low momenta, consistent with a propagator mass [12]. At least qualitatively, we need an analytic understanding of these lattice results. Indeed, a number of papers have analyzed the Schwinger-Dyson equations of QCD with a view to showing that the gluon self-energy is nonvanishing at zero momentum, along with attempts to extract quantitative predictions from it [13,14]. The appearance of a possible gauge-invariant mass term in our analysis provides a parallel track of viewing such analyses.
Perhaps the most striking and qualitatively new feature of our analysis is the appearance of the WZW action S wzw (H). With the above given argument for a nonzero mass term (70), it is then possible to consider a kinematic regime of momenta ≪ µ Ren where we can neglect the massive components a i ,āī and consider a reduced theory where The volume element then takes the form (Here C 1 is the renormalized value of the coefficient of the term in (71), after the log ǫ-divergence is eliminated.) In the last line of (74) we have neglected the term quartic in the derivatives as it is less significant for long wavelength modes compared to S wzw (H). The theory defined by (74) should be applicable for λ ≪ µ 2 Ren , with λr 2 ≫ 1. This theory is the four-dimensional WZW theory on CP 2 . So our conclusion is that we expect that for fields of modes of wavelength small compared to µ Ren , the four dimensional Yang-Mills theory can be approximated by a 4d-WZW theory for the field H ∈ G C /G = SL(N, C)/SU(N).
The 4d-WZW theory, we may note, also has a history going back to the 1980s, appearing first in the work of Donaldson in the context of antiself-dual instantons [9].
The same theory is obtained in the Kähler-Chern-Simons theory [10] which attempted to generalize the 2d-WZW theory to four dimensions, a paradigm similar to the WZW-CS relation in two and three dimensions [15]. As shown in [10], and elaborated in [16,17], this action also leads to a holomorphically factorized current algebra, analogous to the case in two dimensions. 4d-WZW theories have also been found in higher dimensional quantum Hall systems [18]. They also describe the target space dynamics of (world-sheet) N = 2 heterotic superstrings [19]. More recently, such theories have been analyzed in [20] in the context of holomorphic field theories on twistor space.
The critical points of the action S wzw (H) are antiself-dual instantons. They are related to holomorphic vector bundles, with M and M † defining the holomorphic frames for the bundle. What is interesting is that there is some evidence, based on lattice simulations, that the correlation functions for gauge fields and hadrons seem to be dominated by instantons at low energies; see, for example, [21,22]. While it is difficult to see instanton dominance analytically for fields on R 4 , the present result that the theory can be approximated by the 4d-WZW theory along the lines argued above provides some analytical evidence for an instanton liquid picture.
Finally, there is another aspect of the 4d-WZW theory which is worth pointing out. In the (2+1)-dimensional analysis considered in [4,5] the expectation value of the Wilson loop operator (in a representation indicated as R) takes the form wzw (H) is the 2d-WZW action for H, c R , c A are the values of the quadratic Casimir operators for the representation R and for the adjoint representation, respectively. e 2 is the coupling constant of the Yang-Mills theory. Notice that, as e 2 → ∞, which is the limit where the integrand in (75) defines the 2d-WZW theory for H, the expectation value of W (C) vanishes for any curve C enclosing any nonzero area. If we consider evaluating W (C) in terms of correlators for the current ∇HH −1 , the leading term due to the two-point function is of the form The UV singularity of this integral is not regularized when e 2 → ∞, and this is the genesis of the vanishing of W (C) .
We see that a similar situation is obtained in the theory defined in the 4d-theory (74). Using the Polyakov-Wiegmann identity we obtain By taking small variations of N, we then find gā a∇ā (∇ a HH −1 ) α (x)gb b∇b (∇ b HH −1 ) β (y) = − 4 πC gā a∇ā ,y ∇ a,y δ(y, x) δ αβ (79) The two-point function for the currents can be obtained from this as where G(y, x) is the propagator for scalars on CP 2 given in [1] as where s = σ 2 y,x is given by (41). We see that the logarithmic term of G(y, x) in (80)  In this Appendix we outline the calculations of the current Ĵ in (60) for Y = −2 fields.
The procedure is analogous to the scalar calculation in [1]. The only difference is in the propagator and the connections of the gradients as given in (35).

The current is
The primed coordinate is given in (48) and we perform an angular integration over w and α with the conditions that |α| 2 = ǫ and σ 2 (x, w) = 1.
As in the scalar calculation (in [1]), we make a coordinate transformation w → w ′ such that w a = x a + (e −1 x ) a b w ′b 1−x·w ′ , where e −1 are the tangent frame fields. Under this transformation σ 2 (x, w) = |w ′ | 2 . Term 1 in (A1) becomes where the extra term − 3 2 F (ǫ) (as compared to the scalar case) comes from the spin connection in ∇ a and the phase factor ofG(x, y ′ ). Including the scale r in (A2), we take ǫ → ǫ/r 2 and F (s) is the IR regulated propagator Term 1 then becomes Performing similar calculations as for Term 1 it's straightforward to find that where E 1 is the exponential integral For Term 3 we need three coordinate transformations: 1) A transformation w → w ′ such that σ 2 (x, w) = 1, as explained above; 2) A transformation z → z ′ such that z a = x a + (e −1 x ) a b z ′b 1−x·z ′ , setting σ 2 (x, z) = |z ′ | 2 ; 3) Finally, a transformation z ′ →z, which can be given in homogenous coordinates as Z/Z 3 =W 3 Z ′ /(W ′ · Z ′ ), whereW ′ = (αW ′ 1 , αW ′ 2 , W ′ 3 ) = W ′ 3 (αw ′ 1 , αw ′ 2 , 1). The first two transformations effectively eliminate x from the integrals by translating the integration variables. The last one is useful because 1 + σ 2 (z ′ , αw ′ ) = (1 + ǫ)(1 + |z| 2 ) which significantly simplifies the integrals inz and w ′ . These transformations are the same as in [1] (where more details are given as well) and, as was the case there, the integration measures remain unchanged. With these changes, Term 3 becomes The exact calculation of I is more complicated than for its scalar counterpart. So instead of calculating it for arbitrary λ, we find it in the two limits, namely, in the case of no infrared cutoff, i.e., λ → 0, and in the case of λr 2 ≫ 1. Including the scale factor r, we find Combining (A4), (A5), and (A8), and taking the appropriate limits, which are the results in (62) and (63).
and C αC β = G scalar = (−∇ · D) −1 scalar and B α B β =G Y=−2 = (−∇ · D) −1 Y=−2 . In carrying out various calculations, we will be expanding G scalar andG Y=−2 in terms of the corresponding free propagators G andG, as in (51). For Y = −2, the UV regularized form of the free propagator is given in (45), (50). The scalar propagator was given in [1] and has the form with the replacement of s by σ 2 (Z, Y ′ ) to take account of regularization.
For the mass term, we keep r finite and perform similar calculations as in Appendix A. The log-divergent terms can be obtained by calculating in the flat space limit, and then upgrading the metric and volume factors to the curved space expressions. Then, expanding each G in ∇HH −1 , we find the following UV divergent terms  which are the results for the mass term and log-divergent terms in (70) and (71).