with and with Looking at the following standard formula for Gaussian integration for bosonic fields we observe that we can get rid of the inverse of the Faddeev-Popov operator in h$_{3}$ ( x ) by introducing new fields. For every index i , defined by . . . a i = . . . ac µ , we can write for h$_{3}$ ( x ) Notice that, as already remarked, the limit θ → 0, in equation (19) has been performed after localization. As one can see from (13), taking this limit is equivalent with setting γ $^{2}$( x ) equal to the constant γ $^{2}$. As at the level of the action, total derivatives are always neglected, S ′ γ becomes From this expression we can easily understand why certain doubts have arisen. Starting from the first horizon function h$_{1}$ ( x ) given in (7) and undertaking the same procedure, we would end up with exactly the same action S ′ $_{γ}$. This can be understood as we have neglected the total derivatives. In conclusion, although the local actions derived from h$_{1}$ ( x ) and h$_{3}$ ( x ) are the same, at the nonlocal level they are clearly different, which also follows from [8]. This is important when one is doing manipulations at the level of the nonlocal action as has been done in [8]. Let us now translate the nonlocal horizon condi- tion (11) into a local version [4]. The local action S$_{GZ}$ and the nonlocal action S$_{YM}$ + S$_{gf}$ + S$_{h}$ are related as follows, whereby we have introduced a pair of complex conju- gate bosonic fields ( ϕ ac µ , ϕ ac µ ) = ( ϕ a $_{i}$, ϕ a $_{i}$). We can then also lift the determinants det( −M ) into the exponential by introducing a pair of Grassmann fields ( ω ac µ , ω ac µ ) = ( ω a $_{i}$, ω a i ). Making use of the standard Gaussian formula for Grassmann variables whereby we set the sources J a ω and J b ω equal to zero, we obtain Next, we take the partial derivative of both sides with respect to γ 2 so we obtain, for both horizon functions h$_{1}$ ( x ) and h$_{3}$ ( x ). We recall that 〈 ∂$_{µ}$ϕ $^{aa}$〉 = 0 and 〈 ∂$_{µ}$ϕ $^{aa}$〉 = 0, meaning that both horizon functions h$_{1}$ and h$_{3}$ give rise to the same local horizon condition. Using these expressions and assuming that γ = 0, we can rewrite the horizon condition (11) The new localized action thus becomes ∫ [d A ][d b ][d c ][d c ]e − ( S$_{YM}$ + S$_{gf}$ + S$_{h}$ ) = ∫ [d A ][d b ][d c ][d c ][d ϕ ][d ϕ ][d ω ][d ω ] e − S$_{GZ}$ . − 2 γ $^{2}$〈 h 〉 = 〈 gf $^{abc}$Aa $_{µ}$( ϕ bc µ + ϕ bc µ ) 〉 , (21) 〈 gf $^{abc}$Aa $_{µ}$( ϕ bc µ + ϕ bc µ ) 〉 + 2 γ $^{2}$d ( N 2 − 1) = 0 . (22) S$_{GZ}$ = S ′ 0 + S ′ γ , (17) C (det A ) exp ( − ∫ d $^{d}$x d $^{d}$y Ja $_{ω}$( x )( A − $^{1}$) $^{ab}$( x, y ) J b $_{ω}$( y ) = ∫ [d ω ][d ω ] exp [∫ d $^{d}$x d $^{d}$y ω$^{a}$( x ) A $^{ab}$ω$^{b}$( y ) + ∫ d $^{4}$x ( J a $_{ω}$( x ) ω $^{a}$( x ) + ω $^{a}$( x ) J a $_{ω}$( x )) ] , (16) exp ( − ∫ d $^{d}$xh$_{3}$ ( x ) = d ( N $^{2}$+1) ∏ i =1 det( −M ) ∫ [d ϕ ][d ϕ ] exp lim θ → 0 [ ∫ d $^{d}$x ∫ d $^{d}$yϕa $_{i}$( x ) M $^{ab}$( x, y ) ϕ b $_{i}$( y ) + ∫ d $^{d}$x ( D a $_{i}$( x ) γ $^{2}$( x ) ϕ a $_{i}$( x ) + ( D a $_{i}$( x ) γ $^{2}$( x ) ϕ a $_{i}$( x ) ] ) , (15) S ′ γ = γ 2 ∫ d $^{d}$xgf $^{abc}$Aa µ ( ϕ bc µ + ϕ bc µ ) . (20) S ′ 0 = S$_{YM}$ + S$_{gf}$ + ∫ d $^{d}$x ( ϕ ac µ ∂$_{ν}$D ab ν ϕ bc µ − ω ac µ ∂$_{ν}$D ab ν ω bc µ ) , (18) S ′ γ = − lim θ → 0 ∫ d $^{d}$x [( D ac µ ( x ) γ $^{2}$( x ) ϕ ac µ ( x ) + ( D ac µ ( x ) γ $^{2}$( x ) ϕ ac µ ( x ) ] = lim θ → 0 ∫ d $^{d}$x γ $^{2}$( x ) D ca µ ( ϕ ac µ ( x ) + ϕ ac µ ( x )) = γ 2 ∫ d $^{d}$x Dca µ ( ϕ ac µ ( x ) + ϕ ac µ ( x )) . (19) C det A − $^{1}$exp ∫ d $^{d}$x d $^{d}$y Ja $_{ϕ}$( x )( A − $^{1}$) $^{ab}$( x, y ) J b $_{ϕ}$( y ) = ∫ [d ϕ ][d ϕ ] exp [ ∫ d $^{d}$x d $^{d}$y − ϕ $^{a}$( x ) A $^{ab}$( x, y ) ϕ $^{b}$( y ) + ∫ d $^{4}$x ( ϕ $^{a}$Ja $_{ϕ}$( x ) + ϕ $^{a}$( x ) J a $_{ϕ}$( x )) ] , (14) 3 exp ( − ∫ d $^{d}$xh$_{3}$ ( x ) = d ( N $^{2}$+1) ∏ i =1 ∫ [d ω ][d ω ][d ϕ ][d ϕ ] exp [ ∫ d $^{d}$x ∫ d $^{d}$y ( ϕ a $_{i}$( x ) M $^{ab}$( x, y ) ϕ b $_{i}$( y ) − ω a $_{i}$( x ) M $^{ab}$( x, y ) ω b $_{i}$( y ) + lim θ → 0 ∫ d $^{d}$x ( D a i ( x ) γ $^{2}$( x ) ϕ a $_{i}$( x ) + ( D a i ( x ) γ $^{2}$( x ) ϕ a $_{i}$( x ) ] .