Adopted rigid $N=4$ Yang–Mills localization background on the round four-sphere

Adopt the following Euclidean source model in the conventions stated here. Let $r>0$ have units of length and let $S_r^4=\{y\in {\mathbb{R}}^5:{\sum }_{j=1}^5{y}_j^2=r^2\}$ have its round metric, standard orientation, and scalar curvature $12\mathrm{/}{r}^2$. In stereographic coordinates $x^{\mu }$ of length, $\mu =1,\dots ,4$, the metric is $d{s}^2=\frac{{\sum }_{\mu =1}^4(d{x}^{\mu }{)}^2}{(1+x^2\mathrm{/}(4r^2){)}^2},x^2={\sum }_{\mu =1}^4(x^{\mu }{)}^2\mathrm{.}$ Use the standard spin structure and the spin connection induced by the Levi-Civita connection of the round metric. Identify the four real scalar directions $5,6,7,8$ with $\mathbb{H}$ in the ordered basis $(1,i,\mathbf{j},i\mathbf{j})$. The source's internal $SU(2{)}_R^L\times SU(2{)}_R^R$ is the two-factor cover of $SO(4)$ acting on these scalar directions by $q\mapsto u_{R}q{u}_L^{-1}$ and fixing scalar directions $0,9$; the labels $L,R$ follow the source's spinor representations. Give the internal four-plane its Euclidean metric and the orientation of the ordered basis above. If $f^5,\dots ,f^8$ is its dual basis, identify a skew matrix $M$ with ${\alpha }_M=\frac{1}{2}{\sum }_{A,B=5}^8{M}_{AB}{f}^A\wedge f^B$ and define its Hodge star on two-forms by $\ast (f^A\wedge f^B)=\frac{1}{2}{\sum }_{C,D=5}^8{\epsilon }_{ABCD}{f}^C\wedge f^D$, extended linearly, where $\epsilon$ is the fully antisymmetric symbol with ${\epsilon }_{5678}=1$. The $SU(2{)}_R^L$ infinitesimal action $q\mapsto -qa$, $a\in \mathrm{Im}\mathbb{H}$, has $\ast {\alpha }_M=-{\alpha }_M$ (anti-self-duality), whereas $SU(2{)}_R^R$ acts by $q\mapsto aq$ with $\ast {\alpha }_M={\alpha }_M$ (self-duality). For example, $a=i$ in the first action gives ${\alpha }_M=f^5\wedge f^6-f^7\wedge f^8$. Thus the source's anti-self-dual $R^Q$ belongs to $\mathrm{Lie}(SU(2{)}_R^L)$ in these conventions. Set $SU(2{)}_R:=SU(2{)}_R^L$, take $P_R=S_r^4\times SU(2{)}_R$ with zero connection, and use the selected $SU(2{)}_R$ generator below. The global gauge group is $G=SU(2)$, not $SO(3)$; its Lie algebra consists of anti-Hermitian $2\times 2$ traceless matrices and has positive invariant form $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, normalized by $\kappa (-i{\sigma }_a\mathrm{/}2,-i{\sigma }_b\mathrm{/}2)={\delta }_{ab}$. Let $\mathrm{\Sigma }$ be the declared set of bundle-sector codes for smooth right principal $SU(2)$-bundles over $S_r^4$, using the nonempty minimum-rank set code of each smooth equivariant isomorphism type. As part of the sectorwise model data, supply a set-indexed family of representative bundles $P_s\to S_r^4$ with $[P_s]=s$ for $s\in \mathrm{\Sigma }$; each $s$ is a separate gauge sector. The family is supplied data, not a choice asserted to exist for an arbitrary declared set $\mathrm{\Sigma }$. The sector set and any sum over it are formal model inputs, not a classification or measure construction. For a chosen sector $s$, abbreviate its supplied representative $P_s$ by $P$ and write $\mathrm{ad}P=P{\times }_{\mathrm{Ad}}\mathfrak{s}\mathfrak{u}(2)$ with ${\mathrm{Ad}}_{u}X=uX{u}^{-1}$. Set $E_{\mathrm{a}\mathrm{u}\mathrm{x}}=S_r^4\times {\mathbb{R}}^7$ with its standard Euclidean inner product and fixed global orthonormal frame $e_1,\dots ,e_7$. The auxiliary field in sector $P$ is $K={\sum }_i{K}_i{e}_i$, where each $K_i$ is a smooth section of $\mathrm{ad}P$; all contractions over $i$ use this fixed fiber metric.

The matter content is the massless four-dimensional $\mathcal{N}=4$ vector multiplet: one gauge connection $A_{\mu }$, six real adjoint scalar fields ${\mathrm{\Phi }}_A$ with $A\in \{0,5,6,7,8,9\}$, and four adjoint Weyl fermions. Equivalently, its flat-space component content is one $\mathcal{N}=1$ vector multiplet and three adjoint $\mathcal{N}=1$ chiral multiplets. Set the topological $\theta$-term to zero in this selected model. Use the ordinary trace ${\mathrm{Tr}}_2$ in the defining representation of $SU(2)$ and a positive dimensionless coupling $g_{\mathrm{Y}\mathrm{M}}$. The current $N=4$ model carrier uses the same global group, $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, the same dimensionless coupling, and connection-normalized adjoint chiral fields, with one vector and three adjoint chiral multiplets; this is an interface match for flat-space component content and normalization. This postulate separately adopts the source’s curved $S^4$ action, curvature coupling, selected-spinor algebra, gauge fixing, contour, and formal regulator inputs; it does not identify that curved action with the supplier’s flat Minkowski superspace action. For the dimensional-reduction notation, in a local gauge trivialization define the Lie-algebra-valued components $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$, $F_{\mu A}=D_{\mu }{\mathrm{\Phi }}_A$, and $F_{AB}=[{\mathrm{\Phi }}_A,{\mathrm{\Phi }}_B]$, extended antisymmetrically to $F_{MN}$. Here $D_{\mu }X={\mathrm{\nabla }}_{\mu }^{\mathrm{L}\mathrm{C}}X+[A_{\mu },X]$ on adjoint tensor fields, with the supplied spin connection on spinor indices, and $D_{A}X=[{\mathrm{\Phi }}_A,X]$; in the fermion term $D_{\mu }\mathrm{\Psi }$ includes the spin and gauge connections. These $F_{MN}$ are Lie-algebra-valued matrices, so ${\mathrm{Tr}}_2$ is taken in the defining representation. Use ${\eta }_{MN}=\mathrm{diag}(-1,1,\dots ,1)$ for $M,N=0,\dots ,9$. With the seven localization auxiliaries included, the adopted classical action input is $S_{\mathrm{o}\mathrm{f}\mathrm{f}}=-\frac{1}{g_{\mathrm{Y}\mathrm{M}}^2}{\int }_{S_r^4}{d}^{4}x\sqrt{g}{\mathrm{Tr}}_2(\frac{1}{2}{\sum }_{M,N}{F}_{MN}{F}^{MN}-\mathrm{\Psi }{\mathrm{\Gamma }}^M{D}_M\mathrm{\Psi }+\frac{2}{r^2}{\sum }_A{\mathrm{\Phi }}_A{\mathrm{\Phi }}^A-{\sum }_{i=1}^7{K}_i{K}_i)\mathrm{.}$ Here $\mathrm{\Psi }$ is the adjoint ten-dimensional Majorana–Weyl fermion used to package the four-dimensional fields; after Euclidean continuation it is complex and has no independent conjugate field. The chiral gamma matrices obey ${\mathrm{\Gamma }}_M{\tilde{\mathrm{\Gamma }}}_N+{\mathrm{\Gamma }}_N{\tilde{\mathrm{\Gamma }}}_M=2{\eta }_{MN}{I}_{16}$, with ${\tilde{\mathrm{\Gamma }}}_0=-{\mathrm{\Gamma }}_0$ and ${\tilde{\mathrm{\Gamma }}}_M={\mathrm{\Gamma }}_M$ for $M=1,\dots ,9$; ${\mathrm{\Gamma }}_{MN}:={\tilde{\mathrm{\Gamma }}}_{[M}{\mathrm{\Gamma }}_{N]}$ acts on $S^{+}$ and ${\tilde{\mathrm{\Gamma }}}_{MN}:={\mathrm{\Gamma }}_{[M}{\tilde{\mathrm{\Gamma }}}_{N]}$ acts on $S^{-}$. Brackets on two indices have weight $1\mathrm{/}2$. The blocks are symmetric and $({\mathrm{\Gamma }}_{MN}{)}^{\mathsf{T}}=-{\tilde{\mathrm{\Gamma }}}_{MN}$ in the source basis. A spinor bilinear $\zeta B\chi$ means ${\zeta }^{\mathsf{T}}B\chi$, with no Hermitian conjugation; the background spinors commute, while fermion components anticommute. The four sphere directions have positive metric. The term with $2\mathrm{/}{r}^2$ is the conformal curvature coupling of the six scalars. The Yang–Mills coupling is dimensionless, $[A_{\mu }]=[{\mathrm{\Phi }}_A]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, $[K_i]=[F_{MN}]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, and $[\mathrm{\Psi }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-3\mathrm{/}2}$.

Let ${\widehat{ϵ}}_s=(0_8,1,0_7{)}^{\mathsf{T}}$ be the normalized sixteen-component chiral spinor in the source gamma convention, with ${\mathrm{\Gamma }}_{5678}{\widehat{ϵ}}_s={\widehat{ϵ}}_s$ and ${\mathrm{\Gamma }}_{1234}{\widehat{ϵ}}_s=-{\widehat{ϵ}}_s$, and set ${\widehat{ϵ}}_c=(2r{)}^{-1}{\mathrm{\Gamma }}_{012}{\widehat{ϵ}}_s$. With $\mathrm{\Omega }(x)=1+x^2\mathrm{/}(4r^2)$, adopt $ϵ(x)=\mathrm{\Omega }(x{)}^{-1\mathrm{/}2}({\widehat{ϵ}}_s+x^i{\tilde{\mathrm{\Gamma }}}_i{\widehat{ϵ}}_c),\widetilde{ϵ}(x)=\mathrm{\Omega }(x{)}^{-1\mathrm{/}2}({\widehat{ϵ}}_c-\frac{x^i}{4r^2}{\mathrm{\Gamma }}_i{\widehat{ϵ}}_s),i=1,\dots ,4.$ They solve the conformal Killing-spinor system ${\mathrm{\nabla }}_{\mu }ϵ={\tilde{\mathrm{\Gamma }}}_{\mu }\widetilde{ϵ},{\mathrm{\nabla }}_{\mu }\widetilde{ϵ}=-\frac{1}{4r^2}{\mathrm{\Gamma }}_{\mu }ϵ\mathrm{.}$ The normalized spinor ${\widehat{ϵ}}_s$ and the resulting $ϵ$ are dimensionless, while ${\widehat{ϵ}}_c$ and $\widetilde{ϵ}$ have mass dimension one. Here $Q_{ϵ}$ denotes the supercharge operator of mass dimension $1\mathrm{/}2$ selected by this normalized spinor; an infinitesimal physical transformation uses a separate odd coefficient of mass dimension $-1\mathrm{/}2$. Accordingly, $Q_{ϵ}^2$, $v^{\mu }{\mathrm{\partial }}_{\mu }$, $R^Q$, and the selected generator $\omega$ have mass dimension one, and the flow parameter for $v$ has mass dimension $-1$. After Euclidean continuation, put ${\mathrm{\Delta }}_{\pm }={\mathbb{C}}^2$ with the defining $\mathrm{Spin}(4)$ representations ${\pi }_{\pm }$, let $V_R={\mathbb{C}}^2$ with the defining representation $\tau$ of $SU(2{)}_R$, and let $V_0={\mathbb{C}}^2$ with the trivial $SU(2{)}_R$ representation. Define the sixteen-dimensional complex chiral module and its smooth representation by $W_{ϵ}=({\mathrm{\Delta }}_{+}\oplus {\mathrm{\Delta }}_{-})\otimes (V_R\oplus V_0),{\rho }_{ϵ}(s,k)=({\pi }_{+}(s)\oplus {\pi }_{-}(s))\otimes (\tau (k)\oplus I_{V_0})\mathrm{.}$ Thus ${\rho }_{ϵ}:\mathrm{Spin}(4)\times SU(2{)}_R\to {\mathrm{GL}}_{\mathbb{C}}(W_{ϵ})$ is the restriction of the continued source chiral module: each four-dimensional spin chirality occurs once with the defining $SU(2{)}_R^L$ doublet and once with a two-dimensional multiplicity space trivial under $SU(2{)}_R^L$. The spinor/R field bundle is $(\tilde{F}{\times }_{S^4}{P}_R){\times }_{{\rho }_{ϵ}}{W}_{ϵ}$ with quotient relation $[(\tilde{p},p_R)h,z]=[(\tilde{p},p_R),{\rho }_{ϵ}(h)z]$ for $h\in \mathrm{Spin}(4)\times SU(2{)}_R$. In local product sections its derivative is $d{ϵ}_i+(d{\rho }_{ϵ}{)}_e(A_i^S,A_i^R){ϵ}_i$, where $(d{\rho }_{ϵ}{)}_e$ is the differential at the identity. The spin connection in $A_i^S$ is induced by the round metric's Levi-Civita connection; because the R bundle is trivial with zero connection, the ${\mathrm{\nabla }}_{\mu }$ in this system has zero R-connection term. The selected spinors obey the source's same-chirality conditions, in particular $\widetilde{ϵ}ϵ=0$, so the dilation term in the supercharge square vanishes. Their bilinear $v^{\mu }=ϵ{\mathrm{\Gamma }}^{\mu }ϵ$ is the source-normalized infinitesimal generator of the $U(1)$ isometry rotating the $(x^1,x^2)$ and $(x^3,x^4)$ planes equally; no further rescaling of this generator is made below. The corresponding internal R-symmetry generator is the skew matrix $R_{AB}^Q=2ϵ{\tilde{\mathrm{\Gamma }}}_{AB}\widetilde{ϵ}$ on the six scalar indices and acts by $({\delta }_R\mathrm{\Phi }{)}_A=R_{AB}^Q{\mathrm{\Phi }}^B$. The fermion action is ${\delta }_R\mathrm{\Psi }=\frac{1}{4}{R}_{AB}^Q{\mathrm{\Gamma }}^{AB}\mathrm{\Psi }$ with the source reduction metric used to raise indices. These formulas adopt the ordering in Pestun (2.9); no interchange of the two commuting background spinors is assumed. In particular the transpose identity would give $2\widetilde{ϵ}{\mathrm{\Gamma }}_{AB}ϵ=-R_{AB}^Q$, so that expression is not substituted with a plus sign. The gauge connection and Faddeev--Popov ghosts are R-singlets.

In the fixed global frame $e_i$ of $E_{\mathrm{a}\mathrm{u}\mathrm{x}}$ above, take seven smooth spinor sections ${\nu }_i$; equivalently, ${\sum }_i{\nu }_i\otimes e^i$ is a section of the spinor bundle tensored with $E_{\mathrm{a}\mathrm{u}\mathrm{x}}^{\ast }$. They satisfy $ϵ{\mathrm{\Gamma }}_M{\nu }_i=0,\frac{1}{2}(ϵ{\mathrm{\Gamma }}^Nϵ){\tilde{\mathrm{\Gamma }}}_{N,\alpha \beta }={\sum }_{i=1}^7{\nu }_{i,\alpha }{\nu }_{i,\beta }+{ϵ}_{\alpha }{ϵ}_{\beta },{\nu }_i{\mathrm{\Gamma }}_M{\nu }_j={\delta }_{ij}ϵ{\mathrm{\Gamma }}_Mϵ\mathrm{.}$ Adopt the off-shell transformations $Q_{ϵ}{A}_M=\mathrm{\Psi }{\mathrm{\Gamma }}_Mϵ,Q_{ϵ}\mathrm{\Psi }=\frac{1}{2}{\mathrm{\Gamma }}^{MN}{F}_{MN}ϵ+\frac{1}{2}{\mathrm{\Gamma }}^{\mu A}{\mathrm{\Phi }}_A{\mathrm{\nabla }}_{\mu }ϵ+{\sum }_i{K}_i{\nu }_i,Q_{ϵ}{K}_i=-{\nu }_i{\mathrm{\Gamma }}^M{D}_M\mathrm{\Psi }\mathrm{.}$ With $v^M=ϵ{\mathrm{\Gamma }}^Mϵ$, the source writes the off-shell square as $Q_{ϵ}^2=-{\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}-{\delta }_R-{\delta }_{\mathrm{\Omega }}$. On adjoint scalars ${\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}X=v^{\mu }{D}_{\mu }X+[v^A{\mathrm{\Phi }}_A,X]$; on the connection it is $({\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}A{)}_{\mu }=v^{\nu }{F}_{\nu \mu }-D_{\mu }(v^A{\mathrm{\Phi }}_A)$. On spinors it includes the geometric spin lift, and on the seven auxiliary components it is the exact operator in Pestun (2.30), with its auxiliary-index rotation. Thus this source operator already contains the gauge connection; it must not be added to a second gauge action. The equivalent split into a separately chosen geometric lift and a gauge transformation is specified next. The dilation parameter is $2\widetilde{ϵ}ϵ=0$. This off-shell closure is for the selected generator; it does not assert finite-auxiliary closure of the full hypermultiplet algebra.

Write ${\alpha }_{\theta }={\mathrm{F}\mathrm{l}}_v^{\theta }$ for the base circle action, parameterized so ${\frac{d}{d\theta }\mid }_0{\alpha }_{\theta }=v$. For every declared gauge sector $s\in \mathrm{\Sigma }$, with supplied representative $P=P_s$, adopt as additional formal model data a smooth lift ${\widehat{\alpha }}_{\theta ,P}:P\to P$ of the circle flow generated by $v$, together with compatible lifts to the spin, R, and auxiliary bundle $E_{\mathrm{a}\mathrm{u}\mathrm{x}}$; the auxiliary lift is orthogonal and has infinitesimal index action fixed by ${\mathrm{\Xi }}_{ij}$ below; each ${\widehat{\alpha }}_{\theta ,P}$ is a principal-bundle automorphism covering the same base isometry. Gauge-chart background connections below are chosen invariant under this lift. This postulate assumes these sectorwise lifts as inputs; it does not prove their existence for every smooth $SU(2)$ bundle. For a field bundle with induced action ${\widehat{\alpha }}_{\theta }^{\ast }$, define ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}\xi ={\frac{d}{d\theta }\mid }_{\theta =0}{\widehat{\alpha }}_{\theta }^{\ast }\xi$. In a local gauge frame let ${\beta }_{v,P}$ be the even infinitesimal vertical coefficient of this fixed lift, characterized on adjoint scalar fields by ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}X=v^{\mu }{\mathrm{\partial }}_{\mu }X+[{\beta }_{v,P},X]$ and on connections by ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}A={\mathcal{L}}_{v}A-D_A{\beta }_{v,P}$. Here the unadorned ${\mathcal{L}}_v$ is the ordinary Lie derivative of a local one-form. Let ${\mu }_{v,A}=v^{\mu }{A}_{\mu }-{\beta }_{v,P}$; this difference is a globally defined adjoint section, equivalently the contraction of the principal connection with the chosen lifted generator in these conventions. Define ${\kappa }_v={\mu }_{v,A}+v^A{\mathrm{\Phi }}_A$ and $\varphi =-{\kappa }_v$. With ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)X=[u,X]$ and ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)A=-D_{A}u$, the source operator decomposes as ${\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}={\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}+{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}({\kappa }_v),Q_{ϵ}^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}-{\delta }_R\mathrm{.}$ Indeed, on scalars the vertical coefficients add to $v^{\mu }{A}_{\mu }+v^A{\mathrm{\Phi }}_A$; on a connection the right side is ${\mathcal{L}}_{v}A-D_A(v^{\mu }{A}_{\mu }+v^A{\mathrm{\Phi }}_A)={\iota }_v{F}_A-D_A(v^A{\mathrm{\Phi }}_A)$. Thus no gauge contribution is counted twice. A lift-adapted local frame has ${\beta }_{v,P}=0$ and recovers the source's local $v^M{A}_M$ parameter. On spinors the spacetime spin lift is used separately from ${\delta }_R$. On the auxiliary bundle set ${\mathrm{\Xi }}_{ij}={\nu }_{[i}{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_{j]},({\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}K{)}_i=v^{\mu }{\mathrm{\partial }}_{\mu }{K}_i+[{\beta }_{v,P},K_i]+{\mathrm{\Xi }}_{ij}{K}_j\mathrm{.}$ The background ${\nu }_i$ are gauge singlets, so their derivative here is the supplied spin derivative; the internal derivatives vanish in this massless model. The auxiliary-index contribution in Pestun (2.30) is included in this lift and there is no further independent $R$ matrix on $K_i$. The antisymmetrization is ${\nu }_{[i}{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_{j]}=\frac{1}{2}({\nu }_i{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_j-{\nu }_j{\mathrm{\Gamma }}^{\mu }{\mathrm{\nabla }}_{\mu }{\nu }_i)$.

On each sectorwise gauge chart, with the chart label suppressed in the notation below, write $\mathrm{ad}P=P{\times }_{\mathrm{Ad}}\mathfrak{s}\mathfrak{u}(2)$ and let $\mathcal{A}(P)$ be the affine space of smooth principal connections on $P$. For a smooth vector bundle $E$, write $\mathrm{\Gamma }(E)=C^{\mathrm{\infty }}(S^4,E)$; the Lie algebra of $\mathcal{G}(P)$ is $\mathrm{\Gamma }(\mathrm{ad}P)$ with its pointwise Lie bracket. Use the real Fréchet chart and $C^1$ conventions of the rigid-background definition. In each sector the even physical configuration space before choosing the contour is ${\mathcal{F}}_{P,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\bar{0}}=\mathcal{A}(P)\times \mathrm{\Gamma }((\mathrm{ad}P{)}_{\mathbb{C}}{)}^6\times \mathrm{\Gamma }((\mathrm{ad}P{)}_{\mathbb{C}}\otimes (E_{\mathrm{a}\mathrm{u}\mathrm{x}}{)}_{\mathbb{C}}),$ where the complex section spaces are regarded as real spaces and carry uniform convergence on compact coordinate subsets of every derivative. Here $(\mathrm{ad}P{)}_{\mathbb{C}}=\mathrm{ad}P{\otimes }_{\mathbb{R}}\mathbb{C}$, and likewise for $E_{\mathrm{a}\mathrm{u}\mathrm{x}}$. The physical odd coordinates are the anticommuting coefficients of $\mathrm{\Psi }\in \mathrm{\Gamma }((\mathrm{ad}P{)}_{\mathbb{C}}\otimes S_{ϵ})$, where $S_{ϵ}$ is the spinor/R bundle with fibre $W_{ϵ}$ defined above. The connection charts are $A=A_{\ast }+a$ with tangent space ${\mathrm{\Omega }}^1(S^4,\mathrm{ad}P)$; the section factors have their own section spaces as tangents. Equip $\mathcal{G}(P)$ with the smooth-section topology and local charts obtained from the fibrewise $SU(2)$ exponential near zero in $\mathrm{\Gamma }(\mathrm{ad}P)$ and group translates. Its action on these fields is the supplied $C^1$ gauge action, with connection orbit derivative $R_{A}u=-D_{A}u$. Ghost, multiplier and finite zero-mode factors below extend this configuration space with the same section topology and the finite-dimensional topology, respectively. Choose an invariant background connection $A_{\ast }$ as fixed sectorwise data, inert under $s$ and $\delta$, and set $a=A-A_{\ast }\in {\mathrm{\Omega }}^1(S^4,\mathrm{ad}P)$; this difference is a globally defined adjoint-valued one-form. The induced covariant derivative is $D_{A}u=du+[A,u]$ in a local trivialization, and $d_{A_{\ast }}^{\ast }$ is the formal $L^2$ adjoint of the exterior covariant derivative for the round metric and $\kappa$. For $u,w\in \mathrm{\Gamma }(\mathrm{ad}P)$ define $(u,w{)}_P={\int }_{S^4}\kappa (u,w){\mathrm{v}\mathrm{o}\mathrm{l}}_g$ and extend this pairing to adjoint-valued forms using the metric. The background-gauge slice is $d_{A_{\ast }}^{\ast }a=0$, and the Faddeev–Popov operator is ${\mathcal{M}}_{P,A}=d_{A_{\ast }}^{\ast }{D}_A:\mathrm{\Gamma }(\mathrm{ad}P)\to \mathrm{\Gamma }(\mathrm{ad}P)$. Define the stabilizer algebra of the background by ${\mathfrak{h}}_P=\{u\in \mathrm{\Gamma }(\mathrm{ad}P):D_{A_{\ast }}u=0\}$. Let $L^2(P)$ be the completion of $\mathrm{\Gamma }(\mathrm{ad}P)$ in $(\cdot ,\cdot {)}_P$, set ${\mathfrak{h}}_P^{\perp }=\{u\in L^2(P):(u,h{)}_P=0\text{ for all }h\in {\mathfrak{h}}_P\}$, and take the smooth nonzero-mode domain to be $\mathrm{\Gamma }(\mathrm{ad}P)\cap {\mathfrak{h}}_P^{\perp }$. Since $S^4$ is connected, evaluation at one point injects ${\mathfrak{h}}_P$ into $\mathfrak{s}\mathfrak{u}(2)$, so this space of parallel sections is finite-dimensional. Since $D_{A_{\ast }}h=0$ for $h\in {\mathfrak{h}}_P$, the image of ${\mathcal{M}}_{P,A}$ is orthogonal to ${\mathfrak{h}}_P$; on the background-gauge slice, ${\mathfrak{h}}_P$ is contained in its kernel. The gauge map is $f(A,\mathrm{\Phi },K)=d_{A_{\ast }}^{\ast }(A-A_{\ast })$ with target $E_f=\mathrm{\Gamma }(\mathrm{ad}P)\cap {\mathfrak{h}}_P^{\perp }$, and its differential on a tangent connection $\dot{A}$ is $d{f}_A(\dot{A})=d_{A_{\ast }}^{\ast }\dot{A}$; hence $-d{f}_A{R}_A={\mathcal{M}}_{P,A}$. Its zero set is the supplied slice submanifold with tangent $\ker d{f}_A$. In each gauge chart, assume the kernel is exactly ${\mathfrak{h}}_P$, the reduced tangent splitting required by the rigid-background definition holds, and ${\mathcal{M}}_{P,A}$ is a continuous linear bijection of $\mathrm{\Gamma }(\mathrm{ad}P)\cap {\mathfrak{h}}_P^{\perp }$ onto itself with continuous inverse; ${\det }^{\mathrm{\prime }}{\mathcal{M}}_{P,A}$ denotes this restricted determinant. This chartwise invertibility, the chart cover, and the determinant are formal gauge-fixing inputs; no global gauge slice or determinant construction is asserted. In the trivial bundle with $A_{\ast }=0$, ${\mathfrak{h}}_P$ is the space of constant $\mathfrak{s}\mathfrak{u}(2)$-valued sections and these definitions reduce to $d^{\ast }A=0$ and $d^{\ast }{D}_A$.

The odd ghost and antighost fields $c,\tilde{c}$ and even multiplier $b$ are smooth sections of $\mathrm{ad}P$; their zero modes lie in ${\mathfrak{h}}_P$. The even zero-mode fields $a_0,{\tilde{a}}_0,b_0$ and odd zero-mode fields $c_0,{\tilde{c}}_0$ lie in ${\mathfrak{h}}_P$, so they need not be globally constant. Their coefficient spaces are complexified when imposing the Euclidean contours; ${\mathfrak{h}}_P$ still denotes the real parallel-section space, and the gauge action and brackets extend complex-linearly to its complexification. The lift above is assumed to fix ${\mathfrak{h}}_P$ pointwise. Assign mass dimensions $[c]=\frac{1}{2}$, $[\tilde{c}]=\frac{3}{2}$, $[b]=[b_0]=2$, $[a_0]=[\varphi ]=1$, $[{\tilde{a}}_0]=3$, $[c_0]=\frac{5}{2}$, and $[{\tilde{c}}_0]=\frac{7}{2}$; here brackets denote numerical mass dimensions. Both $s$ and $\delta$ have mass dimension $1\mathrm{/}2$. Take positive dimensionless ${\xi }_1,{\widehat{\xi }}_2$ and set ${\xi }_2={\widehat{\xi }}_2\mathrm{/}{r}^2$, of mass dimension two. With the unnormalized four-dimensional pairing $(\cdot ,\cdot {)}_P$, each term of the following odd gauge fermion has mass dimension $-1\mathrm{/}2$, so its $\delta$ and $Q$ variations are dimensionless. In each gauge chart, adopt the sectorwise BRST functional $S_{\mathrm{g}\mathrm{f}}^{\delta }[P]=\delta ((\tilde{c},i{d}_{A_{\ast }}^{\ast }a+\frac{{\xi }_1}{2}b+i{b}_0{)}_P-(c,{\tilde{a}}_0-\frac{{\xi }_2}{2}{a}_0{)}_P)\mathrm{.}$ Let ${\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}[P]$ be the odd functional inside this variation. In the sectorwise gauge-fixed formulas, $\omega$ denotes the same source-normalized generator as $v=ϵ\mathrm{\Gamma }ϵ$. Define ${\delta }_{U(1)}(\omega ):=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}-{\delta }_R,{\rho }_{\mathrm{g}\mathrm{h}}(\omega ):=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}\text{on adjoint-valued ghosts}\mathrm{.}$ The second operator restricts to the stated zero-mode domains, where it vanishes by the chosen lift; the first includes the declared auxiliary-index action on $K$. The exact sectorwise completion is $S_{\mathrm{g}\mathrm{f}}^Q[P]=Q{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}[P]$, with $Q=s+\delta$ and $\psi =sA\in {\mathrm{\Omega }}^1(S^4,\mathrm{ad}P)$ odd. In the conventions displayed here its expansion is $S_{\mathrm{g}\mathrm{f}}^Q[P]=S_{\mathrm{g}\mathrm{f}}^{\delta }[P]-(\tilde{c},i{d}_{A_{\ast }}^{\ast }\psi +\frac{{\xi }_1}{2}{\rho }_{\mathrm{g}\mathrm{h}}(\omega )\tilde{c}{)}_P-(\varphi ,{\tilde{a}}_0-\frac{{\xi }_2}{2}{a}_0{)}_P\mathrm{.}$ In particular the factor $i$ in the first term follows from the chosen gauge fermion and $\psi =sA$. The invariant background $A_{\ast }$, the metric pairing, and the fixed zero-mode algebra make ${\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$ invariant under ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)+{\delta }_{U(1)}(\omega )$, so the adopted algebra gives $Q{S}_{\mathrm{g}\mathrm{f}}^Q=Q^2{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}=0$. For nontrivial $P$, the lift, background-gauge chart, zero-mode complex and this completion are additional formal model inputs; Pestun's trivialization-chart calculation does not construct them. The gauge-fixed functional integral is the formal expression $Z_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}={\sum }_{[P]\in \mathrm{\Sigma }}\frac{1}{\mathrm{vol}(\mathcal{G}(P),g_{\mathrm{Y}\mathrm{M}})}{\int }_{{\mathcal{C}}_P}[DAD\mathrm{\Phi }D\mathrm{\Psi }DKDcD\tilde{c}DbD{a}_{0}D{\tilde{a}}_{0}D{b}_{0}D{c}_{0}D{\tilde{c}}_0]e^{-S_{\mathrm{o}\mathrm{f}\mathrm{f}}[P]-S_{\mathrm{g}\mathrm{f}}^Q[P]}\mathrm{.}$ Here $D\mathrm{\Phi }={\prod }_{A\in \{0,5,6,7,8,9\}}D{\mathrm{\Phi }}_A$ and $DK={\prod }_{i=1}^{7}D{K}_i$. The contour ${\mathcal{C}}_P$ concerns the full independent tuple $(A,\mathrm{\Phi },\mathrm{\Psi },K,c,\tilde{c},b,a_0,{\tilde{a}}_0,b_0,c_0,{\tilde{c}}_0)$; odd factors denote formal Berezin integration over their anticommuting coefficients. The $c,\tilde{c},b$ factors include their own ${\mathfrak{h}}_P$ modes; the five explicitly added zero-mode variables are additional fields, each integrated once. The composite $\varphi =-{\kappa }_v$ and the background spinors are not additional independent integration variables, and there is no independent conjugate of $\mathrm{\Psi }$. This notation remains formal and constructs no continuum measure. In this sum the index is the sector code $s=[P]\in \mathrm{\Sigma }$ and $P$ denotes its supplied representative $P_s$. Here $\mathcal{G}(P)$ is the smooth gauge-automorphism group of $P$, ${\mathcal{C}}_P$ is the declared contour in that sector, and $\mathrm{vol}(\mathcal{G}(P),g_{\mathrm{Y}\mathrm{M}})$ is the source's formal gauge-group volume with its invariant measure scaled by the gauge coupling; it is not an actual infinite-dimensional Haar measure. In the Euclidean contour, gauge fields and ${\mathrm{\Phi }}_5,\dots ,{\mathrm{\Phi }}_9$ lie in the real Lie algebra, ${\mathrm{\Phi }}_0=i{\mathrm{\Phi }}_0^E$ with ${\mathrm{\Phi }}_0^E$ real, the seven localization auxiliaries satisfy $K_i=i{K}_i^E$ with $K_i^E$ real, and $a_0=i{a}_0^E$ with $a_0^E$ real.

Use ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)$ with the infinitesimal gauge-action convention specified above, and use the combined circle/R operator ${\delta }_{U(1)}(\omega )$ and ghost action ${\rho }_{\mathrm{g}\mathrm{h}}(\omega )$ defined above. The supersymmetry differential acts by $sX=X^{\mathrm{\prime }}$, $s{X}^{\mathrm{\prime }}={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )X+{\delta }_{U(1)}(\omega )X$, $sc=\varphi$, $s\varphi =0$, $s{a}_0=0$, $s\tilde{c}=s{\tilde{a}}_0=s{b}_0=s{\tilde{c}}_0=s{c}_0=0$, and $sb={\rho }_{\mathrm{g}\mathrm{h}}(\omega )\tilde{c}$. The Faddeev–Popov differential acts on an adjoint-valued tensor field $X$ by $\delta X=-[c,X]$ and on its opposite-parity partner by $\delta X^{\mathrm{\prime }}=-[c,X^{\mathrm{\prime }}]$. On the gauge connection it acts by $\delta A_{\mu }=D_{\mu }c=-{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(c)A_{\mu }$, using the earlier convention ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)A_{\mu }=-D_{\mu }u$; the connection's one-form superpartner $\psi$ transforms as $\delta \psi =-[c,\psi ]$. The remaining rules are $\delta c=-a_0-\frac{1}{2}[c,c]$, $\delta a_0=0$, $\delta \varphi =-[c,\varphi ]+{\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$, $\delta \tilde{c}=b$, $\delta b=[a_0,\tilde{c}]$, $\delta {\tilde{a}}_0={\tilde{c}}_0$, $\delta {\tilde{c}}_0=[a_0,{\tilde{a}}_0]$, $\delta b_0=c_0$, and $\delta c_0=[a_0,b_0]$. Here $X$ ranges over homogeneous adjoint-valued physical and auxiliary doublet coordinates other than the connection and the unpaired $\varphi$, $X^{\mathrm{\prime }}$ has parity opposite to $X$, $c,\tilde{c},c_0,{\tilde{c}}_0$ are odd, and $a_0,{\tilde{a}}_0,b,b_0,\varphi$ are even. The total localization differential is $Q=s+\delta$, with $s$ the supersymmetry differential and $\delta$ the Faddeev–Popov differential. These rules give, on adjoint-valued tensor fields and their superpartners, $s^2{X}^{\mathrm{\prime }}={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )X^{\mathrm{\prime }}+{\delta }_{U(1)}(\omega )X^{\mathrm{\prime }},{\delta }^{2}X={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)X,\{s,\delta \}X=-{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )X,$ and the same graded algebra on the connection and its one-form superpartner when the gauge action on the connection is interpreted as ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)A_{\mu }=-D_{\mu }u$. On the gauge-fixing ghosts other than the unpaired $\varphi$, $\{s,\delta \}={\rho }_{\mathrm{g}\mathrm{h}}(\omega )$ and $s^2=0$; the displayed ghost transformations give ${\delta }^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)$. Thus $Q^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(a_0)+{\delta }_{U(1)}(\omega )$ on fields and ghosts: a gauge transformation generated by the zero-mode section $a_0\in {\mathfrak{h}}_P$ plus the selected combined spacetime/R action. The affine rule for $\varphi$ is important: $\delta \varphi =-[c,\varphi ]+{\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$, equivalently $-[c,\varphi +\omega ]$ in semidirect-product notation, not $-[c+\omega ,\varphi ]$. For example, $s\delta c=-[\varphi ,c]$ and $\delta sc=\delta \varphi$ give $\{s,\delta \}c={\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$; together with the graded Jacobi identity this gives $Q^{2}c=[a_0,c]+{\rho }_{\mathrm{g}\mathrm{h}}(\omega )c$. This convention agrees with Pestun (4.12); the differently ordered entry printed in (4.10) is not used as an identity. These are the finite-dimensional geometric symmetry data of the background; this assertion does not identify the infinite-dimensional field space with a smooth finite-dimensional manifold.

Adopt as a formal regulator the supersymmetry-compatible noncommutative deformation used by the source for gauge-field regularization on ${\mathbb{R}}^4$. Its $\mathrm{\Omega }$-background is the $U(1{)}^2$ action rotating the two coordinate planes in ${\mathbb{R}}^4\cong {\mathbb{C}}^2$, with angular parameters ${\epsilon }_1={\epsilon }_2=r^{-1}$, each of mass dimension one. For the four-sphere, assume an extension exists and agrees near the north and south poles with the stated ${\mathbb{R}}^4$ regulator. This extension is not explicitly constructed in the source and is an additional unresolved model assumption. The regulated measure, contour, and gauge-fixed action are likewise formal inputs; any use of $S_t=S+tQV$ is conditional on measure and contour invariance, no anomalous Jacobian, justified differentiation under the integral, and no uncontrolled boundary contribution. This postulate is a source-qualified modeling choice, not a proof of a continuum path integral or a result about nature.

Pestun, §§1–4 and Appendices B–C

Peelaers, §1

Festuccia and Seiberg, §2