Adopted $N=2^{\ast }$ U(2) Model on the Round Four-Sphere

Use natural units $c=\mathrm{\hslash }=1$. For $r>0$, let

$S_r^4=\{y\in {\mathbb{R}}^5:{\sum }_{j=1}^5{y}_j^2=r^2\}$

with its standard orientation, round metric, scalar curvature $12\mathrm{/}{r}^2$, and standard spin structure. A spin structure here is a smooth principal $\mathrm{Spin}(4)$ bundle $\tilde{F}\to S_r^4$ with an equivariant two-sheeted covering $\lambda :\tilde{F}\to F_g^{+}(S_r^4)$ of the oriented orthonormal frame bundle, as in the supplied rigid-background definition. The radius has dimension $[r]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^{-1}$. In stereographic length coordinates $x^{\mu }$, $\mu =1,\dots ,4$, use

$d{s}^2=\frac{{\sum }_{\mu =1}^4(d{x}^{\mu }{)}^2}{(1+x^2\mathrm{/}(4r^2){)}^2},x^2={\sum }_{\mu =1}^4(x^{\mu }{)}^2\mathrm{.}$

Let $G=U(2)$, with Lie algebra $\mathfrak{u}(2)=\{X\in M_2(\mathbb{C}):X^{†}=-X\}$ and positive invariant form $(X,Y)=-{\mathrm{tr}}_{\mathbf{2}}(XY)$. A gauge sector is a smooth right principal $U(2)$ bundle $P\to S_r^4$ up to smooth bundle isomorphism, with a smooth connection $A$ and curvature $F_A=dA+A\wedge A$ in a local trivialization. Fields in the adjoint representation are smooth sections of the associated adjoint bundle. This postulate does not classify the bundle sectors.

Use Pestun's dimensional-reduction notation only as a field and spinor convention: $M,N=0,\dots ,9$ are ten-dimensional indices, $\mu =1,\dots ,4$ are the S4 coordinate indices, $A\in \{0,5,\dots ,9\}$ labels the six reduced scalars, and $i,j,k=5,\dots ,8$ label the four hypermultiplet scalars. Independently, $a,b=1,\dots ,7$ label localization auxiliaries $K_a$ and their background spinors ${\nu }_a$; these labels are not reduced spacetime directions. Write the reduced ten-dimensional connection as $A_M=(A_{\mu },{\mathrm{\Phi }}_A)$; ${\mathrm{\Phi }}_0,{\mathrm{\Phi }}_9$ and the fermion components in the ${\mathrm{\Gamma }}_{5678}=+1$ multiplet form the $N=2$ vector multiplet, while ${\mathrm{\Phi }}_5,\dots ,{\mathrm{\Phi }}_8$ and the complementary fermion components form one adjoint hypermultiplet. There is no other matter. The ten-dimensional indices do not enlarge the four-dimensional spacetime.

In the source's reduced $(9,1)$ Clifford convention, let $g_{MN}=\mathrm{diag}(-1,1,\dots ,1)$ and let ${\gamma }^M$ be the $32\times 32$ matrices satisfying ${\gamma }^M{\gamma }^N+{\gamma }^N{\gamma }^M=2g^{MN}I$. Their two chiral blocks are ${\mathrm{\Gamma }}^M:S^{+}\to S^{-}$ and ${\tilde{\mathrm{\Gamma }}}^M:S^{-}\to S^{+}$, satisfying ${\mathrm{\Gamma }}^{(M}{\tilde{\mathrm{\Gamma }}}^{N)}={\tilde{\mathrm{\Gamma }}}^{(M}{\mathrm{\Gamma }}^{N)}=g^{MN}I$, where parentheses and brackets denote symmetrization and antisymmetrization with factor $1\mathrm{/}2$. Set ${\mathrm{\Gamma }}^{MN}={\tilde{\mathrm{\Gamma }}}^{[M}{\mathrm{\Gamma }}^{N]}$ on $S^{+}$ and ${\tilde{\mathrm{\Gamma }}}^{MN}={\mathrm{\Gamma }}^{[M}{\tilde{\mathrm{\Gamma }}}^{N]}$ on $S^{-}$, as in Pestun Appendix A (A.7). Let ${\rho }_{\pm }$ be the restrictions of the two chiral $\mathrm{Spin}(9,1)$ representations to the embedded $\mathrm{Spin}(4)$ rotating directions $1,\dots ,4$, and set ${\mathcal{S}}_{\pm }=\tilde{F}{\times }_{{\rho }_{\pm }}{S}^{\pm }$. Thus, before Euclidean continuation, $\mathrm{\Psi }\in C^{\mathrm{\infty }}(S_r^4,\mathrm{ad}P\otimes {\mathcal{S}}_{+})$, $ϵ$ and each ${\nu }_a$ are smooth sections of ${\mathcal{S}}_{+}$, and $\widetilde{ϵ}$ is a smooth section of ${\mathcal{S}}_{-}$. For a local oriented orthonormal coframe $e^{\widehat{\mu }}$ on $S_r^4$, the spacetime Clifford maps are ${\mathrm{\Gamma }}_{\mu }=e_{\mu }{}^{\widehat{\nu }}{\mathrm{\Gamma }}_{\widehat{\nu }}$ and ${\tilde{\mathrm{\Gamma }}}_{\mu }=e_{\mu }{}^{\widehat{\nu }}{\tilde{\mathrm{\Gamma }}}_{\widehat{\nu }}$. Pestun's convention identifies $S^{-}$ with $(S^{+}{)}^{\ast }$ and $S^{+}$ with $(S^{-}{)}^{\ast }$; accordingly, when a chiral gamma block is used as a bilinear it is the symmetric lowered-index form obtained from that duality. In the chosen bases, a bilinear $\zeta B\chi$ means ${\zeta }^{\mathsf{T}}B\chi$, with no Hermitian conjugation; the background spinors commute and dynamical fermions are odd. After Euclidean continuation $\mathrm{\Psi }$ is a section of the complexification of $\mathrm{ad}P\otimes {\mathcal{S}}_{+}$ and its conjugate is not an independent field. The operators $D_M$ and $F_{MN}$ below are the covariant derivatives and curvature obtained by dimensional reduction of the ten-dimensional connection, with the mass twist in the 0 direction specified below.

The hypermultiplet has real Euclidean mass $m_E$ of mass dimension one, supplied by a nondynamical background flavor-vector multiplet whose fields are fixed and not integrated. In the source's orthonormal basis of the four hypermultiplet scalar directions, set $M_{ij}=i{m}_E{J}_{ij}$, where $J$ is a real antisymmetric self-dual endomorphism of the oriented $i=5,\dots ,8$ space, $J^2=-I$, and $J_{ij}{J}^{ij}=4$. Thus $M$ is imaginary, antisymmetric, and self-dual, as required by the source. This is the $N=2^{\ast }$ deformation; $m_E=0$ is its $N=4$ limit. Its Scherk--Schwarz twist is

$D_0{\mathrm{\Phi }}_i=[{\mathrm{\Phi }}_0,{\mathrm{\Phi }}_i]+M_{ij}{\mathrm{\Phi }}_j,D_0\mathrm{\Psi }=[{\mathrm{\Phi }}_0,\mathrm{\Psi }]+\frac{1}{4}{M}_{ij}{\mathrm{\Gamma }}^{ij}\mathrm{\Psi }\mathrm{.}$

For $D_{\mu }$, use the Levi-Civita derivative on spacetime/spin indices and the gauge covariant derivative ${\mathrm{\nabla }}_{\mu }^{A}X={\mathrm{\nabla }}_{\mu }X+[A_{\mu },X]$ on adjoint fields. In reduced scalar directions, $D_{A}X=[{\mathrm{\Phi }}_A,X]$ before the displayed 0-direction twist. Define $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$ and $F_{\mu A}=D_{\mu }{\mathrm{\Phi }}_A$. Before the twist, $F_{AB}=[{\mathrm{\Phi }}_A,{\mathrm{\Phi }}_B]$; afterward, $F_{0i}=D_0{\mathrm{\Phi }}_i$, $F_{i0}=-F_{0i}$, and the other reduced internal components retain their commutator values. The vector multiplet is neutral under this flavor twist. The fields have $[A_{\mu }]=[{\mathrm{\Phi }}_A]=[m_E]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $[\mathrm{\Psi }]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^{3\mathrm{/}2}$, and $[K_a]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$; the seven $K_a$ are smooth adjoint-valued scalars carrying the source's auxiliary $SO(7)$ index.

Let $R_{ij}$ be the source's real antisymmetric anti-self-dual matrix on the $i=5,\dots ,8$ scalar directions, normalized by $R_{ij}{R}^{ij}=4$. Adopt the source's curvature-coupled Scherk--Schwarz action

$S_{\mathcal{N}=2^{\ast }}=-\frac{1}{g_{\mathrm{Y}\mathrm{M}}^2}{\int }_{S_r^4}{d}^{4}x\sqrt{g}{\mathrm{tr}}_{\mathbf{2}}\text{\hspace{0.17em}⁣}(\frac{1}{2}{F}_{MN}{F}^{MN}-\mathrm{\Psi }{\mathrm{\Gamma }}^M{D}_M\mathrm{\Psi }+\frac{2}{r^2}{\mathrm{\Phi }}_A{\mathrm{\Phi }}^A-\frac{1}{4r}(R_{ki}{M}_{kj}){\mathrm{\Phi }}^i{\mathrm{\Phi }}^j-{\sum }_{a=1}^7{K}_a{K}_a)\mathrm{.}$

Here $g_{\mathrm{Y}\mathrm{M}}>0$ is dimensionless, repeated ten-dimensional indices use the source's $(9,1)$ reduction metric before Euclidean continuation, and the trace is in the defining representation. In particular its four-dimensional gauge term is

$S_{\mathrm{Y}\mathrm{M}}=\frac{1}{2g_{\mathrm{Y}\mathrm{M}}^2}{\int }_{S_r^4}{d}^{4}x\sqrt{g}(F_{\mu \nu },F^{\mu \nu })\mathrm{.}$

The displayed $S_{\mathcal{N}=2^{\ast }}$ is the local curvature-coupled action; the theta angle $\theta$ is real and dimensionless and enters the source's sector weights. The source's sector-weight convention is

$\tau =\frac{4\pi i}{g_{\mathrm{Y}\mathrm{M}}^2}+\frac{\theta }{2\pi },q=e^{2\pi i\tau },$

so a positive-instanton sector with formal label $k\in {\mathbb{Z}}_{\ge 0}$ carries weight $q^k$, and the corresponding anti-instanton factor uses ${\bar{q}}^k$ on the physical real slice. This adopts the source's convention without classifying bundle sectors or constructing the path-integral measure.

The rigid background is the fixed round metric and spin structure, the trivial $SU(2{)}_R$ background bundle with zero connection, the mass background, and a nonzero smooth conformal Killing-spinor pair $(ϵ,\widetilde{ϵ})$ satisfying

${\mathrm{\nabla }}_{\mu }ϵ={\tilde{\mathrm{\Gamma }}}_{\mu }\widetilde{ϵ},{\mathrm{\nabla }}_{\mu }\widetilde{ϵ}=-\frac{1}{4r^2}{\mathrm{\Gamma }}_{\mu }ϵ\mathrm{.}$

Choose the source's $N=2$ projection ${\mathrm{\Gamma }}_{5678}ϵ=ϵ$, the source's constant-unit-norm solution, and $\widetilde{ϵ}ϵ=0$. Use the same anti-self-dual matrix $R_{ij}$ as in the action to set ${\mathrm{\Lambda }}_{+}=\frac{1}{4}{\mathrm{\Gamma }}^{ij}{R}_{ij}$ on ${\mathcal{S}}_{+}$. On the $N=2$ subbundle $\ker ({\mathrm{\Gamma }}_{5678}-I)$ use the source normalization ${\mathrm{\Lambda }}_{+}^2=-I$. The reduced time direction supplies the $\mathrm{Spin}(4)$-equivariant isomorphism $C={\mathrm{\Gamma }}^0:{\mathcal{S}}_{+}\to {\mathcal{S}}_{-}$, with inverse $-{\tilde{\mathrm{\Gamma }}}^0$ by the Clifford relation. In Pestun's Appendix A (A.11) bases its matrix is $I_{16}$; the source's numerical formula suppresses this identification. With bundle types explicit, the selected spinor pair satisfies $\widetilde{ϵ}=\frac{1}{2r}C{\mathrm{\Lambda }}_{+}ϵ$. Thus its equivalent first equation is ${\mathrm{\nabla }}_{\mu }ϵ=\frac{1}{2r}{\tilde{\mathrm{\Gamma }}}_{\mu }C{\mathrm{\Lambda }}_{+}ϵ$, preserving the source's (2.16) and (2.24) conventions. Choose seven smooth auxiliary spinors ${\nu }_a$, $a=1,\dots ,7$, in the source's chiral spinor bundle satisfying

$ϵ{\mathrm{\Gamma }}^M{\nu }_a=0,\frac{1}{2}(ϵ{\mathrm{\Gamma }}_Nϵ){\tilde{\mathrm{\Gamma }}}_{\alpha \beta }^N={\sum }_{a=1}^7{\nu }_{\alpha }^a{\nu }_{\beta }^a+{ϵ}_{\alpha }{ϵ}_{\beta },{\nu }^a{\mathrm{\Gamma }}^M{\nu }^b={\delta }^{ab}ϵ{\mathrm{\Gamma }}^Mϵ\mathrm{.}$

The selected supersymmetry transformation $s={\delta }_{ϵ}$ on the reduced physical and auxiliary fields is

$s{A}_M=\mathrm{\Psi }{\mathrm{\Gamma }}_Mϵ,s\mathrm{\Psi }=\frac{1}{2}{\mathrm{\Gamma }}^{MN}{F}_{MN}ϵ+\frac{1}{2}{\mathrm{\Gamma }}^{\mu A}{\mathrm{\Phi }}_A{\mathrm{\nabla }}_{\mu }ϵ+{\sum }_{a=1}^7{K}_a{\nu }_a,s{K}_a=-{\nu }_a{\mathrm{\Gamma }}^M{D}_M\mathrm{\Psi }\mathrm{.}$

These are the off-shell transformations for this one selected generator; the finite set $K_a$ does not assert off-shell closure of the full $N=2$ hypermultiplet algebra. Their closure on the auxiliaries is

$s^2{K}_a=-(ϵ{\mathrm{\Gamma }}^Mϵ)D_M{K}_a-{\sum }_{b=1}^7({\nu }_{[a}{\mathrm{\Gamma }}^M{D}_M{\nu }_{b]})K_b-4(\widetilde{ϵ}ϵ)K_a\mathrm{.}$

Use the bilinear in exactly the source's order, $R_{AB}^Q=2ϵ{\tilde{\mathrm{\Gamma }}}_{AB}\widetilde{ϵ},({\delta }_R\mathrm{\Phi }{)}_A=R_{AB}^Q{\mathrm{\Phi }}^B,{\delta }_R\mathrm{\Psi }=\frac{1}{4}{R}_{AB}^Q{\mathrm{\Gamma }}^{AB}\mathrm{\Psi }\mathrm{.}$ Indices are raised with the reduction metric. The fixed normalized matrix $R_{ij}$ entering the mass action and ${\mathrm{\Lambda }}_{+}$ above is distinguished from this dimensionful supercharge-square matrix $R^Q$. No interchange of $ϵ$ and $\widetilde{ϵ}$ is used: the stated transpose relation instead gives $2\widetilde{ϵ}{\mathrm{\Gamma }}_{AB}ϵ=-R_{AB}^Q$. The source square is $s^2=-{\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}-{\delta }_R-{\delta }_{\mathrm{\Omega }}-{\delta }_M$, with $v^M=ϵ{\mathrm{\Gamma }}^Mϵ$, and ${\delta }_{\mathrm{\Omega }}=0$ because $\widetilde{ϵ}ϵ=0$. Here ${\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}$ already contains the gauge connection: on an adjoint scalar $X$ it is $v^{\mu }{D}_{\mu }X+[v^A{\mathrm{\Phi }}_A,X]$, on the connection it is ${\iota }_v{F}_A-D_A(v^A{\mathrm{\Phi }}_A)$, and on spinors it includes the geometric spin lift. The positive-sign mass operator is ${\delta }_M{\mathrm{\Phi }}_i=v^0{M}_{ij}{\mathrm{\Phi }}_j$ and ${\delta }_M\chi =\frac{1}{4}{v}^0{M}_{ij}{\mathrm{\Gamma }}^{ij}\chi$ on the adjoint hypermultiplet Dirac fermion $\chi$, and zero on the vector multiplet; consequently $-{\delta }_M$ gives exactly Pestun (2.25).

For a chosen sector, supply a one-parameter principal-bundle lift of the base flow and its induced spin lift, and let ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}$ denote its infinitesimal pullback. In a local gauge frame write its vertical coefficient as ${\beta }_{v,P}$, so on adjoint scalars ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}X=v^{\mu }{\mathrm{\partial }}_{\mu }X+[{\beta }_{v,P},X]$, and on connections ${\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}A={\mathcal{L}}_{v}A-D_A{\beta }_{v,P}$. The difference ${\mu }_{v,A}=v^{\mu }{A}_{\mu }-{\beta }_{v,P}$ is a global adjoint section. Set ${\kappa }_v={\mu }_{v,A}+v^A{\mathrm{\Phi }}_A$ and $\varphi =-{\kappa }_v$, and use ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)X=[u,X]$, ${\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(u)A=-D_{A}u$. Then ${\mathcal{L}}_v^{\mathrm{c}\mathrm{o}\mathrm{v}}={\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}+{\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}({\kappa }_v),s^2={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}-{\delta }_R-{\delta }_M\mathrm{.}$ On scalars the vertical coefficients add to $v^{\mu }{A}_{\mu }+v^A{\mathrm{\Phi }}_A$; on connections, ${\mathcal{L}}_{v}A-D_A(v^{\mu }{A}_{\mu }+v^A{\mathrm{\Phi }}_A)={\iota }_v{F}_A-D_A(v^A{\mathrm{\Phi }}_A)$. This explicitly reconciles the split with Pestun's gauge-covariant Lie derivative, rather than counting a gauge action twice. The auxiliary bundle is a rank-seven real orthogonal bundle tensored with $\mathrm{ad}P$; its combined spin/flavor/index lift is fixed by the displayed exact formula for $s^2{K}_a$. Equivalently its nongauge generator is $\omega K_a=s^2{K}_a-[\varphi ,K_a]$, with $s^2{K}_a$ exactly as above; a further independent scalar $R^Q$ or hypermultiplet $M_{ij}$ action is not added to its seven index labels. In this formula $D_{\mu }{K}_a={\mathrm{\partial }}_{\mu }{K}_a+[A_{\mu },K_a]$ and $D_A{K}_a=[{\mathrm{\Phi }}_A,K_a]$; the auxiliary flavor rotation is placed in the index matrix rather than added to $D_0{K}_a$. On the gauge-singlet background spinors set $D_{\mu }{\nu }_a={\mathrm{\nabla }}_{\mu }{\nu }_a$, $D_0{\nu }_a=\frac{1}{4}{M}_{ij}{\mathrm{\Gamma }}^{ij}{\nu }_a$ and $D_A{\nu }_a=0$ in the other reduced directions. This allocation follows directly from the mass term in $s{K}_a=-{\nu }_a{\mathrm{\Gamma }}^M{D}_M\mathrm{\Psi }$: its $K_b$ contribution on applying $s$ again is $-\frac{1}{4}{\nu }_a{\mathrm{\Gamma }}^0{M}_{ij}{\mathrm{\Gamma }}^{ij}{\nu }_b{K}_b$, the antisymmetric auxiliary-index rotation encoded by ${\nu }_{[a}{\mathrm{\Gamma }}^0{D}_0{\nu }_{b]}$. It fixes the induced auxiliary action even when the background spinors vary, without assigning the four-dimensional hypermultiplet index range to the seven auxiliaries. The spacetime component $v^{\mu }$ rotates the two orthogonal coordinate planes with equal angular magnitudes $r^{-1}$, so ${ϵ}_1={ϵ}_2=r^{-1}$ at the poles. These sectorwise lifts are formal model inputs, not existence theorems for arbitrary bundles.

For gauge fixing in each declared sector, choose a lift-invariant background connection $A_{\ast }$, set $a=A-A_{\ast }\in {\mathrm{\Omega }}^1(S^4,\mathrm{ad}P)$, and use the background slice $d_{A_{\ast }}^{\ast }a=0$, with Faddeev–Popov operator $d_{A_{\ast }}^{\ast }{D}_A$ on smooth adjoint-valued scalar sections. Here $d_{A_{\ast }}^{\ast }$ is the metric adjoint of the exterior covariant derivative. Let ${\mathfrak{h}}_P=\ker D_{A_{\ast }}$ be the finite-dimensional stabilizer algebra, give sections the $L^2$ pairing from the round metric and $-{\mathrm{tr}}_{\mathbf{2}}(XY)$, and take nonzero modes in the smooth part of ${\mathfrak{h}}_P^{\perp }$. The chosen gauge charts assume the kernel of the Faddeev–Popov operator is exactly ${\mathfrak{h}}_P$ and the reduced operator is invertible; a primed determinant denotes only that restriction. These are formal chart/determinant inputs, not a global gauge-slice theorem. In the trivial sector with $A_{\ast }=0$ they reduce to the source's $d^{\ast }A=0$ and $d^{\ast }{D}_A$, with constant zero modes.

Let $c,\tilde{c}$ be odd adjoint-valued smooth ghost/antighost scalar fields and $b$ an even multiplier of the same bundle type. The even variables $a_0,{\tilde{a}}_0,b_0$ and odd $c_0,{\tilde{c}}_0$ lie in ${\mathfrak{h}}_P$, which the lift fixes pointwise. They are parallel background zero modes, and are literally constant only in the trivial background frame. In the cohomological change of variables, $X$ is a homogeneous physical/auxiliary doublet coordinate, $X^{\mathrm{\prime }}=sX$ has the opposite parity, and the even unpaired parameter is $\varphi =-{\kappa }_v$ above. On tensor fields set $[\omega ,X]=-({\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}+{\delta }_R+{\delta }_M)X$, with the auxiliary action fixed by its exact square as above. On ghosts and multipliers set $[\omega ,c]=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}c$ and likewise for the other adjoint scalar ghost fields; there is no R/flavor charge. On connections $[\omega ,A]=-{\mathcal{L}}_v^{\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}}A$. The source's supersymmetry complex is $sX=X^{\mathrm{\prime }}$, $s{X}^{\mathrm{\prime }}=[\varphi +\omega ,X]$, $s\varphi =0$, and for the connection $sA=\psi \in {\mathrm{\Omega }}^1(S^4,\mathrm{ad}P)$ with $s\psi ={\delta }_{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}(\varphi )A+[\omega ,A]$. Brackets with two homogeneous matrix fields are parity-graded, and brackets with $\omega$ denote the just-defined linear symmetry operator.

The source's BRST-like differential $\delta$ and its supersymmetry differential $s$ act on this combined complex by

$\begin{array}{rlrlrlrlrlrl}{\displaystyle \delta X}& {\displaystyle =-[c,X],}& {\displaystyle \delta X^{\mathrm{\prime }}}& {\displaystyle =-[c,X^{\mathrm{\prime }}],}& {\displaystyle \delta A_{\mu }}& {\displaystyle =-[c,{\mathrm{\nabla }}_{\mu }],}& {\displaystyle \delta c}& {\displaystyle =-a_0-\frac{1}{2}[c,c],}& {\displaystyle \delta \tilde{c}}& {\displaystyle =b,}& {\displaystyle \delta a_0}& {\displaystyle =0,}\\ {\displaystyle \delta \varphi }& {\displaystyle =-[c,\varphi ]+[\omega ,c],}& {\displaystyle \delta {\tilde{a}}_0}& {\displaystyle ={\tilde{c}}_0,}& {\displaystyle \delta b_0}& {\displaystyle =c_0,}& {\displaystyle \delta b}& {\displaystyle =[a_0,\tilde{c}],}& {\displaystyle \delta {\tilde{c}}_0}& {\displaystyle =[a_0,{\tilde{a}}_0],}& {\displaystyle \delta c_0}& {\displaystyle =[a_0,b_0],}\\ {\displaystyle s{a}_0}& {\displaystyle =0,}& {\displaystyle sX}& {\displaystyle =X^{\mathrm{\prime }},}& {\displaystyle s{X}^{\mathrm{\prime }}}& {\displaystyle =[\varphi +\omega ,X],}& {\displaystyle s\varphi }& {\displaystyle =0,}& {\displaystyle sc}& {\displaystyle =\varphi ,}& {\displaystyle s\tilde{c}}& {\displaystyle =0,}\\ {\displaystyle s{\tilde{a}}_0}& {\displaystyle =0,}& {\displaystyle s{b}_0}& {\displaystyle =0,}& {\displaystyle sb}& {\displaystyle =[\omega ,\tilde{c}],}& {\displaystyle s{\tilde{c}}_0}& {\displaystyle =0,}& {\displaystyle s{c}_0}& {\displaystyle =0.}\end{array}$

Here $X$ ranges over physical and auxiliary fields, and $\delta A_{\mu }$ is the infinitesimal gauge action on a connection. Summing the two displayed differentials gives $QX=X^{\mathrm{\prime }}-[c,X]$, $Q{X}^{\mathrm{\prime }}=[\varphi +\omega ,X]-[c,X^{\mathrm{\prime }}]$, $Q{A}_{\mu }=s{A}_{\mu }-[c,{\mathrm{\nabla }}_{\mu }]$, $Qc=\varphi -a_0-\frac{1}{2}[c,c]$, $Q\tilde{c}=b$, $Q\varphi =-[c,\varphi ]+[\omega ,c]$, $Q{a}_0=0$, $Q{\tilde{a}}_0={\tilde{c}}_0$, $Q{b}_0=c_0$, $Qb=[a_0+\omega ,\tilde{c}]$, $Q{\tilde{c}}_0=[a_0,{\tilde{a}}_0]$, and $Q{c}_0=[a_0,b_0]$. The source's mixed squares are ${\delta }^2=[a_0,\cdot ]$, $s^2=[\varphi +\omega ,\cdot ]$, and $\{s,\delta \}=-[\varphi ,\cdot ]$ on physical/auxiliary fields; on gauge-fixing ghosts other than the unpaired $\varphi$, $s^2=0$ and $\{s,\delta \}$ is the $\omega$-action. Thus $Q^2=[a_0+\omega ,\cdot ]$, the gauge transformation by $a_0\in {\mathfrak{h}}_P$ together with the selected combined spacetime/R/flavor action. Here $\delta \varphi =-[c,\varphi ]+[\omega ,c]=-[c,\varphi +\omega ]$ in semidirect-product notation follows the total complex (4.12), rather than the differently ordered entry printed in (4.10). For example $s\delta c=-[\varphi ,c]$ and $\delta sc=\delta \varphi$ give $\{s,\delta \}c=[\omega ,c]$, and the graded Jacobi identity yields $Q^{2}c=[a_0,c]+[\omega ,c]$. The chosen $\omega$ action fixes all zero-mode parameters. The sectorwise zero-mode and gauge-fixing prescription remains formal.

For completeness the invariant gauge-fixed insertion is $S_{\mathrm{g}\mathrm{f}}^Q=Q{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$, not the Faddeev–Popov part alone. With positive gauge parameters ${\xi }_1,{\xi }_2$ and the declared $L^2$ pairing, take ${\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}=(\tilde{c},i{d}_{A_{\ast }}^{\ast }a+\frac{{\xi }_1}{2}b+i{b}_0)-(c,{\tilde{a}}_0-\frac{{\xi }_2}{2}{a}_0)\mathrm{.}$ The background and zero-mode invariance give $Q^2{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}=[a_0+\omega ,{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}]=0$, so $Q{S}_{\mathrm{g}\mathrm{f}}^Q=0$ in the adopted algebra. With $\psi =sA$, its supersymmetry correction to $\delta {\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$ is $-(\tilde{c},i{d}_{A_{\ast }}^{\ast }\psi +\frac{{\xi }_1}{2}[\omega ,\tilde{c}])-(\varphi ,{\tilde{a}}_0-\frac{{\xi }_2}{2}{a}_0)$. These are the explicit sectorwise formal completion data; no equivalence of constructed infinite-dimensional measures is inferred.

On the Euclidean contour, gauge and hypermultiplet fields take values in the real Lie algebra $\mathfrak{u}(2)$; the scalar descending from the time component is continued as ${\mathrm{\Phi }}_0=i{\mathrm{\Phi }}_0^E$, and the seven localization auxiliaries as $K_a=i{K}_a^E$, with ${\mathrm{\Phi }}_0^E,K_a^E$ real $\mathfrak{u}(2)$-valued fields. The finite-dimensional Coulomb matrix $a_{\mathrm{C}}$ is the zero mode ${\mathrm{\Phi }}_0^E$ integrated over $\mathfrak{u}(2)$, while the continued physical scalar is ${\mathrm{\Phi }}_0=i{a}_{\mathrm{C}}$. The real mass $m_E$ and pole weights ${ϵ}_i$ have mass dimension one. In the source's finite-dimensional Coulomb convention, $a_E=(a_1^E,a_2^E)\in {\mathbb{R}}^2$ parametrizes the skew-Hermitian Cartan element $a_{\mathrm{C}}=i\mathrm{diag}(a_1^E,a_2^E)$; the one-loop and Nekrasov factors use the continued arguments $i{a}_E$ and $a^{\mathrm{P}}=i{a}_E$, respectively. This fixes the contour relation but constructs no field-space measure. Let $\mathrm{\Sigma }$ denote the chosen noncommutative ${\mathbb{R}}^4$ regulator for the instanton sectors, with its $U(1{)}^2$ rotations and the above ${ϵ}_i$. Adopt the additional source assumption that a supersymmetry-compatible noncommutative deformation on $S_r^4$ exists and agrees near each pole with Nekrasov's noncommutative ${\mathbb{R}}^4$ deformation. Pestun explicitly does not construct this $S^4$ deformation. The gauge-fixed functional measure and this regulator extension remain formal model inputs; this postulate proves no measure existence, anomaly cancellation, regulator-removal limit, continuum Q-exact identity, or empirical claim.

Pestun, §§1–5 and Appendix C

Peelaers, Lecture 1 §§1.1–1.2 and Lecture 2

Rigid supersymmetric background and localization data