RNS N=(1,1) Worldsheet Superspace Data

Let $\mathrm{\Sigma }$ be a compact connected Riemann surface with complex structure $J$ satisfying $J^2=-\mathrm{id}$, and give it the orientation induced by $J$. Choose a smooth positive-definite metric $h$ such that $h(Ju,Jv)=h(u,v)$ for all $p\in \mathrm{\Sigma }$ and $u,v\in T_p\mathrm{\Sigma }$. Supply a Spin structure on the oriented Euclidean bundle $(T\mathrm{\Sigma },h)$: a principal $\mathrm{Spin}(2)$-bundle $P\to \mathrm{\Sigma }$ with an equivariant two-fold map $\tilde{\lambda }:P\to P_{\mathrm{S}\mathrm{O}}(T\mathrm{\Sigma },h)$ over ${\mathrm{id}}_{\mathrm{\Sigma }}$, satisfying $\tilde{\lambda }(pg)=\tilde{\lambda }(p){\lambda }_2(g)$ for the standard double cover ${\lambda }_2:\mathrm{Spin}(2)\to \mathrm{SO}(2)$. Let $S_{+}$ and $S_{-}$ be the associated complex line bundles for the chiral representations ${\rho }_{\pm }(e^{\frac{t}{2}{e}_1{e}_2})=e^{\pm it\mathrm{/}2}$, where $(e_1,e_2)$ is a positive orthonormal frame and $t$ is its rotation angle. The two representations are complex conjugates, so $S_{-}\cong \overline{S_{+}}$. Let ${\mathcal{S}}_{\mathbb{R}}$ be the real rank-two bundle associated to the underlying real representation of ${\rho }_{+}$ on $\mathbb{C}$; then ${\mathcal{S}}_{\mathbb{R}}{\otimes }_{\mathbb{R}}\mathbb{C}\cong S_{+}\oplus S_{-}$. These are supplied data; no existence theorem for a Spin structure is asserted.

A local $(1,1)$ worldsheet superspace $\mathcal{W}$ is the split real smooth supermanifold of dimension $2\mathrm{\mid }2$ with body $\mathrm{\Sigma }$ and structure sheaf locally $C_{\mathrm{\Sigma },\mathbb{R}}^{\mathrm{\infty }}{\otimes }_{\mathbb{R}}{\mathrm{\Lambda }}^{\bullet }{\mathcal{S}}_{\mathbb{R}}^{\ast }$. Thus its coefficient field is $\mathbb{R}$ and its odd tangent bundle has real rank two. After complexifying this sheaf to use chiral notation, a local oriented chart has coordinates $({\sigma }^0,{\sigma }^1\mid {\theta }^{+},{\theta }^{-})$, where ${\sigma }^0,{\sigma }^1$ are real even coordinates and ${\theta }^{\pm }$ are the odd coordinate generators in the dual chiral lines; they anticommute. The real structure exchanges the two chiral lines and, in compatible local frames, $\overline{{\theta }^{+}}={\theta }^{-}$. The body transitions preserve the orientation, and the odd transitions are induced by the supplied spin bundle. “$(1,1)$” means one odd tangent direction of each chirality after complexification, related by this real structure.

For component bookkeeping extend coefficients to a fixed supercommutative real algebra $\mathbb{A}={\mathbb{A}}_{\bar{0}}\oplus {\mathbb{A}}_{\bar{1}}$, with $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$ for homogeneous elements, and complexify it for chiral components. Conjugation on $\mathbb{A}{\otimes }_{\mathbb{R}}\mathbb{C}$ is the antilinear multiplicative involution $\overline{a\otimes z}=a\otimes \bar{z}$ for $a\in \mathbb{A}$ and $z\in \mathbb{C}$; it preserves the written order of factors. In a separate flat Lorentzian light-cone superspace (not a chart on the Euclidean $\mathrm{\Sigma }$), take independent real odd coordinates ${\theta }^{\pm }$ and $x^{++},x^{--}$ with ${\mathrm{\partial }}_{\pm \pm }:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^{\pm \pm }$. The left superderivative generators $Q_{\pm }={\mathrm{\partial }}_{{\theta }^{\pm }}+i{\theta }^{\pm }{\mathrm{\partial }}_{\pm \pm }$ satisfy $Q_{\pm }^2=i{\mathrm{\partial }}_{\pm \pm }$ and $\{Q_{+},Q_{-}\}=0$. These are the plus-sign operators denoted $Q_{\pm }$ by Hull, Papadopoulos, and Townsend in §5, equation (5.3); no flat covariant derivative is denoted by $D_{\pm }$ here. An even scalar superfield $X:\mathcal{W}\to N$ into a smooth real manifold $N$ has local expansion $X^{\mu }=x^{\mu }+{\theta }^{+}{\mathrm{\Xi }}_{+}^{\mu }+{\theta }^{-}{\mathrm{\Xi }}_{-}^{\mu }+{\theta }^{+}{\theta }^{-}{A}^{\mu }\mathrm{.}$ Here $T{N}_{\mathbb{C}}:=TN{\otimes }_{\mathbb{R}}\mathbb{C}$, $x:\mathrm{\Sigma }\to N$ is a smooth body map, and ${\mathrm{\Xi }}_{\pm }$ are smooth odd sections with coefficients in ${\mathbb{A}}_{\bar{1}}{\otimes }_{\mathbb{R}}\mathbb{C}$ of $S_{\pm }^{\ast }\otimes x^{\ast }T{N}_{\mathbb{C}}$. The signs on ${\mathrm{\Xi }}_{\pm }$ label the coordinate monomials they multiply, not the associated-bundle chirality. Intrinsically the linear term is a section of ${\mathcal{S}}_{\mathbb{R}}^{\ast }\otimes x^{\ast }TN$. We retain this dual-bundle convention. The invariant real spin metric is $g_S(u,v)=\mathrm{Re}(u\bar{v})$ in a unit real spin frame. Its complex-bilinear extension pairs $S_{+}$ with $S_{-}$ and vanishes on either summand paired with itself; consequently $g_S^{\mathrm{♭}}$ maps $S_{-}$ to $S_{+}^{\ast }$ and $S_{+}$ to $S_{-}^{\ast }$. Thus raising the spin index sends ${\mathrm{\Xi }}_{+}$ to $S_{-}$, and ${\mathrm{\Xi }}_{-}$ to $S_{+}$; it does not preserve the written sign. The coefficient $A$ is a local even component with coefficients in ${\mathbb{A}}_{\bar{0}}{\otimes }_{\mathbb{R}}\mathbb{C}$. For a real even superfield the induced conjugation, with this coefficient order, is $\bar{x}=x$, $\overline{{\mathrm{\Xi }}_{+}}={\mathrm{\Xi }}_{-}$, and $\bar{A}=-A$. Under a smooth target-coordinate change $y^{\alpha }=f^{\alpha }(x)$ between real charts, the displayed coefficient order gives ${\mathrm{\Xi }}_{\pm }^{\mathrm{\prime }}{}^{\alpha }=({\mathrm{\partial }}_{\mu }{f}^{\alpha }){\mathrm{\Xi }}_{\pm }^{\mu }$ and $A^{\mathrm{\prime }}{}^{\alpha }=({\mathrm{\partial }}_{\mu }{f}^{\alpha })A^{\mu }-({\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }{f}^{\alpha }){\mathrm{\Xi }}_{+}^{\mu }{\mathrm{\Xi }}_{-}^{\nu }$. Thus $A$ alone need not be a section of $x^{\ast }T{N}_{\mathbb{C}}$; when ${\mathrm{\Xi }}_{\pm }=0$, the condition $A=0$ is invariant under target-coordinate changes. A covariant global auxiliary component away from that locus would require a supplied target connection and the associated splitting. No equations of motion or field existence result is asserted.

The curved data below are constrained geometric data; arbitrary unconstrained supervielbeins are not called an RNS supergravity geometry. Choose a holomorphic coordinate-disc cover with transitions $z_j=f_{ij}(z_i)$ and spin lifts $a_{ij}^2=f_{ij}^{\mathrm{\prime }}$ satisfying the cocycle law. Use the split supercoordinate transitions $z_j=f_{ij}(z_i),{\theta }_j^{+}=a_{ij}{\theta }_i^{+},{\bar{z}}_j={\bar{f}}_{ij}({\bar{z}}_i),{\theta }_j^{-}={\bar{a}}_{ij}{\theta }_i^{-}\mathrm{.}$ The odd coordinate generators belong to the dual of the supplied smooth real spin bundle, with the conformal/orthonormal identification described below. These are conformal spin coordinates, not orthonormal spin coordinates. The body $s=S{\mathrm{\mid }}_{\theta =0}$ is fixed by the supplied metric $h=e^{-4s}\mathrm{\mid }dz{\mathrm{\mid }}^2$. On a chart write $z=x+iy$, ${\mathrm{\partial }}_z=({\mathrm{\partial }}_x-i{\mathrm{\partial }}_y)\mathrm{/}2$, ${\mathrm{\partial }}_{\bar{z}}=({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)\mathrm{/}2$, and choose a real even smooth superfunction $S$. Its overlap law is $S_j=S_i+\frac{1}{2}\log \mathrm{\mid }{f}_{ij}^{\mathrm{\prime }}\mathrm{\mid },$ where supercoordinate pullback on the right is understood. A conformal presentation consists of these local functions with these laws. More general supplied presentations may be obtained by invertible parity-preserving supercoordinate changes and local Spin rotations of such data; the construction below is covariant under these operations. No global conformal-gauge theorem for every conceivable supergeometry is asserted.

Let $M$ be the complexified Lorentz generator acting on frame-weight $w$ components by multiplication by $w$, where $w\in \frac{1}{2}\mathbb{Z}$. Thus the component-index generator $J_0=iM$ acts by $iw$; its real matrix on component columns is $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. This fixes $M=-i{J}_0$. The component-index convention is contragredient to the moving-bundle-frame convention: on the bundle $S_{+}^{\ast }$ the moving-frame representation has infinitesimal character $-i\mathrm{/}2$, while its local coefficient has frame weight $+1\mathrm{/}2$. The scalar connection $\mathrm{\Omega }$ below is the negative of the moving-vector-frame connection $\omega$, so $d-i\omega \mathrm{/}2=d+i\mathrm{\Omega }\mathrm{/}2$ is exactly the induced connection on $S_{+}^{\ast }$, with conjugate formula on $S_{-}^{\ast }$. No identification of opposite chiral representations is implicit in $M=-i{J}_0$. An operator with label $+$ raises frame weight by $1\mathrm{/}2$, one with label $-$ lowers it by $1\mathrm{/}2$, and labels $++$ and $--$ change it by $1$ and $-1$. These conventions specify the connection on indexed components, including when two derivatives are composed. Define left odd derivatives $D_{+}={\mathrm{\partial }}_{{\theta }^{+}}+{\theta }^{+}{\mathrm{\partial }}_z,D_{-}={\mathrm{\partial }}_{{\theta }^{-}}+{\theta }^{-}{\mathrm{\partial }}_{\bar{z}},u=D_{+}S,v=D_{-}S,f=e^S\mathrm{.}$ The symbols $D_{\pm }$ refer only to this Euclidean presentation and are distinct from the flat Lorentzian $Q_{\pm }$ above. Define ${\mathrm{\nabla }}_{+}=f(D_{+}+2uM),{\mathrm{\nabla }}_{-}=f(D_{-}-2vM),$ ${\mathrm{\nabla }}_{++}=f^2({\mathrm{\partial }}_z+2u{D}_{+}+2({\mathrm{\partial }}_{z}S)M),{\mathrm{\nabla }}_{--}=f^2({\mathrm{\partial }}_{\bar{z}}+2v{D}_{-}-2({\mathrm{\partial }}_{\bar{z}}S)M)\mathrm{.}$ Their conventional torsion/curvature constraints, as equalities on every frame-weight component, are $\{{\mathrm{\nabla }}_{+},{\mathrm{\nabla }}_{+}\}=2{\mathrm{\nabla }}_{++},\{{\mathrm{\nabla }}_{-},{\mathrm{\nabla }}_{-}\}=2{\mathrm{\nabla }}_{--},\{{\mathrm{\nabla }}_{+},{\mathrm{\nabla }}_{-}\}=R_{M}M,R_M=4f^2{D}_{-}{D}_{+}S\mathrm{.}$ In particular the mixed odd bracket has no derivative-valued torsion term, and $[{\mathrm{\nabla }}_{+},{\mathrm{\nabla }}_{++}]=[{\mathrm{\nabla }}_{-},{\mathrm{\nabla }}_{--}]=0$. These identities are proved below, rather than imposed to infer the component reduction.

Writing ${\mathrm{\nabla }}_A=E_A{}^I({\mathrm{\partial }}_I+{\mathrm{\Omega }}_I{J}_0)$ defines the inverse supervielbein and scalar connection coefficient. The ordered indices are $I=(z,\bar{z},{\theta }^{+},{\theta }^{-})$ and $A=(++,--,+,-)$. In left-coefficient convention their full finite component expressions are $E_A{}^I=\left(\begin{array}{cccc}{f}^2(1+2u{\theta }^{+})& 0& 2f^{2}u& 0\\ 0& f^2(1+2v{\theta }^{-})& 0& 2f^{2}v\\ f{\theta }^{+}& 0& f& 0\\ 0& f{\theta }^{-}& 0& f\end{array}\right),$ $E_I{}^A=\left(\begin{array}{cccc}{f}^{-2}& 0& -2f^{-1}u& 0\\ 0& f^{-2}& 0& -2f^{-1}v\\ -f^{-2}{\theta }^{+}& 0& f^{-1}(1-2u{\theta }^{+})& 0\\ 0& -f^{-2}{\theta }^{-}& 0& f^{-1}(1-2v{\theta }^{-})\end{array}\right),$ ${\mathrm{\Omega }}_z=-2i{\mathrm{\partial }}_{z}S,{\mathrm{\Omega }}_{\bar{z}}=2i{\mathrm{\partial }}_{\bar{z}}S,{\mathrm{\Omega }}_{{\theta }^{+}}=-2i{\mathrm{\partial }}_{{\theta }^{+}}S,{\mathrm{\Omega }}_{{\theta }^{-}}=2i{\mathrm{\partial }}_{{\theta }^{-}}S\mathrm{.}$ Every entry is a smooth polynomial in the two odd coordinates after expanding $e^S$; coefficients may themselves lie in the component algebra $\mathbb{A}$. In particular these are explicit components, not an instruction to solve unspecified torsion equations. The scalar connection $\mathrm{\Omega }$ is conjugation compatible. For components of a frame of weight $w$, the connection is ${\mathrm{\partial }}_I+iw{\mathrm{\Omega }}_I$. In the bosonic restriction its coefficient is the negative of the connection acting on the moving vector basis; this is the usual opposite sign between basis and component connections.

For an even block supermatrix $T=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ with invertible $D$, use $\mathrm{Ber}T=\det (A-B{D}^{-1}C)\mathrm{/}\det D$. Set $\mathcal{E}=\mathrm{Ber}(E_A{}^I),\mu ={\mathcal{E}}^{-1}=\mathrm{Ber}(E_I{}^A)=e^{-2S}\mathrm{.}$ Thus the measure is the Berezinian of the coframe, not its reciprocal. Use the real oriented surface measure $d^{2}z=dxdy$; equivalently its even complex-coordinate normalization is $(i\mathrm{/}2)dz\wedge d\bar{z}$. To fix the real odd orientation, write ${\theta }^{+}=({\eta }^1+i{\eta }^2)\mathrm{/}\sqrt{2}$, ${\theta }^{-}=({\eta }^1-i{\eta }^2)\mathrm{/}\sqrt{2}$, with ${\eta }^1,{\eta }^2$ real odd variables. Write each local superfunction uniquely as $F=f_0+{\eta }^1{f}_1+{\eta }^2{f}_2+{\eta }^1{\eta }^2{f}_{12}$, with coefficients in $\mathbb{A}{\otimes }_{\mathbb{R}}\mathbb{C}$ on the right, and define $I_E(F):=f_{12}$, coefficientwise over the even coordinates. Equivalently, $I_E(1)=I_E({\eta }^1)=I_E({\eta }^2)=0$ and $I_E({\eta }^1{\eta }^2)=1$, so $I_E({\theta }^{+}{\theta }^{-})=-i,I_E({\theta }^{-}{\theta }^{+})=i\mathrm{.}$ This real Berezin integral is $i$ times the purely formal ordered chiral rule $I_0({\theta }^{-}{\theta }^{+})=1$. The change is essential: a real even superfunction has a purely imaginary ${\theta }^{+}{\theta }^{-}$ coefficient under the specified conjugation. Integrals are ${\int }_{\mathrm{\Sigma }}{d}^{2}z{I}_E(\mu \mathcal{L})$ in such charts, using an ordinary partition of unity if needed. Their transition law is proved below.

Let $F_J$ denote the scalar coefficient of $J_0$ in the mixed frame curvature. It is defined invariantly by $\{{\mathrm{\nabla }}_{+},{\mathrm{\nabla }}_{-}\}=F_J{J}_0$, so $F_J=-i{R}_M$. Equivalently it is the contraction of $d\mathrm{\Omega }$ with the two odd frame vectors, with the signs of the graded differential-form evaluation included. Define the real even scalar supercurvature used by the RNS model to be $\mathcal{R}=4F_J=-16i{e}^{2S}{D}_{-}{D}_{+}S\mathrm{.}$ The factor $4$ is the normalization relative to the coefficient of the real rotation generator, with the present $D_{\pm }$ and real odd integration convention. Its ordinary-curvature interpretation is derived below; it is not specified by assuming the desired integral identity.

The bosonic truncation means, in a conformal presentation, $S=s(z,\bar{z})$ with no independent odd or top component, and $X=x$ with ${\mathrm{\Xi }}_{+}={\mathrm{\Xi }}_{-}=A=0$. In a general local expansion $S=s+{\theta }^{+}{\xi }_{+}+{\theta }^{-}{\xi }_{-}+{\theta }^{+}{\theta }^{-}b$, these conditions are ${\xi }_{+}={\xi }_{-}=b=0$; define the conformal-gauge gravitino to be the odd spinor-valued one-form with local components ${\psi }_z{}^{+}=E_z{}^{+}\mathrm{\mid }=-2e^{-s}{\xi }_{+}$, ${\psi }_{\bar{z}}{}^{-}=E_{\bar{z}}{}^{-}\mathrm{\mid }=-2e^{-s}{\xi }_{-}$ and other components zero, and define the real curvature auxiliary scalar to be $A_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}:=\mathcal{R}\mathrm{\mid }=-16i{e}^{2s}b$. Thus the truncation removes precisely these independent component fields. They do not set the odd-coordinate-dependent coframe coefficients to zero. For the supplied body metric write $h=e^{-4s}(d{x}^2+d{y}^2)\mathrm{.}$ The odd coordinates in orthonormal spin frames are ${\vartheta }^{\pm }=e^{-s}{\theta }^{\pm }$; this identifies the conformal spin sheaf with the earlier smooth spin-bundle sheaf. If an original superfield expansion is written in the orthonormal generators as $X=x+{\vartheta }^{+}{\mathrm{\Xi }}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},+}+{\vartheta }^{-}{\mathrm{\Xi }}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},-}+{\vartheta }^{+}{\vartheta }^{-}{A}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}}$, then its conformal-coordinate coefficients are ${\mathrm{\Xi }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},\pm }=e^{-s}{\mathrm{\Xi }}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},\pm }$ and $A_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}=e^{-2s}{A}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}}$. They are the local components of the same bundle-valued data in nonunit conformal spin frames; they are not numerically identified with unrescaled orthonormal components. Both reality conditions and the zero-component truncation are preserved by these positive real rescalings. On overlaps $f_{ij}^{\mathrm{\prime }}=\mathrm{\mid }{f}_{ij}^{\mathrm{\prime }}\mathrm{\mid }{e}^{i\tau }$, ${\vartheta }_j^{\pm }=e^{\pm i\tau \mathrm{/}2}{\vartheta }_i^{\pm }$ and ${\mathrm{\Xi }}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},j,\pm }=e^{\mp i\tau \mathrm{/}2}{\mathrm{\Xi }}_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h},i,\pm }$. In conformal frames ${\theta }_j^{+}=a_{ij}{\theta }_i^{+}$, ${\theta }_j^{-}={\bar{a}}_{ij}{\theta }_i^{-}$, ${\mathrm{\Xi }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},j,+}=a_{ij}^{-1}{\mathrm{\Xi }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},i,+}$ and ${\mathrm{\Xi }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},j,-}={\bar{a}}_{ij}^{-1}{\mathrm{\Xi }}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f},i,-}$. These are dual-spin section components, consistently with the intrinsic linear term. At this truncation the full geometry is $E_I{}^A=\left(\begin{array}{cccc}{e}^{-2s}& 0& -2e^{-s}{\theta }^{+}{s}_z& 0\\ 0& e^{-2s}& 0& -2e^{-s}{\theta }^{-}{s}_{\bar{z}}\\ -e^{-2s}{\theta }^{+}& 0& e^{-s}& 0\\ 0& -e^{-2s}{\theta }^{-}& 0& e^{-s}\end{array}\right),\mathrm{\Omega }=-2s_{y}dx+2s_{x}dy,$ $\mu =e^{-2s},F_J=4i{e}^{2s}{\theta }^{+}{\theta }^{-}{s}_{z\bar{z}},\mathcal{R}=16i{e}^{2s}{\theta }^{+}{\theta }^{-}{s}_{z\bar{z}}\mathrm{.}$ For every smooth body map $x:\mathrm{\Sigma }\to N$ and every smooth real target scalar $\mathrm{\Phi }$, the resulting weighted component identity is $I_E\text{\hspace{0.17em}⁣}(\mu \mathrm{\Phi }(x)\mathcal{R})=16\mathrm{\Phi }(x)s_{z\bar{z}}=\sqrt{\det h}\mathrm{\Phi }(x)R^{(2)}(h)\mathrm{.}$ Here $R^{(2)}=2K_h$ is positive on the round sphere. The identity is pointwise in every oriented chart; no integration by parts of $\mathrm{\Phi }$ is used. Consequently it gives the corresponding global integral for arbitrary such $\mathrm{\Phi }$, not merely for constant dilaton or modulo a total derivative. This construction asserts neither a quantum measure nor a classification of all supergeometries.

Riemann surfaces and holomorphic atlases

Given Spin and Spin-c structures and their spinor bundles

The flat Lorentzian plus-sign generators $Q_{\pm }$ retain their original separate convention. The curved Euclidean geometry above gives an explicit constrained formulation. The construction and component identities are mathematical; identifying the ensuing sigma-model action as a physical model remains the consumer Postulate's adopted assumption.

The new primary source, Knutt-Wehlau–Mann, uses Lorentzian flat derivatives $D_{\pm }^L={\mathrm{\partial }}_{{\theta }_L^{\pm }}+i{\theta }_L^{\pm }{\mathrm{\partial }}_{\pm \pm }$ and generator $M_L$ in its equations (4.2)–(4.4). The displayed Euclidean formulas are obtained in the complexified algebra by ${\theta }_E^{\pm }=e^{i\pi \mathrm{/}4}{\theta }_L^{\pm }$, so $D_{\pm }^L=e^{i\pi \mathrm{/}4}{D}_{\pm }$, then by identifying the complexified even derivative labels with ${\mathrm{\partial }}_z,{\mathrm{\partial }}_{\bar{z}}$ and $M_L=M=-i{J}_0$. In particular $-2i(D_{+}^{L}S)D_{+}^L=2(D_{+}S)D_{+}$, $R_M^L=i{R}_M$, and the source coframe (4.4), after the corresponding odd frame/coordinate phase changes, becomes exactly the matrix above. The new Euclidean reality structure exchanges the chiral variables; it is not an identification with the source's independent real Lorentzian odd coordinates. The real odd orientation and the factor $4$ in $\mathcal{R}$ are specified and checked here. The source's $E=\mathrm{sdet}(E_A{}^M)$, its $E^{-1}=e^{-2S}$, and its component discussion (B.3)–(B.4) support this choice of inverse-frame notation and the interpretation of vanishing independent spinor/auxiliary components. This records an explicit algebraic continuation and mathematical normalization, not a claim that the Lorentzian and Euclidean quantum theories or their functional measures have been identified.