Curvature-Current and Fradkin-Tseytlin Source Contacts

Use Curved Type II Pure-Spinor Model and Quantum BV Data and Intrinsic Pure-Spinor Volume and Current Cocycle. In the Euclidean source-contact chart $z=x+iy$, use $d^{2}z=dxdy$, ${\mathrm{\partial }}_z=\frac{1}{2}({\mathrm{\partial }}_x-i{\mathrm{\partial }}_y)$ and ${\bar{\mathrm{\partial }}}_z=\frac{1}{2}({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)$. Let ${\delta }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}(z-w)$ be the Dirac distribution with $⟨{\delta }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}(\cdot -w),\sigma ⟩=\sigma (w)$ for $\sigma \in C_c^{\mathrm{\infty }}(\mathbb{C})$. These are contact-chart derivatives, separate from the real canonical-time notation. For this contact channel additionally impose the source's conventional dimension-zero data: for odd frame indices $A,B$ and any odd output $\delta$, $T_{AB}{}^{\delta }=0$ (using the corresponding left or right connection); all components of $H$ with three odd indices vanish; $T_{\alpha \bar{\beta }}{}^a=0$; and $T_{\alpha \beta a}=H_{\alpha \beta a}=({\gamma }_a{)}_{\alpha \beta }$ and ${\tilde{T}}_{\bar{\alpha }\bar{\beta }a}=-H_{\bar{\alpha }\bar{\beta }a}=({\tilde{\gamma }}_a{)}_{\bar{\alpha }\bar{\beta }}$. These are additional conventional background assumptions, not consequences of the pure-spinor-contracted constraints alone. Use the displayed canonical transformation as the specified source-convention input on this branch. For fixed left odd index $\alpha$, its field-dependent Lorentz parameter and transformation are

${\mathrm{\Sigma }}_{\alpha }{}^{\beta }={\lambda }^{\gamma }{\mathrm{\Omega }}_{\gamma \alpha }{}^{\beta }+{\tilde{\lambda }}^{\bar{\gamma }}{\mathrm{\Omega }}_{\bar{\gamma }\alpha }{}^{\beta },Q_c{d}_{\alpha }={\mathrm{\Sigma }}_{\alpha }{}^{\beta }{d}_{\beta }+({\gamma }_a\lambda {)}_{\alpha }{\mathrm{\Pi }}^a+V_{\alpha }{}^{\delta }{w}_{\delta },$

where the actual curvature-current coefficient is

$V_{\alpha }{}^{\delta }={\lambda }^{\beta }{R}_{\beta \alpha \gamma }{}^{\delta }{\lambda }^{\gamma }+{\tilde{\lambda }}^{\bar{\beta }}{R}_{\bar{\beta }\alpha \gamma }{}^{\delta }{\lambda }^{\gamma }\mathrm{.}$

Here $R_{AB\alpha }{}^{\beta }$ is the curvature of the left spinor connection and $r_{AB}$ its scale-curvature component. As in Curved Type II Pure-Spinor Model and Quantum BV Data, ${\gamma }^{mn}$ is the Weyl block of $\frac{1}{2}[{\mathrm{\Gamma }}^m,{\mathrm{\Gamma }}^n]$; thus $R_{BA\alpha }{}^{\delta }=r_{BA}{\delta }_{\alpha }{}^{\delta }+\frac{1}{4}{R}_{BA,mn}({\gamma }^{mn}{)}_{\alpha }{}^{\delta }$. Set $t_{\alpha \beta }=R_{\alpha \gamma \beta }{}^{\gamma }$. Then

${\mathrm{div}}_{\mathrm{\Omega }}{V}_{\alpha }={\lambda }^{\beta }(t_{\alpha \beta }+8r_{\beta \alpha })+8{\tilde{\lambda }}^{\bar{\beta }}{r}_{\bar{\beta }\alpha }\mathrm{.}$

On the branch ${\mathrm{\Omega }}_{\bar{\beta }}=0$, the supplier's directional-derivative convention gives $r_{\bar{\beta }\alpha }=D_{\bar{\beta }}{\mathrm{\Omega }}_{\alpha }-c_{\bar{\beta }\alpha }{}^C{\mathrm{\Omega }}_C$, where $[E_{\bar{\beta }},E_{\alpha }]=c_{\bar{\beta }\alpha }{}^C{E}_C$ is the graded frame commutator. With $D_{\alpha }\mathrm{\Phi }=4{\mathrm{\Omega }}_{\alpha }$, the opposite-ghost term is $2{\tilde{\lambda }}^{\bar{\beta }}{D}_{\bar{\beta }}{D}_{\alpha }\mathrm{\Phi }-8{\tilde{\lambda }}^{\bar{\beta }}{c}_{\bar{\beta }\alpha }{}^C{\mathrm{\Omega }}_C$. The simpler first term is recovered if $c_{\bar{\beta }\alpha }{}^C{\mathrm{\Omega }}_C=0$ is additionally imposed; vanishing mixed torsion alone does not imply this condition. The source conjugate has local components $w_i={\beta }_i$. Since the current convention of Intrinsic Pure-Spinor Volume and Current Cocycle is $j_{V_{\alpha }}=-:\text{\hspace{0.17em}⁣}{\beta }_i{V}_{\alpha }{}^i\text{\hspace{0.17em}⁣}:$, the actual Qd term $+w_i{V}_{\alpha }{}^i$ is $-j_{V_{\alpha }}$, so its third-pole coefficient is $-{\alpha }^{\mathrm{\prime }}{\mathrm{div}}_{\mathrm{\Omega }}{V}_{\alpha }$. The original left-odd simple poles are $\mathrm{\Phi }(z)d_{\beta }(w)\sim -{\alpha }^{\mathrm{\prime }}{D}_{\beta }\mathrm{\Phi }\mathrm{/}(z-w)$ and $\mathrm{\Phi }(z){\mathrm{\Pi }}^a(w)\sim +{\alpha }^{\mathrm{\prime }}{D}^a\mathrm{\Phi }\mathrm{/}(z-w)$. With $U_{\alpha }=({\gamma }_a\lambda {)}_{\alpha }{D}^a\mathrm{\Phi }-{\mathrm{\Sigma }}_{\alpha }{}^{\beta }{D}_{\beta }\mathrm{\Phi }$, the left Fradkin–Tseytlin insertion $T_{\mathrm{F}\mathrm{T},\mathrm{L}}={\mathrm{\partial }}^2\mathrm{\Phi }$ contributes $2{\alpha }^{\mathrm{\prime }}{U}_{\alpha }\mathrm{/}(z-w{)}^3$. The computed partial third-pole coefficient is therefore ${\alpha }^{\mathrm{\prime }}[-{\mathrm{div}}_{\mathrm{\Omega }}{V}_{\alpha }+2U_{\alpha }]$; the opposite-chirality insertion also gives the distribution $\pi {\alpha }^{\mathrm{\prime }}{U}_{\alpha }{\bar{\mathrm{\partial }}}_z{\delta }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}(z-w)$, whose pairing with a test function $\sigma$ is $-\pi {\alpha }^{\mathrm{\prime }}{U}_{\alpha }\bar{\mathrm{\partial }}\sigma (w)$ under $\bar{\mathrm{\partial }}(1\mathrm{/}z)=\pi {\delta }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$. These are only the specified curvature/FT source contacts.

Curved Type II Pure-Spinor Model and Quantum BV Data

Intrinsic Pure-Spinor Volume and Current Cocycle

Bedoya–Chandía, original curved-model conventions

Nekrasov, local constrained-current pole

Chandía–Vallilo, type-II canonical transformation and full $Q{d}_{\alpha }$

The full source insertions also contain ${\mathrm{\Sigma }}_{\alpha }{}^{\beta }{d}_{\beta }$ and $({\gamma }_a\lambda {)}_{\alpha }{\mathrm{\Pi }}^a$; only their stated FT simple-pole contractions are evaluated here. Other pole orders, target-coefficient contractions, canonical endpoint/momentum partners, the $C,\tilde{C},S$ interaction vertices, the opposite right source, and the full regulated BV density remain unevaluated. The inspected research note constrained-ghost-weyl-current.md separately derives the classical scalar-current source cancellation after independent-coupling pullback. That redundant-source identity does not establish the one-loop QME, cone boundary prescription, or restored eight-slot BV comparison and is not used to cancel this partial sum.