Intrinsic Pure-Spinor Volume and Current Cocycle

For $X=X^{\rho }{\mathrm{\partial }}_{\rho }+{\sum }_{a<b}{X}^{ab}{\mathrm{\partial }}_{u_{ab}}$, the coordinate formula is ${\mathrm{div}}_{\mathrm{\Omega }}X={\rho }^{-7}{\mathrm{\partial }}_{\rho }({\rho }^7{X}^{\rho })+{\sum }_{a<b}{\mathrm{\partial }}_{u_{ab}}{X}^{ab}$. Hence ${\mathrm{div}}_{\mathrm{\Omega }}(\rho {\mathrm{\partial }}_{\rho })={\rho }^{-7}{\mathrm{\partial }}_{\rho }({\rho }^8)=8$.

Write $\mathrm{\Omega }=v(\gamma )d^{11}\gamma$. The double contraction of $-{\alpha }^{\mathrm{\prime }-1}:\text{\hspace{0.17em}⁣}{\beta }_i\mathrm{\partial }{\gamma }^i\text{\hspace{0.17em}⁣}:(z)$ with $-:\text{\hspace{0.17em}⁣}{\beta }_j{V}^j\text{\hspace{0.17em}⁣}:(w)$ contributes ${\alpha }^{\mathrm{\prime }}{\mathrm{\partial }}_i{V}^i\mathrm{/}(z-w{)}^3$ under [A1]. The contraction of $-\frac{1}{2}{\mathrm{\partial }}^2\log v(\gamma (z))$ with the current contributes ${\alpha }^{\mathrm{\prime }}{V}^i{\mathrm{\partial }}_i\log v\mathrm{/}(z-w{)}^3$. Adding them gives ${\alpha }^{\mathrm{\prime }}[{\mathrm{\partial }}_i{V}^i+V^i{\mathrm{\partial }}_i\log v]={\alpha }^{\mathrm{\prime }}{v}^{-1}{\mathrm{\partial }}_i(v{V}^i)$, which is the Lie-derivative definition of divergence. Since this expression is coordinate invariant and the overlap preserves $\mathrm{\Omega }$, the third-pole coefficient agrees on the compatible overlaps.

From the Clifford action in Pure-Spinor Cone and Finite BV Semidensity Geometry, the differential spin representation sends a bivector $v\wedge w$ to $\frac{1}{4}[c(v),c(w)]$. In the isotropic splitting $V=W\oplus W^{\ast }$ its three summands are ${\mathrm{\Lambda }}^2{W}^{\ast }$, $\mathfrak{g}\mathfrak{l}(W)$ and ${\mathrm{\Lambda }}^{2}W$. Acting on $\lambda =\rho e^{B_2}$, the creation summand shifts $u$ by a constant and fixes $\rho$. The middle summand $A\in \mathfrak{g}\mathfrak{l}(W)$ acts on the scalar component by $\delta \rho =-\frac{1}{2}(\mathrm{tr}A)\rho$ and on the two-form coordinates by $\delta u=Au+u{A}^t$. The linear map $u\mapsto Au+u{A}^t$ has trace $4\mathrm{tr}A$ on ${\mathrm{\Lambda }}^{2}W$; its weighted radial contribution is $8(-\frac{1}{2}\mathrm{tr}A)=-4\mathrm{tr}A$. Thus each summand has zero $\mathrm{\Omega }$-divergence.

For a lowering generator $C\in {\mathrm{\Lambda }}^{2}W$, the Fock action is $\frac{1}{2}{C}^{ab}{\iota }_{e_a}{\iota }_{e_b}$, with repeated indices summed. Acting on the degree-zero, degree-two and degree-four pieces of $\rho e^{B_2}$ gives the chart vector $X_C^u=-uCu$ and $X_C^{\rho }=\frac{1}{2}\mathrm{tr}(Cu)\rho$. For an independent coordinate $u_{ab}$, varying $u_{ab}$ by $+1$ and $u_{ba}$ by $-1$ gives ${\mathrm{\partial }}_{u_{ab}}(-uCu{)}_{ab}=-{\sum }_{\mathrm{\ell }}{C}_{b\mathrm{\ell }}{u}_{\mathrm{\ell }b}-{\sum }_k{u}_{ak}{C}_{ka}$. Summing these ten derivatives in dimension five gives ${\sum }_{a<b}{\mathrm{\partial }}_{u_{ab}}(-uCu{)}_{ab}=8{\sum }_{a<b}{C}_{ab}{u}_{ab}=-4\mathrm{tr}(Cu)$, since $\mathrm{tr}(Cu)=-2{\sum }_{a<b}{C}_{ab}{u}_{ab}$. The radial term is $8X_C^{\rho }\mathrm{/}\rho =4\mathrm{tr}(Cu)$, so total divergence is zero. These three types span $\mathfrak{s}\mathfrak{o}(10,\mathbb{C})$, and the connected Spin group is generated by its one-parameter subgroups; consequently its local flows preserve $\mathrm{\Omega }$.

For $g\in \mathrm{Spin}(10,\mathbb{C})$ mapping the displayed patch to another patch, put ${\rho }_g=(g\lambda {)}^{+}$ and $u_{ab}^g=(g\lambda {)}_{ab}\mathrm{/}{\rho }_g$ on the overlap where ${\rho }_g\mathrm{\ne }0$. These are the scalar and two-form coordinates by the chart definition. Since the group action preserves $\mathrm{\Omega }$ by steps 2.1 and 3.1, the overlap obeys ${\rho }_g^{7}d{\rho }_g\wedge {\bigwedge }_{a<b}d{u}_{ab}^g={\rho }^{7}d\rho \wedge {\bigwedge }_{a<b}d{u}_{ab}$, equivalently ${\rho }_g^7\det \text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\partial }({\rho }_g,u^g)}{\mathrm{\partial }(\rho ,u)}\right)={\rho }^7$. Thus the local forms agree on each compatible patch overlap.

The same coordinate formula gives ${\mathrm{div}}_{\mathrm{\Omega }}(fX)=f{\mathrm{div}}_{\mathrm{\Omega }}X+X(f)$ by expanding $v^{-1}{\mathrm{\partial }}_i(vf{X}^i)$. Finally, varying the affine term at fixed coordinates gives ${\delta }_{\sigma }[-\frac{{\alpha }^{\mathrm{\prime }}}{2}{\mathrm{\Gamma }}_z{\mathrm{div}}_{\mathrm{\Omega }}V]=-{\alpha }^{\mathrm{\prime }}{\mathrm{div}}_{\mathrm{\Omega }}V\mathrm{\partial }\sigma$. The sign is tied to the positive third-pole convention in the statement; reversing the operator order also reverses the displayed pole.