Pure-Spinor Cone and Finite BV Semidensity Geometry

Let $W$ be a five-dimensional complex vector space with ordered basis $e_1,\dots ,e_5$ and dual coframe $e^1,\dots ,e^5$. On $V=W\oplus W^{\ast }$ set

$q(x+\xi )=\xi (x),B(x+\xi ,y+\eta )=\frac{1}{2}(\eta (x)+\xi (y)),$

so $q(v)=B(v,v)$ and $q$ is nondegenerate and split. The complex Clifford algebra $\mathrm{C}\mathrm{l}(V,q)$ is generated by $V$ with relation $c(v)c(v^{\mathrm{\prime }})+c(v^{\mathrm{\prime }})c(v)=2B(v,v^{\mathrm{\prime }})1$; in particular $c(v{)}^2=q(v)1$. On $S={\mathrm{\Lambda }}^{\bullet }{W}^{\ast }$ use

$c(x+\xi )\omega ={\iota }_x\omega +\xi \wedge \omega \mathrm{.}$

Write $S^{+}={\mathrm{\Lambda }}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}{W}^{\ast }$ and $S^{-}={\mathrm{\Lambda }}^{\mathrm{o}\mathrm{d}\mathrm{d}}{W}^{\ast }$. For a homogeneous $k$-form put $\tau (\omega )=(-1{)}^{k(k-1)\mathrm{/}2}\omega$, extend $\tau$ linearly, and let $[\omega {]}_{\mathrm{v}\mathrm{o}\mathrm{l}}$ mean the coefficient of $\mathrm{v}\mathrm{o}\mathrm{l}=e^1\wedge \cdots \wedge e^5$. The invariant pairing between $S^{-}$ and $S^{+}$ and the chiral gamma bilinear are fixed by

$⟨s_{-},t_{+}⟩=[\tau (s_{-})\wedge t_{+}{]}_{\mathrm{v}\mathrm{o}\mathrm{l}},\gamma (s_{+},t_{+})(v)=⟨c(v)s_{+},t_{+}⟩(v\in V)\mathrm{.}$

The nondegenerate form $B$ identifies $V$ with $V^{\ast }$, so the components of $\gamma$ in a $B$-orthonormal basis are the symmetric ten-dimensional matrices ${\gamma }_{\alpha \beta }^m$ in the convention used here; the displayed pairing fixes their overall normalization. Define $\mathrm{Spin}(V,q)$ as the subgroup of ${\mathrm{C}\mathrm{l}}^0(V,q{)}^{\times }$ generated by even products of vectors $v$ with $q(v)=1$. It acts on $S^{+}$ by left Clifford multiplication and on $V$ by $g\cdot v=gv{g}^{-1}$. This is the complex group $\mathrm{Spin}(10,\mathbb{C})$; it preserves $q$, the spinor pairing, and $\gamma$. A nonzero $\lambda \in S^{+}$ is pure when

$\mathrm{Ann}(\lambda )=\{v\in V:c(v)\lambda =0\}$

has dimension five, the maximal possible dimension of an isotropic subspace. Equivalently, $\gamma (\lambda ,\lambda )(v)=0$ for every $v\in V$, or $\lambda {\gamma }^m\lambda =0$ in the stated component convention.

The equivalence and the complete scalar-component chart can be checked directly. Write $\lambda =\rho +\eta +\chi$ with $\eta \in {\mathrm{\Lambda }}^2{W}^{\ast }$, $\chi \in {\mathrm{\Lambda }}^4{W}^{\ast }$ and $\rho \in \mathbb{C}$. If $\rho \mathrm{\ne }0$, the $W^{\ast }$ components of $\gamma (\lambda ,\lambda )=0$ are $2\rho \xi \wedge \chi -\xi \wedge \eta \wedge \eta =0$ for every $\xi \in W^{\ast }$, hence $\chi =\eta \wedge \eta \mathrm{/}(2\rho )$. The $W$ components then vanish because ${\eta }^3=0$ in dimension five. This is exactly $\lambda =\rho \exp (\eta \mathrm{/}\rho )$; with $\eta \mathrm{/}\rho =\frac{1}{2}{u}_{ab}{e}^a\wedge e^b$ and summation over $a,b$, the chart is

$\lambda =\rho \exp \text{\hspace{0.17em}⁣}(\frac{1}{2}{u}_{ab}{e}^a\wedge e^b),\rho \in {\mathbb{C}}^{\ast },u_{ab}=-u_{ba}\mathrm{.}$

There are ten independent coordinates $u_{ab}$ with $a<b$. With $B_2=\frac{1}{2}{u}_{ab}{e}^a\wedge e^b$, the annihilator is exactly $\{x-{\iota }_x{B}_2:x\in W\}$, spanned by $e_a-u_{ab}{e}^b$ acting as ${\iota }_{e_a}-u_{ab}{e}^b\wedge$. Conversely, if $\rho \mathrm{\ne }0$ and $\dim \mathrm{Ann}(\lambda )=5$, projection of the annihilator to $W$ is injective (a pure covector would act on the scalar term as $\rho \xi$), hence is an isomorphism. Its isotropic graph has the form $x-{\iota }_x{B}_2$ for a unique two-form $B_2$; the degree-one and degree-three components of $c(x-{\iota }_x{B}_2)\lambda =0$ give $B_2=\eta \mathrm{/}\rho$ and $\chi ={\eta }^2\mathrm{/}(2\rho )$. Thus the displayed coordinates exhaust the whole domain $\rho \mathrm{\ne }0$, rather than merely providing pure points.

For each even subset $I\subseteq \{1,\dots ,5\}$ let ${\lambda }_I$ be the coefficient of $e^I$ in $S^{+}$ and put $U_I=\{\lambda \in C:{\lambda }_I\mathrm{\ne }0\}$, where $C$ denotes the punctured pure-spinor cone. Every $\lambda \in C$ lies in some $U_I$ because a nonzero vector in $S^{+}$ has a nonzero component. These are Spin-translated domains: for $a=1,\dots ,5$, $v_a=e_a+e^a$ has $q(v_a)=1$, and $c(v_a)={\iota }_{e_a}+e^a\wedge$ toggles whether $a$ occurs in a basis wedge $e^J$. For even $I=\{i_1,\dots ,i_{2k}\}$ the even Clifford product $g_I=v_{i_1}\cdots v_{i_{2k}}$ belongs to $\mathrm{Spin}(V,q)$, sends each basis line $\mathbb{C}{e}^J$ to $\mathbb{C}{e}^{J\mathrm{△}I}$, and sends $1$ to $\pm e^I$. Consequently the $I$-component of $g_I\lambda$ is $\pm {\lambda }_{\mathrm{\varnothing }}$, so $U_I=g_I{U}_{\mathrm{\varnothing }}$. On $U_I\cap U_J$ the chart change is the spin action of $g_J^{-1}{g}_I$, followed by division by its nonzero scalar component; the four-form coordinates are fixed by the same quadratic relation above. This proves the sixteen-domain cover and the claimed translated chart domains. The same scalar-chart calculation now proves the equivalence globally: for any nonzero spinor, a nonzero component is moved to the scalar component by a suitable $g_I^{-1}$; the Spin action preserves both $\gamma (\lambda ,\lambda )=0$ and $\dim \mathrm{Ann}(\lambda )$. The coordinates agree with Nekrasov's sixteen nonzero-component charts (hep-th/0511008, §§3.3.1–3.3.2, equations (3.34)–(3.44)); his Euclidean $SO(10)$ module is complexified, which is valid here because all nondegenerate ten-dimensional complex quadratic spaces are isometric.

On the scalar chart fix the ordering $a<b$ of the ten $d{u}_{ab}$ and set

${\mathrm{\Omega }}_{\mathrm{\varnothing }}={\rho }^{7}d\rho \wedge {\bigwedge }_{a<b}d{u}_{ab}\mathrm{.}$

On $U_I$ let $({\rho }_I,u_{ab}^I)$ be the scalar and two-form coordinates of $g_I^{-1}\lambda$, and use the spin-translated expression ${\mathrm{\Omega }}_I={\rho }_I^{7}d{\rho }_I\wedge {\bigwedge }_{a<b}d{u}_{ab}^I$ with the same ordering. These are the local volume expressions whose transition identity is proved by Intrinsic Pure-Spinor Volume and Current Cocycle; Nekrasov's nowhere-zero invariant form is $e^{8\phi }d\phi \wedge {\bigwedge }_{a<b}d{u}_{ab}$ with $\rho =e^{\phi }$, equal to the displayed expression. This is a holomorphic top form on the punctured cone only; it makes no assertion at the apex or about an integration boundary.

The ambient conjugate $w\in (S^{+}{)}^{\ast }$ restricts to a cotangent covector on $C$. If $F_v(\lambda )=\frac{1}{2}\gamma (\lambda ,\lambda )(v)$, then $d{F}_v{\mathrm{\mid }}_{\lambda }$ is conormal to $C$. On $U_{\mathrm{\varnothing }}$ the five independent components $2\rho \chi -{\eta }^2=0$ have derivative $2\rho \mathrm{I}\mathrm{d}$ in the $\chi$ directions, so the covectors $d{F}_v{\mathrm{\mid }}_{\lambda }$ span the full five-dimensional conormal space; Spin translates give the same fact on every $U_I$. Thus for a fixed basis $v_1,\dots ,v_{10}$ of $V$, $w$ and $w+{\sum }_{m=1}^{10}{\mathrm{\Lambda }}^{m}d{F}_{v_m}{\mathrm{\mid }}_{\lambda }$ represent the same cotangent vector; in components this is $w_{\alpha }\sim w_{\alpha }+({\gamma }_m\lambda {)}_{\alpha }{\mathrm{\Lambda }}^m$. Locally write the cone coordinates as $q^i=(\rho ,u_{ab})$, $i=1,\dots ,11$, with cotangent coordinates $p_i$ and canonical one-form $p_{i}d{q}^i$; in the source convention $p_i=w_i$. For the local holomorphic $\beta \gamma$ OPE used in this pair, set ${\gamma }^i=q^i$, ${\beta }_i=p_i=w_i$, take ${\alpha }^{\mathrm{\prime }}\in {\mathbb{C}}^{\times }$ as the formal OPE normalization, and use ${\beta }_i(z){\gamma }^j(w)\sim -{\alpha }^{\mathrm{\prime }}{\delta }_i{}^j\mathrm{/}(z-w)$ and the current $j_V=-:\text{\hspace{0.17em}⁣}{\beta }_i{V}^i\text{\hspace{0.17em}⁣}:$; thus a source insertion $+w_i{V}^i$ is $-j_V$. The dimensional interpretation of ${\alpha }^{\mathrm{\prime }}$ is specified by any consuming physical model. No sixteen-component unconstrained-pair measure is part of this definition.

For a finite graded coordinate space with coordinates $x^I$ of parity ${ϵ}_I$ and ghost number $g_I$, introduce antifields $x_I^{\ast }$ of parity ${ϵ}_I+1$ and ghost number $-g_I-1$. Use the canonical bracket of Classical BV Antibracket and Yang–Mills Master-Action Data. A compatible odd Laplacian is an odd operator $\mathrm{\Delta }$ with $\mathrm{\Delta }1=0$, ${\mathrm{\Delta }}^2=0$, and

$\mathrm{\Delta }(FG)=\mathrm{\Delta }FG+(-1{)}^{\mathrm{\mid }F\mathrm{\mid }}F\mathrm{\Delta }G+(-1{)}^{\mathrm{\mid }F\mathrm{\mid }}(F,G)\mathrm{.}$

Let ${\rho }_0$ be the coordinate Berezinian. In these Darboux coordinates, ${\mathrm{\Delta }}_0$ denotes the canonical coordinate BV Laplacian associated with the bracket and ${\rho }_0$; explicitly,

${\mathrm{\Delta }}_{0}F={\sum }_I(-1{)}^{{ϵ}_I}\frac{{\mathrm{\partial }}_l}{\mathrm{\partial }{x}^I}\left(\frac{{\mathrm{\partial }}_{l}F}{\mathrm{\partial }{x}_I^{\ast }}\right)\mathrm{.}$

Thus ${\mathrm{\Delta }}_{0}f=0$ for every function $f$ independent of all antifields. Let $\mu (x)$ be a nowhere-zero even density factor, independent of antifields and admitting a chosen logarithm on the chart. The BV Berezinian is ${\rho }_{\mathrm{B}\mathrm{V}}={\mu }^2{\rho }_0$, its semidensity is locally a function times $\sqrt{{\rho }_0}$ with the square-root Berezinian transition law, and its compatible operator is ${\mathrm{\Delta }}_{\rho }={\mathrm{\Delta }}_0+(\log \mu ,-)$. These are finite chart data, not an infinite-dimensional path-integral measure.

A finite odd-symplectic split into low and high dual pairs uses the product reference semidensity. A high integration cycle is a specified maximal isotropic subspace or smooth deformation on which the formal or convergent integral is defined, together with its contour/continuation and determinant branch. BV pushforward is integration of the semidensity over that cycle. Its boundary remainder is the actual integral of ${\mathrm{\Delta }}_H$ of the integrand and is not set to zero by definition. For metric-dependent spectral subspaces, the chosen odd-symplectic transport pulls them back to fixed Darboux coordinates before differentiating. This definition makes no global cone-cycle or no-boundary assertion.

Classical BV Antibracket and Yang–Mills Master-Action Data

Nekrasov, pure-spinor charts and holomorphic volume

Gomis, París, and Samuel, BV and semidensity context