Finite Matched Curved Non-Minimal Quartet Determinant

Use the finite integration conventions of the Pure-Spinor Cone and Finite BV Semidensity Geometry definition. Fix a finite exterior coefficient algebra $\mathcal{A}={\mathrm{\Lambda }}_{\mathbb{C}}({\xi }_1,\dots ,{\xi }_q)$, where the generators anticommute and have odd parity; its body map kills all ${\xi }_i$. Let $E={\mathbb{C}}^n$ have a specified basis. The variables $x^i,{\bar{b}}_i$ are independent even variables, and $r^i,{\bar{\eta }}_i$ are independent odd generators; bars label dual variables, not complex conjugation. All products lie in $\mathcal{A}[x,\bar{b}]\otimes \mathrm{\Lambda }(r,\bar{\eta })$ with the Koszul sign rule $uv=(-1{)}^{\mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }}vu$ for homogeneous elements. Pair a row dual with a column in the displayed order.

Let $D\in {\mathrm{Mat}}_n({\mathcal{A}}_{\bar{0}})$ have invertible body, so its nilpotent expansion gives an inverse. It acts identically in the even and odd sectors. Let $M$ be a finite graded complex parameter space with homogeneous basis of parities ${ϵ}_j$, and hold fixed coefficients $m^j\in {\mathcal{A}}_{{ϵ}_j}$. Choose matrices $B,C$ with $B_{ij}\in {\mathcal{A}}_{{ϵ}_j}$ and $C_{ij}\in {\mathcal{A}}_{{ϵ}_j+1}$, and $K\in {\mathrm{Mat}}_n({\mathcal{A}}_{\bar{1}})$. Thus $Bm$ is even, while $Cm$ and $Kx$ are odd; the lower-left block has odd coefficients even though the full supermatrix is even. With a specified complex Gaussian contour/continuation for the body and coefficientwise nilpotent expansion, consider

$S=\bar{b}Dx+\bar{b}Bm+\bar{\eta }Dr+\bar{\eta }Kx+\bar{\eta }Cm,$

The paired Gaussian factor is exactly one, including its determinant-line phase, while the fixed $m$-dependent minimal action and minimal-only antifield sources are unchanged. This is a finite mode calculation; it proves neither convergence of a noncompact integral nor BV Stokes at the pure-spinor cone apex/infinity.

Pure-Spinor Cone and Finite BV Semidensity Geometry

Chandía and Vallilo, curved non-minimal quartet

Here is the local curved-quartet realization to which this finite model applies. In the non-minimal sector the additional bosonic spinor ${\mathrm{\ell }}_{\alpha }$ satisfies $\mathrm{\ell }{\gamma }^a\mathrm{\ell }=0$ and $\mathrm{\ell }\mathrm{\ne }0$, and its odd partner obeys $\mathrm{\ell }{\gamma }^{a}r=0$, the linearized constraint. Thus $r$ is a parity-reversed tangent vector to the non-minimal cone, and the conjugates $s,\omega$ are the corresponding odd and even cotangent variables modulo conormal gauge. On an eleven-coordinate cone chart, $x,r,\bar{\eta },\bar{b}$ are respectively finite mode coefficients of the coordinate fluctuation of $\mathrm{\ell }$, its tangent partner $r$, and the dual variables $s,\omega$. The cone is the one in [F1], with the dual Weyl block used for $\mathrm{\ell }$; exchanging the two Weyl modules gives the same eleven-coordinate construction. No sixteen unconstrained pairs are used.

More explicitly, fix an open worldsheet coordinate patch $U\subseteq \mathbb{C}$, a smooth background map ${\mathrm{\ell }}_0:U\to C$ into that chart, and its coordinate frame of the rank-eleven bundle $V={\mathrm{\ell }}_0^{\ast }TC$. Choose linearly independent modes $u_j\in C_c^{\mathrm{\infty }}(U,V)$ and specified complex-linear maps $p^i:C^{\mathrm{\infty }}(U,V)\to \mathbb{C}$ with $p^i(u_j)={\delta }_j^i$, extended coefficientwise over $\mathcal{A}$. Specify a smooth even matrix $a_{\bar{z}}:U\to {\mathrm{Mat}}_{11}({\mathcal{A}}_{\bar{0}})$; the operator $u\mapsto ({\mathrm{\partial }}_{\bar{z}}u+a_{\bar{z}}u)d\bar{z}$ is a connection in the $(0,1)$ direction in this frame. Its common finite kinetic matrix, after using $d\bar{z}$ to identify the output coefficients, is $D_{ij}=p^i({\mathrm{\partial }}_{\bar{z}}{u}_j+a_{\bar{z}}{u}_j)$ in both parity sectors. Retain only a chosen block with invertible body. Let $q$ be the odd left derivation on the minimal coefficient algebra induced by the minimal BRST variation, and let $\sigma$ be the odd matrix of its induced frame rotation. The linear quartet rules used in the gauge fermion are $Qx=-r+\sigma x$ and $Q\bar{\eta }=\bar{b}+\bar{\eta }\sigma$, with $Q$ acting as $q$ on fixed minimal coefficients. For a fixed even column $b=Bm$, the graded Leibniz rule gives the exact finite identity

$Q(\bar{\eta }(Dx+b))=\bar{b}Dx+\bar{b}b+\bar{\eta }Dr+\bar{\eta }(\sigma D-qD-D\sigma )x+\bar{\eta }(\sigma b-qb)\mathrm{.}$

Thus the affine gauge-fermion realization identifies $K=\sigma D-qD-D\sigma$ and $Cm=\sigma b-qb$; independent fixed odd source columns $Cm$ are also allowed in the Gaussian above. These are odd coefficients, as required. Chandía–Vallilo (3.14), (3.17), and (3.23) provide the curved non-minimal rules and the gauge fermion $\int s\mathrm{\nabla }\mathrm{\ell }$ motivating this linear mode realization. The calculation here is conditional on the declared common truncation and linear quartet rules: it does not assert that an arbitrary cutoff of their complete interacting action preserves them, or discard additional nonlinear vertices. Minimal-only antifield sources are precisely fixed coefficients multiplying functions of $m$ alone, so they factor out of the quartet integration.

Unmatched zero modes, unequal operators, incompatible contours, noncompact convergence and a cone boundary contribution are outside this finite example.