Derived self-intersection of the zero section

The cotangent plane has free cotangent complex generated by $dx$ and $dp$ in degree zero. Its form $dx\wedge dp$ has invertible pairing matrix $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),$ so it is a closed $0$-shifted symplectic form. The zero section pulls it back to zero. The relative cotangent complex is the cofiber of $i^{\ast }{L}_F\to L_X$, the map $k[x]dx\oplus k[x]dp\to k[x]dx$ sending $dx\mapsto dx$ and $dp\mapsto 0$; hence $L_{X\mathrm{/}F}\simeq k[x]dp[1]$. The isotropic structure induces $T_X\to L_{X\mathrm{/}F}[-1]$, sending ${\mathrm{\partial }}_x$ to $dp$ up to the sign fixed by $\omega$. It is an isomorphism of free rank-one $k[x]$-modules, so each zero-section map is Lagrangian.

To compute the derived self-intersection, resolve the second copy of $k[x]=k[x,p]\mathrm{/}(p)$ over $k[x,p]$ by the Koszul dg algebra $K=k[x,p,\eta ]$ with $\mathrm{\mid }\eta \mathrm{\mid }=-1$, ${\eta }^2=0$, and $d\eta =p$. The two-term complex $k[x,p]\eta \stackrel{p}{\to }k[x,p]$ is exact away from its cokernel $k[x,p]\mathrm{/}(p)$ because multiplication by $p$ is injective. Thus $K$ is a cofibrant resolution. Tensoring it with the first $k[x]$ over $k[x,p]$ sets $p=0$, so $K{\otimes }_{k[x,p]}k[x]=k[x,\eta ]$ with zero differential. This proves the stated model of $Z$.

In the tangent fiber triangle for $Z\to X\times X\to F$, the degree-zero kernel is generated by the diagonal vector $({\mathrm{\partial }}_x,{\mathrm{\partial }}_x)$, while the degree-one cokernel class comes from the normal vector ${\mathrm{\partial }}_p$ and is identified with ${\mathrm{\partial }}_{\eta }$ by the Koszul model in step 2.1. In the cotangent comparison of [F4], each zero-section Lagrangian map sends ${\mathrm{\partial }}_x$ to its relative conormal class $dp$, and the ambient symplectic adjoint sends ${\mathrm{\partial }}_p$ to $-dx$. The comparison therefore pairs ${\mathrm{\partial }}_x$ with $d_{\mathrm{d}\mathrm{R}}\eta [-1]$ and ${\mathrm{\partial }}_{\eta }$ with $d_{\mathrm{d}\mathrm{R}}x[-1]$, up to the Koszul sign convention; its underlying 2-form is $d_{\mathrm{d}\mathrm{R}}x\wedge d_{\mathrm{d}\mathrm{R}}\eta$. The construction in [F4] makes this a point of the closed-form space, not merely an ordinary pairing.

In the displayed cdga, $L_Z$ is free on $dx$ in degree $0$ and $d_{\mathrm{d}\mathrm{R}}\eta$ in degree $-1$. Its tangent complex is dual, with ${\mathrm{\partial }}_x$ in degree $0$ and ${\mathrm{\partial }}_{\eta }$ in degree $1$. The residue pairing maps ${\mathrm{\partial }}_x$ to $d_{\mathrm{d}\mathrm{R}}\eta [-1]$ and ${\mathrm{\partial }}_{\eta }$ to $dx[-1]$, up to the standard Koszul sign. These are degree-preserving isomorphisms between the two free rank-one summands of $T_Z$ and $L_Z[-1]$. Hence the adjoint $T_Z\to L_Z[-1]$ is a quasi-isomorphism, proving nondegeneracy.

Step 3.1 gives the closed residue form, and step 4.1 proves its nondegeneracy. Therefore $Z$ has the claimed canonical $(-1)$-shifted symplectic structure. The calculation is a mathematical derived intersection; it asserts no corresponding physical phase space or path-integral construction.