Transgression to shifted symplectic mapping stacks

For every derived stack $G$, $\mathcal{O}$-compactness gives a natural map of graded mixed complexes ${\kappa }_{G,X}:\mathrm{D}\mathrm{R}(G\times X\mathrm{/}k)⟶\mathrm{D}\mathrm{R}(G\mathrm{/}k){\otimes }_{k}C(X,{\mathcal{O}}_X),$ where $C(X,{\mathcal{O}}_X)$ has weight zero and zero mixed differential. On derived affines $\mathrm{Spec}B\to G$ and $\mathrm{Spec}C\to X$ this is constructed from the K"unneth equivalence $\mathrm{D}\mathrm{R}(B{\otimes }_{k}C)\simeq \mathrm{D}\mathrm{R}(B){\otimes }_k\mathrm{D}\mathrm{R}(C)$, the weight-zero projection $\mathrm{D}\mathrm{R}(C)\to {\mathcal{O}}_X$, and the projection formula $C(X,\mathrm{D}\mathrm{R}(B){\otimes }_k{\mathcal{O}}_X)\simeq \mathrm{D}\mathrm{R}(B){\otimes }_{k}C(X,{\mathcal{O}}_X)$. The projection formula is an equivalence because ${\mathcal{O}}_X$ is compact; this tensor factor is perfect over $k$, so tensoring with it commutes with the limits that extend the construction from derived affines to derived stacks. The map therefore respects the internal and mixed differentials. Applying weighted negative cyclic complexes and then $1\otimes [X]$ defines integration ${\int }_X:{\mathrm{N}\mathrm{C}}^w(G\times X)\to {\mathrm{N}\mathrm{C}}^w(G)[-d]$, still a map of graded complexes.

For an $A$-point $x:\mathrm{Spec}A\to M$ corresponding to $f:X_A\to F$, a square-zero extension of $A$ by a perfect $A$-module $N$ identifies the tangent complex at $x$ with $T_{M,x}\simeq C(X_A,f^{\ast }{T}_F)\mathrm{.}$ Indeed, the infinitesimal lifts of $x$ are maps from the pullback of $\mathrm{Spec}(A\oplus N)$ to $F$ over $X_A$; the derivation representing $L_F$ identifies their controlling complex with the global sections of $f^{\ast }{T}_F$. The derived base-change condition in [A1] makes this identification compatible with $A$-linear mapping spaces.

Pull back $\omega \in {\mathcal{A}}^{2,\mathrm{c}\mathrm{l}}(F,n)$ along evaluation $\mathrm{e}\mathrm{v}:X\times M\to F$, then apply the integration map of step 1.1. Because integration is induced by a map of graded mixed complexes, it sends this point to a closed-form point ${\omega }_M\in {\mathcal{A}}^{2,\mathrm{c}\mathrm{l}}(M,n-d)$.

The underlying pairing of ${\omega }_M$ at $x$ is obtained by applying $C(X_A,-)$ to the symplectic pairing $f^{\ast }{T}_F{\otimes }_A{f}^{\ast }{T}_F\to {\mathcal{O}}_{X_A}[n]$, then composing with $[X{]}_A$. The construction of $\kappa$ identifies this with the underlying form of the integrated closed form in step 2.1. Since ${\mathrm{\Theta }}_{\omega }$ is a quasi-isomorphism, it identifies $f^{\ast }{T}_F$ with its dual shifted by $n$; the orientation duality [A1] then identifies $C(X_A,f^{\ast }{T}_F)$ with its $A$-linear dual shifted by $n-d$. Thus the adjoint $T_{M,x}\to L_{M,x}[n-d]$ is a quasi-isomorphism.

The nondegeneracy check in step 3.1 holds at every affine test point $x$. Since quasi-isomorphisms of quasi-coherent complexes are local on a smooth atlas, ${\mathrm{\Theta }}_{{\omega }_M}:T_M\to L_M[n-d]$ is a quasi-isomorphism on $M$. Thus the closed form from step 2.1 is $(n-d)$-shifted symplectic by [F2].