  We propose a general strategy to derive null-homotopy operators for
differential complexes based on the Bernstein-Gelfand-Gelfand (BGG)
construction and properties of the de Rham complex. Focusing on the elasticity
complex, we derive path integral operators $\mathscr{P}$ for elasticity
satisfying $\mathscr{D}\mathscr{P}+\mathscr{P}\mathscr{D}=\mathrm{id}$ and
$\mathscr{P}^{2}=0$, where the differential operators $\mathscr{D}$ correspond
to the linearized strain, the linearized curvature and the divergence,
respectively. In general we derive path integral formulas in the presence of
defects. As a special case, this gives the classical Ces\`{a}ro-Volterra path
integral for strain tensors satisfying the Saint-Venant compatibility
condition.
