Poincar\'e path integrals for elasticity

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.