The geometric string structure conjecture for Chern–Simons enhancements

Let $K$ be a Lie group and let $P_K:M\to\mathrm{B}K$ classify a principal $K$-bundle. Let $\mathrm{B}^2\mathrm{U}(1)\to\mathrm{String}(K)\to K$ be the string extension, enhanced to $\mathrm{B}\,\mathrm{B}_\nabla\mathrm{U}(1)\to\mathrm{String}_{\rm CS}(K)\to K$, where $\mathrm{B}_\nabla\mathrm{U}(1)$ classifies $\mathrm{U}(1)$-bundles with connection. Define $\operatorname{Hom}_v$ to be the full substack in which the pulled-back connection 1-forms are vertical, meaning that they have form legs only along $M$. Geometric string structure conjecture. The stacks $\operatorname{Hom}_v(M,\mathrm{B}^2\mathrm{B}_\nabla\mathrm{U}(1))/\mathsf{Diff}(M)$ and $\operatorname{Hom}_v(M,\mathrm{B}\mathrm{String}_{\rm CS}(K))/\mathsf{Diff}(M)$ have possibly quasi-coherent deformation theory; a lift of $P_K$ to $\mathrm{B}\mathrm{String}_{\rm CS}(K)$ exists if and only if a lift to $\mathrm{B}\mathrm{String}(K)$ exists; and the $L_\infty$-algebra

$$T\big(* / (\operatorname{Hom}_v(M,\mathrm{B}\mathrm{String}_{\rm CS}(K))/\mathsf{Diff}(M))\big)$$

has a canonical map to $TM$, whose $C^\infty(M)$-linear splittings as differential graded $C^\infty(M)$-modules correspond precisely to a geometric string structure on $P_K$ together with a symmetric tensor $\gamma\in\Gamma(M,\vee^2T^*M)$. These assertions are proposed as a package relating the Chern–Simons enhancement to geometric string structures; the cited constructions establish the underlying extension, while the conjectural deformation-theoretic and splitting statements remain open.