The skein module–DT sheaf conjecture

Let $M$ be a connected closed oriented $3$-manifold with a basepoint $x\in M$, and let $G$ be a reductive algebraic group equipped with the extra data described in the source. For generic quantum parameters, let $\mathrm{Sk}^{\mathrm{gen}}_G(M)$ be the skein module, a $\mathbf C(q^{1/d})$-vector space for some integer $d$, and let

$$R_G(M)=\operatorname{Hom}(\pi_1(M,x),G)$$

be the representation variety, viewed as a d-critical locus. Let $\phi_{R_G(M)}$ denote its DT sheaf. The skein module–DT sheaf conjecture. There is an isomorphism of $\mathbf C(q^{1/d})$-vector spaces

$$\mathrm{Sk}^{\mathrm{gen}}_G(M)\cong \mathrm{H}^0(R_G(M),\phi_{R_G(M)})\otimes_{\mathbf C}\mathbf C(q^{1/d}).$$

This conjecture proposes a relationship between skein modules and the DT sheaf; for $G=\mathrm{SL}_2$, the resulting hypercohomology is related to framed complexified instanton Floer homology. Its status is not determined by the supplied source context.