Wilson lines from representations of \(NQ\)-manifolds (Q2929682)

An \(NQ\) manifold is a non-negatively graded supermanifold with a degree \(1\) homological vector field \(Q\). Its representation is a graded vector bundle \(\mathcal{E}\) endowed with a lift of the \(Q\)-structure linear in the fiber coordinates. In a previous work the last two authors constructed Wilson loops on \(NQ\) manifolds, and obtained new weight systems for knots in \(3\)-manifolds by evaluating their expectation values. This paper is aimed at clarifying the issue of trivialization invariance when transition functions of \(\mathcal{E}\) have components of positive degree. The main result is that the Wilson loops are trivialization invariant, and moreover homotopy invariant. In contrast, Wilson lines are not trivialization covariant. They can be made so by restricting to a subcategory of \(NQ\) manifolds, but that restriction discards a vast amount of information about the space of supercurves.NEWLINENEWLINELet \(T[1]M\) denote the tangent bundle of a \(C^\infty\)-manifold \(M\) with coordinates of the fiber assigned degree \(1\). Let \(\mathcal{M}\) be an \(NQ\) manifold, and let \(\mathcal{L}\) be a test manifold. The authors look at superpaths \(T[1][0,1]\times\mathcal{L}\to\mathcal{M}\) parametrized by \(\mathcal{L}\). When \(\mathcal{L}\) is specialized to \(n\)-cubes they define the singular cubical complex of \(\mathcal{M}\), which as a presheaf is representable by an \(NQ\) manifold \(\overline{S}\). Wilson lines are obtained from the parallel transport in \(\mathcal{E}\) (dependent on \(\overline{S}\)) through the Chen's iterated integral. Similarly, Wilson loops are functions on \(\overline{S}_0\), a closed loop analog of \(\overline{S}\). This construction is an alternative to the more traditional integration of \(NQ\) manifolds up to homotopy, which is based on the singular simplicial complex instead of the cubical one, but the results have significant overlap.