The main result of this paper establishes an isomorphism between the transverse string algebra \(A_{St}(K)\) – defined in this work – and the cord algebra \(A_{Ng}(K)\) introduced by Ng, which are both given for a framed submanifold \(K\) of codimension two in \(M\).
The transverse string algebra \(A_{St}(K)\) is constructed from the space of transverse open strings \(\mathcal{ST}\). A transverse open string \(\omega\) starts and ends transversely on \(K\). There can be further transverse intersections of \(\omega\) with \(K\). Splitting open strings at intersections and removing intersections by small local displacements induces two nilpotent maps \(\Delta\) and \(R\) on the reduced tensor algebra \(T^{red}(C_*(\mathcal{ST}))\) of the singular chain complex \((C_*(\mathcal{ST}),\partial)\). Then the degree zero homology \(H_0(T^{red}(C_*(\mathcal{ST})))\) with respect to the differential \(\partial + R + \Delta\) is – modulo a certain ideal – the transverse string algebra \(A_{St}(K)\). The cord algebra \(A_{Ng}(K)\) has been introduced by L. Ng [Geom. Topol. 9, 1603–1637 (2005; Zbl 1112.57001)] and [Duke Math. J. 141(2), 365–406 (2008; Zbl 1145.57010)].
In the context of a knot \(K\) in \(\mathbb{R}^3\) the cord algebra \(A_{Ng}(K)\) encodes non-trivial knot invariants distinct from other – maybe more conventional – knot invariants [L. Ng, Geom. Topol. 9, 247–297 (2005; Zbl 1111.57011)]. Thus – stated as a corollary – the established isomorphism to \(A_{St}(K)\) determines cord algebra knot invariants as well.
Studying knots in \(S^3\) in open topological string theory by means of local mirror symmetry assigns a spectral curve to each knot \(K\), c.f., M. Aganagic et al. [Adv. Theor. Math. Phys. 18, No. 4, 827–956 (2014; Zbl 1315.81076)] and J. Gu et al. [Commun. Math. Phys. 336, No. 2, 987–1051 (2015; Zbl 1323.57008)], which agrees with the augmentation variety of the cord algebra \(A_{Ng}(K)\) L. Ng [Duke Math. J. 141, No. 2, 365–406 (2008; Zbl 1145.57010)]. Thus the isomorphism of algebras readily indicates a connection between the spectral curve of \(K\) and the transverse string algebra \(A_{St}(K)\). This is intriguing as the definition of \(A_{St}(K)\) has similarities to the construction of open topological strings [E. Witten, Prog. Math. 133, 637–678 (1995; Zbl 0844.58018)].