On the \(k\)th higher Nash blow-up derivation Lie algebras of isolated hypersurface singularities

Many physical questions such as \(4d\) \(N=2\) superconformal field theories, the Coulomb branch spectrum, and the Seiberg–Witten solutions are related to singularities. In this paper, we introduce some new invariants \(\mathcal L^k_n(V)\), \(\rho_n^k\), and \(d_n^k(V)\) of isolated hypersurface singularities \((V,0)\). We give a new conjecture for the characterization of simple curve singularities using the \(k\)th higher Nash blow-up derivation Lie algebra \(\mathcal L^k_n(V)\). This conjecture is verified for small \(n\) and \(k\). A inequality conjecture for \(\rho_n^k\) and \(d_n^k(V)\) is proposed. These two conjectures are verified for binomial singularities.