Derivation Lie algebras of singular locus moduli algebras for singularities

Singularities play a central role in various areas of physics, including 4dN=2 superconformal field theories, Coulomb-branch spectra, and Seiberg–Witten solutions. Ma, Yau, and Zuo introduced the singular-locus moduli algebra NMlk1,…,kl(V) and its derivation Lie algebras Llk1,…,kl(V) for any isolated hypersurface singularity (V,0)⊂(Cn,0). In this paper, we first compute L21,1(V), L21,2(V), L22,1(V), and L22,2(V) for isolated binomial singularities, and L12(V) for trinomial singularities. We then formulate a conjecture that provides a sharp upper bound for dimLlk1,…,kl(V) in the weighted homogeneous case, and verify it for a large class of singularities.