Derivation Lie algebras of singular locus moduli algebras for singularities

Singularities play a central role in various areas of physics, including 4 d N = 2 superconformal field theories, Coulomb-branch spectra, and Seiberg-Witten solutions. Ma, Yau, and Zuo introduced the singular-locus moduli algebra (Formula presented) and its derivation Lie algebras (Formula presented) for any isolated hypersurface singularity (V,0)⊂( Cⁿ,0). In this paper, we first compute L₂^1,1(V), L₂^1,2(V), L₂^2,1(V), and L₂^2,2(V) for isolated binomial singularities, and L₁²(V) for trinomial singularities. We then formulate a conjecture that provides a sharp upper bound for (Formula presented) in the weighted homogeneous case, and verify it for a large class of singularities.