Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic invariants

In this article we are defining a refinement of Kool-Thomas invariants of local surfaces via the equivariant $K$-theoretic invariants proposed by Nekrasov and Okounkov. Kool and Thomas defined the reduced obstruction theory for the moduli of stable pairs ${\mathcal{P}}_{\chi }(X,i_{\ast }\beta )$ as the degree of the virtual class ${[{\mathcal{P}}_{\chi }(S,\beta )]}^{red}$ after we apply $\tau ([pt]{)}^m\in H^{\ast }({\mathcal{P}}_{\chi }(X,i_{\ast }\beta ),\mathbb{Z})$. $\tau ([pt])$ contain the information of the incidence of a point and a curve supporting a $(\mathcal{F},s)$. $\mathbf{\text{Keywords:}}$Kool-Thomas invariants, $K$-theoretic invariants, G"ottsche Shende invariants