Rota-Baxter Operators on Vertex Algebras in Integrated $\lambda$-Bracket Formalism and Their Associated 2-Cocycles

We study Rota–Baxter operators on vertex algebras using the integrated $\lambda$-bracket formalism. A Rota–Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields a two-cocycle in vertex algebra cohomology. This generalizes the classical relation between Rota–Baxter operators and Hochschild two-cocycles. We also characterize when this two-cocycle is trivial, showing that non-scalar operators give rise to non-trivial cohomology classes.