Explore advanced mathematical concepts in this conference talk examining integration by parts techniques within random conformal geometry and their practical applications. Delve into the variational formulas of prominent stochastic processes like the Gaussian free field and Schramm-Loewner evolutions under local conformal transformations, discovering how these lead to elegant integration by parts formulas that can be expressed as representations of the Virasoro algebra on L2-spaces. Learn how this algebraic structure yields significant probabilistic insights, including a novel characterization of Schramm-Loewner Evolution (SLE) and an innovative approach to the conformal welding of quantum surfaces, bridging abstract mathematical theory with concrete applications in random conformal geometry.