Optical Wave Turbulence: The Conformal Symmetry Transformations of Statistics of the Quantum Fluids of Light

We use the quantum fluid analogy [Nazarenko et al., Phys. Rev. E, 92, 2015] for coherent structures (vortices or solitons) interacting among themselves and with the random wave component. This is performed for 2D defocusing media based on quantum fluid approximation of the two-dimensional nonlinear Schr"{o}dinger equation in the statistical frame. With this, the Lundgren-Monin-Novikov infinite chain of equations for the $n$-point density function $f_n$ for the vorticity field is used. The conformal group of symmetry transformations calculated [Grebenev et. al., Theor. Math. Phys., 217(2), 2023] is applied to implement several elements of a gauge theory in the conformal transformation optics. Finally, we demonstrate how to use the variational generalized Brenier principle [Brenier, J. Am. Math. Soc., 2, 1989] together with the conformal invariance of statistics to close the infinite chain of Landgren-Monin-Novikov equations.