  Recently, a concept of generalized multifractality, which characterizes
fluctuations and correlations of critical eigenstates, was introduced and
explored for all ten symmetry classes of disordered systems. Here, by using the
non-linear sigma-model field theory, we extend the theory of generalized
multifractality to boundaries of systems at criticality. Our numerical
simulations on two-dimensional (2D) systems of symmetry classes A, C, and AII
fully confirm the analytical predictions of pure-scaling observables and Weyl
symmetry relations between critical exponents of surface generalized
multifractality. This demonstrates validity of the non-linear sigma-model field
theory for description of Anderson-localization critical phenomena not only in
the bulk but also on the boundary. The critical exponents strongly violate
generalized parabolicity, in analogy with earlier results for the bulk,
corroborating the conclusion that the considered Anderson-localization critical
points are not described by conformal field theories. We further derive
relations between generalized surface multifractal spectra and linear
combinations of Lyapunov exponents of a strip in quasi-one-dimensional
geometry, which hold under assumption of invariance with respect to a
logarithmic conformal map. Our numerics demonstrate that these relations hold
with an excellent accuracy. Taken together, our results indicate an intriguing
situation: the conformal invariance is broken but holds partially at critical
points of Anderson localization.
