  We study generalized multifractality that characterizes eigenstate
fluctuations and correlations in disordered systems of chiral symmetry classes
AIII, BDI, and CII. By using the non-linear sigmamodel field theory, we
construct pure-scaling composite operators and eigenfunction observables that
satisfy Abelian fusion rules. The observables are labelled by two multi-indices
$\lambda$, $\lambda'$ referring to two sublattices, at variance with other
symmetry classes, where a single multi-index $\lambda$ (that can be viewed as a
generalized version of a Young diagram) is needed. Further, we analyze Weyl
symmetries of multifractal exponents, which are also peculiar in chiral
classes, in view of a distinct root system associated with the sigma-model
symmetric space. The analytical results are supported and complemented by
numerical simulations that are performed for a 2D lattice Hamiltonian of class
AIII, both in the metallic phase and at the Anderson transition. Both in the
metallic phase and at the transition, the numerically obtained exponents
satisfy Weyl symmetry relations, confirming that the sigma-model is the right
theory of the problem. Furthermore, in the metallic phase, we observe the
generalized parabolicity (proportionality to eigenvalues of the quadratic
Casimir operator), as expected in the one-loop approximation. On the other
hand, the generalized parabolicity is strongly violated at the metal-insulator
transition, implying violation of local conformal invariance.
