  The main goal of this paper is to prove the following two conjectures for
genus up to two:
  1. Witten's conjecture on the relations between higher spin curves and
Gelfand--Dickey hierarchy.
  2. Virasoro conjecture for target manifolds with conformal semisimple
Frobenius manifolds.
  The main technique used in the proof is the invariance of tautological
equations under loop group action.
