Quantum Bundle Description of Quantum Projective Spaces

We realise Heckenberger and Kolb's canonical calculus on quantum projective (N - 1)-space C (q) [C p (N-1)] as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group C (q) [SU (N) ]. We introduce a calculus on the quantum sphere C (q) [S (2N-1)] in the same way. With respect to these choices of calculi, we present C (q) [C p (N-1)] as the base space of two different quantum principal bundles, one with total space C (q) [SU (N) ], and the other with total space C (q) [S (2N-1)]. We go on to give C (q) [C p (N-1)] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb's calculus as an associated vector bundle to the principal bundle with total space C (q) [SU (N) ]. Finally, we construct strong connections for both bundles.