PROJECTIVE STRUCTURES AND -CONNECTIONS

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

Quaternionic-like manifolds and homogeneous twistor spaces

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the ‘quaternionic-like manifolds’. These contain, as particular subclasses, the CR quaternionic and the ρ -quaternionic manifolds. Moreover, the notion of ‘heaven space’ finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a ρ -quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.

QUATERNIONIC CONNECTIONS, INDUCED HOLOMORPHIC STRUCTURES AND A VANISHING THEOREM

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. In particular, we show that any self-dual quaternionic connection D of (M, Q) induces an holomorphic structure [Formula: see text] on Θ. We construct a Penrose transform which identifies solutions of the Penrose operator PD on (M, Q) defined by D with the space of [Formula: see text]-holomorphic purely imaginary sections of Θ. We prove that the tensor powers Θs (for any s ∈ ℕ\{0}) have no global non-trivial [Formula: see text]-holomorphic sections, when (M, Q) is compact and has a compatible quaternionic-Kähler metric of negative (respectively, zero) scalar curvature and the quaternionic connection D is closed (respectively, closed but not exact).

ON THE KAUFFMAN BRACKET SKEIN MODULE OF THE QUATERNIONIC MANIFOLD

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over ℤ[A±1] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the ℤ2-homology of the manifold, we determine that they are linearly independent.

U(N) GAUGED ${\mathcal N} = 2$ SUPERGRAVITY AND PARTIAL BREAKING OF LOCAL ${\mathcal N} = 2$ SUPERSYMMETRY

We study a U (N) gauged [Formula: see text] supergravity model with one hypermultiplet parametrizing SO (4, 1)/ SO (4) quaternionic manifold. Local [Formula: see text] supersymmetry is known to be spontaneously broken to [Formula: see text] in the Higgs phase of U (1) graviphoton × U (1). Several properties are obtained of this model in the vacuum of unbroken SU (N) gauge group. In particular, we derive mass spectrum analogous to the rigid counterpart and put the entire resulting potential on this vacuum in the standard superpotential form of [Formula: see text] supergravity.