Twisted Real Structures in Noncommutative Geometry

Twisted real structures are a generalisation of real structures for spectral triples which are motivated as a way to implement the conformal transformation of a Dirac operator without needing to twist the noncommutative 1-forms. Taking inspiration from this example, in this thesis, we study further applications of twisted real structures, in particular those pertaining to commutative or almost-commutative geometries. We investigate how a reality operator can implement a noncommutative Clifford algebra Morita equivalence bimodule and find that the corresponding real structure on a commutative spectral triple must be twisted. We also investigate how the presence of a twisted real structure affects the implementation of the C*-algebra self-Morita equivalence bimodule which gives the gauge transformations of a spectral triple and find that the twist operators must be tightly constrained to yield meaningful physical action functionals. The form of the resulting action functionals suggests that the twist operator may implement a Krein structure, which often appears in pseudo-Riemannian generalisations of spectral triples. Thus we further investigate if twisted real structures can implement Wick rotations, and though we do not find a fully satisfactory construction, our preliminary attempts are encouraging and suggest that the possibility cannot yet be ruled out. Lastly we identify from the literature that the twisted spectral triple for kappa-Minkowski space admits a reality operator which gives a twisted real structure. This indicates that twisted real structures are compatible with twisted spectral triples as had been previously conjectured, opening up a whole new range of potential applications.