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We continue the investigation of twisted homology theories in the context ofdimension drop phenomena. This work unifies previous equivariant indexcalculations in twisted cyclic cohomology. We do this by proving the existenceof the resolvent cocycle, a finitely summable analogue of the JLO cocycle,under weaker smoothness hypotheses and in the more general setting of `modular'spectral triples. As an application we show that using our twisted resolventcocycle, we can obtain a local index formula for the Podles sphere. Theresulting twisted cyclic cocycle has non-vanishing Hochschild class which is indimension 2.

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere