Modular spectral triples and deformed Fredholm modules

In the setting of non-type $\ty{II_1}$ representations, we propose a definition {\it deformed Fredholm module} $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct$ for modular spectral triple $\ct$, where $D_\ct$ is Dirac operator. assumed to be invertible sake simplicity, and its domain an "essential" operator system $\ce_\ct$. According such definition, obtain $\big[D_\ct|D_\ct|^{-1}\,,\,{\bf\cdot}\,\big]_\ct=|D_\ct|^{-1}d_\ct(\,{\bf\cdot}\,)+d_\ct(\,{\bf\cdot}\,)|D_\ct|^{-1}$, $d_\ct$ derivation associated $D_\ct$. Since "quantum differential" $1/|D_\ct|$ appears in symmetric position, module differs from usual one even undeformed case, that tracial case. Therefore, it seems more suitable investigation noncommutative manifolds which nontrivial structure might play crucial role. We show all models \cite{FS} representations 2-tori indeed provide triples, addition modules according proposed present paper. detailed knowledge spectrum plays fundamental role geometry, characterisation eigenvalues eigenvectors terms periodic solutions particular class eigenvalue Hill equations.