Modular Curvature and Morita Equivalence

Abstract

The curvature of the noncommutative torus $T^2_\theta$ ($\theta$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_\theta = C(T^2_\theta)$ with other toric algebras $A_{\theta'}$. In the enlarged context the conformal metric on $T^2_\theta$ is exchanged with an arbitrary Hermitian metric on the Heisenberg $(A_\theta, A_{\theta'})$-bimodule $E'$ for which ${\rm End}_{A_{\theta'}}(E') = A_\theta $. We prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on $E'$, attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule $E'$ such that $A_\theta = {\rm End}_{A_{\theta'}}(E')$, and in this sense it is Morita invariant. To prove the above results we extend Connes' pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.