Modular Curvature for Noncommutative Two-Tori

Details Modular Curvature for Noncommutative Two-Tori Modular Curvature for Noncommutative Two-Tori

2011-10-16

arxiv.org/abs/1110.3500v1

Mathematics

Quantum Algebra

43 pages, 5 figures

Scientific paper

Starting from the description of the conformal geometry of noncommutative 2-tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a closed formula for the Ray-Singer analytic torsion in terms of the Dirichlet quadratic form and the generating function for Bernoulli numbers applied to the modular operator. The gradient of the Ray-Singer analytic torsion is then expressed in terms of these functionals, and yields the analogue of scalar curvature. Computing this gradient in two ways elucidates the meaning of the complicated two variable functions occurring in the formula for the scalar curvature. Moreover, the corresponding evolution equation for the metric produces the appropriate analogue of Ricci curvature. We prove the analogue of the classical result which asserts that in every conformal class the maximum value of the determinant of the Laplacian on metrics of a fixed area is attained only at the constant curvature metric.

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