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Curvature of the determinant line bundle for the noncommutative two torus. (English) Zbl 1413.46063

Summary: We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen’s original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes’ algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation \(\delta_w\delta_{\bar{w}}\log \det(\Delta)\).

MSC:

46L87 Noncommutative differential geometry
58B34 Noncommutative geometry (à la Connes)
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