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2009

The Gauss-Bonnet Theorem for the noncommutative two torus

Connes, Alain, Tretkoff, Paula

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In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state.

We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state.
We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.

The Gauss-Bonnet Theorem for the noncommutative two torus · Around

Built on

P. Gilkey, The Index Theorem and the Heat Equation
1974
Earlier work this paper cites.
A. Connes, C ∗ C^{*} -algèbres et géométrie différentielle
1980
Earlier work this paper cites.
S. Baaj, Calcul pseudo-différentiel et produits croisés de C ∗ C^{*} -algèbres. I et II
1988
Earlier work this paper cites.

Similar

A. Connes, J. Cuntz, Quasi homomorphismes, cohomologie cyclique et positivité
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P. B. Cohen, On the Non-commutative Torus of real dimension Two
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Then

A. Chamseddine, A. Connes, The Spectral action principle
1997
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S. Rosenberg, The Laplacian in a Riemannian manifold
1997
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A. Chamseddine, A. Connes, Scale invariance in the spectral action
2006
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