JournalsjncgVol. 6, No. 3pp. 457–480

The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

  • Farzad Fathizadeh

    The University of Western Ontario, London, Canada
  • Masoud Khalkhali

    The University of Western Ontario, London, Canada
The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure cover
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Abstract

In this paper we give a proof of the Gauss–Bonnet theorem of Connes and Tretkoff for noncommutative two tori Tθ2\mathbb{T}_{\theta}^2 equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number τ\tau in the upper half plane, representing the conformal class of a metric on Tθ2\mathbb{T}_{\theta}^2, and a Weyl factor given by a positive invertible element kC(Tθ2)k \in C^{\infty}(\mathbb{T}_{\theta}^2), the value at the origin, ζ(0)\zeta (0), of the spectral zeta function of the Laplacian \triangle\mkern-.5mu ' attached to (Tθ2,τ,k)(\mathbb{T}_{\theta}^2, \tau, k) is independent of τ\tau and kk.

Cite this article

Farzad Fathizadeh, Masoud Khalkhali, The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure. J. Noncommut. Geom. 6 (2012), no. 3, pp. 457–480

DOI 10.4171/JNCG/97