The Gauss-Bonnet theorem for the noncommutative two torus (Q2904908)

There is no better way to stun a freshman class than talking about the Gauss-Bonnet Theorem. It is an elementary and truly amazing fact of differential geometry of compact surfaces immersed into the three-dimensional euclidean space. The theorem says that the integral of the gaussian curvature (i.e., product of the maximal and minimal curvature of surface at a point) over all points of the surface is equal to \(2\pi\) times an integer; the integer does not depend on the immersion, it is a topological invariant of the surface. Simply put, it does not matter how badly a tennis ball is squashed (as long as there are no wrinkles) -- the integral of its gaussian curvature is always \(4\pi\)! There are numerous generalizations of this remarkable theorem; one of these is a spectral ``Gauss-Bonnet''. Recall that immersion of a surface into \({\mathbb R}^3\) turns the surface into a Riemann surface \(\Sigma\) (with a metric induced by the euclidean metric of \({\mathbb R}^3\)). Let \(\Delta\) be the Laplacian operator on \(\Sigma\) and \(\lambda_j\) its eigenvalues; consider the zeta function \(\zeta(s)=\sum\lambda_j^s\), \(\mathrm{Re} (s)>0\). The spectral Gauss-Bonnet theorem asserts that \(\zeta(0)={1\over 6}\chi(\Sigma)\), where \(\chi(\Sigma)\) is the Euler characteristic of \(\Sigma\). The spectrum of linear self-adjoint operators on a Hilbert space is a natural subject of noncommutative geometry; it is interesting, therefore, to have a ``noncommutative'' analog of the Gauss-Bonnet theorem. The paper under review solves the problem in the case \(\Sigma=T^2\) is a two-torus; clearly, in this case \(\zeta(0)=0\), since \(\chi(T^2)=0\).NEWLINENEWLINEA legitimate replacement for \(T^2\) is the so-called noncommutative torus \(T^2_{\theta}\), i.e., the \(C^*\)-algebra generated by the unitaries \(U\) and \(V\) satisfying the commutation relation \(VU=e^{2\pi i\theta}UV\), where \(\theta\in {\mathbb R}\setminus {\mathbb Q}\). A conformal metric on \(T^2_{\theta}\) can be defined by a pair of operators \(\partial=\delta_1+i\delta_2\) and \(\partial^*=\delta_1-i\delta_2\), where \(\delta_1\) and \(\delta_2\) are derivations on the algebra \(T^2_{\theta}\) satisfying the equations \(\delta_1(U)=U\), \(\delta_1(V)=0\), \(\delta_2(U)=0\) and \(\delta_2(V)=V\). The Laplacian \(\Delta\) on \(T^2_{\theta}\) can be defined as \(\Delta=\partial^*\partial=\delta_1^2+\delta_2^2\). An analog of the conformal class of a metric on \(T^2_{\theta}\) is represented by a family of modular automorphisms \(\Delta_h=k\Delta k\), where \(k=e^{h\over 2}\) and \(h\in T^2_{\theta}\). The main theorem of the paper states that if \(\theta\) is an irrational number and \(k\) an invertible positive element of \(T^2_{\theta}\), then the value of \(\zeta(0)\) of the zeta function of the operator \(k\Delta k\) does not depend on \(k\). The proof is based on a calculus of the pseudo-differential operators and explicit calculations (sometimes scarily long!)NEWLINENEWLINE The text is clearly written; a gentle introduction and lucid motivation are provided. It can easily be recommended to graduate students as an introduction to the ideas and methods of noncommutative geometry, which is co-written by the founder of the field.NEWLINENEWLINEFor the entire collection see [Zbl 1243.14002].