A note on nonlinear \(\sigma\)-models in noncommutative geometry (Q2797889)

The spectral triple \((A,H,D)\) consists of an involutive algebra \(A\) represented as bounded operators on a Hilbert space \(H\) and a self-adjoint operator \(D\) with compact resolvent such that the commutators \([D,a]\) are bounded for all \(a\in A\). A spectral triple \((A,H,D)\) is said to be even if the Hilbert space \(H\) is endowed with a super-grading which commutes with all \(a \in A\) and anti-commutes with \(D\). The spectral triple \((A,H,D)\) is \((2,\infty)\)-summable, means roughly \(Tr_\omega(a|D|^{-2}) < \infty\), where \(Tr_\omega\) is the Dixmier trace.NEWLINENEWLINEA (noncommutative) sigma-model consists of homomorphisms \(\phi : B \rightarrow A\) with the given even \((2,\infty)\)-spectral triple \((A,H,D)\) as the target and a positive element \(G\in \Omega^2(B)\) in the space of universal 2-forms on \(B\).NEWLINENEWLINEThe paper under review considers the sigma-models based on the noncommutative torus \(A_\theta\). It considers a quantum group as a noncommutative space-time as well as two points, a circle, and a (irrational) noncommutative torus. A source space or a world-sheet is fixed as the noncommutative torus for an irrational \(\theta\) and for the various target spaces the Euler-Lagrange equations are observed. Using the established results the paper under review shows that the trivial harmonic unitaries of the noncommutative chiral model, which are known as local minima, are not global minima by comparing those with the symmetric unitaries coming from instanton solutions of the non-commutative Ising model, which corresponds to the two points target space.