Spontaneous symmetry breaking and geometry from the spectral viewpoint
This is an introduction to the main ideas of noncommutative geometry, developed by the author. It's main object, the space, is a triple \((A, H, D)\), where \(A\) is an involutive operator algebra in the Hilbert space \(H\) and \(D\) is a self-adjoint unbounded operator on \(H\). The algebra \(A\) is the set of ``continuous functions on a noncommutative space'', and the operator \(D^{-1} = ds\) corresponds to the infinitesimal length of Riemannian geometry. In this general setting, a fiber bundle is given by an \(A\)-projective module of finite type, a de Rham homology class is given by a cyclic cocycle of \(A.\) The right notion of infinitesimal calculus is developed: for example, a real variable is given by a self-adjoint operator, an infinitesimal is given by a compact operator, the integral of an infinitesimal \(T\) of order one is given by the coefficient of the logarithmic divergence of the trace of \(T\), the differential of \(g\) is given by the bracket \([F,g]\), where \(F\) is a fixed operator such that \(F^2= \text{Id}\) and \([F,a]\) is compact for any \(a\in A\). There is a pairing between \(K_*(A)\) and cyclic cohomology \(HC_*(A)\), generalizing the Chern-Weil construction of classical differential geometry. When one pairs a class \(e \in K_*(A)\) with the Chern character of \((A, H, D)\) one gets the index of a Fredholm operator and thus an integer. This result explains the quantum Hall effect in physics. A local formula in cyclic cohomology for the Chern character of \((A, H, D)\) generalizes the local index theorem for Dirac-type operators. The notion of noncommutative Poincaré duality allows to extend to finite type noncommutative spaces satisfying Poincaré duality the eigenvalue inequalities of Vafa and Witten. There are ``noncommutative manifolds'' - examples are provided by noncommutative tori and the ``Standard Model'', which proved to be useful in physics.
local index theorem for Dirac-type operators
eigenvalue inequalities of Vafa and Witten
scientific article; zbMATH DE number 1112279
scientific article