Spectral Action on SU q (2)

Abstract

The spectral action on the equivariant real spectral triple over A SU q (2) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3 .

Heat Trace and Spectral Action on the Standard Podleś Sphere

“…Yet, the latter is not a single number but rather a dimension spectrum [13,29], which is a (possibly infinite) discrete subset of the complex plane with finite multiplicities allowed. The dimension spectrum has been computed for various commutative [3,13,14,29,30,36] and noncommutative spectral triples [5,18,21,24,27,28,30,43] including the quantum group SU q (2) [12,19] and some of the Podleś spheres [18] (see also [16]). One of the significant features of almost all spectral triples for the q-deformations of manifolds is the discrepancy between the homological and metric dimensions.…”

Tadpoles and commutative spectral triples

“…When M has a boundary, some a 0 k are non-zero, the dimension spectrum can be non simple (even if it is simple for the Dirac operator, see for instance [40].) On a spectral triple .A; H ; D/, the fact to change the product on A may or not affect the dimension spectrum: for instance, there is no change when one goes from the commutative torus to the noncommutative one (see [19]), while the dimension spectrum of SU q .2), which is bounded from below, does not coincide with the dimension spectrum of the sphere S 3 corresponding to q D 1; see [33], Corollary 4.10.…”

“…This map is of course a trace on A for a commutative geometry. But for the triple associated to SU q .2/ this not a trace since (see [33]) Note that the notion of pseudodifferential operator is modified as ‰.A/ now includes J AJ 1 ; see [19]. …”

“…For examples of spectral action in the real noncommutative setting, see [9], [6], [37] for the case of almost commutative instances which pops up in particle physics, [21] for the Moyal plane (and few points for non-compact manifolds [22]), [23], [19] for the noncommutative torus, and [33] for the quantum group SU q .2/. In the latter case there are tadpoles.…”

“…This proposition does not survive in noncommutative spectral triples, see for instance [33], Table 1.…”