Heat Trace and Spectral Action on the Standard Podleś Sphere

Abstract

We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podleś spheres S 2 q for 0 < q < 1, study its behavior when q → 1 and fully compute its exact spectral action for an explicit class of cut-off functions.

Asymptotic and Exact Expansions of Heat Traces

“…Having an asymptotic expansion at hand, a natural question one may pose is that of its convergence. This issue has not been studied in full generality, even in the context of P being a differential operator (see however [29,46]). We show how the conditions on P shall be refined in order to get an exact formula for the heat trace valid on some open interval (0, T ).…”

“…In fact, the existence of an asymptotic expansion of the heat trace is assumed whenever needed in applications (see for instance [19,Section 11], [21, Section 2.1]) and has been proven rigorously only for a few specific examples [29,37,71]. Whence the results of [37,71] essentially used the methods of pseudodifferential calculus, the casus of the standard Podleś sphere [29] required completely different tools (see Section 4.2).…”

“…Clearly, htr P (t) decays exponentially as t grows to infinity, whereas the RHS of (49) seems to diverge to infinity because of the log t terms. However, it turns out that the sum over n in the RHS of (49) compensates for large t the divergent terms (log t) j , the constant terms and the oscillatory part to yield an exponential decay (compare [35,Example 12] and [29,Section 4.1…”

“…We shall also assume that the multiplicities are given by a polynomial. This type of operators appears naturally in the context of quantum groups [50,51] and their homogeneous spaces [27,29,60] (see also [52] and [35,Example 12]), as well as in the framework of fractal spaces [11-13, 41, 42].…”

“…The large energies expansion of the latter is based on the asymptotic expansion of the heat trace associated with the relevant Dirac operator. Recently, also the exact computations of the spectral action gained interest [10,28,29,57,58,61,68] due to their possible application to the study of cosmic topology.…”