Nonperturbative spectral action of round coset spaces of \(\mathrm{SU}(2)\) (Q380270)

Recall that the spectral action on a spectral triple \(({\mathcal A}, {\mathcal H}, {\mathcal D})\) is a functional \(\mathrm{Tr}~f({\mathcal D}/\Lambda)\), where \(f: {\mathbb R}\to {\mathbb R}\) is a test function and \(\Lambda>0\) is a constant.NEWLINENEWLINE The main result of the paper says that for any finite subgroup \(\Gamma\) of \(\mathrm{SU}(2)\) and the canonical Dirac operator on \(\mathrm{SU}(2)/\Gamma\) with a round metric and trivial spin, the spectral action satisfies \(\mathrm{Tr}~(f({\mathcal D}/\Lambda)={1\over |\Gamma|}(\Lambda^3\hat f^{(2)}(0)-{1\over 4} \Lambda\hat f(0))+O(\Lambda^{-\infty})\). The spectral triples considered are commutative, i.e. \({\mathcal A}=C^{\infty}(M)\) and \({\mathcal H}=L^2(M, \Sigma_n)\), where \(\Sigma_n\) is the spinor.