Frobenius manifolds and variance of the spectral numbers (Q2754592)

The spectral numbers \(\alpha_1,\dots,\alpha_\mu\) of a quasihomogeneous singularity satisfy the symmetry condition \(\alpha_i+\alpha_{\mu+1-i}=n-1\), which makes the number \((n-1)/2\) their ``expectation value'' and \(\mu^{-1}\sum_{i=1}^\mu(\alpha_i-(n-1)/2)^2\) their ``variance''. The author proves that this variance is equal to \((\alpha_\mu-\alpha_1)/12\). NEWLINENEWLINENEWLINEThe method of proof is an application of the theory of the \(G\)-function on Frobenius manifolds. The \(G\)-function was defined by \textit{B. Dubrovin} and \textit{Y. Zhang} [Commun. Math. Phys. 198, 311-361 (1998; Zbl 0923.58060); Sel. Math., New Ser. 5, 423-466 (1999; Zbl 0963.81066)] as \(G(t)=\log\tau_I-{1\over {24}}\log J\), where \(J\) is the determinant of the base change matrix between flat and idempotent vector fields and \(\tau_I\) is the \(\tau\)-function of isomonodromic deformations in the sense of Miwa, Jimbo, et al. (Frobenius manifolds can be viewed as isomonodromic deformations of restrictions to a slice.) NEWLINENEWLINENEWLINEThe author suggests some other possible uses of the \(G\)-function in singularity theory and conjectures that for isolated hypersurface singularities the variance of the spectral numbers is \(\leq {1\over{12}}(\alpha_\mu-\alpha_1)\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].