Kähler potentials of the Weil-Petersson metric (Q2769208)

Let \(X\) be a compact oriented Riemann surface of genus \(\gamma> 1\) and \(\Upsilon_{\gamma}\) the corresponding Teichmüller space. Its holomorphic tangent and cotangent spaces at a point \([X]\in \Upsilon_{\gamma}\) can be identified, respectively, with the space of harmonic forms \(H^{0,1}(X,TX)\) and \(H^{1,0} (X,T^*X).\) The corresponding pairing \((. , .): H^{0,1}(X,TX) \bigotimes H^{1,0} (X,T^*X)\to \mathbb{C}\) is given by the integral \((p q)=\int_Xpq\), where \(p\in H^{0,1}(X,TX)\), \(q\in H^{1,0}(X,T^*X)\), and the integration is with respect to the Poincaré metric. The Weil-Petersson Hermitian metric is defined as \(\langle . ,.\rangle_{W-P}=(. ,*.)\) where the Hodge star \(*\) corresponds to the Poincaré metric on \(X.\) The corresponding symplectic form is denoted by \(\omega_{W-P}.\)NEWLINENEWLINENEWLINEIt is a problem of some interest to search for an explicit Kähler potential of the Weil-Petersson metric, i.e., a complex function \(K\) such that \(\omega_{W_P}= \partial\overline \partial K\) either locally or globally. In this paper, the author gives a survey of the answers obtained for this geometrical problem by A. J. Tromba, S. T. Yau, P. G. Zograf, L. A. Takhtajan and J. Jost, and briefly explains its physical interpretation in the context of bosonic string theory.