A global and stochastic analysis approach to bosonic strings and associated quantum fields (Q1197338)

In this long and clearly written paper the authors construct a family of Gibbs probability measures for the free Euclidean bosonic string which is a rigorous mathematical realization of the Polyakov Ansatz. The following steps in this construction are presented. 1. The space of Riemannian metrics \(g\) on a compact two-dimensional manifold \(\Lambda\) is parameterized by \((f,\varphi,t)\) where \(f\) is a diffeomorphism of \(\Lambda\), \(\varphi\) is a scalar function describing conformal transformation of \(g\) and \(t\) is an element of Teichmüller space. 2. The Jacobian associated with a change of variables from \(g\) to \((f,\varphi,t)\) is obtained and its determinant is computed using heat kernel regularization. 3. The ``Liouville measure'' is defined by adopting to the case of a general \(\Lambda\) previous methods developed for the Høegh-Krohn model on flat space \(\mathbb{R}^ 2\). 4. The full probability measure for the bosonic string is constructed using the ingredients described above for the space-time dimension \(3\leq d\leq 13\).