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Polyakov’s formulation of \(2d\) bosonic string theory. (English) Zbl 1514.81216

Summary: Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus \(\text bf{g} \geq 2 \) and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov’s partition function of noncritical bosonic string theory (Polyakov in Phys. Lett. B 103:207,1981) (also called 2d bosonic string theory) and to Liouville quantum gravity. More specifically, we show the convergence of Polyakov’s partition function over the moduli space of Riemann surfaces in genus \(\mathbf{g} \geq 2\) in the case of D\(\leq 1\) boson. This is done by performing a careful analysis of the behavior of the partition function at the boundary of moduli space.An essential feature of our approach is that it is probabilistic and non perturbative. The interest of our result is twofold. First, to the best of our knowledge, this is the first mathematical result about convergence of string theories. Second, our construction describes conjecturally the scaling limit of higher genus random planar maps weighted by Conformal Field Theories: we make precise conjectures about this statement at the end of the paper.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
81V17 Gravitational interaction in quantum theory
83C47 Methods of quantum field theory in general relativity and gravitational theory
14D22 Fine and coarse moduli spaces
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