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Topological strings and loop equations. (English) Zbl 0783.58088

Random surfaces and quantum gravity, Proc. NATO Adv. Res. Workshop, Cargèse/Fr. 1990, NATO ASI Ser., Ser. B 262, 53-76 (1991).
[For the entire collection see Zbl 0745.00026.]
The paper deals with the recently established topological formulation of ordinary quantum gravity by E. Witten [On the structure of the topological phase of two-dimensional gravity, Nucl. Phys., B 340, 281-332 (1990)]. The topological field theories are shown to provide a very useful starting point for studying the symmetries of string theory. The paper starts with a description of several ingredients of a general topological string theory following very closely the usual discussions of the standard string. The \(d=0\) string model is solved through a set of recursion relations which relate correlation functions at different genera, and the corresponding symmetry structure is exhibited. It is demonstrated that the partition function satisfies Virasoro constraints. These constraints are shown to correspond to the loop equation, which arise in matrix models. Finally, the obtained results are generalized to \(d<1\) models. A more detailed description of the general structure of the matter sector in topological string theory is also given. The authors claim that for minimal models this can be encoded in a Landau-Ginzburg formulation.

MSC:

58Z05 Applications of global analysis to the sciences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83C45 Quantization of the gravitational field
83E30 String and superstring theories in gravitational theory

Citations:

Zbl 0745.00026
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