[摘要] Two-dimensional models of quantum gravity have been solved using matrix model techniques. Furthermore, these solutions have turned out to be encoded in integrable nonlinear PDEs belonging to the KdV hierarchy. This thesis presents a new KdV recursion relation, distinct from one found previously by Dijkgraaf and Witten, for a certain class of theories known as the two-matrix models. The two recursion relations together are used to relate arbitrary correlation functions containing a puncture operator P (at any genus) to the three basic correlators
, , and by unique algebraic expressions. (Q is the dilaton operator.) The derivation requires assuming a certain scaling law, whose justification is discussed.Other KdV recursion relations, given by Virasoro or W-algebra constraints, are possible for multi-matrix models when an infinite number of couplings are added. These constraints have been presented for A_n-type models by Fukuma et al. and Dijkgraaf et ai. We derive analogous Virasoro constraints for the multi-matrix models associated with the other simply-laced Lie algebras D_(2n+1), E_6, E_7, and E_8. As a check, it is verified that the proposed constraints imply operator scaling dimensions identical to those found by Kostov. It is then demonstrated that these Virasoro constraints (or, more generally, W -algebra constraints) can be used to derive expressions for correlation functions containing a non-primary operator in terms of correlation functions that only contain primary operators.The second subject of this thesis concerns the underlying symmetries of string theory as probed by fixed-angle scattering at very high energy. The asymptotic behavior depends sensitively on the choice of the string vacuum. Therefore, we examine the effect of modifying the vacuum on the behavior of high-energy scattering amplitudes. In particular, high-energy fixed-angle elastic scattering of open-string tachyons is studied explicitly. Tadpole corrections to the tree-level formulas are included. The main conclusion of the analysis is that symmetry relations among amplitudes at high energy seem to be unaffected by modifications of the vacuum, even though the amplitudes themselves do change.
