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Extra info for Matrix Quantum Mechanics and 2-D String Theory. thesis]
The compactification restricts the allowed tachyon momenta to discrete values pn = n/R so that we have only a discrete set of vertex operators. Besides, depending on the radius, the compactification can create or destroy the discrete states. Whereas for rational values of the radius some discrete states are present in the spectrum, for general irrational radius there are no discrete states. 23 Chapter I: String theory But the compactification also leads to the existence of new physical string states.
The critical behaviour arises when the highest filled level reaches an extremum of the effective potential. Generically, this happens as shown in fig. 4. By fine tuning of the coupling constants, one can get more special configurations which will describe the critical behaviour with m > 1. 43 Chapter II: Matrix models z B1 A1 Ak Bk Fig. 5: Riemann surface associated with the solution of 1MM. 5 General solution and complex curve So far we considered the case where the density is concentrated on one interval.
We will solve this model by the method of orthogonal polynomials. This approach can be also applied to 1MM where it looks even simpler. However, we would like to illustrate the basic features of 2MM and the technique of orthogonal polynomials is quite convenient for this. 1 Reduction to eigenvalues Similarly to the one-matrix case, one can diagonalize the matrices and rewrite the integral in terms of their eigenvalues. However, now one has to use two unitary matrices, which are in general different, for the diagonalization B = Ω†B yΩB .