Functional Evaluation of the Dual Partition Functions: How Do We "Hear the Shape of a Drum" in Dual Models?

how many states with a definite mass squared of the resonances the dual model has, and for instance it readily informs us how the degeneracy increases as the energy does.*' It is true that any dual model is not determined solely by the partition of the mass spectra, but any dual partition function well features the mathematical characteristics of its respective model: We may be thus allowed to say that the variety of the partition functions well reflects the variety of the dual models. It is therefore always preferable to have a well-established foundation for the formulation of dual partition functions. Our aim in this paper is to give a consistent way of describing and calculating the partition functions of the known dual models in the framework of the so-called functional formalism. We share thus our objective in common with those in Refs. 2) and 3), but it will turn out that our line of calculation still has some value of its own. In § 2, we review a lemma due to Kac4' on a characteristic of annulus domains, which will form a starting background for our way of description. In § 3, we begin to deal with the partition function of Veneziano's orbital model. In the first subsection it is defined as a functional integral and is converted to be com pactly written by the values of the zeta-function along the line of Hawking. 5' In the second subsection the values are explicitly evaluated by the use of several known formulae including Kronecker's first limit formula. The partition function of the orbital string proves without overcharge to be a modular form of weight - (] /2, where (] denotes the effective dimension of space-time. The divergence problem of the "zero-point energies" is also automatically solved. Section 4 is