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A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions
It is conjectured that a class of n-fold integral transformations
{I(alpha)|alpha in {C}} forms a mutually commutative family, namely, we have
I(alpha) I(beta)=I(beta) I(alpha) for all alpha, beta in {C}. The commutativity
of I(alpha) for the two-fold integral case is proved by using several summation
and transformation formulas for the basic hypergeometric series. An explicit
formula for the complete system of the eigenfunctions for n=3 is conjectured.
In this formula and in a partial result for n=4, it is observed that all the
eigenfunctions do not depend on the spectral parameter alpha of I(alpha).Comment: Basic parameters are replaced to make the notation consistent with
the standard Macdonald polynomials: q is replaced by t, and p^{1/2} is
replaced by
Wilson Loops in Open String Theory
Wilson loop elements on torus are introduced into the partition function of
open strings as Polyakov's path integral at one-loop level. Mass spectra from
compactification and expected symmetry breaking are illustrated by choosing the
correct weight for the contributions from annulus and M\"obius strip. We show
that Jacobi's imaginary transformation connects the mass spectra with the
Wilson loops. The application to thermopartition function and cosmological
implications are briefly discussed.Comment: 5 pages, no figur
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