Abstract
Previously the exact solution of the planar sector of the self-dual Φ4-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant λ > − frac{1}{uppi} , the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension 4 − 2 frac{arcsin left(uplambda uppi right)}{uppi} for |λ| < frac{1}{uppi} . It is this dimension drop which for λ > 0 avoids the triviality problem of the matricial {varPhi}_4^4 -model. We also establish the power series approximation of the Fredholm solution to all orders in λ. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters 0 and −1. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.
Highlights
The exact solution of the planar sector of the self-dual Φ4-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation
As main result of this paper we prove that (1.3) is solved by a hypergeometric function, J (x) = x 2F1 αλ, 1 − αλ 2 x − μ2
The solution is given by hypergeometric functions 2F1 f (x) = 2F1 αλ, −αλ − x 1 x g (x) = αλ2 f (x) = x2F1
Summary
Let 0(x)dx be the spectral measure of the operator E in the initial action (1.1). The main discovery of [8]. Effectively modifies the spectral measure to λ(x)dx. What before, when expressed in terms of 0(x)dx, was intractable became suddenly exactly solvable in terms of the deformation λ(x)dx. Moyal space one has 0(x) = x and λ(x) = J(x). The explicit solution (1.4) shows that the deformation is drastic: it changes the spectral dimension D defined by D =. The change of spectral dimension is important. If instead of (1.3) the function J was given by J(x) = x − λx. 0(x) = x this function Jis bounded above. The dimension drop down to arcsin(λπ) π avoids this (triviality) problem
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