Supersymmetric large-order perturbation with the Nicolai map

TL;DR

This paper demonstrates that the Nicolai map in supersymmetric quantum theories admits a convergent perturbative expansion, providing a non-perturbative existence proof and insights into large-order behavior.

Contribution

It introduces a recursive construction of the Nicolai map's perturbative expansion and proves its convergence, extending understanding of supersymmetric theories.

Findings

The perturbative expansion of the Nicolai map converges with a finite radius.

Tree diagram growth rate is estimated as $n^{-3/2} imes 4.967^n$.

Results suggest extension to higher-dimensional supersymmetric theories.

Abstract

In rigidly supersymmetric quantum theories, the Nicolai map allows one to turn on a coupling constant (from zero to a finite value) by keeping the (free) functional integration measure but subjecting the fields to a particular nonlocal and nonlinear transformation. A recursive perturbative construction of the Nicolai-transformed field configuration expresses it as a power series in the coupling, with its coefficient function at order $n$ being a sum of particular tree diagrams. For a quantum-mechanical example, the size of these tree diagrams (under a certain functional norm) is estimated by the $(n{+}1)$st power of the field size, and their number grows like $n^{-3/2}\times4.967^{\;n}$. Such an asymptotic behavior translates to a finite convergence radius for the formal perturbative expansion of the Nicolai map, which establishes its non-perturbative existence. The known factorial growth of the number of Feynman diagrams for quantum correlators is reproduced by the combinatorics of free-field Wick contractions as usual. We expect our results to extend to higher dimensions, including super Yang-Mills theory.