Monte Carlo Approach to M-Theory

TL;DR

This work investigates the finite, high-dimensional partition functions ${\cal Z}_{D,N}$ of dimensionally reduced supersymmetric Yang–Mills theory for gauge groups ${\rm SU}(N)$, focusing on $D=3,4,6,10$. By integrating out fermions, the authors obtain a Pfaffian ${\cal P}_{D,N}$ multiplying the bosonic action and analyze the structure of these integrals, including exact SU(2) results and conjectured forms extending Green and Gutperle’s ideas to higher $N$ with a dimension-independent normalization ${\cal F}_N$. They develop a Monte Carlo methodology—employing importance sampling, radial compactification to $S^d$, and a ratio technique—to evaluate ratios of partition functions and test the conjectures for $N=2$ and $N=3$, obtaining precise agreement with the predicted values (e.g., $Z_{D,2}$ and the SU(3) ratios). The results provide nonperturbative checks of the Green–Gutperle conjecture and support the IKKT model’s proposed nonperturbative framework, while highlighting the numerical challenges posed by Pfaffians and high-dimensional valley structures. Overall, the paper demonstrates viable Monte Carlo strategies for probing supersymmetric matrix models beyond analytic reach and strengthens connections to string theoretic constructs.

Abstract

We discuss supersymmetric Yang-Mills theory dimensionally reduced to zero dimensions and evaluate the SU(2) and SU(3) partition functions by Monte Carlo methods. The exactly known SU(2) results are reproduced to very high precision. Our calculations for SU(3) agree closely with an extension of a conjecture due to Green and Gutperle concerning the exact value of the SU(N) partition functions.