Supersymmetry, Vacuum Statistics, and the Fundamental Theorem of Algebra

TL;DR

This paper offers a physical interpretation of the fundamental theorem of algebra using supersymmetry and the Witten index, explaining zero distributions of polynomials and analyzing vacuum state statistics in supersymmetric theories.

Contribution

It provides a novel physical perspective on the fundamental theorem of algebra and investigates the statistical properties of vacua in supersymmetric models.

Findings

Real polynomials of degree n may lack n real roots.

Complex polynomials of degree n always have n roots in the complex plane.

Vacuum state statistics are analyzed in a model-independent manner.

Abstract

I give an interpretation of the fundamental theorem of algebra based on supersymmetry and the Witten index. The argument gives a physical explanation of why a real polynomial of degree $n$ need not have $n$ real zeroes, while a complex polynomial of degree $n$ must have $n$ complex zeroes. This paper also addresses in a general and model-independent way the statistics of the perturbative ground states (the states which correspond to classical vacua) in supersymmetric theories with complex and with real superfields.