Computational Complexity of Vacua and Near-Vacua in Field and String Theory

TL;DR

This paper shows that finding stable vacua in field and string theories is computationally hard, involving NP-hard problems, while finding near-vacua is computationally feasible, with implications for string theory and cosmology.

Contribution

It establishes the computational complexity of vacuum-finding problems in field and string theories, revealing NP-hardness and exponential difficulty under certain assumptions.

Findings

Vacua-finding problems are NP-hard and co-NP-hard.

Near-vacua problems are solvable in polynomial time.

Implications for string theory and cosmology are discussed.

Abstract

We demonstrate that the problems of finding stable or metastable vacua in a low energy effective field theory requires solving nested NP-hard and co-NP-hard problems, while the problem of finding near-vacua is in P. Multiple problems relevant for computing effective potential contributions from string theory are shown to be instances of NP-hard problems. If P $\neq$ NP, the hardness of finding string vacua is exponential in the number of scalar fields. Cosmological implications, including for rolling solutions, are discussed in light of a recently proposed measure.