Abstract
We demonstrate that the problems of finding stable or metastable vacua in a low energy effective field theory requires solving nested nondeterministically polynomial (NP)-hard and co-NP-hard problems, while the problem of finding near-vacua can be solved in polynomial (P) time. Multiple problems relevant for computing effective potential contributions from string theory are shown to be instances of NP-hard problems. If $\mathrm{P}\ensuremath{\ne}\mathrm{NP}$, 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.
Highlights
The enormous landscape of string theory realizes rich and diverse physical phenomena, but poses critical practical obstacles to its study
We take the original result and impose box constraints by replacing the general constraints Ax ≥ b with li ≤ xi ≤ ui. This may be done by taking the final instance of QPLOC in the Appendix proof and putting it on f0 ≤ xi ≤ uig with ui > 0; the proof version was on f0 ≤ xig, but the upper bounds just imposed do not change the validity of the relationship between cliques and whether or not yà 1⁄4 0 is a local minimum
We have shown that the problem of finding global and local minima of a scalar potential are both co-nondeterministically polynomial (NP)-hard
Summary
The enormous landscape of string theory realizes rich and diverse physical phenomena, but poses critical practical obstacles to its study. Immediately imply intractability, since exponential time may be affordable for concrete instances of problems at parameter values relevant in string theory (see, e.g., [12]) Beyond these practical issues, complexity is known to affect the dynamics of physical systems that realize a landscape of metastable states, for example in spin glasses and protein folding; see [10,13] for discussions. JAMES HALVERSON and FABIAN RUEHLE determining contributions to the effective scalar potential, and from the hardness of finding metastable minima; i.e., one is faced with nested computationally hard problems This explains the dearth of concrete studies of string vacua at large numbers of scalar fields (e.g., CalabiYau compactifications with h11 or h21 of Oð10Þ or higher), despite the expectation that most of the landscape lies in such regions. Appendix contains a proof relevant for the hardness of minimizing the scalar potential
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
