Abstract
Finding vacua for the four-dimensional effective theories for supergravity which descend from flux compactifications and analyzing them according to their stability is one of the central problems in string phenomenology. Except for some simple toy models, it is, however, difficult to find all the vacua analytically. Recently developed algorithmic methods based on symbolic computer algebra can be of great help in the more realistic models. However, they suffer from serious algorithmic complexities and are limited to small system sizes. In this paper, we review a numerical method called the numerical polynomial homotopy continuation (NPHC) method, first used in the areas of lattice field theories, which by construction findsallof the vacua of a given potential that is known to have only isolated solutions. The NPHC method is known to suffer from no major algorithmic complexities and isembarrassingly parallelizable, and hence its applicability goes way beyond the existing symbolic methods. We first solve a simple toy model as a warm-up example to demonstrate the NPHC method at work. We then show that all the vacua of a more complicated model of a compactified M theory model, which has anSU(3)structure, can be obtained by using a desktop machine in just about an hour, a feat which was reported to be prohibitively difficult by the existing symbolic methods. Finally, we compare the various technicalities between the two methods.
Highlights
A lot of current research in string phenomenology is focused on developing methods to find and analyze vacua of four-dimensional effective theories for supergravity descended from flux compactifications
We have reviewed a novel method, called the numerical polynomial homotopy continuation NPHC method, which can find all the string vacua of a given potential
It does not suffer from any major algorithmic complexities compared to the existing symbolic algebra methods based on the Grobner basis techniques, which are known to suffer from exponential space complexity
Summary
A lot of current research in string phenomenology is focused on developing methods to find and analyze vacua of four-dimensional effective theories for supergravity descended from flux compactifications. One can indirectly obtain this information by computing the so-called primary decomposition of the ideal still using the Grobner basis technique internally This was a remarkable success as it allowed one to work on nontrivial models and extract a lot of information using a regular desktop machine only. Stringvacua is a Mathematica interface to Singular and has string phenomenology-specific utilities which makes the package quite useful to the users Even with such tricks, there are a few problems with the symbolic methods: the BA is known to suffer from exponential space complexity, that is, the memory Random Access Memory required by the machine blows up exponentially with the number of variables, equations, terms in each polynomial, and so forth. After mentioning a few other important aspects of the NPHC method in the Frequently Asked Questions section, we conclude the paper
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