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Numerical Polynomial Homotopy Continuation Method and String Vacua

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 article, 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 finds \textit{all} of 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 is \textit{embarrassingly 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 and compare the results with the available results from the symbolic methods. We then show that all the vacua of a more complicated model of M theory compactified on the coset $\frac{SU(3)\times U(1)}{U(1)\times U(1)}$, which has an SU(3) structure, can be obtained by the NPHC method using a desktop machine in just about one 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.