Numerical algebraic geometry: a new perspective on gauge and string theories

Abstract

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

Energy landscape of the finite-size spherical three-spin glass model

“…(8), the so-called numerical polynomial homotopy continuation (NPHC) method [12] is particularly suitable. This method has the virtue of being able to find all the solutions of a given system of polynomial equations, and has been applied in the past to a variety of problems in particle theory and statistical mechanics [13][14][15][16][17][18][19][20][21][22][23][24]. A drawback of the NPHC method is that it is restricted to fairly small system sizes.…”

Stability ofRparity in supersymmetric models extended byU(1)B−L

“…The Gröbner basis method can be used to find all the minima of a polynomial function, as has recently seen some discussion in the literature [80,81]. A less well-known method is that of homotopy continuation [82,83], which has found use in several areas of physics [84][85][86], in particular finding string theory vacua [87,88] and extrema of extended Higgs sectors [89], where the authors investigated a system of two Higgs doublets with up to five singlet scalars in a general tree-level potential. In contrast, the Gröbner basis method is deemed prohibitively computationally expensive for systems involving more than five or six degrees of freedom [80], while for our purposes eight to ten degrees of freedom are necessary, and the solutions for thousands of parameter points needed to be calculated.…”

Potential energy landscape of the two-dimensional XY model: Higher-index stationary points

“…5 Polynomial equations are more amenable to numerical techniques such as the polynomial homotopy continuation method, 16 a method that has been applied to compute the stationary points of a variety of models in statistical mechanics and particle physics. [17][18][19][20][21][22][23][24][25] By applying this method to the polynomial form of the XY model, numerical results for the stationary points of the two-dimensional XY model were reported in Refs. 26 and 27 for small lattices of 3 × 3 sites.…”