Lattice Landau Gauge and Algebraic Geometry

Details Lattice Landau Gauge and Algebraic Geometry Lattice Landau Gauge and Algebraic Geometry

2009-12-02

arxiv.org/abs/0912.0450v1

PoS QCD-TNT09::025,2009

Physics

High Energy Physics

High Energy Physics - Lattice

Talk given at International Workshop on QCD Green's Functions, Confinement, and Phenomenology - QCD-TNT09, Trento, Italy, Sept

Scientific paper

Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.

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