Computational Algebraic Geometry in String and Gauge Theory
Yang‐Hui He, Philip Candelas, Amihay Hanany, André Lukas, Burt A. Ovrut · Advances in High Energy Physics · 2012
The last few years have witnessed a rapid development in algebraic geometry, computer algebra, and string and field theory, as well as fruitful cross-fertilization amongst them.The dialogue between geometry and gauge theory is, of course, an old and rich one, leading to tools crucial to both.The introduction of algorithmic and computational algebraic geometry, however, is relatively new and is tremendously facilitated by the rapid progress in hardware, software as well as theory.Applications of once specialized mathematical topics such as Gr öbner bases, sheaf cohomology, scheme theory, and Hilbert series are quickly becoming indispensible tools in theoretical physics, from topics ranging from AdS/CFT to string phenomenology, from supersymmetric gauge theory to Calabi-Yau compactifications, and so forth.In this special issue, we have invited many international experts, culminating in 17 papers on related subjects, which we order below alphabetically according to title.The paper "A simple Introduction to Gröbner basis methods in string phenomenology" by J. Gray is a review on the most important subject in computational and algorithmic algebraic geometry: the Gr öbner basis.It illustrates how this can be used in string phenomenology and gives some concrete examples ranging from flux parameter to vacuum spaces.