BPS counting for knots and combinatorics on words
Abstract
We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum Apolynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (HilbertPoincaré series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the branch of mathematics referred to as combinatorics on words. We illustrate these results in the framework of colored extremal knot polynomials: among others we determine dual quantum extremal Apolynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal LabastidaMariñoOoguriVafa invariants) and discuss their integrality.
 Publication:

Journal of High Energy Physics
 Pub Date:
 November 2016
 DOI:
 10.1007/JHEP11(2016)120
 arXiv:
 arXiv:1608.06600
 Bibcode:
 2016JHEP...11..120K
 Keywords:

 ChernSimons Theories;
 NonCommutative Geometry;
 Topological Field Theories;
 Topological Strings;
 High Energy Physics  Theory;
 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra
 EPrint:
 41 pages, 1 figure, a supplementary Mathematica file attached