BPS counting for knots and combinatorics on words
We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincaré 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 A-polynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal Labastida-Mariño-Ooguri-Vafa invariants) and discuss their integrality.
KeywordsChern-Simons Theories Non-Commutative Geometry Topological Field Theories Topological Strings
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- M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications volume 17, Addison Wesley, Reading U.S.A. (1983).Google Scholar
- M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications volume 90, Cambridge University Press, Cambridge U.K. (2002).Google Scholar
- V. Vologodsky, Integrality of instanton numbers, arXiv:0707.4617.
- RISC Combinatorics group, A. Riese, qZeil.m, http://www.risc.jku.at/research/combinat/software/qZeil/.