Year: 2026
Type: preprint
Abstract: We consider quantum A-periods of local ${\mathbb P}^1 \times {\mathbb P}^1$, or equivalently the generating function of constant terms for powers of the Laurent polynomial $x + x^{-1} + y + y^{-1}$ where $x$ and $y$ are $q$-commuting variables. We derive a number of new representations for these rich combinatorial and algebraic objects. First, we find that the well-known TBA equation encodes not only the A-period itself but also higher diagonal modes of the corresponding noncommutative Laurent polynomials. Second, we obtain new formulas for the period in terms of continued fractions and related orthogonal polynomials, as well as a nontrivial explicit recursion allowing its efficient computation. Finally, we compute quantum corrections to the classical Picard-Fuchs equation, and observe that holonomicity of the classical period does not generalise to $q$-holonomicity of its quantum counterpart. Applications to random walks on the Cayley graph of the Heisenberg group, and related new approaches to more efficient generative AI systems, are outlined.
Source: arXiv (Cornell University)
Institutions: Institut des Maladies Génétiques Imagine, Institut des Hautes Études Scientifiques, City, University of London, City St George's, University of London, Centre National de la Recherche Scientifique, Institute for Information Transmission Problems, Laboratoire Angevin de Recherche en Mathématiques, Université d'Angers, ITMO University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G7087920317
OpenAlex ID: W7222935914