Fiberwise Gromov-Witten theory, quantum spectra of flag bundles, and prime factorization of integers
Giordano Cotti
TL;DR
The paper develops a fiberwise Gromov–Witten framework for flag bundles, defining vertical quantum cohomology and the vertical quantum spectrum and proving functorial and inductive properties that connect vertical and absolute theories. It establishes precise spectral degeneracy criteria for Grassmann and flag bundles in terms of prime factorization of ranks, linking geometry to arithmetic via smallest prime divisors $p_1(n)$. It then introduces three double sequences lcyr, tlcyr, and ell, analyzes their ordinary and Dirichlet generating functions, and uncovers deep connections to primes, zeta zeros, and Goldbach type problems through Pascal-type recursions, graph-walk models, and spectra in semiclassical limits. The results reveal a rich interplay between enumerative geometry, quantum spectral degeneracies, and classical prime number theory, including density, continuation, and potential reformulations of major conjectures. Overall, the work forges new bridges between algebraic geometry and analytic number theory with concrete combinatorial and spectral consequences for flag varieties and their fibrations.
Abstract
This work investigates the vertical quantum cohomology and quantum spectra of flag bundles, uncovering new links between the Gromov-Witten theory of homogeneous fibrations and analytic number theory. Building on previous constructions by Astashkevich and Sadov (arXiv:hep-th/9401103) and by Biswas, Das, Oh, and Paul (arXiv:2408.06616), we establish functorial and inductive properties of vertical quantum cohomology, and relate the vertical and absolute quantum spectra. We show that the degeneracy of the small vertical quantum spectrum of a Grassmann bundle - namely, the appearance of eigenvalues with unexpectedly high multiplicities - is controlled by the prime factorization of the involved ranks. This extends earlier results for Grassmannians to the relative setting and applies, in particular, to partial flag varieties viewed as total spaces of Grassmann bundles. We then introduce three families of double sequences that classify partial flag varieties according to distinct quantum spectral and combinatorial features. Their recursive and arithmetic behavior is studied through ordinary and Dirichlet generating functions. One of these sequences satisfies a Pascal-type recursion, allowing a precise analysis of its partial Dirichlet series, whose analytic continuation exhibits logarithmic singularities determined by the nontrivial zeros of the Riemann zeta function. Moreover, for any fixed integer shift, the diagonal subsequences display eventual polynomial behavior, which admits a natural interpretation in terms of weighted walks on graphs. Finally, we examine the vanishing pattern of one of these sequences and derive equivalent formulations of Goldbach's conjecture. Overall, the paper reveals a rich correspondence between enumerative geometry, quantum spectral degeneracy, and classical problems in prime number theory.