Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

Abstract

The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.

26 pages, 1 figure; corrected typos, added references

References in corpus (6)

• Local Moduli of Semisimple Frobenius Coalescent Structures
• Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues
• Helix Structures in Quantum Cohomology of Fano Varieties
• Quantum cohomology and the Satake isomorphism
• Semisimple quantum cohomology of some Fano varieties
• On almost duality for Frobenius manifolds

Cited by in corpus (6)

• Local Moduli of Semisimple Frobenius Coalescent Structures
• Helix Structures in Quantum Cohomology of Fano Varieties
• Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities
• Notes on Non-Generic Isomonodromy Deformations
• Quantum differential equations and helices
• A quantum number theory