On the spectral properties of the quantum cohomology of odd quadrics

TL;DR

The paper analyzes the spectral properties of the quantum multiplication operators in the specialized quantum cohomology $H^\bullet(\mathrm{OG})$ of the odd quadric, deriving the characteristic polynomials for $A(\tau_p)$ and a complete description of their spectra. It establishes that all $A(\tau_p)$ are simultaneously diagonalizable and provides explicit eigenvalues and eigenvectors in terms of $p$, $n$, and $d=\gcd(p,2n-1)$, linking these to the Frobenius-Perron dimensions $\mathrm{FPdim}(\tau_p)$. It also proves Galkin's lower bound conjecture for $\mathrm{OG}$ by analyzing the anticanonical class $c_1=(2n-1)\tau_1$ and the function $f(x)=4^x-x-1$. Collectively, the results yield a precise spectral picture for the quantum cohomology of the odd quadric and contribute to the understanding of Frobenius-algebra structures arising from quantum cohomology.

Abstract

Let $H^\bullet(\mbox{OG})$ be the quantum cohomology (specialized at $q=1$) of the $2n-1$ dimensional quadric $\mbox{OG}$. We will calculate the characteristic polynomial of the linear operators induced by quantum multiplication in $H^\bullet(\mbox{OG})$ and the Frobenius-Perron dimension. We also check that Galkin's lower bound conjecture holds for $\mbox{OG}$.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 2.1
• Example 2.2
• Proposition 2.3
• Proposition 3.1
• proof
• Corollary 3.2
• Theorem 3.3