Monodromy conjecture for the quantum connection and Fourier–Mukai transformation

Establish that, once the Fourier–Mukai kernel for the moduli of stable trace-free rank-r Higgs bundles is extended to the full Hitchin base, the quantum connection and the Fourier–Mukai transformation have the same monodromy.

Background

The paper compares the quantum cohomology of the moduli space X of stable trace-free rank-r Higgs bundles with the twisted orbifold quantum theory of the quotient stack [X/Γ], where Γ=Pic0(C)[r]. It proves compatibility of the Fourier–Mukai transformation with quantum products by divisors only under explicitly stated kernel, coefficient-theory, and convolution hypotheses.

The authors explain that this conditional compatibility would imply the Monodromy conjecture: the quantum connection associated with the Higgs-bundle moduli space and the Fourier–Mukai transformation should have identical monodromy. The required extension of the Fourier–Mukai kernel to the full Hitchin base is not established in the paper, so the monodromy statement remains conditional.

References

Theorem \ref{thm:F_Z_intro} implies that the Fourier-Mukai transform $\text{FM}_{\mathcal{P}$ is compatible with the quantum product. Hence this implies the Monodromy conjecture saying that the quantum connection and the Fourier-Mukai transformation have the same Monodromy once the Fourier-Mukai kernel can be extended to the full Hitchin base $$, see , .

— Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison  (2610.03520 - Jiang et al., 2 Oct 2026) in Section 1, subsection “Fourier–Mukai comparison,” Remark following Theorem 1.4