Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison
Abstract
Let X be the moduli space of stable trace-free rank r Higgs bundles over a smooth projective curve C with fixed determinant Λ∈Pic(C) and gcd(r,degΛ)=1. We prove that the cc*-equivariant quantum product of a divisor on X is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in X× X. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to X under evaluation map. For Γ=Pic0(C)[r], we give a character decomposition for Γ-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For X̂=[X/Γ], which is the moduli space of stable PGLr-Higgs bundles over C. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory KU*(X) and twisted K-theory KU*([X/Γ],α) by the lifting μr-gerbe α of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.
Related
- Ricci flow of Integrable Sigma Models, Hitchin Systems and Quantum Seiberg-Witten theory
- A Fourier expansion principle for coherent cohomology of powers of Tate curves
- Bounded functor cohomology and comparison theorem
- A comparison of artificial neural networks and partial least squares modelling for the rapid detection of the microbial spoilage of beef fillets based on Fourier transform infrared spectral fingerprints
- Cohomology, Extensions and Deformations of restricted Lie triple systems
BibTeX
@misc{jiang2026quantum,
title = {{Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison}},
author = {Yunfeng Jiang and Hsian-Hua Tseng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03520},
eprint = {2610.03520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03520},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison
AU - Yunfeng Jiang
AU - Hsian-Hua Tseng
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03520
UR - https://arxiv.org/abs/2610.03520
AB - Let X be the moduli space of stable trace-free rank r Higgs bundles over a smooth projective curve C with fixed determinant Λ∈Pic(C) and gcd(r,degΛ)=1. We prove that the cc^*-equivariant quantum product of a divisor on X is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in X× X. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to X under evaluation map. For Γ=Pic^0(C)[r], we give a character decomposition for Γ-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For X̂=[X/Γ], which is the moduli space of stable PGL_r-Higgs bundles over C. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory KU^*(X) and twisted K-theory KU^*([X/Γ],α) by the lifting μ_r-gerbe α of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.
ER -