SU(N) QCD, four-manifolds and Segre numbers
Abstract
We study correlation functions of topologically twisted 𝒩=2 supersymmetric QCD with gauge group SU(N) on a compact four-manifold X. We derive the Coulomb branch integral for all N, including new couplings to background fluxes for the flavor symmetry. For X of simple type with b2+>1, the correlators become sums over a finite number of strong-coupling singularities on the Coulomb branch, and can be expressed using a number of Coulomb branch couplings. The sum contains the 𝒩=1 vacua, which become the supersymmetric vacua when the 𝒩=2 theory is broken to 𝒩=1 by an adjoint mass deformation, and whose contribution we attribute to the instanton branch. We focus on asymptotically free SU(N) QCD with Nf<2N equal mass hypermultiplets, and conjecture that its instanton contribution to topological correlators agrees with the generating function of virtual Segre numbers of rank N, which are intersection numbers on the moduli space of semistable sheaves on an algebraic surface. We present a precise dictionary for all N between universal series for Segre numbers and Coulomb branch couplings at the 𝒩=1 vacua. The calculation of these couplings requires new higher-rank techniques, involving Siegel modular forms, an exact ℤN symmetry, charge conjugation symmetry, factorisation of the Seiberg--Witten curve, and the blowup formula. For pure SU(N), our result agrees with earlier work by Mariño and Moore, and the twisted correlators give the SU(N) Donaldson invariants of X. For SU(3) QCD, we determine closed-form expressions for all 13 Coulomb branch couplings for each Nf=1,…, 5, and find precise agreement with results of Göttsche and Kool for virtual rank 3 Segre numbers.
Related
- Finite holomorphic pullbacks, Segre numbers, and generic Lê numbers
- Linking numbers and special differential characters in arithmetic hyperbolic manifolds
- Symplectic four-manifolds and hyperbolic π-orbifolds
- Simple homotopy types of smooth four-manifolds with finite fundamental group
- Mapping class groups of negatively curved four-manifolds
BibTeX
@misc{furrer2026fourmanifolds,
title = {{SU($N$) QCD, four-manifolds and Segre numbers}},
author = {Elias Furrer and Ranveer Kumar Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03874},
eprint = {2610.03874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03874},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - SU(N) QCD, four-manifolds and Segre numbers
AU - Elias Furrer
AU - Ranveer Kumar Singh
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03874
UR - https://arxiv.org/abs/2610.03874
AB - We study correlation functions of topologically twisted 𝒩=2 supersymmetric QCD with gauge group SU(N) on a compact four-manifold X. We derive the Coulomb branch integral for all N, including new couplings to background fluxes for the flavor symmetry. For X of simple type with b_2^+>1, the correlators become sums over a finite number of strong-coupling singularities on the Coulomb branch, and can be expressed using a number of Coulomb branch couplings. The sum contains the 𝒩=1 vacua, which become the supersymmetric vacua when the 𝒩=2 theory is broken to 𝒩=1 by an adjoint mass deformation, and whose contribution we attribute to the instanton branch. We focus on asymptotically free SU(N) QCD with N_f<2N equal mass hypermultiplets, and conjecture that its instanton contribution to topological correlators agrees with the generating function of virtual Segre numbers of rank N, which are intersection numbers on the moduli space of semistable sheaves on an algebraic surface. We present a precise dictionary for all N between universal series for Segre numbers and Coulomb branch couplings at the 𝒩=1 vacua. The calculation of these couplings requires new higher-rank techniques, involving Siegel modular forms, an exact ℤ_N symmetry, charge conjugation symmetry, factorisation of the Seiberg--Witten curve, and the blowup formula. For pure SU(N), our result agrees with earlier work by Mariño and Moore, and the twisted correlators give the SU(N) Donaldson invariants of X. For SU(3) QCD, we determine closed-form expressions for all 13 Coulomb branch couplings for each N_f=1,…, 5, and find precise agreement with results of Göttsche and Kool for virtual rank 3 Segre numbers.
ER -