SU($N$) QCD, four-manifolds and Segre numbers

Source abstract

We study correlation functions of topologically twisted $\mathcal N=2$ supersymmetric QCD with gauge group $\mathrm{SU}(N)$ 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 $\mathcal N=1$ vacua, which become the supersymmetric vacua when the $\mathcal N=2$ theory is broken to $\mathcal N=1$ by an adjoint mass deformation, and whose contribution we attribute to the instanton branch. We focus on asymptotically free $\mathrm{SU}(N)$ 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 $\mathcal N=1$ vacua. The calculation of these couplings requires new higher-rank techniques, involving Siegel modular forms, an exact $\mathbb Z_N$ symmetry, charge conjugation symmetry, factorisation of the Seiberg--Witten curve, and the blowup formula. For pure $\mathrm{SU}(N)$, our result agrees with earlier work by Mariño and Moore, and the twisted correlators give the $\mathrm{SU}(N)$ Donaldson invariants of $X$. For $\mathrm{SU}(3)$ QCD, we determine closed-form expressions for all 13 Coulomb branch couplings for each $N_f=1,\dots, 5$, and find precise agreement with results of Göttsche and Kool for virtual rank 3 Segre numbers.