Unstable Instantons in A-model Localization

TL;DR

This work develops a real-contour localization framework for the A-model vector multiplet on $S^{2}$, deriving integral expressions for partition functions and correlators that preserve unstable instanton contributions. By solving the localization locus and computing the one-loop determinants with monopole harmonics, the authors obtain a clean correlator formula in terms of $u$ and the GNO flux $\mathfrak{m}$: $\langle \mathcal{O} \rangle = \frac{(-1)^r}{|W|} \sum_{\mathfrak{m}} \int d u^r \mathcal{O}(u) e^{4\pi \widetilde{W}'(u)\cdot \mathfrak{m}} (-1)^{\sum_{\alpha\in\Delta_{+}} \alpha(\mathfrak{m})} \prod_{\alpha\in\Delta} \alpha(u)$. These results are then used to recover the two-dimensional Yang-Mills partition function as a sum over Weyl-averaged data, expressing the final result in terms of irreps via the Weyl dimension formula and Casimir $C_2(R)$. The approach provides an alternative to JK-residue localization and elucidates how non-abelian localization captures unstable instanton sectors, with potential extensions to chiral multiplets and higher-genus surfaces. Overall, the work bridges A-model localization with YM$_2$ physics and contributes to unifying different localization frameworks in supersymmetric gauge theories.

Abstract

We apply localization techniques to $A$-twisted $\mathcal{N}=(2,2)$ theories of vector multiplets on $S^{2}$. We derive formulae for $A$-model partition functions and correlators as integrals along a real contour, as opposed to a complex one. Using the correlator formula, we successfully recover the unstable instanton partition function of pure two-dimensional Yang-Mills theory from the vacuum expectation value of an $A$-model operator.