The BPS Spectra and Superconformal Points in Massive N=2 Supersymmetric QCD

TL;DR

This work analyzes the BPS spectra and superconformal points of massive ${N}=2$ ${SU}(2)$ gauge theories with ${N_f}=1,2,3$ flavors by leveraging Seiberg–Witten curves expressed as elliptic curves. The authors compute period integrals $a(u)$ and $a_D(u)$ via Weierstrass uniformization and elliptic integrals, enabling precise determination of curves of marginal stability and the existence domains of BPS states. A key result is the detailed RG-flow picture: as quark masses deform the theory, the spectrum reorganizes smoothly, and at specific superconformal points (classified by an integer $k=1,2,3$) mutually non-local states become massless, with monodromies realizing exact dualities that constrain the SCFT data. The paper also establishes a rational relation between the beta-function slope at the fixed point and the anomalous dimension of ${\langle\,\mathrm{tr}\,\phi^2\rangle}$ via ${\omega = 2(\alpha-1)}$, and provides a robust framework for predicting the SCFT content from the maximal BPS spectrum. Overall, the work gives a coherent, calculable picture of stable BPS spectra across RG flows and clarifies the nature of 4D ${N}=2$ SCFTs emerging in massive Seiberg–Witten theories, with implications for exact dualities and scaling dimensions in these theories.

Abstract

We present a detailed study of the analytic structure, BPS spectra and superconformal points of the $ N=2 $ susy $ SU(2) $ gauge theories with $ N_f=1,2,3 $ massive quark hypermultiplets. We compute the curves of marginal stability with the help of the explicit solutions for the low energy effective actions in terms of standard elliptic functions. We show that only a few of these curves are relevant. As a generic example, the case of $ N_f=2 $ with two equal bare masses is studied in depth. We determine the precise existence domains for each BPS state, and show how they are compatible with the RG flows. At the superconformal point, where two singularities coincide, we prove that (for $ N_f=2 $) the massless spectrum consists of four distinct BPS states and is S-invariant. This is due to the monodromy around the superconformal point being S, providing strong evidence for exact S-duality of the SCFT. For all $ N_f $, we compute the slopes $ ω$ of the $ β$-functions at the fixed point couplings and show that they are related to the anomalous dimensions $ α$ of $ u= < tr φ^2 > $ by $ ω= 2 (α-1) $.

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