Seiberg-Witten and "Polyakov-like" magnetic bion confinements are continuously connected

TL;DR

This work establishes a continuous connection between two confinement mechanisms in mass‑perturbed Seiberg–Witten theory: monopole/dyon condensation at large circle size and a Polyakov‑like magnetic bion mechanism at small circle size. By exploiting a Poisson duality between 3d instanton towers and 4d dyon towers, the authors show that the nonperturbative dynamics governing confinement at small and large ${ m S}^1$ are two faces of a single underlying structure. The analysis spans both supersymmetric and some non‑supersymmetric gauge theories, revealing a robust mechanism for center symmetry realization and chiral symmetry breaking via topological molecules. The results yield concrete mass gaps and string tensions, clarify the role of fermionic zero modes, and provide a framework that connects abelian confinement in controlled settings to more general, lattice‑constrained gauge dynamics. Overall, the work deepens our understanding of confinement in locally four‑dimensional gauge theories and offers tools applicable to QCD‑like theories.

Abstract

We study four-dimensional N=2 supersymmetric pure-gauge (Seiberg-Witten) theory and its N=1 mass perturbation by using compactification S**1 x R**3. It is well known that on R**4 (or at large S**1) the perturbed theory realizes confinement through monopole or dyon condensation. At small S**1, we demonstrate that confinement is induced by a generalization of Polyakov's three-dimensional instanton mechanism to a locally four-dimensional theory - the magnetic bion mechanism - which also applies to a large class of nonsupersymmetric theories. Using a large- vs. small-L Poisson duality, we show that the two mechanisms of confinement, previously thought to be distinct, are in fact continuously connected.

Figures (5)

• Figure 1: Taking different paths in the $u$-$L$ plane. The horizontal direction, $u$, is the modulus of Seiberg-Witten theory and the vertical, $L$, is the size of ${\mathbb S}^1$. Ref. Seiberg:1996nz studied the softly broken ${\cal N}=2$ theory on ${\mathbb R}^3 \times {\mathbb S}^1$ by using elliptic curves through path ${A}$. In this work, we reexamine the same theory along the path $BCD$ in moduli space. The $CD$ branch always remains semi-classical and allows us to understand the relation between the topological defects responsible for confinement at small-$L$ and large-$L$ in detail.
• Figure 2: a)The spectrum of charges of a monopole and its dyonic tower obtained by quantizing the $U(1)_e$ zero mode. b) The mass spectrum. In the semi-classical regime, $\Delta E\equiv E_{(1, \pm 1)} - E_{(1,0)} \ll M$. This tower of states, labelled by electric charge, is pertinent to large-$L$ and is Poisson-dual to the 3d BPS monopole-instanton and its tower, characterized by the winding number, and pertinent to small-$L$.
• Figure 3: Monopole-instanton solutions for $n_m = +1$ are represented by a line with an arrow pointing from the vev of $a_{4,5}$ at the center of the monopole, denoted by a circle, to the vev at infinity, denoted by a square (the vev at the center vanishes and is, on both pictures, taken to be the origin of coordinates in the $a_{4}/a_5$-plane). Since $a_4$ is an angular variable on ${\mathbb R}^3 \times {\mathbb S}^1$, a tower of instanton-monopoles of "winding numbers" labelled by $n_w$ exists. The length of the arrow equals the distance between the vevs at the center of the monopole and infinity, $\sqrt{a_5^2 + a_4(n_w)^2}$, and is proportional to the action of the corresponding topological defect. The $n_m = +1$ tower, shown in the upper figure, is composed of deformations (obtained by turning on $a_5$) of BPS monopole-instantons and $\overline {\rm KK}$ twisted anti-instantons, shown in the lower figure (BPS monopoles have their arrows pointing to the right and $\overline{\rm KK}$-monopoles to the left). The $n_m = -1$ tower, not shown above, obtained by reverting all the arrows, is composed of deformations of $\overline{\rm BPS}$ anti-monopole-instantons and KK twisted monopole-instantons.
• Figure 4: 3d-instantons in the magnetic charge +1 tower in the regime $a_4 \ll a_5 \ll a_4/g_4^2$. The tower is composed of deformation of BPS monopole instantons and $\overline {\rm KK}$ twisted anti-instantons. The properties of fermionic zero modes is dictated by the leading $a_5$ dependence for the low winding number instantons.
• Figure 5: The ${\cal N}=2$ theory broken down to ${\cal N}=1$ exhibits confinement. At small $m$ and/or small $L$ (in units of $\Lambda$) the dynamics abelianizes at large distances and the theory exhibits abelian confinement. The phase diagram in the $m$-$L$ plane for the small-$N$ theory is shown on the left figure, where the shaded areas indicate the calculable regimes at small and large $L$. The third ($u$) direction---which allows to smoothly connect the topological excitations responsible for confinement at large and small $L$ via Poisson duality---is also indicated. At large-$N$, shown on the right figure, the calculable semi-classical confinement regime shrinks to a narrow sliver both in $m$ and $L$, in a correlated manner, as explained in the text.