Domain walls in high-T SU(N) super Yang-Mills theory and QCD(adj)

TL;DR

This work analyzes high-temperature domain walls in $4$-D $SU(N)$ SYM and QCD(adj), identifying the $k$-wall worldvolume as a $2$-D gauge theory with gauge group $SU(N-k) imes SU(k) imes U(1)$ and massless bifundamental fermions. It reveals a nontrivial interplay of anomalies: a 1-form center symmetry and a discrete 0-form chiral symmetry on the wall exhibit a mixed 't Hooft anomaly, matched by inflow from the bulk, and the wall dynamics screen fundamental charges, allowing strings to end on the wall. Through 2D bosonization and gauged WZW analysis, the discrete chiral symmetry is expected to break in the IR, with the anomaly saturated by a topological quantum field theory, and the wall hosting $N$ vacua in SYM (and a parallel structure in QCD(adj) with $n_f$ flavors). The work draws connections between high-temperature domain walls and chiral-symmetry-domain walls in other regimes, highlighting a unified anomaly-saturation mechanism across temperatures and compactifications. Overall, it advances a microscopic, field-theoretic understanding of wall/bulk interplay, screening phenomena, and IR topological phases in gauge theories with adjoint matter.

Abstract

We study the domain walls in hot $4$-D $SU(N)$ super Yang-Mills theory and QCD(adj), with $n_f$ Weyl flavors. We find that the $k$-wall worldvolume theory is $2$-D QCD with gauge group $SU(N-k)\times SU(k) \times U(1)$ and Dirac fermions charged under $U(1)$ and transforming in the bi-fundamental representation of the nonabelian factors. We show that the DW theory has a $1$-form $\mathbb Z_{N}^{(1)}$ center symmetry and a $0$-form $\mathbb Z_{2Nn_f}^{dχ}$ discrete chiral symmetry, with a mixed 't Hooft anomaly consistent with bulk/wall anomaly inflow. We argue that $\mathbb Z_{N}^{(1)}$ is broken on the wall, and hence, Wilson loops obey the perimeter law. The breaking of the worldvolume center symmetry implies that bulk $p$-strings can end on the wall, a phenomenon first discovered using string-theoretic constructions. We invoke $2$-D bosonization and gauged Wess-Zumino-Witten models to suggest that $\mathbb Z_{2Nn_f}^{dχ}$ is also broken in the IR, which implies that the $0$-form/$1$-form mixed 't Hooft anomaly in the gapped $k$-wall theory is saturated by a topological quantum field theory. We also find interesting parallels between the physics of high-temperature domain walls studied here and domain walls between chiral symmetry breaking vacua in the zero temperature phase of the theory (studied earlier in the semiclassically calculable small spatial circle regime), arising from the similar mode of saturation of the relevant 't Hooft anomalies.

Figures (1)

• Figure 1: Two DW vacua (\ref{['condensate']}) separated by a fundamental quark worldline (Euclidean). As explained in Section \ref{['Screening and strings ending on walls']}, $W$ can be viewed as the end of a confining string worldsheet extending into the ${\mathbb R}^3$ bulk. The picture holds in the high-$T$ DW on ${\mathbb R}^3\times S^1_\beta$, associated with center symmetry breaking. It also applies in the zero-$T$${\mathbb R}^3 \times {\mathbb S}^1_L$, in the semiclassically calculable $\Lambda N L \ll 1$ regime, where the DW is associated with chiral symmetry breaking. In both the small-$\beta$ and small-$L$ case, the DW worldvolume is $2$-D. In the small-$L$ case, the $N$$P$-vacua are represented by distinct semiclassical DW solutions ($N$ such solutions are known to exist for $k=1$), each carrying one-half the fundamental quark flux, see Anber:2015keaAnber:2015whaPoppitz:2017iviAnber:2017tug for details. The resemblance between the small-$\beta$ and small-$L$ cases is because the relevant 't Hooft anomalies on the DW are saturated in a similar mode. Note that on ${\mathbb R}^3\times {\mathbb S}^1_L$, confinement in the ${\mathbb R}^3$ bulk is abelian Unsal:2007jx, in contrast to the small-$\beta$ case.