Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory

TL;DR

<3-5 sentence high-level summary> This work analyzes 4d SU(2)$_{\theta=\pi}$ Yang-Mills as a boundary of 5d gapped topological states, uncovering new higher 't Hooft anomalies captured by 5d bordism invariants and requiring a 5d bulk to realize symmetry protection. It classifies four UV-complete 4d theories, labeled by $(K_1,K_2)\in(\mathbb{Z}_2,\mathbb{Z}_2)$, and shows that dynamically gauging the 1-form center symmetry yields 4d SO(3)$_{\theta=\pi}$ YM coupled to 5d higher-SETs with emergent 2-form gauge fields. The paper introduces novel 5d link invariants for extended objects, derives lattice regularizations for 4d/5d SPTs and SETs, and discusses implications for infrared dynamics via anomaly matching and higher symmetry-extension methods. These results connect non-supersymmetric 4d YM physics with higher-categorical topological phases, offering a framework to constrain dynamics and bridge to lattice spin liquids and topological link invariants in five dimensions.

Abstract

We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at $θ=π$ (SU(2)$_{θ=π}$ YM), by turning on background fields for both the time-reversal (i.e., on unorientable manifolds) and 1-form center global symmetry. We find four classes of time-reversal and Lorentz symmetry-enriched SU(2)$_{θ=π}$ YM, labeled by $(K_1, K_2)$: $K_1=0,1$ specifies Kramers singlet/doublet Wilson line and new mixed higher 't Hooft anomalies; $K_2=0,1$ specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-like counterterm. Higher anomalies indicate that to realize all higher $n$-global symmetries locally on $n$-simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (TQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter). By dynamically gauging the 1-form symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)$_{θ=π}$ YM-5d LRE-higher-SETs coupled system with higher-form gauge fields. We further derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, physically characterizing the 5d SETs. We discover triple and quadruple link invariants associated with the 5d higher-gauge TQFTs, hinting at a relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide 4d-5d lattice simplicial complex regularizations and bridge to 4d quantum spin liquids. We constrain gauge dynamics by higher anomalies and a higher symmetry-extension method.

Figures (15)

• Figure 1: (a) Schematic illustration of 4d-5d coupled system: 4d SU(2)$_{\theta=\pi}$ YM and 5d SRE higher-SPTs coupled systems. There are Four Siblings of such systems with bosonic UV completion, summarized in Table \ref{['table:TQFTlink']}. We use $x,y,z$ to label the spatial coordinates of 4d (3+1D) YM, and we introduce an extra coordinate $w$ to label the additional dimension of 5d higher-SPTs. (b) Schematic illustration of 4d-5d coupled system: 4d SO(3)$_{\theta=\pi}$ YM-5d LRE higher-SETs coupled systems via gauging 1-form $\mathbb{Z}_{2,[1]}^e$ center symmetry in Fig. \ref{['Fig-SU(2)-SPT']} (a). There are Four Siblings of such 5d SET systems with bosonic UV completion, summarized in Table \ref{['table:TQFTlink']}. We use $x,y,z$ to label the spatial coordinates of 4d (3+1D) YM, and we introduce an extra coordinate $w$ to label the additional dimension of 5d higher-SETs. See also Fig. \ref{['Fig:curve-SU(2)-SO(3)']}.
• Figure 2: $S^5=\partial D^6=\partial(D^4\times D^2)=S^3\times D^2\cup D^4\times S^1=S^3\times D^2\cup D^2\times D^2\times S^1$, the intersection of the two copies of $D^2\times S^1$ in the second piece ($D^2\times 0_{\text{pt}}\times S^1$ and $0_{\text{pt}}\times D^2\times S^1$) is $0_{\text{pt}}\times 0_{\text{pt}}\times S^1=0_{\text{pt}}\times S^1$, this $0_{\text{pt}}\times S^1$ and $S^3\times0_{\text{pt}}$ in the first piece are linked. In this figure, $\Sigma^3_X=S^3\times0_{\text{pt}}$, $\Sigma^2_{U_{\bf (i)}}=\partial(D^2\times 0_{\text{pt}}\times S^1)$, $\Sigma^2_{U_{\bf (ii)}}=\partial(0_{\text{pt}}\times D^2\times S^1)$.
• Figure 3: Following the last Fig. \ref{['fig:link-inv-wPB-1']}, $V^4_X=D^4\times0_{\rm pt}$ which bounds $\Sigma^3_X$, $V^3_{U_{\bf (i)}}=D^2\times 0_{\text{pt}}\times S^1$ which bounds $\Sigma^2_{U_{\bf (i)}}$, $V^3_{U_{\bf (ii)}}=0_{\text{pt}}\times D^2\times S^1$ which bounds $\Sigma^2_{U_{\bf (ii)}}$. The intersection of $V^3_{U_{\bf (i)}}$ and $V^3_{U_{\bf (ii)}}$ is $0_{\text{pt}}\times S^1$, the intersection of $V^4_X$ and this $0_{\text{pt}}\times S^1$ is a point which is the point in black in this figure.
• Figure 4: $S^5=\partial D^6=\partial(D^3\times D^3)=S^2\times D^3\cup D^3\times S^2$. Put a 2-torus (denoted by (1)) in $D^3\times 0_{\text{pt}}$, and put a Hopf link (the two circles are denoted by (2) and (3) respectively) in the solid 2-torus. Put two circles (denoted by $S^1_{(1)}$ and $S^1_{(3)}$ respectively) which intersect in only one point in $0_{\text{pt}}\times S^2$ (denoted by $S^2_{(2)}$). In this figure, $\Sigma^3_{X_{\bf (i)}}$ is the cartesian product of the 2-torus (1) and $S^1_{(1)}$, $\Sigma^3_{X_{\bf (ii)}}$ is the cartesian product of the circle (2) and $S^2_{(2)}$, $\Sigma^2_U$ is the cartesian product of the circle (3) and $S^1_{(3)}$.
• Figure 5: Following the last Fig. \ref{['fig:link-inv-3']}, if we fill in $\Sigma^3_{X_{\bf (i)}}$ and $\Sigma^3_{X_{\bf (ii)}}$, we get $V^4_{X_{\bf (i)}}=D^2\times S^1\times S^1$ and $V^4_{X_{\bf (ii)}}=D^2\times S^2$, $V^4_{X_{\bf (i)}}$, $V^4_{X_{\bf (ii)}}$ and $\Sigma^2_U$ will intersect in only one point which is the point in black in this figure.