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New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and $\mathbb{CP}^{\mathrm{N}-1}$ Sigma Model

Zheyan Wan, Juven Wang, Yunqin Zheng

TL;DR

The paper develops a cobordism-based framework to classify and quantify higher 't Hooft anomalies for 4d SU(N) Yang–Mills theories at θ=π and 2d CP^{N−1} models at θ=π. By mapping 5d Yang–Mills invariants to 3d CP^{N−1} invariants through twisted T^2 reductions, it identifies new higher anomalies for N=2 and N=4 and clarifies their physical meaning via higher SPTs and symmetry extensions. It also analyzes how these anomalies constrain low-energy dynamics, including symmetry-extended gapped phases and potential spontaneous symmetry breaking, and discusses the implications for a higher-symmetry analogue of Lieb–Schultz–Mattis theorems. The work thus strengthens the bridge between high-energy QFT, topology, and condensed-matter motivated SPT/topological terms, with potential implications for lattice realizations and IR behavior. Overall, it provides a structured, cobordism-driven account of complete (or near-complete) anomaly matching across 4d YM and 2d CP^{N−1} theories for key values of N and elucidates the role of higher-form symmetries in constraining quantum dynamics.

Abstract

We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic verification. First, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric pure SU(N)-Yang-Mills (YM) gauge theory with a second-Chern-class topological term at $θ=π$, via 5d cobordism invariants (higher symmetry-protected topological states), with N = $2, 3, 4$ and others. Second, we propose a set of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models with a first-Chern-class topological term at $θ=π$, by enlisting all possible 3d cobordism invariants and selecting the matched terms. Based on algebraic/geometric topology, QFT analysis, manifold generator correspondence, condensed matter inputs such as stacking PSU(N)-generalized Haldane quantum spin chains, and additional physics criteria, we derive a correspondence between 5d and 3d new invariants. Thus we broadly prove a potentially complete anomaly-matching between 4d SU(N) YM and 2d $\mathbb{CP}^{\mathrm{N}-1}$ models at N = 2, and suggest new (but maybe incomplete) anomalies at N = 4. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis theorem" to constrain the low-energy dynamics.

New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and $\mathbb{CP}^{\mathrm{N}-1}$ Sigma Model

TL;DR

The paper develops a cobordism-based framework to classify and quantify higher 't Hooft anomalies for 4d SU(N) Yang–Mills theories at θ=π and 2d CP^{N−1} models at θ=π. By mapping 5d Yang–Mills invariants to 3d CP^{N−1} invariants through twisted T^2 reductions, it identifies new higher anomalies for N=2 and N=4 and clarifies their physical meaning via higher SPTs and symmetry extensions. It also analyzes how these anomalies constrain low-energy dynamics, including symmetry-extended gapped phases and potential spontaneous symmetry breaking, and discusses the implications for a higher-symmetry analogue of Lieb–Schultz–Mattis theorems. The work thus strengthens the bridge between high-energy QFT, topology, and condensed-matter motivated SPT/topological terms, with potential implications for lattice realizations and IR behavior. Overall, it provides a structured, cobordism-driven account of complete (or near-complete) anomaly matching across 4d YM and 2d CP^{N−1} theories for key values of N and elucidates the role of higher-form symmetries in constraining quantum dynamics.

Abstract

We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic verification. First, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric pure SU(N)-Yang-Mills (YM) gauge theory with a second-Chern-class topological term at , via 5d cobordism invariants (higher symmetry-protected topological states), with N = and others. Second, we propose a set of 't Hooft anomalies of 2d -sigma models with a first-Chern-class topological term at , by enlisting all possible 3d cobordism invariants and selecting the matched terms. Based on algebraic/geometric topology, QFT analysis, manifold generator correspondence, condensed matter inputs such as stacking PSU(N)-generalized Haldane quantum spin chains, and additional physics criteria, we derive a correspondence between 5d and 3d new invariants. Thus we broadly prove a potentially complete anomaly-matching between 4d SU(N) YM and 2d models at N = 2, and suggest new (but maybe incomplete) anomalies at N = 4. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis theorem" to constrain the low-energy dynamics.

Paper Structure

This paper contains 60 sections, 4 theorems, 216 equations, 6 figures.

Table of Contents

  1. Introduction and Summary
  2. Comments on QFTs: Global Symmetries and Topological Invariants
  3. 4d Yang-Mills Gauge Theory
  4. SU(${\rm N}$)-YM theory: Mixed higher-anomalies
  5. Global symmetry and preliminary
  6. YM theory coupled to background fields
  7. $\theta$ periodicity and the vacua-shifting of higher SPTs
  8. Time reversal ${\cal T}$ transformation
  9. Mixed time-reversal and 1-form-symmetry anomaly
  10. Charge conjugation ${\cal C}$, parity ${\cal P}$, reflection ${\cal R}$, ${\cal C} {\cal T}$, ${\cal C} {\cal P}$ transformations, $\mathbb{Z}_2^{CT} \times (\mathbb{Z}_{{\rm N},[1]}^e \rtimes \mathbb{Z}_2^C )$ and $\mathbb{Z}_2^{CT} \times \mathbb{Z}_{2,[1]}^e$ -symmetry, and their higher mixed anomalies
  11. 2d $\mathbb{CP}^{{\rm N}-1}$-sigma model
  12. Related Models
  13. Global symmetry: $\mathbb{Z}_2^{CT} \times {\rm PSU}(2) \times \mathbb{Z}_2^{C}$ and $\mathbb{Z}_2^{CT} \times ({\rm PSU}({\rm N}) \rtimes \mathbb{Z}_2^{C'})$
  14. Cobordisms, Topological Terms, and Manifold Generators: Classification of All Possible Higher 't Hooft Anomalies
  15. Mathematical preliminary and co/bordism groups
  16. ...and 45 more sections

Key Result

Lemma 6.1

The map $F$ is defined as sending $(N^3,E)\in \Omega_3^{{\rm O}}(\mathrm{B}{\rm O}(3))$ to $[(M^5,B)]\in \Omega_5^{{\rm O}}(\mathrm{B}^2\mathbb{Z}_2)/\Omega_5^{{\rm SO}}(\mathrm{B}^2\mathbb{Z}_2)$. $F$ is well defined.

Figures (6)

  • Figure 1: An interpretation of 't Hooft anomaly ${w_1(E)({w_2(V_{\mathrm{SO}(3)})}+{w_1(TM)^2})}$ (eq. (\ref{['eq:anom-stacking-Haldane']})) for 2d $\mathbb{CP}^1$ or O(3) NLSM model, is obtained from the 2d dangling spin-1/2 gapless modes living on the 2d boundary a 3d layer-stacking system of the 2d spin-1 Haldane chain. Each vertical solid line represents a 2d spin-1 Haldane chain eq. (\ref{['eq:Haldane']}). The 2d boundary combined from the dangling spin-1/2 gapless modes, encircled by the dashed-line rectangle, on the top and on the bottom, are effectively the 2d $\mathbb{CP}^1$ model. The ${w_1(E){w_2(V_{\mathrm{SO}(3)})}}$ has been identified by Metlitski2017fmd1707.07686. The same trick of Metlitski2017fmd1707.07686 applies to eq. (\ref{['eq:Haldane']}) teaches us the more complete anomaly eq. (\ref{['eq:anom-stacking-Haldane']}). The $\mathbb{Z}_2^x \equiv \mathbb{Z}_2^C$-translational nature of Heisenberg anti-ferromagnet (AFM) is purposefully emphasized by the spin-1/2 orientation ($\uparrow$ or $\downarrow$) and the two different sizes of the black dots ($\bullet$). Notice that understand the 3d bulk nature obtained from this stacking, inform us that the 3d bulk SPTs also includes a secretly hidden $A_x \cup A_x \cup A_x \equiv A_x^3$ or the ${w_1(E)^3 \equiv w_1(\mathbb{Z}_2^x)^3}$-anomaly WangSantosWen1403.5256Metlitski2017fmd1707.07686.
  • Figure 2: Follow the setup of the twisted boundary condition induced 't Hooft boundary condition tHooft1979rtgEM along the 2-torus $T^2_{zx} \equiv S^1_z \times S^1_x$, and the twisted compactification Yamazaki:2017ulcYamazaki:2017dra, we examine that the higher anomaly of 4d SU(N) YM theory at $\theta=\pi$ induces the anomaly of 2d $\mathbb{CP}^{\mathrm{N}-1}$ model at $\theta=\pi$. The 4d YM on ${S^1_x \times S^1_y \times S^1_z \times \mathbb{R}}$ is compactified along the small size of $T^2_{yz} \equiv S^1_y \times S^1_z$, whose moduli space of flat connections becomes the target space $\mathbb{CP}^{\mathrm{N}-1}$Looijenga1976FriedmanWitten1997yq9701162, while the remained $S^1_x \times \mathbb{R}$ becomes the 2d spacetime of 2d $\mathbb{CP}^{\mathrm{N}-1}$ model. Our goal, in Sec. \ref{['sec:rule']}, \ref{['sec:newSUn']} and \ref{['sec:newCPn']} is to identify the underlying 't Hooft anomalies of 4d SU(N) YM and 2d $\mathbb{CP}^{\mathrm{N}-1}$ model, namely identifying the theories living on the boundary ($\equiv$ bdry) of 5d and 3d (higher) SPTs when all the (higher) global symmetries needed to be regularized strictly onsite and local (e.g. [Wen2013oza1303.18031405.7689Wang2017locWWW1705.06728]). (Higher global symmetries can be regularized, instead on the 0-simplex, on the higher-simplices: 1-simplex, 2-simplex, etc.) The twisted boundary condition of 4d YM for 1-form $\mathbb{Z}_\mathrm{N}$-center symmetry (as a higher symmetry twist of 1405.7689) can be dimensionally reduced to the 0-form $\mathbb{Z}_\mathrm{N}$-flavor symmetry twisted Dunne2012aeUnsal1210.2423 in the 2d $\mathbb{CP}^{\mathrm{N}-1}$ model. In fact, the twisted boundary conditions can be designed numerically, and the twisted boundary conditions can result in a fractional instanton number in a gauge theory, see, for example, recent numerical attempts and Reference therein Itou2018wkm1811.05708.
  • Figure 3: A main result of our work obtains the higher 't Hooft anomalies of 4d SU(N)$_{\theta=\pi}$ YM and 't Hooft anomalies of 2d $\mathbb{CP}^{{\mathrm{N}}-1}_{\theta=\pi}$ at ${\mathrm{N}}=2$. We express the 4d and 2d anomalies in terms of 5d and 3d SPT/cobordism invariants respectively. We related the 4d and 2d anomalies by compactifying a 2-torus with twisted boundary conditions. Some useful formulas are proved in Sec. \ref{['sec:newSUn']}, \ref{['sec:newCPn']} and Ref. [W2] --- Note that on a closed 5-manifold, we rewrite: $B_2{\mathrm{Sq}}^1B_2+{\mathrm{Sq}}^2{\mathrm{Sq}}^1B_2=\frac{1}{4} \delta(\mathcal{P}_2(B_2))=\beta_{(2,4)}\mathcal{P}_2(B_2)= \frac{1}{2} \tilde{w}_1(TM) \mathcal{P}_2(B_2)$, $\mathrm{Sq}^2\mathrm{Sq}^1B_2 = (w_2(TM) +w_1(TM)^2) \mathrm{Sq}^1 B_2 =(w_3(TM) +w_1(TM)^3) B_2$, and $B_2{\mathrm{Sq}}^1B_2=(\frac{1}{2} \tilde{w}_1(TM) \mathcal{P}_2(B_2)-(w_3(TM) +w_1(TM)^3) B_2)$.
  • Figure 4: A main result of our work obtains the higher 't Hooft anomalies of 4d SU(N)$_{\theta=\pi}$ YM and 't Hooft anomalies of 2d $\mathbb{CP}^{{\rm N}-1}_{\theta=\pi}$ at ${\rm N}=4$. We express the 4d and 2d anomalies in terms of 5d and 3d SPT/cobordism invariants respectively. We related the 4d and 2d anomalies by compactifying a 2-torus with twisted boundary conditions. Some useful formulas are proved in Sec. \ref{['sec:newSUn']}, \ref{['sec:newCPn']} and Ref. [W2] --- Note that on a closed 5-manifold, we rewrite: $B_2\beta_{(2,4)}B_2 =\frac{1}{4}\tilde{w}_1(TM)\mathcal{P}_2(B_2)$.
  • Figure 5: One $\Delta$-complex structure of Klein bottle
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • Theorem 6.3
  • proof
  • Theorem 6.4
  • proof