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Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

Zheyan Wan, Juven Wang

TL;DR

The paper develops a cobordism-based framework to study $QCD_4$ quark matter in the $\mu$–$T$ phase diagram, incorporating higher-form symmetries and higher 't Hooft anomalies to constrain the four principal phases: QGP, ChSB, 2SC, and CFL. It introduces an approximate higher anomaly mixing a discrete axial symmetry with a 1-form color-flavor-locked center symmetry, captured by a 5d iTQFT term, and demonstrates how the four phases can realize anomaly matching through symmetry breaking. The analysis combines non-perturbative cobordism classifications (via $\Omega_d^G$, MTG, and spectral sequences) with time-reversal enrichments (Spin/Pin structures) to classify SPT/SET terms for gauged $SU(2)$ or $SU(3)$ matter theories across dimensions. The results provide a structured, topological constraint on the QCD phase diagram and offer a pathway to generalize to higher dimensions and broader gauge groups, potentially informing UV–IR dualities and beyond-Landau phase structure in strongly interacting matter.

Abstract

We explore QCD$_4$ quark matter, the $μ$-T (chemical potential-temperature) phase diagram, possible 't Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, 't Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCD$_d$-like matter theories in general in any spacetime dimensions $d=2,3,4,5$ via cobordism.

Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

TL;DR

The paper develops a cobordism-based framework to study quark matter in the – phase diagram, incorporating higher-form symmetries and higher 't Hooft anomalies to constrain the four principal phases: QGP, ChSB, 2SC, and CFL. It introduces an approximate higher anomaly mixing a discrete axial symmetry with a 1-form color-flavor-locked center symmetry, captured by a 5d iTQFT term, and demonstrates how the four phases can realize anomaly matching through symmetry breaking. The analysis combines non-perturbative cobordism classifications (via , MTG, and spectral sequences) with time-reversal enrichments (Spin/Pin structures) to classify SPT/SET terms for gauged or matter theories across dimensions. The results provide a structured, topological constraint on the QCD phase diagram and offer a pathway to generalize to higher dimensions and broader gauge groups, potentially informing UV–IR dualities and beyond-Landau phase structure in strongly interacting matter.

Abstract

We explore QCD quark matter, the -T (chemical potential-temperature) phase diagram, possible 't Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from or time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, 't Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCD-like matter theories in general in any spacetime dimensions via cobordism.

Paper Structure

This paper contains 21 sections, 75 equations, 28 figures, 10 tables.

Table of Contents

  1. Introduction and Summary
  2. Physics in QCD quark matter
  3. Approximate higher anomaly constraint on the QCD phase diagram
  4. Mathematical Primer
  5. QCD Symmetries, Anomalies and Topological Terms Without Time-Reversal
  6. Chiral symmetry breaking ${\frac{ ( [{\rm SU}(3)_V] \times {\rm SU}(3)_V \times {\rm U}(1)_V) }{ \mathbb{Z}_{3_c, V} \times \mathbb{Z}_{3_f, V} }}$ as $({\rm Spin}(d) \times_{\mathbb{Z}_2} \frac{{\rm U}(3)}{\mathbb{Z}_3})$
  7. 3-Color-Flavor locking superconductivity ${\rm SU}(3)_{C+L+R} \times \mathbb{Z}_{2,V}^F$ as ${\rm Spin}\times{\rm SU}(3)$
  8. 2-Color Superconductivity: $\frac{[{\rm SU}(2_c)_{V,rg}] \times {\rm SU}(2_f)_{L,ud} \times {\rm SU}(2_f)_{R,ud} {\times {\rm U}(1_f)_{V,s}} {\times {\rm U}(1_c)_{V,b}}}{{\mathbb{Z}_{2,V}^F}}$ as ${\rm Spin}\times_{\mathbb{Z}_2}{\rm Spin}(4)\times{\rm U}(1)\times{\rm U}(1)$
  9. Quark Gluon Plasma/Liquid $\frac{[{\rm SU}(3)_V] \times {\rm SU}(3)_L \times {\rm SU}(3)_R \times {\rm U}(1)_V}{\mathbb{Z}_{3, V} \times \mathbb{Z}_{3, V}}$ as ${\rm Spin}(d)\times_{\mathbb{Z}_2}\frac{{\rm U}(3)_L \times {\rm U}(3)_R }{(\mathbb{Z}_{3,V} \times \mathbb{Z}_{3,A} \times {\rm U}(1)_A)}$
  10. QCD Symmetries, Anomalies and Topological Terms With Time-Reversal
  11. Chiral symmetry breaking ${\frac{ ( [{\rm SU}(3)_V] \times {\rm SU}(3)_V \times {\rm U}(1)_V) \rtimes \mathbb{Z}_4^T }{ \mathbb{Z}_{3_c, V} \times \mathbb{Z}_{3_f, V} \times \mathbb{Z}_{2,V}^F}}$
  12. ${\frac{ ( [{\rm SU}(3)_V] \times {\rm SU}(3)_V \times {\rm U}(1)_V) \times \mathbb{Z}_4^{TF} }{ \mathbb{Z}_{3_c, V} \times \mathbb{Z}_{3_f, V} \times \mathbb{Z}_{2,V}^F}}$ as $(\mathrm{Pin}^+(d) \times_{\mathbb{Z}_2} \frac{{\rm U}(3)}{\mathbb{Z}_3})$ and ${\frac{ ( [{\rm SU}(3)_V] \times {\rm SU}(3)_V \times {\rm U}(1)_V) \times \mathbb{Z}_2^T }{ \mathbb{Z}_{3_c, V} \times \mathbb{Z}_{3_f, V} }}$ as $(\mathrm{Pin}^-(d) \times_{\mathbb{Z}_2} \frac{{\rm U}(3)}{\mathbb{Z}_3})$
  13. 3-Color-Flavor locking superconductivity ${\rm SU}(3)_{C+L+R} \rtimes \mathbb{Z}_{4}^{T}$
  14. ${\rm SU}(3)_{C+L+R} \times \mathbb{Z}_{4}^{TF}$ as $\mathrm{Pin}^+\times{\rm SU}(3)$
  15. ${\rm SU}(3)_{C+L+R} \times \mathbb{Z}_{2,V}^F \times \mathbb{Z}_{2}^T$ as $\mathrm{Pin}^- \times{\rm SU}(3)$
  16. ...and 6 more sections

Figures (28)

  • Figure 1: We revisit the QCD$_4$ matter phase in the $\mu$-T (chemical potential-temperature) phase diagram: The high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We do not attempt to address the nature of phase transitions in this figure, thus we make some of the phase boundaries blur. The $\cancel{\mathbb{Z}_{6,A}}$ or $\cancel{\mathbb{Z}_{4,A}}$ means that part of the discrete axial symmetry $(A)$ is broken: $\cancel{\mathbb{Z}_{2N_f,A}}$. In general, if the $G_{\text{sym}}$ group is broken, we denote it as $\cancel{G_{\text{sym}}}$.
  • Figure 2: Follow Fig. \ref{['fig:qcd-phase-1']}, but now we include possible time-reversal symmetries, which can be any reasonable $\mathbb{Z}_2$-reflection symmetry by putting the Euclidean QCD$_4$ path integral on an unorientable spacetime. Here we choose a semi-direct product of $\rtimes \mathbb{Z}_4^T$, where $\mathbb{Z}_4^T \supset \mathbb{Z}_2^F$. The semi-direct product is more general, which also includes the direct product case of $\times \mathbb{Z}_4^{CT}$ as the $CT$ symmetry.
  • Figure 3: Follow Fig. \ref{['fig:qcd-phase-1']} and Fig. \ref{['fig:qcd-phase-T']}, QCD$_4$ matter phases shown here include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. The four phases can be matched and cancelled by our approximate anomaly (\ref{['eq:approximateanomalyBcf']}), ${\mathbf{Z}} \xrightarrow[]{\mathbb{Z}_{2N_f,A} \text{transformation}} {\mathbf{Z}} \cdot {\exp( \frac{ { -\space\mathrm{i}\space N }}{2 \pi } \int\limits_{M_4} ( B_{cf}^{(2)} \wedge B_{cf}^{(2)} ))}$. The four phases can also be matched and cancelled by the anomaly (\ref{['eq:approximateanomalyBcBf3d']}), ${\mathbf{Z}} \xrightarrow[]{\mathbb{Z}_{2N_f,A} \text{transformation}} {\mathbf{Z}} \cdot {\exp( \frac{ { -\space\mathrm{i}\space N }}{2 \pi } \int\limits_{M_4} ( B_{c}^{(1)} \wedge B_{f}^{(2)}))}$ introduced in Ref. 1706.06104Yonekura. We use the triple data $(\mathbb{Z}_{2N_f,A}; \mathbb{Z}_{N_{c\text{-shift}}}^{(0)}, {\mathbb{Z}^{{(1)}}_{N_{cf}}})$ to label the discrete 0-form axial symmetry $\mathbb{Z}_{2N_f,A}$, a dimensionally reduced color-shift $\mathbb{Z}_{N_{c\text{-shift}}}^{(0)}$ symmetry introduced in Ref. 1706.06104Yonekura , and the 1-form mixed chiral-flavor-locked center symmetry ${\mathbb{Z}^{{(1)}}_{N_{cf}}}$ that we introduce.
  • Figure 4: Serre spectral sequence for the fibration $\mathrm{B}{\rm U}(3)\to\mathrm{B}(\frac{{\rm U}(3)}{\mathbb{Z}_3})\to\mathrm{B}^2\mathbb{Z}_3$. The arrow from (0,4) to (5,0) is a nontrivial differential by comparison with the Serre spectral sequence of the fibration $\mathrm{B}{\rm SU}(3)\to\mathrm{B}{\rm PSU}(3)\to\mathrm{B}^2\mathbb{Z}_3$. There is no differential from (0,2) to (3,0) since the 3d $\mathbb{Z}_3$ survives the spectral sequence in Figure \ref{['fig:SSS-1']}.
  • Figure 5: Serre spectral sequence for the fibration $\mathrm{B}{\rm PSU}(3)\to\mathrm{B}(\frac{{\rm U}(3)}{\mathbb{Z}_3})\to\mathrm{B}{\rm U}(1)$.
  • ...and 23 more figures