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Infrared phases of 2d QCD

Diego Delmastro, Jaume Gomis, Matthew Yu

TL;DR

This work delivers a complete classification of massless-fermion 2d QCD theories into gapped (IR TQFT) and gapless (IR CFT) phases using both lightcone and temporal Hamiltonians. The authors derive a concrete gap criterion: a theory is gapped iff both decoupled left and right chiral cosets satisfy $T_{ ext{so}( ext{dim }R)_1}-T_{ rak g_{I(R)}}=0$, which is equivalent to a conformal-embedding/conformal-symmetry condition. The gapped theories are enumerated via symmetric-space and conformal-embedding data, with abelian-factor couplings via non-singular charge matrices enabling tensor-product constructions; the gapless cases flow to coset CFTs (often WZW) that encode 't Hooft and gravitational anomalies. The infrared dynamics of many concrete theories are worked out, including adjoint QCD, bifundamental and rank-2 quivers, and fundamental-matter cases, connecting to WZW models, minimal models, diagonal cosets, and Kazama-Suzuki theories. The results provide a unifying IR picture for 2d QCD and reveal rich connections to known CFTs and topological phases, illustrating how anomalies and one-form symmetries shape the low-energy landscape.

Abstract

We derive the necessary and sufficient conditions for a 2d QCD theory of massless gluons and left and right chiral quarks in arbitrary representations of a gauge group G to develop a mass gap. These results are obtained from spectral properties of the lightcone and temporal QCD Hamiltonians. The conditions can be explicitly solved, and we provide the complete list of all 2d QCD theories that have a quantum mechanical gap in the spectrum, while any other theory not in the list is gapless. The list of gapped theories includes QCD models with quarks in vector-like as well as chiral representations. The gapped theories consist of several infinite families of classical gauge groups with quarks in rank 1 and 2 representations, plus a finite number of isolated cases. We also put forward and analyze the effective infrared description of QCD - TQFTs for gapped theories and CFTs for gapless theories - and exhibit several interesting features in the infrared, such as the existence of non-trivial global 't Hooft anomalies and emergent supersymmetry. We identify 2d QCD theories that flow in the infrared to celebrated CFTs such as minimal models, bosonic and supersymmetric, and Wess-Zumino-Witten and Kazama-Suzuki models.

Infrared phases of 2d QCD

TL;DR

This work delivers a complete classification of massless-fermion 2d QCD theories into gapped (IR TQFT) and gapless (IR CFT) phases using both lightcone and temporal Hamiltonians. The authors derive a concrete gap criterion: a theory is gapped iff both decoupled left and right chiral cosets satisfy , which is equivalent to a conformal-embedding/conformal-symmetry condition. The gapped theories are enumerated via symmetric-space and conformal-embedding data, with abelian-factor couplings via non-singular charge matrices enabling tensor-product constructions; the gapless cases flow to coset CFTs (often WZW) that encode 't Hooft and gravitational anomalies. The infrared dynamics of many concrete theories are worked out, including adjoint QCD, bifundamental and rank-2 quivers, and fundamental-matter cases, connecting to WZW models, minimal models, diagonal cosets, and Kazama-Suzuki theories. The results provide a unifying IR picture for 2d QCD and reveal rich connections to known CFTs and topological phases, illustrating how anomalies and one-form symmetries shape the low-energy landscape.

Abstract

We derive the necessary and sufficient conditions for a 2d QCD theory of massless gluons and left and right chiral quarks in arbitrary representations of a gauge group G to develop a mass gap. These results are obtained from spectral properties of the lightcone and temporal QCD Hamiltonians. The conditions can be explicitly solved, and we provide the complete list of all 2d QCD theories that have a quantum mechanical gap in the spectrum, while any other theory not in the list is gapless. The list of gapped theories includes QCD models with quarks in vector-like as well as chiral representations. The gapped theories consist of several infinite families of classical gauge groups with quarks in rank 1 and 2 representations, plus a finite number of isolated cases. We also put forward and analyze the effective infrared description of QCD - TQFTs for gapped theories and CFTs for gapless theories - and exhibit several interesting features in the infrared, such as the existence of non-trivial global 't Hooft anomalies and emergent supersymmetry. We identify 2d QCD theories that flow in the infrared to celebrated CFTs such as minimal models, bosonic and supersymmetric, and Wess-Zumino-Witten and Kazama-Suzuki models.

Paper Structure

This paper contains 53 sections, 7 theorems, 227 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $\boldsymbol{2d}$ QCD theories
  3. Symmetries, 't Hooft Anomalies and Gaplessness
  4. Towards Gapped QCD Theories
  5. Mass spectrum and QCD Hamiltonians
  6. Lightcone Hamiltonian
  7. Temporal gauge Hamiltonian
  8. Classification of gapped theories
  9. Infrared dynamics of $\boldsymbol{2d}$ QCD
  10. Gapped spectrum.
  11. Continuous symmetries.
  12. 't Hooft anomalies.
  13. One-form symmetry.
  14. Central charge.
  15. Some simple examples.
  16. ...and 38 more sections

Key Result

Lemma 1

A $2d$ QFT with a continuous chiral global symmetry is symmetry preserving and gapless.

Figures (3)

  • Figure 1: Diagram describing infrared dynamics of QCD with massless quarks in a representation $R$ of the gauge group $G$. The theory without quarks is expected to be gapped, and is gapless for large enough $R$. The intermediate regime where the representation $R$ is small remains an open problem.
  • Figure 2: The first few complete graphs $K_1,K_2,\dots,K_5$. If we associate to each node a gauge group $U(n_i),SO(n_i),Sp(n_i)$ (with the global $U(1)$ modded out in the unitary case), and to each edge a bifundamental quark, then the quiver gauge theory associated to $K_n$ has a Kazama-Suzuki model as its effective low energy description.
  • Figure 3: Hermite normal form of an $N_F\times N_c$ integral matrix $Q$. The gray region represents the non-zero entries. This decomposition is unique if we impose some further restrictions, such as positivity of pivots; this shall play no role in this work.

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7