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Global Structures from the Infrared

Michele Del Zotto, Iñaki García Etxebarria

TL;DR

The paper develops a field-theoretic route to determine the global form of 4d and higher-dimensional QFTs by deriving a symmetry TFT from infrared Coulomb-phase data. It shows that the allowed flux sectors and line operators organize into a bulk BF-type theory in one higher dimension, whose boundary data reproduce the observed 1-form defect groups and centers for various theories, including non-Lagrangian Argyres-Douglas theories. The approach reproduces known results from geometric engineering and extends them to a purely field-theoretic setting, with consistent generalizations to 6d and 5d theories. This provides a unifying framework connecting IR Coulomb dynamics, anomaly inflow, and global structure, and points toward richer symmetry TFTs and higher-form/ non-invertible structures to explore in future work.

Abstract

Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Such choices can be reformulated in terms of a topological field theory in one dimension higher, the symmetry TFT. In this paper we show that this TFT can be reconstructed from a careful analysis of the infrared Coulomb-like phases. In particular, the TFT matches between the UV and the IR. This provides a purely field theoretical counterpart of several recent results obtained via geometric engineering in various string/M/F theory setups for theories in four and five dimensions that we confirm and extend.

Global Structures from the Infrared

TL;DR

The paper develops a field-theoretic route to determine the global form of 4d and higher-dimensional QFTs by deriving a symmetry TFT from infrared Coulomb-phase data. It shows that the allowed flux sectors and line operators organize into a bulk BF-type theory in one higher dimension, whose boundary data reproduce the observed 1-form defect groups and centers for various theories, including non-Lagrangian Argyres-Douglas theories. The approach reproduces known results from geometric engineering and extends them to a purely field-theoretic setting, with consistent generalizations to 6d and 5d theories. This provides a unifying framework connecting IR Coulomb dynamics, anomaly inflow, and global structure, and points toward richer symmetry TFTs and higher-form/ non-invertible structures to explore in future work.

Abstract

Quantum field theories with identical local dynamics can admit different choices of global structure, leading to different partition functions and spectra of extended operators. Such choices can be reformulated in terms of a topological field theory in one dimension higher, the symmetry TFT. In this paper we show that this TFT can be reconstructed from a careful analysis of the infrared Coulomb-like phases. In particular, the TFT matches between the UV and the IR. This provides a purely field theoretical counterpart of several recent results obtained via geometric engineering in various string/M/F theory setups for theories in four and five dimensions that we confirm and extend.
Paper Structure (16 sections, 45 equations, 4 figures, 1 table)

This paper contains 16 sections, 45 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Genuine and non-genuine lines from the infrared
  3. Line operators from the Coulomb phase
  4. An example: the $\mathcal{N}=2$ $\mathfrak{su}(2)$ theory on the Coulomb branch
  5. Symmetry TFT and global structure
  6. A quick review of BF theory in $D+1$ dimensions
  7. Global structures from the infrared
  8. Examples and consistency checks from $\mathcal{N}=2$ theories
  9. Pure $\mathcal{N}=2$ SYM theories with algebra $\mathfrak{g}$
  10. Example of $\mathcal{N}=2^*$ theories with algebra $\mathfrak{g}$
  11. Adding matter in various representations
  12. Non-Lagrangian theories
  13. Generalization to other dimensions
  14. The case of 6d (2,0) theories
  15. The case of 5d theories
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Left: Inflow picture of anomaly matching for intrinsic QFTs: the anomaly is an invertible TFT $\boldsymbol{A}$ in one dimension higher. Right: TFT matching for QFTs with global structure. The choices of global structure are encoded by non-invertible TFT on an interval with a boundary condition interface $\boldsymbol{B}$ to an anomaly theory. This symmetry TFT must match for the theories in the UV and the IR, thus generalizing the anomaly matching procedure.
  • Figure 2: Aharonv-Bohm effect for lines: monodromies can give rise to phases proportional to the corresponding Dirac pairings. Only lines which have charges satisfying the Dirac quantization condition can be simultaneously genuine. The presence of non-genuine lines is the hallmark of a relative QFT -- see e.g. section 2 of Bhardwaj:2021mzl for a nice review.
  • Figure 3: Top: action of a $U(1)^{(1)}$-form symmetry on a line operator. Bottom: Constraint on the higher form symmetry in presence of charged particles (red dot at the end of the line). The presence of a particle of charge $q\in\mathbb Z$ enforces $e^{i\theta q} = 1$ which in turns entail that only rotations with phase $\theta = 2 \pi k / q$ are allowed, thus breaking $U(1)^{(1)}$ down to $\mathbb Z^{(1)}_q$.
  • Figure 4: The BPS quiver for $SU(4)$ with matter in the $\mathbf{6}\oplus \mathbf{10}$