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Generalised anomalies, QCD$_4$, and holography

Mohammad Akhond, Shigeki Sugimoto

TL;DR

The paper provides a top-down holographic derivation of generalized anomalies in QCD$_4$, including Tanizaki’s mixed axial–flavour and axial–baryon anomalies and the axial self-anomaly, by embedding the problem in the D4–D8 holographic setup. It develops a dual formulation of type IIA fluxes, couples it to smeared D8-branes, and reduces the bulk topological action on $S^4$ to a 6d TFT that encodes anomaly inflow. The resulting inflow action reproduces both continuous and discrete anomalies, including the coupling-constant anomaly for massive QCD$_4$, thereby illustrating how holography captures intricate global structure and topological data of the QCD$_4$ global symmetry. This work extends the holographic toolkit for generalized global anomalies and opens paths to gravitational and other generalized anomaly analyses in related holographic duals.

Abstract

During the last decade, the notion of an 't Hooft anomaly has been generalised to the case of discrete symmetries. An interesting instance, discussed by Tanizaki, is the mixed anomaly between the discrete axial symmetry and the flavour and baryonic symmetries in massless QCD$_4$. The goal of this note is to provide a derivation of this anomaly from a top-down holographic dual of QCD$_4$. It is found that the topological couplings in the bulk supergravity dual of the D4-D8 system encode Tanizaki's anomaly, once fluctuations around the bulk gauge fields are turned on. A technical challenge for this computation is the difficulty in maintaining gauge invariance of supergravity theories in the presence of D-branes. To overcome this issue, a compact formulation of the flux sector of (massive) type IIA supergravity in the presence of D8 branes is presented. A crucial ingredient in the analysis is the need for smearing the D8 branes, in order to impose $τ$-shift invariance on the fluctuation ansatz.

Generalised anomalies, QCD$_4$, and holography

TL;DR

The paper provides a top-down holographic derivation of generalized anomalies in QCD, including Tanizaki’s mixed axial–flavour and axial–baryon anomalies and the axial self-anomaly, by embedding the problem in the D4–D8 holographic setup. It develops a dual formulation of type IIA fluxes, couples it to smeared D8-branes, and reduces the bulk topological action on to a 6d TFT that encodes anomaly inflow. The resulting inflow action reproduces both continuous and discrete anomalies, including the coupling-constant anomaly for massive QCD, thereby illustrating how holography captures intricate global structure and topological data of the QCD global symmetry. This work extends the holographic toolkit for generalized global anomalies and opens paths to gravitational and other generalized anomaly analyses in related holographic duals.

Abstract

During the last decade, the notion of an 't Hooft anomaly has been generalised to the case of discrete symmetries. An interesting instance, discussed by Tanizaki, is the mixed anomaly between the discrete axial symmetry and the flavour and baryonic symmetries in massless QCD. The goal of this note is to provide a derivation of this anomaly from a top-down holographic dual of QCD. It is found that the topological couplings in the bulk supergravity dual of the D4-D8 system encode Tanizaki's anomaly, once fluctuations around the bulk gauge fields are turned on. A technical challenge for this computation is the difficulty in maintaining gauge invariance of supergravity theories in the presence of D-branes. To overcome this issue, a compact formulation of the flux sector of (massive) type IIA supergravity in the presence of D8 branes is presented. A crucial ingredient in the analysis is the need for smearing the D8 branes, in order to impose -shift invariance on the fluctuation ansatz.

Paper Structure

This paper contains 24 sections, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review: Anomalies in field theory
  3. Anomalies of QCD$_4$
  4. Continuous anomaly
  5. Discrete anomalies
  6. Dual formulation of type IIA fluxes
  7. Without D-branes
  8. Compact notation
  9. Inclusion of D8 branes
  10. Anomaly matching in holographic QCD
  11. Truncation of the topological action on $S^4$
  12. Discrete anomalies
  13. Coupling constant anomaly of massive QCD$_4$
  14. Conclusions and outlook
  15. Acknowledgements:
  16. ...and 9 more sections

Figures (2)

  • Figure 1: Schematic of a holographic spacetime $M$. The QFT degrees of freedom live at the boundary $\partial M$, while the holographic coordinate $U$ extends into the bulk spacetime $M$. In this setting it is natural to identify the bulk spacetime of the anomaly TFT with the holographic spacetime $M$.
  • Figure 2: Pictorial representation of the brane system for QCD$_4$, and its holographic dual. (a) Before backreaction the chiral symmetry is realised as the gauge symmetry on the separate stacks of $N_f$ D8, and $\overline{\mathrm{D8}}$ branes. (b) After backreaction of the $N$ D4 branes, the geometry ends at the tip of the cigar, where the $N_f$ D8 and $\overline{\mathrm{D8}}$ branes meet to for a single continuous stack of D8 branes. This phenomenon is interpreted as the holographic manifestation of chiral symmetry breaking.