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Higher-form anomalies and state-operator correspondence beyond conformal invariance

Stathis Vitouladitis

TL;DR

This work develops a non-conformal generalization of the CFT state-operator correspondence for QFTs with a mixed higher-form anomaly between ${\mathrm{U}(1)}^{[0]}$ and ${\mathrm{U}(1)}^{[d-2]}$, realised universally as a relativistic superfluid. A central achievement is the explicit construction of an infinite tower of conserved charges forming an abelian current algebra with a central extension, which organizes states and local operators into Verma modules and underpins a robust state-operator map. The formalism is illustrated in free realizations via compact scalars and superfluid phonons, where vertex operators become primaries and descendants are generated by currents, with the vacuum prepared by a Bogoliubov squeezing operator rather than by a trivial path integral. The results suggest a universal, integrability-inspired structure in these gapless anomalous phases and open avenues for applications to entanglement, large-charge analyses, and holography. Overall, the paper provides a concrete, symmetry-based framework for correlating states and local operators in non-conformal, higher-form anomalous QFTs through a superfluid EFT perspective.

Abstract

We establish a state-operator correspondence for a class of non-conformal quantum field theories with continuous higher-form symmetries and a mixed anomaly. Such systems can always be realised as a relativistic superfluid. The symmetry structure induces an infinite tower of conserved charges, which we construct explicitly. These charges satisfy an abelian current algebra with a central extension, generalising the familiar Kac-Moody algebras to higher dimensions. States and operators are organised into representations of this algebra, enabling a direct correspondence. We demonstrate the correspondence explicitly in free examples by performing the Euclidean path integral on a $d$-dimensional ball, with local operators inserted in the origin, and matching to energy eigenstates on $S^{d-1}$ obtained by canonical quantisation. Interestingly, in the absence of conformal invariance, the empty path integral prepares a squeezed vacuum rather than the true ground state.

Higher-form anomalies and state-operator correspondence beyond conformal invariance

TL;DR

This work develops a non-conformal generalization of the CFT state-operator correspondence for QFTs with a mixed higher-form anomaly between and , realised universally as a relativistic superfluid. A central achievement is the explicit construction of an infinite tower of conserved charges forming an abelian current algebra with a central extension, which organizes states and local operators into Verma modules and underpins a robust state-operator map. The formalism is illustrated in free realizations via compact scalars and superfluid phonons, where vertex operators become primaries and descendants are generated by currents, with the vacuum prepared by a Bogoliubov squeezing operator rather than by a trivial path integral. The results suggest a universal, integrability-inspired structure in these gapless anomalous phases and open avenues for applications to entanglement, large-charge analyses, and holography. Overall, the paper provides a concrete, symmetry-based framework for correlating states and local operators in non-conformal, higher-form anomalous QFTs through a superfluid EFT perspective.

Abstract

We establish a state-operator correspondence for a class of non-conformal quantum field theories with continuous higher-form symmetries and a mixed anomaly. Such systems can always be realised as a relativistic superfluid. The symmetry structure induces an infinite tower of conserved charges, which we construct explicitly. These charges satisfy an abelian current algebra with a central extension, generalising the familiar Kac-Moody algebras to higher dimensions. States and operators are organised into representations of this algebra, enabling a direct correspondence. We demonstrate the correspondence explicitly in free examples by performing the Euclidean path integral on a -dimensional ball, with local operators inserted in the origin, and matching to energy eigenstates on obtained by canonical quantisation. Interestingly, in the absence of conformal invariance, the empty path integral prepares a squeezed vacuum rather than the true ground state.

Paper Structure

This paper contains 21 sections, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Anomalies and infinite-dimensional current algebra
  4. Infinitely many charges in the superfluid EFT
  5. Superfluidisation
  6. Compact scalars in d dimensions
  7. The states
  8. The operators
  9. The correspondence
  10. Superfluid phonons
  11. Symmetry considerations
  12. Canonical quantisation
  13. The other side
  14. Conclusions and future directions
  15. Large charge
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Charges act on a state created by a local operator, $\mathcal{O}$, by radially evolving inwards and linking with $\mathcal{O}$.