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Trace and diffeomorphism anomalies of the classical Liouville theory, Virasoro algebras, Weyl-gauging and all that

Pavel Haman, Alfredo Iorio

TL;DR

The paper reveals a genuine classical Virasoro center in Liouville theory on flat space, c = 48π/β^2, arising from an improved conformal symmetry and imprinting on the energy-momentum tensor through non-tensorial transformations. It then analyzes curved backgrounds where Weyl gauging introduces a Weyl field W^μ and, via Ricci/curvature couplings, trades Weyl freedom for geometric terms, moving the center into EMT transformations and often breaking diffeomorphism invariance. The work shows that preserving Weyl invariance typically conflicts with diffeomorphism invariance in local constructions, while certain global symmetries survive in flat space; it also connects these classical centers to trace-like structures and Polyakov-type nonlocal actions. Overall, it advances the understanding of classical central charges, their relation to anomalies, and the distinct roles of Weyl and diffeomorphism invariances in two-dimensional Liouville theory. The results pave the way for exploring classical centers in other models and potential links to gravitational or Unruh/Hawking-type phenomena.

Abstract

To fully clarify the invariance of the classical Liouville field theory under the Virasoro algebra, we first elucidate in detail the concept of classical anomaly, discuss the occurrence of two symmetry algebras associated to this problem, and provide some new formulae to compute the classical center in a general fashion. We apply this to the study of the symmetries of the free boson in two dimensions. Moving to Liouville, we see how this gives rise to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and a non-vanishing trace, in curved space. We provide a variety of improvements of the (local) theory, that restore Weyl invariance. With explicit computations, we show that the covariant conservation of the Weyl-invariance-improved energy-momentum tensor is lost, in general, and relate the chosen improvement with the corresponding subset of preserved diffeomorphisms. The non-tensorial transformation rule of the Weyl-invariance-improved energy-momentum tensor in curved space is explicitly back-traced to the Virasoro center.

Trace and diffeomorphism anomalies of the classical Liouville theory, Virasoro algebras, Weyl-gauging and all that

TL;DR

The paper reveals a genuine classical Virasoro center in Liouville theory on flat space, c = 48π/β^2, arising from an improved conformal symmetry and imprinting on the energy-momentum tensor through non-tensorial transformations. It then analyzes curved backgrounds where Weyl gauging introduces a Weyl field W^μ and, via Ricci/curvature couplings, trades Weyl freedom for geometric terms, moving the center into EMT transformations and often breaking diffeomorphism invariance. The work shows that preserving Weyl invariance typically conflicts with diffeomorphism invariance in local constructions, while certain global symmetries survive in flat space; it also connects these classical centers to trace-like structures and Polyakov-type nonlocal actions. Overall, it advances the understanding of classical central charges, their relation to anomalies, and the distinct roles of Weyl and diffeomorphism invariances in two-dimensional Liouville theory. The results pave the way for exploring classical centers in other models and potential links to gravitational or Unruh/Hawking-type phenomena.

Abstract

To fully clarify the invariance of the classical Liouville field theory under the Virasoro algebra, we first elucidate in detail the concept of classical anomaly, discuss the occurrence of two symmetry algebras associated to this problem, and provide some new formulae to compute the classical center in a general fashion. We apply this to the study of the symmetries of the free boson in two dimensions. Moving to Liouville, we see how this gives rise to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and a non-vanishing trace, in curved space. We provide a variety of improvements of the (local) theory, that restore Weyl invariance. With explicit computations, we show that the covariant conservation of the Weyl-invariance-improved energy-momentum tensor is lost, in general, and relate the chosen improvement with the corresponding subset of preserved diffeomorphisms. The non-tensorial transformation rule of the Weyl-invariance-improved energy-momentum tensor in curved space is explicitly back-traced to the Virasoro center.
Paper Structure (18 sections, 281 equations, 1 figure, 2 tables)

This paper contains 18 sections, 281 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Symmetries in Lagrangian formalism and Noether theorem
  3. Symmetries in Hamiltonian formalism
  4. Classical anomalies, a tale of two algebras
  5. Conformal transformations in two dimensions
  6. The free scalar in two dimensions and its affine center
  7. Liouville theory in flat space and its genuine center
  8. On the general case
  9. Liouville theory on curved backgrounds
  10. The extra term $W^\mu W_\mu$
  11. Energy-momentum tensor
  12. Breaking diffeomorphism invariance
  13. Conclusions
  14. Passive and active transformations
  15. On singular Lagrangians
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Cylindrical manifold