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Light-Cone Gauge Quantization of 2D Sigma Models

Robert Rudd

TL;DR

The paper develops a manifestly ghost-free light-cone gauge quantization for general 2D sigma models describing bosonic strings in nontrivial background fields. It identifies a sufficient background condition—a covariantly constant null vector symmetry—along with Weyl invariance, to render X^+, γ^{00} gauge-fixing accessible and to render X^-, γ^{01} auxiliary, thereby eliminating unphysical ghosts. Through a background-field expansion and dimensional regularization, it derives the standard Weyl beta functions in the light-cone gauge, including a novel treatment of the dilaton beta function and the critical dimension within a ghost-free framework. The work further demonstrates how string interactions can be formulated via Mandelstam diagrams and constructs exactly solvable light-cone models with quadratic transverse dynamics, suggesting viable paths toward light-cone string field theory in nontrivial backgrounds. Overall, it shows that a wide class of critical string backgrounds admits a consistent, unitary light-cone quantum mechanics with a flat transverse measure and a tractable interaction structure, expanding the toolkit beyond covariant formulations.

Abstract

This work describes the formulation of the manifestly ghost-free (spacetime) light-cone gauge for bosonic string theory with non-trivial spacetime metric, antisymmetric tensor, dilaton and tachyon fields. The action is a general two-dimensional sigma model, corresponding to a closed string theory with a second order action in the Polyakov picture. The spacetime fields must have a symmetry generated by a null, covariantly constant spacetime vector in order for the light-cone gauge to be accessible. Also, the theory must be Weyl invariant. The conditions for Weyl invariance are computed within the light-cone gauge, reproducing the usual beta functions. The calculation of the dilaton beta function and the critical dimension is somewhat novel in this ghost-free theory. Some exactly solvable light-cone theories are discussed.

Light-Cone Gauge Quantization of 2D Sigma Models

TL;DR

The paper develops a manifestly ghost-free light-cone gauge quantization for general 2D sigma models describing bosonic strings in nontrivial background fields. It identifies a sufficient background condition—a covariantly constant null vector symmetry—along with Weyl invariance, to render X^+, γ^{00} gauge-fixing accessible and to render X^-, γ^{01} auxiliary, thereby eliminating unphysical ghosts. Through a background-field expansion and dimensional regularization, it derives the standard Weyl beta functions in the light-cone gauge, including a novel treatment of the dilaton beta function and the critical dimension within a ghost-free framework. The work further demonstrates how string interactions can be formulated via Mandelstam diagrams and constructs exactly solvable light-cone models with quadratic transverse dynamics, suggesting viable paths toward light-cone string field theory in nontrivial backgrounds. Overall, it shows that a wide class of critical string backgrounds admits a consistent, unitary light-cone quantum mechanics with a flat transverse measure and a tractable interaction structure, expanding the toolkit beyond covariant formulations.

Abstract

This work describes the formulation of the manifestly ghost-free (spacetime) light-cone gauge for bosonic string theory with non-trivial spacetime metric, antisymmetric tensor, dilaton and tachyon fields. The action is a general two-dimensional sigma model, corresponding to a closed string theory with a second order action in the Polyakov picture. The spacetime fields must have a symmetry generated by a null, covariantly constant spacetime vector in order for the light-cone gauge to be accessible. Also, the theory must be Weyl invariant. The conditions for Weyl invariance are computed within the light-cone gauge, reproducing the usual beta functions. The calculation of the dilaton beta function and the critical dimension is somewhat novel in this ghost-free theory. Some exactly solvable light-cone theories are discussed.

Paper Structure

This paper contains 15 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gauge Fixing
  3. Fixing $X^+$ and $\gamma ^{00}$
  4. The Auxiliary Fields $X^-$ and $\gamma ^{01}$
  5. The Geometry of Light-Cone Sigma Models
  6. Additional Constraints and Weyl Invariance
  7. The Light-Cone Action
  8. Light-Cone Quantum Mechanics
  9. Sigma Model Perturbation Theory
  10. The Hilbert Space of States
  11. Renormalization and the Beta Functions
  12. Interactions and Exactly Solvable Models
  13. String Interactions
  14. Exactly Solvable Models
  15. Conclusion

Figures (1)

  • Figure :