[Paper Review] Lectures on 2D gravity and 2D string theory (TASI 1992)

Paul Ginsparg, Gregory Moore|ArXiv.org|Apr 5, 1993

Black Holes and Theoretical PhysicsPhysics and Astronomy93 references161 citations

TL;DR

This seminal 1993 lecture series by Ginsparg and Moore provides a comprehensive theoretical framework for 2D gravity and 2D string theory, focusing on the c=1 conformal field theory coupled to gravity. It establishes the equivalence between matrix models and non-perturbative 2D quantum gravity via orthogonal polynomials, fermionic representations, and collective field theory, achieving exact results for string scattering amplitudes and the c=1 matrix model spectrum.

ABSTRACT

Emphasis is on 2d target space (c=1 coupled to gravity). Contents: 0. Introduction, Overview, and Purpose 1. Loops and States in Conformal Field Theory 2. 2D Euclidean Quantum Gravity I: Path Integral Approach 3. Brief Review of the Liouville Theory 4. 2D Euclidean Quantum Gravity II: Canonical Approach 5. 2D Critical String Theory 6. Discretized surfaces, matrix models, and the continuum limit 7. Matrix Model Technology I: Method of Orthogonal Polynomials 8. Matrix Model Technology II: Loops on the Lattice 9. Matrix Model Technology III: Free Fermions from the Lattice 10. Loops and States in Matrix Model Quantum Gravity 11. Loops and States in the $c=1$ Matrix Model 12. Fermi Sea Dynamics and Collective Field Theory 13. String scattering in two spacetime dimensions 14. Vertex Operator Calculations and Continuum Methods 15. Achievements, Disappointments, Future Prospects "if you read only one set of lecture notes this year, don't read these."

Motivation & Objective

• To provide a self-contained, pedagogical introduction to 2D quantum gravity and string theory for graduate students and researchers.
• To bridge the gap between conformal field theory, matrix models, and non-perturbative quantum gravity in two dimensions.
• To establish the connection between discrete lattice models and the continuum limit in 2D gravity using matrix model technology.
• To derive exact results for string scattering amplitudes and the spectrum of the c=1 matrix model through collective field theory and fermionic formulations.
• To clarify the role of the c=1 critical point in 2D string theory and its relation to Liouville theory and Euclidean quantum gravity.

Proposed method

• Utilizes the path integral approach to 2D Euclidean quantum gravity, emphasizing the role of conformal symmetry and moduli spaces.
• Applies the canonical quantization approach to 2D gravity, linking it to the Liouville theory and its conformal properties.
• Employs matrix models as a non-perturbative regularization of 2D quantum gravity, using orthogonal polynomials to compute correlation functions.
• Introduces lattice formulations of matrix models and derives their continuum limit via loop equations and collective field theory.
• Represents the matrix model as a free fermion gas, enabling exact computation of the partition function and correlation functions.
• Uses collective field theory to map the fermionic system to a bosonic field theory, allowing derivation of string scattering amplitudes.

Experimental results

Research questions

• RQ1 How does the c=1 conformal field theory coupled to 2D gravity behave in the non-perturbative regime?
• RQ2 What is the precise relationship between matrix models and the continuum limit of 2D quantum gravity?
• RQ3 How can string scattering amplitudes be computed exactly in 2D string theory using matrix model techniques?
• RQ4 What is the role of the fermionic representation in the c=1 matrix model and how does it relate to the collective field theory?
• RQ5 How do the loop equations and orthogonal polynomial methods lead to exact solutions in the matrix model framework?

Key findings

• The c=1 matrix model is shown to be equivalent to a free fermion gas, enabling exact computation of the partition function and correlation functions.
• The continuum limit of the matrix model reproduces the Liouville theory with central charge c=1, confirming the consistency of the non-perturbative formulation.
• String scattering amplitudes in 2D string theory are computed exactly using vertex operator techniques and collective field theory, yielding explicit expressions for n-point functions.
• The method of orthogonal polynomials allows exact evaluation of loop operators and correlation functions in the matrix model, providing a powerful computational tool.
• The fermionic formulation leads to a clear picture of the ground state as a Fermi sea, with collective excitations corresponding to string states.
• The full structure of the c=1 matrix model is derived, including the spectrum and dynamics, through the interplay of lattice discretization, orthogonal polynomials, and collective field theory.