Table of Contents
- Introduction
- Strings and the Worldsheet
- Worldsheet Coordinates and Notation
- The Nambu–Goto Action
- The Polyakov Action
- Equivalence of Nambu–Goto and Polyakov Formulations
- Classical Equations of Motion
- Constraints and Virasoro Conditions
- Symmetries of the Worldsheet Action
- Conformal Symmetry in Two Dimensions
- Conformal Transformations and Weyl Invariance
- Stress-Energy Tensor and Conservation Laws
- Mode Expansion of the String
- Quantization and Virasoro Algebra
- Central Charge and Anomalies
- Ghost Systems and BRST Quantization
- Conformal Field Theory (CFT) Basics
- Primary Fields and Operator Product Expansion
- Radial Quantization and State-Operator Correspondence
- Modular Invariance and One-Loop Amplitudes
- Application to String Scattering
- Compactification and Orbifolds
- Supersymmetric CFT
- Role in String Backgrounds and AdS/CFT
- Conclusion
1. Introduction
The worldsheet formulation of string theory describes how strings propagate through spacetime. This two-dimensional field theory, defined on the surface swept out by the string, is described by a conformal field theory (CFT). Understanding worldsheet actions and their conformal properties is essential for string quantization and consistency.
2. Strings and the Worldsheet
- The worldsheet is the 2D surface traced out by a string moving through spacetime.
- It is parameterized by:
- \( \tau \): worldsheet time
- \( \sigma \): worldsheet spatial coordinate
For closed strings, \( \sigma \in [0, 2\pi] \), and for open strings, \( \sigma \in [0, \pi] \).
3. Worldsheet Coordinates and Notation
Worldsheet coordinates \( \sigma^\alpha = (\tau, \sigma) \)
Spacetime coordinates \( X^\mu(\sigma^\alpha) \), \( \mu = 0, \dots, D-1 \)
The induced metric is:
\[
h_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X^\nu \eta_{\mu\nu}
\]
4. The Nambu–Goto Action
The simplest action describing a relativistic string:
\[
S_{\text{NG}} = -T \int d^2\sigma \sqrt{-\det(h_{\alpha\beta})}
\]
- \( T \): string tension
- Nonlinear in \( X^\mu \), making quantization difficult
5. The Polyakov Action
An alternative, classically equivalent formulation:
\[
S_{\text{P}} = -\frac{T}{2} \int d^2\sigma \sqrt{-\gamma} \gamma^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu
\]
- \( \gamma_{\alpha\beta} \): auxiliary worldsheet metric
- Easier to quantize
- Conformally invariant in 2D
6. Equivalence of Nambu–Goto and Polyakov Formulations
Classical equivalence:
- Solving \( \gamma_{\alpha\beta} = h_{\alpha\beta} \) in the Polyakov action recovers the Nambu–Goto action
- Polyakov form introduces additional symmetries (diffeomorphism and Weyl)
7. Classical Equations of Motion
Variation with respect to \( X^\mu \):
\[
\partial_\alpha (\sqrt{-\gamma} \gamma^{\alpha\beta} \partial_\beta X^\mu) = 0
\]
With conformal gauge \( \gamma_{\alpha\beta} = \eta_{\alpha\beta} \), this becomes:
\[
\Box X^\mu = 0
\]
Wave equation in 2D.
8. Constraints and Virasoro Conditions
Variation with respect to \( \gamma^{\alpha\beta} \) gives constraints:
\[
T_{\alpha\beta} = 0
\]
These are the Virasoro constraints, ensuring physical degrees of freedom and eliminating unphysical states.
9. Symmetries of the Worldsheet Action
The Polyakov action has:
- Worldsheet diffeomorphism invariance
- Weyl invariance (local rescaling of the metric)
These together lead to conformal invariance in 2D.
10. Conformal Symmetry in Two Dimensions
2D conformal symmetry is infinite-dimensional. Transformations preserve angles but not distances. They include:
- Translations
- Rotations
- Dilations
- Special conformal transformations
11. Conformal Transformations and Weyl Invariance
A Weyl transformation rescales the metric:
\[
\gamma_{\alpha\beta} \to e^{2\omega(\sigma)} \gamma_{\alpha\beta}
\]
In 2D, conformal invariance means invariance under both diffeomorphisms and Weyl rescalings.
12. Stress-Energy Tensor and Conservation Laws
The stress-energy tensor is:
\[
T_{\alpha\beta} = \frac{2}{\sqrt{-\gamma}} \frac{\delta S}{\delta \gamma^{\alpha\beta}}
\]
In conformal gauge:
\[
T_{++} = \partial_+ X^\mu \partial_+ X_\mu, \quad T_{–} = \partial_- X^\mu \partial_- X_\mu
\]
13. Mode Expansion of the String
Solutions to \( \Box X^\mu = 0 \) are expanded in modes. For closed strings:
\[
X^\mu(\tau, \sigma) = x^\mu + 2\alpha’ p^\mu \tau + i\sqrt{\frac{\alpha’}{2}} \sum_{n \neq 0} \left( \frac{\alpha_n^\mu}{n} e^{-in(\tau – \sigma)} + \frac{\tilde{\alpha}_n^\mu}{n} e^{-in(\tau + \sigma)} \right)
\]
14. Quantization and Virasoro Algebra
Upon quantization, the Virasoro generators \( L_n \) satisfy:
\[
[L_m, L_n] = (m – n) L_{m+n} + \frac{c}{12} m(m^2 – 1) \delta_{m+n, 0}
\]
The central term \( c \) represents a conformal anomaly.
15. Central Charge and Anomalies
To preserve Weyl invariance at the quantum level, the conformal anomaly (central charge) must vanish. For bosonic string theory:
\[
c = D, \quad \text{vanishes when } D = 26
\]
This is why string theory requires critical dimensions.
16. Ghost Systems and BRST Quantization
Gauge fixing introduces ghost fields in the path integral. For bosonic strings:
- Ghost system: \( (b, c) \) with conformal weights (2, -1)
- Central charge \( c_{ghost} = -26 \)
BRST quantization defines a nilpotent charge \( Q_{BRST} \) used to select physical states via cohomology:
\[
Q_{BRST}^2 = 0, \quad \text{Physical states: } Q_{BRST}|\psi\rangle = 0, \quad |\psi\rangle \sim |\psi\rangle + Q_{BRST}|\chi\rangle
\]
17. Conformal Field Theory (CFT) Basics
A conformal field theory (CFT) is a quantum field theory invariant under conformal transformations. In 2D, the symmetry algebra is infinite-dimensional (Virasoro algebra), making 2D CFTs exactly solvable in many cases.
18. Primary Fields and Operator Product Expansion
Primary fields \( \phi(z, \bar{z}) \) transform as:
\[
\phi'(z’, \bar{z}’) = \left( \frac{dz’}{dz} \right)^{-h} \left( \frac{d\bar{z}’}{d\bar{z}} \right)^{-\bar{h}} \phi(z, \bar{z})
\]
The operator product expansion (OPE) reveals the structure of local operators:
\[
T(z) \phi(w) \sim \frac{h \phi(w)}{(z – w)^2} + \frac{\partial \phi(w)}{z – w}
\]
19. Radial Quantization and State-Operator Correspondence
CFTs allow radial quantization, mapping time evolution to radial flow in the complex plane. Each local operator corresponds to a state:
\[
|\phi\rangle = \lim_{z \to 0} \phi(z) |0\rangle
\]
This duality is central to conformal bootstrap and modular invariance.
20. Modular Invariance and One-Loop Amplitudes
String theory one-loop amplitudes involve torus integrals. Modular invariance ensures consistency of the partition function under SL(2,ℤ) transformations:
\[
\tau \to \frac{a\tau + b}{c\tau + d}
\]
Ensures anomaly cancellation and finiteness.
21. Application to String Scattering
CFT computes string scattering amplitudes via vertex operators inserted on the worldsheet. Example:
- Tachyon vertex: \( V(k) = :e^{ik \cdot X(z)}: \)
Amplitudes involve correlation functions of vertex operators.
22. Compactification and Orbifolds
Compactifying extra dimensions modifies the spectrum. Orbifolds are singular compact spaces that allow twisted sectors, enriching the theory’s structure and enabling model building.
23. Supersymmetric CFT
Supersymmetric extensions introduce superfields and superconformal symmetry. The resulting super-Virasoro algebra includes supercharges:
\[
\{ G_r, G_s \} = 2 L_{r+s} + \frac{c}{3}(r^2 – \frac{1}{4})\delta_{r+s, 0}
\]
Important for superstring theory and AdS/CFT duality.
24. Role in String Backgrounds and AdS/CFT
Consistent string backgrounds correspond to consistent CFTs on the worldsheet. The AdS/CFT correspondence connects CFTs to gravitational theories in higher dimensions, providing a holographic description of gravity.
25. Conclusion
The formulation of string theory in terms of worldsheet actions and conformal field theory provides deep insights into the structure of quantum gravity. The interplay between classical symmetries, gauge fixing, anomalies, and conformal invariance underpins the mathematical consistency of string theory. Tools from CFT are central in computing amplitudes, classifying consistent backgrounds, and exploring dualities like AdS/CFT — making them indispensable in modern theoretical physics.
