string theory

1. From Points to Strings

In the Standard Model, particles are idealized as points.
But attempts to quantize gravity in this framework run into non-renormalizable divergences.

String theory replaces points with one-dimensional strings.
Their vibrations correspond to different particle species.

Particle → worldline
String → worldsheet

2. Classical Dynamics of Strings

Nambu–Goto Action

The simplest description is the area of the worldsheet:

S_{\text{NG}} = -T \int d^2\sigma \, \sqrt{-\det \left( \partial_a X^\mu \partial_b X_\mu \right)} ,

where $T = 1/(2\pi \alpha')$ is the string tension.

Polyakov Action

Equivalent but more convenient:

S_{\text{P}} = -\frac{T}{2} \int d^2\sigma \, \sqrt{-h}\, h^{ab} \partial_a X^\mu \partial_b X_\mu .

$X^\mu(\sigma^a)$ : embedding of the string in $D$ -dimensional spacetime
$h_{ab}$ : worldsheet metric

This action enjoys diffeomorphism and Weyl invariance, making it a 2D conformal field theory (CFT).

3. Quantization and Mode Expansion

Choose conformal gauge $h_{ab} = \eta_{ab}$ .
Equations of motion reduce to the 2D wave equation:

\left( \partial_\tau^2 - \partial_\sigma^2 \right) X^\mu(\tau,\sigma) = 0.

Mode Expansion (closed string):

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).

Virasoro Constraints

From worldsheet diffeomorphism invariance:

T_{ab} = 0 \quad \Rightarrow \quad L_n = 0, \;\; \tilde{L}_n = 0 \quad (n \geq 0).

$L_n, \tilde{L}_n$ : Virasoro generators
Ensure physical states are consistent with conformal invariance

4. Mass Spectrum

The string Hamiltonian yields:

M^2 = \frac{4}{\alpha'} \left( N + \tilde{N} - 2a \right),

$N, \tilde{N}$ : oscillator numbers
$a$ : normal ordering constant

Special states:

Closed strings: contain a universal massless spin-2 graviton
Open strings: yield gauge bosons
Higher excitations: infinite Regge tower $M^2 \sim n/\alpha'$

Thus, quantum gravity and gauge interactions appear naturally.

5. The Five Superstring Theories

The bosonic string (26D) suffers from tachyons.
Introducing worldsheet supersymmetry reduces the critical dimension to $D=10$ and removes tachyons (after GSO projection).
This yields five consistent superstring theories:

(1) Type I

Open + closed unoriented strings
Gauge group: $SO(32)$
Supersymmetry: $N=1$ in 10D

(2) Type IIA

Closed oriented strings
Left- and right-moving fermions of opposite chirality
Massless sector: graviton, dilaton, antisymmetric $B_{\mu\nu}$ , plus $p$ -form RR fields (even rank)