String theory replaces the point-particle ontology of quantum field theory with one-dimensional extended objects—strings. This seemingly modest change has profound consequences.
The Classical String
A string sweeps out a 2-dimensional worldsheet \(\Sigma\) embedded in spacetime \(M\). The embedding is described by functions \(X^\mu(\tau, \sigma)\) where \((\tau, \sigma)\) are worldsheet coordinates and \(\mu = 0, 1, \ldots, D-1\) indexes spacetime dimensions.
The dynamics are governed by the Nambu-Goto action, which extremises the worldsheet area:
\[ S_{NG} = -T \int_\Sigma d\tau \, d\sigma \sqrt{-\det(h_{ab})} \]
where \(T = \frac{1}{2\pi\alpha'}\) is the string tension (with \(\alpha'\) the Regge slope) and \(h_{ab} = \partial_a X^\mu \partial_b X_\mu\) is the induced metric on the worldsheet.
The square root is analytically inconvenient, so one typically uses the classically equivalent Polyakov action:
\[ S_P = -\frac{T}{2} \int_\Sigma d^2\sigma \sqrt{-\gamma} \, \gamma^{ab} \partial_a X^\mu \partial_b X_\mu \]
Here \(\gamma_{ab}\) is an independent worldsheet metric. This action has manifest worldsheet diffeomorphism and Weyl invariance.
Quantization and Critical Dimensions
Upon quantization, Weyl invariance becomes anomalous unless one works in the critical dimension. For the bosonic string:
\[ D = 26 \]
For the superstring (which includes worldsheet fermions \(\psi^\mu\)):
\[ D = 10 \]
This arises from requiring the cancellation of the conformal anomaly, which in the path integral formalism appears as the central charge condition \(c = 0\) for the total worldsheet CFT (matter + ghosts).
The String Spectrum
Strings can be open (with endpoints) or closed (loops). The mass spectrum comes from the oscillator modes. For a closed bosonic string:
\[ M^2 = \frac{2}{\alpha'}\left(N + \tilde{N} - 2\right) \]
where \(N, \tilde{N}\) are the left/right-moving oscillator number operators, subject to the level-matching constraint \(N = \tilde{N}\).
Key features of the spectrum:
- Tachyon: The ground state (\(N = \tilde{N} = 0\)) has \(M^2 < 0\), signalling an instability in the bosonic string. Supersymmetry removes this.
- Massless states: At \(N = \tilde{N} = 1\), the closed string contains a symmetric traceless tensor (the graviton \(g_{\mu\nu}\)), an antisymmetric tensor (\(B_{\mu\nu}\)), and a scalar (the dilaton \(\phi\)).
The graviton’s emergence is why string theory necessarily includes gravity—it’s not optional.
Superstrings and the Five Theories
Adding worldsheet supersymmetry and imposing consistent boundary conditions (GSO projection) yields five consistent 10-dimensional superstring theories:
Dualities and M-Theory
These five theories are related by dualities:
- T-duality: Relates Type IIA \(\leftrightarrow\) IIB and the two heterotic theories. Compactification on a circle of radius \(R\) is equivalent to radius \(\alpha'/R\).
- S-duality: Relates strong and weak coupling, e.g., Type I \(\leftrightarrow\) Heterotic \(SO(32)\).
All five theories appear to be limits of a single 11-dimensional framework called M-theory, whose low-energy limit is 11-dimensional supergravity. The full non-perturbative definition of M-theory remains elusive.
Compactification
To connect to 4-dimensional physics, one compactifies the extra dimensions on a compact manifold \(K\):
\[ M^{10} = M^4 \times K^6 \]
For \(\mathcal{N}=1\) supersymmetry in 4D, \(K\) must be a Calabi-Yau threefold—a Kähler manifold with \(SU(3)\) holonomy. The topology of \(K\) (Hodge numbers, etc.) determines the particle spectrum.
Would you like me to go deeper into any aspect—perhaps the worldsheet CFT structure, the mathematics of Calabi-Yau compactifications, or the role of D-branes?