String Theory

String theory is a framework in which the point-like particles of quantum field theory are replaced by one-dimensional extended objects, strings, whose different vibrational states appear as different particles. Its defining feature is that a massless spin-2 state — the graviton — appears automatically in the spectrum of the closed string, so the theory is a candidate for a consistent quantum theory of gravity unified with gauge forces and matter. Quantum consistency fixes the spacetime dimension (26 for the bosonic string, 10 for the superstring) and, at weak coupling, allows exactly five supersymmetric theories, now understood as limits of a single framework often called M-theory. No experiment has yet tested a prediction specific to string theory; its established impact so far is on mathematics, black-hole physics, and quantum field theory (notably through holography).

This hub covers the foundations: the motivation, the classical and quantum theory of a single string, the superstring, and the five theories. Three companion pages continue:

Motivation

General relativity and quantum field theory are each extremely well tested, but they do not combine naively. Quantizing the metric as a field around flat space gives a theory whose coupling, Newton’s constant $G$, has mass dimension $-2$ in four dimensions; each additional loop brings more powers of energy, and the divergences cannot be absorbed into finitely many parameters. Perturbative quantum gravity is therefore non-renormalizable: it works as an effective field theory below the Planck scale $M_{\text{Pl}} \approx 1.2 \times 10^{19}$ GeV but says nothing about what happens above it.

String theory addresses this by changing the objects that interact. A point particle traces a worldline; a string traces a two-dimensional worldsheet. In a Feynman diagram, particles meet at a vertex — a sharp point where the short-distance divergences originate. A string interaction is a smooth surface (the “pair of pants” below) with no distinguished point at which the interaction happens, and the ultraviolet behaviour of string amplitudes is soft: in perturbation theory, string amplitudes are free of the ultraviolet divergences of point-particle gravity.

The second reason string theory is taken seriously is that gravity is not put in by hand. The closed string always contains a massless symmetric spin-2 state, and consistency of the string propagating in a curved background requires, at leading order in $\alpha’$, that the background obey Einstein’s equations. Gauge fields, chiral fermions, and supersymmetry arise from the same structure.

Historical development

Strings and Scales

A string is characterized by a single dimensionful parameter, the Regge slope $\alpha’$, equivalently the string tension $T$ or the string length $\ell_s$ (natural units, $\hbar = c = 1$):

A second parameter, the string coupling $g_s$, controls interactions. It is not a free input: it is the expectation value of a dynamical scalar, the dilaton, $g_s = e^{\langle \phi \rangle}$.

Strings come in two topologies:

• Closed strings are loops. Their spectrum always contains the graviton.
• Open strings have two endpoints. Their massless states include gauge bosons, and the endpoints may be confined to hypersurfaces called D-branes.

Every consistent string theory contains closed strings, because two open-string endpoints can join.

How large is a string?

The string scale is often quoted as “about the Planck length, $10^{-35}$ m”, but the two are related through the coupling and the volume of any extra dimensions. For a ten-dimensional theory compactified on a six-dimensional space of volume $V_6$, the four-dimensional Planck mass satisfies, up to numerical factors,

With $g_s \lesssim 1$ and $V_6 \sim \ell_s^6$ this gives $M_s$ within one or two orders of magnitude of $M_{\text{Pl}}$ — the conventional expectation. Large volumes, strongly warped geometries, or very small $g_s$ can lower $M_s$ substantially, in principle as far as the TeV scale; collider and gravity experiments constrain those scenarios (see experimental signatures).

Classical String Theory

Worldsheets

As it moves, a string sweeps out a worldsheet parametrized by a time-like coordinate $\tau$ and a space-like coordinate $\sigma$. The embedding functions $X^\mu(\tau, \sigma)$ describe where each point of the string sits in $D$-dimensional spacetime.

Nambu–Goto and Polyakov actions

The natural action for a relativistic string is proportional to the area of its worldsheet — the direct analogue of a point particle’s action being proportional to proper time. This is the Nambu–Goto action:

where $h_{ab}$ is the metric induced on the worldsheet. Classical strings therefore sweep out surfaces of extremal area.

The square root makes quantization awkward. The Polyakov action introduces an independent worldsheet metric $\gamma_{ab}$ and is classically equivalent (eliminating $\gamma_{ab}$ by its equation of motion returns $S_{\text{NG}}$):

$S_{\text{P}}$ is a two-dimensional field theory of $D$ free scalars $X^\mu$ coupled to 2D gravity. It has three symmetries:

Diffeomorphisms and Weyl invariance together allow the conformal gauge $\gamma_{ab} = \eta_{ab}$. The equations of motion then become the free wave equation, supplemented by constraints from the $\gamma_{ab}$ equation of motion (the vanishing of the worldsheet stress tensor):

The general solution splits into left- and right-moving waves, $X^\mu = X_L^\mu(\tau + \sigma) + X_R^\mu(\tau - \sigma)$. That the two sectors are independent is what later allows the heterotic string to treat them differently.

Boundary conditions

A string with Neumann conditions in $p+1$ directions (including time) and Dirichlet conditions in the rest ends on a D$p$-brane. For many years Dirichlet conditions were treated as a curiosity because they break translation invariance; in 1995 Polchinski showed that D-branes are dynamical objects of the theory itself (see D-branes).

Mode expansion

For the closed string, with $\sigma \in [0, 2\pi)$,

The zero modes $x^\mu, p^\mu$ describe the centre-of-mass motion; the oscillators $\alpha_n^\mu$ (right-moving) and $\tilde{\alpha}_n^\mu$ (left-moving) describe vibrations. The open string has a single set of oscillators, because the boundary conditions reflect left-movers into right-movers.

Quantum String Theory

Oscillators and the Virasoro algebra

Quantization promotes the modes to operators obeying

so each $\alpha_{-n}^\mu$ ($n > 0$) is a creation operator for a vibration with frequency $n$. The Fourier modes $L_n$ of the worldsheet stress tensor generate the Virasoro algebra:

The critical dimension

The number of spacetime dimensions is an output of the quantum theory. In the covariant (BRST) treatment, each free boson $X^\mu$ contributes $c = 1$ to the central charge and the Faddeev–Popov ghosts from gauge-fixing contribute $c = -26$. The Weyl symmetry survives quantization only if the total vanishes:

In light-cone gauge the same result appears differently: only the $D-2$ transverse oscillators are physical, the normal-ordering constant is $a = (D-2)/24$ (from the regularized sum $\sum n = -1/12$), and Lorentz invariance of the quantum theory requires $a = 1$, hence $D = 26$. If $D \neq 26$ the theory either loses Lorentz symmetry or contains negative-norm states. For the superstring, worldsheet fermions add $c = D/2$ and superconformal ghosts contribute $+11$ (for a total ghost contribution of $-15$), giving $\tfrac{3}{2}D = 15$, i.e. $D = 10$.

The mismatch with four observed dimensions is the origin of compactification: the extra dimensions must be small, curved, or otherwise hidden, and their geometry determines the four-dimensional physics (see Compactification).

Bosonic string spectrum

With $N = \sum_{n>0} \alpha_{-n} \cdot \alpha_n$ the oscillator level, the mass-shell conditions are

The closed-string condition $N = \tilde{N}$ is level matching.

The massive states lie on linear Regge trajectories, $J_{\max} = \alpha’ M^2 + 1$ for the open string — the feature that originally connected strings to hadron physics. The tachyon signals that the bosonic string’s flat vacuum is unstable, and the theory has no fermions; it is a laboratory, not a candidate description of nature.

The superstring

Adding worldsheet fermions $\psi^\mu$, superpartners of $X^\mu$ under two-dimensional supersymmetry, gives the RNS superstring. Fermions on the closed string can be periodic or antiperiodic, giving two sectors:

The GSO projection keeps states of definite worldsheet fermion parity. It removes the tachyon, makes the numbers of bosonic and fermionic states match at every level, and yields spacetime supersymmetry. The open superstring’s lightest states are a massless vector and a massless Majorana–Weyl spinor — the ten-dimensional super-Yang–Mills multiplet — followed by a massive tower with $\alpha’ M^2 = N$ for positive integer $N$. The alternative Green–Schwarz formalism makes spacetime supersymmetry manifest instead. Details of both are on the Graduate Formalism page.

The Five Superstring Theories

Quantum consistency — cancellation of gravitational and gauge anomalies, modular invariance, and tadpole cancellation — leaves exactly five supersymmetric string theories in ten flat dimensions:

A few structural points explain the table:

• Type II theories apply the GSO projection to left- and right-movers independently. Choosing opposite chiralities on the two sides gives the non-chiral IIA; the same chirality gives the chiral IIB, whose Ramond–Ramond sector includes a four-form with self-dual field strength.
• Type I is obtained from IIB by gauging worldsheet orientation reversal (an orientifold). Consistency then requires 32 D9-branes, which supply the $SO(32)$ gauge group; Green and Schwarz’s 1984 anomaly-cancellation argument singled out exactly this group.
• Heterotic strings combine a ten-dimensional superstring in one chiral sector with a 26-dimensional bosonic string in the other. The 16 surplus bosonic dimensions are compactified on an even self-dual lattice; the only two such lattices in 16 dimensions give $\mathrm{Spin}(32)/\mathbb{Z}_2$ and $E_8 \times E_8$.

The low-energy limit of each theory is the corresponding ten-dimensional supergravity. Their perturbative spectra differ — but in the mid-1990s it became clear that all five, together with eleven-dimensional supergravity, are limits of one theory, connected by dualities:

flowchart TB
    M["M-theory<br/>(11D; low energy: 11D supergravity)"]
    IIA["Type IIA"]
    IIB["Type IIB"]
    I["Type I"]
    HO["Heterotic SO(32)"]
    HE["Heterotic E8 x E8"]
    M -->|"compactify on circle S¹"| IIA
    M -->|"compactify on interval S¹/Z₂"| HE
    IIA <-->|"T-duality"| IIB
    HO <-->|"T-duality"| HE
    I <-->|"S-duality"| HO
    IIB -->|"S-duality: self-dual"| IIB
    IIB -->|"orientifold"| I

T-duality relates a theory on a circle of radius $R$ to another on radius $\alpha’/R$; S-duality relates strong coupling $g_s$ to weak coupling $1/g_s$; and the strong-coupling limits of Type IIA and heterotic $E_8 \times E_8$ each grow an eleventh dimension. These dualities, the D-branes that make them possible, and the compactifications and holographic dualities built on them are developed in D-Branes, Dualities & M-Theory.

Current Status

As of 2026, string theory is best described as a mathematically consistent framework for perturbative quantum gravity with a rich but incompletely understood non-perturbative structure. Its main established results are internal and theoretical: finite perturbative graviton scattering, the microscopic counting of entropy for supersymmetric black holes, and the AdS/CFT correspondence, which is now a standard tool in quantum field theory and quantum-information approaches to gravity. Its main open problems are the absence of a complete non-perturbative definition, the enormous number of candidate vacua and the question of whether any describe our accelerating universe, and the lack of experimental signatures at accessible energies. These are discussed in Criticisms & Research Frontiers.

See Also

• D-Branes, Dualities & M-Theory — the non-perturbative structure connecting the five theories.
• Criticisms & Research Frontiers — the landscape, the Swampland, holography, and experimental prospects.
• Graduate Formalism — worldsheet CFT, BRST quantization, and the full mathematical treatment.
• Quantum Field Theory — the point-particle framework that string theory extends.
• Gauge Theory and the Standard Model — the low-energy physics string compactifications must reproduce.
• Quantum Gravity — string theory in the context of other approaches.
• Black Holes — the geometry whose entropy string theory accounts for microscopically.
• Statistical Mechanics — the entropy counting behind black-hole thermodynamics.
• Physics Hub — all physics topics.