In 1984, Michael Green and John Schwarz ignited the First Superstring Revolution by proving that chiral gauge anomalies in ten-dimensional supergravity cancel identically if and only if the gauge group is ( ext{SO}(32)) or (E_8 imes E_8). Suddenly, string theory was recognized not as a mathematical curiosity, but as the only known candidate for a fully unified Theory of Everything. Remarkably, there are precisely five mathematically self-consistent, anomaly-free superstring theories in ten dimensions. In this treatise, we systematically dissect their symmetries, spectra, and field contents.
1. The Type II Superstring Theories: IIA vs. IIB
Type II theories contain only closed oriented strings with maximal (mathcal{N}=2) spacetime supersymmetry (32 supercharges):
- Type IIA Superstring (Non-Chiral): The left-moving and right-moving GSO projections are chosen with opposite chiralities in the Ramond sectors (( ext{GSO}_L = +1, ext{GSO}_R = -1)). The theory is non-chiral:
[
ext{Massless Bosons: } quad ext{NS-NS} , (g_{mu
u}, B_{mu
u}, Phi) quad oplus quad ext{R-R} , (C_1, C_3)
]
The Ramond-Ramond (R-R) gauge potentials are odd differential forms ((C_1, C_3)), coupling to stable odd D-branes (( ext{D0, D2, D4, D6, D8})). - Type IIB Superstring (Chiral): Both left and right GSO projections choose identical chiralities (( ext{GSO}_L = ext{GSO}_R = +1)), producing a strictly chiral theory with (mathcal{N}=(2,0)) supersymmetry:
[
ext{Massless Bosons: } quad ext{NS-NS} , (g_{mu
u}, B_{mu
u}, Phi) quad oplus quad ext{R-R} , (C_0, C_2, C_4^+)
]
The R-R 4-form potential (C_4^+) has a self-dual 5-form field strength ((F_5 = *F_5)), which plays a monumental role in D3-branes and the AdS/CFT correspondence.
2. Type I Superstring: Open Strings and Unoriented Worlds
Type I string theory is constructed from Type IIB by performing an unoriented Orientifold Projection (Omega: sigma o 2pi – sigma), which flips worldsheet parity, identifying left-moving and right-moving sectors. To cancel tadpole anomalies, open strings must be introduced:
[
ext{Type I} = ( ext{Type IIB}) / Omega quad oplus quad ext{Open Strings Ending on 32 D9-Branes}
]
The Chan-Paton topological charges at the endpoints of open strings produce an ( ext{SO}(32)) Yang-Mills gauge theory coupled to (mathcal{N}=1) supergravity.
3. The Heterotic Superstring Theories
Conceived by the “Princeton String Quartet” (Gross, Harvey, Martinec, and Rohm) in 1985, Heterotic Strings are brilliant relativistic hybrids: the right-moving modes are a 10D superstring, while the left-moving modes are a 26D bosonic string!
The extra 16 left-moving dimensions are compactified on an internal flat torus (T^{16} = mathbb{R}^{16}/Lambda). Modular invariance of the 1-loop torus partition function requires the lattice (Lambda) to be Even and Self-Dual. In 16 dimensions, Lie algebra classification permits precisely two such lattices:
- Heterotic ( ext{SO}(32)): Lattice (Gamma_{16} = ext{Spin}(32)/mathbb{Z}_2).
- Heterotic (E_8 imes E_8): Lattice (Gamma_8 imes Gamma_8), where (Gamma_8) is the root lattice of the exceptional Lie group (E_8).
Heterotic (E_8 imes E_8) is the prime candidate for particle phenomenology: compactifying on a Calabi-Yau threefold breaks one (E_8) into (E_6 supset ext{SU}(3) imes ext{SU}(2) imes ext{U}(1)), producing chiral matter in the fundamental (mathbf{27}) representation with three generations matching the Standard Model.
4. The Green-Schwarz Anomaly Cancellation Mechanism
In 10 dimensions, chiral gauge and gravitational anomalies are governed by a 12-form polynomial (I_{12}). In 1984, Green and Schwarz showed that the irreducible hexagonal anomaly factorizes as:
[
I_{12} = ( ext{tr} R^2 – ext{tr} F^2 ) wedge X_8
]
if and only if (dim(G) = 496), which is uniquely satisfied by ( ext{SO}(32)) and (E_8 imes E_8) ((dim E_8 = 248 implies 248 + 248 = 496)). Modifying the 2-form gauge transformation (delta B = ext{tr}(Lambda d A) – ext{tr}(Theta d omega)) cancels the anomaly completely via classical-quantum tree-level exchange, cementing the 5 string theories as mathematical masterpieces.
