VitorMSAlmeida.pt

Contrary to popular belief, String Theory is actually five theories.

The five consistent string theories (superstrings) in 10 dimensions are:
– Type I
$S=12κ102∫d10x −G e−2Φ(R+4∂μΦ∂μΦ−112HμνρHμνρ)−12g102∫d10x −G Tr(FμνFμν)+⋯$
Where:
$Gμν$ : spacetime metric (graviton)
Φ : dilaton (scalar field that controls string coupling)
H=dB+H = dB +H = dB + Chern-Simons terms (antisymmetric field of 3 forms, coming from $Bμν$ )
$Fμν$: Yang-Mills force field of the SO(32) group
The terms “$\cdots$” include fermionic interactions, Chern-Simons terms and higher-order corrections (α′ ).
– Type IIA
$SIIA, bosonic=12κ2∫d10x−g e−2ϕ[R+4∂μϕ∂μϕ−112HμνρHμνρ]S_{\text{IIA, bosonic}} = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-g} \, e^{-2\phi} \left[ R + 4 \partial_\mu \phi \partial^\mu \phi – \frac{1}{12} H_{\mu\nu\rho} H^{\mu\nu\rho} \right]S_{\text{IIA, bosonic}} = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-g} \, e^{-2\phi} \left[ R + 4 \partial_\mu \phi \partial^\mu \phi – \frac{1}{12} H_{\mu\nu\rho} H^{\mu\nu\rho} \right] −14κ2∫d10x−g[12F2,μνF2μν+124F~4,μνρσF~4μνρσ]−14κ2∫B∧F4∧F4$
Where:
$gμν$ — metric (gravity)
ϕ — dilaton (scalar field)
$Bμν$ — 2-way field (NS-NS), with H3=dB
C1 (or $Aμ$ ) — 1-form potential RR → F2=dC1
$C_3$ — 3-form potential RR → $F4=dC3$
$F~4=F4$ –$C1∧H3$ (modified force field)
κ — gravitational coupling constant
– Type IIB
$SIIB, bosonic=12κ2∫d10x−g e−2ϕ[R+4∂μϕ∂μϕ−12∣H∣2]S_{\text{IIB, bosonic}} = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-g} \, e^{-2\phi} \left[ R + 4 \partial_\mu \phi \partial^\mu \phi – \frac{1}{2} |H|^2 \right]S_{\text{IIB, bosonic}} = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-g} \, e^{-2\phi} \left[ R + 4 \partial_\mu \phi \partial^\mu \phi – \frac{1}{2} |H|^2 \right] −14κ2∫d10x−g[∣F1∣2+∣F~3∣2+12∣F~5∣2]−14κ2∫C4∧H∧F3$
Where:
ϕ: dilaton
H = dB: 3-form Kalb-Ramond field (NS-NS)
F₁ = dC₀: 1-form (axion)
F₃ = dC₂: 3-form RR
F₅ = dC₄: 5-form RR F̃₃ = F₃ − C₀ ∧ H (modified field) F̃₅ = F₅ − ½ C₂ ∧ H + ½ B ∧ F₃ (modified field, with F̃₅ = ⋆ F̃₅ imposed)
– Heterotic SO(32)
$S=12κ102∫d10x−G e−2Φ[R+4∂μΦ∂μΦ−112HμνρHμνρ]−14gYM2∫d10x−G Tr(FμνFμν)+…$
Where:
$G_{μν}$ is the metric (graviton).
Φ is the dilaton (scalar field that controls the coupling of strings $g_s = e^⟨Φ⟩$).
H = dB + (Chern-Simons terms with F ∧ F) is the 3-form field (antisymmetrization of B_{μν}).
F is the Yang-Mills curvature (gauge strength) with group SO(32).
κ₁₀ and g_{YM} are related coupling constants.
There are also fermions (gravitino, dilatino, gaugino) to complete the N=1 supersymmetry in 10D.
– Heterotic E₈ × E₈
There is no single simple “equation” that defines the entire theory (it is defined by an action on the worldsheet + quantization + quantum consistency), but the main equations arise from the effective low-energy action (N=1 supergravity theory in 10D coupled with Yang-Mills) and the anomaly cancellation conditions.
For Effective Low Energy Action (in a string frame, up to order α′):
$S=12κ102∫d10x−G e−2Φ[R+4∂μΦ∂μΦ−13∣H∣2−α′4Tr∣F∣2+O(α′2)]$
Main fields:
$G_{μν}$: Metric (graviton)
Φ: Dilaton (scalar)
B_{μν}: 2-form antisymmetric field (NS-NS)
A_μ (or gauge field): Gauge connection with E₈ × E₈ group (or SO(32) in the other heterotic)
$H = dB + (α’/4) (ω_{CS}^L – ω_{CS}^{YM})$: Modified 3-form (includes gravitational and Yang-Mills terms from Chern-Simons)
H incorporates the Green-Schwarz mechanism for anomaly cancellation.

They are unified by M-Theory (in 11 dimensions) via dualities. There is no single simple “equation” like in General Relativity $(G_{\mu\nu} = 8\pi T_{\mu\nu})$, but they are defined by actions on the worldsheet (two-dimensional surface swept by the string) and by their effective low-energy actions (supergravities in 10D).

1. Fundamental action in the worldsheet (common basis)
The quantum description starts from the Polyakov (or Nambu-Goto) action with supersymmetry in the worldsheet.
For the bosonic string (26 dimensions, not supersymmetric):
S = -T²∫d²σ -hhab ∂aXμ/∂bXμS = -T/2 √(-h) √(hab) ∂a X√(m) √(b X√(m)) (or in the conformal gauge).
Where:
– (T): string tension (related to α′’ , Regge’s parameter).
– $Xμ(σ,τ)$ : spacetime coordinates (embeddings).
– $h_{ab}$ : auxiliary metric in the sheet-world (2D).
– σ,τ : coordinates in the sheet-world.
For superstrings (10 dimensions), fields are added Fermionics (spinors) and supersymmetry in the worldsheet (RNS or GS formalism).
The basic action includes bosonic + fermionic + supersymmetry terms. The differences between the five theories arise from:
$S=Sbosoˆnico+i2πα′∫d2σ ψˉμρa∂aψμ$.
Orientability (with or without orientable fold). Ramond (R) and Neveu-Schwarz (NS) sectors. Gauge groups (for heterotics).

2. Effective low-energy actions (Supergravity in 10D)
In the low-energy limit (lengths >> string size), each theory reduces to a specific supergravity.
Here are the main features:
– Type IIA (closed strings, non-chiral N=(1,1) supersymmetry):
Fields: metric $GμνG_{\mu\nu}G_{\mu\nu} , dilaton Φ$ , B-field (2-form), Ramond-Ramond forms C1C_1C_1 and C3C_3C_3 .
Effective action includes Einstein terms, dilaton, H = dB, and RR terms.
Type IIB (closed strings, chiral N=(2,0) supersymmetry):
Similar fields, but RR: C0C_0C_0 (scalar), C2C_2C_2 , C4C_4C_4 (with self-duality). Richer in D-branes (odd/even pairs).
– Type I (open + closed strings, non-oriented, gauge group SO(32)):
Combines elements of Type IIB with an orientifold. Includes gauge fields at the end of open strings.
– Heterotic (closed strings, asymmetric: 10D right-hand supersymmetric + 26D bosonic left-hand, with gauge currents): SO(32) and E₈ × E₈. Effective action includes Yang-Mills term for gauge group:
$Sgauge∼∫Tr(F∧∗F)S_{\text{gauge}} \sim \int \text{Tr}(F \wedge *F)S_{\text{gauge}} \sim \int \text{Tr}(F \wedge *F)$
They differ in gauge group, but have very similar actions. The common low-energy bosonic action (N-NS sector, present in all) has the approximate form:
S=12κ2∫d10x−G e−2Φ(R+4∂μΦ∂μΦ−112HμνρHμνρ)+…$S = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-G} \, e^{-2\Phi} \left( R + 4 \partial_\mu \Phi \partial^\mu \Phi – \frac{1}{12} H_{\mu\nu\rho} H^{\mu\nu\rho} \right) +.S = \frac{1}{2\kappa^2} \int d^{10}x \sqrt{-G} \, e^{-2\Phi} \left( R + 4 \partial_\mu \Phi \partial^\mu \Phi – \frac{1}{12} H_{\mu\nu\rho} H^{\mu\nu\rho} \right) + \dots$ (more specific terms for each theory for Ramond-Ramond or gauge fields).

Summary of condensation characteristics:
– Type I The only one that contains open strings (besides closed ones). It has the lowest level supersymmetry (N=1). It is self-dual with the SO(32) heterotic via S-duality.
– Type IIA Closed string theory with type IIA supersymmetry (opposite chirality). Reduced to 11 dimensions gives 11D supergravity.
– Type IIB Very rich in symmetries (has S and T duality). It is the theory that contains the famous D3-brane, widely used in AdS/CFT (gauge/gravity correspondence).
– Heterotic SO(32) Closed strings where the left and right vibration modes are different (“heterotic”). Uses the SO(32) group to cancel anomalies.
– Heterotic E₈×E₈ Considered the most “physical” for a long time, as it allows the construction of Standard Model particle models with three generations of quarks and leptons in a relatively natural way.

Important observations
The equations of motion are obtained by varying these actions (very complex, involving superfields, etc.). Theories are equivalent via dualities (T-duality, S-duality, U-duality), so they are not really five independent theories. For precise calculations, consult classic books such as Polchinski (String Theory), Green-Schwarz-Witten, or notes by David Tong.

Conclusion:
Now I hope the need for M-Theory is understood, as it is the only one that can (by adding one toroidal dimension) unify these five equations into one.