{"id":1341,"date":"2026-07-22T08:58:53","date_gmt":"2026-07-22T07:58:53","guid":{"rendered":"https:\/\/vitormsalmeida.pt\/?p=1341"},"modified":"2026-07-22T09:38:17","modified_gmt":"2026-07-22T08:38:17","slug":"the-five-string-theories-and-the-need-for-m-theory","status":"publish","type":"post","link":"https:\/\/vitormsalmeida.pt\/index.php\/2026\/07\/22\/the-five-string-theories-and-the-need-for-m-theory\/","title":{"rendered":"The five string theories and the need for M-theory."},"content":{"rendered":"\n

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

The five consistent string theories (superstrings) in 10 dimensions are:
– Type I <\/strong>
S<\/mi>=<\/mo>12<\/mn>\u03ba<\/mi>102<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2009<\/mtext>\u2212<\/mo>G<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow>(<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2212<\/mo>112<\/mn>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>)<\/mo>\u2212<\/mo>12<\/mn>g<\/mi>102<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2009<\/mtext>\u2212<\/mo>G<\/mi>\u2009<\/mtext>T<\/mi>r<\/mi>(<\/mo>F<\/mi>\u03bc<\/mi>\u03bd<\/mi>F<\/mi>\u03bc<\/mi>\u03bd<\/mi>)<\/mo>+<\/mo>\u22ef<\/mo><\/mrow>S=12\u03ba102\u222bd10x\u2009\u2212G\u2009e\u22122\u03a6(R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1)\u221212g102\u222bd10x\u2009\u2212G\u2009Tr(F\u03bc\u03bdF\u03bc\u03bd)+\u22ef<\/annotation><\/semantics><\/math>
Where:
G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow>G\u03bc\u03bd<\/annotation><\/semantics><\/math> : spacetime metric (graviton)
\u03a6 : dilaton (scalar field that controls string coupling)
H=dB+H = dB +H = dB + Chern-Simons terms (antisymmetric field of 3 forms, coming from B<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow>B\u03bc\u03bd<\/annotation><\/semantics><\/math> )
F<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow>F\u03bc\u03bd<\/annotation><\/semantics><\/math>: Yang-Mills force field of the SO(32) group
The terms \u201c$\\cdots$\u201d include fermionic interactions, Chern-Simons terms and higher-order corrections (\u03b1\u2032 ).
– Type IIA <\/strong>
S<\/mi>I<\/mi>I<\/mi>A<\/mi>,<\/mo>b<\/mi>o<\/mi>s<\/mi>o<\/mi>n<\/mi>i<\/mi>c<\/mi>=<\/mo>12<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>g<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi>\u03d5<\/mi>\u2212<\/mo>112<\/mn>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>]<\/mo>S<\/mi>IIA, bosonic<\/mtext><\/msub>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>g<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi><\/mrow><\/msup>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03d5<\/mi>\u2212<\/mo>1<\/mn>12<\/mn><\/mfrac>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msub>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msup>]<\/mo><\/mrow>S<\/mi>IIA, bosonic<\/mtext><\/msub>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>g<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi><\/mrow><\/msup>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03d5<\/mi>\u2212<\/mo>1<\/mn>12<\/mn><\/mfrac>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msub>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msup>]<\/mo><\/mrow>\u2212<\/mo>14<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>g<\/mi>[<\/mo>12<\/mn>F<\/mi>2<\/mn>,<\/mo>\u03bc<\/mi>\u03bd<\/mi>F<\/mi>2<\/mn>\u03bc<\/mi>\u03bd<\/mi>+<\/mo>124<\/mn>F<\/mi> <\/mtext>4<\/mn>,<\/mo>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>\u03c3<\/mi>F<\/mi> <\/mtext>4<\/mn>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>\u03c3<\/mi>]<\/mo>\u2212<\/mo>14<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>B<\/mi>\u2227<\/mo>F<\/mi>4<\/mn>\u2227<\/mo>F<\/mi>4<\/mn><\/mrow>SIIA, bosonic=12\u03ba2\u222bd10x\u2212g\u2009e\u22122\u03d5[R+4\u2202\u03bc\u03d5\u2202\u03bc\u03d5\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1]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] \u221214\u03ba2\u222bd10x\u2212g[12F2,\u03bc\u03bdF2\u03bc\u03bd+124F~4,\u03bc\u03bd\u03c1\u03c3F~4\u03bc\u03bd\u03c1\u03c3]\u221214\u03ba2\u222bB\u2227F4\u2227F4<\/annotation><\/semantics><\/math>
Where:
g<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow>g\u03bc\u03bd<\/annotation><\/semantics><\/math> \u2014 metric (gravity)
\u03d5 \u2014 dilaton (scalar field)
B<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow>B\u03bc\u03bd<\/annotation><\/semantics><\/math> \u2014 2-way field (NS-NS), with H3=dB
C1 (or A<\/mi>\u03bc<\/mi><\/mrow>A\u03bc<\/annotation><\/semantics><\/math> ) \u2014 1-form potential RR \u2192 F2=dC1
C<\/mi>3<\/mn><\/msub>C_3<\/annotation><\/semantics><\/math> \u2014 3-form potential RR \u2192 F<\/mi>4<\/mn>=<\/mo>d<\/mi>C<\/mi>3<\/mn><\/mrow> F4=dC3<\/annotation><\/semantics><\/math>
F<\/mi> <\/mtext>4<\/mn>=<\/mo>F<\/mi>4<\/mn><\/mrow>F~4=F4<\/annotation><\/semantics><\/math> –C<\/mi>1<\/mn>\u2227<\/mo>H<\/mi>3<\/mn><\/mrow>C1\u2227H3<\/annotation><\/semantics><\/math> (modified force field)
\u03ba \u2014 gravitational coupling constant
– Type IIB <\/strong>
S<\/mi>I<\/mi>I<\/mi>B<\/mi>,<\/mo>b<\/mi>o<\/mi>s<\/mi>o<\/mi>n<\/mi>i<\/mi>c<\/mi>=<\/mo>12<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>g<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi>\u03d5<\/mi>\u2212<\/mo>12<\/mn>\u2223<\/mtext>H<\/mi>\u2223<\/mtext>2<\/mn>]<\/mo>S<\/mi>IIB, bosonic<\/mtext><\/msub>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>g<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi><\/mrow><\/msup>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03d5<\/mi>\u2212<\/mo>1<\/mn>2<\/mn><\/mfrac>|<\/mi>H<\/mi>|<\/mi>2<\/mn><\/msup>]<\/mo><\/mrow>S<\/mi>IIB, bosonic<\/mtext><\/msub>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>g<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03d5<\/mi><\/mrow><\/msup>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03d5<\/mi>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03d5<\/mi>\u2212<\/mo>1<\/mn>2<\/mn><\/mfrac>|<\/mi>H<\/mi>|<\/mi>2<\/mn><\/msup>]<\/mo><\/mrow>\u2212<\/mo>14<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>g<\/mi>[<\/mo>\u2223<\/mtext>F<\/mi>1<\/mn>\u2223<\/mtext>2<\/mn>+<\/mo>\u2223<\/mtext>F<\/mi> <\/mtext>3<\/mn>\u2223<\/mtext>2<\/mn>+<\/mo>12<\/mn>\u2223<\/mtext>F<\/mi> <\/mtext>5<\/mn>\u2223<\/mtext>2<\/mn>]<\/mo>\u2212<\/mo>14<\/mn>\u03ba<\/mi>2<\/mn>\u222b<\/mo>C<\/mi>4<\/mn>\u2227<\/mo>H<\/mi>\u2227<\/mo>F<\/mi>3<\/mn><\/mrow>SIIB, bosonic=12\u03ba2\u222bd10x\u2212g\u2009e\u22122\u03d5[R+4\u2202\u03bc\u03d5\u2202\u03bc\u03d5\u221212\u2223H\u22232]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] \u221214\u03ba2\u222bd10x\u2212g[\u2223F1\u22232+\u2223F~3\u22232+12\u2223F~5\u22232]\u221214\u03ba2\u222bC4\u2227H\u2227F3<\/annotation><\/semantics><\/math>
Where:
\u03d5: dilaton
H = dB: 3-form Kalb-Ramond field (NS-NS)
F\u2081 = dC\u2080: 1-form (axion)
F\u2083 = dC\u2082: 3-form RR
F\u2085 = dC\u2084: 5-form RR F\u0303\u2083 = F\u2083 \u2212 C\u2080 \u2227 H (modified field) F\u0303\u2085 = F\u2085 \u2212 \u00bd C\u2082 \u2227 H + \u00bd B \u2227 F\u2083 (modified field, with F\u0303\u2085 = \u22c6 F\u0303\u2085 imposed)
– Heterotic SO(32) <\/strong>
S<\/mi>=<\/mo>12<\/mn>\u03ba<\/mi>102<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>G<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2212<\/mo>112<\/mn>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi>]<\/mo>\u2212<\/mo>14<\/mn>g<\/mi>Y<\/mi>M<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>G<\/mi>\u2009<\/mtext>T<\/mi>r<\/mi>(<\/mo>F<\/mi>\u03bc<\/mi>\u03bd<\/mi>F<\/mi>\u03bc<\/mi>\u03bd<\/mi>)<\/mo>+<\/mo>\u2026<\/mo><\/mrow>S=12\u03ba102\u222bd10x\u2212G\u2009e\u22122\u03a6[R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1]\u221214gYM2\u222bd10x\u2212G\u2009Tr(F\u03bc\u03bdF\u03bc\u03bd)+\u2026<\/annotation><\/semantics><\/math>
Where:
G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>G_{\u03bc\u03bd}<\/annotation><\/semantics><\/math> is the metric (graviton).
\u03a6 is the dilaton (scalar field that controls the coupling of strings g<\/mi>s<\/mi><\/msub>=<\/mo>e<\/mi>\u27e8<\/mo><\/msup>\u03a6<\/mi><\/mrow>\u27e9<\/mo><\/mrow>g_s = e^\u27e8\u03a6\u27e9<\/annotation><\/semantics><\/math>).
H = dB + (Chern-Simons terms with F \u2227 F) is the 3-form field (antisymmetrization of B_{\u03bc\u03bd}).
F is the Yang-Mills curvature (gauge strength) with group SO(32).
\u03ba\u2081\u2080 and g_{YM} are related coupling constants.
There are also fermions (gravitino, dilatino, gaugino) to complete the N=1 supersymmetry in 10D.
– Heterotic E\u2088 \u00d7 E\u2088 <\/strong>
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 \u03b1\u2032):
S<\/mi>=<\/mo>12<\/mn>\u03ba<\/mi>102<\/mn>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>G<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow>[<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2202<\/mi>\u03bc<\/mi>\u03a6<\/mi><\/mrow>\u2212<\/mo>13<\/mn>\u2223<\/mtext>H<\/mi>\u2223<\/mtext>2<\/mn>\u2212<\/mo>\u03b1<\/mi>\u2032<\/mtext>4<\/mn>T<\/mi>r<\/mi>\u2223<\/mtext>F<\/mi>\u2223<\/mtext>2<\/mn>+<\/mo>O<\/mi>(<\/mo>\u03b1<\/mi>\u2032<\/mtext>2<\/mn>)<\/mo>]<\/mo><\/mrow>S=12\u03ba102\u222bd10x\u2212G\u2009e\u22122\u03a6[R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u221213\u2223H\u22232\u2212\u03b1\u20324Tr\u2223F\u22232+O(\u03b1\u20322)]<\/annotation><\/semantics><\/math>
Main fields:
G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>G_{\u03bc\u03bd}<\/annotation><\/semantics><\/math>: Metric (graviton)
\u03a6: Dilaton (scalar)
B_{\u03bc\u03bd}: 2-form antisymmetric field (NS-NS)
A_\u03bc (or gauge field): Gauge connection with E\u2088 \u00d7 E\u2088 group (or SO(32) in the other heterotic)
H<\/mi>=<\/mo>d<\/mi>B<\/mi>+<\/mo>(<\/mo>\u03b1<\/mi>\u2032<\/mo><\/msup>\/<\/mi>4<\/mn>)<\/mo>(<\/mo>\u03c9<\/mi>C<\/mi>S<\/mi><\/mrow>L<\/mi><\/msubsup>\u2212<\/mo>\u03c9<\/mi>C<\/mi>S<\/mi><\/mrow>Y<\/mi>M<\/mi><\/mrow><\/msubsup>)<\/mo><\/mrow>H = dB + (\u03b1’\/4) (\u03c9_{CS}^L – \u03c9_{CS}^{YM})<\/annotation><\/semantics><\/math>: 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 (<\/mo>G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>=<\/mo>8<\/mn>\u03c0<\/mi>T<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>)<\/mo><\/mrow>(G_{\\mu\\nu} = 8\\pi T_{\\mu\\nu})<\/annotation><\/semantics><\/math>, 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 = -T\u00b2\u222bd\u00b2\u03c3 -hhab \u2202aX\u03bc\/\u2202bX\u03bcS = -T\/2 \u221a(-h) \u221a(hab) \u2202a X\u221a(m) \u221a(b X\u221a(m)) (or in the conformal gauge).
Where:
– (T): string tension (related to \u03b1\u2032’ , Regge’s parameter).
– X<\/mi>\u03bc<\/mi>(<\/mo>\u03c3<\/mi>,<\/mo>\u03c4<\/mi>)<\/mo><\/mrow>X\u03bc(\u03c3,\u03c4)<\/annotation><\/semantics><\/math> : spacetime coordinates (embeddings).
– h<\/mi>a<\/mi>b<\/mi><\/mrow><\/msub>h_{ab}<\/annotation><\/semantics><\/math> : auxiliary metric in the sheet-world (2D).
– \u03c3,\u03c4 : 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<\/mi>=<\/mo>S<\/mi>b<\/mi>o<\/mi>s<\/mi>o<\/mi>\u02c6<\/mtext>n<\/mi>i<\/mi>c<\/mi>o<\/mi>+<\/mo>i<\/mi>2<\/mn>\u03c0<\/mi>\u03b1<\/mi>\u2032<\/mtext>\u222b<\/mo>d<\/mi>2<\/mn>\u03c3<\/mi>\u2009<\/mtext>\u03c8<\/mi>\u02c9<\/mtext>\u03bc<\/mi>\u03c1<\/mi>a<\/mi>\u2202<\/mi>a<\/mi>\u03c8<\/mi>\u03bc<\/mi><\/mrow>S=Sboso\u02c6nico+i2\u03c0\u03b1\u2032\u222bd2\u03c3\u2009\u03c8\u02c9\u03bc\u03c1a\u2202a\u03c8\u03bc<\/annotation><\/semantics><\/math>.
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 G<\/mi>\u03bc<\/mi>\u03bd<\/mi>G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>G<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>,<\/mo>d<\/mi>i<\/mi>l<\/mi>a<\/mi>t<\/mi>o<\/mi>n<\/mi>\u03a6<\/mi><\/mrow><\/mrow>G\u03bc\u03bdG_{\\mu\\nu}G_{\\mu\\nu} , dilaton \u03a6<\/annotation><\/semantics><\/math> , 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 E\u2088 \u00d7 E\u2088. Effective action includes Yang-Mills term for gauge group:
S<\/mi>g<\/mi>a<\/mi>u<\/mi>g<\/mi>e<\/mi>\u223c<\/mo>\u222b<\/mo>T<\/mi>r<\/mi>(<\/mo>F<\/mi>\u2227<\/mo>\u2217<\/mo>F<\/mi>)<\/mo>S<\/mi>gauge<\/mtext><\/msub>\u223c<\/mo>\u222b<\/mo>Tr<\/mtext>(<\/mo>F<\/mi>\u2227<\/mo>\u2217<\/mo>F<\/mi>)<\/mo>S<\/mi>gauge<\/mtext><\/msub>\u223c<\/mo>\u222b<\/mo>Tr<\/mtext>(<\/mo>F<\/mi>\u2227<\/mo>\u2217<\/mo>F<\/mi>)<\/mo><\/mrow>Sgauge\u223c\u222bTr(F\u2227\u2217F)S_{\\text{gauge}} \\sim \\int \\text{Tr}(F \\wedge *F)S_{\\text{gauge}} \\sim \\int \\text{Tr}(F \\wedge *F)<\/annotation><\/semantics><\/math>
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=12\u03ba2\u222bd10x\u2212G\u2009e\u22122\u03a6(R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1)+\u2026S<\/mi>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>G<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow><\/mrow><\/msup>(<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03a6<\/mi><\/mrow>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03a6<\/mi><\/mrow>\u2212<\/mo>1<\/mn>12<\/mn><\/mfrac>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msub>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msup>)<\/mo><\/mrow>+<\/mo>.<\/mi>S<\/mi>=<\/mo>1<\/mn>2<\/mn>\u03ba<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac>\u222b<\/mo>d<\/mi>10<\/mn><\/msup>x<\/mi>\u2212<\/mo>G<\/mi><\/mrow><\/msqrt><\/mspace>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow><\/mrow><\/msup>(<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03bc<\/mi><\/msub>\u03a6<\/mi><\/mrow>\u2202<\/mi>\u03bc<\/mi><\/msup>\u03a6<\/mi><\/mrow>\u2212<\/mo>1<\/mn>12<\/mn><\/mfrac>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msub>H<\/mi>\u03bc<\/mi>\u03bd<\/mi>\u03c1<\/mi><\/mrow><\/msup>)<\/mo><\/mrow>+<\/mo>\u2026<\/mo><\/mrow>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<\/annotation><\/semantics><\/math> (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 E\u2088\u00d7E\u2088 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

Contrary to popular belief, String Theory is actually five theories. The five consistent string theories (superstrings) in 10 dimensions are: – Type I S=12\u03ba102\u222bd10x\u2009\u2212G\u2009e\u22122\u03a6(R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1)\u221212g102\u222bd10x\u2009\u2212G\u2009Tr(F\u03bc\u03bdF\u03bc\u03bd)+\u22efS=12\u03ba102\u222bd10x\u2009\u2212G\u2009e\u22122\u03a6(R+4\u2202\u03bc\u03a6\u2202\u03bc\u03a6\u2212112H\u03bc\u03bd\u03c1H\u03bc\u03bd\u03c1)\u221212g102\u222bd10x\u2009\u2212G\u2009Tr(F\u03bc\u03bdF\u03bc\u03bd)+\u22efWhere: G\u03bc\u03bdG\u03bc\u03bd : spacetime metric (graviton) \u03a6 : dilaton (scalar field that controls string coupling) H=dB+H = dB +H = dB + Chern-Simons terms (antisymmetric field of 3 forms, coming from B\u03bc\u03bdB\u03bc\u03bd […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-1341","post","type-post","status-publish","format-standard","hentry","category-my-thoughts"],"_links":{"self":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/comments?post=1341"}],"version-history":[{"count":2,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1341\/revisions"}],"predecessor-version":[{"id":1345,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1341\/revisions\/1345"}],"wp:attachment":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/media?parent=1341"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/categories?post=1341"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/tags?post=1341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}