Hypothetical framework in which elementary particles are one-dimensional strings whose vibrational modes yield the particle spectrum (including massless spin-2 states interpretable as gravitons). Internally consistent only in 10 dimensions…
string-theory
Bosonic string action
Nambu–Goto action S = –T ∫ d²σ √(–det h_ab) with induced worldsheet metric h_ab = ∂_aX^μ∂_bX_μ (Polyakov form introduces auxiliary…
Superstring theory
Worldsheet supersymmetry (or spacetime SUSY via GSO projection) eliminates the tachyon and drops critical dimension from 26 to 10. Five…
M-theory
11-dimensional theory whose strong-coupling limits yield the five 10D superstring theories via compactification/dualities. Fundamental…
Compactification
Six of the 10 spatial dimensions wrapped on a compact internal manifold (Calabi–Yau threefold for N=1 SUSY preservation). Shape/size moduli…
Calabi–Yau manifold
Compact Ricci-flat Kähler manifold with trivial canonical bundle and holonomy SU(n). The 6-D three-fold case is the preferred internal…
AdS/CFT correspondence
Maldacena's conjecture: type IIB string theory on AdS₅×S⁵ is dual to N=4 SU(N) super Yang–Mills in four dimensions. First concrete…
T-duality
Compactifying a closed string on a circle of radius R is physically equivalent to compactifying on radius α'/R, exchanging momentum and…
String landscape
Discretuum of metastable vacua arising from flux compactifications of type IIB on Calabi–Yau orientifolds (KKLT). Estimated cardinality…
Calabi–Yau quintic threefold (string-theory compactification)
String-theory application of L0 Hodge theory and projective varieties. The generic smooth degree-5 hypersurface X ⊂ ℙ⁴ is the prototypical…
Mirror symmetry Hodge-diamond swap (string-theory duality)
String-theory application of L0 sheaf cohomology and moduli spaces. Mirror symmetry posits, for every Calabi–Yau n-fold X, a mirror…
Monstrous moonshine: j-function coefficients as Monster irreducible dimensions (string-theory)
String-theory application of L0 modular-form and Euler-product. The Klein j-function j(τ) is the fundamental SL(2,ℤ)-invariant modular…
K3 elliptic genus and Mathieu-M₂₄ moonshine (string-theory)
String-theory application of L0 elliptic-curve and Dirichlet-L-function. The elliptic genus of the K3 complex surface, φ_{K3}(τ, z), is a…
Polyakov action S = (T/2) int sqrt(-h) h^{ab} d_a X^mu d_b X_mu d^2 sigma; CY compactification
Polyakov action and bosonic critical dimension framework (Polyakov 1981 Phys Lett B 103, 207; Brink-Di Vecchia-Howe 1976). Setup: the…
T-duality R -> alpha'/R; self-dual radius R = sqrt(alpha'); Day-convolution Fourier-Mukai
T-duality framework for string compactification on a circle (Kikkawa-Yamasaki 1984 Phys Lett B 149, 357; Sakai-Senda 1986). Setup: closed…
Theorem: D - 26 = 0 (bosonic-string critical dimension)
Theorem (bosonic-critical canonical): the bosonic string's central charge is c = D_{target} for the matter sector and c_{ghost} = -26 for…
Theorem: R^2 - alpha' = 0 at R = sqrt(alpha') (T-duality self-dual radius)
Theorem (T-duality self-dual canonical): R = sqrt(alpha') squared gives R² = alpha', residual R² - alpha' = 0 identically. Canonical sympy…
AdS/CFT (Maldacena 1997)
J Maldacena 1997 N=4 SYM_4 = type-IIB on AdS5 x S5; foundational gauge-gravity duality; modern holographic-RG + entanglement-spacetime.
String revolutions
Green-Schwarz 1984 1st-revolution anomaly cancellation; Witten 1995 M-theory 2nd-revolution; modern landscape + swampland.
Veneziano amplitude (1968)
G Veneziano 1968 dual-resonance-model B(s,t)=Γ(-α(s))Γ(-α(t))/Γ(-α(s)-α(t)); birth of string theory.
Scherk-Schwarz (1974)
J Scherk-J Schwarz 1974 closed-string spin-2 graviton; reinterpretation of strings as quantum-gravity rather than hadrons.
GS anomaly (Green-Schwarz 1984)
M Green-J Schwarz 1984 SO(32)/E8×E8 anomaly-cancellation; first superstring revolution; phenomenologically-viable string.
M-theory (Witten 1995)
E Witten 1995 11D M-theory; modern unification of 5 superstring theories + heterotic; 3rd revolution non-perturbative.
AdS/CFT (Maldacena 1997)
J Maldacena 1997 N=4 SYM duality; modern holographic-QCD + condmat-strange-metals + 100k+ citations.
F-theory (Vafa 1996)
C Vafa 1996 12D F-theory; modern elliptic-Calabi-Yau compactifications + GUT model-building landscape.