A class of theories that attempt to explain all existing particles (including force carriers) as vibrational modes of extended objects, such as the 1-dimensional fundamental string. PLEASE DO NOT USE THIS TAG for non-relativistic material strings, such as, e.g., a guitar string.

A class of theories that attempt to explain all existing particles (including force carriers) as vibrational modes of extended objects, such as the 1-dimensional fundamental string. String theory in its original form arose as a theory of hadrons, but soon failed due to its inability to explain fermions and its unstable tachyonic ground state. It was consistent only in 26 dimensions. However, with the discovery of , fermions were included in the theory and the discovery of the GSO projection projected out the tachyonic states.

Soon, it was discovered, in the first superstring revolution, that there are at least 5 consistent supersymmetric string theories, or in short, superstring theories, namely, the Type IIB, the Type IIA, the Type I, the Type HO and the Type HE. These are all consistent only in 10 dimensions.

The Type IIB and Type IIA differed in terms of the choice of the GSO Projection, and thus in chirality. The Type I was an orientifold projection of the Type IIB along with the open string sectors. The Type HO and Type HE were discovered upon tensoring the left-moving state of the non-supersymmetric, bosonic string theory (with added Majorana-Weyl fermions, but no supersymmetry) and the right-moving state of the Type II (it doesn't matter whether the Type IIA or the Type IIB are taken, since the left- and right- movers hardly interact at all.) string theory and then compactifying the 16 mismatched dimensions on an even, unimodular lattice. There are two consistent choices of this lattice, $\frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}^2}$ and $E_8\otimes E_8$, which lead to the Type HO and Type HE respectively.

The Type HE string theory was generally preferred due to its gauge group, $E_8\otimes E_8$, having the standard model gauge group as its subgroups.

However, until now, string theory was perturbative, and thus full of . Also, perturbation techniques are actually pretty much of an approximation. Both of these are fine in theories such as QED, QCD, etc. and such QFTs, but very unattractive features for a potential candidate for a Theory of Everything. Ideally, one would like to have a non-perturbative formulation of a Theory of Everything.

During the second superstring revolution, dualities were discovered between string theories. For example, the Type IIA and Type IIB happened to exhibit T-duality, and so did the Type HE and the Type HO. Similarly, the Type I and the Type HO exhibited S-duality, and the Type IIB with a fundamental string exhibited S-duality with Type IIB with a D1 brane. Another discovery was that of D-branes. T-duality mapped between Neumann boundary conditions and Dirichlet boundary conditions, so free open strings (Neumann) in Type IIA, for example, would result in open strings bound to D-branes (Dirichlet). Another requirement for D-branes was to ensure that something coupled to Ramond-Ramond potentials, and not only Ramond-Ramond fields.

Due to the dualities, instead of 5 different 10-dimensional string theories, there were only 2; the Type IIA and the Type HE. It was postulated that these are actually S-dual to a more fundamental 11-dimensional M-theory compactified on a circle and a line segment respectively.

Nearing the end of the second superstring revolution, a fully non-perturbative formulation of M-theory was found, called M(atrix) theory. This relies on the AdS/CFT correspondence, which is a result of the holographic principle. This correspondence states that a string theory in $D+1$ - dimensional Anti-de-Sitter (AdS) Space is completely described by the (non-relativisitc) $D$ - dimensional Conformal Field theory (CFT) on its boundary. The CFT corresponding to 11-dimensional M-theory in AdS Space is 10-dimensional Supersymmetric Yang-Mills theory with gauge group $\operatorname{lim}\limits_{N\rightarrow\infty}U(N)$.

It was later proposed that the case when $N$ does not approach $\infty$ is actually non-perturbative Type IIA String theory. The reasoning is that uncompactified M-theory, or M-theory compactified on a circle of infinite radius, needs to have $N$ to approach infinity because the momenta is quantised as $\frac{N}{R}$ so if $R\rightarrow \infty$, $N\rightarrow \infty$ in order to allow momenta to be non-zero, but for Type IIA string theory, which is M-theory compactified on a circle of finite $R$, $N$ must be finite too to ensure that the momenta is finite. This is known as Matrix string theory.

After the second superstring revolution, attention went to obtaining a 4-dimensional compactification of 11-dimensional M-theory, to reconcile with our daily lives. M-theory has supersymmetry $\mathcal N=8$; however, $\mathcal N=1$ is a rather elegant supersymmetry.

Compactifying on a $G_2$ manifold, i.e. a manifold whose holonomy group is $G_2$, preserves $\frac18$ of the supersymmetry, so compactifying M-theory on them would yield an $\mathcal N=1$ supersymmetry. Furthermore, $G_2$ manifolds are 7-dimensional, so compactifying M-theory on them would result in a 4-dimensional theory. Thus, $G_2$ manifolds are an ideal manifold to compactify on.

There are, however, on the order of $10^{500}$ to $10^{520}$ $G_2$ manifolds in existence, which means that there are between $10^{500}$ and $10^{520}$ $\mathcal N = 1$ 4-dimensional M-theories, called the 4-dimensional $\mathcal N=1$ M-theory vacua. These 4-dimensional $\mathcal N=1$ M-theories form what is known as the 4-dimensional $\mathcal N=1$ M-theory landscape. There is still no complete classification of this landscape and thus it is still not known which is the M-theory that actually describes our universe, or whether any of them do.

Note: In string theory, it is quite common that natural units $\ell_s=c_0=1$ are used.

Prerequisites to learn String Theory:

Phys: Quantum Field Theory (and all its math-phys prerequisites); Path Integral Formalism; Standard Model (SM) and Higg's Mechanism; Sigma Models; Regularization and Renormalization; Renormalization Group (RG); General Relativity (GR); Thermodynamics and Statistical Physics; Conformal Field Theory (CFT); Faddeev-Popov and Becchi-Rouet-Stora-Tyutin (BRST) Formalism; Matrix Models; Supersymmetry and Superfield Formalism; Instantons; Anomalies; S-Matrix Theory. Maybe also: Topological Field Theory (TFT); Batalin-Vilkovisky (BV) Formalism.

Math: Complex Function Theory; Real and Complex Differential Geometry; Homology and Cohomology Theory; Infinite-Dimensional Lie algebras; Clifford Algebras; Algebraic Topology; Algebraic Geometry; Superalgebras and Supermanifolds. Maybe also: Atiyah-Singer Index Theory; K-Theory; Non-Commutative Geometry (NCG); Category Theory.

Good Resources:

Polchinski (1,2), Becker, Becker, Schwarz, McMahon, Kaku (1), Kaku (2), Mohaupt, Szabo, Wray,