String theory replaces point particles by one-dimensional objects whose histories are two-dimensional worldsheets.
Bosonic string theory quantizes a relativistic string using only bosonic worldsheet fields. Lorentz-invariant quantization requires 26-dimensional target spacetime and contains a tachyonic ground state.
The Polyakov action introduces an independent worldsheet metric :Its metric equation makes conformal to the induced worldsheet metric, reducing it classically to the Nambu–Goto action.
A worldsheet diffeomorphism is a smooth reparameterization of the worldsheet coordinates under which is a scalar and is a rank-two tensor.
Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as for a string.
A Weyl transformation rescales the worldsheet metric locally, , without changing . The classical Polyakov action is Weyl invariant.
After conformal gauge fixing, independent reparameterizations and remain, accompanied by a compensating Weyl transformation.
Light-cone gauge uses residual conformal transformations to make the target-space coordinate linear in worldsheet time. The Virasoro constraints then determine from the transverse fields.
The worldsheet metric equation in conformal gauge sets the stress tensor to zero, . In oscillator language these are the Virasoro constraints.
Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost and a symmetric traceless antighost . In complex coordinates they form holomorphic and antiholomorphic systems.
A string worldsheet is the two-dimensional surface swept out by a string in spacetime. Local coordinates are commonly denoted .
A two-dimensional conformal field theory is invariant under local holomorphic and antiholomorphic conformal transformations.
A primary operator of weights transforms covariantly under local conformal maps and has a universal operator-product expansion with the stress-energy tensor.
The conformal weights determine the response of a primary operator to holomorphic and antiholomorphic coordinate rescaling. Its scaling dimension is and its spin is .
An operator product expansion expresses the short-distance product of local operators as a sum of local operators multiplied by singular coefficient functions.
The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.
Normal ordering removes self-contractions from a composite operator. For free fields it places annihilation modes to the right of creation modes.
Wick theorem expresses a product of free fields as its normal-ordered product plus all possible contractions.
The Fourier modes and of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.
Closed-string periodicity and the residual spatial constraint require equal left- and right-moving excitation levels, .
The normal-ordering constant is the zero-point shift in the quantum Virasoro generator. Closure of the target-space Lorentz algebra fixes and the spacetime dimension for the bosonic string.
The critical dimension is the target-spacetime dimension in which quantum anomalies obstructing Lorentz or Weyl symmetry cancel. It is 26 for the bosonic string.
Gauge/gravity duality relates a quantum field theory without dynamical gravity to a gravitational or string theory in one additional emergent dimension.
The AdS/CFT correspondence identifies a conformal field theory with quantum gravity on an asymptotically anti-de Sitter spacetime. Boundary sources couple to CFT operators and equal boundary values of bulk fields.
A Euclidean conformal transformation pulls the flat metric back to a position-dependent positive multiple of itself. Its infinitesimal generator obeys the conformal Killing equation.
Three scalar primaries have a conformally fixed position dependence, leaving one theory-dependent coefficient .
A Witten diagram is a position-space perturbative diagram in anti-de Sitter spacetime whose external bulk-to-boundary propagators end at boundary operator insertions.
In a large-rank gauge theory, a single-trace operator is dual at leading order to a single bulk field, while multi-trace operators correspond to multiparticle bulk states.
The holographic dictionary identifies leading near-boundary bulk-field coefficients with CFT sources and subleading normalizable coefficients with operator expectation values.
For a bulk gauge field, the leading boundary value of its time component is the dual chemical potential, while the normalizable radial coefficient determines the charge density.
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String theory is a theoretical framework in physics that attempts to reconcile quantum mechanics and general relativity, two fundamental but seemingly incompatible theories that describe how the universe works at very small and very large scales. The core idea of string theory is that the fundamental building blocks of the universe are not point-like particles, as traditionally thought, but rather tiny, vibrating strings of energy.
