Massless free scalar bosons are a family of two-dimensional conformal field theories, whose symmetry is described by an abelian affine Lie algebra.
Via the Coulomb gas formalism, they lead to exact results in interacting CFTs such as minimal models.
Moreover, they play an important role in the worldsheet approach to string theory.
In a free bosonic CFT, the Virasoro algebra's central charge can take any complex value.
, there exist compactified free bosonic CFTs with arbitrary values of the compactification radius.
This permits the presence of a non-vanishing background charge, and is at the origin of the theory's conformal symmetry.
This provides realizations of correlation functions as expected values of random variables.
, and the algebra is a direct sum of mutually commuting subalgebras of dimension 1 or 2: For any value of
For special values of the central charge and/or of the radius of compactification, free bosonic theories can have not only their
, for special values of the radius of compactification, there may appear non-abelian affine Lie algebras, supersymmetry, etc.
If they coincide, the affine primary field is called diagonal and written as
Normal-ordered exponentials of the free boson are affine primary fields.
Due to the affine symmetry, momentum is conserved in free bosonic CFTs.
In this context, a non-compact free bosonic CFT is called a linear dilaton theory.
is the free bosonic CFT where the left and right momentums take the values The integers
From a sigma model point of view, this equivalence is called T-duality.
, the compactified free boson CFT exists on any Riemann surface.
This partition function is the sum of characters of the Virasoro algebra over the theory's spectrum of conformal dimensions.
The remaining constant factors are signs that depend on the fields' momentums and winding numbers.
If the radius is irrational, the additional boundary states are parametrized by a number
massless free scalar bosons, it is possible to build a product CFT with the symmetry algebra
-dimensional torus (with Neveu–Schwarz B-field) gives rise to a family of CFTs called Narain compactifications.
These CFTs exist on any Riemann surface, and play an important role in perturbative string theory.
The idea is to perturb the free CFT using screening operators of the form
In spite of its perturbative definition, the technique leads to exact results, thanks to momentum conservation.
Correlation functions in minimal models can be computed using these screening operators, giving rise to Dotsenko–Fateev integrals.
[11] Residues of correlation functions in Liouville theory can also be computed, and this led to the original derivation of the DOZZ formula for the three-point structure constant.
free bosons, the introduction of screening charges can be used for defining nontrivial CFTs including conformal Toda theory.
The symmetries of these nontrivial CFTs are described by subalgebras of the abelian affine Lie algebra.
[14] The Coulomb gas formalism can also be used in two-dimensional CFTs such as the q-state Potts model and the