Massless free scalar bosons are a family of two-dimensional conformal field theories, whose symmetry is described by an abelian affine Lie algebra.
Since they are free i.e. non-interacting, free bosonic CFTs are easily solved exactly. 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. However, the value is sometimes implicitly assumed. For , there exist compactified free bosonic CFTs with arbitrary values of the compactification radius.
Lagrangian formulation
The action of a free bosonic theory in two dimensions is a functional of the free boson ,
where is the metric of the two-dimensional space on which the theory is formulated, is the Ricci scalar of that space. The parameter is called the background charge.
What is special to two dimensions is that the scaling dimension of the free boson vanishes. This permits the presence of a non-vanishing background charge, and is at the origin of the theory's conformal symmetry.
In probability theory, the free boson can be constructed as a Gaussian free field. This provides realizations of correlation functions as expected values of random variables.
Symmetries
Abelian affine Lie algebra
The symmetry algebra is generated by two chiral conserved currents: a left-moving current and a right-moving current, respectively
which obey . Each current generates an abelian affine Lie algebra . The structure of the left-moving affine Lie algebra is encoded in the left-moving current's self-OPE,
Equivalently, if the current is written as a Laurent series about the point , the abelian affine Lie algebra is characterized by the Lie bracket
The center of the algebra is generated by , and
the algebra is a direct sum of mutually commuting subalgebras of dimension 1 or 2:
Conformal symmetry
For any value of , the abelian affine Lie algebra's universal enveloping algebra has a Virasoro subalgebra with the generators
The central charge of this Virasoro subalgebra is
and the commutation relations of the Virasoro generators with the affine Lie algebra generators are
If the parameter coincides with the free boson's background charge, then the field coincides with the free boson's energy-momentum tensor. The corresponding Virasoro algebra therefore has a geometrical interpretation as the algebra of infinitesimal conformal maps, and encodes the theory's local conformal symmetry.
Extra symmetries
For special values of the central charge and/or of the radius of compactification, free bosonic theories can have not only their symmetry, but also additional symmetries. In particular, at , for special values of the radius of compactification, there may appear non-abelian affine Lie algebras, supersymmetry, etc.
Affine primary fields
In a free bosonic CFT, all fields are either affine primary fields or affine descendants thereof. Thanks to the affine symmetry, correlation functions of affine descendant fields can in principle be deduced from correlation functions of affine primary fields.
Definition
An affine primary field with the left and right -charges is defined by its OPEs with the currents,
These OPEs are equivalent to the relations
The charges are also called the left- and right-moving momentums. If they coincide, the affine primary field is called diagonal and written as .
Normal-ordered exponentials of the free boson are affine primary fields. In particular, the field
is a diagonal affine primary field with momentum . This field, and affine primary fields in general, are sometimes called vertex operators.
An affine primary field is also a Virasoro primary field with the conformal dimension
The two fields and have the same left and right conformal dimensions, although their momentums are different.
OPEs and momentum conservation
Due to the affine symmetry, momentum is conserved in free bosonic CFTs. At the level of fusion rules, this means that only one affine primary field can appear in the fusion of any two affine primary fields,
Operator product expansions of affine primary fields therefore take the form
where is the OPE coefficient, and the term is the contribution of affine descendant fields. OPEs have no manifest dependence on the background charge.
Correlation functions
According to the affine Ward identities for -point functions on the sphere,