Massless free scalar bosons in two dimensions
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 formula_1 is sometimes implicitly assumed. For formula_1 , 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 formula_3, where formula_5 is the metric of the two-dimensional space on which the theory is formulated, formula_6 is the Ricci scalar of that space. The parameter formula_7 is called the background charge. What is special to two dimensions is that the scaling dimension of the free boson formula_3 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 formula_10. Each current generates an abelian affine Lie algebra formula_11. 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 formula_13 about the point formula_14 , the abelian affine Lie algebra is characterized by the Lie bracket The center of the algebra is generated by formula_16, and the algebra is a direct sum of mutually commuting subalgebras of dimension 1 or 2: Conformal symmetry. For any value of formula_7, 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 formula_22 coincides with the free boson's background charge, then the field formula_23 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 formula_11 symmetry, but also additional symmetries. In particular, at formula_1, 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 formula_26 with the left and right formula_11-charges formula_28 is defined by its OPEs with the currents, These OPEs are equivalent to the relations The charges formula_28 are also called the left- and right-moving momentums. If they coincide, the affine primary field is called diagonal and written as formula_32. Normal-ordered exponentials of the free boson are affine primary fields. In particular, the field formula_33 is a diagonal affine primary  field with momentum formula_34. 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 formula_36 and formula_37 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 formula_40 is the OPE coefficient, and the term formula_41 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 formula_42-point functions on the sphere, Moreover, the affine symmetry completely determines the dependence of sphere formula_42-point functions on the positions, Single-valuedness of correlation functions leads to constraints on momentums, Models. Non-compact free bosons. A free bosonic CFT is called non-compact if the momentum can take continuous values. Non-compact free bosonic CFTs with formula_47 are used for describing  non-critical string theory. In this context, a non-compact free bosonic CFT is called a linear dilaton theory. A free bosonic CFT with formula_48 i.e. formula_1 is a sigma model with a one-dimensional target space. Compactified free bosons. The compactified free boson with radius formula_54 is the free bosonic CFT where the left and right momentums take the values The integers formula_56 are then called the momentum and winding number. The allowed values of the compactification radius are formula_57 if formula_48 and formula_59 otherwise. If formula_48, free bosons with radiuses formula_54 and formula_62 describe the same CFT. From  a sigma model point of view, this equivalence is called T-duality. If formula_48, the compactified free boson CFT exists on any Riemann surface. Its partition function on the torus formula_64 is where formula_66, and formula_67 is the Dedekind eta-function. This partition function is the sum of characters of the Virasoro algebra over the theory's spectrum of conformal dimensions. As in all free bosonic CFTs, correlation functions of affine primary fields have a dependence on the fields' positions that is determined by the affine symmetry. The remaining constant factors are signs that depend on the fields' momentums and  winding numbers. Boundary conditions in the case c=1. Neumann and Dirichlet boundary conditions. Due to the formula_68 automorphism formula_69 of the abelian affine Lie algebra there are two types of boundary conditions that preserve the affine symmetry, namely If the boundary is the line formula_71, these conditions correspond respectively to the Neumann boundary condition and Dirichlet boundary condition for the free boson formula_72. Boundary states. In the case of a compactified free boson, each type of boundary condition leads to a family of boundary states, parametrized by formula_73. The corresponding one-point functions on the upper half-plane formula_74 are  In the case of a non-compact free boson, there is only one Neumann boundary state, while Dirichlet boundary states are parametrized by a real parameter. The corresponding one-point functions are where formula_77 and formula_78 for a Euclidean boson. Conformal boundary conditions. Neumann and Dirichlet boundaries are the only boundaries that preserve the free boson's affine symmetry. However, there exist additional boundaries that preserve only the conformal symmtry. If the radius is irrational, the additional boundary states are parametrized by a number formula_79. The one-point functions of affine primary fields with formula_80 vanish. However, the Virasoro primary  fields that are affine descendants of the affine primary field with formula_81 have nontrivial one-point functions. If the radius is rational formula_82, the additional boundary states are parametrized by the manifold formula_83. Related theories and generalizations. Multiple bosons and orbifolds. From formula_42 massless free scalar bosons, it is possible to build a product CFT with the symmetry algebra formula_85. Some or all of the bosons can be compactified. In particular, compactifiying formula_42 bosons without background charge on an formula_42-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. Due to the existence of the automorphism formula_69 of the affine Lie algebra formula_11, and of more general automorphisms of formula_85, there exist orbifolds of free bosonic CFTs. For example, the formula_68 orbifold of the compactified free boson with formula_48 is the critical two-dimensional Ashkin-Teller model. Coulomb gas formalism. The Coulomb gas formalism is a technique for building interacting CFTs, or some of their correlation functions, from free bosonic CFTs. The idea is to perturb the free CFT using screening operators of the form formula_93 , where formula_94 is an affine primary field of conformal dimensions formula_95. In spite of its perturbative definition, the technique leads to exact results, thanks to momentum conservation. In the case of a single free boson with background charge formula_22, there exist two diagonal screening operators formula_97, where formula_98. Correlation functions in minimal models can be computed using these screening operators, giving rise to Dostenko-Fateev integrals. 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. In the case of formula_42 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. Depending on the screenings, these subalgebras may or may not be W-algebras. The Coulomb gas formalism can also be used in two-dimensional CFTs such as the q-state Potts model and the formula_100 model. Various generalizations. In arbitrary dimensions, there exist conformal field theories called generalized free theories. These are however not generalizations of the free bosonic CFTs in two dimensions. In the former, it  is the conformal dimension which is conserved (modulo integers). In the latter, it is the momentum. In two dimensions, generalizations include: