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 c = 1 {\displaystyle c=1} is sometimes implicitly assumed. For c = 1 {\displaystyle c=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 ϕ {\displaystyle \phi } ,
S [ ϕ ] = 1 4 π ∫ d 2 x g ( g μ ν ∂ μ ϕ ∂ ν ϕ + Q R ϕ ) , {\displaystyle S[\phi ]={\frac {1}{4\pi }}\int d^{2}x{\sqrt {g}}(g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi +QR\phi )\ ,}
where g μ ν {\displaystyle g_{\mu \nu }} is the metric of the two-dimensional space on which the theory is formulated, R {\displaystyle R} is the Ricci scalar of that space. The parameter Q ∈ C {\displaystyle Q\in \mathbb {C} } is called the background charge. What is special to two dimensions is that the scaling dimension of the free boson ϕ {\displaystyle \phi } 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
J = ∂ ϕ and J ¯ = ∂ ¯ ϕ {\displaystyle J=\partial \phi \quad {\text{and}}\quad {\bar {J}}={\bar {\partial }}\phi }
which obey ∂ J ¯ = ∂ ¯ J = 0 {\displaystyle \partial {\bar {J}}={\bar {\partial }}J=0} . Each current generates an abelian affine Lie algebra u ^ 1 {\displaystyle {\hat {\mathfrak {u}}}_{1}} . The structure of the left-moving affine Lie algebra is encoded in the left-moving current's self-OPE,
J ( y ) J ( z ) = − 1 2 ( y − z ) 2 + O ( 1 ) {\displaystyle J(y)J(z)={\frac {-{\frac {1}{2}}}{(y-z)^{2}}}+O(1)}
Equivalently, if the current is written as a Laurent series J ( z ) = ∑ n ∈ Z J n z − n − 1 {\displaystyle J(z)=\sum _{n\in \mathbb {Z} }J_{n}z^{-n-1}} about the point z = 0 {\displaystyle z=0} , the abelian affine Lie algebra is characterized by the Lie bracket
[ J m , J n ] = 1 2 n δ m + n , 0 {\displaystyle [J_{m},J_{n}]={\frac {1}{2}}n\delta _{m+n,0}}
The center of the algebra is generated by J 0 {\displaystyle J_{0}} , and the algebra is a direct sum of mutually commuting subalgebras of dimension 1 or 2:
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