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
c=1
The action of a free bosonic theory in two dimensions is a functional of the free boson
\phi
S[\phi]=
1 | |
4\pi |
\intd2x\sqrt{g}(g\mu\partial\mu\phi\partial\nu\phi+QR\phi) ,
g\mu
R
Q\inC
What is special to two dimensions is that the scaling dimension of the free boson
\phi
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.
The symmetry algebra is generated by two chiral conserved currents: a left-moving current and a right-moving current, respectively
J=\partial\phi and \bar{J}=\bar\partial\phi
\partial\barJ=\bar\partialJ=0
\hat{ak{u}}1
J(y)J(z)= |
| |||
(y-z)2 |
+O(1)
J(z)=\sumn\inZ
-n-1 | |
J | |
nz |
z=0
[Jm,Jn]=
12 | |
n\delta |
m+n,0
J0
\hat{ak{u}}1=Span(J0) ⊕
infty | |
oplus | |
n=1 |
Span(Jn,J-n)
For any value of
Q\inC
\begin{align} Ln&=-\summ\in{Z
c=1+6Q2
[Lm,Jn]=-nJm+n-
Q | |
2 |
m(m+1)\deltam+n,0
Q
T(z)=\sumn\inZLnz-n-2
For special values of the central charge and/or of the radius of compactification, free bosonic theories can have not only their
\hat{ak{u}}1
c=1
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.
An affine primary field
V\alpha,(z)
\hat{ak{u}}1
\alpha,\bar\alpha
J(y)V\alpha,(z)=
\alpha | |
y-z |
V\alpha,(z)+O(1) , \barJ(y)V\alpha,(z)=
\bar\alpha | |
\bary-\barz |
V\alpha,(z)+O(1)
Jn>0V\alpha,(z)=\barJn>0V\alpha,(z)=0 , J0V\alpha,(z)=\alphaV\alpha,(z) , \barJ0V\alpha,(z)=\bar\alphaV\alpha,(z)
\alpha,\bar\alpha
V\alpha(z)=V\alpha,\alpha(z)
Normal-ordered exponentials of the free boson are affine primary fields. In particular, the field
:e2\alpha\phi(z):
\alpha
An affine primary field is also a Virasoro primary field with the conformal dimension
\Delta(\alpha)=\alpha(Q-\alpha)
V\alpha(z)
VQ-\alpha(z)
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,
V | |
\alpha1,\bar\alpha1 |
x
V | |
\alpha2,\bar\alpha2 |
=
V | |
\alpha1+\alpha2,\bar\alpha1+\bar\alpha2 |
V | |
\alpha1,\bar\alpha1 |
(z1)V
\alpha2,\bar\alpha2 |
(z2)=C(\alphai,\bar\alphai)(z1-z
-2\alpha1\alpha2 | |
2) |
(\barz1-\bar
-2\bar\alpha1\bar\alpha2 | |
z | |
2) |
\left(
V | |
\alpha1+\alpha2,\bar\alpha1+\bar\alpha2 |
(z2)+O(z1-z2)\right)
C(\alphai,\bar\alphai)
O(z1-z2)
According to the affine Ward identities for
N
N | |
\left\langle\prod | |
i=1 |
V | |
\alphai,\bar\alphai |
(zi)\right\rangle ≠ 0 \implies
N | |
\sum | |
i=1 |
\alphai=
N\bar | |
\sum | |
i=1 |
\alphai=Q
N
N | |
\left\langle\prod | |
i=1 |
V | |
\alphai,\bar\alphai |
(zi)\right\rangle\propto \prodi<j(zi-z
-2\alphai\alphaj | |
j) |
(\barzi-\bar
-2\bar\alphai\bar\alphaj | |
z | |
j) |
\Delta(\alphai)-\Delta(\bar\alphai)\in
12Z | |
A free bosonic CFT is called non-compact if the momentum can take continuous values.
Non-compact free bosonic CFTs with
Q ≠ 0
A free bosonic CFT with
Q=0
c=1
\alpha=\bar\alpha\iniR
\Delta(\alpha)\geq0
\alpha=\bar\alpha\inR
\Delta(\alpha)\leq0
The compactified free boson with radius
R
(\alpha,\bar\alpha)=\left(
i | \left[ | |
2 |
n | |
R |
+Rw\right],
i | \left[ | |
2 |
n | |
R |
-Rw\right]\right) with (n,w)\inZ2
n,w
R\inC*
Q=0
R\in | 1 |
iQ |
Z
If
Q=0
R
1 | |
R |
If
Q=0
C | |
Z+\tauZ |
ZR(\tau)=
Z | ||||
|
(\tau)=
1 | |
|η(\tau)|2 |
\sumn,w\inZ
| ||||||||
q |
| ||||||||
\bar{q} |
q=e2\pi
η(\tau)
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.
Due to the
Z2
J\to-J
J=\bar{J} or J=-\bar{J}
z=\bar{z}
\phi
In the case of a compactified free boson, each type of boundary condition leads to a family of boundary states, parametrized by
\theta\in
R | |
2\piZ |
\{\Imz>0\}
\begin{align} \left\langleV(n,w)(z)\right\rangleDirichlet,&=
ein\theta\deltaw,0 | |||||||||
|
\\ \left\langleV(n,w)(z)\right\rangleNeumann,&=
eiw\theta\deltan,0 | |||||||||
|
\end{align}
\begin{align} \left\langleV\alpha(z)\right\rangleDirichlet,&=
e\alpha\theta | |
|z-\barz|2\Delta(\alpha) |
\\ \left\langleV\alpha(z)\right\rangleNeumann&=\delta(i\alpha) \end{align}
\alpha\iniR
\theta\inR
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 symmetry.
If the radius is irrational, the additional boundary states are parametrized by a number
x\in[-1,1]
(n,w) ≠ (0,0)
(n,w)=(0,0)
If the radius is rational
R= | p |
q |
SU(2) | |
Zp x Zq |
Conformal boundary conditions at arbitrary
c
From
N
N | |
\hat{ak{u}} | |
1 |
In particular, compactifying
N
N
Due to the existence of the automorphism
J\to-J
\hat{ak{u}}1
N | |
\hat{ak{u}} | |
1 |
Z2
Q=0
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
style{\int}d2zO(z)
O(z)
(\Delta,\bar\Delta)=(1,1)
In the case of a single free boson with background charge
Q
style{\int}Vb,
style{\int}V | |
b-1 |
Q=b+b-1
In the case of
N
The Coulomb gas formalism can also be used in two-dimensional CFTs such as the q-state Potts model and the
O(n)
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: