% File: vertex.tex
% Section: CFT2
% Title: Vertex algebras
% Last modified: 05.11.2003
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\begin{document}

\begin{center}
\large\textbf{VERTEX ALGEBRAS}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Formal distributions and locality}
\end{center}

\vspace{.1cm}

Vertex algebras encode mathematical content of 2-dimensional
conformal quantum field theory (CFT${}_2$). Roughly speaking, chiral
quantum fields depending on a single complex variable $z$ are replaced
by operator-valued formal power series in an abstract variable $z$\,.
We begin with a brief exposition of the corresponding formalism.

Formal power series of the following type
($m,\,n\in\mathbb{Z}\,, \ a_n,\,a_{m,n}\in A$),
\begin{equation}
\label{distrib}
a(z) = \sum_{n}\frac{a_n}{z^{n+1}}\,, \ \ \ \
a(z,\,w) = \sum_{m,\,n}\frac{a_{m,n}}{w^{m+1}\,z^{n+1}}\,,
\end{equation}
are called $A$-valued formal distributions in one, two, or more
indeterminates $z,\,w,\,...$\,.
We can freely multiply formal distributions by polynomials,
but a product of two distributions in the same variable(s)
is not generally defined. Also, there is a problem in treating
negative powers of $(z-w)$ etc. as formal distributions.
To do it unambiguously, we will use (when needed) the notation like
\begin{equation}
\label{expanzw}
(z-w)_{w}^{n} \doteq \sum_{k\geqslant0}(-)^k\binom{n}{k}w^k z^{n-k}\,, \ \
(z-w)_{z}^{n} \doteq \sum_{k\geqslant0}(-)^{n+k}\binom{n}{k}w^{n-k}z^k
\end{equation}
which indicates, in positive powers of what variable ($w$ or $z$ in this
case) the expansion is performed. So, e.g., $(z-w)_{w}^{n}$ is analogous to
$(z-w)_{|z|>|w|}^{n}$ in terms of complex plane.

Of crucial importance in the present formalism is the $\delta$-function:
\begin{equation}
\label{delta}
\delta(z-w) = \delta(w-z) = \sum_{n}w^n z^{-n-1}
 = \ldots +\frac{z}{w^2}+\frac{1}{w}+\frac{1}{z}+\frac{w}{z^2}+\ldots\,.
\end{equation}
It exhibits standard-looking properties
\begin{equation}
\label{}
f(z)\,\delta(z-w) = f(w)\,\delta(z-w)\,, \ \
(z-w)\,\delta(z-w) = 0\,, \ \
\text{Res}_z f(z)\,\delta(z-w) = f(w)
\end{equation}
where $\text{Res}_z$ stands for a coefficient of $z^{-1}$\,,
being an analog of $(2\pi i)^{-1}\!\oint\!dz$ on a complex plane.
Note $\text{Res}_z\!\circ\partial_z = 0$\,.
Evidently, the definition (\ref{delta}) mimics the well-known
formulas like
$\delta(x-y) = \sum_n e^{in(x-y)} = \sum_n (e^{ix})^n (e^{iy})^{-n}$\,.

Useful relations with the derivatives of the $\delta$-function
are listed below:
\begin{gather}
\label{deriv}
\frac{1}{n!}\,\partial_{w}^{n}\,\delta(z-w)
 = \frac{(-)^n}{n!}\,\partial_{z}^{n}\,\delta(z-w)
 = \sum_{m}\binom{m}{n}w^{m-n}z^{-m-1} \ \ \ \ \ \ \
(n\geqslant0) \\
(z-w)^m\,\partial_{w}^{n}\,\delta(z-w) = 0 \ \ \ \ \ \ (m>n\geqslant0) \\
(z-w)^m\frac{1}{n!}\,\partial_{w}^{n}\,\delta(z-w)
 = \frac{1}{(n\!-\!m)!}\,\partial_{w}^{n-m}\delta(z-w) \ \ \ \ \ \
(0\leqslant m\leqslant n)  \label{dd} \\
\text{Res}_z\,f(z)\,\partial_{w}^{n}\,\delta(z-w)
 = \partial_{w}^{n}\,f(w)\,, \ \ \
\text{Res}_z\,\frac{(z-w)^m}{n!}\,\partial_{w}^{n}\,\delta(z-w)
 = \delta_{m,n} \ \ \  (m\,,n\geqslant0) \label{rd}
\end{gather}
The following formula may also be of some use:
\begin{equation}
\label{grenze}
(z-w)_w^{-n}-(z-w)_z^{-n}
 = \frac{1}{(n-1)!}\,\partial_{w}^{n-1}\delta(z-w)
\ \ \ \ \ \ (n>0)\,.
\end{equation}

Another principal notion here is locality. For the (most
important) case of two variables the locality condition is
\begin{equation}
\label{local}
(z-w)^n a(z,w)=0 \ \ \ \ \ \ \ (n\geqslant N\geqslant0)\,.
\end{equation}
It imposes severe restrictions on $a(z,w)$\,. Consider first the
simplest case
\begin{gather}
(z-w)\,b(z,w)=0 \ \ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ \
b_{m+1,n}=b_{m,n+1} \notag\\
\ \ \ \ \ \ \ \Rightarrow \ \ \ \ \
b(z,w) = c(w)\,\delta(z-w)\,, \ \
c(w) = \text{Res}_z b(z,w)\,.
\label{loc1}
\end{gather}
If now only $(z-w)^2 a(z,w)=0$\,, we consecutively find that
\begin{gather*}
(z-w)\,a(z,w)=c_1(w)\,\delta(z-w)
\equiv c_1(w)\,(z-w)\,\partial_w\delta(z-w)\,, \\
c_1(w) = \text{Res}_z (z-w)\,a(z,w)\,, \\
a(z,w) - c_1(w)\,\partial_w\delta(z-w) = c_0(w)\,\delta(z-w)\,, \\
c_0(w) = \text{Res}_z [a(z,w)-c_1(w)\,\partial_w\delta(z-w)]
 = \text{Res}_z a(z,w)
\end{gather*}
and, therefore,
\begin{equation}
\label{loc2}
a(z,w)=c_0(w)\,\delta(z-w)+c_1(w)\,\partial_w\delta(z-w)\,.
\end{equation}
By induction, using (\ref{dd}) and (\ref{rd})\,, we show that
$(z-w)^n a(z,w)=0$ entails
\begin{equation}
\label{locn}
a(z,w)=\sum_{m=0}^{n-1}c_m(w)\frac{1}{m!}\partial_{w}^{m}\delta(z-w)\,,
\ \ \ \ c_m(w)=\text{Res}_z (z-w)^m a(z,w)\,.
\end{equation}
It is easily seen from (\ref{loc1}) or (\ref{locn}) that
a local formal distribution should contain infinitely many nonzero
terms of positive and negative degree in both variables, otherwise
it is identically zero. It is also clear that $a(w,z),\,
\partial_z a(z,w), \,\partial_w a(z,w),\,
f(z)\,a(z,w)$ and $f(w)\,a(z,w)$ are local if $a(z,w)$ is.

One interesting example of a local distribution is provided
by the product of two series with all unity coefficients:
\begin{equation}
\label{delta1}
(\sum_n z^{-n-1})(\sum_m w^{-m-1}) = \delta(z-1)\,\delta(w-1)
= \delta(z-w)\,\delta(w-1)
\end{equation}
(setting one argument of $\delta$-function to a nonzero number
makes sense)\,.

One can verify (directly, or by induction) the following analog of
the Taylor formula:
\begin{equation}
\label{Taylor}
[f(z) - \sum_{n=0}^{N}\,\frac{(z-w)^n}{n!}\,\partial_{w}^{n}\,f(w)\,]
\,\partial_{w}^{N}\,\delta(z-w) = 0\,.
\end{equation}
It is precisely in this way we may consider an expression inside square
brackets in (\ref{Taylor}) as something of the order $(z-w)^{N+1}$\,.

The following formulas with binomial coefficients are widely used
in the calculations performed (or implicitly meant) in the present text:
\begin{gather}
\binom{-n-1}{m} = (-)^m \binom{n+m}{m} \ \ \ \ \ (m\geqslant0)\,,
\ \ \ \ \ \ \ \ \binom{n}{m}=0 \ \ \ \ \ (m>n\geqslant0)\,, \\
\sum_{k=m}^{n}(-)^k \binom{n}{k}\binom{k}{m}=(-)^n\delta_{m,n}\,,
\ \ \ \  \sum_{k=0}^{n}(-)^k k^m\binom{n}{k}=(-)^n\,n!\,\delta_{m,n}\,,
\ \ \ \ (n\geqslant m\geqslant0) \\
\sum_{k=0}^{n}(-)^k \binom{n}{k}\binom{p-k}{m}=\binom{p-n}{m-n}\,,
\ \ \ \ \ \ \
\sum_{k=0}^{n}(-)^k \binom{n}{k}\binom{p+k}{m}=(-)^n\binom{p}{m-n}
\end{gather}

\newpage

\begin{center}
\large\textbf{Operator product expansion}
\end{center}

\vspace{.1cm}

A single-variable formal distribution $a(z)$ is called a field
on a vector space $V$ if $a_n$ are operators on $V$\,, and for
any $v\in V$ a series
$a(z)\,v$ contains only finitely many negative powers of $z$\,.
In other words, $a_n v = 0$ for $n\geqslant N(v)$\,.
Fields $a$ and $b$ are called (mutually) local if for some
$N(a,b)\geqslant0$
\begin{equation}
\label{locf}
(z-w)^N [a(z),b(w)]=0 \ .
\end{equation}
In what follows, we assume all fields to be pairwise local.

For each $n\in\mathbb{Z}$\,, an important notion of $n$-product of
two fields $a$ and $b$ is introduced as follows:
\begin{equation}
\label{nprod}
(a_n b)(w) = \sum_{m}\frac{(a_n b)_m}{w^{m+1}}\,, \ \ \ \
(a_n b)_m \doteq \sum_{k\geqslant0}(-)^k\binom{n}{k}
(a_{n-k}b_{m+k} -(-)^n b_{m+n-k}a_k)\,.
\end{equation}
It is seen from (\ref{nprod}) that $(a_n b)(w)$ is also a local field,
because for given $n$ and $v$ (or $c(z)$) we are able to find such $N$
that $(a_n b)_m v = 0$ (resp. $(a_n b)_m c = 0$) for $m\geqslant N$\,.

Let us discuss the definition of the $n$-product in more details.
For $n\geqslant0$ it is equal to
\begin{equation}
\label{n>0}
(a_n b)(w)=\text{Res}_z (z-w)^n [a(z),b(w)]\,.
\end{equation}
Locality condition (\ref{locf}) implies $(a_n b)(w)=0 \ (n\geqslant N)$
and, due to (\ref{locn}),
\begin{equation}
\label{commf}
[a(z),b(w)]=\sum_{n=0}^{N-1}(a_n b)(w)\frac{1}{n!}
\partial_{w}^{n}\delta(z-w)\,.
\end{equation}
Therefore, a commutator of local fields and (a finite number of)
their $n$-products with non-negative $n$ are closely related.
This is also seen at the component level ($n\geqslant0$)\,:
\begin{equation}
\label{commm}
(a_n b)_m = \sum_{k=0}^{n}(-)^k\binom{n}{k}[a_{n-k},b_{m+k}]\,, \ \ \ \ \
[a_p,\,b_q] = \sum_{k\geqslant0}\binom{p}{k}(a_k b)_{p+q-k}
\end{equation}
(the second sum is also finite due to locality)\,.

A special case $n=-1$ corresponds to the normal (or normally ordered)
product,
\begin{equation}
\label{normw}
(a_{-1}b)_m = \sum_{k<0}a_k b_{m-k-1} + \sum_{k\geqslant0}b_{m-k-1}a_k
\ \ \ \Rightarrow \ \ \
(a_{-1}b)(w) = \ :\!a(w)b(w)\!:\, \ \equiv \,\ :\!ab\!:\!(w)\,,
\end{equation}
where
\begin{equation}
\label{normzw}
:\!a(z)b(w)\!:\, \doteq a_+(z)b(w)+b(w)a_-(z)\,, \ \ \
a_+(z) = \sum_{n<0}\frac{a_n}{z^{n+1}}\,, \ \ \
a_-(z) = \sum_{n\geqslant0}\frac{a_n}{z^{n+1}}\ .
\end{equation}
Usually $a_+(z)$ is said to contain creation and $a_-(z)$
annihilation operators. We can now verify that the following general
formula for negative $n$ agrees with the definition (\ref{nprod})\,:
\begin{equation}
\label{n<0}
(a_n b)(w) = \ \frac{1}{(-n-1)!}:\!\partial^{-n-1}a(w)\cdot b(w)\!:
\ \ \ \ \ \ \ \ (n<0)
\end{equation}
There also exists a universal form of the $n$-product definition:
\begin{equation}
\label{resprod}
(a_n b)(w)=\text{Res}_z\bigl(a(z)b(w)(z-w)^n_w-b(w)a(z)(z-w)^n_z\bigr)\,.
\end{equation}

Remarkably, $n$-products play the role of coefficient functions of the
operator product expansion (OPE). In CFT${}_2$\,, the latter is formulated for
the radially ordered products
\begin{equation}
\label{rad}
\mathcal{R}\,a(z)\,b(w) \doteq
\begin{cases}
a(z)\,b(w) & \qquad |z|>|w| \\ b(w)\,a(z) & \qquad |w|>|z|
\end{cases}
\end{equation}
Within the present formalism, OPE relations are expressed
in terms of the definitions (\ref{expanzw})\,:
\begin{equation}
\label{ope}
a(z)\,b(w) = \sum_n \frac{(a_n b)(w)}{(z-w)_{w}^{n+1}}\ , \ \ \ \
b(w)\,a(z) = \sum_n \frac{(a_n b)(w)}{(z-w)_{z}^{n+1}}
\end{equation}
or, equivalently,
\begin{gather}
\label{opew}
a(z)\,b(w) = \sum_{n=0}^{N-1} \frac{(a_n b)(w)}{(z-w)_{w}^{n+1}}
\ + :\!a(z)\,b(w)\!:\, \\
\label{opez}
b(w)\,a(z) = \sum_{n=0}^{N-1} \frac{(a_n b)(w)}{(z-w)_{z}^{n+1}}
\ + :\!a(z)\,b(w)\!:\, \\
\label{ope<0}
:\!a(z)\,b(w)\!:\, = \sum_{n\geqslant0}(a_{-n-1}b)(w)\ (z-w)^n
\end{gather}
Eqs. (\ref{ope<0}) and (\ref{ope}) are to be read not literally,
but in the sense of Taylor's expansion (\ref{Taylor}),
its coefficients given by (\ref{n<0})\,.
At the same time, (\ref{opew}) and (\ref{opez}) are derived from
(\ref{grenze}) and
\begin{gather}
a(z)\,b(w) \ - :\!a(z)\,b(w)\!:\ =\, [a_-(z),b(w)]\,, \\
b(w)\,a(z) \ - :\!a(z)\,b(w)\!:\ =\, [b(w),a_+(z)]
\end{gather}
by extracting negative and non-negative
powers, respectively, from the $z$-expansion of (\ref{commf})\,,
and are identically true. We see that a singular part of OPE is given
by $n$-products with non-negative $n$\,, or by the commutator of fields.

It seems to be instructive to rewrite this derivation of OPE using the
CFT${}_2$ language. Let
$\mathcal{R}\,a(z)\,b(w) = \sum_n \varphi_n(w)(z-w)^{-n-1}$\,.
Then
\begin{multline}
\label{derivope}
\varphi_n(w) = \text{Res}_{z-w}\bigl((z-w)^n\,\mathcal{R}\,a(z)\,b(w)\bigr)\\
= \frac{1}{2\pi i}\oint_w dz (z-w)^n\,\mathcal{R}\,a(z)\,b(w)
= \frac{1}{2\pi i}\left(\oint_{|z|>|w|}dz - \oint_{|z|<|w|}dz\right)
 (z-w)^n\,\mathcal{R}\,a(z)\,b(w) \\
= \frac{1}{2\pi i}\oint dz
 \left(a(z)\,b(w)\,(z-w)^n_{|z|>|w|}-b(w)\,a(z)\,(z-w)^n_{|z|<|w|}\right) \\
= \text{Res}_z
 \left(a(z)\,b(w)\,(z-w)^n_{|z|>|w|}-b(w)\,a(z)\,(z-w)^n_{|z|<|w|}\right)
= (a_n b)(w)\,.
\end{multline}

It should be noted that from (\ref{nprod}) and (\ref{commm})
(i.e., from the definition of the $n$-product plus the locality condition),
the following useful formula (Borcherds identity) can be deduced. Namely,
for $l,m,n\in\mathbb{Z}$\,, and $a,b,c$ being fields, the following relations
hold:
\begin{equation}
\label{Borch}
\sum_{k\geqslant0}\binom{m}{k}(a_{l+k}b)_{m+n-k}c
= \sum_{k\geqslant0}(-)^k\binom{l}{k}
\bigl(a_{m+l-k}(b_{n+k}c) -(-)^l b_{n+l-k}(a_{m+k}c)\bigr)\,.
\end{equation}

In CFT${}_2$\,, it's common practice to construct composite operators
as normal products of two (or more) fields. To find then the OPE relations
with these composite fields, the so-called Wick formulas are used. In the
present formalism the latter are obtained from (\ref{Borch})\,,
or (\ref{commm})\,, and look like
\begin{equation}
\label{Wick}
a_n(b_{-1}c) = b_{-1}(a_n c)
 + \sum_{k\geqslant0}\binom{n}{k}(a_k b)_{n-k-1}c\ .
\end{equation}

\newpage

\begin{center}
\large\textbf{Vertex algebras}
\end{center}

\vspace{.1cm}

A vertex algebra consists of a vector space $V$ (whose elements
are called `states'), a vacuum state $\Omega\in V$\,, and a
`state-field correspondence' linear map $Y$ which
associates a field $a(z)\equiv Y(a,z)=\sum_n a_n z^{-n-1}$
(vertex operator) to each
$a\in V$ in such a way that $Y(a_n b,z)=(a_nb)(z)$
are given by (\ref{nprod}), all fields are mutually local
($a_nb$\,=\,0 for $n\geqslant N(a,b)$)\,, and $Y(\Omega,z)$ is equal
to unity ($\Omega_n=\delta_{n,-1}\,\mathbf{1}$)\,. In addition,
a shift operator $D$ is defined on $V$ by
$(Da)(z) \doteq \partial_z a(z)$\,.
Note that an expression $a_nb$ is used here in two meanings:
it denotes a state (an element in $V$ obtained by the action of
the operator $a_n$ upon $b\in V$), and serves simultaneously as a name of
the field $(a_nb)(z)$ (associated to this state via the map $Y$)\,,
which is exactly the $n$-product of two fields $a(z)$ and $b(z)$\,.

Now let us study the consequences of the above definitions.
Due to locality of vertex operators, we may use not only
(\ref{nprod})\,, but also Borcherds' relation (\ref{Borch})\,. Evidently,
\begin{equation}
\label{vac}
a_{-1}\,\Omega=a\,, \ \ \ \ \ \
a_n\,\Omega=0 \ \ \ (n\geqslant0)\,, \ \ \ \ \ \ \
D\,\Omega=0\,, \ \ \ \ \ \
(Da)_n=-na_{n-1}\,.
\end{equation}
Further, for $k\geqslant0$\,,
\begin{equation}
\label{exp}
(D^k a)_n = (-)^k\,k!\,\binom{n}{k}\,a_{n-k}\,, \ \ \ \
a_{-k-1} = \frac{1}{k!}\,(D^k a)_{-1} \ \ \ \ \ \
\Rightarrow \ \ \ \ Y(a,z)\,\Omega = e^{zD}a\,.
\end{equation}
Using (\ref{Borch}) with $m=0, \,n=-2, \, c=\Omega$\,, we obtain
$D(a_lb)=(Da)_lb+a_l(Db)\,,$ whence
\begin{equation}
\label{Leib}
[D,\,a_n]=(Da)_n=-na_{n-1} \ \ \ \ \ \ \Rightarrow \ \ \ \
[D,\,Y(a,z)]=Y(Da,z)=\partial_z Y(a,z)\,.
\end{equation}
As an immediate generalization we find
\begin{equation}
\label{transl}
e^{zD}Y(a,w)e^{-zD} = Y(e^{zD}\!a,w) = Y(a,w+z)_z\ .
\end{equation}
For $m=-1, \,n=0, \, c=\Omega$ \ formula (\ref{Borch}) yields
a so-called skew symmetry property
\begin{equation}
\label{skew}
b_la = \sum_{k\geqslant0}\,\frac{(-)^{l+k+1}}{k!}\,D^k(a_{l+k}b)
\ \ \ \ \ \ \Rightarrow \ \ \ \
Y(b,z)\,a = e^{zD}\,Y(a,-z)\,b\,.
\end{equation}
The OPE relation assumes here the following form:
\begin{equation}
\label{opey}
Y(a,z)Y(b,w) = Y(Y(a,z-w)_w\,b,w)\,, \ \ \ \
Y(b,w)Y(a,z) = Y(Y(a,z-w)_z b,w)\,.
\end{equation}

If $V={\scriptstyle\bigoplus_{k\geqslant0}}V_k$ is a graded space
equipped with a vertex algebra structure (so that $\Omega\in V_0\,,
\ D:V_k\rightarrow V_{k+1}$\,, \,all $a_n$ homogeneous),
one usually renumbers the components of the field $a(z)$ with
$a\in V_\Delta$ in such a way (shifting index $n\rightarrow n-\Delta+1$)
that
\begin{equation}
\label{grad}
a(z)=\sum_n\,a_n\,z^{-n-\Delta}, \ \ \
a_{-n}:V_k\rightarrow V_{k+n}\,, \ \ \
a_{-\Delta}\Omega= a\,, \ \ \
a_n\Omega=0 \ \ (n>-\Delta)\,.
\end{equation}
As the famous example, the Virasoro algebra with central charge $c$
can be reconstructed in this way from a single `conformal' vector
$\omega\in\!V_2$\ :
\begin{gather}
\label{Vir}
\omega(z)\,\equiv\, Y(\omega,z)=\sum_n\,\omega_n\,z^{-n-1} \
\equiv \,T(z) =\sum_n\,L_n\,z^{-n-2} \ \ \ \ \ (\omega_n=L_{n-1})\ , \\
\label{Vir1}
L_{-2}\,\Omega=\omega\,, \ \ \
L_{-1}=D\,, \ \ \
L_0\,a = \Delta\,a \ \ \,(a\in V_{\Delta})\,, \ \ \
L_1\,\omega=0\,, \ \ \
L_2\,\omega=\frac{c}{2}\ \Omega\,.
\end{gather}
These definitions prove to be consistent, agree with (\ref{Borch})\,,
and ultimately result in
\begin{gather}
[L_m,L_n] =
 (m-n)L_{m+n}+\frac{c}{12}\,m(m^2-1)\,\delta_{n,-m} \ , \\
\mathcal{R}\,T(z)\,T(w) = \frac{c}{2}\,(z-w)^{-4}+2\,(z-w)^{-2}\,T(w)
 +(z-w)^{-1}\,\partial\, T(w)+\,\ldots\ .
\end{gather}

\newpage

\begin{center}
\large\textbf{Free fermions}
\end{center}

\vspace{.1cm}

Below we describe the (Sugawara-like) construction of a conformal vector
as a bilinear combination of `more elementary' objects: in this (simplest)
case, neutral fermions. The corresponding vertex algebra is generated by the
vacuum state $\Omega$ and one more vector $\phi$ subject to
\begin{equation}
\label{neutral}
\phi_0\phi = \Omega\,, \ \ \ \ \ \
\phi_n\phi = 0 \ \ \ \ \ (n>0)\ .
\end{equation}
This exactly corresponds to
\begin{equation}
\label{phiz}
\phi(z) = \sum_n\,\phi_n\,z^{-n-1}\,, \ \ \ \ \
[\phi_m,\phi_n]_+ = \delta_{m+n,-1}\ , \ \ \ \ \
\phi(z)\,\phi(w) \sim \frac{1}{(z-w)_w}\ .
\end{equation}
Since $\phi(z)$ is a fermionic field, we deal with anticommutators
instead of commutators, and all formulas like (\ref{nprod}), (\ref{commm}),
and so on, should be modified accordingly.

Let us now introduce a vector
\begin{equation}
\label{omff}
\omega \doteq \frac{1}{2}\,\phi_{-2}\phi
 \equiv \frac{1}{2}\,(D\phi)_{-1}\phi
 \equiv \frac{1}{2}\,\phi_{-2}\phi_{-1}\Omega
\end{equation}
which is to be recognized as a conformal one. Eq.\;(\ref{Wick}) yields
\begin{equation}
\label{}
\phi_0\omega = -\frac{1}{2}\,D\phi\,, \ \ \ \ \
\phi_1\omega = \frac{1}{2}\,\phi\,, \ \ \ \ \
\phi_n\omega = 0 \ \ \ \ \ (n>1)
\end{equation}
and (\ref{skew}) then gives
\begin{equation}
\label{omphi}
\omega_0\phi = D\phi\,, \ \ \ \ \
\omega_1\phi = \frac{1}{2}\,\phi\,, \ \ \ \ \
\omega_n\phi = 0 \ \ \ \ \ (n>1)
\end{equation}
Analogously, we come to
\begin{equation}
\label{}
\omega_0 D\phi = D^2\phi\,, \ \ \ \ \
\omega_1 D\phi = \frac{3}{2}\,D\phi\,, \ \ \ \ \
\omega_2 D\phi = \phi\,,
\end{equation}
and, finally, to
\begin{equation}
\label{omom}
\omega_0\omega = D\omega\,, \ \ \ \ \
\omega_1\omega = 2\omega\,, \ \ \ \ \
\omega_3\omega = \frac{1}{4}\,\Omega
\end{equation}
(only nonzero terms are displayed). So $\omega$ (\ref{omff})
is a genuine conformal vector, with central charge $c=1/2$\,,
and $\phi$ a \textit{primary} element of conformal weight $\Delta=1/2$
(that means literally (\ref{omphi}))\,,
with the OPE as follows:
\begin{gather}
\label{OPEomf}
\omega(z)\,\phi(w) = \frac{\phi(w)}{2(z-w)^2_w}
 + \frac{\partial\phi(w)}{(z-w)_w} + \ldots\ , \\
\label{OPEomom}
\omega(z)\,\omega(w) = \frac{1}{4(z-w)^4_w}
 + \frac{2\omega(w)}{(z-w)^2_w}
 + \frac{\partial\omega(w)}{(z-w)_w}\,+\,\ldots\ .
\end{gather}
Note that in this case, unlike (\ref{grad}) and (\ref{Vir}),
we prefer not to shift indices of the field components in (\ref{phiz}).

As for the vertex algebra as a whole, it is spanned, according to
the Pauli principle, by the (finite) monomials of the form
\begin{equation}
\label{span}
\ldots \phi_{-m}^{i_m} \ldots \phi_{-1}^{i_1}\Omega\ \ \ \ \ \
(m>0, \ \ i_m = 0,1)\ .
\end{equation}
The second equality in (\ref{commm}) gives
\begin{equation}
\label{}
[\omega_0,\phi_n] = (D\phi)_n = -n\phi_{n-1}\,, \ \ \ \ \
[\omega_1,\phi_n] = (-\frac{1}{2}-n)\phi_n\,,
\end{equation}
and one readily verifies that
\begin{equation}
\label{omgen}
\omega_0 a = Da\,, \ \ \ \ \ \ \ \omega_1 a = \Delta a
\end{equation}
for any $a$ of the type (\ref{span})\,, in accordance with the definition
(\ref{Vir1}) of conformal vector.

\newpage

Let us now proceed with a slightly more involved example related to charged
fermions. Here the vertex algebra is generated by the vacuum state $\Omega$
and two fermionic (odd) states $\psi^+\ (\equiv \psi)$
and $\psi^-\ (\equiv \bar\psi)$ subject to
\begin{equation}
\label{charged}
\psi^+_0\psi^- = \psi^-_0\psi^+ = \Omega\,, \ \ \ \ \ \ \
\psi^+_0\psi^+ = \psi^-_0\psi^- =
\psi^{\pm}_n\psi^{\pm} = \psi^{\pm}_n\psi^{\mp} = 0 \ \ \ \ (n>0)
\end{equation}
or, in other words,
\begin{gather}
\label{psiz}
\psi^{\pm}(z) = \sum_n\,\psi^{\pm}_n\,z^{-n-1}\,, \ \ \ \ \
[\psi^{\pm}_m,\psi^{\pm}_n]_+ = 0\,, \ \ \ \ \
[\psi^{\pm}_m,\psi^{\mp}_n]_+ = \delta_{m+n,-1}\ , \\
\label{psiOPE}
\psi^{\pm}(z)\,\psi^{\pm}(w) \sim 0\,, \ \ \ \ \ \
\psi^+(z)\,\psi^-(w) \sim \psi^-(z)\,\psi^+(w) \sim \frac{1}{(z-w)_w}\ .
\end{gather}
Now we can construct a bilinear (bosonic) current
\begin{equation}
\label{J}
J = \psi^+_{-1}\psi^- = \psi^+_{-1}\psi^-_{-1}\Omega
\end{equation}
and, for arbitrary complex number $\lambda$\,,
a conformal vector
\begin{equation}
\label{omlam}
\omega_\lambda = \frac{1}{2}J_{-1}J + (\frac{1}{2}-\lambda)DJ
 = (1-\lambda)\,\psi^+_{-2}\psi^- + \lambda\,\psi^-_{-2}\psi^+
 = [(1-\lambda)\,\psi^+_{-2}\psi^-_{-1}
 + \lambda\,\psi^-_{-2}\psi^+_{-1}]\Omega\,.
\end{equation}
We will see that $\psi^+$ and $\psi^-$ are primary states of conformal weights
$\lambda$ and $1-\lambda$\,, respectively, whereas $J$ has $\Delta=1$ and
becomes primary only for $\lambda=1/2$ (when $c=1$).

First, we check that two definitions of $\omega_\lambda$ (in terms of $J$ and
$\psi^{\pm}$) do agree. Using (\ref{nprod}), (\ref{psiz}), and properties of
$D$\,, we find
$$
J_{-1}J = (\psi^+_{-1}\psi^-)_{-1}J
 = \sum_{k\geqslant 0}(\psi^+_{-1-k}\psi^-_{-1+k} - \psi^-_{-2-k}\psi^+_k)
  \,\psi^+_{-1}\psi^-_{-1}\Omega
 = (\psi^+_{-2}\psi^-_{-1} + \psi^-_{-2}\psi^+_{-1})\,\Omega\,,
$$
$$
DJ = (D\psi^+)_{-1}\psi^- + \psi^+_{-1}D\psi^-
 = (\psi^+_{-2}\psi^-_{-1} + \psi^+_{-1}\psi^-_{-2})\,\Omega
 = (\psi^+_{-2}\psi^-_{-1} - \psi^-_{-2}\psi^+_{-1})\,\Omega\,,
$$
which verifies (\ref{omlam})\,.

Now we are ready to obtain, quite straightforwardly, the $n$-products
(with $n\geqslant0$), (anti)commutators, and singular parts of the OPE
for $\psi^{\pm}, J$ and $\omega_\lambda$ in all the relevant combinations
(as usual, only nonzero contributions are listed)\,:
\begin{gather}
\label{Jnpsi}
\psi_0^{\pm}J = \mp \psi^{\pm}\,, \ \ \ \ \
J_0\psi^{\pm} = \pm \psi^{\pm}\,, \ \ \ \ \
[J_m,\psi_n^{\pm}] = \pm\psi^{\pm}_{m+n}\,, \\
\label{Jzpsiw}
\psi^{\pm}(z)J(w) \sim \mp\frac{\psi^{\pm}(w)}{(z-w)_w}\,, \ \ \ \ \
J(z)\,\psi^{\pm}(w) \sim \pm\frac{\psi^{\pm}(w)}{(z-w)_w}\,, \\
\label{JJ}
J_1 J = \Omega\,, \ \ \ \ [J_m,J_n] = m\delta_{m,-n}\,, \ \ \ \
J(z)J(w) \sim \frac{1}{(z-w)^2_w}\,, \\
\label{ompsi}
\omega_\lambda{}_0 \psi^{\pm} = D\psi^{\pm}\,, \ \ \ \ \
\omega_\lambda{}_1 \psi^+ = \lambda\psi^+\,, \ \ \ \ \
\omega_\lambda{}_1 \psi^- = (1-\lambda)\,\psi^-\,, \\
\label{ompsiOPE}
\omega_\lambda(z)\,\psi^{\pm}(w) \sim
\frac{[1\pm(2\lambda-1)]\psi^{\pm}(w)}{2(z-w)^2_w}
 + \frac{\partial\psi^{\pm}(w)}{(z-w)_w}\,, \\
\label{Jom}
J_1\omega_\lambda = J\,, \ \ \ \
J_2\omega_\lambda = (1-2\lambda)\,\Omega\,, \ \ \ \
J(z)\,\omega_\lambda(w) \sim \frac{1-2\lambda}{(z-w)^3_w}
 + \frac{J(w)}{(z-w)^2_w}\,, \\
\label{omJ}
\omega_\lambda{}_0 J = DJ\,, \ \ \ \ \
\omega_\lambda{}_1 J = J\,, \ \ \ \ \
\omega_\lambda{}_2 J = (2\lambda-1)\,\Omega\,, \\
\label{omJOPE}
\omega_\lambda(z)J(w) \sim \frac{2\lambda-1}{(z-w)^3_w}
 + \frac{J(w)}{(z-w)^2_w} + \frac{\partial J(w)}{(z-w)_w}\,, \\
\label{omlaomla}
\omega_\lambda{}_0\omega_\lambda = D\omega_\lambda\,, \ \ \ \ \
\omega_\lambda{}_1\omega_\lambda = 2\omega_\lambda\,, \ \ \ \ \
\omega_\lambda{}_3\omega_\lambda = (-6\lambda^2+6\lambda-1)\,\Omega\,, \\
\label{omomOPE}
\omega_\lambda(z)\,\omega_\lambda(w) \sim
 \frac{-6\lambda^2+6\lambda-1}{(z-w)^4_w}
 + \frac{2\omega_\lambda(w)}{(z-w)^2_w}
 + \frac{\partial\omega_\lambda(w)}{(z-w)_w}\,,
\end{gather}
the central charge here being \,$c=-12\lambda^2+12\lambda-2
=1-3(2\lambda-1)^2$\,.

\pagebreak[1]

Let us now consider an important set of primary states $V^m$\,:
\begin{equation}
\label{mvac}
V^m = \psi_{-m}^{+}\psi_{-m+1}^{+}\ldots\psi_{-1}^{+}\Omega\,, \ \ \ \ \
V^{-m} = \psi_{-m}^{-}\psi_{-m+1}^{-}\ldots\psi_{-1}^{-}\Omega \ \ \ \ \ \
(m\geqslant 0)\,.
\end{equation}
Of course, \,$V^0 = \Omega\,, \ V^{\pm 1}=\psi^{\pm}$\,.
Using (\ref{psiz}) and (\ref{Jnpsi}), one easily checks that
\begin{equation}
\label{Jmvac}
J_{n>0}V^m = 0\,, \ \ \ \ \ J_0 V^m = m V^m\,, \ \ \ \ \
m J_{-1}\!V^m = D V^m\,.
\end{equation}
The last equality is verified by induction (say, for $m>0$)\,:
\begin{gather}
\notag
J_{-1}V^1 = J_{-1}\psi_{-1}^+\Omega = \psi_{-2}^+\Omega + \psi_{-1}^+ J
 = (D\psi^+)_{-1}\Omega + \psi_{-1}^+\psi_{-1}^+\psi^- = DV^1, \\
\notag
\psi_{-m-1}^+ J_{-1}V^m = 0\,, \ \ \ \
J_{-1}V^{m+1} = \psi_{-m-2}^+ V^m = \frac{1}{m+1}[D,\psi_{-m-1}^+]V^m
 = \frac{D V^{m+1}}{m+1}\,.
\end{gather}
Now, from
\begin{equation}
\label{}
\omega_{\lambda n} = \frac{1}{2}\,\left(\,\sum_{k<0} J_k J_{n-k-1} +
\sum_{k\geqslant 0} J_{n-k-1}J_k\right) + (\lambda
 - \frac{1}{2})\,n J_{n-1}
\end{equation}
and, in particular,
\begin{equation}
\label{}
\omega_{\lambda 0} = \sum_{k\geqslant 0} J_{-k-1}J_k\,, \ \ \ \ \
\omega_{\lambda 1} = \frac{1}{2}\,J_0 J_0 + (\lambda - \frac{1}{2})J_0
 + \sum_{k>0} J_{-k}J_k\,,
\end{equation}
we see that
\begin{equation}
\label{primV}
\omega_{\lambda 0}V^m = J_{-1}J_0 V^m = m J_{-1}V^m = DV^m\,, \ \ \ \ \
\omega_{\lambda 1}V^m = \left(\frac{m^2}{2} + m\lambda
 - \frac{m}{2}\right)V^m\,.
\end{equation}
These relations, together with $\omega_{\lambda n}V^m = 0$ for $n\!>\!1$,
characterize $V^m$ as a primary state of conformal weight
$(\frac{m^2}{2}+m\lambda-\frac{m}{2})$\,.

There exists a remarkable representation for the field $V^m(z)$ in terms of
$J_n$ and an additional invertible operator $Y$ (not to be confused with the
state-field map itself!) defined by
\begin{equation}
\label{Y}
Y\psi_n^+ = \psi_{n-1}^+Y\,, \ \ \ Y\psi_n^- = \psi_{n+1}^-Y\,,
\ \ \ Y\Omega = V^1 \ \ \Longrightarrow \ \ \ YV^m = V^{m+1},
\ \ \ V^m = Y^m\Omega\,.
\end{equation}
From
\begin{equation}
\label{}
J_{n\neq0} = \sum_k \psi_k^+ \psi_{n-k-1}^-\,, \ \ \ \ \ \
J_0 = \sum_{k\geqslant0} (\psi_{-k-1}^+\psi_k^- - \psi_{-k-1}^-\psi_k^+)
\end{equation}
one immediately concludes that
\begin{equation}
\label{YJ}
J_n Y = Y J_n \ \ (n\neq0)\,, \ \ \ \ \ \ [J_0,Y] = Y \ \ \ \ \
\Longrightarrow \ \ \ \ \ [J_0,Y^m] = m Y^m\,.
\end{equation}
A lengthy argument based on the Schur lemma shows that
\begin{equation}
\label{Vz}
V^m(z) = \ :\!e^{m\varphi(z)}\!: \ = Y^m \exp(-m\sum_{n<0}\frac{J_n}{nz^n})
\ z^{mJ_0}\, \exp(-m\sum_{n>0}\frac{J_n}{nz^n})
\end{equation}
where
\begin{equation}
\label{phi}
\varphi(z) = \ln Y + J_0\ln z - \sum_{n\neq0}\frac{J_n}{nz^n}\,, \ \ \ \ \ \
\varphi\,'(z) = J(z)\,.
\end{equation}
We treat $\ln Y$ as a creation operator in accordance with
$[J_0,\ln Y]=1$\,. Differentiating $V^m(z)$ is in agreement with
(\ref{Jmvac})\,: \ $\partial_z V^m(z)= \ m:\!J(z)V^m(z)\!: \
\equiv m(J_{-1}V^m)(z)$\,. Another identity stemming from (\ref{Vz})\,,
$V^m(z)\,\Omega=e^{zD}V^m=\exp(-m\sum_{n<0}\frac{J_n}{nz^n})V^m$, is also
valid due to the properties of Schur polynomials.

\end{document}