% Title: Conformal Features of Integrable Equations
% Last modified: 12.02.2004
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\begin{center}
\large\textbf{CONFORMAL FEATURES OF INTEGRABLE EQUATIONS}
\end{center}

\vspace{.0cm}

\begin{center}
\large\textbf{Virasoro algebra in KdV}
\end{center}

\vspace{.1cm}

The famous Korteweg\,-\,de\,Vries (KdV) equation
\begin{equation}
\label{KdV}
u_t = \frac{1}{4}\,(u_{xxx} + 6uu_x)
\end{equation}
is a (first non-trivial) member of an integrable hierarchy
\begin{equation}
\label{hierKdV}
L_{t_n} = [L_{+}^{(2n-1)/2},L] \ \ \ \ \ \ \ \ \ (t_1=x\,, \ t_2=t)
\end{equation}
generated by the Lax operator
\begin{equation}
\label{L}
L = \partial^2 + u\,, \ \ \ \ \ u = u(t,x)\,, \ \ \
\partial = \partial_x\,.
\end{equation}
The KdV equation corresponds to $n=2$\,, and so is governed by
the $M$-operator
\begin{equation}
\label{L3/2}
L_{+}^{3/2} = \partial^3 + \frac{3}{2}u\partial + \frac{3}{4}u_x
 = \partial^3 + \frac{3}{4}(u\partial + \partial\circ u)\,.
\end{equation}
The hierarchy (\ref{hierKdV}) may be also seen as a reduction of the
Kadomtsev-Petviashvili (KP) hierarchy which reduces immense KP
phase space to the (unique) square root of $L$\,, with a single unknown
function $u$\,:
\begin{equation}
\label{L1/2}
L^{1/2} = \partial + \frac{1}{2}u\partial^{-1} - \frac{1}{4}u_x\partial^{-2}
 + \frac{1}{8}(u_{xx}-u^2)\,\partial^{-3} + \mathcal{O}\,(\partial^{-4})\,.
\end{equation}

The following \textit{modified} KdV equation (mKdV) is closely related
to (\ref{KdV})\,:
\begin{equation}
\label{mKdV}
q_t = \frac{1}{4}\,(q_{xxx} - 6q^2 q_x)\,.
\end{equation}
Namely, it is obtained from (\ref{KdV}) by the Miura transformation
\begin{equation}
\label{Miura}
u = q_x - q^2 \ \ \ \ \ \Rightarrow \ \ \ \
L = \partial^2 + u = (\partial - q)\,(\partial + q)
\end{equation}
which sends (\ref{KdV}) to
$$
(\partial - 2q)\,(q_t - \frac{1}{4}q_{xxx} + \frac{3}{2}q^2 q_x) = 0\,.
$$

Let us now look at these integrable equations from the Hamiltonian standpoint.
Here
\begin{gather}
\label{brac}
f_t=\{H,f\}\,, \ \ \ \ \
\{F,G\}\doteq\int\!dx\,dy \frac{\delta F}{\delta u(x)}\{u(x),u(y)\}
 \frac{\delta G}{\delta u(y)}\,, \\
\label{var}
F(u+\varepsilon v)\doteq F(u)
 + \varepsilon\!\int\!dx\,v(x)\,\frac{\delta F}{\delta u(x)}
 + \mathcal{O}(\varepsilon^2)
\ \ \ \ \Rightarrow \ \ \ \
\frac{\delta F}{\delta u(x)}
 = \sum_{n=0}^{\infty}(-\partial)^n\frac{\partial F}{\partial u^{(n)}}\,,
\end{gather}
and $u^{(n)}\equiv \partial^n u$\,. Assuming locality of the Poisson bracket,
\begin{equation}
\label{u-brac}
\{u(x),u(y)\} = -\Lambda(x)\,\delta(x-y)\,,
\end{equation}
we come to
\begin{equation}
\label{brac2}
\{G,F\}=\int\!dx \frac{\delta F}{\delta u(x)}\Lambda(x)
 \frac{\delta G}{\delta u(x)}\,, \ \ \ \ \
u_t = \Lambda\frac{\delta H}{\delta u}\,.
\end{equation}

Hamiltonians $H_m$ of the KdV hierarchy are known to be of the form
$\sim\!\!\int dx\text{Res}L^{m+\frac{1}{2}}$ ( $\text{Res}=\text{Res}_\partial$
is a coefficient of $\partial^{-1}$)\,. For example,
\begin{equation}
\label{H012}
H_0 = \int\!dx\,u\,, \ \ \
H_1 = \frac{1}{4}\int\!dx\,u^2\,, \ \ \
H_2 = \frac{1}{16}\int\!dx\,(2u^3-u_x^2)\,.
\end{equation}
Moreover, KdV is a bi-Hamiltonian system, possessing (at least) two local
Poisson structures of the type (\ref{u-brac})\,:
\begin{equation}
\label{Lambda12}
\Lambda_1 = 2\partial\,, \ \ \ \ \ \
\Lambda_2 = \frac{1}{2}\partial^3 + 2u\partial + u_x
 = \frac{1}{2}\partial^3 + u\partial + \partial\circ u\,.
\end{equation}
Acting on adjacent Hamiltonians, these two $\Lambda$ operators produce
the same result:
\begin{gather}
u_x \equiv u_{t_1} = \Lambda_1\frac{\delta H_1}{\delta u}
 = \Lambda_2\frac{\delta H_0}{\delta u} = u_x \ \ \ \ \ \
(\text{identity}) \\
u_t \equiv u_{t_2} = \Lambda_1\frac{\delta H_2}{\delta u}
 = \Lambda_2\frac{\delta H_1}{\delta u} =  \frac{1}{4}(u_{xxx} + 6uu_x)
\ \ \ \ \ (\text{KdV}) \\
u_{t_m} = \Lambda_1\frac{\delta H_m}{\delta u}
 = \Lambda_2\frac{\delta H_{m-1}}{\delta u} = \ \
\text{$m$-th member of KdV hierarchy}
\end{gather}

It is the (symmetry) properties of the second Poisson bracket provided by
$\Lambda_2$ that appear to be the main subject of this section. To begin
with, we use the Miura transformation (\ref{Miura}) to discover that
\begin{equation}
\label{}
\Lambda_2 = \frac{1}{2}\partial^3 + 2u\partial + u_x
 = (\partial - 2q)\,(-\frac{\partial}{2})\,(-\partial - 2q)\,.
\end{equation}
Then, from (\ref{brac2}) and
$$
\frac{\delta u(x)}{\delta q(y)} = (\partial_x - 2q)\,\delta(x-y)\,,
\ \ \ \ \ \ \frac{\delta G}{\delta q(x)}
 = \int\!dy\,\frac{\delta u(y)}{\delta q(x)}\,\frac{\delta G}{\delta u(y)}
 = (-\partial_x - 2q)\,\frac{\delta G}{\delta u(x)}
$$
we see that $-\frac{1}{2}\partial$ is nothing but a Poisson structure of mKdV:
\begin{equation}
\label{q-brac}
\{q(x),q(y)\} = \frac{1}{2}\partial_x\delta(x-y)\,.
\end{equation}
Indeed, the second KdV bracket immediately follows from (\ref{q-brac})
and (\ref{Miura})\,. It is also readily verified that
\begin{equation}
\label{}
q_t = -\frac{1}{2}\partial\frac{\delta H_1(u(q))}{\delta q}
 = \frac{1}{4}(q_{xxx} - 6q^2q_x)\,.
\end{equation}

Our next observation is a striking similarity between the Poisson brackets
of (m)KdV and appropriate field commutators in CFT${}_2$\,. Let us rewrite
the second KdV bracket as
\begin{equation}
\label{u-brac2}
\{u(x),u(y)\} = -(\frac{c}{12}\partial_x^3
 +2u(x)\partial_x+u_x(x))\,\delta(x-y)\,, \ \ \ \ \ \ \ \ c=6\,.
\end{equation}
Compare (\ref{u-brac2}) and (\ref{q-brac}) with local commutators of the
energy-momentum tensor and free bosonic field, respectively:
\begin{align}
\label{TT}
 [T(z),\,T(w)] &= -(\frac{c}{12}\,\partial_z^3+2T(z)\,\partial_z
 +T_z(z))\,\delta(z-w)\,, \\
\label{aa}
[a(z),\,a(w)] &= -\partial_z\,\delta(z-w)\,.
\end{align}
Now it looks only natural to mimic the corresponding CFT${}_2$ definitions
as follows:
\begin{equation}
\label{defs}
u(x) = \sum_{n}x^{-n-2}L_n\,, \ \ \ \ \
q(x) = \sum_{n}x^{-n-1}a_n\,, \ \ \ \ \
\delta(x-y) = \sum_{n}x^n y^{-n-1},
\end{equation}
to reproduce the Virasoro and Heisenberg algebras, up to inessential overall
factors:
\begin{align}
\label{-Vir}
\{L_m,L_n\} &= (m-n)L_{m+n} + \frac{c}{12}\,m(m^2-1)\,\delta_{n,-m}\,, \\
\label{aman}
\{a_m,a_n\} &= -\frac{m}{2}\,\delta_{n,-m}\,.
\end{align}
To recall the techniques, we show how to deduce (\ref{q-brac}) from
(\ref{aman})\,:
\begin{multline}
\label{}
\{q(x),q(y)\} = \sum_{mn}x^{-m-1}y^{-n-1}\{a_m,a_n\}
 = -\frac{1}{2}\sum_m m x^{-m-1}y^{m-1} \\
= -\frac{1}{2}\partial_y\,\delta(y-x)
 = \frac{1}{2}\partial_x\delta(x-y)\,.
\end{multline}
By the way, we have shown that the Miura map (\ref{Miura}) is a kind of
Sugawara construction: it builds the stress-tensor-like object $u(x)$
out of the ``free scalar field" $q(x)$\,.

The first link between the KdV Poisson bracket (\ref{u-brac2})
and the Virasoro algebra is thus established. Let us try to elaborate
it further. Recall that the Virasoro algebra (Vir)
\begin{equation}
\label{Vir}
[L_m,L_n] = (m-n)L_{m+n} + \frac{c}{12}\,m(m^2-1)\,\delta_{n,-m}
\end{equation}
can be realized as a centrally extended algebra of vector fields
$f(z)\,\partial_z$ on a circle,
\begin{equation}
\label{Vir2}
[f\partial,g\partial] \doteq (fg'-f'g)\,\partial
 + \frac{c}{12}\,\text{Res}_z (fg''')\,, \ \ \ \ \ \ \ f'\equiv\partial f\,,
\end{equation}
with $L_m=z^{-m+1}\partial_z$\,. Here $\text{Res}_z$ (coefficient of $z^{-1}$)
may be thought of as $\oint\!\frac{dz}{2\pi i}$\,, which we will denote
simply by $\int\!dz$\,. Note the property $\text{Res}_z f'\!=\!0$\,.
Now let us show that in terms of structure constants of the algebra
(\ref{Vir2})\,, the bracket (\ref{u-brac2}) assumes the form of Lie-Poisson
(Kirillov) bracket $\{\lambda_i,\lambda_j\}=c_{ij}^{k}\lambda_k$\,.
Viewing (\ref{Vir2}) as an expression for a component of the commutator
in a Lie algebra, like $[p,q]^k=c_{ij}^{k}p^i q^j$\,, we may cast it into
functional form, reserving a single discrete index 0 for central terms:
\begin{equation}
\label{funcVir}
[f,g](x) = \int\!dy\,dz\,C(x,y,z)f(y)g(z)\,, \ \ \ \ \ \
[f,g]_0 = \int\!dy\,dz\,C_0(y,z)f(y)g(z)\,,
\end{equation}
whence
\begin{equation}
\label{C}
C(x,y,z) = \delta(x-z)\,\partial_y\delta(y-z)
 - \delta(x-y)\,\partial_z\delta(y-z)\,, \ \ \ \ \ \
C_0(y,z) = - \frac{c}{12}\,\partial_z^3\delta(y-z)\,.
\end{equation}
Now a functional analog of the Lie-Poisson bracket arises:
\begin{equation}
\label{Kir}
\{\lambda(y),\lambda(z)\}
 = \lambda_0 C_0(y,z) + \int\!dx\,C(x,y,z)\lambda(x)
 = (\frac{\lambda_0 c}{12}\,\partial_y^3
 +(\lambda(y)+\lambda(z))\,\partial_y)\,\delta(y-z)\,.
\end{equation}
We assume $\lambda_0=1$\,, and observe that
the bracket (\ref{u-brac2}) may also be written as
\begin{equation}
\label{u-brac3}
\{u(x),u(y)\} = -(\frac{c}{12}\partial_x^3
 +(u(x)+u(y))\partial_x)\,\delta(x-y)
\end{equation}
owing to
\begin{equation}
\label{}
u(y)\partial_x\,\delta(x-y)
 = u(x)\partial_x\,\delta(x-y) + (y-x)u_x\,\partial_x\,\delta(x-y)
 = (u(x)\partial_x + u_x(x))\,\delta(x-y)\,.
\end{equation}
Therefore, it is of the Lie-Poisson form.

Lie-Poisson brackets are closely connected with the coadjoint action
of a Lie algebra on its dual. Let us now find a counterpart of the Poisson
structure $\Lambda_2$ (\ref{Lambda12}) in $\text{Vir}^*$\,, a dual space to
Vir. Proceeding a bit more accurately than above, we denote
a general element of the latter by a pair $(f,a)\equiv f\partial+ca$\,, with
$a$ being a number, and rewrite (\ref{Vir2}) as
\begin{equation}
\label{Vir3}
[(f,a),(g,b)] = (fg'-f'g\,,\ \frac{1}{12}\text{Res}_z\,fg''')\,.
\end{equation}
Let us also denote the adjoint action of an infinitesimal element
$(\varepsilon,\omega)$ (that is, a commutator
$[(\varepsilon,\omega),\,\cdot\ ]$) simply by $\delta_\varepsilon$\,. Then
\begin{equation}
\label{eps}
\text{ad}_{(\varepsilon,\omega)}(f,a)
 = (\delta_\varepsilon f,\delta_\varepsilon a)\,,
\ \ \ \ \ \delta_\varepsilon f = \varepsilon f' - f\varepsilon'
 = (-f\partial + f')\,\varepsilon,
\ \ \ \delta_\varepsilon a = \frac{1}{12}\text{Res}_z\, f'''\varepsilon.
\end{equation}
Further, let $(\lambda,\alpha) \in \text{Vir}^*$, \
$\alpha$ be a number, and the contraction be chosen in the form
\begin{equation}
\label{contr}
\langle(f,a)\,,(\lambda,\alpha)\rangle \doteq a\alpha
 + \text{Res}_z\,f\lambda\,.
\end{equation}
The invariance condition for (\ref{contr}) is
\begin{equation}
\label{invar}
\alpha\,\delta_\varepsilon a + a\,\delta_\varepsilon\alpha
 + \text{Res}_z\,(\lambda\,\delta_\varepsilon\!f
  + f\delta_\varepsilon\!\lambda)=0\,.
\end{equation}
This implies the following form of the coadjoint action
$\delta_\varepsilon(\lambda,\alpha)
\equiv \text{ad}_{(\varepsilon,\omega)}^{*}(\lambda,\alpha)$\,:
\begin{equation}
\label{epsdual}
\delta_\varepsilon\alpha = 0\,, \ \ \ \ \
\delta_\varepsilon\lambda = (\frac{\alpha}{12}\,\partial^3
 + 2\lambda\,\partial + \lambda')\,\varepsilon\,.
\end{equation}
Thus, $\Lambda_2$ (\ref{Lambda12}) corresponds (for $\alpha=6$) to
coadjoint action in Vir${}^*$.
This is in agreement with our previous observation that
the bracket (\ref{u-brac2}) is Lie-Poisson.

Moreover, eq.\,(\ref{epsdual}) provides additional evidence of an intimate
relation between the KdV solutions $u(x)$ and the energy-momentum tensor
$T(z)$ in CFT${}_2$\,. Recall the transformation rule of a primary field
$\varphi(z)$ of conformal dimension $\Delta$\,:
\begin{equation}
\label{prim}
\tilde{\varphi}(\tilde{z})(d\tilde{z})^\Delta = \varphi(z)(dz)^\Delta\,,
\ \ \ \ \ \
\tilde{z} = z + \varepsilon(z)\,.
\end{equation}
For infinitesimal $\varepsilon$\,, this reads
\begin{equation}
\label{eps2}
\tilde{\varphi}(z)-\varphi(z) \
 \simeq \ -(\Delta\varphi\partial + \varphi')\,\varepsilon\,.
\end{equation}
Analogous formulas for the stress tensor (which corresponds to $\Delta=2$
but is not a primary field due to an anomaly) look as follows:
\begin{equation}
\label{T'}
\tilde{T}(\tilde{z})(d\tilde{z})^2 = T(z)(dz)^2
 - \frac{c}{12}(dz)^2\,\{\tilde{z};z\}
\end{equation}
with $\{\tilde{z};z\}$ being the Schwarzian derivative
\begin{equation}
\label{Schwarz}
\{f(z)\,;z\} \doteq \frac{f'''}{f'}
 - \frac{3}{2}\left(\frac{f''}{f'}\right)^2
\end{equation}
with the properties
\begin{equation}
\label{}
\{t;z\}\,(dz)^2 = -\{z;t\}\,(dt)^2 = \{t;w\}\,(dw)^2 + \{w;z\}\,(dz)^2\,.
\end{equation}
In the infinitesimal form,
\begin{equation}
\label{epsT}
 \tilde{T}(z) - T(z) \
 \simeq \ -(\frac{c}{12}\partial^3 + 2T\partial + T')\,\varepsilon\,.
\end{equation}
Comparing $\delta_\varepsilon f$ in (\ref{eps}) with (\ref{eps2})\,,
and $\delta_\varepsilon\lambda$ in (\ref{epsdual}) with (\ref{epsT})\,,
we see that vector fields (elements of Vir) belong to $\Delta=-1$
whereas their duals to $\Delta=2$\,. Intrinsic duality of $\Delta$
and $1\!-\!\Delta$ spaces is well known in CFT${}_2$\,. It is due to
\begin{equation}
\label{dual}
\int\!dz\,f(z)\,\varphi(z) =
 \int\!f(z)(dz)^{1-\Delta}\,\varphi(z)(dz)^\Delta
\end{equation}
being a natural invariant contraction.

All the reasoning given above suggests that the KdV solutions
$u(x)$ also live in $\text{Vir}^*$ and so belong to the class $\Delta=2$
(with the central charge $c=6$)\,. Probably, the most direct evidence of
this fact is provided by the following observation. The Lax
operator~(\ref{L}) reveals the covariant properties generalizing
(\ref{prim}) with $\Delta=2$\,,
\begin{equation}
\label{covar}
\tilde{L}(\tilde{x}) = \partial_{\tilde{x}}^{2} + \tilde{u}(\tilde{x})
 = (\tilde{x}')^{-\frac{3}{2}}\,(\partial_x^2+u(x))\circ
  (\tilde{x}')^{-\frac{1}{2}}\,, \ \ \ \ \ \ \
\tilde{x}' \equiv \frac{d\tilde{x}}{dx}\,,
\end{equation}
if $u$ transforms abnormally, like a conformal stress tensor:
\begin{equation}
\label{u'}
\tilde{u}(\tilde{x})(d\tilde{x})^2 = u(x)(dx)^2
 - \frac{1}{2}(dx)^2\,\{\tilde{x};x\}\,.
\end{equation}
To show this, one uses
\begin{gather}
\label{}
\partial_{\tilde{x}}^{2} = \partial_{\tilde{x}}\circ\partial_{\tilde{x}}
 = (\tilde{x}')^{-1}\partial_x\circ(\tilde{x}')^{-1}\partial_x
 = (\tilde{x}')^{-2}\partial_x^2
  - \tilde{x}''(\tilde{x}')^{-3}\partial_x\,, \\
\partial_x^2\circ(\tilde{x}')^{-\frac{1}{2}}
 = -\frac{1}{2}\,(\tilde{x}')^{-\frac{1}{2}}\,\{\tilde{x};x\}
 - \tilde{x}''(\tilde{x}')^{-\frac{3}{2}}\partial_x
 + (\tilde{x}')^{-\frac{1}{2}}\,\partial_x^2\,.
\end{gather}
We conclude that not only the Poisson structure $\Lambda_2$\,,
but also the Lax operator of the KdV hierarchy, as well as all its
solutions, exhibit apparent features of conformal covariance, being
thus related with certain representations of the Virasoro algebra.


\end{document}