# Ngô Quốc Anh

## August 11, 2010

### Two ways to present SU(3) Toda systems

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 4:13

This entry is concerned with SU(3) Toda systems in non-abelian relativistic self-dual gauge theory. Usually, they are given by

$\displaystyle\begin{gathered} - {\Delta _g}{u_1} = 2{\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) - {\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ - {\Delta _g}{u_2} = - {\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) + 2{\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ \end{gathered}$

on $M$ with

$\displaystyle\int_M {{u_1}d{v_g}} = \int_M {{u_2}d{v_g}} = 0$

where $(M,g)$ denotes a compact Riemannian surface and $\lambda_1$, $\lambda_2$ are non-negative constants.

We now consider function $h$ satisfying

$\displaystyle - \Delta h =\frac{1}{|M|} \quad \text{ in } M$

The existence of such a $h$ will be considered later.

We now introduce

$\displaystyle\begin{gathered} {v_1} = {u_1} - \log \int_M {{e^{{u_1}}}d{v_g}} - (2{\lambda _1} - {\lambda _2})h, \hfill \\ {v_2} = {u_2} - \log \int_M {{e^{{u_2}}}d{v_g}} - ( - {\lambda _1} + 2{\lambda _2})h. \hfill \\ \end{gathered}$

It is then obvious to see that

$\displaystyle\begin{gathered} - \Delta {v_1} = - \Delta {u_1} + (2{\lambda _1} - {\lambda _2})\Delta h = - \Delta {u_1} + \frac{{2{\lambda _1} - {\lambda _2}}}{{|M|}}, \hfill \\ - \Delta {v_2} = - \Delta {u_2} + ( - {\lambda _1} + 2{\lambda _2})\Delta h = - \Delta {u_2} + \frac{{ - {\lambda _1} + 2{\lambda _2}}}{{|M|}}. \hfill \\ \end{gathered}$

Besides,

$\displaystyle\begin{gathered} {e^{{v_1}}} = \frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }}{e^{ - (2{\lambda _1} - {\lambda _2})h}}, \hfill \\ {e^{{v_2}}} = \frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }}{e^{ - ( - {\lambda _1} + 2{\lambda _2})h}}, \hfill \\ \end{gathered}$

which implies

$\displaystyle\begin{gathered} \frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} = {e^{{v_1}}}{e^{(2{\lambda _1} - {\lambda _2})h}}, \hfill \\ \frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} = {e^{{v_2}}}{e^{( - {\lambda _1} + 2{\lambda _2})h}}. \hfill \\ \end{gathered}$

Thus

$\displaystyle\begin{gathered} - \Delta {v_1} = - \Delta {u_1} + \frac{{2{\lambda _1} - {\lambda _2}}}{{|M|}} \hfill \\ \qquad= 2{\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) - {\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) + \frac{{2{\lambda _1} - {\lambda _2}}}{{|M|}} \hfill \\ \qquad= \frac{{2{\lambda _1}{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{{{\lambda _2}{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} \hfill \\ \qquad= 2{e^{\log {\lambda _1} + (2{\lambda _1} - {\lambda _2})h}}{e^{{v_1}}} - {e^{\log {\lambda _1} + ( - {\lambda _1} + 2{\lambda _2})h}}{e^{{v_2}}}. \hfill \\ \end{gathered}$

Similarly, we get

$\displaystyle - \Delta {v_2} = - {e^{\log {\lambda _1} + (2{\lambda _1} - {\lambda _2})h}}{e^{{v_1}}} + 2{e^{\log {\lambda _1} + ( - {\lambda _1} + 2{\lambda _2})h}}{e^{{v_2}}}$.

Therefore we have the following form

$\displaystyle\begin{gathered} - \Delta {v_1} = 2{a_1}{e^{{v_1}}} - {a_2}{e^{{v_2}}}, \hfill \\ - \Delta {v_2} = - {a_1}{e^{{v_1}}} + 2{a_2}{e^{{v_2}}}. \hfill \\ \end{gathered}$