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}

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