Nonlinear integrable flow on the framed moduli space of instantons (Q806122)

The author considers the connection between gauge theory and nonlinear integrable systems. It is introduced a set of matrices \((\alpha_ 1,\alpha_ 2,a,b)\in {\mathbb{C}}^{k\times k}\times {\mathbb{C}}^{k\times k}\times {\mathbb{C}}^{n\times k}\times {\mathbb{C}}^{k\times n}\) satisfying (i) \([\alpha_ 1,\alpha_ 2]+ab=0\) (ii)\(\begin{pmatrix} \lambda-\alpha_1 \\ \mu-\alpha_2 \\ a \end{pmatrix}\) is injective and \((\lambda -\alpha_ 1,\mu -\alpha_ 2,b)\) is surjective for all \((\lambda,\mu)\in {\mathbb{C}}^ 2,\) by the action of \[ GL((k,{\mathbb{C}}): (\alpha_ 1,\alpha_ 2,a,b)\to (p\alpha_ 1p^{-1},p\alpha_ 2p^{-1},ap^{-1},pb),\text{ where } p\in GL(k,{\mathbb{C}}). \] Using the standard theory of generalized Toda equations [\textit{W. W. Symes}, Physica D 1, 339-373 (1980)] the author shows that the definite factorization induces a flow on the space of matrices \(\{(\alpha_ 1,\alpha_ 2,a,b)\}\) characterized by a system of nonlinear equations (SNE). At the definite values of the parameter involved this SNE can be reduced to the complex cyclic-Toda hierarchy which completely parametrizes the moduli space of SU(2) k-monopoles. The received SNE is integrable provided \([\alpha_ 1,\alpha_ 2]=0\). The author proves that for any initial value \((\alpha_{1\sigma},\alpha_{2\sigma},a_{\sigma},b_{\sigma})\) of \(\tilde H^ 0(SU(n),k)\) the SNE induces a flow on the subspace \(\tilde H^0(SU(n),k)\) of the framed moduli space of SU(n) k-instantons.