Euler type equations in Yang-Mills and chiral field models (Q1204630)

1. Let \(G\) be a semisimple Lie group with generators \(J_ i\), \(i = 1,\dots,N\), and \(A(x)\) be the 1-form of connectivity in the \(G\)-fiber bundle over \(\mathbb{R}^ 4\). Writing \(A(x) = A_ \mu^ i(x) J_ i dx^ \mu\), \(\mu = 1,\dots,4\), we introduce a gauge field \(A_ \mu^ i(x)\). Respective 2-form of curvature is \(F(A) = 2^{-1}F_{\mu\nu}^ i J_ i dx^ \mu \wedge dx^ \nu\). Then the Euclidean Yang-Mills field in \(\mathbb{R}^ 4\) with the gauge group \(G\) solves the equation \(\partial \mu F_{\mu\nu}^ j + f_{jk}^ \ell A^ j_ \mu F^ k_{\mu\nu} = 0\) where \(f^ \ell_{jk}\) are structure constants of \(G\). Consider also the selfduality equations \(F_{\mu\nu}^ i = 2^{-1} \varepsilon_{\mu\nu\sigma \kappa}F_{\sigma \kappa}^ i\) where \(\varepsilon_{\mu \nu \sigma \kappa}\) is the unit antisymmetric tensor in \(\mathbb{R}^ 4\). The gauge field of the form \(A_ \mu^ i = 2\eta^ a_{\mu\nu} x^ \nu T(\tau)\) is called the generalized t'Hooft ansatz. Here \(\eta^ a_{\mu\nu} = -\eta^ a_{\nu\mu}\), \(\eta^ a_{b c} = \varepsilon^ a_{bc}\), \(\eta^ a_{bu} = \delta^ a_ b = 1,2,3\). Consider now the equation \(T' + 2^{-1} \varepsilon_{abc}f^ \ell_{jk} T^ j_ b T^ k_ c = 0\) known as the Nahm equation. Theorem 1. Any solution of the Nahm equation solves the selfduality equation. 2. Let \(K\) be a semisimple Lie group, \(H\) its subgroup and \({\mathcal K}\) and \({\mathcal H}\) be their Lie algebras. Denote by \({\mathcal M}\) the tangent space to \(K/H\) at \(x_ 0 \in \mathbb{R}^ 2\) in the \(K\)-foliation over \(\mathbb{R}^ 2\). Then \({\mathcal K}= {\mathcal H}\oplus {\mathcal M}\) and if \(J_ i\) and \(J_{i'}\) are respective generators then \(f_{ik}^ j\), \(f_{i'k'}^{j'}\) and \(f^{j'}_{ii'}\) are respective structure constants. For \(g(x)\), \(g \in K\), \(x \in \mathbb{R}^ 2\) introduce the vector field \(L_ \alpha(x) = g^{-1}\partial_ \alpha g = A^ i_ \alpha J_ i + B^{i'}_ \alpha J_{i'}\).The \(K/H\) chiral model solves the equations: \(\partial_ \alpha B_ \alpha^{i'} + f^{i'}_{jk'} A^ j_ aB^{k'}_ \alpha = 0\), \(F_{\alpha\beta}^ i = -f_{j'k'}^ i B^{j'}_ \alpha B^{k'}_ \beta\), \(\partial_{[\alpha} B^{j'}_{\beta]}+ f^{i'}_{jk'} A^ j_{[\alpha} B^ i_{\beta]} = 0\). If \(K = G^{\mathbb{C}} \equiv G \otimes_ R \mathbb{C}\), \(H = G\) then these equations are called \(G^ \mathbb{C}/G\)-chiral model equations. To solve these equations the author considers the rotation invariant ansatz \(A^ i_ \alpha = 2\tau^{-1}\varepsilon_{ab} x_ \beta \psi^ i_ 3\), \(B^ i_ \alpha = 2\tau^{-1}(x_ \alpha \psi_ 2^ j - \varepsilon_{ab}x_ \beta \psi^ i_ 1)\), \(\tau = x^ 2_ 1 + x^ 2_ 2\), \(\psi_ a = \psi^ i_ a(\ln \tau)\). Theorem 2. Any solution of the Nahm equations gives a solution of \(G^ \mathbb{C}/G\)-chiral model of the above form. 3. Let \({\mathcal P}\) be a semisimple Lie algebra and \({\mathcal P}^*\) its dual space. Introduce in \({\mathcal P}^*\) the commutative multiplication operation as \(X \circ Y = Q^{\beta\gamma} Y_ \gamma\) where \(Q_ \alpha^{\beta\gamma} = Q_ \alpha^{\gamma\beta} = \text{const.}\) Then the Euler equation is \(\dot \varphi = \varphi \circ \varphi\) [see \textit{V. Trofimov} and \textit{A. T. Fomenko}, J. Sov. Math. 39, No. 3, 2683- 2746 (1987); translation from Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. 29, 3-108 (1987; Zbl 0664.58013)]. The author discusses a number of cases when the Euler equations reduce to the Nahm equation and as a result give rise to new explicit solutions of the Yang-Mills equations and the equations of chiral fields.