Quantization for a nonlinear Dirac equation (Q2817027)

Let \(\sum M\) be the spinor bundle over a closed Riemann surface, endowed with a Hermitian metric \(<\cdot,\cdot>_{\sum M}\) and a compatible spin connection. The author considers the Dirac operator \(D\) on \(\Gamma\left(\sum M\right)\), the set of smooth sections on \(\sum M\). Then, the author studies properties of solutions on \(M\), for NEWLINE\[NEWLINED\psi=H_{jkl}<\psi^j,\psi^k>_{\sum M}\psi^l,\eqno{(*)}NEWLINE\]NEWLINE such that \(\psi=\left(\psi^l\right)_{1\leq l\leq d}\), and \(H_{jkl}=\left(H_{jkl}^{r}\right)_{1\leq r\leq d}\) is a family of \(\mathbb C^d\)-valued smooth functions on \(M\). Next, the author states two meaningful results. Regarding the first one (see Theorem 1.1), the author assumes that the energy of \(\left(\psi_n\right)_{n\in\mathbb N}\), a sequence of smooth solutions for \((*)\) are uniformely bounded by a positive constant \(\Lambda\), i.e., \(\displaystyle E\left(\psi_n,M\right):=\int_M|\psi_n|^4d\mu_g\leq \Lambda\). Then there exist finitely many blow-up points \(\{x_1,\dots,x_I\}\), a solution \(\psi\) for \((*)\) and finitely many solutions \(\zeta^{i,l}\) on \(\mathbb S^2\) for \((*)\) with \(H_{jkm}\equiv H_{jkm}(x_i)\) for \((i,l)\in \{1,\dots,I\}\times\{1,\dots,L_i\}\) such that \(\psi_n\) converges to \(\psi\), in the \(C^\infty_{\mathrm{loc}}\left(M\setminus\{x_1,\dots,x_I\}\right)\)-topology. Furthermore, the energy identity is satisfied, i.e., NEWLINE\[NEWLINE\lim_{n\to\infty}E(\psi_n,M)=E(\psi,M)+\sum_{1\leq i\leq I,1\leq l\leq L_i}E(\zeta^{i,l},M).\eqno{(**)}NEWLINE\]NEWLINE The proof of \((**)\) is done by induction thus the author shows it just for the case when \(I=1\) and \(L_1=1\). In regard to the second result, the author enhances the previous one by proving that the energy identity remains true for a noncompact spin surface (Theorem 1.2).