Classification of \(\mathrm{U}(1)\)-vortices with target \(\mathbb C^N\) (Q2788777)

The author defines the notion of a twisted holomorphic map \((P,A,u)\) from a Riemann surface \(\Sigma\) to a symplectic manifold \((M,\omega)\) with a Hamiltonian group action of a compact Lie group \(G\). The twisted holomorphic map consists of a principle \(G\)-bundle \(P\), a smooth connection \(A\) of \(P\) and a smooth section \(u\) of the associated bundle \(Y=P \times_G M\) over \(\Sigma\). The section \(u\) satisfies symplectic vortex equations and is called an affine vortex if \(\Sigma\) is \(\mathbb C\).NEWLINENEWLINENEWLINEWhen the target \(M\) is \(\mathbb C^N\), \(Y\) is a vector bundle and \(u\) is a holomorphic section of the holomorphic structure determined by the anti holomorphic part of \(A\). The author characterizes the affine vortices of the diagonal \(\mathrm{U}(1)\) action on \(\mathbb C^N\) of finite energy and Maslov index \(d>0\) (the energy functional and Maslov index are defined in text). He links them with the \(N\)-product of polynomials of maximum degree \( d\) modeled out by the diagonal \(\mathbb C-0\) action.NEWLINENEWLINENEWLINEThe paper is well written and most things are developed from near scratch.