Determinant line bundle

Summary

The determinant line bundle of a continuous family of Fredholm operators is a complex line bundle over the parameter space. Its fibre at a parameter $y$ is built from the top exterior powers of the kernel and the cokernel of the operator $T_y$. The kernels and cokernels jump in dimension, but these lines fit together into a locally trivial line bundle. Quillen used it for families of Cauchy–Riemann operators and gave it a metric through zeta-regularized determinants. It is basic in the study of anomalies in physics.