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  "body": "\n## Summary\n\nThe determinant line bundle of a continuous family of [[Fredholm operators|8HWGKWD6ZM]] is a complex line bundle over the parameter space.\nIts fibre at a parameter $y$ is built from the top exterior powers of the kernel and the cokernel of the operator $T_y$.\nThe kernels and cokernels jump in dimension, but these lines fit together into a locally trivial line bundle.\nQuillen used it for families of Cauchy–Riemann operators and gave it a metric through [[zeta-regularized determinants|YT3ZQ2F7FG]].\nIt is basic in the study of anomalies in physics.\n\n## Explanations\n\n### L1 human\n\nFor an invertible matrix one has a number, the determinant.\nFor a family of operators that are sometimes not invertible, one cannot have a single number, but one can have a line at each parameter where a determinant would live.\nWhen the operator is invertible, the line has a canonical nonzero element, the determinant itself.\nThe twisting of this family of lines over the parameter space is a topological invariant of the family.\n",
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