---
tag: 5DST3VKEMX
kind: concept
name: Determinant line bundle
msc: [58J52, 47A53]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant_line_bundle
same_as:
  wikidata: Q135648941
  wikipedia: Determinant line bundle
---

## Summary

The determinant line bundle of a continuous family of [[Fredholm operators|8HWGKWD6ZM]] 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|YT3ZQ2F7FG]].
It is basic in the study of anomalies in physics.

## Explanations

### L1 human

For an invertible matrix one has a number, the determinant.
For 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.
When the operator is invertible, the line has a canonical nonzero element, the determinant itself.
The twisting of this family of lines over the parameter space is a topological invariant of the family.