---
tag: R72NAFZ722
kind: concept
name: Seeley–DeWitt coefficients
msc: [58J35, 58J50, 81T20]
sources:
  - url: https://en.wikipedia.org/wiki/Heat_kernel
---

## Summary

The Seeley–DeWitt coefficients are the coefficients $a_k(x)$ of the small-time asymptotic expansion of the heat kernel of a Laplace-type operator on the diagonal.
They are local invariants: each is a universal polynomial in the curvature, the endomorphism part of the operator, and their covariant derivatives.
Their integrals are spectral invariants, and in index theory and quantum field theory they compute indices, anomalies and divergences.
For the scalar Laplacian they are the coefficients of the [[Minakshisundaram–Pleijel expansion|45P2SEGGR1]].

## Explanations

### L1 human

Heat spreading for a short time on a curved space behaves almost as in flat space.
The deviation is a series in powers of the time, and its coefficients are the Seeley–DeWitt coefficients.
The first one is $1$, the second involves the scalar curvature, and later ones involve more and more curvature terms.
Physicists use them to read off the short-distance behaviour of quantum fields on curved spacetimes.