# Why are topological properties described by surface terms?

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An example are the anomalies in abelian and non-abelian gauge quantum field theories.

For example, the abelian anomaly is $\tilde {F}_{\mu\nu}F^{\mu\nu}$ and the integral over this quantity is a topological invariant which measures a topological characteristic of the gauge field $A_\mu$.

All such quantities can be rewritten as total derivatives and then, using Gauss' law transformed into a surface integral.

**What's the intuitive reason that quantities which describe topological properties can always be written as surface integrals?**

Formulated a bit differently: Why are topological properties always completely encoded in the boundary of the system?

Locally (i.e., in the bulk), topology cannot enter as all manifolds are locally diffeomorphic to the unit ball.

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