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In Symmetry Breaking through Bell-Jackiw Anomalies G. 't Hooft, Phys. Rev. Lett. 37, 8 – Published 5 July 1976,

't Hooft said that the topological quantum number $n$

$n$ is an integer for all field configuration in Euclidean space that have the vacuum (or a gauge transformation thereof) at the boundary.

I understand this $n$ is a quantized Euclidean soliton number.

But what does he mean to say "the vacuum (or a gauge transformation thereof) at the boundary"?

  • What is the vacuum at the boundary ?

  • What is the gauge transformation at the boundary ?

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Марина Marina S
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1 Answers1

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By `vacuum' he means a vanishing gauge field $A$ and by gauge transformation thereof he means some arbitrary $A$ which is related to the vanishing configuration by a non-Abelian gauge transformation. i.e. the formula for $A$ is something like (there are various conventions), $$A_\mu=i U^\dagger\partial_\mu U.$$ where $U$ is a Euclidean spacetime dependent $SU(2)$ matrix.

The boundary of 4D Euclidean space is a 3-sphere, and mappings of the 3-sphere to $SU(2)$ break up into homotopy classes indexed by the integer $n$ in formula (2).

octonion
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