# Is the Chern-Simons integral of gauge fields over black hole singularities zero?

Suppose we have an evaporating black hole and a nonabelian Yang-Mills theory with a $\theta$ topological term. This counts the total number of instantons minus antiinstantons. Consider the total number of instantons minus antiinstantons inside the black hole, i.e. between the event horizon and the singularity. This is equal to the difference between Chern-Simons integral between the singularity and event horizon. According to black hole complementarity, we replace the interior with a stretched horizon. Do we have to add an effective Chern-Simons term to the stretched horizon action? If so, is this equivalent to saying the CS integral over the singularity has to be zero? Otherwise, we get decoherence by CS singularity value, converting pure states to mixed states?

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ok, no downvote, maybe this is sincere. The singularity isn't space--- r is time inside the BH. Instanton count is in imaginary time. – Ron Maimon Jul 30 '12 at 15:39