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I am currently working with (2+1) gravity in Chern-Simons formulation and I have a question about thermal AdS. The way I understand that one retrieves thermal AdS is by Wick rotation and compactification in the time direction. However, I am not sure how you find the temperature in Chern-Simons direction.

For example, when you consider BTZ black hole, the holonomy in the (euclidean) time-cycle will give you a constaint between the asymptotic charges and the temperature. Thus the temperature is encoded this way in the connection.

Now, in thermal AdS in three dimensions, I cannot see how to encode the temperature in the gauge potential.

Do you have any references or information about it? Thanks in advance!

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