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We investigate the Teichm\"{u}ller parameters for a Euclidean multiple BTZblack hole spacetime. To induce a complex structure in the asymptotic boundaryof such a spacetime, we consider the limit in which two black holes are at alarge distance from each other. In this limit, we can approximately determinethe period matrix (i.e., the Teichm\"{u}ller parameters) for the spacetimeboundary by using a pinching parameter. The Teichm\"{u}ller parameters areessential in describing the partition function for the boundary conformal fieldtheory (CFT). We provide an interpretation of the partition function for thegenus two extremal boundary CFT proposed by Gaiotto and Yin that it is relevantto double BTZ black hole spacetime.

Teichm\"{u}ller parameters for multiple BTZ black hole spacetime