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I have a simple question about Lewkowycz and Maldacena's paper

In section 2, they consider the scalar field in BTZ background ground and require boundary condition of the scalar field,

$\phi \sim e^{i\tau}$ . This boudary condition breaks the U(1) isomentry of BTZ background. My question is what kinds of boudary condition will respect the U(1) isometry? phi does not depend on tau?

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