Oscillator Level for Black Holes and Black Rings

Microscopic calculations of the Bekenstein-Hawking entropy of supersymmetric black holes in string theory are typically based on the application to a dual 2D CFT of Cardy’s formula, S = 2π √ c q̂0/6, where c is the central charge and q̂0 is the oscillator level. In the CFT, q̂0 is non-trivially related to the total momentum. We identify a Komar integral that equals q̂0 when evaluated at the horizon, and the total momentum when evaluated at asymptotic infinity, thus providing a gravitational dual of the CFT result.