Logarithm of charge ratio in black hole entropy

Abstract

Logarithmic correction to BPS black hole entropy, computed from microscopic description, often contains terms involving large ratios of charges, besides the logarithmic terms involving the overall scale of the charges. If the electric charges are much larger than the magnetic charges, then the attractor value of the string coupling is small and one might hope to use weakly coupled string theory to compute logarithmic corrections involving ratios of charges from the macroscopic side. We compute these for black holes in flat space-time, preserving four supercharges, in $\mathcal{N} = 2$, $\mathcal{N}=4$ and $\mathcal{N}=8$ supersymmetric string compactifications in four dimensions. We find perfect agreement with the microscopic results in $\mathcal{N}=4$ and $\mathcal{N}=8$ theories, for which the microscopic results are known. Various stringy and statistical mechanical effects become important in this analysis, including 1) use of the correct ultra-violet cut-off (string scale instead of Planck scale), 2) correct path integral measure (ultra-local measure with appropriate dilaton dependent metric), 3) use of the correct path integral variable (Kalb-Ramond 2-form instead of the dual axion) and 4) change of ensemble (from grand canonical to microcanonical). We also verify that the measure we use is consistent with what follows from the BV formalism of string field theory.