Dipole Moments of Black Holes and String States

TL;DR

The authors test the conjectured equivalence between massive string states and extremal black holes by computing the electric and magnetic dipole moments for four-dimensional $N=4$ supergravity with 22 vector multiplets and comparing them to heterotic strings on $T^6$ or Type IIA strings on $K3\times T^2$. By generating full black hole supermultiplets from a bosonic Kerr-like state using fermion zero modes, they identify the Kerr angular momentum $L$ with the superspin and derive gyromagnetic and gyroelectric ratios, finding exact agreement with string states not only for BPS short multiplets but also for intermediate multiplets, and even for certain non-supersymmetric extremal states. The intermediate multiplets, in particular, reveal that gyromagnetic ratios are not fixed by supersymmetry alone, providing a stringent test of the correspondence; duality to Type II strings clarifies the electromagnetic structure and resolves ambiguities in left-handed dipole moments. Additionally, electric dipole moments arise for graviphoton couplings in several cases, with duality-mapped results consistent with Type II/heterotic pictures, suggesting a rich underlying worldsheet structure and a broader, M-theory–level interpretation of black holes as elementary string states.

Abstract

As a further test of the conjectured equivalence of string states and extremal black holes, we compute the dipole moments of black holes with arbitrary spin and superspin in D=4,N=4 supergravity coupled to 22 vector multiplets and compare them with the dipole moments of states in the heterotic string on $T^6$ or the Type IIA string on $K3 \times T^2$. Starting from a purely bosonic black hole with Kerr angular momentum L, the superpartners are generated by acting with fermion zero modes, thus filling out the complete supermultiplet. $L$ is then identified with the superspin. On the heterotic side, elementary states belong only to short to long multiplets, but Type IIA elementary states can belong to intermediate multiplets as well. We find that the black hole gyromagnetic ratios are in perfect agreement with the string states not only for the BPS states belonging to short multiplets but also for those belonging to intermediate multiplets. In fact, these intermediate multiplets provide a stronger test of the black-hole/string-state equivalence because the gyromagnetic ratios are not determined by supersymmetry alone, in contrast to those of the short multiplets. We even find agreement between the non-supersymmetric (but still extremal) black holes and non-BPS string states belonging to long supermultiplets. In addition to magnetic dipole moments we also find electric dipole moments even for purely electrically charged black holes. The electric dipole moments of the corresponding string states have not yet been calculated directly but are consistent with heterotic/Type IIA duality.