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Abstract
We derive exact relations to all orders in the α′ expansion for the charges of a bound system of heterotic strings, solitonic 5-branes and, optionally, a Kaluza-Klein monopole. The expressions, which differ from those of the zeroth-order supergravity approximation, coincide with the values obtained when only the corrections of quadratic order in curvature are included. Our computation relies on the consistency of string theory as a quantum theory of gravity; the relations follow from the matching of the Wald entropy with the microscopic degeneracy. In the heterotic frame, the higher-curvature terms behave as delocalized sources that introduce a shift between near-horizon and asymptotic charges. On the other hand, when described in terms of lower-dimensional effective fields, the solution carries constant charges over space which coincide with those of the asymptotic heterotic fields. In addition, we describe why the Gauss-Bonnet term, which only captures a subset of the relevant corrections of quadratic order in curvature, in some cases succeeds to reproduce the correct value for the Wald entropy, while fails in others.
Highlights
We consider a bound state of a fundamental string (F1) wrapping S1 with winding number w and momentum n, N solitonic 5-branes (NS5) wrapping T4 × S1 and a Kaluza-Klein monopole (KK) of charge W associated with the circle S1
We identify why the inclusion of only a partial subset of corrections, like the Gauss-Bonnet term, is unable to reproduce the relevant properties of the solution for the three-charge system [34], while it succeeds for the four-charge system [35]
The gravitational instanton number acts as a negative source of magnetic charge for the Kalb-Ramond field strength
Summary
A perturbative solution to first order in α of the equations (2.5)–(2.8) was found in [23, 24]. The non-vanishing terms responsible for this effect occur at the Bianchi identity (2.5) and the uu component of the Einstein equation (2.6), which produce deviations of the functions Z0,+ from the leading harmonic term. They introduce solitonic 5-brane and string momentum charge densities distributed in the exterior of the black hole. Additional higher-curvature corrections will behave as new delocalized charge sources, modifying the explicit expressions of the functions in (2.11) and, presumably, the asymptotic charges Qi and ADM mass M.
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