Abstract
Dilaton gravity with the form fields is known to possess dyon solutions with two horizons for the discrete “triangular” values of the dilaton coupling constant a=n(n+1)/2. This sequence first obtained numerically and then explained analytically as consequence of the regularity of the dilaton, should have some higher-dimensional and/or group theoretical origin. Meanwhile, this origin was explained earlier only for n=1,2 in which cases the solutions were known analytically. We extend this explanation to n=3,5 presenting analytical triangular solutions for the theory with different dilaton couplings a,b in electric and magnetic sectors in which case the quantization condition reads ab=n(n+1)/2. The solutions are derived via the Toda chains for B2 and G2 Lie algebras. They are found in the closed form in general D space–time dimensions. Solutions satisfy the entropy product rules indicating on the microscopic origin of their entropy and have negative binding energy in the extremal case.
Highlights
Einstein-Maxwell-dilaton (EMD) theory in four dimensions may originate from different supersymmetric higher-dimensional theories with various values of the dilaton coupling constant a in the Maxwell term e−2aφF 2 in the Lagrangian
A = 3, corresponds to dimensionally reduced D = 5 gravity [3, 4], which have different supersymmetric extensions. In all these cases analytical solutions are known for static black holes possessing both electric and magnetic charges with two horizons between which the dilaton exhibits n oscillations
We give solutions for n = 3, 5 in the theory where the dilaton couples to electric and magnetic sectors with different coupling constants a, b, in which case the quantization rule generalizes to ab = n(n + 1)/2
Summary
Einstein-Maxwell-dilaton (EMD) theory in four dimensions may originate from different supersymmetric higher-dimensional theories with various values of the dilaton coupling constant a in the Maxwell term e−2aφF 2 in the Lagrangian. A = 3, corresponds to dimensionally reduced D = 5 gravity [3, 4], which have different supersymmetric extensions In all these cases analytical solutions are known for static black holes possessing both electric and magnetic charges (dyons) with two horizons between which the dilaton exhibits n oscillations. For which dyons (known for higher n only numerically) exhibit similar behavior of the dilaton This triangle quantization rule was rederived in [7] as condition of regularity of the dilaton in the case of coinciding horizons, i.e. for extremal dyons. The solutions satisfy the entropy product rule: the product of the entropies of the internal and internal horizons are functions of their charges only [9, 10] This property is considered to be an indication on possibility of statistical interpretation of the entropy. Our results have some overlap with the recent preprints [8, 16]
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