Higher-$n$ triangular dilatonic black holes

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 $a = \sqrt{n (n + 1)/2}$. From this sequence only $n = 1,\, 2$ members were known analytically so far. We present two new $n = 3,\, 5$ triangular solutions for the theory with different dilaton couplings $a,\, b$ in electric and magnetic sectors in which case the quantization condition reads $a b = n (n + 1)/2$. These are derived via the Toda chains for $B_2$ and $G_2$ Lie algebras. Solutions are found in the closed form in general $D$ space-time dimensions. They satisfy the entropy product rules and have negative binding energy in the extremal case.

supergravity. The value a = 1 corresponds to N = 4, D = 4 supergravity [1,2]. The value 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. Such solutions have non-singular extremal limits with AdS 2 × S 2 horizons contrary the one-charged dilatonic black hole which have singular horizons in the extremal limit [1,2,5] for generic coupling constant.
As it was first noticed by Poletti, Twamley and Wiltshire [6], the values a = 1, √ 3 are just the two lowest members n = 1, 2 of the "triangular" sequence of dilaton couplings a n = n(n + 1)/2, for which dyons (known for higher n only numerically) exhibit similar behavior of the dilaton.
Analytically 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 same rule was recently shown to arise from the linearized dilaton equation on the Reissner-Nordtström dyonic background [8] as condition of existence of dilaton bound states. However, exact higher-n dyonic solutions were not known analytically. Here 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. We also give generalization to arbitrary dimensions. These new solutions are derived via Toda chains for B 2 and G 2 algebras.
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. Another interesting feature is that the extremal dyons have negative binding energy like the SL(n, R) multi-scalar Toda black holes [11].

A. Setup
Consider Einstein-Maxwell-dilaton system in D dimensions with two different dilaton coupling constants a, b for the magnetic (D − 2)-form F [D−2] and the electric 2-form F [2] . Assume the static spherically symmetric ansatz for the metric and the following solution of the equations of motion for the form fields: All the unknown functions A, B, C, φ depend on a single variable r. One convenient gauge Denoting then e C(r) = R(r), from (2) one can derive the following four equations for three functions B, R, φ (one of which following from the other three by the Bianchi identity):

B. Triangular quantization
The dilaton coupling quantization can be obtained analytically for extremal dyons as condition of regularity of the dilaton at the degenerate event horizon r = r 0 , where the expansion of e 2B in x ≡ (r − r 0 )/r 0 starts with x 2 : The leading power index n of the dilaton is assumed to be integer to ensure analyticity of the solution. The second and the third equations in (5) yield the following expressions for R h and The other equations lead to the following relations aP 2 e aφ h − bQ 2 e −bφ h = 0, Using the first equation to simplify the ratio of the second and the third ones, one finally obtains the necessary condition for the product of coupling constants: Redefining them as we getãb which for a = b coincides with (1).

C. Toda representation
Passing to another gauge: one obtains e 2A = e − 2B D−3 f − D−4 D−3 , so the metric reduces to The function f , often called the blackening factor, has one zero corresponding to the black-hole horizon and located at r = r + , r D−3 + = 2µ. The inner horizon resides at r = 0, and for the extremality parameter µ = 0 the event horizon coincides with the inner horizon. Switching to the new radial variable ρ, and denoting the derivative ∂ ρ by dot, one arrives at the following lagrangian: which belongs to the Toda type. This Lagrangian has a discrete S-duality symmetry: With this in mind, we can restrict to magnetically dominated solutions P > Q, obtaining the electrically dominated Q > P via this duality.
The general two-component Toda system for the variables χ j , j = 1, 2 is known to be integrable if the matrix C ij is the Cartan matrix of some Lie algebra and the matrix K := B −1 C T is diagonal [17]. For irreducible rank two algebras one has three options: In all these cases the Cartan matrix can be represented as where To put our system into this form one makes the substitution arriving at (17) with and the matrix B given by Imposing the conditions of diagonality on the matrix K := B −1 C T we find: Substituting the values of C 21 for Lie algebras A 2 , B 2 , G 2 one gets: Obviously one can interchangeã,b due to the symmetry. From (23) wee see that these values satisfy the quantization condition (9) for n = 2, 3, 5 respectively. The equations of motion for where we denoted To proceed further we make the following decomposition: Here H 1 , H 2 are new unknown functions and γ 1 , γ 2 are rational powers. For non-extreme solutions we have to impose the conditions g tt ∝ f 1 (non-degeneracy of the horizon) and e φ ∝ f 0 (regularity of the horizon) leading to the following expressions for γ 1 , γ 2 : Note that the parameters γ = (γ 1 , γ 2 ) are nothing but the components of the dual Weyl vector for given Lie algebra with the Cartan matrix (18). They may be represented in a explicitly symmetric form with respect to the discrete S-duality (16): In terms of the new variables the metric of the general solution reads: where h 1 = 2/κ 1 , h 2 = 2/κ 2 , so one has the following relation between the parameters intro- The extremal solutions correspond to the limit µ → 0.
The unknown functions H 1 , H 2 obey the following system of equations written in terms of the original radial coordinate r: Assuming H i → 1 as r → ∞, we look for the polynomial solutions of the form where the number of terms depends on a given algebra. The polynomials H 1 , H 2 are interchanged by S-duality as follows: Consider behavior of the metric as r → 0. The dominant terms in the polynomials . The line element in (29) for µ = 0 with account for (30) reduces to , which is obviously AdS 2 × S D−2 . This is the geometry of the extremal horizon as expected. The spatial section of the hypersurface r = 0 therefore is the sphere S D−2 .
For µ = 0 the interval in the vicinity of r = 0 is so ̺ > 0 is T -region. The D−2 sphere ̺ = 0 is the internal horizon, which is again non-singular.
In the non-extreme case it is easier to work in another representation. Specifically, for µ = 0 one can use the following map [18]: where z ≡ f = 1 − 2µ/r D−3 . Then the functions H i obey the equations provided the following relation holds for P i : We will use this condition to establish relations between charges P, Q and coefficients p 1 , q 1 .
The functions H i may also be represented as polynomials of z of degrees p, q: In the Eqs. (37, 38) P i may be set arbitrary. We fix P i = (2γ 1 p 1 , 2γ 2 q 1 ) to ensure the simplest form of the polynomials H i . Note again that the map (36) is valid only as long as µ = 0, so the extreme case has to be considered separately.

D. Physical parameters
The mass of the solutions in our gauge can be extracted from the asymptotic g tt = 1 − 2M/r D−3 + . . . as r → ∞. It depends on the extremality parameter µ and the first two coefficients of the polynomials P 1 , Q 1 as follows: Similarly, from the asymptotic of the dilaton function e λφ = 1 + 2Σ/r D−3 + . . . we find: From the Lagrangian (15) one can derive the hamiltonian which is constant by virtue of the equations of motion: where dot denotes derivative with respect to the variable ρ introduced in (14). We can equate , corresponding to the horizon r = r + , and H ∞ = H(ρ = 0), corresponding to r → ∞). Using the polynomial form of e B , e λφ and transforming to ρ, one finds: On the horizon one has: e 2B = 0,φ = 0,Ḃ = −µ(D −3) entailing H + = µ 2 (D −3) 2 . Thus from H + = H ∞ one obtains the following relation between the physical parameters of the solution which reduces to the familiar no-force condition [6,7] for µ = 0 (note different choices of units: 16πG = 1 here and 4πG = 1 in [6,7]). We will use the equation (44) to check our results presented below.
The Bekenstein-Hawking entropy associated with the outer r = r + and inner r = 0 horizons is computed as the quarter of their areas (as we have seen, the inner horizon is not a point but a (D − 2)-sphere): where r D−3 + = 2µ and r − = 0. Actually, S − is expressed through the last polynomial coefficients P p and Q q : The Hawking temperature is computed in terms of the surface gravity of the event horizon: The relation between the physical charges and coefficients p 1 , q 1 or, equivalently, P 1 , Q 1 may be obtained with help of the Eq. (38).
In the extreme limit µ → 0 the event horizon merges with the surface r = 0 while the polynomials H 1 , H 2 are such that g tt ∼ r 2(D−3) . Then the temperature vanishes, but the entropy is still finite and is given again by (46). In this case the relevant coefficients are proportional to the powers of the charges: P p ∼ P p , Q q ∼ Q q , therefore For n = 2 the following entropy rule is known to hold [11]: so the intriguing question is whether this is also true for B 2 and G 2 solutions. We will see that this is indeed the case. This property is regarded as indication on the microscopic origin of the entropies involved [9,10].

III. SOLUTIONS IN D = 4
In this section we present solutions for A 2 , B 2 , G 2 algebras in D = 4 with non-trivial dilaton profile. The first case is well-known [3], we include it here just as illustration of our representation. The dilaton oscillates between the two horizons, exhibiting n zeroes. This behavior is in agreement with the results of [6] obtained numerically for a = b. Note that due to our parametrization of the metric (13) the same formulas will be valid for any D, see the last subsection. Since the solutions were obtained with fixed values of the coupling constants a, b in accord with (23), the S-duality under the transformation (33) is not manifest explicitly.
It is worth noting that our equations also have trivial φ = const solutions for charges obeying the relation aP 2 = bQ 2 , which are Reissner-Nordstrom dyons.

A. A 2 ≃ sl(3, R)
In this well-known case we have p = q = 2 and the polynomials H i have the following structure: One can also obtain expressions of p 1 , q 1 via the coefficients P 1 , Q 1 for H 1 (r), H 2 (r): so the charges by virtue of the Eqs. (38) are In the extreme case (µ = 0) we work with polynomials H i instead of H i . We can directly express coefficients P i , Q i in polynomials H i through values of charges: From here one finds the following expansion of the dilaton field φ near the horizon in the extremal case: Typical behavior of the shifted dilaton φ + φ h and e 2B for the extremal A 2 , B 2 and G 2 solutions is shown in Figs. 1, 2, where φ h is the horizon value in the extremal limit (note that it is different from φ(r + ) in the non-extreme case) The mass and the residual entropy of the extremal solutions are: The product of entropy of inner horizon S − and entropy of outer horizon S + obeys the following relation The mass of dyons and the normalized mass difference (57) of extremal solutions are shown in Fig. 3. Black holes related to SL(n, R), n ≥ 3 Toda chains were shown to have with negative binding energy (defined as −∆M) in [11]. Such dyons could split into a pair of singly charged black holes. For non-extremal solutions the sign of the binding energy depends on the charges. The dependence of the dilaton charge on parameter ξ = Q/P in both extreme and nonextreme cases is shown in Fig. 4. Expressions for the polynomials H i read H 1 = 1 + 3p 1 z + 3p 1 q 1 z 2 + p 2 1 q 1 z 3 , The relation between the pairs of the coefficients p 1 , q 1 and P 1 , Q 1 is , and the constraint (38) gives In the extreme case we were able to express the coefficients P 1 , Q 1 through the charges in terms of some parameter ζ instead of ξ = Q/P : We obtain: The near-horizon expansion of the dilaton in the extreme case is with φ h being constant (nonzero) value. This is exactly what we expected from (6).
The entropy of the extreme solution is given by the following formula: The product of entropy of inner horizon S − and entropy of outer horizon S + obeys the following relation S − S + = π 2 r 4 + p 6/5 1 q 4/5 1 The numerical calculations confirms that this solution also has negative binding energy in the extreme case (see Fig. 3).

C. G 2 solution
The polynomials H i are: The constraint (38) gives For G 2 algebra the expressions relating p 1 , q 1 to P 1 , Q 1 are rather lengthy, so we do not give them here.
In the extreme G 2 -case we were not able to obtain short expressions using the auxiliary parameter ξ (or ζ), so we give here just the explicit relations between the coefficients P i , Q i : −63Q 2 (P 2 1 + P 2) + 2Q 2 2 , P 5 = 1 60 −7P 2 Q 3 + 12(P 1 P 4 + P 2 P 3 ) , To find the relation between P, Q and P 1 , Q 1 , one should notice that the system of algebraic equations is over-determined, so there are additional relations between the coefficients. Using the no-force condition (44) one finds from the relations (68) the following: In terms of P 1 , Q 1 the no-force condition (44) reads Solving these equations one has to take the solution corresponding to the positive values of P 2 , Q 2 and positive values of P 1 , Q 1 . Lacking an analytic relation between the charges and P 1 , Q 1 , we cannot present the near-horizon expansion of the dilaton field, but we checked numerically that φ ext = φ h + O(r 5 ) and e 2B ∝ r 2 + O(r 3 ) indeed.
Evaluation of mass confirms that G 2 black holes also have negative binding energy in the extreme case (see Fig. 3). The product of entropy of inner horizon S − and entropy of outer horizon S + obeys the following relation

IV. DISCUSSION
We have constructed new analytic solutions of dilaton gravity with different couplings of the dilaton to electric/magnetic sectors. They correspond to n = 3, 5 sequence of triangular coupling and are presented as B 2 and G 2 Toda solutions respectively. The metric functions are expressed in terms of polynomials whose rank depends on the group and which were found explicitly for the non-extreme and extremal cases. As expected, the dilaton function has n zeroes between the horizons in the non-extreme case while in the extremal case the integer n is the power of the first Taylor expansion term of the dilaton at the degenerate horizon. The binding energy depends on the ratio of electric/magnetic charges and it becomes negative for all charges in the extremal case. New dyons satisfy the entropy product rule, indicating on possibility of statistical explanation of their entropy.
It would be interesting to find the higher-dimensional origin of the presented solutions.
Clearly, the underlying group structure may serve the key. Indeed, it is believed that the Toda solutions are related to one-dimensional subspaces of the three-dimensional sigma models target spaces arising in dimensional reduction of higher-dimensional theories [19][20][21]. This is so for a = b [22]. The simplest sigma-model underlying to B 2 = SO(2, 3) could be the fourdimensional Einstein-Maxwell-Dilaton-Theory [23], while the G 2 is encountered in the D = 5 minimal supergravity [24]. These theories, however, do not seem to be relevant. Rather, our Toda black holes could correspond to some higher-dimensional oxidation of the threedimensional cosets [25,26], in particular, different embeddings of G 2 are known [27]. We leave this question to future work.