Generalized Freudenthal duality for rotating extremal black holes

Freudenthal duality (FD) is a non-linear symmetry of the Bekenstein-Hawking entropy of extremal dyonic black holes (BHs) in Maxwell-Einstein-scalar theories in four space-time dimensions realized as an anti-involutive map in the symplectic space of electric-magnetic BH charges. In this paper, we generalize FD to the class of rotating (stationary) extremal BHs, both in the under- and over-rotating regime, defining a (generalized) rotating FD (generally, non-anti-involutive) map (RFD), which also acts on the BH angular momentum. We prove that the RFD map is unique, and we compute the explicit expression of its non-linear action on the angular momentum itself. Interestingly, in the non-rotating limit, RFD bifurcates into the usual, non-rotating FD branch and into a spurious branch, named"golden"branch, mapping a non-rotating (static) extremal BH to an under-rotating (stationary) extremal BH, in which the ratio between the angular momentum and the non-rotating entropy is the square root of the golden ratio. Finally, we investigate the possibility of inducing transitions between the under- and over-rotating regimes by means of RFD, obtaining a no-go result.


Introduction
Extremal black holes (BHs) are objects of great interest to the string theorists, though they concern about a pretty ideal situation, as they lack temperature T .On the other hand, astrophysical BHs are never exactly extremal, but, for instance, the BH GRS1915+105 observed through X-ray and a radio telescope is likely within 1% of the extremal value of its angular momentum [1]; in other words, this BH is near-extremal, i.e. not far from saturating the extremality bound.
For asymptotically flat and static solutions, Freudenthal duality (FD) is an intrinsically nonlinear and anti-involutive map between the BH dyonic charges, which, unexpectedly, keeps the Bekenstein-Hawking BH entropy invariant [2][3][4].Interestingly, under FD, the attractor configurations of the scalar fields at the electric-magnetic (e.m.) dyonic BH event horizon also remain invariant [3].A suitable extension of FD has been formulated for Maxwell-Einstein theories in the presence of gaugings of the isometries of the vector multiplets' or hypermultiplets' scalar manifolds (at least as far as Abelian gaugings are concerned [5]).Furthermore, in recent years FD has turned out to characterize a number of directions of investigation within the Maxwell-Einstein (super)gravity theories coupled to non-linear sigma models of scalar fields [6][7][8][9][10][11].A crucial point to stress is that FD is inherently different from the electro-magnetic duality (aka U -duality in string theory 1 ), since the later act linearly on dyonic BH charges, contrary to the former [15][16][17].
Recently, in [18] and [19], FD has been studied and further extended to the class of nearextremal (non-rotating) BHs.As mentioned above, for extremal, (asymptotically flat) nonrotating BHs, FD is an anti-involutive non-linear map acting on the e.m.BH charges, and it is a symmetry of the Bekenstein-Hawking BH entropy, which is a homogeneous function of degree two in the charges themselves.Since the T -dependent entropy of a near-extremal BH is no longer a degree-two homogeneous function of charges [18], a consistent generalization of FD has been achieved in [19] only at the price of transforming the temperature T .The resulting map, which is no longer anti-involutive (but nevertheless is analytical and unique), has been termed nearextremal (generalized2 ) FD : two near-extremal BHs, whose charges and (small) temperatures are connected by the near-extremal FD, have the same entropy.
In this paper, we aim to further extend the notion of FD to the class of extremal (stationary) rotating BHs, within the same spirit as [19].In section 2, we show that in general the definition of the Freudenthal dual for an extremal (asymptotically flat) over-(or under-)rotating BH is not consistent while keeping its angular momentum fixed.To overcome this issue, in section 3 we define a -generally, non-anti-involutive -(generalized) rotating FD (RFD) map: interestingly, within our formulation, RFD uniquely maps the dyonic e.m. charges and the angular momentum J of an extremal rotating BH to the ones of another extremal rotating BH, while keeping the Bekenstein-Hawking BH entropy fixed.
The plan of the paper is as follows.After a first, naïve generalizing approach and some preliminary considerations in Sec. 2, we present the generalization of FD to RFD in Sec. 3, analyzing in detail its non-rotating limit in Sec.3.1.Then, after presenting numerical evidence in Sec. 4, we prove the uniqueness of the RFD map in Sec. 5, and determine its analytic form in Sec. 6.Finally, in Sec.7 we investigate the possibility of inducing transitions between the (stationary) under-and over-rotating regimes by means of the RFD map, obtaining a no-go result.Final remarks and hints for further developments are given in the concluding Sec. 8.
Apps.A and B, containing details concerning the treatment given in Sec. 7, conclude the paper.

Freudenthal duality (FD) and rotating BHs
In this section, we analyse the prospect of having an F-dual of an extremal, over-(or under)rotating BH, such that both of them have the same angular momentum and entropy whereas their e.m. charges are related by FD.
Before proceeding further, we first briefly review the Freudenthal duality for extremal, nonrotating black holes.For such black holes with e.m. charges collected into the symplectic vector Q and entropy S 0 (Q), the Freudenthal duality acts non-trivially on Q as follows Ω is a symplectic matrix with Ω T = −Ω and Ω 2 = −I.In other words, two non-rotating, extremal black holes with e.m. charge vectors Q and Q related to each other by (2.1) have the same value for their entropy.The non-rotating, extremal entropy is a degree-two homogeneous function in the e.m. charge Q i.e.
and this property plays a fundamental role for the invariance of the entropy under (2.1).
It is worth noticing that, in this case, the transformation (2.1) is anti-involutive, i.e.Q = −Q.
The study of the Freudenthal duality was further extended in a non-anti-involutive way for the near-extremal, non-rotating black holes in [18,19].Due to the presence of the temperature T in the form, the entropy of a near-extremal, non-rotating black hole is no more a degreetwo homogeneous function of e.m. charges.Hence, the obvious way of charge transformation following (2.1), i.e.Q(Q; T ) := Ω ∂S N E (Q;T )

∂Q
, called as "on-shell" FD does not keep the nearextremal entropy invariant.In spite of this, it has been shown that two different near-extremal, non-rotating black holes with charge and temperature respectively (Q, T ) and ( Q, T + δT ) have the same entropy when with a unique solution for δT .
With this brief recap, we now establish the Freudenthal duality for extremal, rotating black holes.
After [20,21], the Bekenstein-Hawking entropy for an extremal, (asymptotically flat) underrotating Kerr-Newman BH with angular momentum J and e.m. charge Q is given by (2.5) whereas the entropy for an extremal, (asymptotically flat) over-rotating Kerr-Newman BH is3 (2.7) In general, within the semi-classical supergravity approximation, the e.m. charges take real values, and so do the angular momentum J and the non-rotating entropy S 0 (Q), which are further constrained to be non-negative and strictly positive, respectively: J ∈ R + , S 0 ∈ R + 0 .Following the usual formulation of FD [3], the action of the "on-shell" FD map on Q reads with Ω denoting the symplectic invariant structure of the e.m. charge representation space (Ω T = −Ω and Ω 2 = −I); the "+" and "−" signs correspond to the under-rotating and overrotating cases respectively.Thus, a naïve generalization of FD (leaving J fixed), in these cases, can be formulated as a map acting on Q as (2.9), such that S under(over) (Q, J) = S under(over) Ω ∂S under(over) (Q, J) ∂Q , J .
(2.10) Using (2.5) and (2.7), one can show that for both the under-rotating and over-rotating extremal Kerr-Newman BHs, the conditions for the existence of a Freudenthal dual boils down to solving a very simple algebraic equation, namely4

Under-rotating
For an under-rotating BH, J 2 − S 2 0 (Q) < 0, and therefore J 2 − 2S 2 0 (Q) < 0 always.In this case, the only possible solution of (2.11) is J = 0.This is a trivial solution, since in such a limit (2.10) reduces to the usual (and anti-involutive) notion of FD for a non-rotating, extremal BH [3].

Over-rotating
For an over-rotating BH, J 2 − S 2 0 (Q) > 0, and this rules out the possibility of the solution J = 0 to (2.11).Still, one could define the F-dual of an over-rotating extremal Kerr-Newman BH, when the condition (2.11) is met as (2.12) Thus, one can conceive the starting extremal BH as a non-rotating BH, for which having a F-dual is obvious [3].
3 Generalized rotating Freudenthal duality (RFD) We have just observed that, by the usual notion of FD, one cannot map two extremal rotating BHs to each other.In this section, we go beyond the usual definition of FD and try to map two extremal rotating BHs (both under-or over-rotating) with different angular momenta.
Namely, motivated by the approach carried out in [19] for a near-extremal BH with the temperature T , here we define the transformation of the dyonic e.m.BH charges as follows : Where the branch "+" and "−" represents the under-rotating (S 2 0 (Q) > (J + δJ) 2 ) and overrotating (S 2 0 (Q) < (J + δJ) 2 ) cases respectively, such that the Bekenstein-Hawking BH entropy remains invariant, i.e. S under(over) (Q, J) = S under(over) Q(Q, J + δJ), J + δJ . (3.2) Therefore, the set of transformations RFD : define the (generalized) rotating FD (RFD) map.By using (2.5), (2.7), for both under-and overrotating BHs, the condition (3.2) boils down to the following sextic algebraic inhomogeneous equation in the variation δJ of the angular momentum : where each coefficients is a function of J and S 0 = S 0 (Q), In order to find out two rotating extremal BHs with different angular momenta but the same entropy, having their dyonic charges related by RFD (3.3), we need to look for a real solution of (3.4), namely for δJ = δJ(J, S 0 ) ∈ R such that the transformed angular momentum reads 3.1 Non-rotating limit of RFD : the spurious, "golden" branch Before dealing with a detailed analysis of the roots of the algebraic equation (3.4), let us investigate the limit J → 0 (+) of the set of solutions to (3.4).In this limit, which is well-defined only for the under-rotating class of solutions, Ĵ → δJ (+) ∈ R + , and Eq.(3.4) reduces to which consistently admits the solution δJ = 0.This is no surprise, since in the J → 0 (+) limit, RFD simplifies down to its usual, non-rotating definition (namely, to FD, [3]).
Interestingly, other two real solutions to the cubic algebraic homogeneous equation (in (δJ) 2 ) (3.6) exist.In fact, Eq. (3.6) can be solved by two more real roots 5 with ϕ : being the so-called golden ratio (see e.g.[22]).Since Ĵ → δJ (+) ∈ R + , only δJ + in (3.7) has a sensible physical meaning.Correspondingly, within the J → 0 + (i.e., nonrotating) limit, the unique non-vanishing, physically sensible solution for the angular momentum transformation reads This analysis shows the existence, in the limit J → 0 + , of a spurious, "golden" branch of RFD, which in the non-rotating limit does not reduce to the usual FD, but rather it allows to map a non-rotating extremal BH to an under-rotating (stationary) one; this can be depicted as where Ĵgolden is defined by (3.8), Qgolden is defined by and RFD J→0 + , golden denotes such a spurious, "golden" branch of the non-rotating limit of RFD Note that in the last step of (3.10) we used the crucial property of the golden ratio ϕ, namely

.12)
5 There exist also two purely imaginary roots with , which we ignore.
Consistently, the total entropy is preserved by the map (RFD) J→0 + , golden (3.11), because where in the last step the crucial property (3.12) has been used again.Moreover, in achieving (3.13), we have also exploited two crucial properties of the static extremal BH entropy S 0 (Q), namely its homogeneity of degree two in the e.m. charges, and its invariance under the Freudenthal duality for static extremal BHs, as given by (2.3) and (2.2) respectively.
Remark 1 It is worth remarking that the stationary extremal BH in the r.h.s. of (3.9), namely the image of an extremal, static BH under the map RFD J→0 + , golden (3.11), is necessarily underrotating, because the non-rotating limit J → 0 + is well-defined only in the under-rotating case.
On the other hand, the BH entropy entering the definition of Qgolden (3.10) is necessarily of the over-rotating type, since which pertains to the over-rotating case.
Remark 2 In a sense, the use of the spurious, "golden" branch (RFD) J→0 + , golden (3.11) of the map RFD (3.3) can be regarded as a kind of solution-generating technique, which generates an under-rotating, stationary extremal BH from a non-rotating, static BH, while keeping the Bekenstein-Hawking entropy fixed.Indeed, while both such BHs are asymptotically flat, their near-horizon geometry changes under (RFD) J→0 + , golden : under-rotating extremal BH [24] . (3.15)

Remark 3
The expression of the Bekenstein-Hawking entropy of under-rotating resp.overrotating stationary extremal BHs, respectively given by (2.5)-(2.6)and (2.7)-(2.8), is suggestive of a representation of the two fundamental quantities S 0 and J characterizing such BHs in terms of elements of the split complex (also named hypercomplex ) numbers C s (see e.g.[25]), i.e. respectively as where i denotes the split imaginary unit (i 2 = 1, not to be confused with ±1), and the bar stands for the hypercomplex conjugation, defined as Z := S 0 − iJ.
Thus, any map acting on S 0 and J, as the RFD map (3.3) itself, can be represented as acting on the hypercomplex (analogue of the) Argand-Gauss plane, denoted by for under-resp.over-rotating BHs.Since C s is not a field (because it contains zero divisors due to its split nature), there are no split analogues of the "wild" automorphisms [26]  or C s [J, S 0 ]; interestingly, since S 0 ∈ R + 0 and J ∈ R + , none (but, trivially, I) of such discrete automorphisms are physically allowed, and the physically sensible quadrant of C s [S 0 , J] resp.
C s [J, S 0 ] is only the first one (with the J axis excluded).
In particular, the RFD map acts on any preserving transformation, because, by definition, it preserves the Bekenstein-Hawking entropy S, as given by the defining condition (3.2).Thus, considering a under-resp.over-rotating (stationary, asymptotically flat) extremal BH with Bekenstein-Hawking entropy S = S ∈ R + , its RFD-dual extremal BH will belong to the arc of the hyperbola 0 = S within the first quadrant (with the J axis excluded) of the hypercomplex Argand-Gauss plane.
In the non-rotating limit, in light of the treatment of Sec.3.1, the action of the RFD map (3.3) on a non-rotating extremal BH (represented as a point of the S 0 axis -with its origin excluded -in the hyper-Argand-Gauss plane C s [S 0 , J], thus with coordinates (S 0 , 0)) may be nothing but the identity I (in the usual, non-rotating branch of RFD J→0 + ) or a point with coordinates ϕS 0 , √ ϕS 0 along the arc of hyperbola defined as (in the spurious, "golden" branch RFD J→0 + , golden of RFD J→0 + , discussed above).

Numerical interlude
Before carrying out an analytical study of the roots of the algebraic Eq. (3.4), we want to present numerical evidence that for J > 0 (thus going beyond the non-rotating limit), only two roots, out of the six roots of the sextic inhomogeneous algebraic Eq. (3.4) (which exist in the algebraically closed field of complex numbers C, by the fundamental theorem of algebra), are real roots.Moreover, as shown in the plots of Fig. 1, out of such two real roots, named δJ 1 and δJ 2 , only one, say δJ 2 , is consistent with the requirement of physical soundness, namely with the condition that Ĵ := J + δJ ⩾ 0. Fig. 1(a) shows the plot6 of J + δJ vs. J for fixed entropy, whereas Fig. 1(b) shows the plot of δJ vs. J, for fixed entropy for the two aforementioned real roots of Eq. (3.4).It is worth here remarking that δJ 2 actually goes into (3.8) in the limit -2 Figure 1: The two real roots of (3.4)

Uniqueness of RFD
Recalling the definition Ĵ := J + δJ ∈ R + of the (physically sensible) RFD-dual angular momentum as, the sextic algebraic Eq. (3.4)-(3.5)can be rewritten as By setting x := Ĵ2 , (5.1) becomes an inhomogeneous cubic algebraic equation, In order to analyse the roots of (5.2), we consider the following algebraic curve7 and look for the turning points on x vs. y = f (x), indeed, counting the number of times f (x) crosses the x-axis provides the number of real roots.
In general, a cubic equation has either one or three real roots, and at most two turning points.
The collective information about the locations of the turning points and the x-intercept of f (x) or the position of f (0) provides us with a clear picture of the situation.In order to compute the number of turning points or the points where the slope goes to zero, one has to solve which generates the following turning points T i := (x i , f (x i )), i = 1, 2 : As S 0 > 0, T 1 always lies in the fourth quadrant of the x − y plane.From the values of f (x) defined in (5.3) at both the turning points T 1 and T 2 , one can conclude the number of real roots for the cubic equation f (x) = 0 (5.2).For further insight, one can look at the second derivatives at the turning points, implying that the convexity of the function y = f (x) always remains opposite at T 1 and T 2 .The case J 2 = 2S 2 0 , which can only arise for over-rotating BHs, will be discussed further below.The same information can be derived from the discriminant of the cubic equation (5.3), which can be computed to read For any cubic equation, if the discriminant ∆ > 0, there are three real roots, whereas if ∆ < 0, there is only one real root.To determine the number of real roots of f (x) = 0, one then needs to check the sign of either the discriminant or the value of f (x) at the turning points.Below, we will consider all possible cases.

J = 0 (non-rotating)
We start and consider the simplest, i.e. the non-rotating, case: J = 0. From (5.4), we find the following turning points : Thus, T 2 always belongs to the second quadrant, with S 1 > 0 and S 2 < 0. Consequently, there exist 3 real roots, as shown in Fig 2(a).This also can be understood by computing that ∆ = 5S 12 0 > 0. Out of the resulting three real roots, one is zero, as f (0) = 0, hence corresponding to the usual (and anti-involutive) notion of FD for a non-rotating, extremal BH [3].
We also observe that T 2 has a negative x coordinate, whereas T 1 has a positive one : this implies that among the two non-zero real roots, one is positive and the other is negative.Since x := Ĵ2 = (J + δJ) 2 , only the positive real root is a physically sensible choice: as discussed in Sec.3.1, it corresponds to RFD mapping a non-rotating, static extremal BH to an underrotating, stationary extremal BH.In order to further clarify the locations of the turning point, as an example in Fig. 2(a) we plot y = f (x) when J = 0 and S 0 = 5.   5.2 J ̸ = 0, A = 0 (under-rotating) Next, we consider J > 0, but A = 0.In this case, ∆ = 0, and therefore there are multiple real roots; since all the coefficients of the cubic equation (5.2) are real, then all roots are real.From (5.5), we find that A = 0 ⇒ J 2 = S 2 0 g, where g : 0 > 0, implying that this is an under-rotating case.The turning points read (5.9) Since there are two different turning points, again with S 1 > 0 and S 2 < 0, there exist two real roots, as depicted in Fig. 2(b) in the case in which S 0 = 5.Inserting the actual value of g, we observe that T 2 has a vanishing y = f (x) coordinate, but it has a negative x coordinate, and thus it must be discarded.On the other hand, T 1 lies on the positive x axis, implying the existence of only one strictly positive real root, which is the physically acceptable one.By setting J 2 = S 2 0 g and using g ≈ 0.110118, one can solve for the f (x) = 0 and find that the multiplicity arises at the negative real root, leaving a single positive real root.In this case, T 1 still belongs to the fourth quadrant, and ∆ > 0 implies the existence of three distinct real roots.As A > 0 ⇒ J 2 < S 2 0 g, T 2 belongs to the second quadrant and this case is under-rotating, too.Again, since f (0) = −J 2 S 4 0 and observing that S 1 > 0 and S 2 < 0, one can realize that only one of the real roots lies on the positive x axis, and is therefore physically acceptable.A prototypical plot for this case is shown in Fig. 3 with J = 2 and S 0 = 7.

J ̸ = 0, A < 0
As evident from the previous analysis, A < 0 ⇔ J 2 > S 2 0 g.In this case, ∆ < 0, hence there exists a single real root.The turning point T 1 is still located in the fourth quadrant, whereas T 2 can be either in the third or fourth quadrant.This situation is rather delicate, as it covers both the under-and over-rotating class of rotating, stationary, extremal BHs.Depending on the various possible values of J 2 there are several situations as shown in Fig. 4(a) and Fig. 4(b).However, although the location of T 1 and T 2 changes, the curve y = f (x) (5.3) always looks the same, and furthermore, it implies the existence of a unique strictly positive real root of the equation f (x) = 0, i.e. of the inhomogeneous cubic equation (5.2), corresponding to the physically sensible solution.

Under-rotating
Since J 2 < S 2 0 ⇒ S 2 0 g < J 2 < S 2 0 , both cases depicted in Fig. 4(a) and Fig. 4(b) are possible, again with S 1 > 0 and S 2 < 0. The existence of a unique and physically sensible root is evident.

Over-rotating
Besides the case depicted in Fig. 4(b) with S 2 0 < J 2 < 2S 2 0 , when J 2 ⩾ 2S 2 0 there exist two other interesting possibilities :  1.The case J 2 = 2S 2 0 , which has been already considered in Sec.2.2, has T 1 = T 2 and S 1 = S 2 = 0, implying that y = 2S 2 0 is the unique (strictly) positive real root.This (overrotating) case trivializes the RFD map, which coincides with the identity, since it leaves both J and S 0 (and thus S over (2.7)) invariant.
2. The case J 2 > 2S 2 0 has, interestingly, S 1 < 0 and S 2 > 0. However, since both T 1 and T 2 swap their order along the x axis, the equation f (x) = 0 ends up having a unique (strictly positive, and thus) physically sensible solution.

To recap
The above-detailed analysis proves that, for any value of J and S 0 physically allowed for stationary (Kerr-Newman, asymptotically flat) extremal BHs (i.e., J ∈ R + and S 0 ∈ R + 0 ), the (generalized) rotating Freudenthal duality (RFD) map defined in (3.3) allows for a unique RFDdual BH.The above analysis is pictorially summarized in Fig. 6, in which we plotted all the real roots of (5.2) by varying J ∈ R + (for S 0 = 2).

Analytic form of RFD
After the qualitative analysis of the previous Section, in which we have proved that RFD (3.3) defines a one-to-one map between two rotating (stationary, Kerr-Newman, asymptotically flat) extremal BHs with the same Bekenstein-Hawking entropy, we now discuss the analytical form of the RFD itself, namely the explicit, analytical form of the solutions to the inhomogeneous cubic equation (5.2), i.e. of f (x) = 0, where the curve y = f (x) is defined in (5.3).
From the general theory of cubic equations, the depressed form of the equation (5.2) reads and the discriminant can be computed to be (cf.(5.7)) We have the following case study.

∆ < 0
In this case, A < 0 ⇔ J 2 > S 2 0 g.Since g < 1, the extremal BH can be under-rotating or over-rotating, depending on whether S 2 0 g < J 2 < S 2 0 or J 2 > S 2 0 , respectively.Moreover, q < 0, and the equation ( 5.2) has one real root and two non-real complex conjugate roots.By further defining by Cardano's method, the unique real root reads For completeness, we mention that the other two non-real complex conjugate roots of (6.1) can be written as with ω and ω being the non-real cube roots of unity, Thus, by recalling that x := Ĵ2 , the unique physically sensible solution for the RFDtransformed (aka RFD-dual) angular momentum reads with t 1 given by (6.6).

∆ = 0 (under-rotating)
In this case, A = 0 ⇔ J 2 = S 2 0 g < S 2 0 (and thus the extremal BH is under-rotating).Then, since its coefficients are all real, the equation ( 5.2) has three real roots, with non-trivial multiplicity.
Since q < 0 and p < 0, in this case the roots indeed read Therefore, the unique physically sensible solution for the RFD-dual angular momentum still formally reads but with t 1 given by (6.10).

∆ > 0 (under-rotating)
This particular situation goes under the name of casus irreducibilis, since at Cardano's time the complex roots were not known to exist.In this case, A > 0 ⇒ J 2 < S 2 0 g, and therefore the extremal BH is under-rotating; from the general theory, there are three distinct real roots, but, as discussed in Sec.5.3, only one of them is positive.Remarkably, the same three roots t 1 , t 2 and t 3 given by Cardano's method (i.e., (6.6) and (6.7)) work, provided one takes the principal value for the cube roots in (6.6) and (6.7), as well as for the square root in (6.9).In fact, in this case t 1 , given by (6.6), still is the unique physically consistent solution, which can now more conveniently be written as (6.13) Since we are taking the principal value here, t 1 is real and positive, as the imaginary parts would cancel out in (6.13).

To recap
Therefore, in all cases, there exists a unique physically sensible root of (5.2), which can be cast in the same form, namely with t 1 which can generally be written as follows : . (6.17) 7 Under-⇄ Over-rotating transitions through RFD? No-Go In the previous Sections, we have proved the uniqueness and presented the explicit analytic form, of the RFD map introduced in (3.3).One might wonder whether RFD may be responsible for a transition from the under-rotating regime to the over-rotating one, or vice versa.In this Section, we will answer such a question.
Hence, the starting BH is "small", i.e. it has a vanishing Bekenstein-Hawking entropy/area of the (unique) event horizon, at least within the two-derivative (Einsteinian) treatment understood in the present treatment.The RFD map (3.3) yields to (cf.(7.4) and (7.8)) where However, since the RFD map (3.3) preserves (by definition) the overall Bekenstein-Hawking BH entropy, the RFD-dual extremal BH is also "small", with S under (Q, J = S 0 ) = 0 = S over (Q, J = S 0 ) → S Q(J + δJ), J + δJ = 0, (7.15) implying that (cf.(7.14)) This is an inhomogeneous algebraic equation of degree 6 in δJ, equivalently obtained by plugging S 0 = J in (3.4) and (3.5) : Solving (7.19), the unique real and positive (and thus physically meaningful) solution reads δJ ≃ 0.324718J; the details are provided in App.A 9 .However, since both the starting and the RFD-dual extremal BHs are "small", this limit case is not interesting, at least at two-derivative level.
Reconsidering the general treatment, by using the explicit form of the physically sensible root of (3.4), namely (6.15) and (6.17), and recalling the definitions (5.5) of A, (7.7) of α, and (7. of S0 , one can compute10 and where we have defined the following non-negative function of α: We can now discuss the possibility of under ⇄ over -rotating transition by means of the RFD map (3.3), by conveniently plotting Ĵ2 /S 2 0 = f (α) and S2 0 /S 2 0 = (1 − f (α)) −2 vs. α in Fig. 7. Indeed, we have to study the sign of depending on the value of α = J 2 /S 2 0 ⩾ 0 (by recalling that for 0 ⩽ α < 1 the extremal BH is under-rotating, while for α > 1 it is over-rotating).By defining z := f (α), it holds that This peculiar cubic equation has been solved in App.A (cf. Eq. (A.2)), giving as unique 11 real non-negative solution 12 (cf.Eq. (A.4)) In App.B we prove that f (α) is monotone increasing for α ⩾ 0; thus, the result (7.26) holds iff α = 1, and the following no-go result is achieved : In other words, no under ⇄ over -rotating transition can be achieved by means of the RFD map (3.3), starting from an under-rotating resp.over-rotating stationary extremal BH.

Conclusion
In this work, we focused on extremal (stationary, asymptotically flat) rotating BHs in four space-time dimensions, and we have shown that there exists a (generally non-anti-involutive) non-linear symmetry of their Bekenstein-Hawking entropy, both in the under-rotating and in the over-rotating regime.We have named such a non-linear symmetry (generalized) rotating Freudenthal duality (RFD): this map generalizes, in the presence of non-vanishing (constant) angular momentum J, the usual Freudenthal duality (FD) for non-rotating (static) extremal BHs introduced in [3], and it is part of the program, started in [18] and [19] dealing with nearextremal non-rotating BHs, to extend FD to various classes of BH solutions beyond the extremal static one.The RFD map has been defined as an intrinsically non-linear map acting both on the e.m. dyonic charges of the BH (collectively denoted by the symplectic vector Q) and on its angular momentum J, and keeping the Bekenstein-Hawking BH entropy S invariant.We have proved the uniqueness of the RFD map, and we also explicitly computed the analytical expression of the transformation of the angular momentum J under such a map, which generally is a quite involved function of the charges of the starting extremal BH (through the non-rotating BH entropy S 0 (Q)) as well as of its angular momentum J itself.
12 From the treatment given in App.A, the value of f (1) can be computed to be f "golden" branch of the RFD map when J → 0 + , which maps a non-rotating (static) extremal BH (with near-horizon geometry AdS 2 ⊗ S 2 , and Bekenstain-Hawking entropy S 0 (Q)) to an underrotating (stationary) extremal BH (with near-horizon geometry AdS 2 ⊗ S 1 ), with a (constant) angular momentum given by √ ϕS 0 (Q), where ϕ is the golden ratio; it is then noteworthy that both such BHs have the same Bekenstein-Hawking entropy!Furthermore, one can state that the space of (generalized) FD functionals undergoes a sort of phase transition in the non-rotating limit J → 0 + , such that J = 0 appears to be a bifurcation point between the non-rotating usual FD branch and the aforementioned "golden" branch.It is suggestive to remark that the duality between a non-rotating extremal BH and an under-rotating extremal BH could be traced back to a common parent five-dimensional extremal BH [27,28].This observation needs further investigation, and we leave that as a future endeavour.B Proof of monotonicity of f (α) We will here prove that both f (α) and (f (α) − 1) −2 are monotone (increasing resp.decreasing) functions ∀α ⩾ 0.
We start and recall Eq. (6.1), t 3 + p t + q = 0, with in C s , but rather only four discrete fundamental automorphisms, namely I, −I, C and −C, where I and C respectively denote the identity map and the aforementioned conjugation map on C s [S 0 , J] J ̸ = 0, A = 0.

J 2 .y − 2 3 .
It will also be interesting to investigate the effect of RFD on the dual CFT 2 of an extremal Kerr-Newman BH, by exploiting the so-called Kerr/CFT correspondence.On the other hand, we should recall that so far we have only formulated the generalization of the Freudenthal duality map for the semi-classical Bekenstein-Hawking entropy of various classes of BHs; the invariance of the quantum-corrected BH entropy under a suitable generalization of Freudenthal duality still stands as a crucial issue, which we hope to address in future investigation.bydefining y = Ĵ2 The depressed form of the equation is then Evidently, (A.3) is the same as (6.1) with the identification that t ≡ t J 2 and p and q evaluated at α = 1.Therefore we have the relation that Ĵ2 f (α) being defined in(7.23).