Freudenthal变换与黑洞：不变性与超重力理论

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"黑洞和一般的Freudenthal变换"是关于E型全局对称的Einstein-Maxwell-Scalar（超）重力理论的研究，探讨了黑洞解的通用Freudenthal变换（GFT）。这篇论文深入剖析了GFT在黑洞物理学中的应用，特别是它如何影响Bekenstein-Hawking熵。 Freudenthal变换是一种数学操作，它扩展了双线性对偶性，用两个实数参数a和b来表示，将电荷向量x映射到新的向量x_F = ax + bx_~，其中x_~是x的Freudenthal对偶。这种变换的关键特性是它可以保持Bekenstein-Hawking熵的不变性，只要参数满足特定条件，即2±b^2 = 1，此时的变换称为弗洛伊登塔尔旋转。在这些旋转下，由电荷向量x和其对偶x_~生成的Freudenthal平面形成一个闭合的系统，通过Freudenthal旋转的轨道进行叶化，展示了其内在的对称性。文章进一步讨论了在超重力背景下的GFT与全局对称性或U对偶的关联。作者考虑了不同的理论模型，包括纯超重力、轴力-狄拉克顿理论以及N = 2，D = 4的超引力理论。这些理论在粒子物理和弦理论中占有重要地位，它们允许研究人员探索高维空间中的黑洞性质和宇宙学问题。通过对Freudenthal变换的详细分析，研究者揭示了这些变换如何在理论框架内影响黑洞的热力学性质，特别是在熵的保护和对称性的展开方面。此外，这些发现可能对理解和统一量子引力理论提供关键洞察，因为它们涉及黑洞的微观结构，这是量子引力研究的核心问题。这篇论文发表在JHEP（Journal of High Energy Physics）上，由SISSA（国际高等研究院）授权Springer发布。从接收、修订到接受，经过了近两个月的时间，最后在2019年7月正式发表，反映了该领域研究的严谨性和深度。作者团队来自爱尔兰、英国、美国、西班牙和意大利的不同机构，体现了国际科研合作的广泛性。 "黑洞和一般的Freudenthal变换"是一个深入探讨黑洞物理学和量子引力理论的重要研究，它不仅深化了我们对Freudenthal变换的理解，也为我们理解黑洞的动态特性和宇宙的宏观结构提供了新的视角。

JHEP07(2019)070

Similarly, we have the following expressions describing the behaviour of the map Υ on

the T-plane:

) = 3T (x, x, x

) + {x, x

}x = −∆

x − 2∆

x = −3∆

x, (3.10)

(ax + bx

) = −3b∆(x)x + 3ax

= 3T (ax + bx

)/(a

+ b

∆

). (3.11)

Further mathematical properties of the T-planes are shown in section A and section B.

T- and F-transformations on the planes. For ˜x well-deﬁned, we rewrite eq. (3.4),

eq. (3.9), by a redeﬁnition of the parameters a, b in terms of the F-dual eq. (3.1) as ( =

sgn ∆(x))

T (ax + b˜x) = 



+ b



| ∆(x) | (−bx + a˜x) , (3.12)

∆(ax + b˜x) =



+ b



∆(x). (3.13)

It is clear from this last expression that a

+b

= ±1 deﬁnes a subset of elements in F

with

ﬁxed entropy: an SO(2) or SO(1, 1) symmetry. Moreover, sgn ∆(x) = sgn ∆(ax + b˜x) = ,

unless a

+ b

= 0. In particular

∆(x ± ˜x) = (1 + )

∆(x). (3.14)

As consequence of eq. (3.12) and eq. (3.13), the F-dual of a linear combination is given

by (∆(x) 6= 0, ∆(ax + b˜x) 6= 0, sgn ∆(x) = sgn ∆(ax + b˜x) = )

ax + b˜x =

T (ax + b˜x)

| ∆(ax + b˜x) |

(3.15)

and, ﬁnally,

ax + b˜x = η (−bx + a˜x) =

(

−bx + a˜x, (∆(x) > 0),

η (bx + a˜x) , (∆(x) < 0),

(3.16)

where  = sgn ∆(x), η = sgn(a

+ b

). Clearly, if  = 1 then η = 1

As discussed in appendix subsection B.4 the F-plane is a, quadratic, two dimensional,

sub-FTS system with suitably restricted operations ∆

, T

, {, }

and I

On the full FTS, for maximal rank elements, one can deﬁne an (Aut(F)-invariant)

“metric” by the (non quadratic) expression

(u, v) ≡

[{˜u, v} + {˜v, u}]. (3.17)

and a “pseudo-norm” by

||u|| = (u, u) =

{˜u, u} = 

|∆(u)|. (3.18)

– 9 –

JHEP07(2019)070

If we ﬁx x, and restrict ourselves to the F

plane we can use the expressions in sub-

section B.4 (see also further properties in ref. [58]) and connect ||u|| with I

(u):

(u, v) ≡

[{˜u, v} + {˜v, u}] (3.19)

η(u){

Su, v} + η(v){

Sv, u}

(3.20)

(η(u) + η(v)){

Sv, u} (3.21)

(η(u) + η(v))I

(v, u) (3.22)

where  = sgn ∆(x), η(x) = sgn I

(x) and

S is a linear map given in section B. In particular

||u|| =

η(u)I

(u) =

 | I

(u) | (3.23)

We arrive at the same conclusions as for the T-plane. For  = 1, the pseudo-norm (·, ·)

(or I

(u)) is positive deﬁnite and the norm-preserving group is SO(2); thus, the F-plane

undergoes a “spherical foliation”. On the other hand, for  = −1, the norm is positive

semi-deﬁnite and the norm-preserving group is SO(1, 1); thus, the F-plane F

undergoes

an “hyperboloid-like foliation”. While the norm is timelike or null, the vector u can be

timelike, spacelike and null according to η(x), the sign of I

(x).

As for the T-operation, the F-duality x → ˜x, change the character of the vector. The

vectors x, ˜x are “I

-orthogonal”, by eq. (B.29), (x, ˜x) = I

(x, ˜x) = 0. ˜x is timelike (resp.

spacelike) if x is spacelike (resp. timelike):

x : lightlike ←→ ˜x : lightlike, ˜x = ±x, (3.24)

x : timelike (spacelike) ←→ ˜x : spacelike (timelike). (3.25)

It is noted that, although the metric I

(x, y) is deﬁned only inside a concrete F-plane,

the character null, time or spacelike of a vector is an intrinsic property, as any given element

belongs to one and only one F-plane, “its” plane, from the disjointness of the F-planes (see

subsection B.2).

4 The orthogonal space F

⊥

and the orthogonal plane F

y⊥x

In general {x, ˜x} = −2

|∆ (x)| 6= 0. The bilinear form {·, ·} is non-degenerate on F

construction, since x is neccesarily of maximal rank (∆(x) 6= 0). Consequently, for a given

x ∈ F

(4)

, the FTS F may be decomposed as

F = F

⊕ F

⊥

, (4.1)

where F

is the 2-dimensional F -plane and F

⊥

is its (dim

F − 2)-dimensional orthogonal

complement w.r.t. the bilinear form {·, ·}:

⊥



y ∈ F : {x, y} = {x

, y} = 0



. (4.2)

– 10 –

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