超弦理论中黑洞熵的精确全息计算

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"这篇研究论文详细探讨了N = 8 和 N = 4 弦理论中的黑洞熵问题，特别是八分之一和四分之一BPS（Bogomol'nyi-Prasad-Sommerfeld）黑洞。研究者在K 3和T 4压缩背景下的所有N = 4 CHL（Calabi-Yau Hyperkähler Lattice）模型中进行了精确计算。主要成果是得到了在AdS 2×S 2 上定位后的超重力有限维积分的度量，这个度量完全由AdS 3 上的超对称 Chern-Simons 理论中的异常决定，并考虑了超重力多重性的全部贡献。在Chern-Simons理论中，这相当于计算紧凑歧管体积的单环依赖关系的异常。对于八分之一BPS黑洞，论文提供了arXiv：1111.1161中提出度量的首次第一原理推导，而对于四分之一BPS黑洞，研究者则精确计算了所有扰动或面积校正。此外，他们认为可以整合瞬时贡献，并通过与微观公式精确匹配的度量提供证据。研究还提出了一种统一的条件，用于截断与微观公式完美符合的有限瞬时数之和。因此，这些结果不仅解决了超重力局域性相关的复杂问题，而且为全息精确性提供了坚实的基础。论文于2017年5月16日提交，6月23日接受，7月5日发表，是开放访问的。" 这篇学术文章深入探讨了弦理论中的一个重要领域，即如何计算高维黑洞的熵，特别是在具有不同对称性的弦理论背景下。N = 8 和 N = 4 弦理论分别代表了不同的超对称性级别，其中BPS黑洞保持了部分超对称性，使得它们的熵可以通过数学上更为精确的方法来计算。在K 3 和 T 4 紧缩背景下的N = 4 CHL 模型中，黑洞的熵计算涉及了AdS/CFT对应（Anti-de Sitter/Conformal Field Theory correspondence），这是一种强弱耦合的对偶理论，允许在特定情况下用低维理论描述高维物理现象。研究者采用了一种称为“定位”的技术，将超重力问题简化为AdS 2×S 2 上的有限维积分，这个过程中的度量由AdS 3 上的超对称 Chern-Simons 理论中的异常决定。Chern-Simons理论是一种拓扑量子场论，其异常与拓扑不变量相关，可以帮助理解和计算某些物理量。对于八分之一BPS黑洞，研究者使用了第一原理方法重新推导了之前提出的结果，而对于四分之一BPS黑洞，他们计算了所有修正项，包括由于黑洞表面的几何变化而引入的扰动。此外，研究者提出了一种新的方法来处理瞬时态的贡献，并且找到一种条件来限制这些瞬时态的组合，使之与微观物理的精确预测相一致。这种统一的条件为黑洞熵的精确计算提供了关键的理论支持，同时解决了超重力理论中的局域性难题。这些发现对于深化对全息原理的理解至关重要，全息原理认为高维空间的信息可以被编码到其边界上的低维理论中。这项工作为弦理论和黑洞物理的研究开辟了新的道路，也为未来更深入的理论探索奠定了基础。

JHEP07(2017)022

One of the goals of this section is to rewrite the microscopic degeneracies in a manner

suitable for a gravitational comparison. We will ﬁnd that the microscopic answer can be

written as a ﬁnite sum of Bessel functions, up to much subleading terms.

We consider IIB string theory on M ×

× S

, with M = K3, T

modded out by

a Z

symmetry group. The orbifold identiﬁcation involves a 1/N shift along the circle

and an order N ˜g transformation on M . The element ˜g commutes with the N = 4

supersymmetry generators and therefore the orbifold preserves all the supersymmetry of

the parent theory. By convention, we take the radius of the circle S

to have size N in the

parent theory. Here N runs over 1, 2, 3, 5, 7 in the K3 case and 2, 3 in the T

case. Under

the orbifold only a subgroup SL(2, Z) × SO(6, r − 6; Z) of the U-duality group survives,

with r = 2k + 8 the total number of U(1) gauge ﬁelds. The integer k is given by

k =

N + 1

− 2, N = 1, 2, 3, 5, 7, for M = K3 (2.6)

k =

N + 1

− 2, N = 2, 3 for M = T

. (2.7)

Let us consider a conﬁguration with a single D5-brane wrapping M ×S

, Q

D1-branes

wrapping S

, a single Kaluza-Klein monopole associated with the circle

, n/N units of

momentum along S

and J units of momentum along

. At low energies this system is

described by a two dimensional (0, 4) SCFT on R ×S

. At the orbifold point in the moduli

space, this theory is described by a symmetric product sigma model and we can compute

the supersymmetric index that counts one-quarter BPS states. The index, which we denote

by d(Q, P ), has the form

d(Q, P ) = (−1)

Q.P +1

dρ dσ dv

−πi(NQ

ρ+P

/Nσ+2Q.P v)

(ρ, σ, v)

(2.8)

where C is a three dimensional contour in the complexiﬁed (ρ, σ, v) = (ρ

+iI

, σ

+iI

, v

) space with

, I

= constant  1 (2.9)

0 ≤ ρ

≤ 1, 0 ≤ σ

≤ N, 0 ≤ v

≤ 1. (2.10)

and Q

= 2n/N , P

= 2Q

and Q.P = J are the T-duality invariants. The function

(ρ, σ, v) is a Sp(2, Z) modular form. In particular, for N = 1 it becomes the Igusa

cusp form: the unique weight ten Siegel modular form, while for other N we obtain Siegel

modular forms of congruence subgroups.

We now study the degeneracy in the regime of large charges. In a saddle point approx-

imation of the integral (2.8) we deform the contour and pick poles of 1/Φ

(ρ, σ, v). The

leading contribution comes from the residue at a quadratic divisor of Φ

(ρ, σ, v) and has

ﬁnal expression [8, 15, 18, 19],

d(Q, P ) '

(−1)

Q.P

4πN

(k+4)/2

dτ



2(k + 3) +

|Q − τ P |



× (2.11)

exp



2τ

|Q − τ P |

− ln g(τ ) − ln g(−τ ) − (k + 2) ln(2τ

)



+ . . .

– 8 –

JHEP07(2017)022

with τ = τ

+ iτ

and τ = τ

− iτ

. The function g(τ) is determined by the pole structure

of 1/Φ

(ρ, σ, v). In the case of M = K3 it is given by

g(τ) = η(τ)

k+2

η(Nτ)

k+2

, (2.12)

while for M = T

it has the form

g(τ) = η(τ)

2N(k+2)

N−1

η(Nτ)

−2

k+2

N−1

, (2.13)

with N, k given as in (2.6).

The formula (2.11) is not in a form that is suitable for a comparison with the local-

ization computation. The reason is that the measure in (2.11) depends on the electric

charges, while from a gravitational point of view they appear only at level of the Wilson

lines. Besides, the contour in (2.11) has to be chosen appropriately. The only requirement

at this point is that it passes near the leading physical saddle.

We show that there is

a choice for which we recover not only the leading Bessel function (1.3) [6] but also an

additional tail of subleading Bessel type corrections. Namely, we choose a contour with

, τ

complex deﬁned as

C : τ

= iτ

u, −1 + δ ≤ u ≤ 1 − δ,

=  + iy, −∞ < y < ∞,  > 0 (2.14)

Here δ is small but positive (we will make precise what we mean by small in due course).

This contour ensures that we always have Im(τ ) and Im(τ ) positive- here τ and τ are not

necessarily complex conjugate. This in turn leads to the exact Bessel function determined

in [7] for the leading asymptotics of one-quarter BPS states in IIB on K3 × T

We proceed with an integration by parts. First we rewrite the expression (2.11) in the

convenient way

d(Q, P ) ' (−1)

Q.P +1

k+4



2(k + 3) + π

|Q − τ P |



|Q−τ P |

−Ω(τ,τ )

(2.15)

with

Ω(τ, τ) = ln g(τ) + ln g(−τ). (2.16)

The exponential is just the entropy function of Sen (3.8)

E =

|Q − τ P |

− Ω(τ, τ). (2.17)

Using the identity

∂

∂τ

|Q − τ P |

= −

|Q − τ P |

+ 2P

(2.18)

we can write the measure in (2.15) as

k+3



2(k + 3)

+ π

|Q − τ P |



k+3



2(k + 3)

− 2

∂

∂τ

E − 2

∂

∂τ

Ω + 2πP



(2.19)

By physical, we mean that it reproduces the attractor background and thus the area formula for the

entropy.

– 9 –

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