五维N=2超重力中带电黑弦的一阶流动方程

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"这篇学术文章详细探讨了五维N = 2阿贝尔计量超重力中的带电黑弦一阶流动方程，这是在开放获取的JHEP01(2017)106期刊上发表的研究成果。研究团队包括Dietmar Klemm、Nicolò Petri和Marco Rabbiosi，他们来自意大利米兰大学和INFN米兰分部。文章于2016年11月10日提交，2017年1月15日接受，并于同年1月25日发表。" 在文章中，作者们运用汉密尔顿-雅各比形式论，首先阐述了如何在仅耦合到矢量多重峰和U（1）Fayet-Iliopoulos（FI）度量的条件下，推导出BPS（Bogomol'nyi-Prasad-Sommerfield）和非BPS状态的一阶流动方程。接着，他们将这个理论推广到包含超多重子集的情况，同时考虑四元超标量目标空间的阿贝尔对称性。这一部分展示了在超重力理论中如何处理更复杂的物理系统。进一步，研究者们利用这些流动方程来分析极端黑弦的近水平几何的吸引子方程。吸引子方程是描述黑洞或类似天体近水平几何的重要工具，它们揭示了这些对象的稳定性和动力学特性。在特定情况下，当Chern-Simons项中的常数满足伴随恒等时，这些方程可以被明确求解。这使得能够计算出二维共形场理论的中心电荷，中心电荷是理解黑线在红外区域行为的关键。最后，作者们将“r-map”方法扩展到测量的超重力，并用它来连接五维和四维的流动方程。r-map是一个重要的理论工具，它允许在不同的引力理论和维度之间建立联系。通过这种方式，他们能够更深入地理解超重力理论中的物理过程和相互作用。这篇文章对五维N = 2阿贝尔计量超重力理论进行了深入研究，特别是在带电黑弦背景下的一阶流动方程及其应用。这些结果不仅深化了我们对超重力理论的理解，也对黑洞物理学、弦理论以及相关的共形场理论产生了重要影响。

JHEP01(2017)106

the Fayet-Iliopoulos (FI)-gauged case, and then generalized to include also hypermultiplets,

where abelian symmetries of the quaternionic hyperscalar target manifold are gauged.

Extremal magnetic black strings interpolate between (so-called ‘magnetic’) AdS

at inﬁnity

and AdS

× Σ (with Σ a two-dimensional space of constant curvature) at the horizon.

Holographically, this corresponds to an RG ﬂow across dimensions from a 4d ﬁeld theory

to a two-dimensional CFT in the infrared [19–23]. By plugging the near-horizon data into

the ﬂow equations, one gets the attractor equations. We solve the latter in full generality

under the additional assumption that the ‘adjoint identity’ (A.3) holds. This enables us

to compute the central charge of the 2d CFT that describes the black strings in the IR, in

terms of the string charges, the FI parameters (or, more generally, the moment maps, if

also hypermultiplets are present), and the constants C

IJK

that appear in the Chern-Simons

term of the supergravity action.

The remainder of this paper is organized as follows: in the next section, we brieﬂy

review N = 2, d = 5 Fayet-Iliopoulos gauged supergravity. In section 3, we derive the BPS

ﬂow equations for static black strings, and generalize them in 4 to the non-supersymmetric

case, by using a simple deformation of the BPS superpotential. After that, in section 5,

the presence of (abelian) gauged hypermultiplets is taken into account as well. In 6, the

ﬂow equations are solved in the near-horizon limit, where the geometry contains an AdS

factor. This leads to the attractor equations for black strings, that we subsequently solve

in full generality, and compute the central charge of the 2d CFT that describes the black

strings in the IR. We conclude in section 7 with some proposals for extensions of our work.

An appendix contains some useful relations in very special geometry and the construction

of the r-map in the gauged case.

2 N = 2, d = 5 Fayet-Iliopoulos gauged supergravity

The bosonic Lagrangian of N = 2, d = 5 FI-gauged supergravity coupled to n

vector

multiplets is given by [24, 25]

−1

L =

R −

∂

−

µν

Jµν

−1

IJK



µνρσλ

µν

ρσ

− g

V , (2.1)

where the scalar potential reads

V = V



∂

− 6h



. (2.2)

Here, V

are FI constants, ∂

denotes a partial derivative with respect to the real scalar

ﬁeld φ

, and h

= h

(φ

) satisfy the condition

V ≡

IJK

= 1 . (2.3)

Moreover, G

and G

can be expressed in terms of the homogeneous cubic polynomial V

which deﬁnes a ‘very special geometry’ [26],

= −

∂

∂h

∂

∂h

log V|

V=1

, G

= ∂

∂

V=1

. (2.4)

The indices I, J, . . . range from 1 to n

+ 1, while i, j, . . . = 1, . . . , n

– 3 –

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