AdS/CFT对应中操作的局部化与超对称解

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"AdS / CFT中操作的本地化 - JHEP10(2019)252" 这篇研究论文详细探讨了AdS（Anti-de Sitter）/ CFT（Conformal Field Theory）对应中的操作本地化原理。AdS / CFT对应是一种强大的工具，它在量子场论和弦理论中建立了重力理论与非重力理论之间的等价关系，特别是在AdS4空间与CFT3理论的对偶中。作者Pietro Benetti Genolini、Juan Manuel Pérez Ipiña和James Sparks展示了如何在最小规范超重力框架下计算任何超对称解的行动。这些解具有超对称 Killing 向量，这是一种特殊的矢量场，保持了空间时间的某些对称性。他们证明了全息重新归一化的操作——即在AdS边界上的CFT中观察到的行动——可以完全由这个Killing向量在其固定点的权重以及一些拓扑数据来表示。这意味着经典重力的 partition function 在几何体内部“本地化”了，集中在这些固定点上。文章中提到，通过这种方式，经典引力的分布函数得以精确确定，无需处理整个AdS空间的复杂度。作者提供了多个具体的例子来验证这个一般公式，其中包括超对称Taub-NUT和Taub-bolt类型的时空，以及黑洞解决方案。这些例子不仅揭示了理论的普适性，还利用了精确的双场理论计算，使得计算更加直观和可靠。此外，由于公式简洁且基于拓扑性质，它可以应用于任何具有相应Killing向量的解决方案，只要这些解决方案存在。这为理解和计算AdS / CFT对应中的复杂物理现象提供了一个简洁而实用的工具，进一步加深了我们对量子引力和强耦合量子场论之间关系的理解。这篇论文为AdS / CFT对应中的操作计算提供了一种新的、拓扑导向的方法，为研究高维引力理论和相关CFT的物理性质开辟了新途径。通过这个局部化的技术，研究人员能够更有效地处理和理解这些理论中的复杂问题，推动理论物理的发展。

JHEP10(2019)252

2.3 Reduction of the bulk action

Upon substitution of (2.15) and (2.19) into the bulk action (2.1), and further integration

along the circle ﬁbre of the U(1) isometry, we ﬁnd an expression for the action on the base

B. We then add to this a Lagrange multiplier term imposing the constraint that H is

conserved (2.11):

S = −

16πG



+ ∇

log V −

hH, Hi

−

−2

hdV, dV i

+ 6V

−1

− 2V

−1

hdϕ, dϕi

− 2ϕ

V hH, Hi

− 2ϕV h∗

H, fi

− V hf, fi

− σ ∇



vol

(2.20)

This expression extends the analogue for pure Einstein gravity found in [19]. The equation

of motion for the Lagrange multiplier σ is obviously just the constraint (2.11) on H, whereas

the equation of motion for a, where recall f = da, is more interesting:

d(ϕV H + V ∗

f) = 0 . (2.21)

This equation implies that locally on B there is an electromagnetic potential function ν

such that

dν = ϕV H + V ∗

f = ξ

∗

F , (2.22)

which notice is gauge invariant under (2.18). Here the second equality follows from com-

paring with (2.19). In terms of the electromagnetic potential, the equation of motion for

ϕ becomes

d ∗



−1

dϕ − νH



= 0 , (2.23)

which in turn tells us that we may locally write

∗

−1

dϕ − ν H) = d$ (2.24)

for some one-form $.

The Lagrange multiplier σ is known as the nut potential [19]. It also appears in the

equation of motion for H, which may be expressed in terms of the electromagnetic potential

ν as

dσ = V

H + 4ϕ dν . (2.25)

The origin of the name nut potential is clearer in the pure gravity case, where the last

term in (2.25) vanishes. With the introduction of an Abelian gauge ﬁeld, the nut potential

is not gauge invariant under (2.18): since ν, H and V are left untouched by the gauge

transformation along dψ, we ﬁnd that σ changes as

σ 7→ σ + 4cν . (2.26)

– 7 –

JHEP10(2019)252

Finally, we note the equations of motion for the scalar V , and the Einstein equation

for the metric γ on the base:

0 = ∇

log V − V

hH, Hi

− 6V

−1

+ 2V

−1

(hdϕ, dϕi

− hdν, dνi

) ,

(2.27)

0 = (Ric

)

−

−2

∇

V ∇

V + 6V

−1

− 2V

−1



∇

ϕ∇

ϕ − ∇

ν∇



(2.28)

Notice that all the equations of motion are written in terms of quantities that are invariant

under (2.18), (2.26).

Substituting these equations of motion inside the expression for the bulk Euclidean

action (2.20), we readily ﬁnd an expression for the bulk on-shell action, which we refer to

as I

bulk

, expressed as an integral over the base:

bulk

16πG

d ∗



d log V − σH + 4V

−1

ϕ dϕ



. (2.29)

This is not immediately gauge invariant, but can be written in a more invariant way using $

bulk

16πG

d [4ϕ d$ + ∗

(d log V + (4ϕν − σ) H)] . (2.30)

The bracketed term contains the gauge invariant combination 4ϕν −σ, and $ is invariant,

so the overall variation of the integrand under (2.18) is d(4cd$) = 0. Indeed, the variation

of the integrand of (2.29) is precisely the equation of motion for ϕ, (2.23).

The expression (2.29) shows that the on-shell action is naturally exact on the base B.

Importantly, we shall see in section 3 that for supersymmetric solutions there is a natural

gauge in which both ϕ and σ are global functions on B. Since also V > 0 is a global

function, and H a global one-form on B, it follows that we may integrate the action (2.29)

by parts using Stokes’ theorem, reducing to an integral over the boundary of B. For an

asymptotically locally Euclidean AdS spacetime, which is the case we are interested in, the

latter consists of two types of boundary term: the conformal boundary at inﬁnity, and the

boundaries of the tubular neighbourhoods surrounding the ﬁxed point loci of the isometry,

where the ﬁbration of spacetime (2.6) degenerates. It follows that (2.29) then reduces to

a contribution from the conformal boundary, together with a sum over contributions from

each connected component of the ﬁxed point set of ξ. We shall evaluate this more explicitly

for supersymmetric solutions in the next section.

3 Supersymmetric solutions

3.1 Supersymmetry equations

In this section we shall assume we are given a supersymmetric solution to the equations

of motion (2.2), (2.3), meaning there is a non-identically zero Dirac spinor  satisfying the

(generalized) Killing spinor equation (2.4).

It will be convenient for what follows to introduce the charge conjugation matrix C

satisfying

C = C

∗

= −C

, C

= −1 , Γ

= C

−1

C . (3.1)

– 8 –

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