3D引力与Chern-Simons全息对偶的带电SYK模型

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"从3D重力和Chern-Simons全息双电荷SYK" 这篇论文探讨了将三维（3D）引力理论通过Kaluza-Klein（KK）还原应用于低能量扇区的SYK模型，特别是那些带有U(1)电荷的SYK模型。Kaluza-Klein理论是一种将额外维度的概念引入经典物理理论的方法，允许我们将高维理论简化为低维理论，通常用于弦理论和超引力理论中。在这个过程中，3D爱因斯坦引力作用和其边界项的KK还原生成了一个2D Jackiw-Teitelboim (JT)模型，这是一种描述近AdS2几何的低能量量子引力模型。KK半径在生成的AdS2几何中与膨胀值相对应。当存在U(1)电荷时，3D模型中还包括U(1) Chern-Simons（CS）行动。Chern-Simons理论是拓扑量子场论的一个关键组成部分，它描述了规范场与空间时间曲率的相互作用。为了描述具有非零化学势的边界理论，论文引入了CS规范场与体积重力之间的耦合。这种耦合使得3D CS行动在二维下减少为BF型行动，加上源项和边界项。 KK还原后的2D理论代表了带电SYK模型的软部分，即模型的低能量行为。伪Nambu-Goldstone模态，即Diff / SL(2, ℝ)和U(1)局部/ U(1)变换的组合，由大衍射和大规格变换表示。这些模态在确定参数后，其有效作用可以通过批量对偶中的作用成本来再现。此外，论文还进行了整体计算，分析了混沌相关器，这是研究系统混沌性质的重要工具。结果再现了“边界光子”对零Lyapunov指数的贡献，Lyapunov指数是衡量系统混沌程度的指标。这一发现对于理解全息对偶中的混沌行为以及黑洞物理学具有重要意义。该研究通过3D引力和Chern-Simons理论的结合，为理解和描述带电SYK模型提供了新的视角，同时揭示了全息对偶在低能级扇区的应用，特别是在量子引力和复杂性理论的研究中。

JHEP02(2020)033

In the section 5, we present derivation of boundary eﬀective action from 2D gravity.

We closely follow [14] to discuss the action of the soft modes from the bulk. The focus of

the section is,

• We discuss two points of view to look at the NAdS

gravity. In this paper we adopt

the ﬁrst point of view (explained in section 5.1). We discuss the physical origin of

the action for the soft modes.

• The soft mode action is eventually a boundary action and thus the boundary behavior

of the ﬁelds matter the most in this computation. We, therefore, present a careful

study of boundary fall oﬀ and boundary conditions on all the ﬁelds and accordingly

propose covariant counter terms so as to keep the on shell action ﬁnite.

• Based on the boundary dynamics, in section 5.2, we obtain an eﬀective action, which

represents the action of pseudo Nambu-Goldstone modes of the charged SYK model.

It is given by a Schwarzian plus a sigma model term. The Schwarzian action is

already known from [14]. The sigma model in the context of charged SYK was

proposed in [12].

• The eﬀective action we derive from the bulk agrees with the ﬁeld theory eﬀective

action given in [12]. We conclude this section with appropriate identiﬁcation of

parameters in the bulk and boundary theory.

In section 6, we compute chaotic correlators from the bulk. We reproduce the result

that there is no contribution from “boundary photons” to the Liapunov exponent.

2 The charged SYK model

The original SYK model involved Majorana fermions. In [12], a generalized SYK model

with Dirac fermions with a global U(1) symmetry was introduced. The model is given by

a hamiltonian

H =

,...,i

†

. . . ψ

†

q/2

q/2+1

. . . ψ

1 ≤ i

< . . . < i

q/2

≤ N & 1 ≤ i

< . . . < i

≤ N ,

where q is an even number and j

,...,i

are complex Gaussian random variables with

,...i

q/2

q/2+1

...,i

= j

∗

q/2+1

...,i

,...i

q/2

, h|j

...i

i =

((q/2)!)

q−1

Eﬀective action and Schwinger-Dyson equations

The Euclidean action for this model is

S =

dτ



†

(∂

− µ)ψ

− H



, (2.1)

– 6 –

JHEP02(2020)033

where µ denotes the chemical potential. The large N limit is characterized by Schwinger-

Dyson equations, which can be derived from the following eﬀective action in terms of the

bilocal ﬁeld

G(τ

, τ

) =

†

(τ

)ψ

(τ

) and an auxiliary ﬁeld

Σ(τ

, τ

)

Σ] = −

Tr ln

δ(τ

− τ

)



−

∂

∂τ

+ µ



−

Σ(τ

, τ

)

(2.2)

−

dτ



Σ(τ

, τ

)

G(τ

, τ

) + J

(−1)

q/2



G(τ

, τ

)



q/2



G(τ

, τ

)



q/2



In fact, the original fermion integral with a quenched average over the disorder reduces,

in the large N limit, to a path integral over the variables

G(τ

, τ

) and

Σ(τ

, τ

) with the

above action.

The large N limit of the above action is characterized by the following

Schwinger-Dyson equations,

Σ(τ) = −(−1)

[G(τ)]

q/2

[G(−τ)]

q/2−1

(2.3)

G(iw

) = [iw

+ µ − Σ(iw

)]

−1

, (2.4)

here w

denotes the Matsubara frequency and τ = τ

− τ

(here Σ and G denote on-shell

ﬁelds).

2.1 Symmetry transformations

In the IR regime deﬁned by w

, µ  J [12] one can drop the (iw

+ µ) term from (2.4),

leading to an emergent time reparameterization and U(1) gauge symmetry.

The symmetry transformations of the charged SYK model, at strong coupling (the IR

regime), consist of (a) diﬀeomorphisms parameterized by f : ˜τ = f(τ) ∈ Diﬀ,

as well as

(b) gauge transformations parameterized by exp[iφ(τ)] ∈ U(1)

gauge

. We will denote the

combined symmetry group as G,

which acts on the bilocal meson variable G(τ

, τ

) =

(1/N ) ψ

†

(τ

)ψ

(τ

) and Σ(τ

, τ

) as follows:

G(˜τ

, ˜τ

)d˜τ

∆

d˜τ

∆

= G(τ

, τ

)dτ

∆

dτ

∆

exp

φ(˜τ

) − i

φ(˜τ

)

Σ(˜τ

, ˜τ

)d˜τ

(1−∆)

d˜τ

(1−∆)

= Σ(τ

, τ

)dτ

(1−∆)

dτ

(1−∆)

exp

φ(˜τ

) − i

φ(˜τ

)

. (2.5)

Inﬁnitesimally, for small φ(τ) (here

φ(˜τ(τ )) = φ(τ )) and small (τ) ≡ f(τ) − τ ,

δG = δ



G + δ

G ,



G(τ

, τ

) =



(τ

)∂

+ ∆

(τ

) + (τ

)∂

+ ∆

(τ

)



G(τ

, τ

) , (2.6)

G(τ

, τ

) = i (φ(τ

) − φ(τ

)) G(τ

, τ

) . (2.7)

Our expression for the eﬀective action diﬀers slightly from that of [12] because of diﬀerent conventions

for the bilocal ﬁelds and the conserved charge.

Diﬀ represents Diﬀ S

or Diﬀ R

, depending on whether the system is at a ﬁnite temperature or zero

temperature.

The action of Diﬀ and U(1)

gauge

do not commute. The parameterization of the combined symmetry

transformation follows here the convention that the gauge transformation is performed ﬁrst, followed by a

Diﬀ transformation, as in (2.5).

– 7 –

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