AdS5×S5中1/4-BPS Wilson环的精确强耦合计算及其分歧解析

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本文主要探讨的是AdS 5×S 5空间中的1/4-BPS Wilson环在N=4超杨-米尔斯理论中的精确计算。N=4 Super Yang-Mills理论是一个重要的理论框架，它在弦理论和量子场论中扮演着关键角色，尤其是在AdS/CFT对偶（Anti-de Sitter空间/Conformal Field Theory对应）的背景下，它提供了研究强耦合现象的有效工具。 1/4-BPS Wilson环，特别是圆形的“纬度”族，是研究理论中的基本对象，它们代表了特定类型的局部化算符，用于测量强耦合下的一类物理过程。在AdS 5×S 5这个背景中，这些环对应于弦理论中的经典解，通过计算其一环修正，我们可以获得关于理论在非perturbative区域的信息。作者采用的方法是超对称定位，这是一种在强耦合极限下提供精确结果的强大工具。然而，传统的sigma模型方法在处理具有Dirichlet边界条件的问题时遇到了困难，这涉及到计算相关的功能决定因子。Gel'fand-Yaglom方法在此过程中被应用，尽管这带来了一些挑战，比如如何避免由于字符串分区函数的绝对归一化问题导致的歧义。作者通过比较通用纬度和最大1/2-BPS圆之间的比率来解决这类歧义，认为任何与度量相关的不确定因素都应该在这种对比中显现并消除。然而，实际的数值评估结果显示，计算结果与超对称局部化的预期答案存在分歧，表现为一个精确的“余数函数”。这一发现引发了关于计算误差来源的讨论，可能是由于尚未完全理解的物理效应、计算技术局限或是理论框架的某些假设。文章着重分析了这种矛盾的可能性，并提出了可能的解决方案，可能包括进一步的理论研究，比如寻找新的计算方法或者深入理解弦理论的非perturbative结构。这篇论文不仅提供了1/4-BPS Wilson环在AdS 5×S 5中的一环修正的计算细节，还揭示了强耦合下理论的复杂性以及当前研究中的挑战，对于深化我们对N=4 Super Yang-Mills理论的理解具有重要意义。

JHEP02(2016)105

Here κ

stands for the geodesic curvature of the boundary at σ = 

and ds is the invariant

line element. With this subtraction, we have the value of the regularized classical area

(0)

(θ

) −

√

λχ

(θ

) = ∓

√

λ cos θ

, (2.13)

The (upper-sign) solution dominates the string path integral and is responsible for the

leading exponential behaviour in (1.2) and so, in the following, we will restrict to the

upper signs in (2.3).

3 One-loop ﬂuctuation determinants

This section focusses on the semiclassical expansion of the string partition function around

the stable classical solution (2.7) (taking upper signs in (2.3)) and the determinants of the

diﬀerential operators describing the semiclassical ﬂuctuations around it. The 2π-periodicity

in τ allows to trade the 2D spectral problems with inﬁnitely-many 1D spectral problems

for the (Fourier-transformed in τ ) diﬀerential operators in σ. Let us call O one of these

one-loop operators. For each Fourier mode ω, the evaluation of the determinant Det

is a one-variable eigenvalue problem on the semi-inﬁnite line σ ∈ [0, ∞) which we solve

using the Gel’fand-Yaglom method, a technique based on ζ-function regularization reviewed

in appendix B. Multiplying over all frequencies ω (which are integers or semi-integers

according to the periodicity of the operator O) gives then the full determinant

DetO =

Det

O. (3.1)

All our worldsheet operators are intrinsically singular on this range of σ, since their prin-

cipal symbol diverges at σ = 0, the physical singularity of the boundary divergence for the

AdS

metric. Moreover the interval is non-compact, making the spectra continuous and

more diﬃcult to deal with. We consequently introduce an IR cutoﬀ at σ = 

(related

to the  = tanh 

cutoﬀ in z) and one at large values of σ = R [7]. While the former is

necessary in order to tame the near-boundary singularity, the latter has to be regarded as

a mere regularization artifact descending from a small ﬁctitious boundary on the tips of

the surfaces in AdS

and S

. Indeed it disappears in the one-loop eﬀective action.

3.1 Bosonic sector

The derivation of the bosonic ﬂuctuation Lagrangian around the minimal-area surface (2.7)

is readily available in section 5.2 of [40]. The one-loop ﬂuctuation Lagrangian in static

gauge is

(2)

≡ Ω

(σ) y

(θ

) y , (3.2)

– 7 –

JHEP02(2016)105

where the diﬀerential operator O

(θ

) acts on the vector of ﬂuctuation ﬁelds orthogonal

to the worldsheet y ≡ (y

)

i=1,...8

. In components it reads

(θ

)]

= −

Ω

(σ)



∂

+ ∂



+ m

+ n

∂

, (3.4)

where the non-vanishing entries of the matrices are

= m

Ω

(σ) sinh

, m

= m

= −

Ω

(σ) cosh

(σ+σ

)

= m

−2 + 3 tanh

(2σ + σ

)

Ω

(σ)

, n

= −n

2 tanh (2σ + σ

)

Ω

(σ)

. (3.5)

The worldsheet excitations decouple in the bosonic sector, apart from y

and y

which

are coupled through a 2 × 2 matrix-valued diﬀerential operator. The determinant of the

bosonic operator is decomposed into the product

DetO

(θ

) = Det

Det

(θ

) DetO

(θ

) . (3.6)

Going to Fourier space (∂

→ iω), formula (3.6) holds for each frequency ω with

≡ −∂

+ ω

sinh

(3.7)

(θ

) ≡ −∂

+ ω

−

cosh

(σ + σ

)

(3.8)

(θ

) ≡

−∂

+ ω

− 2 + 3 tanh

(2σ + σ

) 2 i tanh (2σ + σ

) ω

−2 i tanh (2σ + σ

) ω −

dσ

+ ω

− 2 + 3 tanh

(2σ + σ

)

(3.9)

The unitary matrix U =

√



i 1

−i 1



diagonalizes the operator (3.9)

(θ

) = U

†

diag{O

, O

3−

}U ,

(θ

) = −∂

+ ω

− 2 + 3 tanh

(2σ + σ

) − 2ω tanh (2σ + σ

) , (3.10)

3−

(θ

) = −∂

+ ω

− 2 + 3 tanh

(2σ + σ

) + 2ω tanh (2σ + σ

) .

We performed a rescaling by

√

h = Ω

(σ) (as in the analogous computations of [7, 14, 23])

which will not aﬀect the ﬁnal determinant ratio (4.1) (see discussions in appendix A of [6]

and in [7, 14, 23]) and is actually instrumental for the analysis in appendices B.2 and B.3.

We rewrite (3.6) as follows

Det

(θ

) = Det

Det

(θ

) Det

(θ

) Det

3−

(θ

) , (3.11)

To compare with [40], and using the notation used therein, notice that the bosonic Lagrangian is

derived as

(2)

= δ

αβ

∂

− δ

αβ



∂

β ij

+ A

α j

∂





αβ

α i

β `j

−

√

γM



, (3.3)

which deﬁnes in an obvious way m

and n

in (3.4).

There would be an overall minus sign in the kinetic and mass term of the y

ﬂuctuation, which we disre-

gard in (3.4) for simplifying the formula, considering that it does not play a practical role in the evaluation

of determinants with Gel’fand-Yaglom and is reabsorbed in the Wick-rotation of the time coordinate t.

– 8 –

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