N=4 SYM中量子光谱曲线的推广与解析表达

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本文主要探讨了尖峰威尔逊线在N=4$$\mathcal{N}=4$$超级杨-米尔斯(Super Yang-Mills, SYM)理论中的量子光谱曲线(QSC)。起初，QSC形式主义是为研究N=4 SYM中所有局部单迹线算符的异常维度谱而设计的，但这一理论被扩展到了更广泛的范围，即对于所有异常尺寸参数的广义尖峰情况。作者们证明，尽管在处理更大的光谱参数时，以及某些特定的分析性质上需要做出修正，但基本的功能关系保持不变。这项工作的一个关键成果是，他们成功地找到了在小|φ±θ|条件下的前两个非平凡项的全环解析表达式，这展示了理论的精确性和适用性。这些解析表达式不仅在理论上具有重要意义，也为深入理解量子场论的非线性行为提供了新的洞察。此外，文章还展示了在一般角度下的非perturbative数值结果，这些结果与四环扰动理论以及经典弦理论的预测完美匹配。这进一步验证了量子光谱曲线作为一种强大的工具，能够处理复杂的量子系统，并且能够在不同层次的理论计算中达成一致。这种一致性是理论物理学中的一个重要成就，因为它强化了理论模型的可靠性和有效性。这篇文章不仅扩展了量子光谱曲线在N=4 SYM中的应用，而且还揭示了其在处理非平凡物理条件下的强大适应性。它为未来的理论研究提供了新的框架和计算方法，有助于深化我们对量子场论特别是超对称理论的理解。

JHEP04(2016)134

Similarly to (2.7), the powers in the asymptotics of Q

contain the SO(4, 2) charges

including the conformal dimension ∆:

∼ B

−1

, Q

∼ B

−

(2.15)

where



∆ − S

− S

+ 2

∆ + S

+ S

−∆ − S

+ S

+ 2

−∆ + S

− S



. (2.16)

From consistency of the above equations we can extract the relation between ∆ and the

leading coeﬃcients A

, A

in P

, P

appearing in (2.7). To do this we expand Q

a|i

, P

etc

as (asymptotic) series at large u, e.g.

a|i

= u

∞

n=0

ai,n

. (2.17)

Then from (2.12) we ﬁnd at leading order

a|j

= −iA

−

(1 + O(1/u)) . (2.18)

Plugging this into (2.12) we see that the coeﬃcients B

cancel and we get

− 1 = i

a=1

−

, j = 1, 2, 3, 4 . (2.19)

These equations ﬁx the values of A

for a

= 1, 2, 3, 4, leading to the important

relations

= i

(

−

)

b6=a

(

−

)

, a

= 1, 2, 3, 4 . (2.20)

We see that while the SO(6) charges are encoded in powers of the asymptotics of

P’s, the leading coeﬃcients A

and A

contain the remaining charges S

, S

, ∆. Equa-

tions (2.7), (2.20) supplement the Pµ system relations (2.1)–(2.4) and ﬁx the conformal

dimension ∆ as a function of λ and of the other charges.

Let us ﬁnally mention that one can close the equations at the level of Q

(together

with Q

which also have one long cut), introducing new functions ω

(u) which are analogs

of µ

. The matrix ω

is antisymmetric and has unit Pfaﬃan, but while µ

are periodic

with long cuts, ω

are periodic with short cuts instead. A related useful property is that

while µ

has powerlike asymptotics at large real u, ω

tends to a constant matrix. The

equations relating ω’s with Q’s read

= ω

, ˜ω

− ω

= Q

−

. (2.21)

where ω

is the inverse matrix to ω

. These are analogs of the Pµ system relations

we described above. When these equations are supplemented with constraints on the

– 7 –

JHEP04(2016)134

asymptotics of Q’s similar to (2.20), they form a closed system of equations alternative to

the Pµ-system.

One can also relate µ’s and ω’s with the help of Q-functions with four indices Q

ab|ij

deﬁned as a determinant,

ab|ij



a|i

a|j

b|i

b|j



. (2.22)

Then we have

−

ab|ij

. (2.23)

In the next section we will discuss how to change the large u asymptotics to describe

the generalized cusp.

2.2 Adapting the QSC for the cusp anomalous dimension

In this section we will discuss the modiﬁcations in the QSC which are needed to describe

the generalized quark-antiquark potential. Below we will only discuss the vacuum state,

i.e. the Wilson line with L scalar insertions at the cusp (the extension for more general

insertions should be straightforward).

The Quantum Spectral Curve equations of [2, 3] in N = 4 SYM can be deduced

from the TBA equations or the corresponding T- and Y-systems with special analyticity

assumptions. In our case the TBA equations for the generalised cusp are almost the same

as the original TBA system. The Y-system and T-system equations are exactly the same

as for the original problem. Thus it is natural to expect that the QSC equations should

also be the same to a large extent. In the TBA there are only two important diﬀerences:

the extra boundary dressing phase supplementing the BES phase, and the twists which

appear as chemical potentials and introduce the angles φ, θ into the problem.

We do not

derive the QSC from the Thermodynamic Bethe ansatz, rather we will put forward and

motivate a proposal which is consistent with several highly nontrivial checks, leaving little

doubt as to its correctness.

First, we expect to have the same set of Q-functions and auxiliary functions such as

as in the original problem. All of them will satisfy the same functional relations, for

instance the Pµ-system equations (2.1)–(2.4) or the QQ relations are unchanged. However

some analyticity properties will change, as we will discuss below, and in particular the

P-functions acquire an extra cut going from u = 0 to inﬁnity. In addition, the large u

asymptotics clearly need to be modiﬁed. Indeed, the twists in the boundary conditions

typically correspond to imposing exponential rather than powerlike asymptotics for the

Q-functions (see e.g. [3] and references therein). In our case the angle θ is naturally related

to the S

part of the geometry, which qualitatively corresponds to the P-functions, so

roughly speaking we expect P

∼ e

±θu

at large u. Similarly, the angle φ is associated to

AdS

leading to Q

∼ e

±φu

. This argument is also supported by the expectation that P’s

and Q’s should be related in the classical limit to the quasimomenta for S

and AdS

There is also an extra symmetry requirement on the Y-functions of the TBA, namely they should be

invariant under the exchange of the two wings of the Y-system with a simultaneous reﬂection u → −u, i.e.

a,s

(u) = Y

a,−s

(−u), see [33, 34] for details.

– 8 –

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