高自旋理论中的偏τ函数：新视角与量子化应用

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关于高自旋理论中的偏τ函数（skew τ-functions），这是一个近年来在三维自旋理论研究中的前沿课题。焦点主要集中在Chern-Simons理论中的威尔逊环上。在经典极限下，这些理论展现出独特之处，表现为在特定的SL(N)表示中，最高权重态与最低权重态之间的特殊角矩阵元素。这些"偏斜"的τ函数与传统的τ函数——即两个最高权重态之间的矩阵元素——有所不同，但它们仍然具有重要的数学结构。尽管它们看起来新颖且不同于传统定义，偏τ函数却遵循着不同基本表示之间的Toda方程组，这是一种在量子力学和数学物理中广泛使用的递归关系。Toda方程在理论物理中扮演着关键角色，尤其是在统计力学和可积系统的研究中。这些函数的出现揭示了一种潜在的统一性，使得它们在理解更深层次的理论结构时显得尤为重要。在最常被提及的示例中，偏τ函数还与矩阵模型和Schur函数建立了联系。矩阵模型是一种重要的工具，它在弦理论、量子重力和拓扑量子场论等领域有着广泛的应用，而Schur函数则是经典代数学中的一个重要概念，它们在组合学和表示论中占有核心地位。通过这样的连接，偏τ函数提供了一个将打结理论（knot theory）和可积性理论（integrability theory）这两个看似不同的领域相互融合的桥梁。这篇发表于《Journal of High Energy Physics》(JHEP05, 2016, 027)的研究论文，由D. Melnikov, A. Mironova和A. Morozova三位作者合作完成，他们在文中详细阐述了这些偏τ函数的性质、计算方法以及它们在理论框架下的应用。这篇开放获取的文章不仅深化了对高自旋理论的理解，也为未来在该领域的进一步探索开辟了新的可能性。

JHEP05(2016)027

The metric and the higher spin ﬁelds are then obtained as traces of symmetrized products

of e

, e.g.

µν

tr T

tr (e

) , φ

µνρ

= tr



(µ

ρ)



, etc. (1.11)

Here T

and T

are the generators of SL(2, R) in SL(N, R). It is important how the

SL(2, R) are embedded in the SL(N, R) and in what follows the principal embedding will

be assumed. We will come back with the details and importance of the embedding in

latter sections.

Thus, ﬂat connections are mapped to solutions of the Einstein equations. The gauge

transformation acting on the connections are consequently mapped to diﬀeomorphisms of

the metric. Speciﬁcally, the inﬁnitesimal transformations

δA = dΛ +



A, Λ



, δ

A = d

Λ +





(1.12)

have the following action on the frame ﬁelds

δe

= e

;µ



, Λ +



, ξ



−



. (1.13)

The class of ﬂat connections relevant for (1.9) can be realized as the Cartan gauge

transformation with b = exp(ρT

) of the constant connections so that the uppercase ﬂat

connections are

A = b

−1

(a + ∂)b ,

A = b(¯a + ∂)b

−1

(1.14)

and the lowercase connections are

a =

−T

−

+MT

s=3

(s)

(s−1)

(dφ+dt) , ¯a =

−

s=3

(s)

−(s−1)

(dφ−dt) ,

(1.15)

where T

(s)

±(s−1)

are the lowest (highest) weight generators of SL(N, R) classiﬁed by the spin

3 ≤ s ≤ N, and M

are the corresponding physical charges, classifying the background

solution. The SL(2, R) charges M and

M will be related, in particular, to the Hamiltonian

of the gravity background.

In the SL(2) case, for M =

M, adopting the choice of the generators

1/2 0

0 −1/2

, T

−

0 0

1 0

, T

0 1

0 0

, (1.16)

one arrives at the following form of the uppercase connections

A =

0 e

−ρ

−e

dw +

1/2 0

0 −1/2

dρ , (1.17)

A =

0 e

−e

−ρ

M 0

d ¯w +

−1/2 0

0 1/2

dρ , (1.18)

where we have introduced complex coordinates

w = φ + iτ , ¯w = φ − iτ . (1.19)

– 8 –

JHEP05(2016)027

Using (1.10) one recovers the Euclidean version of metric (1.9). The Minkowski version is

obtained through the Wick rotation iτ → t.

In the Poincar´e coordinates the gauge transformation (1.14) is generated by

b = u

−T

. (1.20)

However, to describe the source in these coordinates the lowercase connections a and ¯a

must be coordinate dependent. The coordinate dependence enters through the charges

M(z),

M(¯z). In the case with no sources M =

M = 0 the gauge connections take the form

A =

0 0

−1/u 0

dz +

−1/2u 0

0 1/2u

du , (1.21)

A =

0 1/u

0 0

d¯z +

1/2u 0

0 −1/2u

du . (1.22)

Metric (1.6) now easily follows from (1.10).

1.2.3 Non-constant connections

The constant lowercase connections (1.15) are not the most general connections corre-

sponding to asymptotically AdS geometries. In fact, the metric still satisﬁes the Einstein

equations (1.1) up to a relevant source term, if one lets the coeﬃcients M, M

be holomor-

phic functions of w and

of ¯w. In the SL(2) case, the connections look like

a = (−T

−

+ J(w)T

) dw ¯a =



−

J( ¯w)T

−



d ¯w , (1.23)

The property

∂J = ∂

J = 0 is consistent with the ﬂatness condition. The metric reads

= dρ

+(e

2ρ

+J +

J +J

−2ρ

)dτ

+(e

2ρ

−J −

J +J

−2ρ

)dφ

−2i(J −

J)dτ dφ , (1.24)

where the real and imaginary part of J are related to the black hole mass and angular

momentum respectively.

The global and Poincar´e coordinates are related through z ∼ e

±iw

. Thus the sources

in the global coordinates, which appeared in the center of the AdS

bulk, are mapped to

the sources at z = 0 (equivalently, at z = ∞). As a result, to introduce a source (conical

defect) in Poincar´e coordinates one has to use a coordinate-dependent connection a. Then,

the metric in the global coordinates is



− uJ(z)



− u

J(¯z)





+ uJ(z)



+ u

J(¯z)



. (1.25)

For example, taking J =

J =

¯z

produces the metric [57–64]



1 −





1 +



dη

, (1.26)

where r and η are the polar coordinates on the z-plane.

– 9 –

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