U(N)矩阵模型与WZW共形场论的关联

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"这篇论文详细探讨了一种与手性WZW共形场理论密切相关的U（N）规范矩阵量子力学模型。在大N极限下，该模型展现了与WZW模型的深层联系。研究中，作者们通过矩阵自由度构建了左移的Kac-Moody代数，并利用Schur和Kostka多项式计算了矩阵模型的分区函数，证明其在大N限制下与WZW模型的分区函数一致。此外，这个矩阵模型也被发现能够描述非阿贝尔量子霍尔态，从而提供了一个理解WZW模型与其关系的框架。" 这篇JHEP08(2016)007的论文由Nick Dorey、David Tong和Carl Turner共同撰写，分别来自剑桥大学应用数学与理论物理系、印度理论物理研究所和斯坦福理论物理研究所。他们提出了一种U(N)规范的矩阵量子力学模型，该模型在N趋向于无穷大时，与手性WZW共形场理论的联系变得尤为紧密。这种联系首先体现在矩阵自由度能够生成左移的Kac-Moody代数，这是一个在共形场论中至关重要的结构，它反映了理论的对称性和规范不变性。其次，通过Schur和Kostka多项式，研究人员计算了矩阵模型的分区函数。Schur多项式是表示理论中不同态权重的工具，而Kostka多项式则在组合优化和代数几何中有重要应用。在大N极限下，矩阵模型的分区函数与WZW模型的分区函数相匹配，这揭示了两者在统计物理和共形场论中的基本一致性。值得注意的是，该矩阵模型还被证明可以用来描述非阿贝尔量子霍尔态，这是凝聚态物理学中的一个重要领域，涉及拓扑有序态和量子临界现象。这种联系使得矩阵模型不仅在理论物理学中具有理论价值，也可能在实验上提供理解和模拟非阿贝尔量子霍尔效应的新途径。这篇论文深入研究了矩阵模型与WZW模型之间的数学和物理联系，为理解和探索高维量子系统提供了新的理论工具。同时，通过与非阿贝尔量子霍尔态的关联，该工作可能对实验物理学家在探索新型量子材料和量子相变方面产生影响。

JHEP08(2016)007

2.1 The currents

It is straightforward to construct generators of the positive graded current algebra in the ma-

trix model. The problem factorises into U(1) and SU(p) parts. The U(1) curre nts are simply

= tr Z

while the SU(p) adjoint-valued currents are

= i



†

−

†



Here i, j, k = 1, . . . , p are ﬂavour indices, while m ≥ 0 denotes the grading.

It is simpl e to show that the commutators (

2.2) imply that these currents give a

representation of half of the Kac-Moody algebra,

[

] = i



m+n

− δ

m+n



(2.5)

while [

] = [

, J

] = 0. This holds for any N . This same expression holds in

both the quantum theory and the classical theory wher e, in the latter, the commutation

relations (

2.2) should be replaced by classical Poisson brackets.

While the result (

2.5) is he ar t ening, our inte r e st re al l y lie s in the full Kac-Moody

algebra and, in particul ar , t he central extension term. Here we will see the diﬀerence

betwe en classical and quantum theories.

The central charge of the U(1) current is harder to pin down due to a possible rescaling.

For this reason, we focus on the SU(p) currents. Here too there is a normalisation issue,

but one that will turn out to be uniquely ﬁxed. To this end, we rescale the positive-grade d

current s



(k + p)N



−m/2

m ≥ 0

Note that these still ob ey the algebra (

2.5) sinc e the overall scaling is a power of m. We

will see that only these rescaled currents will give rise to the full Kac-Moody algebra. We

then deﬁne the negative graded currents as

= J

|m| †

m < 0

and similarly for

J. These too obey the graded Lie algebra (2.5) if we restrict to m, n < 0.

Of course, the central term only arises when we consider mixed commutators of the

form [J

, J

−n

] with m, n > 0. These are trickier to c ompu te because now the con-

straint (

2.1) comes into play. However, things simplify somewhat in the N → ∞ limit. We

will show that the currents obey the Kac-Moody algebra

, J

] ∼ i(δ

m+n

− δ

m+n

) + km δ

m+n,0



−



(2.6)

Here ∼ means up to 1/N corrections. Moreover, the operators in this eq uat i on should

act on states that are constructed from the vacuum |0i by acting with fewer than O(N)

creation operators.

The rest of this section is de voted to the derivation of (

2.6). (We also show this struc-

ture arises perturbatively, in a sense made clear in the appendix, in the Poisson bracke ts

of the classi cal theory. In that setting we obtain an algebra at the unshifted level k + p.)

– 5 –

JHEP08(2016)007

2.2 Deriving the Kac-Moody algebra

The novelty in deriving (

2.6) arises from the c ommutator [Z, Z

†

] terms between curr e nts

travelling in opposite directions. We take m, n > 0 and look at

[

−n

] = [ϕ

†

, ϕ

†

†n

]

= δ

†

†n

− δ

†

†n

+ ϕ

†

, Z

†n

]ϕ

= δ

†

, Z

†n

]ϕ

+ δ

†

†n

−δ

†

†n

+ ϕ

†

, Z

†n

]ϕ

(2.7)

All U(N) group indices are contracted in the obvious manner apart from in the ﬁnal term

where we’ve written them explicitly.

Our ﬁrst goal is to simplify the two commutators in this expression. We deal with

them in turn. For the ﬁrst, we write

, Z

†n

] =

m−1

r=0

n−1

s=0

†s

[Z, Z

†

†n−1−s

m−1−r

(2.8)

We’re going to replace the factor [Z, Z

†

] appearing here with some combination of ϕ

†

using Gauss’ law (

2.3). However, Gauss’ law is not an identity between operators; instead

it holds only when evaluated on physical states |physi. If we don’t include the normal

ordering in (

2.3), then the constraint is written as



[Z, Z

†

] + ϕ

†



|physi = (k + p + N)|physi (2.9)

To this end, we consider the operator ϕ

†

, Z

†n

]ϕ

acting on a phy s i cal state. Then, after

some manipulation, we can use (

2.9) to write

†

, Z

†n

]ϕ

|physi =

m−1

r=0

n−1

s=0

†

†s

[Z, Z

†

†n−1−s

m−1−r

|physi

m−1

r=0

n−1

s=0

†

†s

(k + p − ϕ

′

†

′

†n−1−s

m−1−r

|physi

= −

m−1

r=0

n−1

s=0

†

†s

′

†

′

†n−1−s

m−1−r

|physi (2.10)

+(k + p)n

m−1

r=0

†

†n−1

m−1−r

|physi

This term above proportional to (k + p)n i s key: it will become the central term in the

algebra. We’ll come back to this shortly. Meanwhile, the ﬁrst term combines nicely with

the second commutator in (

2.7). Using the expansi on (2.8), it can be written as

†

, Z

†n

]ϕ

= ϕ

†

m−1

r=0

n−1

s=0

†s

)

†n−1−s

m−1−r

)

m−1

r=0

n−1

s=0

(ϕ

†

†s

)(ϕ

†

†n−1−s

m−1−r

) + δ



···



– 6 –

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