分数θ角与电荷q多味Schwinger模型的非摄动动力学

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"这篇学术文章详细探讨了电荷q多味Schwinger模型中的分数θ角度，以及它在非摄动量子场论（QFT）动力学中的作用。研究使用了离散的't Hooft异常，结合圆紧化和玻色化的方法，深入分析了二维量子场论的特性。主要关注点是电荷q的N味Schwinger模型和Wess-Zumino-Witten模型。" 在二维QFT的研究中，离散的't Hooft异常被用来揭示理论的内在性质，尤其是在对R2理论及其压缩到R×S1空间的研究中。作者首先介绍了这一领域的最新进展，并将这些概念应用于电荷q的N味Schwinger模型。't Hooft异常匹配技术在这里发挥了关键作用，因为它能够帮助理解和比较理论的非摄动动力学。接下来，通过构造本征态并计算具有特定边界条件（周期性和加味扭曲）的圆柱体时空上的物理量，研究人员将't Hooft异常与模型的动态行为进行了对比。研究发现，不同类型的边界条件会导致不同的异常表现，特别是在扭曲边界条件下，与离散手性对称性破缺相关的Nq个真空态出现。在这种情况下，手性凝聚物的θ依赖性呈现出分数形式，即eiθ/ Nq，导致了Nq分支结构和软费米子质量的出现。值得注意的是，这种分数θ依赖性在小圆周上的行为无法仅通过传统的瞬态来解释，而需要引入“量子”瞬态的概念。这涉及到BPS饱和，即经典作用和量子诱导的有效势之间的平衡。在半经典近似范围内，量子瞬态的影响与通过玻色化方法得到的精确结果相符。此外，研究还探讨了具有扭曲边界条件的Schwinger模型在大N极限下的体积独立性，这是QFT中一个重要的性质。这些发现深化了我们对二维量子场论的理解，并为探索量子场论的非平凡动力学提供了新的视角。该研究展示了离散't Hooft异常在理解电荷q多味Schwinger模型和Wess-Zumino-Witten模型中复杂物理现象的重要性，特别是在量子瞬态和分数θ角度的上下文中。这些成果不仅扩展了我们对量子场论基础的理解，也为未来的理论研究和实验验证提供了宝贵的理论框架。

JHEP07(2019)018

Combined with the fermion kinetic term (2.20), we obtain the gauged action S

gauged

. Fixing

a UV regularization scheme, we can compute the partition function as

[A] =

DaDψDψ exp (−S

gauged

) . (2.25)

Since the background gauge ﬁelds A are chiral, the fermion path integral potentially suﬀers

from non-Abelian chiral anomaly.

2.2.2 Computation of anomaly by Stora-Zumino procedure

To ﬁnd the chiral anomaly, the easiest way is to use the descent equation of Stora-Zumino

chain [61, 62]. It starts from computing 4-dimensional Abelian anomaly density,

Ω

4π

tr[(qda +

+ F

)

] −

4π

tr[(qda +

)

]

4π

tr[(

+ F

)

−

] +

2π

qda ∧ (tr[

+ F

] − tr[

])

4π

tr[(

+ F

)

−

] +

2π

da ∧ d(B

(1)

+ NA

(1)

). (2.26)

Here,

R/L

= d

R/L

+ i

R/L

, and F

= dA

(1)

. Since q(NA

(1)

+ B

(1)

) = dA

(0)

, the second

term of the last line vanishes identically. Therefore, we obtain

Ω

4π

tr[(

+ F

)

−

]. (2.27)

Let us apply the descent procedure to Ω

. We should ﬁnd Ω

, which satisﬁes dΩ

= Ω

Ω

is not uniquely determined, and diﬀerent ones correspond to diﬀerent regularization

scheme of symmetry generators, so we just have to pick up a favorite one. It is easy to

check that either of the following ones satisﬁes the equation;

Ω

4π



(

+ A

(1)



+ A

(1)

) +



−





, (2.28)

Ω

4π







−





2π

(1)

∧ (qNB

(2)

+ NB

(2)

) +

4π

(1)

∧ dA

(1)

. (2.29)

In the ﬁrst expression (2.28), we take the L-R scheme for the whole expression. In the

second one (2.29), we take the L-R scheme for the non-Abelian part, and take the V-A

scheme for the linear term in terms of A

(1)

. The diﬀerence between them is expressed by

the total derivative.

The ﬁnal step of the descent procedure shows that the three-dimensional topologi-

cal action,

[A] =

Ω

, (2.30)

– 10 –

JHEP07(2019)018

satisﬁes the anomaly-inﬂow mechanism so that

[A] exp(−iS

[A]) (2.31)

is gauge invariant for ∂M

= M

, when we take the consistent regularizations. There-

fore, eq. (2.28), or (2.29), characterizes the ’t Hooft anomaly of the charge-q N-ﬂavor

Schwinger model.

2.2.3 Discrete ’t Hooft anomaly and four-fermion interaction

In this part, let us pay attention to a subgroup G

sub

of the symmetry G:

sub

= Z

[1]

SU(N)

)

× (Z

)

⊂ G. (2.32)

That is, continuous chiral symmetry SU(N )

× SU(N)

is restricted to its diagonal sub-

group SU(N )

, while we keep the discrete axial symmetry (Z

)

. Any fermion bilinear

operators ψ

, ψ

break the discrete chiral symmetry completely, but we can consider

four-fermion operators which are invariant under G

sub

(ψ

)(ψ

−1

a=1

(ψ

)(ψ

), (2.33)

where T

are generators of SU(N). By adding this four-fermion interaction to the La-

grangian, we can explicitly break G down to G

sub

. In the context of 4-dimensional QCD

with fundamental fermions, this operator was important to discuss the exotic scenario of

chiral symmetry breaking, called Stern phase [32, 63–66]. Also, this restriction of symme-

try is important to discuss the application of 2-ﬂavor Schwinger model (' SU(2) level-1

Wess-Zumino-Witten (WZW) model) to (1 + 1)-dimensional anti-ferromagnetic quantum

spin chain in the context of Haldane conjecture [67–70]. Its generalization to N -ﬂavor

case is important when we consider the generalization of Haldane conjecture to SU(N)

anti-ferromagnetic spin chain [54, 71–74].

We can readily ﬁnd the anomaly of G

sub

. We have to set A

= A

≡ A

for SU(N )

gauge ﬁelds, and B

= 0. This sets

≡

with the constraint tr[

] = B

(1)

and qNA

(1)

= dA

(0)

, and we substitute it into (2.29). As a consequence, the anomaly is

characterized by

sub

] =

2π

(1)

∧ B

(2)

. (2.34)

This action is quantized to Z

phase, and thus we ﬁnd the discrete anomaly of G

sub

Let us make a remark on a related anomaly, which is found in previous studies [24–

26]. In those papers, authors only perform gauging of Z

[1]

and not of SU(N)

. As

a consequence, anomalous breaking of discrete chiral symmetry occurs as Z

→ Z

refs. [24–26]. In our case, we ﬁnd a stronger discrete anomaly, since (2.34) says that the

discrete chiral symmetry Z

is completely anomalously broken by background gauge ﬁelds.

– 11 –

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分数θ角与电荷q多味Schwinger模型的非摄动动力学

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