N=1到N=2 SCFTs的变形与RG流：非阿贝尔风味对称性

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"该论文探讨了四维N=1 $$ \mathcal{N}=1 $$超保形场理论（SCFTs）在保持非阿贝尔风味对称性情况下的特定变形，以及这些变形导致的重整化群（RG）流。通过在风味对称性的伴随表示中添加N=1 $$ \mathcal{N}=1 $$手性多重峰并赋予其零幂真空期望值，引发红外SCFT的重规范流。研究中发现了N=2 $$ \mathcal{N}=2 $$超对称性在红外端得到增强的不动点理论，包括广义的Argyres-Douglas理论和一些具有非阿贝尔风味对称性的第一秩SCFT。特别地，观察到了从归一化的共形SQCD到（A 1，A n）Argyres-Douglas理论的RG流。此外，还研究了N个理论，如TN和R 0，S类$$ \mathcal{S} $$类的一系列情况，变形产生了真实流。" 这篇学术论文详细分析了四维N=1 $$ \mathcal{N}=1 $$超对称场理论在保留非阿贝尔风味对称性时的变形。这些变形通过在风味对称群的伴随表示中引入N=1 $$ \mathcal{N}=1 $$手性多重峰并为其设定零幂真空期望值来实现。这种操作会触发一个重整化群流，将理论带入红外区域的新的SCFT状态。作者们特别指出，在这些变形过程中，存在一类理论在红外端展现出增强的N=2 $$ \mathcal{N}=2 $$超对称性，形成所谓的不动点理论。这些理论包括广义的Argyres-Douglas理论和第一秩SCFT，它们具有非阿贝尔风味对称性。一个显著的发现是，当从归一化的共形SQCD（超量子色动力学）出发进行变形时，存在一个RG流至（A 1，A n）Argyres-Douglas理论。这扩展了我们对这些理论的理解，特别是对于它们的超保形指数的计算，这些计算结果与之前的工作相吻合。此外，论文还涵盖了TN和R 0，S类$$ \mathcal{S} $$类的N个理论，以及一些第一秩SCFT的情况，变形导致了真实世界的物理现象。这篇论文的贡献在于深入研究了N=1 $$ \mathcal{N}=1 $$ SCFTs的非平凡变形，揭示了这些变形如何导致新的N=2 $$ \mathcal{N}=2 $$超对称性增强的红外固定点理论，以及它们与Argyres-Douglas理论的关系。这样的研究不仅深化了我们对超保形场论的理解，也为理解更复杂的量子场论系统提供了新的视角和工具。对于理论物理学家和数学物理学家来说，这些发现可能为探索超对称性和重整化群流的新领域打开大门。

JHEP02(2017)075

f su(N) so(2N + 1) sp(N) so(2N) e

pr,f

N(N

−1)

N(N+1)(2N+1)

N(4N

−1)

N(N−1)(2N−1)

156 399 1240 156 28

Table 2. The embedding indices associated to the principal embeddings for all simple Lie alge-

bras [38].

Central charges. The central charges, a

and c

, and the ﬂavor central charge k

an N = 2 SCFT T can be written in terms of the ’t Hooft anomaly coeﬃcients of the

R-symmetries as follows [37]:

TrJ

= TrJ

= 0,

TrJ

−

= TrJ

−

= 48(a

− c

TrJ

−

= 8(2a

− c

TrJ

−

= 0,

TrJ

−

= −

(2.15)

where T

are the generators of the ﬂavor symmetry F . After the N = 1 deformation, the

TrJ

−

anomaly coeﬃcient shifted due to (2.9) as

TrJ

−

→ TrJ

−

+ 12TrJ

−

ρ(σ

)

, (2.16)

while all the other anomalies remain ﬁxed. The second term of (2.16) can be computed

by remembering

TrJ

−

ρ(σ

)

= I

TrJ

−

= −

, (2.17)

where the ﬂavor index a is not summed and I

is the embedding index. When F = SU(N)

where the embedding is given by the partition N =

, the embedding index I

given as

k=1

k(k

− 1)n

. (2.18)

For the principal embedding associated with the partition n

= 1, this gives I

N(N

−1)

In the case of the other semi-classical group, the embedding index for the principal embed-

ding is given in table 2.

By adding the contribution of the remaining chiral multiplet M

j,−j,f

one gets the

anomalies of the deformed theory. When ρ is given by the principal embedding into f, the

anomalies are given by

TrJ

= TrJ

= −r, TrJ

−

= 48(a

− c

) +

i=1

(2d

− 1),

TrJ

−

= 48(a

− c

) − 6k

i=1

(2d

− 1)

TrJ

−

= 8(2a

− c

) +

i=1

(2d

− 1), TrJ

−

= −

i=1

(2d

− 1)

(2.19)

– 7 –

JHEP02(2017)075

For the F = SU(N) case, they are

TrJ

= TrJ

= −N + 1,

TrJ

−

= 48(a

− c

) + N

− 1,

TrJ

−

= 48(a

− c

) − k

SU(N)

N(N

− 1) + N

(2N

− 1) −1,

TrJ

−

= 8(2a

− c

) + N

− 1,

TrJ

−

= −

N(4N

− 1)

+ 1.

(2.20)

By assuming that there is no accidental global symmetry in the IR, the IR U(1)

symmetry is a combination of two U(1)

which is determined by using a-maximization [3].

Other U(1) global symmetries in T or those may come from the subgroup of F cannot be

mixed with R-symmetry. This is because TrR

F = 0 for any global symmetry F in an

N = 2 SCFT [39], and it also implies TrF = 0 so that the ﬂavor symmetry F is “baryonic”.

Baryonic symmetries cannot be mixed with the R-symmetry [3]. Therefore, we pick the

trial U(1)

symmetry as

R =

1 + 

1 − 

−

(2.21)

and compute the trial central charge a() =

(3TrR

− TrR). Maximizing a() gives a

solution of . However there is a caveat here: one should check that all the dimensions of

the operators, which is given by ∆(O

) =

R(O

), satisfy the unitarity bound. If it hits the

bound, the operator becomes free and U(1) symmetry under which only the free multiplet

is charged appears. One thus has to subtract the contribution of this operator from the

trial central charge and re-maximize, as in [19]. This process has to be repeated until all

the chiral operators satisfy the unitarity bound.

3 Generalized Argyres-Douglas theories

In this section we apply the general argument in the previous section to a family of SCFTs

of Argyres-Douglas type with an SU(N) ﬂavor symmetry. After reviewing the SCFTs, we

consider the deformations of the so-called (A

, D

) theory with an SU(2) ﬂavor symmetry,

and the (I

N,k

, F ) theory with an SU(N) ﬂavor symmetry. Interestingly, for all these cases

we will see the supersymmetry is enhanced to N = 2 in the IR.

3.1 N = 2 Argyres-Douglas theories

An N = 2 SCFT with Coulomb branch operators with fractional dimensions is called as

(a generalized) Argyres-Douglas theory. The simplest example is the H

theory found

in [10, 15] which has the single dimension-

operator. In the following, we will review

the four classes of the Argyres-Douglas theories, collecting the results necessary for the

computation in the subsequent subsection.

We would like to thank Ken Intriligator for instructing this to us.

– 8 –

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