三维非相对论N = 4超场的动力学实现与伽利略超粒子

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"这篇学术论文探讨了从N = 4伽利略超粒子到三维非相对论N = 4超场的转化。作者Sergey Fedoruk、Evgeny Ivanov和Jerzy Lukierski深入研究了一般具有任意中心电荷的N = 4，d = 3伽利略超代数的动力学实现。他们采用了非线性实现技术，为N = 4三维非相对论超粒子设计了一类作用，使其在中心电荷的Maurer-Cartan一式中保持线性。文章还涉及了量子化前模型的相空间约束结构分析，特别是第一类约束如何生成规范变换，包括包含费米子κ规范变换的这部分。通过量子化过程，这些模型产生了自由的N = 4，d = 3伽利略超场，这些自由场可以进一步用于构建和理解三维非相对论N = 4超对称理论。" 本文的主要知识点包括： 1. **伽利略超代数**：这是一个在非相对论物理学中出现的扩展代数，它包含了伽利略变换和超对称。在N = 4的情况下，这意味着存在四个超对称生成元。 2. **中心电荷**：在超代数中，中心电荷是与代数生成元关联的标量量，可以影响超粒子的性质。在本文中，作者考虑了具有任意中心电荷的超代数。 3. **非线性实现**：这是一种处理物理系统中守恒律的方法，通过非线性地表示代数的生成元，使得物理系统的动态行为变得复杂但可控。 4. **N = 4三维非相对论超粒子**：这是具有四个超对称生成元的非相对论粒子模型。通过引入特定的作用，可以确保在中心电荷的Maurer-Cartan一式中的线性行为。 5. **Maurer-Cartan一式**：在 Lie群或Lie代数的背景下，Maurer-Cartan一式是关于群参数的导数形式，通常用于描述曲率和规范变换。 6. **第一类约束**：在经典力学中，第一类约束产生由哈密顿函数定义的守恒律，并生成规范变换。在本文中，这些约束导致了κ规范变换。 7. **κ规范变换**：这是一种涉及费米子的规范变换，它是超对称理论中的一种基本变换，可以影响理论中的费米子部分。 8. **量子化**：将经典理论转化为量子理论的过程，涉及波函数、算符和不确定性原理等概念。在这个过程中，模型的相空间约束结构分析是至关重要的。 9. **自由超场**：在量子场论中，自由场是指不与其他场相互作用的场。在本文的上下文中，量子化的结果是生成了自由的N = 4伽利略超场。 10. **三维非相对论N = 4超对称理论**：这是利用自由超场构建的理论，它在三维空间中具有非相对论的对称性和四个超对称生成元。这些知识点展示了超对称理论在非相对论框架下的研究，对于理解非相对论量子场论和可能的应用，如凝聚态物理或弦理论的低能量有效理论，具有重要意义。

JHEP05(2018)019

Each of the real central charges X

and Y

transform in the adjoint of O(4) ∝

(a)

. The U(4)/O(4) coset generators properly mix X

and Y

The general N = 4, D = 4 super Poincar´e anticommutation relations are

βB

} = 2(σ

)

, (2.3)

, Q

} = 2

αβ

= 2

αβ



+ i Y



[6pt]{

˙αA

βB

} = 2

˙α

= 2

˙α



− i Y



(2.4)

and the remaining non-zero commutation relations read

µν

, Q

] = −

(σ

µν

)

, [M

µν

˙αA

] =

(˜σ

µν

)

˙α

βA

, (2.5)

, Q

] = [P

˙αA

] = 0 , (2.6)

, Q

] = δ

−

, [T

˙αC

] = −δ

˙αB

˙αC

, (2.7)

[A, Q

] = α Q

, [A,

˙αA

] = −α

˙αA

, (2.8)

[A, Z

] = 2α Z

, [A,

] = −2α

. (2.9)

Here α is some real parameter. If we choose α = 1, it deﬁnes the chirality of supercharges

(see (2.8)) and so identiﬁes A as the generator of axial symmetry.

In order to perform the non-relativistic contraction of N = 4, D = 4 Poincar´e super-

algebra to the limit describing N = 4, d = 3 Galilean superalgebra one should rewrite the

superalgebra (2.3)–(2.8) in the new fermionic Weyl basis

±A

√



± 

αβ

Ω

βB



˙αA

√



˙αA

∓ 

˙α

Ω



, (2.10)

where the real 4×4 matrix Ω

= −Ω

is the symplectic metric, Ω

= −Ω

, Ω

Ω

. In this paper we choose the following explicit form of Ω:

Ω



0 

˜a

, Ω



0 

˜a

, (2.11)

where 

0 1

−1 0

, 

0 −1

1 0

, a = 1, 2, ˜a = 1, 2.

The relations (2.10) break manifest Lorentz symmetry O(3, 1) to O(3) (spinorial scalar

product a

˙α

is U(2)-invariant) and the internal symmetry U(4) is broken to its subgroups

which depend on the choice of central charges [41].

The supercharges (2.10) by deﬁnition satisfy the subsidiary symplectic-Majorana con-

ditions [49]

±A

)

†

˙αA

= ∓

˙α

Ω

±B

. (2.12)

In the cases of N = 4, D = 4 superconformal symmetry and the corresponding d = 3 Galilean

superconformal algebra the generator A fully decouples, so implying α = 0 (see [47]).

The basis (2.10) was introduced in [48] and further used in [28, 30, 47] in order to obtain Galilean limit

for D = 4 Majorana supercharges.

– 7 –

JHEP05(2018)019

The full set of supercharges



±a

˙αa

; Q

±˜a

˙α˜a



can be split into the holomorphic

sector



±a

, Q

±˜a



and the antiholomorphic one



˙αa

˙α˜a



; these both sectors are related

by the subsidiary conditions (2.12), thus revealing the quaternionic structure of the pairs

of complex supercharges related by Hermitian conjugation (see [47, 50]). Due to the con-

straints (2.12) one can choose as unconstrained sets of linearly independent supercharges

the generators from either holomorphic or antiholomorphic sectors. The N = 4 superalge-

bra spanned by the generators from holomorphic sector is however not self-conjugate. In

order to deﬁne the complete self-conjugated Hermitian basis one should choose the full set

of pairs of supercharges, which are related by Hermitian conjugation. For the choice (2.11)

of the matrix Ω these self-conjugated pairs are

= Q

±1

˙α

˙α1

;

= Q

˙α

. (2.13)

In this paper we will use the supercharges belonging to the holomorphic sector, i.e. Q

±A

(A = (a, ˜a) = 1, . . . , 4). They transform linearly under the USp(4) ' O(5) R-symmetry,

which deﬁnes the compact R-symmetry sector of N = 4, d = 3 supersymmetry with one

central charge Z corresponding to the following choice of 4×4 central charge matrix (1.10)

= Z Ω

. (2.14)

In the holomorphic basis the non-vanishing relations (2.3), (2.4) can be represented as

±A

, Q

±B

} = 2

αβ



± P

Ω

+ X



, (2.15)

, Q

−B

} = 2



(σ

)

αβ

Ω

+ i

αβ



, (2.16)

where (σ

)

αβ

= (σ

)

βα

= 

αγ

(σ

)

β ˙γ

, i = 1, 2, 3 and



− Ω

Ω



, Y



+ Ω

Ω



, (2.17)





†

= Ω

Ω





†

= Ω

Ω

. (2.18)

The relations inverse to (2.17) are

= X

+ iY

= X

− iY

. (2.19)

Note that X

and Y

do not coincide with X

and Y

deﬁned in (2.2): while

the latter are real in the ordinary sense, the former are subject to the pseudo-reality

conditions (2.18). The ﬁrst kind of reality is preserved by the subgroup O(4) ⊂ SU(4) with

both X

and Y

in the adjoint representation of O(4); the pseudoreality is consistent

with the subgroup USp(4) ' O(5) ⊂ SU(4), such that each of X

and Y

encompasses

an O(5) vector and an O(5) scalar.

– 8 –

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