N=4超对称d=1sigma模型在群流形上的构造

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"这篇论文是关于N=4超对称d=1sigma模型在群流形U（2）和SU（3）上的构建，利用了谐波d=1的方法。研究中提到了（4,4,0）和（4,4,0）⊕（4,4,0）多重峰，它们分别由一个满足非线性谐波约束的超场q+来描述。论文展示了这两个情况下的不变作用，并且在超场和分量级别上验证了其不变性。还讨论了U（2）和SU（3）的等距超场实现，这些实现实际上扩大到了U（2）×SU（2）和SU（3）×U（2）。此外，文中给出了积分d=1 sigma模型的行动，特别是对于U（2）的情况，对比了HKT模型的一般谐波方法，并建立了一个特定作用与非线性多重态（3,4,1）的对应关系。最后，提出了将U（2）模型推广到矩阵U（2n）情况的途径。" 这篇论文详细探讨了N=4超对称性的应用，特别是在一维弦理论（即sigma模型）中的应用。超对称性是一种物理现象，它允许粒子与反粒子的性质相联系，而sigma模型则用于描述弦在不同背景几何中的运动。在本文中，作者利用谐波方法来处理群流形U（2）和SU（3）上的问题，这种方法允许他们在超对称性保持不变的情况下构建模型。（4,4,0）和（4,4,0）⊕（4,4,0）多重峰是超对称理论中的重要概念，它们代表了一种特定的粒子组合方式，其中的数字表示超对称代数的生成元数量。这里，这些多重峰由满足特定非线性约束的超场q+来描述，这使得模型具有N=4的超对称性，即存在四个独立的超对称生成元。在U（2）和SU（3）的等距超场实现中，它们被进一步扩展到更大的对称群，即U（2）×SU（2）和SU（3）×U（2），这提供了更丰富的对称结构。作者还通过积分操作展示了sigma模型的行动，包括在U（2）情况下与HKT模型的比较，HKT（Hyperkähler with torsion）模型是研究超kahler几何的一种特殊方法，这里的比较有助于理解两者之间的联系。此外，文章还提出了一种将U（2）模型推广到矩阵U（2n）情况的方法，这是对现有模型的一次重要扩展，可能开启研究更高维度和复杂结构的新途径。这种推广方法对于理解和探索更复杂的超对称系统以及群流形的几何性质具有重要意义。这项工作在理论物理学，特别是超弦理论和超对称性领域，提供了深入的研究成果，对于理论物理学家和数学家都具有很高的价值，因为它不仅提供了新的数学工具，还深化了我们对群流形上物理过程的理解。

6 F. Delduc, E. Ivanov / Nuclear Physics B 949 (2019) 114806

The bosonic component action following from the superﬁeld one (2.5)is

bos



dtdu



˙σ

−2

+σ ˙ω





dtduL(t, u) =



dt L(t). (2.16)

After substituting the solutions (2.13)into L(t, u), integrating by parts with respect to ∂

and

∂

, as well as using the constraint for σ

−2

, the Lagrangian L(t, u) can be presented in the

following form

L(t, u) =

+ic

˙a

+−

(1 + a

+−

)

−( ˙a

+−

)

3 + a

+−

(1 + a

+−

)

˙a

−



(1 + a

+−

)

2 ˙a

+−

(1 + a

+−

)



. (2.17)

It remains to calculate the harmonic integrals in (2.14) and (2.17). Using the formulas (B.2)-

(B.5), after some algebra (2.14) can be reduced to

c =−2i



1 +



b :=b +



1 +



. (2.18)

Next, as explained in Appendix A , the ﬁrst term in the square bracket in (2.17)gives zero con-

tribution to the harmonic integral in (2.16). The harmonic integral of the second term can be

represented as

∼a

˙a



−

(1 + a

+−

)

and



−

(1 + a

+−

)

(mn)

(ik)

) + (ε

+ε

which is the most general structure compatible with the SU(2) covariance. Substituting the sec-

ond expression into the ﬁrst one, we immediately ﬁnd that the latter is also vanishing. Thus only

the ﬁrst line in (2.17) contributes. Using the formulas (B.12)-(B.14), it is easy to compute the

remaining harmonic integrals and to present the ﬁnal answer for the Lagrangian L(t

) in (2.16)as

L = (

⎡

⎢

⎣

( ˙a ·˙a)

1 +

−

( ˙a ·a)



1 +



⎤

⎥

⎦

. (2.19)

Here, the ﬁeld

b is invariant under the nonlinear realization of SU(2) acting on the second

(sigma-model) piece of (2.19), while a

(ik)

are just Goldstone ﬁelds supporting this nonlin-

ear realization. Thus (2.19) describes a nonlinear d = 1sigma model on the group manifold

SU(2)×U(1), parametrized by the d = 1 ﬁelds a

(ik)

(t) and

b(t). The U(1) factor acts as constant

shifts of

b(t), while nonlinear SU(2) transformations of a

can be found by considering the

component-ﬁeld realization of the SU(2) symmetry originally deﬁned on the superﬁelds ω and

Starting from the transformation of n

δn

=λ

−2λ

+−

+λ

−−

)

, (2.20)

and using the ﬁrst constraint in (2.12), one obtains an equation which determines δa

(ik)

F. Delduc, E. Ivanov / Nuclear Physics B 949 (2019) 114806 7

δa

(1 + a

+−

) − δa

+−

=(1 +a

+−

)

−2λ

+−

(1 + a

+−

) + λ

−−

)

(2.21)

After some algebra, making use of the completeness relation for harmonics, one ﬁnds

δa

=λ

(λ · a)a

(ik)

+λ

(il

. (2.22)

It is easy to check that the Lie bracket of these transformations is again of the same form, i.e. we

deal with a realization of SU(2) on itself by left (or right) multiplications. The ﬁelds a

(ik)

(t) just

provide a particular parametrization of the SU(2) group element.

Analogously, starting from the SU(2) transformation of ω

δω

=λ

+−

−λ

−−

=λ

+−

−λ

−−

1 + a

+−

, (2.23)

and rewriting this variation in terms of the solution for ω

in (2.13),

δω

=δb +

δa

+−

1 + a

+−

, (2.24)

one ﬁnds the SU(2) transformation law of the ﬁeld b(t):

δb =−

(λ · a) . (2.25)

Now it is easy to show that the ﬁeld

b(t) deﬁned in (2.18) and entering the bosonic Lagrangian

(2.19)is indeed inert under SU(2)

b =0 . (2.26)

We ﬁnally note that in fact the nonlinear sigma model part of the Lagrangian (2.19)is invariant

under two independent SU(2) symmetries, one being just (2.22), while another is the linearly

realized SU(2) induced by the SU(2)

rotations of the harmonic variables (2.10):

=2τ

(il

. (2.27)

This SU(2)

does not commute with (2.22). However, one can choose another basis for the

generators of these two SU(2) groups, such that the new SU(2) commutes with (2.22). This

redeﬁned SU(2) is given by the following transformations

δa

=−τ

−

(τ ·a)a

(ik)

+τ

(il

, (2.28)

where the signs chosen ensure that the relevant Lie bracket parameter is composed in the same

way as in the cases (2.22) and (2.27). We denoted the transformation parameter in (2.28)by the

same letter as in (2.27), hoping that this will not give rise to confusion. The mutually commuting

SU(2) transformations (2.22)

and (2.28)in fact amount to the left and right shifts on the same

SU(2) group manifold.

It is easy to see that the diagonal SU(2) subgroup in the product of these tw

o commuting

nonlinearly realized SU(2) is just SU(2)

(2.27), so the bosonic sigma-model in (2.19) describes

a nonlinear realization of this product on the coset manifold SU(2)× SU(2)/ SU(2)

, i.e., the

3-sphere S

. The ﬁeld b transforms under the transformations (2.28)as

δb =

(τ ·a), (2.29)

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