基于可积性的p-Brane与D-Brane超代数构造

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本文探讨了在超弦理论中至关重要的p-布雷恩（p-brane）和D-布雷恩（D-brane）作用的数学基础——可积的p-布雷恩超级代数。这些超级代数不仅包含了超庞加莱代数的额外生成器，即玻色子和费米子，而且它们在理论构建中扮演着关键角色。理论学家们已经通过两种主要途径理解这些代数：一是“自下而上”（bottom-up）的方法，通过分析Noether守恒量来揭示这些代数结构；二是“自上而下”（top-down）的方式，通过构造显式保持超对称性的Wess-Zumino作用。 Wess-Zumino项在超弦理论中负责确保超对称性的保留，尤其是在构建p-brane和D-brane的有效动作时，其存在至关重要。本文作者，D.T.格拉索（D.T. Grasso）和I.N.麦克阿瑟（I.N. McArthur），从一个全新的角度出发，即基于超对称变换的可积性进行研究。可积性意味着这些变换满足一定的线性关系，使得它们能够在不破坏理论一致性的情况下，自然地结合在一起。这种积分性质有助于解决与构造Wess-Zumino项相关的同调问题，这是构建有效场论时需要处理的重要数学挑战。通过这种方法，作者提供了一个简洁且深刻的证明，展示了如何从超对称变换的内在结构出发，推导出这些关键的p-布雷恩超级代数，从而进一步加深了我们对超弦理论中这些复杂对象的理解。这一工作发表于《核物理B》（Nuclear Physics B）杂志的926期，2018年，是开放获取的，可供全球科研人员查阅。本文的研究不仅扩展了我们对超弦理论中p-布雷恩和D-布雷恩作用数学框架的认识，而且也为未来探索更深层次的理论物理问题，如量子引力和宇宙学，提供了新的思考路径。同时，它强调了数学在理论物理学中的核心地位，特别是通过整合和应用高级数学概念如可积性来解决实际物理问题的能力。

242 D.T. Grasso, I.N. McArthur / Nuclear Physics B 926 (2018) 239–255

where Z

···a

are bosonic central charges. This was based on solutions to the “descent

equations”

(p+2)

= db

(p+1)

,δ



(p+1)

= dδ



(p)

(2.17)

(where δ



denotes a supersymmetry variation with parameter 

, and the second equation follows

from δ



(p+2)

= 0) in which b

(p+1)

and δ



(p)

were chosen to be invariant under spacetime

translations. More general solutions which are not invariant under spacetime translations lead to

additional fermionic topological charges in the algebra of Noether charges [20,22]. In the case

p = 1 (superstrings), the enlarged algebra is the Green algebra [1].

The same enlar

ged supersymmetry algebras involving additional bosonic and fermionic gen-

erators have also emerged via an alternative route. As was pointed out by Siegel in the case of the

superstring [7], and generalised to p-branes by others, it is possible to construct Wess–Zumino

Lagrangians that are manifestly invariant under spacetime supersymmetry transformations. The

form b

(p+1)

in standard superspace with coordinates (x, θ) cannot be chosen to be supersym-

metric (i.e. expressed in terms of left-invariant forms); however, by introducing an enlarged

superspace with additional coordinates, it is possible to introduce additional terms into the ac-

tion to make it supersymmetric. The enlarged superspace is related by a coset construction [12,

15] to the enlar

ged supersymmetry algebras that emerge via “quasi-invariance” and modiﬁed

Noether currents. Stated technically, h

(p+2)

belongs to a trivial Chevalley–Eilenberg cohomol-

ogy class for the enlarged superspace, in that it can be expressed as d(b

(p+1)

− dA

(p)

) where

the additional superspace p-form A

(p)

is chosen so that b

(p+1)

− dA

(p)

is invariant under su-

persymmetry transformations – and indeed can be expressed in terms of left-invariant forms on

the enlarged superspace. The additional left-invariant forms have been determined by a variety of

means by different authors, including using the concept of a free differential algebra [8,10,11,15]

and requiring that the left-invariant forms satisfy as consistent set of Maurer–Cartan equations

(equi

valent to the Jacobi identities of the corresponding algebra) [12,13]. The Jacobi identities

rely on the gamma matrix identities (2.9), (2.10) – the same identities required to ensure closure

of the superspace forms (2.8) giving rise to the Wess–Zumino terms.

The supersymmetry in

variant Wess–Zumino term is then of the form



∗

(p+1)

− dA

(p)

). (2.18)

This still satisﬁes the requirements of a Wess–Zumino term, namely that d(b

(p+1)

− dA

(p)

) is

closed and supersymmetry invariant; the fact that the p-form A

(p)

appears as a total derivative

means that the degrees of freedom it contains (including the additional superspace coordinates)

do not contribute to the dynamics.

A similar issue is in

volved in the construction of D-brane actions. As originally formulated,

D-branes include a world-volume U(1) gauge ﬁeld A whose ﬁeld strength F = dA gives rise to

a Born–Infeld term in the action [29–32]; speciﬁcally,

S =



(p+1)



det (G

+ F

) + S

. (2.19)

As with p-branes, G

is the pullback to the world-volume of the spacetime metric. F is the

world-volume two-form F = F −σ

∗

(2)

, where db

(2)

= h

(3)

is a supersymmetry invariant closed

superspace three-form given by (2.8) with p = 1. The Wess–Zumino term is again of the form



∗

(p+1)

, (2.20)

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