3D N=8超共形场论的复兴与Leibniz代数

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"Reviving 3D N=8 Superconformal Field Theories" 这篇研究论文探讨了三维空间中N=8超共形场论的恢复，这是一个涉及到高维规范代数的复杂理论。N=8超共形场论是物理学中的一种理论，它结合了超对称性和共形不变性，提供了对物理现象更深入的理解，特别是在量子场论和弦理论中扮演重要角色。在描述中，作者提出了一种拉格朗日公式，这种公式设计得足够通用，能够包容超越传统李代数的无限维规范代数。他们利用基于Leibniz代数的 Chern-Simons 理论来实现这一目标。Chern-Simons理论是一种拓扑量子场论，它通过特定的耦合方式将规范场与拓扑结构联系起来，而Leibniz代数则是一种非交换的代数结构，允许元素之间的乘法不满足交换律。 Leibniz代数的Chern-Simons理论生成了L∞代数，这是一种更广泛的结构，可以描述非平凡的相互作用和无穷维的对称性。为了定义这个理论，研究者通过嵌入张量映射ϑ将李代数g的对偶空间g*映射回g本身。特别地，他们展示了对于三维流形上的体积保持微分（如流形的变形不影响其体积）的李代数sdiff3，存在一个自然的嵌入张量，它定义了一个空间上的Leibniz代数。进一步的研究揭示，任何具有体积形式的三维流形的余切束都具有（广义）库兰特代数的特性。库兰特代数是包含导数运算、 Lie 导数和泊松括号的代数结构，是研究共形场论和几何的重要工具。这些发现暗示N=8超共形场论与Bandos-Townsend理论有等价性，Bandos-Townsend理论是另一种探讨多膜物理的理论。此外，基于S3（三维球面）的理论被证明是Bagger-Lambert-Gustavsson模型的一个特例。Bagger-Lambert-Gustavsson模型试图构建一个描述M2-brane（M2膜）相互作用的理论，是膜理论研究中的一个重要进展。这项工作不仅深化了我们对3D N=8超共形场论的理解，还提供了一种处理无限维规范代数的新方法，这对于理解和探索高维物理现象，尤其是量子场论中的超对称性和共形不变性，具有重要意义。通过将这些理论与拓扑量子场论和非交换代数结构联系起来，研究者开辟了新的研究途径，可能有助于解决当前理论物理中的挑战。

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JHEP04(2019)047

there is a conventional Lie algebra formulation, but one has to abandon manifest locality,

as we will discuss in more detail. The route taken here, which is common in physics, is

to work with ﬁelds living on the larger space but to impose a gauge redundancy. Then

there is a perfectly local formulation, but with a gauge structure that is governed by an L

∞

algebra rather than a Lie algebra. For the case at hand, we will show that the cotangent

bundle T

∗

M for any 3-manifold M equipped only with a volume form provides (a mild

generalization of) a Courant algebroid, which in turn gives rise to an L

∞

structure [19].

Thus, the gauge algebras of the SCFTs based on sdiff

are governed in particular by a

non-trivial ‘3-bracket’ encoding the failure of the ‘2-bracket’, the antisymmetric part of

the Leibniz product (1.1), to satisfy the Jacobi identity. Taking R to be the space of

functions on the 3-manifold, there is also the 4-tensor (1.4) and hence a trilinear bracket

on R, which turns out to be the Nambu bracket [39], but it should be emphasized that

the latter bracket is not directly related to the 3-bracket deﬁning the gauge algebra; in

fact, they are not even deﬁned on the same space. To illuminate this point, consider the

BLG model, for which the trilinear bracket, the ‘3-algebra’ of [2], is given by the invariant

epsilon tensor of SO(4), i.e., {φ

, φ

}

= ε

abcd

. The gauge algebra, on the

other hand, is the Lie algebra structure on so(4)

∗

that is transported via the embedding

tensor ϑ(

) =

abcd

from the Lie algebra so(4), where t

and

are the generators

and dual generators, respectively. Being a Lie algebra, this trivially deﬁnes an L

∞

algebra

whose 3-bracket is identically zero.

The rest of this paper is organized as follows. In section 2 we develop the general

formulation of N = 8 SCFTs, starting from a Lie algebra g and constructing, via an

embedding tensor, a Leibniz-Loday algebra that is suﬃcient in order to deﬁne gauge the-

ories. In particular, we show that these general structures are suﬃcient in order to prove

N = 8 supersymmetry. In section 3 we specialize to the Lie algebra of volume-preserving

diﬀeomorphisms on a 3-manifold, and we show that there is a natural embedding tensor

satisfying all constraints. We then show that the resulting theories are equivalent, modulo

some topological assumptions, to the Bandos-Townsend theories. In section 4 we con-

sider truncations and deformations of the general framework. Speciﬁcally, we consider the

mode expansion of the sdiff

SCFTs for S

and show that the BLG model is a consistent

(generalized Scherk-Schwarz) truncation. Conversely, the full Bandos-Townsend theory for

provides an inﬁnite-dimensional generalization of the BLG model. We also review the

×S

model, whose Scherk-Schwarz reduction on S

yields a 3D N = 8 super-Yang-Mills

theory with gauge group SDiﬀ

, and we discuss massive deformations that preserve N = 8

supersymmetry. In the conclusion section we discuss possible generalizations.

2 Maximal 3D superconformal ﬁeld theories

We deﬁne the 3D N = 8 superconformal ﬁeld theory for the general data given in the

introduction. In the ﬁrst subsection we review how an embedding tensor satisfying the

quadratic constraint deﬁnes a Leibniz-Loday algebra and a gauge invariant Chern-Simons

action. In the second subsection we introduce a representation R with invariant metric

and show that with these data one can universally deﬁne an invariant 4-tensor, which in

– 5 –

JHEP04(2019)047

turn satisﬁes a quadratic identity as a consequence of the quadratic constraint. With these

structures we are able to deﬁne in the third subsection the complete N = 8 Lagrangian

and prove supersymmetry.

2.1 Embedding tensor and Leibniz-Chern-Simons theory

Our starting point is a (generally inﬁnite-dimensional) Lie algebra g, with Lie brackets

[·, ·], and its dual space g

∗

. As mentioned in the introduction, we only need a weak notion

of dual space, meaning that it is a representation space (coadjoint representation) with

a pairing between covector and vector (giving a real number) that is invariant under the

combined adjoint and coadjoint action. Writing v, w ∈ g for vectors and ω, η ∈ g

∗

for

covectors, the pairing is written as ω(v) ∈ R. The coadjoint representation is denoted by

∗

, and denoting collectively by δ

the adjoint and coadjoint action we have

(ω(v)) ≡ (ad

∗

ω)(v) + ω([w, v]) = 0 . (2.1)

Moreover, being a representation, the coadjoint action also satisﬁes [ad

∗

, ad

∗

] = ad

∗

[v,w]

We now assume the existence of an embedding tensor, a linear map

ϑ : g

∗

→ g , (2.2)

that in general is not invertible. Generally, since ϑ is not an isomorphism, the Lie algebra

structure on g cannot be transported to a Lie algebra structure on g

∗

. There is, however,

a natural bilinear operation deﬁned in (1.1),

ω ◦ η ≡ ad

∗

ϑ(ω)

η . (2.3)

We will now show that this product deﬁnes a Leibniz-Loday algebra, which means that the

following Jacobi-type identity holds

(ω

◦ ω

) ◦ ω

+ ω

◦ (ω

◦ ω

) − ω

◦ (ω

◦ ω

) = 0 ,

(2.4)

provided the embedding tensor satisﬁes the quadratic constraint (1.3),

ϑ(ω ◦ η) =



ϑ(ω), ϑ(η)



. (2.5)

The proof proceeds by noting that the left-hand side of the Leibniz relation (2.4) can be

rewritten as

∗

ϑ(ω

◦ω

)

−



∗

ϑ(ω

)

, ad

∗

ϑ(ω

)



= ad

∗

ϑ(ω

◦ω

)−



ϑ(ω

),ϑ(ω

)



= 0 ,

(2.6)

using that ad

∗

is a representation of g.

The Leibniz relation (2.4) can be interpreted as saying that the action of g

∗

on itself,

deﬁned by

η ≡ ω ◦η , (2.7)

– 6 –

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3D N=8超共形场论的复兴与Leibniz代数

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