膜二重性：从视场方程到广义几何的SL(3)×SL(2)对称性揭示
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本文主要探讨了膜的二重性（Membraneduality）在M理论中的重要概念，特别是在高维时空背景下，特别是与弦理论的T-duality相对应。作者提到，M理论中的U对偶性（类似于弦理论中的T-duality，是通过转换场方程到膜世界上的Bianchi恒等式来理解的）并非总是适用，除非目标空间的维度满足特定条件，如D=p+1。这里的挑战在于，U对偶变换在非理想的维度下可能不具有可积性。
在具体示例中，作者以D=3的M2-brane（二维膜）为例，展示了广义几何如何体现SL(3)×SL(2)的双重性。这个发现揭示了在特定维度下，膜的物理性质和对称性的复杂交织。在D=4的M2-branes情景中，尽管有额外的目标空间坐标，实际实现的U-对偶对称仅限于预期的SL(5,R)对称性的GL(4,R)⋉R4子群，这意味着实际对称性降级到了一个较小的子集。
文章还提到了研究背景，即该工作是在 Imperial College London 的 Blackett Laboratory、中国科学技术大学的 Interdisciplinary Center for Theoretical Study、意大利 SISSA 和 INFN Trieste 分别进行的，以及Texas A&M University的 Mitchell Institute for Fundamental Physics and Astronomy 和英国 Cambridge University 的 DAMTP 中心的参与。研究者们通过将弦理论中的T-duality原理应用到M理论中，进一步探索了高维空间中基本物理对称性的本质。
这篇论文深入探讨了膜的U对偶性和其在不同维度下的具体表现，以及与弦理论T-duality的比较，对理解M理论中的对称性结构和空间维度的依赖性提供了新的见解。读者可以通过 ScienceDirect 或 www.elsevier.com/locate/nuclphysb 获取该篇开放获取的文章，发表于2015年的《核物理学B》期刊上。
M.J. Duff et al. / Nuclear Physics B 901 (2015) 1–21 5
2. Topological membranes
Although p-branes in p + 1 dimensions carry no dynamical degrees of freedom [66], they
are nevertheless of considerable interest. In the present context, they provide us with a setting
in which we can get a handle on duality symmetries that transform field equations and Bianchi
identities into each other. Everything we will do here applies to topological p-branes for gen-
eral p, bu
t for simplicity in notation as well as our special interest in M2-branes, we shall focus
on topological membranes.
The standard action for the closed membrane is
I =
d
3
σ
−
1
2
√
−γγ
ij
∂
i
X
μ
∂
j
X
ν
g
μν
+
1
6
ε
ij k
∂
i
X
μ
∂
j
X
ν
∂
k
X
ρ
B
μνρ
+
1
2
√
−γ
, (2.1)
where in our conventions ε
012
=+1. For the topological membrane we take μ = 0, 1, 2.
3
The
SL(3) is manifest. For simplicity, we shall take the metric tensor g
μν
and 3-form potential B
μνρ
in the 3-dimensional target space to be constant.
4
In this case, using the algebraic field equation γ
ij
=∂
i
X
μ
∂
j
X
ν
g
μν
, we have the identity
√
−γγ
ij
g
μν
∂
j
X
ν
=−
1
2
√
−gε
μνρ
ε
ij k
∂
j
X
ν
∂
k
X
ρ
. (2.2)
Therefore, letting B
μνρ
=
√
−gε
μνρ
B, the action can be written as
I =
1
3!
d
3
σ
√
−g(1 + B)ε
ij k
∂
i
X
μ
∂
j
X
ν
∂
k
X
ρ
ε
μνρ
. (2.3)
We shall, however, use the form (2.1) of the action below, motivated by the fact that this form
will make it easier to compare with what happens in the case of the non-topological membrane.
The resulting equations are
∂
i
P
i
μ
=0 ,γ
ij
=∂
i
X
μ
∂
j
X
ν
g
μν
, (2.4)
where
P
i
μ
≡−
√
−γγ
ij
∂
j
X
ν
g
μν
+
1
2
ε
ij k
∂
j
X
ν
∂
k
X
ρ
B
μνρ
. (2.5)
There is also a conserved topological current:
∂
i
J
iμν
=0 ,J
iμν
=ε
ij k
∂
j
X
μ
∂
k
X
ν
. (2.6)
Because we are considering a target space that is three dimensional, we can make the definitions
J
i
μ
=
1
2
ε
μνρ
J
iνρ
,B
μνρ
=B |g|
1/2
ε
μνρ
. (2.7)
3
This form of the action is in accordance with the terminology of ‘topological membrane’ we are using here. It should
be noted, however, that the characterization of membranes as ‘topological’ also arises in the context of membranes that
propagate in dimensions higher than three but with action that consists of Wess–Zumino term and no kinetic term. See,
for example [67]. In general, branes with pure Wess–Zumino terms exhibit a huge symmetry enhancement; see, for
example [68].
4
This is an important assumption. Otherwise the modification in equation (2.4) will obstruct the sought after duality
symmetry. The spacetime background here can be viewed as being a subsector a time-like dimensional reduction of the
bosonic sector of D = 11 supergravity down to 8 dimensions, where only the fields (g
μν
, B
μνρ
) are kept, and the 8
dimensional Euclidean coordinates are taken to be constants. In a spacelike dimensional reduction, the signature of the
metric g
μν
would be Euclidean, and we would take the 8 dimensional spacetime coordinates to be constant. All of our
considerations apply for this case as well.
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