川口拉格朗日公式：弦论中的场与参数对称性
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本文主要探讨了川口拉格朗日公式在弦论中的应用，特别是针对Y. Hosotani提出的弦标量对偶性和A. Sugamoto提出的膜标量对偶性。川口拉格朗日公式的一大特点是其对字段和参数的独立性，这意味着这两个概念的角色是可以互换的，这为验证和证明这些理论的对偶性提供了新的视角。在川口指标的研究中，作者发现了一种简单的关系，揭示了双重相关理论之间的内在联系。
作者Ryoko Yahagi和Akio Sugamoto将焦点放在超弦模型上，通过具体实例来展示如何将费米子场中的Neveu-Schwarz-Ramond（NSR）费米子的第9个分量替换为二维超空间中的Grassmann参数。这种操作展示了参数与场之间的交换机制，使得理论在保持基本原理不变的同时，展现出不同的物理现象，例如压缩效应。这种灵活性在弦论中是至关重要的，因为它促进了不同理论结构间的相互理解，并可能推动理论的进一步发展。
论文的核心内容涵盖了以下几个方面：
1. **川口拉格朗日公式背景下的对偶性检验**：川口拉格朗日形式为检验和探索弦标量和膜标量对偶性提供了一个统一的框架，强调了其公式在处理此类对称变换时的优越性。
2. **参数与场的交换性**：由于公式特性，允许参数和场在理论中的角色互换，这简化了证明对偶性的过程。
3. **Kawaguchi几何与双重理论的关系**：研究了双重相关理论中的川口指标，揭示出它们之间的数学联系，这对于理论一致性至关重要。
4. **NSR费米子与超空间参数的交互作用**：通过NSR费米子与Grassmann参数的交换，展示了费米子场的多样性及其在模型中的实际应用。
5. **压缩效应的讨论**：在模型中，参数替换带来的影响不仅体现在理论结构上，还可能涉及到物理现象，如压缩现象的出现。
这篇Open Access文章在弦论的理论探索中引入了川口拉格朗日公式，通过实例展示了参数与场的对称性在理解和验证理论对偶性上的关键作用，以及这些对称性如何影响超弦模型的具体性质。这对于深入理解弦论的基本原理和拓展其可能性具有重要意义。
Physics Letters B 750 (2015) 294–297
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Application of Kawaguchi Lagrangian formulation to string theory
Ryoko Yahagi
∗
, Akio Sugamoto
Physics Department, Ochanomizu University, 2-1-1 Ohtsuka Bunkyo-ku, Tokyo 112-8610, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
12 August 2015
Received
in revised form 11 September
2015
Accepted
11 September 2015
Available
online 19 September 2015
Editor:
M. Cveti
ˇ
c
Keywords:
Lagrangian
formalism
Kawaguchi
geometry
Field
theory
String
theory
Duality
String-scalar duality proposed by Y. Hosotani and membrane-scalar duality by A. Sugamoto are
reexamined in the context of Kawaguchi Lagrangian formulation. The characteristic feature of this
formulation is the indifferent nature of fields and parameters. Therefore even the exchange of roles
between fields and parameters is possible. In this manner, dualities above can be proved easily. Between
Kawaguchi metrics of the dually related theories, a simple relation is found. As an example of the
exchange between fermionic fields and parameters, a replacement of the role of Grassmann parameters
of the 2-dimensional superspace by the 9th component of Neveu–Schwarz–Ramond (NSR) fermions is
studied in superstring model. Compactification is also discussed in this model.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Recently, one of the authors (RY) proposed in collaboration with
Ootsuka, Ishida and Tanaka, a covariant Lagrangian formalism for
field theories with the aid of Kawaguchi geometry [1]. Kawaguchi
space [2], also known as areal space, is defined by a set of a man-
ifold
M and an areal metric called Kawaguchi metric K . In case of
field theories, M includes fields y
i
(i = 1, ···, n) and parameters
x
μ
(μ = 0, 1, ···, D − 1) of space and time, on the equal footing,
and is called extended configuration space. We label them as Z
a
(a = 0, ···, n + D − 1). Kawaguchi metric K is a kind of Lagrangian
density, depending on the coordinates of point in M and its deriva-
tives,
K = K (Z
a
, dZ
a
1
a
2
···a
D
), where (1)
dZ
a
1
a
2
···a
D
≡ dZ
a
1
∧ dZ
a
2
∧···∧dZ
a
D
. (2)
Integral of K over a given “sheet” S (or a higher dimensional sub-
manifold
depending on the theory) gives an area (or volume) of
the “sheet”. This gives an action of field theories:
*
Corresponding author.
E-mail
addresses: yahagi@hep.phys.ocha.ac.jp (R. Yahagi),
sugamoto.akio@ocha.ac.jp (A. Sugamoto).
Action =
S
K
= the area or volume of a sub-manifold S. (3)
To guarantee the reparametrization invariance, the homogeneity
condition is imposed on the Kawaguchi metric, namely
K (Z,λdZ
a
1
a
2
···a
D
) = λK (Z , dZ
a
1
a
2
···a
D
). (4)
In the formulation, fields and parameters are not identified.
If we assign {y
i
} as fields and {x
μ
} as parameters among {Z
a
},
the fields become functions of the parameters, such as σ : y
i
=
y
i
(x
0
, ···, x
D−1
). After the parametrization σ is fixed, we have
σ
∗
dx
0···μ−1
∧ dy
i
∧ dx
μ+1···D−1
dx
0···D−1
= ∂
μ
y
i
(x), (5)
and the usual description of field theories appears. This operation
σ
∗
is sometimes called pullback of a parametrization σ . However
it is important to note that there are a number of different ways
for the parametrization.
In
[1], it is proved that every known action can be an area
(a volume) of a certain subspace, so that every field theory can be
reformulated à la Kawaguchi. Nambu–Goto action is the prototype
of this formulation. The equal treatment of fields and parameters
http://dx.doi.org/10.1016/j.physletb.2015.09.034
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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