12维超引力与T-对偶：F理论的动力学与维度降维

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本文主要探讨的是十二维有效作用与T-对偶在F-theory低能动力学中的应用。作者Kang-Sin Cho在文章《Twelve-dimensional effective action and T-duality》中提出了一种新的理论框架，该理论基于Eur.Phys.J.C（2015）75:202的发表，具有开放获取权限。文章的核心内容围绕着12维超重力模型的构建，它描述了F-theory在较低维度下的行为，即当理论被降维至11维、IIA超重力以及IIB超重力时的动态特性。在IIB超重力理论中，特别关注的是四形式场的自对偶性，这是理论中的关键概念。自对偶性意味着该四形式场在特定条件下可以互相转化，这种性质对于理解高维度理论到低维度的对称性转换至关重要。然而，为了与11维超重力保持物理自由度的一致性，文章强调必须放弃原始的12维庞加莱对称性。这涉及到一个维度的选择性紧缩，即在一维结构上进行非平凡的几何变形，使之在参数空间的不同区域中变得不连续，从而实现维度的压缩。这个过程并非随意，而是有选择地进行，目的是确保理论在降维后的物理效应与11维超重力保持一致。这一操作使得T-对偶性得以清晰展现，即不同的维度压缩方案之间存在着内在的联系和相互转化的关系，即在不同的物理条件下，通过T-对偶可以将一个紧凑化方案转换成另一个等效的方案。 T-对偶是弦理论中的一个重要概念，它揭示了在某些情况下，物理系统可以通过维度变换和空间结构的微小调整来保持基本的物理性质不变。这种对偶性对于理解高维理论的性质、验证理论的统一性和探索额外维度的可能结构有着重要意义。总结来说，这篇文章不仅提供了一个新的视角来探讨F-theory的动力学，还通过T-对偶的概念深化了我们对多维理论中对称性和空间结构的理解。通过将12维超重力与T-对偶相结合，作者展示了如何通过理论的降维和维度压缩来实现理论的简化，并且揭示了物理现象在不同维度压缩方案下的不变性。这对于深入研究弦理论、M理论以及高能物理的其他分支具有重要的理论价值。

Eur. Phys. J. C (2015) 75:202

DOI 10.1140/epjc/s10052-015-3423-8

Regular Article - Theoretical Physics

Twelve-dimensional effective action and T -duality

Kang-Sin Choi

Scranton Honors Program, Ewha Womans University, Seoul 120-750, Korea

Received: 1 December 2014 / Accepted: 17 April 2015 / Published online: 9 May 2015

Abstract We propose a 12-dimensional supergravity

action which describes low-energy dynamics of F-theory.

Dimensional reduction leads the theory to become 11-

dimensional, IIA, and IIB supergravities. Self-duality of the

four-form ﬁeld in IIB supergravity is understood. It is neces-

sary to abandon 12-dimensional Poincaré symmetry by mak-

ing one dimension compact, which is to be decompactiﬁed

in some region of parameter space, such that the physical

degrees of freedom are the same as those of 11-dimensional

supergravity. This makes T -duality explicit as a relation

between different compactiﬁcation schemes.

1 Introduction

The ideas of Kaluza and Klein (KK) [1,2], generalized to

higher dimensions, are beautiful ones that translate the known

ﬁeld degrees of freedom and their interactions into geometry

of extra dimensions. Most of the supergravity theories, which

one hopes to have an intimate connection to our world, can be

obtained by dimensional reduction of an 11-dimensional one

[3]. However, it does not directly give type IIB supergravity

in ten dimensions, although their relation is well understood

in the context of string theory.

Eleven-dimensional supergravity is a low-energy descrip-

tion of M-theory [4,5]. It has also been suggested that type

IIB superstring theory is obtained by a reduction of F -theory

on a torus, with its complex structure identiﬁed by the axion–

dilaton, and the latter is shown to be T -dual to M-theory [6].

Thus, the effective ﬁeld theory of F-theory should be 12-

dimensional; however, it is not easy to write down the action.

One crucial difﬁculty might be that the 12-dimensional mini-

mal fermion with Lorentzian signature (11, 1), which must be

the case for F -theory, should have superpartner components

with spin higher than two in the four-dimensional language,

whose interacting theory would be inconsistent [7]. Another

e-mail: kangsin@ewha.ac.kr

obstacle is that, if F-theory is dual to M-theory, there should

be no surplus ﬁeld degrees of freedom, although the former

is a higher-dimensional theory.

An important hint comes from a careful look at the deriva-

tion of F -theory [6,8,9]. Although it is T -dual to M-theory,

F-theory has one more dimension than the latter. Now, this

extra dimension is a dual dimension to one of the dimension

shared by the two theories. In other words, F -theory has two

redundant dimensions whose radii are inverse to each other.

Although we cannot maintain 12-dimensional Poincaré sym-

metry fully, each of the 10- and 11-dimensional theories

can be symmetric on its own. There is no contradiction if

we cannot see both at once. Therefore, it is natural to keep

both dimensions. In this picture, M-theory looks like a com-

pactiﬁcation F-theory on a circle, as schematically shown in

Fig. 1.

We propose for the bosonic part a desired 12-dimensional

effective action, whose dimensional reductions lead to those

of all known supergravities in 11 and 10 dimensions, found

in the standard textbooks [10]. Since we follow and make

use of the duality relation between M- and F-theory from

the 11-dimensional supergravity, this theory shall provide

the effective ﬁeld theory for F-theory.

Supergravity is powerful enough in the sense that many

of the new results here, like the existence of a three-brane

and generalized T -duality, are obtained without referring to

string theory. Of course, the effective ﬁeld description of F-

theory is timely in realistic model building, because we have

so far borrowed descriptions, for instance of the gauge ﬁelds,

from M-theory [11–13].

2 The bosonic action of 12-dimensional supergravity

We start with the fundamental bosonic degrees of freedom of

11-dimensional supergravity: the graviton G

and the rank

three antisymmetric tensor ﬁeld C

mnp

. The latter is promoted

to a four-form ﬁeld,

123

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