麦克斯韦超代数在D=4和D=11超引力的自由微分代数中的隐藏作用资源-CSDN文库

Open

Access

124 浏览量 2020-03-28 20:10:03 上传评论收藏 595KB PDF 举报

资源推荐

资源详情

资源评论

Eur. Phys. J. C (2018) 78:211

https://doi.org/10.1140/epjc/s10052-018-5673-8

Regular Article - Theoretical Physics

Hidden role of Maxwell superalgebras in the free differential

algebras of D = 4 and D = 11 supergravity

Lucrezia Ravera

INFN, Sezione di Milano, Via Celoria 16, 20133 Milan, Italy

Received: 6 February 2018 / Accepted: 23 February 2018

Abstract The purpose of this paper is to show that the so-

called Maxwell superalgebra in four dimensions, which nat-

urally involves the presence of a nilpotent fermionic gener-

ator, can be interpreted as a hidden superalgebra underlying

N = 1, D = 4 supergravity extended to include a 2-form

gauge potential associated to a 2-index antisymmetric ten-

sor. In this scenario, the theory is appropriately discussed in

the context of Free Differential Algebras (an extension of

the Maurer–Cartan equations to involve higher-degree dif-

ferential forms). The study is then extended to the Free Dif-

ferential Algebra describing D = 11 supergravity, showing

that, also in this case, there exists a super-Maxwell algebra

underlying the theory. The same extra spinors dual to the

nilpotent fermionic generators whose presence is crucial for

writing a supersymmetric extension of the Maxwell algebras,

both in the D = 4 and in the D = 11 case, turn out to be

fundamental ingredients also to reproduce the D = 4 and

D = 11 Free Differential Algebras on ordinary superspace,

whose basis is given by the supervielbein. The analysis of

the gauge structure of the supersymmetric Free Differential

Algebras is carried on taking into account the gauge trans-

formations from the hidden supergroup-manifold associated

with the Maxwell superalgebras.

1 Introduction

It is well known that supergravity theories in D ≥ 4 space-

time dimensions contain gauge potentials described by p-

forms, of various p > 1, associated to p-index antisym-

metric tensors. In this scenario, the Free Differential Alge-

bras framework, that is an extension of the Maurer–Cartan

equations to involve higher-degree differential forms, is par-

ticularly well suited for studying supergravity models. The

concept of Free Differential Algebra (FDA in the sequel)

e-mail: lucrezia.ravera@mi.infn.it

was introduced in [1] and subsequently applied to the study

of supergravity theories (see, for instance, Ref. [2]).

A review of the standard procedure for the construction

of a minimal FDA (namely a FDA where the differential of

any p-form does not contain forms of degree greater than p)

starting from an ordinary Lie algebra can be found in [3].

In [2], the authors considered the D = 11 supergravity

theory of [4], introducing and investigating the supersym-

metric FDA describing the theory (using the so-called super-

space geometric approach) in order to see whether the FDA

formulation could be interpreted in terms of an ordinary Lie

superalgebra (in its dual Maurer–Cartan formulation). This

was proven to be true, and the existence of a hidden super-

algebra underlying the D = 11 supergravity theory was pre-

sented for the ﬁrst time. It includes the D = 11 Poincaré

superalgebra as a subalgebra, but it also contains two extra,

almost-central, bosonic generators, which were lately under-

stood as p-brane charges, sources of the dual potentials A

(3)

and B

(6)

appearing in the (complete) FDA of [2] (see Refs.

[5,6]).

Furthermore, a nilpotent fermionic generator must be

included to close the superalgebra and in order for the same

superalgebra to reproduce the D = 11 FDA on ordinary

superspace, whose basis is given by the supervielbein. Rel-

evant contributions concerning the physical role played by

this extra fermionic generator were given ﬁrst in [7] and

then in particular in [8,9], where the results presented in

[2] were further analyzed and generalized. Finally, its group-

theoretical and physical meaning was recently clariﬁed in

[3] (and subsequently further discussed in [10]): in [3]itwas

shown that the spinor 1-form dual to the nilpotent fermionic

charge is not a physical ﬁeld in superspace, rather behaving

as a cohomological BRST ghost, since its supersymmetry

and gauge transformations exactly cancel the non-physical

contributions coming from the extra tensor ﬁelds, guarantee-

123

211 Page 2 of 12 Eur. Phys. J. C (2018) 78:211

ing that the extra bosonic 1-forms dual to the almost-central

charges are genuine abelian gauge ﬁelds.

As shown in Ref. [3], where the authors analyzed also the

FDA of the minimal N = 2, D = 7 supergravity theory, this

interpretation is valid for any supergravity theory containing

antisymmetric tensor ﬁelds, and any supersymmetric FDA

can always be traded for a hidden Lie superalgebra containing

fermionic nilpotent generators (see also [11] for the study of

a particular D = 4 FDA case).

In the ﬁrst part of this paper, we will consider the FDA of

N = 1, D = 4 supergravity containing a 2-form potential

under the same perspective of [3]. Let us mention, here, that

supergravity in D = 4 space-time dimensions is often formu-

lated as a theory of gravity coupled to scalar-vector multiplets

only, that is to say 1-form gauge ﬁelds. On the other hand,

when we think of the theory as obtained by Kaluza-Klein

compactiﬁcation from eleven-dimensional supergravity, then

it naturally contains also 2-form ﬁelds (tensor multiplets). In

four dimensions, if these 2-form ﬁelds are massless, then

they can be dualized, through Hodge duality of their ﬁeld

strengths, to scalars (this is the reason why they often do not

explicitly appear in the formulation). However, when they are

massive

such dualization does not (at least directly) apply

and, in this case, the 2-form gauge ﬁelds must be made man-

ifest [12] (see also Refs. [13–15] for more details on the role

of 2-forms in four-dimensional supergravity theories).

The aim of the present paper is to show that the so-called

minimal Maxwell superalgebra (or minimal super-Maxwell

algebra) in four dimensions (a non-semisimple superalgebra

naturally endowed with a nilpotent fermionic generator), can

be interpreted as a hidden superalgebra underlying the FDA

of D = 4 supergravity that includes a 2-form potential A

(2)

This will be done by studying the parametrization of A

(2)

and the hidden gauge structure of the FDA on the same lines

of what was done in the D = 11 (and D = 7) case in [2,3].

Then, we will extend our discussion to the FDA introduced in

[2], which describes D = 11 supergravity, showing that, also

in this case, there exists a Maxwell superalgebra underlying

the theory.

The extra spinors dual to the nilpotent fermionic

generators whose presence is crucial for writing a supersym-

metric extension of the Maxwell algebras, both in the D = 4

and in the D = 11 case, will turn out to be fundamental

also to reproduce the D = 4 and D = 11 FDAs on ordinary

superspace.

Actually, as it was lately pointed out in [10], the extra spinor 1-form

dual to the nilpotent fermionic generator can be parted into two different

spinors, whose integrability conditions close separately.

This happens, for instance, in the case in which the higher-

dimensional theory is reduced via a ﬂux compactiﬁcation.

Actually, we will consider the D = 11 FDA just containing a 3-form

potential A

(3)

. We leave the study of the complete FDA containing also

a 6-form potential B

(6)

(see Refs. [2,3]) to future works.

This work is organized as follows: in Sect. 2,weﬁrst

recall the main features of the Maxwell superalgebra; then,

we move to the analysis of the hidden gauge structure of

the supersymmetric FDA of N = 1, D = 4 supergrav-

ity (containing a 2-form potential A

(2)

), showing that the

Maxwell superalgebra can be viewed as a hidden superal-

gebra underlying the theory. Subsequently, in Sect. 3,we

extend our study and results to the FDA describing D = 11

supergravity (which, in its minimal cohomology formulation,

contains just a 3-form potential A

(3)

), introducing a (hidden)

Maxwell superalgebra underlying the theory. Finally, Sect. 4

contains the conclusions and possible future developments.

In the Appendix we collect our conventions and some useful

formulas.

2 Minimal super-Maxwell algebra and hidden gauge

structure of the D = 4 supergravity FDA

After the discovery of the cosmic microwave background

and the mysterious dark energy, it appears interesting to con-

sider some ﬁeld densities uniformly ﬁlling space-time. One

such modiﬁcation of empty Minkowski space can be obtained

by adding a constant electromagnetic ﬁeld background,

parametrized by additional degrees of freedom related to

tensorial almost-central charges. The presence of a constant

electromagnetic ﬁeld modiﬁes the Poincaré symmetries into

the so-called Maxwell symmetries. On the other hand, since

the advent of supersymmetry, there has been a great interest

in superalgebras going beyond the super-Poincaré one.

In particular, the (minimal) Maxwell superalgebras are

(minimal) super-extensions of the Maxwell algebra, which

in turn is a non-central extension of the Poincaré algebra

involving an extra, bosonic, abelian generator (along the lines

of non-commutative geometry).

Speciﬁcally, the D = 4 Maxwell algebra is obtained by

replacing the commutator [P

, P

]=0(a = 0, 1, 2, 3) of

the Poincaré algebra with [P

, P

]=Z

, where Z

−Z

are abelian generators commuting with translations

and behaving like a tensor with respect to Lorentz trans-

formations (i.e. Z

are tensorial central charges). Setting

= 0 one gets back to the Poincaré algebra.

The Maxwell algebra arises when one considers symme-

tries of systems evolving in ﬂat Minkowski space ﬁlled in

by a constant electromagnetic background [16,17]. Indeed,

an action for a massive particle which is invariant under the

Maxwell symmetries satisﬁes the equations of motion of a

charged particle interacting with a constant electromagnetic

ﬁeld via the Lorentz force. In particular, in order to interpret

the Maxwell algebra and the corresponding Maxwell group,

a Maxwell group-invariant particle model was studied on an

extended space-time with coordinates (x

,φ

μν

), where the

translations of φ

μν

are generated by Z

μν

[18–21]. The inter-

123

剩余11页未读，继续阅读

评论收藏

内容反馈

weixin_38530995

粉丝: 0
资源: 891

麦克斯韦超代数在D = 4和D = 11超引力的自由微分代数中的隐藏作用

最新资源