关于超对称E11例外场论资源-CSDN文库

Open

Access

7 浏览量 2020-04-17 20:51:33 上传评论收藏 1.35MB PDF 举报

资源推荐

资源详情

资源评论

JHEP10(2019)165

Published for SISSA by Springer

Received: August 4, 2019

Accepted: September 24, 2019

Published: October 14, 2019

On supersymmetric E

exceptional ﬁeld theory

Guillaume Bossard,

Axel Kleinschmidt

b,c

and Ergin Sezgin

Centre de Physique Théorique, CNRS, Institut Polytechnique de Paris,

91128 Palaiseau cedex, France

Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),

Am Mühlenberg 1, DE-14476 Potsdam, Germany

International Solvay Institutes,

ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium

Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University,

College Station, TX 77843, U.S.A.

E-mail: guillaume.bossard@polytechnique.edu,

axel.kleinschmidt@aei.mpg.de, sezgin@tamu.edu

Abstract: We construct an inﬁnite system of non-linear duality equations, including

fermions, that are invariant under global E

and gauge invariant under generalised diﬀeo-

morphisms upon the imposition of a suitable section constraint. We use ﬁnite-dimensional

fermionic representations of the R-symmetry K(E

) to describe the fermionic contribu-

tions to the duality equations. These duality equations reduce to the known equations

of E

exceptional ﬁeld theory or eleven-dimensional supergravity for appropriate (partial)

solutions of the section constraint. Of key importance in the construction is an indecompos-

able representation of E

that entails extra non-dynamical ﬁelds beyond those predicted

by E

alone, generalising the known constrained p-forms of exceptional ﬁeld theories. The

construction hinges on the tensor hierarchy algebra extension of e

, both for the bosonic

theory and its supersymmetric extension.

Keywords: Global Symmetries, M-Theory, Extended Supersymmetry, Space-Time

Symmetries

ArXiv ePrint: 1907.02080

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP10(2019)165

JHEP10(2019)165

9 Conclusions 66

A Representations of e

A.1 Level decomposition into gl(11) 68

A.2 Level decomposition under gl(3) ⊕e

B Variations of the fermionic bilinears under K(E

) 73

B.1 Bilinears of the form ¯ψ

B.2 Bilinears in ψ

C Eleven dimensional supergravity 75

D The gauge parameter representation 76

K(E

) fermions under Spin(1, 3) × SU(8) 78

1 Introduction

Exceptional ﬁeld theories [1–4] are based on generalised exceptional geometries in which

diﬀeomorphisms are uniﬁed with tensor gauge transformations in such a way that the

closure of the local transformations require constraints on the ﬁelds, known as section

constraints [1, 5, 6]. These theories live on a space which is locally a direct product of

D-dimensional ‘external’ space-time with an ‘internal’ space whose coordinates are in a

representation of a split real form of the exceptional group E

and are subject to the

covariant section constraint. Here, E

is the usual hidden Cremmer-Julia symmetry

group of ungauged maximal supergravity in D = 11 − n space-time dimensions [7], which

also governs gauged maximal supergravity through the embedding tensor formalism [8, 9]

and the associated tensor hierarchy [10, 11]. The tensor hierarchy ﬁelds play a central

role in constructing exceptional ﬁeld theory. Solving the section constraint amounts to

restricting the dependence on the extra coordinates so that the dynamics of an appropriate

supergravity theory emerges.

The study of exceptional ﬁeld theories is interesting for several reasons. Besides provid-

ing a uniﬁed description of supergravity theories that are related by duality transformations

(like D = 11 and type IIB supergravity [2]), they allow for the derivation of uplift formulæ

for solutions of gauged supergravity [12–15] and the construction of gauged supergravi-

ties via a generalised Scherk-Schwarz mechanism [16–21]. They are also instrumental in

studying non-geometric string theory solutions [22–28]. Further aspects of exceptional ﬁeld

theory have been discussed in the recent overview [29].

It is a remarkable fact that the bosonic sector of exceptional ﬁeld theory is completely

determined by generalised diﬀeomorphisms without the use of supersymmetry. Another

key property is that these theories typically require extra p-forms of rank p ≥ D −2 beyond

those present in the usual tensor hierarchy of D-dimensional maximal gauged supergravity.

These obey constraints that are similar to the section constraints. They are related to the

physical ﬁelds by ﬁrst-order equations and do not themselves describe new physical degrees

of freedom.

– 1 –

JHEP10(2019)165

So far, E

exceptional ﬁeld theories have been constructed explicitly for n = 6, 7, 8

in [2–4] and in [1, 30–33] for smaller n. The cases beyond n = 8 involve inﬁnite dimensional

groups and bring in formidable new challenges. A dent has been made recently in the case

of E

[34]. The present paper studies the case of E

It has been proposed by West long ago and prior to the development of exceptional

ﬁeld theory that the D = 11 supergravity equations of motion should emerge from an

invariant theory formulated in the framework of a non-linear realisation of E

with

coordinates in the ‘vector’ representation, such that the dynamics would follow from an

invariant set of duality equations [35–37].

It has been realised recently that these ﬁrst

order duality equations can only hold modulo certain equivalence relations [41–43]. These

ambiguities are argued to be liftable by passing to equations of motion that are eventually

of arbitrarily high order in derivatives. These equivalence relations may potentially be

interpreted as arising from additional gauge symmetries, although their precise form has

not been determined. The section constraint was not used in those references in connection

with the gauge invariance of the equations of motion, but only in connection with the

description of 1/2-BPS states [44]. E

does capture the supergravity tensor hierarchy ﬁeld

content in D dimensions [45, 46], but not the extra constrained p-forms of the exceptional

ﬁeld theories mentioned above.

In [47], it was explained that constructing linearised gauge invariant ﬁrst order ﬁeld

equations with E

symmetry requires the ﬁelds to satisfy the section constraint as well

as the introduction of additional ﬁelds that do not appear in the E

coset space. This

construction is based on an inﬁnite-dimensional super-algebra T (e

), that includes e

a subalgebra and that generalises the tensor hierarchy algebra T (e

) introduced in [48]

for n ≤ 8 to the Kac-Moody case. The tensor hierarchy algebra then includes a non-

semi-simple extension T

) of the algebra e

that entails the introduction of extra ﬁelds

already in the linearised theory. A gauge invariant linearised duality equation can be written

in this formulation for a ﬁeld strength that transforms covariantly under E

provided one

introduces these extra ﬁelds in the corresponding indecomposable representation. The extra

ﬁelds are necessary to write a gauge invariant duality equation for the graviton in eleven

dimensions [47]. We exhibit here that our E

duality equation, including the extra ﬁelds,

gives in the linearised approximation an inﬁnite tower of gauge invariant duality equations

of the type described in [49]. Gauge invariance of these equations as described in [50, 51]

is only satisﬁed in the presence of the extra ﬁelds.

The primary goal of this paper is to construct the E

and gauge invariant non-linear

duality equation that captures all the duality equations of all E

exceptional ﬁelds theories.

We will show that one can generalise the duality equation constructed in [47] to a non-linear

equation invariant under generalised diﬀeomorphism. The key observation that facilitates

this construction is that the derivative of the extra ﬁelds found in [47] at the linearised

level are the cohomologically trivial part of extra ﬁelds that turn out to underlie the extra

The idea of extended Kac-Moody symmetries, in particular in connection with low-dimensional gravi-

tational systems, was ﬁrst expressed in [38]. An explicit trace of E

symmetry was found in a Belinskii-

Khalatnikov-Lifshitz analysis of eleven-dimensional supergravity [39] and was later generalised to a cosmo-

logical E

model [40].

– 2 –

JHEP10(2019)165

constrained p-forms ﬁelds mentioned above in the GL(11 − n, R) × E

decomposition and

subsequent truncation of the E

invariant theory. In analogy with what happens in lower-

dimensional exceptional ﬁeld theories, and as mentioned above, these extra p-forms are

related to the propagating ﬁelds by ﬁrst order equations but they do not themselves describe

new physical degrees of freedom. These ﬁrst order equations for the constrained ﬁelds are

sourced by bilinear terms in the derivatives of the ﬁelds parametrizing E

. Therefore they

cannot follow from the E

variations of the duality equation we construct in this paper,

and they must be derived separately by requiring gauge invariance and integrability of the

equations. We shall not attempt to determine these ﬁrst order equations in this paper, and

will only make some comments on their expected structure.

Besides the investigation of a non-linear bosonic theory based on the tensor hierarchy

algebra, an important part of the present paper is the study of its supersymmetric extension.

Fermions are introduced here — as for maximal supergravity and other exceptional ﬁeld

theories — as representations of (the double cover of) the involution invariant subgroup

K(E

) that plays the role of a generalised R-symmetry group. As noticed in [52–56], this

subgroup admits ﬁnite-dimensional (a.k.a. unfaithful) spinor representations in the case

of Kac-Moody groups K(E

) with n ≥ 9. In particular, these representations were con-

structed for the gravitino and supersymmetry parameter of K(E

) in [57] with beginnings

of the supersymmetry parameter representation already given in [58]. The compatibility

of local K(E

) symmetry with supersymmetry, i.e., whether the supersymmetry generator

transforms correctly as a spinor under K(E

) was investigated in [56, 59, 60] for n = 10

where it was found that there was an inconsistency in the transformation arising for the

bosonic ﬁelds beyond the six-form, i.e., starting from the so-called dual graviton. Based

on [58] and [54–56], fermions in K(E

) were introduced in [61] and a similar calculation

was carried out up to the level of the six-form.

In the present paper, we resolve this inconsistency starting from the dual graviton by

considering not only e

but its non-semi-simple extension T

) that appears in the tensor

hierarchy algebra. As already emphasised above, one important consequence of the tensor

hierarchy algebra is that it introduces additional ﬁelds into the theory beyond those of the

standard E

/K(E

) symmetric space. These ﬁelds will resolve the inconsistencies with

the supersymmetry transformations, because the supersymmetry transformation of ﬁelds in

)  K(e

) can be written consistently with K(E

). We shall use this construction

to write linearised supersymmetry transformation rules and equations of motion for the

(unfaithful) gravitino ﬁeld. We also show that one obtains a closed supersymmetry algebra

at linearised order. These results will be derived explicitly at low levels, including the dual

graviton. At present, we do not have a complete algebraic proof to all levels.

After establishing the linearised supersymmetry and the equations of motion for the

Fermi ﬁelds, we investigate their non-linear extension and their compatibility with the non-

linear duality equation proposed in section 3. We present some ﬁrst steps in this direction

by introducing the non-linear K(E

) connection and a Pauli coupling to the E

ﬁeld

strength. Although we do not have the complete expression of the non-linear equations,

the ﬁrst few levels exhibit promising cancellations that lead to the desired couplings of

eleven-dimensional supergravity.

– 3 –

剩余87页未读，继续阅读

评论收藏

内容反馈

weixin_38656662

粉丝: 2
资源: 898

关于超对称E 11例外场论

最新资源

关于超对称E 11例外场论

关于超对称Lifshitz场论

关于非超对称共形流形：场论与全息术

关于10和11维的超对称AdS 6解决方案

从例外场论到通过K3的异质双场论

在E 7（+7）例外场论和N = 8 $$ \ mathcal {N} = 8 $$中心电荷超空间中的超粒子的截面条件下

E7（7）超空间中的特殊场论

E 8（8）特殊场论：几何，费米子和超对称

使用SL（5）例外场论，7维N $$ \ mathcal {N} $$ = 2个一致的截断

来自特殊场论的广义可并行空间

在特殊场论中消除多达三个回路的发散

石墨烯和边界中心电荷的超保形模型

软引导和超对称

ℝ+×E 3（3）的几何场理论和F理论

超弦理论介绍引论

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

XCP协议的规范文档

GD32替换STM32注意事项.pdf

NPPJSONViewer.zip

蓝牙BLE协议中文版.pdf

CANoe通过CAPL脚本实现自动测试

AD20官方中文教程.pdf

完整版 Microsoft.ACE.OLEDB.12.0 驱动下载.rar

电路分析基础第二版PDF电子书免费下载

Tangent免费.rar

qt样式表一键生成（花狗Fdog）

VS2015安装证书，JavaScript_ProjectSystem.msi，JavaScript_LanguageService.msi

最新资源

Qt 5实现串口调试助手（源工程文件、0积分下载）