E11超对称例外场论的非线性对偶方程

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"这篇研究论文深入探讨了超对称E11例外场论，这是一个复杂的数学物理理论，涉及非线性对偶方程组、全局E11不变性、广义微分态下的规范不变性以及R-对称性的应用。文章通过R-对称性K(E11)的有限维铁电离子表示来阐述二重性方程，并揭示这些方程如何简化为E8特殊场论或十一维超引力方程。作者在构建过程中引入了E11的不可分解表示，拓展了特殊场论中已知的受约束p形式，同时包含了额外的非动力学场。理论的构建基于e11的张量层次代数扩展，这适用于玻色子理论及其超对称版本。该研究发表于JHEP期刊，由Springer出版，并由SISSA支持。" 这篇学术文章主要讨论了超对称E11例外场论，这是一种理论物理中的高级概念，涉及到高维空间和对称性的深刻理解。E11是一个连续对称群，它在超引力理论中扮演着关键角色，特别是在理解和统一不同物理现象方面。文章的核心是建立一个无限系统的非线性对偶方程，这些方程不仅在全局E11下保持不变，而且在考虑广义微分变形时依然具有规范不变性。这种对偶性是理论物理中重要的工具，它允许不同物理描述之间的转换。在文章中，作者使用了R-对称性K(E11)的有限维铁电离子表示，这是一种特殊的对称性操作，有助于解释二重性方程的贡献。通过对R-对称性的利用，他们能够将这些复杂的二重性方程简化，使其与已知的E8特殊场论或十一维超引力方程相联系。这种简化过程通常需要满足特定的截面约束，这些约束条件决定了理论的具体形式。 E11的不可分解表示是本文的另一个关键点，它们超越了E11自身预测的场，引入了新的非动力学场。这些新的场扩展了特殊场论的框架，特别是对于受约束的p形式有显著影响。p形式通常代表场论中的势，而受约束的p形式意味着这些势受到特定条件的限制，这在理论的构建和解算中具有重要意义。最后，作者的构建方法依赖于e11的张量层次代数扩展，这一方法对于处理玻色子理论及其超对称扩展非常有用。张量层次代数是一种抽象的数学结构，它允许物理学家在高维空间中描述各种场的相互作用和动力学。这篇论文提供了对超对称E11例外场论的深入洞察，展示了如何通过非线性对偶方程组、R-对称性和不可分解表示来理解和构建这一理论。这种方法对于进一步探索高维物理学、对称性破缺以及可能的宇宙学和量子引力理论的统一具有潜在的重要性。

JHEP10(2019)165

where η

is a symmetric non-degenerate K(E

) invariant bi-linear form on T

−1

and Ω

the E

invariant symplectic form on T

−1

In what follows we propose a non-linear extension of this duality equation. One im-

portant step will be to replace the partial derivative of the extra ﬁelds (X

˜α

, ζ

, . . .) by

constrained ﬁelds that are familiar from exceptional ﬁeld theory, in particular for E

[34],

in which case the tensor hierarchy algebra gives rise to the non-semi-simple Virasoro exten-

sion e

h hL

−1

i of e

3.3 Proposal for the non-linear duality equations

In this section we shall argue that the construction of a non-linear E

exceptional ﬁeld

theory can be achieved by deﬁning the non-linear duality equation

= M

Ω

, (3.9)

where M

is the exceptional metric, a function of the ﬁelds in E

/K(E

), in the ﬁeld

strength representation, and F

is a non-linear ﬁeld strength whose deﬁnition needs to

incorporate the extra constrained ﬁelds that are expected to arise from what we already

know from the structure of the E

exceptional ﬁeld theories for lower n. In constructing

this ﬁeld strength, we shall use the tensor hierarchy algebra extension of e

introduced in

section 2.2.

Before deﬁning the non-linear F

we shall give more details on the deﬁnition of the

exceptional metric M

. Let V(z) be a coset representative of E

/K(E

) transforming as

V(z) → k(z)V(gz)g with g ∈ E

a global element and k(z) a local K(E

) element. ‘Local’

here refers to the dependence on the extended space-time with coordinates z

where M

labels the R(Λ

) representation of E

occurring at level p = +1 in the tensor hierarchy

algebra. The subgroup K(E

) is deﬁned as the subgroup of elements k that preserve a non-

Euclidean metric η such that k

†

ηk = η in a suitable highest (or lowest) weight representation

where the Hermitian conjugate can be deﬁned [63]. The non-Euclidean nature means for

example that K(E

) ∩ GL(11) = SO(1, 10) where GL(11) denotes the regular GL(11)

subgroup of E

that appears in the level decomposition relevant for describing D = 11

supergravity. In other words, η is the standard Minkowski metric of eleven-dimensional

space-time extended to the whole extended space-time.

As usual in exceptional ﬁeld theory, it is convenient to work with the exceptional metric

in order to avoid the introduction of the K(E

) gauge invariance and its gauge-ﬁxing,

M(z) = V(z)

†

ηV(z) → g

†

M(gz)g . (3.10)

This deﬁnition is in complete analogy with non-linear realisations of ﬁnite-dimensional

groups but requires some care in the case of inﬁnite-dimensional Kac-Moody groups. We

As we do not know whether T

−1

is completely reducible, it is possible that η

is only well-deﬁned and

non-degenerate on some (maximal) completely reducible submodule. For simplicity, we shall only refer to

−1

as this (maximal) completely reducible submodule.

As shown in [71], there are other D = 11 signatures embedded in K(E

) that relate to so-called exotic

forms of supergravity [72] that have more than one time-like direction.

– 13 –

JHEP10(2019)165

shall make more comments on this subtlety when we discuss the vielbein and its gauge

transformation in section 3.5.

A key building block for the duality equations is the current deﬁned as

αβ

= M

−1

∂

M ∈ R(Λ

) ⊗ e

. (3.11)

This expression makes sense for the so-called small group in any integrable module [73].

Since M only involves E

/K(E

), the combination M

−1

∂

M can be expanded in the

adjoint of E

and transforms covariantly in the tensor product of R(Λ

) with the adjoint

with, where the former factor is due to the partial derivative.

We use the current (3.11) to deﬁne the non-linear ﬁeld strength F

= C

+ C

˜α

+ C

+ . . . (3.12)

where the structure coeﬃcients are the same as in (3.5), but ∂

has been promoted to

the non-linear current J

, while the partial derivative ∂

˜α

, ∂

, . . . are promoted

to constrained ﬁelds χ

˜α

, ζ

, . . . that are not total derivatives, but satisfy the section

constraint (3.4) on their index M . It is important to stress that the structure coeﬃcients

are deﬁned such that the potential φ

ˆα

is in T

−2

and not in T

, such that one cannot simply

extend J

Mα

to a current in the extended algebra T

. Nonetheless, because e

is simple one

can raise the index of the E

/K(E

) current to get J

. The indecomposable structure

of the module T

−2

implies that the ﬁeld strength must necessarily involve an additional ﬁeld

in R(Λ

) ⊗ R(Λ

), and because J

is not a total derivative, this additional ﬁeld χ

˜α

cannot be a total derivative either. It is nevertheless consistent with the indecomposable

representation to require that it satisﬁes the strong section constraint. We shall therefore

introduce the constrained ﬁelds χ

˜α

, ζ

, . . . so that they transform under E

according

to the indecomposable representation and the ﬁeld strength F

is indeed an E

tensor

in T

−1

. Even though the additional ﬁeld ζ

in R(Λ

) ⊗ R(Λ

) is not required by E

covariance, we shall see that both χ

˜α

and ζ

are necessary to write down a twisted

selfduality equation (3.9) covariant under generalised diﬀeomorphisms. In the current paper

we assume implicitly that we can consistently truncate to this known part of T

−2

but in

principle an extension to additional modules in T

−2

can be envisaged as indicated by the

ellipsis, which will be dropped for short in the following.

We now describe in more detail why (3.12) deﬁnes an E

covariant object due to the

indecomposable structure of the module T

−2

. Under rigid E

transformations one has as

in (2.4)

→ g

−1N

, (3.13a)

˜α

→ g

−1N



˜α

+ ω

˜α

(g)J



. (3.13b)

Recalling (2.9), it follows that

+ C

˜α

→ g



+ C

˜α



(3.14)

The ﬁeld ζ

associated with R(Λ

) was not considered in the linearised analysis in [47]. Including

it here has the beneﬁt of making the ﬁrst order duality equation gauge invariant while [47] only had gauge

invariant second order equations.

– 14 –

JHEP10(2019)165

transforms covariantly. The cocycle appearing in the indecomposable representation is

crucial in this calculation. The inﬁnitesimal transformations under g = exp(Λ

) corre-

sponding to (3.13) are

= Λ



−f

γα

− T

γN



, (3.15a)

˜α

= Λ



γ ˜α

− T

γN

˜α



+ Λ

γ ˜α

. (3.15b)

3.4 Gauge invariance of the non-linear ﬁeld equations

Having introduced an E

covariant tensor ﬁeld strength F

, the next step is to compute

how it transforms under E

generalised diﬀeomorphism. The gauge parameter ξ

of the

generalised Lie derivative transforms also in R(Λ

) just as the coordinates z

. Note that

additional gauge transformations with constrained gauge parameters are required for the

closure of the algebra of generalised Lie derivatives for e

with n ≥ 8 [4, 68, 69, 74]. We

shall not check the closure of the algebra of generalised Lie derivatives for e

, but we will

comment on these additional transformations in section 3.6.

The dynamical degrees of freedom of the theory appear through the representative M.

Therefore we start by deﬁning the generalised Lie derivative with parameter ξ acting on

M. The formula, as for all exceptional ﬁeld theories, can be deﬁned as (see e.g. [34])

M = ξ

∂

M + κ

αβ

αM

∂



+ t

β†



. (3.16)

This formula reproduces the unique linearised gauge transformation studied in [37, 47] in

the linearised approximation and provides a non-linear extension of it.

Combining the deﬁnition of J

in (3.11) and the transformation (3.16) it follows that

= ξ

∂

− T

βN

∂

βγ

+ ∂

+ T

αN



∂

+ M

P R

∂



, (3.17)

where the third term in the ﬁrst line originates from the derivative in the current acting

on the generalised diﬀeomorphism parameter in the variation of M. Using the section

constraint (3.4) on this term, one can recognise the ﬁrst line as the expected generalised Lie

derivative, including the transport term, an inﬁnitesimal e

transformation plus a weight

term, i.e. introducing the notation T

≡ κ

αβ

βM

= ξ

∂

+ T

∂

βα

+ T

∂

βN

∂

+ T

αN



∂

+ M

P R

∂



. (3.18)

The inhomogeneous terms in the second line are non-covariant variations that resemble the

linearised gauge transformation of the linearised current. At this level the variation of the

current is identical in structure to what one would obtain for any E

exceptional ﬁeld theory.

In order to obtain a consistent transformation of the ﬁeld strength F

, it is necessary

that all the components in T

−2

transform according to the indecomposable representation

of E

. To this end, it is useful to introduce the notation J

bα

≡ (J

, χ

˜α

, ζ

, . . .)

– 15 –

JHEP10(2019)165

that includes the additional constrained ﬁelds in a single object transforming as an element

bα

ˆα

of T

−2

. In this way one can deﬁne the following ansatz for the gauge transformation

of J

bα

that is manifestly consistent with the indecomposable representation

bα

= ξ

∂

bα

+ T

∂

β bα

bγ

+ T

∂

βN

bα

∂

bα

+ T

bαN



∂

+ M

P R

∂



+ Π

bα

∂

. (3.19)

This formula extends (3.18) to all components bα with structure coeﬃcients invariant under

for the indecomposable module, and we have added one extra term at the very end of

the equation with the understanding that Π

= 0, while Π

˜α

and Π

are the highest

weight projectors from R(Λ

) ⊗R(Λ

) to R(Λ

) and R(Λ

), respectively. These exist due

to (2.1). We again restrict only to R(Λ

) and R(Λ

); additional E

representations in

−2

could be accommodated in the same way, as long as they would appear in the tensor

product R(Λ

) ⊗ R(Λ

Assuming the uniform gauge transformation (3.19) we can read oﬀ the gauge transfor-

mations of the constrained ﬁelds to be

˜α

= ξ

∂

˜α

− T

∂

α˜α

+ ∂

˜α

+ T

∂

α˜α

+ T

˜αN



∂

+ M

P R

∂



+ Π

˜α

∂

, (3.20a)

= ξ

∂

− T

∂

αΛ

+ ∂

+ T

ΛN



∂

+ M

P R

∂



+ Π

∂

. (3.20b)

As explained above, this form is by construction in agreement with the indecomposable

structure of the module: the gauge transformation of χ

˜α

must include the same gauge

transformation as J

with the index ˜α instead, so that the gauge transformation of J

bα

is written in terms of invariant tensors. The indecomposable structure is such that one

has the freedom to add any transformation of χ

˜α

in the R(Λ

) module, which gives the

freedom to add the term involving the projector Π

˜α

. It will turn out that this additional

transformation is indeed necessary for the duality equations to be gauge invariant under

generalised diﬀeomorphisms.

Equipped with the formulas above, we can now compute the variation of the non-linear

ﬁeld strength (3.12) under gauge transformations and ﬁnd

= ξ

∂

− T

∂

αI

∂

+ C

bα



bαN



∂

+ M

P R

∂



+ Π

bα

∂



= ξ

∂

− T

∂

αI

∂





αR

+ C

˜α

˜αR

+ C

ΛR



+ C

˜α

+ C



∂

. (3.21)

Here, we have recombined the terms into the generalised Lie derivative of F

in the ﬁrst

lines of the two equations. The transport term and the e

transformation term recom-

bine by invariance of the structure coeﬃcients C

bα

. The last term of the ﬁrst lines

– 16 –

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