异常Sigma模型：U-对偶与弦论的10维构造

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"本文详细探讨了在异常场论框架下的‘扩展时空’中的杰出弦模型（Exceptional Sigma Model），这是一种与U-duality相对应的双倍弦模型，类似于T-duality的关系。该模型的核心是利用对称性来确定Weyl不变的拉格朗日量，这是构建弦理论的基础。作者首先概述了模型的构造过程，特别强调了如何通过理论上的独特对称性来保证其有效性，并且展示了这一模型是如何精确地转化为10维弦论的正确拉格朗日形式。在深入研究中，作者将Fradkin-Tseytlin（或者说是广义dilaton）耦合纳入模型，这是弦理论中一个重要的概念，它涉及到额外维度中的能量流动和弦的尺度变化。这种耦合反映了弦在不同能量状态下的行为，对于理解弦理论的物理性质至关重要。此外，文章还讨论了动态张力的引入，这意味着在模型中，弦的张力不再是固定不变的，而是作为一个动态变量参与进来。这种动态张力的概念进一步丰富了弦模型的复杂性和物理内涵，使得模型能够展现出更为丰富的物理现象和可能的理论预测。值得注意的是，这篇论文发表在《Journal of High Energy Physics》（JHEP）的2018年4月期上，是开放获取的，并得到了SCOAP3项目的资金支持。引用信息中提到的ArXiv预印本编号为1802.00442，表明该研究在正式发表之前已经进行了同行评审并得到了认可。总结来说，本文是一篇深入研究异常弦模型的重要学术贡献，不仅展示了理论模型的构造和验证，还探讨了关键的耦合项和弦模型的动态特性，为M理论和弦双极性提供了新的洞见。"

JHEP04(2018)064

The strategy for the construction of the exceptional sigma model will be to begi n with the

EFT generalisations of the transformations (

2.13), (2.17), (2.18) for the background ﬁelds,

and check what sigma model action is compati bl e with these. There we will also check the

covariance requirement under external diﬀeomorphisms, which allow us to completely ﬁx

all relati ve coeﬃcients in the action (our approach will b e quite general and so also applies

to the doubled sigma model, hence one can use the results of section 4 to conﬁrm the

covariance of the latter under external diﬀeomorphisms, which we have not discussed here).

2.3 Tensor hierarchy and gener al is ation s

We now ﬁnish our review of the doubled sigma model by pointing out the ingredients

that generalise naturally (if surprisingly) to the excepti onal c ase . Let us focus on the

Wess-Zumino term

W Z

= −

F 1

σǫ

αβ



µν

∂

+ η

∂

+ η

∂



(2.24)

The leading term involves the DFT generalised two-form B

µν

, which is what we expe c t

a string to c oupl e to. Both B

µν

and the generalised one-form A

transform under

generalised diﬀeomorphisms and gauge tran sfor mat i ons, while B

µν

appears in the ﬁeld

strength (

2.19) for A

. The pair (A

, B

µν

) are the “tensor hierarchy” of O(d , d) DFT.

Similar tensor hierarchies appear in EFT, and reﬂect t he fact that the O(d, d) or E

D(D)

co-

variant ﬁelds incorporate components from the same supergravity ﬁelds, hence their gauge

transformations, ﬁeld strengths and Bianchi identities are linked in a systematic fashion.

Part of the systematisation is that we can always associated the generalised p-form

ﬁelds to a sequenc e of repres entations R

of O(d, d) or E

D(D)

. The DFT one-for m is in the

representation R

= 2d of O(d, d) while the DFT two-form is in the representation R

= 1.

Given objects A

, A

∈ R

, there is a map • : R

⊗ R

→ R

, which for DFT is given by

• A

= η

. The above Wess-Zumino ter m clearly involves:

µν

∂

+ ∂

• D

Y + ∂

Y • V

. (2.25)

A natural conjecture would be that the same formula should hold for the groups and

representations of EF T. In general though, R

will not be the trivial representation. In

order to obtain a quantity that can be integrated, we will need to introduce a charge q ∈

and deﬁne

W Z

= −

σǫ

αβ

q · (B

µν

∂

+ ∂

• D

Y + ∂

Y • V

) , (2.26)

where q · B ∈ 1. This charge will encode the tension of the string action. Clearly for the

doubled sigma model, we just have q = T

F 1

Remarkably, the guess (

2.26) turns out to be correct, as long as the charge q obeys a

constraint (

1.4). This constraint comes about when one checks the gauge invariance of the

Wess-Zumino term.

– 8 –

JHEP04(2018)064

n = 10 −d d G = E

D,D

, D = d + 1 H R

9 1 SL(2) ×R

SO(2) 2

⊕ 1

−1

8 2 SL(3) × SL(2) SO(3) ×SO(2) (3, 2) (

3, 1)

7 3 SL(5) SO(5) 10 5

6 4 SO(5, 5) SO(5) ×SO(5) 16 10

5 5 E

USp(8) 27 27

4 6 E

SU(8) 56 ∗

3 7 E

SO(16) 248 ∗

Table 1. Generalised di ﬀe omorp his m groups and tensor hierarchy representations for EFT. n is

the e xt er n al dimension while d is the internal dimension from the point of view of type II sections.

3 Exceptional ﬁeld theory

In this section, we will introduce the cor e elements of exceptional ﬁeld theory, focusing

on the ﬁelds to which the exceptional sigma model coup l es , and their symmetries. After

presenting the general details, we will focus on the gr oup E

, for which some extra details

have to be ﬁlled in.

3.1 Field content and symmetries

Exceptional ﬁeld theory is a reformulation of supergravity with a formal manifest E

D(D)

symmetry, realised on an extended set of coordinates (X

, Y

). The coordinates Y

lie

in a representation of E

D(D)

denoted by R

. The ﬁelds of the theory are also assigned to

various E

D(D)

representations, and include the following: an external metric, g

µν

, in the

trivial representation, a generalised metric, M

, in the coset E

D(D)

where H

the maximal c ompac t subgroup of E

D(D)

, and a tensor hierarchy of generalised form ﬁelds,

∈ R

, B

µν

∈ R

, C

µνρ

∈ R

, . . . .

The exceptional string will c oupl e to the generalised two-form B

µν

, as well as to the ex-

ternal and generalised metric, and the generalised one-form. The list of the representations

and R

for the groups E

D(D)

, D = 2, . . . 8 is displayed in table

Note that in p r act i ce , not al l the ﬁelds appearing in the tensor hierarchy are needed

in formulating the dynamics of EFT, as in many case s they will contain dualisations of

the physical degrees of freedom. Additionally, we may also have to includ e addi ti onal

“constrained compensator ﬁelds” which also drop out of the dynamics, but are important

in formulating the complete gauge invariant set of ﬁeld strengths. Such compensator ﬁelds

appear when R

, and are re l ate d to th e appearance of components of the dual

graviton. These extra ﬁelds are necessary to construct the ﬁeld strength of the of t he

generalised one-form in E

[

14], and are needed already at the level of the generalised Lie

derivative for E

[

15]. In this pap er , they will become relevant when consider i ng the E

exceptional sigma model, appearing in the ﬁeld strength for the generalised two-form to

which the exceptional string couples.

– 9 –

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异常Sigma模型：U-对偶与弦论的10维构造

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