非关联微分几何与非几何重力：扭结与R-磁通量修正

108 浏览量更新于2024-07-16 收藏 1.02MB PDF 举报

"Nonassociative Differential Geometry and Gravity with Non-Geometric Fluxes" 这篇研究论文深入探讨了非关联微分几何及其在非几何闭合弦真空背景下的应用，特别是涉及引力理论的方面。作者们引入了一种抛物线相空间模型，该模型专门设计用于处理恒定局部非几何通量的问题。非关联性在这里指的是代数结构不再遵循传统的乘法规则，这在传统几何中是基础的，但在非几何设置中变得复杂。首先，他们发展了非缔合黎曼几何的度量理论，这是对经典黎曼几何的扩展，其中几何性质不再仅由光滑流形的局部坐标描述，而是由更复杂的结构决定。他们得到了扭转、曲率、Ricci张量以及Levi-Civita连接的显式表达式，这些都是理解时空几何的重要工具。Ricci张量作为度量和连接的函数，通常用于衡量时空的曲率，而Levi-Civita连接则确保了平行移动过程中向量的长度和方向不变。接下来，作者们利用这种形式主义构建了非缔合重力理论的初步框架。他们提出了一个针对时空的Ricci张量的修正项，即R-磁通量校正，这是对标准爱因斯坦场方程的延伸。爱因斯坦场方程是广义相对论的核心，描述了物质能量分布如何影响时空的几何。R-磁通量校正值的引入可能会影响这个关系，为理解和模拟非几何环境下的引力效应提供新的视角。此外，论文还讨论了这些结构在非几何弦理论和双场理论中的潜在意义。非几何弦理论是对弦理论的一种扩展，其中弦的动力学不再与某个背景几何密切关联，而双场理论则涉及到两个相互作用的标量场，可能对应于不同的宇宙学或凝聚态物理现象。这些理论在理解量子引力和早期宇宙的某些方面可能非常有用。这项工作提供了对非几何环境中的引力理论和微分几何的新洞察，对于深化对弦理论和相关物理现象的理解具有重要意义。通过引入非关联性，研究者开辟了探索传统理论无法触及的物理领域的道路。

JHEP02(2018)036

2.3 Double ﬁeld theory formulation

Before deforming the geometry of M into a nonassociative diﬀere ntial ge ome tr y with the

cochain twist F and the associated quasi-Hopf algebra U Vec

(M), let us desc r i be the

general extent to which our results will be applicable, particularly from the perspective of

double ﬁeld theory, as they wi l l mostly be suppressed in the following in order to streamline

our presentation and formulas.

Firstly, if globally non-geometric Q-ﬂux is also present [

1, 17, 19, 25], then it has the

eﬀect of modifying the twist element (

2.3) to

F = F F

, (2.26)

where

= exp



−

i κ

µν

∂

⊗ ∂

− ∂

⊗ w

∂

)



(2.27)

with w

closed string winding coordinates which may be regarded as mom enta ˆp

conjugate

to coordinates ˆx

that are T-dual to the spacetime variables x

. The twist F

is an

abelian 2-cocycle, and the vector ﬁelds w

∂

commute wi th the other vector ﬁel ds ∂

and p

∂

generating the twists F and F

, so the Hopf coboundary of (

2.26) is sti l l the

associator (

2.10); indeed, unlike the R-ﬂux, the Q-ﬂux only sources noncommutativity. In

this setting the Q-ﬂux and R-ﬂux are independent deformation parameter s, while coexistent

constant Q-ﬂux and R-ﬂux in string t he or y are constrai ned in the absence of geometric

ﬂuxes by the Bianchi identities

µ[νρ

λτ ]

= 0 and Q

[νρ

λ]µ

= 0 (2.28)

which are easily imposed as additional constraints on the twist element (

2.26).

In fact, analogously to [

8] one can exte nd the twist element (2.1) to the full phase

space M ×

M of double ﬁeld theory in the R-ﬂux frame as

F = F

F , (2.29)

where

F = exp



−

i ~

(

∂

⊗

∂

−

∂

⊗

∂

)



(2.30)

implements the deformation of the canonic al T-dual phase space

M with coordinates

ˆx

, ˆp

= w

. The cochain twist (

2.29) is O(2d, 2d)-invari ant so that it can be rotated

to any other T-duality frame by using an O(2d, 2d) transformation on M ×

M, and in

this way one can write down a nonassociative theory which is manifestly invariant un-

der O(2d, 2d) rotations. In particular, by restricting to rotations which also preserve the

canonical symplectic 2-form of double phase space, we obtain a formulation that is invariant

under a subgroup

O(d, d) ⊆ O(2d, 2d) ∩ Sp(2d) . (2.31)

– 8 –

JHEP02(2018)036

As the inclusion of

F aﬀects neit he r the commutation nor the association relations on the

original phase space M, while yielding the standard Moyal-Weyl deformation of the T-dual

phase space

M, we will regard the phase space

M as implicitly hidden in the background

in all of our subsequent treatments, with the understanding that all of our formalism can be

rotated to any T-duality frame by including a depe nde nc e on the T-dual coordi nate s of

and suitably inserting

F in formulas. In this way we obtain a m ani fe st l y O(d, d)-invar i ant

formulation of the gravity theory which follows.

From this per s pective, there are als o natural modiﬁcations of our formalism, analogous

to those of Moyal-Weyl spaces [3], which ﬁt nicely into the ﬂux formulation of double ﬁeld

theory appropriate to curved backgrounds [

17]. The vector ﬁelds

= ∂

∂

and X

µν

= p

∂

− p

∂

(2.32)

on M, deﬁning the twist deformation of ﬂat space M = R

, represent a ni l potent subalgebra

k of the Lie algebra iso(2d) with the nonvanishing Lie brackets



, X

νρ



= δ

− δ

. (2.33)

For any collection of vector ﬁ el d s {X

, X

µν

, µ, ν = 1, . . . , d} satisfying these Lie

bracket relations on an arbitrary manifold M, the cochain twist element

= exp



−

i ~

⊗

−

⊗ X

) −

i κ

µνρ

νρ

⊗ X

− X

⊗ X

νρ

)



(2.34)

of the Hopf algebra Uiso(2d) ⊂ UVec( M ) provides a nonassociative deformation of M, all

of whose features ﬁt into the framework we develop in the following.

For e x amp l e, in the cases considered in the pre se nt paper we will see that there is

a preferred basis ∂

, dx

of vector ﬁelds and 1-forms on M that is invariant under the

action of the associator and which greatly simpliﬁes calculations. In particular, the Cartan

structure equations expressin g torsion and c ur vature as operators on vector ﬁelds will be

established by checking that the se operators d eﬁn e tensor ﬁe l ds , and by showing that

in the preferred basis they coincide with the tor si on and curvature coeﬃcients. These

simpliﬁcations can be carried out as well for the more gener al twist F

, by considering as

basis the commuting vector ﬁelds X

and their dual 1-forms; more generally, if the

vector ﬁelds X

do not form a basis (e.g. they become degenerate), this can be achieved

by completing them to a basis that is still invariant under the action of the associator with

the methods described in ([

3], section 4).

3 Nonassociative deformation of tensor calculus

3.1 Principles of twist deformation

The tensor algebra on M is covariant under the action of the universal enveloping algebra

of inﬁnitesimal diﬀeomorphisms UVec(M). We have seen how the R-ﬂux induc es a twist

deformation of the Hopf algebra U Vec(M) into the quasi-Hopf algebra U Vec

(M). We

construct a nonassoc i ati ve diﬀerential geometry on M by requiring it to be covariant with

respect to the quasi-Hopf al geb r a UVec

(M).

– 9 –

JHEP02(2018)036

Every time we have an algebra A that carries a representation of the Hopf alge-

bra U Vec(M), and where vector ﬁelds u ∈ Vec(M) act on A as derivati ons : u(a b) =

u(a) b + a u(b), i.e., every time we have a UVec(M)-module algebra A, then deforming the

multiplication in A into the star-multiplication

a ⋆ b =

(a) f

(b) (3.1)

yields a noncommutative and nonassociative al ge br a A

⋆

that carries a rep r es entation of

the quasi-Hopf algeb ra U Vec

(M), where

ξ(a ⋆ b) = ξ

(1)

(a) ⋆ ξ

(2)

(b) , (3.2)

for all ξ ∈ UVec

(M) and a, b ∈ A; in particular, the vector ﬁelds ∂

and

∂

act as

deformed derivations according t o t he Leibniz rule implied by the coproduct (

2.7). We

say that A

⋆

is a U Vec

(M)-module algebr a because of the compatibility (

3.2) of the

action of UVec

(M) with the product in A

⋆

. For later use we recall t he p r oof of the key

property (

3.2):

ξ(a ⋆ b) = ξ



(a) f

(b)



= ξ

)



(a)



)



(b)





(1)

(a)





(2)

(b)



= ξ

(1)

(a) ⋆ ξ

(2)

(b) , (3.3)

where we used the notation ∆(ξ) = ξ

)

⊗ ξ

)

for the unde for me d coproduct together

with ∆(ξ) F

−1

= F

−1

∆

(ξ).

If the algebra A is commutative then the noncommutati v i ty of A

⋆

is controlled by the

R-matrix as

a ⋆ b = R

(b) ⋆ R

(a) =:

b ⋆

a , (3.4)

where in the last equality we used the notation

a := R

(a) and

a := R

(a); the ex-

pression (

3.4) is easil y proven by recalling that R = F

−2

and that a ⋆ b = f

(a) f

(b) =

(b) f

(a). If the algebra A is associative then the nonassociativi ty of A

⋆

is controlled

by the associator Φ as

(a ⋆ b) ⋆ c =

a ⋆ (

b ⋆

c) , (3.5)

where we denote

a := φ

(a); an explic i t proof can be found in ([5], section 4.2).

In the following we de for m the algebra of functions, the exterior algebra of diﬀerential

forms and the algebra of tensor ﬁelds on M according to this pre sc r i pt i on.

3.2 Functions

The action of a vector ﬁeld u ∈ Vec(M) on a function f ∈ C

∞

(M) is via the Lie derivative

(f) = u(f ), which is indeed a derivation. The action of the Lie algebra Vec(M) on

functions is ex t e nde d to an action of the universal enveloping algebra UVec(M ) by deﬁning

the L i e derivative on products of vector ﬁelds as L

··· u

:= L

◦ L

◦ · · · ◦ L

and

– 10 –

剩余52页未读，继续阅读

weixin_38627826

粉丝: 5
资源: 939

非关联微分几何与非几何重力：扭结与R-磁通量修正

非几何字符串和M理论背景下的非缔合性，八进制的代数和缺少的动量模式

M理论R磁通背景的非几何Kaluza-Klein单极子和磁对偶

G 2-非几何M理论背景的结构和量化

疏水缔合型和非疏水缔合型驱油聚合物的结构与溶液特征 (2013年)

宇宙常数，超对称性，非缔合性和大数

非缔合量子力学中的小磁性电荷和单极子

非缔合Snyder ϕ4理论中的UV-IR混合

行业分类-设备装置-包含酸单体、缔合性单体和非离子性单体的纸张涂布浆液添加剂.zip

疏水缔合聚合物溶液的临界缔合浓度 (2008年)

分子问高缔合比例疏水缔合聚合物研究(Ⅲ)* (2007年)

最新资源