协变非线性对偶作用下的kappa-symmetric D3膜在双曲对称下的构造与验证

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本文主要探讨了κ对称D3-膜在双曲几何框架下的协变非线性对偶对称作用的构造与对偶表示。研究的核心内容围绕着Zwanziger作用的PST（Proca-Sigma-Tonin）协变化和非线性化展开，这是一种关键的理论工具，它在量子场论中有着重要的应用，尤其是在弦理论和膜理论中。作者Pichet Vanichchapongjaroen从The Tah Poe Academia Institute, Naresuan University出发，引入了三个辅助标量场，通过巧妙的协变化过程将非线性对称性融入到理论模型中。这种构造方法与标准公式构建方式相似，使得理论不仅保持了局部PST对称性，还能与外部场相互作用，从而增加了理论的适用性和复杂性。文章重点展示了如何将这种对偶对称性应用于著名的DBI（Dirac-Born-Infeld）理论和D3-brane模型。在这个过程中，作者详细展示了如何从对偶对称性角度出发，通过扭曲自对偶条件，揭示了场强与单势行为之间的对偶关系，以及对偶对称形式与单势形式之间的关系。这一成果有助于深化我们对弦理论中膜行为的理解，特别是在动力学和对称性的相互作用方面。此外，文章的核心部分还包括了对协变非线性对偶对称D3膜作用的κ对称性的验证。κ对称性是D-brane理论中的一个重要特性，它确保了理论的物理量在空间维度上的正确统计性质。作者在这里证明了在双曲对偶框架下，这种对称性得到了保持，进一步增强了该理论的完备性和一致性。总结来说，这篇论文不仅提供了对协变非线性对偶对称作用在D3-膜理论中的具体实现，还展示了如何在双曲对偶框架下进行有效构造和应用。这对于理解高维量子场论和弦理论中的对称性、相互作用及其在实际物理现象中的应用具有重要意义。

JHEP02(2018)116

where g

µν

and g are, respectively, inverse an d determinant of the 4D spacetime metric g

µν

This gives the identity •• = −1. With these deﬁnitions, we may write eq. (

2.2) as

(•F

¯r

)

ρσ

= −

√

−g

δS

δF

ρσ

. (2.5)

Let us call this equation as ‘duality relation’.

Next, the gauge ﬁelds A

and their duals A

¯r

can be c ol l ec t ed into

A ≡ (A

) ≡ (A

, A

¯r

), i = 1, 2, ··· , 2N. (2.6)

Likewise,

F ≡ (F

) ≡ (F

, F

¯r

). (2.7)

Note that an arrow on the top of a symbol is introduced to indicate that it is a 2N −tuple.

Denote ~u

as a 2N−tuple whose ith entry equals to 1 but the rest are 0. This allows us

to express the 2N−tuples of gauge ﬁelds and ﬁeld strengths respe ct i vely as

A = A

F =

. For deﬁniteness, let us denote the vector space of 2N −tuples as V. The vector space

V is ac t ed linearly by the elements of the duali ty group G ⊂ Sp(2N, R). The couple to

gravity, scalars and fermi ons can be introduced into the theory.

In particular, the scalars φ parametrise a coset space G/H of a principal H−bundle

G → G/H, where H i s the maximal compact subgroup of G. The vielbeins on G/H allows

the introduction of scalar-dependent invertible metric M

(φ), through a symmetric non-

degenerate bilinear map M on V such that

M(~u

, ~u

) = M (~u

, ~u

), (2.8)

or M

= M

. Since G is a subgroup of Sp(2N, R), one can deﬁne an antisymmetric non-

degenerate bilinear map Ω on V such that

Ω(~u

, ~u

) = −Ω(~u

, ~u

), (2.9)

or simply Ω

= −Ω

, and that Ω

= Ω

¯r¯s

= 0, Ω

r¯r

= δ

r¯r

. One also deﬁnes Ω

= −Ω

such that Ω

Ω

= −δ

. This allows one to deﬁne a complex structure, which is a linear

map J : V → V deﬁned via

Ω(~v, J ~w) = −M (~v, ~w), M (J~v, J ~w) = M(~v, ~w), Ω(J~v, J ~w) = Ω(~v, ~w), (2.10)

for all ~v, ~w ∈ V.

As was the case of duality-symmetric theory in standard formulation studied in [11],

the study of duality-symmetric theory in dual formulation also beneﬁts f rom the use of

diﬀerential form language. As a direct extension to the 2N−tuple of gauge ﬁelds and ﬁeld

strength, one can consider the 2N −tuple of p−forms. In other words, for A

(p)

, a p−form

with internal index i, the associated 2N −tuple is gi ven by

(p)

= A

(p)

. (2.11)

– 4 –

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