4N维时空中的对偶不变阿贝尔p型场非线性相互作用解析

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本文探讨了在任意维度D=4N的时空中，对于2N形式的麦克斯韦场的非线性相互作用。作者采用Pasti-Sorokin-Tonin (PST) 方法，这是一种在保持对偶不变性和洛伦兹不变性的前提下研究动力学的一致性条件的有效工具。该方法的核心是寻找能够同时体现SO(2)对称性和动力学一致性条件的理论框架。通过PST方法，作者得到了一种动态的一般一致性条件，这一条件确保了理论在高维空间中的对称性和有效性。他们特别关注的是如何将已知的非线性对偶不变麦克斯韦理论扩展到更高维度D=4N，这涉及到经典解的精确分类。这些解不仅代表了一种理论的推广，还提供了一个基础，用于理解对称性在不同维度下的行为。在D=8这样的特定维度，作者不仅解决了PST的一致性条件，而且还发现了新的非规范的明显对偶不变四次相互作用。这种发现进一步丰富了对偶不变理论的结构，因为它展示了对偶性不仅仅是运动方程的内在特性，还可以通过新的非平凡相互作用形式表现出来。文中还讨论了在有电荷的强子对p布雷源存在的背景下，这两种方法（PST方法和GZGR方法）的物理不等式，这揭示了它们在实际物理情境中的应用和区别。PST方法的优势在于其通用性，使得构建非线性对偶不变麦克斯韦理论的过程简化为纯粹的代数问题，从而降低了理论构造的复杂性。这篇研究论文不仅深化了对高维对称性理论的理解，还提出了新的理论模型，这对于探索多维度物理学，尤其是在量子引力和弦理论等领域具有重要意义。此外，通过比较不同的方法和理论框架，它还展示了理论物理中的对称性和相互作用之间的深刻联系。这项工作对于未来的研究者来说，提供了宝贵的参考和新的研究方向。

JHEP09(2019)022

, so that (2.7) gives F

= cos ϕ F

−sin ϕ

. The right hand side of equation (2.10)

for the transformed Lagrangian can then be computed to be

) =

ϕM

, meaning

that the SO(2) map leaves the free Lagrangian unaltered,

( · ) =

L( · ), as expected.

Introducing the ﬁctitious Lagrangian

G(F ) = L(F ) −

∂L

∂F

, (2.12)

the map (2.10), which deﬁnes the transformed Lagrangian, can be recast in the alterna-

tive form

) = G(F ). (2.13)

Notice that, despite the formal similarity, the Lagrangian G(F ) (2.12) diﬀers from the

Legendre transform of L(F ) by a factor of 1/2. For the free Lagrangian (2.3) one has

G(F ) = 0.

2.2 The GZGR condition

Eventually, we are interested in Lagrangians that remain unaltered under a generic SO(2)

transformation (2.10). We say that a Maxwell theory based on a Lagrangian L such that

( · ) = L( · ), for all ϕ, (2.14)

is duality invariant. Recalling equation (2.10), this means that under a duality rotation

the Lagrangian L(F ) itself is not invariant, but must rather obey the transformation law,

ﬁrst given for D = 4 in [45],

L(F

) = L(F ) − sin

ϕ F L +

sin ϕ cos ϕ



F + L



. (2.15)

Using the equivalent deﬁnition of L

( · ) (2.13), this law can also be expressed as the

condition that the Lagrangian (2.12) is invariant under the duality transformation of the

ﬁeld (2.7)

G(F

) = G(F ). (2.16)

Given the Lie-group structure of the transformation (2.6), it is suﬃcient to satisfy equa-

tion (2.15), or (2.16), for inﬁnitesimal transformations. From the transformation law of

the ﬁeld (2.7) we obtain L(F

) = L(F ) + ϕ(L

L) + O(ϕ

), so that (2.15), at ﬁrst order in

ϕ, amounts to the GZGR condition

F − L

L = 0. (2.17)

In summary, in the GZGR formulation, a Maxwell theory is duality invariant if and only

if the Lagrangian satisﬁes, equivalently, the conditions (2.15), (2.16), or (2.17).

As in the GZGR formulation SO(2) is not realized as a Noether symmetry, there is no

“canonical” local conserved current associated with it. However, if we admit a mild non-

locality, then a conserved current can be constructed. In fact, the equation of motion for

···µ

n−1

in (2.5) can be “solved” in terms of the S-dual potential A

d µ

···µ

n−1

, see below,

by setting

...µ

= n ∂

[µ

d µ

···µ

]

. (2.18)

– 7 –

JHEP09(2019)022

A conserved current is then given by

= (

A) − (L

), (2.19)

where we adopted the notation (2.11). Using Maxwell’s equations (2.5), and the GZGR

condition (2.17), one veriﬁes that ∂



F F − L



= 0.

2.3 S-duality

Although equations (2.15) and (2.17) are equivalent formulations of the same physical

condition on the dynamics of the electromagnetic ﬁeld, the former entails some properties

which appear rather hidden in the latter. Choosing the angle ϕ = π/2, and denoting the

dual ﬁeld by F

π/2

≡ F

, from formula (2.7) and the invariance condition (2.15) we obtain

L(F

) = L(F ) − F

∂L

∂F

, F

∂L(F )

∂F

. (2.20)

These relations, in D = 4, are sometimes interpreted by saying that in a duality invariant

theory the Lagrangian is invariant under a Legendre transform. However, this is not the

case, since the dual ﬁeld F

as a function of F

is not the derivative ∂L/∂F

, but rather

its Hodge dual. It is, in fact, a known result that the only Lagrangians that are ﬁxed

points of the Legendre transform are the free ones L(q) =

, see for instance [46].

The

actual content of the relations (2.20) is, rather, the S-duality invariance of the theory, as

a consequence of its SO(2) duality invariance. To perform the S-duality transformation of

the action I[F ] =

L(F ) d

x we start from the equivalent functional of F and A

I[F, A

] =



L(F ) + n

···µ

∂

d µ

···µ



x, (2.21)

where A

is the S-dual potential, and F

is now considered as an independent ﬁeld.

Varying I[F, A

] with respect to A

, one obtains the constraint ∂

[µ

···µ

n+1

]

= 0 ↔

···µ

= n ∂

[µ

···µ

]

, leading back to the original action I[F]. Varying I[F, A

] instead

with respect to F , one obtains

∂L

∂F

= n (∂A

)

≡ F

. Inverting this relation to compute

= F

), and using this expression to eliminate from the action (2.21) F in favor of

, one obtains the action for the S-dual potential

I[A

] =



L(F (F

)) − F

∂L

∂F



x =

L(F

) d

x, (2.22)

where in the last step we have used (2.20). The Lagrangian for the S-dual ﬁeld strength

has, thus, the same functional form as the original Lagrangian. Finally, evaluating the

transformation law (2.8) at ϕ = π/2 we obtain L

π/2

, i.e. (see (2.9))

= −

∂L(F

)

∂F

. (2.23)

Whether the invariance condition under the pseudo-Legendre transform (2.20), implied by the GZGR

condition (2.17), is equivalent to the latter is, to our knowledge, not known and deserves further investiga-

tion.

– 8 –

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