四维1/2-BPS单极子驱动的六维非阿贝尔自对称弦构造

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本文主要探讨了在六维空间中的一种非阿贝尔自对偶弦的新构造，这是基于四维1/2-BPS单极子的理论研究。非阿贝尔自对偶弦是物理学中一个重要的概念，它们满足特定的自对偶方程，这种方程是由[1]中的工作提出。这些自对偶弦的特性与四维BPS单极子紧密相连，单极子的电荷决定了自对偶弦的电荷量。文章首先回顾了之前发现的SO(4)不变的自对偶弦解决方案，这是一类特殊的自对称解，其对称性体现在空间结构上。作者进一步扩展了这一研究，通过't Hooft-Polyakov单极子模型，构建了一个奇异的自对偶弦解。这个解特别地描述了两个处于有限距离分离的M5膜（M5-branes）在空间中相遇的情景，这是一个非平凡且具有物理意义的示例，因为它展示了自对偶弦如何在多模态宇宙中扮演角色。值得注意的是，由于BPS单极子通常可以通过Nahm方程进行构造，这意味着这个新的构造可能揭示了一种更广泛的Nahm变换的概念，不仅限于阿贝尔自旋网络，而是适用于非阿贝尔情境。这种潜在的广义Nahm变换可能是理解自对偶弦及其在量子引力理论中的作用的关键。整个研究过程既包含了理论分析，也涉及了实际数学建模，它在弦理论和高维物理学领域具有重要的理论价值。该成果发表在《核物理学B》杂志的882期，是开放获取资源，读者可以在ScienceDirect和Elsevier的网站上获取全文。该研究不仅深化了我们对自对偶弦的认识，也为未来的理论发展提供了新的探索方向。

C.-S. Chu / Nuclear Physics B 882 (2014) 289–302 291

Evidence that this self-duality equation describes the physics of multiple M5-branes was

provided in [1], and further in [21–23].In[21,22], non-abelian self-dual string solutions were

constructed and a precise agreement [22] of the ﬁeld theory results and the supergravity descrip-

tions [24] was found. Moreover, it was found that the constant c is ﬁxed by quantization condition

of the self-dual strings solution of the theory. This is satisfying as otherwise c would be a free

dimensionless constant in the theory and hence contradicts with what we know about M5-branes

in ﬂat space. In [23], non-abelian wave conﬁgurations which are supported by Yang–Mills in-

stanton were constructed and they were found to match up nicely with the description of M-wave

on the worldvolume of M5-branes system.

One thing interesting about the self-dual string solutions constructed in [21,22] is that the

auxiliary gauge ﬁeld is

always given by a magnetic monopole which gives rise to the charge of

the self-dual string. This was shown to be case for the original Perry–Schwarz self-dual string and

the Wu–Yang self-dual string [21], as well as for the generalized Wu–Yang self-dual string [22],

with the corresponding monopole conﬁgurations given by the Dirac monopole, the Wu–Yang

monopole and the generalized Wu–Yang monopole. It is natural to ask if this connection with

monopole is a general feature of the non-abelian self-dual string. In this paper, we show that this

is indeed the case. This result is potentially interesting as, given this rather explicit connection

between BPS monopole and self-dual string, one may be able to provide a Nahm like construction

for non-abelian self-dual string, which has been speculated and analyzed by other authors [25].

In the next section, we provide a general formalism for the construction of non-abelian self-

dual string starting from an 1/2

-BPS monopole solution in four dimensions. In Section 3,we

show that one can recover the previously constructed self-dual string solutions with this new for-

malism. We also construct, for the ’t Hooft–Polyakov monopole [26,27], a singular solution that

describes two ﬁnitely separated M5-branes meeting midway in between. The paper is concluded

with some further discussions in Section 4.

2. A general construction of self-dual strings in terms of BPS monopoles

In this section, we give a general construction for self-dual strings solutions to the non-abelian

self-duality equations (1), (2). We will be interested in static conﬁgurations with the self-dual

string being inﬁnite long straight line, say, in the x

-direction. As a result, physical properties of

the system are independent of x

and x

Let us consider an ansatz with the following non-vanishing components of the B-ﬁeld:

:=−φ. (6)

The non-vanishing components of H are

ij k

jk]

5ij

=∂

, (7)

04i

=−D

φ, H

045

=−∂

φ, (8)

and the self-duality equation (1) reads

∂

φ =−



ij k

, (9)

φ =



kij

∂

. (10)

We remark that for U(1) gauge group, the self-duality equations (9), (10) are precisely the

same as the BPS equations of Howe–Lambert–West [28] for the (2, 0) M5-branes theory in

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