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我们将解释[1]中提出的非阿贝尔二形式自对偶方程的自对偶字符串解的新构造。 此类自对偶弦由四维BPS单极子确定,而自对偶弦电荷由单极子的电荷给出。 我们的构造涵盖了先前发现的SO(4)不变自对偶字符串解决方案。 我们还基于't Hooft-Polyakov单极子构造了一个奇异解,该解描述了两个有限分开的M5大脑在中间相遇。 我们评论说,由于BPS单极子通常由Nahm构造给出,因此我们的构造表明,对于非阿贝尔自对偶弦,可能存在广义的Nahm变换。
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Available online at www.sciencedirect.com
ScienceDirect
Nuclear Physics B 882 (2014) 289–302
www.elsevier.com/locate/nuclphysb
Non-abelian self-dual strings in six dimensions
from four dimensional 1/2-BPS monopoles
Chong-Sun Chu
a,b
a
Department of Physics and National Center for Theoretical Sciences, National Tsing Hua University,
Hsinchu 30013, Taiwan
b
Centre for Particle Theory and Department of Mathematics, Durham University, Durham, DH1 3LE, UK
Received 18 November 2013; received in revised form 5 March 2014; accepted 7 March 2014
Available online 15 March 2014
Editor: Stephan Stieberger
Abstract
We explain a new construction of self-dual string solutions to the non-abelian two-form self-duality equa-
tion proposed in [1]. This class
of self-dual strings is determined by the BPS monopoles in four-dimensions
and the self-dual string charge is given by the charge of the monopole. Our construction covers the SO(4)
invariant self-dual string solutions found previously. We have also constructed, based on the ’t Hooft–
Polyakov monopole, a singular solution that describes two finitely separated M5-branes meeting midway
in between. We comment that as BPS monopoles are generally given by the Nahm construction, our con-
struction suggests that a generalized Nahm transform may exist for the non-abelian self-dual strings.
© 2014 Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
Keywords: M-Theory; D-branes; M-branes; Gauge symmetry
1. Introduction
The theory of N coincident M5-branes in a flat spacetime is given by an interacting (2, 0)
superconformal theory in six dimensions [2]. The understanding of the dynamics of this system
is of utmost importance. It will not only improve our understanding of the AdS/CFT correspon-
dence for the AdS
7
× S
4
background [3]; in addition, as the problem involves a mathematical
formulation of a self-duality equation for a non-abelian 3-form gauge field strength, one may
http://dx.doi.org/10.1016/j.nuclphysb.2014.03.006
0550-3213/© 2014 Published by Elsev
ier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP
3
.
290 C.-S. Chu / Nuclear Physics B 882 (2014) 289–302
suspect that it may have an impact on mathematical physics in a way similar to it’s lower dimen-
sional cousin, the self-dual Yang–Mills equation [4].
On general grounds, the theory of multiple M5-branes does not have a free dimensionless
parameter
and is inherently non-perturbative.
It does not mean that an action does not exist,
though it does mean that the action will be of limited use, probably no more than giving the
corresponding equation of motion. This is still very interesting since one can expect that non-
trivial spacetime physics of M-theory could be learned from the physics of the solitonic objects
of the worldvolume theory of M5-branes, much like the cases of M2-branes and D-branes. See
for example, [5].
In a recent paper [1],
a consistent self-duality equation of motion
for a non-abelian tensor
gauge field in six dimensions has been constructed and proposed to be the low energy equation of
motion of the self-dual tensor field living on the worldvolume of a system of multiple M5-branes.
The self-dual equation of motion proposed in [1] is meant to be an effective description for the
M5-branes, just like the supergravity equation of motion provides an effective description for the
M-theory. Recently it has been conjectured [6,7] that the 5d supersymmetric Yang–Mills theory
can be used to provide a fundamental definition of the (2, 0) theory. This conjecture has been
checked quite recently in [8] and it was shown that the 5d supersymmetric Yang–Mills theory is
divergent at six loops and hence extra degrees of freedom is needed to provide a UV completion.
Having a workable fundamental definition for the M5-branes system is highly desirable but very
difficult, see also [9–11] for other proposals. We will be restricting ourselves with the effective
description and hope something useful can be learned.
The non-abelian self-duality equation constructed in [1] generalizes
the equation of motion
for a single
M5-brane of [12–15]. It was constructed in the gauge B
5μ
= 0(μ = 0,...,4) and
is a non-abelian generalization of the Henneaux–Teitelboim–Perry–Schwarz construction for the
U(1) case [13,16]. The construction of [1] involves the introduction of a set of non-propagating
non-abelian 1-form gauge fields which was motivated originally by the boundary analysis in
[17] and further analyzed [18]. This aspect is very similar to the BLG [19] and ABJM model
[20] of multiple M2-branes where a set of non-propagating Chern–Simons gauge fields was
introduced in order to allow for a simple representation of the highly non-linear and non-local
self-interactions of the matter fields of the theory.
The proposed self-duality equation reads
˜
H
μν
=∂
5
B
μν
, (1)
where the gauge field A
μ
is constrained to be given by
F
μν
=c
dx
5
˜
H
μν
. (2)
Here
H
μνρ
=D
[μ
B
νρ]
=∂
[μ
B
νρ]
+[A
[μ
,B
νρ]
], (3)
˜
H
μν
=
1
6
μνρσ τ
H
ρστ
,
01234
=−1, (4)
F
μν
=∂
μ
A
ν
−∂
ν
A
μ
+[A
μ
,A
ν
]. (5)
All fields are in the adjoint representation of the Lie algebra of the gauge group G, and c is
a free parameter. Our convention for the Lie algebra are: [T
a
,T
b
]=if
abc
T
c
, F
μν
= iF
a
μν
T
a
,
A
μ
=iA
a
μ
T
a
and F
a
μν
=∂
μ
A
a
ν
−∂
ν
A
a
μ
−f
abc
A
b
μ
A
c
ν
.
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
4
-direction. As a result, physical properties of
the system are independent of x
0
and x
4
.
Let us consider an ansatz with the following non-vanishing components of the B-field:
B
ij
,B
04
:=−φ. (6)
The non-vanishing components of H are
H
ij k
=D
[i
B
jk]
,H
5ij
=∂
5
B
ij
, (7)
H
04i
=−D
i
φ, H
045
=−∂
5
φ, (8)
and the self-duality equation (1) reads
∂
5
φ =−
1
2
ij k
D
i
B
jk
, (9)
D
k
φ =
1
2
kij
∂
5
B
ij
. (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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