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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

Department of Physics and National Center for Theoretical Sciences, National Tsing Hua University,

Hsinchu 30013, Taiwan

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.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

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

× S

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

ier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

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

μν

=∂

μν

, (1)

where the gauge ﬁeld A

is constrained to be given by

μν



μν

. (2)

Here

μνρ

[μ

νρ]

=∂

[μ

νρ]

+[A

[μ

νρ]

], (3)

μν



μνρσ τ

ρστ

,

01234

=−1, (4)

μν

=∂

−∂

+[A

]. (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

]=if

abc

, F

μν

= iF

μν

=iA

and F

μν

=∂

−∂

−f

abc

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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