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Physics Letters B 798 (2019) 134948

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

(2, 0) lagrangian structures

Neil Lambert

Department of Mathematics, King’s College London, WC2R 2LS, UK

a r t i c l e i n f o a b s t r a c t

Article history:

Received

11 September 2019

Accepted

16 September 2019

Available

online 20 September 2019

Editor:

M. Cveti

By including an additional self-dual three-form we construct a Lorentz invariant lagrangian for the

abelian (2, 0) tensor supermultiplet. The extra three-form is a supersymmetry singlet and decouples

from the (2, 0) tensor supermultiplet. We also present an interacting non-abelian generalization which

reproduces the equations of motion of [1]and can describe some aspects of two interacting M5-branes.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

There are good reasons to believe that there is no lagrangian

formulation for the six-dimensional (2, 0)-theories that describe

M5-branes. Some of the various arguments can be summarised as

follows:

• Reduce the (2, 0)-Theory on a compact four-manifold M.

The presence of the self-dual 3-form would lead to σ (M) =

(M) − b

−

(M) chiral bosons in the resulting two-dimen-

sional

theory. If one had an action then one would expect

there to be a modular invariant partition function coming

from the SL(2, Z) of large diffeomorphisms in the remaining

two-dimensions, which we take to be a torus of ﬁnite size.

However such a partition function only exists if σ is a multi-

ple

of 8. In particular one expects to be able to embed C P

which has σ (M) = 1, in to M-theory and the resulting two-

dimensional

theory cannot have a modular invariant partition

function [2].

• Reducing the su(2n)(2, 0)-Theory on S

, along with an outer

automorphism twist, leads to ﬁve-dimensional maximally su-

persymmetric

Yang-Mills with gauge group so(2n + 1). But

so(2n + 1) is not a subalgebra of su(2n) for generic n [3].

• The standard M-theory dictionary states that reducing the

(2, 0)-Theory on S

of radius R we should ﬁnd ﬁve-dimen-

sional

maximally supersymmetric Yang-Mills with coupling

proportional to R, meaning that the ﬁve-dimensional action is

inversely proportional to R. However dimensional reduction of

an action in six-dimensions naively leads to a ﬁve-dimensional

action that is directly proportional to R [4].

E-mail address: neil.lambert@kcl.ac.uk.

• There are no satisfactory deformations of the free Lagrangian

[5]. In addition there is no sequence of interacting six-

dimensional

superconformal ﬁeld theories that converge to a

free theory [6].

• There are no interacting, power counting renormalizable, la-

grangians

in six-dimensions with an energy-momentum tensor

that is bounded from below [7]. However the (2, 0) theory is

a conformal ﬁeld theory and should have no dimensionful pa-

rameters

and be UV complete.

These

arguments are quite convincing and hence we don’t ex-

pect

to ﬁnd a deﬁnitive lagrangian for the interacting (2, 0) theory.

However it is worth exploring what lagrangian structures exist and

what they can do. Furthermore by hunting for unicorns we may

ﬁnd other creatures that are useful in understanding the theory

more generally. In particular there are some possible ways out of

the ﬁrst two points:

• Rokhlin’s theorem states that for any compact four-dimen-

sional

spin manifold σ (M) is a multiple of 16 and hence the

dimensionally reduced theory can have a modular invariant

partition function. The problem could be that the action must

be coupled to background ﬁelds in a non-standard way so as

to allow for a non-spin manifold such as C P

• For n =1 so(3) = su(2) so this objection fails. This is reminis-

cent

of the M2-brane story for which a lagrangian with all the

supersymmetries manifest only exists for two M2’s.

This

work was inﬂuenced by Sen who has introduced a method

to formulate an action for self-dual abelian ﬁelds in 4n + 2di-

mensions

[8,9]by including a second self-dual form which then

decouples. This construction has the feature that the coupling to

gravity is somewhat non-standard so that diffeomorphisms act dif-

ferently

from usual and hence provides hope that the ﬁrst and

https://doi.org/10.1016/j.physletb.2019.134948

0370-2693/

SCOAP

2 N. Lambert / Physics Letters B 798 (2019) 134948

third issues can be overcome, although we will not discuss this

here.

Thus

the purpose of this paper is twofold. The ﬁrst is to con-

struct

a new action for the abelian (free) (2, 0) multiplet. We do

this by introducing an additional self-dual three form which is a

supersymmetry singlet. The second is to explore how one might

generalise it to a non-abelian theory, at least for two M5-branes,

and see how far we get. In the latter case we must be willing to be

suitably creative. We will postpone for later the issue of whether

or not the resulting dynamical theories are well-deﬁned and how

much of the (2, 0) theory they capture. We are more interested in

exploring the possible structures with a hope that they will lead to

additional insights that will be fruitful, even without a lagrangian.

would also like to mention other related work. Using the

notion of tensor hierarchy a class of six-dimensional (1, 0) La-

grangians

was obtained in [10]but the self-duality condition was

imposed ‘by hand’ on the equations of motion. The use of an

additional self-dual three-form to construct actions for self-dual

three-forms has appeared in the Twistor approach of [11], [12]

and

was generalised to a non-abelian but ﬂat gauge ﬁelds in [13].

Mathematically focused discussions of lagrangian structures for the

(2, 0)-theory also recently appeared in [14,15]. Even more recently

an alternative construction of self-dual forms was given in [16].

The

rest of this paper is organised as follows. In section 2 we

will review the construction of Sen for the particular case of a self-

dual

three-form in six-dimensional Minkowski space. In section 3

will adapt this to the case of an abelian supersymmetry (2, 0)

multiplet, including potential external interactions. In section 4 we

will examine how we might introduce an interacting (2, 0) theory,

leading to an action (or more precisely a family of actions) which

reproduces the equation of motion of the (2, 0) theory of [1]. Fi-

nally

in section 5 we state our results and conclusions.

2. The (2, 0) multiplet and sen’s prescription

The linearised equations of motion for the (2, 0) tensor multi-

plet

can be written as

∂

= 0



∂

 =0

= 0 , (1)

where H = H is a self-dual 3-form and  is a chiral spinor

with 8 real on-shell degrees of freedom: 

012345

 =−. We use

conventions where μ, ν = 0, 1, 2, 3, 4, 5, I = 6, 7, 8, 9, 10, η

μν

diag(−1, 1, 1, 1, 1, 1), ε

012345

= 1 and (

, 

) form a real rep-

resentation

of the Spin(1, 10) Clifford algebra and all spinors are

real.

These

equations are invariant under the on-shell (2, 0) super-

symmetries:

δ X

= i





δH

μνλ

= 3i



[μν

∂

λ]



δ =



∂

 +

2 · 3!



μνλ

 , (2)

where 

012345

 = . These close, on-shell, onto translations. Alter-

natively

one often introduces a two-form b so that H = db with

δb

μν

= i



μν

.

Let

us now review the action proposed in [8,9]:

S =





∧dB − H ∧dB + L

int

(H) + L

, )



. (3)

Here we have relabelled ﬁelds so as to conform more closely to the

standard (2, 0) literature. In particular, in the notation of [8,9] B =

√

2P and H =−Q /

√

2. We have also split the interaction term

that appears in [8,9]into one that depends on H and the rest

which includes the kinetic terms for the remaining ﬁelds to

facilitate our discussion.

We use a convention where, for a p-form ω,

ω =

...μ

∧...dx

dω =

∂

...μ

∧dx

∧...dx

. (4)

The Hodge dual acts on p-form components as

(ω)

..μ

6−p

...μ

6−p

...ν

. (5)

This satisﬁes 

= 1on odd forms, 

=−1on even forms and

ω ∧ χ =χ ∧ω for two p-forms.

we observe that the equations of motion for B and H that

result from this action can be written as



(dB +dB) + H



−

(dB −dB) − R = 0 , (6)

where the anti-self-dual 3-form R is deﬁned by

δL

int

=−



δH ∧ R . (7)

Note that R =−  R by construction.

Thus we see that there

are two self-dual 3-forms

(dB + dB) and H. The ﬁrst one has

the wrong sign kinetic term but, as shown in [9], the combina-

tion

(dB + dB) + H is free (closed) and decouples. This is most

transparently seen in the Hamiltonian formulation. Thus it can be

safely discarded from any physical quantities. The physically rele-

vant

3-form is H which is not closed but rather has a source:

d  H = J ⇔ dH =− J , (8)

where J =−  dR is a 2-form current. Of course these two equa-

tions

are equivalent and can also be written more succinctly as

d(H +R) = 0but the above form seems more suggestive. Note that

since R =−  R one can’t simply solve this by taking H =−R +dC

and

imposing dC = dC.

3. An abelian (2,0) action

Our ﬁrst task is to extend the action (3)to the free (2, 0) mul-

tiplet

by setting L

int

= 0 and choosing a suitable L

. Thus we

consider

S =





ηdB ∧dB − H ∧dB −

∂



∂





(9)

where η is a constant to be determined. In particular the usual

sign kinetic term for B requires η < 0. We postpone interaction

terms to the next section. Here we wish to establish supersymme-

try

of this free action. To this end we consider the ansatz:

We have also rescaled R →−2R compared to [8,9].

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