The M5-brane action revisited

Abstract

We construct an alternative form of the M5-brane action in which the sixdimensional worldvolume is subject to a covariant split into 3+3 directions by a triplet of auxiliary fields. We consider the relation of this action to the original form of the M5-brane action and to a Nambu-Poisson 5-brane action based on the Bagger-Lambert-Gustavsson model with the gauge symmetry of volume preserving diffeomorphisms.

Published for SISSA by Springer

Received: September 5, 2013

Accepted: October 21, 2013 Published: November 8, 2013

The M5-brane action revisited

Sheng-Lan Ko,a Dmitri Sorokinb and Pichet Vanichchapongjaroena

aCentre for Particle Theory and Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, U.K.

bINFN Sezione di Padova,via F. Marzolo 8, 35131 Padova, Italia

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We construct an alternative form of the M5-brane action in which the six-dimensional worldvolume is subject to a covariant split into 3+3 directions by a triplet of auxiliary elds. We consider the relation of this action to the original form of the M5-brane action and to a Nambu-Poisson 5-brane action based on the Bagger-Lambert-Gustavsson model with the gauge symmetry of volume preserving di eomorphisms.

Keywords: p-branes, M-Theory

c

JHEP11(2013)072

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP11(2013)072

Web End =10.1007/JHEP11(2013)072

Contents

1 Introduction 1

2 Notation and conventions 4

3 M5-brane actions 43.1 Original M5-brane action 53.2 New M5-brane action 6

4 Derivation of the new M5-brane action 84.1 Non-linear self-duality of the M5-brane in the superembedding approach 104.1.1 3+3 splitting 114.1.2 The M5-brane action in terms of G and F ia 124.1.3 Self-duality relations in particular cases 12

5 Comparison of the two M5-brane actions 15

6 Relation to M2-branes 16

7 Conclusion 17

A Exact check of the M5-brane action non-linear self-duality from superem-bedding 18A.1 Matrix notation 18A.2 Outline of computation 19

1 Introduction

The construction of duality-symmetric actions has been an active topic of research since the 1970s [1, 2]. It has recently seen a revival of interest in relation with the discussion of possible niteness of N = 8, D = 4 supergravity [310], and in connection with attempts of making progress in understanding the non-Abelian (2,0) 6d superconformal gauge theory [11] on the worldvolume of N coincident M5-branes [1234]. For a single M5-brane the complete set of equations of motion was derived in [35] and considered in detail in [36] using the superembedding approach put forward in [37] (see [38] and e.g. [39, 40] for review and a detailed list of references). A complete M5-brane action was constructed in [41, 42] as a result of a step-by-step generalization [4347] of a self-dual action for a free chiral 2-form gauge eld [48, 49]. It was then shown that the non-linear self-duality relation [50] and the complete set of the equations of motion [51] derived from the M5-brane action are equivalent to the manifestly covariant equations obtained from superembedding.

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It is well known that to lift the duality symmetry to the level of the action one should deal with the issue of space-time covariance of the theory. In the non-manifestly SO(1, 5) Lorentz-invariant construction of the 6d chiral 2-form action by Henneaux and Teitelboim [48, 49] only an SO(5) or SO(1, 4) [44] subgroup of SO(1, 5) is manifest. The construction can be made space-time covariant (di eomorphism invariant) by introducing into the action a normalized gradient of an auxiliary scalar eld a(x) [45, 52, 53]. The manifestly covariant formulation signicantly simplies the construction of the consistent couplings of the self-dual eld action to gravity and other elds, and its non-linear deformations. Di erent gauge xings of the value of a(x) using an associated local symmetry (or its dualization [54]) results in di erent non-covariant forms of the self-dual action. On the other hand, the self-duality equations obtained from the action can be cast into the manifestly covariant form which does not contain the auxiliary eld a(x), thus the latter completely disappears on the mass-shell without imposing any gauge xing condition.

With the advent of the Bagger-Lambert-Gustavsson (BLG) model [5557], an alternative construction of a 5-brane action based on the BLG action with the gauge symmetry of volume preserving di eomorphisms was put forward in [58, 59] (see [60, 61] for a review and references and [62, 63] for a related work). The space-time and duality symmetries of this construction were analyzed in detail in [64, 65]. The equivalence of this model to the M5-brane description of [41, 42] is still to be proved, though some steps have already been undertaken in [64, 66] and various checks via comparison of classical solutions on the both sides have been carried out (see [61] and references therein).

The relation between the two actions is not obvious, rst, since the original non-linear M5-brane action is of a Dirac-Born-Infeld type whose chiral 2-form gauge eld transforms under the usual Abelian gauge transformations, while the action of [59] is a polynomial of up to six order in the elds and has a Nambu-Poisson 3-algebra structure associated with an un-conventional gauge invariance under volume preserving di eomorphisms. In [58, 59] it was conjectured that the Nambu-Poisson (NP) M5-brane model is related to the conventional description of the M5-brane in a constant C3-eld background through a transformation analogous to the Seiberg-Witten map [67]. Such a map between the elds and gauge transformations of the two models was constructed in [61], however the relation between the two actions still remains to be established. The second reason which hampers the resolution of this issue is that in the NP M5-brane model the manifest SO(1, 5) 6d Lorentz symmetry is naturally broken by the presence of multiple M2-branes and the C3-

eld to SO(1, 2)SO(3), which corresponds to a 3+3 = 6 splitting of the six dimensions

of the M5-brane worldvolume. In the original M5-brane action, as we mentioned above, the six dimensions split into 1+5. In [64] it was shown that, even when reduced to the second order in the elds, the two duality-symmetric actions are not equivalent o the mass shell, though both produce the same self-duality equation for the 2-form gauge eld.

The M5-brane case exemplied the fact that the Lagrangian description of the self-dual elds and duality-symmetric elds in general is not unique (see also [68, 69]), and various free (quadratic) duality-symmetric actions in D dimensions with di erent splittings of D = p + q + r + . . . corresponding to various ways of breaking manifest space-time symmetry have been constructed [70, 71]. These di erent o -shell formulations may be

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useful for studying issues of the quantization of the self-dual elds in topologically nontrivial backgrounds [68, 69, 7277].

As far as the M5-brane is concerned, it is advisable for a better understanding of the relation between the original M5-brane descriptions and the NP 5-brane, to see whether the quadratic self-dual action of [58] with 3+3 splitting can be extended to a full nonlinear action which is invariant under the conventional gauge transformations of the gauge eld and which would produce the same equations of motion as the ones obtained from the superembedding [35] and the action of [41, 42, 51]. This is the main goal of this paper.

Our strategy to achieve this goal is as follows.1 We will start with the covariant form [64] of the quadratic self-dual action of [58] for a 2-form chiral gauge eld in six-dimensions. In addition to the conventional invariance under the gauge transformations of the chiral eld, the covariant action possesses two more local symmetries. One of them ensures that the auxiliary elds, which make the action covariant, are non-dynamical and another one guarantees that the self-duality condition on the eld strength of the chiral eld is the general solution of its equations of motion. We will add to this quadratic action a generic non-linear function of components of the chiral eld strength and derive conditions on the form of this function imposed by the two local symmetries. It is known that the conditions obtained in this way may have more than one solution (see e.g. [4, 7, 44, 78, 79]), so to single out the solution which describes the M5-brane we will look for the one which is equivalent to the non-linear self-duality relation of the superembedding approach. More concretely, we will check that the non-linearly self-dual eld strength of the superembedding formulation satises the condition imposed on the non-linear part of the self-dual action and, as a result, will derive an explicit form of the M5-brane action in which the 6d di eomorphism invariance is subject to 3+3 splitting.

As is known from an extensive literature (see e.g. [47, 9, 79] and references therein), in general, the functionals of gauge-eld strengths which determine non-linear self-duality conditions are constructed order-by-order as perturbative series expansions in powers of the eld strength and in general their explicit form is unknown except for the Born-Infeld-type actions and few other examples (see e.g. [10, 80]). Our construction is a new example of an explicit (closed) form of the non-linearly self-dual action which di ers from the canonical form of the Born-Infeld-type actions by additional terms and factors.

The paper is organized as follows. In section 2 we introduce main notation and conventions. In section 3 we review the original action and present the structure of the novel action for the M5-brane. The derivation of the new action is explained in section 4. In section 5 we show that on-shall values of the two actions are equal and in section 6 briey discuss the dimensional reduction of the novel M5-brane action to that of the M2-brane. The results are summarized in Conclusion, where we also discuss open issues and possible directions of further research. In the appendix we give more details of the check of the form of the new M5-brane action by comparing the self-duality relations which follow from the action with those obtained in the superembedding description of the M5-brane.

1For analogous procedures of getting manifestly duality-symmetric non-linear actions see e.g. [4, 7, 44].

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2 Notation and conventions

The 6d and the D = 11 Minkowski metrics have the almost plus signature, x ( =

0, 1, , 5) stand for the worldvolume coordinates of the M5-brane which carries the chiral

gauge eld B2(x) = 12 dxdx B (x). The D = 11 bulk superspace is parametrized by ZM = (XM, ), where XM are eleven bosonic coordinates and are 32 real fermionic coordinates. The geometry of the D = 11 supergravity are described by tangent-space vector supervielbeins EA(Z) = dZMEMA(Z) (A = 0, 1, 10) and Majorana-spinor su

pervielbeins E (Z) = dZMEM (Z) ( = 1, 32).

The vector supervielbein satises the following essential torsion constraint, which is required for proving the kappa-symmetry of the M5-brane action,

T A = DEA = dEA + EB BA = iE A E , (2.1)

where BA(Z) is the one-form spin connection in D = 11, A = A are real symmetric gamma-matrices and the external di erential acts from the right.

The induced metric on the M5-brane worldvolume is constructed with the pull-backs of the vector supervielbeins EA(Z)

g (x) = EAEB AB, EA = ZN EN A(Z(x)). (2.2)

The M5-brane couples to the D = 11 supergravity 3-form gauge supereld C3(Z) =

13! dZM1dZM2dZM3CM3M2M1 and its C6(Z) dual, their eld strengths are constrained as follows

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dC3 =

i 2EAEBE E ( BA) +

14!EAEBECEDF (4)DCBA(Z) ,

dC6 C3dC3 =

2i

5! EA1 EA5E E ( A5A1) +

17!EA1 EA7F (7)A7A1(Z) (2.3)

F (7)A1A7 = 1

4!A1A11F (4)A8A11 , 0...10 = 0...10 = 1.

The generalized eld strengths of B2(x) which appears in the M5-brane action is

H3 = dB2 + C3 , (2.4)

where C3(Z(x)) is the pullback on the M5-brane worldvolume of the 3-form gauge eld.

3 M5-brane actions

We start by briey reviewing the original form of the M5-brane action and then will present our main result, namely, the alternative worldvolume action for the M5-brane in a generic D = 11 supergravity background.

4

3.1 Original M5-brane action

In this case to ensure the 6d worldvolume covariance of the M5-brane action one uses a normalized gradient of the auxiliary scalar eld a(x) which can be chosen to be time-like or space-like, e.g.

v(x) = a

p a g (x) a, vv = 1 (3.1) The both choices are equivalent since in the action a appears only in the projector of rank one

P (x) = a a(a)2 , P P = P, (a)2 a g a = a a . (3.2)

This projector singles out one worldvolume direction from the six, i.e. makes the 1+5 covariant splitting of the 6d worldvolume directions.

The M5-brane action in a generic D = 11 supergravity superbackground constructed in [41, 42, 47] has the following form:

S = +2ZM6 d6x qdet(g + i ) + g4(a)2 a H a

ZM6 (C6 + H3 C3) , (3.3)

a

p(a)2

05 = 05 = 1 .

In addition to the conventional abelian gauge symmetry for the chiral 2-form, the action (3.3) has also the following two local gauge symmetries:

B = 2[a ](x), a(x) = 0, (3.5)

as well as

a = (x), B = (x)

p(a)2 (H V ), (3.6)

where

p(a)2 , (3.7) with (x) and (x) being arbitrary local functions on the woldvolume. The rst symmetry (3.5) ensures that the equation of motion of B2 reduces to the non-linear self-duality condition

H = V (

) , (3.8)

while the second symmetry (3.6) is responsible for the auxiliary nature of the scalar eld a(x) and the 6d covariance of the action.

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with

1 6g

H ,

, g = det g , (3.4)

where

V (

) 2

qdet( + i )

, H H

a(x)

The action (3.3) is also invariant under the local fermionic kappa-symmetry transformations with the parameter (x) which act on the pullbacks of the target-space supervilebeins and the B2 eld strength as follows

i E ZME M =

1

2(1 +

) , i EA ZMEAM = 0. (3.9)

g = 4iE (( )) i E , H(3) = i dC(3), a(x) = 0 ,

where (1 +

)/2 is the projector of rank 16 with having the following form

qdet( + i ) = (6) 12 P 116g1612 34P 56,

2 = 1 , tr = 0, (3.10)

where = EA A , (6) = 1

6!g

16 16 . (3.11)

3.2 New M5-brane action

For this case, to ensure worldvolume covariance of the construction, instead of the single scalar eld we need to introduce a triplet of auxiliary scalar elds as(x) with the index (s = 1, 2, 3) labeling a 3-dimensional representation of GL(3) which is an internal global symmetry of the action. The partial derivatives of the scalars are used to construct the projector matrices [64]

P = arY 1rs as, = P , a = 0 (3.12)

with Y 1rs being the inverse matrix of

Y rs arasg . (3.13)

The projectors identically satisfy the following di erential condition

[ ]D P = 0 = [ ]D (3.14)

where D is the worldvolume covariant derivative with respect to the induced metric g .

Note that the projectors (3.12) have rank 3 and thus e ectively split the 6d directions into 3+3 ones orthogonal to each other.

The new M5-brane action coupled to a curved superbackground has the following form

S =ZM6 d6x

g

6 (

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G + 3 ~F F ) + 2LM5(F, G) ZM6

(C6 + H C3) ,

(3.15)

where

136(1 + G2)123456G123F4 F5 F6

+ 1

1 + G2

s det

g + 12(F + G)(F + G) ) (3.16)

6

LM5 =

and F and G are components of the eld strength H projected as follows

F H P , G H , G2

16H H , (3.17)

, . (3.18)

The action enjoys the following two local gauge symmetries analogous to eqs. (3.5) and (3.6). The rst one is

B = P P (x), as = 0, (3.19)

where (x) are arbitrary parameters. Note that in view of the conditions (3.14) it follows that the projected eld strengths (3.17), and hence LM5(G, F ), are invariant under

this symmetry G = F = 0, (3.20)

while their dual (3.18) are not.The second symmetry ensures the triplet of the scalar elds as(x) to be auxiliary

as = s(x), B = 12rY 1rsas

~F

P ,

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g ~F

LM5 F

, (3.21)

where s(x) are local parameters.2

This symmetry allows one to gauge x as(x) to coincide with three world-sheet coordinates, e.g. xa (a = 0, 1, 2) or xi (i = 3, 4, 5), thus getting a non-covariant but non-manifestly worldsheet di eomorphism invariant M5-brane action. For instance, let us impose the gauge xing condition

as = sa xa , (3.22)

identifying as with xa. Then the following combination of the worldvolume di eomorphism x = (x) and the local symmetry (3.21) leaves this gauge condition intact

as(x) = (x)as + s(x) = s(x) + s(x) = 0, s(x) = s(x).

Under the local transformation combined of the 6d di eomorphism x = (x) and the

local variation (3.21) with a(x) = a(x) the gauge eld B transforms as follows

B = B

1

2a(x) a

g ~F

LM5 F

,

while the other M5-brane elds XM(x) and (x) being transformed in the conventional way as worldvolume scalars. In the gauge (3.22) the action (3.15), (3.16) is non-manifestly invariant under the modied worldvolume di eomorphisms of the above form.

2In what follows we will use a normalization of the functional derivative, denoted by @

L(F )@F ... , which di ers

from the one dened in (3.7). Namely, by denition the variation of a p-form F1p and the corresponding functional derivatives are dened as follows F1p = F 1 p F1p F

1 p = F 1 p 1p! @F1p@F 1 p . So that

@L @F 1 p

p! L

F 1 p .

7

Upon tedious computations we have checked that the action is invariant under the kappa-symmetry transformations (3.9) but with a projector which has the following form

11 + G2

sdet

+ 12(F + G)(F + G)

=

= (6) + 16(6)(3F + G) +

1

2(1 + G2)(6)F F

+ 1

6(1 + G2)(6) 3(F F G) + (F F F )

, (3.23)

where

(F F G) FF G , (F F F ) FF F . (3.24)

Note that the term multiplying on the left hand side of (3.23) is equal (modulo

p det g ) to the last term of the non-linear part (3.16) of the M5-brane Lagrangian. Finally, the non-linear self-duality condition which is obtained from action (3.15) as

the consequence of the equations of motion of B2 (see eq. (4.7) of the next section) has the following form

= 1g

LM5 G

, ~F[ ] = 1g LM5 F

[ ]. (3.25)

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As we will show, this self-duality condition is related to eq. (3.8) via the manifestly covariant self-duality relation which comes from the superembedding approach [35].

4 Derivation of the new M5-brane action

To get the new M5-brane action (3.15) we start from the covariant form [64] of the quadratic action [58] for the 6d chiral eld. It is obtained from (3.15) by truncating the latter to the second order in the chiral eld strength H3,3

S = 16 Z

d6x

p det g (H ) ( + 3 P)H

1 6

Z

d6x

p det g [(G ) G + 3(F ~F) F ] , (4.1)

The action is invariant under the symmetry (3.19) and under the linearized counterpart of (3.21)

as = s(x), B = 1

2sY 1srar g

~F F

. (4.2)

The quadratic action leads to the equation of motion

g(G ) + 3g(F ~F)[ ]

= 0, (4.3)

3For simplicity, but without loss of generality, we consider (for a moment) the pullbacks of the 11D gauge elds be zero.

8

which has the general solution

g(G

) + 3g(F

~F)[ ] = 1

2

hg~ P P

i

, (4.4)

with an arbitrary tensorial function ~

. This function can be compensated by a gauge transformation of the equation of motion under (3.19) with the gauge parameter

~

.

Hence, in view of the denition of the projected components of the eld strength (3.17), the solution of the dynamical equation is equivalent to the self-duality conditions

(G

) = 0, (F

~F)[ ] = 0. (4.5)

We are now looking for a non-linear generalization of the action (4.1) which would respect the both symmetries (3.19) and (3.21). Note that the second symmetry should be deformed by the non-linear terms, since the form of its transformation is associated with the form of the non-linear self-duality condition. In the case of the M5-brane these are (3.6)(3.8), and (3.21) and (3.25).

Since the eld strength components F and G are invariant under the transformations (3.19) (see eqs. (3.20)), while their dual (3.18) are not, the non-linear terms in the action should only depend on F and G. So the general form of the non-linear action which respects the symmetry (3.19) is obtained by replacing the quadratic terms F F and GG in (4.1) by an arbitrary function L(F, G)

S =ZM6 d6x

g

6 (

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G + 3 ~F F ) + 2L(F, G)

. (4.6)

The variation of this action with respect to the gauge potential B2 produces the equations of motion

"

L

G

g + 3

L F

[ ] 3g ~F[ ]

#

= 0. (4.7)

In view of (3.20) and the fact that L only depends on F and G, we can integrate the above

equation of motion with the help of the symmetry (3.19) along the same lines as in free theory. The integration produces the non-linear self-duality relations

= 1g

L

G

, ~F[ ] = 1g L F

[ ]. (4.8)

We should now nd conditions on the form of L(F, G) imposed by the requirement that

the action is invariant under

as = s(x), B = 1

2sY 1srar

g ~F

L

F

. (4.9)

Upon somewhat lengthy calculations using, in particular, the properties of the projectors (3.12)(3.14) and the form of their variation under (4.9)

'P = 2 (rY 1rs )as (4.10)

9

we get the following condition on L(F, G)

Y 1rs as g L G

F gG L F

= 0. (4.11)

This condition is analogous to those found in other instances of models with non-linear (twisted) self-duality, e.g in D = 6 [44] and D = 4 [4, 7]. It is well known that these conditions may have di erent solutions leading to di erent non-linear generalizations of quadratic duality-symmetric actions (see e.g. [4, 7, 44, 78, 79]). We are interested in a particular solution of the above equation, i.e. in the form of L(F, G) which describes the M5-brane. To

nd this form we assume that, as in the case of the self-duality condition (3.8) obtained from the original M5-brane action, also the self-duality conditions (3.25) (or (4.8)) should be equivalent to the self-duality conditions appearing in the superembedding formulation of the M5-brane [35]. Exploring these conditions we shall derive the form (3.16) of the non-linear M5-brane Lagrangian.

4.1 Non-linear self-duality of the M5-brane in the superembedding approach

In the superembedding description of the M5-brane [35, 36] the eld strength H3 of the chiral eld B2 is expressed in terms of a self-dual tensor h3 = h3 as follows4

14H = m1 h ,

1 4

1

2

g2 F F

L F

L F

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1 11 = 1

61 11 m1 h = Q1m1 h 11 (4.12)

where m1 is the inverse matrix of

m = 2k , m1 = Q1(2 m ), k = h h (4.13)

and

Q = 1

2

3tr k2 . (4.14)

As was shown in [50], by splitting the indices in eqs. (4.12) into 1+5 and expressing components of h3 in terms of 5 one gets the duality relation (3.8).5

We shall now carry out a similar procedure, but splitting the 6d indices into 3+3, and upon a somewhat lengthy algebra will arrive at the self-duality condition in the form of (3.25), thus getting the non-linear function LM5(F, G) (3.16) which enters the M5-brane

action (3.15).

The 3+3 splitting can be performed with the use of the projectors (3.12), but for computational purposes we have found it more convenient to pass to a local tangent-space frame using 6d vielbeins em (emmnen = g ) and to write the 3+3 tangent space indices explicitly. So the three directions singled out by the projector Pmn emP en , which we

4Our normalization of the eld strength di ers from that in [50] by the factor of 14 in front of H3.

5This splitting is amount to projecting the tensor elds along the direction of @a and orthogonal to it.

10

assume to contain the time direction, will be labeled by the indices a, b, c, and the three spacial directions singled out by mn em en will be labeled by i, j, k:

Pmn ba , mn ji , a, b, c = 0, 1, 2; i, j, k = 3, 4, 5 , (4.15) while the 6d Levi-Civita tensor splits as follows

abcijk, 012 = 012 = 1, 345 = 1 . (4.16)

We are now ready to split the indices of H3 and h3 in (4.12).

4.1.1 3+3 splitting

As h3 is self-dual, we pick its 10 independent components in the local Lorentz frame as follows

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hija, hijk (4.17)

and dene6

fka

1

2ijkhija, g

16ijkhijk. (4.18)

In view of the self-duality

hmnp = 1

3!mnpl1l2l3hl1l2l3, (4.19)

we have

hjab = abcfjc, habc = gabc, (4.20)

or

fic = 1

2abchiab g =

16abchabc. (4.21)

The corresponding components of H3 are dened as

F ka

1

2ijkHija, G

16ijkHijk. (4.22)

The duals of F and G are

~Fic

1

2abcHiab,

16abcHabc. (4.23)

Note that the tensors (4.22) and (4.23) are counterparts of (3.17) and (3.18) in the local Lorentz frame (4.15).

Our nal goal is to write ~F, in terms of F, G using the relations (4.12). To this end, using (4.12) we rst nd the expressions for F, G, ~F and in terms of g and fai

14F ai = Q1 f(1 + 4g2 4trf2) + 8f3 8gf1 det f

a i

= Q1

12(g2 + trf2) 1 16Q

, (4.24)

14G = Q1 g + 4g3 + 4gtrf2 8 det f

= Q1

g

12(g2 + trf2) 1 16Q

fia

, (4.25)

6One should not confuse the eld g(x) with the determinant of the induced metric g .

11

and

1 4

~Fai = Q1 f(1 4g2 + 4trf2) 8f3 + 8gf1 det f

a i

= Q1

12(g2 + trf2) +1 16Q

, (4.26)

1 4

fia

= Q1 g 4g3 4gtrf2 + 8 det f

= Q1

12(g2 + trf2) +1 16Q

, (4.27)

g

where

Q = 1 16g4 + 16(trf2)2 32g2trf2 32trf4 + 128 g det f, (4.28) trf2 faifbjijab , det f

16ijkabcfiafjbfkc , (f1)ai det f

1

2ijkabcfjbfkc .

(4.29)

4.1.2 The M5-brane action in terms of G and F ia

For the elds (4.22) and (4.23) the M5-brane action (3.15) takes the following form

S3+3 = Z

d6x

p

det g (F ia ~Fai + G) 2LM5 Z

M6 (C6 + H C3) , (4.30)

JHEP11(2013)072

where the term LM5 is

LM5 =

G det F

1 + G2 +

qdet(ji(1 + G2) + F aiF ja) 1 + G2

p det g

, (4.31)

and the non-linear self-duality relations (3.25) become

= 1

p det g

LM5 G ,

~Fai = 1

LM5

p det g

F ia . (4.32)

4.1.3 Self-duality relations in particular cases

To guess the form (4.31) of the function LM5 in the M5-brane action we rst consider a

number of simple cases.

f = 0 case. The relations (4.24)(4.28) reduce to

Fia = ~Fia = 0 , Q = 1 16g4, 14G =

g + 4g3 1 16g4

= g

, (4.33)

1 4

1 4g2

= g 4g3

1 16g4

= g

1 + 4g2 . (4.34)

g = 1 + G2 1

2G . (4.35)

12

We now solve eq. (4.33) for g

Since, due to (4.33), in the linear approximation G/4 = g, we should pick up only the solution with the upper sign. Substituting this solution into (4.34) we get the relation between and G

= G

1 + G2 =

1 + G2G . (4.36)

We see that eq. (4.36) is exactly the same as (4.32) when in (4.31) we put F ai = 0 =

F . This demonstrates how the (square root of) factor 1 + G2 appears in the function

LM5(F, G) (3.16) or (4.31) of the M5-brane action (3.15).

g = det f = 0 case. Now the relations (4.24)(4.28) reduce to

G = = 0 , Q = 1 + 16(trf2)2 32trf4, 14F ai = Q1 f(1 4trf2) + 8f3

ai = Q1 fia

JHEP11(2013)072

12trf2 1 16Q

, (4.37)

1 4

~Fai = Q1 f(1 + 4trf2) 8f3

ai = Q1 fia

12trf2 +1 16Q

, (4.38)

Let us simplify things even further by considering a solution of the non-linear self-duality equation such that the only non-zero components of fai are f1i. Then the above equations further reduce to

G = = 0 , Q = 1 16(f2)2, f2 f1if1i , (4.39)

14F 1i = Q1 1 + 4f2

f1i = f1i

, (4.40)

1 4

1 4f2

1 + 4f2 , (4.41)

From these equations we nd that

1 4f2 =

2(1 1 + F 2)F 2 , 1 + 4f2 =

~F1i = f1i

21 + F 2

F 2 (

p1 + F 2 1) ,

p1 + F 2 1).

Since, due to (4.40), in the linear approximation F ai/4 = fai, in the above relation we should pick the upper sign and upon substituting it into (4.38) we get the duality relation

~F1i = F 1i

1 + F 2 =

f1i = F 1i

2F 2 (

1 + F 2 F i1

. (4.42)

We see that this relation coincides with (4.32) for G = 0 and F ai having only the non-zero components F 1i.

Self-dual string soliton (g 6= 0, det f = 0 case)

Let us now consider a more complicated particular case of a string soliton solution of [44]. A similar consideration is applicable to the BPS self-dual string of [81]. For the string aligned

13

along the x2-coordinate, in terms of elds (4.22) and (4.23) the string soliton solution of [44] has the following form:

G =

x1

xk

4 , F k1 =

4 , (4.43)

=

x1

,

~Fk1 =

xk

. (4.44)

where k = 3, 4, 5, :=

px21 + x23 + x24 + x25, is a constant and

() =

p2 + 6 . (4.45)

In this form the string soliton solution was considered in [21]. It naturally splits the 6d worldvolume into 3+3 directions.

The form (4.43) of G and F suggests that in (4.24) and (4.25) g 6= 0 and the non-zero

components of fai are f1i. So the equations (4.24)(4.28) reduce to

Q = 1 16(g2 + f2)2. (4.46)

14F 1i =

f1i1 4(g2 + f2)

, 1

4G =

g1 4(g2 + f2)

, (4.47)

JHEP11(2013)072

and

= g

1 + 4(g2 + f2). (4.48)

Carrying out the same analysis as in the previous examples, from (4.46)(4.48) we get the duality relations

~F1i = F 1i

1 + G2 + F 2 =

1 4

~F1i = f1i

1 + 4(g2 + f2) ,

1 4

1 + G2 + F 2

F i1

, = G

1 + G2 + F 2 =

1 + G2 + F 2

G

(4.49)

which are again a particular case of (4.32). One can then guess that in the manifestly covariant formulation the expression under the square root combines into the determinant of the matrix formed by the bilinear combinations of G and F as in eq. (3.16) or (4.31).

To see that this is indeed so and that (4.31) should also contain the term G det F let us consider the case in which G = 0 while F ai is (otherwise) generic.

G = 0 case. We have

G = 0 = g + 4g3 + 4gtrf2 8 det f

, (4.50)

1 4

= 2Q1g , (4.51)

Q = 1 16g4 + 16(trf2)2 32g2trf2 32trf4 + 128g det f

= 1 + 16g2 + 16(trf2)2 + 48g4 + 32g2trf2 32trf4, (4.52)

14F ai = Q1 f(1 + 4g2 4trf2) + 8f3 8gf1 det f

ai (4.53)

14

and

Now, the direct computation of det F using (4.53) and (4.50) gives (see also eq. (A.15) of the appendix)

Comparing this equation with (4.51) we get

= det F , (4.56)

which is exactly the relation that we get by varying the term (4.31) of the M5-brane action (3.15) or (4.30) with respect to G and setting G = 0 afterwards. This explains the appearance of the term G det F in the M5-brane action.

On the other hand, upon expressing the right-hand side of (4.54) in terms of F ai and performing somewhat lengthy computations using Mathematica one gets the duality relation for ~F which coincides with eq. (4.32) evaluated at G = 0.

Finally, by a direct check using Mathematica one can verify that also in the generic case the components F , ~F, G and of the eld strength H3 determined by the superembedding relations (4.24)(4.28) satisfy the non-linear duality relations (4.32) which follow from the M5-brane action (3.15). Main steps of the calculation are described in the appendix.

The last point that one should check is that the function (3.16) satises eq. (4.11) which insures the invariance of the M5-brane action under the local transformations (3.21). The direct calculation shows that this is indeed so. Actually, (3.16) satises even stronger relation, namely, it makes to vanish the expression under the derivative in (4.11).

5 Comparison of the two M5-brane actions

As was discussed in [64] duality symmetric actions corresponding to di erent splittings of space-time di er from each other by terms that vanish on-shell, i.e. when (an appropriate part of) the self-duality relations is satised. In [64] this was discussed for the free chiral 2-form in 6d.

We shall now confront the two M5-brane actions (3.3) and (3.15) by comparing their values for the 3-form eld strength satisfying the non-linear self-duality equation. As we have seen, the non-linear self-duality relations that follow from these actions are similar and are equivalent to the self-duality condition that follows from the superembedding formulation. Therefore, to compute the on-shell values of the M5-brane actions we will substitute into them the expressions of the components of H3 and3 in terms of the components of the self-dual tensor h3.

In the case of the novel action these are eqs. (4.24)(4.27). Substituting them into the action (3.15) (or (4.30)) and using Mathematica we nd that the on-shell value of the self-dual M5-brane action is

Son-shellM5 = 4

Z

1 4

~Fai = Q1 f(1 4g2 + 4trf2) 8f3 + 8gf1 det f

ai . (4.54)

det F = 8Q1g . (4.55)

JHEP11(2013)072

p det g Q1 ZM6(C6 + H C3) . (5.1)

15

d6x

Notice that the Lagrangian of this action is the functional of Q(h) dened in (4.14). We thus see that the on-shell action is manifestly 6d covariant and does not depend on the auxiliary elds ar(x) (3.12).

To compute the corresponding on-shell value of the original M5-brane action (3.3) we perform the 1+5 splitting of the duality relations (4.12) which take the following form

ab5 = 4Q1

(1 2trf2)f + 8f3 b5 , Hab5 = 4Q1 (1 + 2trf2)f 8f3 b5 ,

where fb = hab5 and,b = 0, 1, 2, 3, 4. Upon substituting the above expressions into the action (3.3) we nd that its value is again given by eq. (5.1). Thus the two forms of the

M5-brane action give rise to the same equations of motion and their on-shell values are equal and are given by the superembedding scalar function Q(h). For the self-dual string soliton considered in section 4.1.3, the value of the action determines the tension of the string, as was discussed in [44].

An interesting open problem that may have important consequences for the issue of quantization of the self-dual elds is the understanding of the o -shell relationship between the di erent self-dual actions.

6 Relation to M2-branes

The new form of the M5-brane action can be useful for studying its relation to the Nambu-Poisson description of the M5-brane in a constant C3 eld originated from the 3d BLG model with the gauge group of volume preserving di eomoprhisms [58, 59]. The BLG model invariant under the volume preserving di eomorphisms describes a condensate of M2-branes which via a Myers e ect may grow into an M5-brane. In [58, 59] it was conjectured that the Nambu-Poisson M5-brane model is related to the conventional description of the M5-brane in a constant C-eld background through a transformation analogous to the Seiberg-Witten map [67]. Such a map between the elds and gauge transformations of the two models was constructed in [61], however the relation between the two actions still remains to be established. We leave the study of this issue for future and will only show that in a at background without C-eld the worldvolume dimensional reduction of the bosonic M5-brane action (3.15) (or (4.30)) directly results in the membrane action. To this end we x the 6d worldvolume di eomorphisms by imposing the static gauge

x = X, XI(x) I = 6, 7, 8, 9, 10

where XI(x) are ve physical scalar elds corresponding to the target-space directions transversal to the M5-brane worldvolume. We perform the dimensional reduction of three worldvolume directions xi (i = 3, 4, 5) assuming that the scalar elds XI and the chiral tensor eld B only depend on the three un-compactied coordinates xa and not on xi.

Then the induced worldvolume metric takes the form

g = (ab + aXIbXI, ij) , gai = 0 . (6.1)

16

JHEP11(2013)072

We use the local gauge symmetry (3.21) to x the values of the three auxiliary scalars ar(x) in such a way that the projectors (3.12) take the form

P = a a , = i i . (6.2)

Then the components G (3.17) of the gauge eld strength vanish and F reduce to

Faij = aBij F ia =

where the dualized components of the gauge eld Bij(xa)

~Xi

q det(ab + aXIbXI + a~Xib ~Xi) , (6.4)

which is the action for a membrane in at D = 11 space-time in the static gauge.

7 Conclusion

Using the non-linear self-duality equation for the 3-form gauge eld strength arising in the superembedding description of the M5-brane we have derived a novel form of the kappa-symmetric M5-brane action with a covariant 3+3 splitting of its 6d worldvolume.

The value of this action on the mass-shell of the non-linear self-dual gauge eld coincides with the on-shell value of the original M5-brane action expressed in terms of the 6d scalar function Q of the self-dual chiral eld h3 appearing in the superembedding description of the M5-brane. It would be interesting and important to better understand the o -shell relation between the two actions.

Having at hand the M5-brane action in the form (3.15), (3.16) one can repeat the steps of [66] towards understanding the link of this action to the Nambu-Poisson 5-brane of [58, 59] by restricting the worldvolume pullback of the 11D gauge eld C3 to be constant and by partial gauge xing local symmetries of (3.15), (3.16) to a group of 3d volume preserving di eomorphisms. The Seiberg-Witten-like map constructed in [61] may be need to relate the elds of the two models. It would be also of interest to relate our construction to a noncommutative M5-brane of [82].

The novel form of the action is also naturally suitable for studying the e ective theory of the M5-brane wrapping a 3d compact Riemann-manifold.

As another direction of study, one may try, using the superembedding form of the self-duality relation, to construct an M5-brane action in the form which exhibits 2+4 splitting of the 6d worldvolume which may be useful for studying M5-branes wrapping 2d and 4d manifolds, and M5-brane instantons wrapping 4d divisors of Calabi-Yau 4-folds in M3CY4 compactications of M-theory as discussed e.g. in [8388].

17

1

2ijkFajk = a

~Xi , (6.3)

1

2ijkBjk

play the role of the additional three scalar uctuations of the membrane associated with D = 11 target-space directions orthogonal to the membrane worldvolume. Indeed, upon the dimensional reduction the M5 brane action (4.30) becomes

SM2 =

Z

JHEP11(2013)072

d3x

Acknowledgements

The authors are grateful to I. Bandos, F. Bastianelli, M. Cederwall, W.-M. Chen, C.-S. Chu, P.-M. Ho, H. Isono, S. Kuzenko, P. Pasti, I. Samsonov, D. Smith, P. Townsend, M. Tonin, P. West and B. Zupnik for useful discussions and comments. Work of Sh-L.K. and D.S. was partially supported by the Padova University Research Grant CPDA119349 and by the INFN Special Initiative TV12. D.S. was also supported in part by the MIUR-PRIN contract 2009-KHZKRX. Sh-L. K. is grateful to INFN Padova section and the Department of Physics and Astronomy for kind hospitality and support during his stay in Padova on June 2 - June 13, 2013. D.S. acknowledges hospitality and support extended to him during the Workshop Symmetry and Geometry of Branes in String/M Theory at Durham University (January 28 - February 1, 2013) at the initial stage of this project, the Galileo Galilei Institute Workshop program Higher spins, strings and dualities (Florence, March 13 - May 10, 2013) and the Benasque Scientic Centre Program Gravity - New perspectives from strings and higher dimensions (July 17-26, 2013) during work in progress. P.V. is supported by a Durham Doctoral Studentship and by a DPST Scholarship from the Royal Thai Government.

A Exact check of the M5-brane action non-linear self-duality from superembedding

To check the form of (3.16) (or, equivalently, (4.31)), using the superembedding relations (4.24)(4.27) we should verify that

(f, g) = 4Q1 g 4g3 4gtrf2 + 8 det f

~F(f, g) = 4Q1 f(1 4g2 + 4trf2) 8f3 + 8gf1 det f

= 1

JHEP11(2013)072

= 1

LM5

p det g

G F (f, g), G(f, g)

, (A.1)

and

LM5

. (A.2)

To verify the above relations, on their right hand sides we should take G- and F -derivatives of LM5 in the form (4.31), substitute into the results the expressions (4.24) and (4.25) for

F and G in terms of f and g, and to see that they coincide with the left hand sides of (A.1) and (A.2), i.e. with and ~F expressed in terms of f and g. In particular, we will need to express tr(F 2), tr(F 4) and det(F ) in terms of f and g.

The algebra is very involved but it is manageable systematically by Mathematica. To this end we used NCAlgebra package which is found in http://math.ucsd.edu/~ncalg/

Web End =http://math.ucsd.edu/ http://math.ucsd.edu/~ncalg/

Web End =ncalg/ .

A.1 Matrix notation

To use Mathematica we should properly dene the matrices we deal with. Let Fai be the

components of the matrix F , ab or ab be the components of the matrix and ij or ij be the component of the matrix . It will be clear from the context whether the indices of and are up or down. To simplify the notation, we drop from all the matrix expressions.

p det g

F F (f, g), G(f, g)

18

For example, FajjkF kbbcFci is denoted as F F T F or just F F T F. This expression is what in previous sections we simply referred to as F 3.

The inverse matrix F 1 has the components (F 1)ia. We will, actually, encounter the adjugate matrix adj(F ) and the cofactor matrix co(F ) adj(F )T more often than F 1

and (F 1)T . The denition of adj(F ) is

adj(F )ia (F 1)ia det F =

where

In the matrix form, the equation (4.24) reads

F = 4Q1 f(1 + 4g2 4tr(fT f)) + 8ffT f 8g co(f)

its transpose is given by

F T = 4Q1 fT (1 + 4g2 4tr(fT f)) + 8fT ffT 8g adj(f)

and

Q = 1 16g4 + 128g det(f) 32g2tr(fT f) + 16(tr(fT f))2 32tr(fT ffT f). (A.7)

We are ready to discuss the computation of the expressions tr(F 2) trF T F , tr(F 4)

trF T F F T F and det(F ) in terms of f and g.

A.2 Outline of computation

To compute F T F, the following identities are useful to simplify the results:

2 = 1, (A.8)

fadj(f) = adj(f)f = det f, fT co(f) = co(f)fT = det f, (A.9)

adj(f)co(f) = (fT f)1 det(fT f) = adj(fT f)

= (fT f)2 + tr(fT f)fT f

1

2[(tr(fT f))2 tr((fT f)2)], (A.10)

where in the last equality we used the Cayley-Hamilton formula for 3 3 matrices. We

also need the Cayley-Hamilton formula of the form

(fT f)3 = tr(fT f)(fT f)2

1

2ijkabcFbjFck, (A.3)

det F

16ijkabcFaiFbjFck. (A.4)

JHEP11(2013)072

(A.5)

. (A.6)

1

2[(tr(fT f))2 tr((fT f)2)]fT f (det f)2. (A.11)

Using these formulas one can see that each term in the expression for F T F is proportional to either

1, or fT f, or (fT f)2. (A.12)

19

Therefore,

tr(F 2)

16Q2 = tr f2

+ 48g det(f) + 16tr f4

+ 8g2tr f2

2

8 trf2

+

64g det(f)tr f2

192g3 det(f) 192(det f)2 + 96g2tr f4

+ 16g4tr f2

64g2 trf2

2 16

trf2

3 + 32tr

f2 tr

f4 ,

(A.13)

where tr(f2) and tr(f4) are shorthand for tr(fT f) and tr((fT f)2).

We compute tr(F 4) and tr(F 6) tr((F T F )3) using the same method. We nally

trade tr(F 6) with det F using the Cayley-Hamilton formula

det F =

r16(tr(F 2)3 3tr(F 4)tr(F 2) + 2tr(F 6)) (A.14)

The explicit expression for det F in terms of f and g looks as follows

164Q3 det(F ) = det(f) + 12g2 det(f) + 48g4 det(f) + 64g6 det(f) + 192g det(f)2

+1280g3 det(f)2 512 det(f)3 4 det(f)tr(f2) 96g2 det(f)tr(f2) 320g4 det(f)tr(f2) + 256g det(f)2tr(f2) + 4g(trf2)2 + 32g3(trf2)2 +64g5(trf2)2 + 16 det(f)(trf2)2 + 320g2 det(f)(trf2)2 64 det(f)(trf2)3

+64g(trf2)4 4gtr(f4) 32g3tr(f4) 64g5tr(f4) 32 det(f)tr(f4) 640g2 det(f)tr(f4) + 128 det(f)tr(f2)tr(f4) 192g (trf2)2tr(f4)

+128g(trf4)2, (A.15)

We can now compute the expression in terms of f and g of the term in (4.31) containing the square root

s1 det(F 2)(1 + G2)2 + (G2 + tr (F 2)) +12((trF 2)2 tr (F 4)) 1 + G2

= Q3(1 + G2)1

v

u

u

t

JHEP11(2013)072

12

Xn=0 an(f)gn

!2, (A.16)

where the argument of the square root in the last line, which turns out to form a perfect square, is a polynomial in g with coe cients an(f) depending on tr(f2), tr(f4) and det(f).

The form of these coe cients is rather cumbersome, and we do not give it here. Using the above expressions we can then check that (A.1) indeed holds.

We now pass to the check of (A.2). In the matrix form it reads

~F = 4Q1 f(1 4g2 + 4trf2) 8ffT f + 8g co(f)

= 1

p det g LM5F T . (A.17)

This is a matrix equation, and we need to compute F , F F T F , and co(F ). To do this, we proceed as above and compute F , F F T F and F F T F F T F and then trade F F T F F T F

20

with co(F ) using the relation

co(F ) =

F F T F F T F + 12F (trF 2)2 tr(F 4)

tr(F 2)F F T F

det F . (A.18)

In the nal result, the matrices F , F F T F and co(F ) are expressed in terms of g, f, ffT f, and co(f). We can then substitute these into LM5/(

pdetg F ) which is

given by

1

p det g LM5 F T =

Gco(F )(1 + G2) +

det(F) co(F)(1+G2)2 +

(F tr(F 2)F F T F )

1+G2 + F

r1 det(F2)(1+G2)2 + (G2 + tr (F 2)) + 12((trF 2)2tr(F 4)) 1+G2,

(A.19)

JHEP11(2013)072

and check that eq. (A.2) does hold.

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