The equations of motion for a self-interacting self-dual tensor in six
dimensions are extracted from the equations describing the M-theory five-brane.
These equations are presented in a self-contained, six-dimensional
Lorentz-covariant form. In particular, it is shown that the field-strength
tensor satisfies a non-linear generalised self-duality constraint. The
self-duality equation is rewritten in five-dimensional notation and shown to be
identical to the corresponding equation in the non-covariant formalism.Comment: 8 pages, late

Aganagic

Bandos

E. Sezgin

Howe

P.C. West

P.S. Howe

Perry

Physics Letters B

English

arXiv

Crossref

The six-dimensional self-dual tensor

Howe, P.S.

Sezgin, E.

West, P.C.

Texas A&M University

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97
KCL-TH-97-11
CTP TAMU-12/97
NI-97008
hep-th/9702111
September 2, 2018
The Six-Dimensional Self-Dual Tensor
P.S. Howe 1, E. Sezgin 2† and P.C. West 3‡
1 Department of Mathematics, King’s College, London, UK
2 Center for Theoretical Physics, Texas A&M University, College Station, TX 77843, USA
3 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
Abstract
The equations of motion for a self-interacting self-dual tensor in six dimensions are extracted
from the equations describing the M -theory five-brane. These equations are presented in a
self-contained, six-dimensional Lorentz-covariant form. In particular, it is shown that the field-
strength tensor satisfies a non-linear generalised self-duality constraint. The self-duality equation
is rewritten in five-dimensional notation and shown to be identical to the corresponding equation
in the non-covariant formalism.
† Research supported in part by NSF Grant PHY-9411543
‡ Permanent Address: Department of Mathematics, King’s College, London, UK
1 Introduction
Super p-branes play a central roˆle in string duality and inM -theory, and it is therefore important
to understand their properties. One aspect of super p-branes which can be studied with available
mathematical tools is the construction of the worldsurface actions that describes their dynamics.
A well-known property of all the existing super p-brane actions is that in addition to being
worldsurface reparametrization invariant, they possess a fermionic gauge symmetry, called κ-
symmetry. Gauge-fixing of these symmetries leads to worldsurface supersymmetric field theories
that describe highly nonlinear self-interactions of matter supermultiplets.
The focus of this paper will be on the M -theory five-brane for which the resulting worldsurface
multiplet contains a chiral two-form. The full equations of motion for this object were given in
[1], using a concise superspace language. More recently, full actions for the same object have
been constructed in [2, 3] in different approaches, and the component version of the results
given in [1] has been obtained [4]. An interesting aspect of these results is the manner in
which the self-duality equation for the interacting chiral two-form arises. In an earlier work
that led to [2], Perry and Schwarz [5] constructed a self-duality equation with manifest five-
dimensional covariance, and they suggested that it was impossible to make manifest the hidden
six-dimensional covariance. On the other hand, the superspace formalism of [1] and the explicit
component results of [4] involved just such a system.
In view of these developments, it is of some interest to study the self-dual tensor in six dimensions
by itself, i.e. extracted from the brane context. This is what we shall do in this paper. We begin
in the next section by extracting these equations in terms of a self-dual three-form field of the
type introduced in [6] and then show that there is a new three-form which satisfies the Bianchi
identity and a generalised, but still manifestly six-dimensional covariant, duality constraint. In
the following section we rewrite our equations with only manifest five-dimensional covariance
in order to study the relationship between our results [1, 4] and those of [5]. The result we
find is that our six-dimensional covariant self-duality equation, when expressed in terms of five-
dimensional fields, reduces precisely to that of Perry and Schwarz. Section 4 contains some
concluding remarks.
2 The 6D Covariant Self-Duality Equation
In the superspace approach to the five-brane in eleven dimensions the antisymmetric tensor
makes its first appearance as a self-dual tensor habc which occurs directly in the embedding.
This field (or rather its leading component in a θ expansion) is not directly related to a two-
form potential, but it can be shown that there is superspace three-form H3 which is; in fact this
three-form satisfies
dH3 = −
1
4
H4 , (1)
where H4 is the pull-back of the four-form of eleven-dimensional supergravity. The purely
vectorial component of this three-form, which we denote by H ′abc, is related to h by
H ′abc = ma
dmb
ehcde (2)
where
ma
b := δba − 2ka
b , (3)
1
and where
ka
b := hacd h
bcd . (4)
We shall extract from the brane equations the equations describing a self-interacting self-dual
tensor field in six-dimensional flat spacetime by setting all the other fields equal to their flat
space values. The equations are self-duality,
habc =
1
3!
ǫabcdef hdef . (5)
with ǫ012345 = +1 and ηab = diag(−1,+1, . . . ,+1), and the equation of motion for h which
becomes
mab∂ahbcd = 0 . (6)
Equations (5) and (6) are equations in ordinary flat six-dimensional spacetime as are all sub-
sequent equations in this paper. The goal is to rewrite these equations in terms of the more
familiar H field. In fact equation (2) has been written in a slightly unusual basis and it is nec-
essary to correct this in order to find the relation between the components of H in a coordinate
basis and the components of h. This turns out to be
Habc = (m
−1)a
d hbcd . (7)
Through this relation the self duality of hbcd imposes a self duality condition on Habc which, as
we will show below, is a rather complicated condition.
Since these equations were derived from the superspace system the Bianchi identity (1) will
ensure that dH = 0 in the truncated theory, whereH is now the spacetime three-form. However,
to be completely self-contained we shall show that the Bianchi identity for H can be derived
from the basic equations for h. It follows from self-duality that the tensor k introduced in (4) is
traceless and that its square is proportional to the unit tensor, i.e., in matrix notation,
k2 = 1
6
tr k2 , (8)
and that
habeh
cde = −δ[a
[c kb]
d] , (9)
Therefore
m−1 = Q−1(1 + 2k) = Q−1(2−m) , (10)
where we have introduced
Q = 1− 2
3
tr k2 . (11)
We note in passing that the above expession for m−1 shows that H defined by (7) is indeed
totally antisymmetric since duality implies that ka
dhbcd is totally antisymmetric and anti-self-
dual (while ka
dkb
ehcde is totally antisymmetric and self-dual). Using this expression for m
−1 we
find
ǫabcdef∂
cHdef = 6∂c(Q−1mc
dhabd) (12)
= 6∂c
(
Q−1mc
d
)
habd , (13)
where in the second step we have used the equation of motion (6). It is not difficult to show,
again using the equations of motion, that
mab∂ambc = 0 . (14)
2
Using (10) in this equation we find
mab∂a(Q(m
−1)bc) = 0 , (15)
and this leads, after a little algebra, to
∂a(Q−1mab) = 0 . (16)
Hence we have established
ǫabcdef∂cHdef = 0 , (17)
as required.
We next translate the self-duality condition on h into a generalised self-duality condition for H.
Splitting H into its self-dual and anti-self-dual parts, H+ and H−, one finds from (7) that
H+abc = Q
−1habc , (18)
H−abc = Q
−1ka
dhdbc . (19)
If we define
Ka
b = H+acdH
+bcd , (20)
then
Kab = Q
−2kab . (21)
Now
H+ ·H− := H+abcH
−abc = Q−2trk2 = Q2trK2 , (22)
and
H− = Q2Ka
dH+bcd , (23)
so that we finally derive
H− = (trK2)−1(H+ ·H−)Ka
dH+bcd , (24)
where we recall that Ka
dH+bcd is totally antisymmetric and anti-self-dual as a consequence of the
self-duality of H+. This is the self-duality condition we were looking for. At first sight it does
not appear to determine H− in terms of H+ since it is clearly invariant under rescalings of H−.
However, the equation has been derived by purely algebraic manipulations and does not take
into account the Bianchi identity which must also be satisfied. When one does this one finds
that the freedom to rescale H− disappears.
3 The Self-dual Tensor in 5D Notation
In this section we will use hatted indices for six dimensions, and unhatted ones for five dimen-
sions. For clarity we shall also put a hat on the six-dimensional m-matrix. Our main purpose
is to analyse (7) in five-dimensional language. We begin by defining
fab := hab5 , (25)
Fab := Hab5 . (26)
Hence
habc =
1
2
ǫabcde f
de . (27)
3
Inverting (7) we get
h
aˆbˆcˆ
= mˆaˆ
dˆH
bˆcˆdˆ
(28)
from which it follows that
fab = ma
c Fcb . (29)
Finally, let us define
H˜ab =
1
3!ǫabcdeH
cde . (30)
Our goal is now to express H˜ in terms of Hab5 := Fab. To this end, we begin by expressing the
matrix m and its inverse in five-dimensional notation. Recalling the definition (3), one finds
ma
b = δa
b(1− 2tr f2) + 8(f2)a
b , (31)
ma
5 = −ǫabcde f
bc fde , (32)
m5
5 = (1 + 2tr f2) . (33)
The components of the inverse matrix can be calculated from (10):
(m−1)a
b = Q−1
[
δa
b(1 + 2tr f2)− 8(f2)a
b
]
, (34)
(m−1)a
5 = Q−1ǫabcde f
bc fde , (35)
(m−1)5
5 = Q−1(1− 2tr f2) , (36)
where
Q = 1 + 4(tr f2)2 − 16tr f4 . (37)
From (7) we have
Habc = (m
−1)a
d hdbc + (m
−1)a
5 fbc . (38)
Taking the dual of this equation, and using the definitions and formulae above, one finds
H˜ab = −Q
−1ma
c fcb , (39)
or, in matrix notation,
H˜ = −Q−1
[
(1− 2tr f2)f + 8f3
]
. (40)
It is straightforward even at this stage to write the matrix part of the above equation in terms
of F . Using the relation f = mF , the identity Fa
cmc
5 = 0, which again follows directly from
equation (7) and the above expressions for m, we can write
H˜ab = −Q
−1Fa
cGcb , (41)
where
Gab := ma
cˆmcˆb (42)
= (2− 4trf2 −Q)ηab + 16(f
2)ab , (43)
which follows from (31), (32) and (37). Multiplying (42) with F 2, using Fa
cmc
5 = 0 and recalling
that f = mF , we derive GF 2 = f2. Using this relation in (43) we can solve for G. Substituting
the result in (41) we find
H˜ = −Q−1(2− 4trf2 −Q)
(
F
1− 16F 2
)
. (44)
4
To find the final result we must express the traces of f in terms of F . At this point, it is
important to recall a well known identity that holds for any antisymmetric 5× 5 matrix X:
X5 = 1
4
[
trX4 − 1
2
(trX2)2
]
X + 1
2
(trX2)X3 . (45)
Applying this identity to the matrix F , we write
F 5 = (y2 −
1
2
y21)F + y1 F
3 , (46)
where
y1 =
1
2
trF 2 ,
y2 =
1
4
trF 4 . (47)
The next step in the analysis of (40) is to express f in terms of F . From the main duality
equation (29), and the general identity (45), it follows that f necessarily has the form
f = aF + b F 3 , (48)
where a and b are functions of F that can be determined from the duality equation (29), repro-
duced here for the reader’s convenience:
f = (1− 2tr f2 + 8f2)F . (49)
The f2 term can easily be computed from (48) and (46):
f2 =
[
a2 + b2(y2 −
1
2
y21)
]
F 2 +
(
2ab+ b2y1
)
F 4 . (50)
All the terms in (49) can be computed in a similar manner. At the end, comparing the coefficients
of the terms proportional to F and F 3, one finds
a+ 1
2
by1 = 1 , (51)
a2 + 2aby1 + b
2(y2 +
1
2
y21) =
1
8
b . (52)
The solution of these equations is given by
a± =
32− y1 ± y1Z
32y2 − 8y21
, b± =
1− 8y1 ± Z
16y2 − 4y21
, (53)
where
Z =
√
1− 16y1 + 128y
2
1 − 256y2 (54)
The equations (51) and (52) will be used frequently in the calculations to simplify the resulting
expressions. It turns out that we will not need to use (53) and therefore the final result is
independent of the sign ambiguity in this equation.
Having determined f explicitly in terms of F , we next evaluate tr f2, f3 and Q, in order to
express (40) in terms of F alone. An expression for tr f2 is easily obtained from (50). The
calculation of f3 requires a bit of work, but is straighforward. With the aid of (51) and (52),
we find
f3 =
(
2y2 − y
2
1 +
1
8
b
)
F + 1
8
[
a+ b(y1 − y
2
1 + 2y2)
]
F 3 . (55)
5
Putting together the results obtained so far, we find that the self-duality equation (40) takes
the simple form
H˜ = − 1
8
bQ−1
[
(1− 16y1)F + 16F
3
]
. (56)
There remains the evaluation of Q. Starting from (37), and using (50) to calculate (tr f2)2 and
tr f4, we find, with the aid of (51) and (52), the result
Q = 1
8
bZ . (57)
Therefore, recalling the definition of Z, and rescaling F → 14F , H → −
1
4H, we deduce the final
result
H˜ =
(1− y1)F + F
3√
1− y1 +
1
2
y21 − y2
=
√
1− y1 +
1
2
y21 − y2
(
F
1− F 2
)
. (58)
This formula is precisely the self-duality equation of Perry and Schwarz [5] (their equation (51)).
4 Conclusions
Our main result is that there does indeed exist a self-interacting self-dual tensor in six dimensions
for which the generalised duality relation and the equations of motion can be written in mani-
festly six-dimensional covariant form. Furthermore, the self-duality equation which arises natu-
rally in the description of the M-theory five-brane reduces precisely to the self-duality equation
of Perry and Schwarz, when expressed in five-dimensional language. Starting from the Perry-
Schwarz version, it may not have been obvious in the past how to obtain manifest six-dimensional
covariance. However, we now know the answer: first, one combines Habc = −
1
2
ǫabcdeH˜
de and Fab
into a six-dimensional field strength as
(Habc, Fab)→ Haˆbˆcˆ , (59)
where aˆ is now a six dimensional index, aˆ = (a, 5), and Fab := Hab5. Next, one defines three-form
h
aˆbˆcˆ
by
h
aˆbˆcˆ
=
(
δdˆaˆ − 2haˆeˆfˆ h
dˆeˆfˆ
)
H
bˆcˆdˆ
, (60)
then, as a consequence of the five-dimensional duality relation, the derived field h
aˆbˆcˆ
is self-dual.
It is interesting that this equation arises naturally in the supersurface embedding approach to
the formulation of super p-branes. The approach is by no means new, and it has been applied
to various p-branes in the past, with varying degrees of success. We refer the reader to [6] for
a brief review of the extensive literature on the subject. The point we wish to emphasize here
is that the approach is very natural, and its power becomes apparent when one gets to terms
with the idea of searching for the equations of motion first and later deriving an action, when
possible.
The beauty of the superembedding approach is that not only it treats the worldsurface and target
space supersymmetry on an equal footing by considering the embedding of a world supersurface
into the target superspace, but it applies to all super p-branes, regardless of the precise nature
of the worldsurface multiplet, in a universal way that has a natural geometrical interpretation.
Indeed, new kinds of super p-branes, e.g. L-branes, which have matter supermultiplets other
than those considered so far have been suggested in [6], and we have already preliminary results
6
which show that things work in much the same way they do in the case of M-theory five-brane.
We should expect, therefore, the uncovering of interesting new results on nonlinear self-couplings
of matter supermultiplets in diverse dimensions which have not been explored so far.
Finally, it would be interesting to investigate various aspects of the M -theory five-brane self-
duality equation discussed in this paper equation further. We have already exhibited the six-
dimensional self-dual tensor equations as part of the full set of equations for the five-brane in an
eleven-dimensional supergravity background [1, 4]. A lot remains to be done, however, and we
will report elsewhere [7] on further aspects of the emerging picture of the M -theory five-brane
which seems to possess many interesting features.
Acknowledgements. We thank Michael Duff and Per Sundell for many stimulating discussions
on super five-branes.
7
References
[1] P.S. Howe and E. Sezgin, D=11,p=5, hep-th/9611008.
[2] I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. Sorokin and M. Tonin, Covariant
action for the super five-brane of M theory, hep-th/9701149.
[3] M. Aganagic, J. Park, C. Popescu and J.H. Schwarz, Worldvolume action of the M theory
five-brane, hep-th/9701166.
[4] P.S. Howe, E. Sezgin and P.C. West, Covariant field equations of the M theory five-brane,
hep-th/9702008.
[5] M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six dimensions and Born-Infeld
theory, hep-th/9611065.
[6] P.S. Howe and E. Sezgin, Superbranes, hep-th/9607227.
[7] P.S. Howe, E. Sezgin and P.C. West, in preparation.
8


Texas A&amp;M Repository

arXiv:hep-th/9702111v1  14 Feb 1997KCL-TH-97-11CTP TAMU-12/97NI-97008hep-th/9702111September 2, 2018The Six-Dimensional Self-Dual TensorP.S. Howe 1, E. Sezgin 2† and P.C. West 3‡1 Department of Mathematics, King’s College, London, UK2 Center for Theoretical Physics, Texas A&M University, College Station, TX 77843, USA3 Isaac Newton Institute for Mathematical Sciences, Cambridge, UKAbstractThe equations of motion for a self-interacting self-dual tensor in six dimensions are extractedfrom the equations describing the M -theory five-brane. These equations are presented in aself-contained, six-dimensional Lorentz-covariant form. In particular, it is shown that the field-strength tensor satisfies a non-linear generalised self-duality constraint. The self-duality equationis rewritten in five-dimensional notation and shown to be identical to the corresponding equationin the non-covariant formalism.† Research supported in part by NSF Grant PHY-9411543‡ Permanent Address: Department of Mathematics, King’s College, London, UK1 IntroductionSuper p-branes play a central roˆle in string duality and inM -theory, and it is therefore importantto understand their properties. One aspect of super p-branes which can be studied with availablemathematical tools is the construction of the worldsurface actions that describes their dynamics.A well-known property of all the existing super p-brane actions is that in addition to beingworldsurface reparametrization invariant, they possess a fermionic gauge symmetry, called κ-symmetry. Gauge-fixing of these symmetries leads to worldsurface supersymmetric field theoriesthat describe highly nonlinear self-interactions of matter supermultiplets.The focus of this paper will be on the M -theory five-brane for which the resulting worldsurfacemultiplet contains a chiral two-form. The full equations of motion for this object were given in[1], using a concise superspace language. More recently, full actions for the same object havebeen constructed in [2, 3] in different approaches, and the component version of the resultsgiven in [1] has been obtained [4]. An interesting aspect of these results is the manner inwhich the self-duality equation for the interacting chiral two-form arises. In an earlier workthat led to [2], Perry and Schwarz [5] constructed a self-duality equation with manifest five-dimensional covariance, and they suggested that it was impossible to make manifest the hiddensix-dimensional covariance. On the other hand, the superspace formalism of [1] and the explicitcomponent results of [4] involved just such a system.In view of these developments, it is of some interest to study the self-dual tensor in six dimensionsby itself, i.e. extracted from the brane context. This is what we shall do in this paper. We beginin the next section by extracting these equations in terms of a self-dual three-form field of thetype introduced in [6] and then show that there is a new three-form which satisfies the Bianchiidentity and a generalised, but still manifestly six-dimensional covariant, duality constraint. Inthe following section we rewrite our equations with only manifest five-dimensional covariancein order to study the relationship between our results [1, 4] and those of [5]. The result wefind is that our six-dimensional covariant self-duality equation, when expressed in terms of five-dimensional fields, reduces precisely to that of Perry and Schwarz. Section 4 contains someconcluding remarks.2 The 6D Covariant Self-Duality EquationIn the superspace approach to the five-brane in eleven dimensions the antisymmetric tensormakes its first appearance as a self-dual tensor habc which occurs directly in the embedding.This field (or rather its leading component in a θ expansion) is not directly related to a two-form potential, but it can be shown that there is superspace three-form H3 which is; in fact thisthree-form satisfiesdH3 = −14H4 , (1)where H4 is the pull-back of the four-form of eleven-dimensional supergravity. The purelyvectorial component of this three-form, which we denote by H ′abc, is related to h byH ′abc = madmbehcde (2)wheremab := δba − 2kab , (3)1and wherekab := hacd hbcd . (4)We shall extract from the brane equations the equations describing a self-interacting self-dualtensor field in six-dimensional flat spacetime by setting all the other fields equal to their flatspace values. The equations are self-duality,habc =13!ǫabcdef hdef . (5)with ǫ012345 = +1 and ηab = diag(−1,+1, . . . ,+1), and the equation of motion for h whichbecomesmab∂ahbcd = 0 . (6)Equations (5) and (6) are equations in ordinary flat six-dimensional spacetime as are all sub-sequent equations in this paper. The goal is to rewrite these equations in terms of the morefamiliar H field. In fact equation (2) has been written in a slightly unusual basis and it is nec-essary to correct this in order to find the relation between the components of H in a coordinatebasis and the components of h. This turns out to beHabc = (m−1)ad hbcd . (7)Through this relation the self duality of hbcd imposes a self duality condition on Habc which, aswe will show below, is a rather complicated condition.Since these equations were derived from the superspace system the Bianchi identity (1) willensure that dH = 0 in the truncated theory, whereH is now the spacetime three-form. However,to be completely self-contained we shall show that the Bianchi identity for H can be derivedfrom the basic equations for h. It follows from self-duality that the tensor k introduced in (4) istraceless and that its square is proportional to the unit tensor, i.e., in matrix notation,k2 = 16tr k2 , (8)and thathabehcde = −δ[a[c kb]d] , (9)Thereforem−1 = Q−1(1 + 2k) = Q−1(2−m) , (10)where we have introducedQ = 1− 23tr k2 . (11)We note in passing that the above expession for m−1 shows that H defined by (7) is indeedtotally antisymmetric since duality implies that kadhbcd is totally antisymmetric and anti-self-dual (while kadkbehcde is totally antisymmetric and self-dual). Using this expression for m−1 wefindǫabcdef∂cHdef = 6∂c(Q−1mcdhabd) (12)= 6∂c(Q−1mcd)habd , (13)where in the second step we have used the equation of motion (6). It is not difficult to show,again using the equations of motion, thatmab∂ambc = 0 . (14)2Using (10) in this equation we findmab∂a(Q(m−1)bc) = 0 , (15)and this leads, after a little algebra, to∂a(Q−1mab) = 0 . (16)Hence we have establishedǫabcdef∂cHdef = 0 , (17)as required.We next translate the self-duality condition on h into a generalised self-duality condition for H.Splitting H into its self-dual and anti-self-dual parts, H+ and H−, one finds from (7) thatH+abc = Q−1habc , (18)H−abc = Q−1kadhdbc . (19)If we defineKab = H+acdH+bcd , (20)thenKab = Q−2kab . (21)NowH+ ·H− := H+abcH−abc = Q−2trk2 = Q2trK2 , (22)andH− = Q2KadH+bcd , (23)so that we finally deriveH− = (trK2)−1(H+ ·H−)KadH+bcd , (24)where we recall that KadH+bcd is totally antisymmetric and anti-self-dual as a consequence of theself-duality of H+. This is the self-duality condition we were looking for. At first sight it doesnot appear to determine H− in terms of H+ since it is clearly invariant under rescalings of H−.However, the equation has been derived by purely algebraic manipulations and does not takeinto account the Bianchi identity which must also be satisfied. When one does this one findsthat the freedom to rescale H− disappears.3 The Self-dual Tensor in 5D NotationIn this section we will use hatted indices for six dimensions, and unhatted ones for five dimen-sions. For clarity we shall also put a hat on the six-dimensional m-matrix. Our main purposeis to analyse (7) in five-dimensional language. We begin by definingfab := hab5 , (25)Fab := Hab5 . (26)Hencehabc =12ǫabcde fde . (27)3Inverting (7) we gethaˆbˆcˆ= mˆaˆdˆHbˆcˆdˆ(28)from which it follows thatfab = mac Fcb . (29)Finally, let us defineH˜ab =13!ǫabcdeHcde . (30)Our goal is now to express H˜ in terms of Hab5 := Fab. To this end, we begin by expressing thematrix m and its inverse in five-dimensional notation. Recalling the definition (3), one findsmab = δab(1− 2tr f2) + 8(f2)ab , (31)ma5 = −ǫabcde fbc fde , (32)m55 = (1 + 2tr f2) . (33)The components of the inverse matrix can be calculated from (10):(m−1)ab = Q−1[δab(1 + 2tr f2)− 8(f2)ab], (34)(m−1)a5 = Q−1ǫabcde fbc fde , (35)(m−1)55 = Q−1(1− 2tr f2) , (36)whereQ = 1 + 4(tr f2)2 − 16tr f4 . (37)From (7) we haveHabc = (m−1)ad hdbc + (m−1)a5 fbc . (38)Taking the dual of this equation, and using the definitions and formulae above, one findsH˜ab = −Q−1mac fcb , (39)or, in matrix notation,H˜ = −Q−1[(1− 2tr f2)f + 8f3]. (40)It is straightforward even at this stage to write the matrix part of the above equation in termsof F . Using the relation f = mF , the identity Facmc5 = 0, which again follows directly fromequation (7) and the above expressions for m, we can writeH˜ab = −Q−1FacGcb , (41)whereGab := macˆmcˆb (42)= (2− 4trf2 −Q)ηab + 16(f2)ab , (43)which follows from (31), (32) and (37). Multiplying (42) with F 2, using Facmc5 = 0 and recallingthat f = mF , we derive GF 2 = f2. Using this relation in (43) we can solve for G. Substitutingthe result in (41) we findH˜ = −Q−1(2− 4trf2 −Q)(F1− 16F 2). (44)4To find the final result we must express the traces of f in terms of F . At this point, it isimportant to recall a well known identity that holds for any antisymmetric 5× 5 matrix X:X5 = 14[trX4 − 12(trX2)2]X + 12(trX2)X3 . (45)Applying this identity to the matrix F , we writeF 5 = (y2 −12y21)F + y1 F3 , (46)wherey1 =12trF 2 ,y2 =14trF 4 . (47)The next step in the analysis of (40) is to express f in terms of F . From the main dualityequation (29), and the general identity (45), it follows that f necessarily has the formf = aF + b F 3 , (48)where a and b are functions of F that can be determined from the duality equation (29), repro-duced here for the reader’s convenience:f = (1− 2tr f2 + 8f2)F . (49)The f2 term can easily be computed from (48) and (46):f2 =[a2 + b2(y2 −12y21)]F 2 +(2ab+ b2y1)F 4 . (50)All the terms in (49) can be computed in a similar manner. At the end, comparing the coefficientsof the terms proportional to F and F 3, one findsa+ 12by1 = 1 , (51)a2 + 2aby1 + b2(y2 +12y21) =18b . (52)The solution of these equations is given bya± =32− y1 ± y1Z32y2 − 8y21, b± =1− 8y1 ± Z16y2 − 4y21, (53)whereZ =√1− 16y1 + 128y21 − 256y2 (54)The equations (51) and (52) will be used frequently in the calculations to simplify the resultingexpressions. It turns out that we will not need to use (53) and therefore the final result isindependent of the sign ambiguity in this equation.Having determined f explicitly in terms of F , we next evaluate tr f2, f3 and Q, in order toexpress (40) in terms of F alone. An expression for tr f2 is easily obtained from (50). Thecalculation of f3 requires a bit of work, but is straighforward. With the aid of (51) and (52),we findf3 =(2y2 − y21 +18b)F + 18[a+ b(y1 − y21 + 2y2)]F 3 . (55)5Putting together the results obtained so far, we find that the self-duality equation (40) takesthe simple formH˜ = − 18bQ−1[(1− 16y1)F + 16F3]. (56)There remains the evaluation of Q. Starting from (37), and using (50) to calculate (tr f2)2 andtr f4, we find, with the aid of (51) and (52), the resultQ = 18bZ . (57)Therefore, recalling the definition of Z, and rescaling F → 14F , H → −14H, we deduce the finalresultH˜ =(1− y1)F + F3√1− y1 +12y21 − y2=√1− y1 +12y21 − y2(F1− F 2). (58)This formula is precisely the self-duality equation of Perry and Schwarz [5] (their equation (51)).4 ConclusionsOur main result is that there does indeed exist a self-interacting self-dual tensor in six dimensionsfor which the generalised duality relation and the equations of motion can be written in mani-festly six-dimensional covariant form. Furthermore, the self-duality equation which arises natu-rally in the description of the M-theory five-brane reduces precisely to the self-duality equationof Perry and Schwarz, when expressed in five-dimensional language. Starting from the Perry-Schwarz version, it may not have been obvious in the past how to obtain manifest six-dimensionalcovariance. However, we now know the answer: first, one combines Habc = −12ǫabcdeH˜de and Fabinto a six-dimensional field strength as(Habc, Fab)→ Haˆbˆcˆ , (59)where aˆ is now a six dimensional index, aˆ = (a, 5), and Fab := Hab5. Next, one defines three-formhaˆbˆcˆbyhaˆbˆcˆ=(δdˆaˆ − 2haˆeˆfˆ hdˆeˆfˆ)Hbˆcˆdˆ, (60)then, as a consequence of the five-dimensional duality relation, the derived field haˆbˆcˆis self-dual.It is interesting that this equation arises naturally in the supersurface embedding approach tothe formulation of super p-branes. The approach is by no means new, and it has been appliedto various p-branes in the past, with varying degrees of success. We refer the reader to [6] fora brief review of the extensive literature on the subject. The point we wish to emphasize hereis that the approach is very natural, and its power becomes apparent when one gets to termswith the idea of searching for the equations of motion first and later deriving an action, whenpossible.The beauty of the superembedding approach is that not only it treats the worldsurface and targetspace supersymmetry on an equal footing by considering the embedding of a world supersurfaceinto the target superspace, but it applies to all super p-branes, regardless of the precise natureof the worldsurface multiplet, in a universal way that has a natural geometrical interpretation.Indeed, new kinds of super p-branes, e.g. L-branes, which have matter supermultiplets otherthan those considered so far have been suggested in [6], and we have already preliminary results6which show that things work in much the same way they do in the case of M-theory five-brane.We should expect, therefore, the uncovering of interesting new results on nonlinear self-couplingsof matter supermultiplets in diverse dimensions which have not been explored so far.Finally, it would be interesting to investigate various aspects of the M -theory five-brane self-duality equation discussed in this paper equation further. We have already exhibited the six-dimensional self-dual tensor equations as part of the full set of equations for the five-brane in aneleven-dimensional supergravity background [1, 4]. A lot remains to be done, however, and wewill report elsewhere [7] on further aspects of the emerging picture of the M -theory five-branewhich seems to possess many interesting features.Acknowledgements. We thank Michael Duff and Per Sundell for many stimulating discussionson super five-branes.7References[1] P.S. Howe and E. Sezgin, D=11,p=5, hep-th/9611008.[2] I. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. Sorokin and M. Tonin, Covariantaction for the super five-brane of M theory, hep-th/9701149.[3] M. Aganagic, J. Park, C. Popescu and J.H. Schwarz, Worldvolume action of the M theoryfive-brane, hep-th/9701166.[4] P.S. Howe, E. Sezgin and P.C. West, Covariant field equations of the M theory five-brane,hep-th/9702008.[5] M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six dimensions and Born-Infeldtheory, hep-th/9611065.[6] P.S. Howe and E. Sezgin, Superbranes, hep-th/9607227.[7] P.S. Howe, E. Sezgin and P.C. West, in preparation.8

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The Six Dimensional Self-Dual Tensor

The Six Dimensional Self-Dual Tensor

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The Six Dimensional Self-Dual Tensor

Abstract

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Texas A&M University

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