According to string/fivebrane duality, the Green-Schwarz factorization of the
$D=10$ spacetime anomaly polynomial $I_{12}$ into $X_4\, X_8$ means that just
as $X_4$ is the anomaly polynomial of the $d=2$ string worldsheet so $X_8$
should be the anomaly polynomial of the $d=6$ fivebrane worldvolume. To test
this idea we perform a fivebrane calculation of $X_8$ and find perfect
agreement with the string one--loop result.Comment: 14 pages, CERN TH-6614/92, CTP-TAMU 60/9

A. Achucarro

A. Dabholkar

A. H. Chamseddine

A. Salam

A. Strominger

C. Callan

E. Bergshoeff

E. Witten

G. F. Chapline

G. Horowitz

I. Pesando

J. A. Dixon

J. C. Plefka

L. Alvarez-Gaumé

M. B. Green

M. J. Duff

M. T. Grisaru

R. Bott

S. J. Gates

English

arXiv

Crossref

Physical Review Letters

Putting string–five-brane duality to the test

Dixon, JA

Duff, Michael J

Plefka, JC

Texas A&M University

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CERN-TH. 6614/92
CTP-TAMU- 60/92
PUTTING STRING/FIVEBRANE DUALITY TO
THE TEST †
J. A. Dixon 2, M. J. Duff 1,2 and J. C. Plefka 2 ‡
1 Theory Division
CERN
CH–1211 Geneva 23
2 Center for Theoretical Physics
Physics Department
Texas A & M University
College Station, Texas 77843
ABSTRACT
According to string/fivebrane duality, the Green-Schwarz factorization of
the D = 10 spacetime anomaly polynomial I12 into X4X8 means that just
as X4 is the anomaly polynomial of the d = 2 string worldsheet so X8 should
be the anomaly polynomial of the d = 6 fivebrane worldvolume. To test this
idea we perform a fivebrane calculation of X8 and find perfect agreement
with the string one–loop result.
PACS numbers: 11.17+y, 11.10.Kk
CERN-TH. 6614/92
August 1992
† Research supported in part by NSF Grant PHY-9106593
‡ Supported by a Fulbright Scholarship
The dual formulations of D = 10 supergravity, one with a 7–form field
strength 1 and the other with a 3–form field strength 2 have long been some-
thing of an enigma from the point of view of superstrings. As field theories,
each seems equally as good. In particular, provided we couple them to E8×E8
or SO(32) super Yang-Mills, then both are anomaly–free 3, 4. Since the 3–
form version corresponds to the field theory limit of the heterotic string, it
was natural to conjecture 5 that the 7–form version corresponds to the field
theory limit of an extended object dual to the string: the “heterotic five-
brane”. Just as the 2–form potential BMN (M = 0, 1, . . . , 9) couples to the
d = 2 string worldsheet via the term
S2 =
1
2πα′
∫
d2ξ
1
2
ǫij∂ix
M∂jx
NBMN =
1
2πα′
∫
B2 (1)
where ξi (i = 1, 2) are the worldsheet coordinates and (2πα′)−1 is the string
tension, so the 6–form potential BMNPQRS (M = 0, 1, . . . , 9) couples to the
d = 6 fivebrane worldvolume via the term
S6 =
1
(2π)3β ′
∫
d6ξ
1
6!
ǫijklmn∂ix
M∂jx
N∂kx
P∂lx
Q∂mx
R∂nx
SBMNPQRS
=
1
(2π)3β ′
∫
B6 (2)
where ξi (i = 1, . . . , 6) are the worldvolume coordinates and [(2π)3β ′]−1 is
the fivebrane tension. Writing H3 = dB2+O(α
′) and H7 = dB6+O(β
′), the
relation
H7 = e
−φ ∗H3 (3)
1
where φ is the D = 10 dilaton and ∗ denotes the Hodge dual using the canon-
ical metric gMN(can), ensures that the roles of field equations and Bianchi
identities of the 3–form version of supergravity are interchanged in the 7–
form version. The main subject of this paper will be the O(α′) corrections
to H3 and the O(β
′) corrections to H7, discussed below.
An existence proof for this heterotic fivebrane was provided by Strominger 6,
who showed that the heterotic fivebrane emerges as a soliton solution of the
3–form version. He went on to suggest that the strong coupling regime of
the string should correspond to the weak coupling regime of the fivebrane;
an idea made quantitatively more precise in Ref. 7, where it was shown
that the σ–model metrics gMN(string) and gMN(fivebrane) are related to the
canonical metric by e−φ/2 gMN(string) = gMN(can) = e
φ/6 gMN(fivebrane)
and hence that g(fivebrane) = g(string)−1/3 where g(string) and g(fivebrane)
are the string and fivebrane loop expansion parameters. The same paper also
established the Dirac quantization rule
2κ2 = n(2π)5α′β ′ , n = integer (4)
where κ2 is the D = 10 gravitational constant. Further evidence was pro-
vided by the complementary discovery that a heterotic string emerges as a
soliton solution of the 7–form version 8. Both the string and fivebrane soli-
ton solutions break half the supersymmetries, both saturate a Bogomol’nyi
bound between the mass and the topological charge, and both go over into
the corresponding elementary string 9 and elementary fivebrane 10 solutions
2
at large distances. These elementary solutions are the extreme mass = charge
limit of the black string and black fivebrane which display event horizons and
a singularity at the origin 11. However, they are mutually non–singular in the
sense that the string is a non–singular solution of the 7–form version and the
fivebrane a non–singular solution of the 3–form version 12. Recent work on
conformal field theories 13 and other exact solutions 14 are also all consistent
with this string/fivebrane duality conjecture.
Crucial to the solution of Ref. 8 was the observation that duality mixes
up string and fivebrane loops: what is a one loop effect for the string might
be a tree level effect for the fivebrane, and vice versa. At higher loop orders,
this leads to an infinite number of non–renormalization theorems (includ-
ing the vanishing of the cosmological term) all of which are consistent with
known string calculations to higher orders both in α′ (worldsheet loops) and
g(string) (spacetime loops) 15. It is this loop mixing which allows us to test
string/fivebrane duality, in spite of our ignorance of how to quantize the five-
brane. If duality is correct, we should be able to reproduce string loop effects
from tree–level fivebranes !
To see this, let us first consider the well–known SO(32) Yang–Mills Chern–
Simons corrections to H3. (E8 × E8 requires a separate treatment and will
be discussed elsewhere). In bosonic formulation, these are obtained 16 by
augmenting the action S2 of (1) by the WZW terms
S2
′ = −n2
6π
N2
∫
∂M3
trAK + n2
2π
N2
∫
M3
trK3 (5)
3
where A = AM dx
M , K = g−1dg and where the gauge fields AM = AM
a T a
and the group elements g are matrices in the fundamental representation of
SO(32). Here n2 = integer is the level of the Kac–Moody algebra and N2 is
a normalization constant given by the general formula 17
N2n−2 =
(2π
i
)n (2n− 1)!
(n− 1)!
(6)
However, in 8k+2 dimensions for which n = 4k+2 we may define Majorana–
Weyl fermions and the corresponding WZW terms are to be divided by 2.
This is the case for the heterotic string, so N2 = −48π
2. This agrees with
Witten 18. Let us define F = dA+ A2 and
I2n =
( i
2π
)n 1
n!
trF n
dω2n−1 = I2n
δω2n−1 = dω
1
2n−2 (7)
then the sum S2 + S2
′ is gauge invariant under δA = dλ + [A, λ] and δK =
dλ+ [K, λ] provided
δB2 =
n2
2
α′(2π)2ω1
2
(8)
and hence the gauge invariant field strength is given by 19
H3 = dB2 −
n2
2
α′(2π)2ω3
dH3 = −
n2
2
α′(2π)2I4 (9)
This modification to the Bianchi identity is thus seen to be a classical string
effect (i.e. tree level in the D = 10 string loop expansion).
4
Next we turn to the Green–Schwarz anomaly cancellation mechanism 3
which is a genuine quantum (string one loop) effect. The Majorana–Weyl
gauginos of the D = 10 theory belong to the dimension 496 adjoint rep-
resentation. The non–abelian anomaly polynomial is given by the abelian
anomaly in d = 12, which is given by 1
2
I12 of (7) with the fundamental tr
replaced by the adjoint Tr. The factor of 1
2
arises because the fermions are
Majorana. As emphasized by Green and Schwarz 3, the miracle of SO(32) is
that since TrF 6 = TrF 2TrF 4 /48− (TrF 2)3 /14, 400, I12 factorizes:
1
2
I12 = X4X8 (10)
In fact, since TrF 4 = 24 trF 4 + 3(trF 2)2 and TrF 2 = 30 trF 2, we have the
remarkable coincidence
X4 =
1
2
I4 X8 = I8 (11)
The consistent anomaly
G =
1
2
· 2π
∫ (1
3
ω1
2
I8 +
2
3
ω1
6
I4
)
(12)
is then cancelled by adding to the effective action
∆Γ2 = −2π
∫ ( 1
n2α′(2π)2
B2 I8 +
1
3
ω3 ω7
)
(13)
and recalling the transformation rule for B2 given in (8). Now for B2 nor-
malized as in (1), its kinetic term is
Γ2 = −
1
2κ2
∫
1
2
e−φH3 ∧ ∗H3 (14)
5
and hence the addition of (13) modifies the field equation to
d(e−φ ∗H3) =
2κ2
n2α′(2π)
I8 (15)
So far, all our considerations started with the string worldsheet. The
acid test for string/fivebrane duality is to reproduce (15) starting from the
fivebrane worldvolume. We begin by augmenting the action S6 of (2) by the
WZW term 20
S6
′ =
70πn6
N6
∫
∂M7
C6 −
2πn6
N6
∫
M7
trK7 (16)
The explicit form for C6 is given in Ref. 20. Here we need only note that it
transforms as
δC6 = 24(2π)
4[ω1
6
(A, λ)− ω1
6
(K, λ)] (17)
Here n6 = integer is the level of the Mickelsson–Faddeev algebra
21 and N6 =
(2π)47!/3! from (6). The sum S6 + S6
′ is gauge invariant provided
δB6 = −n6β
′(2π)4ω1
6
(18)
and hence the gauge invariant field strength is given by 19
H7 = dB6 + n6β
′(2π)4ω7 (19)
dH7 = n6β
′(2π)4I8 (20)
Using (3), this is identical to (15) provided
2κ2 = n2n6(2π)
5α′β ′ (21)
But this is just the Dirac quantization rule (4) with n = n2n6 !
6
Having established that the classical fivebrane correctly reproduces the
quantum string result in the pure Yang–Mills sector, we now turn to the grav-
itational and mixed anomalies. Now one must include the D = 10 gravitino
contribution to I12, but again it factorizes as in (10) where now
22
X4 =
1
2
·
1
(2π)2
[
−
1
2
trF 2 +
1
2
trR2
]
X8 =
( 1
2π
)4[ 1
24
trF 4 −
1
192
trF 2 trR2 +
1
768
(trR2)2 +
1
192
trR4
]
(22)
Since we have already obtained the correct overall normalization of X4 and
X8 it remains only to explain the relative coefficients. (We set n2 = n6 = 1
from now on). This is most easily done by changing from the bosonic WZW
formalism to the fermionic one where I4 and I8 are the anomaly polyno-
mials arising via the index theorem from the chiral fermions on the d = 2
worldsheet and d = 6 worldvolumes respectively. We should mention that
although we have the luxury of choosing a bosonic or fermionic formulation
on the d = 2 worldsheet, there is probably no such bose–fermi equivalence
in d = 6 and so ultimately one will have to choose between the bosonic
and fermionic formulations. This choice must await a complete covariant κ–
symmetric Green–Schwarz action for the heterotic fivebrane which, to date,
is still lacking 23. Fortunately, for the present purposes of calculating anoma-
lies, either way will do. In the Yang–Mills case, this may be seen explicitly
via the Euclidean identities for the index of the Dirac operator 24
(ind iD/)2n−2 =
∫
I2n =
( i
2π
)n 1
n!
∫
trF n
7
= (−1)n−1
( i
2π
)n (n− 1)!
(2n− 1)!
∫
trK2n−1 (23)
which provide an independent check on the equivalence of the WZW and chi-
ral fermion calculations. For the gravitational anomalies, the WZW method
is unknown to us and we shall follow the fermionic approach where 24
I4 =
1
(2π)2
[i2
2
trF 2 +
r
48
trR2
]
I8 =
1
(2π)4
[ i4
24
trF 4 +
i2
96
trF 2 trR2 +
r
4608
(trR2)2 +
r
5760
trR4
]
(24)
Here the Yang–Mills trace is in whatever representation the fermions are in
and r is its dimensionality and the Lorentz trace is in the vector represen-
tation. In the Green–Schwarz formalism the heterotic string is described by
superspace coordinates (XM , θα) where the θ’s are in the 16 of SO(1, 9) and
the SO(32) quantum numbers are carried by Majorana–Weyl fermions in the
32–dimensional fundamental representation. Although the θ’s are worldsheet
scalars, they are anticommuting and obey a first order Lagrangian. However,
because of κ symmetry, only 8 of the θ’s are physical (equal to 10 − 2, the
number of transverse X ’s). The net effect of integrating out the θ’s is thus
equivalent to 8 Majorana–Weyl fermions but with the opposite chirality to
the 32 gauge fermions 25. In summary we calculate X4 by dividing I4 by
two (because the fermions are Majorana), taking the tr in the fundamental
representation and setting r = 32 − 8 = 24. This agrees with (22). In the
case of the fivebrane we again have the same (XM , θα) and we again take the
gauge fermions in the 32 of SO(32). Because we are in d = 6, however, the
8
fermions are no longer Majorana–Weyl (which exist only in 2 mod 8 dimen-
sions) and so we do not divide I8 by 2. Moreover in d = 6, the number of
physical θ’s is only 4 (equal to 10 − 6 , the number of transverse X ’s) since
κ–symmetry halves the number and (unlike d = 2) going on–shell halves it
again. The net effect of integrating out the θ’s is thus equivalent to only two
d = 6 fermions 25 and hence r = 32− 2 = 30. Thus
I8 =
1
(2π)4
[ 1
24
trF 4 −
1
96
trF 2 trR2 +
15
2304
(trR2)2 +
1
192
trR4
]
(25)
This is not quite in agreement with X8 of (22) but the difference is propor-
tional to trR2X4 and hence the discrepancy is easily remedied. Up until now
we have been assuming that the fivebrane B6 which appears in (2) was identi-
cal to theD = 10 supergravity B6 which satisfies (3). However, this may need
to be modified when we include the gravitational Chern–Simons corrections
which are of higher order in the low–energy expansion than those of Yang–
Mills. The Green–Schwarz result is dH7(supergravity) = β
′(2π)4X8, whereas
our fivebrane calculations have lead us to dH7(fivebrane) = β
′(2π)4 I8. So if
we define
B6(fivebrane) = B6(supergravity)− c
β ′
α′
trR2B2 (26)
then the gauge–invariant field strength H7(supergravity) satisfies
dH7(supergravity) = β
′(2π)4 (I8 −
c
(2π)2
trR2X4) (27)
on using (9). Comparing X8 of (22) with I8 of (25) and using X4 of (22),
we find perfect agreement with the Green–Schwarz result with the choice
9
c = 1/48.
In their discussion of anomalies, Green, Schwarz and Witten 22 say “The
case where something really new happens is that of 4k+2 dimensions and this
is the case of interest to superstring theory since the worldsheet has dimension
two and the spacetime has dimension ten !”. The results of the present
paper might be summarized by adding “. . . and the fivebrane worldvolume
has dimension six !”.
Acknowledgements
Conversations with L. Alvarez–Gaume´, E. Bergshoeff, B. Campbell, M.
Grisaru, R. Stora and P. Townsend are gratefully acknowledged.
10
References
[1] A. H. Chamseddine, Nucl. Phys. 185 (1981) 403.
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1208 (1983) 105.
[3] M. B. Green and J. H. Schwarz, Phys. Lett. 49B (1984) 117.
[4] S. J. Gates and H. Nishino, Phys. Lett. B173 (1986) 52; A. Salam and
E. Sezgin, Physica Scripta 32 (1985) 283.
[5] M. J. Duff, Class. Quantum Grav. 5 (1988) 189.
[6] A. Strominger, Nucl. Phys. B343 (1990) 167.
[7] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 129.
[8] M. J. Duff and J. X. Lu, Phys. Rev. Lett. 66 (1991) 1402; Class. Quan-
tum Grav. 9 (1991) 1.
[9] A. Dabholkar, G.W. Gibbons, J. A. Harvey and F. Ruiz–Ruiz, Nucl.
Phys. B340 (1990) 33.
[10] M. J. Duff and J.X. Lu, Nucl. Phys. B354 (1991) 141.
[11] G. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 409.
[12] M.J. Duff, R.R. Khuri and J. X. Lu, Nucl. Phys. B377 (1992) 281.
11
[13] C. Callan, J. Harvey and A. Strominger, Nucl. Phys. B359 (1991) 611;
Nucl. Phys. B367 (1991) 60; preprint EFI–91–66.
[14] I. Pesando and A. K. Tollsten, Phys. Lett. B274 (1992) 374.
[15] M. J. Duff and J. X. Lu, Nucl. Phys. B357 (1991) 354.
[16] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Lett. B163 (1985)
343; M. J. Duff, B. E. W. Nilssson, C. N. Pope and N. P. Warner, Phys.
Lett. B171 (1986) 170; E. Bergshoeff, F. Delduc and E. Sokatchev,
Phys. Lett. B262 (1991) 444.
[17] R. Bott and R. Seeley, Comm. Math. Phys. 62 (1987) 235.
[18] E. Witten, Comm. Math. Phys. 92 (1984) 455.
[19] These results imply that previous versions of these Chern–Simons cor-
rections quoted in Refs. [6,8,13,15,20] for H3 and Refs. [8,15,20] for H7
corresponded to the choice n2 = n6 = 8.
[20] J. A. Dixon, M. J. Duff and E. Sezgin, Phys. Lett. B279 (1992) 265.
[21] J. A. Dixon and M. J. Duff, preprint CTP–TAMU–45/92.
[22] M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory”,
C.U.P. 1987.
12
[23] The κ–symmetric action for the superfivebrane, without internal degrees
of freedom, is given in E. Bergshoeff, P. K. Townsend and E. Sezgin,
Phys. Lett. B189 (1987) 75.
[24] L. Alvarez–Gaume´ and P. Ginsparg, Ann. Phys. 161 (1985) 423.
[25] For both string and fivebrane, the θ contribution to the gravitational
anomaly may be calculated from their equations of motion (Γi)αβ∇i θ
β+
O(θ2) = 0 in the gauge Γ θ = −θ, where Γi are the pulled–back Dirac
matrices Γi = ΓM ∂X
M/∂ξi and for a p–brane Γ = Γ0Γ1 . . .Γp. These
equations are given in A. Achucarro, J. M. Evans, P. K. Townsend and D.
L. Wiltshire, Phys. Lett. B198 (1987) 441. In the string case, Feynman
diagram calculations in Green–Schwarz formalism have also been carried
out for both Yang–Mills anomalies by M. T. Grisaru, H. Nishino and D.
Zanon, Nucl. Phys. B314 (1989) 363 and gravitational anomalies by P.
E. Haagensen, Mod. Phys. Lett. 6 (1991) 431.
13


OAKTrust Digital Repository (Texas A&M Univ)

arXiv:hep-th/9208055v1  21 Aug 1992CERN-TH. 6614/92CTP-TAMU- 60/92PUTTING STRING/FIVEBRANE DUALITY TOTHE TEST †J. A. Dixon 2, M. J. Duff 1,2 and J. C. Plefka 2 ‡1 Theory DivisionCERNCH–1211 Geneva 232 Center for Theoretical PhysicsPhysics DepartmentTexas A & M UniversityCollege Station, Texas 77843ABSTRACTAccording to string/fivebrane duality, the Green-Schwarz factorization ofthe D = 10 spacetime anomaly polynomial I12 into X4X8 means that justas X4 is the anomaly polynomial of the d = 2 string worldsheet so X8 shouldbe the anomaly polynomial of the d = 6 fivebrane worldvolume. To test thisidea we perform a fivebrane calculation of X8 and find perfect agreementwith the string one–loop result.PACS numbers: 11.17+y, 11.10.KkCERN-TH. 6614/92August 1992† Research supported in part by NSF Grant PHY-9106593‡ Supported by a Fulbright ScholarshipThe dual formulations of D = 10 supergravity, one with a 7–form fieldstrength 1 and the other with a 3–form field strength 2 have long been some-thing of an enigma from the point of view of superstrings. As field theories,each seems equally as good. In particular, provided we couple them to E8×E8or SO(32) super Yang-Mills, then both are anomaly–free 3, 4. Since the 3–form version corresponds to the field theory limit of the heterotic string, itwas natural to conjecture 5 that the 7–form version corresponds to the fieldtheory limit of an extended object dual to the string: the “heterotic five-brane”. Just as the 2–form potential BMN (M = 0, 1, . . . , 9) couples to thed = 2 string worldsheet via the termS2 =12πα′∫d2ξ12ǫij∂ixM∂jxNBMN =12πα′∫B2 (1)where ξi (i = 1, 2) are the worldsheet coordinates and (2πα′)−1 is the stringtension, so the 6–form potential BMNPQRS (M = 0, 1, . . . , 9) couples to thed = 6 fivebrane worldvolume via the termS6 =1(2π)3β ′∫d6ξ16!ǫijklmn∂ixM∂jxN∂kxP∂lxQ∂mxR∂nxSBMNPQRS=1(2π)3β ′∫B6 (2)where ξi (i = 1, . . . , 6) are the worldvolume coordinates and [(2π)3β ′]−1 isthe fivebrane tension. Writing H3 = dB2+O(α′) and H7 = dB6+O(β′), therelationH7 = e−φ ∗H3 (3)1where φ is the D = 10 dilaton and ∗ denotes the Hodge dual using the canon-ical metric gMN(can), ensures that the roles of field equations and Bianchiidentities of the 3–form version of supergravity are interchanged in the 7–form version. The main subject of this paper will be the O(α′) correctionsto H3 and the O(β′) corrections to H7, discussed below.An existence proof for this heterotic fivebrane was provided by Strominger 6,who showed that the heterotic fivebrane emerges as a soliton solution of the3–form version. He went on to suggest that the strong coupling regime ofthe string should correspond to the weak coupling regime of the fivebrane;an idea made quantitatively more precise in Ref. 7, where it was shownthat the σ–model metrics gMN(string) and gMN(fivebrane) are related to thecanonical metric by e−φ/2 gMN(string) = gMN(can) = eφ/6 gMN(fivebrane)and hence that g(fivebrane) = g(string)−1/3 where g(string) and g(fivebrane)are the string and fivebrane loop expansion parameters. The same paper alsoestablished the Dirac quantization rule2κ2 = n(2π)5α′β ′ , n = integer (4)where κ2 is the D = 10 gravitational constant. Further evidence was pro-vided by the complementary discovery that a heterotic string emerges as asoliton solution of the 7–form version 8. Both the string and fivebrane soli-ton solutions break half the supersymmetries, both saturate a Bogomol’nyibound between the mass and the topological charge, and both go over intothe corresponding elementary string 9 and elementary fivebrane 10 solutions2at large distances. These elementary solutions are the extreme mass = chargelimit of the black string and black fivebrane which display event horizons anda singularity at the origin 11. However, they are mutually non–singular in thesense that the string is a non–singular solution of the 7–form version and thefivebrane a non–singular solution of the 3–form version 12. Recent work onconformal field theories 13 and other exact solutions 14 are also all consistentwith this string/fivebrane duality conjecture.Crucial to the solution of Ref. 8 was the observation that duality mixesup string and fivebrane loops: what is a one loop effect for the string mightbe a tree level effect for the fivebrane, and vice versa. At higher loop orders,this leads to an infinite number of non–renormalization theorems (includ-ing the vanishing of the cosmological term) all of which are consistent withknown string calculations to higher orders both in α′ (worldsheet loops) andg(string) (spacetime loops) 15. It is this loop mixing which allows us to teststring/fivebrane duality, in spite of our ignorance of how to quantize the five-brane. If duality is correct, we should be able to reproduce string loop effectsfrom tree–level fivebranes !To see this, let us first consider the well–known SO(32) Yang–Mills Chern–Simons corrections to H3. (E8 × E8 requires a separate treatment and willbe discussed elsewhere). In bosonic formulation, these are obtained 16 byaugmenting the action S2 of (1) by the WZW termsS2′ = −n26πN2∫∂M3trAK + n22πN2∫M3trK3 (5)3where A = AM dxM , K = g−1dg and where the gauge fields AM = AMa T aand the group elements g are matrices in the fundamental representation ofSO(32). Here n2 = integer is the level of the Kac–Moody algebra and N2 isa normalization constant given by the general formula 17N2n−2 =(2πi)n (2n− 1)!(n− 1)!(6)However, in 8k+2 dimensions for which n = 4k+2 we may define Majorana–Weyl fermions and the corresponding WZW terms are to be divided by 2.This is the case for the heterotic string, so N2 = −48π2. This agrees withWitten 18. Let us define F = dA+ A2 andI2n =( i2π)n 1n!trF ndω2n−1 = I2nδω2n−1 = dω12n−2 (7)then the sum S2 + S2′ is gauge invariant under δA = dλ + [A, λ] and δK =dλ+ [K, λ] providedδB2 =n22α′(2π)2ω12(8)and hence the gauge invariant field strength is given by 19H3 = dB2 −n22α′(2π)2ω3dH3 = −n22α′(2π)2I4 (9)This modification to the Bianchi identity is thus seen to be a classical stringeffect (i.e. tree level in the D = 10 string loop expansion).4Next we turn to the Green–Schwarz anomaly cancellation mechanism 3which is a genuine quantum (string one loop) effect. The Majorana–Weylgauginos of the D = 10 theory belong to the dimension 496 adjoint rep-resentation. The non–abelian anomaly polynomial is given by the abeliananomaly in d = 12, which is given by 12I12 of (7) with the fundamental trreplaced by the adjoint Tr. The factor of 12arises because the fermions areMajorana. As emphasized by Green and Schwarz 3, the miracle of SO(32) isthat since TrF 6 = TrF 2TrF 4 /48− (TrF 2)3 /14, 400, I12 factorizes:12I12 = X4X8 (10)In fact, since TrF 4 = 24 trF 4 + 3(trF 2)2 and TrF 2 = 30 trF 2, we have theremarkable coincidenceX4 =12I4 X8 = I8 (11)The consistent anomalyG =12· 2π∫ (13ω12I8 +23ω16I4)(12)is then cancelled by adding to the effective action∆Γ2 = −2π∫ ( 1n2α′(2π)2B2 I8 +13ω3 ω7)(13)and recalling the transformation rule for B2 given in (8). Now for B2 nor-malized as in (1), its kinetic term isΓ2 = −12κ2∫12e−φH3 ∧ ∗H3 (14)5and hence the addition of (13) modifies the field equation tod(e−φ ∗H3) =2κ2n2α′(2π)I8 (15)So far, all our considerations started with the string worldsheet. Theacid test for string/fivebrane duality is to reproduce (15) starting from thefivebrane worldvolume. We begin by augmenting the action S6 of (2) by theWZW term 20S6′ =70πn6N6∫∂M7C6 −2πn6N6∫M7trK7 (16)The explicit form for C6 is given in Ref. 20. Here we need only note that ittransforms asδC6 = 24(2π)4[ω16(A, λ)− ω16(K, λ)] (17)Here n6 = integer is the level of the Mickelsson–Faddeev algebra21 and N6 =(2π)47!/3! from (6). The sum S6 + S6′ is gauge invariant providedδB6 = −n6β′(2π)4ω16(18)and hence the gauge invariant field strength is given by 19H7 = dB6 + n6β′(2π)4ω7 (19)dH7 = n6β′(2π)4I8 (20)Using (3), this is identical to (15) provided2κ2 = n2n6(2π)5α′β ′ (21)But this is just the Dirac quantization rule (4) with n = n2n6 !6Having established that the classical fivebrane correctly reproduces thequantum string result in the pure Yang–Mills sector, we now turn to the grav-itational and mixed anomalies. Now one must include the D = 10 gravitinocontribution to I12, but again it factorizes as in (10) where now22X4 =12·1(2π)2[−12trF 2 +12trR2]X8 =( 12π)4[ 124trF 4 −1192trF 2 trR2 +1768(trR2)2 +1192trR4](22)Since we have already obtained the correct overall normalization of X4 andX8 it remains only to explain the relative coefficients. (We set n2 = n6 = 1from now on). This is most easily done by changing from the bosonic WZWformalism to the fermionic one where I4 and I8 are the anomaly polyno-mials arising via the index theorem from the chiral fermions on the d = 2worldsheet and d = 6 worldvolumes respectively. We should mention thatalthough we have the luxury of choosing a bosonic or fermionic formulationon the d = 2 worldsheet, there is probably no such bose–fermi equivalencein d = 6 and so ultimately one will have to choose between the bosonicand fermionic formulations. This choice must await a complete covariant κ–symmetric Green–Schwarz action for the heterotic fivebrane which, to date,is still lacking 23. Fortunately, for the present purposes of calculating anoma-lies, either way will do. In the Yang–Mills case, this may be seen explicitlyvia the Euclidean identities for the index of the Dirac operator 24(ind iD/)2n−2 =∫I2n =( i2π)n 1n!∫trF n7= (−1)n−1( i2π)n (n− 1)!(2n− 1)!∫trK2n−1 (23)which provide an independent check on the equivalence of the WZW and chi-ral fermion calculations. For the gravitational anomalies, the WZW methodis unknown to us and we shall follow the fermionic approach where 24I4 =1(2π)2[i22trF 2 +r48trR2]I8 =1(2π)4[ i424trF 4 +i296trF 2 trR2 +r4608(trR2)2 +r5760trR4](24)Here the Yang–Mills trace is in whatever representation the fermions are inand r is its dimensionality and the Lorentz trace is in the vector represen-tation. In the Green–Schwarz formalism the heterotic string is described bysuperspace coordinates (XM , θα) where the θ’s are in the 16 of SO(1, 9) andthe SO(32) quantum numbers are carried by Majorana–Weyl fermions in the32–dimensional fundamental representation. Although the θ’s are worldsheetscalars, they are anticommuting and obey a first order Lagrangian. However,because of κ symmetry, only 8 of the θ’s are physical (equal to 10 − 2, thenumber of transverse X ’s). The net effect of integrating out the θ’s is thusequivalent to 8 Majorana–Weyl fermions but with the opposite chirality tothe 32 gauge fermions 25. In summary we calculate X4 by dividing I4 bytwo (because the fermions are Majorana), taking the tr in the fundamentalrepresentation and setting r = 32 − 8 = 24. This agrees with (22). In thecase of the fivebrane we again have the same (XM , θα) and we again take thegauge fermions in the 32 of SO(32). Because we are in d = 6, however, the8fermions are no longer Majorana–Weyl (which exist only in 2 mod 8 dimen-sions) and so we do not divide I8 by 2. Moreover in d = 6, the number ofphysical θ’s is only 4 (equal to 10 − 6 , the number of transverse X ’s) sinceκ–symmetry halves the number and (unlike d = 2) going on–shell halves itagain. The net effect of integrating out the θ’s is thus equivalent to only twod = 6 fermions 25 and hence r = 32− 2 = 30. ThusI8 =1(2π)4[ 124trF 4 −196trF 2 trR2 +152304(trR2)2 +1192trR4](25)This is not quite in agreement with X8 of (22) but the difference is propor-tional to trR2X4 and hence the discrepancy is easily remedied. Up until nowwe have been assuming that the fivebrane B6 which appears in (2) was identi-cal to theD = 10 supergravity B6 which satisfies (3). However, this may needto be modified when we include the gravitational Chern–Simons correctionswhich are of higher order in the low–energy expansion than those of Yang–Mills. The Green–Schwarz result is dH7(supergravity) = β′(2π)4X8, whereasour fivebrane calculations have lead us to dH7(fivebrane) = β′(2π)4 I8. So ifwe defineB6(fivebrane) = B6(supergravity)− cβ ′α′trR2B2 (26)then the gauge–invariant field strength H7(supergravity) satisfiesdH7(supergravity) = β′(2π)4 (I8 −c(2π)2trR2X4) (27)on using (9). Comparing X8 of (22) with I8 of (25) and using X4 of (22),we find perfect agreement with the Green–Schwarz result with the choice9c = 1/48.In their discussion of anomalies, Green, Schwarz and Witten 22 say “Thecase where something really new happens is that of 4k+2 dimensions and thisis the case of interest to superstring theory since the worldsheet has dimensiontwo and the spacetime has dimension ten !”. The results of the presentpaper might be summarized by adding “. . . and the fivebrane worldvolumehas dimension six !”.AcknowledgementsConversations with L. Alvarez–Gaume´, E. Bergshoeff, B. Campbell, M.Grisaru, R. Stora and P. Townsend are gratefully acknowledged.10References[1] A. H. Chamseddine, Nucl. Phys. 185 (1981) 403.[2] E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl.Phys. B195 (1982) 97; G. F. Chapline and N. S. Manton, Phys. Lett.1208 (1983) 105.[3] M. B. Green and J. H. Schwarz, Phys. Lett. 49B (1984) 117.[4] S. J. Gates and H. Nishino, Phys. Lett. B173 (1986) 52; A. Salam andE. Sezgin, Physica Scripta 32 (1985) 283.[5] M. J. Duff, Class. Quantum Grav. 5 (1988) 189.[6] A. Strominger, Nucl. Phys. B343 (1990) 167.[7] M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 129.[8] M. J. Duff and J. X. Lu, Phys. Rev. Lett. 66 (1991) 1402; Class. Quan-tum Grav. 9 (1991) 1.[9] A. Dabholkar, G.W. Gibbons, J. A. Harvey and F. Ruiz–Ruiz, Nucl.Phys. B340 (1990) 33.[10] M. J. Duff and J.X. Lu, Nucl. Phys. B354 (1991) 141.[11] G. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 409.[12] M.J. Duff, R.R. Khuri and J. X. Lu, Nucl. Phys. B377 (1992) 281.11[13] C. Callan, J. Harvey and A. Strominger, Nucl. Phys. B359 (1991) 611;Nucl. Phys. B367 (1991) 60; preprint EFI–91–66.[14] I. Pesando and A. K. Tollsten, Phys. Lett. B274 (1992) 374.[15] M. J. Duff and J. X. Lu, Nucl. Phys. B357 (1991) 354.[16] M. J. Duff, B. E. W. Nilsson and C. N. Pope, Phys. Lett. B163 (1985)343; M. J. Duff, B. E. W. Nilssson, C. N. Pope and N. P. Warner, Phys.Lett. B171 (1986) 170; E. Bergshoeff, F. Delduc and E. Sokatchev,Phys. Lett. B262 (1991) 444.[17] R. Bott and R. Seeley, Comm. Math. Phys. 62 (1987) 235.[18] E. Witten, Comm. Math. Phys. 92 (1984) 455.[19] These results imply that previous versions of these Chern–Simons cor-rections quoted in Refs. [6,8,13,15,20] for H3 and Refs. [8,15,20] for H7corresponded to the choice n2 = n6 = 8.[20] J. A. Dixon, M. J. Duff and E. Sezgin, Phys. Lett. B279 (1992) 265.[21] J. A. Dixon and M. J. Duff, preprint CTP–TAMU–45/92.[22] M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory”,C.U.P. 1987.12[23] The κ–symmetric action for the superfivebrane, without internal degreesof freedom, is given in E. Bergshoeff, P. K. Townsend and E. Sezgin,Phys. Lett. B189 (1987) 75.[24] L. Alvarez–Gaume´ and P. Ginsparg, Ann. Phys. 161 (1985) 423.[25] For both string and fivebrane, the θ contribution to the gravitationalanomaly may be calculated from their equations of motion (Γi)αβ∇i θβ+O(θ2) = 0 in the gauge Γ θ = −θ, where Γi are the pulled–back Diracmatrices Γi = ΓM ∂XM/∂ξi and for a p–brane Γ = Γ0Γ1 . . .Γp. Theseequations are given in A. Achucarro, J. M. Evans, P. K. Townsend and D.L. Wiltshire, Phys. Lett. B198 (1987) 441. In the string case, Feynmandiagram calculations in Green–Schwarz formalism have also been carriedout for both Yang–Mills anomalies by M. T. Grisaru, H. Nishino and D.Zanon, Nucl. Phys. B314 (1989) 363 and gravitational anomalies by P.E. Haagensen, Mod. Phys. Lett. 6 (1991) 431.13

Putting string���five-brane duality to the test

According to string/fivebrane duality, the Green-Schwarz factorization of the $D=10$ spacetime anomaly polynomial $I_{12}$ into $X_4\, X_8$ means that just as $X_4$ is the anomaly polynomial of the $d=2$ string worldsheet so $X_8$ should be the anomaly polynomial of the $d=6$ fivebrane worldvolume. To test this idea we perform a fivebrane calculation of $X_8$ and find perfect agreement with the string one--loop result.According to string/fivebrane duality, the Green-Schwarz factorization of the $D=10$ spacetime anomaly polynomial $I_{12}$ into $X_4\, X_8$ means that just as $X_4$ is the anomaly polynomial of the $d=2$ string worldsheet so $X_8$ should be the anomaly polynomial of the $d=6$ fivebrane worldvolume. To test this idea we perform a fivebrane calculation of $X_8$ and find perfect agreement with the string one--loop result

Dixon, John A.

Duff, M.J.

Plefka, J.C.

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Putting string/fivebrane duality to the test

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Putting String/Fivebrane Duality to the Test

Putting String/Fivebrane Duality to the Test

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