- Email: [email protected]
N=(4,2) chiral supergravity in six dimensions and solvable Lie algebras
26 February 1998
Physics Letters B 420 Ž1998. 289–299
N s ž4,2 / chiral supergravity in six dimensions and solvable Lie algebras Riccardo D’Auria a
a,c
, Sergio Ferrara
b,c
, Costas Kounnas
b,1
Dipartimento di Fisica, Politecnico di Torino, C.so degli Abruzzi, 24, I-10129 Torino, Italy b Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland c Department of Physics and Astronomy, UniÕersity of California, Los Angeles, USA Received 6 November 1997 Editor: L. Alvarez-Gaume´
Abstract Decomposition of the solvable Lie algebras of maximal supergravities in D s 4, 5 and 6 indicates, at least at the geometrical level, the existence of an N s Ž4,2. chiral supergravity theory in D s 6 dimensions. This theory, with 24 supercharges, reduces to the known N s 6 supergravity after a toroidal compactification to D s 5 and D s 4. Evidence for this theory was given long ago by B. Julia. We show that this theory suffers from a gravitational anomaly equal to 4r7 of the pure N s Ž4,0. supergravity anomaly. However, unlike the latter, the absence of N s Ž4,2. matter to cancel the anomaly presumably makes this theory inconsistent. We discuss the obstruction in defining this theory in D s 6, starting from an N s 6 five-dimensional string model in the decompactification limit. The set of massless states necessary for the anomaly cancellation appears in this limit; as a result the N s Ž4,2. supergravity in D s 6 is extended to N s Ž4,4. maximal supergravity theory. q 1998 Elsevier Science B.V.
1. Introduction Supersymmetry algebras in any dimension and their representations are closely related to properties of spinors. Spinors of O Ž D y 1,1. are real for D s 2,3,9 mod 8, pseudoreal for D s 5,6,7 mod 8, and complex in D s 4,8 mod 8. The above reality properties refer to spinors of a given chirality when the dimension is even. The corresponding supersymme-
1
On leave from Ecole Normale Superieure, 24 rue Lhomond, ´ F-75231, Paris Cedex 05, France. E-mail address: [email protected].
try algebras have automorphism groups H Žsometimes called R-symmetry., which fall in three categories w1x:
i. O Ž N . ,
i . USp Ž N .
and
iii . U Ž N . ,
N being the number of spinorial charges in the appropriate dimension. We note that in D s 2 and 6 Žmod 8. algebras with different numbers of chiral charges NL , NR exist with O Ž NL . = O Ž NR . and USpŽ NL . = USpŽ NR . as H-groups respectively. If one considers compactifications of higher-dimensional theories on smooth manifolds the number of
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 0 8 - 6
290
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
supersymmetry charges N in lower dimensions is always 2 n for some n. This leaves aside certain theories, which can be constructed in lower dimensions, where the number of supercharge components is w2,3x: Ø In D s 4: 12, 20 and 24, corresponding to N s 3, 5 and 6 supergravity with H s UŽ3., UŽ5. and UŽ6. respectively. Ø In D s 5 and D s 6: 24, corresponding to N s 6 with H s USpŽ6. and N s Ž4,2. with H s USpŽ4. = USpŽ2. respectively. 2 Theories in D s 4 and 5, have been shown to exist as consistent type II string compactifications on asymmetric orbifolds w3,4x. The N s 6 supergravity is special because it is the only one that can be lifted to six dimensions where it becomes chiral w5,6x. In D s 4 the N s 6 supergravity is also special when is defined in anti de Sitter space. In fact OspŽ6 <4. gauge supergravity is the only one with a zero-center representation. w7x In Section 2 of this work we explore some properties of N s Ž4,2. supermultiplets in D s 6 by means of the chain decomposition N s Ž4,4. ™ N s Ž4,2. ™ N s Ž0,2.. This allow a derivation of the structure of the N s Ž4,2. supergravity, and in particular a relation of its scalar manifold to the decomposition of the maximal supergravity into N s 2 sub-theories. We show that this theory suffers from a gravitational anomaly w6x, which is 4r7 of the N s Ž4,0. pure supergravity anomaly. In Section 3 we derive the D s 5, N s Ž4,2. string ground state and show why the four- and fivedimensional descendants of the anomalous six-dimensional theory are not affected by this anomaly when one includes massive degrees of freedom dictated by string theory. In Section 4 we analyse the two decompactification limits R ™ `, R ™ 0 and we show the restoration of N s Ž4,4. in D s 6. Our conclusions are in Section 5.
2. N s 2 and N s 6 decomposition of maximal supergravity in D s 4, 5 and 6 dimensions. Let us first consider the general decomposition of maximal supergravities in terms of N s 2 multiplets w8–10x. The important fact in this decomposition is that the spin 3r2 multiplet does not contain scalars. If we associate to the scalar manifold M s GrH a solvable group with a Lie algebra S w11x, then we find that the S vector space decomposes into S V q SH , where S V and SH are solvable Lie algebras associated to Jmax s 1 and Jmax s 12 respectively. The Jmax s 12 w9,10x, multiplets are hypermultiplets and SH is thus related to a quaternionic space with holonomy SUŽ2. = H; H is the automorphism group of the supersymmetry algebra in the corresponding dimension. Note that in D s 6, since the vector multiplet has no scalars, the Jmax s 1 multiplet is a tensor multiplet. Our starting point is to decompose the maximal D s 6 supermultiplet into N s 2 ones. This amounts to decomposing the N s Ž4,4. representation into N s Ž0,2. massless representations. The N s Ž0,2. representations are: Ø graÕiton; G ™ Ž1 L ,1 R . q 2Ž1 L , 12 R . q Ž1 L ,0 R .; Ø graÕitino; g 1 ™ 2Ž1 L , 12 R . q 4Ž1 L ,0 R .; Ø graÕitino; g 2 ™ 2Ž 12 L ,1 R . q 4Ž 12 L , 12 R . q 2Ž 12 L ,0 R .; Ø Õector; V ™ Ž 12 L , 12 R . q 2Ž 12 L ,0 R .; Ø tensor; T ™ Ž0 L ,1 R . q 2Ž0 L , 12 R . q Ž0 L ,0 R .; Ø hyper; H ™ 2Ž0 L , 12 R . q 4Ž0 L ,0 R .. All the above multiplets are obtained by tensoring Clifford algebra states Ž0 L , 12 R ., 2Ž0 L ,0 R . by a Ž JL , JR ., SUŽ2.L = SUŽ2.R representation, with a doubling Žwhen required. by CPT. The N s Ž4,4. G graviton multiplet decomposes into N s Ž0,2. as follows: N s Ž 4,4 .
q16 2 Instead of the notation, N s1 for the minimal and N s 2 for the maximal supersymmetry in Ds6, we are using the chiral notation N s Ž2,0. for the minimal and N s Ž4,4. for the maximal supersymmetry; this notation is more convenient in Ds6 since the chiral spinors in Ds6 are Majorana symplectic w2x.
q20
ž ž
Gs
1
,
ž
Ž 1 L ,1 R . q 4 1 L ,
1
2L 2R 1 2L
/
1 2R
/ ž q4
1 2L
,1 R
q 5 Ž 1 L ,0 R . q 5 Ž 0 L ,1 R .
/ ž
,0 R q 20 0 L ,
1 2R
/
q 25 Ž 0 L ,0 R . .
s 1G q 1 g 1 q 2 g 2 q 8V q 5T q 5H.
/
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
On the other hand we may decompose N s Ž4,4. ™ N s Ž0,2. through the chain N s Ž4,4. ™ N s Ž4,2. ™ N Ž0,2.. We first get w N s Ž4,4.xG ™ w N s Ž4,2.xG q w N s Ž4,2.x g , Ž256 ™ 128 q 128 states.. Then we can further decompose w N s Ž4,2.xG and w N s Ž4,2.x g into N s Ž0,2. multiplets to finally obtain: - N s Ž 4,2 .
Gs
Ž 1 L ,1 R . q 8
1
ž
,
ž
1 2R
/ ž
q 10 0 L ,
s N s Ž 0,2 .
G q2
q 5 N s Ž 0,2 . - N s Ž 4,2 .
g s2
ž
q10 0 L ,
ž
1 2R
/
2L 2R 1
1 2R
2L
,1 R
q4
N s Ž 0,2 .
1
2L
,0 R
g2
T,
1
1
1
/ ž / / ž /
1L ,
2R
q 16
q8 1
2L
,
2L 2R
,0 R
q4 Ž 1 L ,0 R . q 20 Ž 0 L ,0 R . s N s Ž 0,2 . q8 N s Ž 0,2 .
V q5
N s Ž 0,2 .
g1
H.
This implies, in particular, the solvable Lie algebra decomposition S
O Ž 5,5 . O Ž 5. = O Ž 5. qS
O Ž 4,5 . O Ž 4. = O Ž 5.
O Ž 5. ,
S
E6Ž6. USp Ž 8 . qS
S
Ž 2.1 .
USp Ž 6 .
F4Ž4.
SU Ž 8 .
sS
Ž 2.2 .
SO ) Ž 12 . U Ž 6.
E6Ž2.
in Ds4.
SU Ž 2 . = SU Ž 6 .
Ž 2.3 .
The rank of the above cosets in the various dimensions decomposes as follows: Ø In D s 6: 5 s 1V q 4 H; Ø In D s 5: 6 s 2V q 4 H; Ø In D s 4: 7 s 3V q 4 H. This shows that the fields that correspond to the compactification radii are vector multiplets; the rank of the hypermultiplet manifold does not change with the dimensions. The rank-4 quaternionic spaces at D s 6,5 and 4, correspond to the following maximal decompositions of the corresponding isometry groups: F4Ž4. ™ O Ž 5,4 . ,
52 ™ 36 q 16;
78 ™ 52 q 26.
The total number of states of the N s Ž4,2. supergravity in six dimensions is 128 s 64 bosons q 64 fermions as in D s 4 and D s 5. However the theory in D s 6 is anomalous since the gravitational anomaly is y360 I , where I stands for the anomaly contribution of one Weyl fermion in six dimensions. The gravitational anomaly contribution of the pure N s Ž4,0. supergravity multiplet is y630 I , so that IŽ4,2. s 47 IŽ4,0. . In the case of the Ž4,0. theory the anomaly is cancelled by adding twenty-one Ž4,0. tensor multiplets, giving a contribution to the anomaly q21 = 30 I w12x. In the present case, owing to the absence of matter multiplets with Jmax less than 32 , the N s Ž4,2. cannot be anomaly-free. If one adds a w N s Ž4,2.x g 1 multiplet then the supersymmetry is extended to N s Ž4,4., which is an 1 2
i.e. algebras with a rank-1 tensor multiplet coset and a rank-4 quaternionic manifold with holonomy SUŽ2. = H, H s USpŽ2. = USpŽ4.. This statement shows that the decomposition of the maximal solvable Lie algebra into N s 2 solvable Lie algebras corresponds to the scalar sector of the two N s Ž4,2. multiplets into which the w N s Ž4,4.xG multiplet decomposes. If we delete the w N s Ž4,2.x g multiplet we obtain a pure N s Ž4,2. supergravity theory with scalar manifold O Ž1,5.rO Ž5.. This result was obtained by Julia in w5x.
in Ds5,
USp Ž 2 . = USp Ž 6 .
E7Ž7.
qS
SU ) Ž 6 .
sS
E6Ž2. ™ F4Ž4. ,
O Ž 1,5 .
sS
Note that a similar procedure in D s 5 and D s 4 gives analogous relations among solvable Lie algebras w9,8x:
q Ž 1 L ,0 R .
ž / / ž /
q5 Ž 0 L ,1 R . q 5 Ž 0 L ,0 R q 4 q2 1 L ,
1
291
1 2
1 2
1 2
292
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
anomaly-free theory. It seems that this is the only consistent way to cancel the anomaly. Another possibility to cancel the anomaly would be to add twelve Ž4,0. tensor multiplets; this solution, however, explicitly violates N s Ž4,2. supersymmetry. It is of interest to understand the reason why the six- dimensional theory with 24 supercharges is anomalous while, in lower dimensions, theories with the same number of supercharges can be constructed in type II string theories w3x. In the next sections we will show the restoration of N s Ž4,4. in six dimensions, starting from a fivedimensional N s 6 string theory and decompactifying one of the dimensions considering either R 6 ™ ` or R 6 ™ 0.
3. N s 6 construction in five dimensions In order to construct in string theory a ground state with N s Ž4L q 2 R . it is necessary to consider the type II theory with maximal supersymmetry N s Ž4L q 4R .; then one has to reduce the supersymmetry asymmetrically by projecting out half of the right moving supercharges. The heterotic or type I construction of N s Ž4L q 2 R . ground states is impossible, since the maximal supersymmetry is N s Ž4L q 0 R . in the heterotic theory and N s 4 in that of type I. We start by presenting the type II N s Ž4L q 2 R . string models and write down their partition function. Models with the same number of supercharges in D s 4 have been obtained by applying projections to the maximal-supersymmetry string theories w3x, which preserve modular invariance as well as the conformal symmetries on the string world-sheet. The maximal-supersymmetry type II model is described in the light-cone gauge, by 8 world-sheet leftrrightmoving bosonic and fermionic coordinates. In our notation, in D s 5, the coordinates cmL, R and XmL, R Ž m s 3,4,5. represent the space-time degrees of freedom Žin the light-cone gauge., whereas the remaining ones correspond to the fermionic x IL, R and bosonic f IL, R , I s 1,2,3,4,5 internal degrees of freedom. This description will be appropriate for the asymmetric orbifold construction w15x, where four of the right-moving internal coordinates are twisted in
order to reduce by a factor of 2 the right-moving supersymmetries: Z2 : Ž x RI , f RI . ™ y Ž x RI , f RI . ,
I s 2,3,4,5.
Ž 3.1 .
f IL, R
Ž Is In the fermionic construction w13,14x, 1,2,3,4,5. are replaced by a pair of Majorana–Weyl spinors yIL, R and v IL, R Ž I s 1,2,3,4,5., by means of the 2d-boson-fermion equivalence: JIL , R s Ef IL , R s yIL , R v IL , R .
Ž 3.2 .
The construction of string models amounts to a choice of boundary conditions for the 2d fermions x IL, R , yIL, R , v IL, R , which satisfies local and global consistency requirements w13x. The N s Ž4L q 4R . maximal supersymmetry model, constructed in this manner, have four space-time supercharges originating from the left-moving sector and another four from the right-moving sector. In the language of the fermionic construction w13x, this model is defined by introducing three basis sets, F, S and S. The first one contains all the left- and the right-moving fermions: F s cmL , x IL , yIL , v IL < cmR , x IR , yIR , v IR = Ž m s 3,4,5; I s 1, . . . ,5 . ,
Ž 3.3 .
and the basis sets S and S, which contain only eight left- or right-moving fermions and generate the GSO projections of the maximal supersymmetry w3x: S s cmL , x IL
S s cmR , x IR .
Ž 3.4 .
Four of the gravitinos of the N s Ž4L q 4R . model belong to the S-sector and the other four to the S-sector. Then, by applying the appropriate projection one obtains the N s Ž4L q 2 R . superstring model. A possible projection for constructing N s Ž4L q 2 R . is specified by a choice of fermion basis b such that b s cmR , x 1R , y 2R,3,4,5 < y 1L , v 1L , y 1R , v 1R .
Ž 3.5 .
Then b generates the desire left–right-asymmetric Z2 projection. Namely, it acts as a twist on x IR and JIR when I s 2,3,4,5. Also it acts non-trivially on the set of fermions T s y 1L , v 1L , y 1R , v 1R .
Ž 3.6 .
This action however keeps the currents J1L, R s Ef 1L, R invariant. In the bosonic f 1L, R language, the non-
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
trivial action on T implies a shifting by 1r2 unit on the f 1 lattice; f 1L, R ™ f 1L, R q p . Therefore, the Z2 defined by b acts as a shift on f 1 and as an asymmetric twist on the remaining four internal supercoordinates x IR , f IR , I s 2,3,4,5. This asymmetric Z2 breaks half of the right-moving supersymmetries by projecting out two of the gravitinos of the S-sector. The resulting model has the desired N s Ž4L q 2 R . supersymmetry. The presence of T fermions in b makes the states coming from the b-twisted sector massive. Indeed, the lowest state in the b-twisted sector, has right-moving conformal dimensions DR s 10 16 , which corresponds to the conformal dimension of a spin field constructed with the ten right-moving R fermions spinw cmR , x 1R , y 2,3,4,5 y 1R , v 1R x. Thus, the lower right-moving mass level in this sector is 1 1 w m R Ž b .x2 s 10 16 y 2 s 8 . The absence of massless twisted sectors in any consistent orbifold construction is expected. Indeed, the the fact that the massless spectrum of N s Ž4L q 2 R . in D s 5 is fully determined by the graviton supermultiplet implies that all the N s Ž4L q 2 R . massless states must be the Z2-invariant states of N s Ž4L q 4R . supergravity. These states are all present in the untwisted sector and therefore all extra states have to be massive. Using for instance b˜ s cmR , x 1R , y 2R,3,4,5
Ž 3.7 .
instead of b, then the b˜ asymmetric projection still projects out the two right-moving gravitinos from the ˜ ˜ S-sector. In this case, however, the b-twisted sector contains some extra massless states, since the conforR x is precisely mal dimension of spinw cmR , x 1R , y 2,3,4,5 ˜ 1r2. Among the extra massless states of the b-twisted sector, there exist two extra graÕitinos with vertex operator
and thus, the only available asymmetric projection is ˜ this projection, however, the one that is based on b; gives rise to massless b˜ twisted sectors, which contain two extra right-moving gravitinos. In the following, we will discuss in more detail the b fermionic model in D s 5. Then this model will be generalized in order to include the compactification radius R 1 modulus associated to f 1. Finally, in the next section, we study the extension of supersymmetry in the two decompactification limits R 1 ™ ` and R 1 ™ 0. We will show that the helicity supertraces B2 n s str w s x 2 n vanish for n s 0,1,2,3 in both decompactification limits, indicating the extension of N s Ž4L q 2 R . ™ N s Ž4L q 4R .. We start with the partition function of N s Ž4L q 2 R . based on b: 1
ZbDs 5 s
3
Imt 2 h 3 h 3 =
=
˜
, q
Ž 3.8 .
and so the N s Ž4L q 2 R . supersymmetry is extended again to the maximal N s Ž4L q 4R .. Already at this point one can guess the difficulties that will arise in constructing a six-dimensional string model with N s Ž4L q 2 R . supersymmetry. Indeed, in six dimensions the coordinate f 1 is non-compact
1
1 4 1 2
Ž y.
Ý
aq bqa b
2 a 4 u ba b
Ýu
Ž y . aqbqa b
h4
a, b , a , bs0
h4 ayh byg
u
u
aqh bqg
h, g
=Z5,5
h g
.
Ž 3.9 . Z5,5 w hg x
In the above expression denotes the contribution of the five compactified coordinates f IL, R , I s 1, . . . ,5 Žfermionized.; four of them are Ž h, g .-twisted while f 1L, R is Ž h, g .-shifted. In terms of fermionic characters, the analytic expression of Z5,5w hg x is: Z5,5
h g
u
1 s 2
Ý
=
gqh d qg
g qh dqg
u
hh
g ,d
u b R V3r2 s CmL spin cmR , x 1R , y 2,3,4,5
293
g 4 d 4
h
2
u w dg x u
Ž y.
g qh dqg 4
h
u
d hq g gqh g
g yh d yg
Ž y.
hg
,
Ž 3.10 . where the term in the first square bracket corresponds to the contribution of the compactified coordinate f 1 written in terms of the fermionic characters of y 1L, R , v 1L, R. In terms of f 1-lattice characters, it corresponds to a fixed S 1 radius at the fermionic
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
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point R 1 s 1r '2 . The term in the second square bracket corresponds to the characters of the Z2 asymmetric orbifold, where four right-moving coordinates JI ™ yJI , I s 2,3,4,5 are twisted. The phase factors Žy. d hq g gqh g , Žy. h g are dictated by modular invariance. As we have already explained, we would like to examine the behaviour of the N s Ž4L q 2 R . model in the decompactification limits R 1 ™ ` and R 1 ™ 0. For this purpose we must deform the above model and move away from the fermionic point by switching on the marginal deformation that is associated to the modulus R 1. Before doing that, let us first examine the moduli space of the above model, namely all possible J L P J R Ž1,1.-deformations which we are able to switch on simultaneously. Due to the left–right asymmetry the only possible deformations are: J1L P J1R ™ R 1
and
JIL P J1R ™ YI , I s 2,3,4,5.
The first deformation corresponds to the R 1 modulus while the remaining four correspond to the Wilson lines YI moduli. Altogether they form the perturbative moduli space constructed from the NS-scalars
Mp s
O Ž 5,1 . O Ž 5.
.
Ž 3.11 .
The perturbative moduli space is extended to Mn p s SU ) Ž6.rUSpŽ6. once we include the dilaton moduli Žsinglet under O Ž5.. and the remaining RR-scalars Žin 15, 10, and 10’ representations of O Ž5... The perturbative string states form the charge lattice G 1,5w hg x associated to the moduli space Mp . The absence of continuous deformations associated to the twisted coordinates f IR , I s 2,3,4,5 implies that all radii R I , I s 2,3,4,5, of the ‘‘twisted’’ four-dimensional space must be frozen to some special values of the moduli space Žthe SOŽ8. fermionic point in our case.. This also means that the asymmetric orbifold projection is well defined only for special Õalues of the moduli of the initially untwisted G4,4 lattice, e.g. the fermionic SO Ž8. symmetric point. Keeping this point in mind we can generalize the model to include the radius and Wilson line deforma-
tions by replacing Z5,5w hg x by Z5,5 Ž R 1 ,YI .; since the non-vanishing Wilson lines do not affect the decompactification limits we will restrict ourselves to YI s 0, keeping the modulus R 1 arbitrary: Z5,5
h g
Ž R1 ,YI s 0 . s
h g
G 1,1
Ž R1 .
hh
Z4,4
h g
.
Ž 3.12 . Z4,4 w hg x
where is the asymmetric orbifold contribution of the four internal coordinates f I , I s 2,3,4,5 Z4,4
h g
s 2
g 4 d 4
u
1
Ý
2
u w dg x u
h
g ,d
g qh d qg 4
u
g yh dyg
h
Ž y.
hg
Ž 3.13 . and
G 1,1 s
h g
Ž R1 .
Ý exp
ip gm q i2pt PL2 yi2pt PR2
m, n
with PL s
PR s
1
m q
2 R1 1
Ž 2 n q h . R1 2
m y
2 R1
Ž 2 n q h . R1 2
,
Ž 3.14 .
is the shifted f 1-lattice w16–18x. The N s Ž4L q 2 R . model constructed above is an asymmetric freely acting orbifold. Following Refs. w16–18x it corresponds to a partial spontaneous breaking w16,17x of supersymmetry N s Ž4L q 4R . ™ N s Ž4L q 2 R .. The massless states of this model consist only of N s Ž4L q 2 R . gravitational multiplet. The remaining massless states of an N s Ž4L q 4R . Žtwo 3r2-multiplets. become massive, with a common mass equal to: 1
Ž m23r2 . 7,8 s R 2
Ž h s 0, < m < s 1, n s 0 . .
1
These states correspond to the ‘‘untwisted’’ sector states with momentum < m < s 1 and 2 n q h s 0. The ‘‘odd’’ winding states h s 1 correspond to the
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
twisted sector. The lowest twisted states correspond to two massive 3r2-multiplets with masses proportional to:
Ž
m23r2 7X ,8X s
.
R 12 4
Ž h s 1, m s 0, <2 n q h < s 1 . .
4. Six-dimensional decompactification of the fivedimensional N s (4 L H 2 R ) string model A six-dimensional model in string theory can be defined from a five-dimensional one in two different ways: either by sending the compactification radius R 1 ™ ` or R 1 ™ 0. In the first case the Kaluza-Klein ŽKK. momenta become the six-dimensional continuous momentum P62 ; m2rR 12 . In the limit R 1 ™ 0, however, the KK states become superheavy while the string windings give rise to the six-dimensional continuous momentum P˜62 ; n 2 R 2 . Starting with the D s 5, N s Ž4L q 2 R . string defined in the previous section, we would like to examine the six-dimensional theories obtained in the two stringy decompactification limits. The five-dimensional spectrum of this model always contains 6 massless gravitinos. There is also a tower of massive gravitinos with the same R symmetry charges having even KK-momentum m and even winding charge nX s 2 n q h Ž h s 0.. When R 1 ™ ` or 0 one obtains 6 massless gravitinos with continuous momenta, either P6 or P˜6 . However, there exist two more extra towers of massive gravitinos with different R-symmetry charges, which can become massless in the two decompactification limits, namely: Ø In the limit R 1 ™ ` the Žtwo. massive gravitinos with odd KK-momentum m become massless, Ž m23r2 . 7,8 ™ 0, while the ‘‘winding’’ gravitinos Ž m 23r2 . 7X ,8X ™ ` become superheavy and decouple from the spectrum. The two extra massless gravitinos Ž m 23r2 . 7,8 s 0 together with the six massless gravitinos of N s Ž4L q 2 R . restore in six dimensions the N s Ž4L q 4R . supersymmetry. Ø In the other limit, R 1 ™ 0, the two gravitinos with odd KK momentum m, Ž m23r2 . 7,8 ™ ` become superheavy and decouple while the gravitinos with winding nX s 2 n q h odd Ž h s 1. become massless Ž m23r2 . 7X ,8X ™ 0.
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The ‘‘field theory’’ analogue of the above model can be obtained by a Scherk-Schwarz w19x supersymmetry breaking mechanism w20,16,17x starting either from N s 2 in D s 10 or from N s 1 in D s 11. The difference from the string model is the absence of the winding states, nX s 2 n q h s 0 Ž n s 0, h s 0.. Therefore, in field theory, the first decompactification limit, R 1 ™ `, gives the same effective supergravity theory as in string theory. On the other hand the second limit, R 1 ™ 0, does not correspond any more to a decompactification but rather to a ‘‘dimensional reduction’’ where all KK states are truncated out from the spectrum. In the limit R 1 ™ 0 the field theory remains five-dimensional. The restoration of the maximal supersymmetry in D s 6 can be checked in the two decompactification limits by constructing the helicity supertrace B2 n w21,22,16–18x as a function of the radius R 1: 2s
w s x 2 n ey p t M s Ž R 1 .
2n
eyp t M s Ž R 1 . .
B2 n Ž R 1 ;t . s tr Ž y . s str w s x
2
2
Ž 4.1 .
The little group of massless particle in in D s 5 is SOŽ3.. By ‘‘helicity’’ s we mean UŽ1. charge of a UŽ1. sub-group of SO Ž3.. M s Ž R 1 . denotes the mass of the state. If there is a restoration of N s Ž4L q 4R ., B6 has to vanish in both decompactification limits due to the N s 8 supertrace identities: B2 n s 0,
n s 0,1,2,3,
in
N s 8.
In N s 6 B6 is not trivial since the supertrace identities for N s 6 are: B2 n s 0,
n s 0,1,2,
in
N s 6.
The supergravity massless sector of the N s 6 supergravity gives a non-trivial B6 ŽSUGRA. s 45r2. If in the decompactification limits the N s 6 supersymmetry is extended to N s 8 due to the presence of extra massless states, then B6 s 0 must be found in both limits. In order to derive B2 n in string theory it is convenient to define the helicity generating partition function w22,16,17,26,25x: Z string Ž Õ,Õ . s Tr q L 0 q L 0 e 2 p i Õ s Ly2 p i Õ s R , q s e 2 p it , q s e 2 p it ,
t s Imt ,
Ž 4.2 .
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
296
where sL and sR denote the left- and right-moving helicities. The physical helicity is given by s s sL q sR . Once Z string Ž Õ,Õ . is defined, then B2 n can be derived from Z string Ž Õ,Õ . by the action of the leftand right-helicity operators Qs
E
1
2p i E Õ
,
Qsy
E
1
2p i E Õ
In the expression for Z string Ž Õ,Õ . we can sum over the indices Ž a,b . and Ž a,b . using the Riemann identity of theta functions 1 2
,
Ž y.
Ý
u
h4
a, b
1
2n
s
so that B2string s Ž Q q Q . Z string Ž Õ,Õ . < Õs Õs0 . n
aq bqa b
h
4
u
Õ
1 1
a b
Ž Õ. u Õ
1 1
a b
a b
u Õ
1 1
a b
u
Õ
1 1
ž /u ž /uw xž /uw xž / 2
2
2
2
Ž 4.3 . The Õ,Õ helicity modifications for the partition function have been studied earlier, in order to obtain exact solutions of string theory in the background of magnetic fields and to investigate the associated phase-transition phenomena w22–24x. In that context the quantities Õ and Õ play the role of the background magnetic field. An explicit expression for Z string Ž Õ,Õ . for the N s Ž4,2. string model of the previous section is given by an expression that is similar to the Õ s Õ s 0 partition function:
j Ž Õ. j Ž Õ. h3 h3 =
1
1
Ý
4 =
1 2
=u
a b
Ž y. aqbqa b
h4
a, b , a , bs0
Ýu
aq bqa b
Ž y.
Ž Õ. u
a b
h4
Ž Õ. u
u
aqh bqg
Z5,5
h g
Ž R1 . ,
Ž 4.4 .
n 2
` 1
Ž1yq .
Ž1 y q
n 2p i Õ
e
sinp Õ
u 1X
p
u 1Ž Õ .
s
. Ž 1 y q n ey2 p i Õ .
2
Ý
h4
a, b
1 s
h
4
u
1 1
Õ
u
a b
Ž Õ. u
Õ
1 1
ž / ž / u
2
2
u
a b
u
aq h bqg
Õ
1y h
ž /
1yg
2
u
u
ayh byg
1qh 1qg
Õ
ž / 2
.
Ž 4.7 .
s 15Q 4 Q 2 Z string Ž Õ,Õ . < Õs Õs0 ,
Ž 4.5 .
Ž 4.8 .
receiving non-vanishing contribution from the N s 6 sector with Ž h, g . / Ž0,0.. Furthermore, using the identities 4 iQ u
1 1
Ž Õr2. s u
0 0
u
0 1
u
1 0
s 2h 3 ,
Ž 4.9 .
the expression for B6 , as a function of R 1 w25,26x, simplifies to: B6 s
,
j Ž Õ . s j Ž yÕ . , j Ž 0 . s 1.
Ž y . aq bqa b
1
6
where the j Ž Õ . and j Ž Õ . modifications is due to the helicity charge of the 2d bosonic oscillators. The Õ, Õ modifications due to the 2d fermionic degrees of freedom give rise to non-zero characteristics to the fermionic u w ab xŽ Õ . and u w ab xŽ Õ . functions w22,16,17,26,25x. The analytic expression of j Ž Õ . is:
j Ž Õ. s Ł
and
B6 s Ž Q q Q . Z string Ž Õ,Õ . < Õs Õs0
a 3 u ba b
h, g ay h byg
Ž 4.6 .
The vanishing of B0 , B1 and B2 follows automatically from the vanishing of u w 11 xŽ Õr2. and u w 11 xŽ Õr2. for Õ s 0 and Õ s 0: Indeed, Z string Ž Õ,Õ . vanishes like Õ 4 Õ 2 in the sector with Ž h, g . / Ž0,0. Ž N s 6 sector., and it vanishes like Õ 4 Õ 2 in the sector with Ž h, g . s Ž0,0. Ž N s 8 sector.. Therefore, the only non-trivial helicity supertrace is
Z string Ž Õ,Õ . s
,
45 4
Ý
x
h g
G 1,1
h g
< R1 ,
Ž 4.10 .
Ž h , g ./ Ž0,0 .
where G 1,1w hg x < R 1 is the radius-dependent shifted lattice and x w hg x w25,26x is a holomorphic function
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
coming from the four left-moving invariant coordinates f IL , I s 2,3,4,5:
x
h g
1 s 2
Ýu
g 4 d
g ,d
Ž 4.11 . In the infrared limit Im t ™ ` only the massless states give a non-zero contribution for any finite R 1. In this limit B6 Ž t ™ ` . s 45r2 s y360 s1r2 ,
Ž 4.12 .
s1r2 s y1r16 denotes the contribution to strw s x6 of a massless spin-1r2 state. Thus, B6 Ž t ™ `. matches the contribution of the massless fields of the N s 6 supergravity multiplet B6Ns 6 ŽSUGRA. s y360 s1r2 . Therefore for any finite R 1 the massless degrees of freedom are precisely those of the N s Ž4L q 2 R . supergravity multiplet in D s 5. Having obtained B6 as a function of R 1 , we can examine the limits R 1 ™ ` and R 1 ™ 0. In the first limit the winding states are superheavy and give exponentially suppressed contributions to B6 . Only the zero winding sector survives: B6 Ž R 1 ™ ` . s
45 4 45
s 4 45 s 2
0 1
x
G w 01 x
u 34 Ž t . q u44 Ž t . u4 u4
it R 12
,
it
B6 Ž R 1 ™ 0 . s
1
;
2
Ž 4.14 .
45 q 4 45 2
45 2
h g
< ms 0
1 0
x
1
qx 1
0
/
u 2 itR12
u 3 itR12
x 1
itR12 y u 2 itR12
Žu3
.,
Ž 4.16 .
where we have used the identity x w 10 x q x w 10 x q x w 11 x s 0, and the leftrright-mass level matching conditions are valid in the m s 0 sector. Performing a Poisson resummation of the above expression for B6 we obtain:
Ž B6 . Ds5 s
ty 2
Ž B6 . Ds6
R1
Ž 4.17 .
with
Ž B6 . Ds6 s ;
iR12
u2
ž
4
s
G
Ž h , g ./ Ž0,0 .
45
with
Ž B6 . Ds6 s
4
h g
x
Ý
1
where the last equality follows from the leftrright mass-level matching condition in the zero winding sector. Performing a Poisson resummation of the last expression we find
45
45
s
R 12
Ž 4.13 .
Ž B6 . Ds5 s R ty 2 Ž B6 . Ds6 ,
In the above equation we renormalize the six-dimensional Ž B6 .Ds6 by dividing out the S 1 volume R 1 and the contribution of the zero modes of a noncompact coordinate ty . Thus we show that B6 vanishes exponentially in the decompactification limit R 1 ™ ` due to the restoration of N s 8 in D s 6. When R 1 ™ 0 the KK momenta become superheavy, so that only the sector with zero KK-momenta Ž m s 0. survives: 1 2
w e ip Ž hqgg . y e ip Ž gqh d . x .
t ' Im t ;
297
45 2 45 2
ž
u3
ž
i tR 12
O 2exp
i
y u4 yp tR 12
/
tR12 ,
/ Ž 4.18 .
t
ž
O 2exp
yp R 12 4t
/
.
Ž 4.15 .
where the normalization factor from D s 5 to D s 6 now involves the dual S 1 volume 1rR1. As in the previous decompactification limit, B6 Ž R 1 ™ 0. van-
298
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
ishes exponentially in the dual decompactification limit Ž R 1 ™ 0. due to the restoration of N s 8 in D s 6.
dimensional anomaly and the B6-helicity supertrace at the same time. The supersymmetry is extended to the maximal one and the theory becomes leftrrightsymmetric.
5. Conclusions Acknowledgements In this paper we have given evidence for the existence of a chiral N s Ž4,2. supergravity in six dimensions, which reduces to known N s 6 supergravity upon dimensional reduction at D s 5 and D s 4. This theory is based on an automorphism group of the supersymmetry algebra, which is H s USpŽ4. = USpŽ2. and a ‘‘duality group’’, which is O Ž1,5.. Therefore the fermions of this theory are in H-representations, the vectors are in the O Ž1,5. spinor representation, the two-forms are in the vector representation of O Ž1,5. and the scalars are coordinates of the O Ž1,5.rO Ž5. coset. This is a consequence of the Ž0,2. decomposition of the Ž4,2. supergravity multiplet, which reveals that the scalars are in five tensor multiplets w27x. This theory, like Ž2,0. and Ž4,0. D s 6 supergravities, is a non-Lagrangian theory, due to the presence of self-dual 2-forms, but it can presumably be defined through a consistent set of equations of motion, by using supersymmetry. However, unlike the other chiral theories, this theory has a gravitational anomaly that cannot be cancelled by adding matter, since there are no N s Ž4,2. multiplets with spin J - 3r2. As a consequence the N s Ž4,2. theory is inconsistent at the quantum level. A way the anomaly cancellation can be achieved is, either by adding another gravitino multiplet, which enlarges the theory to N s Ž4,4., or by adding twelve N s Ž4,0. matter multiplets, which are expected to explicitly break two supersymmetries and then make the theory inconsistent. It is interesting to understand why the analogous theories at D s 4 and D s 5 are consistent string backgrounds while the D s 6 case is not. If we define the D s 6 theory by decompactifying one dimension from the D s 5 case, then some extra massive states become massless in the decompactification limits R 6 ™ ` or 0. In both limits the extra massless states correspond to an extra N s Ž4,2. gravitino multiplet, which cancels precisely the six-
We would like to thank L. Andrianopoli, M. Flato and C. Fronsdal for interesting discussions. This work is supported in part by EEC under TMR contracts ERB-4061-PL-95-0789, ERBFMRX-CT960045 and by DOE grant DE-GG03-91ER40662. References w1x J. Strathdee, Int. J. Mod. Phys. A 2 Ž1987. 273. w2x See for instance: A. Salam, E. Sezgin, Supergravities in Diverse Dimensions, vol. 1, North-Holland, Word Scientific, 1989. w3x S. Ferrara, C. Kounnas, Nucl. Phys. B 328 Ž1989. 406. w4x S. Ferrara, P. Fre, ´ Int. J. Mod. Phys. A 5 Ž1990. 989. w5x B. Julia, in: S.W. Hawking, M. Rocek ŽEds.., Superspace and Supergravity, Cambridge, 1981, p. 331 w6x E. Bergshoeff, T.W. Kephart, Abdus Salam, E. Sezgin, Mod. Phys. Lett. A 1 Ž1986. 267. w7x M. Flato, C. Fronsdal, UCLA-84-TEP-20, Dec 1984; in: C. Fronsdal, D. Reidel ŽEds.., Essays On Supersymmetry, Publishing Company, 1986. w8x M. Gunaydin, G. Sierra, P. Townsend, Phys. Lett. B 133 Ž1983. 72. w9x L. Andrianopoli, R. D’Auria, S. Ferrara, hep-thr9705024. w10x L. Andrianopoli, R. D’Auria, S. Ferrara, P. Fre’, M. Trigiante, hep-thr9707087. w11x L. Andrianopoli, R. D’Auria, S. Ferrara, P. Fre’, M. Trigiante, Nucl. Phys. B 496 Ž1997. 617. w12x N. Seiberg, Nucl. Phys. B 303 Ž1988. 286. w13x I. Antoniadis, C. Bachas, C. Kounnas, Nucl. Phys. B 289 Ž1987. 87. w14x H. Kawai, D. Lewellen, S.H.H. Tye, Nucl. Phys. B 288 Ž1987. 1. w15x K.S. Narain, M.H. Sarmadi, C. Vafa, Nucl. Phys. B 288 Ž1987. 551. w16x C. Kounnas, Nucl. Phys. ŽProc. Suppl.. 58 Ž1997. 57, hepthr9703198. w17x E. Kiritsis, C. Kounnas, hep-thr9703059. w18x E. Kiritsis, C. Kounnas, P.M. Petropoulos, J. Rizos, Phys. Lett. B 385 Ž1996. 87, hep-thr9606087. w19x J. Scherk, J.H. Schwarz Nucl. Phys. B 153 Ž1979. 61; Phys. Lett. B 82 Ž1979. 60. w20x R. Rohm, Nucl. Phys. B 237 Ž1984. 553; C. Kounnas, M. Porrati, Nucl. Phys. B 310 Ž1988. 355; S. Ferrara, C. Kounnas, M. Porrati, F. Zwirner, Nucl. Phys. B 318 Ž1989. 75; C. Kounnas, B. Rostand, Nucl. Phys. B 341 Ž1990. 641; I.
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299 Antoniadis, C. Kounnas, Phys. Lett. B 261 Ž1991. 369; I. Antoniadis, Phys. Lett. B 246 Ž1990. 377. w21x S. Ferrara, C. Savoy, B. Zumino, Phys. Lett. B 110 Ž1981. 393. w22x C. Bachas, E. Kiritsis, Nucl. Phys. B ŽProc. Suppl.. 55B Ž1997. 194, hep-thr9611205. w23x E. Kiritsis, C. Kounnas, Nucl. Phys. B 456 Ž1995. 699, hep-thr9507051; hep-thr9508078; hep-thr9509043.
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w24x E. Kiritsis, C. Kounnas, Nucl. Phys. B ŽProc. Suppl.. 45BC Ž1996. 207, hep-thr9509017. w25x C. Kounnas, A. Kumar, CERN-TH-97-194, August 1997, to appear in Nucl. Phys. B , hep-thr9709061. w26x A. Gregori, E. Kiritsis, C. Kounnas, N.A. Obers, P.M. Petropoulos, B. Pioline, CERN-TH-97-103, August 1997, to appear in Nucl. Phys. B , hep-thr9708062. w27x L. Romans, Nucl. Phys. B 276 Ž1986. 71.
Physics Letters B 420 Ž1998. 289–299
N s ž4,2 / chiral supergravity in six dimensions and solvable Lie algebras Riccardo D’Auria a
a,c
, Sergio Ferrara
b,c
, Costas Kounnas
b,1
Dipartimento di Fisica, Politecnico di Torino, C.so degli Abruzzi, 24, I-10129 Torino, Italy b Theory DiÕision, CERN, CH-1211 GeneÕa 23, Switzerland c Department of Physics and Astronomy, UniÕersity of California, Los Angeles, USA Received 6 November 1997 Editor: L. Alvarez-Gaume´
Abstract Decomposition of the solvable Lie algebras of maximal supergravities in D s 4, 5 and 6 indicates, at least at the geometrical level, the existence of an N s Ž4,2. chiral supergravity theory in D s 6 dimensions. This theory, with 24 supercharges, reduces to the known N s 6 supergravity after a toroidal compactification to D s 5 and D s 4. Evidence for this theory was given long ago by B. Julia. We show that this theory suffers from a gravitational anomaly equal to 4r7 of the pure N s Ž4,0. supergravity anomaly. However, unlike the latter, the absence of N s Ž4,2. matter to cancel the anomaly presumably makes this theory inconsistent. We discuss the obstruction in defining this theory in D s 6, starting from an N s 6 five-dimensional string model in the decompactification limit. The set of massless states necessary for the anomaly cancellation appears in this limit; as a result the N s Ž4,2. supergravity in D s 6 is extended to N s Ž4,4. maximal supergravity theory. q 1998 Elsevier Science B.V.
1. Introduction Supersymmetry algebras in any dimension and their representations are closely related to properties of spinors. Spinors of O Ž D y 1,1. are real for D s 2,3,9 mod 8, pseudoreal for D s 5,6,7 mod 8, and complex in D s 4,8 mod 8. The above reality properties refer to spinors of a given chirality when the dimension is even. The corresponding supersymme-
1
On leave from Ecole Normale Superieure, 24 rue Lhomond, ´ F-75231, Paris Cedex 05, France. E-mail address: [email protected].
try algebras have automorphism groups H Žsometimes called R-symmetry., which fall in three categories w1x:
i. O Ž N . ,
i . USp Ž N .
and
iii . U Ž N . ,
N being the number of spinorial charges in the appropriate dimension. We note that in D s 2 and 6 Žmod 8. algebras with different numbers of chiral charges NL , NR exist with O Ž NL . = O Ž NR . and USpŽ NL . = USpŽ NR . as H-groups respectively. If one considers compactifications of higher-dimensional theories on smooth manifolds the number of
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 0 8 - 6
290
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
supersymmetry charges N in lower dimensions is always 2 n for some n. This leaves aside certain theories, which can be constructed in lower dimensions, where the number of supercharge components is w2,3x: Ø In D s 4: 12, 20 and 24, corresponding to N s 3, 5 and 6 supergravity with H s UŽ3., UŽ5. and UŽ6. respectively. Ø In D s 5 and D s 6: 24, corresponding to N s 6 with H s USpŽ6. and N s Ž4,2. with H s USpŽ4. = USpŽ2. respectively. 2 Theories in D s 4 and 5, have been shown to exist as consistent type II string compactifications on asymmetric orbifolds w3,4x. The N s 6 supergravity is special because it is the only one that can be lifted to six dimensions where it becomes chiral w5,6x. In D s 4 the N s 6 supergravity is also special when is defined in anti de Sitter space. In fact OspŽ6 <4. gauge supergravity is the only one with a zero-center representation. w7x In Section 2 of this work we explore some properties of N s Ž4,2. supermultiplets in D s 6 by means of the chain decomposition N s Ž4,4. ™ N s Ž4,2. ™ N s Ž0,2.. This allow a derivation of the structure of the N s Ž4,2. supergravity, and in particular a relation of its scalar manifold to the decomposition of the maximal supergravity into N s 2 sub-theories. We show that this theory suffers from a gravitational anomaly w6x, which is 4r7 of the N s Ž4,0. pure supergravity anomaly. In Section 3 we derive the D s 5, N s Ž4,2. string ground state and show why the four- and fivedimensional descendants of the anomalous six-dimensional theory are not affected by this anomaly when one includes massive degrees of freedom dictated by string theory. In Section 4 we analyse the two decompactification limits R ™ `, R ™ 0 and we show the restoration of N s Ž4,4. in D s 6. Our conclusions are in Section 5.
2. N s 2 and N s 6 decomposition of maximal supergravity in D s 4, 5 and 6 dimensions. Let us first consider the general decomposition of maximal supergravities in terms of N s 2 multiplets w8–10x. The important fact in this decomposition is that the spin 3r2 multiplet does not contain scalars. If we associate to the scalar manifold M s GrH a solvable group with a Lie algebra S w11x, then we find that the S vector space decomposes into S V q SH , where S V and SH are solvable Lie algebras associated to Jmax s 1 and Jmax s 12 respectively. The Jmax s 12 w9,10x, multiplets are hypermultiplets and SH is thus related to a quaternionic space with holonomy SUŽ2. = H; H is the automorphism group of the supersymmetry algebra in the corresponding dimension. Note that in D s 6, since the vector multiplet has no scalars, the Jmax s 1 multiplet is a tensor multiplet. Our starting point is to decompose the maximal D s 6 supermultiplet into N s 2 ones. This amounts to decomposing the N s Ž4,4. representation into N s Ž0,2. massless representations. The N s Ž0,2. representations are: Ø graÕiton; G ™ Ž1 L ,1 R . q 2Ž1 L , 12 R . q Ž1 L ,0 R .; Ø graÕitino; g 1 ™ 2Ž1 L , 12 R . q 4Ž1 L ,0 R .; Ø graÕitino; g 2 ™ 2Ž 12 L ,1 R . q 4Ž 12 L , 12 R . q 2Ž 12 L ,0 R .; Ø Õector; V ™ Ž 12 L , 12 R . q 2Ž 12 L ,0 R .; Ø tensor; T ™ Ž0 L ,1 R . q 2Ž0 L , 12 R . q Ž0 L ,0 R .; Ø hyper; H ™ 2Ž0 L , 12 R . q 4Ž0 L ,0 R .. All the above multiplets are obtained by tensoring Clifford algebra states Ž0 L , 12 R ., 2Ž0 L ,0 R . by a Ž JL , JR ., SUŽ2.L = SUŽ2.R representation, with a doubling Žwhen required. by CPT. The N s Ž4,4. G graviton multiplet decomposes into N s Ž0,2. as follows: N s Ž 4,4 .
q16 2 Instead of the notation, N s1 for the minimal and N s 2 for the maximal supersymmetry in Ds6, we are using the chiral notation N s Ž2,0. for the minimal and N s Ž4,4. for the maximal supersymmetry; this notation is more convenient in Ds6 since the chiral spinors in Ds6 are Majorana symplectic w2x.
q20
ž ž
Gs
1
,
ž
Ž 1 L ,1 R . q 4 1 L ,
1
2L 2R 1 2L
/
1 2R
/ ž q4
1 2L
,1 R
q 5 Ž 1 L ,0 R . q 5 Ž 0 L ,1 R .
/ ž
,0 R q 20 0 L ,
1 2R
/
q 25 Ž 0 L ,0 R . .
s 1G q 1 g 1 q 2 g 2 q 8V q 5T q 5H.
/
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
On the other hand we may decompose N s Ž4,4. ™ N s Ž0,2. through the chain N s Ž4,4. ™ N s Ž4,2. ™ N Ž0,2.. We first get w N s Ž4,4.xG ™ w N s Ž4,2.xG q w N s Ž4,2.x g , Ž256 ™ 128 q 128 states.. Then we can further decompose w N s Ž4,2.xG and w N s Ž4,2.x g into N s Ž0,2. multiplets to finally obtain: - N s Ž 4,2 .
Gs
Ž 1 L ,1 R . q 8
1
ž
,
ž
1 2R
/ ž
q 10 0 L ,
s N s Ž 0,2 .
G q2
q 5 N s Ž 0,2 . - N s Ž 4,2 .
g s2
ž
q10 0 L ,
ž
1 2R
/
2L 2R 1
1 2R
2L
,1 R
q4
N s Ž 0,2 .
1
2L
,0 R
g2
T,
1
1
1
/ ž / / ž /
1L ,
2R
q 16
q8 1
2L
,
2L 2R
,0 R
q4 Ž 1 L ,0 R . q 20 Ž 0 L ,0 R . s N s Ž 0,2 . q8 N s Ž 0,2 .
V q5
N s Ž 0,2 .
g1
H.
This implies, in particular, the solvable Lie algebra decomposition S
O Ž 5,5 . O Ž 5. = O Ž 5. qS
O Ž 4,5 . O Ž 4. = O Ž 5.
O Ž 5. ,
S
E6Ž6. USp Ž 8 . qS
S
Ž 2.1 .
USp Ž 6 .
F4Ž4.
SU Ž 8 .
sS
Ž 2.2 .
SO ) Ž 12 . U Ž 6.
E6Ž2.
in Ds4.
SU Ž 2 . = SU Ž 6 .
Ž 2.3 .
The rank of the above cosets in the various dimensions decomposes as follows: Ø In D s 6: 5 s 1V q 4 H; Ø In D s 5: 6 s 2V q 4 H; Ø In D s 4: 7 s 3V q 4 H. This shows that the fields that correspond to the compactification radii are vector multiplets; the rank of the hypermultiplet manifold does not change with the dimensions. The rank-4 quaternionic spaces at D s 6,5 and 4, correspond to the following maximal decompositions of the corresponding isometry groups: F4Ž4. ™ O Ž 5,4 . ,
52 ™ 36 q 16;
78 ™ 52 q 26.
The total number of states of the N s Ž4,2. supergravity in six dimensions is 128 s 64 bosons q 64 fermions as in D s 4 and D s 5. However the theory in D s 6 is anomalous since the gravitational anomaly is y360 I , where I stands for the anomaly contribution of one Weyl fermion in six dimensions. The gravitational anomaly contribution of the pure N s Ž4,0. supergravity multiplet is y630 I , so that IŽ4,2. s 47 IŽ4,0. . In the case of the Ž4,0. theory the anomaly is cancelled by adding twenty-one Ž4,0. tensor multiplets, giving a contribution to the anomaly q21 = 30 I w12x. In the present case, owing to the absence of matter multiplets with Jmax less than 32 , the N s Ž4,2. cannot be anomaly-free. If one adds a w N s Ž4,2.x g 1 multiplet then the supersymmetry is extended to N s Ž4,4., which is an 1 2
i.e. algebras with a rank-1 tensor multiplet coset and a rank-4 quaternionic manifold with holonomy SUŽ2. = H, H s USpŽ2. = USpŽ4.. This statement shows that the decomposition of the maximal solvable Lie algebra into N s 2 solvable Lie algebras corresponds to the scalar sector of the two N s Ž4,2. multiplets into which the w N s Ž4,4.xG multiplet decomposes. If we delete the w N s Ž4,2.x g multiplet we obtain a pure N s Ž4,2. supergravity theory with scalar manifold O Ž1,5.rO Ž5.. This result was obtained by Julia in w5x.
in Ds5,
USp Ž 2 . = USp Ž 6 .
E7Ž7.
qS
SU ) Ž 6 .
sS
E6Ž2. ™ F4Ž4. ,
O Ž 1,5 .
sS
Note that a similar procedure in D s 5 and D s 4 gives analogous relations among solvable Lie algebras w9,8x:
q Ž 1 L ,0 R .
ž / / ž /
q5 Ž 0 L ,1 R . q 5 Ž 0 L ,0 R q 4 q2 1 L ,
1
291
1 2
1 2
1 2
292
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
anomaly-free theory. It seems that this is the only consistent way to cancel the anomaly. Another possibility to cancel the anomaly would be to add twelve Ž4,0. tensor multiplets; this solution, however, explicitly violates N s Ž4,2. supersymmetry. It is of interest to understand the reason why the six- dimensional theory with 24 supercharges is anomalous while, in lower dimensions, theories with the same number of supercharges can be constructed in type II string theories w3x. In the next sections we will show the restoration of N s Ž4,4. in six dimensions, starting from a fivedimensional N s 6 string theory and decompactifying one of the dimensions considering either R 6 ™ ` or R 6 ™ 0.
3. N s 6 construction in five dimensions In order to construct in string theory a ground state with N s Ž4L q 2 R . it is necessary to consider the type II theory with maximal supersymmetry N s Ž4L q 4R .; then one has to reduce the supersymmetry asymmetrically by projecting out half of the right moving supercharges. The heterotic or type I construction of N s Ž4L q 2 R . ground states is impossible, since the maximal supersymmetry is N s Ž4L q 0 R . in the heterotic theory and N s 4 in that of type I. We start by presenting the type II N s Ž4L q 2 R . string models and write down their partition function. Models with the same number of supercharges in D s 4 have been obtained by applying projections to the maximal-supersymmetry string theories w3x, which preserve modular invariance as well as the conformal symmetries on the string world-sheet. The maximal-supersymmetry type II model is described in the light-cone gauge, by 8 world-sheet leftrrightmoving bosonic and fermionic coordinates. In our notation, in D s 5, the coordinates cmL, R and XmL, R Ž m s 3,4,5. represent the space-time degrees of freedom Žin the light-cone gauge., whereas the remaining ones correspond to the fermionic x IL, R and bosonic f IL, R , I s 1,2,3,4,5 internal degrees of freedom. This description will be appropriate for the asymmetric orbifold construction w15x, where four of the right-moving internal coordinates are twisted in
order to reduce by a factor of 2 the right-moving supersymmetries: Z2 : Ž x RI , f RI . ™ y Ž x RI , f RI . ,
I s 2,3,4,5.
Ž 3.1 .
f IL, R
Ž Is In the fermionic construction w13,14x, 1,2,3,4,5. are replaced by a pair of Majorana–Weyl spinors yIL, R and v IL, R Ž I s 1,2,3,4,5., by means of the 2d-boson-fermion equivalence: JIL , R s Ef IL , R s yIL , R v IL , R .
Ž 3.2 .
The construction of string models amounts to a choice of boundary conditions for the 2d fermions x IL, R , yIL, R , v IL, R , which satisfies local and global consistency requirements w13x. The N s Ž4L q 4R . maximal supersymmetry model, constructed in this manner, have four space-time supercharges originating from the left-moving sector and another four from the right-moving sector. In the language of the fermionic construction w13x, this model is defined by introducing three basis sets, F, S and S. The first one contains all the left- and the right-moving fermions: F s cmL , x IL , yIL , v IL < cmR , x IR , yIR , v IR = Ž m s 3,4,5; I s 1, . . . ,5 . ,
Ž 3.3 .
and the basis sets S and S, which contain only eight left- or right-moving fermions and generate the GSO projections of the maximal supersymmetry w3x: S s cmL , x IL
S s cmR , x IR .
Ž 3.4 .
Four of the gravitinos of the N s Ž4L q 4R . model belong to the S-sector and the other four to the S-sector. Then, by applying the appropriate projection one obtains the N s Ž4L q 2 R . superstring model. A possible projection for constructing N s Ž4L q 2 R . is specified by a choice of fermion basis b such that b s cmR , x 1R , y 2R,3,4,5 < y 1L , v 1L , y 1R , v 1R .
Ž 3.5 .
Then b generates the desire left–right-asymmetric Z2 projection. Namely, it acts as a twist on x IR and JIR when I s 2,3,4,5. Also it acts non-trivially on the set of fermions T s y 1L , v 1L , y 1R , v 1R .
Ž 3.6 .
This action however keeps the currents J1L, R s Ef 1L, R invariant. In the bosonic f 1L, R language, the non-
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
trivial action on T implies a shifting by 1r2 unit on the f 1 lattice; f 1L, R ™ f 1L, R q p . Therefore, the Z2 defined by b acts as a shift on f 1 and as an asymmetric twist on the remaining four internal supercoordinates x IR , f IR , I s 2,3,4,5. This asymmetric Z2 breaks half of the right-moving supersymmetries by projecting out two of the gravitinos of the S-sector. The resulting model has the desired N s Ž4L q 2 R . supersymmetry. The presence of T fermions in b makes the states coming from the b-twisted sector massive. Indeed, the lowest state in the b-twisted sector, has right-moving conformal dimensions DR s 10 16 , which corresponds to the conformal dimension of a spin field constructed with the ten right-moving R fermions spinw cmR , x 1R , y 2,3,4,5 y 1R , v 1R x. Thus, the lower right-moving mass level in this sector is 1 1 w m R Ž b .x2 s 10 16 y 2 s 8 . The absence of massless twisted sectors in any consistent orbifold construction is expected. Indeed, the the fact that the massless spectrum of N s Ž4L q 2 R . in D s 5 is fully determined by the graviton supermultiplet implies that all the N s Ž4L q 2 R . massless states must be the Z2-invariant states of N s Ž4L q 4R . supergravity. These states are all present in the untwisted sector and therefore all extra states have to be massive. Using for instance b˜ s cmR , x 1R , y 2R,3,4,5
Ž 3.7 .
instead of b, then the b˜ asymmetric projection still projects out the two right-moving gravitinos from the ˜ ˜ S-sector. In this case, however, the b-twisted sector contains some extra massless states, since the conforR x is precisely mal dimension of spinw cmR , x 1R , y 2,3,4,5 ˜ 1r2. Among the extra massless states of the b-twisted sector, there exist two extra graÕitinos with vertex operator
and thus, the only available asymmetric projection is ˜ this projection, however, the one that is based on b; gives rise to massless b˜ twisted sectors, which contain two extra right-moving gravitinos. In the following, we will discuss in more detail the b fermionic model in D s 5. Then this model will be generalized in order to include the compactification radius R 1 modulus associated to f 1. Finally, in the next section, we study the extension of supersymmetry in the two decompactification limits R 1 ™ ` and R 1 ™ 0. We will show that the helicity supertraces B2 n s str w s x 2 n vanish for n s 0,1,2,3 in both decompactification limits, indicating the extension of N s Ž4L q 2 R . ™ N s Ž4L q 4R .. We start with the partition function of N s Ž4L q 2 R . based on b: 1
ZbDs 5 s
3
Imt 2 h 3 h 3 =
=
˜
, q
Ž 3.8 .
and so the N s Ž4L q 2 R . supersymmetry is extended again to the maximal N s Ž4L q 4R .. Already at this point one can guess the difficulties that will arise in constructing a six-dimensional string model with N s Ž4L q 2 R . supersymmetry. Indeed, in six dimensions the coordinate f 1 is non-compact
1
1 4 1 2
Ž y.
Ý
aq bqa b
2 a 4 u ba b
Ýu
Ž y . aqbqa b
h4
a, b , a , bs0
h4 ayh byg
u
u
aqh bqg
h, g
=Z5,5
h g
.
Ž 3.9 . Z5,5 w hg x
In the above expression denotes the contribution of the five compactified coordinates f IL, R , I s 1, . . . ,5 Žfermionized.; four of them are Ž h, g .-twisted while f 1L, R is Ž h, g .-shifted. In terms of fermionic characters, the analytic expression of Z5,5w hg x is: Z5,5
h g
u
1 s 2
Ý
=
gqh d qg
g qh dqg
u
hh
g ,d
u b R V3r2 s CmL spin cmR , x 1R , y 2,3,4,5
293
g 4 d 4
h
2
u w dg x u
Ž y.
g qh dqg 4
h
u
d hq g gqh g
g yh d yg
Ž y.
hg
,
Ž 3.10 . where the term in the first square bracket corresponds to the contribution of the compactified coordinate f 1 written in terms of the fermionic characters of y 1L, R , v 1L, R. In terms of f 1-lattice characters, it corresponds to a fixed S 1 radius at the fermionic
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294
point R 1 s 1r '2 . The term in the second square bracket corresponds to the characters of the Z2 asymmetric orbifold, where four right-moving coordinates JI ™ yJI , I s 2,3,4,5 are twisted. The phase factors Žy. d hq g gqh g , Žy. h g are dictated by modular invariance. As we have already explained, we would like to examine the behaviour of the N s Ž4L q 2 R . model in the decompactification limits R 1 ™ ` and R 1 ™ 0. For this purpose we must deform the above model and move away from the fermionic point by switching on the marginal deformation that is associated to the modulus R 1. Before doing that, let us first examine the moduli space of the above model, namely all possible J L P J R Ž1,1.-deformations which we are able to switch on simultaneously. Due to the left–right asymmetry the only possible deformations are: J1L P J1R ™ R 1
and
JIL P J1R ™ YI , I s 2,3,4,5.
The first deformation corresponds to the R 1 modulus while the remaining four correspond to the Wilson lines YI moduli. Altogether they form the perturbative moduli space constructed from the NS-scalars
Mp s
O Ž 5,1 . O Ž 5.
.
Ž 3.11 .
The perturbative moduli space is extended to Mn p s SU ) Ž6.rUSpŽ6. once we include the dilaton moduli Žsinglet under O Ž5.. and the remaining RR-scalars Žin 15, 10, and 10’ representations of O Ž5... The perturbative string states form the charge lattice G 1,5w hg x associated to the moduli space Mp . The absence of continuous deformations associated to the twisted coordinates f IR , I s 2,3,4,5 implies that all radii R I , I s 2,3,4,5, of the ‘‘twisted’’ four-dimensional space must be frozen to some special values of the moduli space Žthe SOŽ8. fermionic point in our case.. This also means that the asymmetric orbifold projection is well defined only for special Õalues of the moduli of the initially untwisted G4,4 lattice, e.g. the fermionic SO Ž8. symmetric point. Keeping this point in mind we can generalize the model to include the radius and Wilson line deforma-
tions by replacing Z5,5w hg x by Z5,5 Ž R 1 ,YI .; since the non-vanishing Wilson lines do not affect the decompactification limits we will restrict ourselves to YI s 0, keeping the modulus R 1 arbitrary: Z5,5
h g
Ž R1 ,YI s 0 . s
h g
G 1,1
Ž R1 .
hh
Z4,4
h g
.
Ž 3.12 . Z4,4 w hg x
where is the asymmetric orbifold contribution of the four internal coordinates f I , I s 2,3,4,5 Z4,4
h g
s 2
g 4 d 4
u
1
Ý
2
u w dg x u
h
g ,d
g qh d qg 4
u
g yh dyg
h
Ž y.
hg
Ž 3.13 . and
G 1,1 s
h g
Ž R1 .
Ý exp
ip gm q i2pt PL2 yi2pt PR2
m, n
with PL s
PR s
1
m q
2 R1 1
Ž 2 n q h . R1 2
m y
2 R1
Ž 2 n q h . R1 2
,
Ž 3.14 .
is the shifted f 1-lattice w16–18x. The N s Ž4L q 2 R . model constructed above is an asymmetric freely acting orbifold. Following Refs. w16–18x it corresponds to a partial spontaneous breaking w16,17x of supersymmetry N s Ž4L q 4R . ™ N s Ž4L q 2 R .. The massless states of this model consist only of N s Ž4L q 2 R . gravitational multiplet. The remaining massless states of an N s Ž4L q 4R . Žtwo 3r2-multiplets. become massive, with a common mass equal to: 1
Ž m23r2 . 7,8 s R 2
Ž h s 0, < m < s 1, n s 0 . .
1
These states correspond to the ‘‘untwisted’’ sector states with momentum < m < s 1 and 2 n q h s 0. The ‘‘odd’’ winding states h s 1 correspond to the
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
twisted sector. The lowest twisted states correspond to two massive 3r2-multiplets with masses proportional to:
Ž
m23r2 7X ,8X s
.
R 12 4
Ž h s 1, m s 0, <2 n q h < s 1 . .
4. Six-dimensional decompactification of the fivedimensional N s (4 L H 2 R ) string model A six-dimensional model in string theory can be defined from a five-dimensional one in two different ways: either by sending the compactification radius R 1 ™ ` or R 1 ™ 0. In the first case the Kaluza-Klein ŽKK. momenta become the six-dimensional continuous momentum P62 ; m2rR 12 . In the limit R 1 ™ 0, however, the KK states become superheavy while the string windings give rise to the six-dimensional continuous momentum P˜62 ; n 2 R 2 . Starting with the D s 5, N s Ž4L q 2 R . string defined in the previous section, we would like to examine the six-dimensional theories obtained in the two stringy decompactification limits. The five-dimensional spectrum of this model always contains 6 massless gravitinos. There is also a tower of massive gravitinos with the same R symmetry charges having even KK-momentum m and even winding charge nX s 2 n q h Ž h s 0.. When R 1 ™ ` or 0 one obtains 6 massless gravitinos with continuous momenta, either P6 or P˜6 . However, there exist two more extra towers of massive gravitinos with different R-symmetry charges, which can become massless in the two decompactification limits, namely: Ø In the limit R 1 ™ ` the Žtwo. massive gravitinos with odd KK-momentum m become massless, Ž m23r2 . 7,8 ™ 0, while the ‘‘winding’’ gravitinos Ž m 23r2 . 7X ,8X ™ ` become superheavy and decouple from the spectrum. The two extra massless gravitinos Ž m 23r2 . 7,8 s 0 together with the six massless gravitinos of N s Ž4L q 2 R . restore in six dimensions the N s Ž4L q 4R . supersymmetry. Ø In the other limit, R 1 ™ 0, the two gravitinos with odd KK momentum m, Ž m23r2 . 7,8 ™ ` become superheavy and decouple while the gravitinos with winding nX s 2 n q h odd Ž h s 1. become massless Ž m23r2 . 7X ,8X ™ 0.
295
The ‘‘field theory’’ analogue of the above model can be obtained by a Scherk-Schwarz w19x supersymmetry breaking mechanism w20,16,17x starting either from N s 2 in D s 10 or from N s 1 in D s 11. The difference from the string model is the absence of the winding states, nX s 2 n q h s 0 Ž n s 0, h s 0.. Therefore, in field theory, the first decompactification limit, R 1 ™ `, gives the same effective supergravity theory as in string theory. On the other hand the second limit, R 1 ™ 0, does not correspond any more to a decompactification but rather to a ‘‘dimensional reduction’’ where all KK states are truncated out from the spectrum. In the limit R 1 ™ 0 the field theory remains five-dimensional. The restoration of the maximal supersymmetry in D s 6 can be checked in the two decompactification limits by constructing the helicity supertrace B2 n w21,22,16–18x as a function of the radius R 1: 2s
w s x 2 n ey p t M s Ž R 1 .
2n
eyp t M s Ž R 1 . .
B2 n Ž R 1 ;t . s tr Ž y . s str w s x
2
2
Ž 4.1 .
The little group of massless particle in in D s 5 is SOŽ3.. By ‘‘helicity’’ s we mean UŽ1. charge of a UŽ1. sub-group of SO Ž3.. M s Ž R 1 . denotes the mass of the state. If there is a restoration of N s Ž4L q 4R ., B6 has to vanish in both decompactification limits due to the N s 8 supertrace identities: B2 n s 0,
n s 0,1,2,3,
in
N s 8.
In N s 6 B6 is not trivial since the supertrace identities for N s 6 are: B2 n s 0,
n s 0,1,2,
in
N s 6.
The supergravity massless sector of the N s 6 supergravity gives a non-trivial B6 ŽSUGRA. s 45r2. If in the decompactification limits the N s 6 supersymmetry is extended to N s 8 due to the presence of extra massless states, then B6 s 0 must be found in both limits. In order to derive B2 n in string theory it is convenient to define the helicity generating partition function w22,16,17,26,25x: Z string Ž Õ,Õ . s Tr q L 0 q L 0 e 2 p i Õ s Ly2 p i Õ s R , q s e 2 p it , q s e 2 p it ,
t s Imt ,
Ž 4.2 .
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
296
where sL and sR denote the left- and right-moving helicities. The physical helicity is given by s s sL q sR . Once Z string Ž Õ,Õ . is defined, then B2 n can be derived from Z string Ž Õ,Õ . by the action of the leftand right-helicity operators Qs
E
1
2p i E Õ
,
Qsy
E
1
2p i E Õ
In the expression for Z string Ž Õ,Õ . we can sum over the indices Ž a,b . and Ž a,b . using the Riemann identity of theta functions 1 2
,
Ž y.
Ý
u
h4
a, b
1
2n
s
so that B2string s Ž Q q Q . Z string Ž Õ,Õ . < Õs Õs0 . n
aq bqa b
h
4
u
Õ
1 1
a b
Ž Õ. u Õ
1 1
a b
a b
u Õ
1 1
a b
u
Õ
1 1
ž /u ž /uw xž /uw xž / 2
2
2
2
Ž 4.3 . The Õ,Õ helicity modifications for the partition function have been studied earlier, in order to obtain exact solutions of string theory in the background of magnetic fields and to investigate the associated phase-transition phenomena w22–24x. In that context the quantities Õ and Õ play the role of the background magnetic field. An explicit expression for Z string Ž Õ,Õ . for the N s Ž4,2. string model of the previous section is given by an expression that is similar to the Õ s Õ s 0 partition function:
j Ž Õ. j Ž Õ. h3 h3 =
1
1
Ý
4 =
1 2
=u
a b
Ž y. aqbqa b
h4
a, b , a , bs0
Ýu
aq bqa b
Ž y.
Ž Õ. u
a b
h4
Ž Õ. u
u
aqh bqg
Z5,5
h g
Ž R1 . ,
Ž 4.4 .
n 2
` 1
Ž1yq .
Ž1 y q
n 2p i Õ
e
sinp Õ
u 1X
p
u 1Ž Õ .
s
. Ž 1 y q n ey2 p i Õ .
2
Ý
h4
a, b
1 s
h
4
u
1 1
Õ
u
a b
Ž Õ. u
Õ
1 1
ž / ž / u
2
2
u
a b
u
aq h bqg
Õ
1y h
ž /
1yg
2
u
u
ayh byg
1qh 1qg
Õ
ž / 2
.
Ž 4.7 .
s 15Q 4 Q 2 Z string Ž Õ,Õ . < Õs Õs0 ,
Ž 4.5 .
Ž 4.8 .
receiving non-vanishing contribution from the N s 6 sector with Ž h, g . / Ž0,0.. Furthermore, using the identities 4 iQ u
1 1
Ž Õr2. s u
0 0
u
0 1
u
1 0
s 2h 3 ,
Ž 4.9 .
the expression for B6 , as a function of R 1 w25,26x, simplifies to: B6 s
,
j Ž Õ . s j Ž yÕ . , j Ž 0 . s 1.
Ž y . aq bqa b
1
6
where the j Ž Õ . and j Ž Õ . modifications is due to the helicity charge of the 2d bosonic oscillators. The Õ, Õ modifications due to the 2d fermionic degrees of freedom give rise to non-zero characteristics to the fermionic u w ab xŽ Õ . and u w ab xŽ Õ . functions w22,16,17,26,25x. The analytic expression of j Ž Õ . is:
j Ž Õ. s Ł
and
B6 s Ž Q q Q . Z string Ž Õ,Õ . < Õs Õs0
a 3 u ba b
h, g ay h byg
Ž 4.6 .
The vanishing of B0 , B1 and B2 follows automatically from the vanishing of u w 11 xŽ Õr2. and u w 11 xŽ Õr2. for Õ s 0 and Õ s 0: Indeed, Z string Ž Õ,Õ . vanishes like Õ 4 Õ 2 in the sector with Ž h, g . / Ž0,0. Ž N s 6 sector., and it vanishes like Õ 4 Õ 2 in the sector with Ž h, g . s Ž0,0. Ž N s 8 sector.. Therefore, the only non-trivial helicity supertrace is
Z string Ž Õ,Õ . s
,
45 4
Ý
x
h g
G 1,1
h g
< R1 ,
Ž 4.10 .
Ž h , g ./ Ž0,0 .
where G 1,1w hg x < R 1 is the radius-dependent shifted lattice and x w hg x w25,26x is a holomorphic function
R. D’Auria et al.r Physics Letters B 420 (1998) 289–299
coming from the four left-moving invariant coordinates f IL , I s 2,3,4,5:
x
h g
1 s 2
Ýu
g 4 d
g ,d
Ž 4.11 . In the infrared limit Im t ™ ` only the massless states give a non-zero contribution for any finite R 1. In this limit B6 Ž t ™ ` . s 45r2 s y360 s1r2 ,
Ž 4.12 .
s1r2 s y1r16 denotes the contribution to strw s x6 of a massless spin-1r2 state. Thus, B6 Ž t ™ `. matches the contribution of the massless fields of the N s 6 supergravity multiplet B6Ns 6 ŽSUGRA. s y360 s1r2 . Therefore for any finite R 1 the massless degrees of freedom are precisely those of the N s Ž4L q 2 R . supergravity multiplet in D s 5. Having obtained B6 as a function of R 1 , we can examine the limits R 1 ™ ` and R 1 ™ 0. In the first limit the winding states are superheavy and give exponentially suppressed contributions to B6 . Only the zero winding sector survives: B6 Ž R 1 ™ ` . s
45 4 45
s 4 45 s 2
0 1
x
G w 01 x
u 34 Ž t . q u44 Ž t . u4 u4
it R 12
,
it
B6 Ž R 1 ™ 0 . s
1
;
2
Ž 4.14 .
45 q 4 45 2
45 2
h g
< ms 0
1 0
x
1
qx 1
0
/
u 2 itR12
u 3 itR12
x 1
itR12 y u 2 itR12
Žu3
.,
Ž 4.16 .
where we have used the identity x w 10 x q x w 10 x q x w 11 x s 0, and the leftrright-mass level matching conditions are valid in the m s 0 sector. Performing a Poisson resummation of the above expression for B6 we obtain:
Ž B6 . Ds5 s
ty 2
Ž B6 . Ds6
R1
Ž 4.17 .
with
Ž B6 . Ds6 s ;
iR12
u2
ž
4
s
G
Ž h , g ./ Ž0,0 .
45
with
Ž B6 . Ds6 s
4
h g
x
Ý
1
where the last equality follows from the leftrright mass-level matching condition in the zero winding sector. Performing a Poisson resummation of the last expression we find
45
45
s
R 12
Ž 4.13 .
Ž B6 . Ds5 s R ty 2 Ž B6 . Ds6 ,
In the above equation we renormalize the six-dimensional Ž B6 .Ds6 by dividing out the S 1 volume R 1 and the contribution of the zero modes of a noncompact coordinate ty . Thus we show that B6 vanishes exponentially in the decompactification limit R 1 ™ ` due to the restoration of N s 8 in D s 6. When R 1 ™ 0 the KK momenta become superheavy, so that only the sector with zero KK-momenta Ž m s 0. survives: 1 2
w e ip Ž hqgg . y e ip Ž gqh d . x .
t ' Im t ;
297
45 2 45 2
ž
u3
ž
i tR 12
O 2exp
i
y u4 yp tR 12
/
tR12 ,
/ Ž 4.18 .
t
ž
O 2exp
yp R 12 4t
/
.
Ž 4.15 .
where the normalization factor from D s 5 to D s 6 now involves the dual S 1 volume 1rR1. As in the previous decompactification limit, B6 Ž R 1 ™ 0. van-
298
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ishes exponentially in the dual decompactification limit Ž R 1 ™ 0. due to the restoration of N s 8 in D s 6.
dimensional anomaly and the B6-helicity supertrace at the same time. The supersymmetry is extended to the maximal one and the theory becomes leftrrightsymmetric.
5. Conclusions Acknowledgements In this paper we have given evidence for the existence of a chiral N s Ž4,2. supergravity in six dimensions, which reduces to known N s 6 supergravity upon dimensional reduction at D s 5 and D s 4. This theory is based on an automorphism group of the supersymmetry algebra, which is H s USpŽ4. = USpŽ2. and a ‘‘duality group’’, which is O Ž1,5.. Therefore the fermions of this theory are in H-representations, the vectors are in the O Ž1,5. spinor representation, the two-forms are in the vector representation of O Ž1,5. and the scalars are coordinates of the O Ž1,5.rO Ž5. coset. This is a consequence of the Ž0,2. decomposition of the Ž4,2. supergravity multiplet, which reveals that the scalars are in five tensor multiplets w27x. This theory, like Ž2,0. and Ž4,0. D s 6 supergravities, is a non-Lagrangian theory, due to the presence of self-dual 2-forms, but it can presumably be defined through a consistent set of equations of motion, by using supersymmetry. However, unlike the other chiral theories, this theory has a gravitational anomaly that cannot be cancelled by adding matter, since there are no N s Ž4,2. multiplets with spin J - 3r2. As a consequence the N s Ž4,2. theory is inconsistent at the quantum level. A way the anomaly cancellation can be achieved is, either by adding another gravitino multiplet, which enlarges the theory to N s Ž4,4., or by adding twelve N s Ž4,0. matter multiplets, which are expected to explicitly break two supersymmetries and then make the theory inconsistent. It is interesting to understand why the analogous theories at D s 4 and D s 5 are consistent string backgrounds while the D s 6 case is not. If we define the D s 6 theory by decompactifying one dimension from the D s 5 case, then some extra massive states become massless in the decompactification limits R 6 ™ ` or 0. In both limits the extra massless states correspond to an extra N s Ž4,2. gravitino multiplet, which cancels precisely the six-
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