University of Groningen
(8,0) locally supersymmetric sigma models with conformal invariance in two
dimensions
Bergshoeff, Eric; Sezgin, E.; Nishino, H.
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Physics Letters B
DOI:
10.1016/0370-2693(87)90274-7
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Bergshoeff, E., Sezgin, E., & Nishino, H. (1987). (8,0) locally supersymmetric sigma models with conformal
invariance in two dimensions. Physics Letters B, 186(2). DOI: 10.1016/0370-2693(87)90274-7
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Volume 186, number 2
PHYSICS LETTERS B
5 March 1987
(8,0) LOCALLY S U P E R S Y M M E T R I C S I G M A M O D E L S W I T H C O N F O R M A L I N V A R I A N C E
IN TWO DIMENSIONS
E BERGSHOEFF, E SEZGIN
Internattonal Centrefor TheoretwalPhysws, 1-34100 Trwste, Italy
and
H NISHINO 1
Departmentof Physwsand Astronomy, Umversttyof Maryland, CollegePark, MD 20742, USA
Received 13 September 1986
The (8,0) conformal supergravlty, and an actton which describes its couphng to an arbitrary number of (8,0) scalar mulUplets
are constructed The 64 + 64 components of the conformal supermultlplet occur as Lagrangemultlphers which lead to dlfferentml
and algebraicconstraints on the fields of the scalar multlplets Solvingthe algebrmcconstraints yields an (8,0) locallysupersymmetric sigma model based on the mamfold SO( 8 + n,m )/SO (8) × SO(n,m), where n,m >10
1 Introduction Actions describing the couphng of
an arbitrary n u m b e r of scalar multlplets to conformal supergrav~ty m two d i m e n s i o n s have been constructed with (p,q) supersymmetry, for p,q<~4 :~,2
Essentmlly, these models are described by locally
supersymmetrlc sigma models wxth W e s s - Z u m m o
term and heterotlc fermlons, i e fermlons without onshell bosomc partners
Having possible s t n n g apphcatlons m m i n d , it is
clearly of interest to m q m r e as to whether one can
construct (p,q) models w~th p,q> 4
F r o m the work of Alvarez-Gaum6 a n d F r e e d m a n
[ 14] it is k n o w n that globally s u p e r s y m m e t n c sigma
models exist only for p,q<~ 4 O n the other hand, the
dimensional reduction of t e n - d i m e n s i o n a l Yang- M i l l s coupled conformal supergrawty [ 15 ] to two
t Present address Department of Physics, Brandeis Umverslty,
Waltham, MA 02254, USA
:~ For the (1,1) model see refs [1-3], for the (1,0) an (2,0)
models see refs [4-7], for the (2,2) model see ref [8], the
(4,4) model was considered m ref [9], the (4,0) model with
Wess-Zummo term is given in ref [ 10]
:2 By (p,q) supersymmetry we mean a supersymmetry algebra
generated by p left-handed and q right-handed supersymmetry
generators [ 11-13 ]
0370-2693/87/$ 03 50 © Elsevier Science Pubhshers B V
( N o r t h - H o l l a n d Physics P u b h s h l n g D i v i s i o n )
d i m e n s i o n s is expected to give a n (8,8) or (8,0) conformal supergravaty coupled to scalar multlplets This
suggests that, m order to construct (p,q) models w~th
p,q> 4, one should consider the local supersymmetry
as an essentml ingredient of the theory This is despite
the fact that the fields of the supergrawty multlplet
do not have dynamics in two d~menslons In fact, a
similar situation arises in three dimensions, where
the construction of N = 8 or N = 16 s~gma models
reqmres the coupling of a n o n - d y n a m i c a l supergravlty multlplet [ 16 ]
In this letter, we indeed construct a n action for an
(8,0) locally supersymmetrlc sigma model Our
strategy in this construction is as follows We first
extract the field content of the (8,0) superconformal
multlplet, a n d the matter scalar mulUplet by dimensional reduction of ten-d~mensxonal conformal
supergravlty [ 15 ], a n d Y a n g - M d l s multlplet [ 17 ],
respectwely We then construct the transformaUon
rules a n d the action of the two-dimensional model
by the Noether procedure
We next observe that the fields of the conformal
supermultlplet occur as Lagrange mulUphers which
lead to dffferentml a n d algebraic constraints on the
167
Volume 186, number 2
PHYSICS LETTERS B
fields o f the scalar multlplets Solwng the algebrmc
constraints, we then find that the 8n independent
scalars o f the theory are described by an (8,0) locally
supersymmetnc sigma model based on the grassm a n m a n coset space SO ( 8 + n,m)/SO (8) × SO (n,m)
where n,m >t 0 Note that both compact and noncompact spaces are allowed It is amusing that for m = 1,
these theories have a global Lorentz symmetry
SO(8+n,1)
Several features o f our constructmn are similar to
those which arise m the coupling o f an arbitrary
number of vector multlplets to N = 4 conformal
supergrawty m four dlmensxons [ 18 ]
In this letter, we focus our attentmn to the constructmn o f the actmn and the transformation rules
of the (8,0) model, and estabhshmg its geometry We
shall nexther try to estabhsh an infinite dxmenslonal
super V~rasoro symmetry, nor compute the anomalies associated wxth the model presented here In the
conclusmns we shall c o m m e n t on these issues, and
d~scuss further propertxes o f the model
A more detailed description o f the model presented m th~s letter will be given elsewhere [ 19 ]
2 Actton and transformatlon rules The field content of the two-dlmensmnal (8,0) conformal supergravity and scalar multlplet can be easily obtained by
a dimensional reduction and subsequent chlral truncation o f the ten-dxmenslonal conformal supergravlty and vector mult~plet A more detailed discussion
o f this dxmensxonal reduction will be given in ref
[ 19 ] Here we only give the result (See table 1 ) Note
that all fermlomc fields carry both a spacetxme splnor index a ( a = 1, 2) and an SO(8) splnor lndexA
( a = 1, , 16) which are suppressed for brevaty In the
table we have given the SO (8) content o f the different fields, their scale weight w, and the spaceUme and
SO (8) chlrallty o f the fermlomc fields
Knowing the field content, it is not difficult to make
an ansatz for the hnearlzed transformatmn rules of
the different fields Once the hnearlzed transformation rules are known one can find their nonhnear
extensaon by r e q m n n g closure o f the c o m m u t a t o r
algebra order by order m the fields Given these nonhnear transformation rules one can then construct a
supersymmetrlc action for one scalar multlplet coupled to (8,0) conformal supergravlty by applying the
standard Noether procedure Here we will only give
168
5 March 1987
the action up to terms quadratic in the fermlOnlC
fields and the transformation rules up to terms billnear in the fermlomc fields The full result will be
given m ref [ 19]
The actxon and the transformation rules are given
by .3
..{. ~ gl.W O p ( ~)t O t ) O v ~O ..{_ 1
l~bC~,OtT/z~/z~
- ~7o,"7"~u,,o.(0'0') - ~ O'¢'R
+ ½e- i ¢,7'O' w ~ ~ ; ~ , o + ¢,z'¢ '
-- ½0'¢SDq + 1 1 0 ' ~ ~/z7/t7tZJ
+ quartlc fermlons,
8 ell m =
( 1)
-- 21(7m ~///z,
8~u = ~ u e + l y u r / + b l l m e a r fermlon terms,
V# ° = -- 41¢7~e ° ~ 2 "1-l~-7.u7 [t)~J]
+ blhnear fermxon terms,
0Z' = - ½e- 1eu~ Vu ,j)a e + D,jTj e _ 7
- trace + blhnear fermmn terms,
OD 'j = - lg7 (' ~)f) + 207 °Z J) - trace
+ bfllnear fermlon terms,
8),=--lyU(Ou~o) e-- 2rl
+ blhnear fermmn terms,
(2)
8~0 = ~-2,
~0'=¢7'~u,
~3 Our
conventmns
are
q.m=(+-),
~',.Y.=r/.~.+Ys~m.,
¢~m.?n = __ ~)mys ~ mn ¢~pq= -- ( (~mpl~nq __ (~mql~np ) ,
y,Ts+ yjy,= 2j,~
Symmetry 2X=~)., ~[?uZ=-~y~2, ~[?. . . . X=~2y . . . . 2 with
~ = + 1 f o r n = l , 4 , 5, 8 a n d e = - I f o r n = 2 , 3, 6, 7
5 March 1987
PHYSICS LETTERS B
V o l u m e 186, n u m b e r 2
Table 1
Fields of (8,0) superconformal mult~plets m two d~menslons The +suffix in the fourth column denotes the SO(8) ch~rallty of the
fermlonlc fields
conformal supergravlty
Field
Type
Restrictions
SO ( 8 )
w
e~ "
~uu
boson
fermmn
ysVu= + ~ u
1
8, ÷
- 1
-1/2
V.'~
boson
ferrnlon
boson
fermlon
boson
28
5635
8~+
1
0
+3/2
+2
+1/2
+1
8.
8~-
0
+ 1/2
Z'
D°
.~
scalar mulnplet
0'
V
boson
fermmn
Va'~= - V.~'
)'~Z'= +X', ~';~'=0
D '~= IY', D" = 0
7~2=-2
Y~ = - V
#~z'= ( G z ' + ½oJ~z' - 1 Gk%tZ ' - G'JZj)
+ b l h n e a r f e r m i o n terms
(3)
In (1), ( 2 ) a n d ( 3 ) we have used the following
definitions The derivatives 9u are c o v a r l a n t with
respect to S O ( 8 ) a n d Lorentz rotations, e g
+½e-~p~vp~vyquu-D'JyJ~,u-7-trace,
(7)
+ 1~'u 7" ~'u ( 0 . q) - 21e - 1~pt7~p~/o"
The spin connection o)u, the curvature scalar R a n d
the S O ( 8 ) curvature tensor Vu'J are defined by
3 The n o n h n e a r theory and the g e o m e t r y The
action (1) is q u a d r a t i c a n d the t r a n s f o r m a t i o n rules
( 2 ) , ( 3 ) are linear in the fields o f the scalar multiplet Therefore we can generalize the results o f the
previous section to an arbitrary n u m b e r o f scalar
multlplets
w u = _ e -1 f Pa(eumOpeam + l~p?/~//a) ,
0z',~z
N u V - O u ~ - ¼V ~ V ? v ¢ - ½O)ugt
(4)
I=1,
,p,
I e the action (1) is generalized to
R=2e-~Eu~Ouo)~ ,
e - l.Lp= ½gm ~ u ~ t ' ~ v ( b f qlJ +
v~'J = O,~ v~'~ - O~ v J - v,,'~ v / ~ + V~'k v, d ~
(5)
The s p m o r p a r a m e t e r s e a n d q characterize a supers y m m e t r y a n d a super Weyl transformation, respectively They are spinors o f S O ( 8 ) a n d satisfy the
following chirahty conditions
)'se=+¢,
75r/=-q,
79E~-- "~-~,
y9~]=
+r/
(8)
(6)
Finally the derivatives ~u are c o v a r l a n t with respect
to Lorentz, S O ( 8 ) , s u p e r s y m m e t r y a n d super Weyl
transformations
,
(9)
where all terms are constructed in the same fashion,
using a constant real metric q u W i t h o u t loss o f generahty we can take r/u dmgonal, with elgenvalues + 1
The action ( 9 ) is linear in the field D U o f the
superconformal multlplet, and its equation o f m o t i o n
imposes restrictions on Or' The field equation o f D °
reads
qlSc,'(9 S = IJ'J ~Jg~tk~j k
(10)
We m u s t solve this equation for the fields 01' This is
possible only lfp~> 8 We solve (10) by the following
ansatz
(~/=½ v/rr L / ,
r=--½rl'1(~tkos k,
(11)
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Volume 186, number 2
PHYSICS LETTERS B
where
/.I ~ ~ ( L / , L # ) ,
/ , A = 1, , n + m + 8 ,
t=l,
,8,
a=l,
,n+m
(12)
is an ( n + 8 ) × ( n + 8 ) matrix, which is a representative
element
of
the
coset
manifold
S O ( 8 + n , m ) / S O ( 8 ) × S O ( n , m ) with n,m>~O The
inverse matrix L-~=r/LVr/ satisfies the following
relations
5 March 1987
cally solvable equations
Substituting the solutions (11 ), (15) o f ~ ' , {/,/l In
terms of (LI A, r) and (~a, Z) respectively, and the
expression (17) for VuU in terms o f (QuU+blhnear
fermlons), into the action (l) and the transformation rules (2), (3), we obtain the following result for
the coupling of 8 (n + m) scalar multlplets to (8,0)
conformal supergravxty
LI'LO=¢~ o, ZzaLh=O,
LlaLzb= ÷h ab,
(13)
where hab=diag (n t i m e s + , m t i m e s - ) is a constant
tensor which can be used to raise and lower or to contract the SO (n,m) indices a, b = 1, , n + m
The equatmn o f motion of the fields X' o f the
superconformal multlplet imposes further restnctlons on ~,~ The X' field equatmn is
L , ' ~ I = ~7'~JLIJ~ I,
- ~27uT~zO.r - ~rR + ½e-I].EP° ~p ~,~
- ~l,~'uy'~uaOu~ ~ V. a' + quartlc ferrnxons,
(19)
8eu m = - 21~-~m~Uu,
~uu= ~u e + 17u r/+ blllnear ferrmon terms,
(14)
where we have used (10) and (11 ) The solution o f
(14) is given by
4- blhnear fermlon terms,
~lli=N//~ Zla~ll a .4- ( l / I v / r ) 7'L,'z,
8Z = - l~,u Oure + blhnear fermlon terms,
X=½x/~rr'L,'~'
(15)
Finally, the equation o f motion o f Vuu is given by
~/[' ~# ~/J] =~l~TttTqql'÷lJgl(l~)k~)kT,uT",~.,
(16)
52 = - i7 ~ Ou9 e - 2r/+ bihnear fermlon terms,
8 r = e%, ~o = ~2
(20)
In (19) and (20) we have used the following definitions The field V,~'a(fb) denotes the 8 ( n + m ) - b e l n
o f the coset manifold It satisfies
and can be used to solve for VuU algebraically
v/~=O/
got# wotat v#b) =¢~u hab '
+ ½1(¢'a~',,~'u~'a+,,r-~e~,yX)
hab Votat vBbJ ÷ ol ~"+fl = l gaB(~tJ ,
-4-(l/2x/~) ,,~7'-q/u
' .4- ~lr-'~qTuTu,E
(17)
Here Q U is a composite S O ( 8 ) gauge field defined
by
Qu u= - L It' OuL/1
(18)
The equations of motion of the other fields o f the
superconformal multlplet do not lead to algebral170
OuV,J"vpbJ+ot,-+fl=[2/(n--m)]g~,ph ab,
(21)
where g,,a(O) is the metric o f the manifold We have
also used the relation
L IaOuLI ' =OuO a Vot at
(22)
The derivatives ~u are covarlant with respect to Lor-
Volume 186, number 2
PHYSICS LETTERS B
5 March 1987
entz, composite SO(8) and composite SO(n,m)
rotations, e g
Ra#ab = / ( 2 ( vaa, Vpb)--Ol ~ fl)t~tj ,
~e-aue+ ½¢oue- ~a~vyvE ,
Rap q ___/(2( Va at vpb) --Ol ~
~# ~/[a= O.~ua _ ½co.~ua
where we have introduced the gravitational coupling
constant x Consequently, in the global supersymmetry limit both curvatures vanish since x goes to
zero In that limit the fields (r,z) decouple from the
physical scalar multlplet fields (0%~u~), and can conSlstently be set equal to (1,0) The lagrangmn then
reduces to free kinetic terms for (0%~ ,a) This
lagranglan coincides with the one given in ref [20]
which describes the field theory of the AdS3 singleton multiplet
(b) We expect that the (8,0) conformal supergravity theory described in this paper corresponds to
the gauge theory of the conformal superalgebra
0 S p ( 2 , 8 ) ~ 0 S p ( 2 , 8 ) [21 ]
(c) In the conformal gauge ~ = 1 and 2 = 0, the
(8,0) model for n = 0 and without the heterotlc fertalons coincides with the model one would obtain by
dimensional reduction and subsequent chIral truncation of the N = 8, d = 3 Poincar6 supergravlty model
of Marcus and Schwarz [16] This is remarkable
because the compensating fields ~ and 2 belong to
the conformal supermultlplet itself, as opposed to
being independent matter fields as is usually the case
in a Brans-Dicke type theory The only other known
conformal supergravitles which contain their own
compensators are the conformal supergravlty in
d = 1 0 [15] and in d = 6 [22]
(d) In view of the previous remark, since an N = 16
Poincar6 supergravlty theory coupled to Es/SO (16)
scalar manifold exists in three dimensions, we expect
that an (16,0) model in two dimensions where the
matter scalars parametnze the coset Es/SO (l 6) and
action similar to that given in section 3 can be constructed However such a model probably only exists
on-shell, since the off-shell (16,0) superconformal
multlplet contains a large number of components
with undesirable canonical dimensions which makes
the construction of an action highly nontrivial
(e) The signature of the metric gap has 8m positive, and 8n negative signs which do not correspond
to a lorentzian signature An intriguing possibility
might be that, say, for m = 0 , the 8n scalars of the theory, together with the unphysical scalars r and ~oform
the coordinates of a special (8n + 2)-dimensional
- ~Qu'Jyu~a W Quab~,b
(23)
Here Q~ab IS the composite gauge field of SO(n,m)
rotations defined by
Qu ab = L zta OuLz b]
(24)
The 8 (n + m)-bein fields V~'4 satisfy the following
vielbein postulate
a a vB at -- { ~B } v~,al - Qa lJ VB aj
+Qaabv#b'=o,
(25)
with Q,~'J=Qu'J(OuO '~) and Q,~b=Quab(OuO°)In
obtaining (19) and (20) we have performed the following field redefinitions
~0 ~ ~0-1n r,
2 ~ 2 - (1/x/~)Z
(26)
Note that the fields (r,z) and (~0,2) do not describe
physical degrees of freedom The field equations of
eu m, ~u imply that r and Z are constant, while ~o and
2 are gauge degrees of freedom
The action (19) can easily be generalized to include
heterotic fermxons as well Up to quartic fermlons we
find
e - ~ Aa (heterotic)
=
½I(llry'u( OuJ rs + A,~r~OuO~')~ ~
+quartlcs,
(27)
+ blhnear fermion terms,
(28)
where A. "s is a composite gauge field for some
Yang-Mllls gauge group
4 Comments
(a) From [ ~,., ~p] V~,'a = 0 it follows that the curvature tensor of the scalar manifold decomposes as
follows
Ra# a''by =Raaab t~q + Rc~aUh ab
(29)
From the usual definition of the Rlemann tensor, and
from (22), one finds
fl)hab,
(30)
171
Volume 186, number 2
PHYSICS LETTERS B
m a m f o l d with lorentzmn signature
(f) We expect that a W e s s - Z u m m o term can be
added to the (8,0) model of this paper, in much the
same way it is added m the (4,0) model [ 10]
(g) It seems conceivable that one can gauge any
subgroup of the global SO (8 + n,m) group by introducing an (8,0) Y a n g - M d l s multlplet (Au, O , where
( Is an elght-spmor o f S O ( 8 ) This would work in a
similar way as m ref [4]
(h) F r o m the (8,0) model, by t r u n c a u o n , one can
obtain a (2,0) model with the field content (e~ m,
~u~',A~,, 2', ~) where t = 1, 2 is an S O ( 2 ) index Here
the vector field A~ has only one gauge transformation It is interesting to compare th~s theory wxth the
one based on the superconformal multlplet (e~ m,
~uu', B~,) where the vector field B~, has two independent gauge transformations [ 8 ]
(1) Concerning the anomahes, the perturbative
Lorentz a n o m a l y m two d i m e n s i o n s will be absent
provided that the n u m b e r ofheterotlc fermions ~ r is
fixed to be 8 ( m + n + 22 ) The sigma model a n o m a l y
[23] can be cancelled by the H u l l - W l t t e n mecham s m [ 11 ], provided that one adds a W e s s - Z u m i n o
term to the acUon Whether the trace anomaly, or the
global anomalies [24], or a n o m a h e s m any one of
the other local symmetries of the theory such as
supersymmetry are absent remmns to be investigated
One of the authors (E B ) would hke to t h a n k M
de Roo for useful discussions on the dimensional
reductxon of t e n - & m e n s l o n a l conformal supergravlty H N would hke to thank the International Atomic
Energy Agency and U N E S C O for the kind hospitallty at the International Centre for Theoretical Physics, Trieste
We also apprecmte stimulating
&scusslons w~th E Sokatchev
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