N = 2 w∞ supergravity

Physics Letters B 278 (1992) 72-78 North-Holland

PHYSICS LETTERS B

N = 2 woo supergravity E. Bergshoeff ~ and M. de Roo 2 Institute for Theoretical Physics, P.O. Box 800, NL-9700 A V Groningen, The Netherlands

Received 23 September 1991; revised manuscript received 27 November 1991

We construct the gauge theory of N= 2 w~ supergravity. The formulation presented here is obtained starting from a realization of the N= 2 super-Woo().) algebra in terms of a supersymmetric B C system. We next apply a superbosonization of the BC superfields in terms of two real scalar superfields and take the classical limit h--.0. Different properties of the theory are discussed.

I. Introduction W~¢ algebras are extensions o f the Virasoro algebra which contain generators with all integer conformal spins 2 ~~ 1 and have been studied from a variety o f viewpoints [ 3,4 ]. A characteristic feature o f the WN algebras with N~> 3 is that they are nonlinear, i.e., the c o m m u t a t o r o f two generators leads to p o l y n o m i a l expressions in the generators. These nonlinearities make the construction o f a gauge theory o f WN gravity *~ a nontrivial task. Some t i m e ago the gauge theory o f chiral [ 5,6 ] a n d nonchiral [ 7 ] w3 gravity was constructed. M o r e recently, w3 gravity has been considered as a theory o f critical W3 strings

[8]. In a separate development, algebras containing an infinite n u m b e r o f higher-spin generators have been discussed in the literature. The simplest example is the ( l i n e a r ) wo~ algebra [ 9,10 ]. A gauge theory o f woo gravity was constructed in ref. [ 11 ]. It turns out that the woo gravity theory can be consistently truncated to a wN gravity theory for any N [ 11 ]. U p o n this t r u n c a t i o n the nonlinearities inherent to the WN algebras are automatically reproduced. The q u a n t u m version o f woo is the Woo algebra o f ref. [ 12]. Indeed, one can show that q u a n t i z a t i o n deforms woo gravity into Woo gravity [ 13 ]. S u p e r s y m m e t r y is clearly an element that could be a d d e d to the above picture. A supersymmetric version o f the woo algebra indeed exists [ 14,15 ]. However, this in itself does not imply that one can straightforwardly s u p e r s y m m e t r i z e the woo gravity theory [ 11 ]. A w3 supergravity theory has been constructed [ 6,16,17 ] but in contrast to the bosonic case the underlying algebra is nonuniversal, i.e., the structure constants d e p e n d on the explicit realization one is using for the currents generating the algebra. In this letter we will show that there exists an N = 2 s u p e r s y m m e t r i z a t i o n o f woo gravity. The underlying algebra is the ( l i n e a r ) N = 2 super-woo algebra [ 15 ] which is a contraction o f the N = 2 super-Woo algebra [ 18 ]. The theory is constructed by starting from a realization o f the N = 2 super-Woo algebra in terms o f a s u p e r s y m m e t r i c t Bitnet address:bergshoeff@hgrrug5. 2 Bitnet address: deroo@hgrrug5. *~ The algebra WNindicates the classical version of W~¢.An algebra is called classical with respect to a given field realization if the algebra can be realized as a Poisson bracket algebra between currents which depend on the fields. The algebra is called quantum if, in order to realise the algebra, one needs to make more than single contractions between the currents (the single contractions correspond to the Poisson brackets). 72

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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BC system, as given in ref. [ 19 ]. Next the BC system is superbosonized ~2 in terms o f two real scalar superfields. We will show that after this superbosonization one can d e f n e a classical limit of the field-theoretic representation, i.e., one can consistently throw away terms o f higher order in h. We thus obtain a field-theoretic realization of the N = 2 super-w~ algebra.

2. T h e N = 2 super-w~ algebra The N = 2 super-w~ algebra [ 15 ] is an extension o f the N = 2 super-Virasoro algebra which is defined by the following operator product ( O P E ) expansions:

w(')(1)w(')(2)~-2

w(3/2)(1)w(3/2)(2)~

0`2w'3/2)ZI2

w(3/2) ( 1 ) w ~'' (2) ~ (012 w(l, ~ '

(~ 012W(3/2) z~2

102w(l'-- 7!- 012 O2w(l)) -712 /I ' z12

2

Zf2

1 D2 - -w(3/2) + 012 02 W(3/2))

2

~,~

z12

-.

(1)

We use here a supeffield notation where every superfield contains a bosonic and fermionic component field. We note that w t~) is commuting and wt3/2) is anticommuting. The superspace coordinates are Z = (z, 0). The superspace differential operator D is defined by D = 0 0 - 0 0 with D 2 = - 0 . We have furthermore defined 012 = 01 --02 and z~2 = z ~ - z2 + 0~02. The notation w tl) (1) is a shorthand notation for w ~1) (Zl, 01 ). From eq. (1) one can recover the commutation relations of the generators of the algebra by multiplying the OPEs by the parameters of the corresponding transformations and integrating over the superspace coordinates. To obtain the N = 2 super-w~ algebra one extends the N = 2 super-Virasoro algebra by additional generators w (~) with s = 2 , ~, 5 3 ..... The generators with integer (half-integer) s are commuting (anticommuting). The N = 2 super-w~ algebra is then defined by the following OPE expansions. The OPE expansion o f two generators w ~s), w u) where both s and t are half-integer or one is integer and the other half-integer is given by

w(S)(l)w(t)(2)~(--)12s+ll2 ( s + t - 3 )

012W(s+t-3/2) z~2

1 D2 W(s+t-3/2) + ( S - 1 ) 012 02w(S--t--3/2)l,+\ 2 z12 zi2 /

(2)

where Isl 2 is equal to zero for s even and 1 for s odd. On the other hand, if both s and t are integer, the OPE expansion is given by w(~)(1)w(t)(2) ~ - 2

012w(S+t-l/2 ) (3)

Z12

The superfields {w ~s), w (s+ 1/2)} with s integer form N = 2 multiplets with respect to the osp (2, 2) subalgebra of the N = 2 super-Virasoro algebra. The osp (2, 2) subalgebra is defined by the s = 1, 3 transformations where the parameters k ~1), k ~3/2) which multiply the currents w (l), w (3/2) satisfy the conditions D3k~ 1) = DSk~3/2) = 0. It turns out that it is possible to extend further the N = 2 super-w~ algebra with an additional s = 1 generator w ¢1/2) with w ~1/2) ( 1 )w ~1/2) (2) ~0. The OPE expansion o f w ~w2) with w Cs) (s integer) is given by

w~l/2)(1)w(S)(2)

w~S-l/2) - - + z12

(

(s-l)

012w ~s-I) z~2

1D2w(s-l))

2

z12

.

(4)

For half-integer s, the OPE expansion is given by #2 The observation that bosonization might lead to a gauge theory ofw~ supergravity was independently made by Sezgin [20]. 73

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w
W(s-3/2) +

½( s - 3 )

PHYSICS LETTERS B

1)

O12w(s--l)z22

21

1 012D2W(s-3/2)

z21-----~ 4

z122

19 March 1992

D2w(s-l) )212 1 02w(S-3/2)~ 4

Zl~

-/"

(5)

3. Superbosonization Our task is to find a field-theoretic representation of the N = 2 super-woo algebra. Our starting point for finding such a realization will be the N = 2 super-W~ algebra [ 18 ]. We find it convenient to use here the formulation of ref. [ 19] where the algebra is called super-Wo~ (2). Here 2 refers to a one-parameter choice of bases of the algebra. The N = 2 super-Woo (2) algebra contains generators W~ s) with s = 1, l, 3 ..... The terms of highest spin occurring in the OPE expansion of W] s) ( 1 ) W] t) (2) exactly coincide with those of the N = 2 super-w~ algebra given above. In addition the OPE expansions contain additional generators of lower spin. The N = 2 superWoo(2) algebra can be contracted to the N = 2 super-wo~ algebra [ 18]. In this contraction only the highest spin generators in the OPE expansion survive ~3. One can view the contraction parameter as playing the role of h. In this sense one can consider the N = 2 super-woo algebra as the classical limit of the quantum N = 2 super-W~ (2) algebra. To summarize, schematically we have W) s) ( 1 ) W] t) (2) ~ as for the classical N = 2 super-w~ algebra + h (nonleading lower-spin generators).

(6)

Our strategy is now the following. A realization of the N = 2 super-Wo~ (2) algebra in terms of a supersymmetric BC system is known [ 19 ]. In order to be able to take the classical limit in this representation we first superbosonize the B, C superfields in terms of two real scalar superfields and then take the limit h - , 0 . Note that this procedure is the reverse of that of ref. [ 13]. We would like to stress that the N = 2 super-Woo algebra is classical with respect to the B, C superfield realization ,4. After the superbosonization the same algebra can be considered as a q u a n t u m algebra with respect to the two scalar superfields realization and only then it is possible to define a classical limit. To be more explicit, we consider the following action [ 22 ]:

S= 1_ f d Z Z B r ) C ,

(7)

7~ d

where B is a commuting superfield of weight 2 and C is an anticommuting superfield of weight ½- 2 . The operator product of B, C is equal to

B ( 1 ) C(2 ) ~ 012/z~2 + regular t e r m s .

(8)

A representation of the quantum N = 2 super-Woo (2) algebra is then given by the OPEs of the following set of conserved currents: 2s-- 1

W]S)=

~ A'(s, 2 ) ( D i B ) ( D 2 ~ - ' - 1 C ) ,

(9)

i=0

with the coefficients ~i(s, ,~) given by [ 19] a3 An exception is the OPE expansion W~~/2)( 1) W~s) (2) where also the first nonleading spin generator survives. ,4 the classical gauge theory corresponding to the N= 2 super-W~ algebra has been given in ref. [ 21 ]. It is interesting in its own right to compare the quantum theory of the N= 2 w~ supergravity theory constructed in this paper with the quantum theory of the N= 2 W~ supergravity theory of ref. [ 21 ]. It is not obvious to us what the exact relationship between the two quantum theories is. 74

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.~i(s,~.)= l + 1 2 s l E l i + l [ x ( - ) H + t i / 2 j + ' 2 s + l ' : l i + l l 2 ( [ s ] - l + [ 2 s l 2 1 i + l l 2 ) 1~i2~2 ( - [-s])[s]-1zsmij~ [i/2] × ( 2 2 - [s] )tilE] + 12s+~miu( - 2 2 - Is] + 1 )t~l- [i/2)-lil2,

(10)

where (a)~-= ( a + n - - 1 )!/(a-- 1 )! and [a] denotes the integer part o f a . In eq. (10) a normal ordering with respect to the modes of the currents is understood (see ref. [ 3 ] for more details on the normal ordering). The BC system can be superbosonized in terms of two real scalar superfields 0, ~as follows [23 ]: B=exp(O),

C = e x p ( - O ) D~.

(11)

The basic operator product expansion ofO, q~is given by 0( 1 )¢7(2) ~ - I n z,2.

(12)

From eqs. (11 ), (12) one can derive that the operator product expansion B( 1 )C(2), including the regular terms, is given by

012 + D 2 ~ ) • B( 1 ) C ( 2 ) ~ exp [0( 1 ) - 0 ( 2 ) ] (\Zl2

(13)

From this one may determine the superbosonized form of the currents W~ s) given in eq. (9). We next obtain a field-theoretic representation for the currents w (~) by taking the classical limit. As an example we consider the first few currents:

W~ 1/2) = B C ,

W~ l) = ( 1 - 2 2 ) ( D B ) C - 2 2 B ( D C ) ,

WJ 3/2) = ½( 1 - 2 2 ) ( O B ) C - ½(DB) (DC) - 2 B ( O C ) .

(14)

Their superbosonized form is given by #5

W~(1/2) = D ~ ,

W~ 1) = 0 0 O ~ ~ ~/h 00+ 22x/OhOqT,

W(~ 3/2) = ½00 Dq~+ ~D0 00+ ½x/~ 0DO-2x/~ 0D~.

(I 5)

The corresponding classical currents w (~) are given by wtS)= lira W~~) h~0

(16)

or

w(~/2)=Dq),

w(~)=DOD~,

W(3/z)=½OOD~+½DO0~.

(17)

Note that the 2 dependence disappears in the classical limit. Since we are interested in the classical limit of the WJ s) currents we only need to determine the terms of highest power in 0, ~in their expressions. Using this fact it is not too difficult to find a closed expression for the classical currents w t~). After some algebra we find

2S--I D2 ~] w(S)= ~ Ai(s, 2) D~D22~-i-l[exp(012D20+z12020) i=O

•

(18)

ZI2~012~0

Using the expression for the coefficients ~i(s, 2) given in eq. (10) we find for half-integer s that

w (s)= (00) s-l/2 Dq~+ ½D[D0 (00) s-3/2 Dq~] .

(19)

~5 In the next two equations we have indicated the explicit factors of h. They can be easily recoveredby dimension counting. The dimension of the ~, 4superfields is x/h. 75

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For integer s we find w(~)=D0 ( 0 ¢ ) ' - ' D~.

(20)

Here we have made use of the fact that for half-integers s >t -32the following identities hold: 2s ~ 1

2s -- 2

i even

i odd

~. ( - ) ' / 2 A ' ( s , 2 ) = + ½ ,

Z (-)'/2-'/2A'(s, 2)=-½ •

(21)

Similarly, for integer s we have 2s-- 1

(-)~+u/zL4~(s, 2) = - 1 .

(22)

i=0

One can verify that taking single contractions between the currents w (s) (or, equivalently, by taking Poisson brackets) leads to the OPEs corresponding to the N = 2 super-w~ algebra given in section 2. The currents w ~) form the basic ingredient in the construction of the gauge theory of N = 2 woo supergravity.

4. N = 2 woo supergravity

Our starting point is a free action for the scalar superfields 0, ~: So= f dEZDODO,

(23)

where I ) = 0 a - 0 0 . This action is invariant under global N = 2 super-w~ transformations with parameters k)0(2)= ~ i

dZi k ( , ) ( 1 ) w ( ' ) ( 1 ) 0 ( 2 ) ,

(24)

and similarly for ~. Note that k m is commuting (anticommuting) for half-integer (integer) s. Using the explicit form of the currents w
~(b=

~

[k(.)(OO)s-'/2-½Dk(s)D(~(OO)'-3/2]+

s=3/2,5/2,..,

6¢7=

~.

~. s=

{-(s-

k(s,D~(OO) s-I ,

1 , 2 ....

l ) D[k(s)(OO)s-3/2D~l+½Dkm (O0)s-3/2D~+½(s - 3 ) D[DkmDO(OO)~-5/2D~I}

s=3/2,5/2,...

+

~

[-k(,)(OO)'-'D~-(s-1)D(K(~)D~)(O~)'-2I)¢)].

(25)

s= 1,2,...

In this section we will not consider the s = ½transformation. To gauge a chiral N = 2 super-w~ symmetry, we allow the parameters k(~) to depend on Z as well as Z, i.e., f)k(s) ~ O. We must also introduce gauge fields A(s) and add gauge field × current terms to the action: Schiral~

f d 2 / ( D~I3~+ s=l ~ A(s)W(')) "

(26)

The Noether procedure now goes as follows. The variation of 0, ~in the kinetic term is cancelled by the leadingorder transformation of the gauge field which is of the form 6,4m = £)k(s)+ .... We next vary the currents using 76

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the OPE expansion of the N = 2 super-woo algebra. This variation is cancelled by adding terms to the transformation rule of the gauge field A t,) so that its total variation 5A (,)= Okts ) + ~A (,) for s half-integer and t integer is given by s--l~2

OA(~)=Ok(s)-2

~, A(~)k(,_t+l/2).

(27)

t= 1,2...

In all other cases we have

5A(~) =f)k(') +(,=~2.... + ~jil/~i)2,...)[ - ( t - ½)A(t) Ok(s-t+3~2) ..~ 1 ( - -

)1202 DA(t) Dk(s_t+ 3/z) + ( s - t + 1 ) c3A(t)k(s_t+ 3/z ) ] •

(28)

We next consider the nonchiral gauging. Following refs. [ 7,16] we introduce gauge fields A(~), A(s) and currents w (~), ~ (s) corresponding to the left- and right-handed symmetries (with parameters k(,), ~(~) ), and auxiliary fields F, F, G, G. We then find that the action for nonchiral N = 2 super-w~ is given by S .... hiral = ~ d 2 Z ( - D 0 f)O-FG+ FG+FO0+ DO ( 7 - F D ~ - O 0 G \

+

[A(s)w(~)(F, G) +A(~)W(s)(/e, G ) ] } .

(29)

/

S=I

The notation w (s) (F, G) (w (s) (F, G) ) indicates that in the expression for w (~) ( vp(s)) everywhere DO, DO (DO, DO) has been replaced by F, G (F, G). The action is invariant under the following nonchiral symmetries:

,~O=,~O(F) +5¢(~e),

a~=a0(V, G)+,~¢(F, G),

OF=D~O(F), ~ f f = I ) ~ 0 ( F ) , ~A(,) =I)k(,) +dA(s),

6G=D~O(F,G),

~-~(s)=DxC(,) +dd(~).

5(7=f)Sfb(f,(7), (30)

We have used here an obvious notation where, e.g., 60(F) indicates that in the expression for 40 everywhere DO has been replaced by F , etc. Furthermore 60(f) is obtained from 40(F) by the replacements k(,), F--,E(,), F, and similarly for 60 (F, G ). Finally, dA(s) is obtained from 8A (,) by the replacements k(,), A (s) --, E(s), A(,).

5. Discussion

In this letter we have constructed the gauge theory of N = 2 woo supergravity. Clearly, there are a number of directions in which our work can be further developed. First of all, it would be interesting to investigate whether the final result can be reformulated in a supergeometrical framework, along the lines discussed in ref. [23 ]. It would also be interesting to investigate whether the N = 2 woo supergravity theory allows a truncation with N = 1 supersymmetry. For the (quantum) supersymmetric BC system such a truncation can be achieved by imposing the condition C=DB (for 2 = 0 ) . It is not obvious whether a similar truncation is possible in the classical case. One way to investigate this would be to first perform the truncation C=DB in the quantum case and then try to superbosonize the B superfield. Since the B superfield contains only one real fermion, it seems that one would need a chiral bosonization. Finally, another application of our result might be the construction of gauge theories of WN supergravity theories with an underlying universal superalgebra. For this to work, it would be necessary to generalize the Stueckelberg symmetries of ref. [ 11 ]. At first sight it looks that the N = 2 woo supergravity theory has no 77

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Stueckelberg s y m m e t r i e s at all. T h e r e a s o n for this is that the c u r r e n t s are n o n l i n e a r i n the superfield 0 b u t l i n e a r in q~ Therefore, at the classical level, o n e c a n n e v e r write the higher-spin c u r r e n t s as p r o d u c t s o f lower-spin currents. H o w e v e r , we s h o u l d n o t e the following. A l t h o u g h the s u p e r s y m m e t r i c BC system has o n l y o n e s = ½ current, for the 0, 0 system, o n e can c o n s t r u c t ( i n the l e f t - m o v i n g sector) two such currents, n a m e l y w ~ / 2 ) = D ~ as well as DO. U s i n g this s e c o n d s = ½ c u r r e n t a Sugawara c o n s t r u c t i o n o f the s = 1, ~ s u p e r c u r r e n t s can be given [ 2 4 ] . I n this context, it is also worth m e n t i o n i n g that it seems possible to express the q u a n t u m c u r r e n t s o f the s u p e r s y m m e t r i c BC system in t e r m s o f ( p r o d u c t s o f ) lower-spin currents. It w o u l d be i n t e r e s t i n g to p u r s u e this d i r e c t i o n further, since it m i g h t lead to a p r o p e r d e f i n i t i o n o f "WN supergravity", and, possibly, "WN superstrings". After the c o m p l e t i o n o f this w o r k we received a p r e p r i n t [25 ] in which a s i m i l a r b o s o n i z a t i o n t e c h n i q u e is used to arrive at two-scalar realizations o f classical woo.

Acknowledgement We w o u l d like to t h a n k K. S c h o u t e n s for e x p l a i n i n g to us s o m e details o f the n o r m a l o r d e r i n g p r e s c r i p t i o n i n t r o d u c e d in ref. [ 3 ]. F o r o n e o f us ( E . B . ) T h i s work has b e e n m a d e possible by a fellowship o f the Royal N e t h e r l a n d s A c a d e m y o f Arts a n d Sciences ( K N A W ) .

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