Volume 214, number 2 PHYSICS LETTERS B 17 November 1988

AUXILIARY SUPERGRAVITY FIELDS, C H E R N - S I M O N S T E R M S AND T H E H E T E R O T I C STRING

Mark EVANS and Burt A. OVRUT Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6396, USA

Received 29 June 1988

The generating functional of target space S-matrix amplitudes is generalized to include auxiliary field and fermionic vertex operators. Using this formalism, a new Green-Schwarz mechanism associated with the auxiliary vector field of the four-dimen- sional, N= 1 supergravity multiplet of the heterotie string is derived.

It has long been known that one needs to include both Yang-Mills [ 1 ] and Lorentz [2 ] Chern-Simons terms in the field strength o f the two-form gauge po- tential in order to obtain a consistent low energy field theory limit o f the superstring. In ref. [ 3 ], it was shown that the existence o f such Chern-Simons terms followed from the cancellation o f sigma model anomalies on the string worldsheet. Questions about whether or not the cancellation o f sigma model anomalies was consistent with (0, 1 ) worldsheet su- persymmetry were resolved in refs. [4,5 ].

Recently, using general results about heterotic string vacua which admit local N = 1 supersymmetry in four dimensions [6] , an irreducible four-dimensional superfield of supergravity vertex operators was con- structed [7]. Three o f these vertex operators are BRST invariant and generate physical gravitino, graviton, and antisymmetric tensor fields. The fourth, and final, vertex operator in this supermultiplet is not BRST invariant and corresponds to an auxiliary real vector field. (Analogously, it has been shown [ 8 ] that BRST violating vertex operators in chiral and vector four-dimensional supermultiplets correspond to F and D auxiliary fields respectively.)

The existence of an extra U ( 1 ) worldsheet current in these theories (which extends the (0, 1 ) world- sheet supersymmetry to (0, 2) ) leads one to conjec- ture that there is an associated abelian gauge

Work supported in part by the Department of Energy Contract Number DOE-AC02-76-ERO-3071.

symmetry in the target space with the auxiliary real vector field as its connection. In this paper we will show that such a symmetry exists as long as the ap- propriate abelian gravitational Chern-Simons term is added to the field strength of the physical antisym- metric tensor field. We will do this by constructing the generating functional for target space S-matrix amplitudes [ 9 ] and showing that it is invariant un- der the U(1 ) gauge and accompanying Chern- Simons transformations. This procedure is equiva- lent to cancelling sigma model anomalies on the string worldsheet.

The method that we describe in this paper is a quite general approach to the discovery of symmetry in string theory. Here we include auxiliary field and gravitino vertex operators in addition to massless physical bosons, but the method has many applica- tions beyond that used in this paper (see, for exam- ple, ref. [10] ) and will be discussed in detail elsewhere [ 11 ].

The four-dimensional spacetime supersymmetry currents in the - ½ picture are given by [ 6 ]

V_ wz,~(z) = e x p ( - ½(b)S,~27(z) ,

V_ ,/2a(z) = e x p ( - ½(P)S&27*(z) , (1)

where exp ( - ½ ¢) is a spin field for the (fl, y) super- conformal ghost system, S . and Sa are spin fields for the free worldsheet fermions ~u ~' which four-dimen- sional Minkowski indices, and 27 and 27* are primary

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fields of the internal superconformal field theory. The supersymmetry charges are

;dz Q~= ~ v_,/z,~(z) ,

r dz Qc~ =~ ~niz V_l/za(z) . (2)

Demanding that Q . , Q~ satisfy the N = 1 supersym- metry algebra

{Q~, Q~}=a~,~/~Pu, {Q,, Q#}=0, (3)

where Pu is the translation operator in the - 1 picture,

p _ ~ d z exp(-¢~) ~,u(z) (4) - J 2hi

implies that 27, 27* have the operator product expan- sions (OPEs)

~(Z)27* (W) = (Z-- W) - 3/4ZWI+ ... .

Z,(z)27(w) = ( z -w)3 /4zwO(w) +... ,

~ t ( Z ) ~ J ' ( W ) = ( Z - - W ) 3 / 4 z w O * ( w ) "~- .. . . ( 5 )

where O is some dimension -3 operator. It was shown in ref. [ 6 ] that N= 1 supersymmetry in four-dimen- sional spacetime implies the existence of a U ( 1 ) cur- rent J(z) on the string worldsheet. This current satisfies the OPEs

J(z)27(w) = ( z - w) - lz . ~£(w) + ...,

J(z)Z,* (w) = - ( z - w)- lz" 327, (w) +...,

Z(z)27* (w) = ( z - w ) - 3 / 4 z w I

+ ( z - w)'/4z. ½J(w) +... (6)

and

J( z ) J ( w ) = ( Z-- W) -2ZW" ½C'~ f i n i t e , (7)

where 6 is the central extension of the U ( 1 ) Kac- Moody algebra. For compactification of the heterotic string to four dimensions

~= 6 . (8)

Note that the single pole term in (7) vanishes. Using J(z) it is possible to construct a four-dimensional ir- reducible superfield of supergravity vertex operators. It was shown in ref. [ 7 ] that these vertex operators a r e

VUo = 2j(z) ( ZOzXU ) exp [ ik. X(z, 5) ] , ( 9 )

V~_,/2~=exp(-½¢)

xS~27(z) (5~zX") exp [ik.X(z, 5) l , (10)

V~"_I = z exp(-q~)

X ~t(~(z) (5~zX/') ) exp [ik.X(z, 5) ] , ( 1 1 )

V f~_, = z e x p ( - O )

× ~t"(z) (5~zX u] ) exp[ik.X(z, 2) ] , (12)

corresponding to an auxiliary real vector field A u, gravitino ~ , graviton Gu~, and physical antisymme- tric tensor field Bu, respectively. Note that (9) is in the 0 picture, ( 1 0) is in the - ½ picture and ( 1 1 ), ( 1 2) in the - 1 picture. It is convenient to convert ( 1 1 ) and ( 1 2) to the 0 picture. We find that

V~o =z[~zX("(z) +ik .~ qt(" (z) ] ( Z ~ X u) )

× exp[ik.X(z, z) ] ,

Vf~ =z[azXt"(z) +ik.~u ~t[~(z) ] (z~zX ~'] )

×exp[ik.X(z, 5) ] . (1 3)

Having exhibited the relevant vertex operators we now discuss the generating functional for ampli- tudes. It was pointed out in ref. [ 9 ] that the partition function for a non-linear sigma model was the gen- erating functional for amplitudes for certain massless string excitations. Differentiating the partition func- tion with respect to the couplings of the sigma model yields amplitudes as vacuum expectation values of vertex operators. This procedure can be used for any vertex operators. The generating functional for am- plitudes involving the operators of eqs. (9), ( 1 0) and (1 3) is the partition function for the generalized sigma model with action

2 z

s= f + ( 1/2z)G~(X)q?[Oz~+F~(OzX/ ')~" ]

+ [ ( 1/2z)z'~(X) exp ( - ½0) S~Z(Oz Xu) +b.c. ]

+ ( i /x /2 z)Au(X)J(OzX u) + OzHO~H}, (14)

where H is defined by

J=ixf3 z ~ H (15)

and F ~ is the usual connection constructed out of

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G.. and Bu.. All fields in (14) are functions o fz and g. Note that we have not included in S the kinetic energy terms for fields, such as q~, that do not corre- late with the current of interest in this paper. Con- verting (14) to worldsheet Minkowski variables x ° and x I gives

S=f d x e, (16)

where

5¢= (1/ 47r){ Gu~( J(~J(~- X' uX' ~) - 2Buj (~X ' ~

- i~ /2 G~y~u~[ tp~-~u'Y + F ~,( J(u- X' U)~ ~ ]

+ [iz~ exp( - ½O) S,~Z(j(u-X'u) +h.c. ]

-A , , J ( J ( " -X ' " ) +/Z/z- H'Z} (17)

and 0o, 0~ are denoted by a dot and prime respec- tively. The hamiltonian density can be found from (17) in the usual way. The relevant part of this den- sity is found to be

oeg = ( 1 /4n) [ Gu,J(~'J( ~ + ( Gu~ + ~ OA~,A,)X'~'X '~

_ ix ~ F,,a~/,tuaX,,~+ 2j2

+(ix '~uexp(-½0) S,~SX +h.c.)-A~,JX ], (18)

where

G,,j("= 2ztHx~, + (Bu~ + ~6A,,A~)X '~

• 4- ( i / N / ~ ) o ) . a b ~ a ~ b "t- I A u J " F ( - - ½ixu"

×exp( - 10) S~27+ h.c. ) . (19)

F~,,,p is the gravitational connection, and o)u ~b is the corresponding spin connection. The torsion part of these connections is

O u r 2 3 = _ ~ ( 0 t u B ~ l l - ~ c g 2 u . ~ ) , ( 2 0 )

g2 is the Chern-Simons three-form

g2u~ =AtuO~A~j • (21)

The terms proportional to 6 are required to restore superconformal invariance in the presence of the central extension of the U ( 1 ) current algebra, eq. (7). The associated generating functional for S-matrix amplitudes can be written as

Z[A;, Z~, Gu., Bu. ]

= l i m Trexp(-f lH[Au,z~J, Gu.,Bu.]) , (22) f l o ~

where

2n

/¢=f (23) 0

To exhibit a symmetry of the theory we must show that the generating functional is invariant under some transformation of fields. In hamiltonian form, this corresponds to an automorphism of the operator al- gebra. I f U is any invertible operator in the Hilbert space then

Z = lim Tr U exp ( - /3H) U- l

= lim Tr exp ( - f lUHU- ~ ) . ( 24 ) ,O~oo

For U infinitesimally close to the identity operator

U= 1 + i h , (25)

and (24) becomes

Z = lim Tr e x p { ' f l ( H + i [ h , H] )}. (26)

In general, for arbitrary choice of h, the commutator [ h, H] bears little relation to H and nothing is learned. However, for certain choices ofh we find that [h, H] retains the form of the hamiltonian, albeit with shifted sigma model couplings. That is

H + i [ h , H] =H[A~ + ~A~,z~ + SZ~, G.~ + SG~,

B~,+~Bu,] . (27)

In this case, it follows from (22) and (26) that

Z[A,,, Z~, Gu., Bu~] =Z[Au + 8Au, Zu + 8Z~,

G~. + fiGht, Bu~ + 8Bu~]. (28)

Therefore, associated with each such operator h is a target space symmetry. This formalism will be devel- oped in full generality in ref. [ 11 ]. In this paper we will explore the consequences of taking

2n

h=f dxl - ~ A(XIJ, (29) 0

where A is an infinitesimally small scalar function of

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worldsheet fields X ~. The canonical equal time com- mutat ion relations (ETCRs) for this theory are

[Hxu(X °, x I ), X ~ ( x °, x l' ) ] = - i2Ou v (~ (x l - -X 1' ) ,

[X~(x °, x ~ ), X~(x °, x " ) ] = 0 ,

[ Hx~,(x °, x I ), Hx~(X °, x " ) ] = 0 . (30)

Also, it follows from (6) and (7) that

[ J ( x 0, X 1 ) , £ ( X 0, X It ) ]

= 3rrfi(x I - - X It ) ~ ( X O, X 1' ) ,

[ J ( x ° , x ~ ) , J ( x ° , x l ' ) ] = i r t d O l ¢ 5 ( x ~ - x ~' ) . (31)

Note that

[ X / Z ( x 0, X 1 ), J ( x °, x 1' ) ] = 0 . (32)

Using ( 3 0 ) - (32) and the ETCRs deduced from them we can evaluate the commuta tor [h, H] with h given in (29) and ~ g i v e n in (18), (20). The result is

2~

fax, i[h, H] = ~ (2Gu~X~{- ldOxAA~,X ' ~ - 2 0 u A J 0

"t- 3 a [~AZu exp(-½q~) S,~X+h.c.]}

- ½~OuAA~X'~X '~ + [ - ~Azu3 ,~

Xexp(-½qb) S,~XY'U+h.c.l+4OuAJX'U) , (33)

where we have used eq. (8). Although it is not man- ifestly obvious, i[h, H] is o f the same form as the hamiltonian. To see this perform an arbitrary varia- tion of the background fields o f H using ( 18 ) - (21 ). We find that

27t

~H= f ~ ( SGUV ( J(uXv + X'uX'v ) + ~ A u A v X ' u X '~ 0

+ 2J(u{~Bu~X '~' + ~ A ( u A ~ , ) X '~'

+ (i/,,/~)acos%,a~u~+ ½aA#+ [ - ~xax~" "

× e x p ( - ½0) S,27+ h.c. ]}

- ix/-2 8Fu,~q/u~uaX '" + [iSz~

×exp( -½O)S ,~XX 'U+h .c . ] -SA~ ,JX ' ) . (34)

Comparing (33) and (34) we find that these expres- sions are identical if one takes

~B~ = 16 (0 tuA)A~ ~ (35)

and

= l g A x u , 6Gu, = 6 G ~ " = 0 , ~Z~ "3 tx

6A~ = - 40uA . (36)

Using the formalism discussed in this paper one can easily show [ 11 ] that the generating functional for S- matrix amplitudes is also invariant under the abelian transformation

~B,,,, = 0[uf~ 1 , ( 37 )

for an arbitrary real vector fieldfi. Choosing

f~= - ¼OAA~ (38)

and combining (35) with (38) yields the transformation

6Bu~ = - ¼~A0tuA~ 1 . (39)

It follows from (28), (36) and (39) that the gener- ating functional for S-matrix amplitudes and, hence, the four-dimensional effective lagrangian is invar- iant under the target space transformations

~)Zu =I~AZu , ~Au= - 4 0 u A , ~Gu~=0 , ~ '3

6Bu~,= - ¼~AOtuA~,l . (40)

The first three terms in (40) correspond to a local U( 1 ) transformation with auxiliary field A u as the gauge connection. However, the existence o f a non- vanishing coefficient 3 in the central extension of the K a c - M o o d y algebra o f J mandates that a compensat- ing Chern-Simons transformation be performed on the physical antisymmetric tensor. This transforma- tion law for Bu~, can be implemented in the effective lagrangian by adding the appropriate Chern-Simons term to the associated field strength n;~uv=OlxBuv 1. We emphasize that, since Au is an auxiliary field, the Chern-Simons terms derived in this paper is differ- ent f rom the Yang-Mills and gravitational Chern- Simons terms previously discussed in string theory.

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