- Email: [email protected]
Structure of new minimal supergravity
ANNALS
189, 318-351 (1989)
OF PHYSICS
Structure
of New Minimal
Supergravity*
S. FERRARA
University
CERN, Geneva, Switzerland; of California, Los Angeles, California
90024
AND S. SABHARWAL University
of California,
Los Angeles,
California
90024
Received March 16, 1988; revised July 20, 1988
We derive the gravitational multiplets of new minimal Poincart: supergravity and obtain a tensor calculus which allows us to compute general matter couplings in arbitrary R-symmetric supergravity theories. In particular, we derive Chern-Simons multiplets and super topological invariants. The relevance of these results for the description of the effective interactions of the massless excitations of four-dimensional heterotic superstrings with N = 1 space-time supersymmetry is stressed. 0 1989 Academic Press, Inc.
1. INTRODUCTION General matter couplings in N = 1 four-dimensional (Poincare) supergravity have been extensively investigated over the last decade. These couplings can be obtained either in superspace [ 1) or in component form using N= 1 tensor calculus [2]. However, since different N= 1 superspace geometry exists, there is actually more than one tensor calculus. The different formulations of Poincare supergravity get unified in the framework of conformal supergravity [2]. In fact in the latter formulation different super-Poincare geometries just reflect different choices of superconformal compensators. The simplest examples are the so-called old and new minimal formulations [3,4] of N= 1 Poincare supergravity each of which contains six bosonic auxiliary degrees of freedom. In the superconformal approach [2], they correspond to a chiral and a real linear multiplet compensator, respectively. In the Poincare framework, the old minimal supergravity has the basic gravitational multiplet given by [3]
* This work is supported in part by the United States Department of Energy under Contract DE AA03-76FOOO34.
318 OOO3-4916/89 $7.50 Copyright 0 1989 by Academx Press, Inc. All rights 01 reproduction in any form reserved.
NEW
while in the new minimal
MINIMAL
formulation eOP’
$ Ir’
319
SUPERGRAVITY
it is A,,
B,,.
Here e,, and 9, denote the physical vierbein and gravitino fields, u is a complex scalar, B,, is an antisymmetric tensor with gauge invariance
and A, is a vector auxiliary field which, in the new minimal associated R gauge invariance
formalism,
has an
In theories with no higher derivative interactions, it was shown that the two formulations become equivalent [S], at least at the classical level since Legendre transformations can be performed which change the linear compensator into a chiral compensator and vice versa. The new minimal theories turn out to be a subset of the old minimal theories, namely those which obey a restriction due to the R symmetry, gauge by the auxiliary A, connection. This implies a restriction on the Kahler potential and on the superpontial for these theories. The situation changes when higher derivative interactions are added to the lowest order supergravity Lagrangian because the dependence on the auxiliary fields is no longer algebraic. In particular it was shown that general theories with R* terms are physically inequivalent in the two-minimal formulations of N= 1 supergravity [6]. The problem of having the correct geometrical formulation of N= 1 supergravity geometry becomes, therefore, an obvious task in the framework of superstring theories [7] which give rise to effective Lagrangians for the massless states containing arbitrary high powers of the gravitational and Yang-Lills curvatures. A recent study [S, 91 of four-dimensional (4D) N= 1 heterotic superstrings [lo] has revealed that the natural superspace geometry for these theories corresponds to the new minimal formulation of N= 1 supergravity. This can be understood by explicit construction of the gravitational supervertices and by studying the superspace Bianchi identities for the vertex supermultiplets [S]. Moreover, there is a deep connection between the U( 1) gauge symmetry of the A, connection and the U( 1) Kac-Moody symmetry of the N = 2 superconformal algebra of the underlying two-dimensional superconformal theory [ll]. This recent investigation is the key motivation for a renewed interest in new minimal supergravity and the explanation of its consequences in the dynamics of the light degree of freedom of four-dimensional superstrings. By new minimal supergravity we mean a formalism in which the Poincare supergravity transformations use a linear rather than a chiral multiplet. Note that this is not in contrast with a recent analysis given by Siegal [9] in which he shows that using superstring field theory the superconformal compensator is chiral rather than linear. This fact would only affect the specific form of the Lagrangian but not
320
FERRARAANDSABHARWAL
the general geometrical framework we are discussing in this paper. Superconformal compensators can be generally duality transformed one into another. Indeed, recently Gates et al. [9] have shown that superstring theories can be formulated in a background using arbitrary compensatory multiplets. As a consequence, Lagrangians having an exact R symmetry are easily formulated in the new minimal formulation. The present paper is devoted to a derivation of the basic geometrical quantities of new minimal Poincart supergravity and general matter couplings. In particular, we derive Chern-Simons multiplets [ 123 which play an important role in the fourdimensional version of the Green-Schwarz mechanism [ 133 occurring in superstring theories. A short version of some of the results contained in this paper were reported in a previous note [14]. This paper is organized as follows: in Section 2, we begin by deriving new minimal supergravity at the linearized level, reviewing the results of Ref. [6]. We then generalize at the nonlinear level giving the algebra supercovariant derivatives and their commutation relations. Finally, we discuss the various supercovariant curvatures, and derive the transformation rules and Bianchi identities. In Section 3, we discuss the multiplets of new minimal supergravity [14]. We first consider general multiplets and then, by imposing suitable constraints, we get other multiplets (chiral, linear, etc....). Explicit expressions for spinor and vector derivatives are given. In Section 4, we derive the gravitational curvature multiplets and their related Bianchi identities [ 141. We then discuss super Yang-Mills theories and give the chiral and Lorentz connection gauge multiplets. The commutation relations of spinor and vector derivatives are obtained. In the last section we give the F- and D-type density formulae and then discuss general matter couplings. The equivalence with standard (old minimal) supergravity is explicitly shown. Finally, all topological invariants and Chern-Simons multiplets are derived. 2. NEW MINIMAL SUPERGRAVITY ALGEBRA
2.1. Linearized
New Minimal
Supergravity
In global supersymmetry, gauge theories are formulated by introducing a real superfield along with superspace gauge transformations. The extra fields are then eliminated by choosing the Wess-Zumino gauge, where we are left with the physical gauge multiplet (V,, ;1, D) and the usual space-time gauge transformations. We now generalize this procedure to describe gravity [6]. The obvious starting point is to introduct a real superfield with a vector index, 4p = C, + exe + 8;1, + O’F, + 0*& + Ba,8(h,,, + B,,) + fm*(A;
+ @*C,)
+ iQ2e($, + +aj,) - if12B($, - ;fix,),
where g,, = qrV +2/r,,
is the graviton and +,, the gravitino.
(1.1)
NEW
MINIMAL
321
SUPERGRAVITY
Then let us define Poincare supergravity
transformations
as
dq&=S,+S,+a,~
(1.2)
with 4 real and S, chiral (i.e., V, S, = 0). These follows from the superconformal transformations, 64, = Va,l - Va,X, by restricting 1 to satisfy V v2A + 0 V2X= 0 [6]. In any case, the above Poincare transformaton allows us to choose a gauge where C = x = F= 0. The remaining transformations are then 6h,, = apt, + a,t,
s*, = a,& 6A, = a,4 6~,, = apt: - a,(;.
(1.3)
The first two are the desired Poincare supergravity transformations and the remaining two are needed to give us a multiplet with equal (12 + 12) off-shell degrees of freedom. The multiplet (e;, 1+9,,,A,, B,,) is precisely the new minimal graviton multiplet with a gauge vector and antisymmetric tensor as auxiliary fields
CL 41. The supersymmetry transformations in the above gauge are derived as usual by adding compensating gauge transformations. They are i+,
= -D;
E
+B,,)=;~Y,~,
(1.4)
w+4” =w cl,tv+ Hp~v W$&”= -@A” - a”hlp) HP,, = a,& + aA,, + ?d,v.
(1.5)
W,,
where
We identify the vector in the new minimal formalism with A, = Al + H,, (H, = -(l/3!) EpvplH,,J instead of AZ. Also note that the vector A; = A: - 2H,, has a covariant (i.e., depends on #,,” = a,$, -a,+,) transformation law, 6Ap- = -6
Similarly,
Ey,y,a,,$,,.
the connection w~;~ = w,,~” -H+
transforms as
JW,,” = - &Y*lCI,“, 595/189/2-6
(1.6)
(1.7)
322
FERRARA
AND
SABHARWAL
where we have neglected a Lorentz transformation. The transformations (1.6) and (1.7) are similar to those for gauge vectors, i.e., SV, = -(i/2) Cy,J.. In Section 4.6 we will indeed find chiral and Lorentz connection multiplets with A; and o~;~ as their gauge vectors. The chiral field strength multiplet invariant under the Poincare supergravity transformation is given by
=~,.-i(fa,,R,:,.+F:,)B+ibJ,.B’+
We call this the Riemann Einstein multiplet,
multiplet.
. ...
Another invariant
E@ = - dEpvpi Oa, v
multiplet
is the real linear
ap4i
=H,-iey,r,-$By,y,e(E,:
-*F&)+
...,
where rp is the Rarita-Schwinger operator and Epy is the Einstein tensor. The relation between these multiplets and the other gravitational multiplets, Weyl and scalar curvature, will be given in detail in Sections 4.1 and 4.2. 2.2. Graviton Mu&let
(1.9)
i.e.,
and Local Algebra
We now consider new minimal supergravity at the nonlinear level [2,4]. It is very convenient to use the supercovariantized field strength of the antisymmetric tensor, H=dB+$$y+
(2.1)
to define three sets of chiral and Lorentz connections. Specifically, Oft = w abc
f
Habc
A; =A,-HP
(2.2)
A; = A, - 3H,,
where a,& = 0 &(e, $) is the usual supercovariantized torsion
Lorentz connection
T”= -f$ya$
with (2.3)
and
Or
H,ea = *H.
(2.4)
NEW
MINIMAL
323
SUPERGRAVITY
The torsion for the connections 02~ is given by T *’ = + H;,ebeC - (i/4) $y”$. The three covariant derivatives are then defined as (2Sa)
D = d+ dL(m,b) + 6,(A)
and (2.5b)
D* =d+6&,:,)+6,(A’),
where 6,((b)@ = in@J (2.6)
6,(A)@=~s,bA~b& For the gravitino Sab = a,,/2 and n = - y5/2. These derivatives are related by Da- = D, - i *SabHb - 3inH, (2.7)
0,’ = D, + i *Sob H, - inHa.
In particular,
for the gravitino Dt E = (D;
+ iy5$y,)c
= (D, - &ysyo4%
The nonlinear supergravity transformations (e;, lLP, A,, B,,) are taken to be [2,4]
(2.8)
of
the
graviton
multiplet
(2.9)
where
+,,” is the
gravitino
curvature
with
plus connections, i.e., tiPV = are simple covariantizations of those at the linear level ((2.1.4) and (2.1.6)). The transformation of the Lorentz connection can then be calculated in the standard manner. It is (2.1.7) D; rjy - 0,’ $P. Note that the above transformations
324
FERRARA
Also the variation
AND
SABHARWAL
of H, is given by 6H, = -;Ey5ra.
(2.11)
The transformations (2.9) form an algebra along with general coordinate, Lorentz, chiral, and B-gauge transformations. The variation of the gauge fields under these transformations is given by
~,U)O,, = -D&b 6‘4(d)A= -4 6,(t’)B=
(2.12)
-d<‘.
All fields except B,, are inert under the B-gauge transformation. The verification of the algebra is quite trivial using differential forms and identifying So,,(<) = YC = di, + i, d. It is also convenient to let E transform as a spinor under Lorentz, as BAc = - (iy,/2) 4.s under chiral (i.e., in two-component notation, E has n = 4 and B has n = - f), and all parameters are assumed to transform covariantly under general coordinate transformations. The nonvanishing commutators are then given by
Cs,(nL SAA’)1 = - SL(C4 A’l) ch3,,(~)~ &icr(~‘)l = &A5 .a<’ - 5’. w
(2.13)
with <” = (i/2) E1y?s, [2, 43. On covariant fields (i.e., those transforming without derivatives of any parameter in the above algebra) inert under the B-gauge transformations, the {Q, Q} anticommutator defines a supercovariant derivative [2],
where fi-
Similarly, (2.7).
B + = D + + 6,(e),
=D-
+8&b).
(2.16)
fi = D + S,(e), and these derivatives are related as in
NEW
MINIMAL
325
SUPERGRAVITY
Finally, we give for reference the supergravity connections
transformation
of the other
6.4+ =iEj4*-iEy5(r+~yi.r)
(2.17)
2.3. Gravitational
Curvatures
The chiral and Lorentz connections defined in the last subsection can be used to define three sets of supercovariant curvatures. Using (2.9), (2.10), and (2.17), these are given by f-
=F-
+itjyy5y.r
(3.1)
For the gravitino there is only one superconvariant curvature and hence we omit the plus sign and hat on cab. These curvatures can be used to explicitly give the commutators of supercovariant derivatives as [2]
326
FERRARA
AND
SABHARWAL
In particular, for a field of spin Sob, chiral weight n, and transforming supersymmetry as 6C = - 4.51,
Another very useful identity derivatives [ 23,
is the supersymmetry
+6, (iGa75y.r)
variation
@++Q(-iy5B7,~)@,
under
of covariant
(3.4)
where in the term 6; (6, @) the derivative does not act on E. Due to the presence of general coordinate transformations in the algebra, covariant fields can only have tangent space indices. When using differential forms, it is very convenient to formally define Be”=0 or explicitly,
(3.5)
Be“ = De” + $60($)ea. Then b *e” = f H&ebec.
Also we define a new supersymmetry SbV=lS
p!
variation
Q
(3.6)
for forms by
V,,...,e”‘...e”p.
(3.7)
The analog of (3.2) and (3.4) in differential forms is B~==s,(ff)+s,(P)+6Q(D+~) b*g*
=d,(R*)+d,(P*)+6p(D+$)
(3.8)
and
(3.9)
NEW
MINIMAL
327
SUPERGRAVITY
The different curvatures are then related as p+ =fi-lj*H
f-
=p-3fi*H
(3.10)
ri& = kab f L?H,,ed
+ HodcHcebedeP
or in components ‘F=,‘b=pab -(&H, P, R;,,
-&Ha)
=pab -3(&H, = R obcd
&if =&b
k
-bbHa)
@cHadb
-
+& abcd&Hd
fidkcb) +
+
%abH:-
(H,ceHedb
-
HadeHecb)
(3.11)
2ffaHb
l?* =&6H;.
Here Rob is the Ricci tensor and R the scalar curvature. The Bianchi identities for the curvatures are trivial to derive using the above formalism. The identity for the three-form H is or
6H=O
&Ha
(3.12)
= 0.
This follows as bH = dH + explicit $ terms = d(dB) + explicit # terms = 0. As H and fiH are supercovariant, the explicit 1(1terms are precisely those needed for supercovariantization and hence can be ignored. Also note that for Ha, fif H, = fiaHb (see (2.7)). Actually for any uncharged vector fi$ V, = 6, V,. Similarly for the gravitino curvature
or E abed%
tied
+
++abHb 6,
rn
=
o
=
0.
(3.13)
The Lorentz curvature satisfies I? abed
+
kcdb
+
pbcd
&$,&
The Lorentz identities,
-
kdbc
=
o
&-bcd
=
k&b
Rofbcd
=
0
&&,
=
f
(3.14) 2(&H,,,
-&&cd).
and chiral curvatures also satisfy the following
differential
Bianchi
328
F’ERRARA
B+E+
AND
SABHARWAL
= -iiEJy5(r+fyy.r)
BP=
-gDfIly,(r+tyy.r)
(3.15)
For the derivation of the Einstein multiplet identities, which are derivable from (3.15),
in Section 4.1, we need the following
(3.16)
The super-symmetry transformations to derive. For example,
of the curvatures are quite straightforward
or
&(D++)=- $-J,&,++
&. >
The explicit $ terms and D + E can be ignored in these derivations, the reason being that, as D + $ is supercovariant and the algebra closes on D +1(/, its variation has to be supercovariant quantity. As another example, consider S’p-
-dJA-
or
S’p-
=iE8+(yysy.r)
or
s’k
In the last equation we have used the supercovariantization namely, fi’y = 0 and b * y = T iy, yfly. Summarizing,
=iEy8-y5y.r.
of the identity Dy = J’,
NEWMINIMAL
329
SUPERGRAVITY
(3.17) sk+abed
=
-&zbr
Finally, let us give the transformation
&b&d.
of the Rarita-Schwinger
operator,
with (3.18)
3. NEW MINIMAL
SUPERGRAVITY
MULTIPLETS
supergravity
[2, 4, 151
3.1. General Multiplet
A general multiplet
of new minimal
I’= (C, x, H, K V,, 1, D),
(1.1)
is specified by the spin and chiral weight of its first component, i.e., 6,C = (i/2) A&s&C and 6,C= iqhC. We also define F= (H+ iK)/2 and F= (H- iK)/2. The supersymmetry transformation rules of this multiplet are given by
6(Hf
iK) = -E- ‘:”
{~s~+~-~-2iy5~~-i~C} (1.2)
330
FERRARA
AND
SABHARWAL
Here
(1.3)
and
&B-v+;F&x =D- (v+;$Y,x)-;dYn or
(1.4) The factor ( - )F accounts for the Fermi or Bose statistics of the first component. These factors disappear if E and II/ are placed to the left of the other fields. Note that [ and A only involve the spin and chiral generators of the first component (e.g., the spin matrix for x is Sab + a,/2). In two component notation JIP, E, 0 have chiral weight f and $,, E, B have - 4. The chiral weights of the other components are determined using this (e.g., x has chiral weight n - 3 and 2 has n + 4). Let us briefly explain the derivation of the above transformation rules. The first equation in (1.2) is the definition of 1. The second is the general form for 6% so that the algebra (2.15) closes on C. The third equation is a definition of I and the fourth is determined by closing the algebra on x. The fifth equation is the general form of 61 to close the algebra on H and K. A term like D’E in 62 is ruled out by requiring the algebra to close on V,. Finally the last equation is determined by closing the algebra on 1. The identities (2.3.3) and (2.3.4) are very useful in these derivations. As E is assigned to transform as a spinor under Lorentz transformations, the Lorentz algebra is manifest. Similarly, the chiral algebra is easily checked by comparing the chiral weights of the left- and right-hand sides of (1.2). The appearance of < and A in the above transformation rules is analogous to the contribution of A and D terms of gauge multiplets in the Wess-Zumino gauge. Indeed c (and A) will be identified with the Iz (and D) component of the chiral and Lorentz connection gauge multiplet in Section 4.5. Let us now give some other transformation rules, which are needed in deriving the constrained multiplets,
NEW
MINIMAL
331
SUPERGRAVITY
;S.“R, +nP- Eysjy > b(iB;
l+Y, A2 Da u-2
C& V,) = -f-
--c- i -lfys
2
2
i lfys -------yy,{y5~-2iy549x-i5C) 2
YAYJ + iK)
6{y,L+#-x-2iy,4/r-i
I
ri-
.V+2H.6-C+$,b0,by~X-i~ys~-AC
The multiplication rules of these multiplets, function F( V), are [ 151,
i.e., the components
I
.
(1.5)
of an arbitrary
CF = F(C) xF = x’F, H, = H’F, + f i’xkFkj K, = KiFj + $ jjiysXkFk,
VaF = ViFj + $ f’y5y,XkFkj (1.6) d,=i’F,+~cia-Cj+y,Ip/-Hj-iy,K’)x*F,, -$
{YsK’(xkK’)-X’(~kY5X’)}F,kj
~,=~‘F,~~(--Vj.Vk-~-Cj.~-Ck+H’Hk+K’K”~F,
--41X-j 21+ysXk-,l-Ys X-i ff;,ncj.
332
FJZRRARA
These multiplication
AND
SABHARWAL
rules are encoded in the superfield representation
v=c-BX-~B{H-iy,K+y,P}B +il300
i
v~n+fb-u-~Ylxi5C}+~(ss)‘(D+~~-C).
(1.7)
3.2. Complex Linear Multiplet We now impose constraints on the general multiplet of the last section to get the various constrained multiple& The complex linear multiple& 9, is defined by requiring F= 0 or V2P’ = 0. (The spinor derivatives will be defined in Section 3.6 [2, 151.) The independent components are C, 1, F, I’,, and 1,. The D and 1, components are given by l=&f+2ijQ-i5C (2.1)
D= -8-C+2H.V+ifi.(V+2CH)-iAC-f$,,d,f+2&. 3.3. Real Linear Multiplet The real linear multiplet, L, is defined by F= F= 0 or V2L =02L = 0 [2, 151. It must have zero chiral weight. The independent components are C, x, V,, with the constraint on the vector b-
.(V+zcx)+~P.:s,c+f$,y,(S.,+~)~=O.
The transformation
(3.1)
rules of the real linear multiplet
are
(3.2)
The I and D components
of this multiplet
are given by
(3.3) D= 4-C+2H.V+&,
(&+y)z
NEW
and the embedding
MINIMAL
333
SUPERGRAVITY
in the general multiplet
is
V(L) = (C, x, 0, 0, v,, 4 D). For S,, = 0 the constraint symmetric tensor, b,, [2, 43,
(3.1) can be explicitly
(3.4)
solved in terms of an anti-
*V=db-ZCdB+-$q and 66 =; Eyyx + f C.cyi+G.
(3.5)
3.4. Chiral Multiplets Chiral multiplets, The transformation
Z, are defined by the constraint xR = 0 or o,.Z=O rules of the chiral multiplet are
[2, 4, 15-J.
z = (z,x,h) sz= i&X
(4.1)
(-))‘SX=i&zE+hs
The embedding
in a general multiplet
is given by
V(Z) = (z, xL, h, -ih, The multiplication
-iB;
z, -irz,
-i AZ).
(4.2)
rule is [4, 151
f(z) = (f(Z), X’fi, h’f-J - f X’X”f kJ ‘1
(4.3)
which implies the chiral superfield representation C=z+8x+82h. From an arbitrary multiplet the chiral projection [4],
V we can form a chiral multiplet n(V)=
The components
(4.4) with weight n + 1 by
-p2v.
(4.5)
of Z7( V) are given by
z, =P Xn =i(&j+2iflj-A-i5C) h, =i
II--i(B-
(4.6) -2iH).(V+i6-C)+idC+f~~~d,X+2~~
>
.
334
Similarly,
FERRARA
the kinetic multiplet,
AND
SABHARWAL
T(Z) = - $V*Z, is [4]
(4.7)
The chiral weight of T(E) is 1 - n (C has weight n and E has -n). 3.5. Gauge Multiplet
From a real scalar multiplet where
we can form the gauge multiplet
(I’;,
1, D) [2,4]
These fields transform among themselves, up to a gauge transformation,
(5.2)
with G = dV’ - (i/Z) $yn. The complete analysis of Abelian and non-Abelian theories will be given in Sections 4.3, 4.4, and 4.5.
gauge
3.6. Spinor and Vector Derivatives
We now discuss the derivative derivative is defined by [ 151
operations C(V u = 2,
on these multiplets.
The spinor (6.1)
i.e., the first component of the multiplet VV is x (or V N - a/N + .I* ). This corresponds to the usual superspace derivative. We will give the explicit component expansion of V, V (two-component notation). The derivative V, V is then determined by complex conjugation, i.e., V, V= ( - )’ (vi V*)*. The multiplet V, V turns out to be complex linear, just like in global supersymmetry. Its components are given by
NEW
C(W)
335
SUPERGRAVITY
=2
@V)=i@-C+
Ip
j(W)
= 2F
F(W)
= i(X + i[C)
V,(OV)=
Here the spinor (i&C+ J%J. The 1, and D should remember From the third
MINIMAL
(6.2)
-a,,~,x+ia,(R-2i~~fir(C)
index of the derivative
is always on the left (e.g., ~~(9~ V) =
components of 0, V are given by 3.2.1. In using this formula, one that Vd V has chiral weight n + f and spin S,, + CJ2. equation in (6.2), we can see that {%~,}=O
(6.3)
exactly like in global supersymmetry. The full set of commutation given in Section 4.6. The vector derivative is defined by [15] C(V, V) = 6, The x component
of this multiplet
c.
relations will be
(6.4)
is
x(Vg V = 6; x - iy5y,Rx + ir,tC.
(6.5)
The second equation in (6.2) implies that this derivatives can also be defined as
v, =; (Vo,0 + 00, V) or (V,, Vi} = -2io$
Vi.
(6.61
Similarly, we can define the other vector derivatives V,, Vz by giving their first components. Due to 2.2.7, these derivatives are related as V, = V, - i *Sab Eb - 3inE, Vz =V, + i *S,,,Eb - inE,, where E, is the Einstein multiplet with first component V, corresponds to the usual superspace derivative.
(6.7) Ha. The vector derivative
336
F'ERRARAANDSABHARWAL
4. CURVATURE MULTIPLETS 4.1. Gravitational
Curvature Multiplets
In new minimal supergravity, there are two fundamental gravitational curvature multiplets, the Einstein multiplet, E,, and the Riemann multiplet, Tzb. The irreducible pieces of the Riemann multiplet are the scalar curvature multiplet, T, and the Weyl multiplet, Wzb. In Section 2.1, we derived the Einstein and Riemann multiplets at the linearized level. Here we take the first component of these expressions, i.e, H, and $,6 respectively, to derive the corresponding multiplets at the nonlinear level. The Einstein multiplet is a real linear multiplet (i.e., o*E, = 0, E, = E,*) with components (see (2.2.11), (2.3.18)) E, = (H,, iysro, ;(I?,‘, -*Pa:)).
(1.1)
The embedding in a general multiplet is given by (3.3.4). It can be shown to satisfy the linear multiplet constraint (3.3.1) by using the Bianchi identities (2.3.16). The Riemann multiplet is chiral (oti T;, =0) with components (see (1.3.17)
Cl, 61) Trrb = tiob -
(’
The scalar curvature multiplet T=;
8+ i &
Gobtl2.
(1.2)
is given by
cabTab
=a.F--f(k Similarly,
i ocdffr+dob+ ifi,+,
the Weyl multiplet
+a,bf=,)9-i&aert32.
is
where 4, = r, + fy,y . r and aabEd is the Weyl tensor.
(1.3)
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The matrix multiplying T,, in the above expression is the projection operator for the 6, traces, i.e., cruWab = 0. Equivalently, in two-component spinor notation w
di
_ =f-@w 4 up
is completely symmetric in CI, /?, and y. The Riemann multiplet can be reconstructed multiplets E,, Wub, and T as Tab = W,, - $0&T-
(1.5)
oby
in terms of the three irreducible
iEabrdgc t7E,.
We can also construct other gravitational multiplets derivatives to the Riemann multiplet. For example,
by applying
(1.6)
the spinorial
F,‘b = +y,T,, = (PA ~ ...) F,
= ; voab T = (P,)
...)
(1.7)
&dab = Rll+bcd = f Ouob Trd = (I?,,,,
...).
These multiplets are related as in Section 2.3, with the replacements 6, -+ V,, &b + Fab, &.cd + &cd, H, + E,, +ob + Tab, d,(@,,) + +Tab v. For example, Fa; = FL - 2(V,Eb
- V,E,).
(1.8)
4.2. Bianchi Identities
The Bianchi identities for the gravitational curvature multiplets are derived by considering the first components and using the Bianchi identities for the curvatures given in Section 2.3. For example, C(V,E,) = b,H, = 0 (see 2.3.12) implies that V,E, = 0. We list here some of the Bianchi identities [l, 61, 595/L89/2-7
338
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AND
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V,E, =0 T= -ic,
‘t7E,,
OE, =; cb *Tba
VT+VT=O
(2.1)
VT,, = -vTa,, VW,, 4.3. Supersymmetric
=2i(V,E,
= 2i(V,E,
-V,E,)
-V,E,)-2
*(V,E,
+ i E,~~~(VC~V -Va,
v)E,
-VbEa)+iVcabT.
Yang-Mills
Gauge theories are formulated exactly as in the global case. We introduce a real superfield V, which transforms under gauge transformations as [ 1,2,4, 161 e-2iV
--* eZe-2iVe-A
(3.1)
,
where A = iA, TA and ?I = i;i, TA are chiral and antichiral, field strength multiplet is defined by
respectively. The chiral
w = - $ v2(e2iV Ve - 2iv),
(3.2)
and it transforms as W + en We-“. In the Wess-Zumino remaining gauge transformations are, A A = ;i, = dA,
gauge, C = x = F = 0, the
6, VP = -DPq5 d&=(252
(3.3)
6,D=4D
and now the derivative includes the Yang-Mills connection, i.e., D, = 8, + V,. The supersymmetry transformations in the Wess-Zumino gauge are given by [16] a,W”(.)=a,(E)+a,(ie~~+e2E(~-~~,N.))
(3.4)
and
E
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SUPERGRAVITY
with (3.5) The component
expansion of the field strength multiplet W=R.fit?()~~~&~~
is
+D)+i@-@2
(3.6)
and its square is given by W2=L2+2i9 +tJ2
;cJ,,~,,
+D
1 >
21i~-L+s6,,6,,~~d,b*8.,-~D2
1
.
(3.7)
Under Yang-Mills transformations, chiral multiplets transform as C + e”C. The minimal coupling to Yang-Mills is then given by replacing E with Ce-“‘“, which gauge, the supertransforms as (Ee-2iv) -+ (Ee-2iv)eCn. In the Wess-Zumino symmetry transformations of the chiral multiplet are given by [16] 62 = t&X 6X=i&zE+hE
(3.8)
6h = +E(i fi ~ + 2fl) x + i$z + iEXz,
where the derivatives now include the Yang-Mills 4.4. Supergravity
and Super- Yang-Mills
connection.
Algebra
In the last subsection we described Yang-Mills theories in the standard manner. In this section we give an alternate method which includes the Yang-Mills algebra directly in the supergravity algebra [2]. We define a gauge multiplet (VP, 1, D) by the transformation rules (4.3.5) and (4.3.3). We now extend the supergravity algebra (2.2.9) by including these. The commutator of two supersymmetry transformations is then given by (2.2.14) with the addition of S,(t . V) on the right-hand side. On covariant fields, this again defines a supercovariant derivative 6-
=d+6&,)+6,(A-)+6,(V)+&&+).
(4.1)
The commutator of two supercovariant derivatives is now given by (2.3.3) with the addition of s,(G,,). Similarly (2.3.4) is modified by adding 6,( -(i/2) Ey,l). Actually all the results of Section 2 are valid if we make the replacements r-t~*.6Sab-iy~y.rn+i~.T (4.2) A-+ -f$$& 2
and BG now includes the Yang-Mills
--id-n+iD.T
connection.
340
FERRARA
AND
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The Yang-Mills field strength multiplet cannot be defined as (4.3.2) (as we only have covariant spinor derivatives). However, we can define it to be a chiral multiplet with first component i. The component expansion is, of course, the same as (4.3.6). These field strength multiplets satisfy the identity -Ow=vw+Olv=o. From the field strength multiplet ponents given by Gob and D,
(4.3)
we can form other multiplets
with first com-
= (Gab,...) (4.4) D=$,
W
= (D, ...). 4.5. Chiral and Lorentz Connection Multiplets There are two important gauge multiplets in the supergravity algebra itself. The Poincare or chiral gauge multiplet is given by (see (2.2.9), (2.3.17), (3.3.5)) [4] VP=&,
-y5y.r,
The field strength of this multiplet
-Q-).
(5.1)
is the scalar curvature multiplet
(3.1.3), i.e.,
T= W(V,). The Lorentz connection gauge multiplet
(5.2)
is given by ((2.2.10), (2.3.17), (3.3.5))
-Q,b = (q&5, $,b, -ES). The field strength of this multiplet
is the Riemann multiplet
(5.3) (3.1.2),
Tab = WQ,,).
(5.4)
4.6. Commutation Relations of Spinor and Vector Derivatives Let us now give the commutation relations of the various derivatives [l, 151, {V,O}=2iV-
(V, , Vi ) = f S,,R,,,
+ inF,
+ iT. Gab - 2E,,, VC- + i SaabV
with Eobc = - ~~~~~ E,. The first equation is exactly analogous to the global case and defines V, . The second can be viewed as a geometric definition of the field strength
NEW MINIMAL
SUPERGRAVITY
341
rnultiplets. The identity VW+ Vm= 0 follows as {V, (y,V, V; )} w (Vi , V; ) = 0. The third equation (and similar equations for Vz and the superspace derivative V,) are derived from (2.3.2) and the substitutions given at the end of Section 4.1. As this equation shows, the Riemann and Einstein multiplets can be interpreted as superspace torsion components. 5. LACRANGIANS
5.1. F-Type Density Formula Chiral multiplets with chiral weight n = 1 can be used to form invariant by the F density formula [2,4]
In superlield notation
actions
this can be written as [l] [L-l, = j d2B b‘z
with b=e
i
l-ief~.~J+~e*~J,ri,$~. I
(1.2)
The restriction n = 1 follows as d6’ has n = - 4 (0 has n = f). The Lagrangian for Yang-Mills theories is then given by (see (4.3.7))
x $coabyrA + i G,, as the field strength multiplet,
*Gab + i e-l ap(eXy5y”A)
(1.3)
W, has n = f.
5.2. D-Type Formula
From a multiplet c2941
with chiral weight n = 0 we can form an invariant
action as
[ VI0 = 2 j d48EV =e
D-i$.yg,i.+
+ surface terms,
(2.1)
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AND
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where
Note that E has no 0*8* component, formulae are related as [2]
i.e., [llD
=0
[l, 43. The P and D-type
[ V] D = 2[ Z7( V)] F + surface terms. To verify the above equation Laplacian [ 21
(2.3)
we need the expression for the supercovariant
(2.4) which is valid for n = S,, = 0. From (2.3), it is easy to see that all the spinor and vector derivatives can be integrated by parts in the D density formula. Also for a chiral multiplet Z and linear multiplet L we have [Z] D = [LID = [ZLID = 0, up to surface terms. The last equation implies that linear multiplets can be duality transformed to chiral multiplets. Finally, the action for Poincare supergravity is given by the Fayet-Illiopoulos term of the chiral gauge multiplet [4],
(2.5) or in differential forms -[~~]D(dx)=4Rnb*eoeb+4~~s~D+~+2A5.3. General Matter Couplings The general supergravity Lagrangian [5,161
containing
dB.
(2.6)
at most two derivatives
is
(3.1) where the superpotential
f has n = 1, tnZ>j
fj
=
f
(3.2)
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and fas, F have n = 0, (nZ)j
fABj
=
0
(3.3)
(nz)j Fj = (nz),. F,. The bosonic components
e-‘SOS
of this Lagrangian
are given by
Al- +;FjD;zj-;FjD,Zj
=;e2”(R+6H;)+2
H,
+ ; FLLs XL . XL’ - $ FLLe VL . VL’
D-.!ij
-F,-D-Zi.
--$Re f,,F;f,F$ +i
Re fABDADB+
+ F,-hihj + hjfj +hjJ; +ihnfirF$
*F$,
(TAz)~ FjDA
(3.4)
a, bil - 2CLHp
(3.5)
with v,”
=
-
; Epvpl
e2rr= 1 - (nz), Fj.
(3.6)
Let us restrict ourselves to the case with only chiral multiplets, as linear multiplets can always be duality transformed to chiral multiplets [2, 51. The field H, can be made unconstrained by adding the Lagrange multiplier, 2H, a,& Then 4 can be eliminated by performing a chiral gauge transformation (i.e., fixing the gauge for chiral transformations). As a result the vectors A,, and HP can be eliminated algebraically [4, 51. The scalar potential, after elimination of the auxiliary fields is V=ed4”fiF,71fi
+$eP4”Re
f;iFi(TAz)i
Fj(TBz)j.
(3.7)
However, F, is no longer the kinetic matrix for the scalars and hence need not be positive definite [S]. Actually the positivity of the scalar kinetic terms requires that F,? ’ i 3ed20(nz)i (G),.
(3.8)
be a positive definite matrix. These results will be derived in detail in Section 5.4. However, the validity of (3.8) can be seen directly by recalling that the negative
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AND
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contribution to the scalar potential in N= 1 supergravity the gravitino mass is given by m$, = e-‘O IfI 2.
[16] is 3m$,, and here
5.4. Equivalence to Standard Supergravity
Matter couplings in N= 1 supergravity theories are specified by two functions, the KHhler potential K and the superpotentialf [ 163. In the presence of Yang-Mills theories, we also have another function fAB. The bosonic Lagrangian in terms of these is
- $Re f >,F;f,Ffb and the scalar potential
+ dImfksF$,
*Ftb - V
(4.1)
is
++Re f>;,-,‘K:.(TAz’)i.
KJs(TBz’)jP,
(4.2)
where G’(z’, Z’) = K’(z’, 5’) + In 1f ‘(z’)12. The negative contribution to the scalar potential is proportional to the gravitino mass squared, rnin = eG’. We now explain how to cast the Lagrangian of Section 5.3 in standard supergravity form, i.e., the explicit relation between K and F. In the old minimal supergravity formalism, the relation between the D term (i.e., f [d],) and the Klhler potential is K= - 3 ln( - 44) [ 161. As 4 is an arbitrary function, so is K. However, in new minimal supergravity, F is subject to the constraint (5.3.3). The Kahler potential can be shown to satisfy a similar constraint. Hence, with new minimal supergravity, we can only write Lagrangians with the Kahler potential and superpotential subject to the constraint (5.3.2) and (5.3.3), or in other words we can only couple R-invariant Lagrangians [5, 161. To put the Einstein term in canonical form, we have to perform the following Weyl scaling on the Lagrangian (5.3.4), e’”P = e”e”P’
(4.3)
As is well known from superconformal tensor calculus [2], the Weyl and chiral weights of chiral multiplets are related as w = 3n. Therefore, we need to perform a Weyl resealing on z as well. However, as c itself depends on z, it is not clear what the transformation should be. We make the ansatz zfI =
e-3n,co+W6)WZi,
(4.4)
where K is for now an arbitrary function of z (it will turn out to be the Kiihler potential). The transformation (4.4) is nonholomorphic, nevertheless we can define a new supersymmetry, Sk = 6, + 6,) such that Sbz’ = 4 s’x’.
NEW
After eliminating choose
MINIMAL
the auxiliary
345
SUPERGRAVITY
fields we find that the 8~’ 8~’ terms cancel if we
K= F-k 3eZa - 60 - 3.
(4.5)
In fact the Lagrangian then reduces to the standard supergravity form (4.11, (4.2) with the function K in (4.5) as the Khhler potential. Note that (4.5) gives K as a function of z. We then have to solve the algebraic equation (4.4) to determine z as a function of z’ and substitute it in K to get the Kahler potential in terms of its natural variable z’ (e.g., the kinetic term for the scalar is KriZ 8~; .a~; with K:,/, = $
$
: ZI
K’(z’, Z’)
and
K’(z’, 2’) = K(z, 2)).
We given a few identities which are needed to explicitly
verify the above results,
1 1 + (,z’)~ K;s
e”’ = 1 - (n~)~ Fj = and ezO+ 3(~z)~ F,(nZ),
= e -20 F,
Fly1
+
3e-'"(nz),
(nF),-
I
1 - 3e20(nz’)i K;3-r(n3)jz
=e20e-3fl,(O+
=
(1/6)K)K,,
e~2ue3ni(a+(1'6'K){Bik
x Ki,.’
+
{ 6, + Kks(n5’)j}
(nz'),Kk.}
e3~(o+(‘/6)K)
(4.7)
f(z) = e 3k7+w6vyyZ~) fi(z)=e
3(~+(1/6)K)~--3n,(o+(1/6)Ktf:,(~1).
Let us explicitly state how to go from F to K (or vice versa). We first calculate 0 by (4.6). Then K (or F) is determined by (4.5). Finally, one solves for the relation between z and z’ by (4.4) and substitutes it in K (or F). Equation (4.4) can always be solved. To see this, let [S] z; = (&)3”’ Zj zi = (z0)3n’ zi.
(4.8)
Then Eq. (4.4) reduces to a single algebraic equation between (&zb) and (Z,z,) which can always be solved. We remark, in passing, that the Klhler transformations K’(z’, 2’) + K’(z’, Z’) + g’(z’) + g’(Z’) and f’(z’) + e-g’@‘)f’(z’) correspond to F(z, 2) -+ F(z, Z) + g(z) + g(Z) and j(z) + f(z). Here g(z) has chiral weight zero. The latter transformation is an invariance, as in the new minimal formalism [g(Z)] D = 0, and the corresponding field strength is W, = -(i/4) ft2 VF. 595118912-a
346
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AND
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There are two examples where the Weyl scaling on the scalars are not needed. The first is when all fields have zero chiral weight. Then the Kahler potential K = F. However, in this case we cannot have a superpotential. The other example is F = ,Z’,,C,F(Z) with C,, Zj having chiral weights n, = f, ni = 0 respectively. In this case, the Klhler potential is given by K = - 3 ln( 1 - fF). We mentioned before that new minimal supergravity can only be used to write R-invariant Lagrangians. However, one can explicitly break the chiral symmetry to write arbitrary Lagrangians. Let us consider the Lagrangian 9 = - f [C,Z;,],, where Co has n = f. Upon eliminating the auxiliary fields and assuming (zO) # 0 (as the Weyl resealing factors are proportional to ZOzo), we get the Poincare supergravity Lagrangian. Similarly, the Lagrangian
9 = $C&&W,
C)l, + 2 WZ$fP’)lF
-&WA,(~)
wAwBIF,
where C have n = 0; upon elimination of all auxiliary fields and Weyl scalings, this gives the general N= 1 supergravity Lagrangians. Finally, let us give the effective Lagrangian for the superstring [17] in the new minimal formalism. As the superpotential is cubic in C, we assign C to have chiral weight f and S, T are assigned chiral weight zero. The Kiihler potential is K’=
-31n(T’+T’-C’C’)(S’+S’)1/3.
We then use (4.6) to determine cr, and substitute between the primed and unprimed fields,
c=
(4.9) it in (4.4) to get the relation
C’ ,/( T’ + T’)(S’ + s)l”
T= T’,
(4.10)
s=s’.
The function F is given by (4.5), F= -3 ln(T+
(4.11)
T)(S+S)“3+3CC(S+S)1’3.
The effective superstring Lagrangian
[17] can then be written as
.9’~=~[F]D+2Re[&CiCjCk]F-[Vp]D -i
Re[SW’], L
+f Re[SiT,,
*TabIF.
(4.12)
L
The last two terms, needed for the anomaly cancellation in the next subsection.
[13], will be discussed
5.5. Topological Invariants and Chern-Simons Multiplets The space-time topological invariants (i.e., Chern, Pontrjagin, and Euler classes) can all be supersymmetrized using the chiral field strength multiplets. The
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imaginary part of the F density for the field strength multiplet derivative (see (5.1.3)), Im[ W’],
squared is a total
(dx) = -G* + @*(Xysv2)
(5.1)
with G'=d{V(G+'*)} = df-2.
(5.2)
The Yang-Mills field strength multiplet gives the Chern class. The Riemann multiplet (i.e., the field strength of the Lorentz connection multiplet) gives the Hirzebruch signature. Formula (5.1) can be generalized by replacing W* by fAB WA WB, with fAB a real, constant, gauge covariant matrix. Using this generalization for the Riemann multiplet, we get the Gauss-Bonnet combination as Im[T,, *Tab], [6, 181. Actually, both the real and imaginary parts of [Tab * TabIF are total derivatives. This can be seen using the Bianchi identities ((4.1.6) implies that [(Tab -i*Tab)*lF N [E;SID + surface terms (ST.)) or directly in components. Incidentally, this implies that the Weyl multiplet also satisfies Im[ Wi,] = ST. (see (4.1.4)). The components of the Gauss-Bonnet multiplet are
(5.3) It can be expressed in terms of the irreducible
multiplets
as
iTab *Tab = W,, W,, - $T* - 2v*(E,E,). The F density is given by
CT,b*TabIF(
-Ra+bRa+b -iRs
=d{w,+,(R+
*R,‘b +2F+*
-$0+*)~~+2,4+F+}
- id{ f.zabcdco,Cb (R + - fco +2)cd} -d{4D+$
(r+fyy.r-yYs@$))
- id{4 D’$
iy,(r + try .r - y54$)}.
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FERRARA
AND
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The space-time Chern-Simons forms, (5.2), can also be extended Chern-Simons multiplets by [ 121
to real
w2 = 217(Q).
(5.5)
If we let Wz = (z, x, h), then Q is ~2 = C”, f,
Re z, - Im z, Vfrl, (5.6)
with the constraint
on the vector fi. ( VR + 2C”H) + $jaby50,f
= -1m h.
As 52 is determined up to a linear multiplet, we can consider Q-L, gauge where CR = CLo, X* = x& [ 121. Then 52-L,=(O,O,Rez,
-Imz,u,,
and choose a
- b5x + $Y~Y~(Y~$ + Re z + iy5 Im z) ti,,
Reh+2H.u++$,(y5tl+Rez+iy51mz)+,) and the constraint is explicitly
(5.7)
(5.8)
solved as d*u = -Im[ u, = Vf-
W’],
(dx)
v,“.
(5.9)
Here we have used the relation d*u = aJet+‘) = (Daua - Tz,ub) e(dx). These Chern-Simons multiplets are needed for the consistent coupling of linear multiplets [12]. The Green-Schwarz mechanism to cancel anomalies requires that linear multiplets should appear in the combination L + s2 [ 12, 313. In terms of the duality transformed theories (L + S), this implies the coupling Re[SW’],. For example, consider the term Re[SiT,, *Tab ] in the superstring Lagrangian (5.4.12). this implies the coupling of a,, Im S with the Lorentz In components, Chern-Simons form (see (5.4)). If we now perform a duality transformation in components, this term modifies the field strength of the antisymmetric tensor by the addition of the Lorentz Chern-Simons form. Also, note that this term contains the coupling of Re S with the Gauss-Bonnet integrand (see (5.4)). The superstring Lagrangian in terms of a linear multiplet is obtained by replacing S + S by U and adding the Lagrange multiplier a[ UL],. Eliminating U we then get Ts=+[-3ln(T+T)-3lng(x)+3xg(x)+lne-V,], (5.10)
g(x) = (1 + (1 + x3)1’2)1’3 - ( - 1 + (1 + x)3)l’2)“3,
(5.11)
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and 2=L-Q+asz’
w2 = 217(Q)
(5.12)
iTab *Tab = 2z7(Q’). 6. CONCLUSIONS In the present paper, we have derived the tensor calculus of new minimal Poincare supergravity and constructed general matter couplings. This investigation was motivated by a renewed interest of new minimal supergravity as the natural framework to describe the dynamics of 4D heterotic superstrings. The results derived here provide to necessary tools to construct any higher order coupling in the gravitational and Yang-Mills curvatures.
APPENDIX:
CONVENTIONS
AND USEFUL FORMULAE
We use a Minkowski metric, g, = ( - + + + ), &0123= + 1. The Dirac matrix conventions are {y,, yb) = -2gab, y5 = iy”y1y2y3. (Tab = (i/2)(?a% Yb), *Oab = = iy5cab, $ = r+K. Left- and right-handed components are dellned by (l/2) Enbcdccd
In a Majorana representation C = y” and the Majorana condition is $ = II/*. The two-component spinor formalism is derived from the following chiral representation of the Dirac matrices,
Some useful Dirac algebra identities are YaYbYc
=
gacYb
ITab acd
=
gac
gbd
and the Fierz identity
&by, -
&dgbc
-
gcbYo -
-EabcdiYSYd
EabcdiY5
for anticommuting
+
ikac
Obd
spinors,
-
&dObc
+
gbdgac
-
gb$ad)
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FERRARA
AND
SABHARWAL
or in two components, II/. v = -f(W)
+ $(Jab(habtu
$ . $’ = - ja&a,lj’). The gravitational
curvature and torsion are given by R; = do; •k w:w;
= fR&,, dx” dx”,
T” = de” + o;eb = f T;, dx” dx’
with w,b = - w bo
=
Oa.?b
dx”.
The Ricci, scalar curvature, Einstein, and Weyl tensors are Rrrt, = Rzdb, W nbcd
=
R = R;,
i(Rabcd
-
Eab = &,
*R:d,b
) -
- &,R
hkc
gbd
-
= *R;;, g,dgbc)R.
The connection is explicitly solved as wabC= CO.bcte) + j(T,bc f Tbac + Tcrrb). Under a variation of the vierbein and torsion, the change in the connection is given by h+b
=
ttXapb
+
Xpnb
+
xb.p)
with x$ = -(D,
he: - D, se;) + 6T;,.
Note that D, only includes connections for the tangent space indices whereas V, includes connections for both the tangent and curved space indices. The gravitino curvature is given by tirv = Dl I), - 0: $,, and +,b = eze;ll/ay. The Rarita-Schwinger operator is ra = 4y5yb *ebo, Some useful identities are
Ydaabrd
=
tiab
+
h5
cab46
*tiaby
=
iray
y .r=ia.,$,
= -371.4.
Finally, to convert our formulae to the Euclidean conventions of Kugo and Uehara or van Nieuwenhuizen [2], the following replacements have to be made: E abed
+
%bcd,
Yu
+ -iy,,
y5 + -y5,
(Tab+ -2ifz&
C+
e~~e~,*,~-*~,EjE,W,pb~Opab,Srrb~-i~.b, A,
+
For a vector multiplet
-1A
2 ,,,n+in,B,,+-iaE,,H,+
-‘BO. 2
LJ
-C,
NEW
MINIMAL
SUPERGRAVITY
351
REFERENCES 1. S. J. GATES, JR., M. T. GRISARU, M. ROCEK, AND W. SIEGEL, “Superspace: One Thousand and One Lessons in Supersymmetry,” Frontiers of Physics, Vol. 58, Benjamin, New York, 1983; M. Mueller, Nucl. Phys. B 264 (1986), 292. 2. T. Kuco AND S. UEHARA, Nucl. Phys. B 226 (1983), 49; P. VAN NIEUWENHUIZEN, Phys. Rep. 68 (1968), 189; A. VAN PROEYEN, in “Supergravity ‘83” (Milewski, Ed.), World Scientific Singapore, 1984; B. DE WIT AND M. ROCEK, Phys. Lert. B 109 (1982), 439. 3. S. FERRARA AND P. VAN NIEUWENHUIZEN, Phys. Left. B 74 (1978), 333; K. S. STELLE AND P. C. WEST, Phys. Lerr. B 74 (1978), 330. 4. M. SOHNIUS AND P. WEST, Nucl. Phys. B 198 (1981), 493; V. P. AKUNLOV, D. V. VOLKOV, AND V. A. SOROKA, Theor. Math. Phys. 31 (1977), 285. 5. S. FERRARA, L. GIRALDELLO, T. Kuco, AND A. VAN PROEYEN, Nucl. Phys. B 222 (1983), 191. 6. S. CECOTTI, S. FERRARA, M. PORRATI, AND S. SABHARWAL, Nucl. Phys. B 306 (1988), 160. 7. M. GREEN, J. H. SCHWARZ, AND E. WITTEN, in “Superstring Theory,” Cambridge Univ. Press, 1986. 8. S. CECOTTI, S. FERRARA, AND L. GIRARDELLO, Phys. Lert. B 198 (1987), 336; Phys. Lerr. B 206, (1988), 451; J. LAUER, D. LUST, AND S. THEISEN, MPIPAEjPth 86/87; see also B. OVRUT, Phys. Lerr. B 205 (1988), 455. 9. W. SIEGAL, Univ. of Maryland preprint UMDEPP88-180; S. GATES, P. MAJUMDAR, R. OERTER, AND A. VAN DE VEN, Univ. of Maryland preprint UMDEPP88-209. 10. K. S. NARAIN, Phys. L&r. B 169 (1986), 71; K. S. NARAIN, M. H. SARMADI, AND E. WRITTEN, Nucl. Phys. B 279 (1987), 369; M. MUELLER AND E. WITTEN, Phys. Lerr. B 182 (1986), 28; D. GEPNER, Princeton Univ. Preprint PUPT-1056 (1987), PUPT-1066 (1987); H. KAWAI, D. LEWELLEN, AND S. TYE, Phys. Rev. Lerr. 57 (1986) 1832; Nucl. Phys. B 288 (1987), 1; I. ANTONIADIS, C. BACHAS, AND C. KOUNNAS, Nucl. Phys. B 289 (1987). 87; W. LERCHE, D. LOST, AND A. N. SCHELLEKENS, Nucl. Phys. B 287 (1987), 477. 11. W. BOUCHER, D. FRIEDAN, AND A. KENT, Phys. Lerr. B 172 (1986), 316; A. SEN, Nucl. Phys. B 278 (1986), 289; Nucl. Phys. B 284 (1987). 423; T. BANKS, L. J. DIXON, D. FRIEDAN, AND E. MARTINEC, preprint SLAC-PUB-4377, 1987.12. S. CECOTTI, S. FERRARA, AND M. VILLASANTE, Inr. J. Mod. Phys. A 2, No. 6 (1987), 1839; B. BAULIEU AND M. BELLON, Phys. Lerr. B 169 (1986). 59; G. GIRARDI AND R. GRIMM, Nucl. Phys. B 292 (1987) 181. 13. M. GREEN AND J. H. SCHWARZ, Phys. Lerr. B 149 (1984), 117. 14. S. FERRARA, S. SABHARWAL, AND M. VILLASANTE, Phys. Lerr. B 205 (1988), 302. 15. T. Kuoo AND S. UEHARA, Prog. Theor. Phys. 73, No. 1 (1985), 235. 16. E. CREMMER, S. FERRARA, L. GIRARDELLO, AND A. VAN PROEYEN, Nucl. Phys. B 212 (1983), 413. 17. E. WITTEN, Phys. Lerr. B 155 (1985), 151. 18. S. FERRARA AND B. ZUMINO, Nucl. Phys. B 134 (1978). 301; P. S. HOWE AND R. W. TUCKER, Phys. Lerr. B 80 (1978), 138; P. K. TOWNSEND AND P. VAN NIEUWENHUIZEN, Phys. Reo. D 19 (1979), 3592; S. CECOTTI, S. FERRARA, L. GIRARDELLO, A. PASQUINUCCI, AND A. PORRATI, Phys. Rev. D 33 (1986), 2504; Inr. J. Mod. Ph. A 3 (1988), 7.
189, 318-351 (1989)
OF PHYSICS
Structure
of New Minimal
Supergravity*
S. FERRARA
University
CERN, Geneva, Switzerland; of California, Los Angeles, California
90024
AND S. SABHARWAL University
of California,
Los Angeles,
California
90024
Received March 16, 1988; revised July 20, 1988
We derive the gravitational multiplets of new minimal Poincart: supergravity and obtain a tensor calculus which allows us to compute general matter couplings in arbitrary R-symmetric supergravity theories. In particular, we derive Chern-Simons multiplets and super topological invariants. The relevance of these results for the description of the effective interactions of the massless excitations of four-dimensional heterotic superstrings with N = 1 space-time supersymmetry is stressed. 0 1989 Academic Press, Inc.
1. INTRODUCTION General matter couplings in N = 1 four-dimensional (Poincare) supergravity have been extensively investigated over the last decade. These couplings can be obtained either in superspace [ 1) or in component form using N= 1 tensor calculus [2]. However, since different N= 1 superspace geometry exists, there is actually more than one tensor calculus. The different formulations of Poincare supergravity get unified in the framework of conformal supergravity [2]. In fact in the latter formulation different super-Poincare geometries just reflect different choices of superconformal compensators. The simplest examples are the so-called old and new minimal formulations [3,4] of N= 1 Poincare supergravity each of which contains six bosonic auxiliary degrees of freedom. In the superconformal approach [2], they correspond to a chiral and a real linear multiplet compensator, respectively. In the Poincare framework, the old minimal supergravity has the basic gravitational multiplet given by [3]
* This work is supported in part by the United States Department of Energy under Contract DE AA03-76FOOO34.
318 OOO3-4916/89 $7.50 Copyright 0 1989 by Academx Press, Inc. All rights 01 reproduction in any form reserved.
NEW
while in the new minimal
MINIMAL
formulation eOP’
$ Ir’
319
SUPERGRAVITY
it is A,,
B,,.
Here e,, and 9, denote the physical vierbein and gravitino fields, u is a complex scalar, B,, is an antisymmetric tensor with gauge invariance
and A, is a vector auxiliary field which, in the new minimal associated R gauge invariance
formalism,
has an
In theories with no higher derivative interactions, it was shown that the two formulations become equivalent [S], at least at the classical level since Legendre transformations can be performed which change the linear compensator into a chiral compensator and vice versa. The new minimal theories turn out to be a subset of the old minimal theories, namely those which obey a restriction due to the R symmetry, gauge by the auxiliary A, connection. This implies a restriction on the Kahler potential and on the superpontial for these theories. The situation changes when higher derivative interactions are added to the lowest order supergravity Lagrangian because the dependence on the auxiliary fields is no longer algebraic. In particular it was shown that general theories with R* terms are physically inequivalent in the two-minimal formulations of N= 1 supergravity [6]. The problem of having the correct geometrical formulation of N= 1 supergravity geometry becomes, therefore, an obvious task in the framework of superstring theories [7] which give rise to effective Lagrangians for the massless states containing arbitrary high powers of the gravitational and Yang-Lills curvatures. A recent study [S, 91 of four-dimensional (4D) N= 1 heterotic superstrings [lo] has revealed that the natural superspace geometry for these theories corresponds to the new minimal formulation of N= 1 supergravity. This can be understood by explicit construction of the gravitational supervertices and by studying the superspace Bianchi identities for the vertex supermultiplets [S]. Moreover, there is a deep connection between the U( 1) gauge symmetry of the A, connection and the U( 1) Kac-Moody symmetry of the N = 2 superconformal algebra of the underlying two-dimensional superconformal theory [ll]. This recent investigation is the key motivation for a renewed interest in new minimal supergravity and the explanation of its consequences in the dynamics of the light degree of freedom of four-dimensional superstrings. By new minimal supergravity we mean a formalism in which the Poincare supergravity transformations use a linear rather than a chiral multiplet. Note that this is not in contrast with a recent analysis given by Siegal [9] in which he shows that using superstring field theory the superconformal compensator is chiral rather than linear. This fact would only affect the specific form of the Lagrangian but not
320
FERRARAANDSABHARWAL
the general geometrical framework we are discussing in this paper. Superconformal compensators can be generally duality transformed one into another. Indeed, recently Gates et al. [9] have shown that superstring theories can be formulated in a background using arbitrary compensatory multiplets. As a consequence, Lagrangians having an exact R symmetry are easily formulated in the new minimal formulation. The present paper is devoted to a derivation of the basic geometrical quantities of new minimal Poincart supergravity and general matter couplings. In particular, we derive Chern-Simons multiplets [ 123 which play an important role in the fourdimensional version of the Green-Schwarz mechanism [ 133 occurring in superstring theories. A short version of some of the results contained in this paper were reported in a previous note [14]. This paper is organized as follows: in Section 2, we begin by deriving new minimal supergravity at the linearized level, reviewing the results of Ref. [6]. We then generalize at the nonlinear level giving the algebra supercovariant derivatives and their commutation relations. Finally, we discuss the various supercovariant curvatures, and derive the transformation rules and Bianchi identities. In Section 3, we discuss the multiplets of new minimal supergravity [14]. We first consider general multiplets and then, by imposing suitable constraints, we get other multiplets (chiral, linear, etc....). Explicit expressions for spinor and vector derivatives are given. In Section 4, we derive the gravitational curvature multiplets and their related Bianchi identities [ 141. We then discuss super Yang-Mills theories and give the chiral and Lorentz connection gauge multiplets. The commutation relations of spinor and vector derivatives are obtained. In the last section we give the F- and D-type density formulae and then discuss general matter couplings. The equivalence with standard (old minimal) supergravity is explicitly shown. Finally, all topological invariants and Chern-Simons multiplets are derived. 2. NEW MINIMAL SUPERGRAVITY ALGEBRA
2.1. Linearized
New Minimal
Supergravity
In global supersymmetry, gauge theories are formulated by introducing a real superfield along with superspace gauge transformations. The extra fields are then eliminated by choosing the Wess-Zumino gauge, where we are left with the physical gauge multiplet (V,, ;1, D) and the usual space-time gauge transformations. We now generalize this procedure to describe gravity [6]. The obvious starting point is to introduct a real superfield with a vector index, 4p = C, + exe + 8;1, + O’F, + 0*& + Ba,8(h,,, + B,,) + fm*(A;
+ @*C,)
+ iQ2e($, + +aj,) - if12B($, - ;fix,),
where g,, = qrV +2/r,,
is the graviton and +,, the gravitino.
(1.1)
NEW
MINIMAL
321
SUPERGRAVITY
Then let us define Poincare supergravity
transformations
as
dq&=S,+S,+a,~
(1.2)
with 4 real and S, chiral (i.e., V, S, = 0). These follows from the superconformal transformations, 64, = Va,l - Va,X, by restricting 1 to satisfy V v2A + 0 V2X= 0 [6]. In any case, the above Poincare transformaton allows us to choose a gauge where C = x = F= 0. The remaining transformations are then 6h,, = apt, + a,t,
s*, = a,& 6A, = a,4 6~,, = apt: - a,(;.
(1.3)
The first two are the desired Poincare supergravity transformations and the remaining two are needed to give us a multiplet with equal (12 + 12) off-shell degrees of freedom. The multiplet (e;, 1+9,,,A,, B,,) is precisely the new minimal graviton multiplet with a gauge vector and antisymmetric tensor as auxiliary fields
CL 41. The supersymmetry transformations in the above gauge are derived as usual by adding compensating gauge transformations. They are i+,
= -D;
E
+B,,)=;~Y,~,
(1.4)
w+4” =w cl,tv+ Hp~v W$&”= -@A” - a”hlp) HP,, = a,& + aA,, + ?d,v.
(1.5)
W,,
where
We identify the vector in the new minimal formalism with A, = Al + H,, (H, = -(l/3!) EpvplH,,J instead of AZ. Also note that the vector A; = A: - 2H,, has a covariant (i.e., depends on #,,” = a,$, -a,+,) transformation law, 6Ap- = -6
Similarly,
Ey,y,a,,$,,.
the connection w~;~ = w,,~” -H+
transforms as
JW,,” = - &Y*lCI,“, 595/189/2-6
(1.6)
(1.7)
322
FERRARA
AND
SABHARWAL
where we have neglected a Lorentz transformation. The transformations (1.6) and (1.7) are similar to those for gauge vectors, i.e., SV, = -(i/2) Cy,J.. In Section 4.6 we will indeed find chiral and Lorentz connection multiplets with A; and o~;~ as their gauge vectors. The chiral field strength multiplet invariant under the Poincare supergravity transformation is given by
=~,.-i(fa,,R,:,.+F:,)B+ibJ,.B’+
We call this the Riemann Einstein multiplet,
multiplet.
. ...
Another invariant
E@ = - dEpvpi Oa, v
multiplet
is the real linear
ap4i
=H,-iey,r,-$By,y,e(E,:
-*F&)+
...,
where rp is the Rarita-Schwinger operator and Epy is the Einstein tensor. The relation between these multiplets and the other gravitational multiplets, Weyl and scalar curvature, will be given in detail in Sections 4.1 and 4.2. 2.2. Graviton Mu&let
(1.9)
i.e.,
and Local Algebra
We now consider new minimal supergravity at the nonlinear level [2,4]. It is very convenient to use the supercovariantized field strength of the antisymmetric tensor, H=dB+$$y+
(2.1)
to define three sets of chiral and Lorentz connections. Specifically, Oft = w abc
f
Habc
A; =A,-HP
(2.2)
A; = A, - 3H,,
where a,& = 0 &(e, $) is the usual supercovariantized torsion
Lorentz connection
T”= -f$ya$
with (2.3)
and
Or
H,ea = *H.
(2.4)
NEW
MINIMAL
323
SUPERGRAVITY
The torsion for the connections 02~ is given by T *’ = + H;,ebeC - (i/4) $y”$. The three covariant derivatives are then defined as (2Sa)
D = d+ dL(m,b) + 6,(A)
and (2.5b)
D* =d+6&,:,)+6,(A’),
where 6,((b)@ = in@J (2.6)
6,(A)@=~s,bA~b& For the gravitino Sab = a,,/2 and n = - y5/2. These derivatives are related by Da- = D, - i *SabHb - 3inH, (2.7)
0,’ = D, + i *Sob H, - inHa.
In particular,
for the gravitino Dt E = (D;
+ iy5$y,)c
= (D, - &ysyo4%
The nonlinear supergravity transformations (e;, lLP, A,, B,,) are taken to be [2,4]
(2.8)
of
the
graviton
multiplet
(2.9)
where
+,,” is the
gravitino
curvature
with
plus connections, i.e., tiPV = are simple covariantizations of those at the linear level ((2.1.4) and (2.1.6)). The transformation of the Lorentz connection can then be calculated in the standard manner. It is (2.1.7) D; rjy - 0,’ $P. Note that the above transformations
324
FERRARA
Also the variation
AND
SABHARWAL
of H, is given by 6H, = -;Ey5ra.
(2.11)
The transformations (2.9) form an algebra along with general coordinate, Lorentz, chiral, and B-gauge transformations. The variation of the gauge fields under these transformations is given by
~,U)O,, = -D&b 6‘4(d)A= -4 6,(t’)B=
(2.12)
-d<‘.
All fields except B,, are inert under the B-gauge transformation. The verification of the algebra is quite trivial using differential forms and identifying So,,(<) = YC = di, + i, d. It is also convenient to let E transform as a spinor under Lorentz, as BAc = - (iy,/2) 4.s under chiral (i.e., in two-component notation, E has n = 4 and B has n = - f), and all parameters are assumed to transform covariantly under general coordinate transformations. The nonvanishing commutators are then given by
Cs,(nL SAA’)1 = - SL(C4 A’l) ch3,,(~)~ &icr(~‘)l = &A5 .a<’ - 5’. w
(2.13)
with <” = (i/2) E1y?s, [2, 43. On covariant fields (i.e., those transforming without derivatives of any parameter in the above algebra) inert under the B-gauge transformations, the {Q, Q} anticommutator defines a supercovariant derivative [2],
where fi-
Similarly, (2.7).
B + = D + + 6,(e),
=D-
+8&b).
(2.16)
fi = D + S,(e), and these derivatives are related as in
NEW
MINIMAL
325
SUPERGRAVITY
Finally, we give for reference the supergravity connections
transformation
of the other
6.4+ =iEj4*-iEy5(r+~yi.r)
(2.17)
2.3. Gravitational
Curvatures
The chiral and Lorentz connections defined in the last subsection can be used to define three sets of supercovariant curvatures. Using (2.9), (2.10), and (2.17), these are given by f-
=F-
+itjyy5y.r
(3.1)
For the gravitino there is only one superconvariant curvature and hence we omit the plus sign and hat on cab. These curvatures can be used to explicitly give the commutators of supercovariant derivatives as [2]
326
FERRARA
AND
SABHARWAL
In particular, for a field of spin Sob, chiral weight n, and transforming supersymmetry as 6C = - 4.51,
Another very useful identity derivatives [ 23,
is the supersymmetry
+6, (iGa75y.r)
variation
@++Q(-iy5B7,~)@,
under
of covariant
(3.4)
where in the term 6; (6, @) the derivative does not act on E. Due to the presence of general coordinate transformations in the algebra, covariant fields can only have tangent space indices. When using differential forms, it is very convenient to formally define Be”=0 or explicitly,
(3.5)
Be“ = De” + $60($)ea. Then b *e” = f H&ebec.
Also we define a new supersymmetry SbV=lS
p!
variation
Q
(3.6)
for forms by
V,,...,e”‘...e”p.
(3.7)
The analog of (3.2) and (3.4) in differential forms is B~==s,(ff)+s,(P)+6Q(D+~) b*g*
=d,(R*)+d,(P*)+6p(D+$)
(3.8)
and
(3.9)
NEW
MINIMAL
327
SUPERGRAVITY
The different curvatures are then related as p+ =fi-lj*H
f-
=p-3fi*H
(3.10)
ri& = kab f L?H,,ed
+ HodcHcebedeP
or in components ‘F=,‘b=pab -(&H, P, R;,,
-&Ha)
=pab -3(&H, = R obcd
&if =&b
k
-bbHa)
@cHadb
-
+& abcd&Hd
fidkcb) +
+
%abH:-
(H,ceHedb
-
HadeHecb)
(3.11)
2ffaHb
l?* =&6H;.
Here Rob is the Ricci tensor and R the scalar curvature. The Bianchi identities for the curvatures are trivial to derive using the above formalism. The identity for the three-form H is or
6H=O
&Ha
(3.12)
= 0.
This follows as bH = dH + explicit $ terms = d(dB) + explicit # terms = 0. As H and fiH are supercovariant, the explicit 1(1terms are precisely those needed for supercovariantization and hence can be ignored. Also note that for Ha, fif H, = fiaHb (see (2.7)). Actually for any uncharged vector fi$ V, = 6, V,. Similarly for the gravitino curvature
or E abed%
tied
+
++abHb 6,
rn
=
o
=
0.
(3.13)
The Lorentz curvature satisfies I? abed
+
kcdb
+
pbcd
&$,&
The Lorentz identities,
-
kdbc
=
o
&-bcd
=
k&b
Rofbcd
=
0
&&,
=
f
(3.14) 2(&H,,,
-&&cd).
and chiral curvatures also satisfy the following
differential
Bianchi
328
F’ERRARA
B+E+
AND
SABHARWAL
= -iiEJy5(r+fyy.r)
BP=
-gDfIly,(r+tyy.r)
(3.15)
For the derivation of the Einstein multiplet identities, which are derivable from (3.15),
in Section 4.1, we need the following
(3.16)
The super-symmetry transformations to derive. For example,
of the curvatures are quite straightforward
or
&(D++)=- $-J,&,++
&. >
The explicit $ terms and D + E can be ignored in these derivations, the reason being that, as D + $ is supercovariant and the algebra closes on D +1(/, its variation has to be supercovariant quantity. As another example, consider S’p-
-dJA-
or
S’p-
=iE8+(yysy.r)
or
s’k
In the last equation we have used the supercovariantization namely, fi’y = 0 and b * y = T iy, yfly. Summarizing,
=iEy8-y5y.r.
of the identity Dy = J’,
NEWMINIMAL
329
SUPERGRAVITY
(3.17) sk+abed
=
-&zbr
Finally, let us give the transformation
&b&d.
of the Rarita-Schwinger
operator,
with (3.18)
3. NEW MINIMAL
SUPERGRAVITY
MULTIPLETS
supergravity
[2, 4, 151
3.1. General Multiplet
A general multiplet
of new minimal
I’= (C, x, H, K V,, 1, D),
(1.1)
is specified by the spin and chiral weight of its first component, i.e., 6,C = (i/2) A&s&C and 6,C= iqhC. We also define F= (H+ iK)/2 and F= (H- iK)/2. The supersymmetry transformation rules of this multiplet are given by
6(Hf
iK) = -E- ‘:”
{~s~+~-~-2iy5~~-i~C} (1.2)
330
FERRARA
AND
SABHARWAL
Here
(1.3)
and
&B-v+;F&x =D- (v+;$Y,x)-;dYn or
(1.4) The factor ( - )F accounts for the Fermi or Bose statistics of the first component. These factors disappear if E and II/ are placed to the left of the other fields. Note that [ and A only involve the spin and chiral generators of the first component (e.g., the spin matrix for x is Sab + a,/2). In two component notation JIP, E, 0 have chiral weight f and $,, E, B have - 4. The chiral weights of the other components are determined using this (e.g., x has chiral weight n - 3 and 2 has n + 4). Let us briefly explain the derivation of the above transformation rules. The first equation in (1.2) is the definition of 1. The second is the general form for 6% so that the algebra (2.15) closes on C. The third equation is a definition of I and the fourth is determined by closing the algebra on x. The fifth equation is the general form of 61 to close the algebra on H and K. A term like D’E in 62 is ruled out by requiring the algebra to close on V,. Finally the last equation is determined by closing the algebra on 1. The identities (2.3.3) and (2.3.4) are very useful in these derivations. As E is assigned to transform as a spinor under Lorentz transformations, the Lorentz algebra is manifest. Similarly, the chiral algebra is easily checked by comparing the chiral weights of the left- and right-hand sides of (1.2). The appearance of < and A in the above transformation rules is analogous to the contribution of A and D terms of gauge multiplets in the Wess-Zumino gauge. Indeed c (and A) will be identified with the Iz (and D) component of the chiral and Lorentz connection gauge multiplet in Section 4.5. Let us now give some other transformation rules, which are needed in deriving the constrained multiplets,
NEW
MINIMAL
331
SUPERGRAVITY
;S.“R, +nP- Eysjy > b(iB;
l+Y, A2 Da u-2
C& V,) = -f-
--c- i -lfys
2
2
i lfys -------yy,{y5~-2iy549x-i5C) 2
YAYJ + iK)
6{y,L+#-x-2iy,4/r-i
I
ri-
.V+2H.6-C+$,b0,by~X-i~ys~-AC
The multiplication rules of these multiplets, function F( V), are [ 151,
i.e., the components
I
.
(1.5)
of an arbitrary
CF = F(C) xF = x’F, H, = H’F, + f i’xkFkj K, = KiFj + $ jjiysXkFk,
VaF = ViFj + $ f’y5y,XkFkj (1.6) d,=i’F,+~cia-Cj+y,Ip/-Hj-iy,K’)x*F,, -$
{YsK’(xkK’)-X’(~kY5X’)}F,kj
~,=~‘F,~~(--Vj.Vk-~-Cj.~-Ck+H’Hk+K’K”~F,
--41X-j 21+ysXk-,l-Ys X-i ff;,ncj.
332
FJZRRARA
These multiplication
AND
SABHARWAL
rules are encoded in the superfield representation
v=c-BX-~B{H-iy,K+y,P}B +il300
i
v~n+fb-u-~Ylxi5C}+~(ss)‘(D+~~-C).
(1.7)
3.2. Complex Linear Multiplet We now impose constraints on the general multiplet of the last section to get the various constrained multiple& The complex linear multiple& 9, is defined by requiring F= 0 or V2P’ = 0. (The spinor derivatives will be defined in Section 3.6 [2, 151.) The independent components are C, 1, F, I’,, and 1,. The D and 1, components are given by l=&f+2ijQ-i5C (2.1)
D= -8-C+2H.V+ifi.(V+2CH)-iAC-f$,,d,f+2&. 3.3. Real Linear Multiplet The real linear multiplet, L, is defined by F= F= 0 or V2L =02L = 0 [2, 151. It must have zero chiral weight. The independent components are C, x, V,, with the constraint on the vector b-
.(V+zcx)+~P.:s,c+f$,y,(S.,+~)~=O.
The transformation
(3.1)
rules of the real linear multiplet
are
(3.2)
The I and D components
of this multiplet
are given by
(3.3) D= 4-C+2H.V+&,
(&+y)z
NEW
and the embedding
MINIMAL
333
SUPERGRAVITY
in the general multiplet
is
V(L) = (C, x, 0, 0, v,, 4 D). For S,, = 0 the constraint symmetric tensor, b,, [2, 43,
(3.1) can be explicitly
(3.4)
solved in terms of an anti-
*V=db-ZCdB+-$q and 66 =; Eyyx + f C.cyi+G.
(3.5)
3.4. Chiral Multiplets Chiral multiplets, The transformation
Z, are defined by the constraint xR = 0 or o,.Z=O rules of the chiral multiplet are
[2, 4, 15-J.
z = (z,x,h) sz= i&X
(4.1)
(-))‘SX=i&zE+hs
The embedding
in a general multiplet
is given by
V(Z) = (z, xL, h, -ih, The multiplication
-iB;
z, -irz,
-i AZ).
(4.2)
rule is [4, 151
f(z) = (f(Z), X’fi, h’f-J - f X’X”f kJ ‘1
(4.3)
which implies the chiral superfield representation C=z+8x+82h. From an arbitrary multiplet the chiral projection [4],
V we can form a chiral multiplet n(V)=
The components
(4.4) with weight n + 1 by
-p2v.
(4.5)
of Z7( V) are given by
z, =P Xn =i(&j+2iflj-A-i5C) h, =i
II--i(B-
(4.6) -2iH).(V+i6-C)+idC+f~~~d,X+2~~
>
.
334
Similarly,
FERRARA
the kinetic multiplet,
AND
SABHARWAL
T(Z) = - $V*Z, is [4]
(4.7)
The chiral weight of T(E) is 1 - n (C has weight n and E has -n). 3.5. Gauge Multiplet
From a real scalar multiplet where
we can form the gauge multiplet
(I’;,
1, D) [2,4]
These fields transform among themselves, up to a gauge transformation,
(5.2)
with G = dV’ - (i/Z) $yn. The complete analysis of Abelian and non-Abelian theories will be given in Sections 4.3, 4.4, and 4.5.
gauge
3.6. Spinor and Vector Derivatives
We now discuss the derivative derivative is defined by [ 151
operations C(V u = 2,
on these multiplets.
The spinor (6.1)
i.e., the first component of the multiplet VV is x (or V N - a/N + .I* ). This corresponds to the usual superspace derivative. We will give the explicit component expansion of V, V (two-component notation). The derivative V, V is then determined by complex conjugation, i.e., V, V= ( - )’ (vi V*)*. The multiplet V, V turns out to be complex linear, just like in global supersymmetry. Its components are given by
NEW
C(W)
335
SUPERGRAVITY
=2
@V)=i@-C+
Ip
j(W)
= 2F
F(W)
= i(X + i[C)
V,(OV)=
Here the spinor (i&C+ J%J. The 1, and D should remember From the third
MINIMAL
(6.2)
-a,,~,x+ia,(R-2i~~fir(C)
index of the derivative
is always on the left (e.g., ~~(9~ V) =
components of 0, V are given by 3.2.1. In using this formula, one that Vd V has chiral weight n + f and spin S,, + CJ2. equation in (6.2), we can see that {%~,}=O
(6.3)
exactly like in global supersymmetry. The full set of commutation given in Section 4.6. The vector derivative is defined by [15] C(V, V) = 6, The x component
of this multiplet
c.
relations will be
(6.4)
is
x(Vg V = 6; x - iy5y,Rx + ir,tC.
(6.5)
The second equation in (6.2) implies that this derivatives can also be defined as
v, =; (Vo,0 + 00, V) or (V,, Vi} = -2io$
Vi.
(6.61
Similarly, we can define the other vector derivatives V,, Vz by giving their first components. Due to 2.2.7, these derivatives are related as V, = V, - i *Sab Eb - 3inE, Vz =V, + i *S,,,Eb - inE,, where E, is the Einstein multiplet with first component V, corresponds to the usual superspace derivative.
(6.7) Ha. The vector derivative
336
F'ERRARAANDSABHARWAL
4. CURVATURE MULTIPLETS 4.1. Gravitational
Curvature Multiplets
In new minimal supergravity, there are two fundamental gravitational curvature multiplets, the Einstein multiplet, E,, and the Riemann multiplet, Tzb. The irreducible pieces of the Riemann multiplet are the scalar curvature multiplet, T, and the Weyl multiplet, Wzb. In Section 2.1, we derived the Einstein and Riemann multiplets at the linearized level. Here we take the first component of these expressions, i.e, H, and $,6 respectively, to derive the corresponding multiplets at the nonlinear level. The Einstein multiplet is a real linear multiplet (i.e., o*E, = 0, E, = E,*) with components (see (2.2.11), (2.3.18)) E, = (H,, iysro, ;(I?,‘, -*Pa:)).
(1.1)
The embedding in a general multiplet is given by (3.3.4). It can be shown to satisfy the linear multiplet constraint (3.3.1) by using the Bianchi identities (2.3.16). The Riemann multiplet is chiral (oti T;, =0) with components (see (1.3.17)
Cl, 61) Trrb = tiob -
(’
The scalar curvature multiplet T=;
8+ i &
Gobtl2.
(1.2)
is given by
cabTab
=a.F--f(k Similarly,
i ocdffr+dob+ ifi,+,
the Weyl multiplet
+a,bf=,)9-i&aert32.
is
where 4, = r, + fy,y . r and aabEd is the Weyl tensor.
(1.3)
NEW
MINIMAL
337
SUPERGRAVITY
The matrix multiplying T,, in the above expression is the projection operator for the 6, traces, i.e., cruWab = 0. Equivalently, in two-component spinor notation w
di
_ =f-@w 4 up
is completely symmetric in CI, /?, and y. The Riemann multiplet can be reconstructed multiplets E,, Wub, and T as Tab = W,, - $0&T-
(1.5)
oby
in terms of the three irreducible
iEabrdgc t7E,.
We can also construct other gravitational multiplets derivatives to the Riemann multiplet. For example,
by applying
(1.6)
the spinorial
F,‘b = +y,T,, = (PA ~ ...) F,
= ; voab T = (P,)
...)
(1.7)
&dab = Rll+bcd = f Ouob Trd = (I?,,,,
...).
These multiplets are related as in Section 2.3, with the replacements 6, -+ V,, &b + Fab, &.cd + &cd, H, + E,, +ob + Tab, d,(@,,) + +Tab v. For example, Fa; = FL - 2(V,Eb
- V,E,).
(1.8)
4.2. Bianchi Identities
The Bianchi identities for the gravitational curvature multiplets are derived by considering the first components and using the Bianchi identities for the curvatures given in Section 2.3. For example, C(V,E,) = b,H, = 0 (see 2.3.12) implies that V,E, = 0. We list here some of the Bianchi identities [l, 61, 595/L89/2-7
338
FERRARA
AND
SABHARWAL
V,E, =0 T= -ic,
‘t7E,,
OE, =; cb *Tba
VT+VT=O
(2.1)
VT,, = -vTa,, VW,, 4.3. Supersymmetric
=2i(V,E,
= 2i(V,E,
-V,E,)
-V,E,)-2
*(V,E,
+ i E,~~~(VC~V -Va,
v)E,
-VbEa)+iVcabT.
Yang-Mills
Gauge theories are formulated exactly as in the global case. We introduce a real superfield V, which transforms under gauge transformations as [ 1,2,4, 161 e-2iV
--* eZe-2iVe-A
(3.1)
,
where A = iA, TA and ?I = i;i, TA are chiral and antichiral, field strength multiplet is defined by
respectively. The chiral
w = - $ v2(e2iV Ve - 2iv),
(3.2)
and it transforms as W + en We-“. In the Wess-Zumino remaining gauge transformations are, A A = ;i, = dA,
gauge, C = x = F = 0, the
6, VP = -DPq5 d&=(252
(3.3)
6,D=4D
and now the derivative includes the Yang-Mills connection, i.e., D, = 8, + V,. The supersymmetry transformations in the Wess-Zumino gauge are given by [16] a,W”(.)=a,(E)+a,(ie~~+e2E(~-~~,N.))
(3.4)
and
E
NEW
MINIMAL
339
SUPERGRAVITY
with (3.5) The component
expansion of the field strength multiplet W=R.fit?()~~~&~~
is
+D)+i@-@2
(3.6)
and its square is given by W2=L2+2i9 +tJ2
;cJ,,~,,
+D
1 >
21i~-L+s6,,6,,~~d,b*8.,-~D2
1
.
(3.7)
Under Yang-Mills transformations, chiral multiplets transform as C + e”C. The minimal coupling to Yang-Mills is then given by replacing E with Ce-“‘“, which gauge, the supertransforms as (Ee-2iv) -+ (Ee-2iv)eCn. In the Wess-Zumino symmetry transformations of the chiral multiplet are given by [16] 62 = t&X 6X=i&zE+hE
(3.8)
6h = +E(i fi ~ + 2fl) x + i$z + iEXz,
where the derivatives now include the Yang-Mills 4.4. Supergravity
and Super- Yang-Mills
connection.
Algebra
In the last subsection we described Yang-Mills theories in the standard manner. In this section we give an alternate method which includes the Yang-Mills algebra directly in the supergravity algebra [2]. We define a gauge multiplet (VP, 1, D) by the transformation rules (4.3.5) and (4.3.3). We now extend the supergravity algebra (2.2.9) by including these. The commutator of two supersymmetry transformations is then given by (2.2.14) with the addition of S,(t . V) on the right-hand side. On covariant fields, this again defines a supercovariant derivative 6-
=d+6&,)+6,(A-)+6,(V)+&&+).
(4.1)
The commutator of two supercovariant derivatives is now given by (2.3.3) with the addition of s,(G,,). Similarly (2.3.4) is modified by adding 6,( -(i/2) Ey,l). Actually all the results of Section 2 are valid if we make the replacements r-t~*.6Sab-iy~y.rn+i~.T (4.2) A-+ -f$$& 2
and BG now includes the Yang-Mills
--id-n+iD.T
connection.
340
FERRARA
AND
SABHARWAL
The Yang-Mills field strength multiplet cannot be defined as (4.3.2) (as we only have covariant spinor derivatives). However, we can define it to be a chiral multiplet with first component i. The component expansion is, of course, the same as (4.3.6). These field strength multiplets satisfy the identity -Ow=vw+Olv=o. From the field strength multiplet ponents given by Gob and D,
(4.3)
we can form other multiplets
with first com-
= (Gab,...) (4.4) D=$,
W
= (D, ...). 4.5. Chiral and Lorentz Connection Multiplets There are two important gauge multiplets in the supergravity algebra itself. The Poincare or chiral gauge multiplet is given by (see (2.2.9), (2.3.17), (3.3.5)) [4] VP=&,
-y5y.r,
The field strength of this multiplet
-Q-).
(5.1)
is the scalar curvature multiplet
(3.1.3), i.e.,
T= W(V,). The Lorentz connection gauge multiplet
(5.2)
is given by ((2.2.10), (2.3.17), (3.3.5))
-Q,b = (q&5, $,b, -ES). The field strength of this multiplet
is the Riemann multiplet
(5.3) (3.1.2),
Tab = WQ,,).
(5.4)
4.6. Commutation Relations of Spinor and Vector Derivatives Let us now give the commutation relations of the various derivatives [l, 151, {V,O}=2iV-
(V, , Vi ) = f S,,R,,,
+ inF,
+ iT. Gab - 2E,,, VC- + i SaabV
with Eobc = - ~~~~~ E,. The first equation is exactly analogous to the global case and defines V, . The second can be viewed as a geometric definition of the field strength
NEW MINIMAL
SUPERGRAVITY
341
rnultiplets. The identity VW+ Vm= 0 follows as {V, (y,V, V; )} w (Vi , V; ) = 0. The third equation (and similar equations for Vz and the superspace derivative V,) are derived from (2.3.2) and the substitutions given at the end of Section 4.1. As this equation shows, the Riemann and Einstein multiplets can be interpreted as superspace torsion components. 5. LACRANGIANS
5.1. F-Type Density Formula Chiral multiplets with chiral weight n = 1 can be used to form invariant by the F density formula [2,4]
In superlield notation
actions
this can be written as [l] [L-l, = j d2B b‘z
with b=e
i
l-ief~.~J+~e*~J,ri,$~. I
(1.2)
The restriction n = 1 follows as d6’ has n = - 4 (0 has n = f). The Lagrangian for Yang-Mills theories is then given by (see (4.3.7))
x $coabyrA + i G,, as the field strength multiplet,
*Gab + i e-l ap(eXy5y”A)
(1.3)
W, has n = f.
5.2. D-Type Formula
From a multiplet c2941
with chiral weight n = 0 we can form an invariant
action as
[ VI0 = 2 j d48EV =e
D-i$.yg,i.+
+ surface terms,
(2.1)
342
FERRARA
AND
SABHARWAL
where
Note that E has no 0*8* component, formulae are related as [2]
i.e., [llD
=0
[l, 43. The P and D-type
[ V] D = 2[ Z7( V)] F + surface terms. To verify the above equation Laplacian [ 21
(2.3)
we need the expression for the supercovariant
(2.4) which is valid for n = S,, = 0. From (2.3), it is easy to see that all the spinor and vector derivatives can be integrated by parts in the D density formula. Also for a chiral multiplet Z and linear multiplet L we have [Z] D = [LID = [ZLID = 0, up to surface terms. The last equation implies that linear multiplets can be duality transformed to chiral multiplets. Finally, the action for Poincare supergravity is given by the Fayet-Illiopoulos term of the chiral gauge multiplet [4],
(2.5) or in differential forms -[~~]D(dx)=4Rnb*eoeb+4~~s~D+~+2A5.3. General Matter Couplings The general supergravity Lagrangian [5,161
containing
dB.
(2.6)
at most two derivatives
is
(3.1) where the superpotential
f has n = 1, tnZ>j
fj
=
f
(3.2)
NEW
MINIMAL
343
SUPERGRAVITY
and fas, F have n = 0, (nZ)j
fABj
=
0
(3.3)
(nz)j Fj = (nz),. F,. The bosonic components
e-‘SOS
of this Lagrangian
are given by
Al- +;FjD;zj-;FjD,Zj
=;e2”(R+6H;)+2
H,
+ ; FLLs XL . XL’ - $ FLLe VL . VL’
D-.!ij
-F,-D-Zi.
--$Re f,,F;f,F$ +i
Re fABDADB+
+ F,-hihj + hjfj +hjJ; +ihnfirF$
*F$,
(TAz)~ FjDA
(3.4)
a, bil - 2CLHp
(3.5)
with v,”
=
-
; Epvpl
e2rr= 1 - (nz), Fj.
(3.6)
Let us restrict ourselves to the case with only chiral multiplets, as linear multiplets can always be duality transformed to chiral multiplets [2, 51. The field H, can be made unconstrained by adding the Lagrange multiplier, 2H, a,& Then 4 can be eliminated by performing a chiral gauge transformation (i.e., fixing the gauge for chiral transformations). As a result the vectors A,, and HP can be eliminated algebraically [4, 51. The scalar potential, after elimination of the auxiliary fields is V=ed4”fiF,71fi
+$eP4”Re
f;iFi(TAz)i
Fj(TBz)j.
(3.7)
However, F, is no longer the kinetic matrix for the scalars and hence need not be positive definite [S]. Actually the positivity of the scalar kinetic terms requires that F,? ’ i 3ed20(nz)i (G),.
(3.8)
be a positive definite matrix. These results will be derived in detail in Section 5.4. However, the validity of (3.8) can be seen directly by recalling that the negative
344
FERRARA
AND
SABHARWAL
contribution to the scalar potential in N= 1 supergravity the gravitino mass is given by m$, = e-‘O IfI 2.
[16] is 3m$,, and here
5.4. Equivalence to Standard Supergravity
Matter couplings in N= 1 supergravity theories are specified by two functions, the KHhler potential K and the superpotentialf [ 163. In the presence of Yang-Mills theories, we also have another function fAB. The bosonic Lagrangian in terms of these is
- $Re f >,F;f,Ffb and the scalar potential
+ dImfksF$,
*Ftb - V
(4.1)
is
++Re f>;,-,‘K:.(TAz’)i.
KJs(TBz’)jP,
(4.2)
where G’(z’, Z’) = K’(z’, 5’) + In 1f ‘(z’)12. The negative contribution to the scalar potential is proportional to the gravitino mass squared, rnin = eG’. We now explain how to cast the Lagrangian of Section 5.3 in standard supergravity form, i.e., the explicit relation between K and F. In the old minimal supergravity formalism, the relation between the D term (i.e., f [d],) and the Klhler potential is K= - 3 ln( - 44) [ 161. As 4 is an arbitrary function, so is K. However, in new minimal supergravity, F is subject to the constraint (5.3.3). The Kahler potential can be shown to satisfy a similar constraint. Hence, with new minimal supergravity, we can only write Lagrangians with the Kahler potential and superpotential subject to the constraint (5.3.2) and (5.3.3), or in other words we can only couple R-invariant Lagrangians [5, 161. To put the Einstein term in canonical form, we have to perform the following Weyl scaling on the Lagrangian (5.3.4), e’”P = e”e”P’
(4.3)
As is well known from superconformal tensor calculus [2], the Weyl and chiral weights of chiral multiplets are related as w = 3n. Therefore, we need to perform a Weyl resealing on z as well. However, as c itself depends on z, it is not clear what the transformation should be. We make the ansatz zfI =
e-3n,co+W6)WZi,
(4.4)
where K is for now an arbitrary function of z (it will turn out to be the Kiihler potential). The transformation (4.4) is nonholomorphic, nevertheless we can define a new supersymmetry, Sk = 6, + 6,) such that Sbz’ = 4 s’x’.
NEW
After eliminating choose
MINIMAL
the auxiliary
345
SUPERGRAVITY
fields we find that the 8~’ 8~’ terms cancel if we
K= F-k 3eZa - 60 - 3.
(4.5)
In fact the Lagrangian then reduces to the standard supergravity form (4.11, (4.2) with the function K in (4.5) as the Khhler potential. Note that (4.5) gives K as a function of z. We then have to solve the algebraic equation (4.4) to determine z as a function of z’ and substitute it in K to get the Kahler potential in terms of its natural variable z’ (e.g., the kinetic term for the scalar is KriZ 8~; .a~; with K:,/, = $
$
: ZI
K’(z’, Z’)
and
K’(z’, 2’) = K(z, 2)).
We given a few identities which are needed to explicitly
verify the above results,
1 1 + (,z’)~ K;s
e”’ = 1 - (n~)~ Fj = and ezO+ 3(~z)~ F,(nZ),
= e -20 F,
Fly1
+
3e-'"(nz),
(nF),-
I
1 - 3e20(nz’)i K;3-r(n3)jz
=e20e-3fl,(O+
=
(1/6)K)K,,
e~2ue3ni(a+(1'6'K){Bik
x Ki,.’
+
{ 6, + Kks(n5’)j}
(nz'),Kk.}
e3~(o+(‘/6)K)
(4.7)
f(z) = e 3k7+w6vyyZ~) fi(z)=e
3(~+(1/6)K)~--3n,(o+(1/6)Ktf:,(~1).
Let us explicitly state how to go from F to K (or vice versa). We first calculate 0 by (4.6). Then K (or F) is determined by (4.5). Finally, one solves for the relation between z and z’ by (4.4) and substitutes it in K (or F). Equation (4.4) can always be solved. To see this, let [S] z; = (&)3”’ Zj zi = (z0)3n’ zi.
(4.8)
Then Eq. (4.4) reduces to a single algebraic equation between (&zb) and (Z,z,) which can always be solved. We remark, in passing, that the Klhler transformations K’(z’, 2’) + K’(z’, Z’) + g’(z’) + g’(Z’) and f’(z’) + e-g’@‘)f’(z’) correspond to F(z, 2) -+ F(z, Z) + g(z) + g(Z) and j(z) + f(z). Here g(z) has chiral weight zero. The latter transformation is an invariance, as in the new minimal formalism [g(Z)] D = 0, and the corresponding field strength is W, = -(i/4) ft2 VF. 595118912-a
346
FERRARA
AND
SABHARWAL
There are two examples where the Weyl scaling on the scalars are not needed. The first is when all fields have zero chiral weight. Then the Kahler potential K = F. However, in this case we cannot have a superpotential. The other example is F = ,Z’,,C,F(Z) with C,, Zj having chiral weights n, = f, ni = 0 respectively. In this case, the Klhler potential is given by K = - 3 ln( 1 - fF). We mentioned before that new minimal supergravity can only be used to write R-invariant Lagrangians. However, one can explicitly break the chiral symmetry to write arbitrary Lagrangians. Let us consider the Lagrangian 9 = - f [C,Z;,],, where Co has n = f. Upon eliminating the auxiliary fields and assuming (zO) # 0 (as the Weyl resealing factors are proportional to ZOzo), we get the Poincare supergravity Lagrangian. Similarly, the Lagrangian
9 = $C&&W,
C)l, + 2 WZ$fP’)lF
-&WA,(~)
wAwBIF,
where C have n = 0; upon elimination of all auxiliary fields and Weyl scalings, this gives the general N= 1 supergravity Lagrangians. Finally, let us give the effective Lagrangian for the superstring [17] in the new minimal formalism. As the superpotential is cubic in C, we assign C to have chiral weight f and S, T are assigned chiral weight zero. The Kiihler potential is K’=
-31n(T’+T’-C’C’)(S’+S’)1/3.
We then use (4.6) to determine cr, and substitute between the primed and unprimed fields,
c=
(4.9) it in (4.4) to get the relation
C’ ,/( T’ + T’)(S’ + s)l”
T= T’,
(4.10)
s=s’.
The function F is given by (4.5), F= -3 ln(T+
(4.11)
T)(S+S)“3+3CC(S+S)1’3.
The effective superstring Lagrangian
[17] can then be written as
.9’~=~[F]D+2Re[&CiCjCk]F-[Vp]D -i
Re[SW’], L
+f Re[SiT,,
*TabIF.
(4.12)
L
The last two terms, needed for the anomaly cancellation in the next subsection.
[13], will be discussed
5.5. Topological Invariants and Chern-Simons Multiplets The space-time topological invariants (i.e., Chern, Pontrjagin, and Euler classes) can all be supersymmetrized using the chiral field strength multiplets. The
NEW
MINIMAL
347
SUPERGRAVITY
imaginary part of the F density for the field strength multiplet derivative (see (5.1.3)), Im[ W’],
squared is a total
(dx) = -G* + @*(Xysv2)
(5.1)
with G'=d{V(G+'*)} = df-2.
(5.2)
The Yang-Mills field strength multiplet gives the Chern class. The Riemann multiplet (i.e., the field strength of the Lorentz connection multiplet) gives the Hirzebruch signature. Formula (5.1) can be generalized by replacing W* by fAB WA WB, with fAB a real, constant, gauge covariant matrix. Using this generalization for the Riemann multiplet, we get the Gauss-Bonnet combination as Im[T,, *Tab], [6, 181. Actually, both the real and imaginary parts of [Tab * TabIF are total derivatives. This can be seen using the Bianchi identities ((4.1.6) implies that [(Tab -i*Tab)*lF N [E;SID + surface terms (ST.)) or directly in components. Incidentally, this implies that the Weyl multiplet also satisfies Im[ Wi,] = ST. (see (4.1.4)). The components of the Gauss-Bonnet multiplet are
(5.3) It can be expressed in terms of the irreducible
multiplets
as
iTab *Tab = W,, W,, - $T* - 2v*(E,E,). The F density is given by
CT,b*TabIF(
-Ra+bRa+b -iRs
=d{w,+,(R+
*R,‘b +2F+*
-$0+*)~~+2,4+F+}
- id{ f.zabcdco,Cb (R + - fco +2)cd} -d{4D+$
(r+fyy.r-yYs@$))
- id{4 D’$
iy,(r + try .r - y54$)}.
348
FERRARA
AND
SABHARWAL
The space-time Chern-Simons forms, (5.2), can also be extended Chern-Simons multiplets by [ 121
to real
w2 = 217(Q).
(5.5)
If we let Wz = (z, x, h), then Q is ~2 = C”, f,
Re z, - Im z, Vfrl, (5.6)
with the constraint
on the vector fi. ( VR + 2C”H) + $jaby50,f
= -1m h.
As 52 is determined up to a linear multiplet, we can consider Q-L, gauge where CR = CLo, X* = x& [ 121. Then 52-L,=(O,O,Rez,
-Imz,u,,
and choose a
- b5x + $Y~Y~(Y~$ + Re z + iy5 Im z) ti,,
Reh+2H.u++$,(y5tl+Rez+iy51mz)+,) and the constraint is explicitly
(5.7)
(5.8)
solved as d*u = -Im[ u, = Vf-
W’],
(dx)
v,“.
(5.9)
Here we have used the relation d*u = aJet+‘) = (Daua - Tz,ub) e(dx). These Chern-Simons multiplets are needed for the consistent coupling of linear multiplets [12]. The Green-Schwarz mechanism to cancel anomalies requires that linear multiplets should appear in the combination L + s2 [ 12, 313. In terms of the duality transformed theories (L + S), this implies the coupling Re[SW’],. For example, consider the term Re[SiT,, *Tab ] in the superstring Lagrangian (5.4.12). this implies the coupling of a,, Im S with the Lorentz In components, Chern-Simons form (see (5.4)). If we now perform a duality transformation in components, this term modifies the field strength of the antisymmetric tensor by the addition of the Lorentz Chern-Simons form. Also, note that this term contains the coupling of Re S with the Gauss-Bonnet integrand (see (5.4)). The superstring Lagrangian in terms of a linear multiplet is obtained by replacing S + S by U and adding the Lagrange multiplier a[ UL],. Eliminating U we then get Ts=+[-3ln(T+T)-3lng(x)+3xg(x)+lne-V,], (5.10)
g(x) = (1 + (1 + x3)1’2)1’3 - ( - 1 + (1 + x)3)l’2)“3,
(5.11)
NEW MINIMAL
349
SUPERGRAVITY
and 2=L-Q+asz’
w2 = 217(Q)
(5.12)
iTab *Tab = 2z7(Q’). 6. CONCLUSIONS In the present paper, we have derived the tensor calculus of new minimal Poincare supergravity and constructed general matter couplings. This investigation was motivated by a renewed interest of new minimal supergravity as the natural framework to describe the dynamics of 4D heterotic superstrings. The results derived here provide to necessary tools to construct any higher order coupling in the gravitational and Yang-Mills curvatures.
APPENDIX:
CONVENTIONS
AND USEFUL FORMULAE
We use a Minkowski metric, g, = ( - + + + ), &0123= + 1. The Dirac matrix conventions are {y,, yb) = -2gab, y5 = iy”y1y2y3. (Tab = (i/2)(?a% Yb), *Oab = = iy5cab, $ = r+K. Left- and right-handed components are dellned by (l/2) Enbcdccd
In a Majorana representation C = y” and the Majorana condition is $ = II/*. The two-component spinor formalism is derived from the following chiral representation of the Dirac matrices,
Some useful Dirac algebra identities are YaYbYc
=
gacYb
ITab acd
=
gac
gbd
and the Fierz identity
&by, -
&dgbc
-
gcbYo -
-EabcdiYSYd
EabcdiY5
for anticommuting
+
ikac
Obd
spinors,
-
&dObc
+
gbdgac
-
gb$ad)
350
FERRARA
AND
SABHARWAL
or in two components, II/. v = -f(W)
+ $(Jab(habtu
$ . $’ = - ja&a,lj’). The gravitational
curvature and torsion are given by R; = do; •k w:w;
= fR&,, dx” dx”,
T” = de” + o;eb = f T;, dx” dx’
with w,b = - w bo
=
Oa.?b
dx”.
The Ricci, scalar curvature, Einstein, and Weyl tensors are Rrrt, = Rzdb, W nbcd
=
R = R;,
i(Rabcd
-
Eab = &,
*R:d,b
) -
- &,R
hkc
gbd
-
= *R;;, g,dgbc)R.
The connection is explicitly solved as wabC= CO.bcte) + j(T,bc f Tbac + Tcrrb). Under a variation of the vierbein and torsion, the change in the connection is given by h+b
=
ttXapb
+
Xpnb
+
xb.p)
with x$ = -(D,
he: - D, se;) + 6T;,.
Note that D, only includes connections for the tangent space indices whereas V, includes connections for both the tangent and curved space indices. The gravitino curvature is given by tirv = Dl I), - 0: $,, and +,b = eze;ll/ay. The Rarita-Schwinger operator is ra = 4y5yb *ebo, Some useful identities are
Ydaabrd
=
tiab
+
h5
cab46
*tiaby
=
iray
y .r=ia.,$,
= -371.4.
Finally, to convert our formulae to the Euclidean conventions of Kugo and Uehara or van Nieuwenhuizen [2], the following replacements have to be made: E abed
+
%bcd,
Yu
+ -iy,,
y5 + -y5,
(Tab+ -2ifz&
C+
e~~e~,*,~-*~,EjE,W,pb~Opab,Srrb~-i~.b, A,
+
For a vector multiplet
-1A
2 ,,,n+in,B,,+-iaE,,H,+
-‘BO. 2
LJ
-C,
NEW
MINIMAL
SUPERGRAVITY
351
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