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Green functions in a super self-dual Yang-Mills background
Nuclear Physics B239 (1984) 93-105 ~ North-Holland Publishing Company
G R E E N F U N C T I O N S 1N A S U P E R S E L F - D U A L YANG-MILLS BACKGROUND I.N. McARTHUR* Lyman l.aboratoo, of Physics, ttaruard Unit,ersit>; Cambridge, MA 02138, USA Received 7 November 1983
In euclidean supersymmetric theories of chiral superfields and vector superficlds coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic part of the action. These Grcen functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the fight chiral Green function in the ftmdamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kerncls associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a supcrfield version of the supcr-indcx theorems for the components of chiral superfields in a self-dual background.
1. Introduction Euclidean q u a n t u m field theories in the presence of a self-dual Yang-Mills b a c k g r o u n d are of major interest in modern physics. A n important feature of these theories which simplifies explicit computations is that the Green functions for fields of different spin in the presence of the background are related to each other [1 ], and can be expressed in terms of the scalar Green function. This Green function m a y be constructed from the general self-dual Yang-Mills solutions provided by the A D H M formalism [1, 2]. There exists an elegant explanation for the relation between Green functions of different spin. If the q u a n t u m fields interacting with the background are the c o m p o n e n t s of a supermultiplet, the theory possesses a global supersymmetry; that is, the self-dual background (or " v a c u u m " ) respects a supersymmetry relating the c o m p o n e n t s of the supermultiplet. This supersymmetry causes the spectrum of nonzero eigenvalues of the Laplace-type operators obtained from the quadratic part of the action to coincide for the fields of different spin [3,4], and thus leads to a relation between the corresponding Green functions [5]. This also has interesting consequences for the one-loop q u a n t u m corrections to the action [3] and is responsi* This research is supported in part by the National Science Foundation under grant no. PI-IY-82-15249. 93
94
LN. McArthur / Self-dual Yang-Mills background
ble for the super-index theorems of Christensen and Duff [6] in a self-dual Yang-Mills background. Similar results apply for quantum fields defined on a euclidean Einstein manifold with self-dual Weyl curvature [4, 6]. Since the above properties follow from the existence of a supersymmetry, it is not surprising that they have a natural superfield extension, which is presented in this paper. We begin in sect. 2 by discussing supersymmetry in a euclidean metric and the super-self-duality condition on the Yang-Mills background. The Green functions for covariantly chiral scalar superfields in the presence of a super-self-dual Yang-Mills background are examined in sect. 3, and a relation between the Green functions on the left and right chiral spaces is obtained. Sect. 4 is devoted to the Green function for a vector superfield coupled to the same background. Using the recently developed supersymmetric version of the A D H M formalism, we present in sect. 5 an explicit expression for the Green function on the space of right chiral superfields which transform under the fundamental representation of an SU(n) gauge group. The Green functions for left chiral and vector superfields in the same representation may be obtained from this. Sect. 6 examines the one-loop quantum corrections to the super-self-dual Yang-Mills action due to chiral and vector superfields, and points out the relation of this work to the super-index theorems. The paper concludes with a short discussion.
2. Euclidean supersymmetry and super-self-dually' On a four-dimensional manifold with a Lorentz metric, the (1,0) and (0,1) components of the Yang-Mills curvature two-form are related by complex conjugation, so one of them cannot be set to zero without the other vanishing as well. In a euclidean metric, however, these components are independent and the (0,1) component can be set to zero to provide a self-dual curvature. To generalize this condition to supersymmetric theories, we must expect to have to work with euclidean supersymmetry. This is not trivial, because the coordinates 0~ and 0~ which parameterize the fermionic directions in euclidean superspacc transform under different SU(2) groups, and are no longer related by complex conjugation as they are in a Lorentz metric. Complex conjugation raises or lowers spinor indices, but does not change dotted indices to undotted indices or vice versa. Thus it seems that euclidean superspace has four spinor coordinates 0~, 0 '~, 0 ~, ~,~ (where 0'~ = (0,,)*) and that the supermultiplets should resemble those of N = 2 Lorentz supersymmetry. This is indeed the case if the euclidean supersymmetry algebra and its representation in superspace are required to be hermitian under complex conjugation, as was done by Zumino [7]. Nicolai [8], however, has pointed out that by abandoning this in favour of hermiticity under the unitary involution operator of Osterwalder and Schrader [9], we can obtain a superspace which retains the supermultiplet structure of the original theory (see also [10], where the corresponding algebra is discussed).
I.N. McArthur / Self-dual Yang-Mills background ~.
95
This superspace is parameterized only by 0,~ and 0% and not by their complex conjugates. We will adopt Nicolai's formulation in this paper, and from now on work in euclidean superspace. If @A (A = a, a, 6) is the supercovariant derivative with respect to a Yang-Mills background [11], then the curvature two-form is defined by @A@R-- (-- 1VR® j ,[email protected] = F~B + torsion terms. The superfields W~ and V/a, in terms of which all nonvanishing components of the curvature can be expressed, are no longer related by complex conjugation as they are in the Lorentz metric. We can thus require that one of them vanishes and still have the other nonvanishing. Since W~ and Wa contain amongst their components the self-dual and anti-self-dual parts of the ordinary Yang-Mills curvature, we obtain a superfield version of self-dual and anti-self-dual backgrounds. For definiteness, we will choose ~'~'a= 07 a super-self-dual background.
3. Green functions for chiral scalar superfields .In this section, we consider theories of covariantly chiral scalar superfields interacting with a super-self-dual Yang-Mills background (covariantly chiral means chiral with respect to the background supercovariant derivatives @A [11]; hereafter, the term "chiral" will mean "covariantly chiral"). We saw in a previous publication [12] that the operators 1 ~ 2 @ 2 and ~g@2@2 are the quadratic Laplace-type differential operators on the spaces of left and right chiral superfields respectively. On their respective spaces, these operators can be rewritten as ~642@ 2 = --6~a6~a q- WC~@a q- l ( 6~aWa) ,
Notice, however, that in a super-self-dual background (I~"a = 0), these expressions become (using ( Q " W , ) = (6)aWa)) -a -a ~6 °~242 = --@a6~'a.
(1) (2)
The simple form of the right chiral operator (2) is responsible for a number of interesting features of this theory. The operator 1~@26~2= --@a@acan be inverted on the space of right chiral superfields, and the corresponding Green function G is defined (with gauge indices suppressed) by -- @a@aG(X, 0, 0; X', 0 t, 0') = ~(4) (.~ R -- )C~ )3(2)(0 -- 0t),
(3)
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I.N. McArthur / Self-dual Yang-Millabackground
where x~' " = x " - t0 ~%a0 " -a is the euclidean version of the usual coordinate adapted to the right chirality constraint. (~ is right chiral in both its arguments (recall that a right covariantly chiral superfield is a function of only ~R and t~ if it is a gauge singlet, because then @a = Da where D4 is the usual flat superspace covariant derivative. A superfield with gauge indices has 0 dependence other than that in ~R from the 0 dependence of the background gauge vector superfield, which is involved in the definition of covariantly chiral superfields [11]). For right chiral superfields which transform under the fundamental representation of an SU(n) gauge group, we will present an explicit expression for G in sect. 5, The situation is different for the operator ~6~-@ 2, which cannot be inverted on the full space of left chiral superfields because it annihilates part of the space. To show this, we observe that the operator PL defined by PL ~---l't?542~6~2'
(4)
has the properties (using (2) and (3)) p 2 = PL, 1 6 ~ 2 @ 2(1L -- P L ) = O.
(5)
(Note: here and throughout the rest of the papg_r, we use operator notation. For example, ~2~c;~2 means C~2G(x,O,O;x',O',O')@ "2, and, acting on a left chiral superfield ~(x, O, O) (i.e. @aq~= 0), , ~, )@2ep(x, ,0', 0 ' ) . &f d4S: d20' _2(7(x,O,O; x., ,0,0
Also, 1 L = ~¢4)(~ L -- - ~ ) 8 ( 2 ) ( 0 -- 0')). T h u s (5) shows t h a t (1L - PL) is the projection operator onto the subspace of left chiral superfields which is annihilated by the operator ~ 2 @ 2 . A Green function (7 can be defined for the operator ~c~ 2@2 by inverting it on the subspace of left chiral superfields determined by the projection operator PL: @2@:G = PL"
(6)
This Green function is left chiral in both its arguments and is related to (~ by (7 : 166~2(~(~@2.
(7)
4. Green function for a vector superfield
In this section, we give an expression for the Green function for the operator which appears in the quadratic part of the superfield Yang-Mills action when it is expanded about a super-self-dual background. After gauge fixing which preserves
1.N. McArthur / Self-dual Yang-Mills background
97
supersymmetry and background gauge invariance [11], this operator for an arbitrary background is Av = - ~,~ 6~, + W ~.~ (this expression corrects a sign error in the first reference of [12]). The operator above acts on vector superfields V(x,O,O) which transform under the adjoint representation of the gauge group and which are self-conjugate under OsterwalderSchrader conjugation [8, 9]. If the background is super-self'dual, then IYd~= 0 and the operator becomes
nv =
-@.@. +
(8)
Note that this operator is of the same form as (1), though itacts on a different space of superfields. This is helpful in analyzing it. The operator (8) cannot be inverted on the full space of vector superfields, because it annihilates part of the space. This follows because, using the definitions (2) and (3), the operator Pv defined by Pv = 1~ 6~-'12~@2 + ~6 ~61==)2@2 -- 1@dG'3~26~°,
(9)
satisfies p2 = Pv, ( _ 6~.
+ W-6~.) Pv = - @6)~ + W-6~) .
These equations show that (1 - Pv) is the projection operator onto the subspace of vector superfields annihilated by A v (note that because we are working with vector superfields, 1 = 6¢4)(x - x')6(2)(0 - 0')6(2)(0 - tJ')). Thus the Green function Gv for the operator Av is obtained by inverting it on the subspace of vector superfields determined by the projection operator Pv, i.e.
( -6~,@a + W%~,~)Gv = Pv. It is not hard to verify that G v has the following expression in terms of the right chiral Green function G:
Gv = ~°~2GG@2 + 16GG@-@-- {@aGG@2@ '~. ~
~
l
~
~
9
~
9
.
.
.
.
.
(10)
Of course, Gv is itself a vector superfield in both its arguments. 5. ADHM construction of (~ As indicated earlier, we can give an explicit expression for the Green function (~ defined by (3) in the presence of a super-self-dual Yang-Mills background for the case in which the gauge group is SU(n) and the right chiral superfields on which
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1.N. McArthur / Se[[-dw21Yang-Mills background
~@2~)2= _c96~~ a acts transform under the fundamental representation of the gauge group. We make use of the recently developed supersymmetric version of the A D H M formalism [13]. This gives an expression for the connection I~1 (defined by 67,A= D4. + f'A, A = a, a, &) which solves the integrability condition I~,,~= 0, in terms of matrices which satisfy quadratic constraints. If c~.4 acts on superfields in the fundamental representation of an SU(n) gauge group, then the general solution to Wa = 0 is provided by
r (x, o, 0): ,.(x, o. O)+O:(x, o. 0),
(11)
where, for a background in which the bosonic component of IV has instanton number k, v(x,O,O) is a (2k + n ) × n matrix of superfields [13]. The matrix c, is defined by t,+v = 1 and v+Aa= 0, where Aa = aa + b"2R,,a + cOa and a,.,, b" and c are constant (2k + n) × k complex matrices satisfying constraints given by Semikhatov [13]. (Note: Semikhatov actually solves the integrability condition W,~= 0. Thus the matrix v used in this paper, which solves l,Va= 0, is obtained from that used by Semikhatov by replacing dotted indices eveLywhere by undotted ones and vice versa. Alternatively, we could retain the notation of [13], and the Green function constructed below would be that for the space of left chiral superfields in the background W,~= 0.) The Green function G in the fundamental representation, defined by (3), is given by
o;< o.
o o>.(x, o.
I&
-
°->2
"
The complex conjugation in vt acts only on the complex numbers in the matrix v, and not on the superspace coordinates x ~a, 0~, 0a. This ansatz was motivated by the construction of the scalar Green function in the presence of a self-dual Yang-Mills background given by Corrigan et al. [21, and it can be checked that (12) satisfies (3) by using the arguments in [2] with minor modifications (see also [14]). As mentioned in sect. 3, the Green function G must be right (covariantly) chiral in both its arguments. Thus, if (12) is to be consistent, v+ must satisfy (@~v*) = 0. We show this here. Using (11) and v%, = 1,
(%,:) = (Do.,) + (:D:),: =
.'DoP,
where P = vv t. However, from [13], P = 1 - AafA+a. Thus, using v~Aa= 0,
v*D~e
= - v*( D,~Ae') fA~ ,
which vanishes because Aa is a function of 2 R and 0 with constant matrix coefficients. So our ansatz is consistent with the chirality constraints.
LN. McArthur / Self-dual Yang-Mills background
99
The expression (12) for (~ allows the construction of G and PL (defined by (4) and (7)) for left chiral superfields in the fundamental representation of an SU(n) gauge group. In particular, the expression for PL permits us to determine explicitly the subspace of the space of left chiral superfields annihilated by ~6~2@2, in the spirit of the corresponding analysis by Osboru [15] of the space of zero modes of the left-handed Weyl operator in the presence of an instanton. This will be reported on elsewhere [16]. There is also no apparent obstruction to the construction of Green functions in other representations of the gauge group by using the analogue of the tensor product construction of Corrigan, Goddard and Templeton [2].
6. One-loop quantum corrections and the super-index theorems If a set of quantum fields which are the components of a supermultiplet interact with a self-dual Yang-Mills background, then at the one-loop level, the theory has the remarkable feature that the quantum correction to the Yang-Mills action is determined only by the zero modes of the operators which appear in the quadratic part of the action [3]. As was mentioned in the introduction, this is because the background (or "vacuum") respects a supersymmetry, so that the spectrum of nonzero eigenvalues is the same for operators associated with both bosonic and fermionic quantum fields in the quadratic part of the action, and the corresponding functional determinants cancel completely in the one-loop functional integral. This pairing of nonzero eigenvalues also gives rise to a super-index theorem [6] relating the number of zero modes of the operators associated with the components of the superfield in the instanton background. Only the zero modes survive in the difference between the suitably regulated total number of bosonic and fermionic eigenmodes, and these total numbers of eigenmodes can be expressed in terms of topological invariants of the background using the b4 coefficients. Here, we use the Green functions developed in the previous section to examine the one-loop quantum corrections to the action of a super-self-dual Yang-Mills background, and obtain superfield formulations of the appropriate super-index theorems. In ordinary field theory., the contribution by a quantum field to the one-loop effective action in the presence of a background is determined by the functional determinant of the associated Laplace-type differential operator. An alternative (but equivalent) procedure is to construct the "heat kernel" for the operator, whose functional trace determines the one-loop effective action [17] (see also [18,19] for modern treatments.) While the former approach to the evaluation of the one-loop effective action has no obvious superspace analogue, the latter does. We use it to examine the contribution from chiral superfields to the one-loop effective action in the presence of a super-self-dual background.
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I.N. McArthur / Self-dual Yang-Mills background
The kernel /((t) has an asymptotic expansion as t ~ 0" which defines the right chiral superfields s-b, [12]:
s-b,(2R,O)t ('-4)/2.
lim(x,0, t}~ x ' , 0 ,' 0 " ) T r K- ( x , O , 0 ; x , 0 , 0 , t ) = t1=0
As a consequence of eq. (16), the superfields s-b~ all vanish in a super-self-dual background. If the background is purely bosonic, then the 02 component of ~ is 2b,,(0,0)-bn(0,½) (see [121), where b,,(A,B) is the ordinary b, coefficient [6] associated with the component field of the quantum fight chiral superfield which transforms under the (A, B) representation of SO(4). Thus we find 2b,,(0,0)b,,(0, ½)= 0. For n =4, this statement combined with the fact that the operator 1L66~z°~L2 can be inverted on the full space of right chiral superfields (there are no "zero-modes"), is the (trivial) super-index theorem for the component fields of a fight chiral superfield in a self-dual Yang-Mills background [6]. The situation is slightly different on the space of left chiral superfields because, as we have already seen, the Laplace-type operator (1) annihilates part of this space in a super-self-dual background. The kernel is defined by
K(x,O,O;
X ,' O , O '
"
;t)=e-(t/16)~)2c'~2~(4}(~ L -
3~_)~(2}(0
-
0')
,
or,
K(t)
= e-('/16)#2"~Zl L ,
and is left chiral in both its arguments. The Green function (6) is related to
(17)
K(t)
by
G = fo°~dtK(t)Pi.. The one-loop correction to the super-self-dual Yang-Mills action from the subspace of left chiral scalar superfields determined by the projection operator PL is thus
(18) To compute str(K(t)PL), we use the definitions (3), (4) and (17) and the cyclic property of operators in the supertrace (this can be checked by writing out the steps with arguments and integrations included). We find: s t r ( K ( t ) PL ) = str(e (t/16)@2@2.?z.6~2"2('~6~ ~ 2)
I.N. McArthur / Self-dual Yang-MilL~background
102
= str(e- {t/16)'~z'~:l R ) = strK(t). Using (16), we have
str(K(t)PL)=O.
(19)
Thus (18) and (19) imply that the one-loop quantum correction to the super-self-dual Yang-Mills action from the subspace of left chiral superfields determined by the projection operator PL vanishes. In a purely bosonic (self-dual) background, this is the result that only zero modes contribute to the one-loop quantum correction to the action [3]. We can obtain further information about the left chiral space by examining strK(t), the supertrace of the full kernel. Using (17), (19), and c~26~2(1L - PL) = 0, we obtain s t r K ( t ) = str(K(t)(1L - P,.))
= str(e
PL))
= str(1L - P t ) .
(20)
The left chiral superfields s-b,, are defined by the following asymptotic expansion in the limit t --+ 0 '- (see [12]): lira(x, 0, 0 ---, x', 0', O')TrK(x, 0, 0; x', 0', 0'; t) = ~ s-bn(~ L, O)t <'-a)/2 . n=0
(21) Thus (20) and (21) imply
str(li - pL)= ~"
,,.-4,/2f d45ctd20s_b,,(2i.,O).
(22)
n=0
Since the left-hand side of (22) is independent of t, we find that for n 4= 4, the
I.N. McArthur / Self-dual Yang-Mills background
103
superfields s-b,, vanish in a super-self-dual background, while for 17= 4 we have str(1L -- PL) = fdaXL d20 s-b4(~I, 0).
(23)
Also, noting from (17) that l i m t ~ o K ( t ) = 1L, (19) implies str(PL) = 0.
(24)
Again, for a Purely bosonic background, eqs. (23) and (24) are statements of familiar results. Eq. (24) is the superfield version of the result that, in a self-dual background, there are equal numbers (after suitable regulation) of bosonic and fermionic nonzero modes of the Laplace-type operators associated with the component fields of a left chiral superfield. The right-hand side of eq. (23) is (see [12]) fd4x(2b4(O,O) - b4(½,0)) , while the left-hand side is 2n(0,0) - n(~, 0), where n(A, B) is the number of zero modes of the Laplace type operator associated with a component field transforming under the representation (A, B) of SO(4) (the fermionic projection operators give a ( - 1 ) when traced because we have to interchange two Fermi fields). Thus (23), combined with (24), gives the super-index theorem of Christensen and Duff [6] for the components of a left chiral superfield in the presence of self-dual Yang-Mills background. It is interesting to note that in the more general super-self-dual background, which may have nonvanishing fermionic components, str(1L - PL) is still an integer. This is because, from the results of ref. [12], s-b4 = (1/16~r2)TrW~W~, so str(l L - PL) = (]/16"lz2)fd4~i. d20Tr(W'~Wa) = (1/32~r2)fd4x(FohFab+ 4i~D~aXa) where D~a here is the ordinary Yang-Mills covariant derivative. The condition I~a = 0 causes Xa to vanish, so str(1 - PL) = u, the instanton number of the bosonic component of the background (which is self dual). So far, we have not considered the contribution from the subspace of left chiral superfields annihilated by 1~6~2@ 2 to the one-loop quantum correction to the action. A full treatment would be complicated, requiring a parameter~ation of this space by supersymmetric collective coordinates. However, we can readily see that part of the contribution from this space is a logarithmically divergent correction to the action, which is characteristic of zero modes in instanton backgrounds [3, 19]. This comes from
f -~str( K( t )(1L -- PL)), which, by (20) and (23), is just f d4.~Ld2 8 s_b4(.~L ' . , r ~ dt
° ) Jo -i- "
104
I.N. McArthur / Self-dual Yang-Mills backgrvund
A quantum vector superfield in a super-self-dual background, which was considered in sect. 4, gives rise to only a collective coordinate contribution to the one-loop effective action, from the space of superfields specified by the projection operator (1 - Pv). This is because the supertrace of the kernel associated with the operator (8) vanishes. To see this, we note that the kernel is defined by K ( x , O , O; x', Ot, O'; l) = e ` ( °'ga°~a-W°C~'~)(~(4)( X -- X ' ) a ( 2 ) ( 0 - - 0 ' ) a ( 2 ) ( 0 - - 0 ' ) .
The factor 6(2)(0- 0')= ( 0 - ~,)2 vanishes in taking the supertrace, because there are no terms involving @,~ in the operator to annihilate it. There will also be one-loop corrections to the super-self-dual action from the three chiral ghosts which result from the supersymmetric gauge fixing. 7. Conclusion
The Green functions associated with chiral scalar superfields and vector superfields coupled to a super-self-dual Yang-Mills background have been shown to be related, and to be obtainable from the Green function on the space of right chiral superfields. By fairly trivial manipulation of operators and superfield kernels, we have demonstrated a number of interesting properties of these theories. For a purely bosonic background, these properties correspond to known results for the component fields of the quantum superfields in the presence of Yang-Mills instantons. We were also able to give an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group by using the supersymmetric version of the ADHM formalism. A similar analysis can be carried out for superfields with spinor indices and coupled to a curved superspace background, and will be presented elsewhere [16]. The corresponding super-self-duality constraint is that ff'~B~= 0, where W~i~ is the superfield containing the anti-self-dual part of the Weyl curvature tensor. If the background is also on-shell, we obtain a superfield formulation of the super-index theorems of Christensen and Duff [6], and of the results of Hawking and Pope [4] on mode cancellations in the presence of self-dual gravitational instantons. The analysis is not as straightforward as that in this paper, because in general the Laplace-t2ype operator on the space of right chiral superfields annihilates part of the space. I wish to thank Dr. R. Brandenberger for a discussion on Osterwalder-Schrader positivity. References [1] G. 't Hooft, Phys. Rev. D14 (1976) 3432; L.S. Brown, R.D. Carlitz, D.B. Crcamcr and C. Lee, Phys. Lctt. 71B (19771) 103: Phys. Rcv. D17 (1978) 1583
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[2] E.F. Corrigan, D.B. Fairlie, S. Templeton and P. Goddard, Nucl. Phys. B140 (1978) 31; E. Corrigan, P. Goddard and S. Templeton, Nucl. Phys. B151 (1979) 93 [3] A. D'Adda and P. Di Vecchia, Phys. Lett. 73B (1978) 162 [4] S.W. ttawking and C.N. Pope, Nucl. Phys. B146 (1978) 381 [5] D.N. Page, Phys. Lett. 85B (1979) 369 [6] S.M. Christensen and M.J. Duff, Nucl. Phys. B154 (1979) 301 [7] B. Zumino, Phys. Lett. 69B (1977) 369 [8] H. Nicolai, Nucl. Phys. B140 (1978) 294; Czech. J. Phys. B29 (1979) 308 [9] K. Osternvalder and R. Schrader, Helv. Phys. Acta 46 (1973) 277 [10] J. Lukierski and A. Nowicki, Trieste preprint SISA 34/82/EP [11] M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429 [12] I.N. McArthur, Phys. Lett. 128B (1983) 194; Harvard University preprint HUTP-83/A052, to be published in Classical and quantum gravity [13] A.M. Semikhatov, Phys. Lett. 120B (1983) 171; I.V. Volovich, Phys. I_ett. 123B (1983) 329 [14] E. Corrigan, Phys. Reports 49 (1979) 95 [15] H. Osborn, Nucl. Phys. B140 (1978) 45 [16] I.N. McArthur, in preparation [17] J. Schwinger, Phys. Rev. 82 (1951) 664 [18] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133 [19] M.J. Duff and D.J. Toms, Proc. second quantum gravity seminar, Moscow, ed. M.A. Markov (1981) [20] G. 't Hooft, Nucl. Phys. B62 (1973) 444
G R E E N F U N C T I O N S 1N A S U P E R S E L F - D U A L YANG-MILLS BACKGROUND I.N. McARTHUR* Lyman l.aboratoo, of Physics, ttaruard Unit,ersit>; Cambridge, MA 02138, USA Received 7 November 1983
In euclidean supersymmetric theories of chiral superfields and vector superficlds coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic part of the action. These Grcen functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the fight chiral Green function in the ftmdamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kerncls associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a supcrfield version of the supcr-indcx theorems for the components of chiral superfields in a self-dual background.
1. Introduction Euclidean q u a n t u m field theories in the presence of a self-dual Yang-Mills b a c k g r o u n d are of major interest in modern physics. A n important feature of these theories which simplifies explicit computations is that the Green functions for fields of different spin in the presence of the background are related to each other [1 ], and can be expressed in terms of the scalar Green function. This Green function m a y be constructed from the general self-dual Yang-Mills solutions provided by the A D H M formalism [1, 2]. There exists an elegant explanation for the relation between Green functions of different spin. If the q u a n t u m fields interacting with the background are the c o m p o n e n t s of a supermultiplet, the theory possesses a global supersymmetry; that is, the self-dual background (or " v a c u u m " ) respects a supersymmetry relating the c o m p o n e n t s of the supermultiplet. This supersymmetry causes the spectrum of nonzero eigenvalues of the Laplace-type operators obtained from the quadratic part of the action to coincide for the fields of different spin [3,4], and thus leads to a relation between the corresponding Green functions [5]. This also has interesting consequences for the one-loop q u a n t u m corrections to the action [3] and is responsi* This research is supported in part by the National Science Foundation under grant no. PI-IY-82-15249. 93
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LN. McArthur / Self-dual Yang-Mills background
ble for the super-index theorems of Christensen and Duff [6] in a self-dual Yang-Mills background. Similar results apply for quantum fields defined on a euclidean Einstein manifold with self-dual Weyl curvature [4, 6]. Since the above properties follow from the existence of a supersymmetry, it is not surprising that they have a natural superfield extension, which is presented in this paper. We begin in sect. 2 by discussing supersymmetry in a euclidean metric and the super-self-duality condition on the Yang-Mills background. The Green functions for covariantly chiral scalar superfields in the presence of a super-self-dual Yang-Mills background are examined in sect. 3, and a relation between the Green functions on the left and right chiral spaces is obtained. Sect. 4 is devoted to the Green function for a vector superfield coupled to the same background. Using the recently developed supersymmetric version of the A D H M formalism, we present in sect. 5 an explicit expression for the Green function on the space of right chiral superfields which transform under the fundamental representation of an SU(n) gauge group. The Green functions for left chiral and vector superfields in the same representation may be obtained from this. Sect. 6 examines the one-loop quantum corrections to the super-self-dual Yang-Mills action due to chiral and vector superfields, and points out the relation of this work to the super-index theorems. The paper concludes with a short discussion.
2. Euclidean supersymmetry and super-self-dually' On a four-dimensional manifold with a Lorentz metric, the (1,0) and (0,1) components of the Yang-Mills curvature two-form are related by complex conjugation, so one of them cannot be set to zero without the other vanishing as well. In a euclidean metric, however, these components are independent and the (0,1) component can be set to zero to provide a self-dual curvature. To generalize this condition to supersymmetric theories, we must expect to have to work with euclidean supersymmetry. This is not trivial, because the coordinates 0~ and 0~ which parameterize the fermionic directions in euclidean superspacc transform under different SU(2) groups, and are no longer related by complex conjugation as they are in a Lorentz metric. Complex conjugation raises or lowers spinor indices, but does not change dotted indices to undotted indices or vice versa. Thus it seems that euclidean superspace has four spinor coordinates 0~, 0 '~, 0 ~, ~,~ (where 0'~ = (0,,)*) and that the supermultiplets should resemble those of N = 2 Lorentz supersymmetry. This is indeed the case if the euclidean supersymmetry algebra and its representation in superspace are required to be hermitian under complex conjugation, as was done by Zumino [7]. Nicolai [8], however, has pointed out that by abandoning this in favour of hermiticity under the unitary involution operator of Osterwalder and Schrader [9], we can obtain a superspace which retains the supermultiplet structure of the original theory (see also [10], where the corresponding algebra is discussed).
I.N. McArthur / Self-dual Yang-Mills background ~.
95
This superspace is parameterized only by 0,~ and 0% and not by their complex conjugates. We will adopt Nicolai's formulation in this paper, and from now on work in euclidean superspace. If @A (A = a, a, 6) is the supercovariant derivative with respect to a Yang-Mills background [11], then the curvature two-form is defined by @A@R-- (-- 1VR® j ,[email protected] = F~B + torsion terms. The superfields W~ and V/a, in terms of which all nonvanishing components of the curvature can be expressed, are no longer related by complex conjugation as they are in the Lorentz metric. We can thus require that one of them vanishes and still have the other nonvanishing. Since W~ and Wa contain amongst their components the self-dual and anti-self-dual parts of the ordinary Yang-Mills curvature, we obtain a superfield version of self-dual and anti-self-dual backgrounds. For definiteness, we will choose ~'~'a= 07 a super-self-dual background.
3. Green functions for chiral scalar superfields .In this section, we consider theories of covariantly chiral scalar superfields interacting with a super-self-dual Yang-Mills background (covariantly chiral means chiral with respect to the background supercovariant derivatives @A [11]; hereafter, the term "chiral" will mean "covariantly chiral"). We saw in a previous publication [12] that the operators 1 ~ 2 @ 2 and ~g@2@2 are the quadratic Laplace-type differential operators on the spaces of left and right chiral superfields respectively. On their respective spaces, these operators can be rewritten as ~642@ 2 = --6~a6~a q- WC~@a q- l ( 6~aWa) ,
Notice, however, that in a super-self-dual background (I~"a = 0), these expressions become (using ( Q " W , ) = (6)aWa)) -a -a ~6 °~242 = --@a6~'a.
(1) (2)
The simple form of the right chiral operator (2) is responsible for a number of interesting features of this theory. The operator 1~@26~2= --@a@acan be inverted on the space of right chiral superfields, and the corresponding Green function G is defined (with gauge indices suppressed) by -- @a@aG(X, 0, 0; X', 0 t, 0') = ~(4) (.~ R -- )C~ )3(2)(0 -- 0t),
(3)
96
I.N. McArthur / Self-dual Yang-Millabackground
where x~' " = x " - t0 ~%a0 " -a is the euclidean version of the usual coordinate adapted to the right chirality constraint. (~ is right chiral in both its arguments (recall that a right covariantly chiral superfield is a function of only ~R and t~ if it is a gauge singlet, because then @a = Da where D4 is the usual flat superspace covariant derivative. A superfield with gauge indices has 0 dependence other than that in ~R from the 0 dependence of the background gauge vector superfield, which is involved in the definition of covariantly chiral superfields [11]). For right chiral superfields which transform under the fundamental representation of an SU(n) gauge group, we will present an explicit expression for G in sect. 5, The situation is different for the operator ~6~-@ 2, which cannot be inverted on the full space of left chiral superfields because it annihilates part of the space. To show this, we observe that the operator PL defined by PL ~---l't?542~6~2'
(4)
has the properties (using (2) and (3)) p 2 = PL, 1 6 ~ 2 @ 2(1L -- P L ) = O.
(5)
(Note: here and throughout the rest of the papg_r, we use operator notation. For example, ~2~c;~2 means C~2G(x,O,O;x',O',O')@ "2, and, acting on a left chiral superfield ~(x, O, O) (i.e. @aq~= 0), , ~, )@2ep(x, ,0', 0 ' ) . &f d4S: d20' _2(7(x,O,O; x., ,0,0
Also, 1 L = ~¢4)(~ L -- - ~ ) 8 ( 2 ) ( 0 -- 0')). T h u s (5) shows t h a t (1L - PL) is the projection operator onto the subspace of left chiral superfields which is annihilated by the operator ~ 2 @ 2 . A Green function (7 can be defined for the operator ~c~ 2@2 by inverting it on the subspace of left chiral superfields determined by the projection operator PL: @2@:G = PL"
(6)
This Green function is left chiral in both its arguments and is related to (~ by (7 : 166~2(~(~@2.
(7)
4. Green function for a vector superfield
In this section, we give an expression for the Green function for the operator which appears in the quadratic part of the superfield Yang-Mills action when it is expanded about a super-self-dual background. After gauge fixing which preserves
1.N. McArthur / Self-dual Yang-Mills background
97
supersymmetry and background gauge invariance [11], this operator for an arbitrary background is Av = - ~,~ 6~, + W ~.~ (this expression corrects a sign error in the first reference of [12]). The operator above acts on vector superfields V(x,O,O) which transform under the adjoint representation of the gauge group and which are self-conjugate under OsterwalderSchrader conjugation [8, 9]. If the background is super-self'dual, then IYd~= 0 and the operator becomes
nv =
-@.@. +
(8)
Note that this operator is of the same form as (1), though itacts on a different space of superfields. This is helpful in analyzing it. The operator (8) cannot be inverted on the full space of vector superfields, because it annihilates part of the space. This follows because, using the definitions (2) and (3), the operator Pv defined by Pv = 1~ 6~-'12~@2 + ~6 ~61==)2@2 -- 1@dG'3~26~°,
(9)
satisfies p2 = Pv, ( _ 6~.
+ W-6~.) Pv = - @6)~ + W-6~) .
These equations show that (1 - Pv) is the projection operator onto the subspace of vector superfields annihilated by A v (note that because we are working with vector superfields, 1 = 6¢4)(x - x')6(2)(0 - 0')6(2)(0 - tJ')). Thus the Green function Gv for the operator Av is obtained by inverting it on the subspace of vector superfields determined by the projection operator Pv, i.e.
( -6~,@a + W%~,~)Gv = Pv. It is not hard to verify that G v has the following expression in terms of the right chiral Green function G:
Gv = ~°~2GG@2 + 16GG@-@-- {@aGG@2@ '~. ~
~
l
~
~
9
~
9
.
.
.
.
.
(10)
Of course, Gv is itself a vector superfield in both its arguments. 5. ADHM construction of (~ As indicated earlier, we can give an explicit expression for the Green function (~ defined by (3) in the presence of a super-self-dual Yang-Mills background for the case in which the gauge group is SU(n) and the right chiral superfields on which
98
1.N. McArthur / Se[[-dw21Yang-Mills background
~@2~)2= _c96~~ a acts transform under the fundamental representation of the gauge group. We make use of the recently developed supersymmetric version of the A D H M formalism [13]. This gives an expression for the connection I~1 (defined by 67,A= D4. + f'A, A = a, a, &) which solves the integrability condition I~,,~= 0, in terms of matrices which satisfy quadratic constraints. If c~.4 acts on superfields in the fundamental representation of an SU(n) gauge group, then the general solution to Wa = 0 is provided by
r (x, o, 0): ,.(x, o. O)+O:(x, o. 0),
(11)
where, for a background in which the bosonic component of IV has instanton number k, v(x,O,O) is a (2k + n ) × n matrix of superfields [13]. The matrix c, is defined by t,+v = 1 and v+Aa= 0, where Aa = aa + b"2R,,a + cOa and a,.,, b" and c are constant (2k + n) × k complex matrices satisfying constraints given by Semikhatov [13]. (Note: Semikhatov actually solves the integrability condition W,~= 0. Thus the matrix v used in this paper, which solves l,Va= 0, is obtained from that used by Semikhatov by replacing dotted indices eveLywhere by undotted ones and vice versa. Alternatively, we could retain the notation of [13], and the Green function constructed below would be that for the space of left chiral superfields in the background W,~= 0.) The Green function G in the fundamental representation, defined by (3), is given by
o;< o.
o o>.(x, o.
I&
-
°->2
"
The complex conjugation in vt acts only on the complex numbers in the matrix v, and not on the superspace coordinates x ~a, 0~, 0a. This ansatz was motivated by the construction of the scalar Green function in the presence of a self-dual Yang-Mills background given by Corrigan et al. [21, and it can be checked that (12) satisfies (3) by using the arguments in [2] with minor modifications (see also [14]). As mentioned in sect. 3, the Green function G must be right (covariantly) chiral in both its arguments. Thus, if (12) is to be consistent, v+ must satisfy (@~v*) = 0. We show this here. Using (11) and v%, = 1,
(%,:) = (Do.,) + (:D:),: =
.'DoP,
where P = vv t. However, from [13], P = 1 - AafA+a. Thus, using v~Aa= 0,
v*D~e
= - v*( D,~Ae') fA~ ,
which vanishes because Aa is a function of 2 R and 0 with constant matrix coefficients. So our ansatz is consistent with the chirality constraints.
LN. McArthur / Self-dual Yang-Mills background
99
The expression (12) for (~ allows the construction of G and PL (defined by (4) and (7)) for left chiral superfields in the fundamental representation of an SU(n) gauge group. In particular, the expression for PL permits us to determine explicitly the subspace of the space of left chiral superfields annihilated by ~6~2@2, in the spirit of the corresponding analysis by Osboru [15] of the space of zero modes of the left-handed Weyl operator in the presence of an instanton. This will be reported on elsewhere [16]. There is also no apparent obstruction to the construction of Green functions in other representations of the gauge group by using the analogue of the tensor product construction of Corrigan, Goddard and Templeton [2].
6. One-loop quantum corrections and the super-index theorems If a set of quantum fields which are the components of a supermultiplet interact with a self-dual Yang-Mills background, then at the one-loop level, the theory has the remarkable feature that the quantum correction to the Yang-Mills action is determined only by the zero modes of the operators which appear in the quadratic part of the action [3]. As was mentioned in the introduction, this is because the background (or "vacuum") respects a supersymmetry, so that the spectrum of nonzero eigenvalues is the same for operators associated with both bosonic and fermionic quantum fields in the quadratic part of the action, and the corresponding functional determinants cancel completely in the one-loop functional integral. This pairing of nonzero eigenvalues also gives rise to a super-index theorem [6] relating the number of zero modes of the operators associated with the components of the superfield in the instanton background. Only the zero modes survive in the difference between the suitably regulated total number of bosonic and fermionic eigenmodes, and these total numbers of eigenmodes can be expressed in terms of topological invariants of the background using the b4 coefficients. Here, we use the Green functions developed in the previous section to examine the one-loop quantum corrections to the action of a super-self-dual Yang-Mills background, and obtain superfield formulations of the appropriate super-index theorems. In ordinary field theory., the contribution by a quantum field to the one-loop effective action in the presence of a background is determined by the functional determinant of the associated Laplace-type differential operator. An alternative (but equivalent) procedure is to construct the "heat kernel" for the operator, whose functional trace determines the one-loop effective action [17] (see also [18,19] for modern treatments.) While the former approach to the evaluation of the one-loop effective action has no obvious superspace analogue, the latter does. We use it to examine the contribution from chiral superfields to the one-loop effective action in the presence of a super-self-dual background.
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I.N. McArthur / Self-dual Yang-Mills background
The kernel /((t) has an asymptotic expansion as t ~ 0" which defines the right chiral superfields s-b, [12]:
s-b,(2R,O)t ('-4)/2.
lim(x,0, t}~ x ' , 0 ,' 0 " ) T r K- ( x , O , 0 ; x , 0 , 0 , t ) = t1=0
As a consequence of eq. (16), the superfields s-b~ all vanish in a super-self-dual background. If the background is purely bosonic, then the 02 component of ~ is 2b,,(0,0)-bn(0,½) (see [121), where b,,(A,B) is the ordinary b, coefficient [6] associated with the component field of the quantum fight chiral superfield which transforms under the (A, B) representation of SO(4). Thus we find 2b,,(0,0)b,,(0, ½)= 0. For n =4, this statement combined with the fact that the operator 1L66~z°~L2 can be inverted on the full space of right chiral superfields (there are no "zero-modes"), is the (trivial) super-index theorem for the component fields of a fight chiral superfield in a self-dual Yang-Mills background [6]. The situation is slightly different on the space of left chiral superfields because, as we have already seen, the Laplace-type operator (1) annihilates part of this space in a super-self-dual background. The kernel is defined by
K(x,O,O;
X ,' O , O '
"
;t)=e-(t/16)~)2c'~2~(4}(~ L -
3~_)~(2}(0
-
0')
,
or,
K(t)
= e-('/16)#2"~Zl L ,
and is left chiral in both its arguments. The Green function (6) is related to
(17)
K(t)
by
G = fo°~dtK(t)Pi.. The one-loop correction to the super-self-dual Yang-Mills action from the subspace of left chiral scalar superfields determined by the projection operator PL is thus
(18) To compute str(K(t)PL), we use the definitions (3), (4) and (17) and the cyclic property of operators in the supertrace (this can be checked by writing out the steps with arguments and integrations included). We find: s t r ( K ( t ) PL ) = str(e (t/16)@2@2.?z.6~2"2('~6~ ~ 2)
I.N. McArthur / Self-dual Yang-MilL~background
102
= str(e- {t/16)'~z'~:l R ) = strK(t). Using (16), we have
str(K(t)PL)=O.
(19)
Thus (18) and (19) imply that the one-loop quantum correction to the super-self-dual Yang-Mills action from the subspace of left chiral superfields determined by the projection operator PL vanishes. In a purely bosonic (self-dual) background, this is the result that only zero modes contribute to the one-loop quantum correction to the action [3]. We can obtain further information about the left chiral space by examining strK(t), the supertrace of the full kernel. Using (17), (19), and c~26~2(1L - PL) = 0, we obtain s t r K ( t ) = str(K(t)(1L - P,.))
= str(e
PL))
= str(1L - P t ) .
(20)
The left chiral superfields s-b,, are defined by the following asymptotic expansion in the limit t --+ 0 '- (see [12]): lira(x, 0, 0 ---, x', 0', O')TrK(x, 0, 0; x', 0', 0'; t) = ~ s-bn(~ L, O)t <'-a)/2 . n=0
(21) Thus (20) and (21) imply
str(li - pL)= ~"
,,.-4,/2f d45ctd20s_b,,(2i.,O).
(22)
n=0
Since the left-hand side of (22) is independent of t, we find that for n 4= 4, the
I.N. McArthur / Self-dual Yang-Mills background
103
superfields s-b,, vanish in a super-self-dual background, while for 17= 4 we have str(1L -- PL) = fdaXL d20 s-b4(~I, 0).
(23)
Also, noting from (17) that l i m t ~ o K ( t ) = 1L, (19) implies str(PL) = 0.
(24)
Again, for a Purely bosonic background, eqs. (23) and (24) are statements of familiar results. Eq. (24) is the superfield version of the result that, in a self-dual background, there are equal numbers (after suitable regulation) of bosonic and fermionic nonzero modes of the Laplace-type operators associated with the component fields of a left chiral superfield. The right-hand side of eq. (23) is (see [12]) fd4x(2b4(O,O) - b4(½,0)) , while the left-hand side is 2n(0,0) - n(~, 0), where n(A, B) is the number of zero modes of the Laplace type operator associated with a component field transforming under the representation (A, B) of SO(4) (the fermionic projection operators give a ( - 1 ) when traced because we have to interchange two Fermi fields). Thus (23), combined with (24), gives the super-index theorem of Christensen and Duff [6] for the components of a left chiral superfield in the presence of self-dual Yang-Mills background. It is interesting to note that in the more general super-self-dual background, which may have nonvanishing fermionic components, str(1L - PL) is still an integer. This is because, from the results of ref. [12], s-b4 = (1/16~r2)TrW~W~, so str(l L - PL) = (]/16"lz2)fd4~i. d20Tr(W'~Wa) = (1/32~r2)fd4x(FohFab+ 4i~D~aXa) where D~a here is the ordinary Yang-Mills covariant derivative. The condition I~a = 0 causes Xa to vanish, so str(1 - PL) = u, the instanton number of the bosonic component of the background (which is self dual). So far, we have not considered the contribution from the subspace of left chiral superfields annihilated by 1~6~2@ 2 to the one-loop quantum correction to the action. A full treatment would be complicated, requiring a parameter~ation of this space by supersymmetric collective coordinates. However, we can readily see that part of the contribution from this space is a logarithmically divergent correction to the action, which is characteristic of zero modes in instanton backgrounds [3, 19]. This comes from
f -~str( K( t )(1L -- PL)), which, by (20) and (23), is just f d4.~Ld2 8 s_b4(.~L ' . , r ~ dt
° ) Jo -i- "
104
I.N. McArthur / Self-dual Yang-Mills backgrvund
A quantum vector superfield in a super-self-dual background, which was considered in sect. 4, gives rise to only a collective coordinate contribution to the one-loop effective action, from the space of superfields specified by the projection operator (1 - Pv). This is because the supertrace of the kernel associated with the operator (8) vanishes. To see this, we note that the kernel is defined by K ( x , O , O; x', Ot, O'; l) = e ` ( °'ga°~a-W°C~'~)(~(4)( X -- X ' ) a ( 2 ) ( 0 - - 0 ' ) a ( 2 ) ( 0 - - 0 ' ) .
The factor 6(2)(0- 0')= ( 0 - ~,)2 vanishes in taking the supertrace, because there are no terms involving @,~ in the operator to annihilate it. There will also be one-loop corrections to the super-self-dual action from the three chiral ghosts which result from the supersymmetric gauge fixing. 7. Conclusion
The Green functions associated with chiral scalar superfields and vector superfields coupled to a super-self-dual Yang-Mills background have been shown to be related, and to be obtainable from the Green function on the space of right chiral superfields. By fairly trivial manipulation of operators and superfield kernels, we have demonstrated a number of interesting properties of these theories. For a purely bosonic background, these properties correspond to known results for the component fields of the quantum superfields in the presence of Yang-Mills instantons. We were also able to give an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group by using the supersymmetric version of the ADHM formalism. A similar analysis can be carried out for superfields with spinor indices and coupled to a curved superspace background, and will be presented elsewhere [16]. The corresponding super-self-duality constraint is that ff'~B~= 0, where W~i~ is the superfield containing the anti-self-dual part of the Weyl curvature tensor. If the background is also on-shell, we obtain a superfield formulation of the super-index theorems of Christensen and Duff [6], and of the results of Hawking and Pope [4] on mode cancellations in the presence of self-dual gravitational instantons. The analysis is not as straightforward as that in this paper, because in general the Laplace-t2ype operator on the space of right chiral superfields annihilates part of the space. I wish to thank Dr. R. Brandenberger for a discussion on Osterwalder-Schrader positivity. References [1] G. 't Hooft, Phys. Rev. D14 (1976) 3432; L.S. Brown, R.D. Carlitz, D.B. Crcamcr and C. Lee, Phys. Lctt. 71B (19771) 103: Phys. Rcv. D17 (1978) 1583
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[2] E.F. Corrigan, D.B. Fairlie, S. Templeton and P. Goddard, Nucl. Phys. B140 (1978) 31; E. Corrigan, P. Goddard and S. Templeton, Nucl. Phys. B151 (1979) 93 [3] A. D'Adda and P. Di Vecchia, Phys. Lett. 73B (1978) 162 [4] S.W. ttawking and C.N. Pope, Nucl. Phys. B146 (1978) 381 [5] D.N. Page, Phys. Lett. 85B (1979) 369 [6] S.M. Christensen and M.J. Duff, Nucl. Phys. B154 (1979) 301 [7] B. Zumino, Phys. Lett. 69B (1977) 369 [8] H. Nicolai, Nucl. Phys. B140 (1978) 294; Czech. J. Phys. B29 (1979) 308 [9] K. Osternvalder and R. Schrader, Helv. Phys. Acta 46 (1973) 277 [10] J. Lukierski and A. Nowicki, Trieste preprint SISA 34/82/EP [11] M.T. Grisaru, M. Ro~ek and W. Siegel, Nucl. Phys. B159 (1979) 429 [12] I.N. McArthur, Phys. Lett. 128B (1983) 194; Harvard University preprint HUTP-83/A052, to be published in Classical and quantum gravity [13] A.M. Semikhatov, Phys. Lett. 120B (1983) 171; I.V. Volovich, Phys. I_ett. 123B (1983) 329 [14] E. Corrigan, Phys. Reports 49 (1979) 95 [15] H. Osborn, Nucl. Phys. B140 (1978) 45 [16] I.N. McArthur, in preparation [17] J. Schwinger, Phys. Rev. 82 (1951) 664 [18] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133 [19] M.J. Duff and D.J. Toms, Proc. second quantum gravity seminar, Moscow, ed. M.A. Markov (1981) [20] G. 't Hooft, Nucl. Phys. B62 (1973) 444