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Unified geometry of antisymmetric tensor gauge fields and gravity
Nuclear Physics B 199 (1982) 482-494 © North-Holland Publishing Company
UNIFIED GEOMETRY OF ANTISYMMETRIC TENSOR GAUGE FIELDS AND GRAVITY* Peter G.O. FREUND and Rafael I. NEPOMECHIEI The Enrico Fermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
Received 29 December 1981
Vector gauge fields are known to be related to connections on principal fibre bundles P(M, G) over space-time M. Here we relate abelian antisymmetric tensor gauge fields of rank r to constrained connections on P(flr_IM, U(1)), the U(I) bundles over the space of all ( r - 1 ) dimensional closed submanifolds of M. In particular, the Kalb-Ramond field ( r = 2) is seen as such a connection on the bundle over the space of all loops in space-time. Moreover, a Kaluza-Jordan construction on P(f~)M,U(I)) leads to the bosonic sector of the N = 1 supergravity action in ten dimensions. This result is highly reminiscent of the a' --, 0 limit of the closed string dual model. 1. Introduction
Antisymmetric tensor gauge fields [1] figure prominently i n theories of strings [2] and also in the formulation of N = 1 supergravity theories in ten and eleven dimensions [3]. However, unlike their vector predecessors, the antisymmetric tensor fields are not very well understood either at the q u a n t u m or classical levels: only recently has their ghost structure been elucidated [4], and their geometric role is not known. Sect. 2 of this paper addresses the latter point. There we argue that an abelian antisymmetric tensor gauge field of rank r is a constrained connection on a U(1) bundle over f~r-1 M, the space of all ( r - 1)-dimensional closed submanifolds of space-time M. In particular, the K a l b - R a m o n d field ( r = 2) [2] becomes such a connection on the bundle over the space of all loops in space-time. In sect. 3 we find the geometric origin of the lagrangians of antisymmetric tensor fields in interaction with gravity and a scalar field, as encountered in supergravity. Thus the U(1) bundle over loop space f~lM is endowed with a Kaluza-Jordan-type metric [5], and a suitably regularized scalar curvature of the bundle is used as a lagrangian. This, when integrated over the portion of the bundle that lies over the point-like loops in f~lM (its base space), reproduces the Bose part of the N - - 1 supergravity action in ten dimensions. Since the same action can also be obtained in * Work supported in part by the NSF: Grant no. PHY-78-23669. i National Science Foundation Predoctoral Fellow. 482
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the a'--, 0 limit of the closed string dual model [6], the geometric framework emphasizing loops developed in this paper may be relevant for string theory. For rank-two antisymmetric tensor fields, a geometry with torsion has been proposed by Scherk and Schwarz [6]. Unlike the geometries considered here, their construction does not generalize to higher rank fields. Thus, in particular, the rank-three field that appears in eleven-dimensional supergravity [3] is not covered by their approach. Questions for future investigations are raised in the final section.
2. Geometry of higher rank antisymmetric tensor gauge fields Ordinary vector gauge fields, the "antisymmetric" tensor fields of rank 1, are related to connections on principal fibre bundles P(M,G) over the space-time manifold M, with structure group G [7]. The pull-back A of the connection one-form o~ on P by the section o:M + P defines the gauge potentials o*to ==-ia = iA~,( x ) d x ~'.
(2.1)
Under a change of section o ( x ) ~ o ' ( x ) = o ( x ) b ( x ) , b ( x ) E G the form A transforms as A' = bAb - l - ibdb -1 ,
(2.2)
which gives the usual gauge transformation law for vector potentials. Can this picture be generalized to antisymmetric tensor gauge fields of higher rank r > 1? We shall provide an affirmative answer to this question and identify the bundles on which these higher rank tensor fields determine a connection. Take the next simplest case r = 2 of the abelian Kalb-Ramond field B~. It is clear that as fibre we are to choose U(1). But unlike the previous case the space-time manifold itself is not a suitable base manifold, for then the connection would again lead to an ordinary vector gauge field. In the case of the vector gauge field the connection associates to any path F between two space-time points x~ and x 2 an element of the structure-group U(1), namely the phase factor e x p ( i f r A z d x ~ ). (See fig. la.) The natural candidate for a phase factor in the Kalb-Ramond theory is e x p ( i f 2 B ~ do~'~), where Y~is any 2-submanifold of the space-time M bounded by the closed curves C I and C2 in M. (See fig. lb.) This suggests that we consider the connection on P(flM, U(1)), a U(1) bundle over the space f~M of loops of the original space-time manifold M. Before discussing the bundle P(fiM, U(1)) we briefly describe the geometry of its base space, the infinite-dimensional loop space tiM. A tangent to tiM at the loop C is [8, 9] a vector field along C (i.e., a set of tangents to M, one at each point of C with the usual differentiability properties). These tangents allow us to construct a
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Fig. 1o
~
2
7. Fig. 1. Geometry of phase factor for (a) ordinary vector gauge fields, (b) Kalb-Ramond fields.
neighboring loop C' from the original loop C. (See fig. 2.) The tangent space t M c of tiM at the loop C E t M is the infinite-dimensional vector space of all vector fields along C. A basis for t M c is (8/6C~(s)}, s ~ [0, 1], with C(s) a parametrization of the loop C. This basis is infinite dimensional as evidenced by the continuous label s. A dual basis (SC~(s)},s E [0, 1] of the space * t M c of 1-forms at C is then defined by
~C~(s)( ~C~(t) )=~(s-t).
(2.3)
The Riemann metric g~ on the space-time manifold M induces a natural metric [8] on tiM. Specifically, for v, w E t M c (i.e., V and w vector fields along the loop C),
=foldt foldsv~(C(s))w~(C(t))g~(C(s))8(s-t).
(2.4)
The natural metric tensor '(~s~t on tiM is thus given by
y~,s~t(C ) = g~( C( s ) )8( s - t ) .
(2.5)
Return now to the bundle P(flM, U(1)). The pull-back 0*60 to M of the connection -
/
C'
A
Fig. 2. A tangent vector to f~M at the loop C E tiM is a vector field on C, that leads to a neighboring loop C'.
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1-form 0~ on P, corresponding to the section a: [~M --, P is
o%: = iB( C ) = ifo' ds B~(s, C) 8C'( s ) .
(2.6)
Under the change of section o ' = o - b with b corresponding to the fibre dependent U(1) element e iA(c), the one-form B transforms as
B'(C) = B(C) - ~A(C).
(2.7)
Consider now a path f" in loop space tiM. Let it be parametrized: t E [0, 1] ~ I't tiM, where f't is the loop Ct~(s) E f~M. Just as for the U(1) bundle in ordinary gauge theory, we associate to this path a phase factor exp(if~B) with
f~B= fol dt fo l dsBl*(s' Ct) aCt~at
(2.8)
Without further specifications the non-local field B,(s, C) cannot as yet be identified with the local Kalb-Ramond field B,~(x). For this reason we impose the constraint B~(s,C = B ~ ( C ( s ) )
dC ~' ds '
(2.9)
where B~,(x)= -B,~(x) is a local field of the space-time point x. This constraint also appears in the work of Marshall and Ramond [2]. The phase (2.8) can now be written as
foldt foldSB.~(Ct(s)) OCt(s) OCt"(s)_½ foldt fo, dSBt..(Ct(s) ) O(C~(s),Ct"(s)) Os
Ot
O(s,t)
=fB..(x)do' '~,
(2.10)
the last integral taken along the 2-manifold Y~generated by the loop I't, t E [0, 1]. In passing, we observe that infinitely many paths in ~ M correspond to the same 2-manifold X in M. To maintain the constraint (2.9), we next restrict A in the gauge transformation ~2.7) to the form
a ( c ) =fo' dth.(c(t)) dC.
dt "
(2.11)
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The gauge transformation law for the form B now becomes
, dC I' , B'(C) = f dsB;,(C(s))--a-;-SC O)
=f
dC~
v
B,,( C(s))--ffF C O)
-Sdssc~[SdtA~,(C(t))-~]SC'(s) = f ds
dC ~ f d,[,O-t)O.A.(C(,))
+ A,(C(t))ds(s-t)]}SC~(s) =f
ds[Bs,,(C(s)) + ~,A,(C(s))-O,A~,(C(s))]-d-~sC'(s).
(2.12)
Writing this law for all loops C, keeping in mind the local nature of the field B~(x), yields B£,(x) = B~,,(x) + 3 z A , ( x ) - 0 , A z ( x ) ,
(2.13)
the usual gauge transformation [1,2] of the abelian Kalb-Ramond field. These considerations are now readily extended to abelian antisymmetric tensors of all ranks. For rank r one need but replace the loop space a M by the space a t - 1 of all dosed connected ( r - 1) submanifolds* of M. (In particular, filM-= f~M.) On f~r_lM erect a U(1) bundle, and on the connection on this bundle impose the obvious generalization of the constraint (2.9): ..... Dr,-,) Bp(SI,...,S#.__I'~D)~-BIII...llr_IIp(D(s I..... St._l)) a - ~ 1 2 ; ? , - ~ _ ~
with
, (2.14)
D~(s~.....s,_0 a parametrizationof the (r- I) submanifold D. Note that for
r = 1 the ordinary vector gauge field case is also covered, since ~o M = M . This concludes our presentation of the general geometric setup. In the next section we discuss a general dynamical principle of geometric origin for these theories.
3. A geometric action principle Having identified the bundles on which the antisymmetric gauge fields serve as connection coefficients, it is natural to ask whether the standard action principles for *Since we deal only with parametrizedsubmanifolds,we can picture f~r- ~Mas the space of all suitably smooth maps from the ( r - 1) sphere s r-1 to space-timeM.
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these fields can be given a simple geometrical meaning. Specifically, we propose to use the natural metric (2.5) on M and combine it with a metric on the fibre to endow P(f~M,U(1)) with a Kaluza-Jordan-type metric [5] and then, roughly speaking, to use an integral of the scalar curvature of the bundle P as action. More precisely, divergences arise in the calculation of the scalar curvature of the infinite-dimensional bundle-space. It is only after coping with these divergences that a meaningful action for a local field theory in ordinary space-time emerges. To start, we assign to the bundle P(flM, U(1)) a metric with components (in the basis {8/8C~(s), O0}, 0 being the coordinate along the fibre)
"~ab(C)-~"( 8(s-t)gt'"(C(s))+e2x2dp2(C)B~'(s'C)B~(t'C)e~cd?2(C)B~(t C) ,
elcdp2(C)B~'(s'C) (3.1)
where es is a coupling constant scaling B~(s,C), and qffC) is a scalar functional counterpart to the Jordan-Brans-Dicke field which is independent of the fibre coordinate* 0. The calculation of the Riemann curvature tensor of the bundle is straightforward. To get from there to a scalar curvature, two index contractions must be performed, tiM being infinite dimensional, this leads a priori to a divergent result. To understand the origin and cure of this divergence, first approximate tiM by the finite-dimensional space of ordered N-tuples of points in M. (This corresponds to approximating each loop in OM by an N-polygon inscribed in it.) There is then a factor N for each index contraction, and thus a factor N 2 in the scalar curvature. In a calculation involving tiM this is reflected by the appearance of a factor 82(0) in the expression for the scalar curvature. To extract a meaningful answer we are therefore not to look at the scalar curvature itself, but at the scalar curvature divided by N 2 in the limit N--, ~ . This leads to the prescription that we perform our formal calculation and in the end, drop an overallfactor 82(0). For the scalar curvature we find
g ( c ) = fo' -- }e2K2q~2(C)fo' ds f01 dt g~'~'( C(s ))g""'(C( t ))F~s., (C)F~,s~,,(C)
+2~-'(C) fo'dsg~"(C(s)) 8 ( 0 ) ~
r ..I ,c,I.s)I--8C.(s)SC~(s) "" , (3.2a)
not
* 0 is the coordinate of a dimension "added" to the already infinite-dimensional loop space, to space-time, and as such the usual arguments about its space-like nature do not apply. The positive (" time-like"7) signature is chosen here so as to yield the proper sign of the F 2 term in the action, eq.
(38).
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488 with
8
F~s~t(C) =SC~(s) B,(t, C)
8
8C'(t) B~(s, C).
(3.2b)
Although the 82(0) factor only appears in the first term, it would be a mistake to throw away the remaining terms at this stage. First of all, they contain functional derivatives that can lead to 8(0) factors; and second, as we shall see, a functional averaging will still have to be performed to get a meaningful action, and this will lead to yet a further 8(0) factor. It is only after all these factors have been tracked down that in the final expression we are to cancel an overall 82(0) factor in the sense of the N--, oo limit discussed above. We now make use of our expression (2.9) for B~ to evaluate the second term of eq. (3.2). Keeping in mind the simple identities (H(x), G(x) are arbitrary functions)
8C~(s) H(C(t)) = O~H(C(t))8(t - s ) , dsG(C(s))H(C(t))
8 ( t - s ) = - -~G(C(t))H(C(t)),
we are led to /e2lc2t~2(c)f01 ds f0 ! dtg~'(C(s))g~v'(C(t))F~s~t(C)F~,,s~,t(C) = ¼e2K2q~2(C)fol ds
8(O)g~'(C(s))g~"'(C(s))
dC 8 d C ~'
× d~- d~-(a,S'~(C(s))+O'B~,(C(sl))r~ '''~'(c('))'
(3.2c)
with -- a , B . , ( x ) + 0 . B , , ( x ) +
(3.2d)
Next we consider the geometrical variational principle. As a lagrangian, we choose the scalar curvature of the bundle P(~M,U(1)) ~t la Kaluza-Jordan. Were we to integrate this lagrangian over the whole bundle space, regions lying over large loops in [~M would be included and a non-local theory would emerge. Being interested in a local field theory, we note that each point of the space-time manifold M is itself also a loop (of zero length). As such, a copy of M sits* inside ~M. Call PM the portion of the bundle P(f~M, U(1)) that lies over this copy of M. We want to restrict *f~M is not in general connected (depending on the homology of the manifold M). For our considerationsthe connectedcomponentthat contains this copy is all that matters.
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to PM the domain of the integration that leads from the lagrangian to the action I. Thus I=-
16~r-----G ~4
dOx~lglq~2/¢"
(3.3)
The integration here is over the (D + 1)-dimensional space PM, and the expression under the square root is the determinant of the metric of this space. Now the crucial factor R should be a local expression on M. Indeed the scalar field ~(C) becomes a local field on M: ~(C)[c_~x ~ qffx), so that, as can be seen from (3.2a-c), only local fields appear in the action (3.3). Note that for C --, x E M the right-hand side of eq. (3.2c) becomes
l eZr28(O),2(x)g"(x)g'~'(x)(O~B,n(x) + O~Bs,(x))F,,,,n,(x)S88"(C),
(3.2e)
where
Snn'(C) =-foI ds Cs(s)CS'(s).
(3.2f)
Since for point-like loops C " = 0, one would expect S 88' to vanish for such loops. Yet a more careful limit is to be taken here on account of the overall e2x28(O) factor multiplying the expression (3.2e). We therefore consider S 88' on a neighborhood of the point-like loop x E t]M consisting of small loops "close" to x. Taking
L:(c)
(3.4)
withp > 0 as a measure of the size ( ~ length squared) of the loop C*, we now define SS~'(x) at the point-like loop x as its average value over all loops of size smaller than ~2 (later we will let ;~ ~ 0):
s~'(x)-[J L 2 ( C ) ~ 2®~C['dsC~(s)CS'(s)/f~ ®xC. dO L (C)~,
(3.5)
Here f6DxC means integration over all small loops "near" x E f~M with a suitable measure. SS~'(x) must be a symmetric tensor at x, and has dimensions of length squared. The size )d of the loops being the only dimensional parameter at our disposal, we expect terms like ~2f(~)g,~#, h4g(ep)R~a,.., to appearin S~#(x). (Here * For simplicity, we have chosen the factor (q~(C)) v in eq. (3.4), although in principle, more general functions of q~ could be used.
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f(q0, g(~) are. some functions of ~.) Thus to lowest order in h we expect S ~ h2f(~)g ~. Indeed, in the appendix we give a formal derivation of the relation Sa#( x ) = ~(O)¼~2~--P( x )gaS( X ) .
(3.6)
We note the appearance of yet another 8(0) factor in the F 2 term (3.2e) of the action, which allows this term to survive. The ~2 factor in eq. (3.6) when brought into eq. (3.2e), requires that the ~ --, 0 limit be accompanied by ex -, ~ so that e21£2~k2
64rr----G- 1.
(3.7)
Putting all this together, the action becomes
The last term in the action is a total divergence. Moreover, we now drop the overall 82(0) factor in accordance with our previous discussion*. Finally, in order to cast the action in a more conventional form, we perform a Weyl rescaling gl~v( X ) ~ gav( X ) ~ ( X ) -2/(0-2),
(3.9a)
followed by the field redefinition
giving
,=_fd xlv (
R
16~rG
½g~0~qoO,~p 6" 8
~s 8
/
-~,8-~,~,8,j,
(3.10a)
with
A=
8¢rG (2D - p D + 2 p ) . (D- 1)(D-2)
(3.10b)
* Alternatively, the metric g ~ can be rescaled in such a way that the factor 82(0) disappear from the first term in eq. (3.8). This correlates the h --, 0 and N --, 8(0) = oo limits and replaces eq. (3.7) by e 2x 2h2(8 (0)) 8/( D- 2) = 64 ~rG.
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For D = 10 a n d p = 1 orp = 4 this action is precisely that of the bosonic sector of N = 1 supergravity in 10 dimensions. It may be worthwhile to recapitulate here the steps that have led to the action (3.10). We defined a U(1) bundle P(flM, U(1)) over the loop space tiM of the space-time manifold M. Subject to the constraint (2.9) the antisymmetric tensor field was found to provide a connection on this bundle. Natural metrics on f~M and U(1) along with the connection on P allowed us to endow the bundle P with a KaluzaJordan-like metric. With a suitable limiting procedure, a finite scalar curvature for P was extracted. This was then used as a lagrangian integrated over the portion Pra of P lying over the copy of space-time M contained in tiM. Certain singular tensors were smoothed out via an averaging over small loops. All these steps suggest that the infinite-dimensional spaces f~M and P were used merely as "geometric crutches." The local field theory is still supported by the finite dimensional space-time M. Yet, it is the surrounding infinite-dimensional manifold f~M that allowed us to use the field B~ in defining a connection. More generally, viewing the points of M as closed submanifolds of dimension 2, 3.... we can surround M by any of the infinite-dimensional spaces f~2M, ~3M .... and treat higher order antisymmetric fields as connections in appropriate bundles over these spaces. In this way geometric action principles for such fields can be obtained, as above.
4. Conclusions and prospects We have provided antisymmetric tensor gauge fields with a geometric basis using the loop spaces f~,M of closed r-submanifolds of the space-time M. To end up with local theories we had to back-track from f~rM to the copy of M that sits inside f~,M by virtue of the fact that the points of M themselves can be thought of as closed r-submanifolds in M of vanishing size. In the case of loop space, this amounts to taking the small-loop limit. This is highly reminiscent of the a' ~ 0 limit in which the dual model of closed bosonic strings also yields the action (3.10) [6]. Could it be that the non-local theory we would encounter by considering the geometrical action on the whole bundle P(flM, U(1)) (rather than just on the restriction PM as we have done in the local case) is the theory of closed strings? Similar questions can be asked for theories built over f~rM with r > 1. Our considerations involved the space of parametrized loops [e.g. the metric (2.4), (2.5) is not repararnetrization invariant]. In the limit 2~~ 0 of small loops this is fully acceptable, but for finite loops (strings) reparametrization invariance is of essence. We would therefore have to make appropriate modifications. Thus the loop-space metric (2.5) would have to be changed either by providing an extra factor that restores reparametrization invariance, or more generally, by allowing this metric to be unconstrained and dynamically determined. The small-loop limit extracted with
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the averaging procedure (3.5), would still lead to an action of the form (3.8), as all extra pieces involve higher powers of the loop size ~. The constraint (2.9) on the loop-space connection figures prominently in our construction. Does this condition have a simple geometric origin? As structure group we chose U(1). Is there a justification for this? Indeed, a natural U(1) bundle lives over tiM. If tiM is the space of loops without a base point, then the loops with all possible choice of base point provides a U(1) bundle over tiM. The additional question as to the meaning and relevance of non-abelian structure groups in bundles over tiM deserves further study. (Such bundles have recently appeared in the literature, but in a rather different context [10].) Our construction provided the bosonic sector of N = 1 supergravity in ten dimensions. Could one reconsider the problem in a supersymmetric context (loops in superspace?) and achieve a geometrization of the complete ten dimensional N = 1 supergravity? Of course the question can be asked for eleven dimensions where t 2 M should enter. We wish to thank Professor I.M. Singer for a valuable discussion, and Professor S.S. Chem for pointing out to us ref. [8]. Discussions with Professor R. Lashoff, M. Rubin and S. Ramaswamy are gratefully acknowledged.
Appendix Here we offer some formal justification for eq. (3.6). First we observe that for H(C) an (arbitrary) functional of the loop C, and for h small
=
f
,®C fo~2d r
- f ®cf% = - 2f®Ce-
f oo _ _dr¢ ~ , ( L• "--oo
2--T)rj_i{r" ~
tzvr)
x2(1 + ½iX2r)-'e"'m C)
2L2<¢~/zI-I(C
),
(A. 1)
where in the last step we have used formula (4), p. 118 of ref. [11]. Thus, using the
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measure (3.4) and eq. (A.1), the quantity
r"#(x)=-fdL2(C)~Tt2®~cf'v0 ds d"(s)d/~(s)
82 =f0 1 ds,/o l ds2 8(s I --$2) 8ja(Sl$-gjfl(S2) Z[
J] s=o'
(A.2)
where we have defined
(A.3) Since the integral is over only loops near the point x. we set g,,(C(s))~g,,(x) and q f f C ) ~ ( x ) as in sect. 3. and the functional integration is then readily performed with the result
Z[ J ] = Z[O]exp[ ~t2~-p( x )g~P( X ) foi dS J~( s )Jp( s ) l.
(A.4)
Hence 82
8Sa(Sl)SJp(s2)
Z[J][
:=o
=Z[O]¼h2e~-P(x)8(s,-s2)g:#(x).
(A.5)
Finally, from (A.2) we have that
T=/J(x ) = Z[O]¼h28(O)ep-P( x )g=tt( x ) .
(A.6)
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The denominator in eq. (3.5) being precisely Z[0], a combination of the eqs. (3.5) and (A.6) yields the result (3.6). References [1] N. Kemmer, Proc. Roy. Soc. A166 (1938) 1127; V. Ogievetsky and I. Polubarinov, Soy. J. Nucl. Phys. 4 (1967) 156; D.Z. Freedman and P.K. Townsend, Nucl. Phys. B177 (1981) 282 [2] M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273; E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117; Y. Nambu, Phys. Reports 23 (1976) 250; C. Marshall and P. Ramond, Nucl. Phys. B85 (1975) 375 [3] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253; E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409; A. Chamseddine, Nucl. Phys. B185 (1981) 403 [4] W. Siegel, Phys, Lett 93B (1980) 170 J. Thierry-Mieg, Harvard University preprint (1980); E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D22 (1980) 301; T. Curtright, private communication (1980) [5] T. Kaluza, Sitzber, Preuss. Akad. Wiss. (1921) 966; P. Jordan, Z. Phys. 157 (1959) 112; C. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 925 [6] J. Scherk and LH. Schwarz, Phys. Lett. 52B (1974) 347 [7] Y.-M. Cho, J. Math. Phys. 16 (1975) 2029 [8] W. Klingenberg, Lectures on closed geodesics (Springer-Verlag, New York, 1978), ch. 1 [9] J. Milnor, Morse theory (Princeton University Press, New Jersey, 1963), sects. 10, 11 [10] A.M. Polyakov, Phys. Lett. 82B (1979) 247; I. Ya. Aref'eva, Lett. Math. Phys. 3 (1979) 241 [11] A. Erd~lyi et al., Tables of integral transforms, vol. I., Bateman Manuscript Project (McGraw-HiU, New York, 1954)
UNIFIED GEOMETRY OF ANTISYMMETRIC TENSOR GAUGE FIELDS AND GRAVITY* Peter G.O. FREUND and Rafael I. NEPOMECHIEI The Enrico Fermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
Received 29 December 1981
Vector gauge fields are known to be related to connections on principal fibre bundles P(M, G) over space-time M. Here we relate abelian antisymmetric tensor gauge fields of rank r to constrained connections on P(flr_IM, U(1)), the U(I) bundles over the space of all ( r - 1 ) dimensional closed submanifolds of M. In particular, the Kalb-Ramond field ( r = 2) is seen as such a connection on the bundle over the space of all loops in space-time. Moreover, a Kaluza-Jordan construction on P(f~)M,U(I)) leads to the bosonic sector of the N = 1 supergravity action in ten dimensions. This result is highly reminiscent of the a' --, 0 limit of the closed string dual model. 1. Introduction
Antisymmetric tensor gauge fields [1] figure prominently i n theories of strings [2] and also in the formulation of N = 1 supergravity theories in ten and eleven dimensions [3]. However, unlike their vector predecessors, the antisymmetric tensor fields are not very well understood either at the q u a n t u m or classical levels: only recently has their ghost structure been elucidated [4], and their geometric role is not known. Sect. 2 of this paper addresses the latter point. There we argue that an abelian antisymmetric tensor gauge field of rank r is a constrained connection on a U(1) bundle over f~r-1 M, the space of all ( r - 1)-dimensional closed submanifolds of space-time M. In particular, the K a l b - R a m o n d field ( r = 2) [2] becomes such a connection on the bundle over the space of all loops in space-time. In sect. 3 we find the geometric origin of the lagrangians of antisymmetric tensor fields in interaction with gravity and a scalar field, as encountered in supergravity. Thus the U(1) bundle over loop space f~lM is endowed with a Kaluza-Jordan-type metric [5], and a suitably regularized scalar curvature of the bundle is used as a lagrangian. This, when integrated over the portion of the bundle that lies over the point-like loops in f~lM (its base space), reproduces the Bose part of the N - - 1 supergravity action in ten dimensions. Since the same action can also be obtained in * Work supported in part by the NSF: Grant no. PHY-78-23669. i National Science Foundation Predoctoral Fellow. 482
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the a'--, 0 limit of the closed string dual model [6], the geometric framework emphasizing loops developed in this paper may be relevant for string theory. For rank-two antisymmetric tensor fields, a geometry with torsion has been proposed by Scherk and Schwarz [6]. Unlike the geometries considered here, their construction does not generalize to higher rank fields. Thus, in particular, the rank-three field that appears in eleven-dimensional supergravity [3] is not covered by their approach. Questions for future investigations are raised in the final section.
2. Geometry of higher rank antisymmetric tensor gauge fields Ordinary vector gauge fields, the "antisymmetric" tensor fields of rank 1, are related to connections on principal fibre bundles P(M,G) over the space-time manifold M, with structure group G [7]. The pull-back A of the connection one-form o~ on P by the section o:M + P defines the gauge potentials o*to ==-ia = iA~,( x ) d x ~'.
(2.1)
Under a change of section o ( x ) ~ o ' ( x ) = o ( x ) b ( x ) , b ( x ) E G the form A transforms as A' = bAb - l - ibdb -1 ,
(2.2)
which gives the usual gauge transformation law for vector potentials. Can this picture be generalized to antisymmetric tensor gauge fields of higher rank r > 1? We shall provide an affirmative answer to this question and identify the bundles on which these higher rank tensor fields determine a connection. Take the next simplest case r = 2 of the abelian Kalb-Ramond field B~. It is clear that as fibre we are to choose U(1). But unlike the previous case the space-time manifold itself is not a suitable base manifold, for then the connection would again lead to an ordinary vector gauge field. In the case of the vector gauge field the connection associates to any path F between two space-time points x~ and x 2 an element of the structure-group U(1), namely the phase factor e x p ( i f r A z d x ~ ). (See fig. la.) The natural candidate for a phase factor in the Kalb-Ramond theory is e x p ( i f 2 B ~ do~'~), where Y~is any 2-submanifold of the space-time M bounded by the closed curves C I and C2 in M. (See fig. lb.) This suggests that we consider the connection on P(flM, U(1)), a U(1) bundle over the space f~M of loops of the original space-time manifold M. Before discussing the bundle P(fiM, U(1)) we briefly describe the geometry of its base space, the infinite-dimensional loop space tiM. A tangent to tiM at the loop C is [8, 9] a vector field along C (i.e., a set of tangents to M, one at each point of C with the usual differentiability properties). These tangents allow us to construct a
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Fig. 1o
~
2
7. Fig. 1. Geometry of phase factor for (a) ordinary vector gauge fields, (b) Kalb-Ramond fields.
neighboring loop C' from the original loop C. (See fig. 2.) The tangent space t M c of tiM at the loop C E t M is the infinite-dimensional vector space of all vector fields along C. A basis for t M c is (8/6C~(s)}, s ~ [0, 1], with C(s) a parametrization of the loop C. This basis is infinite dimensional as evidenced by the continuous label s. A dual basis (SC~(s)},s E [0, 1] of the space * t M c of 1-forms at C is then defined by
~C~(s)( ~C~(t) )=~(s-t).
(2.3)
The Riemann metric g~ on the space-time manifold M induces a natural metric [8] on tiM. Specifically, for v, w E t M c (i.e., V and w vector fields along the loop C),
=foldt foldsv~(C(s))w~(C(t))g~(C(s))8(s-t).
(2.4)
The natural metric tensor '(~s~t on tiM is thus given by
y~,s~t(C ) = g~( C( s ) )8( s - t ) .
(2.5)
Return now to the bundle P(flM, U(1)). The pull-back 0*60 to M of the connection -
/
C'
A
Fig. 2. A tangent vector to f~M at the loop C E tiM is a vector field on C, that leads to a neighboring loop C'.
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1-form 0~ on P, corresponding to the section a: [~M --, P is
o%: = iB( C ) = ifo' ds B~(s, C) 8C'( s ) .
(2.6)
Under the change of section o ' = o - b with b corresponding to the fibre dependent U(1) element e iA(c), the one-form B transforms as
B'(C) = B(C) - ~A(C).
(2.7)
Consider now a path f" in loop space tiM. Let it be parametrized: t E [0, 1] ~ I't tiM, where f't is the loop Ct~(s) E f~M. Just as for the U(1) bundle in ordinary gauge theory, we associate to this path a phase factor exp(if~B) with
f~B= fol dt fo l dsBl*(s' Ct) aCt~at
(2.8)
Without further specifications the non-local field B,(s, C) cannot as yet be identified with the local Kalb-Ramond field B,~(x). For this reason we impose the constraint B~(s,C = B ~ ( C ( s ) )
dC ~' ds '
(2.9)
where B~,(x)= -B,~(x) is a local field of the space-time point x. This constraint also appears in the work of Marshall and Ramond [2]. The phase (2.8) can now be written as
foldt foldSB.~(Ct(s)) OCt(s) OCt"(s)_½ foldt fo, dSBt..(Ct(s) ) O(C~(s),Ct"(s)) Os
Ot
O(s,t)
=fB..(x)do' '~,
(2.10)
the last integral taken along the 2-manifold Y~generated by the loop I't, t E [0, 1]. In passing, we observe that infinitely many paths in ~ M correspond to the same 2-manifold X in M. To maintain the constraint (2.9), we next restrict A in the gauge transformation ~2.7) to the form
a ( c ) =fo' dth.(c(t)) dC.
dt "
(2.11)
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The gauge transformation law for the form B now becomes
, dC I' , B'(C) = f dsB;,(C(s))--a-;-SC O)
=f
dC~
v
B,,( C(s))--ffF C O)
-Sdssc~[SdtA~,(C(t))-~]SC'(s) = f ds
dC ~ f d,[,O-t)O.A.(C(,))
+ A,(C(t))ds(s-t)]}SC~(s) =f
ds[Bs,,(C(s)) + ~,A,(C(s))-O,A~,(C(s))]-d-~sC'(s).
(2.12)
Writing this law for all loops C, keeping in mind the local nature of the field B~(x), yields B£,(x) = B~,,(x) + 3 z A , ( x ) - 0 , A z ( x ) ,
(2.13)
the usual gauge transformation [1,2] of the abelian Kalb-Ramond field. These considerations are now readily extended to abelian antisymmetric tensors of all ranks. For rank r one need but replace the loop space a M by the space a t - 1 of all dosed connected ( r - 1) submanifolds* of M. (In particular, filM-= f~M.) On f~r_lM erect a U(1) bundle, and on the connection on this bundle impose the obvious generalization of the constraint (2.9): ..... Dr,-,) Bp(SI,...,S#.__I'~D)~-BIII...llr_IIp(D(s I..... St._l)) a - ~ 1 2 ; ? , - ~ _ ~
with
, (2.14)
D~(s~.....s,_0 a parametrizationof the (r- I) submanifold D. Note that for
r = 1 the ordinary vector gauge field case is also covered, since ~o M = M . This concludes our presentation of the general geometric setup. In the next section we discuss a general dynamical principle of geometric origin for these theories.
3. A geometric action principle Having identified the bundles on which the antisymmetric gauge fields serve as connection coefficients, it is natural to ask whether the standard action principles for *Since we deal only with parametrizedsubmanifolds,we can picture f~r- ~Mas the space of all suitably smooth maps from the ( r - 1) sphere s r-1 to space-timeM.
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these fields can be given a simple geometrical meaning. Specifically, we propose to use the natural metric (2.5) on M and combine it with a metric on the fibre to endow P(f~M,U(1)) with a Kaluza-Jordan-type metric [5] and then, roughly speaking, to use an integral of the scalar curvature of the bundle P as action. More precisely, divergences arise in the calculation of the scalar curvature of the infinite-dimensional bundle-space. It is only after coping with these divergences that a meaningful action for a local field theory in ordinary space-time emerges. To start, we assign to the bundle P(flM, U(1)) a metric with components (in the basis {8/8C~(s), O0}, 0 being the coordinate along the fibre)
"~ab(C)-~"( 8(s-t)gt'"(C(s))+e2x2dp2(C)B~'(s'C)B~(t'C)e~cd?2(C)B~(t C) ,
elcdp2(C)B~'(s'C) (3.1)
where es is a coupling constant scaling B~(s,C), and qffC) is a scalar functional counterpart to the Jordan-Brans-Dicke field which is independent of the fibre coordinate* 0. The calculation of the Riemann curvature tensor of the bundle is straightforward. To get from there to a scalar curvature, two index contractions must be performed, tiM being infinite dimensional, this leads a priori to a divergent result. To understand the origin and cure of this divergence, first approximate tiM by the finite-dimensional space of ordered N-tuples of points in M. (This corresponds to approximating each loop in OM by an N-polygon inscribed in it.) There is then a factor N for each index contraction, and thus a factor N 2 in the scalar curvature. In a calculation involving tiM this is reflected by the appearance of a factor 82(0) in the expression for the scalar curvature. To extract a meaningful answer we are therefore not to look at the scalar curvature itself, but at the scalar curvature divided by N 2 in the limit N--, ~ . This leads to the prescription that we perform our formal calculation and in the end, drop an overallfactor 82(0). For the scalar curvature we find
g ( c ) = fo' -- }e2K2q~2(C)fo' ds f01 dt g~'~'( C(s ))g""'(C( t ))F~s., (C)F~,s~,,(C)
+2~-'(C) fo'dsg~"(C(s)) 8 ( 0 ) ~
r ..I ,c,I.s)I--8C.(s)SC~(s) "" , (3.2a)
not
* 0 is the coordinate of a dimension "added" to the already infinite-dimensional loop space, to space-time, and as such the usual arguments about its space-like nature do not apply. The positive (" time-like"7) signature is chosen here so as to yield the proper sign of the F 2 term in the action, eq.
(38).
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488 with
8
F~s~t(C) =SC~(s) B,(t, C)
8
8C'(t) B~(s, C).
(3.2b)
Although the 82(0) factor only appears in the first term, it would be a mistake to throw away the remaining terms at this stage. First of all, they contain functional derivatives that can lead to 8(0) factors; and second, as we shall see, a functional averaging will still have to be performed to get a meaningful action, and this will lead to yet a further 8(0) factor. It is only after all these factors have been tracked down that in the final expression we are to cancel an overall 82(0) factor in the sense of the N--, oo limit discussed above. We now make use of our expression (2.9) for B~ to evaluate the second term of eq. (3.2). Keeping in mind the simple identities (H(x), G(x) are arbitrary functions)
8C~(s) H(C(t)) = O~H(C(t))8(t - s ) , dsG(C(s))H(C(t))
8 ( t - s ) = - -~G(C(t))H(C(t)),
we are led to /e2lc2t~2(c)f01 ds f0 ! dtg~'(C(s))g~v'(C(t))F~s~t(C)F~,,s~,t(C) = ¼e2K2q~2(C)fol ds
8(O)g~'(C(s))g~"'(C(s))
dC 8 d C ~'
× d~- d~-(a,S'~(C(s))+O'B~,(C(sl))r~ '''~'(c('))'
(3.2c)
with -- a , B . , ( x ) + 0 . B , , ( x ) +
(3.2d)
Next we consider the geometrical variational principle. As a lagrangian, we choose the scalar curvature of the bundle P(~M,U(1)) ~t la Kaluza-Jordan. Were we to integrate this lagrangian over the whole bundle space, regions lying over large loops in [~M would be included and a non-local theory would emerge. Being interested in a local field theory, we note that each point of the space-time manifold M is itself also a loop (of zero length). As such, a copy of M sits* inside ~M. Call PM the portion of the bundle P(f~M, U(1)) that lies over this copy of M. We want to restrict *f~M is not in general connected (depending on the homology of the manifold M). For our considerationsthe connectedcomponentthat contains this copy is all that matters.
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to PM the domain of the integration that leads from the lagrangian to the action I. Thus I=-
16~r-----G ~4
dOx~lglq~2/¢"
(3.3)
The integration here is over the (D + 1)-dimensional space PM, and the expression under the square root is the determinant of the metric of this space. Now the crucial factor R should be a local expression on M. Indeed the scalar field ~(C) becomes a local field on M: ~(C)[c_~x ~ qffx), so that, as can be seen from (3.2a-c), only local fields appear in the action (3.3). Note that for C --, x E M the right-hand side of eq. (3.2c) becomes
l eZr28(O),2(x)g"(x)g'~'(x)(O~B,n(x) + O~Bs,(x))F,,,,n,(x)S88"(C),
(3.2e)
where
Snn'(C) =-foI ds Cs(s)CS'(s).
(3.2f)
Since for point-like loops C " = 0, one would expect S 88' to vanish for such loops. Yet a more careful limit is to be taken here on account of the overall e2x28(O) factor multiplying the expression (3.2e). We therefore consider S 88' on a neighborhood of the point-like loop x E t]M consisting of small loops "close" to x. Taking
L:(c)
(3.4)
withp > 0 as a measure of the size ( ~ length squared) of the loop C*, we now define SS~'(x) at the point-like loop x as its average value over all loops of size smaller than ~2 (later we will let ;~ ~ 0):
s~'(x)-[J L 2 ( C ) ~ 2®~C['dsC~(s)CS'(s)/f~ ®xC. dO L (C)~,
(3.5)
Here f6DxC means integration over all small loops "near" x E f~M with a suitable measure. SS~'(x) must be a symmetric tensor at x, and has dimensions of length squared. The size )d of the loops being the only dimensional parameter at our disposal, we expect terms like ~2f(~)g,~#, h4g(ep)R~a,.., to appearin S~#(x). (Here * For simplicity, we have chosen the factor (q~(C)) v in eq. (3.4), although in principle, more general functions of q~ could be used.
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f(q0, g(~) are. some functions of ~.) Thus to lowest order in h we expect S ~ h2f(~)g ~. Indeed, in the appendix we give a formal derivation of the relation Sa#( x ) = ~(O)¼~2~--P( x )gaS( X ) .
(3.6)
We note the appearance of yet another 8(0) factor in the F 2 term (3.2e) of the action, which allows this term to survive. The ~2 factor in eq. (3.6) when brought into eq. (3.2e), requires that the ~ --, 0 limit be accompanied by ex -, ~ so that e21£2~k2
64rr----G- 1.
(3.7)
Putting all this together, the action becomes
The last term in the action is a total divergence. Moreover, we now drop the overall 82(0) factor in accordance with our previous discussion*. Finally, in order to cast the action in a more conventional form, we perform a Weyl rescaling gl~v( X ) ~ gav( X ) ~ ( X ) -2/(0-2),
(3.9a)
followed by the field redefinition
giving
,=_fd xlv (
R
16~rG
½g~0~qoO,~p 6" 8
~s 8
/
-~,8-~,~,8,j,
(3.10a)
with
A=
8¢rG (2D - p D + 2 p ) . (D- 1)(D-2)
(3.10b)
* Alternatively, the metric g ~ can be rescaled in such a way that the factor 82(0) disappear from the first term in eq. (3.8). This correlates the h --, 0 and N --, 8(0) = oo limits and replaces eq. (3.7) by e 2x 2h2(8 (0)) 8/( D- 2) = 64 ~rG.
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For D = 10 a n d p = 1 orp = 4 this action is precisely that of the bosonic sector of N = 1 supergravity in 10 dimensions. It may be worthwhile to recapitulate here the steps that have led to the action (3.10). We defined a U(1) bundle P(flM, U(1)) over the loop space tiM of the space-time manifold M. Subject to the constraint (2.9) the antisymmetric tensor field was found to provide a connection on this bundle. Natural metrics on f~M and U(1) along with the connection on P allowed us to endow the bundle P with a KaluzaJordan-like metric. With a suitable limiting procedure, a finite scalar curvature for P was extracted. This was then used as a lagrangian integrated over the portion Pra of P lying over the copy of space-time M contained in tiM. Certain singular tensors were smoothed out via an averaging over small loops. All these steps suggest that the infinite-dimensional spaces f~M and P were used merely as "geometric crutches." The local field theory is still supported by the finite dimensional space-time M. Yet, it is the surrounding infinite-dimensional manifold f~M that allowed us to use the field B~ in defining a connection. More generally, viewing the points of M as closed submanifolds of dimension 2, 3.... we can surround M by any of the infinite-dimensional spaces f~2M, ~3M .... and treat higher order antisymmetric fields as connections in appropriate bundles over these spaces. In this way geometric action principles for such fields can be obtained, as above.
4. Conclusions and prospects We have provided antisymmetric tensor gauge fields with a geometric basis using the loop spaces f~,M of closed r-submanifolds of the space-time M. To end up with local theories we had to back-track from f~rM to the copy of M that sits inside f~,M by virtue of the fact that the points of M themselves can be thought of as closed r-submanifolds in M of vanishing size. In the case of loop space, this amounts to taking the small-loop limit. This is highly reminiscent of the a' ~ 0 limit in which the dual model of closed bosonic strings also yields the action (3.10) [6]. Could it be that the non-local theory we would encounter by considering the geometrical action on the whole bundle P(flM, U(1)) (rather than just on the restriction PM as we have done in the local case) is the theory of closed strings? Similar questions can be asked for theories built over f~rM with r > 1. Our considerations involved the space of parametrized loops [e.g. the metric (2.4), (2.5) is not repararnetrization invariant]. In the limit 2~~ 0 of small loops this is fully acceptable, but for finite loops (strings) reparametrization invariance is of essence. We would therefore have to make appropriate modifications. Thus the loop-space metric (2.5) would have to be changed either by providing an extra factor that restores reparametrization invariance, or more generally, by allowing this metric to be unconstrained and dynamically determined. The small-loop limit extracted with
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the averaging procedure (3.5), would still lead to an action of the form (3.8), as all extra pieces involve higher powers of the loop size ~. The constraint (2.9) on the loop-space connection figures prominently in our construction. Does this condition have a simple geometric origin? As structure group we chose U(1). Is there a justification for this? Indeed, a natural U(1) bundle lives over tiM. If tiM is the space of loops without a base point, then the loops with all possible choice of base point provides a U(1) bundle over tiM. The additional question as to the meaning and relevance of non-abelian structure groups in bundles over tiM deserves further study. (Such bundles have recently appeared in the literature, but in a rather different context [10].) Our construction provided the bosonic sector of N = 1 supergravity in ten dimensions. Could one reconsider the problem in a supersymmetric context (loops in superspace?) and achieve a geometrization of the complete ten dimensional N = 1 supergravity? Of course the question can be asked for eleven dimensions where t 2 M should enter. We wish to thank Professor I.M. Singer for a valuable discussion, and Professor S.S. Chem for pointing out to us ref. [8]. Discussions with Professor R. Lashoff, M. Rubin and S. Ramaswamy are gratefully acknowledged.
Appendix Here we offer some formal justification for eq. (3.6). First we observe that for H(C) an (arbitrary) functional of the loop C, and for h small
=
f
,®C fo~2d r
- f ®cf% = - 2f®Ce-
f oo _ _dr¢ ~ , ( L• "--oo
2--T)rj_i{r" ~
tzvr)
x2(1 + ½iX2r)-'e"'m C)
2L2<¢~/zI-I(C
),
(A. 1)
where in the last step we have used formula (4), p. 118 of ref. [11]. Thus, using the
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measure (3.4) and eq. (A.1), the quantity
r"#(x)=-fdL2(C)~Tt2®~cf'v0 ds d"(s)d/~(s)
82 =f0 1 ds,/o l ds2 8(s I --$2) 8ja(Sl$-gjfl(S2) Z[
J] s=o'
(A.2)
where we have defined
(A.3) Since the integral is over only loops near the point x. we set g,,(C(s))~g,,(x) and q f f C ) ~ ( x ) as in sect. 3. and the functional integration is then readily performed with the result
Z[ J ] = Z[O]exp[ ~t2~-p( x )g~P( X ) foi dS J~( s )Jp( s ) l.
(A.4)
Hence 82
8Sa(Sl)SJp(s2)
Z[J][
:=o
=Z[O]¼h2e~-P(x)8(s,-s2)g:#(x).
(A.5)
Finally, from (A.2) we have that
T=/J(x ) = Z[O]¼h28(O)ep-P( x )g=tt( x ) .
(A.6)
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P.G.O. Freund, R.I. Nepomechie / Gaugefields and gravity
The denominator in eq. (3.5) being precisely Z[0], a combination of the eqs. (3.5) and (A.6) yields the result (3.6). References [1] N. Kemmer, Proc. Roy. Soc. A166 (1938) 1127; V. Ogievetsky and I. Polubarinov, Soy. J. Nucl. Phys. 4 (1967) 156; D.Z. Freedman and P.K. Townsend, Nucl. Phys. B177 (1981) 282 [2] M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273; E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117; Y. Nambu, Phys. Reports 23 (1976) 250; C. Marshall and P. Ramond, Nucl. Phys. B85 (1975) 375 [3] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253; E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409; A. Chamseddine, Nucl. Phys. B185 (1981) 403 [4] W. Siegel, Phys, Lett 93B (1980) 170 J. Thierry-Mieg, Harvard University preprint (1980); E. Sezgin and P. van Nieuwenhuizen, Phys. Rev. D22 (1980) 301; T. Curtright, private communication (1980) [5] T. Kaluza, Sitzber, Preuss. Akad. Wiss. (1921) 966; P. Jordan, Z. Phys. 157 (1959) 112; C. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 925 [6] J. Scherk and LH. Schwarz, Phys. Lett. 52B (1974) 347 [7] Y.-M. Cho, J. Math. Phys. 16 (1975) 2029 [8] W. Klingenberg, Lectures on closed geodesics (Springer-Verlag, New York, 1978), ch. 1 [9] J. Milnor, Morse theory (Princeton University Press, New Jersey, 1963), sects. 10, 11 [10] A.M. Polyakov, Phys. Lett. 82B (1979) 247; I. Ya. Aref'eva, Lett. Math. Phys. 3 (1979) 241 [11] A. Erd~lyi et al., Tables of integral transforms, vol. I., Bateman Manuscript Project (McGraw-HiU, New York, 1954)