View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server LOCAL SUPERSYMMETRY IN ONE-LOOP QUANTUM COSMOLOGY Giampiero Esp osito Istituto Nazionale di Fisica Nucleare Mostra d'Oltremare Padiglione 20, 80125 Napoli, Italy; Dipartimento di Scienze Fisiche Mostra d'Oltremare Padiglione 19, 80125 Napoli, Italy. Abstract. The contribution of physical degrees of freedom to the one-lo op amplitudes of Euclidean sup ergravity is here evaluated in the case of at Euclidean backgrounds b ounded by a three-sphere, recently considered in p erturbative quantum cosmology. In processed by the SLAC/DESY Libraries on 28 Jun 1995. 〉 3 Euclidean sup ergravity, the spin- p otential has the pair of indep endent spatial comp o- 2 PostScript 0 A A e nents ; . Massless gravitinos are here sub ject to the following lo cal b oundary i i p 0 0 0 A A A A 3 e 2 n = , where n is the Euclidean normal to the conditions on S : e e i i A A three-sphere b oundary. The physical degrees of freedom (denoted by PDF) are picked out i A imp osing the sup ersymmetry constraints and cho osing the gauge condition e =0, 0 i AA 0 i A e e =0. These lo cal b oundary conditions are then found to imply the eigenvalue 0 AA i h i h i 2 2 GR-QC-9506065 condition J (E ) J (E ) =0; 8n0, with degeneracy (n + 4)(n + 1). One n+2 n+3 1 can thus apply again a zeta-function technique previously used for massless spin- elds. 2 1 Local Supersymmetry in One-Loop Quantum Cosmology 289 The PDF contribution to the full (0) value is found to b e = . Remarkably, for the 360 massless gravitino eld the PDF metho d and lo cal b oundary conditions lead to a result for (0) which is equal to the PDF value one obtains using sp ectral b oundary conditions 3 on S . PACS numb ers: 03.70.+k, 04.60.+n, 98.80.Dr 2 Local Supersymmetry in One-Loop Quantum Cosmology 1. Intro duction The problem of one-lo op niteness of sup ergravity theories in the presence of b oundaries 110 is still receiving careful consideration in the current literature. As emphasized in Refs. 9,11-12, one can p erform one-lo op calculations paying attention to: (1) S-matrix elements; (2) top ological invariants; (3) presence of b oundaries. For example, in the case of pure gravity with vanishing cosmological constant:=0,itisknown that one-lo op on-shell S-matrix elements are nite. This prop erty is shared by N = 1 sup ergravity when = 0, and in that theory two-lo op on-shell niteness also holds. However, when 6= 0, b oth pure gravity and N = 1 sup ergravity are no longer one-lo op nite in the sense (1) and (2), 11 b ecause the non-vanishing on-shell one-lo op counterterm is given by 1 2BGS S = A : (1:1) (1) 3 In equation (1.1), = n 4 is the dimensional-regularization parameter, is the Euler 87 106 9;11 ;B = for pure numb er, S is the classical on-shell action, and one nds : A = 45 10 41 77 gravity, and A = ;B = for N = 1 sup ergravity.Thus, B 6= 0 is resp onsible for lack 24 12 of S-matrix one-lo op niteness, and A 6= 0 do es not yield top ological one-lo op niteness. If any theory of quantum gravity can b e studied from a p erturbative p oint of view, b oundary e ects playa key role in understanding whether it has interesting and useful niteness prop erties. It is therefore necessary to analyze in detail the structure of the one- lo op b oundary counterterms for elds of various spins. This problem has b een recently studied within the framework of one-lo op quantum cosmology, where the b oundary is 3 Local Supersymmetry in One-Loop Quantum Cosmology usually taken to b e a three-sphere, and the background is at Euclidean space or a de 210 Sitter four-sphere or a more general curved four-geometry. 3 Our pap er describ es one-lo op prop erties of spin- elds to present a calculation which 2 5;9 was previously studied in research b o oks but not in physics journals (see, however, re- 3 marks at the end of Ref. 10). In the Euclidean-time regime, the spin- eld is represented, 2 using two-comp onent spinor notation, by a pair of indep endent spinor-valued one-forms 0 0 4;9 A A A A e e . After imp osing the gauge conditions with spatial comp onents ; ; i i (hereafter e is the tetrad) 0 AA 0 i A i A e e =0 ; e =0 : (1:2) 0 0 AA i AA i 0 A A e and the linearized sup ersymmetry constraints, the expansion of ; on a family of i i 4;9 three-spheres centred on the origin takes the form (n+1)(n+4) 3 1 X X 2 0 0 A nq AB B pq nq AB B 0 = + re ( ) e ; (1:3) m ( ) np BB i np i n 2 n=0 p;q =1 (n+1)(n+4) 3 1 X 0 0 X 2 0 0 0 nq B A B A nq B A B pq e 0 = me ( ) + r ( ) e : (1:4) np np BB i i n 2 n=0 p;q =1 With our notation, is the radial distance from the origin in at Euclidean four-space, 1 1 pq the matrix is blo ck-diagonal in the indices pq , with blo cks . Note also that n 1 1 the mo des me ( ); re ( ) are not the complex conjugates of m ( );r ( ) resp ectively. np np np np 0 0 0 0 0 0 0 4;9 nq AB B nq (AB C ) B nq B A B nq (A B C ) B n n Moreover, one has = ; = where the 0 C C 0 0 0 nq (AB C ) nq (A B C ) harmonics and are symmetric in their three spinor indices and have 4 Local Supersymmetry in One-Loop Quantum Cosmology 1 5 3 4 p ositive eigenvalues n + of the intrinsic three-dimensional Dirac op erator on S , 2 2 0 3 3;9 CB is the Lorentzian normal to S . and n 3 3 Sec. 2 studies lo cally sup ersymmetric b oundary conditions on S for the spin- po- 2 tential, and the equation ob eyed by the eigenvalues by virtue of these b oundary conditions is derived. Sec. 3 uses zeta-function regularization and obtains the contribution of phys- ical degrees of freedom (hereafter referred to as PDF) to the full (0) value. Concluding remarks and op en research problems are presented in Sec. 4. 3 2. Lo cal Boundary Conditions for the Spin- Potential 2 In Euclidean sup ergravity, the mathematical description of the gravitino leads to the intro- 0 A A e duction of the indep endent spinor-valued one-forms ; with spatial comp onents 0 A A e ; .We are here interested in a generalization to simple sup ergravity of the cal- i i 1 culations in Ref. 3 for the spin- eld. Thus, we consider a at Euclidean background, 2 3 requiring on the b ounding S that p 0 0 A A A e 2 n = ; (2:1) e A i i where = 1. The consideration of (2.1) is suggested by the work in Ref. 1, where it is p 0 0 0 A A AA A e 2 n transform shown that the spatial tetrad e and the pro jection e A i i i into each other under half of the lo cal sup ersymmetry transformations at the b oundary, 5 Local Supersymmetry in One-Loop Quantum Cosmology and that after adding a suitable b oundary term, the sup ergravity action is invariant under 3;9 these lo cal sup ersymmetry transformations. Indeed, from Sec. 1 we already know that, imp osing the sup ersymmetry constraints 3 and cho osing the gauge condition (1.2), the spin- p otential nally assumes the form (1.3)- 2 (1.4). It is therefore useful to derive identities relating barred to unbarred harmonics, generalizing the technique in Ref. 13. This is achieved by using the relations Z 0 0 0 np mq CC BB AA nm pq n n d n = H ; (2:2) 0 0 0 n AB C A B C Z np mq AD BE CF nm pq d = A ; (2:3) n AB C DE F and the expansion of the totally symmetric eld strength (n+1)(n+4) 1 X X np np b (x) : (2:4) (x)= ba (x)+b AB C np np AB C AB C n=0 p=1 Thus, we can express the ba co ecients in two equivalentways using (2.4), and (2.2) or np (2.3). The equality of the two resulting formulae leads to (n+1)(n+4) (n+1)(n+4) X X 0 0 0 qp qp nq nq 1 AD BE CF AA BB CC 1 H = n n n A ; (2:5) 0 0 0 n n A B C DE F q =1 q =1 which is nally cast in the form (n+1)(n+4) X qp nq np 1 F E D A H n : (2:6) n = 8n 0 0 0 0 0 0 n n F E D DE F D E F q =1 6 Local Supersymmetry in One-Loop Quantum Cosmology In a similar way,we obtain (n+1)(n+4) X 0 0 0 qp np nq D E F 1 = 8n n n A H : (2:7) 0 0 0 n D E F n DE F D E F q =1 pq pq The form of the matrices A and H is obtained taking the complex conjugate of (2.6), n n np and then inserting the form of so obtained into the right-hand side of (2.6).