•' ••'••'** \. IC/82/18U
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS v
CCMPOSITE GRAVITY MD COMPOSITE SUPERGRAVITY
Jerzy Lukierski
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1982MIRAMARE-TRIESTE
IC/82/1&U
International Atomic Energy Agency and United nations Educational Scientific and Cultural Organization
INTERNATIOHAL CESTKE FOR THEORETICAL PHYSICS
COMPOSITE GRAVITY. AHD COMPOSITE SUPERGHAVITY *
Jeray Lukierski *• International Centre for Theoretical Physics, Trieste, Italy.
- TRIESTE September 1982
• Talk given at XIth International Colloquium on Group-Theoretic Methods in Physics, Istanbul, Turkey, 23-28 August 1982. •* Permanent address: Institute for Theoretical Physics, University of Wroclaw, ul. Cybulskiego 36, Wroclav, Poland.
The known examples of such a construction are the composite U(n) ( 3) 5 potentials obtained from lj(n" "^jn,) o-fields " >, sp(l) « SU{3)
ABSTRACT composite gauge fields in HP(n) a-raodels °>'T\ or eo(8) gauge fields constructed from ^Trhrr o-fields describing scalar sector of N = 8 It is known that the composite YM K-t;aui;e theory can supergravity be constructed from a-flelds taking values in a sym- metric Riemannian space — . We extend such a frame- h 2. Such a scheme can be supersymmetrized in three ways . The first work to graded o-fields taking values in supercosets. already known way is obtained by supersymmetric extension of the space- We show that from supereoset a-fields one can con- time co-ordinates, i.e. by the replacement of o-fields by a-superfields.
struct composite gravity, and from supercoset a- In such a formulation the composite gauge superfields are defined 1 superfields the composite- supergravity models. again by the formula (l), but the differentials dK and one-forms A are expanded in even and odd differentials. For D = k "flat" Salam- • 1, The composite gauge potentials describing the internal symmetry group Strathdee superfields one can write H were constructed from the a-fields which take values in larger internal symmetry group G OH 1)'2'. Usually one assumes that (H,K = ~) is the d8 (3) Riemannian symmetric pair, i.e. the algebra g = h © It is Zg- graded. The composite H-connections A. are defined by the following Cartan one-form; and dK = A + E = U) where A = G, A, = If we u n _ii_ _ Ii ISvT. —-5 - •. If ve o^ervobservee thathatt deie" = - £r d9 i d9 the Using Cartan-Maurer equation for G one gets the formula for composite formulae for field strength components (3) look as follows: B-curvature
F = U-|A*A = J E^E • (2) (5a)
V (5b) If we observe that A = A^ dx and F = F^ a/ A dx" one can conclude- F = D A - 3 A - [A ,k ] that (5c)
£ a-fields composite YM H-gauge (A) theory where
Because the coset structure implies F = 0 , using (2) and (5a), one gets
(7) A = -W )
-1- Substituting (j) in (5b) one obtains the composite field strength In general the supergroup QSp(l;l+) can be replaced by a larger one superfield F (x,6) which provides the composite E3YM Yang-Hills £. 0 = GSp(l»;N})- We iiave therefore the relation action (see,e.g. Ref.10"). The formulation of D = if composite LJSY.M theory has been proposed independently in Refs.ll ar.d 12. The relation — •£ . o-field;; —•> composite gravity With (A) is generalized as follows: 13 ^'J' fermionic fundamental fields
— o-superfields composite SSYM H-gauge theory (3) U. The composite K = 1 D = It aupergravity io obtained by inserting the H composite achtbein superfields E into the well-known supergravity action formula 3. In order to describe the composite D = It gravity we should con- It U struct the composite vierbein e which transforms in tangent space d x d 9 (11) as a four-vector under local SL(2,C) spin group and is a world vector under general co-ordinate transformations. Such a composite vierbein The composite supervierbeins are obtained as functions of Goldstone we obtain from the Cartan one-form on the supereoset superfields by considering the Cartan one-form on the supercoset
K = SL(2,C) SL(2,C) K = SL(2,C~ ) SL(2,C) Osp(l;li
which we parametrize as follows (see also Refs.13-16); parametrized as follows:
K* = exp Va(x) P exp 0™(x)' Q (8) K = exp(V (x,8) P exp (12) a. C(
wherere where (V ,n ) parametrize the local superspace co-ordinates, i a G - V describes local space-time co-ordinates, describe Goldstone superfields living in the supercoset S = ost>(l-U) ' a - i|j describes the spinor Goldstones. The Cartan one-form The composite SL(2,C) connections u and composite vierbein e are _1 defined in a very analogous way to (l) K dK (13)
defines the composite achtbein superfields EC (A = (a,a), B M (9) through the expansion into the covariant differentials d6 , e where M are Lorenta generators. The Cartan structure equations ab
T = dE - Cl A E (10)
provide after inserting £1 and E from (9) the formulae for composite SL(2,C) curvature and torsion. The gravity described \>y any action For the choice G = 0Sp(2;M the composite achtbeins and composite which is the function of vierbeins, SL(2,C) curvature and torsion is supergravity have been introduced in Ref.1T- made composite by inserting respective composite formulae for e^ , R and T (see Hef.1T)- -3- CURRKBT
IC/62/63 U.S. CRAIGIE, V.K. DOBHEV and I.T. TODOROV - Conformal techniques for OPE in asymptotically free quantum field theory. In order to obtain composite supergravity it is necessary to consider supercoset a-superfields as fundamental ones, i.e. IC/82/6U A. FRXDHYSZAK and J. LUKLEBSKl - H = 2 massive matter multiplet from quantization of extended classical mechanics.
IC/82/65 TAHIR ABBAS - Study of the atomic ordering in the alloys MI-IR using (D) diffuse X-ray scattering and pseudopotentials. IC/82/66 E.G. HJAU - An analytic examination of distortions in power spectra due IHT.EKP.* to sampling errors. The composite gravity and composite supergrn,vity satisfy the torsion IC/82/67 E.G. HJAU - Power estimation on sinusoids mounted upon n.C. background: constraints modified in comparison with "elementary" gravity and super- IBT.REP.* Conditional problems. gravity models. We expect that torsion constraints describe only the IC/82/68 E.G. HJAU - Distortions in power spectra of signals with short components. effective theory valid at sufficiently large distances. For the support IST.REP.* of this conjecture we mention that in "non-geometric" composite gravity IC/82/69 E.C. SJAU - Distortions in two- and three-dimensional power spectra. 18) of Aiaati and Veneziano the non-vanishing torsion can be neglected IHT.RSP.* in the sufficiently low energy region. IC/82/70 L. SCHWARTZ and A. PAJA - A note on the electrical conductivity of disordered alloys in the muffin-tin model. IC/82/71 D.G. FAKIHOV - Mass and form factor effects in spectrum and width of the IHT.EEP.* semi-leptonie decays of charmed mesons. REFERENCES IC/82/72 T. MISHONOV and T. SARIISKI - Acoustic plasma vaves in inversion layers 1) M.A. :'emenov-Tjan-Snenski and L.D. Fad&eev, Westnik Lenin^r. Univ. IHT.REP.* and sandwich structures. 13, Si (1977) (in Russian). IC/82/73 T. MISHINOV - An exactly averaged conductivity in a disordered electronic 2) A,P. Balachandran, A. Stern and 0. Trahern, Phys. Rev. D19, 2^16 IKT.REF. model. (1978). 3) J, Frohlich, Lecture at Bielefeld Symposium, December 1978, IHES IC/82M S.M. MUJIBUH RAHMAN - Structural energetics of noble metals. preprint (1979). IC/82/75 E. SEZGIN and P. van NIEUWEHHUIZEN - Ultraviolet finiteness of B = 8 h) M. Dubois-Violette and Y. Georgelin, Phys.Lett. 32B, 251 (1979). supergravity, spontaneously broken ny dimensional reduction. 5) I. Bars in Proc. of 8th Int. Colloquium on Group Theor. Methods in Slem. Particle Phys., Kiriet Amavin, Israel, March 1979, published IC/82/76 JEBZX RAYSKI and JACEK HAYSK1, Jr. - On a fusion of supersymmetries with in Ann. of Isr. Phys. Soc. 3, 5S (1930). gauge theories. 6) J. Lukierski in Proc. of Summer Inst. of Field-Theor. Methods in IC/82/77 A. BOKHARI and A. QADIB - A prescription for n-dimensional vierteins. Elem. Particle Phys., August 1979, Kaserslautern, published by IHT.REP.* Springer, p.309 (1980). T) F. Giirsey and H.C. Tze, Ann. Phys. 128, 29 (1979). IC/82/T8 A. QADIH and J. QUAMAR - Relativistic generalization of the Newtonian force. 3) E. Cremmer and B. Julia, Hucl.Phys. E159, l^l (1.979). INT.REP.• 9) J. Lukierski in Proc. of Symposium on DLffsrential-CieOEietric Methods, IC/82/79 B.E. BAAQUIE - Evolution kernel for the Dirac field. Aix-en-Provence, September 1979, published by Epringer-Verlag; Lecture notes in Hath. Vol. 335, 225 (-980). IC/82/80 S. RAJPOOT AND J.G. TAYLOR - Broken supersymmetries in high-energy physics. 1 10) M. Rocek in Superspace and Supe-rcravit,-j , Ms. S.V. , Hawking and IC/82/81 JAE HYUBG YEE - Photon propagators at finite temperature. M. Rocek (Cambridge Jniv. Fre^2, ±°:M} , p.Ti- 11) I'. Bars and M. Gunaydin, Phy3.;