D. Bailin, A. Love/Physics Reports 315 (1999) 285}408 285 ORBIFOLD COMPACTIFICATIONS OF STRING THEORY D. BAILIN!, A. LOVE" !Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK "Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20-0EX, UK AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO Physics Reports 315 (1999) 285}408 Orbifold compacti"cations of string theory D. Bailin!, A. Love" !Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK "Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 0EX, UK Received October 1998; editor: J.A. Bagger Contents 1. Orbifold constructions 288 3.7. 3-point function for Z orbifold 338 1.1. Introduction 288 3.8. B "eld backgrounds 340 1.2. Toroidal compacti"cations 289 3.9. Classical part of 4-point function from 1.3. Point groups and space groups 293 conformal "eld theory 341 1.4. Orbifold compacti"cations 296 3.10. Quantum part of the 4-point function 343 1.5. Matter content of orbifold models 302 3.11. Factorisation of the 4-point function 1.6. Lattices 303 to 3-point functions 345 1.7. Asymmetric orbifolds 309 3.12. Yukawa couplings involving excited 2. Orbifold model building 314 twisted sector states 349 2.1. Introduction 314 3.13. Quark and lepton masses and 2.2. Wilson lines 315 mixing angles 351 2.3. Modular invariance for toroidal 4. KaK hler potentials and string loop compacti"cation 316 threshold corrections to gauge coupling 2.4. orbifold modular invariance 318 constants 353 2.5. GSO projections 320 4.1. Introduction 353 2.6. Modular invariant Z orbifold 4.2. KaK hler potentials for moduli 355 compacti"cations 322 4.3. KaK hler potentials for untwisted 2.7. Untwisted sector massless states 324 matter "elds 358 2.8. Twisted sector massless states 325 4.4. KaK hler potentials for twisted sector 2.9. Anomalous ;(1) factors 327 matter "elds 363 2.10. Continuous Wilson lines 327 4.5. String loop threshold corrections 3. Yukawa couplings 328 to gauge coupling constants 365 3.1. Introduction 328 4.6. Evaluation of string loop threshold 3.2. Vertex operators for orbifold corrections 367 compacti"cations 329 4.7. Modular anomaly cancellation and 3.3. Space group selection rules 331 threshold corrections to gauge 3.4. H-momentum conservation 332 coupling constants 371 3.5. Other selection rules 334 4.8. Threshold corrections with reduced 3.6. 3-point functions from conformal modular symmetry 375 "eld theory 335 4.9. Uni"cation of gauge coupling constants 378 0370-1573/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 573(98)00126-4 D. Bailin, A. Love / Physics Reports 315 (1999) 285}408 287 5. The e!ective potential and supersymmetry 5.4. Supersymmetry breaking 395 breaking 380 5.5. Cosmological constant 397 5.1. Introduction 380 5.6. A-terms and B-terms 398 5.2. Non-perturbative superpotential due to 5.7. Further considerations 400 gaugino condensate(s) 382 6. Conclusions and outlook 401 5.3. E!ective potential 385 References 404 Abstract The compacti"cation of the heterotic string theory on a six-dimensional orbifold is attractive theoretically, since it permits the full determination of the emergent four-dimensional e!ective supergravity theory, including the gauge group and matter content, the superpotential and KaK hler potential, as well as the gauge kinetic function. This review attempts to survey all of these calculations, covering the construction of orbifolds which yield (four-dimensional space}time) supersymmetry; orbifold model building, including Wilson lines, and the modular symmetries associated with orbifold compacti"cations; the calculation of the Yukawa couplings, and their connection with quark and lepton masses and mixing; the calculation of the KaK hler potential and its string loop threshold corrections; and the determination of the non-perturbative e!ective potential for the moduli arising from hidden sector gaugino condensation, and its connection with supersymmetry breaking. We conclude with a brief discussion of the relevance of weakly coupled string theory in the light of recent developments on the strongly coupled theory. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.-w; 12.10.-g; 12.60.Jv 288 D. Bailin, A. Love / Physics Reports 315 (1999) 285}408 1. Orbifold constructions 1.1. Introduction It is well known that the construction of a consistent quantum string theory is possible only for speci"c dimensionalities of the (target) space}time. For the bosonic string the required dimension is D"26, while for the superstring dimension D"10 is required. Thus from the outset we are forced to consider the `compacti"cationa of the (spatial) dimensions which are surplus to the d"4 dimensions of the world that we inhabit, if we are to have any chance of connecting the string theory with experimental (particle) physics. The string theory which is best placed to generate such a connection is the heterotic string [117], a theory of closed strings, in which the right-moving degrees of freedom of the superstring are adjoined to the twenty-six left-moving degrees of freedom of the bosonic string. To endow such a construction with a geometrical interpretation sixteen of the left-movers are compacti- "ed by associating them with a 16-dimensional torus, with radii of order the Planck length & \ (l. 10 m). Just as the compacti"cation of one dimension onto a circle in the (original) "ve-dimensional Kaluza}Klein theory [135,141] generates a gauge boson, so here the compacti"ca- tion generates gauge "elds, including some of a stringy origin which derive from the possibility of the string winding around the torus. In this way, the 16 left-movers generate an `internala gauge ; symmetry with the (rank 16) gauge group E E being consistent with the cancellation of gauge and gravitational anomalies which is essential for a satisfactory quantum theory [113]. Although this scenario explains in a satisfying way how a gauge symmetry can emerge from string theory, there are serious problems which remain. Firstly, there is the fact that the symmetry ; ; ; group E E is far larger than the (rank 4) SU(3) SU(2) ;(1) gauge symmetry which we observe. Secondly, there remains a ten-dimensional space}time, six of whose dimensions must be compacti"ed before we even contemplate questions like gauge symmetries and matter generations. The orbifolds [79,80], which are the subject of this review are one method of compactifying the unobserved six dimensions. An orbifold is obtained when a six-dimensional torus (¹) is quo- tiented by a discrete (`pointa) group (P), as we shall see shortly. The identi"cation of points on ¹ under the action of the point group generates a "nite number of "xed points where the orbifold is singular. At all other points the orbifold is (Riemann) #at. It is for this reason that we are able to calculate rather easily all of the parameters and functions of the emergent supergravity theory: the gauge group and matter content; the Yukawa couplings and KaK hler potential, which determine the quark and lepton masses and mixing angles; the gauge kinetic function, including string loop threshold corrections, which in turn determine the uni"cation scale of the gauge coupling con- stants. We shall see also how modular invariance constrains the e!ective potential, and hence determines the actual value of the coupling constants at uni"cation, as well as the nature of the supersymmetry breaking mechanism. There are, of course, other methods of string compacti"cation including Calabi}Yau manifolds [43,115,116], free fermion models [139,3], and N"2 superconformal "eld theories [107,108,140], and (some) orbifold models are connected to some of these models [138,98,13,14,24]. However, none of the alternatives has so far been as fully worked out as the orbifold theories, and it is for this reason that we have focused upon them. If for no other reason, they illustrate the sort of predictive power which we should eventually like string theory to have (even if it should transpire that nature does not in fact utilize orbifolds!) D. Bailin, A. Love / Physics Reports 315 (1999) 285}408 289 1.2. Toroidal compactixcations The construction of the ten-dimensional heterotic string has been fully described elsewhere (see, for example, [114,132,35]) and we need not review it here. As already noted, to have any chance of a realistic theory it is obviously essential that six of the (nine) spatial dimensions have to be compacti"ed to a su$ciently small scale as to be unobservable at current accelerators. The simplest way to do this is to compactify on a torus. This ensures that the simple linear string (wave) equations of motion are una!ected, since the torus is #at. We work in the light-cone gauge. Then there are eight transverse bosonic degrees of freedom denoted by XG(q,p) where i"1,2 labels the two transverse four-dimensional space}time coordinates, and XI(q,p) where k"3,2,8 labels the remaining six spatial degrees of freedom. (q, p with 0)p)p are the world sheet parameters.) XG and XI are split into left and right moving components in the standard manner G(I) q p " G(I) q!p # G(I) q#p X ( , ) X0 ( ) X* ( ) . (1.1) WG q!p In addition there are eight right-moving transverse fermionic degrees of freedom 0( ), WI q!p ' q#p " 0( ), and the 16 (internal) left-moving bosonic degrees of freedom X*( )(I 1,2,16) ; which generate the E E gauge group of the ten-dimensional heterotic string.