Index

affine 235, 254 bulk 33, 74 grade 254 level 254, 256 χ˜(θm,θn) 192 anomaly 159 Calabi–Yau manifold 351, 352, 356 6D 99 Calabi–Yau space 9, 351 cancellation from modular invariance Cartan generators 191 eigenvalue 206 cancellation in Z3 orbifold 169 139, 164, 236, 394 fixed point 95 Cartan subalgebra 143 gravitational 8 central charge 124 higher dimension 8 central extension 255 anomaly polynomial 360 centralizer 186 asymmetric orbifold 191, 245, 246 Chan–Paton factor 373 automorphism 158 charge conjugated spinor 65 inner 237, 239 charge conjugation outer 239 on weights P 282 axion 281 charged current 1 model-dependent 228 “V–A” current 2 model-independent 229, 284 chiral basis 63, 64 axionic symmetry 178, 181 chiral field fifteen chiral fields 18 baryon chiral superfield 27 ducaplet 14 chiral theory 80 octet 14 chirality 18, 38, 80, 128, 165, 166, 282, Betti number 284 300 bino 30 Z3 168 bosonization 141 gaugino 166, 300 bottom-up approach 4, 377 chirality operator 128 boundary condition closed string 107 Neumann(open string) 108 cohomology class 355 periodic(closed string) 107 Coleman-Mandula no-go theorem 26 branching rule 250, 253 color 13 brane 74 compactification 7, 33 p-brane 74 Narain 178 400 Index

torus 39 Dirac γ matrices complex representation 23 5D 78 complex structure 116 Dirac–Clifford algebra 64 complex structure modulus 181 5D 78 conformal gauge 106 discrete action 45 conformal symmetry 109, 110 discrete symmetry 282 conformal weight 112, 113 double covering group 63 conjugacy class 46, 186 doublet-triplet splitting 23, 32, 85, 290 more fundamental concept 47 dual lattice 248 conjugate 46 duality correlation function T -duality 180, 374 consistency 217 Dynkin basis 236 factorization 219 396 four point 215 technique in orbifolding 252 normalization 219 Dynkin label 311 three point 212 cosmological constant η function 119 self-tuning 34 elecroweak scale 4 covariant derivative 2 electroweak CP 286 covariant gauge electroweak theory 15 globally 117 embedding 157 Coxeter element 240, 241 energy momentum tensor Coxeter label 235, 256 as constraints 106 Coxeter lattice 58 world-sheet 106 Coxeter number 236, 240 equivalence relation 46 CP Euler number 302, 355 286 exotics 267, 280 CP violation 17 extended Dynkin diagram 236 in 10D 283 extended root 236 CPT theorem 282 extra dimensions 33 critical radius 180 crosscap 49 factor orbifolds 185 crystallographic lattice 58 family 16 crystallography 55 three families from intersection 143 number 376 current-current interaction 2 number of 171 cycle 45 three families 16, 272 cyclic group ZN 45 three families from number of compact dimensions 372 D-brane 74, 374 family problem 6, 10, 18 119 family symmetry 6 degeneracy factor 192, 387, 388 fermion mass 4, 203 Z4 197 fermion number number of fixed points 189 R sector 127, 128 degenerate mass fermionic construction 144, 295, 367, from U secor 222 370 di-quark 20 Z2 × Z2 371 diagonal embedding 263 Z2 × Z2 272 dimensional reduction 78 fermionic string 103 Index 401

field localization 74 GUT 5, 18 fine tuning E6 23 of the curvature 29 family unification 271 fixed point 46, 152 flipped SU(5) 24, 372 conjugacy class 47 Pati-Salam model 24 Lefschetz theorem 48, 248 proton decay 6 number in asymmetric orbifold 249 semi-simple group 24 number of 48, 189 SO(10) 19 fixed point algebra 237 SO(10) breaking 23 flat directions 289 SO(10) subgroups 88 flatness 44 SO(14) 23, 271 flavor problem 268 SO(18) 23 flipped SU(5) 292, 297 SU(11) 24 freely acting 46 SU(5) 19 fundamental domain 48 SU(N) 24 fundamental reflection 240 SUSY– 31 fundamental region 48–51, 73, 84 trinification 24 fundamental weight 236, 248 GUT multiplet 279 fundamental weight of SU(n) 256 H-momentum 204, 211 G2 lattice 54 helicity 165 gamma matrix heterotic string 133 10D 122 compactification 135 2D 120 E8×E8 140, 369 6D 64 mass formula 137, 142 gauge anomaly 23 massless states 142 gauge boson 161 no anomaly 143 gauge coupling Spin(32)/Z2 140 running 20, 21 hierarchy gauge group 211 gauge– 22 gauge hierarchy Higgs supersymmetry 30 no-adjoint theorem 256 gauge principle 2 Higgs doublet 4 gauge symmetry one pair 290 enhancement 179 Higgs mechanism 4, 75 gaugino 30 Higgsino 30 generalized momenta 180 highest root 235 ghost charges 204 highest weight 236, 253, 311 global monodromy condition 216 Hodge diamond 356 global symmetry 282 holomorphic 27, 150 in string theory 282 holomorphic transformation 112 gluino 30 holomorphicity 28 grand unification 5 holonomy 61, 230, 352 gravitational anomaly 66, 133 homology 354 graviton 113 humor 292 GSO projection 127, 368 hypercharge normalization 20 fermionic construction 144 hypercharge quantization 32, 268, 291 generalized for twisted string 192 generalized with Wilson lines 201 identical particle 35 402 Index index Lorentz group generators 125 SU(2) representation 278 Lorentz symmetry 125 index l 20 linkage field 293 inheritance principle 228, 269 localized group 94 instanton 17 localized wave function 79 small size 290 Lorentz group 62 integrability condition 256 Lorentz invariance 211 integral lattice 138 Lorentzian lattice 178 intermediate vector boson IVB 2 M-theory 36, 295 intersecting brane 376 M¨obius strip 44 irreducible representation 236 Majorana condition 66 symplectic 67 Jacobi abstruse identity 130, 384 Majorana mass Jacobi elliptic function 139 symplectic 78 Jarlskog invariant 286 Majorana spinor 66 symplectic 81 K¨ahler modulus 181 mass matrix ansatz Kac–Moody algebra 254 Weinberg-Fritzsch 226 level 254 massless up-quark 285 level k 267, 270 matter representations 161 Kaluza–Klein 33, 135 maximal subalgebra 236 compactification 6 meson octet 14 KK mode 80 minimal coupling 2 KK spectrum 7 modding 44 zero mode 72 mode expansion 72, 74 kink mass 79 of 5D fermion 79 Klein bottle 44 model-dependent axion 284 model-independent axion 284 L-R symmetry modular form 231 asymmetric 3 modular invariance 52, 100, 115, 116, symmetric 3 158 ladder operators 143 condition 139, 191 large extra dimension 37 fundamental region 117, 118 lattice 54 gauge group 139 dual lattice Γ∗ 137 modular transformation 117, 189 lattice Γ 137 modular parameter τ 116 Niemeier 140 moduli 117, 179, 180, 230, 231, 289 self-dual 140 complex structure 181 SO(32), E8×E8 140 K¨ahler modulus 181 SU(3) 52 volume modulus 181 lattice embedding 297 moduli problem 289 Lefschetz fixed point theorem 48, 60, moduli space 179 248 momentum lepto-quark 20 with Wilson lines 180 Lie algebra monodromy 214 graded 26 MSSM 30 light-cone coordinate 113 from superstring 268 light-cone gauge 113, 114, 124 problems 32, 290 Index 403 mu(µ) problem 289 higher dimensional– 57 from string 289 in fermionic construction 370 multiplicity 192, 302, 387 non-prime 273 Z4 196 string theory 100 group space 243 orbifold compactification 37 spacetime 244 orbifold Hilbert space 185 orbit 43 N=2 supermultiplet 81 order 45, 57, 173, 237 hyper-multiplet 81, 85 orientifold 377 vector-multiplet 81, 84 oscillator 208 Narain compactification 178 oscillator number 159 neutral current 16 Neveu-Shwarz boundary condition parity violation 1 121 partition function 118, 185 no-adjoint theorem 256, 262 R sector 129 no-ghost theorem 116 twisted sector 189 non-contractible loop around fixed Pati-Salam group 88, 293 point 54 Peccei-Quinn symmetry 281 non-renormalization theorem 28 photino 30 stringy version 231 picture changing 205 nonabelian gauge 2 Planck mass 1, 4 nonprime orbifold 185 Poincar´e invariance 105 nonrenormalization theorem point group 45 nonperturbative 231 positive root 312 normal ordering 112 projective plane 44 NS sector 125 proton decay R sector 126 in PS model 24 NS sector proton lifetime 22 bosonic sector 122 pseudo-Majorana 67 NSR string quark 13 consistency 127 quintessential axion 285 constraints 121 number of families 359 R parity 294 Calabi-Yau space 274 trinification 292, 293 Rsector operator product expansion 144, 218 fermionic sector 122 orbifold 9, 39 R-symmetry 212 2 T /ZN 59 Ramond boundary condition 121 6 T /Z3 59 Ramond–Ramond tadpole cancellation Z2 × Z2 84 132 Z2 × Z2 in 5D 86 rank reduction 244 Z3 with 3 families 272 reducibility 314 Z4 195 Regge slope 107 2D lattices 56 reparametrization invariance 105 Abelian 60 representation problem definition 43, 45 adjoint 267, 270 equivalent class 52 field theory 7 field theoretic 71 matter 6 field theoretic Z2 × Z2 83 rho(ρ) parameter 16 404 Index

Riemann and Roch theorem 106 4D 158 Riemann surface 207 experimental window 37 right-mover mass 141 first quantization 108 root 236 heterotic 9, 36, 133 rotation matrix 62 mass formula 115 commutation relation 62 no global symmetry 374 running gauge coupling 232 NSR string 120 ultraviolet finiteness 118 S modular transformation (S) 117, world-sheet Hamiltonian 108 236 string tension 104 Sakharov conditions 17 string theory Scherk-Schwarz phase 171 no global symmetry 282 selection rules 209 strong CP problem 17, 283 selectron 30 SU(3) self-dual 138, 393 color 15 self-dual lattice 393 flavor 13 sfermion 30 SU(3) holonomy 9 shift vector 258 SU(5) simple root 312 flipped 88 absolute simple roots 312 Georgi-Glashow 88 2 sin θW 16 highest weights 299 sneutrino 30 root vectors 299 SO(d)62 simple roots 299 SO(10) 87 SU(6) 14 generators 89 super- 122 SO(8) weights 141 supercharge 26, 212 soliton background 79 supergravity 28 space group selection 210 cosmological constant 28 Spin (32)/Z2 141 supergravity Lagrangian spin field 207 10D 227 spin structure 130, 131 superpotential Spin(1,d− 1) 64 orbifold 231 Spin(1,3) 63 superspace 25 spin-statistics theorem 120 superstring 7, 119, 120 spinor mass formula 126 5D 78 superstring;Type IIA 132 Dirac 63 superstring;Type IIB 132 in arbitrary dimensions 62, 63 supersymmetry 25 spinorial basis 64 extended supersymmetry 27 Weyl 64 global supersymmetry 26 spinor state 65 Hamiltonian 26 squark 30 number of 59 standard embedding 162, 357 SUSY breaking 287 standard model 13 world-sheet 126 uniqueness problem 34 supersymmetry breaking standard-like model 269 explicit 82 features 270 supersymmetry from orbifold 59 flaws 270 survival hypothesis 23 string symmetry breaking Index 405

spontaneous 3 weak and electromagnetic interactions symmetry principle 5 5 unorientedness 132 T modular transformation (T ) 117 untwisted sector 46, 151, 162 T-duality 136 untwisted string 150 tachyon 112, 116 tachyon free 142 vacuum energy 166 tension 74 table for left movers 389 theme of unification 19 table for right movers 390 theta function 139 twisted sector 387 theta(θ) vacuum 17 vacuum energyc ˜ 153 threshold correction 232, 296 vacuum phase 191 top-bottom unification 87 vector representation 63 top-down approach 4 vector-like representation toroidal compactification 72, 80 N =2 27 triality 141 Veneziano amplitude 213 triality relation 162 vertex operator 112, 204, 205, 254 triangular decomposition 254 Virasoro algebra 109–111 trinification 24 super 122, 123 from Z3 orbifold 292 Virasoro operator 110 neutrino mass 293 normal ordered product 110 R parity 292 volume modulus 181 spectrum 292 two problems 292 weak mixing angle 15, 16 twist at GUT scale 277 irreducible 58 weight 236 reducible 58 Weyl field 18 twist field 207 238 twisted algebra 258 Weyl reflection 238, 316 twisted Hilbert space 186 Weyl scaling 110 twisted sector 46 Weyl symmetry 105 divided into subsectors by fixed Wilson line 171, 283 points 47 conditions 200 twisted sector spectrum 235 continuous 244 twisted string 150, 151 winding 180 wino 30 U(1) world sheet momentum 158 anomalous 281 world-sheet 105 charges 259, 274 world-sheet action 120 D-term from anomalous U(1) 281 world-sheet boson 57 eigenvalue of any weight 275 world-sheet supercurrent 121 more than one anomalous 378 world-sheet SUSY 120 normalization problem 206, 277 unification Yukawa coupling coupling constants 5, 269, 295 TTT 222 families 6 TTU type 225 of couplings in MSSM 32 area rule 223 of families 23, 33 box closing rule 226 theme of 5 selection rules 209 406 Index

Z2 orbifold 48 fixed points with a G2 lattice 55 fixed point 49 Zm×Zn orbifold 293 no fixed point 49 ZN 45 on 5D fermion 79 Z12 orbifold 297 SUSY breaking 87 zero mode Z2×Z2 orbifold 39, 294 Z2×Z2 84 out of S1 50 zero point energy 166, 187 Z3 orbifold 37, 52, 309 c,˜ c 153, 398 covering space 53 zeta function regularization 115 Z6 orbifold 273 zino 30 Lecture Notes in Physics For information about earlier volumes please contact your bookseller or Springer LNP Online archive: springerlink.com

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