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000197 | 4. VOA and Q-system classification
000198 | 
000199 | The hypotheses of Huang, Kirillov and Lepowsky apply to U=V_K:
000200 | 
000201 |   * U is simple and of CFT type, with U_0=C1.
000202 |   * Positive definite lattice VOAs are regular, hence their modules are
000203 |     completely reducible, and their module category has the required vertex
000204 |     tensor category structure.
000205 |   * All irreducible module weights are nonnegative, and only U has weight
000206 |     zero.
000207 |   * The object direct_sum_(a in H) M_a is haploid because M_0 occurs once.
000208 |   * Its twist is trivial because all four conformal weights are integral.
000209 | 
000210 | HKL Theorem 3.2 therefore identifies the required VOA extensions with
000211 | haploid commutative associative algebra objects of trivial twist. Section 3
000212 | proves that there is exactly one such algebra on the fixed labelled object.
000213 | 
000214 | Existence and identification are concrete. For the even overlattice L=E8,
000215 | the standard lattice decomposition is
000216 | 
000217 |   V_L = direct_sum_(a in H) V_(K+a)                       (4.1)
000218 | 
000219 | as a V_K-module, and the lattice vertex operators give the multiplication.
000220 | Thus the unique extension is V_E8, as an extension of the marked U.
000221 | 
000222 | The unitary statement is equally exact. The unitary representation category
000223 | of a positive definite lattice VOA is a C*-tensor category. The algebra in
000224 | (4.1) is rigid, connected and commutative. Carpi, Gaudio, Giorgetti and
000225 | Hillier, Theorem 3.9, proves that algebra equivalence classes of rigid haploid
000226 | algebras correspond one to one to unitary equivalence classes of normalized
000227 | haploid Q-systems, and that an equivalence between normalized haploid
000228 | Q-systems is automatically unitary. Hence the single algebra class above is
000229 | also exactly one normalized local Q-system class.
000230 | 
000231 | For the isomorphism over the fixed base U, use their Theorem 4.7 directly.
000232 | Positive definite lattice VOAs and their tensor products are regular and
000233 | completely unitary (their Table 1 and Corollary 4.11). Theorem 4.7 says that
000234 | a VOA isomorphism of simple unitary extensions restricting to the identity
000235 | on such a base is unitary; the extending scalar product is unique. Thus the
000236 | algebraic isomorphism fixing U obtained above really is an isomorphism in
000237 | the stated unitary extension class. Their Theorem 4.10 and Corollary 4.11
000238 | also identify these normalized local Q-systems with the corresponding
000239 | irreducible local conformal-net extensions over A_U.
000240 | 
000241 | 
000242 | 5. No spectator, generation and holomorphicity
000243 | 
000244 | No spectator is an automatic conclusion only inside Ext_H(U). Equation
000245 | (0.1) fixes the entire underlying U-module and its multiplicities. An extra
000246 | tensor factor, extra sector or repeated sector would change that object and
000247 | leave the stated class.
000248 | 
000249 | The Q-system multiplication has nonzero components
000250 | 
000251 |   M_(ah) tensor M_(bh) -> M_((a+b)h).
000252 | 
000253 | In the explicit lattice representative these are the nonzero lattice
000254 | intertwiners. Uniqueness transports this property to every object of the
000255 | class. Therefore U together with M_h generates M_2h, M_3h and M_0 and hence
000256 | the whole extension. Since U is generated by its weight-one currents and the
000257 | three nontrivial sector tops all have weight one, V_E8 is generated by
000258 | (V_E8)_1.
000259 | 
000260 | There is also a categorical completeness check. The pointed category has
000261 | global dimension 16 and the algebra has categorical dimension 4. The local
000262 | module category has
000263 | 
000264 |   Dim(C_A^local) = 16/4^2 = 1.                            (5.1)
000265 | 
000266 | Thus the extension is holomorphic. Neither (5.1) nor the no-spectator
000267 | statement follows if one merely asks for an extension containing the four
000268 | sectors without fixing the whole module object and multiplicities.
000269 | 
000270 | 
000271 | 6. Canonical state space, vacuum, conformal vector and L0
000272 | 
000273 | For the now forced lattice L=E8, the standard state space is
000274 | 
000275 |   V_L = M(1) tensor C_epsilon[L],                          (6.1)
000276 | 
000277 | where M(1) is the rank-eight Heisenberg Fock space and C_epsilon[L] is the
000278 | twisted group algebra
000279 | 
000280 |   e^alpha e^beta = epsilon(alpha,beta)e^(alpha+beta).
000281 | 
000282 | The normalized lattice cocycle satisfies
000283 | 
000284 |   epsilon(0,alpha)=epsilon(alpha,0)=1,
000285 |   epsilon(alpha,beta)/epsilon(beta,alpha)
000286 |     = (-1)^(<alpha,beta>).                                (6.2)
000287 | 
000288 | Two normalized cocycles with the same commutator (6.2) differ by a
000289 | coboundary. Rescaling e^alpha by the corresponding 1-cochain gives a VOA
000290 | isomorphism. Thus epsilon is gauge data, not another extension choice. The
000291 | original supplied epsilon is one fixed representative and remains unchanged.
000292 | 
000293 | The unit of the connected algebra forces the vacuum
000294 | 
000295 |   1 = 1_M(1) tensor e^0.                                  (6.3)
000296 | 
000297 | For an orthonormal basis u_1,...,u_8 of L tensor R, the full-rank inclusion
000298 | K subset L forces the same Heisenberg conformal vector on U and V_L:
000299 | 
000300 |   omega = (1/2) sum_(j=1)^8 u_j(-1)u_j(-1)1.              (6.4)
000301 | 
000302 | It has central charge 8. On a monomial state
000303 | 
000304 |   u_(i1)(-n1)...u_(ir)(-nr)e^alpha,
000305 | 
000306 | the conformal Hamiltonian is
000307 | 
000308 |   L0 = n1+...+nr + <alpha,alpha>/2.                       (6.5)
000309 | 
000310 | Equations (6.3)-(6.5) leave no independent vacuum, conformal vector or L0
000311 | choice in the marked lattice reconstruction class.
000312 | 
000313 | 
000314 | 7. Exact vacuum n-point functions
000315 | 
000316 | Let alpha_1,...,alpha_n be lattice charges in L and take the radial domain
000317 | 
000318 |   |z_1| > |z_2| > ... > |z_n| > 0.
000319 | 
000320 | For the exponential ground fields the vacuum function is
000321 | 
000322 |   <1, Y(e^alpha_1,z_1)...Y(e^alpha_n,z_n)1>
000323 |     = delta_(alpha_1+...+alpha_n,0)
000324 |       C_epsilon(alpha_1,...,alpha_n)
000325 |       product_(i<j) (z_i-z_j)^(<alpha_i,alpha_j>),         (7.1)
000326 | 
000327 | where the ordered cocycle factor is
000328 | 
000329 |   C_epsilon(alpha_1,...,alpha_n)
000330 |     = product_(r=1)^(n-1)
000331 |         epsilon(alpha_r,alpha_(r+1)+...+alpha_n).          (7.2)
000332 | 
000333 | If a bimultiplicative representative is used, (7.2) becomes
000334 | 
000335 |   product_(i<j) epsilon(alpha_i,alpha_j).
000336 | 
000337 | Charge conservation in (7.1) is exact. Since L is integral, all exponents
000338 | are integers and mutual locality has no branch ambiguity. All ground fields
000339 | in all four H sectors are covered by taking alpha_i in the corresponding
000340 | cosets K+kh. Linear combinations follow by linearity. Correlators with
000341 | Heisenberg oscillator descendants follow uniquely by Wick contractions and
000342 | derivatives. Therefore (7.1), the Heisenberg two-point function and the VOA
000343 | identities determine every vacuum n-point function; there is no additional
000344 | state functional in this class.
000345 | 
000346 | 
000347 | 8. Unitarity, radial reflection and the strong-local net
000348 | 
000349 | The positive definite lattice construction has its normalized invariant
000350 | Hermitian form and PCT operator Theta. In a compatible cocycle gauge
000351 | 
000352 |   Theta(e^alpha)=kappa(alpha)e^(-alpha),  |kappa(alpha)|=1.
000353 | 
000354 | For a primary lattice vector e^alpha of weight
000355 | 
000356 |   d=<alpha,alpha>/2,
000357 | 
000358 | the radial adjoint relation is
000359 | 
000360 |   Y(e^alpha,z)^dagger
000361 |     = (-1)^d conjugate(z)^(-2d) kappa(alpha)
000362 |       Y(e^(-alpha),1/conjugate(z)).                       (8.1)
000363 | 
000364 | The gauge-independent general identity is the invariant adjoint formula
000365 | 
000366 |   Y(a,z)^dagger
000367 |     = Y(exp(conjugate(z)L1)
000368 |         (-conjugate(z)^(-2))^L0 Theta(a),
000369 |         1/conjugate(z)),                                  (8.2)
000370 | 
000371 | interpreted on the finite-energy domain. For any radially ordered product F
000372 | of fields supported in the unit disk,
000373 | 
000374 |   <1,F^dagger F 1> = ||F1||^2 >= 0.                       (8.3)
000375 | 
000376 | This is the precise radial-reflection condition. Raw complex correlator
000377 | values are not individually required to be positive.
000378 | 
000379 | V_E8 is generated by its weight-one currents, as proved in Section 5.
000380 | Carpi, Kawahigashi, Longo and Weiner, Theorem 8.3, states that a simple
000381 | unitary VOA generated by weight-one fields, with the allowed Virasoro
000382 | generators, is strongly local. Hence V_E8 produces its canonical irreducible
000383 | strongly local conformal net A_(V_E8). This is a chiral conformal net result;
000384 | it is not a 3+1 dimensional field theory statement.
000385 | 
000386 |