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ISAS T r i e s t e preprint 1 6 / 8 7 / E . P . Gepner, On the spectrum of 2D conformai f i e l d t h e o r i e s . Phys. B287 (1987) 1 1 1 - 1 3 0 . Gepner, Z . Q i u , Modular i n v a r i a n t p a r t i t i o n functions for parafermionic f i e l d t h e o r i e s , N u c l . Ρ h y s . B285 [ F S 1 9 ] ( 1987 ) 4 2 3 - 4 5 3 . Witten, S t r i n g theory on group manifolds. N u c l . P h y s . B278 (1986) 493-549. von Gehlen, V . R i t t e n b e r g , Superconformai invariance in the A s h k i n - T e l l e r quantum chain with free boundary c o n d i t i o n s , J .

8)). 4) b) Toroidal boundary condition Γ 241 The lowest excitations are known since a long time here we give the complete picture. We denote by [13] the sector with Z n charge Q corresponding to the boundary condition jfÖ (see Eq. 7)). In this case we can give the operator content 54 in terms of irreps. of the U(l) Kac-Moody algebra. We have: 2 Q . 5) includes the older results [ 1 3 ] , is compatible with numerical estimates [24] and with extended modular invariance [ 3 2 ] . c. (Q=0) and without splitting into sectors according to the Zn charge is <^f ®V° Q=0 Q B C Φ ft B a^6Z (g+4hß) 16h 2 ( [ ] fr-4M ) [ ' 16h 2 7 ] ) ' .

Given by the twisted U(l) Kac-Moody algebra. This completes the description of the finite-size scaling spectra of the nstates (n£5) Hamiltonians in the c=l region. VIII. 1) ( 0 < γ < π) where σ χ , σ ο Y and ζ are Pauli matrices. This model has again a long history in statistical mechanics [ 3 3 ] . The properties of its finite-size scaling spectrum have been investigated by several authors using both analytic and numerical methods [ 3 4 ] . For γ real the system is critical with c=l. W e give here a conjecture for the whole operator content of the model for various boundary conditions.