Minimal models - Alchetron, The Free Social Encyclopedia

In theoretical physics, a minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models are parameterized by two integers p,q. A minimal model is unitary if | p − q | = 1 .

Classification

c = 1 − 6 ( p − q ) 2 p q

h = h r , s ( c ) = ( p r − q s ) 2 − ( p − q ) 2 4 p q

These conformal field theories have a finite set of conformal families which close under fusion. However, generally these will not be unitary. Unitarity imposes the further restriction that q and p are related by q=m and p=m+1.

c = 1 − 6 m ( m + 1 ) = 0 , 1 / 2 , 7 / 10 , 4 / 5 , 6 / 7 , 25 / 28 , …

for m = 2, 3, 4, .... and h is one of the values

h = h r , s ( c ) = ( ( m + 1 ) r − m s ) 2 − 1 4 m ( m + 1 )

for r = 1, 2, 3, ..., m−1 and s= 1, 2, 3, ..., r.

The first few minimal models correspond to central charges and dimensions:

m = 3: c = 1/2, h = 0, 1/16, 1/2. These 3 representations are related to the Ising model at criticality. The three operators correspond to the identity, spin and energy density respectively.

m = 4: c = 7/10. h = 0, 3/80, 1/10, 7/16, 3/5, 3/2. These 6 give the scaling fields of the tri critical Ising model.

m = 5: c = 4/5. These give the 10 fields of the 3-state Potts model.

m = 6: c = 6/7. These give the 15 fields of the tri critical 3-state Potts model.

References

Minimal models Wikipedia

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