Minimal model (physics)

In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. Minimal models can also be defined based on algebras that are larger than the Virasoro algebra, such as W-algebras.

Classification

• $c=1-6{(p-q)^{2} \over pq}$
• $h=h_{r,s}(c)={{(pr-qs)^{2}-(p-q)^{2}} \over 4pq}$

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.

$c=1-{6 \over m(m+1)}=0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28,\ldots$

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

$h=h_{r,s}(c)={((m+1)r-ms)^{2}-1 \over 4m(m+1)}$

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

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York City, 1997. ISBN 0-387-94785-X.