Minimal models

In theoretical physics, the minimal models are a very concrete well-defined type of rational conformal field theory. The individual minimal models are parameterized by two integers p,q that are moreover related for the unitary minimal models.

Classification

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

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.