Minimal model (physics)

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.

$c=1-6{(p-q)^{2} \over pq}$

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

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 element of the sequence of the minimal models is the critical behavior of the Ising model, followed by the tricritical Ising model. There also exist supersymmetric minimal models.

References

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