I am currently studying reducible Verma module of minimal models(by chapter $7.3.3$ of di Francesco) and I seem to arrive to contradiction at some point.
Here is the setup:
Highest-weight states $h_{r,s} (c)$ defined in Kac determinant are supposed to exist for $r,s \geq 1$

$$det M^{(l)} = \alpha_l \prod_{r,s \geq 1; r,s \leq l} [h -h_{r,s}(c)]^{p(l-rs)}$$ where $l$ defines corresponding level, $\alpha_l$ is some function of r and s and $p(l-rs)$ is number of partitions of the integer $l -rs$. Explicit form of $h_{r,s}$ may be parametrised in following way:
$$c=13-6 \big(t+\frac{1}{t} \big)$$ $$h_{r,s}(t) =\frac{1}{4}(r^2-1)t+\frac{1}{4}(s^2-1)\frac{1}{4}-\frac{1}{2}(rs-1)$$ where $t$ is some number defining value of central charge.
Now(as I understand it) minimal model may be defined by following constraint: $$h_{r,s}=h_{r+p',s+p}$$ where $p',p$ are two positive numbers.
Taking into account above property one can show that following statement holds: $$h_{r,s} = h_{p'-r,p-s}$$ (If I am right, then) this should constrain values of $r$ and $s$: $$r < p' \quad s < p$$ otherwise we will have states like $h_{-1,-2}$ which are(?) forbidden to exist by Kac formula. Yet this statement seems to be in contradiction with following expression derived in the same chapter. Later seems to require existence of states like $h_{-|r|,-|s|}$: $$h_{r,s}+rs=h_{p'+r,p-s} = h_{p'-r,p+s}$$ How to resolve this contradiction?


There is no contradiction. Verma modules with arbitrary dimensions $h$ exist as representations of the Virasoro algebra. The Kac table, defined as the dimensions $h_{r,s}$ with $1\leq r \leq p'-1$ and $1\leq s\leq p-1$, tells you which representations do appear in the minimal model with parameters $p,p'$.

The representations that appear in minimal models are not quite Verma modules, but cosets thereof. The relevant coset of the Verma module with dimension $h_{r,s}$ is defined by removing the two null vectors at levels $rs$ and $(p'-r)(p-s)$, together with their descendents.

You seem to be missing the identity $h_{-r,-s}=h_{r,s}$, which may add to the confusion.

  • $\begingroup$ Yes, I believe that this is exactly what I was looking for. Thank you for your answer! $\endgroup$ – Yaroslav Shustrov Mar 3 '17 at 11:37
  • $\begingroup$ We remove the null states at level rs and (p'-r)(p-s). Now, the conformal family of the null states associated with these objects might contain some common states, and hence, if we subtract the conformal family associated with both of these null states, then there might be extra 'subtraction', and we would need to add back those states common to both the families. How is this done? $\endgroup$ – Tushar Gopalka Oct 27 '18 at 6:45
  • $\begingroup$ Just as you state it. You determine the states common to both families by looking for their singular vectors. (When two singular vectors have the same level, they actully coincide.) Iterating, you find an infinite series of additions and subtractions. This is seen in particular in the character of the representation. If you just want to write the irreducible coset, no need to worry about this subtlety though: just take the coset of the reducible Verma module by the sum of the two submodules generated by the subvectors, never mind that the sum is not direct. $\endgroup$ – Sylvain Ribault Oct 28 '18 at 11:16

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