The Im-flip condition in the two-dimensional Potts model

I have been using this blog for publishing the reviewer reports that I write for journals, since the journals typically do not publish the reports. However, the new journal SciPost Physics does publish the reports for accepted articles. I have recently reviewed an article by Gorbenko, Rychkov and Zan for SciPost Physics, and written about the experience: it would seem that I need not blog about that article, since my report is already online.

However, not everything that I have to say on the article made it into the report. I will now write on two calculations that I did: the first one is a test of one of the article’s main predictions in more general cases, the second one is a direct derivation and generalization of a technical result that they obtain in a roundabout way.

On the Im-flip condition

 

The Im-flip condition is a relation between structure constants and conformal dimensions of the fields, which Gorbenko et al predict as a consequence of their ideas on renormalization group flows. They study these flows in the neighbourhood of the central charge $c=1$. The flows are driven by the field $V_{(3,1)}$, which becomes exactly marginal (i.e. has conformal dimension one) at $c=1$. The condition states that the ratio $$\rho_{IF}(V) = \lim_{c\to 1}\frac{\Im \Delta(V)}{\left<VVV_{(3,1)}\right>}$$ does not depend on the field $V$. In words, this is the ratio between the imaginary part of the conformal dimension of the primary field $V$, and a three-point structure constant that involves $V$ and $V_{(3,1)}$, all taken at $c=1$.

In the article, the Im-flip condition is tested for a few fields in the Potts model. Here I would like to explore it for more general fields, which may or may not belong to that model. We do not need to be specific about the model, because the three-point structure constant $\left<VVV_{(3,1)}\right>$ is a universal quantity, and can be computed as a consequence of $V_{(3,1)}$ being a degenerate field with a singular vector at level $3$. This is the technical result that I will discuss later. For the moment let me focus on the limit of $\Im \Delta(V)$. To define this limit, we need to specify how the dimension $\Delta(V)$ depends on the central charge. In the Potts model as in minimal models, there are fields of two types:

• Diagonal fields of the type $V^D_{(r,s)}$, which has spin zero and conformal dimension $$\Delta_{(r,s)}= \frac12(1-r^2)b^2 +\frac12 (1-s^2)b^{-2} + 1-rs$$ where $r,s$ are fixed i.e $c$-independent. This includes degenerate fields, for which $r,s\in \mathbb{N}^*$. The parameter $b^2$ is related to the central charge $c$ by $$c= 1 + 6\left(b+b^{-1}\right)^2$$ Let $b^2=-1+\epsilon$, then $c=1-6\epsilon^2 + O(\epsilon^3)$ and $$\Delta_{(r,s)} = \frac12(r-s)^2 -\frac12 \epsilon (r^2-s^2) + O(\epsilon^2)$$ If $\epsilon$ is not real, $\Im \Delta_{(r,s)}$ is proportional to the first subleading term of $\Delta_{(r,s)}$.
• Non-diagonal fields $V^N_{(r,s)}$, with left and right dimensions $\frac12\Delta_{(r,\pm s)}$ (and therefore spin $rs$). The total dimension is $\frac12(\Delta_{(r,s)}+\Delta_{(r,-s)})$, whose first subleading term is formally the same as for a diagonal field. (The leading term differs, though.)

Therefore, we find $$\lim_{c\to 1} \Delta(V^N_{(r,s)}) = \lim_{c\to 1} \Delta(V^D_{(r,s)}) \propto r^2-s^2$$ As we will see, the structure constant $\left<V^N_{(r,s)}V^N_{(r,s)}V_{(3,1)}\right>$ is also proportional to $r^2-s^2$, provided $r\pm s\notin\mathbb{Z}$. So we can normalize the Im-flip ratio such that its value for non-diagonal fields is one, $$\rho_{IF}(V^N_{(r,s)}) = 1$$ The situation is different for diagonal fields: $$\rho_{IF}(V^D_{(r,s)}) \underset{r-s\notin\mathbb{Z}}{=} \frac{r+s}{r-s}$$ This is one if and only if $s=0$. But then $V^D_{(r,0)}$ is indistinguishable from $V^N_{(r,0)}$. In the special case $r-s\in\mathbb{N}^*$, we find $$\rho_{IF}(V^D_{(r,s)}) \underset{r-s\in\mathbb{N}^*}{=} \frac{r+s}{r-s}\cdot\frac{r-1}{r+1}$$ which is one if $s=1$.

To summarize, we find that the fields with the correct Im-flip ratio (which we normalized to one) are the non-diagonal fields $V^N_{(r,s)}$ (provided $r\pm s\notin\mathbb{Z}$), and the diagonal fields of the type $V^D_{(r,1)}$ with $r> 1$. The particular fields that were studied by Gorbenko et al are all of this type, so they all have the same Im-flip ratio. Remarkably, the diagonal fields in the Potts model are all of the type $V^D_{(r,1)}$.

However, according to Jacobsen and Saleur Section 5.5, some diagonal fields have $r\leq 1$, and some non-diagonal fields have $r\pm s\in \mathbb{Z}$: such fields apparently violate the Im-flip condition. This might be due to the logarithmic nature of the offending fields, as logarithmic fields would probably not be subject to the Im-flip condition. Whether the fields are logarithmic or not is however unknown, as this information cannot be deduced from the partition function.

Structure constants of degenerate fields

 

We want to compute structure constants of the type $\left<VVV_{(3,1)}\right>$. The trick, sometimes called Teschner’s trick, is to consider four-point correlation functions that involve the degenerate field $V_{(2,1)}$. Let us assume we want to compute the structure constants $$c_{(2,1)} = \left< V_{(2,1)} V_{(2,1)} V_{(3,1)} \right> \quad , \quad c_{(3,1)} = \left< V_{(3,1)} V_{(3,1)} V_{(3,1)} \right>$$ We normalize the fields so that $\left<V_{(1,1)} V V \right>=1$ for any field $V$. Let us write the s-channel decompositions of the following four-point functions: $$\left<V_{(2,1)} V_{(2,1)} V_{(2,1)}V_{(2,1)} \right> = \left|\mathcal{F}_-\right|^2 + c_{(2,1)}^2 \left|\mathcal{F}_+\right|^2$$ $$\left<V_{(2,1)} V_{(2,1)} V_{(3,1)}V_{(3,1)} \right> = \left|\mathcal{G}_-\right|^2 + c_{(2,1)}c_{(3,1)} \left|\mathcal{G}_+\right|^2$$ where $\mathcal{F}_\pm,\mathcal{G}_\pm$ are some hypergeometric conformal blocks. Single-valuedness of these four-point functions determines $c_{(2,1)}^2$ and $c_{(2,1)}c_{(3,1)}$. They are given by eq. (2.3.46) of my review article, where $A,B, C$ are given by (2.3.32) with $\alpha_{(2,1)}=-\frac{b}{2}$ and $\alpha_{(3,1)}=-b$: $$c_{(2,1)}^2 = \frac{\gamma(-b^2)\gamma(-1-3b^2)}{\gamma(-2b^2)\gamma(-1-2b^2)} \quad ,\quad c_{(2,1)}c_{(3,1)} = \frac{\gamma(-b^2)^2 \gamma(1+2b^2)\gamma(-1-4b^2)}{\gamma(-2b^2)\gamma(-1-2b^2)}$$ This leads to $$c_{(3,1)} = 2^{-2-4b^2}(1+2b^2)\gamma(\tfrac12+b^2)\gamma(-\tfrac12-2b^2)\sqrt{-\frac{\gamma(-b^2)}{\gamma(-1-3b^2)}}$$ where we used the formulas $\gamma(x+1)=-x^2\gamma(x)$ and $\gamma(2x)= 2^{4x-1}\gamma(x)\gamma(x+\tfrac12)$. In particular, for $c=1$ i.e. $b^2=-1$, we have $$c_{(2,1)}\underset{c=1}{=}\frac{\sqrt{3}}{2} \quad , \quad c_{(3,1)} \underset{c=1}{=} \frac{4}{\sqrt{3}}$$ Let us generalize the calculation to diagonal fields $V_\alpha$, and compute the structure constants $c_\alpha = \left<V_\alpha V_\alpha V_{(3,1)}\right>$. We find $$c_{(2,1)}c_\alpha = \frac{\gamma(-b^2)^2 \gamma(1-2b\alpha)\gamma(-1-2b^2+2b\alpha)}{\gamma(-2b^2)\gamma(-1-2b^2)}$$ In the generic case where $\gamma(1-2b\alpha)$ and $\gamma(-1-2b^2+2b\alpha)$ stay finite, the limit is simply $$c_{(2,1)}c_\alpha \underset{c=1}{=} -\alpha^2 = \frac12 \Delta_\alpha$$ In particular this formula works for the spin field, whose dimension is $\Delta_{(\frac12,0)}\underset{c=1}{=}\frac18$, so that $c_{(\frac12,0)} \underset{c=1}{=} \frac{1}{8\sqrt{3}}$. Things are a bit more complicated if these two gamma factors go to zero or infinity. In particular, if $\alpha = \alpha_{(r,s)} = \frac12((1-r)b+(1-s)b^{-1})$ where $r,s$ are fixed numbers such that $r-s\in\mathbb{Z}$. By invariance under $(r,s)\to (-r,-s)$, we assume $r>s$, and find $$\renewcommand{\arraystretch}{1.3} c_{(2,1)} c_{(r,s)} \underset{c=1}{=} \frac12\Delta_{(r,s)} \frac{r+1}{r-1}$$ In particular this reproduces our previous formulas for $c_{(2,1)},c_{(3,1)}$.
For a non-diagonal field with left and right dimensions $\frac12(\Delta\pm S)$, we only need to take the geometric mean of the results for two diagonal fields of dimensions $\frac12(\Delta\pm S)$. In particular, for a field $V^N_{(r,s)}$ whose left and right dimensions are of the type $\frac12\Delta_{(r,\pm s)}$ with $r\pm s\notin \mathbb{Z}$, the result is $$c_{(2,1)}c^N_{(r,s)} \underset{c=1}{=} \frac12 (r^2-s^2)$$