Reading operators from Kac tables and performing operator product expansions with them

Question

I don't know much CFT, but I'm hoping to learn some of its main results. I'm particularly interested in minimal models in the context of quantum spin chains. I tried reading BPZ, but I'm realizing that I have a fundamental confusion over how to interpret Kac tables and operator product expansions.

Part of my confusion is that BPZ note that primary fields have two labels of $\Delta$ and $\bar{\Delta}$, but they spend a significant fraction of the paper, particularly sections 5 and 6, only writing the labels of $\Delta$. In the minimal models, $\Delta$ itself is characterized by two integers, $m$ and $n$, which are often used to label the operators instead. When reading sections 5 and 6, I was taking $\psi_{(m,n)}$ to denote a purely holomorphic primary field $\psi_{(m,n)}(z)$ with $\Delta = \Delta_{(m,n)}$ and $\bar{\Delta} = 0$.

However, it seems like $\psi_{(m,n)}$ might be doing overtime for both holomorphic and antiholomorphic components of fields. For example, I'm quite confused by equations like 6.14 on page 39, which state things like $\sigma = \psi_{(1,2)} = \psi_{(2,2)}$:

Appendix E appears to use the notation $\sigma(z, \bar{z})$ and notes $\Delta_{\sigma} = \bar\Delta_{\sigma} = 1/16$, so should I be interpreting such equations as something like $\sigma(z,\bar{z}) = \psi_{(1,2)}(z) \bar{\psi}_{(1,2)}(\bar{z})$?

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It is also not entirely clear to me how to interpret the OPE. Equation 6.7 reads

$\psi_{(n_1, m_1)} \psi_{(n_2, m_2)} = \sum_{k \equiv |n_1 - n_2|+1 \pmod 2}^{n_1 + n_2-1} \sum_{l\equiv|m_1 - m_2|+l \pmod 2}^{m_1 + m_2-1} [\psi_{(k,l)}]$

where some of the terms on the right drop out (for example, there's "truncation from above" mentioned in BPZ, and there's further restrictions in the fusion rules on Wikipedia. However, the meaning of these terms isn't really clear to me, especially because I'm not sure how whether this is for purely holomorphic and antiholomorphic fields. Should I imagine that I run the OPE separately for holomorphic and antiholomorphic parts?

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To give a little bit of my perspective, I think part of the trouble is that I struggle to see a consistent way to answer the two questions above. For example, if one does the OPE on the holomorphic and antiholomorphic parts separately using the equation above, I would want to say something like $\sigma(z, \bar{z}) \sigma(0, 0) = \psi_{(1,2)}(z) \psi_{(1,2)}(0) \bar{\psi}_{(1,2)}(\bar{z}) \bar{\psi}_{(1,2)}(0) \sim ([\psi_{(1,1)}(z)] + [\psi_{(1,3)}(z)])([\bar{\psi}_{(1,1)}(\bar{z})] + [\bar{\psi}_{(1,3)}(\bar{z})])$.

However, I'm confused by this, as it seems to include too many terms -- I only want to see terms like $[I]+[\epsilon]$, where $\Delta_I = \bar{\Delta}_I = 0$ and $\Delta_\epsilon = \bar{\Delta}_\epsilon = 1/2$. However, I seem to have cross terms of the kind $[\psi_{(1,3)}(z) \bar{\psi}_{(1,1)}(\bar{z})]$ and $[\psi_{(1,1)}(z) \bar{\psi}_{(1,3)}(\bar{z})]$ corresponding to conformal families with $\Delta = 1/2, \bar \Delta = 0$ and $\Delta = 0, \bar \Delta = 1/2$ respectively.

I hope my confusions are clear, and please comment if you have any questions.

Answer 1

In 2d the conformal algebra factorizes into two Virasoro algebras, left and right. So primary fields have two dimensions $\Delta,\bar\Delta$. However, single-valuedness of correlation functions (and other assumptions) imply $\Delta-\bar\Delta \in \mathbb{Z}$. A simple way to satisfy this is to consider diagonal fields i.e. $\Delta=\bar\Delta$.

A-series minimal models such as the Ising minimal model are diagonal, so all their fields are diagonal and characterized by their left = right conformal dimension. Or by their Kac indices $(m,n)$, which are the same on the left and on the right. This does not mean that the fields factorize as products of objects that depend on $z$ and $\bar z$.

A relation such as $\sigma\times\sigma = I + \epsilon$ can have two meanings. It can be a statement about fusion products of representations of the Virasoro algebra. Or it can be a statement about OPEs of diagonal fields in the Ising model. In this context, the cross terms that you write are not present, as they involve fields that violate $\Delta-\bar\Delta\in\mathbb{Z}$. If you run OPEs separately on the left and on the right you get all the terms that are allowed by conformal symmetry, but then some of them get eliminated by other constraints such as single-valuedness.