Braiding matrix from CFT first principles

Question

Various CFT models are known to produce representations of braid groups.

A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps on $S^2$ is

$$ B = \exp \left( \frac{4 \pi i}{k + 2} T^a \otimes T^a \right). $$

I only know this formula because WZW models are holographically dual to the 3-dimensional gauge theory with Chern-Simons action, which is known to produce knot polynomials related to quantum groups (the Jones polynomial). So I guessed the expression above by simply writing the $R$-matrix of $U_q(\mathfrak{su_2})$ for two fundamental irreps.

However, I would very much like to understand how the action of $B$ on conformal blocks of the WZW model can be explicitly calculated (at least for the simplest possible case of two fundamental irreps on $S^2$). The calculation should only use the first principles of CFT, not clever connections to Chern-Simons and quantum groups (as I already know how to obtain the answer from them).

I also admit to not fully understand the conformal blocks construction, so please add any information that you think is relevant.

As a follow-up question, which CFT models produce representations of the braid group? Is there a specific criterion?

Answer 1

This question is really about the conformal blocks of two-dimensional CFT, not specific to WZW models. An $n$-point conformal block on the sphere is a locally holomorphic function of $n$ points $(z_1,\dots,z_n)$, which is singular when some points coincide. Elements of the braid group act on conformal blocks by moving the $n$ points so that they never collide, and that they end up in a permutation of $(z_1,\dots,z_n)$.

The braiding matrix describes how conformal blocks behave under a transposition, for example the exchange of $z_1$ with $z_2$. The matrix can be diagonalized using a basis of blocks that behave as $$\mathcal{F}_{\Delta}(z_1, z_2,\dots)\underset{z_1\to z_2}{=} (z_1-z_2)^{\Delta-\Delta_1-\Delta_2}(1 + O(z_1-z_2))$$ where $\Delta_1,\Delta_2$ are the conformal dimensions of the primary fields at $z_1,z_2$. Under braiding, such a block indeed just picks the factor $$B_\Delta = e^{\pi i(\Delta-\Delta_1-\Delta_2)}$$ To relate this to your WZW example, let us write $T^a_1,T^a_2$ the Lie algebra elements acting on the fields $1$ and $2$. By the Sugawara construction you have $\Delta_i = \frac{T^a_i T^a_i}{2(k+2)}$. Moreover, for our particular bases of blocks, we also have $\Delta = \frac{(T^a_1+T^a_2)(T^a_1+T^a_2)}{2(k+2)}$ thus $\Delta-\Delta_1-\Delta_2 = \frac{T^a_1T^a_2}{k+2}$. This explains your braiding matrix except for the factor $4$, which I do not see what to do with at the moment.

For more on conformal blocks in 2d CFT, see Wikipedia https://en.wikipedia.org/wiki/Virasoro_conformal_block . For all the details, see Moore-Seiberg, "Classical and Quantum Conformal Field Theory".

Answer 2

The representation of the $n$-strand braid group arises from integrating the $n$-variable Knizhnik-Zamolodchikov (KZ) equations, which are Ward identities for the movement of the insertion points for the $n$-point correlation function. Wikipedia has a nice description, and there is a short book by T. Kohno which explains the connection between 2D CFT and 3D TQFT as a clever encoding of the KZ equations.

All rational CFTs will produce braid group representations. Beyond that I don't know.