Differential equations from fusion rules in 2d CFT

In two-dimensional conformal field theory, correlation functions are partly (and sometimes completely) determined by the properties of the fields under symmetry transformations. In particular, correlation functions of primary fields are relatively simple, because by definition primary fields are killed by the annihilation modes of the symmetry algebra. On top of that, there exist degenerate primary fields that are killed not only by the annihilation modes, but also by some combinations of creation modes. As a result, correlation functions that involve degenerate primary fields sometimes obey nontrivial differential equations, for example BPZ equations. Usually, these equations are deduced from the relevant combinations of creation modes, called null vectors.
Determining null vectors in representations of a symmetry algebra is often complicated, as the algebraic structures of the relevant algebras and of their representations can themselves be complicated. Even in the case of the Virasoro algebra, it is not easy to explicitly determine null vectors. It is however much easier to determine which representations do have null vectors, using the fusion product. For example, if we know degenerate representations $R_{(1,1)}$ and $R_{(2,1)}$ with null vectors at levels $1$ and $2$ respectively, we can deduce that the fusion product $R_{(2,1)}\times R_{(2,1)}$ is degenerate and contains $R_{(1,1)}$. The remainder of $R_{(2,1)}\times R_{(2,1)}$ must therefore be a degenerate representation, which can be identified as $R_{(3,1)}$, and has a null vector at level $3$. (See Section 2.3.1 of my review article for more details.)
An important idea is therefore that it is not the structures of the algebras and representations that matter, but rather the structure of the category of representations, in other words their fusion products. This idea has in particular been developed in the works of Fuchs, Runkel and Schweigert. But how does this help us compute correlation functions, and determine the differential equations that they obey? In other words, can we determine differential equations from fusion products, without computing null vectors?

Fusion rules and characteristic exponents

A proposed answer can be found in a recent article by Mukhi and Muralidhara. The basic idea is that fusion rules determine characteristic exponents of differential equations, and that this is sometimes enough for deducing the equations themselves. For example, the fusion rule $$R_{(2,1)}\times R_{(2,1)} = R_{(1,1)} + R_{(3,1)}$$ implies that any correlation function that involves two copies of the corresponding primary field $V_{(2,1)}$, behaves as $$\Big< V_{(2,1)}(z_1)V_{(2,1)}(z_2) \cdots \Big> \underset{z_1\to z_2}{=} a (z_1-z_2)^{h_{(1,1)}-2h_{(2,1)}}\Big[1+O(z_1-z_2)\Big] + a' (z_1-z_2)^{h_{(3,1)}-2h_{(2,1)}}\Big[1+O(z_1-z_2)\Big]$$ for some coefficients $a,a'$, where the conformal dimension associated to the representation $R_{(r,s)}$ is given in terms of the central charge $c$ of the Virasoro algebra by $$h_{(r,s)} = \frac14\Big((b+b^{-1})^2 - (br + b^{-1}s)^2\Big)\ ,\quad \text{where} \quad c=1+6(b+b^{-1})^2$$ Actually, such a correlation function obeys a second-order differential equation in $z_1$, whose characteristic exponents at $z_1=z_2$ are $h_{(1,1)}-2h_{(2,1)}$ and $h_{(3,1)}-2h_{(2,1)}$.
So, consider a four-point function of degenerate primary fields $\Big< V_1(z_1)V_2(z_2)V_3(z_3)V_4(z_4) \Big>$. Assume that fusion rules allow $n$ primary fields to appear in products of any two of these fields, so that our four-point function obeys a differential equation of order $n$. We want to determine this equation from the knowledge of the characteristic exponents $\lambda^{(z)}_i$ for $i\in \{1,\dots, n\}$ and $z\in\{z_2,z_3,z_4\}$. Writing this equation as $$\sum_{k=0}^n (-1)^{n-k} W_k \partial_{z_1}^k f = 0$$ the coefficients $W_k$ are Wronskians of a basis of $n$ solutions, see eq. (2.3) of the article. (Solutions are called conformal blocks.) Ratios of Wronskians $\frac{W_k}{W_l}$, and their logarithmic derivatives $\frac{\partial_{z_1}W_k}{W_k}$, have trivial monodromies around the singularities $z_2,z_3,z_4$, and are therefore meromorphic functions of $z_1$, with poles at $z_2,z_3,z_4$. It is possible to compute their residues at these poles from the characteristic exponents, and in some cases to deduce the differential equation. (See the article for the details.)
As a simple test of these ideas, let us focus on relations between characteristic exponents. The function $\frac{\partial_{z_1}W_n}{W_n}$ is meromorphic, with simple poles at $z_2,z_3,z_4$, and the residues $$r(z_j) = -\frac{n(n-1)}{2} + \sum_{i=1}^n \lambda^{(z_j)}_i$$ Since the primary field $V_1(z_1)$ with dimension $h_1$ behaves at infinity as $V_1(z_1)\underset{z_1\to \infty}{=} O(z_1^{-2h_1})$, we also have a pole at infinity with the residue $$r(\infty) = -n(n-1) -2nh_1$$ (The term $-\frac{n(n-1)}{2}$ is the number of derivatives in the Wronskian $W_n$. The residue at infinity involves twice this number, because at infinity all the solutions have the same characteristic exponent, so infinity should be considered a singularity with exponents $-2h_1,-2h_1-1,\dots,-2h_1-(n-1)$.) The sum of the residues, where the residue at infinity counts with a minus sign, must vanish, $$S = -r(\infty)+ \sum_{j=2}^4 r(z_j) =0$$ The resulting relation between characteristic exponents is $$S= -\frac{n(n-1)}{2} + 2nh_1 + \sum_{i=1}^n\sum_{j=2}^4 \lambda^{(z_j)}_i =0$$

 

Simple examples

The four-point function $\Big<V_{(2,1)}(z_1)V_{(2,1)}(z_2)V_{(2,1)}(z_3)V_{(2,1)}(z_4)\Big>$ obeys a BPZ equation of order $n=2$. The characteristic exponents are the same at all singularities, $$\lambda_1^{(z_j)} = h_{(1,1)}-2h_{(2,1)} \quad , \quad \lambda_2^{(z_j)} = h_{(3,1)}-2h_{(2,1)}$$ We therefore find $$S = -1 + 3h_{(1,1)} -8 h_{(2,1)} + 3h_{(3,1)}$$ Since $h_{(1,1)}=0, h_{(2,1)}=-\frac12 -\frac34 b^2$ and $h_{(3,1)}=-1-2b^2$, this vanishes as expected.
More generally, the four-point function $\Big<V_{(n,1)}(z_1)V_{(n,1)}(z_2)V_{(n,1)}(z_3)V_{(n,1)}(z_4)\Big>$ obeys an equation of order $n$, whose characteristic exponents follow from the fusion rule $$V_{(n,1)}\times V_{(n,1)} = V_{(1,1)} + V_{(3,1)} + \cdots + V_{(2n-1,1)}$$ In this case we have $$S = -\frac{n(n-1)}{2} -4n h_{(n,1)} +3\sum_{i=1}^n h_{(2i-1,1)}$$ We compute $h_{(n,1)} = -\frac12(n-1) -\frac14(n^2-1)b^2$ and $\sum_{i=1}^n h_{(2i-1,1)} = -\frac12 n(n-1) -\frac13 n(n^2-1)b^2$, and find $S=0$.
Let us discuss an example based on the algebra $W_3$. (See this article for some background.) We consider degenerate representations of $W_3$ that correspond to finite-dimensional representations of the Lie algebra $s\ell_3$, and have the fusion rules $$3\times \bar 3 = 1 + 8 \quad , \quad 3\times 3 = \bar 3 + 6$$ Their conformal dimensions are $$h_1 = 0 \quad , \quad h_3 = h_{\bar 3} = -1-\frac43 b^2 \quad , \quad h_6 = -2 - \frac{10}{3}b^2 \quad , \quad h_8 = -2-3b^2$$ The four-point function $\Big< V_3(z_1)V_{\bar 3}(z_2) V_3(z_3) V_{\bar 3}(z_4)\Big>$ obeys a differential equation of order $n=2$, whose characteristic exponents obey $\lambda_i^{(z_2)}=\lambda_i^{(z_4)}\neq \lambda_i^{(z_3)}$. We find $$S = -1 - 8h_3 + 2h_1 + 2h_8 + h_{\bar 3} + h_6 = 0$$
 

The problem with multiple components

Consider a theory with an affine $s\ell_2$ Lie algebra at a level $k\in\mathbb{C}$. An affine primary field $\Phi^j$ with spin $j$ has the dimension $$h_j = \frac{j(j+1)}{k+2}$$ For $j\in\frac12 \mathbb{N}$ this field is degenerate, and the fusion rule of the corresponding representation is $$R_j \times R_j = \sum_{i=0}^{2j} R_i$$ Therefore, the four-point function $\Big< \Phi^j(z_1) \Phi^j(z_2)\Phi^j(z_3)\Phi^j(z_4)\Big>$ obeys a differential equation of order $n=2j+1$. If we naively computed characteristic exponents from conformal dimensions of primary fields as we did before, then the combination that we would expect to vanish would be $$S_0 = - j(2j+1) -4(2j+1)h_j + 3\sum_{i=0}^{2j} h_i$$ Using $\sum_{i=0}^{2j} h_i = \frac{4}{3(k+2)}j(j+1)(2j+1)$, we however find $$S_0 = -j(2j+1) \neq 0$$ In the article by Mukhi and Muralidhara, such discrepancies are attributed to primary fields having multiple components, in other words to irreducible representations containing several primary fields. And indeed, there are actually $2j+1$ primary fields of spin $j$, labelled by a conserved number $m=-j,-j+1,\dots,j$ and denoted $\Phi^j_{m}$. In the operator product expansion $\Phi^\frac12_{\frac12}\Phi^\frac12_{\frac12}$, we can have only fields with $m=1$. So the primary field $\Phi^0_0$ cannot appear, and we expect the representation $R_0$ to be represented by a level one affine descendent field instead. The corresponding exponent is therefore $h_0 -2h_\frac12 + 1$ instead of $h_0-2h_\frac12$. In the case of the four-point function $\Big< \Phi^\frac12_\frac12(z_1) \Phi^\frac12_{-\frac12}(z_2)\Phi^\frac12_\frac12(z_3)\Phi^\frac12_{-\frac12}(z_4)\Big>$, we therefore do find $S=0$ rather than the naive result $S_0=-1$, which was computed with an incorrect exponent.
It would be nice to have a more conceptual understanding of the discrepancy, and a prediction of $S$ from features of the four representations, without having to invoke specific states in these representations.

Conclusion

The Wronskian method is a promising method for deriving differential equations for four-point conformal blocks and correlation functions. It appears technically simpler than the direct method of first determining null vectors. As a bonus, the Wronskian method yields an ordinary differential equation, whereas the direct method yields a partial differential equation, which then has to be reduced to an ordinary differential equation using global conformal invariance.

 

Multiply degenerate fields, and unphysical singularities (added on September 12th)

Let us return to the case of the Virasoro symmetry algebra. For rational values of the central charge, fields can have multiple null vectors.
For example, let us assume $h_{(3,1)}=h_{(1,2)}$. This occurs in the Ising model ($c=\frac12$) and also for $c=28$. The corresponding doubly degenerate field $V=V_{(3,1)}=V_{(1,2)}$ should obey $$V \times V = V_{(1,1)} + V_{(3,1)} + V_{(5,1)} = V_{(1,1)}+V_{(1,3)}$$ If $c=28$, then $h_{(5,1)}=h_{(1,3)}$. We conclude that $V\times V$ has two terms, with the coefficient of the term $V_{(3,1)}$ being zero. The relation between the characteristic exponents of the four-point function $\Big<V_{(1,2)}(z_1)V_{(1,2)}(z_2)V_{(1,2)}(z_3)V_{(1,2)}(z_4)\Big>$ that holds for generic $c$ also holds for $c=28$ by continuity. The relation for $\Big<V_{(3,1)}(z_1)V_{(3,1)}(z_2)V_{(3,1)}(z_3)V_{(3,1)}(z_4)\Big>$ also holds for $c=28$, and these two relations are mutually compatible. (Notice that they do not involve the same order $n$ of the differential equation.)
In the case of the Ising model, the only term that is allowed by both null vectors is $$V\times V = V_{(1,1)}$$ The characteristic exponents of the first-order differential equation for $\Big<V (z_1)V(z_2)V(z_3)V(z_4)\Big>$ are $\lambda_1^{(z_j)} =\lambda_1^{(\infty)} = -1$, and their combination is $$S = -2$$ The reason why $S\neq 0$ is now that the four-point function does not have singularities at $z_1\in\{z_2,z_3,z_4,\infty\}$ only: it also has two zeros. Its expression is indeed $$\Big<V (z_1)V(z_2)V(z_3)V(z_4)\Big> \propto \frac{1}{(z_1-z_2)(z_3-z_4)} + \frac{1}{(z_1-z_3)(z_4-z_2)} + \frac{1}{(z_1-z_4)(z_2-z_3)}$$ This suggests that the singularities of the conformal blocks are not always limited to $z_1\in\{z_2,z_3,z_4,\infty\}$. If there are extra singularities, the combination $S$ of the characteristic exponents at $z_1\in\{z_2,z_3,z_4,\infty\}$ does not have to vanish. (An extra singularity whose characteristic exponents are not all integer was found in the large $c$ limit of certain W-algebra conformal blocks.)

Earlier reference (added on September 21st)

The idea of deriving differential equations from fusion rules is also explained in Section 14 of these 1997 lecture notes by Jürgen Fuchs.

Later reference (added on December 31st)

A more systematic treatment of the derivation of differential equations from fusion rules, based on the Katz theory of Fuchsian rigid systems, has recently appeared in this article by Belavin, Haraoka and Santachiara. This article mainly deals with theories with W-algebra symmetries.

The theory of Fuchsian rigid systems is powerful, but it does not know about fusion. In a CFT with Virasoro symmetry, $R_{(3,1)}$ can be deduced from $R_{(2,1)}$ by fusion. But the Fuchsian system for a four-point function with $V_{(2,1)}$ is rigid, while the system for a four-point function with $V_{(3,1)}$ is not. It would be nice to have some extra constraints that would allow us to determine the differential equation in the latter case.