Virasoro conformal blocks in closed form

In a recent article, Perlmutter investigated closed-form expressions for Virasoro conformal blocks. As a complement to that article, let me discuss what is known on such expressions, and what they are good for.

The definition of conformal blocks is the subject of an interesting discussion at Physics.StackExchange. Basically, conformal blocks are the universal building blocks of correlation functions, and are determined by conformal symmetry.
In other words, conformal blocks are particular solutions of the conformal Ward identities which correlation functions must obey. The simplest conformal blocks are the characters of Virasoro representations, which are the building blocks of torus partition functions. When it comes to the sphere, three-point conformal blocks are trivial, and the the simplest nontrivial conformal blocks are the four-point blocks.

A closed-form expression from the AGT relation

Let me first point out that a closed-form expression for four-point blocks has been known since the discovery of the Alday--Gaiotto--Tachikawa relation between two-dimensional conformal field theory and four-dimensional gauge theory. That expression is a sum over infinitely many terms, which are parametrized by pairs of Young diagrams. Besides having such a large number of terms, the expression has spurious poles -- poles of individual terms, which cancel when taking the sum. And individual terms are not invariant under reflections of momentums, whereas conformal blocks are invariant by virtue of being functions of conformal dimensions.

So the expression from the AGT relation has some disadvantages when it comes to computing conformal blocks or understanding their properties. On the other hand, it has an interpretation as a sum over descendent states in a representation of the Virasoro algebra.

Zamolodchikov's recursion formulas, and their solutions

Long before the AGT relation, conformal blocks were characterized by two distinct recursion formulas, which lend themselves to efficient numerical implementations. In his article, Perlmutter solved these recursion formulas, and derived two independent closed-form expressions for conformal blocks. Let me discuss one of these recursion formulas, in the notations of my review article where more details can be found. The recursion formula can be written as
$$
H(q, P^2) = 1 + \sum_{m,n=1}^\infty \frac{q^{mn} R_{(m,n)}}{P^2-P^2_{(m,n)}} H(q, P^2_{(m,-n)})
$$
where

• $q$ is an elliptic function of the cross-ratio of the positions of our four fields, and $P$ is the $s$-channel momentum,
• $H(q, P^2)$ is the nontrivial factor of a four-point conformal block, which depends not only on $q$ and $P$, but also on the central charge, and on the momentums of our four fields $P_1,P_2,P_3,P_4$,
• $P_{(m,n)} = \frac{i}{2}(mb+nb^{-1})$ is a degenerate momentum, where the parameter $b$ is related to the central charge,
• the prefactor $R_{(m,n)}$ is
   $$
   R_{(m,n)} = \frac{-2P_{(0,0)} P_{(m,n)}}{\prod_{r=1-m}^m \prod_{s=1-n}^n 2P_{(r,s)}}
   \prod_{r\stackrel{2}{=}1-m}^{m-1} \prod_{s\stackrel{2}{=}1-n}^{n-1} \prod_\pm (P_2\pm P_1 + P_{(r,s)}) (P_3\pm P_4+P_{(r,s)})
   $$

The recursion can be solved,
$$
H(q, P^2) = \sum_{k=0}^\infty \prod_{j=1}^k \sum_{m_j,n_j=1}^\infty \frac{q^{m_jn_j} R_{(m_j,n_j)}}{P^2_{(m_{j-1},-n_{j-1})} - P^2_{(m_j,n_j)}}
$$
with the convention $P^2_{(m_0,-n_0)} = P^2$. This closed-form formula has no spurious poles, and is invariant under the reflections $P_i\to -P_i$.

Attempting a resummation

The closed-form formula involves summing over arbitrary large numbers of integers, just like the formula from the AGT relation. It would be much better to have only finitely many sums or integrals. Let me try to take steps in this direction.

For generic values of $b$, the quantities $P_{(m,-n)}^2$ and $P_{(m',n')}^2$ live in two different sectors $S$ and $S'$ of the complex plane, which are separated by the contour
$$
C = -b^2(0,\infty) \cup -b^{-2}(0,\infty)
$$
Let a kernel $K(\Delta,\Delta')$ be a function of two complex numbers $\Delta\in S$ and $\Delta'\in S'$.
Let us define

• the product $ (K_1\star K_2)(\Delta, \Delta') = \int_C d\Delta''\ K_1(\Delta,\Delta'')K_2(\Delta'', \Delta')$,
• the identity kernel $K_0 (\Delta, \Delta') = \frac{1}{\Delta-\Delta'}$, which obeys in particular $K_0\star K_0 = K_0$,
• the limit $K(\Delta,\infty) = \lim_{\Delta'\to \infty} \Delta' K(\Delta,\Delta')$.

Let us now write
$$
\frac{1}{P^2_{(m_{j-1},-n_{j-1})} - P^2_{(m_j,n_j)}} = (K_0\star K_0)(P^2_{(m_{j-1},-n_{j-1})}, P^2_{(m_j,n_j)})
$$
and introduce the kernel
$$
K(\Delta,\Delta') = \sum_{m,n=1}^\infty K_0(\Delta, P^2_{(m,n)}) q^{mn} R_{(m,n)} K_0(P^2_{(m,-n)}, \Delta')
= \sum_{m,n=1}^\infty\frac{q^{mn} R_{(m,n)} }{(\Delta - P^2_{(m,n)})(P^2_{(m,-n)}-\Delta') }
$$
This allows us to rewrite
$$
H(q, P^2) = \sum_{k=0}^\infty K^{\star k}(P^2, \infty) = (K_0-K)^{\star -1}(P^2,\infty)
$$
This is the result of a formal calculation, and it might not be possible to actually compute the kernel $K$, let alone the inverse of $K_0-K$. Nevertheless, this suggests that there might exist relatively simple expressions for conformal blocks.