Accessory parameters in confluent Heun equations and classical irregular conformal blocks

Abstract

Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its confluent versions. We extend this relation to another class of accessory parameter functions that are defined by inverting all-order Bohr–Sommerfeld periods for confluent and biconfluent Heun equation. The relevant conformal blocks involve Nagoya irregular vertex operators of rank 1 and 2 and conjecturally correspond to partition functions of a 4D ${\mathscr {N}}=2$, $N_f=3$ gauge theory at strong coupling and an Argyres–Douglas theory.

Appendices

Heun accessory parameter from continued fractions

It will be convenient for us here to transform the Heun Eq. (1.1) into its canonical form by the substitution

$$\begin{aligned} \psi \left( z\right) =z^{\frac{1}{2}-\theta _0}\left( z-t\right) ^{\frac{1}{2}-\theta _t}\left( z-1\right) ^{\frac{1}{2}-\theta _1}\phi \left( z\right) . \end{aligned}$$

(A.1)

The resulting equation for $\phi \left( z\right)$ is given by

$$\begin{aligned} \phi ''\left( z\right) +\left( \frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{\epsilon }{z-t}\right) \phi '\left( z\right) +\frac{\alpha \beta z-q}{z\left( z-1\right) \left( z-t\right) }\phi \left( z\right) =0, \end{aligned}$$

(A.2)

with

$$\begin{aligned}&\alpha =1-\theta _0-\theta _1-\theta _t-\theta _\infty ,\qquad \beta =1-\theta _0-\theta _1-\theta _t+\theta _\infty , \end{aligned}$$

(A.3a)

$$\begin{aligned}&\gamma =1-2\theta _0,\qquad \delta =1-2\theta _1,\qquad \epsilon =1-2\theta _t, \end{aligned}$$

(A.3b)

$$\begin{aligned}&q=\left( 1-t\right) {\mathscr {E}}+\frac{\gamma \epsilon +2\alpha \beta t-\left( \gamma +\delta \right) \epsilon t}{2}. \end{aligned}$$

(A.3c)

Obviously, we have a relation $\alpha +\beta +1=\gamma +\delta +\epsilon$. In spite of loosing a symmetry $\theta _k\mapsto -\theta _k$ ($k=0,1,t,\infty$) present in the normal form (1.1), the canonical form turns out to be better adapted for perturbative calculation below. A similar and very much related phenomenon is known from the AGT correspondence: 4D instanton partition functions have simpler series representations than 2D conformal blocks but loose certain of their manifest symmetries because of the presence of the so-called $\mathrm {U}\left( 1\right)$ factor.

Choose a basis in the space of solutions of (A.2) which diagonalizes the composite monodromy around the pair of singular points 0, t. Assuming that $0<|t|<1$, its elements can be represented inside the annulus $|t|<|z|<1$ in the Floquet form

$$\begin{aligned} \phi \left( z\right) =\sum _{n\in {\mathbb {Z}}}c_n z^{n+\omega }. \end{aligned}$$

(A.4)

Substituting this series into (A.2), one obtains a linear three-term recurrence relation for the coefficients,

$$\begin{aligned} A_n c_{n-1}-B_n c_n+C_n t c_{n+1}=0, \end{aligned}$$

(A.5)

where

$$\begin{aligned} A_n&=\,\left( \omega +n-1+\alpha \right) \left( \omega +n-1+\beta \right) , \end{aligned}$$

(A.6a)

$$\begin{aligned} B_n&=\,q+\left( \omega +n\right) \left( \epsilon +\delta t+\left( \omega +n-1+\gamma \right) \left( 1+t\right) \right) , \end{aligned}$$

(A.6b)

$$\begin{aligned} C_n&=\,\left( \omega +n+1\right) \left( \omega +n+\gamma \right) . \end{aligned}$$

(A.6c)

The recurrence relation above is the main reason we started with (A.2) instead of (1.1). As we will see in a moment, the crucial point for the computation of the small t expansion of the accessory parameter,

$$\begin{aligned} q\left( t\right) =\sum _{n=0}^{\infty }q_nt^n, \end{aligned}$$

(A.7)

is how the time t appears in (A.5).

For $n\ge 0$, define $u_n=\displaystyle \frac{c_{n+1}}{c_n}$. This change of variables transforms (A.5) into a nonlinear two-term “Riccati” equation $u_{n-1}=\frac{A_n}{B_n-tC_n u_n}$. Under assumption that $B_{n}=O\left( 1\right)$ as $t\rightarrow 0$ for $n\ge 1$, one can then express $u_0$ as an infinite continued fraction

$$\begin{aligned} u_0=\frac{A_1}{B_1-\frac{t C_1A_2}{B_2-\frac{tC_2A_3}{B_3-\ldots }}}. \end{aligned}$$

(A.8)

For $n\le 0$, we first define rescaled coefficients $d_{-n}=c_nt^{n}$ and rewrite the 3-term relation (A.5) as

$$\begin{aligned} tA_{-n} d_{n+1}-B_{-n} d_n+C_{-n} d_{n-1}=0, \end{aligned}$$

(A.9)

Introducing $v_n=\displaystyle \frac{d_{n+1}}{d_n}$, we have $v_{n-1}= \frac{C_{-n}}{B_{-n}-tA_{-n} v_n}$, so that

$$\begin{aligned} v_0=\frac{C_{-1}}{B_{-1}-\frac{tA_{-1}C_{-2}}{B_{-2}-\frac{tA_{-2}C_{-3}}{B_{-3}-\ldots }}}, \end{aligned}$$

(A.10)

under assumption that $B_n=O\left( 1\right)$ as $t\rightarrow 0$ for $n\le -1$. From $A_0c_{-1}-B_0c_0+tC_0c_1=0$ it follows that $t\left( C_0u_0+A_0v_0\right) =B_0$, which ultimately gives an equation determining the accessory parameter q as a function of t for given $\omega$:

$$\begin{aligned} \quad \frac{tC_0A_1}{B_1-\frac{t C_1A_2}{B_2-\frac{tC_2A_3}{B_3-\ldots }}}+\frac{tA_0C_{-1}}{B_{-1}-\frac{tA_{-1}C_{-2}}{B_{-2}-\frac{tA_{-2}C_{-3}}{B_{-3}-\ldots }}}=B_0. \end{aligned}$$

(A.11)

Note that $\left\{ A_n\right\}$ and $\left\{ C_n\right\}$ are just some monodromy-dependent constants; thus, q enters into (A.11) only through $\left\{ B_n\right\}$. Substituting into (A.6b) the expansion (A.7), the coefficients $q_k$ can be recursively determined from (A.11) by truncating the continued fraction ladder at the desired order in t.

For example, at order $O\left( 1\right)$ we have $B_0=O\left( t\right)$, and therefore

$$\begin{aligned} q_0=-\omega \left( \omega + \gamma +\epsilon -1\right) . \end{aligned}$$

(A.12a)

At order $O\left( t\right)$, one has $B_0=t\left( \frac{C_0A_1}{B_1}+\frac{A_0C_{-1}}{B_{-1}}\right) +O\left( t^2\right)$, which gives

$$\begin{aligned} q_1&=-\omega \left( \omega +\gamma +\epsilon -1\right) +\frac{\left( \omega +1\right) \left( \omega +\alpha \right) \left( \omega +\beta \right) \left( \omega +\gamma \right) }{2\omega +\gamma +\epsilon } \nonumber \\&\quad -\frac{ \omega \left( \omega +\alpha -1\right) \left( \omega +\beta -1\right) \left( \omega +\gamma -1\right) }{2\omega +\gamma +\epsilon -2}, \end{aligned}$$

(A.12b)

and so on. Substituting these coefficients into (A.3c), we obtain the expansion of ${\mathscr {E}}={\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {VI}}\left( t|\,\sigma \right)$:

$$\begin{aligned} \begin{aligned}&\,{\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {VI}}\left( t|\,\sigma \right) =\left( \delta _\sigma -\delta _0-\delta _t\right) +{\mathscr {W}}_1\left( \left\{ \delta _k\right\} \right) t + 2{\mathscr {W}}_2\left( \left\{ \delta _k\right\} \right) t^2+O\left( t^3\right) , \end{aligned} \end{aligned}$$

(A.13)

where ${\mathscr {W}}_{1,2}\left( \left\{ \delta _k\right\} \right)$ are given by (2.6a)–(2.6b), $\left\{ \delta _k\right\}$ are defined by (1.10), $\left\{ \theta _k\right\}$ are related to $\alpha ,\beta ,\gamma ,\delta ,\epsilon$ by (A.3a)–(A.3b), and the Floquet exponent appears only in ${\sigma =\omega -\theta _0-\theta _t+\frac{1}{2}}$. One thus easily recognizes in (A.13) the expansion of the logarithmic derivative $t\frac{\partial }{\partial t}{\mathscr {W}}\left( t\,\bigl |\left\{ \delta _k\right\} \right)$ of the classical regular conformal block (1.8).

Floquet characteristics for confluent Heun equations

Throughout this section, we refer to the notations of Table 1. The perturbative computation of accessory parameter functions of Floquet type presented here allows to check confluent Conjecture B (1.12) at any desired order in t. Assuming the conjecture is true, this technique provides the most elementary method of computing classical CBs of the first kind.

1.1 Equation $\hbox {H}_\mathrm {V}$

The change of variables $\psi \left( z\right) =z^{\frac{1}{2}-\theta _0}\left( z-t\right) ^{\frac{1}{2}-\theta _t}e^{\frac{z}{2}}\phi \left( z\right)$ transforms the confluent HE into the canonical form,

$$\begin{aligned} \phi ''\left( z\right) +\left( \frac{\beta }{z}+\frac{\gamma }{z-t}+1\right) \phi '\left( z\right) +\frac{\alpha z-q}{z\left( z-t\right) }\phi \left( z\right) =0, \end{aligned}$$

(B.1)

with

$$\begin{aligned}&\alpha =1-\theta _0-\theta _t-\theta _*,\qquad \beta =1-2\theta _0,\qquad \gamma =1-2\theta _t, \end{aligned}$$

(B.2a)

$$\begin{aligned}&q=-{\mathscr {E}}+\alpha t-\tfrac{1}{2}{\left( \beta +t\right) \gamma }. \end{aligned}$$

(B.2b)

Looking for the solutions of (B.1) in the Floquet form (A.4), we arrive at the same continued fraction equation (A.11) for $q\left( t\right)$, except that the coefficients $\{A_n\}$, $\{B_n\}$, $\{C_n\}$ are now given by

$$\begin{aligned} A_n&=\, \omega +n-1+\alpha , \end{aligned}$$

(B.3a)

$$\begin{aligned} B_n&=\,q-\left( \omega +n\right) \left( \omega +n-1+\beta +\gamma -t\right) , \end{aligned}$$

(B.3b)

$$\begin{aligned} C_n&=\,-\left( \omega +n+1\right) \left( \omega +n+\beta \right) . \end{aligned}$$

(B.3c)

The expansion of $q\left( t\right)$ can now be computed to any desired order. Its first terms read

$$\begin{aligned} q\left( t\right)&=\omega \left( \omega +\beta +\gamma -1\right) \nonumber \\&\quad +\left[ -\omega +\frac{\left( \omega +1\right) \left( \omega +\alpha \right) \left( \omega +\beta \right) }{2\omega +\beta +\gamma }-\frac{\omega \left( \omega +\alpha -1\right) \left( \omega +\beta -1\right) }{2\omega +\beta +\gamma -2}\right] t+O\left( t^2\right) . \end{aligned}$$

(B.4)

The expansion of the Floquet characteristic ${\mathscr {E}}$ is then obtained from (B.2b). (Note that, in contrast with non-confluent HE, the coefficients of expansions of ${\mathscr {E}}$ and $-q$ coincide starting from the quadratic term.) If we denote $\sigma =\omega -\theta _0-\theta _t+\frac{1}{2}$ as before, then

$$\begin{aligned} \begin{aligned} {\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {V}}\left( t|\,\sigma \right)&=\,\left( \delta _\sigma -\delta _0-\delta _t\right) -\frac{\theta _*\left( \delta _\sigma -\delta _0+\delta _t\right) }{2\delta _\sigma } t\\&\quad +\,\left[ \frac{\theta _*^2\left( \delta _\sigma ^2-\left( \delta _0-\delta _t\right) ^2\right) }{8\delta _\sigma ^3}-\frac{\left( 3\theta _*^2+\delta _\sigma \right) \left( \delta _\sigma ^2+2\delta _\sigma \left( \delta _0+\delta _t\right) -3\left( \delta _0-\delta _t\right) ^2\right) }{8\delta _\sigma ^2\left( 3+4\delta _\sigma \right) }\right] t^2\\&\quad +O\left( t^3\right) . \end{aligned} \end{aligned}$$

(B.5)

It is instructive to check that this indeed agrees with the limit (1.11a) and reproduces the expansion of $t\frac{\partial }{\partial t}{\mathscr {W}}_{N_f=3}\left( t\right)$, cf (2.14).

1.2 Equation $\hbox {H}_{\mathrm {III}_1}$

After the transformation $\psi \left( z\right) =\sqrt{z}\,e^{\frac{z}{2}+\frac{t}{2z}}\phi \left( z\right)$, the equation $\hbox {H}_{\mathrm {III}_1}$ becomes

$$\begin{aligned} \phi ''\left( z\right) +\left( 1+\frac{1}{z}-\frac{t}{z^2}\right) \phi '\left( z\right) +\left( \frac{t\left( \frac{1}{2}-\theta _\star \right) }{z^3}+\frac{{\mathscr {E}}-\frac{t}{2}-\frac{1}{4}}{z^2}+\frac{\frac{1}{2}-\theta _*}{z}\right) \phi \left( z\right) =0. \end{aligned}$$

(B.6)

This is not the canonical form of doubly confluent Heun equation given in http://dlmf.nist.gov/31.12.E2, which is obtained by a slightly different change of variables; yet it is more convenient for us to continue with (B.6). The Floquet substitution (A.4) with $\omega =\sigma$ yields the equation (A.11) with

$$\begin{aligned} A_n=\theta _*-\sigma -n+\tfrac{1}{2} ,\qquad B_n={\mathscr {E}}-\tfrac{t}{2}-\tfrac{1}{4}+\left( \sigma +n\right) ^2 ,\qquad C_n=\theta _\star +\sigma +n+\tfrac{1}{2}, \end{aligned}$$

(B.7)

which then allows to compute the small t expansion of ${\mathscr {E}}$. Its first few terms are given by

$$\begin{aligned} {\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {III}_1}\left( t|\,\sigma \right) = \delta _\sigma +\frac{\theta _*\theta _\star }{2\delta _\sigma } t+\left( \frac{\left( 3\theta _*^2+\delta _\sigma \right) \left( 3\theta _\star ^2+\delta _\sigma \right) }{8\delta _\sigma ^2\left( 3+4\delta _\sigma \right) }-\frac{\theta _*^2\theta _\star ^2}{8\delta _\sigma ^3}\right) t^2+O\left( t^3\right) . \end{aligned}$$

(B.8)

1.3 Equation $\hbox {H}_{\mathrm {III}_2}$

In this case, the relevant change of variables is $\psi \left( z\right) =\sqrt{z}\,e^{\frac{z}{2}}\phi \left( z\right)$. It transforms the equation $\hbox {H}_{\mathrm {III}_2}$ into

$$\begin{aligned} \phi ''\left( z\right) +\left( 1+\frac{1}{z}\right) \phi '\left( z\right) +\left( -\frac{t}{z^3}+\frac{{\mathscr {E}}-\frac{1}{4}}{z^2}+\frac{\frac{1}{2}-\theta _*}{z}\right) \phi \left( z\right) =0. \end{aligned}$$

(B.9)

The Floquet substitution (A.4) with $\omega =\sigma$ again gives the equation (A.11), whose coefficients are now given by

$$\begin{aligned} A_n=\theta _*-\sigma -n+\tfrac{1}{2} ,\qquad B_n={\mathscr {E}}-\tfrac{1}{4}+\left( \sigma +n\right) ^2 ,\qquad C_n=1, \end{aligned}$$

(B.10)

The expansion of accessory parameter function reads

$$\begin{aligned} {\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {III}_2}\left( t|\,\sigma \right)&= \delta _\sigma +\frac{\theta _*}{2\delta _\sigma } t+\frac{\left( 5\delta _\sigma -3\right) \theta _*^2+3\delta _\sigma ^2}{8\delta _\sigma ^3\left( 3+4\delta _\sigma \right) } t^2 \nonumber \\&\quad +\frac{\theta _*\left( \left( 7\delta _\sigma -6\right) \delta _\sigma ^2+\left( 9\delta _\sigma ^2-19\delta _\sigma +6\right) \theta _*^2\right) }{16\delta _\sigma ^5\left( 3+4\delta _\sigma \right) \left( 2+\delta _\sigma \right) }t^3+O\left( t^4\right) . \end{aligned}$$

(B.11)

1.4 Equation $\hbox {H}_{\mathrm {III}_3}$

In this last case (recall that it is equivalent to Mathieu equation), the analog of the above calculations is rather well known. There is no need to transform the equation. Making the Floquet ansatz $\psi \left( z\right) =\sum \nolimits _{n\in {\mathbb {Z}}}c_n z^{\frac{1}{2}+\sigma +n}$ directly in the normal form, we arrive at a three-term recurrence relation and Eq. (A.11) for ${\mathscr {E}}$ with

$$\begin{aligned} A_n=C_n=1,\qquad B_n={\mathscr {E}}-\tfrac{1}{4}+\left( \sigma +n\right) ^2 , \end{aligned}$$

(B.12)

This gives the expansion of the Mathieu characteristic value:

$$\begin{aligned} {\mathscr {E}}^{[\mathrm {F}]}_{\mathrm {III}_3}\left( t|\,\sigma \right) = \delta _\sigma +\frac{t}{2\delta _\sigma }+\frac{ 5\delta _\sigma -3}{8\delta _\sigma ^3\left( 3+4\delta _\sigma \right) } t^2+\frac{ 9\delta _\sigma ^2-19\delta _\sigma +6}{16\delta _\sigma ^5\left( 3+4\delta _\sigma \right) \left( 2+\delta _\sigma \right) }t^3+O\left( t^4\right) . \end{aligned}$$

(B.13)

To facilitate comparison with the literature, note that the parameters $\nu$ and q in, e.g., [1, Eq. 20.3.15] and [17, Eq. 28.15.E1] correspond to our $2\sigma$ and $4\sqrt{t}$ so that $\delta _\sigma =\frac{1-\nu ^2}{4}$.