New Properties of Large-$c$ Conformal Blocks from Recursion Relation

Abstract

We study large $c$ conformal blocks outside the known limits. This work seems to be hard, but it is possible numerically by using the Zamolodchikov recursion relation. As a result, we find new some properties of large $c$ conformal blocks with a pair of two different dimensions for any channel and with various internal dimensions. With light intermediate states, we find a Cardy-like asymptotic formula for large $c$ conformal blocks and also we find that the qualitative behavior of various large $c$ blocks drastically changes when the dimensions of external primary states reach the value $c/32$. And we proceed to the study of blocks with heavy intermediate states $h_p$ and we find some simple dependence on heavy $h_p$ for large $c$ blocks. The results in this paper can be applied to, for example, the calculation of OTOC or Entanglement Entropy. In the end, we comment on the application to the conformal bootstrap in large $c$ CFTs.

Figures (24)

• Figure 1: The sketch of behaviors of $c_n= \mathrm{sgn}(c_n) n^{\alpha} \mathrm{e}^{A \sqrt{n}}$ for various values of $(h_A,h_B)$.
• Figure 2: The behaviors of the coefficients $c_n$ of AABB blocks with $h_A=\frac{c}{24}$. The left is for $(h_B,h_p)=(\frac{c}{16},\frac{c}{24} \times \mathrm{e}^{\frac{5}{2}})$ and the right is for $(h_B,h_p)=(\frac{c}{240},\frac{c}{24}\times \mathrm{e}^{\frac{5}{2}})$. The blue dots are the numerical values of $\log c_n$. The red lines are $B n^{\alpha} \mathrm{e}^{A\sqrt{n}}$ with the constant $B$ determined by the fit. We now set $c=30.01$ and, to fit $A$ and ${\alpha}$, we use the numerical values of $c_n$ at $n=500\sim1000$.
• Figure 3: The plots of the values of $A$ (left) and ${\alpha}$ (right) for various values of ($h_B,h_p$) with $h_A=\frac{c}{24}$. Some strange behaviors near the line $h_B=\frac{c}{32}$ could be resolved by using the values $c_n$ for higher $n$ to fit $A$ and ${\alpha}$ (see Appendix \ref{['subsec:app1']}). Here we set $c=30.01$ and to fit $A$ and ${\alpha}$, we use the numerical values of $c_n$ at $n=500\sim1000$.
• Figure 4: The $h_p$ dependence of ${\alpha}$. The left is for $(h_A,h_B)=(\frac{c}{24},\frac{c}{240})$ and the right is for $(h_A,h_B)=(\frac{c}{24},\frac{c}{16})$. Red dots are fitted by $c_n$ for $n=100\sim 200$ and black dots are fitted by $c_n$ for $n=500\sim 1000$. One can find that the $h_p$ dependence of ${\alpha}$ approaches to constant as we use higher $n$ to fit the values of ${\alpha}$.
• Figure 5: The plots of $c_n$. The upper two plots are for $h_A=h_B=\frac{c}{8}$. We can see that the left of two shows a linear behavior, which suggest that $c_n$ grows polynomially. The lower two plots are for $h_A=h_B=\frac{c}{240}$ and we can find a linear dependence in the right, which suggest that $c_n$ grows exponentially.