Ramanujan's $1/π$ series and conformal field theories

TL;DR

The paper reveals a physics origin for Ramanujan's remarkable $1/\pi$ series by embedding them in 2D logarithmic CFTs, notably the $c=-2$ LCFT with twist operators. It develops a stringy dispersive framework that rewrites LCFT four-point functions in terms of their logarithmic discontinuities, yielding a rapidly convergent conformal-block basis controlled by a parameter $\lambda$. A striking result is that, in the large-$\lambda$ limit, the entire Legendre relation reduces to the log-identity operator, suggesting a universal property of LCFT correlators and linking the mathematics of modular equations to holographic interpretations. The work also uncovers new, fast-converging representations for $1/\pi$, a deeper connection to Ramanujan-Orr series, and a broader landscape of LCFTs with other central charges where similar structures persist.

Abstract

In 1914, Ramanujan unveiled 17 extraordinary infinite series for $1/π$. In this work, we uncover their physics origin by relating them to 2D logarithmic conformal field theories (LCFTs), which emerge in diverse settings such as the fractional quantum Hall effect, percolation, polymers, and even holography. Through this LCFT connection, we reinterpret such infinite series in terms of fundamental CFT data -- the operator spectrum and OPE coefficients. This perspective leads to novel physics-inspired approximations for $1/π$. Drawing lessons from Ramanujan's formulae, we construct a new family of bases for expanding LCFT correlators that converge far more rapidly than the standard conformal block decomposition. This is achieved using recently developed stringy/parametric crossing-symmetric dispersion relations. Remarkably, when working with these new expansions, the action of a certain differential operator (which arises naturally from the Ramanujan connection) dramatically enhances convergence, with the entire contribution collapsing to that of the logarithmic identity operator. This striking simplification hints at a universal property of LCFTs. Finally, we discuss a new holographic interpretation of this unexpected mathematics-physics connection.

Figures (4)

• Figure 1: Slices in the $z,\bar{z}$ plane corresponding to solutions of the modular equations for $n=1,2,3,7$ from right to left. Red crosses indicate the singular values $z_0$.
• Figure 2: Heat plot showing ratio of sum of blocks ($\Delta=2n+\ell$, $n\leq 2,\ell\leq 4$) to exact answer. Left: CBD, Right: Dispersive representation ($\lambda=500$).
• Figure 3: Plot showing the contribution of the log-identity operator in the dispersive block expansion of $1/\pi$ as $\lambda$ is increased for $\sigma =1/2$ and various $z$. We observe that at large $\lambda$, the log-identity operator captures the full Legendre relation, giving $1/\pi$.
• Figure 4: Contour plot of the ratio of $G_L(z,\bar{z})/G_R(z,\bar{z})$. The blue solid lines and red dashed show the $n=1,2,3,7$ as in fig.\ref{['slice']}. The thickened red-lines shade the regions between $1.9$ (lower), $2$ (upper) and $2.9$ (lower), $3$ (upper).