Toda 3-Point Functions From Topological Strings

TL;DR

This work solves the longstanding problem of Toda CFT 3-point functions by leveraging the AGT-W correspondence and topological-string technology. It provides a concrete dictionary linking the 4D partition functions of TN theories on $S^4$ to undeformed Toda 3-point structure constants, obtained as the $\beta\to 0$ limit of the 5D index on $S^4\times S^1$ after isolating decoupled factors. The authors extend the formalism to the $q$-deformed setting, define explicit $\Upsilon_q$-based building blocks, and reveal enhanced Weyl symmetries (SU(4) for Liouville at $N=2$ and $E_6$ for $N=3$). They demonstrate slicing invariance, connect the index to the Weyl-invariant part of the structure constants, and provide explicit checks in the Liouville case, establishing a practical framework for computing 4D TN structure constants and advancing the interface between Toda CFT, gauge theories, and topological strings.

Abstract

We consider the long-standing problem of obtaining the 3-point functions of Toda CFT. Our main tools are topological strings and the AGT-W relation between gauge theories and 2D CFTs. In arXiv:1310.3841 we computed the partition function of 5D $T_N$ theories on $S^4 \times S^1$ and suggested that they should be interpreted as the three-point structure constants of q-deformed Toda. In this paper, we provide the exact AGT-W dictionary for this relation and rewrite the 5D $T_N$ partition function in a form that makes taking the 4D limit possible. Thus, we obtain a prescription for the computation of the partition function of the 4D $T_N$ theories on $S^4$, or equivalently the undeformed 3-point Toda structure constants. Our formula, has the correct symmetry properties, the zeros that it should and, for $N=2$, gives the known answer for Liouville CFT.

Figures (6)

• Figure 1: This figure depicts the identification of the $\boldsymbol{\alpha}$ weights appearing on the Toda CFT side with the position of the flavor branes on the $T_N$ side, here drawn for the case $N=4$.
• Figure 2: The figure shows the $E_6$ Dynkin diagram together with our labeling of the simple roots.
• Figure 3: The parametrization and Kähler parameters of the $T_2$ and $T_3$ junctions. The external "mass" parameters are shown in red, the "face" moduli in blue and the "edge" ones in black.
• Figure 4: The left part of the figure shows the strip diagram, while the right one depicts the dissection of the $T_N$ diagram into $N$ strips. The partitions associated with the horizontal, diagonal and vertical lines are $\nu_{i}^{(j)}$, $\mu_{i}^{(j)}$ and $\lambda_{i}^{(j)}$ with $j=1,\ldots, N-1$, $i=1,\ldots, N-j$ respectively. The Kähler parameters of the horizontal, diagonal and vertical lines are $Q_{n;i}^{(j)}$, $Q_{m;i}^{(j)}$, $Q_{l;i}^{(j)}$ respectively with the same range of indices.
• Figure 5: This figure shows the three different possible preferred directions for the $T_3$ junction. Each one is labeled by the Kähler moduli of the non-full spin content that is factorized. We also indicate the names of the partitions entering the instanton sums and to avoid clutter, we only do it for the middle one.