Toda 3-Point Functions From Topological Strings II

TL;DR

The paper advances the program of computing Toda CFT 3-point structure constants by showing that the semi-degenerate case (one primary with a level-1 null vector) arises from a specific Higgsing of the non-Lagrangian $T_N$ theories within the AGT-W/topological-string framework. By analyzing contour choices (flopping frames) and employing a Kaneko-Macdonald-Warnaar sl$(N)$ hypergeometric summation identity, the authors perform explicit residue evaluations, obtaining the $q$-deformed Fateev-Litvinov 3-point functions for $W_N$ Toda and their $q o1$ limit, thereby validating their general proposal for generic Toda 3-point functions. The work also clarifies how Higgsing on the gauge-theory side maps to semi-degeneration on the Toda side, and it extends the analysis from $W_3$ to general $W_N$, with careful treatment of parameter domains, contour integrals, and decoupled contributions. This strengthens the bridge between Toda CFT, 4D/5D gauge theories via AGT, and topological strings, while suggesting avenues to treat more intricate semi-degenerate cases and conformal blocks. The results provide a concrete, nontrivial check of the authors’ overarching formula for Toda 3-point functions and pave the way for broader explorations of q-deformed structures and their 4D/5D limits.

Abstract

In arXiv:1409.6313 we proposed a formula for the 3-point structure constants of Toda field theory, derived using topological strings and the AGT-W correspondence from the partition functions of the non-Lagrangian $T_N$ theories on $S^4$. In this article, we show how the semi-degeneration of one of the three primary fields on the Toda side corresponds to a particular Higgsing of the $T_N$ theories and derive the well-known formula by Fateev and Litvinov.

Figures (14)

• 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 $N=5$.
• Figure 2: The figure illustrates the desired Higgsing procedure for the general $T_N$ diagram. We denote $7$-branes by crossed circles. The left part of the figure shows the original $T_N$ 5-brane web diagram, while the right one depicts the web diagram obtained by letting $N-1$ of the left 5-branes terminate on the same 7-brane.
• Figure 3: On the left we depict the sphere with three full punctures that corresponds to the un-Higgsed $T_N$ with $\text{SU}(N)^3$ global symmetry. On the right we show the sphere with two full punctures and one L-shaped $\{N-1,1\}$ puncture. This particular Higgsing of $T_N$ leads to a theory with with $\text{SU}(N)\times \text{SU}(N)\times \text{U}(1)$ global symmetry. The partition function of this theory will lead to the Toda 3-point function with one semi-degenerate primary insertion.
• Figure 4: On the left part of this figure, we see $N$ 5-branes ending on $n$ 7-branes in bunches of $\ell_1,\ldots ,\ell_n$ 5-branes each. On the right side of the figure, we depict the Young diagram $\{\ell'_1,\ell'_2,\dots,\ell'_n\}$ that gives the flavor symmetry of the corresponding puncture. Having $n$ bunches of 5-branes, each ending of a 7-brane leads to a puncture in the Gaiotto curve with flavor symmetry $S(\text{U}(k_1)\times \cdots \times \text{U}(k_r))$, where the widths $k_i$ of the boxes are equal to the numbers of stacks with the same number of branes per stack.
• Figure 5: In this figure we present the dot diagrams of $T_4$ with three different Higgsings. On the left we have the un-Higgsed dot diagram with three full punctures, $\text{SU}(4)^3$ global symmetry and three Coulomb moduli. In the middle, the four D5 branes end on two D7 branes with two D5 branes on each, which corresponds to the Young diagram $\{2,2\}$. This theory has apparent global symmetry $\text{SU}(4)^2\times \text{SU}(2)$ and one closed polygon corresponding to one leftover Coulomb modulus. Finally, on the right we have the fully-Higgsed theory with three D5 branes on the first D7 brane and one D5 brane on the second D7. This theory has no Coulomb moduli left.

Theorems & Definitions (2)

• proof
• proof