Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
Shoichi Kanno, Yutaka Matsuo, Shotaro Shiba
TL;DR
This work tests the AGT-W correspondence between 4d ${ m N}=2$ ${ m SU}(3)$ quiver gauge theories and the ${A_2}$ Toda CFT by constructing explicit lower-order ${5}$-point Toda correlators and matching them to Nekrasov partition functions. It develops a contour-based recursion for ${W}$-algebra descendants that reduces calculations to a tractable set of 3-point vertices and inverse Shapovalov matrices, enabling concrete checks of the duality through both the ${ m SU}(3) imes{ m SU}(3)$ and ${ m SU}(3) imes{ m SU}(2)$ quivers. For the ${ m SU}(3) imes{ m SU}(3)$ case, the authors verify the 1-loop factor and low-instanton contributions up to total descendant level $ig|Y_1ig|+ig|Y_2ig| leq 3$, after identifying the Toda and gauge-theory parameters and incorporating the ${U(1)}$ factor. The ${ m SU}(3) imes{ m SU}(2)$ case reveals subtleties such as vanishing ${ m Upsilon}(Q)$ factors and the need to potentially modify 3-point data when multiple punctures are degenerate, signaling directions for future refinement of the AGT-W framework in mixed-rank quivers.
Abstract
We give some evidences of the AGT-W relation between SU(3) quiver gauge theories and A_2 Toda theory. In particular, we derive the explicit form of 5-point correlation functions in the lower orders and confirm the agreement with Nekrasov's partition function for SU(3)xSU(3) quiver gauge theory. The algorithm to derive the correlation functions can be applied to general n-point function in A_2 Toda theory which will be useful to establish the relation for more generic quivers. Partial analysis is also given for SU(3)xSU(2) case and we comment on some technical issues which need clarification before establishing the relation.