Shao-Ping Hu | Searcharxiv

arXiv2024

Proof of 5D $A_n$ AGT conjecture at $β=1$

In this paper, we give a proof of 5D $A_n$ AGT conjecture at $β=1$, where the gauge theory side is one dimension higher than the original 4D case, and corresponds to the q-deformation of the 2D conformal field theory side. We define a q-deformed $A_n$ Selberg integral, which generalizes the $A_n$ Selberg integral and the q-deformed $A_1$ Selberg integral in the literature. A q-deformed $A_n$ Selberg average formula with $n+1$ Schur polynomials is proposed and proved to complete the proof.

hep-th↗

arXiv2024

Proof of $A_{n}$ AGT conjecture at $β=1$

AGT conjecture reveals a connection between 4D $\mathcal{N}=2$ gauge theory and 2D conformal field theory. Though some special instances have been proven, others remain elusive and the attempts on its full proof never stop. When the $Ω$ background parameters satisfy $-ε_1/ε_2\equiv β=1$, the story simplifies a bit. A proof of the correspondence in the case of $A_{1}$ gauge group was given in 2010 by Mironov et al., while the $A_{n}$ extension is verified by Matsuo and Zhang in 2011, with an assumption on the Selberg integral of $n+1$ Schur polynomials. Then in 2020, Albion et al. obtained the rigorous result of this formula. In this paper, we show that their result is equivalent to the conjecture on Selberg integral of Schur polynomials, thus leading to the proof of the $A_{n}$ case at $β=1$. To perform a double check, we also directly start from this formula, and manage to show the identification between the two sides of AGT correspondence.

hep-th↗