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Algebraic Proof of S-Duality Formula in Refined Topological Vertex

福田, 真之 東京大学 DOI:10.15083/0002004705

2022.06.22

概要

We deal with the five-dimensional N = 1 super Yang-Mills theories. Let Z (AN ,AM ) inst. be the instanton partition function of the AM quiver gauge theory with AN gauge group with NF = 2(N + 1) matters. We set all the Chern-Simons levels to be zero. Then, the S -duality claims the invariance of Z under the exchange (AN ,AM ) inst. between N and M . The main ob ject of the present thesis is to prove this claim. By rewriting the equality in terms of the topological vertex, we obtain the duality formula under changing the preferred directions. The key ingredient of the proof is the operator realization of the topological vertex. This is achieved by the intertwiners of the Ding-Iohara-Miki algebra. By gluing the intertwiners, we can realize what we call the Mukad´e operator. The matrix elements of the Mukad´e operator factorize as the products of the Nekrasov factors. This formula proves the claim. Moreover, the Mukad´e operator reduces to the primary fields of the Virasoro algebra, under the q , t → 1 limit. In the gauge theory terminology, this limit corresponds to the reduction to the four dimensions. Then, the matrix elements formula of the Mukad´e operator can be interpreted as the proof of the five-dimensional analogue of the Alday-Gaiotto-Tachikawa correspondence.

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