On 5-point conformal block with level 2 degenerate field insertion and its AGT dual
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arXiv
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| Natura: | Preprint |
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2025
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| author | Poghosyan, Hasmik Poghossian, Rubik |
| author_facet | Poghosyan, Hasmik Poghossian, Rubik |
| contents | In this paper, we develop and explore recursive methods to investigate the 2d CFT 5-point conformal block with a level 2 degenerate insertion, as well as its AGT dual, by solving the BPZ differential equation. First, we represent the solution of the differential equation as a double series expansion. On the 2-node quiver gauge theory side, this corresponds to the instanton series. We then demonstrate that the expansion coefficients are uniquely determined by a recursion relation. Inspired by the approach initiated in a paper by D. Gaiotto and J. Teschner, we partially resum this series and show that the result can be elegantly expressed in terms of a single hypergeometric function and its derivative. This new representation makes it straightforward to relate different asymptotic regions. As a by-product, this provides us a simple derivation of fusion and braiding coefficients.
We describe the subtle procedure of merging the degenerate field with the outgoing state, thereby obtaining a generic 4-point block, which on the gauge theory side corresponds to the partition function of $SU(2)$ gauge theory with four massive hypermultiplets in the $Ω$-background.
Finally, we performed several nontrivial checks that confirm our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11921 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On 5-point conformal block with level 2 degenerate field insertion and its AGT dual Poghosyan, Hasmik Poghossian, Rubik High Energy Physics - Theory In this paper, we develop and explore recursive methods to investigate the 2d CFT 5-point conformal block with a level 2 degenerate insertion, as well as its AGT dual, by solving the BPZ differential equation. First, we represent the solution of the differential equation as a double series expansion. On the 2-node quiver gauge theory side, this corresponds to the instanton series. We then demonstrate that the expansion coefficients are uniquely determined by a recursion relation. Inspired by the approach initiated in a paper by D. Gaiotto and J. Teschner, we partially resum this series and show that the result can be elegantly expressed in terms of a single hypergeometric function and its derivative. This new representation makes it straightforward to relate different asymptotic regions. As a by-product, this provides us a simple derivation of fusion and braiding coefficients. We describe the subtle procedure of merging the degenerate field with the outgoing state, thereby obtaining a generic 4-point block, which on the gauge theory side corresponds to the partition function of $SU(2)$ gauge theory with four massive hypermultiplets in the $Ω$-background. Finally, we performed several nontrivial checks that confirm our results. |
| title | On 5-point conformal block with level 2 degenerate field insertion and its AGT dual |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2504.11921 |