Violation of S-duality in classical $Q$-cohomology
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Chang, Chi-Ming Lin, Ying-Hsuan |
| author_facet | Chang, Chi-Ming Lin, Ying-Hsuan |
| contents | We study the cohomology of a chiral supercharge $Q$ in the $\mathcal{N}=4$ super-Yang-Mills (SYM) theory at tree level. The cohomology classes correspond one-to-one to the $\frac1{16}$ Bogomol'nyi-Prasad-Sommerfield (BPS) states at one-loop. We argue that monotone classes on the Coulomb branch respect the S-duality between the theories with $\mathrm{SO}(2N+1)$ and $\mathrm{USp}(2N)$ gauge groups, but find an explicit example of a pair of cohomology classes that "violate" the S-duality in the sense that the tree-level $Q$-cohomologies are not isomorphic between the neighborhoods near the two free points. Within this pair, one is a fortuitous class and the other is a monotone chiral ring element. Assuming the non-perturbative validity of S-duality, our results disprove a long-standing conjecture on the one-loop exactness of the $\frac1{16}$-BPS spectrum (including the $\frac1{8}$-BPS chiral ring spectrum) in the $\mathcal{N}=4$ SYM. Mathematically, this shows that, the relative Lie algebra cohomology $H^\bullet(\mathfrak{g}[A],\mathfrak{g})$ is generally not graded-isomorphic to $H^\bullet({}^L\mathfrak{g}[A],{}^L\mathfrak{g})$, where $\mathfrak{g}$ and ${}^L\mathfrak{g}$ are a pair of Langlands dual Lie algebras and $A=\mathbb{C}[z^+,z^-]\otimesΛ(θ_1,θ_2,θ_3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Violation of S-duality in classical $Q$-cohomology Chang, Chi-Ming Lin, Ying-Hsuan High Energy Physics - Theory Mathematical Physics Representation Theory We study the cohomology of a chiral supercharge $Q$ in the $\mathcal{N}=4$ super-Yang-Mills (SYM) theory at tree level. The cohomology classes correspond one-to-one to the $\frac1{16}$ Bogomol'nyi-Prasad-Sommerfield (BPS) states at one-loop. We argue that monotone classes on the Coulomb branch respect the S-duality between the theories with $\mathrm{SO}(2N+1)$ and $\mathrm{USp}(2N)$ gauge groups, but find an explicit example of a pair of cohomology classes that "violate" the S-duality in the sense that the tree-level $Q$-cohomologies are not isomorphic between the neighborhoods near the two free points. Within this pair, one is a fortuitous class and the other is a monotone chiral ring element. Assuming the non-perturbative validity of S-duality, our results disprove a long-standing conjecture on the one-loop exactness of the $\frac1{16}$-BPS spectrum (including the $\frac1{8}$-BPS chiral ring spectrum) in the $\mathcal{N}=4$ SYM. Mathematically, this shows that, the relative Lie algebra cohomology $H^\bullet(\mathfrak{g}[A],\mathfrak{g})$ is generally not graded-isomorphic to $H^\bullet({}^L\mathfrak{g}[A],{}^L\mathfrak{g})$, where $\mathfrak{g}$ and ${}^L\mathfrak{g}$ are a pair of Langlands dual Lie algebras and $A=\mathbb{C}[z^+,z^-]\otimesΛ(θ_1,θ_2,θ_3)$. |
| title | Violation of S-duality in classical $Q$-cohomology |
| topic | High Energy Physics - Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2510.24008 |