Dimensional Uplift in Conformal Field Theories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909982873616384 |
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| author | Gliozzi, Ferdinando |
| author_facet | Gliozzi, Ferdinando |
| contents | The n-point functions of any Conformal Field Theory (CFT) in $d$ dimensions can always be interpreted as spatial restrictions of corresponding functions in a higher-dimensional CFT with dimension $d'> d$. In particular, when a four-point function in $d$ dimensions has a known conformal block expansion, this expansion can be easily extended to $d'=d+2$ due to a remarkable identity among conformal blocks, discovered by Kaviraj, Rychkov, and Trevisani (KRT) as a consequence of Parisi-Sourlas supersymmetry and confirmed to hold in any CFT with $d > 1$. In this note, we provide an elementary proof of this identity using simple algebraic properties of the Casimir operators. Additionally, we construct five differential operators, $Λ_i$, which promote a conformal block in $d$ dimensions to five conformal blocks in $d+2$ dimensions. These operators can be normalized such that $\sum_i Λ_i = 1$, from which the KRT identity immediately follows. Similar, simpler identities have been proposed, all of which can be reformulated in the same way. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimensional Uplift in Conformal Field Theories Gliozzi, Ferdinando High Energy Physics - Theory The n-point functions of any Conformal Field Theory (CFT) in $d$ dimensions can always be interpreted as spatial restrictions of corresponding functions in a higher-dimensional CFT with dimension $d'> d$. In particular, when a four-point function in $d$ dimensions has a known conformal block expansion, this expansion can be easily extended to $d'=d+2$ due to a remarkable identity among conformal blocks, discovered by Kaviraj, Rychkov, and Trevisani (KRT) as a consequence of Parisi-Sourlas supersymmetry and confirmed to hold in any CFT with $d > 1$. In this note, we provide an elementary proof of this identity using simple algebraic properties of the Casimir operators. Additionally, we construct five differential operators, $Λ_i$, which promote a conformal block in $d$ dimensions to five conformal blocks in $d+2$ dimensions. These operators can be normalized such that $\sum_i Λ_i = 1$, from which the KRT identity immediately follows. Similar, simpler identities have been proposed, all of which can be reformulated in the same way. |
| title | Dimensional Uplift in Conformal Field Theories |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2504.15904 |