Perturbative 4D conformal field theories and representation theory of diagram algebras
Abstract The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to pertur...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-05-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | http://link.springer.com/article/10.1007/JHEP05(2020)020 |
| _version_ | 1818282475780046848 |
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| author | Robert de Mello Koch Sanjaye Ramgoolam |
| author_facet | Robert de Mello Koch Sanjaye Ramgoolam |
| author_sort | Robert de Mello Koch |
| collection | DOAJ |
| description | Abstract The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N $$ \mathcal{N} $$ = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U ★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described. |
| first_indexed | 2024-12-13T00:21:37Z |
| format | Article |
| id | doaj.art-0d0b6a96e9d243d8855b2aabd6d7f327 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-13T00:21:37Z |
| publishDate | 2020-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-0d0b6a96e9d243d8855b2aabd6d7f3272022-12-22T00:05:33ZengSpringerOpenJournal of High Energy Physics1029-84792020-05-012020516010.1007/JHEP05(2020)020Perturbative 4D conformal field theories and representation theory of diagram algebrasRobert de Mello Koch0Sanjaye Ramgoolam1Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal UniversityNational Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the WitwatersrandAbstract The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N $$ \mathcal{N} $$ = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U ★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.http://link.springer.com/article/10.1007/JHEP05(2020)020AdS-CFT CorrespondenceConformal Field Theory |
| spellingShingle | Robert de Mello Koch Sanjaye Ramgoolam Perturbative 4D conformal field theories and representation theory of diagram algebras Journal of High Energy Physics AdS-CFT Correspondence Conformal Field Theory |
| title | Perturbative 4D conformal field theories and representation theory of diagram algebras |
| title_full | Perturbative 4D conformal field theories and representation theory of diagram algebras |
| title_fullStr | Perturbative 4D conformal field theories and representation theory of diagram algebras |
| title_full_unstemmed | Perturbative 4D conformal field theories and representation theory of diagram algebras |
| title_short | Perturbative 4D conformal field theories and representation theory of diagram algebras |
| title_sort | perturbative 4d conformal field theories and representation theory of diagram algebras |
| topic | AdS-CFT Correspondence Conformal Field Theory |
| url | http://link.springer.com/article/10.1007/JHEP05(2020)020 |
| work_keys_str_mv | AT robertdemellokoch perturbative4dconformalfieldtheoriesandrepresentationtheoryofdiagramalgebras AT sanjayeramgoolam perturbative4dconformalfieldtheoriesandrepresentationtheoryofdiagramalgebras |