Conformal correlators as simplex integrals in momentum space
Abstract We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2021-01-01
|
| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP01(2021)192 |
| _version_ | 1818877792556679168 |
|---|---|
| author | Adam Bzowski Paul McFadden Kostas Skenderis |
| author_facet | Adam Bzowski Paul McFadden Kostas Skenderis |
| author_sort | Adam Bzowski |
| collection | DOAJ |
| description | Abstract We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations. |
| first_indexed | 2024-12-19T14:03:55Z |
| format | Article |
| id | doaj.art-c9b9e1409f55405a82df8c87442942ca |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-19T14:03:55Z |
| publishDate | 2021-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-c9b9e1409f55405a82df8c87442942ca2022-12-21T20:18:23ZengSpringerOpenJournal of High Energy Physics1029-84792021-01-012021114610.1007/JHEP01(2021)192Conformal correlators as simplex integrals in momentum spaceAdam Bzowski0Paul McFadden1Kostas Skenderis2Department of Physics and Astronomy, Uppsala UniversitySchool of Mathematics, Statistics & Physics, Newcastle UniversitySTAG Research Center & Mathematical Sciences, University of SouthamptonAbstract We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.https://doi.org/10.1007/JHEP01(2021)192AdS-CFT CorrespondenceConformal and W SymmetryConformal Field Theory |
| spellingShingle | Adam Bzowski Paul McFadden Kostas Skenderis Conformal correlators as simplex integrals in momentum space Journal of High Energy Physics AdS-CFT Correspondence Conformal and W Symmetry Conformal Field Theory |
| title | Conformal correlators as simplex integrals in momentum space |
| title_full | Conformal correlators as simplex integrals in momentum space |
| title_fullStr | Conformal correlators as simplex integrals in momentum space |
| title_full_unstemmed | Conformal correlators as simplex integrals in momentum space |
| title_short | Conformal correlators as simplex integrals in momentum space |
| title_sort | conformal correlators as simplex integrals in momentum space |
| topic | AdS-CFT Correspondence Conformal and W Symmetry Conformal Field Theory |
| url | https://doi.org/10.1007/JHEP01(2021)192 |
| work_keys_str_mv | AT adambzowski conformalcorrelatorsassimplexintegralsinmomentumspace AT paulmcfadden conformalcorrelatorsassimplexintegralsinmomentumspace AT kostasskenderis conformalcorrelatorsassimplexintegralsinmomentumspace |