5d 2-Chern-Simons Theory and 3d Integrable Field Theories

Schenkel, Alexander; Vicedo, Benoît

Abstract

The 4-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of 2-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of 3-dimensional integrable field theories by considering a 5-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection (A, B) on R 3 × CP 1. The input data for this theory are the choice of a meromorphic 1-form ω on CP 1 and a strict Lie 2-group with cyclic structure on its underlying Lie 2-algebra. Integrable field theories on R 3 are constructed by imposing suitable boundary conditions on the connection (A, B) at the 3-dimensional defects located at the poles of ω and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for 3-dimensional integrable field theories, including a 2-form component B which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's (2 + 1)-dimensional integrable chiral model from a suitable choice of data in the 5-dimensional theory.

Citation

Schenkel, A., & Vicedo, B. (2024). 5d 2-Chern-Simons Theory and 3d Integrable Field Theories. Communications in Mathematical Physics, 405, Article 293. https://doi.org/10.1007/s00220-024-05170-9

• Journal Article Type: Article
• Acceptance Date: Oct 21, 2024
• Online Publication Date: Nov 21, 2024
• Publication Date: Nov 21, 2024
• Deposit Date: Nov 22, 2024
• Publicly Available Date: Nov 22, 2024
• Journal: Communications in Mathematical Physics
• Print ISSN: 0010-3616
• Electronic ISSN: 1432-0916
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 405
• Article Number: 293
• DOI: https://doi.org/10.1007/s00220-024-05170-9
• Public URL: https://nottingham-repository.worktribe.com/output/42217478
• Publisher URL: https://link.springer.com/article/10.1007/s00220-024-05170-9
• Additional Information: Received: 23 May 2024; Accepted: 21 October 2024; First Online: 21 November 2024; : ; : The authors have no Conflict of interest to declare that are relevant to the content of this article.