Computational homological methods for integrable field theories

Marco Benini; Ryan A. Cullinan; Alexander Schenkel; Benoit Vicedo

theoretical physics

Computational homological methods for integrable field theories

arXiv:2607.12142

summary

The paper develops explicit computational tools using homotopy transfer of cyclic L∞‑algebras to construct 2‑dimensional integrable field theories from 4‑dimensional semi‑holomorphic Chern‑Simons theory, and applies the method to derive the principal chiral model with a Wess‑Zumino term.

Abstract

We develop explicit computational tools for the recent homological approach to the construction of $2$ -dimensional integrable field theories on $I^£$ from $4$ -dimensional semi-holomorphic Chern-Simons theory on $I^£ \times C$ . In this framework, the operation of integrating out the spectral curve $C$ is realized by homotopy transfer of a cyclic $L_{\infty}$ -algebra associated with the $4$ -dimensional theory with prescribed singularities and boundary conditions. We construct explicit strong deformation retracts for divisor-twisted Dolbeault complexes on $C = C P^{1}$ and use them to make the transferred $L_{\infty}$ -structure computationally accessible. As an application, we study the choice of meromorphic $1$ -form corresponding to the principal chiral model with a Wess-Zumino term. We compute the transferred Maurer-Cartan action and the associated Lax connection, showing that the former resums to the standard principal chiral model action with a Wess-Zumino term and that the latter reproduces the usual Lax connection.

Topics & keywords

#integrable field theory #homological methods #L-infinity algebras #Chern-Simons theory #principal chiral modelhomotopy transfercyclic L∞ algebraDolbeault complexspectral curveWess-Zumino termLax connection