arXiv subjects

Ryan A. Cullinan

Publications and source records attributed to Ryan A. Cullinan.

4 recordsLinked to original sources

arXiv2026

Computational homological methods for integrable field theories

We develop explicit computational tools for the recent homological approach to the construction of $2$-dimensional integrable field theories on $Σ$ from $4$-dimensional semi-holomorphic Chern-Simons theory on $Σ\times C$. 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 $C=\mathbb{C}P^1$ 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.

hep-th↗

arXiv2026

On the structure of higher-dimensional integrable field theories

We propose a general framework for integrable field theories in arbitrary spacetime dimension $d+1$ which is based on $d$-term $L_\infty$-algebras. Specifically, we introduce cyclic $L_\infty$-algebras describing topological-holomorphic higher Chern-Simons theories on $M \times \mathbb{C}P^1$ with suitable singularity structures and boundary conditions, controlled by a meromorphic $1$-form on $\mathbb{C}P^1$. Using homological perturbation theory and homotopy transfer, we construct weakly equivalent models describing $(d+1)$-dimensional field theories on $M$. Their integrability is witnessed by a natural map to an $L_\infty$-algebra describing higher Lax connections, yielding conserved charges associated with higher-dimensional cycles in $M$. The resulting theories admit natural action functionals and recover the Costello-Yamazaki construction in $2$ dimensions.

hep-th↗

arXiv2024

Integrable Deformations from Twistor Space

Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons in 6 dimensions. We provide the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. Starting from 6d holomorphic Chern-Simons theory on twistor space with a particular meromorphic 3-form $Ω$, we construct the defect theory to find a novel 4d integrable field theory, whose equations of motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry reducing, we find a multi-parameter 2d integrable model, which specialises to the $λ$-deformation at a certain point in parameter space. The same model is recovered by first symmetry reducing, to give 4d Chern-Simons with generalised boundary conditions, and then constructing the defect theory.

hep-th↗

arXiv2024

Gauging The Diamond: Integrable Coset Models from Twistor Space

Recent work has shown that certain integrable and conformal field theories in two dimensions can be given a higher-dimensional origin from holomorphic Chern-Simons in six dimensions. Along with anti-self-dual Yang-Mills and four-dimensional Chern-Simons, this gives rise to a diamond correspondence of theories. In this work we extend this framework to incorporate models realised through gaugings. As well as describing a higher-dimensional origin of coset CFTs, by choosing the details of the reduction from higher dimensions, we obtain rich classes of two-dimensional integrable models including homogeneous sine-Gordon models and generalisations that are new to the literature.

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