Integrable models from 4d holomorphic BF theory

We show how to construct 2d field theories with holomorphic integrability from defect setups in 4d holomorphic BF. In a simple example setup, we explicitly construct the 2d theory and perform an initial classical analysis. Making use of the symmetries, we are able to write down an infinite family of solutions to the equations of motion. Comparing with more typical integrable systems, we explain how 2d holomorphic integrability sits between the standard notions of integrability in one and two dimensions. These 2d theories are designed as toy models for integrable theories in three and four dimensions, many of which can be understood as partially or totally holomorphic. We comment on the implications for higher-dimensional integrability and aspects of quantization in the concluding remarks.

💡 Research Summary

The paper investigates a new class of two‑dimensional integrable field theories (2d IFTs) that arise from defect configurations in four‑dimensional holomorphic BF theory (4d hBF). The authors begin by defining 4d hBF on a complex two‑manifold M with coordinates (z,w). The fundamental fields are a (0,1)-form gauge field A and an adjoint‑valued scalar b, with action
SₕBF = (1/2πi)∫ₘ dz∧dw ⟨b, F(A)⟩,
where F(A)=dA+A∧A and ⟨·,·⟩ is a non‑degenerate invariant bilinear form on the Lie algebra g. Varying the action yields two equations of motion: the (0,2) component of the curvature vanishes, F₀,₂(A)=0, implying that the partial connection ∇ₐ=∂̄+ A is holomorphic, and the covariant constancy condition ∇ₐ b=0. By fixing the gauge Ā_z=0, the authors show that Ā_w and b become holomorphic functions of the auxiliary complex coordinate z, while their dependence on the physical coordinate w encodes the dynamics of a 2d theory. In this set‑up, z plays the role of a spectral parameter and w is the spacetime coordinate of the emergent 2d model.

To generate non‑trivial dynamics, the paper introduces two types of defects. The first, “disorder defects,” modifies the action by inserting a meromorphic (2,0)-form ω₂=φ(z) dz∧dw. The meromorphic function φ(z) has poles and zeros; its exterior derivative dω₂ is a distribution supported at the poles. The variation of the action then produces boundary terms that break gauge invariance at the poles, giving rise to edge modes localized at the defect points. These edge modes become the dynamical fields of the 2d integrable theory. The second, “order defects,” adds localized terms at fixed points z=η, z=κ, such as ∫{z=η} dw ⟨ϕ, h⁻¹∇ₐ h⟩ and ∫{z=κ} d²w P(b), where ϕ and h are fields living on the defect and P is an invariant polynomial. These terms source singularities in the bulk fields and can be interpreted as coadjoint‑orbit realizations of holomorphic Wilson lines together with polynomial insertions of b.

A central conceptual result is that 4d hBF can be obtained as a singular limit of four‑dimensional Chern‑Simons theory (4d CS). The latter is defined on C×Σ with a meromorphic (1,0)-form ω₁=φ(z)dz and action S_CS = (k/4πi)∫ ω₁∧⟨A, dA+⅓