Integrable models from 4d holomorphic BF theory

Abstract

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.

35 pages. Minor updates

References in corpus (45)

• Perturbative Gauge Theory As A String Theory In Twistor Space
• Gauge Theory and Integrability, II
• Twistor actions for non-self-dual fields; a new foundation for twistor-string theory
• Gauge Theory and Integrability, I
• Supersymmetric gauge theory and the Yangian
• A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory
• The Topological B-model on a Mini-Supertwistor Space and Supersymmetric Bogomolny Monopole Equations
• Gauge Theory And Integrability, III
• Alleviating the non-ultralocality of coset sigma models through a generalized Faddeev-Reshetikhin procedure
• M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
• 4-dimensional Chern-Simons theory and integrable field theories
• Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory
• Topological Chern-Simons/Matter Theories
• Local charges in involution and hierarchies in integrable sigma-models
• Quantizing the non-linear graviton
• On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection
• Canonical quantization of the boundary Wess-Zumino-Witten model
• Sigma models as Gross-Neveu models
• Infinite Family of Integrable Sigma Models Using Auxiliary Fields
• The Faddeev-Reshetikhin model from a 4D Chern-Simons theory
• Semi-Chiral Operators in 4d $\mathcal{N}=1$ Gauge Theories
• Ultralocal Lax connection for para-complex ${\mathbb{Z}}_T$-cosets
• Monopole Operators and Bulk-Boundary Relation in Holomorphic Topological Theories
• Minitwistors and 3d Yang-Mills-Higgs theory
• Integrable Deformations from Twistor Space
• New T-duality for Chern-Simons Theory
• $T\bar{T}$ and root-$T\bar{T}$ deformations in four-dimensional Chern-Simons theory
• 4D Chern-Simons theory with auxiliary fields
• Higher Gauge Theory and Integrability
• Gauging The Diamond: Integrable Coset Models from Twistor Space
• The Courant-Hilbert construction in 4D Chern-Simons theory
• 3-dimensional mixed BF theory and Hitchin's integrable system
• Homotopical analysis of 4d Chern-Simons theory and integrable field theories
• Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory
• S-algebra in Gauge Theory: Twistor, Spacetime and Holographic Perspectives
• Hyperbolic Monopoles, (Semi-)Holomorphic Chern-Simons Theories, and Generalized Chiral Potts Models
• Integrable Sigma Models and Universal Root $T\bar{T}$ Deformation via Courant-Hilbert Approach
• Root-$T\bar{T}$ Flows Unify 4D Duality-Invariant Electrodynamics and 2D Integrable Sigma Models
• The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system
• Relating auxiliary field formulations of $4d$ duality-invariant and $2d$ integrable field theories
• 5d 2-Chern-Simons theory and 3d integrable field theories
• Non-Commutative Gauge Theory at the Beach
• Integrable Higher-Spin Deformations of Sigma Models from Auxiliary Fields
• Towards a quadratic Poisson algebra for the subtracted classical monodromy of Symmetric Space Sine-Gordon theories
• Gauge Theory and the Analytic Form of the Geometric Langlands Program