We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0) form Ω. If Ω is nowhere vanishing, the theory is equivalent to anti-self-dual Yang-Mills on spacetime. The twistor action yields space-time actions including the Chalmers & Siegel action for ASD Yang-Mills in terms of an adjoint scalar. Under symmetry reduction, these spacetime actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to the 4d Chern-Simons theory with disorder defects studied by Costello & Yamazaki. Finally we show that similar reductions by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations. Contents 1 Introduction 2 From holomorphic to 4d Chern-Simons theory 2.1 Holomorphic Chern-Simons theory on twistor space 2.2 Effective description as a 4d WZW model 2.3 Symmetry reduction to a principal chiral model with WZW term 2.4 Symmetry reduction to 4d Chern-Simons theory 3 Twistor actions 3.1 Overview 3.2 Chalmers-Siegel action 3.3 Trigonometric action 3.4 4d integrable coupled σ-models 4 Real structures 4.1 Lorentzian and ultrahyperbolic signatures 4.2 Real forms of the gauge group 5 Reductions by 1-dimensional groups of translations 5.1 Reduction to the Bogomolny equations 5.2 The 1+2 dimensional chiral model 6 Conclusions A Notation and conventions for spinors B The twistor correspondence in Euclidean signature C The anti-self-dual Yang-Mills equations D Partially holomorphic structures in 4d and 5d Chern-Simons theory E Calculations for trigonometric actions F Novel actions for anti-self-dual Yang-Mills theory G The twistor correspondence in Lorentzian and ultrahyperbolic signature 1 Here ē0 Ā0 is the component along the base while êA ÂA are the components along the fibres of PT ∼ = O(1) ⊕ O(1) → CP 1 .

We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ × C)/Z 2 , obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The Z 2 -action on Σ × C fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RT T presentation of the boundary Yang-Baxter equation. Introduction The Yang-Baxter EquationInteractions of an integrable spin chain are determined by an object known as an R-matrix. This is a linear mappair of complex vector spaces and z, z ′ complex spectral parameters on which the R-matrix depends meromorphically. The spin chain is integrable if the R-matrix obeys the Yang-Baxter equation (YBE)(1.1) around the identity 1 V ⊗V ′ . R-matrices admitting such an expansion are called quasiclassical, and r(z, z ′ ) is known as the classical r-matrix. This classical r-matrix takes values in g ⊗ g for g a finite dimensional, complex, simple Lie algebra, acting in representations associated to V ⊗ V ′ . Expanding the YBE to second order in shows that the classical r-matrix obeys the classical Yang-Baxter equation,Solutions of the classical Yang-Baxter equations were classified by Belavin & Drinfeld [4], under the (mild) assumption that the r-matrix is non-degenerate. The solutions can be separated into three families, distinguished by whether the classical r-matrix can be written in terms of rational, trigonometric, or elliptic functions. In this work we will concentrate on the rational case. Each family of solutions to is associated with an algebra, which in the rational case is the Yangian Y(g). Indeed, in [5] it was demonstrated that all rational solutions of the YBE of the form (1.1) determine a representation of the Yangian on V , and are themselves determined by such a representation. Accordingly, the line operators in Costello's theory are actually associated with representations of Y(g). These include ordinary Wilson lines in certain representations of g itself, but also more general line operators. As shown in [1-3], the more general line operators arise even in the OPE of two ordinary Wilson lines.Since the YBE is homogeneous, its solutions are defined only up to multiplication by a function f (z 1 , z 2 ) of the spectral parameters. This degeneracy can be removed by required that the R-matrix obeys a constraint known as the quantum determinant condition. In the rational case, for V the defining vector representation of g = sl n (C), the quantum determinant [6] condition is enough to guarantee existence of a unique quasi-classical Rmatrix in this representation for a given classical r-matrix. Similar results e...

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We consider the mixed topological-holomorphic Chern-Simons theory introduced by Costello, Yamazaki & Witten on a Z 2 orbifold. We use this to construct semiclassical solutions of the boundary Yang-Baxter equation in the elliptic and trigonometric cases. A novel feature of the trigonometric case is that the Z 2 action lifts to the gauge bundle in a z-dependent way. We construct several examples of K-matrices, and check that they agree with cases appearing in the literature.

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. We analyse a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra, however at 1-loop its operator products are corrected by the effective graviton vertex. We argue that the chiral algebra can be interpreted as the universal holomorphic surface defect in the twistor uplift of self-dual gravity, and show that the same correction is induced by an anomalous diagram in the bulk-defect system. The 1-loop holomorphic collinear singularities do not form a consistent chiral algebra. The failure of associativity can be traced to the existence of a recently discovered gravitational anomaly on twistor space. It can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.