Contents

Contents

Idea

In mathematical/theoretical physics, semi-holomorphic 4d Chern-Simons theory is a variant of ordinary 3d Chern-Simons theory where, roughly, one of the three coordinate functions is promoted from an ordinary real-valued coordinate to a holomorphic coordinate.

Hence semi-holomorphic 4d Chern-Simons theory is a field theory defined on spacetimes of the form $\Sigma^2 \times C$ where $\Sigma^2$ is a real smooth manifold, while $C$ is a complex curve, and whose Lagrangian density is schematically the ordinary Chern-Simons form, but with one of the wedge factors promoted from $d x^i$ to $d z$.

Accordingly, this field theory is a topological field theory (only) with respect to $\Sigma^2$, while it behaves like a conformal field theory with respect to $C$, whence “semi-holomorphic” Chern-Simons theory: If one instead promotes all three coordinates of ordinary 3d Chern-Simons theory to holomorphic coordinates, one obtains full 6d holomorphic Chern-Simons theory.

But beware that in the literature this is often referred to just as 4d Chern-Simons theory.

Properties

Wilson lines and relation to integrable lattice models

Due to this particular property of the theory, Wilson line observables that form a network on $\Sigma^2$ but with each line located at a distinct point in the complex curve $C$ play a special role: Arranging them along the lines of a rectangular grid makes these observables behave much like those of a lattice model with the the their intersection points as vertices, but the topological invariance of the theory with respect to $\Sigma^2$ implies certain symmetry operations on these observables, coming from invariance of the theory under moving Wilson lines across each other. These symmetries turn out to behave like R-matrices? making the corresponding lattice model an integrable system.

Yangian symmetry

Yangian

Consider the example when the spacetime is of the form $\mathbb{C}_w \times \mathbb{C}_z$ with subscripts to denote the respective coordinates. Pushing forward the factorization algebra for gives a locally constant factorization algebra on $\mathbb{C}_w$.

Applying Koszul duality gives a Hopf algebra because it is augmented as an $E_1$ algebra. This is a result of Tamarkin. In this case the Hopf algebra is the linear dual of the completed Yangian. This is seen by evaluating the classical limit and the first order bialgebra structure and then invoking a uniqueness result of Drinfeld.

The theory and its relation to Yangians and integrable lattice models is due to

Discussion of realizations of semi-holomorphic 4d Chern-Simons theory as the worldvolume theory of intersecting D-brane/NS5-brane models

and further relating to the quantum geometric Langlands correspondence: