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\begin{document}

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\section*{semi-holomorphic 4d Chern-Simons theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{wilson_lines_and_relation_to_integrable_lattice_models}{Wilson lines and relation to integrable lattice models}\dotfill \pageref*{wilson_lines_and_relation_to_integrable_lattice_models} \linebreak
\noindent\hyperlink{yangian_symmetry}{Yangian symmetry}\dotfill \pageref*{yangian_symmetry} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[mathematical physics|mathematical]]/[[theoretical physics]], \emph{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 function|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 \emph{[[4d Chern-Simons theory]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{wilson_lines_and_relation_to_integrable_lattice_models}{}\subsubsection*{{Wilson lines and relation to integrable lattice models}}\label{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]].

\hypertarget{yangian_symmetry}{}\subsubsection*{{Yangian symmetry}}\label{yangian_symmetry}

\ldots{} [[Yangian]] \ldots{}

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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The theory and its relation to [[Yangians]] and [[integrable system|integrable]] [[lattice models]] is due to

\begin{itemize}%
\item [[Kevin Costello]], \emph{Supersymmetric gauge theory and the Yangian} (\href{https://arxiv.org/abs/1303.2632}{arXiv:1303.2632})


\item [[Edward Witten]], \emph{Integrable Lattice Models From Gauge Theory}, Adv.Theor.Math.Phys. 21 (2017) 1819-1843 (\href{https://arxiv.org/abs/1611.00592}{arXiv:1611.00592})


\item [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], \emph{Gauge Theory and Integrability, I}, ICCM Not. 6, 46-119 (2018) (\href{https://arxiv.org/abs/1709.09993}{arXiv:1709.09993})


\item [[Kevin Costello]], [[Edward Witten]], [[Masahito Yamazaki]], \emph{Gauge Theory and Integrability, II}, ICCM Not. 6, 120-146 (2018) (\href{https://arxiv.org/abs/1802.01579}{arXiv:1802.01579})


\item [[Kevin Costello]], [[Masahito Yamazaki]], \emph{Gauge Theory And Integrability, III} (\href{https://arxiv.org/abs/1908.02289}{https://arxiv:1908.02289})



\end{itemize}
Discussion of realizations of semi-holomorphic 4d Chern-Simons theory as the [[worldvolume]] theory of [[intersecting brane|intersecting]] [[D-brane]]/[[NS5-brane]] models

\begin{itemize}%
\item [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], Qin Zhao, \emph{Branes and Categorifying Integrable Lattice Models} (\href{https://arxiv.org/abs/1806.02821}{arXiv:1806.02821})

\end{itemize}
and further relating to the [[quantum geometric Langlands correspondence]]:

\begin{itemize}%
\item [[Meer Ashwinkumar]], [[Meng-Chwan Tan]], \emph{Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory} (\href{https://arxiv.org/abs/1910.01134}{arXiv:1910.01134})

\end{itemize}


\end{document}