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\begin{document}

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\section*{topologically twisted D=4 super Yang-Mills theory}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{for__supersymmetry}{For $N = 4$ supersymmetry}\dotfill \pageref*{for__supersymmetry} \linebreak
\noindent\hyperlink{for__supersymmetry_2}{For $N = 2$ supersymmetry}\dotfill \pageref*{for__supersymmetry_2} \linebreak
\noindent\hyperlink{table_of_relations_via_holographic_compactifications_and_twists}{Table of relations via holographic, compactifications and twists}\dotfill \pageref*{table_of_relations_via_holographic_compactifications_and_twists} \linebreak
\noindent\hyperlink{Formalization}{Formalization}\dotfill \pageref*{Formalization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

A deformation of [[super Yang-Mills theory]] that yields a [[topological field theory]] in 4 [[dimensions]]. This is in higher dimensional analogy to how the [[topological string]]-[[topological twist|twisting]] of the [[superstring]] yields the topological [[A-model]] and [[B-model]] 2d topological field theories.

\hypertarget{for__supersymmetry}{}\subsubsection*{{For $N = 4$ supersymmetry}}\label{for__supersymmetry}

Given [[N=4 D=4 super Yang-Mills theory]], the twisting is induced by a choice of [[subgroup]] inclusion of the [[special orthogonal group]] $SO(4)$ into the [[R-symmetry]] group $SO(6)$. Then choose a [[supersymmetry]] $Q$ which is invariant under the resulting combined action of $SO(4)$ on [[spacetime]] and via [[R-symmetry]] and consider the subspace of [[quantum states]]/[[quantum observables]] which are in the [[kernel]] of $Q$. This subspace defines a [[topological field theory]] which is called the corresponding twisted topological super Yang-Mills theory.

Before the twisting super [[Yang-Mills theory]] depends on the complex [[coupling constant]]

\begin{displaymath}
\tau = \frac{\theta}{2\pi} + \frac{4 \pi i }{g_{YM}^2}
\end{displaymath}
(with $\theta$ the \emph{[[theta angle]]}), after the twisting there is an additional complex parameter $t$ encoding the choice of topological supercharge. The twisted theory however only depends on the combination

\begin{displaymath}
\Psi \coloneqq \frac{\theta}{2 \pi}  + 
  \frac{4 \pi i}{g_{YM}^2} 
  \frac{t - t^{-1}}{t + t^{-1}}
  \,.
\end{displaymath}
(\hyperlink{Witten11}{Witten 11, p. 29})

The resulting [[4d TQFT]] is also called the \emph{[[Kapustin-Witten TQFT]]}.

\hypertarget{for__supersymmetry_2}{}\subsubsection*{{For $N = 2$ supersymmetry}}\label{for__supersymmetry_2}

For [[N=2 D=4 super Yang-Mills theory]] the twisting follows the same idea, but is a little but more intricate (\hyperlink{Witten11}{Witten 11, section 5.1.1})

\hypertarget{table_of_relations_via_holographic_compactifications_and_twists}{}\subsubsection*{{Table of relations via holographic, compactifications and twists}}\label{table_of_relations_via_holographic_compactifications_and_twists}

[[!include gauge theory from AdS-CFT -- table]]

\hypertarget{Formalization}{}\subsection*{{Formalization}}\label{Formalization}

A formalization of the topological twisting in the framework of [[perturbation theory|perturbative]] [[BV-quantization]] of field theory via [[factorization algebras]] of local [[quantum observables]] is proposed in (\hyperlink{Costello11}{Costello 11, section 15, 16, \ldots{}}).

The definition there essentially amounts to saying that a choice of topological twisting is a choice of [[action]] of the [[semidirect product]] [[supergroup]]

\begin{displaymath}
\mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}
\end{displaymath}
of the [[multiplicative group]] acting on the odd-shifted [[additive group]] via the given [[super Poincare Lie algebra]].

We notice that this group is the \href{odd%20line#TheAutomorphismSuperGroup}{automorphism group of the odd line}

\begin{displaymath}
\mathbf{Aut}(\mathbb{R}^{0|1}) \simeq \mathbb{G}_m \ltimes \Pi \mathbb{G}_{ad}
\end{displaymath}
for which it is well known that an [[action]] of it is equivalent to a choice of [[differential]] $Q$ and corresponding grading.

This chosen differential $Q$ among the supersymmetry generators in the [[super Poincare Lie algebra]] is the choice of what in the physics literature is called the twisting ``BRST operator''.

The twisted theory itself is then defined to be given by the [[factorization algebra]] of observables which is essentially the [[homotopy fixed points]] of this $\mathbf{Aut}(\mathbb{R}^{0|1})$-[[infinity-action]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[topological Yang-Mills theory]]


\item [[topological twist]]


\item [[N=1 D=4 super Yang-Mills theory]]


\item [[N=2 D=4 super Yang-Mills theory]]


\item [[N=4 D=3 super Yang-Mills theory]]


\item [[N=4 D=4 super Yang-Mills theory]]


\item [[D=5 super Yang-Mills theory]]


\item [[S-duality]], [[geometric Langlands duality]]


\item [[chiral ring]], [[quantum cohomology]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

The idea of topological twisting of [[supersymmetry|supersymmetric]] quantum field theory goes back to

\begin{itemize}%
\item [[Edward Witten]], \emph{Topological quantum field theory}, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386 (\href{http://projecteuclid.org/euclid.cmp/1104161738}{Euclid})

\end{itemize}
often referred to as ``[[cohomological field theory]]''

\begin{itemize}%
\item [[Edward Witten]], \emph{Introduction to cohomological field theory}, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

\end{itemize}
where it is [[N=2 D=4 super Yang-Mills theory]] that is twisted and related to [[Donaldson theory]]. The analogous twisting of [[N=4 D=4 super Yang-Mills theory]] is due to

\begin{itemize}%
\item [[Cumrun Vafa]], [[Edward Witten]], \emph{A Strong Coupling Test of S-Duality}, Nucl. Phys. B431:3-77,1994 (\href{http://arxiv.org/abs/hep-th/9408074}{arXiv:hep-th/9408074})

\end{itemize}
The $N=4$-case and one of the possible $N=2$-twists yield [[instanton]] invariants captured by the [[Seiberg-Witten theory]] generalization of [[Donaldson theory]]. Another variant of the $N=2$ twist was described in

\begin{itemize}%
\item Neil Marcus, \emph{The other topological twisting of N=4 Yang-Mills}, Nucl.Phys. B452 (1995) 331-345 (\href{http://arxiv.org/abs/hep-th/9506002}{arXiv:hep-th/9506002})

\end{itemize}
and yields a geometric interpretation of the [[geometric Langlands correspondence]], as found in

\begin{itemize}%
\item [[Anton Kapustin]], [[Edward Witten]], \emph{Electric-Magnetic Duality And The Geometric Langlands Program} (\href{http://arxiv.org/abs/hep-th/0604151}{arXiv:hep-th/0604151})

\end{itemize}
A detailed analysis of the three twists of the $N=4$ theory is in

\begin{itemize}%
\item Carlos Lozano, \emph{Duality in Topological Quantum Field Theories}, PhD thesis (\href{http://arxiv.org/abs/hep-th/9907123}{arXiv:hep-th/9907123})

\end{itemize}
Discussion of generalization of the twisting to [[quantum field theory on curved spacetime]] is in

\begin{itemize}%
\item Jeong-Hyuck Park, Dimitrios Tsimpis, \emph{Topological twisting of conformal supercharges}, Nucl. Phys. B776:405-430,2007 (\href{http://arxiv.org/abs/hep-th/0610159}{arXiv:hep-th/0610159})

\end{itemize}
Section 2.2.1 of

\begin{itemize}%
\item [[Edward Witten]], \emph{Fivebranes and knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216})

\end{itemize}
briefly recalls the topological twisting of [[N=4 D=4 super Yang-Mills theory]]. Section 5.1.1 there discusses the twisting of [[N=2 D=4 super Yang-Mills theory]] (induced from the [[6d (2,0)-superconformal QFT]] on the [[M5-brane]]), which was introduced in section 3.1.2 of

\begin{itemize}%
\item [[Davide Gaiotto]], [[Gregory Moore]] and [[Andrew Neitzke]], \emph{Wall-crossing, Hitchin Systems, and the WKB Approximation}, (\href{http://arxiv.org/abs/0907.3987}{arXiv:0907.3987})

\end{itemize}
For more on this see the references listed at \emph{\href{N%3D2+D%3D4+super+Yang-Mills+theory#ReferencesConstructionFrom5Branes}{N=2 D=4 super Yang-Mills theory -- References -- Construction from 5-branes}}.

More mathematically formalized discussion of topologically twisted supersymmetric theories in the framework of [[BV-BRST formalism]] [[perturbation theory]] (and with an eye towards the [[factorization algebra]] formulation) is in

\begin{itemize}%
\item [[Kevin Costello]], from section 15 on in \emph{Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4} (\href{http://arxiv.org/abs/1111.4234}{arXiv:1111.4234})

\end{itemize}
More in

\begin{itemize}%
\item [[Chris Elliott]], [[Pavel Safronov]], \emph{Topological twists of supersymmetric observables} (\href{https://arxiv.org/abs/1805.10806}{arXiv:1805.10806})

\end{itemize}
In

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

\end{itemize}
is discussed that the holomorphically twisted $N=1$ theory is controled by the [[Yangian]] in analogy to how [[Chern-Simons theory]] is controled by a [[quantum group]].

[[!redirects topologically twisted D=4 super Yang-Mills theories]]

[[!redirects topologically twisted super Yang-Mills theory]] [[!redirects topologically twisted super Yang-Mills theories]]

[[!redirects topologically twisted N=2 D=4 super Yang-Mills theory]] [[!redirects topologically twisted N=2 D=4 super Yang-Mills theories]]

[[!redirects topologically twisted N=4 D=4 super Yang-Mills theory]] [[!redirects topologically twisted N=4 D=4 super Yang-Mills theories]]



\end{document}