functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
superalgebra and (synthetic ) supergeometry
A 2d superconformal field theory in which the holomorphic sector (the “right movers”) has $N=2$ supersymmetry, while the antiholomorphic sector (the “left movers”) has no supersymmetry (or the other way around), is traditionally denoted as $N = (2,0)$ supersymmetry. The topological twist of this is also called the half-twisted model.
In contrast, the heterotic string worldsheet 2d SCFT a priori has only $N = (1,0)$ supersymmetry, while the type II superstring similarly has $N = (1,1)$. But for suitable target spaces and background gauge fields the supersymmetry of the heterotic string is enhanced.
In (BDFF 88) it was shown that this enhancement of the $N=(1,0)$-superconformal worldsheet theory of the heterotic string to an $N=(2,0)$-superconformal theory happens precisely when the corresponding target space effective quantum field theory itself has $N=1$ supersymmetry. This in turn is the case notably when the target is a Calabi-Yau manifold, for more on this see at supersymmetry and Calabi-Yau manifolds.
So to the extent that $N=1$ target space supersymmetry is regarded as important in string phenomenology (it used to be regarded as highly important but with the failure of the LHC experiment to detect evidence reflecting this assumption, this may be changing), then 2d $(2,0)$-superconfromal QFTs define the relevant heterotic string theory vacua.
The enhanced supersymmetry of the $2d$ $(2,0)$-models allows to consider the corresponding topological twist (Kapustin 05, Witten 05). The resulting topologically twisted 2d (2,0)-model is on the one hand amenable to detailed mathematical analysis, since its quantum observables for the chiral de Rham complex sheaf of vertex operator algebras (see (Malikov 06) for a mathematical treatment) and at the same time is argued to still detect important properties of the original geometric model, such as notably the Witten genus.
gauged linear sigma model?
The relevance of 2d $(2,0)$-superconformal worldsheet theories for string phenonemology (see also at supersymmetry and Calabi-Yau manifolds) was first highligted in
Further discussion of 2d $(2,0)$-superconformal QFTs in the contest of heterotic string theory vacua includes
Jacques Distler, Brian Greene, Aspects Of $(2,0)$ String Compactifications, Nucl. Phys. B304 (1988)
Eva Silverstein, Edward Witten, Criteria for Conformal Invariance of (0,2) Models, Nucl.Phys.B444:161-190,1995 (arXiv:hep-th/9503212)
Jock McOrist, The Revival of $(0,2)$ Linear Sigma Models, Int.J.Mod.Phys.A26:1-41,2011 (arXiv:1010.4667)
The topological twist leading to the half-twisted model is due to
The original articles in the physics literature identifying the chiral de Rham complex in the topologically twisted 2d $(2,0)$ model are
Anton Kapustin, Chiral de Rham complex and the half-twisted sigma-model (arXiv:hep-th/0504074)
Edward Witten, Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects (arXiv:hep-th/0504078)
Approaches formalizing this mathematically in terms of sheaves of vertex operator algebras and the chiral de Rham complex is in
and in terms of factorization algebras in
A quick review with emphasis on the Stolz conjecture is in
Relation to F/M-theory on elliptically fibered Calabi-Yau 5-folds:
Sakura Schäfer-Nameki, Timo Weigand, F-theory and 2d (0,2) Theories, J. High Energ. Phys. 2016, 59 (2016) (arXiv:1601.02015)
Sakura Schäfer-Nameki, F-theory and $2d$ $(0,2)$ Theories, 2016 (pdf)