Types of quantum field thories
In string theory a spacetime vaccuum is encoded by a sigma-model 2-dimensional SCFT. In heterotic string theory that SCFT is assumed to be the sum of a supersymmetric chiral piece and a non-supersymmetric piece (therefore “heterotic”).
An effective target space quantum field theory induced from a given heterotic 2d CFT sigma model that has a spacetime of the form for the 4-dimensional Minkowski space that is experimentally observed locally (say on the scale of a particle accelerator) has global supersymmetry precisely if the remaining 6-dimensional Riemannian manifold is a Calabi-Yau manifold. See the references below.
Since global supersymmetry for a long time has been considered a promising phenomenological model in high energy physics, this fact has induced a lot of interest in heterotic string background with a Yalabi-Yau factor.
|partition function in -dimensional QFT||index/genus in cohomology theory|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||K-theory index|
|endpoint of 2d Poisson-Chern-Simons theory string||space of quantum states of boundary phase space/Poisson manifold|
|endpoint of type II superstring||D-brane charge|
|2||type II superstring||elliptic genus|
|heterotic string||Witten genus|
Heterotic strings were introduced in
David Gross, J. A. Harvey, E. Martinec and R. Rohm,
Heterotic string theory (I). The free heterotic string Nucl. Phys. B 256 (1985), 253.
Heterotic string theory (I). The interacting heterotic string , Nucl. Phys. B 267 (1986), 75.
Textbook accounts include
Eric D'Hoker, String theory – lecture 8: Heterotic strings in part 3 (p. 941 of volume II) of
Pierre Deligne, P. Etingof, Dan Freed, L. Jeffrey, David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. . Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Compactified on an elliptic curve or, more generally, elliptic fibration, heterotic string compactifictions are controled by a choice holomorphic stable bundle on the compact space. Dually this is an F-theory compactification on a K3-bundles.
The basis of this story is discussed in
A more formal discussion is in
The heterotic/F-theory duality is discussed for instance in