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.
A priori the worldsheet 2d SCFT describing the quantum heterotic string has supersymmetry. Precisely if the corresponding target space effective field theory has supersymmetry does the worldsheet theory enhance to supersymmetry. See at 2d (2,0)-superconformal QFT and at Calabi-Yau manifolds and supersymmetry for more on this.
|partition function in -dimensional QFT||supercharge||index in cohomology theory||genus||logarithmic coefficients of Hirzebruch series|
|0||push-forward in ordinary cohomology: integration of differential forms|
|1||spinning particle||Dirac operator||KO-theory index||A-hat genus||Bernoulli numbers|
|endpoint of 2d Poisson-Chern-Simons theory string||Spin^c Dirac operator twisted by prequantum line bundle||space of quantum states of boundary phase space/Poisson manifold||Todd genus||Bernoulli numbers|
|endpoint of type II superstring||Spin^c Dirac operator twisted by Chan-Paton gauge field||D-brane charge||Todd genus||Bernoulli numbers|
|2||superstring||Dirac-Ramond operator||superstring partition function||elliptic genus/Witten genus||Eisenstein series|
|self-dual string||M5-brane charge|
heterotic string theory
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)
For more mathematically precise discussion in the context of elliptic cohomology and the Witten genus see also the references at Witten genus – Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models.
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