Types of quantum field thories
In string theory a spacetime vacuum 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 Calabi-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.
Some duality in string theory involving the heterotic string:
For the moment see at Horava-Witten theory.
For duality between F-theory and heterotic string theory see there and see references below.
|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||orientation|
|1||spinning particle||Dirac operator||KO-theory index||A-hat genus||Bernoulli numbers||Atiyah-Bott-Shapiro orientation|
|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||Atiyah-Bott-Shapiro orientation|
|endpoint of type II superstring||Spin^c Dirac operator twisted by Chan-Paton gauge field||D-brane charge||Todd genus||Bernoulli numbers||Atiyah-Bott-Shapiro orientation|
|2||type II superstring||Dirac-Ramond operator||superstring partition function in NS-R sector||Ochanine elliptic genus||SO orientation of elliptic cohomology|
|heterotic superstring||Dirac-Ramond operator||superstring partition function||Witten genus||Eisenstein series||string orientation of tmf|
|self-dual string||M5-brane charge|
|3||w4-orientation of EO(2)-theory|
The traditional construction of the worldsheet theory of the heterotic string produces via the current algebra of the left-moving worldheet fermions only those E8-background gauge fields which are reducible to -principal connections (Distler-Sharpe 10, sections 2-4). But is in known that instance the duality between F-theory and heterotic string theory produces more general gauge backgrounds (Distler-Sharpe 10, section 5).
In (Distler-Sharpe 10, section 7), following (Gates-Siegel 88), it is argued that the way to fix this is to consider parameterized WZW models, parameterized over the E8-principal bundle over spacetime. This does allow to incorporate all -background gauge fields, and the Green-Schwarz anomaly (and its cancellation) of the heterotic string now comes out as being equivalently the obstruction (and its lifting) for such a parameterized WZW term to exist.
Moreover, where the traditional construction only produces level-1 current algebras, this construction accomodates all levels, and it is argued (Distler-Sharpe 10, section 8.5) that the elliptic genus of the resulting parameterized WZW models are the equivariant elliptic genera found by Liu and Ando (Ando 07)
However, presently questions remain concerning formulating a sigma-model for strings propagating on the total space of the bundle, as it is only the chiral part of the geometric WZW model that appears in the heterotic string. (…)
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
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, vol 3 (which is part 6) of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
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)
Discussion of heterotic suprgravity in terms of superspace includes the following.
One solution of the heterotic superspace Bianchi identities is due to
Joseph J. Atick, Avinash Dhar, and Bharat Ratra, Superspace formulation of ten-dimensional N=1 supergravity coupled to N=1 super Yang-Mills theory, Phys. Rev. D 33, 2824, 1986 (doi.org/10.1103/PhysRevD.33.2824)
Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B Volume 266, Issue 2, 17 March 1986
These two solutions are supposed to be equivalent under field redifinition.
See also at torsion constraints in supergravity.
Further references include these:
Loriano Bonora, ; Paolo Pasti, Mario Tonin , Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335–339 (doi:10.1016/0370-2693(87)91392-X91392-X))
Loriano Bonora, M. Bregola, Kurt Lechner, Paolo Pasti, Mario Tonin, , Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B Volume 296, Issue 4, 25 January 1988 (10.1016/0550-3213(88)90402-690402-6))
L. Bonora, M. Bregola, R. D’Auria, P. Fré K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. Tonin and D. Zanon, Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in supergravity theories, Physics Letters B 277 (1992) (pdf)
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.
Jim Gates, Strings, superstrings, and two-dimensional lagrangian field theory, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) Functional integration, geometry, and strings, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkhäuser, 1989.
and is further expanded on in
The relation of this to equivariant elliptic cohomology is amplified in
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 original conjecture is due to
More details are then in
The duality between F-theory and heterotic string theory originates in
Björn Andreas, Heterotic/F-theory duality PhD thesis (pdf)