heterotic string theory


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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 M 4×Y 6M^4 \times Y^6 for M 4M^4 the 4-dimensional Minkowski space that is experimentally observed locally (say on the scale of a particle accelerator) has N=1N= 1 global supersymmetry precisely if the remaining 6-dimensional Riemannian manifold Y 6Y^6 is a Calabi-Yau manifold. See the references below.

Since global N=1N=1 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.

Enhanced supersymmetry

A priori the worldsheet 2d SCFT describing the quantum heterotic string has N=(1,0)N=(1,0) supersymmetry. Precisely if the corresponding target space effective field theory has N=1N=1 supersymmetry does the worldsheet theory enhance to N=(2,0)N=(2,0) supersymmetry. See at 2d (2,0)-superconformal QFT and at Calabi-Yau manifolds and supersymmetry for more on this.



Partition function and Witten genus

partition functions in quantum field theory as indices/genera in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential forms
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbers
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbers
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbers
2superstringDirac-Ramond operatorsuperstring partition functionelliptic genus/Witten genusEisenstein series
self-dual stringM5-brane charge



Heterotic strings were introduced in

Textbook accounts include

In elliptic cohomology

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.

As higher super-GS-WZW type σ\sigma-models

Discussion from the point of view of Green-Schwarz action functional-∞-Wess-Zumino-Witten theory is in


Subtleties in the realization of general E8 background gauge fields for the heterotic string are discussed in

On elliptic fibrations

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

  • B. Andreas and D. Hernandez Ruiperez, Adv. Theor. Math. Phys. Volume 7, Number 5 (2003), 751-786 Comments on N = 1 Heterotic String Vacua (project Euclid)


With FF-theory

The heterotic/F-theory duality is discussed for instance in

  • Björn Andreas, N=1N=1 Heterotic/F-theory duality PhD thesis (pdf)

“Open” heterotic string

A kind of unusual boundary condition for heterotic strings, (analogous to open M5-branes ending in Yang monopoles on M9-branes) is discussed in

Revised on March 21, 2014 06:02:57 by Urs Schreiber (