heterotic string theory


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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 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 Calabi-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.


Some duality in string theory involving the heterotic string:

Relation to M-theory

For the moment see at Horava-Witten theory.

Relation to F-theory, type II and type I superstring theory

For duality between F-theory and heterotic string theory see there and see references below.

Partition function and Witten genus

in as // in :

dd in dd-dimensional QFT in logarithmic coefficients of
0push-forward in :
1 MSpinKOM Spin \to KO
endpoint of string twisted by of boundary / MSpin cKUM Spin^c \to KU
endpoint of twisted by MSpin cKUM Spin^c \to KU
2 in NS-R sector

General gauge backgrounds and parameterized WZW models

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 Spin(16)/ 2Spin(16)/\mathbb{Z}_2-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 E 8E_8-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. (…)

Superspace formulation

The gauge field strength:

F αβ=0F_{\alpha \beta} = 0 (Witten 86, Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.13)).

F aα=Γ aαβχ βF_{a \alpha} = \Gamma_{a \alpha \beta} \chi^\beta (Witten 86 (8), Atick-Dhar-Ratra 86, (4.14), Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)).

Here χ α\chi^\alpha is the gaugino.

F ab=14(Γ ab) α βD βχ αF_{a b} = \tfrac{1}{4} (\Gamma_{a b})_\alpha{}^\beta D_\beta \chi^\alpha (Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28))

equations of motion:

(D aΓ a) αβχ β=0 (D^a \Gamma_a)_{\alpha\beta} \chi^\beta =0 (Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.30))

D bF ba+T a bcF bc=(Γ a) αβχ αχ βχ αL αaD^b F_{b a} + T_a{}^{b c} F_{b c} = - (\Gamma_a)_{\alpha \beta} \chi^\alpha \chi^\beta - \chi^\alpha L_{\alpha a} (Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)) (where L αaL_{\alpha a} is defined by (2.20) there…)


The B-field strength:

H αβγ=0H_{\alpha \beta \gamma} = 0 (Atick-Dhar-Ratra 86, (4.2), Bonora-Pasti-Tonin 87, (15), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.14))

H aαβ=ϕΓ aαβH_{a \alpha \beta} = \phi \Gamma_{a \alpha \beta} (Atick-Dhar-Ratra 86, (4.19), Bonora-Pasti-Tonin 87, (15), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.15))

ρD αϕ\rho \coloneqq D_\alpha \phi (Atick-Dhar-Ratra 86, (4.20))

H abα=12Γ abα βρ βH_{a b \alpha} = -\tfrac{1}{2} \Gamma_{a b }_\alpha{}^\beta \rho_\beta (Atick-Dhar-Ratra 86, (4.21))

H abc=32ϕT abc+c 14(Γ abc) αβtr(χ αχ β)H_{a b c} = - \tfrac{3}{2} \phi T_{a b c} + \tfrac{c_1}{4} (\Gamma_{a b c})_{\alpha \beta} tr(\chi^\alpha \chi^\beta) (Atick-Dhar-Ratra 86, (4.22))

According to (Bonora-Bregola-Lechner-Pasti-Tonin 90) in fact all these constraints follow from just T αβ aΓ αβ aT^a_{\alpha \beta} \propto \Gamma^a_{\alpha \beta}, up to field redefinition.

See also at torsion constraints in supergravity.



Heterotic strings were introduced in

Textbook accounts include

Superspace formulation of Heterotic supergravity

Discussion of heterotic supergravity in terms of superspace includes the following.

One solution of the heterotic superspace Bianchi identities is due to

  • Joseph 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 (

  • Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B Volume 266, Issue 2, 17 March 1986

A second solution is due to Bengt Nilsson, Renata Kallosh and others

These two solutions are supposed to be equivalent under field redefinition.

See also at torsion constraints in supergravity.

Further references include these:

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.

General flux backgrounds and parameterized WZW models

Discussion of heterotic strings whoe current algebra-sector is parameterized by a principal bundle originates with

  • Jim Gates, Warren Siegel, Leftons, Rightons, Nonlinear σ\sigma-Models, and Superstrings, Phys.Lett. B206 (1988) 631 (spire)

  • 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.

  • Jim Gates, S. Ketov, S. Kozenko, O. Solovev, Lagrangian chiral coset construction of heterotic string theories in (1,0)(1,0) superspace, Nucl.Phys. B362 (1991) 199-231 (spire)

and is further expanded on in

reviewed in

The relation of this to equivariant elliptic cohomology is amplified 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 type I superstring theory

The original conjecture is due to

More details are then in

With FF-theory

The duality between F-theory and heterotic string theory originates in

Reviews include

“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

Last revised on April 22, 2018 at 10:50:19. See the history of this page for a list of all contributions to it.