nLab 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 theory on CY3-manifolds.

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.

Gauge groups

Precisely two (isomorphism classes of) gauge groups are consistent (give quantum anomaly cancellation) while preserving supersymmetry: one is the direct product group E 8×E 8E_8 \times E_8 of the exceptional Lie group E8 with itself, the other is in fact the semi-spin group SemiSpin ( 32 ) SemiSpin(32) .

If the supersymmetry requirement is dropped, then there is a third option, which locally looks like the product of the special orthogonal group SO(16)×SO(16)SO(16)\times SO(16), as first described in (Alvarez-Gaumé, Ginsparg, Moore, and Vafa (1986)) and in (Dixon, and Harvey (1986)). It was suggested in (Schellekens, and Warner (1987)) that this theory could be regarded as some sort of “difference” (what is better understood as a correspondence, or duality) between the E 8×E 8E_8\times E_8 and SemiSpin ( 32 ) \text{SemiSpin}(32) theories, by looking at how the relevant representations of these two theories assemble into those appearing in the SO(16)×SO(16)SO(16)\times SO(16) theory.

The global character of the gauge group is not SO(16)×SO(16)SO(16)\times SO(16), however. In (McInnes (1999)) it is proposed that the actual gauge group is a quotient of spin groups Spin(16)×Spin(16)\text{Spin}(16) \times \text{Spin}(16) by a 2\mathbb{Z}_2-action (not corresponding to SemiSpin(16)×SemiSpin(16)\text{SemiSpin}(16)\times \text{SemiSpin}(16)). This proposal allows to identify Spin(16)×Spin(16)/ 2\text{Spin} (16) \times \text{Spin}(16)/ \mathbb{Z}_2 as a subgroup of both E 8×E 8E_8\times E_8 and SemiSpin(32)\text{SemiSpin}(32), where the author’s purpose for this is identifying the Spin(16)×Spin(16)/ 2\text{Spin} (16) \times \text{Spin} (16)/\mathbb{Z}_2 theory as the realization of the T-duality between the supersymmetric heterotic strings.


Some dualities in string theory involving the heterotic string:

Duality with type I string theory

Duality with type II string theory

See duality between heterotic and type II string theory.

Duality with M-theory

See duality between heterotic string theory and M-theory

Duality with F-theory

See duality between heterotic string theory and F-theory

and see references below.

Partition function and Witten genus

partition functions in quantum field theory as indices/genera/orientations 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 formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
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 numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

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 worldsheet 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 it is known that, for 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 the incorporation of 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 accommodates 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

Relation to Niemeier lattices:

Textbook accounts:

See also:

Relation to Donaldson-Thomas theory and quiver gauge theory:

Discussion of higher curvature corrections:

  • Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic α\alpha'-corrections (arXiv:2104.09545)

  • Hao-Yuan Chang, Ergin Sezgin, Yoshiaki Tanii, Dimensional reduction of higher derivative heterotic supergravity (arXiv:2110.13163)

On non-supersymmetric branes in heterotic string theory:

Orbifold and orientifold compactifications

Heterotic strings on orbifolds:

Specifically on ADE-singularities:

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

Heterotic string phenomenology

The historical origin of all string phenomenology is the top-down GUT-model building in heterotic string theory due to

Review and exposition:

The E 8×E 8E_8 \times E_8-heterotic string

The following articles claim the existence of exact realization of the gauge group and matter-content of the MSSM in heterotic string theory on orbifolds (not yet checking Yukawa couplings):

A computer search through the “landscape” of Calabi-Yau varieties showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU(5)-GUTs):

general computational theory:

using heterotic line bundle models:

The resulting database of heterotic line bundle models is here:

Review includes

Computation of metrics on these Calabi-Yau compactifications (eventually needed for computing their induced Yukawa couplings) is started in

and via machine learning:

This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:

The issue of moduli stabilization in these kinds of models is discussed in

Principles singling out heterotic models with three generations of fundamental particles are discussed in:

Discussion of non-supersymmetric: GUT models:

  • Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, Classification of Non-Supersymmetric Pati-Salam Heterotic String Models (arXiv:2011.04113)

See also:

The SemiSpin(32)SemiSpin(32)-heterotic string

Discussion of string phenomenology for the SemiSpin(32)-heterotic string (see also at type I phenomenology):

On heterotic line bundle models:

The “SO(16)×SO(16)SO(16)\times SO(16)”-heterotic string

This non-supersymmetric string theory was first described in:

A proposal on what the correct global character of the gauge group is appears in:

A suggestion that the SO(16)×SO(16)SO(16)\times SO(16) heterotic string is a the E 8×E 8E_8\times E_8 string “minus” the semispin group Spin ( 32 ) / 2 \text{Spin}(32)/\mathbb{Z}_2 :

Higher gauge theory of the Green-Schwarz mechanism

Discussion of higher gauge theory modeling the Green-Schwarz mechanisms for anomaly cancellation in heterotic string theory, on M5-branes, and in related systems in terms of some kind of nonabelian differential cohomology (ordered by arXiv time-stamp):

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

SU(2)SU(2)-flavor symmetry on heterotic M5-branes

Emergence of SU(2) flavor-symmetry on single M5-branes in heterotic M-theory on ADE-orbifolds (in the D=6 N=(1,0) SCFT on small instantons in heterotic string theory):

Argument for this by translation under duality between M-theory and type IIA string theory to half NS5-brane/D6/D8-brane bound state systems in type I' string theory:

Reviewed in:

  • Santiago Cabrera, Amihay Hanany, Marcus Sperling, Section 2.3 of: Magnetic Quivers, Higgs Branches, and 6d 𝒩=(1,0)\mathcal{N}=(1,0) Theories, JHEP06(2019)071, JHEP07(2019)137 (arXiv:1904.12293)

The emergence of flavor in these half NS5-brane/D6/D8-brane bound state systems, due to the semi-infinite extension of the D6-branes making them act as flavor branes:

Reviewed in:

  • Fabio Apruzzi, Marco Fazzi, Section 2.1 of: AdS 7/CFT 6AdS_7/CFT_6 with orientifolds, J. High Energ. Phys. (2018) 2018: 124 (arXiv:1712.03235)

See also:

Last revised on July 10, 2024 at 08:01:43. See the history of this page for a list of all contributions to it.