nLab heterotic string theory

Idea

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”).

Properties

Compactifications

An effective target space quantum field theory induced from a given heterotic 2d CFT sigma model that has a spacetime of the form $M^4 \times Y^6$ for $M^4$ the 4-dimensional Minkowski space that is experimentally observed locally (say on the scale of a particle accelerator) has $N= 1$ global supersymmetry precisely if the remaining 6-dimensional Riemannian manifold $Y^6$ is a Calabi-Yau manifold. See the references below.

Since global $N=1$ 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 $N=(1,0)$ supersymmetry. Precisely if the corresponding target space effective field theory has $N=1$ supersymmetry does the worldsheet theory enhance to $N=(2,0)$ 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 $E_8 \times E_8$ of the exceptional Lie group E8 with itself, the other is in fact the semi-spin group $SemiSpin(32)$.

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

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

Dualities

Some dualities in string theory involving the heterotic string:

Duality with type I string theory

• duality between heterotic and 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:

$d$	partition function in $d$-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 $M Spin \to KO$
	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 $M Spin^c \to KU$
	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 $M Spin^c \to KU$
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

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 $Spin(16)/\mathbb{Z}_2$-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 $E_8$-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:

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

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

Here $\chi^\alpha$ is the gaugino.

$F_{a b} = \tfrac{1}{4} (\Gamma_{a b})_\alpha{}^\beta D_\beta \chi^\alpha$ (Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28))

equations of motion:

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

$D^b F_{b a} + T_a{}^{b c} F_{b c} = - (\Gamma_a)_{\alpha \beta} \chi^\alpha \chi^\beta - \chi^\alpha L_{\alpha a}$ (Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)) (where $L_{\alpha a}$ is defined by (2.20) there…)

$\,$

The B-field strength:

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

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

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

$H_{a b \alpha} = -\tfrac{1}{2} \Gamma_{a b }_\alpha{}^\beta \rho_\beta$ (Atick-Dhar-Ratra 86, (4.21))

$H_{a b c} = - \tfrac{3}{2} \phi T_{a b c} + \tfrac{c_1}{4} (\Gamma_{a b c})_{\alpha \beta} tr(\chi^\alpha \chi^\beta)$ (Atick-Dhar-Ratra 86, (4.22))

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

See also at torsion constraints in supergravity.

Related concepts

• SemiSpin(16), SemiSpin(32)

• small instanton

• E-string

• string theory

  • heterotic string theory

    • 2d SCFT,

      • 2-spectral triple, Dirac-Ramond operator

    • 2d (2,0)-superconformal QFT

    • Green-Schwarz mechanism

    • dual heterotic string theory

    • heterotic string theory on CY3-manifolds

    • Witten genus, (2,1)-dimensional Euclidean field theories and tmf

  • heterotic line bundle

  • type II string theory

  • type 0 string theory

  • landscape of string theory vacua

  • supersymmetry and Calabi-Yau manifolds

• string field theory

• 11-dimensional supergravity

  • Hořava-Witten theory

  • M-theory

• AdS-CFT correspondence

• string theory FAQ – Does string theory predict supersymmetry?

References

General

Heterotic strings were introduced in

• David Gross, Jeffrey Harvey, Emil Martinec, Ryan Rohm:

  Heterotic string theory (I). The free heterotic string Nucl. Phys. B 256 (1985), 253 (doi:10.1016/0550-3213(85)90394-3)

  Heterotic string theory (II). The interacting heterotic string , Nucl. Phys. B 267 (1986), 75 (doi:10.1016/0550-3213(86)90146-X)

• Philip Candelas, Gary Horowitz, Andrew Strominger, Edward Witten, Vacuum configurations for superstrings, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) (doi:10.1016/0550-3213(85)90602-9)

• Bert Schellekens, Classification of Ten-Dimensional Heterotic Strings, Phys.Lett. B277 (1992) 277-284 (arXiv:hep-th/9112006)

Relation to Niemeier lattices:

• Wolfgang Lerche, Dieter Lüst, Adrian Norbert Schellekens, Ten-dimensional heterotic strings from Niemeier lattices, Physics Letters B 181 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, inspire:233203]

Textbook accounts:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, vol 3 (which is part 6) of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Joseph Polchinski, volume II, section 11 of String theory,

• 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)

See also:

• Eric Sharpe, Recent developments in heterotic compactifications, AMS/IP Stud. Adv. Math. 44 (2008) 209-230 (arXiv:0801.4080)

• Andrea Fontanella, Tomas Ortin, On the supersymmetric solutions of the Heterotic Superstring effective action (arxiv:1910.08496)

• Ioannis Florakis, John Rizos: Free Fermionic Constructions of Heterotic Strings [arXiv:2407.07034]

Relation to Donaldson-Thomas theory and quiver gauge theory:

• Yang-Hui He, Seung-Joo Lee, Quiver Structure of Heterotic Moduli, J. High Energ. Phys. (2012) 2012: 119 (arXiv:1208.3004)

Discussion of higher curvature corrections:

• Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic $\alpha'$-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:

• Justin Kaidi, Kantaro Ohmori, Yuji Tachikawa, Kazuya Yonekura, Non-supersymmetric heterotic branes [arXiv:2303.17623]

• Justin Kaidi, Non-Supersymmetric Heterotic Branes, talk at TH String Theory Seminar (Nov 2023) [cds:2881994]

• Justin Kaidi, Yuji Tachikawa, Kazuya Yonekura: On non-supersymmetric heterotic branes [arXiv:2411.04344]

Orbifold and orientifold compactifications

Heterotic strings on orbifolds:

• Lance Dixon, Jeff Harvey, Cumrun Vafa, Edward Witten, Strings on orbifolds, Nuclear Physics B Volume 261, 1985, Pages 678-686 (doi:10.1016/0550-3213(85)90593-0)

• Lance Dixon, Jeff Harvey, Cumrun Vafa, Edward Witten, Strings on orbifolds (II), Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (doi:10.1016/0550-3213(86)90287-7)

• Joel Giedt, Heterotic Orbifolds (arXiv:hep-ph/0204315)

• Kang-Sin Choi, Spectra of Heterotic Strings on Orbifolds, Nucl. Phys. B708: 194-214, 2005 (arXiv:hep-th/0405195)

Specifically on ADE-singularities:

• Paul Aspinwall, David Morrison, Point-like Instantons on K3 Orbifolds, Nucl. Phys. B503 (1997) 533-564 (arXiv:hep-th/9705104)

• Edward Witten, Heterotic String Conformal Field Theory And A-D-E Singularities, JHEP 0002:025, 2000 (arXiv:hep-th/9909229)

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

• Joseph Polchinski, Open Heterotic Strings, JHEP 0609 (2006) 082 (arXiv:hep-th/0510033)

Heterotic string phenomenology

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

• Philip Candelas, Gary Horowitz, Andrew Strominger, Edward Witten, Vacuum configurations for superstrings, Nuclear Physics B Volume 258, 1985, Pages 46-74 Nucl. Phys. B 258, 46 (1985) [doi:10.1016/0550-3213(85)90602-9]

Review and exposition:

• Edward Witten, Quest For Unification, Heinrich Hertz lecture at SUSY 2002 at DESY, Hamburg [arXiv:hep-ph/0207124]

• Hans-Peter Nilles, Strings, Exceptional Groups and Grand Unification, talk at Planck 2011 [pdf, pdf]

• Saul Ramos-Sanchez, Michael Ratz, Heterotic Orbifold Models, in Handbook of Quantum Gravity, Springer (2024) [doi:10.1007/978-981-19-3079-9,

  arXiv:2401.03125]

The $E_8 \times E_8$-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):

• Volker Braun, Yang-Hui He, Burt Ovrut, Tony Pantev, A Heterotic Standard Model, Phys. Lett. B618 : 252-258 2005 (arXiv:hep-th/0501070)

• Wilfried Buchmuller, Koichi Hamaguchi, Oleg Lebedev, Michael Ratz, Supersymmetric Standard Model from the Heterotic String, Phys. Rev. Lett. 96 121602 (2006) ([doi:varXiv:hep-ph/0511035]

• Volker Braun, Yang-Hui He, Burt Ovrut, Tony Pantev, The Exact MSSM Spectrum from String Theory, JHEP 0605:043, 2006 (arXiv:hep-th/0512177)

• Vincent Bouchard, Ron Donagi, An SU(5) Heterotic Standard Model, Phys. Lett. B633:783-791,2006 (arXiv:hep-th/0512149)

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:

• Lara Anderson, Yang-Hui He, Andre Lukas, Heterotic Compactification, An Algorithmic Approach, JHEP 0707:049, 2007 (arXiv:hep-th/0702210)

using heterotic line bundle models:

• Lara Anderson, James Gray, Andre Lukas, Eran Palti, Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds, Phys. Rev. D 84, 106005 (2011) (arXiv:1106.4804)

• Lara Anderson, James Gray, Andre Lukas, Eran Palti, Heterotic Line Bundle Standard Models JHEP06(2012)113 (arXiv:1202.1757)

• Lara Anderson, Andrei Constantin, James Gray, Andre Lukas, Eran Palti, A Comprehensive Scan for Heterotic $SU(5)$ GUT models, JHEP01(2014)047 (arXiv:1307.4787)

• Yang-Hui He, Seung-Joo Lee, Andre Lukas, Chuang Sun, Heterotic Model Building: 16 Special Manifolds, J. High Energ. Phys. 2014, 77 (2014) (arXiv:1309.0223)

• Stefan Groot Nibbelink, Orestis Loukas, Fabian Ruehle, Patrick K.S. Vaudrevange, Infinite number of MSSMs from heterotic line bundles?, Phys. Rev. D 92, 046002 (2015) (arXiv:1506.00879)

• Andreas Braun, Callum R. Brodie, Andre Lukas, Heterotic Line Bundle Models on Elliptically Fibered Calabi-Yau Three-folds, JHEP04 (2018) 087 (arXiv:1706.07688)

• Andrei Constantin, Yang-Hui He, Andre Lukas, Counting String Theory Standard Models, Physics Letters B

  Volume 792, 10 May 2019, Pages 258-262 (arXiv:1810.00444)

• Alon E. Faraggi, Glyn Harries, Benjamin Percival, John Rizos, Towards machine learning in the classification of $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold compactifications (arXiv:1901.04448)

• Magdalena Larfors, Robin Schneider, Explore and Exploit with Heterotic Line Bundle Models, Fortschritte der Physik Vol 86 Nr. 5 (arXiv:2003.04817)

The resulting database of heterotic line bundle models is here:

• Lara Anderson, James Gray, Andre Lukas, Eran Palti, Heterotic standard model database (web)

Review includes

• Lara Anderson, New aspects of heterotic geometry and phenomenology, talk at Strings2012, Munich 2012 (pdf)

• Yang-Hui He, Deep-learning the landscape, talk at String and M-Theory: The new geometry of the 21st century (pdf slides, video recording)

• Yang-Hui He, Calabi-Yau Spaces in the String Landscape (arXiv:2006.16623)

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

• Volker Braun, Tamaz Brelidze, Michael Douglas, Burt Ovrut, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 0805:080, 2008 (arXiv:0712.3563)

and via machine learning:

• Andrei Constantin, Cristofero S. Fraser-Taliente, Thomas R. Harvey, Andre Lukas, Burt Ovrut, Computation of Quark Masses from String Theory [arXiv:2402.01615]

• Per Berglund, Giorgi Butbaia, Tristan Hübsch, Vishnu Jejjala, Damián Mayorga Peña, Challenger Mishra, Justin Tan: Precision String Phenomenology [arXiv:2407.13836]

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

• Volker Braun, Yang-Hui He, Burt Ovrut, Supersymmetric Hidden Sectors for Heterotic Standard Models (arXiv:1301.6767)

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

• Michele Cicoli, Senarath de Alwis, Alexander Westphal, Heterotic Moduli Stabilization (arXiv:1304.1809)

• Lara Anderson, James Gray, Andre Lukas, Burt Ovrut, Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories (arXiv:1304.2704)

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

• Philip Candelas, Xenia de la Ossa, Yang-Hui He, Balazs Szendroi, Triadophilia: A Special Corner in the Landscape, Adv.Theor.Math.Phys.12:2,2008 (arXiv:0706.3134)

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:

• Carlo Angelantonj, Ioannis Florakis, GUT Scale Unification in Heterotic Strings, Physics Letters B 789 (2019) 496-501 [doi:10.1016/j.physletb.2018.12.054arXiv:1812.06915]

The $SemiSpin(32)$-heterotic string

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

• Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, Heterotic $SO(32)$ model building in four dimensions, JHEP 0412:063, 2004 (arXiv:hep-th/0410232)

• Hans-Peter Nilles, Saul Ramos-Sanchez, Patrick K.S. Vaudrevange, Akin Wingerter, Exploring the $SO(32)$ Heterotic String, JHEP 0604:050 (2006) [doi:10.1088/1126-6708/2006/04/050arXiv:hep-th/0603086]

• Saul Ramos-Sanchez, Towards Low Energy Physics from the Heterotic String, Fortsch. Phys. 10 (2009) 907-1036 [doi:10.1002/prop.200900073arXiv:0812.3560]

  (heterotic SemiSpin(32) R-parity MSSM vacua)

• Naoki Yamatsu, String-Inspired Special Grand Unification, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (arXiv:1708.02078, doi:10.1093/ptep/ptx135)

• Jihn E. Kim, Grand unfication models from $SO(32)$ heterotic string (arXov:2008.00367)

On heterotic line bundle models:

• Hajime Otsuka, $SO(32)$ heterotic line bundle models, (arXiv:1801.03684)

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

This non-supersymmetric string theory was first described in:

• Luis Alvarez-Gaumé, Paul Ginsparg, Gregory Moore and Cumrun Vafa, An $O(16)\times O(16)$ heterotic string, Phys. Lett. B 171 (1986) 155 [doi:10.1016/0370-2693(86)91524-8]

• Lance Dixon and Jeffrey Harvey, String Theories in Ten-Dimensions Without Space-Time Supersymmetry, Nucl. Phys. B 274 (1986) 93 [doi:10.1016/0550-3213(86)90619-X]

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

• Brett McInnes, The Semispin Groups in String Theory, J. Math. Phys. 40 (1999) 4699-4712 [arXiv:hep-th/9906059, doi:10.1063/1.532999]

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

• Adrian Norbert Schellekens, and Nicholas Warner Anomalies, characters and strings, Nuclear Physics B 287 (1987) 317-361 [doi:10.1016/0550-3213(87)90108-8]

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):

• Hisham Sati, Urs Schreiber, Jim Stasheff, pp. 13 in: $L_\infty$-algebra connections and applications to String- and Chern-Simons $n$-transport, in Quantum Field Theory, Birkhäuser (2009) 303-424 [arXiv:0801.3480, doi:10.1007/978-3-7643-8736-5_17]

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Comm. Math. Phys. 315 (2012) 169-213 (arXiv:0910.4001, doi:10.1007/s00220-012-1510-3)

  (via adjusted Weil algebras, see there for more)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, §3.7, §3.8 in: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229-321 [arXiv:1201.5277, euclid:atmp/1414414836]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The $E_8$ moduli 3-stack of the C-field in M-theory, Comm. Math. Phys. 333 1 (2015) 117-151 [arXiv:1202.2455, doi:10.1007/s00220-014-2228-1]

• Clay Cordova, Thomas Dumitrescu, Kenneth Intriligator: Exploring 2-Group Global Symmetries, J. High Energ. Phys. 2019 184 (2019) (arXiv:1802.04790, doi:10.1007/JHEP02(2019)184)

• Francesco Benini, Clay Cordova, Po-Shen Hsin, On 2-Group Global Symmetries and Their Anomalies, J. High Energ. Phys. 2019 118 (2019) (arXiv:1803.09336, doi:10.1007/JHEP03(2019)118)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation, Reviews in Mathematical Physics 34 05 (2022) 2250013 [doi:10.1142/S0129055X22500131, arXiv:2008.08544]

• Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, J. High Energ. Phys. 2021 252 (2021) [doi:10.1007/JHEP04(2021)252, arXiv:2009.00138]

• Michele Del Zotto, Kantaro Ohmori, 2-Group Symmetries of 6D Little String Theories and T-Duality, Annales Henri Poincaré 22 (2021) 2451–2474 [arXiv:2009.03489, doi:10.1007/s00023-021-01018-3]

• Hisham Sati, Urs Schreiber, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes [arXiv:2011.06533]

• Hisham Sati, Urs Schreiber, §2.9 in: M/F-Theory as Mf-Theory, Reviews in Mathematical Physics 35 10 (2023) [doi:10.1142/S0129055X23500289, arXiv:2103.01877]

• Yasunori Lee, Kantaro Ohmori, Yuji Tachikawa, Matching higher symmetries across Intriligator-Seiberg duality, J. High Energ. Phys. 2021 114 (2021) [arXiv:2108.05369, doi:10.1007/JHEP10(2021)114]

• Monica Jinwoo Kang, Sungkyung Kang, Central extensions of higher groups: Green-Schwarz mechanism and 2-connections [arXiv:2311.14666]

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 (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

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

• Bengt Nilsson, Simple 10-dimensional supergravity in superspace, Nucl. Phys. B188 (1981) 176 (doi:10.1016/0550-3213(81)90111-5)

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

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-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 296 4 (1988) [doi:10.1016/0550-3213(88)90402-6, inspire:247764]

• Loriano Bonora, M. Bregola; Kurt Lechner, Paolo Pasti, Mario Tonin, A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (doi:10.1142/S0217751X90000222)

• Castellani-D’Auria-Fre 91, vol 3, part 6

• L. Bonora, M. Bregola, R. D’Auria, P. Fré Kurt Lechner, Paolo Pasti, I. Pesando, M. Raciti, F. Riva, Mario Tonin and D. Zanon, Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories, Physics Letters B 277 (1992) (pdf)

• Kurt Lechner, Mario Tonin, Superspace formulations of ten-dimensional supergravity, JHEP 0806:021,2008 (arXiv:0802.3869)

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 $\sigma$-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 $(1,0)$ superspace, Nucl.Phys. B362 (1991) 199-231 (spire)

and is further expanded on in

• Jacques Distler, Eric Sharpe, Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)

reviewed in

• Eric Sharpe, Recent developments in heterotic compactifications, in Eric Sharpe, Arthur Greenspoon, Advances in String Theory: The First Sowers Workshop in Theoretical Physics, AMS 2008 (pdf slides (39-49))

The relation of this to equivariant elliptic cohomology is amplified in

• Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)

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

• Robert Friedman, John Morgan, Edward Witten, Vector Bundles And F-Theory (arXiv:hep-th/9701162)

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)

Dualities

With type I superstring theory

The original conjecture is due to

• Edward Witten, section 5 of String Theory Dynamics In Various Dimensions, Nucl.Phys.B443:85-126 (1995) (arXiv:hep-th/9503124)

More details are then in

• Joseph Polchinski, Edward Witten, Evidence for Heterotic - Type I String Duality, Nucl.Phys.B460:525-540,1996 (arXiv:hep-th/9510169)

With $F$-theory

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

• Ashoke Sen, F-theory and Orientifolds (arXiv:hep-th/9605150)

• Robert Friedman, John Morgan, Edward Witten, Vector Bundles And F Theory (arXiv:hep-th/9701162)

Reviews include

• Ron Donagi, ICMP lecture on heterotic/F-theory duality (arXiv:hep-th/9802093)

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

$SU(2)$-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):

• Abhijit Gadde, Babak Haghighat, Joonho Kim, Seok Kim, Guglielmo Lockhart, Cumrun Vafa, Section 4.2 of: 6d String Chains, J. High Energ. Phys. 2018, 143 (2018) (arXiv:1504.04614, doi:10.1007/JHEP02(2018)143)

• Kantaro Ohmori, Section 2.3.1 of: Six-Dimensional Superconformal Field Theories and Their Torus Compactifications, Springer Theses 2018 (springer:book/9789811330919)

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 $\mathcal{N}=(1,0)$ 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:

• Amihay Hanany, Alberto Zaffaroni, Branes and Six Dimensional Supersymmetric Theories, Nucl.Phys. B529 (1998) 180-206 (arXiv:hep-th/9712145)

• Ilka Brunner, Andreas Karch, Branes at Orbifolds versus Hanany Witten in Six Dimensions, JHEP 9803:003, 1998 (arXiv:hep-th/9712143)

Reviewed in:

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

See also:

• Amihay Hanany, Noppadol Mekareeya, The Small $E_8$ Instanton and the Kraft Procesi Transition, JHEP07 (2018) 098 (arXiv:1801.01129)