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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 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 theory on CY3-manifolds.
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.
Precisely two (isomorphism classes of) gauge groups are consistent (give quantum anomaly cancellation) while preserving supersymmetry: one is the direct product group of the exceptional Lie group E8 with itself, the other is in fact the semi-spin group .
If the supersymmetry requirement is dropped, then there is a third option, which locally looks like the product of the special orthogonal group , 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 and theories, by looking at how the relevant representations of these two theories assemble into those appearing in the theory.
The global character of the gauge group is not , however. McInnes 1999 proposed that the actual gauge group is a quotient of spin groups by a -action (not corresponding to ). This proposal allows to identify as a subgroup of both and , where the author’s purpose for this is identifying the theory as the realization of the T-duality between the supersymmetric heterotic strings.
Some dualities in string theory involving the heterotic string:
See duality between heterotic and type II string theory.
See duality between heterotic string theory and M-theory
See duality between heterotic string theory and F-theory
and see references below.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
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 -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 -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. (…)
The gauge field strength:
(Witten 86, Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.13)).
(Witten 86 (8), Atick-Dhar-Ratra 86, (4.14), Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)).
Here is the gaugino.
(Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.28))
(Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.30))
(Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.31)) (where is defined by (2.20) there…)
(Atick-Dhar-Ratra 86, (4.2), Bonora-Pasti-Tonin 87, (15), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.14))
(Atick-Dhar-Ratra 86, (4.19), Bonora-Pasti-Tonin 87, (15), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.15))
According to (Bonora-Bregola-Lechner-Pasti-Tonin 90) in fact all these constraints follow from just , up to field redefinition.
See also at torsion constraints in supergravity.
heterotic string theory
string theory FAQ – Does string theory predict supersymmetry?
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:
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:
Discussion of higher curvature corrections:
Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic -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]
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):
The historical origin of all string phenomenology is the top-down GUT-model building in heterotic string theory due to
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,
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:
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 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 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:
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
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:
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:
Discussion of non-supersymmetric: GUT models:
See also:
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 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 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]
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 heterotic string (arXov:2008.00367)
On heterotic line bundle models:
This non-supersymmetric string theory was first described in:
Luis Alvarez-Gaumé, Paul Ginsparg, Gregory Moore and Cumrun Vafa, An 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:
A suggestion that the heterotic string is a the string “minus” the semispin group :
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: -algebra connections and applications to String- and Chern-Simons -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 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) (arXiv:2009.00138, doi:10.1007/JHEP04(2021)252)
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]
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
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 SUGRA-SYM in 10-D, International Journal of Modern Physics A, February 1990, Vol. 05, No. 03 : pp. 461-477 (doi:10.1142/S0217751X90000222)
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 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)
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.
Discussion of heterotic strings whoe current algebra-sector is parameterized by a principal bundle originates with
Jim Gates, Warren Siegel, Leftons, Rightons, Nonlinear -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 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
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
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, Heterotic/F-theory duality PhD thesis (pdf)
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:
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:
See also:
Last revised on September 21, 2024 at 09:40:43. See the history of this page for a list of all contributions to it.