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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 $M^4 \times Y^6$ 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 background with a Calabi-Yau factor.
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.
Some duality in string theory involving the heterotic string:
For the moment see at Horava-Witten theory.
For duality between F-theory and heterotic string theory see there 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 worldheet 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 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 $E_8$-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. (…)
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))
$(D^a \Gamma_a)_{\alpha\beta} \chi^\beta =0$ (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…)
$\,$
$H_{\alpha \beta \gamma} = 0$ (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.
heterotic string theory
string theory FAQ – Does string theory predict supersymmetry?
Heterotic strings were introduced in
David Gross, J. A. Harvey, E. Martinec and R. Rohm,
Heterotic string theory (I). The free heterotic string Nucl. Phys. B 256 (1985), 253.
Heterotic string theory (I). The interacting heterotic string , Nucl. Phys. B 267 (1986), 75.
Philip Candelas, Gary Horowitz, Andrew Strominger, Edward Witten, Nucl. Phys. B258 (1985) 46
Bert Schellekens, Classification of Ten-Dimensional Heterotic Strings, Phys.Lett. B277 (1992) 277-284 (arXiv:hep-th/9112006)
Textbook accounts include
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)
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 Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)
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)
L. Bonora, M. Bregola, R. D’Auria, P. Fré K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. 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)
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 $\sigma$-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
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, $N=1$ Heterotic/F-theory duality PhD thesis (pdf)
A kind of unusual boundary condition for heterotic strings, (analogous to open M5-branes ending in Yang monopoles on M9-branes) is discussed in