Contents

supersymmetry

# Contents

## 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.

### Dualities

Some duality in string theory involving the heterotic string:

#### Duality with F-theory

See duality between heterotic string theory and F-theory?

and see references below.

### Partition function and Witten genus

$d$partition function in $d$-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 $M Spin \to KO$
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 $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
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 $Spin(16)/\mathbb{Z}_2$-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

$F_{\alpha \beta} = 0$ (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…)

$\,$

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.

## References

### General

Heterotic strings were introduced in

Textbook accounts:

Relation to Donaldson-Thomas theory and quiver gauge theory:

### Orbifold and orientifold compactifications

Heterotic strings on orbifolds:

### Phenomenology

The historical origin of all string phenomenology is the top-down approach in the heterotic string due to (Candelas-Horowitz-Strominger-Witten 85).

A brief review of motivations for GUT models in heterotic string theory is in

Specifically 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 (arXiv:hep-th/0603086)

• Saul Ramos-Sanchez, Towards Low Energy Physics from the Heterotic String, Fortsch. Phys. 10:907-1036, 2009 (arXiv:0812.3560)

The following articles establish the existences of exact realization of the gauge group and matter-content of the MSSM in heterotic string theory (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)

The resulting database of compactifications is here:

Review includes

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

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:

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

• Carlo Angelantonj, Ioannis Florakis, GUT Scale Unification in Heterotic Strings (arXiv:1812.06915)

### 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

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

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 $\sigma$-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

### 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)

### Dualities

#### With type I superstring theory

The original conjecture is due to

More details are then in

#### With $F$-theory

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

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)

### “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 September 28, 2019 at 14:46:24. See the history of this page for a list of all contributions to it.