string phenomenology

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### Context
#### String theory
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#### Physics
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[[!include physicscontents]]
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# Contents #
* table of contents
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## Idea

_String phenomenology_ is [[phenomenology]] in [[particle physics]] and [[cosmology]] based on [[model (in particle physics)|models]] that are derived or at least motivated from [[string theory]] (as [[effective QFTs]] from string [[vacua]]).

Broadly speaking, _string phenomenology_ refers to investigations of the connection of [[string theory]] to experimentally observed [[physics]]. More restrictively it refers to constructions of string theory [[vacua]] whose [[effective field theory]] reproduces the [[standard model of particle physics]] and/or the [[standard model of cosmology]].

String theory models naturally match the _general_ conceptual structure of the [[standard model of particle physics]] plus [[gravity]] (which is what drives the interest in string theory in the first place): for instance the standard model is a four dimensional [[QFT]] with a [[gauge theory|non-Abelian gauge symmetry]], several [[family of elementary particles|families]] of [[chiral fermions]] and hierarchical [[Yukawa couplings]] -- and the same is true for the generic [[Kaluza-Klein mechanism|compactification]] of the [[effective QFT]] that describes [[heterotic string theory]] on a 6-dimensional compact space ([CHSW85](#CHSW85)) as well as for [[11-dimensional supergravity]]/[[M-theory]] compactified on a [[G2-manifold]] ([AW01](#AW01)).

This structure alone already implies a variety of 3-body decays of the heavier fermions into the lighter ones and the existence of massive [[vector bosons]] coupling to [[charged currents]], which in the observed [[standard model of particle physics]] are the [[W-boson]], etc. (See section III of [AKK12](#AKK12) for an exposition.)

Therefore it is not hard to find string theory compactifications that resemble the observed particle physics in _broad_ strokes. Under some simplifying assumptions many string models have been built that very closely resemble also the fine-structure of the standard model.

A central technical issue with string [[model (in particle physics)|model building]] is that of the [[Kaluza-Klein mechanism]] involved: the [[moduli stabilization]]. Historically there had been the hope that the consistency condition of moduli stabilization on string models is so strong that it strongly reduces the number of models that look like the standard model. Arguments that the number is still "not small" even with various extra assumptions lead to the term of a [[landscape of string theory vacua|landscape]] ([[moduli space]]) of string theory models, which remains, however, poorly understood. Arguments for properties of low-energy [[effective QFTs]] that rule out a possible string-theoretic model have been brought forward for instance in ([Vafa05](http://arxiv.org/abs/hep-th/0509212)). A review of what is known about the space of possibilities is in ([Taylor11](#Taylor11)).

While all this remains poorly understood, a noteworthy difference of string phenomenology to [[model (in particle physics)|model building]] in bare [[QFT]] is that a) there is a larger framework at all in which to search for models and b) with every model automatically comes a [[UV-completion]], which is the basic motivation for embedding the standard model of particle physics in a broader theory of [[quantum gravity]] in the first place.

A good account of what it means to have a realistic string theory model and what the subtleties are, and in which sense they have already been found abundantly or not at all, is in the introduction of ([Dolan-Krippendorf-Quevedo](#DolanKrippendorfQuevedo11)).

## Top-down models and bottom-up models
 {#TopDownAndBottomUpModels}

Since a realistic string theoretic model is, by design, a unification of the [[standard model of particle physics]] with [[quantum gravity]] aspects and hence  at least with aspects of the [[standard model of cosmology]], there are more constraints on such a model than are usually imposed on model building in particle physics alone: the model is not only supposed to reproduce the [[fundamental particle]] content but also address [[moduli stabilization]], the [[cosmological constant]] and [[dark matter]] (see e.g. [Dolan-Krippendorf-Quevedo 11, p. 3](#DolanKrippendorfQuevedo11)).

Accordingly one strategy to build models is to first aim for the correct fundamental particle content, and then incrementally adjust to account for the global gravitational constraints. For instance in [type II intersecting brane models](#ModelsInTypeIIWithIntersectingBranes) people often consider just an [[open neighbourhood]] of a singular point in a [[Kaluza-Klein mechanism|KK-compactification space]], adjust the model there, and then later ask about embedding this local construction into an actually globally defined compactification space (typically a [[Calabi-Yau manifold]] for compactifications aiming for $N=1$ low energy supersymmetry in the effective 4d model).

This approach is known as the **[[bottom-up model building|bottom-up approach]]** to string model building ([AIQU 00](#AIQU00)).

Contrary to this is the historically older **[[top-down model building|top-down approach]]** (usually attributed to ([Candelas-Horowitz-Strominger-Witten 85](#CHSW85))) in the [[heterotic string theory]] compactification models (see [below](#HeteroticStandardModels)).

## Semi-realistic models in string theory
 {#SemiRealisticModels}

Examples of [[model (in particle physics)|models]] in string phenomenology include

* _[Models in heterotic string theory](#HeteroticStandardModels)_

* _[Models in type II string theory with intersecting branes](#ModelsInTypeIIWithIntersectingBranes)_

* _[Models in M-theory](#ModelsInMTheory)_

See at _[References - Models](#ReferencesModels)_ below.

### Models in heterotic string theory
 {#HeteroticStandardModels}

The models in [[heterotic string theory]] follow the historically original and hence oldest strategy of finding semi-realistic [[GUT]] models in string theory (see ([Witten 02](#Witten02)) for a brief list of motivations for these models): one considers a [[Kaluza-Klein compactification]] of [[heterotic string theory]]/[[heterotic supergravity]] on a [[closed manifold]] of dimension 6 with a non-trivial [[gauge field]] configuration on it. By choosing different values of the [[holonomy]] of this gauge field around non-trivial [[singular homology|singular 1-cycles]] in the compact space (usually referred to as "[[Wilson lines]]" in this context) one obtains different [[effective quantum field theory|effective physics]] in the remaining 4-dimensional space.

Since most of string model building was aimed for reproducing the minimally [[supersymmetry|supersymmetric]] extension of the [[standard model of particle physics]], these approaches usually take that compact 6-manifold to be a complex 3-dimensional [[Calabi-Yau manifold]].

In more detail, the paradigm of this approach is compactification of
the [[E8]]$\times$ [[E8]] [[heterotic string theory]] on a [[Calabi-Yau manifold]] of [[Euler characteristic]] $\chi = \pm 6$, leading to a three-generation [[E6]]-model. Further gauge [[spontaneous symmetry breaking]] may be achieved e.g. by
the addition of [[Wilson lines]] and a final breakdown of $D = 4$, $N = 1$ supersymmetry is assumed to take place due to some field-theoretical [[non-perturbative effects]].

See at _[References - Models in heterotic string theory](#ReferencesHeteroticModels)_

The lift of these heterotic CY3-compactifications to [[M-theory]] is _[[M-theory on G2-manifolds]]_, discussed [below](#ModelsInMTheory).

### Models in type II string theory / F-theory
 {#ModelsInTypeIIWithIntersectingBranes}

In contrast to the construction of "heterotic standard models" [above](#HeteroticStandardModels), which are basically plain variants of the old [[Kaluza-Klein compactification]] mechanism where the effective [[gauge fields]] in 4-dimensional [[spacetime]] arise as components of the field of [[gravity]] in higher dimensions, in [[type II string theory]] with [[D-branes]] there are [[open strings]] whose massless excitations yield [[gauge fields]] "directly". The precise nature of these gauge fields and their couplings depends on the precise [[boundary conditions]] of these open strings, hence on the choice of [[D-branes]] that they end on.

Therefore in what are called "[[intersecting D-brane models]]" one considers [[Kaluza-Klein compactifications]] of [[type II string theory]] with [[D-branes]] that [[brane intersection|intersect]] in an intricate pattern in the compactification space. By choosing this [[brane intersection|intersection]] geometry suitably, one obtains various different realizations of [[gauge theory]] in the [[effective field theory|effective]] 4-dimensional physics.

The [[intersecting branes]] of main relevance in [[type IIA string theory]] are [[D6-branes]] (e.g. [Lüst 04](#Lust04), [Ibanez-Uranga 12, section 10](#IbanezUranga12)), which, under [[T-duality]], correspond in [[type IIB string theory|type IIB]] to [[D7-branes]]. These are precisely the ones whose lift to [[M-theory]] correspond to [[conifold]]/[[ADE orbifold]] [[singularities]] of [[KK-monopoles]], see also at _[[F-branes -- table]]_.

One way to neatly reorganize the required data for such type II compactifications is to formulate them in terms of "[[F-theory]]", which is why some of this type II model building now goes by names like "F-theory phenomenology" or similar.

The [[moduli stabilization]] in these type of models can be achieved by choosing the various [[RR-field]] and [[B-field]] [[field strength]] (the "fluxes") on the compactification space such that its [[curvature]] forms have certain specified [[periods]] on non-trivial [[singular cohomology|singular cycles]] of the compactification space. See ([Denef 08](#Denef08)) for introduction and review of such type IIB [[flux compactification]].

Since there are only finitely many -- but many -- such choices, it is in this context that people first tried to count the number of possibilities of building models (under all these assumptions, though) and found these large finite numbers such as the meanwhile proverbial number $10^{500}$ (coming from an estimate of the number of non-trivial cycles in a generic Calabi-Yau and the number of choices of periods of the "flux" fields) which then led them to speak of the "[[landscape of string theory vacua]]". (Which of course without making a bunch of assumptions is vastly bigger, even.)

Due to the relation between [[supersymmetry and Calabi-Yau manifolds]], of particular interest is the case of [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]], see there for more.

For references see below at _[References - Models in type II string theory](#ReferencesTypeIIModels)_

#### Computer scan of Gepner-model compactifications
 {#ComputerScanOfGepnerModelCompactifications}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

### Models in M-theory
 {#ModelsInMTheory}

The lift of the [heterotic models](#HeteroticStandardModels) compactified on [[Calabi-Yau manifolds]] to  [[11-dimensional supergravity]] with some of its "[[M-theory]]"-corrections taken into account is _[[M-theory on G2-manifolds]], hence M-theory  [[Kaluza-Klein compactification|KK-compactified]] on [[G2-manifolds]] (or rather: [[orbifolds]]) of, necessarily, [[dimension]] 7.

Accordingly, models in this context go by the name _[[G2-MSSM]]_.

See at _[References - Models in M-theory](#ReferencesMTheoryModels)_.

### Non-supersymmetric models
 {#NonsusyModels}

All of the above models aim for $N = 1$ [[supersymmetry]] in the low-energy [[effective field theory]], because it was a wide-spread thought that this is what describes the observable world at [[electroweak symmetry breaking]]-scale. However, new experimental results at the [[LHC]] make this low energy supersymmetry scenario increasingly unlikely (even if not fully ruled out yet). Accordingly people start to look for string models now that do not display low energy supersymmetry (of course all of them have high energy local supersymmetry, in that they are theories of [[supergravity]]).

See for instance ([MRS 09](#MRS09)) and citations given there.

## Related concepts

* [[string inspired model]]

* [string theory FAQ -- Is string theory testable?](string+theory+FAQ#IsStringTheoryTestable)

* [string theory FAQ - Did string theory provide any insight relevant in experimental particle physics?](string+theory+FAQ#InsightInParticlePhysics)

* [[Brandenberger-Vafa mechanism]]

## References

### Surveys

Useful broad survey is in

* Fernando Marchesano, _String phenomenology today_, talk at _[StringPheno14](http://stringpheno2014.ictp.it)_ ([pdf](https://indico.cern.ch/event/366801/contributions/1784430/attachments/729154/1000461/MarchesanoHEP.pdf))

Technical surveys on particle physics string phenomenology include

* {#Nilles04} [[Hans-Peter Nilles]], _String phenomenology_ (2004) ([pdf](http://www.th.physik.uni-bonn.de/nilles/db/HPtalks/1204desy.pdf))

* [[Michael Dine]], _Supersymmetry and string theory: beyond the standard model_, Cambridge University Press (2006, 2007) [[ISBN:9781009290920](https://www.cambridge.org/ae/academic/subjects/physics/particle-physics-and-nuclear-physics/supersymmetry-and-string-theory-beyond-standard-model-2nd-edition?format=HB), [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/dine.pdf)]

* [[Ralph Blumenhagen]], Boris Kors, [[Dieter Lüst]], Stephan Stieberger, _Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes_, Phys.Rept.445:1-193,2007 ([arXiv:hep-th/0610327](http://arxiv.org/abs/hep-th/0610327))

* {#DouglasEtAl07} [[Michael Douglas]] et. al. (eds.), _[[String theory and the real world]]_, Les Houches  Session LXXXVII 2007

* Tatsuo Kobayashi, _String phenomenology_ 2009 ([pdf slides](http://www-tap.scphys.kyoto-u.ac.jp/sc2/Kobayashi.pdf))

* [[Washington Taylor]], _TASI Lectures on Supergravity and String Vacua in Various Dimensions_ ([arXiv:1104.2051](http://arxiv.org/abs/1104.2051))

* {#AKK12} [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_ ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

* {#Wijnholt14} [[Martin Wijnholt]], _String compactification_, [PITP 2014](https://pitp2014.ias.edu) lecture notes ([[WijnholtCompactification14.pdf:file]], [slides for lecture 1](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf), [slides for lecture 2](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P2_wijnholt.pdf), [slides for lecture 3](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf))

* [[Eran Palti]], _Review of Model Building in String Theory_, talk at [String Phenomenology 2014](http://stringpheno2014.ictp.it/program.html) ([pdf](http://stringpheno2014.ictp.it/lecturenotes/Palti.pdf))

* [[Fernando Marchesano]], [[Gary Shiu]], [[Timo Weigand]], *The Standard Model from String Theory: What Have We Learned?* [[arXiv:2401.01939](https://arxiv.org/abs/2401.01939)]

Technical surveys on cosmological string phenomenology include

* S. F. King, J. P. Roberts, _Natural Dark Matter from Type I String Theory_ ([arXiv:hep-ph/0608135](http://arxiv.org/abs/hep-ph/0608135))

The "[[bottom-up model building|bottom-up approach]]" to string model building is attributed to

* {#AIQU00} G. Aldazabal, [[Luis Ibáñez]], F. Quevedo, [[Angel Uranga]], _D-Branes at Singularities : A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067](http://xxx.lanl.gov/abs/hep-th/0005067))

See also

* [[Hans-Peter Nilles]], Patrick K.S. Vaudrevange, _Geography of Fields in Extra Dimensions: String Theory Lessons for Particle Physics_, Perspectives on String Phenomenology" (World Scientific) ([arXiv:1403.1597](http://arxiv.org/abs/1403.1597))

For [[Gepner models]]:

* Christian Reppel, _Phenomenological Aspects of Gepner Models_, 2007 ([pdf](https://www.ru.nl/publish/pages/760962/christiaan_reppel.pdf))

### Original articles

* {#CHSW85} [[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)

* {#AW01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152))

* [[Cumrun Vafa]], _The String Landscape and the Swampland_ ([arXiv:hep-th/0509212](http://arxiv.org/abs/hep-th/0509212))

[[!include heterotic string phenomenology -- references]]

### Further Models
 {#ReferencesModels}

#### Type II string theory models
 {#ReferencesTypeIIModels}

The canonical textbook for [[type II superstring]] phenomenology via [[intersecting D-brane models]] is

* [Ibanez-Uranga 12](#IbanezUranga12)

The [[bottom-up model building|bottom-up]] [[intersecting D-brane model building]] originates with

* [Aldazabal-Ibáñez-Quevedo-Uranga 00](#AIQU00)

See also

* {#Denef08} [[Frederik Denef]], _Les Houches Lectures on Constructing String Vacua_, in _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194))

Reviews of [[intersecting D-brane model]] in [[type II string theory]] (in [[orientifold]] [[flux compactifications]]) include

* {#Lust04} [[Dieter Lüst]], _Intersecting Brane Worlds -- A Path to the Standard Model ?_, Class. Quant. Grav.21 : S1399-1424, 2004 ([arXiv:hep-th/0401156](http://arxiv.org/abs/hep-th/0401156))

* [[Ralph Blumenhagen]], [[Volker Braun]], Boris Kors, [[Dieter Lüst]], _The Standard Model on the Quintic_, Summary of Talks at SUSY02, 1st Intl. Conference on String Phenomenology in Oxford, Strings 2002 and 35th Ahrenshoop Symposium. ([arXiv:hep-th/0210083](http://arxiv.org/abs/hep-th/0210083))

* [[Ralph Blumenhagen]], [[Mirjam Cvetic]], [[Paul Langacker]], [[Gary Shiu]], *Towards Realistic Intersecting D-Brane Models*, Ann. Rev. Nucl. Part. Sci. **55** (2005) 71-139 [[arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005), [doi:10.1146/annurev.nucl.55.090704.151541](https://doi.org/10.1146/annurev.nucl.55.090704.151541)]

* Ching-Ming Chen, Tianjun Li, Dimitri V. Nanopoulos, _Standard-Like Model Building on Type II Orientifolds_, Nucl.Phys.B732:224-242,2006 ([arXiv:hep-th/0509059](http://arxiv.org/abs/hep-th/0509059))

* [[Angel Uranga]], _The standard model from D-branes in string theory_, talk at Padova, January 2008 ([pdf](http://active.pd.infn.it/g4/seminars/2008/files/uranga.pdf))

* {#DolanKrippendorfQuevedo11} Matthew J. Dolan, Sven Krippendorf, Fernando Quevedo, _Towards a Systematic Construction of Realistic D-brane Models on a del Pezzo Singularity_, JHEP 1110 (2011) 024 ([arXiv:1106.6039](http://arxiv.org/abs/1106.6039))

* Anshuman Maharana, [[Eran Palti]], _Models of Particle Physics from Type IIB String Theory and F-theory: A Review_ ([arXiv:1212.0555](http://arxiv.org/abs/1212.0555))

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

Computer scan of [[Gepner model]]-[[KK-compactifications]] of [[intersecting D-brane models]]:

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

Computer scan of [[toroidal orbifold]]-[[KK-compactification]] of [[intersecting D-brane models]]:

* [[Ralph Blumenhagen]], [[Florian Gmeiner]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _The Statistics of Supersymmetric D-brane Models_, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173))

* {#GBHLW05} [[Florian Gmeiner]], [[Ralph Blumenhagen]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _One in a Billion: MSSM-like D-Brane Statistics_, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170))

Realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys.Rev.D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))

* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, [[Van Eric Mayes]], _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))

* Tianjun Li, Adeel Mansha, Rui Sun, _Revisiting the Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1910.04530](https://arxiv.org/abs/1910.04530))

* Tianjun Li, Adeel Mansha, Rui Sun, _Generalized Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1912.11633](https://arxiv.org/abs/1912.11633))

See also

* Erik Parr, Patrick K.S. Vaudrevange, Martin Wimmer, _Predicting the orbifold origin of the MSSM_ ([arXiv:2003.01732](https://arxiv.org/abs/2003.01732))

#### Type I string theory model
 {#ReferencesTypeIStringTheorymodels}

Discussion for [[type I string theory]]:

* {#MMRB86} H.S. Mani, A. Mukherjee, R. Ramachandran, A.P. Balachandran, _Embedding of $SU(5)$ GUT in $SO(32)$ superstring theories_, Nuclear Physics B Volume 263, Issues 3–4, 27 January 1986, Pages 621-628 (arXiv:10.1016/0550-3213(86)90277-4)

* {#IbanezMunozRigolin98} [[Luis Ibáñez]], C. Muñoz, S. Rigolin, _Aspects of Type I String Phenomenology_, Nucl.Phys. B553 (1999) 43-80 ([arXiv:hep-ph/9812397](https://arxiv.org/abs/hep-ph/9812397))

* {#Dudas00} Emilian Dudas, _Theory and Phenomenology of Type I strings and M-theory_, Class. Quant. Grav.17:R41-R116, 2000 ([arXiv:hep-ph/0006190](https://arxiv.org/abs/hep-ph/0006190))

* {#Yamatsu17} Naoki Yamatsu, _String-Inspired Special Grand Unification_ ([arXiv:1708.02078](https://arxiv.org/abs/1708.02078))

#### F-Theory models

Discussion of [[GUT]] [[model (physics)|models]] via [[F-theory]] is in

* [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], _GUTs and Exceptional Branes in F-theory - I_ ([arxiv:0802.3391](http://arxiv.org/abs/0802.3391)), _II: Experimental Predictions_ ([arxiv:0806.0102](http://arxiv.org/abs/0806.0102))

A direct [[geometric engineering]] of the [[MSSM]] within [[F-theory]] is claimed in

* {#CLLO18} [[Mirjam Cvetič]], Ling Lin, Muyang Liu, Paul-Konstantin Oehlmann, _An F-theory Realization of the Chiral MSSM with $\mathbb{Z}_2$-Parity_ ([arXiv:1807.01320](https://arxiv.org/abs/1807.01320))

Discussion of the _exact_ [[gauge group]] of the [[standard model of particle physics]], $G = \big( SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6 $ including its $\mathbb{Z}_6$-[[quotient group|quotient]] (see [there](standard+model+of+particle+physics#GaugeGroup)) and the exact [[fermion]] field content, realized in F-theory is in

* Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, _F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches_, JHEP01(2015)142 ([arXiv:1408.4808](https://arxiv.org/abs/1408.4808))

* {#CveticLin17} [[Mirjam Cvetic]], Ling Lin, section 3.3 of _The global gauge group structure of F-theory compactifications with $U(1)$s_ ([arXiv:1706.08521](https://arxiv.org/abs/1706.08521))

Based on this, a large number of realizations of the exact field content of the [[standard model of particle physics]] (or rather the [[MSSM]]) in [[F-theory]] is claimed to be realized in

* [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

* {#TaylorTurner19} [[Washington Taylor]], Andrew P. Turner, _Generic construction of the Standard Model gauge group and matter representations in F-theory_ ([arXiv:1906.11092](https://arxiv.org/abs/1906.11092))

Realization of [[E7|$E_7$]]-[[GUT]] models in F-theory:

* Shing Yan Li, [[Washington Taylor]], *Towards natural and realistic $E_7$ GUTs in F-theory* [[arXiv:2401.00040](https://arxiv.org/abs/2401.00040)]

Review:

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

#### M-theory models
 {#ReferencesMTheoryModels}

A comprehensive account on models in [[M-theory on G2-manifolds]] and the [[G2-MSSM]] is in

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics_ ([arXiv:0801.0478](http://arxiv.org/abs/0801.0478))

* [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_,  Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

* {#Kane16} [[Gordon Kane]], _String/M-theories About Our World Are Testable in the traditional Physics Way_ ([arXiv:1601.07511](http://arxiv.org/abs/1601.07511), [video recording](https://videoonline.edu.lmu.de/en/node/7485))

with comments on comparison to more recent experiments in

* [[Gordon Kane]], Ran Lu, Bob Zheng, _Review and Update of the Compactified M/string Theory Prediction of the Higgs Boson Mass and Properties_, Int. J. Mod. Phys. A Volume 28 (2013) 1330002 ([arXiv:1211.2231](http://arxiv.org/abs/1211.2231))

Original articles include

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _Explaining the Electroweak Scale and Stabilizing Moduli in M Theory_ ([arXiv:hep-th/0701034](http://xxx.lanl.gov/abs/hep-th/0701034))

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, _An M theory Solution to the Hierarchy Problem_ ([arXiv:hep-th/0606262](http://arxiv.org/abs/hepth/0606262))

* [[Bobby Acharya]], Konstantin Bobkov, _Kähler Independence of the $G_2$-MSSM_, JHEP ([arXiv:0810.3285](http://arxiv.org/abs/0810.3285))

See also

* {#Anderson08} [[Lara Anderson]], _Heterotic and M-theory Compactifications for String Phenomenology_ ([arXiv:0808.3621](https://arxiv.org/abs/0808.3621))

Alternatively, discussion in [[Hořava-Witten theory]]:

* {#Ovrut18} [[Burt Ovrut]], _Vacuum Constraints for Realistic Heterotic M-Theories_ ([arXiv:1811.08892](https://arxiv.org/abs/1811.08892))

#### Heterotic M-theory models

Discussion in [[heterotic M-theory]]:

* {#DOPW99} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Models from Heterotic M-theory_, Adv. Theor. Math. Phys. 5 (2002) 93-137 ([arXiv:hep-th/9912208](https://arxiv.org/abs/hep-th/9912208))

* {#DOPW00} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Model Vacua in Heterotic M-Theory_, talk at [Strings '99](http://strings99.aei.mpg.de/), Potsdam, Germany, 19 - 24 Jul 1999 ([arXiv:hep-th/0001101](https://arxiv.org/abs/hep-th/0001101))

and with emphasis on [[heterotic line bundle]]-models:

* [[Sebastian Dumitru]], *The strongly coupled $E_8 \times E_8$ heterotic string: Geometry & Phenomenology* [[arXiv:2206.12310](https://arxiv.org/abs/2206.12310)]

See also:

* Cédric Deffayet, [[Burt Ovrut]], [[Paul Steinhardt]], *Stable Vacua with Realistic Phenomenology and Cosmology in Heterotic M-theory Satisfying Swampland Conjectures* [[arXiv:2401.04828](https://arxiv.org/abs/2401.04828)]

#### Non-supersymmetric models

A survey of string model buidling without low energy susy is in

* [[Emil Martinec]], Daniel Robbins, Savdeep Sethi, _Non-Supersymmetric String Theory_ ([arXiv:0904.3498](http://arxiv.org/abs/0904.3498))
 {#MRS09}

An old observation on string models without low energy susy, recently re-appreciated, is

* [[Keith Dienes]], _How Strings Make Do without Supersymmetry: An Introduction to Misaligned Supersymmetry_ ([arXiv:hep-th/9409114](http://arxiv.org/abs/hep-th/9409114))

A newer observation that received much more attention is

* [[Nima Arkani-Hamed]], Savas Dimopoulos, _Supersymmetric Unification Without Low Energy Supersymmetry And Signatures for Fine-Tuning at the LHC_ ([arXiv:hep-th/0405159](http://arxiv.org/abs/hep-th/0405159))

### String Phenomenology conferences

* String Phenomenology 2002 ([home page](http://www-thphys.physics.ox.ac.uk/users/AlonFaraggi/sp2002/conference.html))

* String Phenomenology 2003 ([home page](http://www.ippp.dur.ac.uk/old/SP03/welcome.shtml))

* String Phenomenology 2004 ([home page](http://www.umich.edu/%7Emctp/SciPrgPgs/events/2004/2004sph/index.html))

* String Phenomenology 2005 ([home page](http://wwwth.mppmu.mpg.de/members/blumenha/main.html))

* String Phenomenology 2006 ([home page](http://www.kitp.ucsb.edu/activities/auto2/?id=352))

* String Phenomenology 2007 ([home page](http://people.roma2.infn.it/%7Estringpheno2007/))

* String Phenomenology 2008 ([home page](http://www.math.upenn.edu/StringPhenom2008/local.html))

* String Phenomenology 2009 ([home page](http://www.fuw.edu.pl/~sp09/))

* String Phenomenology 2010 ([home page](http://stringpheno.cpht.polytechnique.fr/))

* String Phenomenology 2011 ([home page](http://conferencing.uwex.edu/conferences/stringpheno2011/), [talks](http://conferencing.uwex.edu/conferences/stringpheno2011/PlenarySessions.cfm))

* String Phenomenology 2012 ([home page](http://www.newton.ac.uk/programmes/BSM/bsmw05.html), [talks](http://www.newton.ac.uk/programmes/BSM/bsmw05p.html))

* String Phenomenology 2013 ([home page](http://stringpheno2013.desy.de/))

* String Phenomenology 2014 ([home page](http://stringpheno2014.ictp.it))

* String Phenomenology 2018 ([home page](http://sp18.fuw.edu.pl/))

* String Phenomenology 2019 ([home page](https://indico.cern.ch/event/782251/))

### String cosmic inflation

In [[string theory]] the [[inflaton]] field for models of [[cosmic inflation]] field can be modeled by various effects, such as

* [[open string]] stretching between [[D-brane]]-[[anti D-brane]] pairs.

For a review and further pointers to the literature see at

* Cliff Burgess, M. Cicoli, F. Quevedo, _String Inflation After Planck 2013_ ([arXiv:1306.3512](http://arxiv.org/abs/1306.3512))

### Axion phenomenology

On stringy [[axion]] [[phenomenology]]:

* Joseph P. Conlon, M.C. David Marsh, _Searching for a 0.1-1 keV Cosmic Axion Background_ ([arXiv:1305.3603](http://arxiv.org/abs/1305.3603))

  > Primordial decays of [[string theory]] [[moduli stabilization|moduli]] at $z \sim 10^{12}$ naturally generate a [[dark radiation]] Cosmic Axion Background (CAB) with $0.1 - 1 keV$ energies. This CAB can be detected through axion-[[photon]] conversion in astrophysical [[magnetic fields]] to give quasi-thermal excesses in the extreme ultraviolet and soft X-ray bands. Substantial and observable luminosities may be generated even for axion-photon couplings $\ll 10^{-11} GeV^{-1}$. We propose that axion-photon conversion may explain the observed excess emission of soft X-rays from galaxy clusters, and may also contribute to the diffuse unresolved cosmic X-ray background. We list a number of correlated predictions of the scenario.

[[!redirects string phenonemology]]
[[!redirects string phenomenology]]
[[!redirects intersecting D-brane standard models]]
[[!redirects heterotic standard models]]

[[!redirects string theory phenomenology]]


string scattering amplitude

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative string theory]] [[scattering amplitudes]] are defined as in [[quantum field theory]], but with [[n-point functions]] of 1-dimensional [[worldline theories]] ([[Feynman diagrams]]) replaced by those of [[worldsheet]] [[2d CFTs]].

<img src="https://ncatlab.org/nlab/files/StringFeynmanDiagrams.png" width="300">

> graphics grabbed from [Jurke 10](https://benjaminjurke.com/content/articles/2010/string-theory/)

## Properties

### Finiteness
 {#Finiteness}

The amplitudes are thought (see the commented [references below](#References)) to come out term-wise (for each "loop order" hence for each [[genus]] and number of punctures of ([[super Riemann surface|super]]-)[[Riemann surfaces]]) finite (at least UV-finite), hence [[renormalization|renormalized]]: the higher string oscillations may be seen as providing canonical counterterms for the massless excitations in the [[effective field theory]]. In this sense string theory provides a [[UV-completion]] of these effective field theories ([[supergravity]] coupled to [[Yang-Mills theory]]).

+-- {: .num_remark}
###### Remark

The full [[perturbation series]] is the [[sum]] of all these (finite) contributions over the [[genus|genera]] of [[Riemann surfaces]] (the "loop orders"). This _sum_ diverges, even if all loop orders are finite. Notice though, that a non-trivial [[perturbation theory|perturbative]] [[QFT]] is not supposed to have a finite [[radius of convergence]] of its scattering amplitudes, since that would imply convergence also for _negative_ [[coupling constant]], which is physically unreasonable. (For the [[bosonic string]] the perturbation series has apparently been explicitly shown not to be [[Borel summation|Borel resummable]].) For more on this see at _[[non-perturbative effect]]_ and _[string theory FAQ -- Is the divergence of the pertubation series fatal?](string+theory+FAQ#NonConvergenceOfPerturbationSeries)_.

=--

+-- {: .num_remark}
###### Remark

A string scattering amplitude is called _UV-finite_ at a given loop order ([[genus]] of a [[Riemann surface]] $\Sigma$ with $n$ marked points/string insertions) if the [[correlation function]] $\langle \phi_1, \cdots, \phi_n\rangle_{\Sigma}$ is finite for every single such [[Riemann surface]]. The actual string amplitude at order $(g,n)$ though is the averaging of this over all possible [[conformal structures]] on $\Sigma$, hence the  [[integration]] of the [[correlation function]], as a function on the conformal structure, over the compactified [[moduli space of conformal structures]] $\mathcal{M}_{g,n}$ (a [[Deligne-Mumford stack]]).

If also this integral is finite, hence if the total [[measure]] on the [[moduli space of conformal structures]] is finite, then one says the amplitude is _IR-finite_.

This distinction between UV-finiteness and IR-finiteness is not always highlighted in all of the articles below. All authors argue that the string is _UV-finite_ to all order. The IR-finiteness is only discussed much more recently at low loop order.

IR non-finiteness is not physically fatal. For instance if a [[perturbation theory|perturbative]] theory of [[quantum gravity]] develops a [[cosmological constant]] perturbatively, then the perturbation series will be IR-divergent, signifying the fact that background spacetime without cosmological constant is no longer a solution to the quantum-corrected [[equations of motion]]. Nevertheless, these potential IR-divergences seem to be absent for the [[superstring]] perturbation series. For the [[cosmological constant]] case this can already be seen from the fact that the [[effective QFT]] of [[type II supergravity]] etc. does not admit a cosmological constant, for that would violate [[supersymmetry]].

=--

### Open-closed scattering duality and KLT relations

The [[open/closed string duality]] implies certain relations in string scattering amplitudes that in the point-particle limit induces relations between [[scattering amplitudes]] in [[Yang-Mills theory]] and in [[gravity]]. These are the _[[KLT relations]]_ in [[QFT]]. See in particular [Mafra-Schlotterer 18a](#MafraSchlotterer18a), [Mafra-Schlotterer 18b](#MafraSchlotterer18b), [Mafra-Schlotterer 18c](#MafraSchlotterer18c).

### Twistor string amplitudes and MHV amplitudes in Yang-Mills theory

The scattering amplitudes in [[twistor string theory]] induce the MHV amplitudes in ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]. See at _[[string theory results applied elsewhere]]_ in the section _[Application to QCD -- Scattering amplitudes](string+theory+results+applied+elsewhere#QCDScatteringAmplitudes)_.

### $p$-Adic formulation

The [[Veneziano amplitude]] (open bosonic string tree-level scattering) has an equivalent formulation as the inverse product over all [[prime numbers]] $p$ of an amplitude computed not by an integral in the real but in the [[p-adic numbers]]. For other open string amplitudes this holds up to some regularization. This is the topic of _[[p-adic string theory]]_, see there for more details.

## Examples

* [[Veneziano amplitude]]

* [[Virasoro-Shapiro amplitude]]

* [[vacuum amplitude]]

## Related entries

* [[scattering amplitude]]

* [[perturbation theory]], [[perturbative string theory]]

* [[non-perturbative effect]], [[M-theory]]

* [[string theory FAQ]], [[string theory results applied elsewhere]]

## References
 {#References}

### General
 {#General}

A comprehensive account of the [[superstring]] [[S-matrix]] may be obtained from combining the general idea presented in

* {#Polchinski01} [[Joseph Polchinski]], section 12.5 of vol 2 of _String theory_, Cambridge Monographs on Mathematical Physics (2001)

with the technical details laid out in

* {#Witten12} [[Edward Witten]], _Superstring Perturbation Theory Revisited_ ([arXiv:1209.5461](https://arxiv.org/abs/1209.5461))

* {#Witten13} [[Edward Witten]], _More On Superstring Perturbation Theory: An Overview Of Superstring Perturbation Theory Via Super Riemann Surfaces_ ([arXiv:1304.2832](https://arxiv.org/abs/1304.2832))

Survey of the tree level string scattering amplitudes includes

* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 ([TOC pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-29497-6%2F1.pdf), [publisher page](http://www.springer.com/gp/book/9783642294969))

* [[Katrin Becker]], [[Melanie Becker]], Ilarion V. Melnikov, Daniel Robbins, Andrew B. Royston, _Some tree-level string amplitudes in the NSR formalism_, JHEP 12 (2015) 010 ([arXiv:1507.02172](http://arxiv.org/abs/1507.02172))

See also

* {#Moore93} [[Gregory Moore]], _Symmetries of the Bosonic String S-Matrix_ ([arXiv:hep-th/9310026](https://arxiv.org/abs/hep-th/9310026))

* Gaston Giribet, Nicholas Labranche, Joan La Madrid, *Comments on the two-point string amplitudes* [[arXiv:2303.15658](https://arxiv.org/abs/2303.15658)]

On string scattering amplitudes in view of the [[S-matrix bootstrap]]:

* Andrea Guerrieri, Joao Penedones, Pedro Vieira, _Where is String Theory?_ ([arXiv:2102.02847](https://arxiv.org/abs/2102.02847))

For more references see also at *[[string theory results applied elsewhere]]*.

### Superstring scattering

A review of superstring scattering amplitudes is in the last section of ([Staessens-Vernocke 10](#StaessensVernocke10)). A general discussion of the problem of superstring amplitudes is in

* [[Eric D'Hoker]] [[Duong Phong]], _Loop amplitudes for the fermionic string_, Nucl. Phys. B 278 (1986) 225;

* [[Greg Moore]], P. Nelson, [[Joseph Polchinski]], _Strings and supermoduli_, Phys. Lett. B 169 (1986) 47-53.

On analycity of the superstring [[S-matrix]]:

* {#dLES18} Corinne de Lacroix, Harold Erbin, [[Ashoke Sen]], _Analyticity and Crossing Symmetry of Superstring Loop Amplitudes_ ([arXiv:1810.07197](https://arxiv.org/abs/1810.07197))

Survey of the presence and role of divergences includes

* [[Ashoke Sen]], _Ultraviolet and Infrared Divergences in Superstring Theory_ ([arXiv:1512.00026](http://arxiv.org/abs/1512.00026))

Discussion of superstring scattering amplitudes in terms of [[pure spinors]] ([[Berkovits superstring]]) with emphasis on [[KLT relations]]:

* {#MafraSchlotterer18a} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics_ ([arXiv:1812.10969](https://arxiv.org/abs/1812.10969))

* {#MafraSchlotterer18b} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics_ ([arXiv:1812.10970](https://arxiv.org/abs/1812.10970))

* {#MafraSchlotterer18c} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure_ ([arXiv:1812.10971](https://arxiv.org/abs/1812.10971))

* [[Carlos R. Mafra]], [[Oliver Schlotterer]], *Tree-level amplitudes from the pure spinor superstring* [[arXiv:2210.14241](https://arxiv.org/abs/2210.14241)]

For more see also at _[[superstring field theory]]_, such as

* Corinne de Lacroix, [[Harold Erbin]], Sitender Pratap Kashyap, [[Ashoke Sen]], Mritunjay Verma, _Closed Superstring Field Theory and its Applications_,  International Journal of Modern Physics AVol. 32, No. 28n29, 1730021 (2017) ([arXiv:1703.06410](https://arxiv.org/abs/1703.06410))

The [[1-loop]] amplitudes in [[type II string theory]] have been discussed in

* [[Michael Green]], [[John Schwarz]], _Supersymmetrical string theories_, Phys. Lett. 109 B (1982) 444-448.

and for [[heterotic string theory]] in

* [[David Gross]], J.A. Harvey, E.J. Martinec and R. Rohm, _Heterotic String Theory (II). The interacting heterotic string_, Nucl. Phys. B267 (1986) 75.

The description of [[2-loop]] amplitudes, including the [[Berezin integral]] over the super-[[moduli space of super Riemann surfaces]] in superstring theory:

* {#DHokerPhong02} [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings I, Main Formulas_, Phys. Lett. B529:241-255, 2002 ([arXiv:hep-th/0110247](http://arxiv.org/abs/hep-th/0110247))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings II, The Chiral Measure on Moduli Space_, Nucl. Phys. B636:3-60, 2002 ([arXiv:hep-th/0110283](http://arxiv.org/abs/hep-th/0110283))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities_, Nucl. Phys. B636:61-79, 2002 ([arXiv:hep-th/0111016](http://arxiv.org/abs/hep-th/0111016))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms_, Nucl. Phys. B639:129-181, 2002 ([arXiv:hep-th/0111040](http://arxiv.org/abs/hep-th/0111040))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function_, Nucl. Phys. B715:91-119, 2005 ([arXiv:hep-th/0501196](http://arxiv.org/abs/hep-th/0501196))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function_,  	Nucl. Phys. B715:3-90, 2005 ([arXiv:hep-th/0501197](http://arxiv.org/abs/hep-th/0501197))

* [[Eric D'Hoker]], [[Duong Phong]], _Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes_, Nucl. Phys. B804:421-506, 2008 ([arXiv:0711.4314](http://arxiv.org/abs/0711.4314))

Review of this work:

* A.Morozov, _NSR Superstring Measures Revisited_, JHEP0805:086,2008 ([arXiv:0804.3167](http://arxiv.org/abs/0804.3167))

Further development in:

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors_ ([arXiv:2006.05270](https://arxiv.org/abs/2006.05270))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality_ ([doi:2008.08687](https://arxiv.org/abs/2008.08687))

The technical issue of the [[moduli space]] of [[super Riemann surfaces]] of higher genus (for higher loop string scattering amplitudes) is discussed in

* [[Edward Witten]], _Notes On Super Riemann Surfaces And Their Moduli_ ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459))

* [[Ron Donagi]], [[Edward Witten]], _Supermoduli Space Is Not Projected_ ([arXiv:1304.7798](http://arxiv.org/abs/1304.7798))

Review of the scattering of massive modes in [[type IIB string theory]]:

* Nicholas Agia, *Massive Type IIB Superstrings Part I: 3- and 4-Point Amplitudes* [[arXiv:2309.11538](https://arxiv.org/abs/2309.11538)]

### Higher order corrections

* [[Christopher Pope]],  _Higher-order corrrections in String and M-theory and Generalized Holonomies_, December 2006 ([pdf](http://www.luth.obspm.fr/IHP06/workshops/cosmo/slides/Pope.pdf))

### On finiteness
 {#ReferencesOnFiniteness}

Here is a commented list of references on the degreewise [finiteness of string scattering amplitudes](#Finiteness).

#### Finiteness of bosonic string scattering

Introductory lecture notes include

* Wieland Staessens, Bert Vercnocke, _Lectures on Scattering Amplitudes in String Theory_, Lecture notes based on lectures given at the fifth Modave School on Mathematical Physics (August 2009) ([arXiv:1011.0456](http://arxiv.org/abs/1011.0456))
 {#StaessensVernocke10}

Discussion of the term-wise finiteness of the bosonic string scattering amplitudes is in

* L. Clavelli, S. T. Jones, _Finiteness of the bosonic string in fewer than 26 dimensions_, Phys. Rev. D 39, 3795–3797 (1989) ([SPIRE](http://inspirehep.net/record/24403))

There are also arguments in

* Simon Davis, _On the domain of string perturbation theory_, Classical and Quantum Gravity, Volume 6, Issue 12, pp. 1791-1803 (1989) ([web](http://adsabs.harvard.edu/abs/1989CQGra...6.1791D))

#### Finiteness of superstring scattering

Finiteness of heterotic and type II superstring $n$-point functions in flat spacetime is argued for in

* Joseph Atick, [[Gregory Moore]], [[Ashoke Sen]], _Catoptrick tadpoles_, Nucl.Phys. B307 (1988) 221 ([spire](http://inspirehep.net/record/252598?ln=en))

General finiteness of superstring amplitudes is discussed in

* [[Stanley Mandelstam]], _The $n$ Loop String Amplitude: Explicit Formulas, Finiteness and Absence of Ambiguities_, Phys. Lett. B277 (1992) 82. ([inspire](http://inspirehep.net/record/319501))

which shows that a certain divergence which could appear does not.

Also

* A. Restuccia, J. Taylor, _Finiteness of type II superstring amplitudes_, Physics Letters B, Volume 187, Issues 3–4, 26 March 1987, Pages 267–272

argues finiteness of the superstring amplitudes at each order.

Also

* [[Eric D'Hoker]], [[Duong Phong]], _Momentum Analyticity and Finiteness of the 1-Loop Superstring Amplitude_,  Phys. Rev. Lett. 70 (1993) 3692-3695 ([arXiv:hep-th/9302003](http://arxiv.org/abs/hep-th/9302003))

* [[Eric D'Hoker]], [[Duong Phong]], _The Box Graph In Superstring Theory_, Nucl.Phys.B440:24-94,1995 ([arXiv:hep-th/9410152](http://arxiv.org/abs/hep-th/9410152))

* [[Eric D'Hoker]], [[Duong Phong]], _Dispersion Relations in String Theory_; Nucl.Phys.B440:24-94,1995 ([arXiv:hep-th/9404128](http://arxiv.org/abs/hep-th/9404128))

Arguments for the finiteness of superstring scattering amplitudes to all loop order based on the [[Berkovits superstring]]-formulation have been promoted in

* [[Nathan Berkovits]], _Multiloop Amplitudes and Vanishing Theorems using the Pure Spinor Formalism for the Superstring_,  JHEP 0409:047,2004 ([arXiv:hep-th/0406055](http://arxiv.org/abs/hep-th/0406055))

(In footnote 2 this article claims that the claimed proofs of the same statement by G.S Danilov in [hep-th/9801013](http://arxiv.org/abs/hep-th/9801013), [hep-th/0312177](http://arxiv.org/abs/hep-th/0312177) are not in fact proofs.)

Discussion of 2-loop amplitudes from holomorphy arguments is in

* [[Edward Witten]], _Notes On Holomorphic String And Superstring Theory Measures Of Low Genus_ ([arXiv:1306.3621](http://arxiv.org/abs/1306.3621))

See also

* [[Ashoke Sen]], _Supersymmetry Restoration in Superstring Perturbation Theory_ ([arXiv:1508.02481](http://arxiv.org/abs/1508.02481))

### Graviton scattering

Computation of [[graviton]] scattering amplitudes ([[perturbative quantum gravity]]):

* [[Taejin Lee]], _Gravitational Scattering Amplitudes and Closed String Field Theory in the Proper-Time Gauge_, EPJ Web of Conferences 168, 07004 (2018) ([doi:10.1051/epjconf/201816807004](https://doi.org/10.1051/epjconf/201816807004))

* [[Taejin Lee]], _Four-Graviton Scattering and String Path Integral in the Proper-time gauge_ ([arXiv:1806.02702](https://arxiv.org/abs/1806.02702))

### Via AdS/CFT
 {#ReferencesViaAdSCFT}

Discussion via [[AdS/CFT]] beyond the [[SCFT]] [[planar limit]], using the [[conformal bootstrap]]:

* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))

### More

On point-particle limit via [[tropical geometry]]

* Piotr Tourkine, _Tropical Amplitudes_ ([arXiv:1309.3551](http://arxiv.org/abs/1309.3551))

Scattering amplitudes of highly excited strings:

* Dimitri P. Skliros, Edmund J. Copeland, Paul M. Saffin, _Highly Excited Strings I: Generating Function_ ([arxiv:1611.06498](https://arxiv.org/abs/1611.06498))

[[!redirects string scattering amplitudes]]

[[!redirects superstring scattering amplitude]]
[[!redirects superstring scattering amplitudes]]

[[!redirects string scattering]]

[[!redirects string perturbation series]]

[[!redirects string amplitude]]
[[!redirects string amplitudes]]


string structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

# Contents #
* table of contents
{:toc}

A _string structure_ on a [[manifold]] is a higher version of a [[spin structure]]. A string structure on a [[manifold]]  with [[spin structure]] given by a [[Spin group]]-[[principal bundle]] to which the [[tangent bundle]] is [[associated bundle|associated]] is a lift $\hat g$ of the classifying map $g : X \to \mathcal{B} Spin(n)$ through the  third nontrivial step $\mathcal{B}String(n) \to \mathcal{B}Spin(n)$ in the [[Whitehead tower]] of $BO(n)$ to a [[String group]]-[[principal bundle]]:

$$
  \array{
    && \mathcal{B}String(n)
    \\
    & {}^{\hat g}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& \mathcal{B}Spin(n)
  }
$$

A lift one further step through the Whitehead tower is a [[Fivebrane structure]].

This has generalizations to the smooth context, where instead of the topological String-group one uses the [[String Lie 2-group]].

Let $X$ be an $n$-dimensional [[topological manifold]].

Its tangent [[bundle]] is canonically associated to a $O(n)$-principal bundle, which is in turn classified by a [[continuous function]]

$$
  X \to B O(n)
$$

from $X$ to the [[classifying space]] of the [[orthogonal group]] $O(n)$.

* A **String structure** on $X$ is the choice of a lift of this map a few steps through the [[Whitehead tower]] of $BO(n)$.

* The manifold "is string" if such a lift exists.

This means the following:

* there is a canonical map $w_1 : B O(n) \to B\mathbb{Z}_2$ from the [[classifying space]] of $O(n)$ to that of $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ that represents the generator of the cohomology $H^1(B O(n), \mathbb{Z}_2)$. The classifying space of the group $SO(n)$ is the [[generalized universal bundle|homotopy pullback]]
  $$
   \array{
    B SO(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    B O(n) &\stackrel{w_1}{\to}& \mathbb{B}\mathbb{Z}_2
   }
  $$

  Namely using the [[homotopy hypothesis]] (which is a theorem, recall), we may identify $B O(n)$ with the one object [[groupoid]] whose space of morphisms is $O(n)$ and similarly for $ B \mathbb{Z}_2$. Then the map in question is the one induced from the group homomorphism that sends orientation preserving elements in $O(n)$ to the identity and orientation reversing elements to the nontrivial element in $\mathbb{Z}_2$.

  * an **[[orientation]]** on $X$ is a choice of lift of the structure group through $B SO(n) \to B O(n)$
  $$
   \array{
    && B SO(n)
    \\
    & {}^{orientation}\nearrow& \downarrow
    \\
    X &\stackrel{}{\to}& B O(n)
   }
   \,.
  $$

* there is a canonical map $w_2 : B SO(n) \to B^2 \mathbb{Z}_2$ representing the generator of $H^2(B SO(n), \mathbb{Z}_2)$. The classifying space of the group $Spin(n)$ is the [[generalized universal bundle|homotopy pullback]]
  $$
   \array{
    B Spin(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    B SO(n) &\stackrel{w_2}{\to}& \mathbb{B}^2\mathbb{Z}_2
   }
  $$

  * a **[[spin structure]]** on an oriented manifold $X$ is a choice of lift of the structure group through $B Spin(n) \to B SO(n)$
  $$
   \array{
    && B Spin(n)
    \\
    & {}^{spin structure}\nearrow& \downarrow
    \\
    X &\stackrel{}{\to}& B SO(n)
   }
   \,.
  $$

* there is a canonical map $B Spin(n) \to B^3 U(1)$ The classifying space of the group $String(n)$ is the [[generalized universal bundle|homotopy pullback]]
  $$
   \array{
    B String(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    B Spin(n) &\stackrel{\frac{1}{2}p_1}{\to}& \mathbb{B}^3 U(1)
   }
  $$

  * a **string structure** on an oriented manifold $X$ is a choice of lift of the structure group through $B String(n) \to B Spin(n)$
  $$
   \array{
    && B String(n)
    \\
    & {}^{string structure}\nearrow& \downarrow
    \\
    X &\stackrel{}{\to}& B Spin(n)
   }
   \,.
  $$

* there is a canonical map $B String(n) \to B^7 U(1)$ The classifying space of the group $Fivebrane(n)$ is the [[generalized universal bundle|homotopy pullback]]
  $$
   \array{
    B Fivebrane(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    B String(n) &\stackrel{\frac{1}{6}p_2}{\to}& \mathbb{B}^7 U(1)
   }
  $$

  * a **[[fivebrane structure]]** on an string manifold $X$ is a choice of lift of the structure group through $B Fivebrane(n) \to B String(n)$
  $$
   \array{
    && B Fivebrane(n)
    \\
    & {}^{fivebrane structure}\nearrow& \downarrow
    \\
    X &\stackrel{}{\to}& B String(n)
   }
   \,.
  $$

## Definition ##

### Topological and smooth string structures

Let the ambient [[(∞,1)-topos]] by $\mathbf{H} = $ [[ETop∞Grpd]] or [[Smooth∞Grpd]]. Write $X$ for a [[topological manifold]] or [[smooth manifold]] of [[dimension]] $n$, respectively.

Write $String(n)$ for the [[string 2-group]], a [[1-truncated]] [[∞-group]] object in $\mathbf{H}$.

+-- {: .num_defn #PlainStringStructures}
###### Definition

The [[2-groupoid]] of (topological or smooth) **string structures** on $X$ is
the hom-space of [[cocycle]]s $X \to \mathbf{B}String(n)$, or equivalently that of (topological or smooth) $String(n)$-[[principal 2-bundle]]s:

$$
  String(X) := String(n) Bund(X) \simeq \mathbf{X}(X,\mathbf{B}String)
  \,.
$$

=--

Write $\frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin(n) \to \mathbf{B}^3 U(1)$ in $\mathbf{H}$ for the topological or smooth refinement of the first fractional Pontryagin class (see [[differential string structure]] for details on this).

+-- {: .num_prop #StringStructuresAsObstructions}
###### Observation

The [[2-groupoid]] of string structure on $X$ is the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1^X$: the [[(∞,1)-pullback]]

$$
  \array{
    String(X) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}Spin(n))
     &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

By definition of the [[string 2-group]] we have the [[fiber sequence]] $\mathbf{B} String \to \mathbf{B}Spin \stackrel{\frac{1}{2}} \mathbf{p_1}{\to} \mathbf{B}^3 U(1)$. The [[hom-functor]] $\mathbf{H}(X,-)$ preserves every [[(∞,1)-limit]], hence preserves this [[fiber sequence]].

=--

+-- {: .num_defn #FiberOverSpinStructure}
###### Definition

Given a [[spin structure]] $S : X \to \mathbf{B} Spin(n)$ we say that the **string structures extending this spin-structure** is the [[homotopy fiber]] $String_S(X)$ of the projection $String(X) \to Spin(X)$ from observation \ref{StringStructuresAsObstructions}:

=--

### Twisted and differential string structures

(...)

The [[2-groupoid]] of string structures is the [[homotopy fiber]] of

$$
  \frac{1}{2}p_1 : Top(X, \mathcal{B}Spin) \to Top(X, \mathcal{B}^4 \mathbb{Z})
$$

over the trivial cocycle. Followowing the general logic of [[twisted cohomology]] the 2-groupoids over a nontrivial cocycle $c : X \to \mathcal{B}^4 \mathbb{Z}$ may be thought of as that of _twisted_ string structures.

The [[Pontryagin class]] $\frac{1}{2}p_1$ refines to the [[smooth first fractional Pontryagin class]] $\frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)$. That leads to [[differential string structure]]s.

(...)

## Properties

### Choices of string structures

+-- {: .num_prop #ChoicesOfSpinStructures}
###### Observation

The space of choices of string structures extending a given spin structure $S$ are as follows

* if $[\frac{1}{2}\mathbf{p}_1(S)] \neq 0$ it is empty: $String_S(X) \simeq \emptyset$;

* if $[\frac{1}{2}\mathbf{p}_1(S)] = 0$ it is $String_S(X) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1))$.

  In particular the set of [[equivalence class]]es of string structures lifting $S$ is the [[cohomology]] set

  $$
    \pi_0 String_S(X) \simeq H^3(X, \mathbb{Z})
    \,.
  $$

=--

+-- {: .proof}
###### Proof

Apply the [[pasting law]] for [[(∞,1)-pullback]]s on the [[diagram]]

$$
  \array{
    String_S(X) &\to& String(X) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    * &\stackrel{S}{\to}& \mathbf{H}(X, \mathbf{B} Spin(n))
    &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \mathbf{H}(X, \mathbf{B}^3 U(1))
  }
  \,.
$$

The outer diagram defines the [[loop space object]] of $\mathbf{H}(X, \mathbf{B}^3 U(1))$. Since $\mathbf{H}(X,-)$ commutes with forming loop space objects (see [[fiber sequence]] for details) we have

$$
  String_S(X) \simeq \Omega \mathbf{H}(X, \mathbf{B}^3 U(1))
   \simeq
  \mathbf{H}(X, \mathbf{B}^2 U(1))
  \,.
$$

=--

### String structures by gerbes on a bundle
 {#ClassesOnTotalSpace}

One can reformulate an

* [[orientation]]-

* [[spin structure|Spin]]-

* [[string structure|String]]-

* [[fivebrane structure|Fivebrane]]-

structure in terms of the existence of a certain class in abelian cohomolgy on the total space of the given [[principal bundle]]. This decomposition is a special case of th general [[Whitehead principle of nonabelian cohomology]].

+-- {: .num_defn #StringStrucDefinitionByTotalSpace}
###### Definition

Let $X$ be a manifolds with [[spin structure]] $S : X \to \mathbf{B}Spin$. Write $P \to X$ for the corresponding [[spin group]]-principal bundle.

Then a **string structure** lifting $S$ is a cohomology class $H^3(P,\mathbb{Z})$ such that the restriction of the class to any fiber $\simeq Spin(n)$ is a generator of $H^3(Spin(n), \mathbb{Z}) \simeq \mathbb{Z}$.

=--

This kind of definition appears in ([Redden, def. 6.4.2](#Redden)).

+-- {: .num_prop #DirectStringStructureImpliesIndirectOne}
###### Proposition

Every string structure in the sense of def. \ref{FiberOverSpinStructure} induces a string structure in the sense of def. \ref{StringStrucDefinitionByTotalSpace}.

=--

+-- {: .proof}
###### Proof

Consider the [[pasting]] diagram of  [[(∞,1)-pullback]]s

$$
  \array{
    String &\to& \hat P &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Spin &\to& P &\to&  B^2 U(1) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& X &\to& B String(n) &\to& B Spin(n)
  }
$$

This uses repeatedly the [[pasting law]] for $(\infty,1)$-pullbacks. The map $P \to B^2 U(1)$ appears by decomposing the homotopy pullback of the point along $X \to B Spin(n)$ into a homotopy pullback first along $B String(n) \to B Spin(n)$ and then along $X \to B String(n)$ using the given String structure. This is the [[cocycle]] for a $\mathbf{B}U(1)$-[[principal 2-bundle]] on the total space $P$ of the $Spin$-principal bundle: a [[bundle gerbe]].

The rest of the diagram is constructed in order to prove the following:

* The class in $H^3(P, \mathbb{Z})$ of this bundle gerbe, represented by $P \to B^2 U(1)$ has the property that restricted to the fibers of the $Spin(n)$-principal bundle $P$ it becomes the generating class in $H^3(Spin(n), \mathbb{Z})$.

=--

## Examples

* On a 3-[[dimension|dimensional]] [[orientation|oriented]] [[manifold]] string structures are equivalently [[2-framings]].

## Related entries

* [[orientation]]

* [[spin structure]], [[twisted spin structure]]

* [[p1-structure]], [[Atiyah 2-framing]]

* **string structure**

  [[differential string structure]]

  [[stringor bundle]]

* [[fivebrane structure]], [[differential fivebrane structure]]

[[!include higher spin structure - table]]

## References ##

The relevance of String structures (like that of Spin structures half a century before) was recognized in the physics of spinning strings, therefore the name.

The article

* [[Timothy Killingback|T.P. Killingback]], _World-sheet anomalies and loop geometry_ Nuclear Physics B Volume 288, 1987, Pages 578-588 [doi:10.1016/0550-3213(87)90229-X](https://doi.org/10.1016/0550-3213%2887%2990229-X)

was (it seems) the first to derive the Green-Schwarz anomaly cancellation condition of the effective background theory as the [[quantum anomaly]] cancellation condition for the worldsheet theory of the heterotic string's [[sigma-model]] by direct generalization of the way the condition of a [[spin structure]] may be deduced from anomaly cancellation for the superparticle.

String stuctures had at that time been discussed in terms of their transgressions to loop spaces

* {#WittenIndexLoopSpace} [[Edward Witten]], _The Index of the Dirac Operator in Loop Space_, In: Landweber, P.S. (eds) Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, N.J., Sep 1986). Lecture Notes in Mathematics **1326**  (1988) Springer, Berlin, Heidelberg. [doi:10.1007/BFb0078045](https://doi.org/10.1007/BFb0078045)

* {#WittenElliptic} [[Edward Witten]], _Elliptic Genera and Quantum Field Theory_, Commun. Math. Phys. **109** (1987) 525–536 [doi:10.1007/BF01208956](https://doi.org/10.1007/BF01208956)

later it was reformulated in terms of the classes down on base space just mentioned in

* [[Stefan Stolz]], [[Peter Teichner]], [What is an elliptic object](http://math.ucr.edu/home/baez/qg-winter2007/Oxford.pdf).

The relation between the two pictures is analyzed for instance in

* A. Asada, _Characteristic classes of loop group bundles and generalized string classes_ , Differential geometry and its applications (Eger, 1989), 33--66, Colloq. Math. Soc. János Bolyai, 56, North-Holland, Amsterdam, 1992. ([[Asada.pdf:file]])

A precise formulation of Killingback's original argument in [[differential K-theory]] appeared in

* [[Ulrich Bunke]], *String structures and trivialisations of a Pfaffian line bundle*, Commun. Math. Phys. **307** (2011) : 675-712 [[arXiv:0909.0846](http://arxiv.org/abs/0909.0846), [doi:10.1007/s00220-011-1348-0](https://doi.org/10.1007/s00220-011-1348-0)]

A review of that is in

* [[Konrad Waldorf]], _Geometric string structure and supersymmetric sigma-models_ ([pdf](http://www.konradwaldorf.de/docs/cardiff.pdf))

The definition of string structures by degree-3 classes on the total space of the spin bundle is used in

* [[Corbett Redden]], _Canonical metric connections associated to string structures_ PhD Thesis, (2006)([pdf](https://www.math.sunysb.edu/~redden/Thesis.pdf))

For discussion of String-structures using 3-classes on total spaces see for instance the work by Corbett Redden and Konrad Waldorf described at

* [[Oberwolfach Workshop, June 2009 -- Friday, June 12]]

Discussion of the [[moduli stack]] of [[twisted differential string structures]]:

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012) 169-213 [[arXiv:0910.4001](http://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3](https://doi.org/10.1007/s00220-012-1510-3)]

and its appearance on [[M5-branes]] under [[schreiber:Hypothesis H]]:

* {#FSS21} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Twisted Cohomotopy implies twisted String structure on M5-branes]]*, J. of Mathematical Physics **62** (2021) 042301 [[arXiv:2002.11093](https://arxiv.org/abs/2002.11093), [doi:10.1063/5.0037786](https://doi.org/10.1063/5.0037786)]

An explicit cocycle construction of the essentially unique [[string 2-group]]-[[principal 2-bundle]] lift of the [[tangent bundle]] of the [[sphere|5-sphere]] is given in

* {#Roberts14} [[David Roberts]], _Explicit string bundles_, talk at _[Workshop on Higher Gauge Theory and Higher Quantization](http://www.christiansaemann.de/higherworkshop2014/index.html)_ 2014, [arXiv:2203.04544](https://arxiv.org/abs/2203.04544),  ([[RobertsStringS5.pdf:file]]).

Discussion for indefinite (Lorentzian) signature is in

* Hyung Bo Shim, _Indefinite string structure_, thesis 2013 ([web](http://d-scholarship.pitt.edu/19620/))

More discussion of relation to spin structures on loop spaces is in

* Alessandra Capotosti, _[[From String structures to Spin structures on loop spaces]]_, Ph.D. thesis, Università degli Studi Roma Tre, Rome, April 2016

* [[Konrad Waldorf]], *String structures and loop spaces*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.12998](https://arxiv.org/abs/2312.12998)]

A study of (flat) string structures encoded in the bicategory of flat 2-group bundles on an oriented surface via weak representations of the fundamental group is in

* [[Daniel Berwick-Evans]], Emily Cliff, Laura Murray, Apurva Nakade, Emma Phillips, _String structures, 2-group bundles, and a categorification of the Freed-Quinn line bundle_ ([arXiv:2110.07571](https://arxiv.org/abs/2110.07571))

[[!redirects String structure]]
[[!redirects String-structure]]
[[!redirects string-structure]]

[[!redirects string structures]]
[[!redirects String structures]]


String Theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

This entry collects links related to the textbook

* [[Joseph Polchinski]]:

  \linebreak

  **String Theory**

  \linebreak

  Cambridge Monographs on Mathematical Physics

  Cambridge University Press (1998)

  * vol 1: **An Introduction to the Bosonic String**

    [doi:10.1017/CBO9780511816079](https://doi.org/10.1017/CBO9780511816079)

    [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/polchinski1.pdf)

  * vol 2: **Superstring Theory and Beyond**

    [doi:10.1017/CBO9780511618123](https://doi.org/10.1017/CBO9780511618123)

    [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/polchinski2.pdf)

on [[string theory]], first [[bosonic string theory]], then [[superstring theory]] and with an outlook on [[M-theory]].

See also:

* [[Matthew Headrick]], *A solution manual for Polchinski's "String Theory"* [[arXiv:0812.4408](https://arxiv.org/abs/0812.4408)]

\linebreak

#Contents#
* table of contents
{:toc}

## Volume 1 An introduction to the bosonic string

### 1 A first look at strings

* [[sigma model]]

* [[string]]

### 2 Conformal field theory

* [[2d CFT]]

### 3 The Polyakov path integral

* [[Polyakov action]]

* [[path integral]]

### 4 The string spectrum

* [[BRST-BV formalism]]

### 5 The string S-matrix

* [[S-matrix]]

* [[string scattering amplitude]]

### 6 Tree-leve amplitudes

* [[string scattering amplitude]]

### 6 One-loop amplitudes

* [[string scattering amplitude]]

### 7 Toroidal compatifications and T-duality

* [[KK-compactification]]

* [[T-duality]]

### 8 Higher order amplitudes

* [[string scattering amplitudes]]

### Appenix A: A short course on path integrals

* [[path integral]]

## Volume 2 Superstring theory and beyond

### 10 Type I and type II superstrings

* [[type II string theory]]

* [[type I string theory]]

### 11 The heterotic string

* [[heterotic string theory]]

### 12 Superstring interaction

(...)

### 13 D-Branes

* [[D-brane]]

### 14 Strings at strong coupling

* [[non-perturbative field theory]]

* [[11-dimensional supergravity]]

* [[M-theory]]

### 15 Advanced CFT

* [[CFT]]

### 16 Orbifold

* [[orbifold]]

### 17 Calabi-Yau compactifictions

* [[Calabi-Yau manifold]]

* [[KK-compactification]]

* [[landscape of string theory vacua]]

### 18 Physics in four dimensions

* [[effective field theory]]

* [[string phenomenology]]

### 19 Advanced topics

* [[mirror symmetry]]

* [[S-duality]]

### Appendix B: Spinors and SUSY in various dimensions

* [[spinor]], [[spin representation]],

* [[super Poincare Lie algebra]]

* [[supersymmetry]]

category: reference

[[!redirects String theory]]