A Simple Introduction to Gröbner Basis Methods in String Phenomenology

Gray, James

doi:10.1155/2011/217035

Cited by 17 publications

(26 citation statements)

References 36 publications

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“…The considerable mathematical and computational tools necessary for such systematic scans have been built up over a number of years [16][17][18][36][37][38][39][40][41][42][43][44][45]. The second reason is related to the nature of the vector bundles V used in the construction.…”

Section: Jhep06(2012)113mentioning

confidence: 99%

Heterotic line bundle standard models

Anderson

Gray

Lukas

et al. 2012

J. High Energ. Phys.

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In a previous publication, arXiv:1106.4804, we have found 200 models from heterotic Calabi-Yau compactifications with line bundles, which lead to standard models after taking appropriate quotients by a discrete symmetry and introducing Wilson lines. In this paper, we construct the resulting standard models explicitly, compute their spectrum including Higgs multiplets, and analyze some of their basic properties. After removing redundancies we find about 400 downstairs models, each with the precise matter spectrum of the supersymmetric standard model, with one, two or three pairs of Higgs doublets and no exotics of any kind. In addition to the standard model gauge group, up to four Green-Schwarz anomalous U(1) symmetries are present in these models, which constrain the allowed operators in the four-dimensional effective supergravity. The vector bosons associated to these anomalous U(1) symmetries are massive. We explicitly compute the spectrum of allowed operators for each model and present the results, together with the defining data of the models, in a database of standard models accessible here. Based on these results we analyze elementary phenomenological properties. For example, for about 200 models all dimension four and five proton decay violating operators are forbidden by the additional U(1) symmetries.

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Section: Jhep06(2012)113mentioning

confidence: 99%

Heterotic line bundle standard models

Anderson

Gray

Lukas

et al. 2012

J. High Energ. Phys.

Self Cite

188

366

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“…Next, this reducible algebraic variety can be primary decomposed to find its irreducible pieces. A discussion of the methods that we use for computations such as this can be found in, for example, [88,89]. We utilized the specific implementations found in [90,91] in this work.…”

Section: )mentioning

confidence: 99%

Jumping spectra and vanishing couplings in heterotic Line Bundle Standard Models

Gray

Wang

2019

J. High Energ. Phys.

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We study two aspects of the physics of heterotic Line Bundle Standard Models on smooth Calabi-Yau threefolds. First, we investigate to what degree modern moduli stabilization scenarios can affect the standard model spectrum in such compactifications. Specifically, we look at the case where some of the complex structure moduli are fixed by a choice of hidden sector bundle. In this context, we study the frequency with which the system tends to be forced to a point in moduli space where the cohomology groups determining the spectrum in the standard model sector jump in dimension. Second, we investigate to what degree couplings, that are permitted by all of the obvious symmetries of the theory, actually vanish due to certain topological constraints associated to their higher dimensional origins. We find that both effects are prevalent within the data set of heterotic Line Bundle Standard Models studied.

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“…The complexity of the problem enforced the use of computational techniques in algebraic geometry such as various Gröbner basis algorithms and led to several, already striking, partial results. (See, for example, [3] for a physicist's introduction to Gröbner bases.) For instance, the appearance of the Veronese surface in the electroweak sector as an almost generic vacuum moduli space geometry was reported in [4,5].…”

Section: Introductionmentioning

confidence: 99%

Veronese geometry and the electroweak vacuum moduli space

Jejjala

Matti

et al. 2014

Physics Letters B

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We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F-and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential. * hey@maths.ox.ac.uk † vishnu@neo.phys.wits.ac.za ‡ Cyril. Matti.1@city.ac.uk

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A Simple Introduction to Gröbner Basis Methods in String Phenomenology

Cited by 17 publications

References 36 publications

Heterotic line bundle standard models

Heterotic line bundle standard models

Jumping spectra and vanishing couplings in heterotic Line Bundle Standard Models

Veronese geometry and the electroweak vacuum moduli space

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