Universe of strings. Image taken from http://turbulentscientist.blogspot.com/
This repository contains implementations of various mathematical tools (written in Mathematica) used in String theory/Supergravity suplemented by examples. The tools are T duality, rotations, boosts, some differential geometry and group theory, etc. So far all the functions are distributed in three packages DiffGeometry, TransformationRules and GroupTheory located in the Packages folder. Usage of each function is demonstrated in various examples located in the Examples folder.
I am not sure if there are any open source libs implementing the desired functionality, but even if there are, it is better to write my own code to get a better understanding of what is going on and eliminate possible errors.
Code from this repository was used in the following papers:
- Y. Chervonyi and O. Lunin, “Generalized lambda-deformations of AdS_p x S^p,” Generalized lambda-deformations of AdSp x Sp, arXiv:1608.06641 [hep-th].
- Y. Chervonyi and O. Lunin, “Supergravity background of the lambda-deformed AdS_3 x S^3 supercoset,” Nucl.Phys. B910 (2016) 685-711, arXiv:1606.00394 [hep-th].
- Y. Chervonyi, “Towards higher dimensional black rings,” Phys.Rev. D92 (2015) 12, 124037, arXiv:1510.06041 [hep-th].
- Y. Chervonyi and O. Lunin, “Killing(-Yano) Tensors in String Theory,” JHEP 1509, 182 (2015) arXiv:1505.06154 [hep-th].
- Y. Chervonyi and O. Lunin, “(Non)-Integrability of Geodesics in D-brane Backgrounds,” JHEP 1402, 061 (2014) arXiv:1311.1521 [hep-th].
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Package TransformationRules implements boosts, rotations, S duality and T duality (the Buscher rules) for NS-NS fields.
A great example of utilizing boosts and T/S dualities is a powerful solution-generating technique (solutions of supergravity equations of motion), in which one starts with a simple solution (e.g. the Schwarzschild black hole) and by performing a series of various transformations obtains something new (e.g. D-branes). This solution-generating technique was used in http://arxiv.org/abs/hep-th/0105136 to construct the fuzzballs.
In GenerateD1braneFromKerr we demonstrate this solution-generating technique. We start with a rotating black hole in 4 dimensions (the Kerr black hole) and perform the following series of transformations: "add flat direction"->"boost"->"T duality"->"S duality", to obtain the black D1 brane. At the end we take the extremal static (no rotation) limit to check that the result is the well-known supersymmetric D1 brane.
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Package DiffGeometry implements computation of connections (Christoffel symbols), Riemann and Ricci tensors, Ricci scalar and spin connections.
One simple example of using the package is AdS2xS2.EinsteinEqs, in which the Einstein equations are checked for AdS2, S2 and AdS2xS2 geometries.
In another example, KillingSpinors.IIB.NearHorizonNS5 we solve the supersymmetry (Killing spinor) equations of type IIB supergravity in the NS-NS sector for one particular background - the near horizon NS5 brane.
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Package GroupTheory contains some group-theoretical functionality, such as computing of adjoint represenation of Lie algebra, perfoming the Cartan-Weyl decomposition, tools needed to work with superalgebras, etc.
In adjoint.su(2) we start with the su(2) Lie algebra in the fundamenatal represenation and compute the corresponding adjoint represenation.
GradedYangBaxter.psu(1,1).Holom demonstrates an example of dealing with psu(1,1|2) Lie superalgebra. We solve the modified graded Yang-Baxter equation for this superalgebra.
- Add more examples.
- Add Sen-Hassan-Cvetic method using large matrices to perform a series of dualities in one rotation arxiv:hep-th/9512127
- Add more functions into GroupTheory