https://github.com/PyryKuusela/GammaMaP
Tip revision: dd15833a71ab6780e9c0e5ad5c7e4395d3880c19 authored by PyryKuusela on 02 May 2019, 16:07:57 UTC
Added newest version of Mathematica package
Added newest version of Mathematica package
Tip revision: dd15833
GammaMaP_examples.nb
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Cell["\<\
Gamma MaP - A Mathematica Package for Clifford Algebras, Gamma Matrices and \
Spinors\
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"We present a Mathematica package, \[OpenCurlyDoubleQuote]Gamma MaP\
\[CloseCurlyDoubleQuote] (",
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" ",
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"trix ",
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"ackage), for performing calculations involving gamma matrices, spinors and \
tensors in any dimension and signature. The main utility of the package is \
making these computations without needing to specify an explicit \
representations, but the package can also be used to perform computations \
with explicit representations. Our approach is substantially different from \
earlier packages, such as [",
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"], in various ways: the approach we use is based on defining basic objects, \
such as spinors, gamma matrices and tensors. Then we define functions \
implementing various operations involving these objects, including index \
contraction, non-commutative multiplication, and more. The leading idea is \
that any function can be used on any expression that can be constructed using \
the basic objects. This allows for a relatively simple and intuitive \
implementation of even complex operations, and ensures that the program can \
be easily extended. Furthermore, this approach allows for user-defined \
preferences for example concerning, where various intertwiner matrices should \
be commuted when simplifying expressions containing these. This makes the \
package ideal for many different types of calculations, since it does not \
make many assumptions about the form in which the user gets the simplified \
output. Instead, the user can modify most elements of this behaviour. \n\nThe \
approach which we have used to create the basic functionality of the package \
is based on essentially defining the commutation properties that the gamma \
matrices and different intertwiners satisfy. Then, based on the preferences \
of the user, matrices are automatically (anti-)commuted with each other, \
resulting in greatly simplified expressions in most cases. This approach \
could also be easily used in other cases that require commuting operators \
with given (anti-)commutation rules to a pre-defined order, such as when \
calculating CFT correlation functions. \n\nIn addition to the basic \
operations on gamma matrices, the package contains various operations \
familiar from many tensor packages (for example [",
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"]) such as upper and lower indices compatible with Einstein summation \
convention, tensors with different symmetries, and partial derivatives. In \
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already exists in Mathematica, but that has been modified to better suit the \
needs of gamma matrix calculations. For example, we have defined new \
functions for making and using assumptions, and dealing with real and \
imaginary parts of expressions, which have been implemented with gamma matrix \
calculations in mind. \n\nBelow we explain how various properties and \
operations are implemented in this package. As examples, we reproduce a few \
results from literature [",
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"]. In section 2, we give instructions for installing and running the \
package, and defining the Clifford algebra, whose representation the gamma \
matrices then form. In the following section 3, we introduce basic objects \
such as antisymmetrised products of gamma matrices, and show how they are \
implemented and used in this package. In section 4, we list basic functions, \
such as multiplication of matrices, and show how to use these. Then, in \
section, 5 we explain how to use explicit representation for the gamma \
matrices, and other objects instead of using just the abstract definitions. \
In section 6 we discuss using features related to subalgebras that are \
useful, for instance, when considering dimensional reduction. Finally, in \
section 7, a few examples are provided by reproducing some lengthy \
calculations from literature."
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"."
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After doing this, we must specify the relevant Clifford algebra, whose \
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Since these anticommute with each other and square to one, these satisfy the \
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As is well known, we can use sigma matrices to build higher dimensional \
representations as well by taking tensor product of these. In particular, \
there is a useful representation that we will in the following call the \
\[OpenCurlyDoubleQuote]special representation\[CloseCurlyDoubleQuote]. For \
the case with no timelike directions, (+ . . . +), it is given by\
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In case with timelike directions, the matrices corresponding to the timelike \
directions can be just multiplied by i to get a representation with the \
correct signature. When speaking of the special representation, we assume \
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",
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In addition to the gamma matrices, there a few other useful matrices, that \
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called intertwiners.\
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A-intertwiner is defined as the product of all timelike gamma matrices\
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and it can be used to represent Hermitian conjugation on the gamma matrices.\
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In addition, it is always possible to define a matrix C, called charge \
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It is then straightforward to verify that the B-intertwiner defined this way \
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"We comment first on how the indices appearing in different objects are \
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More generic derivatives or variations, such as gauge covariant derivatives \
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This is particularly useful when defining custom values for the derivative, \
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Writing a product of intertwiners, gamma matrices and spinors works as \
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Spinor bilinears are also recognised automatically, and they acts as \
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For the following examples, let us set up a seven-dimensional representation \
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As was the case with the gamma matrices, some intertwiners cannot be commuted \
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After having previously defined the correspondence between the two notations \
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\>", "Text",
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This works also the other way round. The gBL notation is changed to product \
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If we want to define a different notation for bilinears that have spinor \
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Using these options is especially useful for simplifying expressions that \
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For example, to reproduce the duality formulae given above for signatures \
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This identity can be multiplied by gamma matrices and spinors to get a set of \
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Assumptions can also be used to define irreducible spinors, i.e. Majorana or \
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Also, after making this assumption the program can use simplifying identities \
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Note that the Majorana condition and reality condition are treated as \
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For example, if we wish to know which spinors are allowed for a \
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This function can also be used to check which spinors are possible for other \
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This far we have only seen the package working with completely abstract \
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Note that the values for A- and B-intertwiners are not needed, since these \
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To set two coordinates x0 and x1 corresponding to two indices 0 and 1, we can \
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For example, making the variables \[Alpha] and \[Beta] take values 0, . . . , \
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The indices that do not appear in the any of the lists, can correspond to any \
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Tensor product can be taken of any objects, such as spinors, matrices or \
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If we wanted to assume that there are terms that contain only indices \
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For the most part, using assumptions with subrepresentations is no different \
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In this section, we give a few more involved examples of computations that \
can be performed with the package. This will both serve as a non-trivial \
demonstration of the properties of objects and functions presented in the \
previous sections, and also shows how these objects can be used in an optimal \
fashion when performing calculations that appear often in practical \
applications of gamma matrices and Clifford algebras.\
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7.1 Torsion Conditions for Supersymmetric SUGRA Compactification\
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"In this example we reproduce the torsion conditions for AdS 3 \[Times] M 7 \
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conditions, the use the decomposition SO(1, 9) \[RightArrow] SO(1, \
2)\[Times]SO(7) to reduce the equations to 7 dimensions. We can use the \
7-dimensional spinors parametrising the supersymmetry transformations to form \
bilinears. The supersymmetry conditions can then be expressed equivalently as \
conditions on the bilinears obtained in this fashion. It can then be shown \
that these in turn impose conditions on so-called torsion tensor. In the \
process we will demonstrate how to use some features related to dimensional \
reduction, and how to use derivatives, bilinears and relations involving \
spinors.\n\nWe start with the 10-dimensional IIB supergravity, whose NS-NS \
sector of consists of a real scalar \[CurlyPhi], called dilaton, the metric \
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The scalars can be combined into a single field, \[Tau], called axio-dilaton\
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which can then be used to define a one-form field P, and a derivative that is \
covariant with respect to both Lorentz and U(1) transformations.\
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We are looking for supersymmetric solutions, so we must require that the \
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After this, we define indices \[Alpha] and \[Beta] to correspond to the first \
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We can now put everything together. By looking at this result and the one \
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Since we are now making computations with spinors that used to form a part of \
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In this section, we prove a few well-known properties of the supersymmetric \
Yang-Mills theories. Specifically, we first show that the 10-dimensional \
Yang-Mills action is invariant under supersymmetry transformations, and then \
we consider supersymmetry transformations of dimensionally reduced N=1 D=4 \
SYM, and show that the supersymmetry transformations
form a correct supersymmetry algebra. In the process we will show how \
examples of how to handle derivatives, Fierz indentities, and different types \
of spinors.\
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In this section, we will consider the four-dimensional supersymmetric \
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