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A Mathematica Package for Clifford Algebras, Gamma Matrices "Gamma MaP" - A Mathematica Package for Clifford Algebras, Gamma Matrices and Spinors Pyry Kuusela Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Oxford, OX2 6GG, UK [email protected] Abstract We present a Mathematica package for doing computations with gamma matrices, spinors, tensors and other objects, in any dimension and signature. The approach we use is based on defining the commutation relations of the relevant matrices, and is thus general and flexible. As examples, we reproduce torsion conditions for AdS3 compactification of type IIB supergravity from literature, and check the vanishing of supersymmetry variations of 10-dimensional supersymmetric Yang-Mills theory. arXiv:1905.00429v1 [hep-th] 1 May 2019 Contents 1 Introduction 1 2 Review of Spinors and Gamma Matrices in Different Dimensions 3 2.1 Gamma Matrices . .3 2.2 A-intertwiner and Hermiticity . .4 2.3 C-intertwiner and Symmetry . .5 2.4 B-intertwiner and Complex Conjugation . .7 2.5 Highest Rank Gamma Matrix γ∗ ........................8 2.6 Clifford Algebra and Lorentz Group . .8 2.6.1 Spinor Bilinears and Tensors . .9 2.7 Irreducible Spinors . 10 2.7.1 Weyl Spinors . 10 2.7.2 Majorana Spinors . 11 2.7.3 Symplectic Majorana Spinors . 12 2.7.4 Majorana-Weyl Spinors . 12 2.7.5 Symplectic Majorana-Weyl Spinors . 12 2.8 Relations for Bilinears . 12 2.9 Basis for Clifford Algebra . 15 2.9.1 Fierz Rearrangement . 16 2.9.2 Expressing Gamma Matrix Products in Terms of the Basis . 17 2.10 Subalgebras . 20 2.10.1 Gamma Matrices . 21 2.10.2 Intertwiners . 21 2.10.3 Representations of Lorentz Algebras . 23 3 Installation and Setup 24 3.1 Downloading and Installing . 24 3.2 Setup . 24 4 Basic Objects 28 4.1 Indices . 28 4.2 Gamma Matrices . 28 4.3 Other Matrices . 29 4.4 Spinors . 30 4.4.1 Dirac Conjugate . 30 4.4.2 Charge Conjugate . 31 4.5 Forms . 33 4.6 Symmetric Tensors . 34 4.7 Other Tensors . 35 4.8 Derivatives . 36 5 Functions 40 5.1 Multiplication . 40 5.1.1 Associated Options . 41 5.2 Products of Gamma Matrices . 50 5.2.1 Cases . 50 5.3 Real and Imaginary Parts . 52 5.4 Conjugations . 54 5.4.1 Complex Conjugation . 54 5.4.2 Transpose . 55 5.4.3 Hermitian Conjugation . 56 5.4.4 Associated Options . 57 5.5 Bilinears . 59 5.5.1 Associated options . 63 5.6 Dual Elements in Clifford Algebra . 65 5.7 Fierz Rearrangement . 67 5.8 Simplifying expressions . 68 5.9 Using Assumptions . 70 5.9.1 Assumptions on Indices . 72 5.9.2 Assumptions on spinors . 74 6 Explicit Representations 77 6.1 Defining Explicit Representation . 77 6.1.1 Gamma Matrices and Intertwiners . 77 6.1.2 Coordinates . 78 6.1.3 Spinors . 78 6.1.4 Tensors . 79 6.2 Using Explicit Representations . 80 7 Subalgebras 84 7.1 Defining Subalgebras . 84 7.2 Decomposing Expressions . 85 7.3 Using Assumptions . 90 7.4 Explicit Representations . 93 7.5 Options and Subrepresentations . 94 8 Other Topics 96 8.1 Naming Conventions . 96 8.2 Limitations and Known Issues . 96 8.3 Bug Reports and Feature Requests . 97 9 Examples 98 9.1 Torsion Conditions for Supersymmetric SUGRA Compactification . 98 9.1.1 Reducing the 10-dimensional Supersymmetry Conditions to 7-dimensions 99 9.1.2 Deriving the 7-dimensional Torsion Conditions . 104 9.1.3 Scalar Bilinear S . 106 9.1.4 One-form K . 107 9.1.5 Higher Forms . 108 9.2 Supersymmetric Yang-Mills Theory . 108 9.2.1 Invariance of 10D SYM-action Under SUSY Transformations . 108 9.2.2 Supersymmetry Transformations of D=4 N = 1 SYM . 111 A List of Objects 114 B List of Functions 115 C List of Options 116 D List of Other Reserved Expressions 116 E Properties of Special Matrices 117 E.1 Properties of Special Matrices . 117 E.1.1 Sigma Matrices . 117 E.1.2 Gamma Matrices . 117 E.1.3 A-intertwiner . 118 E.1.4 C-intertwiner . 118 E.1.5 B-intertwiner . 119 E.1.6 Highest Rank Gamma Matrix γ∗ .................... 119 E.2 Commutation Relations of Intertwiners . 119 E.3 Bilinear Relations . 120 E.4 Decomposition to Subrepresentations . 121 E.4.1 Odd → Even × Odd . 121 E.4.2 Even → Odd × Odd . 122 E.4.3 Even → Even × Even . 122 1 1. Introduction We present a Mathematica package, "Gamma MaP" (Gamma Matrix Package), for per- forming 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 [1–3], 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 oper- ations 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 imple- mentation 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. The 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. In addition to the basic operations on gamma matrices, the package contains various opera- tions familiar from many tensor packages (for example [4]) such as upper and lower indices compatible with Einstein summation convention, tensors with different symmetries, and partial derivatives. In addition to the new functionality, there is also some functionality that 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. Below we will present a brief review of relevant properties of gamma matrices, spinors and tensors following mainly [5], and explain how various properties and operations are imple- mented in this package. As examples, we reproduce a few results from literature [6, 7]. The structure of this paper is as follows. We begin in section 2 with a review of the most impor- tant properties of spinors and gamma matrices in various dimensions. In section 3 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 4 we introduce basic objects such as antisymmetrised products of gamma matrices, and show how they are imple- mented and used in this package. In section 5 we list basic functions, such as multiplication of matrices, and show how to use these. Then, in section 6, we explain how to use explicit representation for the gamma matrices, and other objects instead of using just the abstract definitions. In section 7 we discuss using features related to subalgebras that are useful, for 2 instance, when considering dimensional reduction. Finally, in section 9, a few examples are provided by reproducing some lengthy calculations from literature. In appendices A, B, C and D, we provide a reference list of objects, functions, options and other reserved expressions. In appendix E, we give a list of properties satisfied by gamma matrices and intertwiners, and summarise the most important formulas from the review of section 2. 3 2. Review of Spinors and Gamma Matrices in Different Dimensions In this section, we review the basic properties.
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