xAct:  Efficient tensor computer algebra for the Wolfram Language

José M. Martín-García, GPL 2002-2026

Developers: Steffen Aksteiner, Thomas Bäckdahl, Will Barker, David Brizuela, Guillaume Faye, Alfonso García-Parrado, Thomas Helpin, Manuel Hohmann, Giorgos Karananas, Kevin Kiely, Michele Levi, Leon Manssur, Juan Margalef-Bentabol, José M. Martín-García, Carlo Marzo, Guillermo Mena Marugán, Salvador Mengual Sendra, Johannes Noller, Teake Nutma, Adrian Ottewill, Cyril Pitrou, Renato Portugal, Claire Rigouzzo, Xavier Roy, Laura Sánchez Cotta, Alessandro Stecchina, Leo C Stein, Jan Steinhoff, Francesco Torsello, Haochen Tu, Obinna Umeh, Hugo D. Wahlquist, Barry Wardell, David Yllanes

Introduction

xAct is a suite of free packages for tensor computer algebra in the Wolfram Language. xAct implements state-of-the-art algorithms for fast manipulations of indices and has been modelled on the current geometric approach to General Relativity. It is highly programmable and configurable. Since its first public release in March 2004, xAct has been intensively tested and has solved a number of hard problems in GR.

There are 4 packages acting as a kernel for the rest:

• xCore: Generic programming tools, by JMMG.
• xPerm: Manipulation of large groups of permutations, by JMMG.
• xTensor: Abstract tensor computations, the flagship of the system, by JMMG.
• xCoba: Component tensor computations, by JMMG and David Yllanes.

27 other packages:

• xPert: High-order perturbation theory in GR, by David Brizuela, JMMG and Guillermo Mena Marugán.
• Harmonics: Tensor spherical harmonics, by David Brizuela, JMMG and Guillermo Mena Marugán.
• Invar: Polynomial invariants of the Riemann tensor, by JMMG, David Yllanes, Renato Portugal and Leon Manssur.
• Spinors: Spinor computations in GR, by Alfonso García-Parrado and JMMG.
• xPrint: Formatted input for xAct, by Alessandro Stecchina.
• SymManipulator: Symmetrized expressions in xAct, by Thomas Bäckdahl.
• AVF: Exterior calculus with algebra-valued forms, by Hugo D. Wahlquist.
• xTras: Additions to xAct, by Teake Nutma.
• TexAct: Tex code to format xAct expressions, by Thomas Bäckdahl, JMMG and Barry Wardell.
• xPand: Cosmological perturbation theory, by Cyril Pitrou, Xavier Roy and Obinna Umeh.
• xTerior: Exterior calculus, by Alfonso García-Parrado and Leo C. Stein.
• SpinFrames: NP and GHP form of spinor equations, by Thomas Bäckdahl and Steffen Aksteiner.
• xIST/COPPER: General scalar-tensor theories and perturbations, by Johannes Noller.
• EFTofPNG:: Effective field theory of post-Newtonian gravity, by Michele Levi and Jan Steinhoff.
• bimEX: 3 + 1 Bimetric relativity, by Francesco Torsello.
• FieldsX: Fermions, gauge fields and BRST cohomology, by Markus B. Fröb.
• xPPN: Parametrized post-Newtonian formalism, by Manuel Hohmann.
• SymSpin: Symmetric spinors, by Steffen Aksteiner and Thomas Bäckdahl.
• HiGGS: Hamiltonian analysis of Poincaré gauge theory, by Will Barker.
• xBrauer: Brauer algebra for tensor calculus, by Thomas Helpin.
• xMAG: Metric-affine gravity framework, by Thomas Helpin.
• PSALTer: Particle spectrum for any tensor Lagrangian, by Will Barker, Giorgos Karananas, Carlo Marzo, Claire Rigouzzo and Haochen Tu.
• TInvar: Canonicalization of Riemann expressions with free indices, by Kevin Kiely, Barry Wardell, Adrian Ottewill and JMMG.
• SpaceSpinors: 3d-space spinors on spacelike hypersurfaces in a 4d-spacetime, by Thomas Bäckdahl.
• xCPS: Covariant Phase Space Formalism for xAct, by Juan Margalef-Bentabol and Laura Sánchez Cotta.
• xIdeal: Identification of exact solutions in xAct, by Alfonso García-Parrado and Salvador Mengual Sendra.
• Hamilcar: Hamiltonian (canonical) analysis toolkit for xAct, by Will Barker.

A single file with the current versions (29 December 2025) of all packages can be downloaded: xAct_1.3.0.tgz for linux/unix/mac, or xAct_1.3.0.zip for windows. See the Installation notes.

Check http://contrib.xact.es for possible updates of some of those packages, or entirely new packages. See also there the page of contributed example notebooks.

Examples: very simple session, check of the Bianchi identities for a connection with torsion, benchmarking examples.

xAct is free software. Please cite the associated publications and this web page if you use it.

For comments or questions, please contact jose@xact.es, or send your questions to the xAct forum.

News

29 December 2025: version 1.3.0 of xAct release.

• New package xBrauer by Thomas Helpin, which depends on the separate packages SymmetricFunctions and BrauerAlgebra.
• New package xMAG, by Thomas Helpin.
• New package PSALTer by Will Barker, Giorgos Karananas, Carlo Marzo, Claire Rigouzzo and Haochen Tu.
• New package TInvar by Kevin Kiely, Barry Wardell, Adrian Ottewill and José M. Martín-García.
• New package SpaceSpinors by Thomas Bäckdahl.
• New package xCPS by Juan Margalef-Bentabol and Laura Sánchez Cotta.
• New package xIdeal by Alfonso García-Parrado and Salvador Mengual Sendra.
• New package Hamilcar by Will Barker.
• Bug fixes.

4 March 2024: xAct is 20 years old!

• Watch a series of talks by xAct developers to celebrate two decades of tensor computation.

22 November 2023: 1000 documents with xAct.

• The articles page has collected 1000 articles or theses acknowledging the use of xAct in their computations. Many thanks to all developers and authors who have contributed to reaching this milestone!

Previous news

CopyLeft

The programs in this page are distributed as free software under the GNU General Public License. They are distributed in the hope that they wil be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU GPL for more details.