I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra computations. Also, I would like this package to be able to do wedge and hodge dual, and other computation relating to forms. Can anyone suggest a specific one? I looked for atlas2, but it seems I have to pay to use it without a trial version.

# [Physics] Mathematica package for supergravity and string theory

clifford-algebraresource-recommendationssoftwaresupergravitytensor-calculus

#### Related Solutions

For supersymmetry and supergravity, my primary recommendation is

*Supergravity* by **Freedman and Van Proeyen**. This book has a very large number of exercises interspersed across the text. The exercises are placed at locations that are relevant to the prose around them, and they vary in their level of difficulty from verifying certain results in the text, to involved problems, so they're well-suited to self-study. As an aside, the prose itself is (at least in the first 10 chapters or so) clear and pedgogical. Although this is not strictly a problem book, it has so many of them that it could effectively function as one for someone who treats it that way. This book will be useful if you have a relatively strong background in QFT and are just getting into research in theoretical high energy.

A secondary recommendation is sections 3.6 and 4.12 of the classic review

"Supersymmetric Gauge Theories and the AdS/CFT Correspondence" by **D'Hoker and Freedman**. These two sections contain five and four problems respectively on SYM and sugra/superstrings. Although there are not a large number of exercises here, the exercises are interesting and relevant. The level is suitable for anyone with a strong background in QFT and especially the mathematics of symmetry in physics (groups, algebras, etc.)

### To answer the confusion between the three sources you list:

Using the signature convention of Figueroa O'Farrill, we have Majorana pinor representations for $p - q \pmod 8 = 0,6,7$ and Majorana spinor representations for $p - q \pmod 8 = 1$. Pinor representations induce spinor representations (that will be reducible in even dimension) and so we get Majorana spinor representations for $p - q \pmod 8 = 0,1,6,7$.

Although $\mathcal{Cl}(p,q)$ is not isomorphic to $\mathcal{Cl}(q,p)$, their even subalgebras are isomorphic and so can be embedded in either signature. This means that Majorana pinor representations in $\mathcal{Cl}(q,p)$ also induce spinor representations in the even subalgebra of $\mathcal{Cl}(p,q)$ and so we also get an induced Majorana spinor representation for $p - q \pmod 8 = 2$ (from $q - p \pmod 8 = 6$; this is often called the pseudo-Majorana representation).

Fecko has his signature convention swapped compared to Figueroa O'Farrill, and so swapping back we see that his $0,2 \pmod 8$ gives us $0,6 \pmod 8$. One can also see from his table (22.1.8) that on the page you reference he was listing signatures with Clifford algebra isomorphisms to a single copy of the real matrix algebra, but his table also gives us $p - q \pmod 8 = 1$, converting signature convention to $p - q \pmod 8 = 7$ which is the isomorphism to two copies of the real matrix algebra and so also yields Majorana pinor representations. He doesn't talk about Majorana (or pseudo-Majorana) spinor representations here and so doesn't list $p - q \pmod 8 = 1,2$.

As for Polchinski, he includes pseudo-Majorana representations (or is signature convention agnostic) and so lists all of $p - q \pmod 8 = 0,1,2,6,7$.

### To answer the question of in which dimensions Majorana spinors (including pseudo-Majorana) exist:

For a signature $(p,q)$ they exist whenever any of $\mathcal{Cl}(p,q)$, $\mathcal{Cl}(q,p)$ or the even subalgebra of $\mathcal{Cl}(p,q)$ are isomorphic to either one or a direct sum of two copies of the real matrix algebra. This means $p - q \pmod 8 = 0,1,2,6,7$. If one discounts pseudo-Majorana spinors, then one removes $\mathcal{Cl}(q,p)$ from the previous statement and this means $p - q \pmod 8 = 0,1,6,7$.

Of course, this does not talk about the naturally quaternionic symplectic and pseudo-symplectic Majorana representations.

One can take the algebra isomorphisms of low-dimensional Clifford algebras ($\mathcal{Cl}(1,0) \cong \mathbb{C}$, $\mathcal{Cl}(0,1) \cong \mathbb{R} \oplus \mathbb{R}$ etc.) and use the isomorphisms between Clifford algebras of different signatures ($\mathcal{Cl}(p+1,q+1) \cong \mathcal{Cl}(p,q) \otimes \mathcal{Cl}(1,1)$ etc.) to bootstrap the equivalent matrix algebra isomorphisms of Clifford algebras (and similarly for their even subalgebras) of arbitrary signature and from there one can see when real forms exist.

## Best Answer

For Mathematica the best I know is RGTC. I Used it a long time ago (briefly) for a calculation in IIA SUGRA in 10 dimensions. It calculate gravitational tensors, manages differential forms (also Lie algebra valued ones), calculates Hodge dualities, etc.

Personal comment:If you are more intrepid (and FLOSS lover), there is a software calledSAGE, which now has a (still developing) package (sagemanifold) for differential geometry calculations... it allows the use of differential form, etc.