Geometric or conceptual way to understand supersymmetry algebra

Question

Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ensuring the supersymmetry invariance and then writing down the supersymmetry algebra? For a geometric or algebraic way, I mean to (at least partially) derive the supersymmetry algebra from pure geometry (spinors, spin group, etc.).

Answer 1

In this paper about a global theory of supermanifolds, Alice Rogers develops the theory of supermanifolds as they underly supersymmetric field theories from a rigorous but physicist-friendly differential-geometric point of view from topological scratch, and also puts some additional structures such as vector fields and tangent spaces on them. She also compares the $G^{\infty}$ or deWitt supermanifolds to the algebro-geometric approach of for example Kostant or Leites.

Alice Roger's 2007 textbook explains the supermathematics needed for doing superphysics (including Grassmann algebras, super Lie groups such as the super Poincare group, etc) and contains more applications to different physics topics such as $N=1$ supersymmetry, supergravity, some aspects of string theory, or Brownian motion from the same nice differential-geometric point of view.

Answer 2

If you are looking for geometric-algebraic interpretations of supersymmetric field theories, then non-commutative geometry -in the sense of A. Connes- seems to be the natural playground. There has been quite a lot of literature on the subject since the '80's. Just to mention some inspiring works:

• Supersymmetry and Non-Commutative Geometry, J. Fröhlich, O. Grandjean, A. Recknagel, Quantum Fields and Quantum Space Time, 1997
• Supersymmetry and noncommutative geometry, W.Kalau, M.Walzeb, Journal of Geometry and Physics, v.22, 1, 1997, 77-102
• Non-commutative geometry and string field theory, E. Witten, Nuclear Physics B, v.268, 2, 1986, 253-294

See also this, which is a more modern work, in the form of a book, discussing (non-commutative) geometrical ideas related to supersymmetry although from a more -imo- phenomenological viewpoint (emphasizing the supersymmetric standard model, supersymmetry breaking, elementary particle interactions , etc).

Edit:
After giving some further thinking on your question and since you are asking for some method

... to derive the supersymmetry algebra from pure geometry ...

i decided to try to provide an example of such a construction. This will still be in the spirit of the non-commutative geometrical approach but in a somewhat simplified manner:

Example: Consider the (usual) complex variables $x_{i}$ and the anticommuting Grassman variables $\xi_{j}$. (The $x_{i}$ commute with each other and with the $\xi_{j}$ as well, while the $\xi_{j}$ anticommute with each other). Next consider the space of polynomials in those variables and their formal derivatives. If we use the notation: $$ \begin{array}{cccc} \begin{array}{ccc} b_{i}^{+} & \leftrightarrow & x_{i} \\ b_{i}^{-} & \leftrightarrow & \frac{\partial}{\partial{x_{i}}} \end{array} & & & \begin{array}{ccc} f_{i}^{+} & \leftrightarrow & \xi_{i} \\ f_{i}^{-} & \leftrightarrow & \frac{\partial}{\partial{\xi_{i}}} \end{array} \end{array} $$ it is easy to see that the multiplication and the differential operators $(x_{i},\frac{\partial}{\partial{x_{i}}})$ acting on (usual) complex variables correspond to boson creation-annihilation operators and that the multiplication and the differential operators $(\xi_{i},\frac{\partial}{\partial{\xi_{i}}})$ acting on Grassmann (anticommuting) variables correspond to fermion creation-annihilation operators, in the sense that the following relations hold: $$ \begin{array}{c} [b_{i}^{-}, b_{j}^{+}]\equiv b_{i}^{-}b_{j}^{+} - b_{j}^{+}b_{i}^{-} = \delta_{ij} I, \ \ \ \ \ \ [b_{i}^{-}, b_{j}^{-}] = [b_{i}^{+}, b_{j}^{+}] = 0 \ \ \ \ \ \ \ \ \ (1) \\ \\ \{f_{i}^{-}, f_{j}^{+}\}\equiv f_{i}^{-}f_{j}^{+} + f_{j}^{+}f_{i}^{-} = \delta_{ij} I, \ \ \ \ \ \ \{f_{i}^{-}, f_{j}^{-}\} = \{f_{i}^{+}, f_{j}^{+}\} = 0 \ \ \ \ \ \ \ \ (2) \\ \\ f_{j}^{\varepsilon}b_{i}^{\eta} = b_{i}^{\eta}f_{j}^{\varepsilon} \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \end{array} $$ where $\varepsilon,\eta=\pm$ and $I$ is the identity operator. This is an algebra defined in terms of generators and relations and it is a quotient algebra of the UEA of the Heisenberg Lie superalgebra. To make things even simpler consider a single bosonic and a single fermionic degree of freedom, i.e. $i=j=1$, thus $\xi^2=0$ (such a grassman variable is frequently called a Clifford element).

Now consider, the following elements of the above algebra: $$ Q=\frac{1}{2}\{b^+,f\}, \ \ \ \ Q^+=\frac{1}{2}\{f^+,b\}, \ \ \ \ H=\frac{1}{4}\{b^+,b\}+\frac{1}{4}[f^+,f] $$

Using relations (1), (2), (3) we can verify after some straightforward algebraic computations that: $$ [H,H]=[H,Q]=[H,Q^+]=0, \ \ \ \ \{Q,Q^+\}=2H, \ \ \ \ \ \ \{Q,Q\}=\{Q^+,Q^+\}=0 \ \ \ \ \ (4) $$ generating thus the simplest supersymmetric toy model (see for example: a supersymmetry primer, for more realistic versions of such algebras). What has actually been shown here is that:

the SUSY algebra (4), emerges as a subalgebra of the algebra (1), (2), (3), of multiplication and differentiation operators acting on polynomial spaces spanned by polynomials in a single complex (commuting) and a single grassman (anticommuting) variable.

I hope that you may find something of interest in this example.

Answer 3

I'm not sure what you mean by "derive".

For a more mathematical and geometric description of the super Poincaré group in general dimension you could check out

• Freed, Lectures on field theory and supersymmetry (Lecture 6);
• Freed, Five lectures on supersymmetry, AMS 1999 (Lecture 3);
• Deligne and Freed, Supersolutions [arXiv:hep-th/9901094] (Section 1.1), which can also be found in Quantum fields and strings: a course for mathematicians;

plus the three additional references given for super-Poincaré group at nLab.

For the 4d case see also

• Costello, e.g. [arXiv:1401.2676] (Section 1.1).

Answer 4

These lecture notes (paragraph 1.5) explain supersymmetry in geometric terms as an extension of the Poincaré group to include translations in the Grassmann directions.