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Wikipedia says:

In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa. There is only indirect evidence for the existence of supersymmetry [...]

I want a mathematical explanation of SUSY.

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up vote 7 down vote accepted

Mathematically, SUSY begins with the supersymmetry algebra, a Lie superalgebra, which is itself a special case of a more general class of algebras called graded Lie algebras. Of central importance is the supersymmetry algebra referred to as the super-Poincare algebra that extends the Poincare algebra to include supersymmetry "charges" and their anticommutators.

In the context of physics, one studies field theories, both classical and quantum, that exhibit invariance under some action of supersymmetry algebras on fields and Hilbert spaces of these theories. As a result, representations of supersymmetry algebras are especially important in physics.

I would highly recommend that you look at THIS set of notes written by Sohnius, one of the original supersymmetry masters and co-discoverers of THIS famous and important theorem which really motivates why supersymemtry is all the rage in physics. The notes talk about representations of supersymmetry algebras in a lot of detail, and the clarity of the prose is top-notch if you ask me.

Addendum. I almost forgot, you also hear the word "superspace" which is a construction that physicists use to, among other things, make constructing manifestly supersymmetric Lagrangians easier. The mathematics behind this is supermanifolds.

Lastly, there is some discussion of these things on math.SE, see for example

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By the way, superspace in the form of super-Minkowski space is right there in the super-Poincare algebra, being the quotient of that by the Lorentz sub-algebra ( – Urs Schreiber Sep 18 '13 at 17:46
@UrsSchreiber Interesting stuff. Thanks for the link! – joshphysics Sep 18 '13 at 18:55

In addition to what Joshua said in his nice answer, may favorite (simplified) point of view is looking at a SUSY transformation as a coordinate transformation (translation) in superspace

$$ x' = x + a + \frac{i}{2}\zeta\sigma^{\mu}\bar{\theta} - \frac{i}{2}\theta\sigma^{\mu}\bar{\zeta}$$

$$ \theta'= \theta + \zeta $$

$$ \bar{\theta}'= \bar{\theta} + \bar{\zeta} $$

with $\theta$ and $\bar{\theta}$ denoting the additional Grassmanian coordinates.

The supersymmetry generators or supercharges, when written down as differential operators, contain momentum operators in both, the "usual" even spacetime coordinates and the odd Grassmann coordinates

$$ Q_a = i\partial_a -\frac{1}{2}(\sigma^{\mu})_{a\dot{b}}\bar{\theta}^{\dot{b}}\partial_{\mu}$$

$$ \bar{Q}^{\dot{a}} = i\bar{\partial}^{\dot{a}} -\frac{1}{2}(\bar{\sigma}^{\mu})^{\dot{a}b}\theta_b\partial_{\mu}$$

Where $\partial_a = \frac{\partial}{\partial\theta^a}$ and $\bar{\partial}^{\dot{a}} = \frac{\partial}{\partial\bar{\theta_{\dot{a}}}}$ are the derivatives along the Grassmanian coordinates.

A nice and very readable introduction to the superspace formalism can for example be found in Ch 11 of this book.

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There are two types of ; worldsheet supersymmetry, and spacetime supersymmetry

Worldsheet supersymmetry

The Ramond-Neveu-Schwarz formalism has explicit worldsheet supersymmetry. Since the RNS Action is given by adding the Polyakov Action to the Dirac action, it is given by:

$${{\mathsf{\mathcal{L}}}_ {RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not{\partial} \psi^\mu \right) g_{\mu\nu}$$

The supersymmetric transformations on the worldsheet can therefore be (almost trivially, by taking variations of this above action) shown to be:

$$\begin{gathered} \delta {X^\mu } \to \bar \epsilon {\psi ^\mu } ; \\ \delta {\psi ^\mu } \to - i \not \partial {X^\mu }\epsilon \\ \end{gathered} $$

Spacetime Supersymmetry

The Green-Schwarz formalism, or the , are with explicit spacetime supersymmetry. The supersymmetric transformations on spacetime are (which is rather intuitive if you compare this to the RNS Worldsheet supersymmetry transformations) given by:

$$\begin{gathered} \delta {\Theta ^{Aa}} \leftrightarrow {\varepsilon ^{Aa}} ; \\ \delta {X^\mu } \leftrightarrow {{\bar \varepsilon }^A}{\gamma ^\mu }{\Theta ^A} ; \\ \end{gathered} $$

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