# SUSY Dirac operator

+ 2 like - 0 dislike
84 views
For a supermanifold, can we define a supersymmetric Dirac operator ?

+ 3 like - 0 dislike

Yes. In physics, we think of a Dirac operator $D$ as a first-order differential operator such that $D^2=\Delta$ is the Laplacian. (Mathematically, its an operator acting on the sections of a spinor bundle, see here for details). In the case of supersymmetry, the interesting question to ask is about the possible supersymmetric decompositions of the Dirac operator, i.e., decompositions that satisfy an appropriate SUSY algebra.

For the sake of pedagogy, consider $d=4$, where we have 2 SUSY decompositions of $D$, one of which is called chiral SUSY, and the other complex SUSY. Here I will consider only the first, since it's well-known and a little easier to construct. For the complex case cf. the ref. below.

Let $(\gamma^m)$ be our Clifford algebra generators (i.e., $\{\gamma^m,\gamma^n\}=2g^{mn}1$), $D_m=\partial_m-iA_m$ the covariant derivative for a vector potential $A_m$, and ${\not}D=\gamma^m D_m$. Then Dirac operator can be decomposed as ${\not}D = Q_+ + Q_-,$

where $Q_+=\frac{1}{2}(1+\gamma^5)\gamma^m D_m, \quad Q_-=\frac{1}{2}(1-\gamma^5)\gamma^m D_m.$

As an easy exercise, you can see that $Q_+$ and $Q_-$ are nilpotent operators, and letting $H=-(\gamma^m D_m)^2$, the usual $\mathcal{N}=1$ SUSY algebra is satisfied: $H = \{Q_+,Q_-\}, \quad [Q_+, H] = [Q_-,H]=0,$

as you learn in a SUSY QM course.

For details and a discussion of the complex SUSY case, cf.

answered Apr 4 by (500 points)

You are right, but I believe that it is not the object of my question. I didn't ask if the Dirac operator is supersymmetric; but I would like to have a Dirac operator for a supermanifold and not only for a spin-manifold. I think that we would have to study the spin supersymmetric Lie groups and the supersymmetric connections.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.