Conformal Killing spinors

Question

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).

If $\epsilon$ and the $\bar{\epsilon}$ are the holomorphic and the anti-holomorphic conformal Killing spinors then apparently the following are true,

• In a conformally flat background it satisfies $D_\mu \epsilon = \gamma _ \mu \kappa$ for arbitrary spinor $\kappa$ and on a flat background it is identically sastified by $\epsilon = \xi_1 - x^\mu \gamma _\mu \xi_2$ for $\xi_1, \xi_2$ being arbitrary spinors.

• On a $S^3$ with radius $r$ there are apparently $4$ independent anti-holomorphic ones (how?)and they split into $2$ satisfying $D_\mu \bar{\epsilon} = \pm \frac{i}{2r} \gamma_\mu \bar{\epsilon}$

• If the $S^3$ above is replaced by $S^2 \times \mathbb{R}$ then apparently the equation changes into $D_\mu \bar{\epsilon} = \pm \frac{1}{2r} \gamma_\mu \gamma_3 \bar{\epsilon}$where $r$ is now the radius of $S^2$.

I would be glad if someone can explain or give a reference which has a quick explanation for this.

Also I want to know how the representations of the isometry group of $S^2$ i.e $S0(3)$ somehow help "classify" the solutions. (like I think the claim is that each of the set of two actually lies in a two dimensional irreducible representation of $SO(3)$). I would like to know of the explanation for this classification and if there is a general representation theory idea which works always (like also may be in the first two cases?)

Answer 1

(By the way, the use of "holomorphic" and "antiholomorphic" is wrong in your question. This may be confusing people, which perhaps explains why nobody has answered this question yet.)

Conformal Killing spinors are also known as twistor spinors in the mathematical literature. There is quite a lot of it, mostly by Helga Baum and collaborators in Berlin.

If you take your equation $D_\mu \varepsilon = \gamma_\mu \kappa$ and you hit it with $\gamma^\mu$, you get depending on your conventions that $\kappa = -\frac1{n} D\varepsilon$ in $n$ dimensions and where $D$ is the Dirac operator. In other words, $\varepsilon$ obeys $$ P_\mu \varepsilon := D_\mu \varepsilon + \frac1{n} \gamma_\mu D\varepsilon = 0 $$ where $P_\mu$ is the so-called Penrose operator: a sort of complement of the Dirac operator. (Notice that $\gamma^\mu P_\mu \eta= 0$ for any spinor field $\eta$, which characterises $P_\mu$.) The Penrose operator behaves well under conformal transformations and if two manifolds are conformally related, the kernel of the Penrose operator will have the same dimension, for instance. In particular, if you are looking at conformally flat manifolds, the twistor spinors are essentially given in terms of the flat twistor spinors via rescaling by the appropriate power of the conformal factor.

A special case of twistor spinor are the so-called Killing spinors, which obey the equation $$D_\mu \varepsilon = \lambda \gamma_\mu \varepsilon$$ for some appropriate constant $\lambda$, whose square is related to the scalar curvature of the manifold. (You can see this easily by hitting the above equation again with $D_\nu$ and skewsymmetrising to get an expression for the curvature acting on $\varepsilon$, which can then $\gamma$-trace to get an equation for the scalar curvature in terms of $\lambda$.) Because $\lambda^2$ is real (being proportional to the scalar curvature), you have two cases: $\lambda$ real or $\lambda$ pure imaginary. This gives rise to two different types of geometries.

The case of real Killing spinors (by which one means $\lambda$ real) in riemannian signature was solved by Christian Bär via his celebrated cone construction, whereas the case of imaginary Killing spinors was solved by Baum. In indefinite signature there are many partial results and the classification problem is still open.

Bär's cone construction says that real Killing spinors (with $\lambda = \pm \frac12$, which you can always achieve via a homothety) on a riemannian spin manifold $(M,g)$ are in one-to-one correspondence with parallel spinors on the metric cone $(\hat M, \hat g)$, where $\hat M = M \times \mathbb{R}^+$ and $\hat g = dr^2 + r^2 g$. Furthermore one can show that this correspondence is equivariance under the natural actions of the isometry group of $(M,g)$. In particular, for the case of the round $S^n$ (whose spinor bundle can be trivialised by real Killing spinors of either sign of the Killing constant $\lambda$), the cone construction says they are in one-to-one correspondence to parallel spinors on the euclidean space $\mathbb{R}^{n+1}$ and this correspondence is equivariant under the action of $\operatorname{Spin}(n+1)$ acting (not faithfully) by isometries on $S^n$ and linearly on $\mathbb{R}^{n+1}$. Relative to flat coordinates and to the associated frame, parallel spinors on $\mathbb{R}^{n+1}$ are actually constant and the action of $\operatorname{Spin}(n+1)$ on the parallel spinors is precisely the relevant spin representation. (I am purposefully vague on what "relevant" means, since there is question of whether $n$ is even or odd, in which the sign of the Killing constant might end up becoming the chirality of the parallel spinor in the euclidean space.)

The book contains quite a lot of basic results and definitions:

@Book{BFGK,
  author =   {H Baum and T Friedrich and R Grunewald and I Kath},
  title =  {Twistor and {K}illing spinors on riemannian
      manifolds},
  publisher =    {Humboldt-Universität},
  year =     {1990},
  number =   {108},
  series =   {Seminarberichte},
  address =  {Berlin}
 }

whereas Bär's original paper is

@Article{Baer,
  author =   {C B{\"a}r},
  title =    {Real {K}illing spinors and holonomy},
  journal =  Comm. Math. Phys.,
  year =     1993,
  volume =   154,
  pages =    {509--521}
}

and you may want to look at this paper of mine for a more Physics-friendly description of this story.