Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) connection $\omega$. This lets us describe a Riemannian metric on $M$ using a vector bundle map

$e: TM\rightarrow T$

We have that

$\omega\in so(4)$ (it's a $SO(4)$ connection) and $e\in \mathbb{R}^4$.

We can introduce now a $SO(5)$ connection $A$ such that

$A=\omega+e$ $\hspace{1.0cm}$(because $so(5)=so(4)\oplus \mathbb{R}^4$)

My question is: if I introduce a spin structure (spin bundle), can I define the following Dirac operator:

$D=\gamma^\mu(\partial_\mu+A_\mu)$ ?

Can I consider $A$ as a "pure" (gauge) connection as in the standard case?

  • $\begingroup$ In order for the isomorphism e : TM -> T to define a Riemannian metric you need to require in addition that the structure group of T is not all of GL(n), but is O(n). (Otherwise the condition is empty: just choose T := TM !) In other words, a Riemannian metric, is given by a smooth reduction of the structure group of the tangent bundle along O(n) --> GL(n) and any such is a vielbein. $\endgroup$ – Urs Schreiber Jul 4 '12 at 14:23

You have to be more precise. $A$ is an SO(5) connection but on what bundle? This is a formula for the Dirac operator which depends on local trivialization. I doubt that this defines a global operator in general (this works for a spin or $spin^c$ structures on TM and doesn't have to work on spin structure on every vetor bundle).

I will recommend you chapters "spin structures on vector bundles" and "dirac opetors" form book D.Salamon,Spin geometry and Seiberg-Witten invariants.

Hope it helped a little.

  • $\begingroup$ Hi! I would like define this Dirac operator only locally on $TM$. Thank you for your suggestion! $\endgroup$ – Gian Jun 20 '12 at 7:43
  • $\begingroup$ I have just another question: if now we consider $D=e^a\gamma_a^\mu(\partial+A)$, can I consider $A$ as an extended spin connection respect to $\omega$? $\endgroup$ – Gian Jun 20 '12 at 21:29
  • $\begingroup$ You need an SO(5)-principal bundle which admits a lift to a Spin(5)-principal bundle and then consider your Dirac operator on that. What remains is that you need to decide how you want that Spin(5)-bundle to be related to the original tangent bundle. Probably you want them to be related by reduction/extension? $\endgroup$ – Urs Schreiber Jul 4 '12 at 14:29

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