# A Conformal gauge theory

1. Aug 9, 2016

### shereen1

Dear all
I am trying to prove that the action resulting from studying conformal gauge theory is invariant under SO(2,4)*diff. Can anyone give me a hint to start from thank. I considering several papers: E.A.Ivanov and J.Niederie and others...

2. Aug 9, 2016

### ohwilleke

What kind of foundation of knowledge do you have already? Can you sketch the reasons that you think this hypothesis might be true and provide links to the papers you've looked at?

3. Aug 10, 2016

### haushofer

You mean N=4 SYM and its invariance under the superconformal algebra?

4. Aug 10, 2016

### shereen1

Hello
In fact i have studied poincare gauge theory and i deduced that the resulted action is the same as the einstein hilbert one. So currently i am trying to study conformal gauge theory.
I am using: E.A.Ivanov and J. Niederle paper
Thank you

5. Aug 10, 2016

### shereen1

no i mean gauging conformal group and studying the resulting action if invariant under diffeomorphism group. Do you have any idea about a refernce that could help me in proving this?
Thank you

6. Aug 10, 2016

### haushofer

You mean superconformal tensor calculus then, I guess. The SUGRA-book by Van Proeyen and his online lecture notes treat this in great detail.

7. Sep 7, 2016

### samalkhaiat

By construction, the resulting action is invariant under diffeomorphsim, just like the Yang-Mills action on curved spacetime. Collect the 15 infinitesimal generators of $SO(2,4)$ as $$J_{A} = \{ P_{a} , M_{ab}, K_{a}, D \} ,$$ where the index $a = 0,1,2,3$ is raised by the inverse Minkowski metric $\eta^{ab}$, then rewrite the Lie algebra $so(2,4)$ in the standard form $$[J_{A},J_{B}] = C_{AB}{}^{C}J_{C} .$$ The Cartan-Killing metric on $so(2,4)$ is given in terms of the structure constants as $$G_{AB} = C_{AE}{}^{D} C_{BD}{}^{E} .$$
In the basis $J_{A} , \ \ A = 1,2, \cdots , 15$ , define an $so(2,4)$-valued connection $$\mathbb{A}_{\mu}(x) = A^{C}_{\mu}(x) J_{C} \equiv e^{a}_{\mu}(x) P_{a} + \omega^{ab}_{\mu}(x) M_{ab} + c^{a}_{\mu}(x) K_{a} + \alpha_{\mu}(x) D .$$ The components of the field tensor $\mathbb{F}_{\mu\nu} = F_{\mu\nu}^{C}J_{C}$ are given as usual by $$F^{C}_{\mu\nu} = \partial_{\mu}A_{\nu}^{C} - \partial_{\nu}A_{\mu}^{C} + C_{BD}{}^{C} A_{\mu}^{B}A_{\nu}^{D} .$$ Now you can write down the following diffeomorphsim-invariant action
$$S = - \frac{1}{2 \alpha^{2}_{YM}} \int d^{4}x \ \sqrt{-g} \ g^{\mu\rho}g^{\nu\sigma} \ G_{AB} F^{A}_{\mu\nu}F^{B}_{\rho\sigma} .$$

8. Sep 7, 2016

### haushofer

One can also obtain the EH-action by writing down the action of a conformal scalar field and gaugefixing this field by a local dilation. See van proeyen his sugrabook.