# Holographic Field Theory

I am trying to read this paper http://arxiv.org/abs/1204.1780 and I don't understand how to get from eqn 91 which is,

$$S_{2} = N^{2} \{V[P^{(1)}_{m}] + (J^{(1)m} - \mathcal{J}^{m})P_{m}^{(1)}\} -NJ^{(1)2'} tr(\phi^{2}) - N m^{2} tr(\tilde{\phi^{2}}) - N \sum_{n=3}^{\infty}tr(\phi + \tilde{\phi})^{n}).$$

Here $\phi$ is a low energy mode and $\tilde{\phi}$ is the high energy modes. The partition function is

$$Z = \int d\phi d\tilde{\phi} dJ^{(1)n}dP_{n}^{(1)} e^{iS_{2}}.$$

I am interested to remove the high energy fields $\tilde{\phi}$ to obtain a low energy effective action for $\phi$. Which is given in the eqns 92 and 93.

But this integral is cannot be evaluated, so if someone knows a technique/method, it would great.

• Similar derivation represented in article "Holographic description of quantum field theory" arxiv.org/abs/0912.5223 , also by Sung-Sik Lee, in Appendix A – sigrlami Jun 20 '13 at 16:59