# CFT Entanglement Entropy - relation between translations and the stress-energy tensor

In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric:

$$ds^2_{H_q^{d-1}} = d\tau^2 + du^2 + \sinh^2 u \; d\Omega_{d-2}^2$$

Here $d\tau$ is probably a Wick-rotated time, $u$ is a radial variable and $d\Omega$ is the spherical area measure.

Then they define a partition function $Z_q = \mathrm{tr}(e^{-2\pi q H_\tau})$ where $H_\tau$ "generates translations along the $S^1$". What does that mean? Could it mean this?

$$H_\tau = \frac{d}{d\tau}$$

This is the generator for translations along the $\tau$-axis.

However, they also say this is related to the stress-energy tensor: $H_\tau = \int_{\mathbb{H}^{d-1}} dx^{d-1} \sqrt{g} T_{\tau\tau}$ This seems like a very complicated way of describing translations. Could there be another meaning for the phrase "generates translations along $S^1$?

1. Jeongseog Lee, Aitor Lewkowycz, Eric Perlmutter, Benjamin R. Safdi
Renyi entropy, stationarity, and entanglement of the conformal scalar

Everything is standard here. Think simply to a classical field on a Minkowski space $M = R_t \times R^3$.
The time translation generator, the hamiltonian, is defined by $H_t = P_0=\int T_{0i} d\sigma^i$, where $d\sigma^i = \epsilon^{ijkl} dx^j \wedge dx^k \wedge dx^l$.
Now, we may, in fact, consider only the $d\sigma^o= d\sigma^t$ component which is equals to $dx^1 \wedge dx^2 \wedge dx^3$, so finally :
$H_t = \int d^3xT_{00} = \int d^3xT_{tt}$