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 :

$H_t = \int d^3xT_{00} = \int d^3xT_{tt}$

  • $\begingroup$ I think what I am missing is that you still need an action to define the CFT, and your equation computes the time evolution Hamiltonian from this action. $\endgroup$ – john mangual Aug 1 '14 at 13:34

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