Reparameterization and conformal transformation in SYK I have a simple question about SYK model.
For SYK in the IR limit, the Schwinger-Dyson equations have some so called rearamerization invariance
$$\psi_i(\tau)\rightarrow \psi_i(f(\tau))=f'(\tau)^{-\Delta}\psi_i(\tau)$$
I want to whether this transformation is related to conformal transformation,
since  I often heard that SYK has IR emergent conformal symmetry. But from my knowledge, the conformal trnasformation is not only related to dimension but also to spin which is not appear here.
 A: You are probably thinking about higher dimensional CFTs, e.g. 2d CFT. In 2d CFT, you have two dilation operators ($L_0$ and $\bar{L}_0$ from two copies of $SL_2$), acting on left and right. One can label the eigenvalues by e.g. $h$ and $\bar{h}$ and define spin and scaling dimension by recombining them. However in the conformal limit of the SYK model, only one copy of $SL_2$ is there, so you only have one eigenvalue to play with. 
A: The SYK model is a (0,1)-dim model. This means that because there is only a time dimension that there is no notion of spin. In one dimension there is also no notion of an angle so that every smooth transformation is conformal (Diff$(\mathbb{R})\cong$ Conf$(\mathbb{R})$). In the IR the solution to the Swinger-Dyson equations $G_c(\tau,\tau')$ spontaneously breaks the Diff$(\mathbb{R})$ symmetry down to a SL$(2,\mathbb{R})$ symmetry. This spontaneous symmetry breaking produces Goldstone modes and it can be shown that their dynamics are governed by an EFT described by a Schwarzian action. 
I can recommend a thesis by Eric Marcus where these properties of the SYK are discussed in quite some detail.
