I was looking at few papers on AdS$_2$/CFT$_1$. It seems to me that this is not as well studied as the higher dimensional dualities (e.g, AdS$_3$/CFT$_2$ or AdS$_5$/CFT$_4$). Can someone summarize what the actual difficulties are. For example, I have some related questions in this setup.
How to define a CFT$_1$? Is it just quantum mechanics with conformal invariance?
How to define central charge of CFT$_1$? What will be its physical meaning (like degrees of freedom, anomaly or Casimir energy)?
I came across the statement that AdS$_2$ is not stable by itself. One needs to have electric flux to make it stable. Why/how is it so? And if it is true then what is its analog in the dual CFT$_1$?
(If someone wishes to answer this, please feel free to discuss other subtleties related to AdS$_2$/CFT$_1$ .)