What does St denote? I am reading a paper and notice that the term St is used without being defined. I suspect it is a common notation that I am not aware of. 

It is used in the following paper: 
https://eprint.iacr.org/2015/482.pdf

The first instance of it appears in Section 2.1, in the formula used in Definition 2. 

What does the St denote? 

Thanks
In this case it denotes arbitrary state passed from $A_0$ to $A_1$. This is a common idiom in interactive security game definitions. 

Conceptually a security definition should involve a single adversary who plays some game. The adversary is modeled as arbitrary algorithm. If there is some query/interaction adversary can do repeatedly, we often model this as giving oracle access to some function.

But suppose the game has a significant event that should happen only once (in this case, asking for and receiving a challenge ciphertext). This is not so easily modeled by providing oracle. Instead, it is common to formalize the situation like the authors have done here. $A_0$ is the part of the adversary that runs up until requesting the challenge ciphertext (it terminates by giving the two chosen plaintexts). $A_1$ is the part of the adversary that runs after receiving the ciphertext. To follow our intuition that there is just a "single adversary" this whole time, we allow $A_0$ to pass arbitrary information (without loss of generality, its entire internal state) to $A_1$. That's what $st$ is here.