I just stumbled across a presentation by Tachikawa about "What is Quantum Field Theory". He has an interesting perspective that we should think of (at least a subset of) quantum field theories as being organized by
- 1-ality $\implies $ A theory with a unique Lagrangian
- 0-ality $\implies$ A non-Lagrangian QFT
- 2-ality $\implies$ Two seemingly distinct Lagrangians that generate the same list of correlation functions
- N-ality $\implies$ There are $N$ distinct Lagrangians that give rise to the same correlators.
I can think of specific examples of $N=0,1,2$, but I'm not familiar with examples of n-ality (defined in the above sense) with $N\neq 0,1,2$. I suppose my question is:
- Are there concrete examples for any other $N$?
- For arbitrary $N$?
- Can there be a countably infinite number?
- A continuum?
I know that certain string compactifications on $T^d$ have, for example, a large T-duality group such as $O(d,d;\mathbb{Z})$, but I'm not sure how that should thought of in the above (proposed) classification scheme. Does it even fit in this way of organizing things? Basically, I just want some help understanding if "N-ality" partitions the space of QFTs in a useful way, or if it was just a cute trick to describe non-Lagrangian QFTs.
I have in mind that we are, e.g., defining a Lagrangian as a top-form so issues of total derivatives are excluded. I'm also not including integrating out a field as being genuinely different because the partition functions are the same in an obvious way. There might be some other "trivial" equivalences I'm not thinking of at the moment, but my current understanding is that there's something genuinely different about N-ality from these kinds of examples I just mentioned. Finally, I'm not as interested in duality in the sense of IR-duality, as I think I already understand that case via universality classes. Maybe this last paragraph just shows I'm being naive, but I would be interested to learn more if that's the case.
