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Gauge theories depend on finite-dimensional symmetry groups like $SU(2)$. Is it possible to construct sensible gauge theories (or at least something similar in spirit) in QFT based on infinite dimensional Lie groups? If it is possible, is it possible to write down a Lagrangian for such a theory? If it is not possible, why not?

Martin C.
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    Definitely possible to write a Lagrangian (take a countably infinite-dimensional algebra such as the Affine Lie algebra etc. and just write the usual Yang-Mills Lagrangian). However, quantization becomes highly non-trivial. Probably related: the large $N$ limit of $SU(N)$ Yang-Mills. – Prof. Legolasov Jun 03 '21 at 12:43
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    similar question, no answer: https://physics.stackexchange.com/q/439087/84967 – AccidentalFourierTransform Jun 03 '21 at 16:14
  • SU(∞) Yang Mills is detailed in the last section. Recall the PB algebra is essentially SU(∞). – Cosmas Zachos Jun 03 '21 at 20:57
  • @CosmasZachos Is PB = 'Poisson Bracket' here? Also, since you appear to have written several papers on this topic, feel free to contribute an answer :) – Martin C. Jun 04 '21 at 07:36
  • Yes, the Poisson Bracket algebra is well-known to be su(∞) . Developing the formalism to describe infinite-dimensional matrices as a toroidal "color sheet" and defining the corresponding gauge theory is outlined in this question but you'll have to be more precise/limited in your question. – Cosmas Zachos Jun 04 '21 at 13:28
  • Thanks @CosmasZachos, I'll have a look at the linked question. – Martin C. Jun 04 '21 at 13:42

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