I was working on my thesis on Conformal Bootstraps. For that, I now have to use the inversion formula to get anomalous dimensions in the $\phi^3$ theory. Can anyone suggest any good reference(s) to learn how to use Inversion Formula in CFT?


Probably you are already familiar with the reviews of conformal bootstrap, but this is the standard list

  1. David Simmons Duffin's TASI lectures 1602.07982
  2. Slava Rychkov's EPFL lectures 1601.05000
  3. Recent numerical bootstrap review 1805.04405

These links don't discuss the inversion formula (except sec IX of 3. very briefly), but the notations and all basic facts about CFTs are in there, so it's important to at least have them in mind. Now for the inversion formula papers:

  1. Original paper by Caron Huot 1703.00278
  2. Formal proof by Witten Stanford and Simmons Duffin 1711.03816
  3. Version in Mellin space 1803.05086
  4. Generalization to arbitrary spin 1805.00098
  5. Explicit results for OPE coefficients in Mean Free Theory 1809.05111

A similar story was developed for CFTs at finite temperature

  1. Seminal paper 1802.10266
  2. Application to Ising 1811.05451

The inversion formula can be applied to the $1/J$ perturbation theory, a.k.a. "Lightcone bootstrap" or "Analytic bootstrap". This subject was developed before the formula had been discovered, so in these papers there is a different language.

  1. Seminal (independent) papers 1212.4103, 1212.3616
  2. Application to 3d Ising 1612.08471
  3. Many other refs. for which I refer you to the intro of 7.
  4. Wilson Fisher with lightcone bootstrap 1712.02314

I emphasize the last one because you will probably need similar techniques if you are studying this $\phi^3$ theory with a sort of $\varepsilon$ expansion.


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