Timeline for Four-point function and OPE in 2d minimal models
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2022 at 8:01 | comment | added | Sylvain Ribault | Your summary is correct. If you discover a 3-point function that is forbidden by minimal model fusion rules, then it does not belong to a minimal model. You may then try to build another CFT with different fusion rules. (Do not think about actions: not every CFT has an action.) You can determine fusion rules from 4pt functions but not from 3pt functions. | |
| Dec 1, 2022 at 16:27 | comment | added | user31415926 | If so, then I would still be puzzled: what happens if one discovers 3-point functions forbidden by the fusion rules? Should one change the fusion rules to incorporate the discovery, or just proceed with the old rules? Or are both valid actions that lead to "different theories"? More generally, what is the rule of determining the fusion rules from studying the correlators? | |
| Dec 1, 2022 at 16:21 | comment | added | user31415926 | Ok this is quite different from what I previously believed. But let me try to summarize: the minimal model structure constants (as appeared in the fusion rules and OPEs) are different from the 3-point functions, where the latter is somewhat "more continuous", and the former is "more vanishing/discrete". Is this the correct summary of your point of view? | |
| Nov 30, 2022 at 10:13 | history | edited | Sylvain Ribault | CC BY-SA 4.0 |
Typo
|
| Nov 30, 2022 at 10:12 | comment | added | Sylvain Ribault | Indeed, the statement around (9.56) in the Big Yellow Book sounds natural but it is probably wrong in general. The statement of vanishing in Zamolodchikov (3.12) is basically a definition of the GMM structure constants: the question is to which extent these structure constants are values of the three-point function $C_M$, which is a smooth function of three variables. In (3.16) we see that $C_M$ does not vanish when we would like it to vanish, which is why we also need the prefactors $f$. | |
| Nov 29, 2022 at 17:13 | comment | added | user31415926 | Besides, eq (3.12) in Zamolodchikov seems to be saying that outside the triangle-relation (specified by 3-inequalities), the structure constant always vanishes (the "otherwise" scenario). This sort of matches my intuition as well: outside some range (or, when the fusion rule is violated), the structure constant automatically vanishes. Unless we are talking about different structure constant, since I notice that there's a continuous one, $C_M(\alpha, ...)$. | |
| Nov 29, 2022 at 17:02 | comment | added | user31415926 | But at eq (9.56) in the book says "This coefficient vanishes if the fusion rule is not allowed", which sounds very natural. Also in section 8.4 where the author discuss fusion rules, it seems that the structure constant $g(h_0, h_1, h) \ne 0$ leads to allowed fusion (8.78). Maybe the textbook statement should be taken with a grain of salt? | |
| Nov 29, 2022 at 9:13 | history | answered | Sylvain Ribault | CC BY-SA 4.0 |