Hi I am wondering what the first published paper on Virasoro modes was? And what about Virasoro algebra?
There is a dedicated SE on the history of science where such questions are better addressed, especially when there is no clear answer.
In that site you'd probably get the mathematicians' narrative, so, then
- I. M. Gel'fandD. B. Fuks,
Functional Analysis and Its Applications ,
October 1968, Volume 2, Issue 4, pp 342–343
"The cohomologies of the lie algebra of the vector fields in a circle", translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 2, No. 4, pp. 92–93, October–December, 1968.
However, you wouldn't understand much of the mathematese. Virasoro's Wisconsin paper has the modes, but not the algebra,
- M. A. Virasoro, Phys. Rev. D1, 2933 – Published 15 May 1970, "Subsidiary Conditions and Ghosts in Dual-Resonance Models".
You will first see the algebra in its modern explicit form (albeit without the center!) in
- S Fubini, G Veneziano, Annals of Physics Volume 63, Issue 1, March 1971, Pages 12-27, "Algebraic treatment of subsidiary conditions in dual resonance models"
in eqn (3.4), so in virtually the modern form. In proof (!), on p 27, they credit with J H Weis with the center, which, however, they dismiss as immaterial, as do most. Within a year, however, it is identified as useful and, indeed, crucial,
- RC Brower, & CB Thorn, (1971), "Eliminating spurious states from the dual resonance model", Nuclear Physics B31 (1), 163-182 .
If you are interested in the "big moment", this is it. And by 1972, everyone is using it, cf. https://doi.org/10.1016/0370-2693(72)90420-0 .
- P Goddard & CB Thorn, Physics Letters B 40, Issue 2, 26 June 1972, Pages 235-238, "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model"
A tangled tale, and what you probably learn from it is not to ask for it...