Edward Frenkel

A centre that is a dual algebra

An affine Kac–Moody algebra &ghat; has a distinguished level, the critical level k = −h^∨, where the Sugawara construction of a Virasoro action breaks down because its normalisation 1/(2(k + h^∨)) is undefined. What replaces the Virasoro algebra there is something larger: the centre of the vertex algebra, which Feigin and Frenkel identified with a classical W-algebra — and the W-algebra is that of the Langlands dual algebra ^Lg, not of g itself. This is the Feigin–Frenkel centre. ch-feigin carries the statement and computes the data it is stated in terms of; this chapter is about the man who carried it into the Langlands program.

The route is local. Opers on a punctured disc — connections of a special form — are the spectrum of that centre, and they are local systems for the dual group, which is the Langlands parameter. Frenkel's 2007 Cambridge book, Langlands Correspondence for Loop Groups, and his later survey work set the centre inside the program, and the opers are the same ones Beilinson and Drinfeld used globally; that chapter is the whole-curve end of this disc-level construction.

A necessary condition, in sixteen types

ch-feigin says plainly that the isomorphism is not checked there, not even for sl₂. It is not checked here either. What is checked is something the isomorphism forces: two vertex algebras that are isomorphic have the same central charge. The W-algebra of g at level k has, with t = k + h^∨,

c_g(t)  =  l  -  12 |rho - t rho^v|^2 / t          (l = rank)

and the duality is r(k+h^∨)(k′ + ^Lh^∨) = 1, where r is the lacing number (1 for simply-laced types, 2 for B, C, F₄, 3 for G₂). So the necessary condition is c_g(t) = c_^Lg(1/(rt)). ch-frenkel-verify.py computes both sides from Cartan matrices alone, in exact rational arithmetic, and compares them at twelve values of t in each of sixteen types. They agree in every one.

The formula's inner product is first tested against the Freudenthal–de Vries strange formula, |ρ|² = h^∨ dim g / 12, in all sixteen types. The formula as a whole is anchored to values it did not produce: at t = 4/3 or 3/4 the Virasoro case gives c = 1/2 (the Ising model); at 5/4, 7/10 (the tricritical Ising model); and the W₃ case at 5/4 gives 4/5 (the three-state Potts model). At t = 1 the simply-laced types all give c = rank; the non-simply-laced ones do not, and the script checks that they don't, because there ρ ≠ ρ^∨ and the fixed point of t → 1/(rt) is elsewhere.

The controls are the part that earns the rest. If the lacing number is forgotten and the dual level is taken to be 1/t, the duality fails in all seven non-simply-laced types tested: so the check can see the factor r. B₃, B₄, C₃ and C₄ have different central charges at the same t, so the check can tell an algebra from its dual; and B₂ (which equals C₂), F₄ and G₂, which are self-dual, have equal ones, which they must.

Rung thirty-two, fourth occupant

This is the chapter the slate was designed to end on, because it is the one that reaches back. ch-feigin already stands on the vertex-algebra route that WP-82's correction of 2026-09-11 retained. What this chapter adds is the man, the dual-level arithmetic, and the connection from the disc to the curve. It adds no construction. WP-130 counted the programme's mathematicians at zero before the slate began. With Langlands, Drinfeld & Beilinson and the provers, they now have four chapters. The floor is a gallery floor: names, dates, one checkable object each, and no claim to build.