1995/06/25 by I. Grojnowski, Grojnowski, I. · 4 citations
Mathematics · #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.alg-geom/9506020
This is the first in a series of papers which describe the action of an affine Lie algebra with central charge n on the moduli space of U(n)-instantons on a four manifold X. This generalises work of Nakajima, who considered the case when X is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (iė\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of U(1)-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of ch2, ch2 = (1)/(2) c1⋅ c1 - c2 becomes precisely the formula for the eigenvalue of the degree operator, iė\. the well known quadratic behaviour of affine Lie algebras.