A realization of the modular functor in the space of differentials and the geometric approximation of the moduli space of \(G\)-bundles
type: Item
instance of: scholarly article
title: A realization of the modular functor in the space of differentials and the geometric approximation of the moduli space of \(G\)-bundles
zbMATH Open document ID: 0857.17021
DOI: 10.1007/BF01076110
publication date: 1996-02-25T00:00:00Z
review text: Let \(G\) be a complex semisimple simply connected Lie group, \({\mathfrak g}={\mathfrak n}_+\oplus{\mathfrak h}\oplus{\mathfrak n}_-\) the Lie algebra of it, and \(k\) a nonnegative integer. The modular functor (in the Wess-Zumino model associated with the group \(G\) at the level \(k\)) assigns a finite-dimensional complex vector space \(M(X)\) to each compact Riemann surface \(X\). For chosen small unit disk \(D\in X\) the loop Lie algebra \(L\mathfrak g\) of analytic maps \(\partial D\to{\mathfrak g}\) contains subalgebras \(L^{\text{in}}{\mathfrak g}\) and \(L^{\text{out}}{\mathfrak g}\) of the boundary values of holomorphic maps \(D\to{\mathfrak g}\) and \(X\setminus D\to{\mathfrak g}\), respectively. The vacuum representation \(V_k\) of the corresponding affine Lie algebra \(\hat{\mathfrak g}=L{\mathfrak g}+\mathbb C\cdot K\) is the irreducible factor of the module \(\text{Ind}^{\hat{\mathfrak g}}_{L^{\text{in}}{\mathfrak g}+\mathbb C\cdot K}\chi_k\) induced from character \(\chi_k\). The modular functor (the space of conformal blocks) can be defined as the dual space of the space of coinvariants \(H_0(L^{\text{out}}{\mathfrak g},V_k)\). In the present paper another construction of the modular functor is proposed. This construction is a globalization from the coordinate disk \(D\) to the Riemann surface of the construction exposed in the authors' previous paper [Funct. Anal. Appl. 28, No. 1, 55--72 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 68--90 (1994; Zbl 0905.17030)] for representations of the Lie algebra \(\hat{\mathfrak g}\). In that paper the space of the vacuum representation \(V_k\) is realized as the union of an increasing sequence of subspaces of the form \(\omega\cdot W\), where \(W=U(\hat{\mathfrak n}_+)v\simeq U(\hat{\mathfrak n}_+)/I\), \(v\) is the vacuum vector in \(V_k\), \(\omega\) is an element of the shift lattice of the affine Weyl group, and the annihilator \(I\) of \(v\) is computed explicitly. Application of this filtr…
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 846550
zbMATH Keywords: moduli space of \(G\)-bundles
zbMATH Keywords: vacuum representation of higher level
zbMATH Keywords: affine Lie algebra
zbMATH Keywords: loop Lie algebra of analytic maps
zbMATH Keywords: Wess-Zumino model
zbMATH Keywords: conformal blocks
zbMATH Keywords: modular functor
author: A. V. Stoyanovskii
author: B. Feigin
reviewed by: Yuri Bespalov
MaRDI profile type: Publication
cites work: Q3805429
cites work: Flat connections and geometric quantization
cites work: Q3754832
cites work: Structure of the Standard Modules for the Affine Lie Algebra 𝐴⁽¹⁾₁
cites work: Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space
cites work: Stable pairs, linear systems and the Verlinde formula
cites work: Generalized SU(2) theta functions
cites work: Q4740026
cites work: Fusion rules and modular transformations in 2D conformal field theory
full work available at URL: https://doi.org/10.1007/bf01076110
OpenAlex ID: W2050011257
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 846550