A realization of the modular functor in the space of differentials and the geometric approximation of the moduli space of \(G\)-bundles
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 filtration is made possible by the central theorem of the present paper, which reduces the evaluation of coinvariants over \(L^{\text{out}}{\mathfrak g}\) to the case of the smaller algebra \(\hat{\mathfrak n}_+^{\text{out}}= \hat{\mathfrak n}_+\cap L^{\text{out}}{\mathfrak g}\). Since the space \(M(X)\) is finite-dimensional it is proved that \(M(X)\simeq H_0({\mathfrak h}\ltimes\hat{\mathfrak n}_+^{\text{out}}, \omega\cdot W)^*\) for \(\omega\) ``sufficiently large''. In particular, for \({\mathfrak g}= {\mathfrak{sl}}_2\), the modular functor is realized in the space of symmetric sections of a certain line bundle over the power of the curve \(X\). Also used is another realization \(W^*=H^0(M,{\mathcal L}_k)\), where \({\mathcal L}_k\) is the Borel-Weyl bundle on the flag manifold \({\mathcal F}\) of the Lie algebra \(\hat{\mathfrak g}\) and \(M\) is the closure of the orbit of the distinguished point in \({\mathcal F}\) under the action of the group \(\hat{\mathfrak N}_+\) corresponding to the Lie algebra \(\hat{\mathfrak n}_+\). The modular functor is described as the space of sections of a line bundle on the quotient variety \((\omega\cdot M)/T\ltimes\hat{\mathfrak N}_+\) for ``sufficiently large'' \(\omega\).
scientific article; zbMATH DE number 846550
scientific article