\(G\)-bundles over elliptic curves for non-simply laced Lie groups and configurations of lines in rational surfaces (Q1951203)

The paper under review is a continuation of previous work by the authors and N. C. Leung. We recall the definition of \textit{ADE-surface} from the work of \textit{N. C. Leung} and \textit{J. Zhang} [J. Lond. Math. Soc., II. Ser. 80, No. 3, 750--770 (2009; Zbl 1188.14025)]: this is a pair \((S,C)\), consisting of a smooth rational surface \(S\) and a smooth rational curve \(C\subset S\), satisfying two properties. First, \(D^2\geq -1\) for any smooth rational curve \(D\subset S\). Second, \(\langle K_S, C\rangle^\perp\subset \text{Pic}(S)\) is an irreducible root lattice of rank \(n=\text{rank }\text{Pic}(S)-2\). As shown in Proposition 8, ibid., the only lattices which can occur are of type \(E_n\), \(n\geq 4\) (if \(C^2=-1\)), \(D_n\), \(n\geq 3\) (if \(C^2=0\)) and \(A_n\), if \(C^2=1\). The \(E_n\)-surfaces are precisely the del Pezzo surfaces \(dP_n\), where for \(n\leq 3\) an \(E_n\)-surface is by definition a \(dP_n\). Leung and Zhang construct a natural bundle of Lie algebras (of the corresponding type) over every ADE-surface. Suppose that the anti-canonical linear system on \(S\) contains an elliptic curve \(\Sigma\in \left| -K_S\right|\). Let \(G\) be a compact, simple, simply-connected Lie group of ADE type. For such a group the character and weight lattices coincide, as do the coroot and cocharacter lattices. One can then identify the moduli space \(M_\Sigma^G\) of flat \(G\)-bundles on \(\Sigma\) with \(\text{Hom}(\Lambda,\Sigma)/W\), after fixing a \(d\)-th root of \(\mathcal{O}_\Sigma\). Here \(\Lambda\) is the root lattice, \(W\) the Weyl group, and \(d\) the order of the fundamental group of the corresponding ADE root system. We also note that by the Narasimhan-Seshadri theorem \(M_\Sigma^G\) can be identified with the coarse moduli space of semi-stable topologically trivial holomorphic \(G_\mathbb{C}\)-bundles on \(\Sigma\). Restricting the above bundle of Lie algebras to \(\Sigma\) then gives a morphism \(S(\Sigma,G)\to M_\Sigma^G\), where \(S(\Sigma,G)\) is the moduli space of pairs \((S,\Sigma\in\left|-K_S\right|)\). Then Theorem 1 of [Zbl 1188.14025] states, in particular, that this is an embedding with open dense image. This embedding identifies \(M_\Sigma^G\) with a natural compactification \(\overline{S(\Sigma,G)}\supset S(\Sigma,G)\), whose points parametrise pairs \((S,\Sigma)\), together with an extra structure, called \textit{\(G\)-configuration}. The main result of the present work is an extension of the above to non-simply laced groups \(G\neq F_4\). The authors prove in Theorem 1 that if \(G\) is simple, compact, simply-connected Lie group of type \(B_n\), \(C_n\) or \(G_2\), then \(S(\Sigma, G)\) embeds into \(M_\Sigma^G\) as an open dense subset. This embedding extends to an isomorphism \( \overline{S(\Sigma,G)}\simeq M_\Sigma^G\), where \( \overline{S(\Sigma,G)}\) is a compactification of \(S(\Sigma, G)\), parametrising rational surfaces with \(G\)-configurations. The starting point is the observation that for any such group there exists a canonically determined \textit{simply-laced} Lie subgroup \(G'\subset G\) of maximal rank, determined by the long roots of \(G\), and sharing with it a maximal torus. Non simply-laced groups have also been treated by \textit{N. C. Leung} and \textit{J. Zhang} in [Int. Math. Res. Not. 2009, No. 24, 4597--4625 (2009; Zbl 1222.14023)] by a different approach: embedding \(G\) as a subgroup of a simply-laced group \(G''\). This line of research is largely motivated by heterotic string/F-theory duality. For mathematical expositions and proofs we refer to \textit{E. Looijenga} [Invent. Math. 38, 17--32 (1976; Zbl 0358.17016)], \textit{R. Y. Donagi} [Asian J. Math. 1, No. 2, 214--223 (1997 Zbl 0927.14006)] and \textit{R. Friedman, J. Morgan} and \textit{E. Witten} [Commun. Math. Phys. 187, No. 3, 679--743 (1997; Zbl 0919.14010)].