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Nonabelianization of Higgs bundles. (English) Zbl 1309.14029

Let \(\Sigma\) be a compact Riemann surface of genus \(g\geq 1\) and \(G^c\) a complex reductive Lie group. A Higgs bundle for the group \(G^c\) is a pair \((E,\Phi)\) where \(E\) is a holomorphic principal \(G^c\)-bundle and \(\Phi\) is a section of \(\mathrm{ad}(E)\otimes K\) – the adjoint bundle, twisted by the canonical line bundle \(K\) of \(\Sigma\). It is well known that the moduli space \(\mathcal M_{G^c}\) of Higgs bundles, over \(\Sigma\), for the group \(G^c\), is a completely integrable Hamiltonian system. In other words, there is a proper map \[ h:\mathcal M_{G^c}\to\mathcal A, \] where \(\mathcal A\) is an affine space of half of the dimension of \(\mathcal M_{G^c}\) whose generic fiber is an abelian variety. This is the so-called Hitchin system (and \(h\) is called the Hitchin map). More precisely, since \(\mathrm{ad}(E)\) is a bundle of Lie algebras, isomorphic to \(\mathfrak{g}\), and if \(p\) is an invariant homogeneous polynomial on \(\mathfrak{g}\) of degree \(d\), it makes sense to evaluate \(p\) at \(\Phi\) and \(p(\Phi)\in H^0(\Sigma, K^d)\). So, by taking a basis \(p_1,\ldots,p_k\) for the ring of invariant homogeneous polynomials on \(\mathfrak{g}\), then \(\mathcal A=\bigoplus_{i=1}^kH^0(\Sigma,K^i)\) and \(h(E,\Phi)\) is given by evaluating all these \(p_i\) at \(\Phi\).
For \(G^c=\mathrm{GL}(n,\mathbb{C})\), a Higgs bundle is given by a rank \(n\) vector bundle \(V\) and by a section \(\Phi\) of \(\mathrm{End}(V)\otimes K\). The Hitchin map takes \((V,\Phi)\) to the coefficients of the characteristic polynomial of \(\Phi\) and the generic fiber is the Jacobian of the so-called spectral curve. In a little more detail, given a generic element \(s=(s_1,\ldots,s_n) \in\bigoplus_{i=1}^nH^0(\Sigma,K^i)\) the spectral curve \(S\) corresponding to \(s\) is a curve inside the total space of \(K\), defined by \[ x^n+\pi^*s_1x^{n-1}+\cdots+\pi^*s_n=0, \] where \(\pi:K\to X\) is the projection and \(x\) is the tautological section of \(\pi^*K\). In other words, \(S\) is defined by the equation \(\det(xI-\Phi)=0\) where \(h(V,\Phi)=s\). Then \(h^{-1}(s)\cong\mathrm{Jac}(S)\) basically because any stable \((V,\Phi)\in h^{-1}(s)\) can be uniquely written as \((\pi_*L,\pi_*x)\) with \(L\in\mathrm{Jac}(S)\). If instead one considers \(G^c=\mathrm{SL}(n,\mathbb{C})\), then only line bundles on the Prym variety of \(S\) are allowed so that the determinant of \(V\) is trivial, so the fiber is \(\mathrm{Prym}(S)\). See [N. Hitchin, Duke Math. J. 54, 91–114 (1987; Zbl 0627.14024)] for other examples of the Hitchin system for other complex groups.
This abelianization process plays a central role in many important aspects of Higgs bundle theory, not only on the integrable systems side, but also, for example, on the study of special representations of surface groups (known as Hitchin representations) or on the study of the Langlands duality phenomena.
The notion of Higgs bundles generalizes however to any real reductive Lie group, and the present paper addresses the question of describing the Hitchin system for the real Lie groups \(\mathrm{SL}(m, \mathbb{H})\), \(\mathrm{SO}(2m,\mathbb{H})\) and \(\mathrm{Sp}(m,m)\). In doing so, the authors discover that the above-mentioned abelianization process does not hold anymore for these real forms, in the sense that the generic fiber of the Hitchin map is now described, not in terms of line bundles over the spectral curve, but in terms of rank 2 vector bundles over that curve (or quotients of it). More precisely, the authors prove that for \(\mathrm{SL}(m, \mathbb{H})\), the fiber consists of the moduli space of semi-stable rank \(2\) vector bundles with fixed determinant on the spectral curve \(S\); for \(\mathrm{SO}(2m,\mathbb{H})\), the fiber has several components, each of which is a moduli space of semi-stable rank \(2\) bundles on a quotient \(\overline S\) of the spectral curve; for \(\mathrm{Sp}(m,m)\) it is a \(\mathbb{Z}_2\)-quotient of a moduli space of semi-stable rank \(2\) parabolic bundles on \(\overline S\).

MSC:

14H70 Relationships between algebraic curves and integrable systems
53C10 \(G\)-structures
14H60 Vector bundles on curves and their moduli
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

Citations:

Zbl 0627.14024
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