The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs \(V\)-bundles and components of parabolic \(\operatorname{Sp}(2n,\mathbb{R})\)-Higgs moduli spaces. (English) Zbl 1464.14016
The Beauville-Narasimhan-Ramanan (BNR) correspondence (Prop. 3.6 in [A. Beauville et al., J. Reine Angew. Math. 398, 169–179 (1989; Zbl 0666.14015)]) provides a very useful tool for studying the spectral curves constructed via the Hitchin fibration. In this paper the authors generalize the BNR correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs \(V\)-bundles. Using this correspondence along with Bott-Morse theoretic techniques they provide an exact component count for moduli spaces of maximal parabolic \(\operatorname{Sp}(2n,\mathbb{R})\)-Higgs bundles with fixed parabolic structure. This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. Section 2, deals with parabolic \(\operatorname{Sp}(2n,\mathbb{R})\)-Higgs bundles and their moduli. In this section, the authors review basic facts about moduli spaces of parabolic Higgs bundles and set notation. Section 3, deals with the Bott-Morse theory on the moduli space \(\mathcal{M}_{par}(2n,\mathbb{R})\). Section 4, deals with parabolic Higgs Bundles vs. Higgs \(V\)-Bundles. In this section the authors review the correspondence between parabolic Higgs bundles and Higgs \(V\)-bundles. Further details may be found in [G. Kydonakis et al., Math. Z. 297, No. 1–2, 585–632 (2021; Zbl 1457.14026)] and the references therein. Section 5, deals with the BNR correspondence. In this section, the authors study the BNR correspondence for twisted Higgs \(V\)-bundles and twisted parabolic Higgs bundles. Section 6, deals with the topological invariants of the moduli space of parabolic \(\operatorname{Sp}(2n,\mathbb{R})\)-Higgs bundles. Section 7, deals with connected components of \(M^{\max}_{par}(\operatorname{Sp}(2n,\mathbb{R}))\) and Section 8 with connected components of \(M^{\max}_{par}(\operatorname{Sp}(2n,\mathbb{R}))\) with fixed weight \(\frac{1}{2}\). Section 9, deals with the case of rational weights. Here the authors can analogously consider maximal \(\operatorname{Sp}(2n,\mathbb{R})\)-Higgs bundles for a more general choice of weights.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 55N32 | Orbifold cohomology |
| 14H40 | Jacobians, Prym varieties |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 14H70 | Relationships between algebraic curves and integrable systems |
Keywords:
Beauville-Narasimhan-Ramanan correspondence; twisted Higgs; parabolic Higgs bundles; moduli spacesReferences:
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