### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 062, 41 pages      arXiv:2012.13389      https://doi.org/10.3842/SIGMA.2022.062
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

### Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

Claudio Meneses
Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany

Received September 21, 2021, in final form July 28, 2022; Published online August 13, 2022

Abstract
We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.

Key words: parabolic Higgs bundle; Hitchin fibration; nilpotent cone.

pdf (1416 kb)   tex (1112 kb)

References

1. Bauer S., Parabolic bundles, elliptic surfaces and ${\rm SU}(2)$-representation spaces of genus zero Fuchsian groups, Math. Ann. 290 (1991), 509-526.
2. Beauville A., Narasimhan M.S., Ramanan S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169-179.
3. Bérczi G., Doran B., Hawes T., Kirwan F., Geometric invariant theory for graded unipotent groups and applications, J. Topol. 11 (2018), 826-855, arXiv:1601.00340.
4. Bérczi G., Jackson J., Kirwan F., Variation of non-reductive geometric invariant theory, in Surveys in Differential Geometry 2017. Celebrating the 50th Anniversary of the Journal of Differential Geometry, Surv. Differ. Geom., Vol. 22, Int. Press, Somerville, MA, 2018, 49-69, arXiv:1712.02576.
5. Biswas I., A criterion for the existence of a parabolic stable bundle of rank two over the projective line, Internat. J. Math. 9 (1998), 523-533.
6. Blaavand J.L., The Dirac-Higgs bundle, Ph.D. Thesis, University of Oxford, 2015.
7. Fredrickson L., Mazzeo R., Swoboda J., Weiss H., Asymptotic geometry of the moduli space of parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles, Proc. London Math. Soc., to appear, arXiv:2001.03682.
8. Gaiotto D., Moore G.W., Neitzke A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239-403, arXiv:0907.3987.
9. Godinho L., Mandini A., Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles, Adv. Math. 244 (2013), 465-532, arXiv:1101.3241.
10. Godinho L., Mandini A., Quasi-parabolic Higgs bundles and null hyperpolygon spaces, Trans. Amer. Math. Soc. 374 (2021), 7411-7447, arXiv:1907.01937.
11. Gothen P.B., Oliveira A.G., Topological mirror symmetry for parabolic Higgs bundles, J. Geom. Phys. 137 (2019), 7-34, arXiv:1707.08536.
12. Hamilton E., Stratifications and quasi-projective coarse moduli spaces for the stack of Higgs bundles, arXiv:1911.13194.
13. Hausel T., Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998), 169-192, arXiv:math.AG/9804083.
14. Heller L., Heller S., Abelianization of Fuchsian systems on a 4-punctured sphere and applications, J. Symplectic Geom. 14 (2016), 1059-1088, arXiv:1404.7707.
15. Heu V., Loray F., Flat rank two vector bundles on genus two curves, Mem. Amer. Math. Soc. 259 (2019), v+103 pages, arXiv:1401.2449.
16. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
17. Kim S., Wilkin G., Analytic convergence of harmonic metrics for parabolic Higgs bundles, J. Geom. Phys. 127 (2018), 55-67, arXiv:1705.08065.
18. Komyo A., Saito M.H., Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces, Kyoto J. Math. 59 (2019), 515-552, arXiv:1611.00971.
19. Loray F., Saito M.H., Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line, Int. Math. Res. Not. 2015 (2015), 995-1043, arXiv:1302.4113.
20. Loray F., Saito M.H., Simpson C.T., Foliations on the moduli space of rank two connections on the projective line minus four points, in Geometric and Differential Galois Theories, Sémin. Congr., Vol. 27, Soc. Math. France, Paris, 2013, 117-170, arXiv:1012.3612.
21. Mehta V.B., Seshadri C.S., Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205-239.
22. Meneses C., Remarks on groups of bundle automorphisms over the Riemann sphere, Geom. Dedicata 196 (2018), 63-90, arXiv:1607.03865.
23. Meneses C., Geometric models for moduli of rank 2 parabolic Higgs bundles in genus 0 and applications, in preparation.
24. Meneses C., Takhtajan L.A., Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere, Comm. Math. Phys. 387 (2021), 649-680, arXiv:1407.6752.
25. Miranda R., The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, 1989.
26. Mukai S., An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, Vol. 81, Cambridge University Press, Cambridge, 2003.
27. Nakajima H., Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, in Moduli of Vector Bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Appl. Math., Vol. 179, Dekker, New York, 1996, 199-208.
28. Rayan S., The quiver at the bottom of the twisted nilpotent cone on $\mathbb{P}^1$, Eur. J. Math. 3 (2017), 1-21, arXiv:1609.08226.
29. Rayan S., Schaposnik L.P., Moduli spaces of generalized hyperpolygons, Q. J. Math. 72 (2021), 137-161, arXiv:2001.06911.
30. Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-770.
31. Simpson C.T., Products of matrices, in Differential Geometry, Global Analysis, and Topology (Halifax, NS, 1990), CMS Conf. Proc., Vol. 12, Amer. Math. Soc., Providence, RI, 1991, 157-185.
32. Thaddeus M., Variation of moduli of parabolic Higgs bundles, J. Reine Angew. Math. 547 (2002), 1-14, arXiv:math.AG/0003222.