Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

Ugo Bruzzo; Dimitri Markushevich; Alexander Tikhomirov

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1232-1245
  • ISSN: 2391-5455

Abstract

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Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).

How to cite

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Ugo Bruzzo, Dimitri Markushevich, and Alexander Tikhomirov. "Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3." Open Mathematics 10.4 (2012): 1232-1245. <http://eudml.org/doc/269108>.

@article{UgoBruzzo2012,
abstract = {Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).},
author = {Ugo Bruzzo, Dimitri Markushevich, Alexander Tikhomirov},
journal = {Open Mathematics},
keywords = {Vector bundles; Symplectic bundles; Instantons; Moduli space; vector bundles; symplectic bundles; instantons; moduli space},
language = {eng},
number = {4},
pages = {1232-1245},
title = {Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3},
url = {http://eudml.org/doc/269108},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Ugo Bruzzo
AU - Dimitri Markushevich
AU - Alexander Tikhomirov
TI - Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1232
EP - 1245
AB - Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).
LA - eng
KW - Vector bundles; Symplectic bundles; Instantons; Moduli space; vector bundles; symplectic bundles; instantons; moduli space
UR - http://eudml.org/doc/269108
ER -

References

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  1. [1] Atiyah M.F., Geometry of Yang-Mills Fields, Accademia Nazionle dei Lincei, Scuola Normale Superiore, Pisa, 1979 Zbl0435.58001
  2. [2] Atiyah M.F., Hitchin N.J., Drinfeld V.G., Manin Yu.I., Construction of instantons, Phys. Lett. A, 1978, 65(3), 185–187 http://dx.doi.org/10.1016/0375-9601(78)90141-X Zbl0424.14004
  3. [3] Atiyah M.F., Ward R.S., Instantons and algebraic geometry, Comm. Math. Phys., 1977, 55(2), 117–124 http://dx.doi.org/10.1007/BF01626514 Zbl0362.14004
  4. [4] Barth W., Lectures on mathematical instanton bundles, In: Gauge Theories: Fundamental Interactions and Rigorous Results, Poiana Brasov, August 25–September 7, 1981, Progr. Phys., 5, Birkhäuser, Boston, 1982, 177–206 
  5. [5] Barth W., Hulek K., Monads and moduli of vector bundles, Manuscripta Math., 1978, 25(4), 323–347 http://dx.doi.org/10.1007/BF01168047 Zbl0395.14007
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  7. [7] Horrocks G., Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc., 1964, 14, 684–713 Zbl0126.16801
  8. [8] Jardim M., Verbitsky M., Trihyperkähler reduction and instanton bundles on ℂℙ3, preprint available at http://arxiv.org/abs/1103.4431 Zbl06382611
  9. [9] McCarthy P.J., Rational parametrisation of normalised Stiefel manifolds, and explicit non-’t Hooft solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for Sp(n), Lett. Math. Phys., 1981, 5(4), 255–261 http://dx.doi.org/10.1007/BF00401473 Zbl0489.58038
  10. [10] Tikhomirov A.S., Moduli of mathematical instanton vector bundles with odd c 2 on projective space, Izv. Ross. Akad. Nauk Ser. Mat., 2012, 76(5) (in press, in Russian) 
  11. [11] Tjurin A.N., On the superposition of mathematical instantons. II, Progr. Math., 36, Birkhäuser, Boston, 1983 
  12. [12] Tyurin A.N., The structure of the variety of pairs of commuting pencils of symmetric matrices, Izv. Akad. Nauk SSSR Ser. Mat., 1982, 46(2), 409–430 (in Russian) 

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