Abstract
We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincaré polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the mathematical framework for the counting of D4-D2-D0 branes bound states on total spaces of the bundles \({\mathcal {O}_{\mathbb {P}^1}(-p)}\) .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramovich D., Vistoli A.: Compactifying the space of stable maps. J. Amer. Math. Soc. 15, 27–75 (2002)
Aganagic M., Ooguri H., Saulina N., Vafa C.: Black holes, q-deformed 2d Yang-Mills, and non- perturbative topological strings. Nucl. Phys. B 715, 304–348 (2005)
Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces. Vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Berlin: Springer-Verlag, second ed., 2004
Bonelli G., Tanzini A.: Topological gauge theories on local spaces and black hole entropy countings. Adv. Theor. Math. Phys. 12, 1429–1446 (2008)
Borne, N.: Fibrés paraboliques et champ des racines. Int. Math. Res. Not. IMRN 2007, no. 16, Art. ID rnm049, 38 pp. (2007)
Bruzzo U., Fucito F., MoralesJ.F. Tanzini A.: Multi-instanton calculus and equivariant cohomology. J. High Energy Phys. 0305, 054 (2003)
Bruzzo, U., Markushevich, D.: Moduli spaces of framed torsion-free sheaves on projective surfaces. http://arXiv.org/abs/0906.1436v2 [math.AG], 2009
Bruzzo, U., Markushevich, D., Tikhomirov, A.: Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces. http://arXiv.org/abs/1009.0856v2 [math.AG], 2010
Fantechi, B., Mann, E., Nironi, F.: Smooth toric DM stacks. http://arXiv.org/abs/0708.1254v2 [math.AG], 2009
Flume R., Poghossian R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A 18, 2541–2563 (2003)
Fucito F., Morales J.F., Poghossian R.: Multi instanton calculus on ALE spaces. Nucl. Phys. B 703, 518–536 (2004)
Fucito F., Morales J.F., Poghossian R.: Instantons on toric singularities and black hole countings. J. High Energy Phys. 0612, 073 (2006)
Gasparim E., Liu C.-C.M.: The Nekrasov conjecture for toric surfaces. Commun. Math. Phys. 293, 661–700 (2010)
Griguolo L., Seminara D., Szabo R.J., Tanzini A.: Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang-Mills theory. Nucl. Phys. B 772, 1–24 (2007)
Huybrechts D., Lehn M.: Framed modules and their moduli. Int. J. Math. 6, 297–324 (1995)
Huybrechts D., Lehn M.: Stable pairs on curves and surfaces. J. Alg. Geom. 4, 67–104 (1995)
Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31, Braunschweig: Friedr. Vieweg & Sohn, 1997
Kronheimer P.B., Nakajima H.: Yang-Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)
Losev A., Nekrasov N., Shatashvili S.L.: Freckled instantons in two and four dimensions. Class. Quant. Grav. 17, 1181–1187 (2000)
Macdonald I.G.: Symmetric functions and Hall polynomials. The Clarendon Press/Oxford University Press, New York (1995)
Maulik D., Nekrasov N., Okounkov A., Pandharipande R.: Gromov-Witten theory and Donaldson-Thomas theory, I. Compos. Math. 142, 1263–1285 (2006)
Nakajima H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76, 365–416 (1994)
Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. Vol. 18 of University Lecture Series, Providence, RI: Amer. Math. Soc., 1999
Nakajima H.: Sheaves on ALE spaces and quiver varieties. Moscow Math. J. 7, 699–722 (2007)
Nakajima, H., Yoshioka, K.: Lectures on instanton counting. In: Algebraic structures and moduli spaces. Vol. 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc., 2004, pp. 31–101
Nakajima H., Yoshioka K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162, 313–355 (2005)
Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003)
Nekrasov, N.A.: Localizing gauge theories. In: Proc. of the 14th International Congress on Mathematical Physics (ICMP 2003), Lisbon, Portugal, 28 Jul - 2 Aug 2003, Hackensaer, NJ: World Scientific, 2005, pp. 645–654
Nekrasov N., Schwarz A.S.: Instantons on noncommutative \({{\mathbb{R}}^4}\) , and (2,0) superconformal six dimensional theory. Commun. Math. Phys. 198, 689–703 (1998)
Nevins T.A.: Representability for some moduli stacks of framed sheaves. Manus. Math. 109, 85–91 (2002)
Ooguri H., Strominger A., Vafa C.: Black hole attractors and the topological string. Phys. Rev. D 70(3), 106007 (2004)
Rava, C.: ADHM data for framed sheaves on Hirzebruch surfaces. PhD thesis, SISSA, Trieste, 2010
Sasaki, T.: \({{\mathcal{O}}(-2)}\) blow-up formula via instanton calculus on \({\widehat{{\mathbb{C}}^2/{\mathbb{Z}}_2}}\) and Weil conjecture. http://arXiv.org/abs/hep-th/0603162v2, 2006
Vafa C., Witten E.: A strong coupling test of S-duality. Nucl. Phys. B 431, 3–77 (1994)
Yoshioka K.: Betti numbers of moduli of stable sheaves on some surfaces. Nucl. Phys. B Proc. Suppl. 46, 263–268 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N.A. Nekrasov
This research was partly supported by the INFN Research Project PI14 “Nonperturbative dynamics of gauge theory”, by PRIN “Geometria delle varietà algebriche”, by an Institutional Partnership Grant of the Humboldt foundation of Germany and European Commission FP7 Programme Marie Curie Grant Agreement PIIF-GA-2008-221571.
Rights and permissions
About this article
Cite this article
Bruzzo, U., Poghossian, R. & Tanzini, A. Poincaré Polynomial of Moduli Spaces of Framed Sheaves on (Stacky) Hirzebruch Surfaces. Commun. Math. Phys. 304, 395–409 (2011). https://doi.org/10.1007/s00220-011-1231-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1231-z