Abstract
The Nekrasov conjecture predicts a relation between the partition function for N = 2 supersymmetric Yang–Mills theory and the Seiberg-Witten prepotential. For instantons on \({\mathbb{R}^4}\), the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov and Nakajima-Yoshioka. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bando, S.: Einstein–Hermitian metrics on non-compact Kähler manifolds. Lect. Notes Pure Appl. Math. 145, In: Einstein matrics and Yang-Mills connections (Sanda, 1990) New York: Marcel Dekker, 1993, pp. 27–33
Ballico E., Gasparim E., Köppe T.: Vector bundles near negative curves: moduli and local Euler characteristic. Comm. Alg. 37(8), 2688–2713 (2009)
Braverman, A.P.: Instanton counting via affine Lie algebras I: equivariant J-functions of (affine) flag manifolds and Whittaker vectors. In: Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes 38, Providence, RI: Amer. Math. Soc., 2004, pp. 113–132
Braverman, A., Etingof, P.: Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg–Witten prepotential. In: Studies in Lie theory, Progr. Math. 243, Boston, MA: Birkhäuser Boston, 2006, pp. 61–78
Buchdahl N.P.: Hermitian–Einstein connections and stable vector bundles over compact algebraic surfaces. Math. Ann. 280, 625–648 (1988)
Bruzzo, U., Fucito, F., Morales, J.F., Tanzini, A.: Multi-instanton calculus and equivariant cohomology. J. High Energy Phys. 2003, no. 5, 054, 24 pp.
Donagi, R.: Seiberg-Witten integrable systems. In: Surveys in Differential Geometry: Integrable Systems, Boston, MA: Int. Press, 1998, pp. 83–129
Donagi R., Witten E.: Supersymmetric Yang-Mills theory and integrable systems. Nucl. Phys. B 460(2), 299–334 (1996)
Donaldson S.K.: Anti-self-dual connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3) 50, 1–26 (1985)
Donaldson S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. 93, 453–460 (1984)
Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford University Press, Oxford (1990)
Ellingsrud G., Göttsche L.: Wall-crossing formulas, the Bott residue formula and the Donaldson invariants of rational surfaces. Quart. J. Math. Oxford Ser. (2) 49(195), 307–329 (1998)
Flume R., Poghossian R.: An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential. Internat. J. Mod. Phys. A 18(14), 2541–2563 (2003)
King, A.: Instantons and Holomorphic Bundles on the Blown-up Plane. D. Phil. Thesis, Worcester College, Oxford, 1998
Gasparim E.: The Atiyah-Jones conjecture for rational surfaces. Adv. Math. 218, 1027–1050 (2008)
Gasparim, E., Köppe, T., Majumdar, P.: Local holomorphic Euler characteristic and instanton decay. Pure Appl. Math. Q. 4(2), Special Issue: In honor of Fedya Bogomolov, Part 1, 161–179 (2008)
Göttsche L., Nakajima H., Yoshioka K.: Instanton Counting and Donaldson invariants. J. Differ. Geom. 80(3), 343–390 (2008)
Göttsche L., Nakajima H., Yoshioka K.: K-theoretic Donaldson invariants via instanton counting. Pure Appl. Math. Q. 5(3), 1029–1111 (2009)
Huybrechts D., Lehn M.: Stable pairs on curves and surfaces. J. Alg. Geom. 4, 67–104 (1995)
Labastida, J., Mariño, M.: Topological Quantum Field Theory and Four Manifolds. Math. Phys. Studies 25, Dordrecht: Springer, 2005
Lübke M., Teleman A.: The Kobayashi–Hitchin Correspondence. World Scientific Publishing Co., Inc., River Edge, NJ (1997)
Maulik D., Nekrasov N., Okounkov A., Pandharipande R.: Gromov-Witten theory and donaldson-thomas theory i. Compos. Math. 142(5), 1263–1285 (2006)
Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. University Lecture Series, 18, Providence, RI: Amer. Math. Soc., 1999
Nakajima, H., Yoshioka, K.: Lectures on instanton counting. In: Algebraic Structures and Moduli Spaces, CRM Proc. Lecture Notes 38, 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(2), 313–355 (2005)
Nakajima H., Yoshioka K.: Instanton counting on blowup. II. K-theoretic partition function. Transform. Groups 10(3-4), 489–519 (2005)
Nekrasov N.A.: Five-dimensional Gauge theories and relativistic integrable systems. Nucl. Phys. B 531(1–3), 323–344 (1998)
Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003)
Nekrasov, N.A.: Localizing Gauge Theories. XIVth International Congress on Mathematical Physics, 645–654, Hackensack, NJ: World Sci. Publ., 2005, pp. 645–654
Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: The Unity of Mathematics, Progr. Math. 244, Boston, MA: Birkhäuser, Boston, 2006, 525–596
Okounkov, A.: Random partitions and instanton counting. In: Sanz-Solé, Marta (ed.) et al., Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS), 2006, pp. 687–711
Seiberg N.: Supersymmetry and non-perturbative beta functions. Phys. Lett. B 206(1), 75–80 (1988)
Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52, 1994; Erratum, Nucl. Phys. B 430 (1994), 485–486
Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39, suppl. S257–S293 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N.A. Nekrasov
Rights and permissions
About this article
Cite this article
Gasparim, E., Liu, CC.M. The Nekrasov Conjecture for Toric Surfaces. Commun. Math. Phys. 293, 661–700 (2010). https://doi.org/10.1007/s00220-009-0948-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0948-4