Abstract
We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in \( \mathcal{N} \) = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.
The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.
We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the \( \mathcal{N} \) = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
M. Bershtein, G. Bonelli, M. Ronzani and A. Tanzini, Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants, J. Geom. Phys. 118 (2017) 40 [arXiv:1606.07148] [INSPIRE].
G. Bonelli et al., Gauge theories on compact toric manifolds, Lett. Math. Phys. 111 (2021) 77 [arXiv:2007.15468] [INSPIRE].
R. Poghossian, Recurrence relations for the W3 conformal blocks and N = 2 SYM partition functions, JHEP 11 (2017) 053 [Erratum ibid. 01 (2018) 088] [arXiv:1705.00629] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
H. Nakajima and K. Yoshioka, Lectures on instanton counting, in the proceedings of CRM workshop on algebraic structures and moduli spaces, (2003) [math/0311058] [INSPIRE].
H. Nakajima, Lectures on Hilbert schemes of points on surfaces, AMS University Lecture Series, American Mathematical Society, U.S.A. (1999) [ISBN:0-8218-1956-9].
L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, J. Diff. Geom. 80 (2008) 343 [math/0606180] [INSPIRE].
M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].
A. Klyachko, Vector bundles and torsion free sheaves on the projective plane, preprint MPI/91-59 (1991).
A. Knutson and E.R. Sharpe, Sheaves on toric varieties for physics, Adv. Theor. Math. Phys. 2 (1998) 873 [hep-th/9711036] [INSPIRE].
G.-N. Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Ann. Inst. Fourier 60 (2010) 1.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, MA, U.S.A. (1994) [ISBN:978-0-8176-3660-9].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2209.14949
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Sysoeva, E., Bykov, A. Recurrence relation for instanton partition function in SU(N) gauge theory. J. High Energ. Phys. 2023, 220 (2023). https://doi.org/10.1007/JHEP03(2023)220
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2023)220