Abstract
We relate Nekrasov partition functions, with arbitrary values of ϵ 1, ϵ 2 parameters, to matrix models for β-ensembles. We find matrix models encoding the instanton part of Nekrasov partition functions, whose measure, to the leading order in ϵ 2 expansion, is given by the Vandermonde determinant to the power β = −ϵ 1/ϵ 2. An additional, trigonometric deformation of the measure arises in five-dimensional theories. Matrix model potentials, to the leading order in ϵ 2 expansion, are the same as in the β = 1 case considered in 0810.4944 [hep-th]. We point out that potentials for massive hypermultiplets include multi-log, Penner-like terms. Inclusion of Chern-Simons terms in five-dimensional theories leads to multi-matrix models. The role of these matrix models in the context of the AGT conjecture is discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg and E. Witten, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [SPIRES].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].
N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I 4-dimensional pure gauge theory, math/0306198 [SPIRES].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II: K-theoretic partition function, math/0505553 [SPIRES].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [SPIRES].
R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].
R. Dijkgraaf and C. Vafa, A Perturbative Window into Non-Perturbative Physics, hep-th/0208048 [SPIRES].
S. Katz, A. Klemm, C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B497 (1997) 173 [hep-th/9609239] [SPIRES].
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Marino and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 hep-th/0312085 [SPIRES].
A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [SPIRES].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix Models, Geometric Engineering and Elliptic Genera, JHEP 03 (2008) 069 [hep-th/0310272] [SPIRES].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [SPIRES].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [SPIRES].
M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [SPIRES].
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [SPIRES].
R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [SPIRES].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [SPIRES].
A. Klemm, M. Marino and M. Rauch, Direct Integration and Non-Perturbative Effects in Matrix Models, arXiv:1002.3846 [SPIRES].
A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].
P. Sulkowski, Matrix models for 2* theories, Phys. Rev. D 80 (2009) 086006 [arXiv:0904.3064] [SPIRES].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].
R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES].
M. Mehta, Random matrices, Pure and Applied Mathematics Series vol. 142, Elsevier B.V., Amsterdam Netherlands (2004).
B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and non- commutative algebraic geometry, JHEP 03 (2009) 094 [arXiv:0809.3367] [SPIRES].
L. Chekhov, B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and quantum algebraic geometry, arXiv:0911.1664 [SPIRES].
B. Eynard, All orders asymptotic expansion of large partitions, J. Stat. Mech. (2008) P07023 [arXiv:0804.0381] [SPIRES].
G. Borot, B. Eynard, M. Mulase and B. Safnuk, A matrix model for simple Hurwitz numbers and topological recursion, arXiv:0906.1206 [SPIRES].
A. Morozov and S. Shakirov, Generation of Matrix Models by W-operators, JHEP 04 (2009) 064 [arXiv:0902.2627] [SPIRES].
A. Morozov and S. Shakirov, On Equivalence of two Hurwitz Matrix Models, Mod. Phys. Lett. A 24 (2009) 2659 [arXiv:0906.2573] [SPIRES].
B. Eynard, A Matrix model for plane partitions and TASEP, J. Stat. Mech. (2009) P10011 [arXiv:0905.0535] [SPIRES].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045.
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].
M.-x. Huang and A. Klemm, Holomorphicity and Modularity in Seiberg-Witten Theories with Matter, arXiv:0902.1325 [SPIRES].
D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal \( \mathcal{N} = 2 \) SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].
A. Mironov and A. Morozov, The Power of Nekrasov Functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [SPIRES].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in \( \mathcal{N} = 2 \) SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [SPIRES].
A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].
H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [SPIRES].
H. Itoyama, K. Maruyoshi and T. Oota, Notes on the Quiver Matrix Model and 2d-4d Conformal Connection, arXiv:0911.4244 [SPIRES].
T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [SPIRES].
R. Schiappa and N. Wyllard, An A r threesome: Matrix models, 2d CFTs and 4d \( \mathcal{N} = 2 \) gauge theories, arXiv:0911.5337 [SPIRES].
A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [SPIRES].
M. Fujita, Y. Hatsuda and T.-S. Tai, Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model, JHEP 03 (2010) 046 [arXiv:0912.2988] [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: Discrete case, Phys. Lett. B 265 (1991) 99 [SPIRES].
I. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [SPIRES].
R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [SPIRES].
R. Dijkgraaf, L. Hollands and P. Sulkowski, Quantum Curves and \( \mathcal{D} - Modules \), JHEP 11 (2009) 047 [arXiv:0810.4157] [SPIRES].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [SPIRES].
Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [SPIRES].
P. Sulkowski, Deformed boson-fermion correspondence, Q-bosons and topological strings on the conifold, JHEP 10 (2008) 104 [arXiv:0808.2327] [SPIRES].
M. Abramowitz and I. Stegun, Handbook of mathematical functions, Applied Mathematics Series vol.55, Dover Publications, New York U.S.A. (1972).
F. Tricomi and A. Erdelyi, The Asymptotic Expansion of a Ratio of Gamma Functions, Pacific J. Math. 1 (1951) 133.
G. Gasper, q-Extensions of Erdelyi’s Fractional Integral Representations for Hypergeometric Functions and Some Summation Formulas for Double q-Kampe de Feriet Series, Contemporary Mathematics vol.254, AMS, Boston U.S.A. (2000).
D. Moak, The q-analogue of Stirling’s formula, Rocky Mountain J. Math. 14 (1984) 403.
M. Mansour, An asymptotic expansion of the q-gamma function Γ q (x), Journal of Nonlinear Math. Phys. 13 (2006) 479.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 0912.5476
On leave from University of Amsterdam and So ltan Institute for Nuclear Studies, Poland. (Piotr Sułkowski)
Rights and permissions
About this article
Cite this article
Sułkowski, P. Matrix models for β-ensembles from Nekrasov partition functions. J. High Energ. Phys. 2010, 63 (2010). https://doi.org/10.1007/JHEP04(2010)063
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2010)063