Skip to main content
SpringerLink
Log in

Quantum Geometry and Quiver Gauge Theories

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study macroscopically two dimensional \({\mathcal{N}=(2,2)}\) supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product \({\mathbb{T}^{d} \times \mathbb{R}^{2}_{\epsilon}}\) of a d-dimensional torus and a two dimensional cigar with \({\Omega}\)-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories and the Yangian \({\mathbf{Y}_{\epsilon}(\mathfrak{g}_{\Gamma})}\), quantum affine algebra \({\mathbf{U}^{\mathrm{aff}}_q(\mathfrak{g}_{\Gamma})}\), or the quantum elliptic algebra \({\mathbf{U}^{\mathrm{ell}}_{q,p}(\mathfrak{g}_{\Gamma})}\) associated to Kac–Moody algebra \({\mathfrak{g}_{\Gamma}}\) for quiver \({\Gamma}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000) arXiv:hep-th/9712241

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Gerasimov A.A., Shatashvili S.L.: Higgs bundles, gauge theories and quantum groups. Commun. Math. Phys. 277, 323–367 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Gerasimov, A.A., Shatashvili, S.L.: Two-Dimensional Gauge Theories and Quantum Integrable Systems. arXiv:0711.1472 [hep-th]

  4. Nekrasov, N., Shatashvili, S.: Bethe Ansatz and supersymmetric vacua. In: AIP Conference Proceedings, vol. 1134, pp. 154–169 (2009)

  5. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. In: XVIth International Congress of Mathematical Physics World Scientific, pp. 265–289 (2012). arXiv:0908.4052 [hep-th]

  6. Nekrasov N.A., Shatashvili S.L.: Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl. 177, 105–119 (2009) arXiv:0901.4748 [hep-th]

    Article  ADS  MATH  Google Scholar 

  7. Nekrasov N.A., Shatashvili S.L.: Supersymmetric vacua and Bethe ansatz. Nucl. Phys. Proc. Suppl. 192193, 91–112 (2009) arXiv:0901.4744 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Nekrasov N., Witten E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 09, 092 (2010) arXiv:1002.0888 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Nekrasov N., Rosly A., Shatashvili S.: Darboux coordinates, Yang–Yang functional, and gauge theory. Nucl. Phys. Proc. Suppl. 216, 69–93 (2011) arXiv:1103.3919 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Witten E.: Gauge theories and integrable lattice models. Nucl. Phys. B 322, 629 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  11. Gorsky A., Nekrasov N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B 436, 582–608 (1995) arXiv:hep-th/9401017

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Gorsky A., Nekrasov N.: Hamiltonian systems of Calogero type and two-dimensional Yang–Mills theory. Nucl. Phys. B 414, 213–238 (1994) arXiv:hep-th/9304047

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Gorsky, A., Nekrasov N.: Elliptic Calogero–Moser system from two-dimensional current algebra. arXiv:hep-th/9401021

  14. Nekrasov, N.: On the BPS/CFT correspondence, 3 Feb 2004. Lecture at the string theory group seminar, University of Amsterdam. http://www.science.uva.nl/research/itf/strings/stringseminar2003-4.html

  15. Nakajima H.: Gauge theory on resolutions of simple singularities and simple lie algebras. Int. Math. Res. Not. 2, 61–74 (1994) http://dx.doi.org/10.1155/S1073792894000085

    Article  MathSciNet  MATH  Google Scholar 

  16. Nakajima H.: Quiver varieties and Kac–Moody algebras. Duke Math. J. 91(3), 515–560 (1998) http://dx.doi.org/10.1215/S0012-7094-98-09120-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Vafa C., Witten E.: A Strong coupling test of S duality. Nucl. Phys. B 431, 3–77 (1994) arXiv:hep-th/9408074

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Losev A., Moore G.W., Nekrasov N., Shatashvili S.: Four-dimensional avatars of two-dimensional RCFT. Nucl. Phys. Proc. Suppl. 46, 130–145 (1996) arXiv:hep-th/9509151

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). (To Arkady Vainshtein on his 60th anniversary) arXiv:hep-th/0206161

  20. Losev, A.S., Marshakov, A., Nekrasov, N.A.: Small instantons, little strings and free fermions. arXiv:hep-th/0302191

  21. Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. arXiv:hep-th/0306238

  22. Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional Gauge theories. Lett. Math. Phys. 91, 167–197 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Nekrasov, N., Pestun, V.: Seiberg–Witten geometry of \({\mathcal{N}=2}\) quiver gauge theories. arXiv:1211.2240 [hep-th]

  24. Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B431, 484–550 (1994) arXiv:hep-th/9408099

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Seiberg N., Witten E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills theory. Nucl. Phys. B426, 19–52 (1994) arXiv:hep-th/9407087

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Losev A., Nekrasov N., Shatashvili S.L.: Issues in topological gauge theory. Nucl. Phys. B534, 549–611 (1998) arXiv:hep-th/9711108

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Losev, A., Nekrasov, N., Shatashvili S.L.: Testing Seiberg–Witten solution. In: Bavlieu, L., et al. (eds.) String, Bianes and Dualities. NATO ASI (Series C = Mathematical and Physical sciences), vol. 570. Springer, Dordrecht (1999). arXiv:hep-th/9801061

  28. Nekrasov N., Pestun V., Shatashvili S.: Quantum geometry and quiver gauge theories. arXiv:1312.6689 [hep-th]

  29. Nekrasov N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B 531, 323–344 (1998) arXiv:hep-th/9609219

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Poghossian R.: Deforming SW curve. JHEP 1104, 033 (2011) arXiv:1006.4822 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Dorey N., Lee S., Hollowood T.J.: Quantization of integrable systems and a 2d/4d duality. JHEP 1110, 077 (2011) arXiv:1103.5726 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Chen H.-Y., Dorey N., Hollowood T.J., Lee S.: A new 2d/4d duality via integrability. JHEP 1109, 040 (2011) arXiv:1104.3021 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Bao L., Pomoni E., Taki M., Yagi F.: M5-branes, toric diagrams and gauge theory duality. JHEP 1204, 105 (2012) arXiv:1112.5228 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A.: Spectral duality between Heisenberg chain and Gaudin model. Lett. Math. Phys. 103(3), 299–329 (2013)

  35. Gorsky A., Krichever I., Marshakov A., Mironov A., Morozov A.: Integrability and Seiberg–Witten exact solution. Phys. Lett. B355, 466–474 (1995) arXiv:hep-th/9505035

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Gorsky A., Marshakov A., Mironov A., Morozov A.: N = 2 supersymmetric QCD and integrable spin chains: rational case \({N_f \leq 2N_c}\). Phys. Lett. B380, 75–80 (1996) arXiv:hep-th/9603140

    Article  ADS  Google Scholar 

  37. Gorsky A., Gukov S., Mironov A.: Multiscale N = 2 SUSY field theories, integrable systems and their stringy/brane origin. 1. Nucl. Phys. B 517, 409–461 (1998) arXiv:hep-th/9707120

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Gorsky A., Gukov S., Mironov A.: SUSY field theories, integrable systems and their stringy / brane origin. 2. Nucl. Phys. B 518, 689–713 (1998) arXiv:hep-th/9710239

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory. 3. The Yang–Baxter relation. Commun. Math. Phys. 200, 297–324 (1999) arXiv:hep-th/9805008

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Dorey P., Dunning C., Tateo R.: The ODE/IM correspondence. J. Phys. A40, R205 (2007) arXiv:hep-th/0703066

    ADS  MathSciNet  MATH  Google Scholar 

  41. Gerasimov A., Kharchev S., Lebedev D., Oblezin S.: On a class of representations of the Yangian and moduli space of monopoles. Commun. Math. Phys. 260, 511–525 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Gerasimov, A., Kharchev, S., Lebedev, D., Oblezin S.: On a Class of Representations of Quantum Groups (2005). ArXiv Mathematics e-prints, arXiv:math/0501473

  43. Galakhov D., Mironov A., Morozov A., Smirnov A., Mironov A. et al.: Three-dimensional extensions of the Alday–Gaiotto–Tachikawa relation. Theor. Math. Phys. 172, 939–962 (2012) arXiv:1104.2589 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  44. Mironov A., Morozov A.: Nekrasov functions and exact Bohr–Zommerfeld integrals. JHEP 1004, 040 (2010) arXiv:0910.5670 [hep-th]

    Article  ADS  MATH  Google Scholar 

  45. Mironov A., Morozov A.: Nekrasov functions from exact BS periods: the case of SU(N). J. Phys. A43, 195401 (2010) arXiv:0911.2396 [hep-th]

    ADS  MATH  Google Scholar 

  46. Teschner J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15, 471–564 (2011) arXiv:1005.2846 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  47. Muneyuki K., Tai T.-S., Yonezawa N., Yoshioka R.: Baxter’s T-Q equation, SU(N)/SU(2)N-3 correspondence and \({\Omega}\)-deformed Seiberg–Witten prepotential. JHEP 1109, 125 (2011) arXiv:1107.3756 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Drinfeld V.G.: A new realization of Yangians and of quantum affine algebras. Dokl. Akad. Nauk SSSR. 296(1), 13–17 (1987)

    Google Scholar 

  49. Knight H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174(1), 187–196 (1995) http://dx.doi.org/10.1006/jabr.1995.1123

    Article  MathSciNet  MATH  Google Scholar 

  50. Frenkel E., Reshetikhin N.: The q-characters of representations of quantum affine algebras and deformations of \({\mathcal{W}}\)-algebras. Recent developments in quantum affine algebras and related topics (Raleigh, NC,1998). Contemp. Math. 248, 163–205 (1999) http://dx.doi.org/10.1090/conm/248/03823

    Article  MATH  Google Scholar 

  51. Chari, V., Pressley, A.: Quantum affine algebras and their representations. arXiv:hep-th/9411145

  52. Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Kirillov, A., Reshetikhin, N.Y.: Representations of the algebra u q (sl 2), orthogonal polynomials and invariants of links. http://www.worldscientific.com/worldscibooks/10.1142/1056

  54. Reshetikhin N., Turaev V.: Invariants of three manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Reshetikhin N.Y., Turaev V.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127, 1–26 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Witten, E.: Fivebranes and Knots. arXiv:1101.3216 [hep-th]

  57. Nekrasov, N., Ooguri, H., Vafa, C.: S-duality and topological strings. JHEP. 10, 009 (2004). arXiv:hep-th/0403167

  58. Nekrasov N.: Z-theory: chasing M-F-theory. Comptes Rendus Phys. 6, 261–269 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  59. Hernandez D.: Quantum toroidal algebras and their representations. Sel. Math. (N.S.) 14(3-4), 701–725 (2009) arXiv:0801.2397. http://dx.doi.org/10.1007/s00029-009-0502-4

    Article  MathSciNet  MATH  Google Scholar 

  60. Cherednik, I.: Introduction to double Hecke algebras (2004). ArXiv Mathematics e-prints, arXiv:math/0404307

  61. Pestun, V.: Integrable Systems for 4d N = 2 ADE Quiver Theories from Instanton Counting (2012). http://people.physik.hu-berlin.de/~ahoop/pestun.pdf

  62. Pestun, V.: Supersymmetric Four-Dimensional Quiver Gauge Theories and Quantum ADE Spin Chains (2012). http://scgp.stonybrook.edu/archives/4709

  63. Nekrasov, N.: Seiberg–Witten Geometry of N = 2 Quiver Theories, and Quantization (2012). http://brahms.mth.kcl.ac.uk/cgi-bin/main.pl?action=seminars&id=1108

  64. Nekrasov, N.: Seiberg–Witten Geometry of N = 2 Superconformal Theories, and ADE Bundles on Curves (2012). http://media.scgp.stonybrook.edu/video/video.php?f=20120510_1_qtp.mp4

  65. Shatashvili, S.: Gauge Theory Angle at Integrability (2012). http://www.kcl.ac.uk/nms/depts/mathematics/research/theorphysics/pastevents/stringgauge.aspx

  66. Shatashvili, S.: Integrability and Quantization (2013). https://indico.desy.de/contributionDisplay.py?contribId=16&confId=6969

  67. Shatashvili, S.: Integrability and Supersymmetric Vacua (IV) (2013). http://cdsagenda5.ictp.trieste.it/full_display.php?ida=a13168

  68. Fucito F., Morales J.F., Pacifici D.R.: Deformed Seiberg–Witten curves for ADE quivers. JHEP 1301, 091 (2013). arXiv:1210.3580 [hep-th]

  69. Intriligator K.A., Morrison D.R., Seiberg N.: Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces. Nucl. Phys. B 497, 56–100 (1997) arXiv:hep-th/9702198

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Tachikawa Y.: Five-dimensional Chern–Simons terms and Nekrasov’s instanton counting. JHEP 02, 050 (2004) arXiv:hep-th/0401184

    Article  ADS  MathSciNet  Google Scholar 

  71. Gottsche L., Nakajima H., Yoshioka K.: K-theoretic Donaldson invariants via instanton counting. Pure Appl. Math. Q. 5, 1029–1111 (2009). arXiv:math/0611945 [math-ag]

    Article  MathSciNet  MATH  Google Scholar 

  72. Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. AMS, AMS University Lecture Series (1999). ISBN 0-8218-1956-9

  73. Nakajima, H., Yoshioka, K.: Lectures on Instanton Counting (2003). ArXiv Mathematics e-prints, arXiv:math/0311058

  74. Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type A n ,D n . http://dx.doi.org/10.1090/conm/325/05669

  75. Shadchin, S.: On certain aspects of string theory/gauge theory correspondence. Ph.D. Thesis. arXiv:hep-th/0502180

  76. Pestun : Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129 (2012) arXiv:0712.2824 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. 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) arXiv:hep-th/9802068

    Article  ADS  MATH  Google Scholar 

  78. Atiyah M., Hitchin N.J., Drinfeld V., Manin Y.: Construction of instantons. Phys. Lett. A65, 185–187 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. Frenkel E., Hernandez D.: Baxters relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015) arXiv:1308.3444 [math.QA]

    Article  MathSciNet  MATH  Google Scholar 

  80. Kozlowski, K., Teschner, J.: TBA for the toda chain. arXiv:1006.2906 [math-ph]

  81. Smirnov F.A.: Structure of matrix elements in the quantum toda chain. J. Phys. A Math. Gen. 31(44), 8953 (1998) arXiv:math/9805011. http://stacks.iop.org/0305-4470/31/i=44/a=019

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. Bazhanov V., Reshetikhin N.: Restricted solid on solid models connected with simply based algebras and conformal field theory. J. Phys. A23, 1477 (1990)

    ADS  MATH  Google Scholar 

  83. Kirillov A., Reshetikhin N.Y.: Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum. J. Phys. A 20, 1565–1585 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  84. Kuniba A., Suzuki J.: Analytic Bethe Ansatz for fundamental representations of Yangians. Commun. Math. Phys. 173, 225–264 (1995) arXiv:hep-th/9406180

    Article  ADS  MathSciNet  MATH  Google Scholar 

  85. Kuniba A., Nakanishi T., Suzuki J.: T-systems and Y-systems in integrable systems. J. Phys. A44, 103001 (2011) arXiv:1010.1344 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  86. Fucito F., Morales J., Pacifici D.R, Poghossian R.: Gauge theories on \({\Omega}\)-backgrounds from non commutative Seiberg–Witten curves. JHEP 1105, 098 (2011) arXiv:1103.4495 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  87. Hernandez D.: The algebra \({U_q(\hat{sl}_\infty)}\) and applications. J. Algebra 329, 147–162 (2011) http://dx.doi.org/10.1016/j.jalgebra.2010.04.002

    Article  MathSciNet  Google Scholar 

  88. Frenkel E., Mukhin E.: The Hopf algebra \({{\rm Rep} \,U_q\widehat{\mathfrak{g} \mathfrak{l}_\infty}}\). Sel. Math. (N.S.) 8(4), 537–635 (2002) http://dx.doi.org/10.1007/PL00012603

    Google Scholar 

  89. Faddeev, L.: How algebraic Bethe ansatz works for integrable model. arXiv:hep-th/9605187

  90. Frenkel, E., Hernandez, D.: Baxter’s relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015). 1308.3444 [math.QA]

  91. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics, Vol. 1 of Series on Advances in Statistical Mechanics. World Scientific Publishing, Singapore (1985)

  92. Frenkel E., Mukhin E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys. 216(1), 23–57 (2001) http://dx.doi.org/10.1007/s002200000323

    Article  ADS  MathSciNet  MATH  Google Scholar 

  93. Hernandez D.: The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond. J. Algebra 279(2), 514–557 (2004) http://dx.doi.org/10.1016/j.jalgebra.2004.02.022

    Article  MathSciNet  MATH  Google Scholar 

  94. Etingof, P., Varchenko, A.: Dynamical Weyl groups and applications. Adv. Math. 167(1), 74–127 (2002). arXiv:math/0011001

  95. Smirnov F.A.: Baxter equations and deformation of abelian differentials. Int. J. Mod. Phys. A 19, 396–417 (2004) arXiv:math/0302014

    Article  ADS  MathSciNet  MATH  Google Scholar 

  96. Chari, V., Pressley, A.: Quantum affine algebras. Commun. Math. Phys. 142(2), 261–283 (1991) http://projecteuclid.org/getRecord?id=euclid.cmp/1104248585

  97. Chari, V., Pressley, A.: Yangians: their representations and characters. Acta Appl. Math. 44(1–2), 39–58 (1996). http://dx.doi.org/10.1007/BF00116515. Representations of lie groups, lie algebras and their quantum analogues

  98. Nakajima H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001) http://dx.doi.org/10.1090/S0894-0347-00-00353-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  99. Nakajima, H.: T-analogue of the q-characters of finite dimensional representations of quantum affine algebras. http://dx.doi.org/10.1142/9789812810007_0009

  100. Nakajima H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. Math. (2) 160(3), 1057–1097 (2004) http://dx.doi.org/10.4007/annals.2004.160.1057

    Article  MathSciNet  MATH  Google Scholar 

  101. Nakajima H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76(2), 365–416 (1994) http://dx.doi.org/10.1215/S0012-7094-94-07613-8

    Article  MathSciNet  MATH  Google Scholar 

  102. Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type E 6,E 7,E 8. http://dx.doi.org/10.1007/978-0-8176-4697-4_10

  103. Hernandez D.: Representations of quantum affinizations and fusion product. Transform. Groups 10(2), 163–200 (2005) http://dx.doi.org/10.1007/s00031-005-1005-9

    Article  MathSciNet  MATH  Google Scholar 

  104. Hernandez D.: Drinfeld coproduct, quantum fusion tensor category and applications. Proc. Lond. Math. Soc. (3) 95(3), 567–608 (2007) http://dx.doi.org/10.1112/plms/pdm017

    Article  MathSciNet  MATH  Google Scholar 

  105. Chari V.: Braid group actions and tensor products. Int. Math. Res. Not. 7, 357–382 (2002) http://dx.doi.org/10.1155/S107379280210612X

    Article  MathSciNet  MATH  Google Scholar 

  106. Chari V., Moura A.A.: Characters and blocks for finite-dimensional representations of quantum affine algebras. Int. Math. Res. Not. 5, 257–298 (2005) http://dx.doi.org/10.1155/IMRN.2005.257

    Article  MathSciNet  MATH  Google Scholar 

  107. Hernandez, D., Jimbo, M.: Asymptotic representations and Drinfeld rational fractions. Compos. Math. 148(5), 1593–1623 (2012). arXiv:1104.1891 [math.QA]

  108. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory. 2. Q operator and DDV equation. Commun. Math. Phys. 190, 247–278 (1997) arXiv:hep-th/9604044

    Article  ADS  MathSciNet  MATH  Google Scholar 

  109. Nakajima, H.: Geometric construction of representations of affine algebras. arXiv:math/0212401

  110. Varagnolo, M., Vasserot, E.: On the K-theory of the cyclic quiver variety. Int. Math. Res. Not. no. 18, 1005–1028 (1999). arXiv:math/9902091. http://dx.doi.org/10.1155/S1073792899000525

  111. Varagnolo M.: Quiver varieties and Yangians. Lett. Math. Phys. 53(4), 273–283 (2000) arXiv:math/0005277. http://dx.doi.org/10.1023/A:1007674020905

    Article  MathSciNet  MATH  Google Scholar 

  112. Guay N.: Cherednik algebras and Yangians. Int. Math. Res. Not. (57), 3551–3593 (2005) http://dx.doi.org/10.1155/IMRN.2005.3551

  113. Guay N.: Affine Yangians and deformed double current algebras in type A. Adv. Math. 211(2), 436–484 (2007) http://dx.doi.org/10.1016/j.aim.2006.08.007

    Article  MathSciNet  MATH  Google Scholar 

  114. Guay N.: Quantum algebras and quivers. Sel. Math. (N.S.) 14(3-4), 667–700 (2009) http://dx.doi.org/10.1007/s00029-009-0496-y

    Article  MathSciNet  MATH  Google Scholar 

  115. Braden H., Gorsky A., Odessky A., Rubtsov V.: Double elliptic dynamical systems from generalized Mukai–Sklyanin algebras. Nucl. Phys. B 633, 414–442 (2002) arXiv:hep-th/0111066

    Article  ADS  MathSciNet  MATH  Google Scholar 

  116. Braden H.W., Hollowood T.J.: The curve of compactified 6-D gauge theories and integrable systems. JHEP 0312, 023 (2003) arXiv:hep-th/0311024

    Article  ADS  MathSciNet  Google Scholar 

  117. Hollowood T.J., Iqbal A., Vafa C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008) arXiv:hep-th/0310272

    Article  ADS  MathSciNet  Google Scholar 

  118. Ginzburg, V., Kapranov, M., Vasserot, É.: Langlands reciprocity for algebraic surfaces. Math. Res. Lett. 2(2), 147–160 (1995). arXiv:q-alg/9502013. http://dx.doi.org/10.4310/MRL.1995.v2.n2.a4

  119. Varagnolo, M., Vasserot, E.: Schur duality in the toroidal setting. Commun. Math. Phys. 182(2), 469–483 (1996). arXiv:q-alg/9506026. http://projecteuclid.org/getRecord?id=euclid.cmp/1104288156

  120. Cherednik, I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and Macdonald’s operators. Int. Math. Res. Not. 9, 171–180 (1992). http://dx.doi.org/10.1155/S1073792892000199

  121. Saito, Y.: Quantum toroidal algebras and their vertex representations. Publ. Res. Inst. Math. Sci. 34(2), 155–177 (1998). arXiv:q-alg/9611030. http://dx.doi.org/10.2977/prims/1195144759

  122. Cherednik, I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483–515 (1995). http://dx.doi.org/10.1155/S1073792895000341

  123. Cherednik I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. (2) 141(1), 191–216 (1995) http://dx.doi.org/10.2307/2118632

    Article  MathSciNet  MATH  Google Scholar 

  124. Varagnolo, M., Vasserot, E.: Double-loop algebras and the Fock space. Invent. Math. 133(1), 133–159 (1998). arXiv:q-alg/9612035. http://dx.doi.org/10.1007/s002220050242

  125. Saito, Y., Takemura, K., Uglov, D.: Toroidal actions on level 1 modules of \({U_q(\widehat{\rm sl}_n)}\). Transform. Groups 3(1), 75–102 (1998). arXiv:q-alg/9702024. http://dx.doi.org/10.1007/BF01237841

  126. Kashiwara, M., Miwa, T., Stern, E.: Decomposition of q-deformed Fock spaces. Sel. Math. (N.S.) 1(4), 787–805 (1995). arXiv:q-alg/9508006. http://dx.doi.org/10.1007/BF01587910

  127. Nagao, K.: K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials. Osaka J. Math. 46(3), 877–907 (2009). http://projecteuclid.org/getRecord?id=euclid.ojm/1256564211. arXiv:0709.1767

  128. Miki K.: Toroidal braid group action and an automorphism of toroidal algebra \({U_q({\rm sl}_{n+1,\rm tor})\ (n\geq 2)}\). Lett. Math. Phys. 47(4), 365–378 (1999) http://dx.doi.org/10.1023/A:1007556926350

    Article  MathSciNet  MATH  Google Scholar 

  129. Berman S., Gao Y., Krylyuk Y.S.: Quantum tori and the structure of elliptic quasi-simple Lie algebras. J. Funct. Anal. 135(2), 339–389 (1996) http://dx.doi.org/10.1006/jfan.1996.0013

    Article  MathSciNet  MATH  Google Scholar 

  130. Miki, K.: A \({(q,\gamma)}\) analog of the \({W_{1+\infty}}\) algebra. J. Math. Phys. 48(12), 123520, 35 (2007). http://dx.doi.org/10.1063/1.2823979

  131. Awata H., Kubo H., Odake S., Shiraishi J.: Quantum \({{\mathcal{W_N}}}\) algebras and Macdonald polynomials. Commun. Math. Phys. 179(2), 401–416 (1996) http://projecteuclid.org/getRecord?id=euclid.cmp/1104286998

  132. Shiraishi J., Kubo H., Awata H., Odake S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38(1), 33–51 (1996) http://dx.doi.org/10.1007/BF00398297

    Article  ADS  MathSciNet  MATH  Google Scholar 

  133. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. (2). 145(2), 379–388 (1997). arXiv:alg-geom/9507012. http://dx.doi.org/10.2307/2951818

  134. Ginzburg V., Vasserot É.: Langlands reciprocity for affine quantum groups of type A n . Int. Math. Res. Not. (3), 67–85 (1993) http://dx.doi.org/10.1155/S1073792893000078

  135. Grojnowski, I.: Instantons and affine algebras. I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3(2), 275–291 (1996). arXiv:alg-geom/9506020 http://dx.doi.org/10.4310/MRL.1996.v3.n2.a12

  136. Baranovsky, V.: Moduli of sheaves on surfaces and action of the oscillator algebra. J. Differ. Geom. 55(2), 193–227 (2000) http://projecteuclid.org/getRecord?id=euclid.jdg/1090340878

  137. Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A 2. Duke Math. J. 162(2), 279–366 (2013). arXiv:0905.2555 [math.QA]

  138. Carlsson, E., Okounkov, A.: Exts and vertex operators. Duke Math. J. 161(9), 1797–1815 (2012). arXiv:0801.2565. http://dx.doi.org/10.1215/00127094-1593380

  139. Carlsson, E., Nekrasov, N., Okounkov, A.: Five dimensional gauge theories and vertex operators (2013). arXiv:1308.2465 [math.RT]

  140. Feigin, B., Frenkel, E.: Quantum W-algebras and elliptic algebras. Commun. Math. Phys. 178(3), 653–678 (1996). arXiv:q-alg/9508009 [q-alg]. http://projecteuclid.org/getRecord?id=euclid.cmp/1104286770

  141. Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate \({\mathbb{CP}^1}\) and Macdonald polynomials. J. Math. Phys. 50(9), 095215 (2009). 42, arXiv:0904.2291. http://dx.doi.org/10.1063/1.3192773

  142. Wyllard, N.: A (N-1) conformal toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories. JHEP. 11, 002 (2009). arXiv:0907.2189 [hep-th]

  143. Awata, H., Yamada, Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 1001, 125 (2010). arXiv:0910.4431 [hep-th]

  144. Fateev V., Litvinov A.: Integrable structure, W-symmetry and AGT relation. JHEP 1201, 051 (2012) arXiv:1109.4042 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  145. Smirnov, A.: On the instanton R-matrix (2013). arXiv:1302.0799 [math.AG]

  146. Awata, H., Feigin, B., Hoshino, A., Kanai, M., Shiraishi, J. et al. Notes on Ding–Iohara Algebra and AGT Conjecture. arXiv:1302.0799 [math-ph]

  147. Maulik, D., Okounkov, A.: Quantum Groups and Quantum Cohomology (2012). arXiv:1211.1287 [math.AG]

  148. Schiffmann, O., Vasserot, E.: Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2. Publ. Math. IHES 118, 213 (2013). arXiv:1202.2756 [math.QA]

  149. Alba V.A., Fateev V.A., Litvinov A.V., Tarnopolskiy G.M.: On combinatorial expansion of the conformal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33–64 (2011) arXiv:1012.1312 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  150. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous \({\mathfrak{gl}_\infty}\): semiinfinite construction of representations. Kyoto J. Math. 51(2), 337–364 (2011). arXiv:1002.3100. http://dx.doi.org/10.1215/21562261-1214375

  151. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous \({\mathfrak{gl}_\infty}\): tensor products of Fock modules and \({\mathcal{W}_n}\)-characters. Kyoto J. Math. 51(2), 365–392 (2011) arXiv:1002.3113. http://dx.doi.org/10.1007/BF01237841

  152. Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. Duke Math. J. 161(7), 1171–1231 (2012). arXiv:math/0505148. http://dx.doi.org/10.1215/00127094-1593263

  153. Schiffmann, O.: Drinfeld realization of the elliptic Hall algebra. J. Algebr. Combin. 35(2), 237–262 (2012). arXiv:1004.2575. http://dx.doi.org/10.1007/s10801-011-0302-8

  154. Feigin, B.L., Tsymbaliuk, A.I.: Equivariant K-theory of Hilbert schemes via shuffle algebra. Kyoto J. Math. 51(4), 831–854 (2011). arXiv:0904.1679. http://dx.doi.org/10.1215/21562261-1424875

  155. Ding J.-t., Iohara K.: Generalization and deformation of Drinfeld quantum affine algebras. Lett. Math. Phys. 41, 181–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  156. Feigin B., Odesskii A.: A family of elliptic algebras. Int. Math. Res. Not. 11, 531–539 (1997) http://dx.doi.org/10.1155/S1073792897000354

    Article  MathSciNet  MATH  Google Scholar 

  157. Enriquez, B.: On correlation functions of Drinfeld currents and shuffle algebras. Transform. Groups 5(2), 111–120 (2000). arXiv:math/9809036. http://dx.doi.org/10.1007/BF01236465

  158. Negut, A.: An isomorphism between the quantum toroidal and shuffle algebras, and a conjecture of Kuznetsov (2013). arXiv:1302.6202 [math.RT]

  159. Okounkov, A., Pandharipande, R.: Quantum cohomology of the Hilbert scheme of points in the plane. Invent. Math. 179(3), 523–557 (2010). arXiv:math/0411210. http://dx.doi.org/10.1007/s00222-009-0223-5

  160. Saito, Y.: Elliptic Ding–Iohara algebra and the free field realization of the elliptic Macdonald operator (2013). arXiv:1301.4912 [math.QA]

  161. Saito, Y.: Elliptic Ding–Iohara algebra and commutative families of the elliptic Macdonald operator (2013). arXiv:1309.7094 [math.QA]

  162. Cherkis S.A., Kapustin A.: Periodic monopoles with singularities and N = 2 super QCD. Commun. Math. Phys. 234, 1–35 (2003) arXiv:hep-th/0011081

    Article  ADS  MathSciNet  MATH  Google Scholar 

  163. Cherkis S.A., Kapustin A.: Hyperkahler metrics from periodic monopoles. Phys. Rev. D 65, 084015 (2002) arXiv:hep-th/0109141

    Article  ADS  MathSciNet  Google Scholar 

  164. Drinfeld V.G.: Hopf algebras and the quantum Yang–Baxter equation. Dokl. Akad. Nauk SSSR 283(5), 1060–1064 (1985)

    MathSciNet  Google Scholar 

  165. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2, pp. 798–820 (Berkeley, CA, 1986) (1987)

  166. Meneghelli C., Yang, G.: Mayer–Cluster expansion of instanton partition functions and thermodynamic Bethe Ansatz. JHEP 1405, 112 (2004). arXiv:1312.4537 [hep-th]

  167. Bourgine, J.-E.: Confinement and Mayer cluster expansions. Int. J. Mod. Phys. A 29, 145077 (2004). arXiv:1402.1626 [hep-th]

  168. Polyakov, A.M.: Gauge Fields and Strings. Harwood, Chur (1987)

  169. Shadchin, S.: Status report on the instanton counting. SIGMA 2, 008 (2006). arXiv:hep-th/0601167

  170. Givental, A.: A mirror theorem for toric complete intersections (1997). arXiv:alg-geom/9701016

  171. Enriquez, B.: Quantum currents realization of the elliptic quantum groups \({E_{\tau,\eta}(\mathfrak{sl}_2)}\). In: Calogero–Moser–Sutherland Models (Montréal, QC, 1997), CRM Series in Mathematical Physics, pp. 161–176. Springer, New York (2000)

Download references

Author information

Authors and Affiliations

  1. Simons Center for Geometry and Physics, Stony Brook, NY, USA

    Nikita Nekrasov

  2. Institut des Hautes Études Scientifique, Bures-sur-Yvette, France

    Vasily Pestun

  3. The Hamilton Mathematics Institute, School of Mathematics, Trinity College Dublin, Dublin, Ireland

    Samson Shatashvili

  4. Chaire Louis Michel, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

    Samson Shatashvili

Authors
  1. Nikita Nekrasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasily Pestun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samson Shatashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasily Pestun.

Additional information

Communicated by X. Yin

Nikita Nekrasov is on leave of absence from Institut des Hautes Etudes Scientifiques, France, Institute of Theoretical and Experimental Physics, Russia and Institute for Information Transmission Problems, Lab. 5, Russia.

Vasily Pestun is on leave of absence from Institute of Theoretical and Experimental Physics, Russia.

Samson Shatashvili is on leave of absence from Euler International Mathematical Institute, St. Petersburg, Russia and Institute for Information Transmission Problems, Lab. 5, Russia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nekrasov, N., Pestun, V. & Shatashvili, S. Quantum Geometry and Quiver Gauge Theories. Commun. Math. Phys. 357, 519–567 (2018). https://doi.org/10.1007/s00220-017-3071-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-3071-y

Search

Navigation