Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 049, 14 pages      arXiv:1711.04893

Jacobi-Trudi Identity in Super Chern-Simons Matrix Model

Tomohiro Furukawa and Sanefumi Moriyama
Department of Physics, Osaka City University, Osaka 558-8585, Japan

Received January 19, 2018, in final form May 10, 2018; Published online May 18, 2018

It was proved by Macdonald that the Giambelli identity holds if we define the Schur functions using the Jacobi-Trudi identity. Previously for the super Chern-Simons matrix model (the spherical one-point function of the superconformal Chern-Simons theory describing the worldvolume of the M2-branes) the Giambelli identity was proved from a shifted version of it. With the same shifted Giambelli identity we can further prove the Jacobi-Trudi identity, which strongly suggests an integrable structure for this matrix model.

Key words: Jacobi-Trudi identity; ABJM theory; Chern-Simons theory; matrix model; integrable system.

pdf (447 kb)   tex (43 kb)


  1. Aganagic M., Dijkgraaf R., Klemm A., Mariño M., Vafa C., Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451-516, hep-th/0312085.
  2. Aharony O., Bergman O., Jafferis D.L., Fractional M2-branes, J. High Energy Phys. 2008 (2008), no. 11, 043, 27 pages, arXiv:0807.4924.
  3. Aharony O., Bergman O., Jafferis D.L., Maldacena J., ${\mathcal N}=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 2008 (2008), no. 10, 091, 38 pages, arXiv:0806.1218.
  4. Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., Classical tau-function for quantum spin chains, J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310.
  5. Bonelli G., Grassi A., Tanzini A., Quantum curves and $q$-deformed Painlevé equations, arXiv:1710.11603.
  6. Borodin A., Olshanski G., Strahov E., Giambelli compatible point processes, Adv. in Appl. Math. 37 (2006), 209-248, math-ph/0505021.
  7. Drukker N., Mariño M., Putrov P., From weak to strong coupling in ABJM theory, Comm. Math. Phys. 306 (2011), 511-563, arXiv:1007.3837.
  8. Drukker N., Trancanelli D., A supermatrix model for ${\mathcal N}=6$ super Chern-Simons-matter theory, J. High Energy Phys. 2010 (2010), no. 2, 058, 21 pages, arXiv:0912.3006.
  9. Fuji H., Hirano S., Moriyama S., Summing up all genus free energy of ABJM matrix model, J. High Energy Phys. 2011 (2011), no. 8, 001, 17 pages, arXiv:1106.4631.
  10. Giambelli G.Z., Alcune proprietà delle funzioni simmetriche caratteristiche, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 38 (1903), 823-844.
  11. Grassi A., Hatsuda Y., Mariño M., Quantization conditions and functional equations in ABJ(M) theories, J. Phys. A: Math. Theor. 49 (2016), 115401, 30 pages, arXiv:1410.7658.
  12. Harnad J., Lee E., Symmetric polynomials, generalized Jacobi-Trudi identities and $\tau$-functions, arXiv:1304.0020.
  13. Hatsuda Y., Honda M., Moriyama S., Okuyama K., ABJM Wilson loops in arbitrary representations, J. High Energy Phys. 2013 (2013), no. 10, 168, 31 pages, arXiv:1306.4297.
  14. Hatsuda Y., Honda M., Okuyama K., Large $N$ non-perturbative effects in ${\mathcal N}=4$ superconformal Chern-Simons theories, J. High Energy Phys. 2015 (2015), no. 9, 046, 54 pages, arXiv:1505.07120.
  15. Hatsuda Y., Mariño M., Moriyama S., Okuyama K., Non-perturbative effects and the refined topological string, J. High Energy Phys. 2014 (2014), no. 9, 168, 42 pages, arXiv:1306.1734.
  16. Hatsuda Y., Moriyama S., Okuyama K., Instanton bound states in ABJM theory, J. High Energy Phys. 2013 (2013), no. 5, 054, 23 pages, arXiv:1301.5184.
  17. Hatsuda Y., Moriyama S., Okuyama K., Instanton effects in ABJM theory from Fermi gas approach, J. High Energy Phys. 2013 (2013), no. 1, 158, 40 pages, arXiv:1211.1251.
  18. Hatsuda Y., Moriyama S., Okuyama K., Exact instanton expansion of the ABJM partition function, Prog. Theor. Exp. Phys. 2015 (2015), 11B104, 35 pages, arXiv:1507.01678.
  19. Hatsuda Y., Okuyama K., Exact results for ABJ Wilson loops and open-closed duality, J. High Energy Phys. 2016 (2016), no. 10, 132, 34 pages, arXiv:1603.06579.
  20. Honda M., Exact relations between M2-brane theories with and without orientifolds, J. High Energy Phys. 2016 (2016), no. 6, 123, 28 pages, arXiv:1512.04335.
  21. Hosomichi K., Lee K.M., Lee S., Lee S., Park J., ${\mathcal N}=5,6$ superconformal Chern-Simons theories and M2-branes on orbifolds, J. High Energy Phys. 2008 (2008), no. 9, 002, 24 pages, arXiv:0806.4977.
  22. Jacobi C.G.J., De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum, J. Reine Angew. Math. 22 (1841), 360-371.
  23. Jing N., Rozhkovskaya N., Vertex operators arising from Jacobi-Trudi identities, Comm. Math. Phys. 346 (2016), 679-701, arXiv:1411.4725.
  24. Kapustin A., Willett B., Yaakov I., Exact results for Wilson loops in superconformal Chern-Simons theories with matter, J. High Energy Phys. 2010 (2010), no. 3, 089, 29 pages, arXiv:0909.4559.
  25. Kiyoshige K., Moriyama S., Dualities in ABJM matrix model from closed string viewpoint, J. High Energy Phys. 2016 (2016), no. 11, 096, 15 pages, arXiv:1607.06414.
  26. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  27. Kuniba A., Ohta Y., Suzuki J., Quantum Jacobi-Trudi and Giambelli formulae for $U_q\big(B^{(1)}_r\big)$ from the analytic Bethe ansatz, J. Phys. A: Math. Gen. 28 (1995), 6211-6226, hep-th/9506167.
  28. Macdonald I.G., Schur functions: theme and variations, in Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av., Vol. 498, University Louis Pasteur, Strasbourg, 1992, 5-39.
  29. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  30. Mariño M., Putrov P., ABJM theory as a Fermi gas, J. Stat. Mech. Theory Exp. 2012 (2012), P03001, 53 pages, arXiv:1110.4066.
  31. Matsuno S., Moriyama S., ABJ fractional brane from ABJM Wilson loop, J. High Energy Phys. 2014 (2014), no. 3, 079, 25 pages, arXiv:1310.8051.
  32. Matsuno S., Moriyama S., Giambelli identity in super Chern-Simons matrix model, J. Math. Phys. 58 (2017), 032301, 12 pages, arXiv:1603.04124.
  33. Moens E.M., van der Jeugt J., A determinantal formula for supersymmetric Schur polynomials, J. Algebraic Combin. 17 (2003), 283-307.
  34. Moriyama S., Nakayama S., Nosaka T., Instanton effects in rank deformed superconformal Chern-Simons theories from topological strings, J. High Energy Phys. 2017 (2017), no. 8, 003, 49 pages, arXiv:1704.04358.
  35. Moriyama S., Nosaka T., Partition functions of superconformal Chern-Simons theories from Fermi gas approach, J. High Energy Phys. 2014 (2014), no. 11, 164, 33 pages, arXiv:1407.4268.
  36. Moriyama S., Nosaka T., ABJM membrane instanton from a pole cancellation mechanism, Phys. Rev. D 92 (2015), 026003, 12 pages, arXiv:1410.4918.
  37. Moriyama S., Nosaka T., Exact instanton expansion of superconformal Chern-Simons theories from topological strings, J. High Energy Phys. 2015 (2015), no. 5, 022, 31 pages, arXiv:1412.6243.
  38. Moriyama S., Nosaka T., Orientifold ABJM matrix model: chiral projections and worldsheet instantons, J. High Energy Phys. 2016 (2016), no. 6, 068, 19 pages, arXiv:1603.00615.
  39. Moriyama S., Nosaka T., Yano K., Superconformal Chern-Simons theories from del Pezzo geometries, J. High Energy Phys. 2017 (2017), no. 11, 089, 40 pages, arXiv:1707.02420.
  40. Moriyama S., Suyama T., Instanton effects in orientifold ABJM theory, J. High Energy Phys. 2016 (2016), no. 3, 034, 19 pages, arXiv:1511.01660.
  41. Moriyama S., Suyama T., Orthosymplectic Chern-Simons matrix model and chirality projection, J. High Energy Phys. 2016 (2016), no. 4, 142, 19 pages, arXiv:1601.03846.
  42. Nakagawa J., Noumi M., Shirakawa M., Yamada Y., Tableau representation for Macdonald's ninth variation of Schur functions, in Physics and Combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 180-195.
  43. Nakai W., Nakanishi T., Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type $C_n$, SIGMA 3 (2007), 078, 20 pages, math.QA/0604158.
  44. Okuyama K., Orientifolding of the ABJ Fermi gas, J. High Energy Phys. (2016), no. 3, 008, 64 pages, arXiv:1601.03215.
  45. Pragacz P., Thorup A., On a Jacobi-Trudi identity for supersymmetric polynomials, Adv. Math. 95 (1992), 8-17.
  46. Sato M., Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds, in Random Systems and Dynamical Systems (Kyoto, 1981), RIMS Kokyuroku, Vol. 439, Kyoto, 1981, 30-46.
  47. Tierz M., Schur polynomials and biorthogonal random matrix ensembles, J. Math. Phys. 51 (2010), 063509, 9 pages.
  48. Tsuboi Z., From the quantum Jacobi-Trudi and Giambelli formula to a nonlinear integral equation for thermodynamics of the higher spin Heisenberg model, J. Phys. A: Math. Gen. 37 (2004), 1747-1758, cond-mat/0308333.

Previous article  Next article   Contents of Volume 14 (2018)