From Jack to Double Jack Polynomials via the Supersymmetric Bridge

Luc Lapointe ; Pierre Mathieu

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

The Calogero–Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero–Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak {sl}}_2}$ algebra.

Détail

Keywords: Jack polynomials; supersymmetry; Calogero–Sutherland model; integrable quantum many-body problem; affine algebra.

Bibliographie

[1] Alarie-Vézina L., Desrosiers P., Mathieu P., “Ramond singular vectors and Jack superpolynomials”, J. Phys. A: Math. Theor., 47 (2014), 035202, 17 pp., arXiv: 1309.7965 | DOI | Zbl

[2] 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 (2011), 33–64, arXiv: 1012.1312 | DOI | MR | Zbl

[3] Alday L. F., Gaiotto D., Tachikawa Y., “Liouville correlation functions from four-dimensional gauge theories”, Lett. Math. Phys., 91 (2010), 167–197, arXiv: 0906.3219 | DOI | MR | Zbl

[4] Awata H., Matsuo Y., Odake S., Shiraishi J., “Collective field theory, Calogero–Sutherland model and generalized matrix models”, Phys. Lett. B, 347 (1995), 49–55, arXiv: hep-th/9411053 | DOI | MR | Zbl

[5] Awata H., Matsuo Y., Odake S., Shiraishi J., “Excited states of the Calogero–Sutherland model and singular vectors of the $W_N$ algebra”, Nuclear Phys. B, 449 (1995), 347–374, arXiv: hep-th/9503043 | DOI | MR | Zbl

[6] Bergeron F., Algebraic combinatorics and coinvariant spaces, CMS Treatises in Mathematics, Canadian Mathematical Society, Ottawa, ON; A.K. Peters, Ltd., Wellesley, MA, 2009 | DOI | MR | Zbl

[7] Bernevig B. A., Gurarie V., Simon S. H., “Central charge and quasihole scaling dimensions from model wavefunctions: toward relating Jack wavefunctions to ${\mathcal W}$-algebras”, J. Phys. A: Math. Theor., 42 (2009), 245206, 30 pp., arXiv: 0903.0635 | DOI | MR | Zbl

[8] Bernevig B. A., Haldane F. D. M., “Properties of non-Abelian fractional quantum Hall states at filling $\nu=k/r$”, Phys. Rev. Lett., 101 (2008), 246806, 4 pp., arXiv: 0803.2882 | DOI

[9] Blondeau-Fournier O., Lapointe L., Mathieu P., “Double Macdonald polynomials as the stable limit of Macdonald superpolynomials”, J. Algebraic Combin., 41 (2015), 397–459, arXiv: 1211.3186 | DOI | MR | Zbl

[10] Blondeau-Fournier O., Lapointe L., Mathieu P., “The supersymmetric Ruijsenaars–Schneider model”, Phys. Rev. Lett., 114 (2015), 121602, 5 pp., arXiv: 1403.4667 | DOI | MR

[11] Cardy J., “Calogero–Sutherland model and bulk-boundary correlations in conformal field theory”, Phys. Lett. B, 582 (2004), 121–126, arXiv: hep-th/0310291 | DOI | MR | Zbl

[12] Desrosiers P., Hallnäs M., “Hermite and Laguerre symmetric functions associated with operators of Calogero–Moser–Sutherland type”, SIGMA, 8 (2012), 049, 51 pp., arXiv: 1103.4593 | DOI | MR | Zbl

[13] Desrosiers P., Lapointe L., Mathieu P., “Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials”, Nuclear Phys. B, 606 (2001), 547–582, arXiv: hep-th/0103178 | DOI | MR | Zbl

[14] Desrosiers P., Lapointe L., Mathieu P., “Jack polynomials in superspace”, Comm. Math. Phys., 242 (2003), 331–360, arXiv: hep-th/0209074 | DOI | MR | Zbl

[15] Desrosiers P., Lapointe L., Mathieu P., “Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals”, Comm. Math. Phys., 316 (2012), 395–440, arXiv: 1109.2832 | DOI | MR | Zbl

[16] Desrosiers P., Lapointe L., Mathieu P., “Superconformal field theory and Jack superpolynomials”, J. High Energy Phys., 2012:9 (2012), 037, 42 pp., arXiv: 1205.0784 | DOI | MR

[17] Doyon B., Cardy J., “Calogero–Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators”, J. Phys. A: Math. Theor., 40 (2007), 2509–2540, arXiv: hep-th/0611054 | DOI | MR | Zbl

[18] Dunfield N. M., Gukov S., Rasmussen J., “The superpolynomial for knot homologies”, Experiment. Math., 15 (2006), 129–159, arXiv: math.GT/0505662 | DOI | MR | Zbl

[19] Estienne B., Santachiara R., “Relating Jack wavefunctions to $WA_{k-1}$ theories”, J. Phys. A: Math. Theor., 42 (2009), 445209, 15 pp., arXiv: 0906.1969 | DOI | MR | Zbl

[20] Feigin B., Jimbo M., Miwa T., Mukhin E., “A differential ideal of symmetric polynomials spanned by Jack polynomials at $\beta=-(r-1)/(k+1)$”, Int. Math. Res. Not., 2002 (2002), 1223–1237, arXiv: math.QA/0112127 | DOI | MR | Zbl

[21] Forrester P. J., Log-gases and random matrices, London Mathematical Society Monographs Series, 34, Princeton University Press, Princeton, NJ, 2010 | DOI | MR | Zbl

[22] Guhr T., Kohler H., “Supersymmetry and models for two kinds of interacting particles”, Phys. Rev. E, 71 (2004), 045102, 4 pp., arXiv: math-ph/0408033 | DOI | MR

[23] Kohler H., Guhr T., “Supersymmetric extensions of Calogero–Moser–Sutherland-like models: construction and some solutions”, J. Phys. A: Math. Gen., 38 (2005), 9891–9915, arXiv: math-ph/0510039 | DOI | MR | Zbl

[24] Kuramoto Y., Kato Y., Dynamics of one-dimensional quantum systems. Inverse-square interaction models, Cambridge University Press, Cambridge, 2009 | Zbl

[25] Lapointe L., Vinet L., “Exact operator solution of the Calogero–Sutherland model”, Comm. Math. Phys., 178 (1996), 425–452 | DOI | MR | Zbl

[26] Lascoux A., Symmetric functions and combinatorial operators on polynomials, Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, 99, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[27] Macdonald I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR

[28] Mimachi K., Yamada Y., “Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials”, Comm. Math. Phys., 174 (1995), 447–455 | DOI | MR | Zbl

[29] Minahan J. A., Polychronakos A. P., “Density-correlation functions in Calogero–Sutherland models”, Phys. Rev. B, 50 (1994), 4236–4239, arXiv: hep-th/9404192 | DOI

[30] Morozov A., Smirnov A., “Towards the proof of AGT relations with the help of the generalized Jack polynomials”, Lett. Math. Phys., 104 (2014), 585–612, arXiv: 1307.2576 | DOI | MR | Zbl

[31] Nekrasov N. A., “Seiberg–Witten prepotential from instanton counting”, Adv. Theor. Math. Phys., 7 (2003), 831–864, arXiv: hep-th/0206161 | MR | Zbl

[32] Olshanetsky M. A., Perelomov A. M., “Classical integrable finite-dimensional systems related to Lie algebras”, Phys. Rep., 71 (1981), 313–400 | DOI | MR

[33] Olshanetsky M. A., Perelomov A. M., “Quantum integrable systems related to Lie algebras”, Phys. Rep., 94 (1983), 313–404 | DOI | MR

[34] Ridout D., Wood S., “From Jack polynomials to minimal model spectra”, J. Phys. A: Math. Theor., 48 (2015), 04520, 17 pp., arXiv: 1409.4847 | DOI | MR

[35] Ridout D., Wood S., “Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models”, Nuclear Phys. B, 894 (2015), 621–664, arXiv: 1501.0731 | DOI | MR

[36] Sakamoto R., Shiraishi J., Arnaudon D., Frappat L., Ragoucy E., “Correspondence between conformal field theory and Calogero–Sutherland model”, Nuclear Phys. B, 704 (2005), 490–509, arXiv: hep-th/0407267 | DOI | MR | Zbl

[37] Sergeev A. N., “Superanalogs of the Calogero operators and Jack polynomials”, J. Nonlinear Math. Phys., 8 (2001), 59–64, arXiv: math.RT/0106222 | DOI | MR | Zbl

[38] Sergeev A. N., Veselov A. P., “Deformed quantum Calogero–Moser problems and Lie superalgebras”, Comm. Math. Phys., 245 (2004), 249–278, arXiv: math-ph/0303025 | DOI | MR | Zbl

[39] Sergeev A. N., Veselov A. P., “Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials”, Adv. Math., 192 (2005), 341–375, arXiv: math-ph/0307036 | DOI | MR | Zbl

[40] Shastry B. S., Sutherland B., “Super Lax pairs and infinite symmetries in the $1/r^2$ system”, Phys. Rev. Lett., 70 (1993), 4029–4033, arXiv: cond-mat/9212029 | DOI | MR | Zbl

[41] Stembridge J. R., “A characterization of supersymmetric polynomials”, J. Algebra, 95 (1985), 439–444 | DOI | MR | Zbl

[42] Sutherland B., “Quantum many-body problem in one dimension: ground state”, J. Math. Phys., 12 (1971), 246–250 | DOI

[43] Sutherland B., “Exact results for a quantum many-body problem in one dimension”, Phys. Rev. A, 4 (1971), 2019–2021 | DOI

[44] Sutherland B., “Exact results for a quantum many-body problem in one-dimension, II”, Phys. Rev. A, 5 (1972), 1372–1376 | DOI

[45] van Diejen J. F., Vinet L. (eds.), Calogero–Moser–Sutherland models, CRM Series in Mathematical Physics, Springer-Verlag, New York, 2000 | DOI | MR | Zbl

[46] Wang W., “Rationality of Virasoro vertex operator algebras”, Int. Math. Res. Not., 1993 (1993), 197–211 | DOI | MR | Zbl