Superbosonization via Riesz superdistributions. (English) Zbl 1315.58008
The authors give a new conceptual interpretation of the superbosonization identity of P. Littelmann et al. [Commun. Math. Phys. 283, No. 2, 343–395 (2008; Zbl 1156.82008)] that links it to harmonic superanalysis of Lie supergroups and symmetric superspaces, and in particular, to a supergeneralization of the Riesz distributions. The proof is reduced to computing the Gindikin gamma function of a Riemannian symmetric superspace, which is determined explicitly.
Reviewer: Yuri E. Gliklikh (Voronezh)
MSC:
| 58C50 | Analysis on supermanifolds or graded manifolds |
| 22E30 | Analysis on real and complex Lie groups |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 46S60 | Functional analysis on superspaces (supermanifolds) or graded spaces |
Keywords:
superbosonization identity; harmonic superanalysis; Lie supergroups; supergeneralization of the Riesz distributionsCitations:
Zbl 1156.82008References:
| [1] | A.Alldridge, ‘The Harish-Chandra isomorphism for reductive symmetric superpairs’, Transform. Groups17 (2012), 889-919. · Zbl 1293.17012 |
| [2] | A.Alldridge and J.Hilgert, ‘Invariant Berezin integration on homogeneous supermanifolds’, J. Lie Theory20 (2010), 65-91. · Zbl 1192.58004 |
| [3] | A.Alldridge, J.Hilgert and W.Palzer, ‘Berezin integration on non-compact supermanifolds’, J. Geom. Phys.62 (2012), 427-448. · Zbl 1243.58006 |
| [4] | A.Alldridge, J.Hilgert and T.Wurzbacher, ‘Singular superspaces’, Math. Z. (2014), under revision. · Zbl 1300.32004 |
| [5] | A.Altland and M. R.Zirnbauer, ‘Nonstandard symmetry classes in mesoscopic normal – superconducting hybrid structures’, Phys. Rev. B55 (1997), 1142-1161. |
| [6] | G. E.Bredon, Sheaf Theory, 2nd Edn, (Springer-Verlag, Berlin, New York, 1997). · Zbl 0874.55001 |
| [7] | C.Carmeli, L.Caston and R.Fioresi, Mathematical Foundations of Supersymmetry, (European Mathematical Society, Zurich, 2011). · Zbl 1226.58003 |
| [8] | P.Deligne and J. W.Morgan, ‘Notes on supersymmetry (following Joseph Bernstein)’, in:Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) (Amer. Math. Soc., Providence, RI, 1999), 41-97. · Zbl 1170.58302 |
| [9] | JDieudonné, Treatise on Analysis. Vol. VI, (Academic Press, New York, 1978). · Zbl 0435.43001 |
| [10] | K.Efetov, Supersymmetry in Disorder and Chaos, (Cambridge University Press, Cambridge, 1997). · Zbl 0990.82501 |
| [11] | K.Efetov, G.Schwiete and K.Takahashi, ‘Bosonization for disordered and chaotic systems’, Phys. Rev. Lett.92 (2004), 026807. |
| [12] | L.Ehrenpreis, ‘Analytic functions and the Fourier transform of distributions. I’, Ann. of Math. (2)63 (1956), 129-159. · Zbl 0139.08004 |
| [13] | J.Faraut and A.Korányi, ‘Function spaces and reproducing kernels on bounded symmetric domains’, J. Funct. Anal.88 (1990), 64-89. · Zbl 0718.32026 |
| [14] | J.Faraut and A.Korányi, Analysis on Symmetric Cones, (Oxford University Press, Oxford, 1994). · Zbl 0841.43002 |
| [15] | D. S.Freed and G. W.Moore, ‘Twisted equivariant matter’, Ann. Henri Poincaré (2013), 1-97. · Zbl 1286.81109 |
| [16] | Y. V.Fyodorov, ‘Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation’, Nuclear Phys. B621 (2002), 643-674. · Zbl 1024.82013 |
| [17] | I. M.Gel’fand and G. E.Shilov, Generalized Functions, Vol. 1, (Academic Press, New York, 1964). · Zbl 0115.33101 |
| [18] | I. M.Gel’fand and G. E.Shilov, Generalized Functions, Vol. 2, (Academic Press, New York, 1977). |
| [19] | T.Guhr, ‘Arbitrary unitarily invariant random matrix ensembles and supersymmetry’, J. Phys. A39 (2006), 13191-13223. · Zbl 1206.82051 |
| [20] | V. W.Guillemin and S.Sternberg, Supersymmetry and Equivariant De Rham Theory (Springer-Verlag, Berlin, New York, 1999). · Zbl 0934.55007 |
| [21] | G.Hackenbroich and H.Weidenmüller, ‘Universality of random-matrix results for non-gaussian ensembles’, Phys. Rev. Lett.74 (1995), 4118-4121. |
| [22] | P.Heinzner, A.Huckleberry and M. R.Zirnbauer, ‘Symmetry classes of disordered fermions’, Comm. Math. Phys.257 (2005), 725-771. · Zbl 1092.82020 |
| [23] | J.Hilgert and K.-H.Neeb, ‘Vector valued Riesz distributions on Euclidian Jordan algebras’, J. Geom. Anal.11 (2001), 43-75. · Zbl 0989.22021 |
| [24] | A. E.Ingham, ‘An integral which occurs in statistics’, Proc. Cambridge Philos. Soc.29 (1933), 271-276. · Zbl 0007.00701 |
| [25] | T.Ishihara, ‘On generalized Laplace transforms’, Proc. Japan Acad.37 (1961), 556-561. · Zbl 0154.13801 |
| [26] | B.Iversen, Cohomology of Sheaves, (Springer-Verlag, Berlin, New York, 1986). · Zbl 1272.55001 |
| [27] | G.Kainz, A.Kriegl and P.Michor, ‘<![CDATA \([C^\infty ]]\)>-algebras from the functional analytic viewpoint’, J. Pure Appl. Algebra46 (1987), 89-107. · Zbl 0621.46046 |
| [28] | M.Kieburg, J.Grönqvist and T.Guhr, ‘Arbitrary rotation invariant random matrix ensembles and supersymmetry: orthogonal and unitary – symplectic case’, J. Phys. A42 (2009), 275205, 31. · Zbl 1167.81020 |
| [29] | M.Kieburg, H.-J.Sommers and T.Guhr, ‘A comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation’, J. Phys. A.42 (2009), 275206, 23. · Zbl 1167.81021 |
| [30] | N.Lehmann, SaherD., V. V.Sokolov and H.-J.Sommers, ‘Chaotic scattering—the supersymmetry method for large number of channels’, Nucl. Phys. A582 (1995), 223-256. |
| [31] | D. A.Leĭtes, ‘Introduction to the theory of supermanifolds’, Uspekhi Mat. Nauk35 (1980), 3-57, 255. · Zbl 0439.58007 |
| [32] | P.Littelmann, H.-J.Sommers and M. R.Zirnbauer, ‘Superbosonization of invariant random matrix ensembles’, Comm. Math. Phys.283 (2008), 343-395. · Zbl 1156.82008 |
| [33] | S.MacLane, Categories for the Working Mathematician, 2nd Edn, (Springer-Verlag, Berlin, New York, 1998). · Zbl 0705.18001 |
| [34] | Y. I.Manin, Gauge Field Theory and Complex Geometry, (Springer-Verlag, Berlin, New York, 1988). · Zbl 0641.53001 |
| [35] | R.Meise and D.Vogt, Introduction to Functional Analysis, (Oxford University Press, Oxford, 1997). · Zbl 0924.46002 |
| [36] | B.Mitiagin, S.Rolewicz and W.Żelazko, ‘Entire functions in <![CDATA \([B_0]]\)>-algebras’, Studia Math.21 (1961/2), 291-306. · Zbl 0111.31001 |
| [37] | P.-E.Paradan, ‘Symmetric spaces of the non-compact type: Lie groups’, in:Géométries à courbure négative ou nulle, groupes discrets et rigidités, Séminaire Congrésvol. 18 (Soc. Math. France, Paris, 2009), 39-76. · Zbl 1197.22002 |
| [38] | H. H.Schaefer, Topological Vector Spaces, (Springer-Verlag, Berlin, New York, 1971). · Zbl 0212.14001 |
| [39] | L.Schäfer and F.Wegner, ‘Disordered system with <![CDATA \([n]]\)> orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes’, Z. Phys. B38 (1980), 113-126. |
| [40] | L.Schwartz, Théorie des distributions. Tome II. Hermann, Paris 1951. · Zbl 0042.11405 |
| [41] | L.Schwartz, ‘Transformation de Laplace des distributions’, Medd. Lunds Univ. Mat. Sem.1952 (1952), 196-206. Tôme Supplémentaire. · Zbl 0047.34903 |
| [42] | C. L.Siegel, ‘Über die analytische Theorie der quadratischen Formen’, Ann. of Math. (2)36 (1935), 527-606. · JFM 61.0140.01 |
| [43] | H.-J.Sommers, ‘Superbosonization’, Acta Phys. Polon. B38 (2007), 4105-4110. · Zbl 1371.82080 |
| [44] | M.Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, (Publish or Perish Inc., Houston, TX, 1979). · Zbl 0439.53002 |
| [45] | F.Trèves, Topological Vector Spaces, Distributions, and Kernels, (Academic Press, New York, London, 1967). · Zbl 0171.10402 |
| [46] | M.Vergne and H.Rossi, ‘Analytic continuation of the holomorphic discrete series of a semi-simple Lie group’, Acta Math.136 (1976), 1-59. · Zbl 0356.32020 |
| [47] | M. R.Zirnbauer, ‘Fourier analysis on a hyperbolic supermanifold with constant curvature’, Comm. Math. Phys.141 (1991), 503-522. · Zbl 0746.58014 |
| [48] | M. R.Zirnbauer, ‘Super Fourier analysis and localization in disordered wires’, Phys. Rev. Lett.69 (1992), 1584-1587. · Zbl 0968.82529 |
| [49] | M. R.Zirnbauer, ‘Riemannian symmetric superspaces and their origin in random-matrix theory’, J. Math. Phys.37 (1996), 4986-5018. · Zbl 0871.58005 |
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