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

SIGMA 14 (2018), 066, 20 pages      arXiv:1705.01755

Quantum Klein Space and Superspace

Rita Fioresi a, Emanuele Latini ab and Alessio Marrani c
a) Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
b) INFN, Sez. di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
c) Museo Storico della Fisica e Centro Studi e Ricerche ''Enrico Fermi'', Via Panisperna 89A, I-00184, Roma, Italy

Received February 23, 2018, in final form June 15, 2018; Published online June 28, 2018

We give an algebraic quantization, in the sense of quantum groups, of the complex Minkowski space, and we examine the real forms corresponding to the signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum metrics and providing an explicit presentation of the quantized coordinate algebras. In particular, we focus on the Kleinian signature $(2,2)$. The quantizations of the complex and real spaces come together with a coaction of the quantizations of the respective symmetry groups. We also extend such quantizations to the $\mathcal{N}=1$ supersetting.

Key words: quantum groups; supersymmetry.

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  1. Ballesteros A., Gutiérrez-Sagredo I., Herranz F.J., Meusburger C., Naranjo P., Quantum groups and noncommutative spacetimes with cosmological constant, J. Phys. Conf. Ser. 880 (2017), 012023, 8 pages, arXiv:1702.04704.
  2. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries, J. Phys. A: Math. Gen. 26 (1993), 5801-5823.
  3. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., Quantum $(2+1)$ kinematical algebras: a global approach, J. Phys. A: Math. Gen. 27 (1994), 1283-1297.
  4. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., A new ''null-plane'' quantum Poincaré algebra, Phys. Lett. B 351 (1995), 137-145, q-alg/9502019.
  5. Ballesteros A., Herranz F.J., Meusburger C., Naranjo P., Twisted $(2+1)$ $\kappa$-AdS algebra, Drinfel'd doubles and non-commutative spacetimes, SIGMA 10 (2014), 052, 26 pages, arXiv:1403.4773.
  6. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  7. Bonechi F., Giachetti R., Sorace E., Tarlini M., Induced representations of the one-dimensional quantum Galilei group, J. Math. Sci. 104 (2001), 1105-1110.
  8. Borowiec A., Lukierski J., Tolstoy V.N., Quantum deformations of $D = 4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries in unified ${\mathfrak o}(4,{\mathbb C})$ setting, Phys. Lett. B 754 (2016), 176-181, arXiv:1511.03653.
  9. Borowiec A., Lukierski J., Tolstoy V.N., Addendum to ''Quantum deformations of $D = 4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries in unified ${\mathfrak o}(4,{\mathbb C})$ setting'', Phys. Lett. B 770 (2017), 426-430, arXiv:1704.06852.
  10. Borowiec A., Lukierski J., Tolstoy V.N., Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak o}^\star(4)$ symmetries, J. High Energy Phys. 2017 (2017), no. 11, 187, 35 pages, arXiv:1708.09848.
  11. Bryant R.L., Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4, Soc. Math. France, Paris, 2000, 53-94, math.DG/0004073.
  12. Carmeli C., Caston L., Fioresi R., Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011.
  13. Cervantes D., Fioresi R., Lledó M.A., On chiral quantum superspaces, in Supersymmetry in Mathematics and Physics, Lecture Notes in Math., Vol. 2027, Springer, Heidelberg, 2011, 69-99, arXiv:1109.3632.
  14. Cervantes D., Fioresi R., Lledó M.A., The quantum chiral Minkowski and conformal superspaces, Adv. Theor. Math. Phys. 15 (2011), 565-620, arXiv:1007.4469.
  15. Cervantes D., Fioresi R., Lledó M.A., Nadal F.A., Quadratic deformation of Minkowski space, Fortschr. Phys. 60 (2012), 970-976, arXiv:1207.1316.
  16. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  17. Chen P., Chiang H.-W., Hu Y.-C., A quantized spacetime based on ${\rm Spin}(3,1)$ symmetry, Internat. J. Modern Phys. D 25 (2016), 1645004, 6 pages, arXiv:1606.01490.
  18. Cianfrani F., Kowalski-Glikman J., Pranzetti D., Rosati G., Symmetries of quantum spacetime in three dimensions, Phys. Rev. D 94 (2016), 084044, 17 pages, arXiv:1606.03085.
  19. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  20. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  21. Douglas M.R., Nekrasov N.A., Noncommutative field theory, Rev. Modern Phys. 73 (2001), 977-1029, hep-th/0106048.
  22. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  23. Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), 1205-1222, math.DG/0102225.
  24. Dunajski M., Einstein-Maxwell dilaton metrics from three-dimensional Einstein-Weyl structures, Classical Quantum Gravity 23 (2006), 2833-2839, gr-qc/0601014.
  25. Dunajski M., West S., Anti-self-dual conformal structures in neutral signature, math.DG/0610280.
  26. Faddeev L.D., Reshetikhin N.Y., Takhtajan L.A., Quantization of Lie groups and Lie algebras, in Algebraic Analysis, Vol. I, Academic Press, Boston, MA, 1988, 129-139.
  27. Fioresi R., Quantizations of flag manifolds and conformal space time, Rev. Math. Phys. 9 (1997), 453-465.
  28. Fioresi R., Quantum deformation of the flag variety, Comm. Algebra 27 (1999), 5669-5685.
  29. Fioresi R., On algebraic supergroups and quantum deformations, J. Algebra Appl. 2 (2003), 403-423, math.QA/0111113.
  30. Fioresi R., Latini E., The symplectic origin of conformal and Minkowski superspaces, J. Math. Phys. 57 (2016), 022307, 12 pages, arXiv:1506.09086.
  31. Fioresi R., Latini E., Marrani A., Klein and conformal superspaces, split algebras and spinor orbits, Rev. Math. Phys. 29 (2017), 1750011, 37 pages, arXiv:1603.09063.
  32. Fioresi R., Lledó M.A., The Minkowski and conformal superspaces. The classical and quantum descriptions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
  33. Fioresi R., Lledó M.A., Varadarajan V.S., The Minkowski and conformal superspaces, J. Math. Phys. 48 (2007), 113505, 27 pages, math.RA/0609813.
  34. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
  35. Freidel L., Livine E.R., Ponzano-Regge model revisited. III. Feynman diagrams and effective field theory, Classical Quantum Gravity 23 (2006), 2021-2061, hep-th/0502106.
  36. Garay L.J., Quantum gravity and minimum length, Internat. J. Modern Phys. A 10 (1995), 145-166, gr-qc/9403008.
  37. Girelli F., Sellaroli G., ${\rm SO}^*(2N)$ coherent states for loop quantum gravity, J. Math. Phys. 58 (2017), 071708, 31 pages, arXiv:1701.07519.
  38. Gromov N.A., Man'ko V.I., Contractions of the irreducible representations of the quantum algebras ${\rm su}_q(2)$ and ${\rm so}_q(3)$, J. Math. Phys. 33 (1992), 1374-1378.
  39. Heckman J.J., Verlinde H., Covariant non-commutative space-time, Nuclear Phys. B 894 (2015), 58-74, arXiv:1401.1810.
  40. Hervik S., Pseudo-Riemannian VSI spaces II, Classical Quantum Gravity 29 (2012), 095011, 16 pages, arXiv:1504.01616.
  41. Hull C., Zwiebach B., Double field theory, J. High Energy Phys. 2009 (2009), no. 9, 099, 53 pages, arXiv:0904.4664.
  42. Klemm D., Nozawa M., Geometry of Killing spinors in neutral signature, Classical Quantum Gravity 32 (2015), 185012, 36 pages, arXiv:1504.02710.
  43. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  44. Lakshmibai V., Reshetikhin N., Quantum flag and Schubert schemes, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math., Vol. 134, Amer. Math. Soc., Providence, RI, 1992, 145-181.
  45. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  46. Lukierski J., Ruegg H., Zakrzewski W.J., Classical and quantum mechanics of free $k$-relativistic systems, Ann. Physics 243 (1995), 90-116, hep-th/9312153.
  47. Maggiore M., A generalized uncertainty principle in quantum gravity, Phys. Lett. B 204 (1993), 65-69, hep-th/9301067.
  48. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  49. Majid S., Ruegg H., Bicrossproduct structure of $\kappa$-Poincaré group and non-commutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9405107.
  50. Manin Yu.I., Gauge fields and holomorphic geometry, J. Sov. Math. 21 (1983), 465-507.
  51. Manin Yu.I., Topics in noncommutative geometry, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991.
  52. Manin Yu.I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, Vol. 289, 2nd ed., Springer-Verlag, Berlin, 1997.
  53. Maślanka P., The $n$-dimensional $\kappa$-Poincaré algebra and group, J. Phys. A: Math. Gen. 26 (1993), L1251-L1253.
  54. Ogievetsky O., Schmidke W.B., Wess J., Zumino B., $q$-deformed Poincaré algebra, Comm. Math. Phys. 150 (1992), 495-518.
  55. Ooguri H., Vafa C., Self-duality and $N=2$ string magic, Modern Phys. Lett. A 5 (1990), 1389-1398.
  56. Penrose R., Twistor algebra, J. Math. Phys. 8 (1967), 345-366.
  57. Penrose R., The twistor programme, Rep. Math. Phys. 12 (1977), 65-76.
  58. Phung H.H., On the structure of quantum super groups ${\rm GL}_q(m|n)$, J. Algebra 211 (1999), 363-383, q-alg/9511023.
  59. Podleś P., Woronowicz S.L., Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990), 381-431.
  60. Rennecke F., $O(d,d)$-duality in string theory, J. High Energy Phys. 2014 (2014), no. 10, 069, 22 pages, arXiv:1404.0912.
  61. Semenov-Tyan-Shanskii M.A., What is a classical $r$-matrix?, Funct. Anal. Appl. 17 (1983), 259-272.
  62. Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  63. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, hep-th/0109162.
  64. Taft E., Towber J., Quantum deformation of flag schemes and Grassmann schemes. I. A $q$-deformation of the shape-algebra for ${\rm GL}(n)$, J. Algebra 142 (1991), 1-36.
  65. Witten E., Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004), 189-258, hep-th/0312171.
  66. Yaglom I.M., A simple non-Euclidean geometry and its physical basis. An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Springer-Verlag, New York - Heidelberg, 1979.
  67. Yang C.N., On quantized space-time, Phys. Rev. 72 (1947), 874.
  68. Zakrzewski S., Quantum Poincaré group related to the $\kappa$-Poincaré algebra, J. Phys. A: Math. Gen. 27 (1994), 2075-2082.
  69. Zhang H., Zhang R.B., Dual canonical bases for the quantum general linear supergroup, J. Algebra 304 (2006), 1026-1058, math.QA/0510186.

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