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

SIGMA 11 (2015), 067, 24 pages      arXiv:1411.1072

Harmonic Analysis and Free Field Realization of the Takiff Supergroup of ${\rm GL}(1|1)$

Andrei Babichenko a and Thomas Creutzig b
a) Department of Mathematics, Weizmann Institut, Rehovot, 76100, Israel
b) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received May 28, 2015, in final form August 01, 2015; Published online August 06, 2015

Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of $\mathfrak{gl}(1\vert 1)$, especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.

Key words: logarithmic CFT; Harmonic analysis; free field realization.

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  1. Alfes C., Creutzig T., The mock modular data of a family of superalgebras, Proc. Amer. Math. Soc. 142 (2014), 2265-2280, arXiv:1205.1518.
  2. Ashok S.K., Benichou R., Troost J., Asymptotic symmetries of string theory on ${\rm AdS}_3\times S^3$ with Ramond-Ramond fluxes, J. High Energy Phys. 2009 (2009), no. 10, 051, 31 pages, arXiv:0907.1242.
  3. Ashok S.K., Benichou R., Troost J., Conformal current algebra in two dimensions, J. High Energy Phys. 2009 (2009), no. 6, 017, 35 pages, arXiv:0903.4277.
  4. Babichenko A., Ridout D., Takiff superalgebras and conformal field theory, J. Phys. A: Math. Theor. 46 (2013), 125204, 26 pages, arXiv:1210.7094.
  5. Benichou R., Troost J., The conformal current algebra on supergroups with applications to the spectrum and integrability, J. High Energy Phys. 2010 (2010), no. 4, 121, 47 pages, arXiv:1002.3712.
  6. Candu C., Creutzig T., Mitev V., Schomerus V., Cohomological reduction of sigma models, J. High Energy Phys. 2010 (2010), no. 5, 047, 39 pages, arXiv:1001.1344.
  7. Candu C., Mitev V., Quella T., Saleur H., Schomerus V., The sigma model on complex projective superspaces, J. High Energy Phys. 2010 (2010), no. 2, 015, 48 pages, arXiv:0908.0878.
  8. Creutzig T., Branes in supergroups, arXiv:0908.1816.
  9. Creutzig T., Geometry of branes on supergroups, Nuclear Phys. B 812 (2009), 301-321, arXiv:0809.0468.
  10. Creutzig T., Hikida Y., Branes in the ${\rm OSP}(1\vert 2)$ WZNW model, Nuclear Phys. B 842 (2011), 172-224, arXiv:1004.1977.
  11. Creutzig T., Hikida Y., Rønne P.B., Supergroup - extended super Liouville correspondence, J. High Energy Phys. 2011 (2011), no. 6, 063, 26 pages, arXiv:1103.5753.
  12. Creutzig T., Quella T., Schomerus V., Branes in the $\rm GL(1\vert 1)$ WZNW model, Nuclear Phys. B 792 (2008), 257-283, arXiv:0708.0583.
  13. Creutzig T., Ridout D., Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46 (2013), 494006, 72 pages, arXiv:1303.0847.
  14. Creutzig T., Ridout D., Relating the archetypes of logarithmic conformal field theory, Nuclear Phys. B 872 (2013), 348-391, arXiv:1107.2135.
  15. Creutzig T., Ridout D., W-algebras extending $\widehat{\mathfrak{gl}}(1\vert 1)$, in Lie Theory and its Applications in Physics, Springer Proc. Math. Stat., Vol. 36, Springer, Tokyo, 2013, 349-367, arXiv:1111.5049.
  16. Creutzig T., Rønne P.B., The ${\rm GL}(1\vert 1)$-symplectic fermion correspondence, Nuclear Phys. B 815 (2009), 95-124, arXiv:0812.2835.
  17. Creutzig T., Rønne P.B., From world-sheet supersymmetry to super target spaces, J. High Energy Phys. 2010 (2010), no. 11, 021, 31 pages, arXiv:1006.5874.
  18. Creutzig T., Schomerus V., Boundary correlators in supergroup WZNW models, Nuclear Phys. B 807 (2009), 471-494, arXiv:0804.3469.
  19. Figueroa-O'Farrill J.M., Stanciu S., Nonsemisimple Sugawara construction, Phys. Lett. B 327 (1994), 40-46, hep-th/9402035.
  20. Figueroa-O'Farrill J.M., Stanciu S., Nonreductive WZW models and their CFTs, Nuclear Phys. B 458 (1996), 137-164, hep-th/9506151.
  21. Flohr M.A.I., Bits and pieces in logarithmic conformal field theory, Internat. J. Modern Phys. A 18 (2003), 4497-4591, hep-th/0111228.
  22. Frenkel I.B., Zhu Y., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
  23. Gaberdiel M.R., An algebraic approach to logarithmic conformal field theory, Internat. J. Modern Phys. A 18 (2003), 4593-4638, hep-th/0111260.
  24. Gainutdinov A.M., Jacobsen J.L., Saleur H., Vasseur R., A physical approach to the classification of indecomposable Virasoro representations from the blob algebra, Nuclear Phys. B 873 (2013), 614-681, arXiv:1212.0093.
  25. Gainutdinov A.M., Read N., Saleur H., Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the ${\mathfrak{gl}}(1|1)$ periodic spin chain, Howe duality and the interchiral algebra, arXiv:1207.6334.
  26. Gainutdinov A.M., Read N., Saleur H., Continuum limit and symmetries of the periodic ${\mathfrak{gl}}(1\vert 1)$ spin chain, Nuclear Phys. B 871 (2013), 245-288, arXiv:1112.3403.
  27. Geoffriau F., Sur le centre de l'algèbre enveloppante d'une algèbre de Takiff, Ann. Math. Blaise Pascal 1 (1994), 15-31.
  28. Götz G., Quella T., Schomerus V., The WZNW model on ${\rm PSU}(1,1\vert 2)$, J. High Energy Phys. 2007 (2007), no. 3, 003, 48 pages, hep-th/0610070.
  29. Guruswamy S., LeClair A., Ludwig A.W.W., ${\mathfrak{gl}}(N\vert N)$ super-current algebras for disordered Dirac fermions in two dimensions, Nuclear Phys. B 583 (2000), 475-512, cond-mat/9909143.
  30. Hikida Y., Schomerus V., Structure constants of the ${\rm OSp}(1\vert 2)$ WZNW model, J. High Energy Phys. 2007 (2007), no. 12, 100, 32 pages, arXiv:0711.0338.
  31. Kausch H.G., Symplectic fermions, Nuclear Phys. B 583 (2000), 513-541, hep-th/0003029.
  32. LeClair A., The ${\mathfrak{gl}}(1\vert 1)$ super-current algebra: the rôle of twist and logarithmic fields, Adv. Theor. Math. Phys. 13 (2009), 259-291, arXiv:0710.2906.
  33. Mathieu P., Ridout D., From percolation to logarithmic conformal field theory, Phys. Lett. B 657 (2007), 120-129, arXiv:0708.0802.
  34. Mitev V., Quella T., Schomerus V., Principal chiral model on superspheres, J. High Energy Phys. 2008 (2008), no. 11, 086, 45 pages, arXiv:0809.1046.
  35. Mitev V., Quella T., Schomerus V., Conformal superspace $\sigma$-models, J. Geom. Phys. 61 (2011), 1703-1716, arXiv:1210.8159.
  36. Mohammedi N., Wess-Zumino-Novikov-Witten models based on Lie superalgebras, Phys. Lett. B 331 (1994), 93-98, hep-th/9404132.
  37. Pearce P.A., Rasmussen J., Zuber J.-B., Logarithmic minimal models, J. Stat. Mech. Theory Exp. 2006 (2006), P11017, 36 pages, hep-th/0607232.
  38. Piroux G., Ruelle P., Logarithmic scaling for height variables in the abelian sandpile model, Phys. Lett. B 607 (2005), 188-196, cond-mat/0410253.
  39. Quella T., Schomerus V., Free fermion resolution of supergroup WZNW models, J. High Energy Phys. 2007 (2007), no. 9, 085, 51 pages, arXiv:0706.0744.
  40. Quella T., Schomerus V., Creutzig T., Boundary spectra in superspace $\sigma$-models, J. High Energy Phys. 2008 (2008), no. 10, 024, 26 pages, arXiv:0712.3549.
  41. Read N., Saleur H., Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions, Nuclear Phys. B 613 (2001), 409-444, hep-th/0106124.
  42. Read N., Saleur H., Associative-algebraic approach to logarithmic conformal field theories, Nuclear Phys. B 777 (2007), 316-351, hep-th/0701117.
  43. Rozansky L., Saleur H., $S$- and $T$-matrices for the super ${\rm U}(1,1)$ WZW model. Application to surgery and $3$-manifolds invariants based on the Alexander-Conway polynomial, Nuclear Phys. B 389 (1993), 365-423, hep-th/9203069.
  44. Saleur H., Schomerus V., On the ${\rm SU}(2\vert 1)$ WZNW model and its statistical mechanics applications, Nuclear Phys. B 775 (2007), 312-340, hep-th/0611147.
  45. Schomerus V., Saleur H., The ${\rm GL}(1\vert 1)$ WZW-model: from supergeometry to logarithmic CFT, Nuclear Phys. B 734 (2006), 221-245, hep-th/0510032.
  46. Takiff S.J., Rings of invariant polynomials for a class of Lie algebras, Trans. Amer. Math. Soc. 160 (1971), 249-262.
  47. Vasseur R., Jacobsen J.L., Saleur H., Logarithmic observables in critical percolation, J. Stat. Mech. Theory Exp. 2012 (2012), L07001, 11 pages, arXiv:1206.2312.
  48. Wilson B.J., A character formula for the category $\tilde{\mathcal O}$ of modules for affine ${\mathfrak{sl}}(2)$, Int. Math. Res. Not. 2008 (2008), Art. ID rnn 092, 29 pages, arXiv:0711.0727.
  49. Wilson B.J., Highest-weight theory for truncated current Lie algebras, J. Algebra 336 (2011), 1-27, arXiv:0705.1203.
  50. Zirnbauer M.R., Conformal field theory of the integer quantum Hall plateau transition, hep-th/9905054.

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