×

Generalized Kähler geometry and current algebras in classical \(N=2\) superconformal WZW model. (English) Zbl 1387.81329

Summary: I examine the Generalized Kähler (GK) geometry of classical \(N = (2, 2)\) superconformal WZW model on a compact group and relate the right-moving and left-moving Kac-Moody superalgebra currents to the GK geometry data using biholomorphic gerbe formulation and Hamiltonian formalism. It is shown that the canonical Poisson homogeneous space structure induced by the GK geometry of the group manifold is crucial to provide \(N = (2, 2)\) superconformal \(\sigma\)-model with the Kac-Moody superalgebra symmetries. Then, the biholomorphic gerbe geometry is used to prove that Kac-Moody superalgebra currents are globally defined.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Gepner, D., Nucl. Phys. B296, 757 (1987).
[2] Gates, S. J., Hull, C. M. and Roček, M., Nucl. Phys. B248, 157 (1984).
[3] M. Gualtieri, Generalized complex geometry, Ph.D. thesis, Oxford University, arXiv:math.DG/0401221. · Zbl 1235.32020
[4] Zabzine, M., Commun. Math. Phys.263, 711 (2006), arXiv:hep-th/0502137.
[5] Bredthauer, A., Lindström, U., Persson, J. and Zabzine, M., Lett. Math. Phys.77, 291 (2006), arXiv:hep-th/0603130.
[6] Zabzine, M., Arch. Math.42, 119 (2006), arXiv:hep-th/0605148v4.
[7] Heluani, R. and Zabzine, M., Adv. Math.223, 1815 (2010), arXiv:0812.4855 [hep-th].
[8] Heluani, R. and Zabzine, M., Commun. Math. Phys.306, 333 (2011), arXiv:1006.2773 [hep-th].
[9] Ph. Spindel, A. Sevrin, W. Troost and A. van Proeyen, Nucl. Phys. B308, 662 (1988); Nucl. Phys. B311, 465 (1988).
[10] Parkhomenko, S. E., J. Exp. Theor. Phys.102, 3 (1992).
[11] Parkhomenko, S. E., Mod. Phys. Lett. A11, 445 (1996), arXiv:hep-th/9503071.
[12] E. Getzler, arXiv:hep-th/9307041.
[13] Kac, V. and Todorov, I., Commun. Math. Phys.102, 337 (1985).
[14] Fuchs, J., Nucl. Phys. B318, 631 (1989).
[15] Sevrin, A. and Troost, J., Nucl. Phys. B492, 623 (1997), arXiv:hep-th/9610102.
[16] Roček, M., Schoutens, K. K. and Sevrin, A., Phys. Lett. B265, 303 (1991).
[17] Hull, C. M., Lindstrom, U., Roc̈ek, M., von Unge, R. and Zabzine, M., J. High Energy Phys.0910, 062 (2009), arXiv:0811.3615 [hep-th].
[18] Sevrin, A., Staessens, W. and Wijns, A., J. High Energy Phys.0909, 105 (2009), arXiv:0908.2756 [hep-th].
[19] A. Sevrin, W. Staessens and D. Terryn, The generalized Kähler geometry of \(N = (2, 2)\) WZW models, arXiv:1111.0551 [hep-th]. · Zbl 1306.81277
[20] Parkhomenko, S. E., Mod. Phys. Lett. A32, 1750076 (2017), arXiv:1701.05229 [hep-th].
[21] Kazama, Y. and Suzuki, H., Mod. Phys. Lett. A4, 235 (1989).
[22] Kazama, Y. and Suzuki, H., Phys. Lett. B216, 112 (1989).
[23] Kazama, Y. and Suzuki, H., Nucl. Phys. B321, 232 (1989).
[24] Parkhomenko, S., JETP Lett.100, 545 (2014), arXiv:1410.2977 [hep-th].
[25] Hull, C. M. and Spence, B., Phys. Lett. B241, 357 (1990).
[26] Di Veccia, P., Knizhnik, V. G., Petersen, J. L. and Rossi, P., Nucl. Phys. B253, 701 (1985).
[27] Polyakov, A. and Wiegmann, P., Phys. Lett. B131, 121 (1983).
[28] R. Heluani, Supersymmetry of the chiral de Rham complex II: Commuting sectors, arXiv:0806.1021v2 [math.QA]. · Zbl 1164.81014
[29] Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, , Vol. 107 (Birkhauser, Boston, 1993). · Zbl 0823.55002
[30] Ekstrand, J., Heluani, R., Kallen, J. and Zabzine, M., Adv. Theor. Math. Phys.13, 1221 (2009), arXiv:0905.4447 [hep-th].
[31] Drinfeld, V. G., Quantum Groups, Proc. Int. Congress of Mathematicians, Vol. 1, Berkeley, California, , 1986, p. 798.
[32] Semenov-Tian-Shansky, M. A., RIMS Kyoto Univ.21, 1237 (1985).
[33] Lu, J.-H. and Weinstein, A., J. Diff. Geom.31, 501 (1990).
[34] Parkhomenko, S. E., Nucl. Phys. B510, 623 (1998), arXiv:hep-th/9706199.
[35] Alekseev, A. Yu. and Malkhin, A. Z., Commun. Math. Phys.162, 147 (1994).
[36] Alekseev, A. and Strobl, T., J. High Energy Phys.0503, 035 (2005).
[37] Malikov, F., Commun. Math. Phys.278, 487 (2008), arXiv:math/0604093.
[38] Zabzine, M., Lett. Math. Phys.90, 373 (2009), arXiv:0906.1056 [math.SG].
[39] A. Alekseev, H. Burstyn and E. Meinrenken, arXiv:0709.1452 [math.DG].
[40] F. Malikov, V. Schechtman and A. Vaintrob, Chiral de Rham complex, arXiv:alg-geom/9803041. · Zbl 0952.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.