Extended Sugawara construction for the superalgebras <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">SU</mi><mn></mn><mo>(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mn></mn><mo>|</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>. II. The third-order Casimir algebra

Peter Bouwknegt; Anna Ceresole; Peter van Nieuwenhuizen; Jim McCarthy

doi:10.1103/PhysRevD.40.415

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Extended Sugawara construction for the superalgebras $SU (M + 1 | N + 1)$ . II. The third-order Casimir algebra

Peter Bouwknegt

Anna Ceresole and Peter van Nieuwenhuizen

Jim McCarthy

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11794

Physics Department, Brandeis University, Waltham, Massachusetts 02254

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Phys. Rev. D 40, 415 – Published 15 July, 1989

DOI: https://doi.org/10.1103/PhysRevD.40.415

Abstract

The machinery developed in paper I is used to compute the operator-product algebra (OPA) of an operator constructed from the third-order Casimir invariant of the superalgebra $SU (m | n)$ . The vertex operator construction of $SU {(m | n)}^{(1)}$ is used to find a realization of the OPA for level $k = 1$ in terms of free bosonic fields only. It turns out that in many respects the conformal structure of the affinized Lie superalgebra $SU {(m | n)}^{(1)}$ is similar to that of the Kač-Moody algebra $SU {(m - n)}^{(1)}$ . An intermediate result suggests the occurrence of extended conformal symmetries in $bc$ systems, to which we will devote a separate discussion.

References (23)

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F. A. Bais, P. Bouwknegt, K. Schoutens, and M. Surridge, Nucl. Phys. B304, 371 (1988)
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P. Bouwknegt, Proceedings of the 1987 Braşov International Summer School on Conformal Invariance and Strings (unpublished)
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P. Bouwknegt, A. Ceresole, J. McCarthy, and P. van Nieuwenhuizen, Phys. Rev. D 39, 2971 (1989), paper I
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[c1] F. A. Bais, P. Bouwknegt, K. Schoutens, and M. Surridge, Nucl. Phys. B304, 348 (1988)

[c2] F. A. Bais, P. Bouwknegt, K. Schoutens, and M. Surridge, Nucl. Phys. B304, 371 (1988)

[c3] J. Fuchs, Nucl. Phys. B286, 455 (1987)

[c4] T. Inami, Y. Matsuo, and I. Yamanaka, Phys. Lett. B 215, 701 (1988)

[c5] P. Bouwknegt, Proceedings of the 1987 Braşov International Summer School on Conformal Invariance and Strings (unpublished)

[c6] E. Verlinde, Nucl. Phys. B300 [FS22], 360 (1988)

[c7] H. Sugawara, Phys. Rev. 170, 1659 (1968)

[c8] P. Goddard, A. Kent, and D. I. Olive, Phys. Lett. 152B, 88 (1985) Commun. Math. Phys. 103, 105 (1986)

[c9] P. Bouwknegt, A. Ceresole, J. McCarthy, and P. van Nieuwenhuizen, Phys. Rev. D 39, 2971 (1989), paper I

[c10] V. G. Kač, Adv. Math. 26, 8 (1977) J. van de Leur, University of Utrecht thesis, 1986

[c11] I. Bars, Lect. Appl. Math. 21, 17 (1985)

[c12] L. Frappat, Int. J. Mod. Phys. A 3, 2545 (1988)

[c13] V. G. Kač and J. W. van de Leur, Ann. Inst. Fourier 37, 99 (1987)

[c14] P. Goddard, D. I. Olive, and G. Waterson, Commun. Math. Phys. 112, 591 (1987)

[c15] P. Goddard and D. I. Olive, Int. J. Mod. Phys. A 1, 303 (1986)

[c16] J. Thierry-Mieg, in Nonperturbative Quantum Field Theory, proceedings of the NATO Advanced Studies Institute, Cargése, France, 1987, edited by G. 't Hooft (NATO ASI Series B: Physics, Vol. 185) (Plenum, New York, 1988)

[c17] D-J. Smit, Utrecht Report No. THU-88/33, 1988 (unpublished)

[c18] V. A. Fateev and A. B. Zamolodchikov, Nucl. Phys. B280, [FS18], 644 (1987)

[c19] M. Wakimoto and H. Yamada, Lett. Math. Phys. 7, 513 (1983)

[c20] D. Friedan, E. Martinec, and S. Shenker, Nucl. Phys. B271, 93 (1986)

[c21] J. L. Cardy, Phys. Rev. Lett. 54, 1354 (1985)

[c22] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B241, 333 (1984)

[c23] P. Bouwknegt, Phys. Lett. B 207, 295 (1988)

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Extended Sugawara construction for the superalgebras $SU (M + 1 | N + 1)$ . II. The third-order Casimir algebra

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Extended Sugawara construction for the superalgebras SU(M+1|N+1). II. The third-order Casimir algebra

Abstract

References (23)

Extended Sugawara construction for the superalgebras $SU (M + 1 | N + 1)$ . II. The third-order Casimir algebra