Ghost systems: A vertex algebra point of view. (English) Zbl 0945.81011
Summary: Fermionic and bosonic ghost systems are defined each in terms of a single vertex algebra which admits a one-parameter family of conformal structures. The observation that these structures are related to each other provides a simple way to obtain character formulae for a general twisted module of a ghost system. The \(U(1)\) symmetry and its subgroups that underlie the twisted modules also define an infinite set of invariant vertex subalgebras. Their structure is studied in detail from a \(W\)-algebra point of view with particular emphasis on \(\mathbb{Z}_\mathbb{N}\)-invariant subalgebras of the fermionic ghost system.
MSC:
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
References:
| [1] | Friedan, D.; Martinec, E.; Shenker, S., Conformal invariance, supersymmetry and string theory, Nucl. Phys. B, 271, 93 (1986) |
| [2] | S. Guruswamy and A.W.W. Ludwig, Relating \(cc\); S. Guruswamy and A.W.W. Ludwig, Relating \(cc\) · Zbl 0945.81058 |
| [3] | L.S. Georgiev and I.T. Todorov, Characters and partition function for the Wen-Wu CFT model of the Haldane-Rezayi quantum Hall state, preprint hep-th/9611084, INRNE-TH-96/13.; L.S. Georgiev and I.T. Todorov, Characters and partition function for the Wen-Wu CFT model of the Haldane-Rezayi quantum Hall state, preprint hep-th/9611084, INRNE-TH-96/13. |
| [4] | Gurarie, V.; Flohr, M.; Nayak, C., The Haldane-Rezayi quantum Hall state and conformal field theory, Nucl. Phys. B, 498, 513 (1997), cond-mat/9701212 · Zbl 0934.81073 |
| [5] | Moore, G.; Read, N., Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B, 360, 362 (1991) |
| [6] | Saleur, H., Polymers and percolation in two dimensions and twisted \(N = 2\) supersymmetry, Nucl. Phys. B, 382, 486 (1992) |
| [7] | Bernard, D., (Perturbed) conformal field theory applied to 2D disordered systems: An introduction, (Baulieu, L.; Kazakov, V.; Picco, M.; Windley, P., Low-Dimensional Applications of Quantum Field Theory (1997), Plenum Press: Plenum Press New York), 19, hep-th/9509137 · Zbl 0915.58114 |
| [8] | Maassarani, Z.; Serban, D., Non-unitary conformal field theory and logarithmic operators for disordered systems, Nucl. Phys. B, 489, 603 (1997), hep-th/9605062 · Zbl 0925.81328 |
| [9] | di Francesco, P.; Mathieu, P.; Sénéchal, D., Conformal Field Theory, Graduate Texts in Contemporary Physics (1997), Springer: Springer New York · Zbl 0869.53052 |
| [10] | de Boer, J.; Fehér, L., Wakimoto realizations of current algebras: An explicit construction, Commun. Math. Phys., 189, 759 (1997), hep-th/9611083 · Zbl 0932.81014 |
| [11] | Rasmussen, J., Free field realizations of affine current superalgebras, screening currents and primary fields, Nucl. Phys. B, 510, 688 (1998), hep-th/9706091 · Zbl 0953.81028 |
| [12] | H.G. Kausch, Curiosities at \(c\); H.G. Kausch, Curiosities at \(c\) |
| [13] | Ginsparg, P., Curiosities at \(c = 1\), Nucl. Phys. B, 295, 153 (1988) |
| [14] | Kiritsis, E. B., Proof of the completeness of the classification of rational conformal theories with \(c = 1\), Phys. Lett. B, 217, 427 (1989) |
| [15] | Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the monster, (Proc. Natl. Acad. Sci. USA, 83 (1986)), 3068 · Zbl 0613.17012 |
| [16] | Kac, V.; Radul, A., Representation theory of the vertex algebra \(W\), Transform. Groups, 1, 41 (1996), hep-th/9512150 · Zbl 0862.17023 |
| [17] | Blumenhagen, R.; Eholzer, W.; Honecker, A.; Hornfeck, K.; Hübel, R., Coset realization of unifying \(W\) algebras, Int. J. Mod. Phys. A, 10, 2367 (1995), hep-th/9406203 · Zbl 1044.81594 |
| [18] | Fehér, L.; O’Raifeartaigh, L.; Ruelle, P.; Tsutsui, I.; Wipf, A., On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep., 222, 1 (1992) · Zbl 0748.35032 |
| [19] | Frenkel, E.; Kac, V.; Wakimoto, M., Characters and fusion rules for \(W\) algebras via quantized Drinfeld-Sokolov reduction, Commun. Math. Phys., 147, 295 (1992) · Zbl 0768.17008 |
| [20] | de Boer, J.; Tjin, T., The relation between quantum \(W\) algebras and Lie algebras, Commun. Math. Phys., 160, 317 (1994), hep-th/9302006 · Zbl 0796.17027 |
| [21] | Schellekens, A. N., Meromorphic \(c = 24\) conformal field theories, Commun. Math. Phys., 153, 159 (1993), hep-th/9205072 · Zbl 0782.17014 |
| [22] | Cappelli, A.; Trugenberger, C. A.; Zemba, G. R., Classification of quantum Hall universality classes by \(W\) symmetry, Phys. Rev. Lett., 72, 1902 (1994), hep-th/9310181 |
| [23] | Flohr, M.; Varnhagen, R., Infinite symmetry in the fractional quantum Hall effect, J. Phys. A: Math. Gen., 27, 3999 (1994), hep-th/9309083 |
| [24] | Kac, V.; Radul, A., Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys., 157, 429 (1993), hep-th/9308153 · Zbl 0826.17027 |
| [25] | Frenkel, E.; Kac, V.; Radul, A.; Wang, W., \(W\) and \(W\) with central charge \(N\), Commun. Math. Phys., 170, 337 (1995), hep-th/9405121 · Zbl 0838.17028 |
| [26] | Awata, H.; Fukuma, M.; Matsuo, Y.; Odake, S., Character and determinant formulae of quasifinite representation of the \(W\) algebra, Commun. Math. Phys., 172, 377 (1995), hep-th/9405093 · Zbl 0853.17023 |
| [27] | Kac, V. G.; Todorov, I. T., Affine orbifolds and rational conformal field theory extensions of \(W\), Commun. Math. Phys., 190 (1997), hep-th/9612078 · Zbl 0904.17021 |
| [28] | Kac, V., Vertex Algebras for Beginners, (University Lecture Series, Vol. 10 (1997), American Mathematical Society: American Mathematical Society Providence, R.I) · Zbl 0861.17017 |
| [29] | Kac, V. G.; Raina, A. K., Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, (Advanced Series in Mathematical Physics, Vol. 2 (1987), World Scientific: World Scientific Singapore) · Zbl 0668.17012 |
| [30] | Schwimmer, A.; Seiberg, N., Comments on the \(N = 2, 3, 4\) superconformal algebras in two dimensions, Phys. Lett. B, 184, 191 (1987) |
| [31] | Feigin, B.; Frenkel, E., Bosonic ghost system and the Virasoro algebra, Phys. Lett. B, 246, 71 (1990) · Zbl 1243.81086 |
| [32] | Honecker, A., Automorphisms of \(W\)-algebras and extended rational conformal field theories, Nucl. Phys. B, 400, 574 (1993), hep-th/9211130 · Zbl 0941.81545 |
| [33] | Weyl, H., The Classical Groups, Their Invariants and Representations (1946), Princeton University Press: Princeton University Press Princeton · Zbl 1024.20502 |
| [34] | de Boer, J.; Fehér, L.; Honecker, A., A class of \(W\)-algebras with infinitely generated classical limit, Nucl. Phys. B, 420, 409 (1994), hep-th/9312049 · Zbl 0990.81525 |
| [35] | Ginsparg, P., Applied conformal field theory, (Brézin, E.; Zinn-Justin, J., Fields, Strings and Critical Phenomena. Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School 1988 (1990), North-Holland: North-Holland Amsterdam), 1 · Zbl 0985.82500 |
| [36] | Zamolodchikov, A. B., Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Math. Phys., 65, 1205 (1986) |
| [37] | Bouwknegt, P.; Ceresole, A.; van Nieuwenhuizen, P.; McCarthy, J., Extended Sugawara construction for the superalgebra \(SU (M + 1 |N + 1)\). II. The third-order Casimir algebra, Phys. Rev. D, 40, 415 (1989) |
| [38] | Kausch, H. G., Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B, 259, 448 (1991) |
| [39] | Eholzer, W.; Honecker, A.; Hübel, R., How complete is the classification of \(W\)-symmetries?, Phys. Lett. B, 308, 42 (1993), hep-th/9302124 · Zbl 1015.81511 |
| [40] | Blumenhagen, R.; Eholzer, W.; Honecker, A.; Hornfeck, K.; Hübel, R., Unifying \(W\)-algebras, Phys. Lett. B, 332, 51 (1994), hep-th/9404113 · Zbl 1044.81594 |
| [41] | W. Wang, \(WW\); W. Wang, \(WW\) |
| [42] | Hornfeck, K., \(W\)-Algebras with set of primary fields of dimensions (3,4,5) and (3,4,5,6), Nucl. Phys. B, 407, 237 (1993), hep-th/9212104 · Zbl 1043.81566 |
| [43] | Kausch, H. G.; Watts, G. M.T., A study of \(W\)-algebras using Jacobi identities, Nucl. Phys. B, 354, 740 (1991) |
| [44] | Gurarie, V., Logarithmic operators in conformal field theory, Nucl. Phys. B, 410, 535 (1993), hep-th/9303160 · Zbl 0990.81686 |
| [45] | Flohr, M. A.I., On fusion rules in logarithmic conformal field theories, Int. J. Mod. Phys. A, 12, 1943 (1997), hep-th/9605151 · Zbl 0985.81738 |
| [46] | Gaberdiel, M. R.; Kausch, H. G., A rational logarithmic conformal field theory, Phys. Lett. B, 386, 131 (1996), hep-th/9606050 |
| [47] | Goddard, P., Meromorphic conformal field theory, (Kac, V. G., Infinite dimensional Lie algebras and groups. Infinite dimensional Lie algebras and groups, Proc. CIRM-Luminy Conference 1988 (1989), World Scientific: World Scientific Singapore), 556 · Zbl 0742.17027 |
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