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From the \(w_\infty\)-algebra to its central extension: a \(\tau\)-function approach. (English) Zbl 0925.58031
Summary: The KP hierarchy, deformations of pseudodifferential operators \(L\) of order one, admits a \(w_\infty\)-algebra of symmetries \(\vec Y_{z^\alpha}(\partial/\partial z)^\beta\), which are vector fields transversal to and commuting with the KP hierarchy. Expressed in terms of \(L\) and another pseudodifferential operator \(M\) (introduced by Orlov and coworkers) satisfying \([ L,M ]=1\), these vector fields act on the wave function \(\Psi\) (a properly normalized eigenfunction of \(L\)) as \[ \vec Y_{z^\alpha}(\partial/\partial z)^\beta\Psi\equiv-(M^\beta L^\alpha)_-\Psi. \] Introducing a generating function \(\vec Y_{N}\Psi{}= N_{-}\Psi \), with \[ N\equiv (\mu-\lambda)\exp [(\mu-\lambda)M]\delta(\lambda,L), \] for the algebra of symmetries \(w_{\infty}\) on \(\Psi\) and taking into account the well-known representation of \[ \Psi(t,z)=[e^{-\eta}\tau(\overline t)/\tau(\overline t)]\exp (\sum_{1}^{\infty}\overline t_{i}z^{i}), \] in terms of the \(\tau\)-function, where \(\eta=\sum_{i=1}^{\infty}(z^{-i}/i)(\partial/\partial t_{i})\). We show a precise relationship between \(\vec Y_{N}\) and the Date-Jimbo-Kashiwara-Miwa vertex operator \[ \vec X(t,\lambda,\mu) \equiv\exp [\sum_{i=1}^{\infty}(\mu^{i}-\lambda^{i})t_{i}] \exp [\sum_{i=1}^{\infty}(\lambda^{-i}-\mu^{-i})(1/ i)(\partial/\partial t_{i})], \] a generating function of the \(W_{\infty}\)-algebra of symmetries (with central extension) on \(\tau\), to wit \(\vec Y_{N}\log \Psi=(e^{-\eta}-1)\vec X \log \tau\), where \(\vec Y_{N}\log\) and \(\vec X \log\) act on \(\Psi\) and \(\tau\) as logarithmic derivatives, with respect to the vector fields \(\vec Y_{N}\) and \(\vec X\) .

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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[1] Adler, M.; Van Moerbeke, P.: Commun. math. Phys.. 147, 25 (1992)
[2] Adler, M.; Van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials. Duke math. J. (1995) · Zbl 0848.17027
[3] Adler, M.; Shiota, T.; Van Moerbeke, P.: A Lax representation for the vertex operator and the central extension. Commun. math. Phys. (1994) · Zbl 0839.35116
[4] Aoyama, S.; Kodama, Y.: Phys. lett. B. 278, 56 (1992)
[5] Brezin, E.; Kazakov, V.: Phys. lett. B. 236, 144 (1990)
[6] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T.: Transformation groups for soliton equations. Proc. RIMS symp. On nonlinear integrable systems 3- classical theory and quantum theory, 39-119 (1983) · Zbl 0571.35098
[7] Dickey, L. A.: Soliton equations and integrable systems. (1991) · Zbl 0753.35075
[8] L.A. Dickey, Lectures on classical W-algebras (Cortona, Italy). · Zbl 0882.58024
[9] Dickey, L. A.: On additional symmetries of the KP hierarchy and Sato’s Bäcklund transformation, HEP-TH/9312015, 1993. Commun. math. Phys. (1994) · Zbl 0813.35106
[10] Duistermaat, J. J.; Grünbaum, F. A.: Commun. math. Phys.. 103, 177 (1986)
[11] Fastré, J.: Boson-correspondence for W-algebras, Bäcklund-Darboux transformations and the equation [L, P] = ln. Doctoral dissertation (1993)
[12] Fuchssteiner, B.: Progr. theor. Phys.. 70, No. No. 6, 1508 (1983)
[13] Fukuma, M.; Kawai, H.; Nakayama, R.: Infinite dimensional Grassmannian structure of two-dimensional quantum gravity, UT 572, KEK-TH-272. KEK preprint 90–165 (November 1990) · Zbl 0757.35076
[14] Grinevich, P. G.; Orlov, A. Yu.; Schulman, E. I.: On the symmetries of the integrable systems. Modem development of the soliton theory (1992) · Zbl 0816.35122
[15] Grinevich, P. G.; Orlov, A. Yu.: Flag spaces in KP theory and Virasoro action on det dj and Segal-Wilson taufunction. Modem problems of quantum field theory (1989)
[16] V.G. Kac, Infinite-dimensional Lie algebras, 3rd Ed. (Cambridge Univ. Press, Cambridge). · Zbl 0716.17022
[17] Kontsevich, M.: Commun. math. Phys.. 147, 1 (1992)
[18] Magri, F.; Zubelli, J. P.: Commun. math. Phys.. 141, No. No. 2, 329 (1991)
[19] Oevel, W.; Fuchssteiner, B.: Phys. lett. A. 88, 323 (1982)
[20] Oevel, G.; Fuchssteiner, B.; Blaszak, M.: Progr. theor. Phys.. 83, No. No. 3, 395 (1990)
[21] Orlov, A. Yu.: Vertex operator, \partial-problem, symmetries, variational identities and Hamiltonian formalism for 2+1 integrable systems. Proc. kiev int. Workshop on non-linear and turbulent processes in physics (1988) · Zbl 0691.35075
[22] Orlov, A. Yu.; Schulman, E. I.: Lett. math. Phys.. 12, 171 (1986)
[23] Sato, M.: Soliton equations and the universal Grassmann manifold. Mathematics lecture note series no. 18 (1984) · Zbl 0541.58001
[24] Sato, M.: The KP hierarchy and infinite-dimensional Grassmann manifolds. Proc. symp. On pure mathematics 49, 51-66 (1989) · Zbl 0688.58016
[25] Sato, M.; Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. Lecture notes in num. Appl. anal. 5, 259 (1982)
[26] Shiota, T.: Invent. math.. 83, 333 (1986)
[27] Takasaki, K.; Takebe, T.: Integrable hierarchies and dispersionless limit. HEP-TH/9405096, university of Tokyo preprint UTMS 94-35 (1994) · Zbl 0838.35117
[28] Van Moerbeke, P.: Integrable foundations of string theory. CIMPA summer school at sophia-Antipolis (1993) · Zbl 0850.81049
[29] Witten, E.: Surv. diff. Geom.. 1, 243 (1991)
[30] Witten, E.: On the kontsevich model and other models of two dimensional gravity. IASSNS-HEP-91/24 preprint (June 1991)
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