STRICT VERSIONS OF VARIOUS MATRIX HIERARCHIES RELATED TO SLn-LOOPS AND THEIR COMBINATIONS
Main Article Content
Abstract
Let t be a commutative Lie subalgebra of sln(C) of maximal
dimension. We consider in this paper three spaces of t-loops that each get deformed in a different way. We require that the deformed generators of each of them evolve w.r.t. the commuting flows they generate according to a certain, different set of Lax equations. This leads to three integrable hierarchies: the (sln(C), t)-hierarchy, its strict version and the combined (sln(C), t)-hierarchy. For n = 2 and t the diagonal matrices, the (sl2(C), t)- hierarchy is the AKNS-hierarchy. We treat their interrelations and show that all three have a zero curvature form. Furthermore, we discuss their linearization and we conclude by giving the construction of a large class of
solutions.
dimension. We consider in this paper three spaces of t-loops that each get deformed in a different way. We require that the deformed generators of each of them evolve w.r.t. the commuting flows they generate according to a certain, different set of Lax equations. This leads to three integrable hierarchies: the (sln(C), t)-hierarchy, its strict version and the combined (sln(C), t)-hierarchy. For n = 2 and t the diagonal matrices, the (sl2(C), t)- hierarchy is the AKNS-hierarchy. We treat their interrelations and show that all three have a zero curvature form. Furthermore, we discuss their linearization and we conclude by giving the construction of a large class of
solutions.
Article Details
How to Cite
HELMINCK, Gerard Franciscus.
STRICT VERSIONS OF VARIOUS MATRIX HIERARCHIES RELATED TO SLn-LOOPS AND THEIR COMBINATIONS.
Quarterly Physics Review, [S.l.], v. 3, n. 2, july 2017.
ISSN 2572-701X.
Available at: <https://esmed.org/MRA/qpr/article/view/1408>. Date accessed: 21 dec. 2024.
Section
Research Articles
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References
References
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[27] Aganagic, Mina; Dijkgraaf, Robbert; Klemm, Albrecht; Mario, Marcos; Vafa, Cumrun
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451-516.
[28] Fukuma, M.; Takebe, T.: The Toda lattice hierarchy and deformations of conformal
field theories. Modern Physics Letters A, Vol. 5, No.7 (1990), 509-518.
[29] Eguchi T.; Yang, S-K.: Deformations of conformal field theories and soliton equations
Phys. Letters B 224 (1989), 373-378.
[30] Drinfeld,V. G.; Sokolov, V. V.: Lie algebras and equations of Korteweg-de Vries type,
Soviet Mathematics 30, p. 1975-2036.
[31] Reyman A.G. and Semenov-Tian-Shansky, M.A. Reduction of Hamiltonian systems,
affine Lie algebras and Lax equations. Invent. Math. 54 (1), pp. 81-100 (1979).
[32] Reyman A.G. and Semenov-Tian-Shansky, M.A. Reduction of Hamiltonian systems,
affine Lie algebras and Lax equations. II Invent. Math. 63 (3), pp. 423-432 (1981).
[33] M. Adler, P. van Moerbeke, P. VanHaecke, Algebraic Integrability, Painlev ́e Geome- try and Lie Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, volume 47,
Springer Verlag.
[34] Bernstein I.N.: The dimensionality of commutative subrings in Rn,k. Functional Anal.
Appl. 1 (1967), no. 4, 325–327.
[35] Flashka,H.; Newell,A. C.; Ratiu, T.: Kac-Moody Lie Algebras and soliton equations II;
Lax equations associated with A(1). Physica 9D, p.300-323. 1
[36] Ablowitz, Mark J.; Kaup, David J.; Newell, Alan C.; Segur, Harvey, The inverse scat- tering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (4) (1974): 249-315.
[37] Helminck, G.F.; Helminck A.G.; Panasenko, E.A.; Integrable deformations in the al- gebra of pseudo differential operators from a Lie algebraic perspective. Theoretical and Mathematical Physics 174(1): 134-153 (2013).
[38] Helminck, G.F.: The Strict AKNS-hierarchy: its Structure and Solutions. Advances in Math. Physics. Volume 2016, Article Id. 3649205, 10 pages.
[39] Helminck A.G., Helminck G.F., Opimakh A.V. , Equivalent forms of multi component Toda hierarchies. Journal of Geometry and Physics 61 (2011), p. 847-873.
[40] Date, E.; Jimbo,M.; Kashiwara, M.; Miwa,T.: Transformation groups for soliton equa- tions, Proceedings of the RIMS symposium on nonlinear integrable systems-Classical Theory and Quantum Theory, ed. M.Jimbo and T.Miwa, World Sci. Publishers, Singa- pore.
[41] Segal,G.; Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 63 (1985), 1–64.
[42] Pressley, A.; Segal, G.:Loop groups, Oxford Mathematical Monographs, Clarendon Press 1986.
[43] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222.
[1] Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243-310, Lehigh Univ., Bethlehem, PA, 1991.
[2] Witten, E.: Algebraic geometry associated with matrix models of two-dimensional grav- ity. Topological methods in modern mathematics (Stony Brook, NY, 1991), 235-269, Publish or Perish, Houston, TX, 1993.
[3] Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147 (1992), no.1, 1-23.
[4] Kharchev, S.; Marshakov, A.; Mironov, A.; Morozov, A., Zabrodin, A. Towards unified theory of 2d gravity. Nuclear Physics B 380 (1992), no. 1-2, 181-240.
[5] Givental, A.: Gromov-Witten invariants and quantizations of quadratic Hamiltonians. Mosc. Math. J. 1 (2001), no.4, 551-568, 645.
[6] Givental, A.: An−1 singularities and n-KdV hierarchies. Mosc. Math. J. 3 (2003), no.2, 475-505, 743.
[7] Faber, C.; Shadrin, S.; Zvonkine, D.: Tautological relations and the r-spin Witten conjecture. Ann. Sci. c. Norm. Supr. (4) 43 (2010), no. 4, 621658.
[8] Gorsky, A.; Krichever, I. M.; Marshakov, A.; Mironov, A.; Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B 355 (1995), no. 3-4, 466-474.
[9] Marshakov, A.; Nekrasov, N. A.: Extended Seiberg-Witten theory and integrable hier- archy. J. High Energy Phys. 2007, no. 1, 104, 39 pp.
[10] Marshakov, A.: Seiberg-Witten theory and integrable systems. World Scientific Publish- ing Co., Inc., River Edge, NJ, 1999. ii+253 pp.
[11] Arutyunov, G.; Frolov, S.; Russo, J.; Tseytlin, A. A. : Spinning strings in AdS5 × S5 and integrable systems. Nuclear Physics B 671 (2003), no. 1-3, 3-50.
[12] Arutyunov, G.; Frolov, S.: Foundations of the AdS5 × S5 superstring. I. J. Phys. A 42 (2009), no. 25, 254003, 121 pp.
[13] Kazakov, V. A.; Marshakov, A.; Minahan, J. A.; Zarembo, K.: Classical/quantum integrability in AdS/CFT. J. High Energy Phys. 2004, no. 5, 024, 55 pp.
[14] Cachazo, F.; Intriligator, K.; Vafa, C.: A large N duality via a geometric transition. Nuclear Physics B 603 (2001), no. 1-2, 341.
[15] Dijkgraaf, R.; Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nuclear Phys. B 644 (2002), no. 1-2, 320.
[16] Dijkgraaf, R.; Vafa, C.: On geometry and matrix models. Nuclear Phys. B 644 (2002), no. 1-2, 2139.
[17] Dijkgraaf,R.; Vafa,C.: A perturbative window on non-perturbative physics. arXiv: hep- th/0208048v1
[18] Candelas, P.; Horowitz, G.; Strominger, A.; Witten, E.: Vacuum configurations for superstrings. Nuclear Physics B. 258: 4674.
[19] Dixon, L.: Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. ICTP Ser. Theoret. Phys. 4: 67-126 (1988).
[20] Lerche, W.; Vafa, C.; Warner, N.: Chiral rings in N = 2 superconformal theories Nuclear Physics B. 324 (2): 427-474 (1989).
[21] Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nuclear Physics B. 340 (23): 281–332.
[22] Witten, E.: Mirror manifolds and topological field theory. Essays on mirror manifolds: 121–160 (1992).
[23] Vafa, C.: Topological mirrors and quantum rings. Essays on mirror manifolds: 96-119 (1992).
[24] Candelas, P.; de la Ossa, X.; Green, P.; Parkes, L.: A pair of CalabiYau manifolds as an exactly soluble superconformal field theory. Nuclear Physics B. 359 (1): 21–74.
[25] Givental, A.;Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Com-
mun. Math. Phys. 168, 609-641 (1995).
[26] Givental, A.: Stationary phase integrals, quantum Toda lattices, flag manifolds and the
mirror conjecture. Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, 180,
103-115, Amer. Math. Soc., Providence, RI, 1997.
[27] Aganagic, Mina; Dijkgraaf, Robbert; Klemm, Albrecht; Mario, Marcos; Vafa, Cumrun
Topological strings and integrable hierarchies. Comm. Math. Phys. 261 (2006), no. 2,
451-516.
[28] Fukuma, M.; Takebe, T.: The Toda lattice hierarchy and deformations of conformal
field theories. Modern Physics Letters A, Vol. 5, No.7 (1990), 509-518.
[29] Eguchi T.; Yang, S-K.: Deformations of conformal field theories and soliton equations
Phys. Letters B 224 (1989), 373-378.
[30] Drinfeld,V. G.; Sokolov, V. V.: Lie algebras and equations of Korteweg-de Vries type,
Soviet Mathematics 30, p. 1975-2036.
[31] Reyman A.G. and Semenov-Tian-Shansky, M.A. Reduction of Hamiltonian systems,
affine Lie algebras and Lax equations. Invent. Math. 54 (1), pp. 81-100 (1979).
[32] Reyman A.G. and Semenov-Tian-Shansky, M.A. Reduction of Hamiltonian systems,
affine Lie algebras and Lax equations. II Invent. Math. 63 (3), pp. 423-432 (1981).
[33] M. Adler, P. van Moerbeke, P. VanHaecke, Algebraic Integrability, Painlev ́e Geome- try and Lie Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, volume 47,
Springer Verlag.
[34] Bernstein I.N.: The dimensionality of commutative subrings in Rn,k. Functional Anal.
Appl. 1 (1967), no. 4, 325–327.
[35] Flashka,H.; Newell,A. C.; Ratiu, T.: Kac-Moody Lie Algebras and soliton equations II;
Lax equations associated with A(1). Physica 9D, p.300-323. 1
[36] Ablowitz, Mark J.; Kaup, David J.; Newell, Alan C.; Segur, Harvey, The inverse scat- tering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (4) (1974): 249-315.
[37] Helminck, G.F.; Helminck A.G.; Panasenko, E.A.; Integrable deformations in the al- gebra of pseudo differential operators from a Lie algebraic perspective. Theoretical and Mathematical Physics 174(1): 134-153 (2013).
[38] Helminck, G.F.: The Strict AKNS-hierarchy: its Structure and Solutions. Advances in Math. Physics. Volume 2016, Article Id. 3649205, 10 pages.
[39] Helminck A.G., Helminck G.F., Opimakh A.V. , Equivalent forms of multi component Toda hierarchies. Journal of Geometry and Physics 61 (2011), p. 847-873.
[40] Date, E.; Jimbo,M.; Kashiwara, M.; Miwa,T.: Transformation groups for soliton equa- tions, Proceedings of the RIMS symposium on nonlinear integrable systems-Classical Theory and Quantum Theory, ed. M.Jimbo and T.Miwa, World Sci. Publishers, Singa- pore.
[41] Segal,G.; Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES 63 (1985), 1–64.
[42] Pressley, A.; Segal, G.:Loop groups, Oxford Mathematical Monographs, Clarendon Press 1986.
[43] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222.