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SIGMA 10 (2014), 007, 19 pages arXiv:1306.3072
https://doi.org/10.3842/SIGMA.2014.007
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa
The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints
Johan van de Leur
Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
Received September 23, 2013, in final form January 09, 2014; Published online January 15, 2014
Abstract
The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal
hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding
W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness.
We construct this principal hierarchy of type D in a different way, viz.
as a reduction of some DKP hierarchy.
This gives a Lax type and a Grassmannian formulation of this hierarchy.
We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov.
These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators.
Key words:
affine Kac-Moody algebra; loop group orbit; Kac-Wakimoto hierarchy; isotropic Grassmannian; total descendent potential; W constraints.
pdf (467 kb)
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References
- Bakalov B., Kac V.G., Twisted modules over lattice vertex algebras, in Lie
theory and its Applications in Physics V, World Sci. Publ., River Edge, NJ,
2004, 3-26, math.QA/0402315.
- Bakalov B., Milanov T., W-constraints for the total descendant
potential of a simple singularity, Compos. Math. 149
(2013), 840-888, arXiv:1203.3414.
- Date E., Jimbo M., Kashiwara M., Miwa T., Solitons, τ functions and
Euclidean Lie algebras, in Mathematics and Physics (Paris, 1979/1982),
Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983,
261-279.
- Delduc F., Fehér L., Regular conjugacy classes in the Weyl group and
integrable hierarchies, J. Phys. A: Math. Gen. 28 (1995),
5843-5882, hep-th/9410203.
- Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de
Vries type, Current Problems in Mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn.
i Tekhn. Inform., Moscow, 1984, 81-180.
- Frenkel E., Givental A., Milanov T., Soliton equations, vertex operators, and
simple singularities, Funct. Anal. Other Math. 3 (2010),
47-63, arXiv:0909.4032.
- Fukuma M., Kawai H., Nakayama R., Infinite-dimensional Grassmannian structure
of two-dimensional quantum gravity, Comm. Math. Phys. 143
(1992), 371-403.
- Givental A., An−1 singularities and nKdV hierarchies,
Mosc. Math. J. 3 (2003), 475-505,
math.AG/0209205.
- Givental A., Milanov T., Simple singularities and integrable hierarchies,
in The breadth of symplectic and Poisson geometry, Progr. Math.,
Vol. 232, Birkhäuser Boston, Boston, MA, 2005, 173-201,
math.AG/0307176.
- Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras,
Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
- Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University
Press, Cambridge, 1990.
- Kac V.G., Kazhdan D.A., Lepowsky J., Wilson R.L., Realization of the basic
representations of the Euclidean Lie algebras, Adv. Math.
42 (1981), 83-112.
- Kac V.G., Peterson D.H., 112 constructions of the basic representation of
the loop group of E8, in Symposium on Anomalies, Geometry, Topology
(Chicago, Ill., 1985), World Sci. Publishing, Singapore, 1985, 276-298.
- Kac V.G., Schwarz A., Geometric interpretation of the partition function of
2D gravity, Phys. Lett. B 257 (1991), 329-334.
- Kac V.G., van de Leur J., The geometry of spinors and the multicomponent BKP
and DKP hierarchies, in The Bispectral Problem (Montreal, PQ, 1997),
CRM Proc. Lecture Notes, Vol. 14, Amer. Math. Soc., Providence, RI,
1998, 159-202.
- Kac V.G., van de Leur J., The n-component KP hierarchy and
representation theory, J. Math. Phys. 44 (2003),
3245-3293,
hep-th/9308137.
- Kac V.G., Wakimoto M., Exceptional hierarchies of soliton equations, in Theta
Functions - Bowdoin 1987, Part 1 (Brunswick, ME, 1987),
Proc. Sympos. Pure Math., Vol. 49, Amer. Math. Soc., Providence, RI,
1989, 191-237.
- Kac V.G., Wang W., Yan C.H., Quasifinite representations of classical Lie
subalgebras of W1+∞, Adv. Math. 139
(1998), 56-140, math.QA/9801136.
- Liu S.-Q., Wu C.-Z., Zhang Y., On the Drinfeld-Sokolov hierarchies of D
type, Int. Math. Res. Not. 2011 (2011), 1952-1996,
arXiv:0912.5273.
- Liu S.-Q., Yang D., Zhang Y., Uniqueness theorem of W-constraints
for simple singularities, Lett. Math. Phys. 103 (2013),
1329-1345, arXiv:1305.2593.
- ten Kroode F., van de Leur J., Bosonic and fermionic realizations of the affine
algebra so2n, Comm. Algebra 20 (1992),
3119-3162.
- Vakulenko V.I., Solution of Virasoro conditions for the DKP-hierarchy,
Theoret. and Math. Phys. 107 (1996), 435-440.
- van de Leur J., The Adler-Shiota-van Moerbeke formula for the BKP
hierarchy, J. Math. Phys. 36 (1995), 4940-4951,
hep-th/9411159.
- van de Leur J., The nth reduced BKP hierarchy, the string equation and
BW1+∞-constraints, Acta Appl. Math. 44 (1996),
185-206, hep-th/9411067.
- Wu C.-Z., A remark on Kac-Wakimoto hierarchies of D-type,
J. Phys. A: Math. Theor. 43 (2010), 035201, 8 pages,
arXiv:0906.5360.
- Wu C.-Z., From additional symmetries to linearization of Virasoro symmetries,
Phys. D 249 (2013), 25-37, arXiv:1112.0246.
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