SIGMA 11 (2015), 005, 19 pages      arXiv:1407.5335      https://doi.org/10.3842/SIGMA.2015.005
Contribution to the Special Issue on New Directions in Lie Theory

Bosonizations of $\widehat{\mathfrak{sl}}_2$ and Integrable Hierarchies

Bojko Bakalov ^a and Daniel Fleisher ^b
a) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
^b) Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel

Received July 22, 2014, in final form January 09, 2015; Published online January 14, 2015

Abstract
We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.

Key words: affine Kac-Moody algebra; Casimir element; Friedan-Martinec-Shenker bosonization; lattice vertex algebra; Virasoro algebra; Wakimoto realization.

pdf (438 kb)   tex (24 kb)

References

1. Adamović D., A construction of admissible $A^ {(1)}_1$-modules of level $-\frac 43$, J. Pure Appl. Algebra 196 (2005), 119-134, math.QA/0401023.
2. 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.
3. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.
4. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation, J. Phys. Soc. Japan 50 (1981), 3806-3812.
5. Date E., Kashiwara M., Miwa T., Transformation groups for soliton equations. II. Vertex operators and $\tau $ functions, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 387-392.
6. Dong C., Twisted modules for vertex algebras associated with even lattices, J. Algebra 165 (1994), 91-112.
7. Feigin B.L., Frenkel E.V., Representations of affine Kac-Moody algebras, bosonization and resolutions, Lett. Math. Phys. 19 (1990), 307-317.
8. Feigin B.L., Semikhatov A.M., ${\mathcal W}^{(2)}_n$ algebras, Nuclear Phys. B 698 (2004), 409-449, math.QA/0401164.
9. Feingold A.J., Frenkel I.B., Classical affine algebras, Adv. Math. 56 (1985), 117-172.
10. Feingold A.J., Frenkel I.B., Ries J.F.X., Spinor construction of vertex operator algebras, triality, and $E^{(1)}_8$, Contemporary Mathematics, Vol. 121, Amer. Math. Soc., Providence, RI, 1991.
11. Frenkel E., Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), 297-404, math.QA/0210029.
12. Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Vol. 88, Amer. Math. Soc., Providence, RI, 2001.
13. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
14. Frenkel I.B., Kac V.G., Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980), 23-66.
15. Friedan D., Martinec E., Shenker S., Conformal invariance, supersymmetry and string theory, Nuclear Phys. B 271 (1986), 93-165.
16. Goddard P., Kent A., Olive D., Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), 88-92.
17. Jing N., Misra K.C., Xu C., Bosonic realization of toroidal Lie algebras of classical types, Proc. Amer. Math. Soc. 137 (2009), 3609-3618, arXiv:0908.0211.
18. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
19. Kac V.G., Vertex algebras for beginners, University Lecture Series, Vol. 10, 2nd ed., Amer. Math. Soc., Providence, RI, 1998.
20. Kac V.G., Peterson D.H., $112$ constructions of the basic representation of the loop group of $E_8$, in Symposium on Anomalies, Geometry, Topology (Chicago, Ill., 1985), World Sci. Publ., Singapore, 1985, 276-298.
21. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Sci. Publ. Co., Inc., Teaneck, NJ, 1987.
22. 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.
23. Kashiwara M., Miwa T., Transformation groups for soliton equations. I. The $\tau $ function of the Kadomtsev-Petviashvili equation, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 342-347.
24. Lepowsky J., Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. USA 82 (1985), 8295-8299.
25. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
26. Lepowsky J., Wilson R.L., Construction of the affine Lie algebra $A_{1}^{(1)}$, Comm. Math. Phys. 62 (1978), 43-53.
27. Liszewski K.T., The charged free boson integrable hierarchy, Ph.D. Thesis, North Carolina State University, 2011, available at http://www.lib.ncsu.edu/resolver/1840.16/7058.
28. Miwa T., Jimbo M., Date E., Solitons. Differential equations, symmetries and infinite-dimensional algebras, Cambridge Tracts in Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2000.
29. Peterson D.H., Kac V.G., Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. USA 80 (1983), 1778-1782.
30. Wakimoto M., Fock representations of the affine Lie algebra $A^{(1)}_1$, Comm. Math. Phys. 104 (1986), 605-609.
31. Wang W., ${\mathcal W}_{1+\infty}$ algebra, ${\mathcal W}_3$ algebra, and Friedan-Martinec-Shenker bosonization, Comm. Math. Phys. 195 (1998), 95-111, q-alg/9708008.