Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 025, 29 pages      arXiv:1504.00715      https://doi.org/10.3842/SIGMA.2016.025
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Loops in SU(2), Riemann Surfaces, and Factorization, I

Estelle Basor a and Doug Pickrell b
a) American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA
b) Mathematics Department, University of Arizona, Tucson, AZ 85721, USA

Received October 24, 2015, in final form March 02, 2016; Published online March 08, 2016

Abstract
In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a triangular factorization if and only if the loop $g$ has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its double, the sphere, are replaced by a based compact Riemann surface with boundary, and its double. One ingredient is the theory of generalized Fourier-Laurent expansions developed by Krichever and Novikov. We show that a ${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup factorization satisfies the condition that the multiloop, viewed as a transition function, defines a semistable holomorphic ${\rm SL}(2,\mathbb C)$ bundle. Additionally, for such a multiloop, there is a corresponding factorization for determinants associated to the spin Toeplitz operators defined by the multiloop.

Key words: loop group; factorization; Toeplitz operator; determinant.

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References

  1. Basor E., Pickrell D., Loops in ${\rm SU}(2)$, Riemann surfaces, and factorization, II: Examples, in progress.
  2. Bott R., Stable bundles revisited, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 1-18.
  3. Böttcher A., Silbermann B., Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990.
  4. Clancey K.F., Gohberg I., Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, Vol. 3, Birkhäuser Verlag, Basel - Boston, Mass., 1981.
  5. Gohberg I., Krupnik N., Einführung in die Theorie der eindimensionalen singulären Integraloperatoren, Mathematische Reihe, Vol. 63, Birkhäuser Verlag, Basel - Boston, Mass., 1979.
  6. Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, New York, 1978.
  7. Grothendieck A., Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121-138.
  8. Hawley N.S., Schiffer M., Half-order differentials on Riemann surfaces, Acta Math. 115 (1966), 199-236.
  9. Helgason S., Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Pure and Applied Mathematics, Vol. 113, Academic Press, Inc., Orlando, FL, 1984.
  10. Helton J.W., Howe R.E., Integral operators: commutators, traces, index and homology, in Proceedings of a Conference Operator Theory (Dalhousie Univ., Halifax, N.S., 1973), Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, 141-209.
  11. Krichever I.M., Novikov S.P., Virasoro-Gelfand-Fuks type algebras, Riemann surfaces, operator's theory of closed strings, J. Geom. Phys. 5 (1989), 631-661.
  12. Peller V.V., Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
  13. Pickrell D., Invariant measures for unitary forms of Kac-Moody groups, Mem. Amer. Math. Soc. 146 (2000), 144 pages, funct-an/9510005.
  14. Pickrell D., Homogeneous Poisson structures on loop spaces of symmetric spaces, SIGMA 4 (2008), 069, 33 pages, arXiv:0801.3277.
  15. Pickrell D., Loops in ${\rm SU}(2)$ and factorization, J. Funct. Anal. 260 (2011), 2191-2221, arXiv:0903.4983.
  16. Pressley A., Segal G., Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986, oxford Science Publications.
  17. Ramanathan A., Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129-152.
  18. Rodin Y.L., The Riemann boundary value problem on closed Riemann surfaces and integrable systems, Phys. D 24 (1987), 1-53.
  19. Sato M., Theory of hyperfunctions. I, J. Fac. Sci. Univ. Tokyo. Sect. I 8 (1959), 139-193.
  20. Segal G., The definition of conformal field theory, in Topology, Geometry and Quantum Field Theory, London Mathematical Society Lecture Note Series, Vol. 308, Cambridge University Press, Cambridge, 2004, 421-577.
  21. Widom H., Asymptotic behavior of block Toeplitz matrices and determinants. II, Adv. Math. 21 (1976), 1-29.

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