Abstract
Conformal field theories were first axiomatized by Segal (2004) as symmetric monoidal functors from a topological category of conformal cobordisms between compact oriented 1-dimensional manifolds to vector spaces. Costello (2007) later expanded the definition of the category to allow for cobordisms between manifolds with boundaries, and was able to use representations of this category to give a mirror partner for Gromov-Witten invariants.
The main goal of this paper is to provide a rigorous definition of the category of open conformal cobordisms. To the best of our knowledge, such a definition does not appear in the literature. Although most results here are probably known to the experts, the proofs are, as far as we can tell, new, and require only elementary results about quasiconformal mappings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973.
L. V. Ahlfors, Complex Analysis, third edn., McGraw-Hill Book Co., New York, 1978.
L. V. Ahlfors, Lectures on Quasiconformal Mappings, second edn., University Lecture Series, Vol. 38, American Mathematical Society, Providence, RI, 2006.
L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Mathematical Series, Vol. 26, Princeton University Press, Princeton, NJ, 1960.
M. G. Arsove, The Osgood-Taylor-Carathéodory theorem, Proceedings of the American Mathematical Society 19 (1968), 38–44.
L. Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bulletin of the American Mathematical Society 83 (1977), 1083–1100.
Shiing-shen Chern, An elementary proof of the existence of isothermal parameters on a surface, Proceedings of the American Mathematical Society 6 (1955), 771–782.
K. Costello, A dual version of the ribbon graph decomposition of moduli space, Geometry and Topology 11 (2007), 1637–1652.
K. Costello, Topological conformal field theories and Calabi-Yau categories, Advances in Mathematics 210 (2007), 165–214.
C. J. Earle and C. McMullen, Quasiconformal isotopies, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), Mathematical Sciences Research Institute Publications, Vol. 10, Springer, New York, 1988, pp. 143–154.
B. Farb and D. Margalit, A Primer on Mapping Class Group, Princeton Mathematical Series, Vol. 59, Princeton University Press, Princeton, NJ, 2012.
A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmüller Theory, Oxford Graduate Texts in Mathematics, Vol. 11, Oxford University Press, Oxford, 2007.
F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, Vol. 76, American Mathematical Society, Providence, RI, 2000.
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
Y. Imayoshi and M. Taniguchi, An Introduction to Teichmüller Spaces, Springer-Verlag, Tokyo, 1992.
R. J. V. Jackson, On the boundary values of Riemann’s mapping function, Transactions of the American Mathematical Society 259 (1980), 281–297.
O. Lehto, Univalent Functions and Teichmüller Spaces, Graduate Texts in Mathematics, Vol. 109, Springer-Verlag, New York, 1987.
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, second edn., Springer-Verlag, New York, 1973.
S. Mac Lane, Categories for the Working Mathematician, second edn., Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998.
S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1988.
M. H. A. Newman, Elements of the Topology of Plane Sets of Points, second edn., Dover Publications Inc., New York, 1992.
G. Segal, 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, pp. 421–577.
W. A. Veech, A Second Course in Complex Analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1967.
A. Zernik, An invitation to topological conformal field theories, MSc. thesis, The Hebrew University of Jerusalem, 2010, also available on Arxiv: arXiv:1011.2700v1 [math.GT].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zernik, A. Open conformal cobordisms. Isr. J. Math. 200, 297–325 (2014). https://doi.org/10.1007/s11856-014-0019-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-014-0019-1