Abstract
We compare homological representations of the braid groups and the monodromy representations of the KZ connection by means of hypergeometric integrals. Then we discuss a relationship to the space of conformal blocks.
This is a preview of subscription content, log in via an institution to check access.
References
K. Aomoto, M. Kita, Theory of Hypergeometric Functions. Springer Monographs in Mathematics (Springer, Berlin, 2011)
S. Bigelow, Braid groups are linear. J. Am. Math. Soc. 14, 471–486 (2001)
E. Date, M Jimbo, A. Matsuo, T. Miwa, Hypergeometric type integrals and the sl(2, C) Knizhnik-Zamolodchikov equations. Int. J. Mod. Phys. B4, 1049–1057 (1990)
B. Feigin, V. Schechtman, A. Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I. Commun. Math. Phys. 163, 173–184 (1994)
B. Feigin, V. Schechtman, A. Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II. Commun. Math. Phys. 170, 219–247 (1995)
V.G. Knizhnik, A.B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions. Nucl. Phys. B247, 83–103 (1984)
T. Kohno, Homology of a local system on the complement of hyperplanes. Proc. Jpn. Acad. Ser. A 62, 144–147 (1986)
T. Kohno, Conformal Field Theory and Topology. Translations of Mathematical Monographs vol. 210 (American Mathematical Society, Providence, RI, 2002)
T. Kohno, Quantum and homological representations of braid groups, in Configuration Spaces - Geometry, Combinatorics and Topology, vol. 14 (Edizioni della Normale, Pisa, 2012), pp. 355–372
T. Kohno, Homological representations of braid groups and KZ connections. J. Singularities 5, 94–108 (2012)
T. Kohno, Local systems on configuration spaces, KZ connections and conformal blocks. Acta Math. Vietnam 39, 575–598 (2014)
D. Krammer, Braid groups are linear. Ann. Math. 155, 131–156 (2002)
P. Orlik, H. Terao, Arrangements and Hypergeometric Integrals. MSJ Memoirs, vol. 9 (Mathematical Society of Japan, Tokyo, 2001)
V. Schechtman, A. Varchenko, Hypergeometric solutions of the Knizhnik-Zamolodchikov equation. Lett. Math. Phys. 20, 93–102 (1990)
R. Silvotti, Local systems on the complement of hyperplanes and fusion rules in conformal field theory, Int. Math. Res. Not. 1994(3), 111 ff., approx. 17 pp. (1994) [electronic]
A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups. Advances in Mathematical Physics, vol. 21 (World Scientific, Singapore, 1995)
Acknowledgements
The author is partially supported by Grant-in-Aid for Scientific Research, Japan Society of Promotion of Science and by World Premier Research Center Initiative, MEXT, Japan.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Kohno, T. (2017). Homological Representations of Braid Groups and the Space of Conformal Blocks. In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds) Perspectives in Lie Theory. Springer INdAM Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-58971-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-58971-8_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58970-1
Online ISBN: 978-3-319-58971-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)