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 SIGMA, 2012, Volume 8, 095, 37 pages (Mi sigma772)

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

Andrey M. Levinab, Mikhail A. Olshanetskyb, Andrey V. Smirnovbc, Andrei V. Zotovb

a Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow, 117312, Russia
b Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia
c Department of Mathematics, Columbia University, New York, NY 10027, USA

Abstract: We describe new families of the Knizhnik–Zamolodchikov–Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes – elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a center of the simple complex Lie group $G$. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.

Keywords: integrable system; KZB equation; Hitchin system; characteristic class

 Funding Agency Grant Number Russian Foundation for Basic Research 09-02-0039309-01-9243709-01-9310612-01-0048212-01-33071 Federal Agency for Science and Innovations of Russian Federation 14.740.11.0347 Ministry of Education and Science of the Russian Federation MK-1646.2011.111.G34.31.0023 The work was supported by grants RFBR-09-02-00393, RFBR-09-01-92437-KEa and by the Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347. The work of A.Z. and A.S. was also supported by the Russian President fund MK-1646.2011.1, RFBR-09-01-93106-NCNILa, RFBR-12-01-00482 and RFBR-12-01-33071 mol a ved. The work of A.L. was partially supported by AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023.

DOI: https://doi.org/10.3842/SIGMA.2012.095

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ArXiv: 1207.4386
MSC: 14H70; 32G34; 14H60
Received: July 14, 2012; in final form November 29, 2012; Published online December 10, 2012
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Citation: Andrey M. Levin, Mikhail A. Olshanetsky, Andrey V. Smirnov, Andrei V. Zotov, “Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles”, SIGMA, 8 (2012), 095, 37 pp.

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, Theoret. and Math. Phys., 177:1 (2013), 1281–1338
2. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Classification of isomonodromy problems on elliptic curves”, Russian Math. Surveys, 69:1 (2014), 35–118
3. Aminov G., Arthamonov S., Smirnov A., Zotov A., “Rational TOP and its Classical R-Matrix”, J. Phys. A-Math. Theor., 47:30 (2014), 305207
4. Levin A., Olshanetsky M., Zotov A., “Relativistic Classical Integrable Tops and Quantum R-Matrices”, J. High Energy Phys., 2014, no. 7, 012
5. Morozov A., Smirnov A., “Towards the Proof of AGT Relations with the Help of the Generalized Jack Polynomials”, Lett. Math. Phys., 104:5 (2014), 585–612
6. Gorsky A., Zabrodin A., Zotov A., “Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems”, J. High Energy Phys., 2014, no. 1, 070, 1–28
7. Levin A., Olshanetsky M., Zotov A., “Planck Constant as Spectral Parameter in Integrable Systems and Kzb Equations”, J. High Energy Phys., 2014, no. 10, 109
8. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero–Moser systems, and KZB equations”, Theoret. and Math. Phys., 188:2 (2016), 1121–1154
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