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SIGMA 8 (2012), 095, 37 pages arXiv:1207.4386
https://doi.org/10.3842/SIGMA.2012.095
Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
Andrey M. Levin a, b, Mikhail A. Olshanetsky b, Andrey V. Smirnov b, c and Andrei V. Zotov b
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
Received July 14, 2012, in final form November 29, 2012; Published online December 10, 2012
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.
Key words:
integrable system; KZB equation; Hitchin system; characteristic class.
pdf (648 kb)
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References
- Aminov G., Arthamonov S., Reduction of the elliptic ${\rm SL}(N,{\mathbb C})$
top, J. Phys. A: Math. Theor. 44 (2011), 075201, 34 pages,
arXiv:1009.1867.
- Aminov G., Arthamonov S., Levin A.M., Olshanetsky M.A., Zotov A.V., Around
Painlevé VI field theory, submitted.
- Atiyah M.F., Vector bundles over an elliptic curve, Proc. London Math.
Soc. 7 (1957), 414-452.
- Axelrod S., Pietra S.D., Witten E., Geometric quantization of Chern-Simons
gauge theory, J. Differential Geom. 33 (1991), 787-902.
- Beauville A., Conformal blocks, fusion rules and the Verlinde formula, in
Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry
(Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan
Univ., Ramat Gan, 1996, 75-96, alg-geom/9405001.
- Beauville A., Laszlo Y., Conformal blocks and generalized theta functions,
Comm. Math. Phys. 164 (1994), 385-419,
alg-geom/9309003.
- Ben-Zvi D., Frenkel E., Geometric realization of the Segal-Sugawara
construction, in Topology, geometry and quantum field theory, London
Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press,
Cambridge, 2004, 46-97, math.AG/0301206.
- Bernard D., On the Wess-Zumino-Witten models on Riemann surfaces,
Nuclear Phys. B 309 (1988), 145-174.
- Bernard D., On the Wess-Zumino-Witten models on the torus,
Nuclear Phys. B 303 (1988), 77-93.
- Bernstein J., Schwarzman O., Chevalley's theorem for complex crystallographic
Coxeter groups, Funct. Anal. Appl. 12 (1978), 308-310.
- Bernstein J., Schwarzman O., Complex crystallographic Coxeter groups and
affine root systems, J. Nonlinear Math. Phys. 13 (2006),
163-182.
- Bourbaki N., Lie groups and Lie algebras, Chapters 4-6, Elements of
Mathematics (Berlin), Springer-Verlag, Berlin, 2002.
- Braden H.W., Dolgushev V.A., Olshanetsky M.A., Zotov A.V., Classical
$r$-matrices and the Feigin-Odesskii algebra via Hamiltonian and
Poisson reductions, J. Phys. A: Math. Gen. 36 (2003),
6979-7000, hep-th/0301121.
- Bulycheva K., Monopole solutions to the Bogomolny equation as three-dimensional
generalizations of the Kronecker series, Theoret. and Math. Phys.
172 (2012), 1232-1242, arXiv:1203.4674.
- Chernyakov Yu.B., Levin A.M., Olshanetsky M.A., Zotov A.V., Elliptic
Schlesinger system and Painlevé VI, J. Phys. A: Math. Gen.
39 (2006), 12083-12101, nlin.SI/0602043.
- Enriquez B., Rubtsov V., Hecke-Tyurin parametrization of the Hitchin and
KZB systems, in Moscow Seminar on Mathematical Physics. II,
Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc.,
Providence, RI, 2007, 1-31, math.AG/9911087.
- Etingof P., Schiffmann O., Twisted traces of intertwiners for Kac-Moody
algebras and classical dynamical $R$-matrices corresponding to generalized
Belavin-Drinfeld triples, Math. Res. Lett. 6 (1999),
593-612, math.QA/9908115.
- Etingof P., Schiffmann O., Varchenko A., Traces of intertwiners for quantum
groups and difference equations, Lett. Math. Phys. 62
(2002), 143-158, math.QA/0207157.
- Etingof P., Varchenko A., Geometry and classification of solutions of the
classical dynamical Yang-Baxter equation, Comm. Math. Phys.
192 (1998), 77-120, q-alg/9703040.
- Faltings G., A proof for the Verlinde formula, J. Algebraic Geom.
3 (1994), 347-374.
- Fehér L., Pusztai B.G., Generalizations of Felder's elliptic dynamical
$r$-matrices associated with twisted loop algebras of self-dual Lie
algebras, Nuclear Phys. B 621 (2002), 622-642,
math.QA/0109132.
- Felder G., The KZB equations on Riemann surfaces, in Symétries Quantiques
(Les Houches, 1995), North-Holland, Amsterdam, 1998, 687-725,
hep-th/9609153.
- Felder G., Gawedzki K., Kupiainen A., Spectra of Wess-Zumino-Witten
models with arbitrary simple groups, Comm. Math. Phys. 117
(1988), 127-158.
- Felder G., Wieczerkowski C., Conformal blocks on elliptic curves and the
Knizhnik-Zamolodchikov-Bernard equations, Comm. Math. Phys.
176 (1996), 133-161, hep-th/9411004.
- Frenkel E., Lectures on the Langlands program and conformal field theory, in
Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin,
2007, 387-533, hep-th/0512172.
- Friedman R., Morgan J.W., Holomorphic principal bundles over elliptic curves,
math.AG/9811130.
- Friedman R., Morgan J.W., Holomorphic principal bundles over elliptic
curves. II. The parabolic construction, J. Differential Geom.
56 (2000), 301-379, math.AG/0006174.
- Friedman R., Morgan J.W., Witten E., Principal $G$-bundles over elliptic
curves, Math. Res. Lett. 5 (1998), 97-118,
alg-geom/9707004.
- Fuchs J., Schweigert C., The action of outer automorphisms on bundles of chiral
blocks, Comm. Math. Phys. 206 (1999), 691-736,
hep-th/9805026.
- Gorsky A., Nekrasov N., Hamiltonian systems of Calogero-type, and
two-dimensional Yang-Mills theory, Nuclear Phys. B 414
(1994), 213-238, hep-th/9304047.
- Gukov S., Witten E., Branes and quantization, Adv. Theor. Math. Phys.
13 (2009), 1445-1518, arXiv:0809.0305.
- Harnad J., Quantum isomonodromic deformations and the
Knizhnik-Zamolodchikov equations, in Symmetries and Integrability of
Difference Equations (Estérel, PQ, 1994), CRM Proc. Lecture
Notes, Vol. 9, Amer. Math. Soc., Providence, RI, 1996, 155-161,
hep-th/9406078.
- Hitchin N.J., Flat connections and geometric quantization, Comm. Math.
Phys. 131 (1990), 347-380.
- Hitchin N.J., Stable bundles and integrable systems, Duke Math. J.
54 (1987), 91-114.
- Hori K., Global aspects of gauged Wess-Zumino-Witten models,
Comm. Math. Phys. 182 (1996), 1-32,
hep-th/9411134.
- Ivanov D.A., Knizhnik-Zamolodchikov-Bernard equations on Riemann
surfaces, Internat. J. Modern Phys. A 10 (1995),
2507-2536, hep-th/9410091.
- Ivanova T.A., Lechtenfeld O., Popov A.D., Rahn T., Instantons and
Yang-Mills flows on coset spaces, Lett. Math. Phys. 89
(2009), 231-247, arXiv:0904.0654.
- Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University
Press, Cambridge, 1990.
- Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands
program, Commun. Number Theory Phys. 1 (2007), 1-236,
hep-th/0604151.
- Knizhnik V.G., Zamolodchikov A.B., Current algebra and Wess-Zumino model
in two dimensions, Nuclear Phys. B 247 (1984), 83-103.
- Korotkin D., Samtleben H., On the quantization of isomonodromic deformations on
the torus, Internat. J. Modern Phys. A 12 (1997),
2013-2029, hep-th/9511087.
- Kuroki G., Takebe T., Twisted Wess-Zumino-Witten models on elliptic
curves, Comm. Math. Phys. 190 (1997), 1-56,
q-alg/9612033.
- Levin A.M., Olshanetsky M.A., Double coset construction of moduli space of
holomorphic bundles and Hitchin systems, Comm. Math. Phys.
188 (1997), 449-466, alg-geom/9605005.
- Levin A.M., Olshanetsky M.A., Hierarchies of isomonodromic deformations and
Hitchin systems, in Moscow Seminar in Mathematical Physics,
Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc.,
Providence, RI, 1999, 223-262.
- Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Calogero-Moser
systems for simple Lie groups and characteristic classes of bundles,
J. Geom. Phys. 62 (2012), 1810-1850, arXiv:1007.4127.
- Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Characteristic classes
and Hitchin systems. General construction, Comm. Math. Phys.
316 (2012), 1-44, arXiv:1006.0702.
- Levin A.M., Olshanetsky M.A., Smirnov A.V., Zotov A.V., Characteristic classes
of ${\rm SL}(N)$-bundles and quantum dynamical elliptic $R$-matrices,
J. Phys. A: Math. Theor., to appear, arXiv:1208.5750.
- Levin A.M., Olshanetsky M.A., Zotov A.V., Hitchin systems - symplectic Hecke
correspondence and two-dimensional version, Comm. Math. Phys.
236 (2003), 93-133, nlin.SI/0110045.
- Levin A.M., Olshanetsky M.A., Zotov A.V., Monopoles and modifications of
bundles over elliptic curves, SIGMA 5 (2009), 065,
22 pages, arXiv:0811.3056.
- Levin A.M., Olshanetsky M.A., Zotov A.V., Painlevé VI, rigid tops and
reflection equation, Comm. Math. Phys. 268 (2006), 67-103,
math.QA/0508058.
- Levin A.M., Zotov A.V., On rational and elliptic forms of Painlevé VI
equation, in Moscow Seminar on Mathematical Physics. II,
Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc.,
Providence, RI, 2007, 173-183.
- Looijenga E., Root systems and elliptic curves, Invent. Math.
38 (1976), 17-32.
- Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A., Spectral duality
between Heisenberg chain and Gaudin model, Lett. Math. Phys., to
appear, arXiv:1206.6349.
- Mironov A., Morosov A., Shakirov Sh., Towards a proof of AGT conjecture by
methods of matrix models, Internat. J. Modern Phys. A 27
(2012), 1230001, 32 pages, arXiv:1011.5629.
- Mironov A., Morozov A., Zenkevich Y., Zotov A., Spectral duality in integrable
systems from AGT conjecture, JETP Lett., to appear, arXiv:1204.0913.
- Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact
Riemann surface, Ann. of Math. (2) 82 (1965), 540-567.
- Nekrasov N., Holomorphic bundles and many-body systems, Comm. Math.
Phys. 180 (1996), 587-603, hep-th/9503157.
- Nekrasov N., Pestun V., Seiberg-Witten geometry of four dimensional $N=2$
quiver gauge theories, arXiv:1211.2240.
- Olshanetsky M.A., Three lectures on classical integrable systems and gauge
field theories, Phys. Part. Nuclei 40 (2009), 93-114,
arXiv:0802.3857.
- Olshanetsky M.A., Zotov A. V., Isomonodromic problems on elliptic curve, rigid
tops and reflection equations, in Elliptic Integrable Systems, Rokko Lectures in Mathematics, Vol. 18, Kobe University, 2005, 149-172.
- Presley A., Segal G., Loop groups, Clarendon Press, Oxford, 1986.
- Reshetikhin N., The Knizhnik-Zamolodchikov system as a deformation of the
isomonodromy problem, Lett. Math. Phys. 26 (1992),
167-177.
- Schweigert C., On moduli spaces of flat connections with non-simply connected
structure group, Nuclear Phys. B 492 (1997), 743-755,
hep-th/9611092.
- Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math.
Soc. 3 (1990), 713-770.
- Smirnov A.V., Integrable ${\rm sl}(N,{\mathbb C})$-tops as
Calogero-Moser systems, Theoret. and Math. Phys. 158
(2009), 300-312, arXiv:0809.2187.
- Takasaki K., Gaudin model, KZ equation and an isomonodromic problem on the
torus, Lett. Math. Phys. 44 (1998), 143-156,
hep-th/9711058.
- Zabrodin A.V., Zotov A.V., Quantum Painlevé-Calogero correspondence,
J. Math. Phys. 53 (2012), 073507, 19 pages,
arXiv:1107.5672.
- Zabrodin A.V., Zotov A.V., Quantum Painlevé-Calogero correspondence for
Painlevé VI, J. Math. Phys. 53 (2012), 073508, 19 pages,
arXiv:1107.5672.
- Zotov A.V., 1+1 Gaudin model, SIGMA 7 (2011), 067,
26 pages, arXiv:1012.1072.
- Zotov A.V., Classical integrable systems and their field-theoretical
generalizations, Phys. Part. Nuclei 37 (2006), 759-842.
- Zotov A.V., Elliptic linear problem for the Calogero-Inozemtsev model and
Painlevé VI equation, Lett. Math. Phys. 67 (2004),
153-165, hep-th/0310260.
- Zotov A.V., Chernyakov Yu.B., Integrable multiparticle systems obtained by means
of the Inozemtsev limit, Theoret. and Math. Phys. 129
(2001), 1526-1542, hep-th/0102069.
- Zotov A.V., Levin A.M., An integrable system of interacting elliptic tops,
Theoret. and Math. Phys. 146 (2006), 45-52.
- Zotov A.V., Levin A.M., Olshanetsky M.A., Chernyakov Yu.B., Quadratic algebras
associated with elliptic curves, Theoret. and Math. Phys.
156 (2008), 1103-1122, arXiv:0710.1072.
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