Elliptic zastava
Authors:
Michael Finkelberg, Mykola Matviichuk and Alexander Polishchuk
Journal:
J. Algebraic Geom. 32 (2023), 183-237
DOI:
https://doi.org/10.1090/jag/803
Published electronically:
May 17, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirković local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskiĭ moduli spaces of $G$-bundles with parabolic structure on an elliptic curve.
References
- Alexander Braverman and Michael Finkelberg, Semi-infinite Schubert varieties and quantum $K$-theory of flag manifolds, J. Amer. Math. Soc. 27 (2014), no. 4, 1147–1168. MR 3230820, DOI 10.1090/S0894-0347-2014-00797-9
- Alexander Braverman, Galyna Dobrovolska, and Michael Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles, Adv. Math. 300 (2016), 451–472. MR 3534838, DOI 10.1016/j.aim.2016.03.024
- A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.) 8 (2002), no. 3, 381–418; and Erratum, Selecta Math. (N.S.) 10 (2004), 429–430.
- Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories, II, Adv. Theor. Math. Phys. 22 (2018), no. 5, 1071–1147. MR 3952347, DOI 10.4310/ATMP.2018.v22.n5.a1
- Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Coulomb branches of $3d$ $\mathcal N=4$ quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23 (2019), no. 1, 75–166. With two appendices by Braverman, Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Nakajima, Ben Webster and Alex Weekes. MR 4020310, DOI 10.4310/ATMP.2019.v23.n1.a3
- Sergey Cherkis and Anton Kapustin, Nahm transform for periodic monopoles and $\scr N=2$ super Yang-Mills theory, Comm. Math. Phys. 218 (2001), no. 2, 333–371. MR 1828985, DOI 10.1007/PL00005558
- Sergey A. Cherkis and Richard S. Ward, Moduli of monopole walls and amoebas, J. High Energy Phys. 5 (2012), 090, front matter+36. MR 3042954, DOI 10.1007/JHEP05(2012)090
- V. Drinfeld, Grinberg–Kazhdan theorem and Newton groupoids, arXiv:1801.01046, 2018.
- B. L. Feĭgin and A. V. Odesskiĭ, Vector bundles on an elliptic curve and Sklyanin algebras, Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, vol. 185, Amer. Math. Soc., Providence, RI, 1998, pp. 65–84. MR 1736164, DOI 10.1090/trans2/185/04
- M. Finkelberg, A. Kuznetsov, N. Markarian, and I. Mirković, A note on a symplectic structure on the space of $G$-monopoles, Commun. Math. Phys. 201 (1999), 411–421; and Erratum, Commun. Math. Phys. 334 (2015), 1153–1155.
- Michael Finkelberg, Alexander Kuznetsov, Leonid Rybnikov, and Galyna Dobrovolska, Towards a cluster structure on trigonometric zastava, Selecta Math. (N.S.) 24 (2018), no. 1, 187–225. MR 3769730, DOI 10.1007/s00029-016-0287-1
- R. Friedman and J. W. Morgan, Holomorphic principal bundles over elliptic curves, arXiv:math/9811130, 1998.
- Robert Friedman, John W. Morgan, and Edward Witten, Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), no. 1-2, 97–118. MR 1618343, DOI 10.4310/MRL.1998.v5.n1.a8
- Evgeny Feigin and Ievgen Makedonskyi, Semi-infinite Plücker relations and Weyl modules, Int. Math. Res. Not. IMRN 14 (2020), 4357–4394. MR 4126304, DOI 10.1093/imrn/rny121
- Michael Finkelberg and Alexander Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras, Representations and nilpotent orbits of Lie algebraic systems, Progr. Math., vol. 330, Birkhäuser/Springer, Cham, 2019, pp. 133–304. MR 3971731, DOI 10.1007/978-3-030-23531-4_{6}
- V. Ginzburg, M. Kapranov, and E. Vasserot, Elliptic algebras and equivariant cohomology I, arXiv:q-alg/9505012, 1995.
- D. Gaitsgory, Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.) 13 (2008), no. 4, 617–659. MR 2403306, DOI 10.1007/s00029-008-0053-0
- Nora Ganter, The elliptic Weyl character formula, Compos. Math. 150 (2014), no. 7, 1196–1234. MR 3230851, DOI 10.1112/S0010437X1300777X
- I. Grojnowski, Delocalised equivariant elliptic cohomology, Elliptic cohomology, London Math. Soc. Lecture Note Ser., vol. 342, Cambridge Univ. Press, Cambridge, 2007, pp. 114–121. MR 2330510, DOI 10.1017/CBO9780511721489.007
- Zheng Hua and Alexander Polishchuk, Shifted Poisson structures and moduli spaces of complexes, Adv. Math. 338 (2018), 991–1037. MR 3861721, DOI 10.1016/j.aim.2018.09.018
- Stuart Jarvis, Euclidean monopoles and rational maps, Proc. London Math. Soc. (3) 77 (1998), no. 1, 170–192. MR 1625475, DOI 10.1112/S0024611598000434
- Stuart Jarvis, Construction of Euclidean monopoles, Proc. London Math. Soc. (3) 77 (1998), no. 1, 193–214. MR 1625471, DOI 10.1112/S0024611598000446
- I. Mirković, Y. Yang, and G. Zhao, Loop Grassmannians of quivers and affine quantum groups, arXiv:1810.10095, 2018.
- H. Nakajima and A. Weekes, Coulomb branches of quiver gauge theories with symmetrizers, arXiv:1907.06552, 2019.
- Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328. MR 3090262, DOI 10.1007/s10240-013-0054-1
- Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153, Cambridge University Press, Cambridge, 2003. MR 1987784, DOI 10.1017/CBO9780511546532
- Pavel Safronov, Poisson-Lie structures as shifted Poisson structures, Adv. Math. 381 (2021), Paper No. 107633, 68. MR 4214399, DOI 10.1016/j.aim.2021.107633
- Simon Schieder, The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld’s compactifications, Selecta Math. (N.S.) 21 (2015), no. 3, 763–831. MR 3366920, DOI 10.1007/s00029-014-0161-y
- T. Spaide, Shifted symplectic and Poisson structures on spaces of framed maps, arXiv:1607.03807, 2016.
- Xinwen Zhu, Affine Demazure modules and $T$-fixed point subschemes in the affine Grassmannian, Adv. Math. 221 (2009), no. 2, 570–600. MR 2508931, DOI 10.1016/j.aim.2009.01.003
References
- Alexander Braverman and Michael Finkelberg, Semi-infinite Schubert varieties and quantum $K$-theory of flag manifolds, J. Amer. Math. Soc. 27 (2014), no. 4, 1147–1168. MR 3230820, DOI 10.1090/S0894-0347-2014-00797-9
- Alexander Braverman, Galyna Dobrovolska, and Michael Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles, Adv. Math. 300 (2016), 451–472. MR 3534838, DOI 10.1016/j.aim.2016.03.024
- A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.) 8 (2002), no. 3, 381–418; and Erratum, Selecta Math. (N.S.) 10 (2004), 429–430.
- Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories, II, Adv. Theor. Math. Phys. 22 (2018), no. 5, 1071–1147. MR 3952347, DOI 10.4310/ATMP.2018.v22.n5.a1
- Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Coulomb branches of $3d$ $\mathcal N=4$ quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23 (2019), no. 1, 75–166. With two appendices by Braverman, Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Nakajima, Ben Webster and Alex Weekes. MR 4020310, DOI 10.4310/ATMP.2019.v23.n1.a3
- Sergey Cherkis and Anton Kapustin, Nahm transform for periodic monopoles and $\mathcal {N}=2$ super Yang-Mills theory, Comm. Math. Phys. 218 (2001), no. 2, 333–371. MR 1828985, DOI 10.1007/PL00005558
- Sergey A. Cherkis and Richard S. Ward, Moduli of monopole walls and amoebas, J. High Energy Phys. 5 (2012), 090, front matter+36. MR 3042954, DOI 10.1007/JHEP05(2012)090
- V. Drinfeld, Grinberg–Kazhdan theorem and Newton groupoids, arXiv:1801.01046, 2018.
- B. L. Feĭgin and A. V. Odesskiĭ, Vector bundles on an elliptic curve and Sklyanin algebras, Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, vol. 185, Amer. Math. Soc., Providence, RI, 1998, pp. 65–84. MR 1736164, DOI 10.1090/trans2/185/04
- M. Finkelberg, A. Kuznetsov, N. Markarian, and I. Mirković, A note on a symplectic structure on the space of $G$-monopoles, Commun. Math. Phys. 201 (1999), 411–421; and Erratum, Commun. Math. Phys. 334 (2015), 1153–1155.
- Michael Finkelberg, Alexander Kuznetsov, Leonid Rybnikov, and Galyna Dobrovolska, Towards a cluster structure on trigonometric zastava, Selecta Math. (N.S.) 24 (2018), no. 1, 187–225. MR 3769730, DOI 10.1007/s00029-016-0287-1
- R. Friedman and J. W. Morgan, Holomorphic principal bundles over elliptic curves, arXiv:math/9811130, 1998.
- Robert Friedman, John W. Morgan, and Edward Witten, Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), no. 1-2, 97–118. MR 1618343, DOI 10.4310/MRL.1998.v5.n1.a8
- Evgeny Feigin and Ievgen Makedonskyi, Semi-infinite Plücker relations and Weyl modules, Int. Math. Res. Not. IMRN 14 (2020), 4357–4394. MR 4126304, DOI 10.1093/imrn/rny121
- Michael Finkelberg and Alexander Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras, Representations and nilpotent orbits of Lie algebraic systems, Progr. Math., vol. 330, Birkhäuser/Springer, Cham, 2019, pp. 133–304. MR 3971731
- V. Ginzburg, M. Kapranov, and E. Vasserot, Elliptic algebras and equivariant cohomology I, arXiv:q-alg/9505012, 1995.
- D. Gaitsgory, Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.) 13 (2008), no. 4, 617–659. MR 2403306, DOI 10.1007/s00029-008-0053-0
- Nora Ganter, The elliptic Weyl character formula, Compos. Math. 150 (2014), no. 7, 1196–1234. MR 3230851, DOI 10.1112/S0010437X1300777X
- I. Grojnowski, Delocalised equivariant elliptic cohomology, Elliptic cohomology, London Math. Soc. Lecture Note Ser., vol. 342, Cambridge Univ. Press, Cambridge, 2007, pp. 114–121. MR 2330510, DOI 10.1017/CBO9780511721489.007
- Zheng Hua and Alexander Polishchuk, Shifted Poisson structures and moduli spaces of complexes, Adv. Math. 338 (2018), 991–1037. MR 3861721, DOI 10.1016/j.aim.2018.09.018
- Stuart Jarvis, Euclidean monopoles and rational maps, Proc. London Math. Soc. (3) 77 (1998), no. 1, 170–192. MR 1625475, DOI 10.1112/S0024611598000434
- Stuart Jarvis, Construction of Euclidean monopoles, Proc. London Math. Soc. (3) 77 (1998), no. 1, 193–214. MR 1625471, DOI 10.1112/S0024611598000446
- I. Mirković, Y. Yang, and G. Zhao, Loop Grassmannians of quivers and affine quantum groups, arXiv:1810.10095, 2018.
- H. Nakajima and A. Weekes, Coulomb branches of quiver gauge theories with symmetrizers, arXiv:1907.06552, 2019.
- Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328. MR 3090262, DOI 10.1007/s10240-013-0054-1
- Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153, Cambridge University Press, Cambridge, 2003. MR 1987784, DOI 10.1017/CBO9780511546532
- Pavel Safronov, Poisson-Lie structures as shifted Poisson structures, Adv. Math. 381 (2021), Paper No. 107633, 68. MR 4214399, DOI 10.1016/j.aim.2021.107633
- Simon Schieder, The Harder-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld’s compactifications, Selecta Math. (N.S.) 21 (2015), no. 3, 763–831. MR 3366920, DOI 10.1007/s00029-014-0161-y
- T. Spaide, Shifted symplectic and Poisson structures on spaces of framed maps, arXiv:1607.03807, 2016.
- Xinwen Zhu, Affine Demazure modules and $T$-fixed point subschemes in the affine Grassmannian, Adv. Math. 221 (2009), no. 2, 570–600. MR 2508931, DOI 10.1016/j.aim.2009.01.003
Additional Information
Michael Finkelberg
Affiliation:
Department of Mathematics, National Research University Higher School of Economics, Russian Federation, Usacheva st. 6, 119048; Skolkovo Institute of Science and Technology; and Institute for Information Transmission Problems of Russian Academy of Science, Moscow, Russia
MR Author ID:
304673
Email:
fnklberg@gmail.com
Mykola Matviichuk
Affiliation:
Department of Mathematics, McGill University, Burnside Hall, 805 Sher- brooke st. W., Montréal, Québec H3A 2K6, Canada
MR Author ID:
1393397
Email:
mykola.matviichuk@gmail.com
Alexander Polishchuk
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403; National Research University Higher School of Economics; and Korea Institute for Advanced Study, Seoul, Korea
MR Author ID:
339630
Email:
apolish@uoregon.edu
Received by editor(s):
December 6, 2020
Published electronically:
May 17, 2022
Additional Notes:
The work of the first and third authors has been funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ‘5-100’. The third author is also partially supported by the NSF grant DMS-2001224.
Dedicated:
To Tony Joseph on his 80th birthday, with admiration
Article copyright:
© Copyright 2022
University Press, Inc.
