Here you can find my major research topics and main publications
(see also Full Publication List):
- Enumerative geometry, quantum integrable systems and q-Langlands correspondence
We investigate the relationship between quantum integrable systems
and enumerative geometry, motivated by the results in the study of
supersymmetric gauge theories.
Peter Koroteev, Anton M. Zeitlin,
"The Zoo of Opers and Dualities", International Mathematics Research Notices (IMRN), Volume 2024, Issue 8, pp. 6850-6878, 2024
(doi: 10.1093/imrn/rnad270),
arXiv:2208.08031.
Anton M. Zeitlin,,
"On Wronskians and qq-systems", Contemporary Mathematics, AMS, Proceedings of "Geometric and Algebraic Aspects of Quantum Groups and Related Topics at the University of South Alabama", Nov 2021,
arXiv:2208.08018.
Ty J. Brinson, Daniel S. Sage, Anton M. Zeitlin,
"Opers on the projective line, Wronskian relations, and the Bethe Ansatz",
arXiv:2112.02711.
Peter Koroteev, Anton M. Zeitlin,
"q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors",
Journal für die reine und angewandte Mathematik, Issue 795, pp. 271-296, 2023
(doi: 10.1515/crelle-2022-0084),
arXiv:2108.04184.
Peter Koroteev, Anton M. Zeitlin,
"3d Mirror Symmetry for Instanton Moduli Spaces",
Communications in Mathematical Physics, Volume 403, Issue 2, pp. 1005-1068, 2023 (doi: 10.1007/s00220-023-04831-5),
arXiv:2105.00588.
Peter Koroteev, Anton M. Zeitlin,
"Toroidal q-Opers",
Journal of the Institute of Mathematics of Jussieu, Volume 22, Issue 2, pp. 581-642, 2023
(doi: 10.1017/S1474748021000220),
arXiv:2007.11786.
Edward Frenkel, Peter Koroteev, Daniel S. Sage, Anton M. Zeitlin,
"q-Opers, QQ-Systems, and Bethe Ansatz",
Journal of the European Mathematical Society
(JEMS), Volume 26, Number 1, pp. 355-405, 2024 (doi: 10.4171/JEMS/1268),
arXiv:2002.07344.
Peter Koroteev, Daniel S. Sage, Anton M. Zeitlin,
"(SL(N),q)-opers, the q-Langlands correspondence, and quantum/classical duality",
Communications in Mathematical Physics, Volume 381, Issue 2, pp. 641-672, 2021
(doi: 10.1007/s00220-020-03891-1),
arXiv:1811.09937.
Peter Koroteev, Anton M. Zeitlin,
"qKZ/tRS Duality via Quantum K-Theoretic Counts", Mathematical Research Letters,
Volume 28, Number 2, pp. 435-470, 2021
(doi: 10.4310/MRL.2021.v28.n2.a5),
arXiv:1802.04463.
Peter Koroteev, Petr P. Pushkar, Andrey Smirnov, Anton M. Zeitlin,
"Quantum K-theory of Quiver Varieties and Many-Body Systems",
Selecta Mathematica New Ser., Volume 27, Issue 5, Article: 87, 2021
(doi: 10.1007/s00029-021-00698-3),
arXiv:1705.10419.
Petr P. Pushkar, Andrey Smirnov, Anton M. Zeitlin,
"Baxter Q-operator from quantum K-theory", Advances in Mathematics, Volume 360, 106919, 2020
(doi: 10.1016/j.aim.2019.106919),
arXiv:1612.08723.
- Higher Teichmüller theory
We construct the analogue of Penner coordinates on the N=1 and N=2 super-Teichmüller spaces.
Future projects based on this work involve super-analogue of cluster algebras, computations of volumes of supermoduli spaces, quantization of (higher) super-Teichmüller spaces.
Albert S. Schwarz, Anton M. Zeitlin,
"Super Riemann surfaces and fatgraphs", Universe, Special Issue "Quantum Theory and Beyond", 9(9), 384, 2023
(doi: 10.3390/universe9090384),
arXiv:2307.02706.
Andrea Bourque, Anton M. Zeitlin,
"Flat GL(1|1)-connections and fatgraphs", Journal of Geometry and Physics, Volume 191, 104880, 2023 (doi: 10.1016/j.geomphys.2023.104880),
arXiv:2208.08033
Yi Huang, Robert C. Penner, Anton M. Zeitlin,
"Super McShane identity", Journal of Differential Geometry, Volume 125, Number 3, pp. 509-551, 2023
(doi: 10.4310/jdg/1701804150),
arXiv:1907.09978.
Anton M. Zeitlin (joint with Ivan C.-H. Ip, Robert C. Penner ),
"Super-Teichmüller spaces and related structures",
Oberwolfach Report No. 40/2018,
Volume 15, Issue 3, pp. 2486-2489, 2018
(doi: 10.4171/OWR/2018/40),
arXiv:1811.09939.
Ivan C.H. Ip, Robert C. Penner, Anton M. Zeitlin,
"On Ramond Decorations", Communications in Mathematical Physics, Volume 371, Issue 1, pp. 145-157, 2019
(doi: 10.1007/s00220-019-03424-5),
arXiv:1709.06207.
Ivan C.H. Ip, Robert C. Penner, Anton M. Zeitlin,
"N=2 Super-Teichmüller Theory", Advances in Mathematics, Volume 336, pp. 409-454, 2018 (doi: 10.1016/j.aim.2018.08.001),
arXiv:1605.08094.
R.C. Penner, Anton M. Zeitlin, "Decorated Super-Teichmüller Space",
Journal of Differential Geometry, Volume 111, no. 3, 527-566, 2019
(doi: 10.4310/jdg/1552442609),
arXiv:1509.06302.
- Quantum groups and Kac-Moody algebras
- Towards the continuous Kazhdan-Lusztig correspondence. Chern-Simons theory
for real reductive Lie groups
We attempt to build the braided tensor category of modules for affine
Kac-Moody algebras, equivalent to the modular double tensor category of
quantum groups. This will lead to the mathematical understanding of WZW
model and Chern-Simons theory for real reductive Lie groups.
Anton M. Zeitlin, "On the unitary representations of the affine $ax+b$-group, $\widehat{sl}(2,\mathbb{R})$
and their relatives",
Proc. Symp. Pure Math., AMS,
Volume 92 "Lie Algebras, Lie Superalgebras,
Vertex Algebras and Related Topics", pp. 325-355, 2016 (doi: 10.1090/pspum/092),
arXiv:1509.06072.
Igor B. Frenkel, Anton M. Zeitlin, "On the continuous series for $\widehat{sl}(2,\mathbb{R})$",
Communications in Mathematical Physics, Volume 326, Issue 1, pp. 145-165, 2014
(doi: 10.1007/s00220-013-1832-9),
arXiv:1210.2135.
Anton M. Zeitlin, "Unitary representations of a loop ax+b group, Wiener measure and Gamma-function",
Journal of Functional Analysis,
Volume 263, Issue 3, pp. 529-548, 2012
(doi: 10.1016/j.jfa.2012.05.004),
arXiv:1012.4826.
- Semi-infinite cohomology, quantum groups and noncommutative geometry
We introduce a new approach to quantum groups as semi-infinite cohomology
rings for certain braided vertex algebras. This leads to surprising
realizations of some constructions in noncommutative geometry.
Igor B. Frenkel, Anton M. Zeitlin,
"Quantum Group GLq(2) and Quantum Laplace Operator via Semi-infinite Cohomology",
Journal of Noncommutative Geometry,
Volume 7, Issue 4, pp. 1007-1026, 2013
(doi: 10.4171/JNCG/142),
arXiv:1110.1696.
Igor B. Frenkel, Anton M. Zeitlin, "Quantum Group as Semi-infinite Cohomology",
Communications in Mathematical Physics, Volume 297, Number 3, pp. 687-732, 2010
(doi: 10.1007/s00220-010-1055-2),
arXiv:0812.1620.
- Homotopy Gerstenhaber algebras, Einstein equations and Courant algebroids
Using the relation to 2-dimensional sigma models, we construct an underlying homotopy Gerstenhaber algebra within Einstein equations with extra fields. Such an algebraic structure is
naturally related to the Courant algebroid.
Martin Rocek, Anton M. Zeitlin, "Homotopy algebras of differential (super)forms in three and four dimensions",
Letters in Mathematical Physics, Volume 108, Issue 12, pp. 2669-2694, 2018
(doi: 10.1007/s11005-018-1109-5),
arXiv:1702.03565.
Anton M. Zeitlin, "Beltrami-Courant Differentials and $G_\infty$-algebras",
Advances in Theoretical and Mathematical Physics,
Volume 19, Number 6, pp. 1249-1275, 2015
(doi: 10.4310/ATMP.2015.v19.n6.a3),
arXiv:1404.3069.
Anton M. Zeitlin,
"Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroid and Gauge Theory",
Communications in Mathematical Physics, Volume 303, Number 2, pp. 331-359, 2011
(doi: 10.1007/s00220-011-1206-0),
arXiv:0910.3652.
Anton M. Zeitlin, "Beta-gamma systems and the deformations of the BRST operator",
Journal of Physics A: Mathematical and Theoretical,
Volume 42, Number 35, 355401, 2009
(doi: 10.1088/1751-8113/42/35/355401),
arXiv:0904.2234.
Anton M. Zeitlin,
"SFT-inspired Algebraic Structures in Gauge Theories",
Journal of Mathematical Physics (JMP), 50, Issue 6, 063501, 2009
(doi: 10.1063/1.3142964),
arXiv:0711.3843.
Anton M. Zeitlin,
"Perturbed Beta-Gamma Systems and Complex Geometry",
Nuclear Physics B,
Volume 794, Issue 3[PM], pp. 381-401, 2008
(doi: 10.1016/j.nuclphysb.2007.09.002),
arXiv:0708.0682.
Andrei S. Losev, Andrei Marshakov, Anton M. Zeitlin,
"On First Order Formalism in String Theory",
Physics Letters B, Volume 633/2-3, pp. 375-381, 2006
(doi: 10.1016/j.physletb.2005.12.010),
hep-th/0510065.
- Geometric Langlands correspondence for supergroups
We define a counterpart of an oper on a supercurve and study the analogue of Geometric Langlands correspondence in the simplest nontrivial case of OSp(1|2) supergroup. In the future we plan to extend it to higher rank case.
Anton M. Zeitlin, "Superopers revisited",
arXiv:2307.02675.
Anton M. Zeitlin, "Superopers on Supercurves",
Letters in Mathematical Physics, Volume 105, Issue 2, pp. 149-167, 2015
(doi: 10.1007/s11005-014-0735-9),
arXiv:1311.5997.
- Homotopy algebras of topological conformal field theories
We introduce nonlocal operators in topological conformal field theories using integration over the compactified moduli spaces and study relations between them.
Anton M. Zeitlin, "On higher order Leibniz identities in TCFT",
Contemporary Mathematics, Volume 623, pp. 267-280, 2014
(doi: 10.1090/conm/623/12458)
arXiv:1301.6382.
Anton M. Zeitlin, "Homotopy Relations for Topological VOA",
International Journal of Mathematics (IJM), Volume 23, Issue 1, 1250012, 2012
(doi: 10.1142/S0129167X11007550),
arXiv:1104.5038.
- Integrable systems and (super)conformal field theory
We investigate integrable structures in superconformal field theory. In
particular, that involves quantization of supersymmetric models of KdV type.
Ivan Chi-Ho Ip, Anton M. Zeitlin,
"Q-operator and fusion relations for Cq(2)(2)",
Letters in Mathematical Physics, Volume 104, Issue 8, pp. 1019-1043, 2014
(doi: 10.1007/s11005-014-0702-5),
arXiv:1312.4063.
A.M. Zeitlin, "Quantization of N=2 supersymmetric KdV Hierarchy",
Theoretical and Mathematical Physics, v. 147, n. 2, pp. 303-314, 2006 (in russian,
(doi: 10.4213/tmf1965),
Engl. transl.: Theoretical and Mathematical Physics v. 147, n. 2, pp. 698-708, 2006
(doi: 10.1007/s11232-006-0071-z),
hep-th/0606129.
Petr P. Kulish, Anton M. Zeitlin, "Quantum supersymmetric Toda-mKdV hierarchies",
Nuclear Physics B, Volume 720, Issue 3, pp. 289-306, 2005
(doi: 10.1016/j.nuclphysb.2005.06.002),
hep-th/0506027.
Petr P. Kulish, Anton M. Zeitlin,
"Superconformal field theory and SUSY N=1 KdV hierarchy II: the Q-operator",
Nuclear Physics B, Volume 709, Issue 3, pp. 578-591,
2005 (pdf)
(doi: 10.1016/j.nuclphysb.2004.12.031),
hep-th/0501019.
Petr P. Kulish, Anton M. Zeitlin,
"Quantum inverse scattering method and (super)conformal field theory",
Theoretical and Mathematical Physics, v. 142, n. 2, pp. 252-264, 2005 (in russian,
doi: 10.4213/tmf1779),
Engl. transl.: Theoretical and Mathematical Physics, v. 142, n. 2, pp. 211-221, 2005
(pdf),
(doi: 10.1007/s11232-005-0054-5),
hep-th/0501018.
Petr P. Kulish, Anton M. Zeitlin,
"Superconformal Field Theory and SUSY N=1 KdV Hierarchy I: Vertex
Operators and Yang-Baxter Equation",
Physics Letters B, Volume 597, Issue 2, pp. 229-236, 2004
(doi: 10.1016/j.physletb.2004.07.019),
hep-th/0407154.
Petr P. Kulish, Anton M. Zeitlin,
"Integrable Structure of Superconformal
Field Theory and Quantum super-KdV Theory",
Physics Letters B, Volume 581, Issues 1-2, pp. 125-132, 2004
(doi: 10.1016/j.physletb.2003.12.008),
hep-th/0312159.
Petr P. Kulish, Anton M. Zeitlin,
"Group Theoretical Structure and Inverse Scattering Method for
super-KdV Equation", Zapiski Nauchnih Seminarov POMI (Steklov Institute), vol. 291,
185-205 (ps.gz),
2002 (in russian);
Engl. transl. :
Journal of Mathematical Sciences (Springer/Kluwer), v. 125, n. 2, pp. 203-214, 2005
(doi: 10.1023/B:JOTH.0000049572.41993.9f),
hep-th/0312158.
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