### Research

 Here you can find my major research topics and main publications (see also Full Publication List): Quantum equivariant K-theory and quantum integrable systems Higher Teichmüller theory Quantum groups and Kac-Moody algebras Towards the continuous Kazhdan-Lusztig correspondence. Chern-Simons theory for real reductive Lie groups Semi-infinite cohomology, quantum groups and noncommutative geometry Homotopy Gerstenhaber algebras, Einstein equations and Courant algebroids Geometric Langlands correspondence for supergroups Homotopy algebras of topological conformal field theories Integrable systems and (super)conformal field theory   Quantum equivariant K-theory and quantum integrable systems 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, "Difference Equations for K-theoretic Vertex Functions of Type-A Nakajima Varieties", arXiv:1802.04463. Peter Koroteev, Petr P. Pushkar, Andrey Smirnov, Anton M. Zeitlin, "Quantum K-theory of Quiver Varieties and Many-Body Systems", arXiv:1705.10419. Petr P. Pushkar, Andrey Smirnov, Anton M. Zeitlin, "Baxter Q-operator from quantum K-theory", 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. Ivan C.H. Ip, Robert C. Penner, Anton M. Zeitlin, "On Ramond Decorations", arXiv:1709.06207. Ivan C.H. Ip, Robert C. Penner, Anton M. Zeitlin, "N=2 Super-Teichmüller Theory", arXiv:1605.08094. R.C. Penner, Anton M. Zeitlin, "Decorated Super-Teichmüller Space", Journal of Differential Geometry, in press, 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, 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. 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 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.