- M. Bershtein, B. Feigin, A. Trufanov, Highest-weight vectors and three-point functions in GKO coset decomposition, Commun. Math. Phys. 406, 142 (2925) [arXiv:2404.14350].
- M. Bershtein, A. Vargulevich, NSR singular vectors from Uglov polynomials, J. Math. Phys. 63, 061706 (2022) [arXiv:2202.11810].
- M. Bershtein, P. Gavrylenko, A. Marshakov, Twist-field representations of W-algebras, exact conformal blocks and character identities, JHEP 1808:108 (2018) [arXiv:1705.00957].
- M. Bershtein, A. Shchechkin, Bilinear equations on Painleve tau functions from CFT, Comm. Math. Phys. 339(3), 1021–1061 (2015) [arXiv:1406.3008].
- M. Bershtein, B. Feigin, A. Litvinov, Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations, Lett. Math. Phys. 106(1), 29–56 (2016) [arXiv:1310.7281].
- A. Belavin, M. Bershtein, B. Feigin, A. Litvinov, G. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Comm. Math. Phys. 319(1), 269–301 (2013) [arXiv:1111.2803].
Research
My research focuses on connections between different branches of Mathematical physics:
- Representation theory, primarily of vertex algebras (algebra of symmetry of 2d conformal field theory) and q-deformed algebras (including quantum groups and toroidal algebras)
- Isomonodromic deformations of linear differential and difference equations (including Painlev\'e equations)
- Alday–Gaiotto–Tachikawa correspondence between 4d and 5d gauge theories and 2d conformal and integrable theories. In the framework of geometric representation theory this is a relation between moduli spaces of sheaves on surfaces and (possibly q-deformed) vertex algebras.
- Cluster varieties, primarily decorated moduli spaces of local systems and Goncharov–Kenyon spaces, more generally Coulomb and Higgs branches of 4d and 5d theories.
The list of my publications is available at Google Scholar and arXiv.
Below, my publications are grouped by theme (some papers may appear in more than one theme).
Conformal field theory and Representation theory of vertex algebras
Quantum groups, toroidal algebras, q-deformed algebras
- M. Bershtein, A. Vargulevich, NSR singular vectors from Uglov polynomials, J. Math. Phys. 63, 061706 (2022), [arXiv:2202.11810].
- M. Bershtein, R. Gonin, Twisted Fock module of toroidal algebra via DAHA and vertex operators, [arXiv:2109.12598].
- M. Bershtein, R. Gonin, Twisted and Non-Twisted Deformed Virasoro Algebra via Vertex Operators of U_q(ĥ sl₂), Lett. Math. Phys. 111, 22 (2021), [arXiv:2003.12472].
- M. Bershtein, R. Gonin, Twisted Representations of Algebra of q-Difference Operators, Twisted q-W Algebras and Conformal Blocks, SIGMA 16, 077 (2020), [arXiv:1906.00600].
- M. Bershtein, A. Tsymbaliuk, Homomorphisms between different quantum toroidal and affine Yangian algebras, J. Pure Appl. Algebra 223, 867–899 (2019), [arXiv:1512.09109].
- M. Bershtein, B. Feigin, G. Merzon, Plane partitions with a "pit": generating functions and representation theory, Sel. Math. New Ser. 24, 21–62 (2018), [arXiv:1512.08779].
- A. Belavin, M. Bershtein, G. Tarnopolsky, Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP 1304:019 (2013), [arXiv:1211.2788].
Cluster algebras and varieties
- M. Bershtein, Cluster Integrable Systems, [arXiv:2503.18573].
- M. Bershtein, P. Gavrylenko, A. Marshakov, M. Semenyakin, Cluster Reductions, Mutations, and q-Painlevé Equations, [arXiv:2411.00325].
- M. Bershtein, P. Gavrylenko, A. Marshakov, Cluster Toda chains and Nekrasov functions, Theor. Math. Phys. 198, 157 (2019), [arXiv:1804.10145].
- M. Bershtein, P. Gavrylenko, A. Marshakov, Cluster integrable systems, q-Painlevé equations and quantization, JHEP 1802:077 (2018), [arXiv:1711.02063].
Isomonodromic deformations, Painlevé equations, correspondence with CFT/gauge theories
- M. Bershtein, P. Gavrylenko, A. Marshakov, M. Semenyakin, Cluster Reductions, Mutations, and q-Painlevé Equations, [arXiv:2411.00325].
- M. Bershtein, A. Grigorev, A. Shchechkin, Hamiltonian reductions in Matrix Painlevé systems, Lett. Math. Phys. 113, 47 (2023), [arXiv:2208.04824].
- M. Bershtein, A. Shchechkin, Folding transformations for q-Painlevé equations, [arXiv:2110.15320].
- M. Bershtein, P. Gavrylenko, A. Grassi, Quantum spectral problems and isomonodromic deformations, Commun. Math. Phys. 393, 347–418 (2022), [arXiv:2105.00985].
- M. Bershtein, A. Shchechkin, Painlevé equations from Nakajima-Yoshioka blow-up relations, Lett. Math. Phys. 109(11), 2359–2402 (2019), [arXiv:1811.04050].
- M. Bershtein, P. Gavrylenko, A. Marshakov, Cluster integrable systems, q-Painlevé equations and quantization, JHEP 1802:077 (2018), [arXiv:1711.02063].
- M. Bershtein, A. Shchechkin, Backlund transformation of Painlevé III(D₈) tau function, J. Phys. A 50, 115205 (2017), [arXiv:1608.02568].
- M. Bershtein, A. Shchechkin, q-deformed Painlevé τ function and q-deformed conformal blocks, J. Phys. A 50, 085202 (2017), [arXiv:1608.02566].
- M. Bershtein, A. Shchechkin, Bilinear equations on Painlevé tau functions from CFT, Comm. Math. Phys. 339(3), 1021–1061 (2015), [arXiv:1406.3008].
AGT correspondence
- M. Bershtein, G. Bonelli, M. Ronzani, A. Tanzini, Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants, J. Geom. Phys. 118, 40–50 (2017), [arXiv:1606.07148].
- M. Bershtein, G. Bonelli, M. Ronzani, A. Tanzini, Exact results for ${\cal N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants, JHEP 2016:23 (2016), [arXiv:1509.00267].
- M. Bershtein, O. Foda, AGT, Burge pairs and minimal models, JHEP 1406:177 (2014), [arXiv:1404.7075].
- M. Bershtein, B. Feigin, A. Litvinov, Coupling of two conformal field theories and Nakajima-Yoshioka blow-up equations, Lett. Math. Phys. 106(1), 29–56 (2016), [arXiv:1310.7281].
- A. Belavin, M. Bershtein, G. Tarnopolsky, Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP 1304:019 (2013), [arXiv:1211.2788].
- A. Belavin, M. Bershtein, B. Feigin, A. Litvinov, G. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Comm. Math. Phys. 319(1), 269–301 (2013), [arXiv:1111.2803].
- A. Belavin, V. Belavin, M. Bershtein, Instantons and 2d Superconformal field theory, JHEP 1109:117 (2011), [arXiv:1106.4001].
Miscellaneous
- M. Bershtein, V. Fateev, A. Litvinov, Selberg integrals and three-point correlation function in parafermionic Liouville field theory, Nuclear Physics B 84, 413–459 (2011) [arXiv:1011.4090].
- A. Belavin, M. Bershtein, G. Tarnopolsky, A remark on the three approaches to 2D Quantum gravity, JETP Lett. 93(2), 47–51 (2011) [arXiv:1010.2222].
- O. Alekseev, M. Bershtein, The ring of physical states in the M(2, 3) minimal Liouville gravity, Theor. Math. Phys. 164(1), 929–946 (2010) [arXiv:0906.1377].
- M. Bershtein, V. Dotsenko, A. Khoroshkin, Quadratic algebras related to the bihamiltonian operad, Int. Math. Res. Notices (2007) [arXiv:math/0607289].
Educational and Expository Publications
- M. Bershtein, Cluster Integrable Systems, [arXiv:2503.18573].
- M. Bershtein, G. Merzon, Young diagrams, lattice paths and reflection method, Matematicheskoe prosveshenie Ser. 3, 18, 112–141 (2014).
Edited Volumes
- M. Bershtein, A. Dzhamay, A. Okounkov (editors), The Versatility of Integrability: In Memory of Igor Krichever, Contemporary Mathematics, 823, 270 pp (2025).