My history of publications can be found below. Each publication is listed in correspondence with the project it belongs to, including items which are currently a work-in-progress. A brief pedagogical summary, outlining the narrative of the project under consideration, can be found by clicking the arrow button on the right.
A full list of my publications is also available on Google Scholar.
Infinite-dimensional representations of Uq(sl2) and the shadow world
Part I: quantum 6j-symbols for Verma modules
[arXiv:2410.18463|submitted to Letters in Mathematical Physics]
Part II: polarized modules
[TBA]
Reshetikhin and Turaev constructed a non-perturbative completion of the Chern-Simons partition function, which is a surgery TQFT that "upgrades" a link invariant given by the Reshetikhin-Turaev functor to a 3-manifold invariant, also known as the Witten-Reshetikhin-Turaev invariant. By analogy, Gukov-Putrov-Pei-Vafa series Ẑ are expected to stand in the same relation to Gukov-Manolescu series. As a step toward a non-perturbative definition of Ẑ, it is natural to first seek a functorial formulation of Gukov-Manolescu series, starting with the identification of a suitable target category for the Reshetikhin-Turaev functor.
In [Kirillov-Reshetikhin, 1988] a model for the colored Jones polynomial was proposed, which is based on graphical representation for q6j-symbols, also known as the shadow world. Since for finite-dimensional representations it is equivalent to the R-matrix approach, it makes sense to initiate the study of this model for infinite-dimensional representations.
In [1] we establish formulae and properties for quantum Clebsch-Gordan coefficients(q3j-symbols) and quantum Racah-Wigner coefficients(q6j-symbols) associated with a braided monoidal version of the q-analog of BGG category O, which includes Verma modules of the highest weight. Guided by the bulk-boundary correspondence, we discuss the equivalence of the R-matrix and the shadow world approaches for Verma modules. We introduce a functional SW∞ and investigate its potential as a model for the Gukov-Manolescu series. Based on this discussion, we formulate a conjecture for a target category for the RT functor and its fusion rules.
In [2] we study the tensor structure of the category of representations of the Chevalley extended Uq(sl2), which we call the polarized category, and prove the conjecture for its Grothendieck ring proposed in [1]. We establish proper graphical calculus for such a category, making it a target category for the Reshetikhin-Turaev functor.
Limit shape phenomena in representation theory of quantum groups at roots of unity
Counting filter restricted paths in ℤ² lattice
with Olga Postnova
[arXiv:2107.09774|Enumerative Combinatorics and Applications 4:4 (2024) Article S2R31]
Congruence for lattice path models with filter restrictions and long steps
[arXiv:2111.13857|Mathematics 10 (22), 4209]
Tensor powers of vector representation of Uq(sl2) at even roots of unity
with Anna Lachowska, Olga Postnova and Nicolai Reshetikhin
[arXiv:2404.03933|submitted to AMS Journal of Representation Theory]
TBA
with Henning Haahr Andersen, Anna Lachowska and Nicolai Reshetikhin
[TBA]
TBA
with Henning Haahr Andersen, Anna Lachowska and Nicolai Reshetikhin
[TBA]
The ultimate goal of this project is to obtain results analogous to the Vershik-Kerov limit shape, but for quantum groups at roots of unity, where the category of representations is non-semisimple. In other words, we want to study central limit theorems describing statistics of indecomposable components in tensor product decomposition of representations of Uq(sln) with respect to Plancherel and character measures, when tensor power goes to infinity, the highest weight of the indecomposable component goes to infinity, and, ideally, rank of algebra also goes to infinity, while root of unity remains fixed or grows proportionally to the rank.
We initiate this project by studying Uq(sl2) with divided powers and the small quantum sl2. We represent relations in the Grothendieck ring of the category of tilting modules of Uq(sl2) with divided powers as a random walk in a lattice path model, where weighted numbers of descending paths satisfy the same recurrsion relations as multiplicities of indecomposable components in tensor product decomposition. We study limiting distributions of these random walks: most probable trajectory and fluctuations in its vicinity. We do so in multiple regimes: in the bulk, for the intermediate scaling, for the critical drift. We restrict Uq(sl2) with divided powers to uq(sl2) to obtain formulas for multiplicities, which we then use to study the limit shape of the Plancherel measure.
Mentioned results for the case of classical Lie algebras enjoy certain universality properties, where limiting distributions don't depend on tensor power of which particular representation is being considered. Since for the non-semisimple case limiting distribution is manifested by a tensor product of two distributions, one of which is discrete, it is important to study whether this universality is still present. In [4] we study the same statistics for tensor powers of arbitrary tilting module T(k).
In [5] we initiate the study of Uq(sl3), which heavily relies on 'inverse' Kazhdan-Lusztig polynomials and their asymptotic behavior along rays in the Weyl chamber.
Foams and link homology
Lectures on SL3 foams and link homology
with Mikhail Khovanov
[arXiv:2507.17119|submitted to Proceedings of the Steklov Institute of Mathematics]
These notes contain an introduction to the evaluation of SL3 foams and the associated topological field theory of planar webs and foam cobordisms. We explore categorification of the Kuperberg quantum invariant and survey other strategies for categorification of quantum invariants of knots and links. We also survey results in fields of research adjacent to foam topological field theories, including relations of the SL3 foam homology to the Kronheimer-Mrowka gauge theory and possible strategies for proving the 4 color theorem.
Others
On string functions and double-sum formulas
with Eric Mortenson and Olga Postnova
[arXiv:2107.06225|Research in the Mathematical Sciences 10 (2), 15]
In this paper we use Appell-Lerch functions and theta-functions as basis functions to express string functions for characters of irreducible representations of the untwisted affine Kac-Moody algebra A1, which allows to deduce new symmetries using modular transformations. Though Appell-Lerch functions are the building blocks for the Ramanujan's mock theta-functions, in the obtained formulas for string functions mock-modular part vanishes, hence it would be interesting to consider such a change of basis for characters of affine Lie superalgebras.
Zero modes of local operators in 2d CFT on a cylinder
with Anatoli Dymarsky and Kirill Pavlenko
[arXiv:1905.10869|Journal of High Energy Physics 2020, 172 (2020)]
This paper provides explicit formula for the quantum KdV generator Q7 in terms of Virasoro algebra operators, increaing the number of explicitly known charges for the qKdV theory up to three. This is needed for further studies of the generalized eigenstate thermolization hypothesis for 2d CFTs and further studies of the qKdV hierarchy.
Global embeddings of BTZ and Schwarzschild-AdS type blackholes in a flat space
with Anton Sheykin and Sergei Paston
[arXiv:1905.10869|Symmetry 11 (7), 841]
This paper constructs explicit global embedding of the static BTZ black hole in 6D flat space-time. Using constructed ansatz we obtain global embedding for the d-dimensional Schwarzschild-AdS blackhole to (d+3)-dimensional flat space-time. We finish the classification of minimal 6D embeddings started in [Sheykin-Paston, 2015].
Isometric global embeddings of GR manifolds to an ambient flat space are needed for studying Regge-Teitelboim embedding theory. Explicit embedding functions are treated as dynamical variables in the mentioned theory. Another application of these embeddings can be found in the study of the thermodynamics of black holes, where Unruh radiation (encountered by the observer moving along the event horizon in the embedded surface in a flat ambient space) can be related to the Hawking radiation(see [Deser-Levin, 1999]).
Modifications of gravity via differential transformations of field variables
with Anton Sheykin, Vladimir Sukhanov and Sergei Paston
[arXiv:2002.01745|Symmetry 12 (2), 240]
This paper provides a pedagogical exploration of how differential field transformations sometimes provide additional, non-physical degrees of freedom to the theory. Mentioned transformations alter the dynamics of the theory when the substitution of new variables to the action results in different EoMs, compared to applying the substitution to the EoMs directly. An archetypical example of the situation where these new degrees of freedom emerge is the Regge-Teitelboim gravity.