arXiv subjects

Aleksei Andreev

Publications and source records attributed to Aleksei Andreev.

2 recordsLinked to original sources

arXiv2026

Homological Topological Quantum Field Theories

We develop a new framework for quantum invariants of $3$-manifolds by extending to cobordisms a homological construction of mapping class group representations. More specifically, we construct a $(2+1)$-dimensional topological quantum field theory (TQFT) that assigns to each surface the twisted homology of its unordered configuration space. The construction requires a choice of local systems on configuration spaces together with additional data. We formulate sufficient conditions on these data that guarantee the TQFT axioms, and we show that there are at least two useful examples satisfying them. One of them yields a homological construction of the projective Kerler--Lyubashenko TQFT, while the other recovers the Frohman--Nicas--Donaldson TQFT. In contrast to the classical algebraic constructions of quantum invariants, our approach is purely topological and relies on multi-trajectory spaces of cobordisms.

math.GT↗

arXiv2024

Abelian TQFTS and Schrödinger local systems

We construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a $q$-deformation of $U(1)$ for any root of unity $q$. We prove that for3-cobor\-disms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT. The full abelian TQFT provides a projective representation of the mapping class group $\mathrm{Mod}(Σ)$ on the Schrödinger representation,which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of $\mathrm{Mod}(Σ)$ on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

math.GT↗