arXiv subjects

Andrew Kuzovchikov

Publications and source records attributed to Andrew Kuzovchikov.

7 recordsLinked to original sources

arXiv2026

Anomalous symmetries in Kähler geometry

We initiate a study of centrally extended ($\textit{anomalous}$) symmetries in Kähler geometry, focusing on the simplest, and most ubiquitous, Abelian case. In particular, we provide a local description of the geometry admitting such isometries. A long-standing no-go theorem asserts that there is an obstruction to gauging such symmetries in the purely Kähler framework. Utilizing the language of sypersymmetry, we then show that these symmetries may be gauged within the setup of generalized Kähler geometry. Our results may be applied to quotients, T-dualities, etc.

hep-th↗

arXiv2025

T-duality for toric manifolds in $\mathcal{N}=(2, 2)$ superspace

We study the situation when the T-dual of a toric Kähler geometry is a generalized Kähler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized Kähler geometry of the $η$-deformed $\mathbb{CP}^{n-1}$ model, relating it to the Kähler geometry of its T-dual.

hep-th↗

arXiv2025

Sigma models from Gaudin spin chains

We solve the classical and quantum problems for the 1D sigma model with target space the flag manifold $\mathrm{U}(3)\over \mathrm{U}(1)^3$, equipped with the most general invariant metric. In particular, we explicitly describe all geodesics in terms of elliptic functions and demonstrate that the spectrum of the Laplace-Beltrami operator may be found by solving polynomial (Bethe) equations. The main technical tool that we use is a mapping between the sigma model and a Gaudin model, which is also shown to hold in the $\mathrm{U}(n)$ case.

hep-th↗

arXiv2025

Isotropic embeddings of coadjoint orbits and magnetic geodesic flows

We consider isotropic and Lagrangian embeddings of coadjoint orbits of compact Lie groups into products of coadjoint orbits. After reviewing the known facts in the case of $\mathrm{SU}(n)$ we initiate a similar study for $\mathrm{SO}$ and $\mathrm{Sp}$ cases. In the second part we apply this to the study of dynamical systems with $\mathrm{SU}(n)$ symmetry, proving equivalence between systems of two types: those describing magnetic geodesic flow on flag manifolds and classical `spin chains' of a special type.

math.DG↗

arXiv2025

Mechanics on flag manifolds

We study the connection between $\mathrm{SU}(n)$ spin chains and one-dimensional sigma models on flag manifolds. Using this connection, we calculate the spectrum of the Laplace-Beltrami operator and geodesics for a particular class of metrics on $\mathbb{CP}^1$ and $\mathcal{F}_3$, which is a manifold of complete flags in $\mathbb{C}^3$.

hep-th↗

arXiv2024

Oscillator Calculus on Coadjoint Orbits and Index Theorems

We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We prove that they are natural truncations of 1D sigma models, whose target spaces are $\mathsf{SU}(n)$ (co)adjoint orbits. As a first application, we compute the Witten indices of these finite-dimensional models showing that they reproduce the Dolbeault and de Rham indices of the target space. The problem of finding the exact spectra of generalized Laplace operators on such orbits is shown to be equivalent to the diagonalization of spin chain Hamiltonians.

hep-th↗

arXiv2024

The classical and quantum particle on a flag manifold

In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace-Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace-Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

hep-th↗