arXiv subjects

Maxim Kurkov

Publications and source records attributed to Maxim Kurkov.

11 recordsLinked to original sources

arXiv2025

Action principle for $\kappa$-Minkowski noncommutative $U(1)$ gauge theory from Lie-Poisson electrodynamics

Lie-Poisson electrodynamics describes a semiclassical approximation of noncommutative $U(1)$ gauge theories with Lie-algebra-type noncommutativities. We obtain a gauge-invariant local classical action with the correct commutative limit for a generic Lie-Poisson gauge model, and present the corresponding deformed Maxwell equations. At the semiclassical level, our results provide a relatively simple solution to the old problem of constructing an admissible Lagrangian formulation for the $U(1)$ gauge theory on the four-dimensional $\kappa$-Minkowski space-time. On the one hand, we derive an explicit expression for the classical action which yields the deformed Maxwell equations previously proposed in JHEP 11 (2023) 200 for this noncommutativity on general grounds. On the other hand, according to our analysis, these Maxwell equations follow from the action proposed in the present paper as the Euler-Lagrange equations for any Lie-algebra-type noncommutativity.

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arXiv2024

Classical Mechanics in Noncommutative Spaces: Confinement and More

We consider a semi-classical approximation to the dynamics of a point particle in a noncommutative space. In this approximation, the noncommutativity of space coordinates is described by a Poisson bracket. For linear Poisson brackets, the corresponding phase space is given by the cotangent bundle of a Lie group, with the Lie group playing the role of a curved momentum space. We show that the curvature of the momentum space may lead to rather unexpected physical phenomena such as an upper bound on the velocity of a free nonrelativistic particle, bounded motion for repulsive central force, and no-fall-into-the-centre for attractive Coulomb potential. We also consider a superintegrable Hamiltonian for the Kepler problem in $3$-space with $su(2)$ noncommutativity. The leading correction to the equations of motion due to noncommutativity is shown to be described by an effective monopole potential.

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arXiv2024

Hamiltonian analysis in Lie-Poisson gauge theory

Lie-Poisson gauge formalism provides a semiclassical description of noncommutative $U(1)$ gauge theory with Lie algebra type noncommutativity. Using the Dirac approach to constrained Hamiltonian systems, we focus on a class of Lie-Poisson gauge models, which exhibit an admissible Lagrangian description. The underlying noncommutativity is supposed to be purely spatial. Analysing the constraints, we demonstrate that these models have as many physical degrees of freedom as there are present in the Maxwell theory.

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arXiv2021

Four-dimensional noncommutative deformations of $U(1)$ gauge theory and $L_{\infty}$ bootstrap

We construct a family of four-dimensional noncommutative deformations of $U(1)$ gauge theory following a general scheme, recently proposed in JHEP 08 (2020) 041 for a class of coordinate-dependent noncommutative algebras. This class includes the $\mathfrak{su}(2)$, the $\mathfrak{su}(1,1)$ and the angular (or $\lambda$-Minkowski) noncommutative structures. We find that the presence of a fourth, commutative coordinate $x^0$ leads to substantial novelties in the expression for the deformed field strength with respect to the corresponding three-dimensional case. The constructed field theoretical models are Poisson gauge theories, which correspond to the semi-classical limit of fully noncommutative gauge theories. Our expressions for the deformed gauge transformations, the deformed field strength and the deformed classical action exhibit flat commutative limits and they are exact in the sense that all orders in the deformation parameter are present. We review the connection of the formalism with the $L_{\infty}$ bootstrap and with symplectic embeddings, and derive the $L_{\infty}$-algebra, which underlies our model.

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arXiv2020

How many surface modes does one see on the boundary of a Dirac material?

We present full expressions for the surface part of polarization tensor of a Dirac fermion confined in a half-space in $3+1$ dimensions. We compare this tensor to the polarization tensor of eventual surface mode (which is a $2+1$ dimensional Dirac fermion) and find essential differences in the conductivities in both Hall and normal sectors. Thus, the interaction with electromagnetic field near the boundary differs significantly in the full model and in the effective theory for the surface mode.

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arXiv2019

Remark on the synergy between the heat kernel techniques and the parity anomaly

In this paper, we demonstrate that not only the heat kernel techniques are useful for computation of the parity anomaly, but also the parity anomaly turns out to be a powerful mean in studying the heat kernel. We show that the gravitational parity anomaly on 4D manifolds with boundaries can be calculated using the general structure of the heat kernel coefficient $a_5$ for mixed boundary conditions, keeping all the weights of various geometric invariants as unknown numbers. The symmetry properties of the $\eta$-invariant allow to fix all the relevant unknowns. As a byproduct of this calculation we get an efficient and independent crosscheck (and confirmation) of the correction of the general structure of $a_5$ for mixed boundary conditions, previously suggested in Ref. [59].

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arXiv2019

Spectral Noncommutative Geometry, Standard Model and all that

We review the approach to the standard model of particle interactions based on spectral noncommutative geometry. The paper is (nearly) self-contained and presents both the mathematical and phenomenological aspects. In particular the bosonic spectral action and the fermionic action are discussed in detail, and how they lead to phenomenology. We also discuss the Euclidean vs. Lorentz issues and how to go beyond the standard model in this framework.

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arXiv2018

Gravitational parity anomaly with and without boundaries

In this paper we consider gravitational parity anomaly in three and four dimensions. We start with a re-computation of this anomaly on a 3D manifold without boundaries and with a critical comparison of our results to the previous calculations. Then we compute the anomaly on 4D manifolds with boundaries with local bag boundary conditions. We find, that gravitational parity anomaly is localized on the boundary and contains a gravitational Chern-Simons terms together with a term depending of the extrinsic curvature. We also discuss the main properties of the anomaly, as the conformal invariance, relations between 3D and 4D anomalies, etc.

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arXiv2017

The Gribov problem in Noncommutative gauge theory

After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

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arXiv2016

Emergent spontaneous symmetry breaking and emergent symmetry restoration in rippling gravitational background

We study effects of a rippling gravitational background on a scalar field with a double well potential, focusing on the analogy with the well known dynamics of the Kapitza's pendulum. The ripples are rendered as infinitesimal but rapidly oscillating perturbations of the scale factor. We find that the resulting dynamics crucially depends on a value of the parameter $ξ$ in the $ξ\,R\, ϕ^2$ vertex. For the time-dependent perturbations of a proper form the resulting effective action is generally covariant, and at a high enough frequency at $ξ<0$ and at $ξ>1/6$ the effective potential has a single minimum at zero, thereby restoring spontaneously broken symmetry of the ground state. On the other side, at $0<ξ< 1/6$ spontaneous symmetry breaking emerges even when it is absent in the unperturbed case.

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arXiv2016

The Gribov problem in Noncommutative QED

It is shown that in the noncommutative version of QED (NCQED) Gribov copies induced by the noncommutativity of space-time appear in the Landau gauge. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

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