arXiv subjects

Andrei Smilga

Publications and source records attributed to Andrei Smilga.

16 recordsLinked to original sources

arXiv2024

Cartan monopoles

The effective Hamiltonians for chiral supersymmetric gauge theories at small spatial volume are generalizations of the Hamiltonians describing the motion of a scalar or a spinor particle in a field of Dirac monopoles (we are dealing in fact with a certain lattice of monopoles supplemented with a periodic singular potential). The gauge fields in such Hamiltonians belong to the Cartan subalgebras of the corresponding gauge algebras. Such a construction exists for all groups admitting complex representations, i.e. for $SU(N \geq 3), \ Spin(4n+2)$ with $n \geq 1$ and $E_6$. We give explicit expressions for these Hamiltonians for $SU(3)$, $SU(4) \simeq Spin(6)$ and for $SU(5)$. The simplified version of such a Hamiltonian, deprived of fermion terms, of the extra scalar potential and when only one node of the lattice is taken into consideration, describe a $3r$-dimensional motion ($r$ being the rank of the group) in the field what we call a {\it Cartan monopole}. As is the case for the ordinary monopole, the Lagrangian of this system enjoys gauge symmetry, rotational symmetry, and the parameter, generalizing the notion of magnetic charge for Cartan monopoles, is quantized.

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arXiv2024

Witten index for weak supersymmetric systems: invariance under deformations

When a $4D$ supersymmetric theory is placed on $S^3 \times \mathbb{R}$, the supersymmetric algebra is necessarily modified to $su(2|1)$ and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index Tr$\{(-1)^F e^{-βH}\}$ is no longer an integer number, but a $β$-dependent function. However, this function stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.

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arXiv2023

Monopole harmonics on $\mathbb{CP}^{n-1}$

We find the spectra and eigenfunctions of both ordinary and supersymmetric quantum-mechanical models describing the motion of a charged particle over the $\mathbb{CP}^{n-1}$ manifold in the presence of a background monopole-like gauge field. The states form degenerate $SU(n)$ multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the well-known isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.

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arXiv2022

Comments on noncommutative quantum mechanical systems associated with Lie algebras

We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations $$ [x_a, x_b] \ =\ iθf_{abc} x_c\,, $$ where $f_{abc}$ are the structure constants of a Lie algebra. We note that this problem can be reformulated as an ordinary quantum problem in a commuting momentum space. The coordinates are then represented as linear differential operators $\hat x_a = -i\hat D_a = -iE_{ab} (p)\, \partial /\partial p_b $. Generically, the matrix $E_{ab}(p)$ represents a certain infinite series over the deformation parameter $θ$: $E_{ab} = δ_{ab} + \ldots$. The deformed Hamiltonian, $\hat H = -\frac 12 \hat D_a^2\,,$ describes the motion along the corresponding group manifolds with the characteristic size of order $θ^{-1}$. Their metrics are also expressed into certain infinite series in $θ$, with $E_{ab}$ having the meaning of vielbeins. For the algebras $su(2)$ and $u(N)$, it has been possible to represent the operators $\hat x_a$ in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on all the spheres $S^n$, on the corresponding projective spaces $RP^n$ and on $U(2)$.

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arXiv2021

Modified Korteweg-de Vries Equation as a System with Benign Ghosts

We consider the modified Korteweg-de Vries equation, $ u_{xxx} + 6u^2 u_x + u_t \ =\ 0 $, and explore its dynamics in {\it spatial} direction. Higher $x$ derivatives bring about the {\it ghosts}. We argue that these ghosts are benign, i.e. the classical dynamics of this system does not involve a blow-up. This probably means that also the associated quantum problem is well defined.

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arXiv2021

Dynamical systems with benign ghosts

We consider finite and infinite-dimensional ghost-ridden dynamical systems whose Hamiltonians involve non positive definite kinetic terms. We point out the existence of three classes of such systems where the ghosts are benign, i.e. systems whose evolution is unlimited in time:(i) systems obtained from the variation of bounded-motion systems; (ii) systems describing motions over certain Lorentzian manifolds and (iii) higher-derivative models related to certain modified Korteweg--de Vries equations.

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arXiv2021

A comment on instantons and their fermion zero modes in adjoint QCD_2

The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admits topologically nontrivial gauge field configurations associated with nontrivial $π_1[SU(N)/Z_N] = Z_N$. The topological sectors are labelled by an integer $k=0,\ldots, N-1$. However, in contrast to $QED_2$ and $QCD_4$, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index. These instantons may admit fermion zero modes, but there is always an equal number of left-handed and right-handed modes, so that the Atiyah-Singer theorem, which determines in other cases the number of the modes, does not apply. The mod. 2 argument suggests that, for a generic gauge field configuration, there is either a single doublet of such zero modes or no modes whatsoever. However, the known solution of the Dirac problem for a wide class of gauge field configurations indicates the presence of $k(N-k)$ zero mode doublets in the topological sector $k$. In this note, we demonstrate in an explicit way that these modes are not robust under a generic enough deformation of the gauge background and confirm thereby the mod. 2 conjecture. The implications for the physics of this theory (screening vs. confinement issue) are briefly discussed.

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arXiv2021

Ultraviolet behavior of 6D supersymmetric Yang-Mills theories and harmonic superspace

We revisit the issue of higher-dimensional counterterms for the N=(1,1) supersymmetric Yang-Mills (SYM) theory in six dimensions using the off-shell N=(1,0) and on-shell N=(1,1) harmonic superspace approaches. The second approach is developed in full generality and used to solve, for the first time, the N=(1,1) SYM constraints in terms of N=(1,0) superfields. This provides a convenient tool to write explicit expressions for the candidate counterterms and other N=(1,1) invariants and may be conducive to proving non-renormalization theorems needed to explain the absence of certain logarithmic divergences in higher-loop contributions to scattering amplitudes in N=(1,1) SYM.

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arXiv2020

Comments on the Newlander-Nirenberg theorem

The Newlander-Nirenberg theorem says that a necessary and sufficient condition for the complex coordinates associated with a given almost complex structure tensor $I_M{}^N$ to exist is the vanishing of the Nijenhuis tensor ${\cal N}_{MN}{}^K$. In the first part of the paper, we give a simple explicit proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. ${\it (i)}$ The condition ${\cal N}_{MN}{}^K = 0$ is necessary for a certain $N=1$ supersymmetric mechanical sigma models to enjoy $N=2$ supersymmetry. ${\it (ii)}$ The sufficiency of this condition for the existence of complex coordinates implies that the representation of the supersymmetry algebra realized by the superfields associated with all the real coordinates and their superpartners can be presented as a direct sum of d irreducible representations (d is the complex dimension of the manifold).

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arXiv2017

Classical and Quantum Dynamics of Higher-Derivative Systems

A brief review of the physics of systems including higher derivatives in the Lagrangian is given. All such systems involve ghosts, i.e. the spectrum of the Hamiltonian is not bounded from below and the vacuum ground state is absent. Usually this leads to collapse and loss of unitarity. In certain special cases, this does not happen, however: ghosts are benign. We speculate that the Theory of Everything is a higher-derivative field theory, characterized by the presence of such benign ghosts and defined in a higher-dimensional bulk. Our Universe represents then a classical solution in this theory, having the form of a 3-brane embedded in the bulk.

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arXiv2015

Comments on HKT supersymmetric sigma models and their Hamiltonian reduction

Using complex notation, we present new simple expressions for two pairs of complex supercharges in HKT supersymmetric sigma models. The second pair of supercharges depends on the holomorphic antisymmetric "hypercomplex structure" tensor which plays the same role for the HKT models as the complex structure tensor for the Kaehler models. When the Hamiltonian and supercharges commute with the momenta conjugate to the imaginary parts of the complex coordinates, one can perform a Hamiltonian reduction. The models thus obtained represent a special class of quasicomplex sigma models introduced recently by Ivanov and Smilga.

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arXiv2013

N=4 mechanics with diverse (4,4,0) multiplets: explicit examples of HKT, CKT, and OKT geometries

We present simple models of N=4 supersymmetric mechanics with ordinary and mirror linear (4,4,0) multiplets that give a transparent description of HKT, CKT, and OKT geometries. These models are treated in the N=4 and N=2 superfield approaches, as well as in the component approach. Our study makes manifest that the CKT and OKT supersymmetric sigma models are distinguished from the more simple HKT models by the presence of extra holomorphic torsions in the supercharges.

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arXiv2013

Noninteger flux - why it does not work

We consider the Dirac operator on a 2-sphere without one point in the case of non-integer magnetic flux. We show that the spectral problem for the Hamiltonian (the square of Dirac operator) can always be well defined, if including in the Hilbert space only nonsingular on 2-sphere wave functions. However, this Hilbert space is not invariant under the action of the Dirac operator; the action of the latter on some nonsingular states produces singular functions. This breaks explicitly the supersymmetry of the spectrum. In the integer flux case, the supersymmetry can be restored if extending the Hilbert space to include locally regular sections of the corresponding fiber bundle. For non-integer fluxes, such an extention is not possible.

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arXiv2004

Background field calculations and nonrenormalization theorems in 4d supersymmetric gauge theories and their low-dimensional descendants

We analyze the structure of multiloop supergraphs contributing to the effective Lagrangians in 4d supersymmetric gauge theories and in the models obtained from them by dimensional reduction. When d=4, this gives the renormalization of the effective charge. For d < 4, the low-energy effective Lagrangian describes the metric on the moduli space of classical vacua. These two problems turn out to be closely related. In particular, we establish the relationship between the 4d nonrenormalization theorems (in minimal and extended supersymmetric theories) and their low--dimensional counterparts.

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arXiv1997

BPS Domain Walls in Supersymmetric QCD: Higher Unitary Groups

We consider the N=1 supersymmetric QCD with SU(N_c) gauge group involving N_f = N_c - 1 pairs of chiral matter multiplets in fundamental and antifundamental color representations. For this theory in the framework of the effective lagrangian approach, we solve the BPS equations for the domain walls interpolating between different vacua. The equations always have a unique solution for the walls interpolating between the chirally symmetric and a chirally asymmetric vacua. For the walls interpolating between different chirally asymmetric vacua, the equations admit two different solutions which exist when the mass of the matter field is below some critical value m*. At m = m*, two branches join together and, at m > m*, no BPS - saturated complex domain walls exist.

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arXiv1995

Really Computing Non-perturbative Real Time Correlation Functions

It has been argued by Grigoriev and Rubakov that one can simulate real time processes involving baryon number non-conservation at high temperature using real time evolution of classical equations, and summing over initial conditions with a classical thermal weight. It is known that such a naive algorithm is plagued by ultraviolet divergences. In quantum theory the divergences are regularized, but the corresponding graphs involve the contributions from the hard momentum region and also the new scale $\sim gT$ comes into play. We propose a modified algorithm which involves solving the classical equations of motion for the effective hard thermal loop Hamiltonian with an ultraviolet cutoff $μ\gg gT$ and integrating over initial conditions with a proper thermal weight. Such an algorithm should provide a determination of the infrared behavior of real time correlation function $ _T$ determining the baryon violation rate. Hopefully, the results obtained in this modified algorithm would be cutoff-independent.

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