Noncommutative Solitons Research Articles

Fourier transform and sigma model solitons on noncommutative tori

We investigate how Fourier transform is involved in the analysis of a twisted group algebra $$L^1(G, \sigma )$$ for $$G={\widehat{\Gamma }}\times \Gamma$$ and $$\sigma :G\times G \rightarrow \mathbb {T}$$ 2-cocycle where $$\Gamma$$ is a locally compact abelian group and $${\widehat{\Gamma }}$$ its Pontryagin dual related to noncommutative tori. We construct the dual Schrodinger representation which is unitarily equivalent to the Schrodinger representation, and thereby the dual bimodule of the Heisenberg bimodule with the application to noncommutative solitons in mind.

Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions

Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters that lead to non-singular 1-soliton and periodic solutions is given. Surprisingly, it is shown that such solutions are never singular when the solution is essentially non-commutative. When a 1-soliton solution is combined with another solution through an iterated Darboux transformation, the result behaves asymptotically like a combination of different solutions. This “non-linear superposition principle” is used to find a formula for the phase shift in the general 2-soliton interaction. A concluding section compares these results with other research on non-commutative soliton equations and lists some open questions.

Matter Chern Simons theories in a background magnetic field

We study large N 2+1 dimensional fermions in the fundamental representation of an SU(N)k Chern Simons gauge group in the presence of a uniform background magnetic field for the U (1) global symmetry of this theory. The magnetic field modifies the Schwinger Dyson equation for the propagator in an interesting way; the product between the self energy and the Greens function is replaced by a Moyal star product. Employing a basis of functions previously used in the study of non-commutative solitons, we are able to exactly solve the Schwinger Dyson equation and so determine the fermion propagator. The propagator has a series of poles (and no other singularities) whose locations yield a spectrum of single particle energies at arbitrary t’ Hooft coupling and chemical potential. The usual free fermion Landau levels spectrum is shifted and broadened out; we compute the shifts and widths of these levels at arbitrary t’Hooft coupling. As a check on our results we independently solve for the propagators of the conjecturally dual theory of Chern Simons gauged large N fundamental Wilson Fisher bosons also in a background magnetic field but this time only at zero chemical potential. The spectrum of single particle states of the bosonic theory precisely agrees with those of the fermionic theory under Bose-Fermi duality.

Twisted sigma-model solitons on the quantum projective line

On the configuration space of projections in a noncommutative algebra, and for an automorphism of the algebra, we use a twisted Hochschild cocycle for an action functional and a twisted cyclic cocycle for a topological term. The latter is Hochschild-cohomologous to the former and positivity in twisted Hochschild cohomology results into a lower bound for the action functional. While the equations for the critical points are rather involved, the use of the positivity and the bound by the topological term lead to self-duality equations (thus yielding twisted noncommutative sigma-model solitons, or instantons). We present explicit nontrivial solutions on the quantum projective line.

Dynamics of Noncommutative Solitons I: Spectral Theory and Dispersive Estimates

We consider the Schr\"odinger equation with a Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We prove pointwise in time decay estimates, with the optimal decay rate $t^{-1}\log^{-2}t$ generically. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.

Structure of Noncommutative Solitons: Existence and Spectral Theory

We consider the Schrödinger equation with a Hamiltonian given by a second-order difference operator with nonconstant growing coefficients, on the half one-dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We construct a ground state soliton for this equation and analyze its properties. In particular, we arrive at \({\ell^{\infty}}\) and \({\ell^{1}}\) estimates as well as a quasi-exponential spatial decay rate.

Cosmological production of noncommutative black holes

We investigate the pair creation of noncommutative black holes in a background with a positive cosmological constant. As a first step we derive the noncommutative geometry inspired Schwarzschild--de Sitter solution. By varying the mass and the cosmological constant parameters, we find several spacetimes compatible with the new solution: positive-mass spacetimes admit one cosmological horizon and two, one, or no black hole horizons, while negative-mass spacetimes have just a cosmological horizon. These new black holes share the properties of the corresponding asymptotically flat solutions, including the nonsingular core and thermodynamic stability in the final phase of the evaporation. As a second step we determine the action which generates the matter sector of gravitational field equations and we construct instantons describing the pair production of black holes and the other admissible topologies. As a result we find that for current values of the cosmological constant the de Sitter background is quantum mechanically stable according to experience. However, positive-mass noncommutative black holes and solitons would have plentifully been produced during inflationary times for Planckian values of the cosmological constant. As a special result we find that, in these early epochs of the Universe, Planck size black holes production would have been largely disfavored. We also find a potential instability for production of negative-mass solitons.

Noncommutative solitons of gravity

We investigate a three-dimensional gravitational theory on a noncommutative space which has a cosmological constant term only. We found various kinds of nontrivial solutions by applying a similar technique which was used to seek noncommutative solitons in noncommutative scalar field theories. Some of those solutions correspond to bubbles of spacetimes or represent dimensional reduction. The solution which interpolates Gμν = 0 and the Minkowski metric is also found. All solutions we obtained are non-perturbative in the noncommutative parameter θ, therefore they are different from solutions found in other contexts of noncommutative theory of gravity and would have a close relation to quantum gravity.

Unitary Transformation in kq Representation

In this paper, we discuss the unitary transformation induced by Z6 rotations of noncommutative space on the states |k,q,⟩, which plays a key role in construction for noncommutative solitons T2/Z6 by GHS method. As a result, we prove a well-known ‘Gauss Sum’ formula in the number theory through a concise way.

Interaction of noncommutative solitons with defects

Interaction of solitons of noncommutative sine-Gordon model with potentials is studied by including a potential through one of the soliton parameters. The bahaviour of non commutative solitons during the interaction are compared with commutative solitons and the differences are explained.

Scattering of noncommutative waves and solitons in a supersymmetric chiral model in 2 + 1 dimensions

Interactions of noncommutative waves and solitons in 2 + 1 dimensions can be analyzed exactly for a supersymmetric and integrable U(n) chiral model extending the Ward model. Using the Moyal-deformed dressing method in an antichiral superspace, we construct explicit time-dependent solutions of its noncommutative field equations by iteratively solving linear equations. The approach is illustrated by presenting scattering configurations for two noncommutative U(2) plane waves and for two noncommutative U(2) solitons as well as by producing a noncommutative U(1) two-soliton bound state.

Noncommutative solitons in a supersymmetric chiral model in 2+1 dimensions

We consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from twistor string theory. By a gauge fixing this model is reduced to a modified U(n) chiral model with 2N ≤8 supersymmetries in 2+1 dimensions. After a Moyal-type deformation of the model, we employ the dressing method to explicitly construct multi-soliton configurations on noncommutative R 2,1 and analyze some of their properties.

Moduli-space dynamics of noncommutative abelian sigma-model solitons

In the noncommutative (Moyal) plane, we relate exact U(1) sigma-model solitons to generic scalar-field solitons for an infinitely stiff potential. The static k-lump moduli space C^k/S_k features a natural K"ahler metric induced from an embedding Grassmannian. The moduli-space dynamics is blind against adding a WZW-like term to the sigma-model action and thus also applies to the integrable U(1) Ward model. For the latter's two-soliton motion we compare the exact field configurations with their supposed moduli-space approximations. Surprisingly, the two do not match, which questions the adiabatic method for noncommutative solitons.

Topological solitons in the noncommutative plane and quantum Hall Skyrmions

We analyze topological solitons in the noncommutative plane by taking a concrete instance of the quantum Hall system with the SU(N) symmetry, where a soliton is identified with a skyrmion. It is shown that a topological soliton induces an excitation of the electron number density from the ground-state value around it. When a judicious choice of the topological charge density $J_{0}(\mathbf{x})$ is made, it acquires a physical reality as the electron density excitation $\Delta \rho ^{\text{cl}}(\mathbf{x})$ around a topological soliton, $\Delta \rho ^{\text{cl}}(\mathbf{x})=-J_{0}(% \mathbf{x})$. Hence a noncommutative soliton carries necessarily the electric charge proportional to its topological charge. A field-theoretical state is constructed for a soliton state irrespectively of the Hamiltonian. In general it involves an infinitely many parameters. They are fixed by minimizing its energy once the Hamiltonian is chosen. We study explicitly the cases where the system is governed by the hard-core interaction and by the noncommutative CP$^{N-1}$ model, where all these parameters are determined analytically and the soliton excitation energy is obtained.

TACHYON DYNAMICS IN OPEN STRING THEORY

In this review we describe our current understanding of the properties of open string tachyons on an unstable D-brane or brane–antibrane system in string theory. The various string theoretic methods used for this study include techniques of two-dimensional conformal field theory, open string field theory, boundary string field theory, noncommutative solitons, etc. We also describe various attempts to understand these results using field theoretic methods. These field theory models include toy models like singular potential models and p-adic string theory, as well as more realistic version of the tachyon effective action based on Dirac–Born–Infeld type action. Finally we study closed string background produced by the "decaying" unstable D-branes, both in the critical string theory and in the two-dimensional string theory, and describe the open string completeness conjecture that emerges out of this study. According to this conjecture the quantum dynamics of an unstable D-brane system is described by an internally consistent quantum open string field theory without any need to couple the system to closed strings. Each such system can be regarded as a part of the "hologram" describing the full string theory.

Position-dependent noncommutative products: Classical construction and field theory

We look in Euclidean R 4 for associative star products realizing the commutation relation [ x μ , x ν ] = i Θ μ ν ( x ) , where the noncommutativity parameters Θ μ ν depend on the position coordinates x. We do this by adopting Rieffel's deformation theory (originally formulated for constant Θ and which includes the Moyal product as a particular case) and find that, for a topology R 2 × R 2 , there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components Θ 12 = − Θ 21 = 0 and Θ 34 = − Θ 43 = θ ( x 1 , x 2 ) , with θ ( x 1 , x 2 ) an arbitrary positive smooth bounded function. In Minkowski space–time, this describes a position-dependent space-like or magnetic noncommutativity. We show how to generalize our construction to n ⩾ 3 arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean λ ϕ 4 field theory on such a noncommutative background. Using a zeta-like regulator, the covariant perturbation method and working in configuration space, we explicitly compute the UV singularities. We find that, while the two-point UV divergences are nonlocal, the four-point UV divergences are local, in accordance with recent results for constant Θ.

Deformation of p-adic string amplitudes in a magnetic field

A new term in the p-adic world-sheet action is proposed, which couples a constant B-field to the boundary of the world-sheet at disk level. The induced deformation of tachyon scattering amplitudes by star-products is derived. This is in agreement with the deformation of effective action postulated in recent investigations of non-commutative solitons in p-adic string theory.

Non-commutative solitons and strong-weak duality

Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either $U(1){x} U(1)$ or $U(1)_{C}$ corresponding to the Lechtenfeld et al. (NCSG$_{1}$) or Grisaru-Penati (NCSG$_{2}$) proposals for the NC versions of the sine-Gordon model, respectively. Besides, the relevant NCMT$_{1, 2}$ models are written for two (four) types of Dirac fields corresponding to the Moyal product extension of one (two) copy(ies) of the ordinary massive Thirring model. The NCATM$_{1,2}$ models share the same one-soliton (real Toda field sector of model 2) exact solutions, which are found without expansion in the NC parameter $\theta$ for the corresponding Toda and matter fields describing the strong-weak phases, respectively. The correspondence NCSG$_{1}$ $\leftrightarrow$ NCMT$_{1}$ is promising since it is expected to hold on the quantum level.

Exact noncommutative solitons in p-adic strings and BSFT

The tachyon field of p-adic string theory is made noncommutative by replacing ordinary products with noncommutative products in its exact effective action. The same is done for the boundary string field theory, treated as the p→1 limit of the p-adic string. Solitonic lumps corresponding to D-branes are obtained for all values of the noncommutative parameter θ. This is in contrast to usual scalar field theories in which the noncommutative solitons do not persist below a critical value of θ. As θ varies from zero to infinity, the solution interpolates smoothly between the soliton of the p-adic theory (respectively BSFT) to the noncommutative soliton.

Transmogrifying fuzzy vortices

We show that the construction of vortex solitons of the noncommutative Abelian-Higgs model can be extended to a critically coupled gauged linear sigma model with Fayet-Illiopolous D-terms. Like its commutative counterpart, this fuzzy linear sigma model has a rich spectrum of BPS solutions. We offer an explicit construction of the degree$-k$ static semilocal vortex and study in some detail the infinite coupling limit in which it descends to a degree$-k$ $\C\Pk^{N}$ instanton. This relation between the fuzzy vortex and noncommutative lump is used to suggest an interpretation of the noncommutative sigma model soliton as tilted D-strings stretched between an NS5-brane and a stack of D3-branes in type IIB superstring theory.