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 $G_{μν}=0$ and 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.
We investigate solitonic black hole solutions in three dimensional noncommutative spacetime. We do this in gravity with negative cosmological constant coupled to a scalar field. Noncommutativity is realized with the Moyal product which is expanded up to first order in the noncommutativity parameter in two spatial directions. With numerical simulation we study the effect of noncommutativity by increasing the value of the noncommutativity parameter starting from commutative solutions. We find that even a regular soliton solution in the commutative case becomes a black hole solution when the nonc…
We consider noncommutative theory of a compact scalar field. The recently discovered projector solitons are interpreted as classical vacua in the model considered. Localized solutions to the projector equation are pointed out and their brane interpretation is discussed. An example of the noncommutative soliton interpolating between such vacua is given. No strong noncommutativity limit is assumed.
We show that in the presence of U(1) noncommutative gauge interaction the noncommutative tachyonic system exhibits solitonic solutions for finite value of the noncommutativity parameter.
We study the relationship of soliton solutions for electron system with those of the sigma model on the noncommutative space, working directly in the operator formalism. We find that some soliton solutions of the sigma model are also the solitons of the electron system and are classified by the same topological numbers.
Noncommutative solitons are easier to find in a noncommutative field theory. Similarly, the one-loop quantum corrections to the mass of a noncommutative soliton are easier to compute, in a real scalar theory in 2+1 dimensions. We carry out this computation in this paper. We also discuss the model with a double-well potential, and conjecture that there is a partial symmetry restoration in a vacuum state.
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 theta. This is in contrast to usual scalar field theories in which the noncommutative solitons do not persist below a critical value of theta. As theta varies from zero to infinity, the solution interpolates smoothly between the soliton …
We construct exact soliton solutions to the Chern-Simons-Higgs system in noncommutative space, for non-relativistic and relativistic models. In both cases we find regular vortex-like solutions to the BPS equations which approach the ordinary selfdual non-topological and topological solitons when the noncommutative parameter $θ$ goes to zero.
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.
We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instanton solutions
Interactions of noncommutative solitons in a modified U(n) sigma model in 2+1 dimensions can be analyzed exactly. Using an extension of the dressing method, we construct explicit time-dependent solutions of its noncommutative field equation by iteratively solving linear equations. The approach is illustrated by presenting bound states and right-angle scattering configurations for two noncommutative solitons.
We construct intersecting D-branes as noncommutative solitons in bosonic and type II string theory. ``Defect'' branes which are D-branes containing bubbles of the closed string vacuum play an important role in the construction.
We investigate the low-energy dynamics of the BPS solitons of the noncommutative CP^1 model in 2+1 dimensions using the moduli space metric of the BPS solitons. We show that the dynamics of a single soliton coincides with that in the commutative model. We find that the singularity in the two-soliton moduli space, which exists in the commutative CP^1 model, disappears in the noncommutative model.We also show that the two-soliton metric has the smooth commutative limit.
We derive the Kadomtsev-Petviashvili (KP) equation defined over a general associative algebra and construct its N-soliton solution. For the example of the Moyal algebra, we find multi-soliton solutions for arbitrary space-space noncommutativity. The noncommutativity of coordinates is shown to obstruct the general construction of a tau function for these solitons. We investigate the two-soliton solution in detail and show that asymptotic observers of soliton scattering are unable to detect a finite spatial noncommutativity. An explicit example shows that a pair of solitons in a noncommutative b…
These lectures provide an introduction to noncommutative geometry and its origins in quantum mechanics and to the construction of solitons in noncommutative field theory. These ideas are applied to the construction of D-branes as solitons of the tachyon field in noncommutative open string theory. A brief discussion is given of the K-theory classification of D-brane charge in terms of the K-theory of operator algebras. Based on lectures presented at the Komaba 2000 workshop, Nov. 14-16 2000.
We discuss the characteristic properties of noncommutative solitons moving with constant velocity. As noncommutativity breaks the Lorentz symmetry, the shape of moving solitons is affected not just by the Lorentz contraction along the velocity direction, but also sometimes by additional `elongation' transverse to the velocity direction. We explore this in two examples: noncommutative solitons in a scalar field theory on two spatial dimension and `long stick' shaped noncommutative U(2) magnetic monopoles. However the elongation factors of these two cases are different, and so not universal.
We construct new exact BPS solitons in various noncommutative gauge theories by the ``gauge'' transformation of known BPS solitons. This ``gauge'' transformation introduced by Harvey, Kraus and Larsen adds localized solitons to the known soliton. These solitons include, for example, the bound state of a noncommutative Abelian monopole and N fluxons at threshold. This corresponds, in superstring theories, to a D-string which attaches to a D3-brane and N D-strings which pierce the D3-brane, where all D-strings are parallel to each other.
We show that U(\infty) symmetry transformations of the noncommutative field theory in the Moyal space are generated by a combination of two W_{1+\infty} algebras in the Landau problem. Geometrical meaning of this infinite symmetry is illustrated by examining the transformations of an invariant subgroup on the noncommutative solitons, which generate deformations and boosts of solitons. In particular, we find that boosts of solitons, which was first introduced in Ref[22], are valid only in the infinite "theta" limit.
We find exact solitons in a large class of noncommutative gauge theories using a simple solution generating technique. The solitons in the effective field theory description of open string field theory are interpreted as D-branes for any value of the noncommutativity. We discuss the vacuum structure of open string field theory in view of our results.
We consider the Dunne-Jackiw-Pi-Trugenberger model of a U(N) Chern-Simons gauge theory coupled to a nonrelativistic complex adjoint matter on noncommutative space. Soliton configurations of this model are related the solutions of the chiral model on noncommutative plane. A generalized Uhlenbeck's uniton method for the chiral model on noncommutative space provides explicit Chern-Simons solitons. Fundamental solitons in the U(1) gauge theory are shaped as rings of charge `n' and spin `n' where the Chern-Simons level `n' should be an integer upon quantization. Toda and Liouville models are genera…
We examine the intersections, fluctuations and deformations of codimension two solitons in field theory on noncommutative $R^4$, in the limit of large noncommutativity. We find that holomorphic deformations are zero modes of flat branes, and we show that there is a zero mode localized at the intersection of two solitons.
Investigating the solitons in the non-commutative $CP^{1}$ model, we have found a new set of BPS solitons which does not have counterparts in the commutative model.
We demonstrate the construction of solitons for a time-space Moyal-deformed integrable U(n) sigma model (the Ward model) in 2+1 dimensions. These solitons cannot travel parallel to the noncommutative spatial direction. For the U(1) case, the rank-one single-soliton configuration is constructed explicitly and is singular in the commutative limit. The projection to 1+1 dimensions reduces it to a noncommutative instanton-like configuration. The latter is governed by a new integrable equation, which describes a Moyal-deformed sigma model with a particular Euclidean metric and a magnetic field.
We find non-BPS solutions of the noncommutative CP^1 model in 2+1 dimensions. These solutions correspond to soliton anti-soliton configurations. We show that the one-soliton one-anti-soliton solution is unstable when the distance between the soliton and the anti-soliton is small. We also construct time-dependent solutions and other types of solutions.
We consider the dynamics of two interacting lumps/solitons in a noncommutative gauge model. We show that equations of motion describing this dynamics can be reduced to ones of a two-dimensional mechanical system which is well studied and was shown to exhibit stochastic behaviour.
In this paper, we construct the common eigenstates of "translation" operators $\{U_{s}\}$ and establish the generalized $Kq$ representation on integral noncommutative torus $T^{2N}$. We then study the finite rotation group $G$ in noncommutative space as a mapping in the $Kq$ representation and prove a Blocking Theorem. We finally obtain the complete set of projection operators on the integral noncommutative orbifold $T^{2N}/G$ in terms of the generalized $Kq$ representation. Since projectors are soliton solutions on noncommutative space in the limit $α^{\prime}B_{ij}\to \infty (Θ_{ij}/α^{\prim…
We investigate and interpret a large class of soliton solutions found in noncommutative tachyon condensation. These constructions make extensive use of the idea that Dp-branes may be built out of lower dimensional branes. Finally we comment on a recently proposed solution generating technique.
We propose an ansatz for the equations of motion of the noncommutative model of a tachyonic scalar field interacting with a gauge field, which allows one to find time-dependent solutions describing decaying solitons. These correspond to the collapse of lower dimensional branes obtained through tachyon condensation of unstable brane systems in string theory.
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 adiabati…
We generalize to noncommutative cylinder the solution generation technique, originally suggested for gauge theories on noncommutative plane. For this purpose we construct partial isometry operators and complete set of orthogonal projectors in the algebra of the cylinder, and an isomorphism between the free module and its direct sum with the Fock module on the cylinder. We construct explicitly the gauge theory soliton and evaluate the spectrum of perturbations about this soliton.
We consider noncommutative gauge theories which have zero mass states propagating along both commutative and noncommutative dimensions. Solitons in these theories generically carry U(m) gauge group on their world-volume. From the point of view of string theory, these solitons correspond to ``branes within branes''. We show that once the world-volume U(m) gauge theory is in the Higgs phase, light states become quasi-localized, rather than strictly localized on the soliton, i.e. they mix with light bulk modes and have finite widths to escape into the noncommutative dimensions. At small values of…
These lectures deal mainly with solitons in three-dimensional Moyal-deformed sigma models. The topics are: static and moving (multi-)solitons of the (integrable) Ward sigma model, with space-space and time-space noncommutativity, their scattering, moduli space dynamics, stability and dimensional reduction, including an integrable deformation of the sine-Gordon system.
I consider a supersymmetric Bogomolny-type model in 2+1 dimensions originating from topological string theory. By a gauge fixing this model is reduced to a supersymmetric U(n) chiral model with a Wess-Zumino-Witten-type term in 2+1 dimensions. After a noncommutative extension of the model, I employ the dressing method to construct explicit multi-soliton configurations on noncommutative R^{2,1|2N}.
We consider 2+1-dimensional classical noncommutative scalar field theory. The general ansatz for a radially symmetric solution is obtained. Some exact solutions are presented. Their possible physical meaning is discussed. The case of the finite $θ$ is discussed qualitatively and illustrated by some numerical results.
We study the finite theta correction to the metric of the moduli space of noncommutative multi-solitons in scalar field theory in (2+1) dimensions. By solving the equation of motion up to order O(theta^{-2}) explicitly, we show that the multi-soliton solution must have the same center for a generic potential term. We examine the condition that the multi-centered configurations are allowed. Under this condition, we calculate the finite theta correction to the metric of the moduli space of multi-solitons and argue the possibility of the non right-angle scattering of two solitons. We also obtain …
The interpretation of closed fundamental strings as solitons in open string field theory is reviewed. Noncommutativity is introduced to facilitate an explicit construction. The tension is computed exactly and the correct spectrum is recovered at long wave length.
We investigate BPS soliton solutions of U(N) Chern-Simons gauge theory coupled to a scalar field in noncommutative plane. With a scalar field in the fundamental representation, we show that the BPS equation becomes that of abelian Chern-Simons theory in the unitary gauge. We also find a class of particular solutions for the BPS equation with scalar field in the adjoint representation.
We investigate the behavior of the noncommutative scalar soliton solutions at finite noncommutative scale $θ$. A detailed analysis of the equation of the motion indicates that fewer and fewer soliton solutions exist as $θ$ is decreased and thus the solitonic sector of the theory exhibits an overall hierarchy structure. If the potential is bounded below, there is a finite $θ_c$ below which all the solitons cease to exist even though the noncommutativity is still present. If the potential is not bounded below, for any nonzero $θ$ there is always a soliton solution, which becomes singular only at…
We find soliton solutions of the noncommutative Maxwell-Chern-Simons theory confined to a finite quantum Hall droplet. The solitons are exactly as hypothesized in \cite{Manu}. We also find new variations on these solitons. We compute their flux and their energies. The model we consider is directly related to the model proposed by Polychronakos\cite{Poly} and studied by Hellerman and Van Raamsdonk\cite{HvR} where it was shown that it is equivalent to the quantum Hall effect.
This is a summary of a series of papers hep-th/9910263,0005283,0010066 written with B. Chen, T. Matsuo and K. Murakami on a p-p', (p<p^{\prime}) open string with B_{ij} field, which has led us to the explicit identification of the Dp-brane with the noncommutative projector soliton via the gaussian damping factor. A lecture given at Summer Institute 2000, FujiYoshida, Yamanashi, Japan, at August 7-14, 2000.
The noncommutative soliton is characterized by the use of the projection operators in non-commutative space. By using the close relation with the K-theory of $C^*$-algebra, we consider the variations of projection operators along the commutative directions and identify their topological charges. When applied to the string theory, it gives the modification of the brane charges due to tachyon background.
The purpose of this talk is to review a few issues concerning noncommutativity arising from String Theory. In particular, it is shown how in Type IIB Theory, the annihilation of a D3-\bar{D3} brane pair yields a D1-string. This object, in the presence of a large B-field and fermions, happens to be a complex noncommutative soliton endowed with superconductivity.
Solitonic objects play a central role in gauge and string theory (as, e.g., monopoles, black holes, D-branes, etc.). Certain string backgrounds produce a noncommutative deformation of the low-energy effective field theory, which allows for new types of solitonic solutions. I present the construction, moduli spaces and dynamics of Moyal-deformed solitons, exemplified in the 2+1 dimensional Yang-Mills-Higgs theory and its Bogomolny system, which is gauge-fixed to an integrable chiral sigma model (the Ward model). Noncommutative solitons for various 1+1 dimensional integrable systems (such as sin…
We find classically stable solitons (instantons) in odd (even) dimensional scalar noncommutative field theories whose scalar potential, $V(\ph)$, has at least two minima. These solutions are bubbles of the false vacuum whose size is set by the scale of noncommutativity. Our construction uses the correspondence between non-commutative fields and operators on a single particle Hilbert space. In the case of noncommutative gauge theories we note that expanding around a simple solution shifts away the kinetic term and results in a purely quartic action with linearly realised gauge symmetries.
In the noncommutative field theory of open strings in a B-field, D-branes arise as solitons described as projection operators or partial isometries in a $C^*$ algebra. We discuss how D-branes on orbifolds fit naturally into this algebraic framework, through the examples of $R^n/G$, $T^n=R^n/Z^n$, and $T^n/G$. We also propose a framework for formulating D-branes on asymmetric orbifolds.
We investigate the equation where the commutation relation in 2-dimensional zero-curvature equation composed of the algebra-valued potentials is replaced by the Moyal bracket and the algebra-valued potentials are replaced by the non-algebra-valued ones with two more new variables. We call the 4-dimensional equation the noncommutative zero-curvature equation. We show that various soliton equations are derived by the dimensional reduction of the equation.
The study of noncommutative solitons is greatly facilitated if the field equations are integrable, i.e. result from a linear system. For the example of a modified but integrable U(n) sigma model in 2+1 dimensions we employ the dressing method to construct explicit multi-soliton configurations on noncommutative R^{2,1}. These solutions, abelian and nonabelian, feature exact time-dependence for any value of the noncommutativity parameter theta and describe various lumps of finite energy in relative motion. We discuss their scattering properties and prove asymptotic factorization for large times.
In this paper we study the noncommutative extension of a modified U(n) sigma model in 2+1 dimensions. Using the method of dressing transformations, an iterative approach for the construction of solutions from a given seed solution, we demonstrate the construction of multi-soliton and soliton-antisoliton configurations for general n. As illustrative examples we discuss U(3) solitons and consider the head-on collision of a U(2) soliton and an antisoliton explicitly, which will result in a 90^{\text{o}} angle scattering. Further we discuss the head-on collision of one U(2) soliton with two antiso…
I explain how to construct noncommutative BPS configurations in four and lower dimensions by solving linear matrix equations. Examples are instantons in D=4 Yang-Mills, monopoles in D=3 Yang-Mills-Higgs, and (moving) solitons in D=2+1 Yang-Mills-Higgs. Some emphasis is on the latter as a showcase for the dressing method.
Following the construction of the projection operators on $T^2$ presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold $T^2/G (G=Z_N, N=2, 3, 4, 6)$ which correspond to a set of solitons on $T^2/Z_N$ in noncommutative field theory. In this way, we derive an explicit form of projector on $T^2/Z_6$ as an example. We also construct a complete set of projectors on $T^2/Z_N$ by series expansions for integral case.
In this paper, we construct a closed form of projectors on the integral noncommutative orbifold $T^2/Z_6$ in terms of elliptic functions by $GHS$ construction. After that, we give a general solution of projectors on $% T^{2}/Z_{6}$ and $T^{2}/Z_{3}$ with minimal trace and continuous reduced matrix $M(k,q_{0})$.The projectors constructed by us possess symmetry and manifest covariant forms under $Z_{6}$ rotation. Since projectors correspond to the soliton solutions of field theory on the noncommutative orbifold, we thus present a series of corresponding manifest covariant soliton solutions.
Coincident D2-branes in open N=2 fermionic string theory with a B-field background yield an integrable modified U(n) sigma model on noncommutative R^{2,1}. This model provides a showcase for an established method (the `dressing approach') to generate solutions for integrable field equations, even in the noncommutative case. We demonstrate the technique by constructing moving U(1) and U(2) solitons and by computing their energies. It is outlined how to derive multi-soliton configurations with arbitrary relative motion; they correspond to D0-branes moving inside the D2-branes.
Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar phi^4 braided noncommutative field theory in Lie-algebraic noncommutative spacetime, [x^i,x^j]=2i kappa epsilon^{ijk}x_k (i,j,k=1,2,3),…
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 N<=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.
We construct an approximation to field theories on the noncommutative torus based on soliton projections and partial isometries which together form a matrix algebra of functions on the sum of two circles. The matrix quantum mechanics is applied to the perturbative dynamics of scalar field theory, to tachyon dynamics in string field theory, and to the Hamiltonian dynamics of noncommutative gauge theory in two dimensions. We also describe the adiabatic dynamics of solitons on the noncommutative torus and compare various classes of noncommutative solitons on the torus and the plane.
We explore quantum states of instanton solitons in five dimensional noncommutative Yang-Mills theories. We start with maximally supersymmetric U(N) theory compactified on a circle S^1, and derive the low energy dynamics of instanton solitons, or calorons, which is no longer singular. Quantizing the low energy dynamics, we find N physically distinct ground states with a unit Pontryagin number and no electric charge. These states have a natural D-string interpretation. The conclusion remains unchanged as we decompactify S^1, as long as we stay in the Coulomb phase by turning on adjoint Higgs exp…
We review recent developments of soliton theories and integrable systems on noncommutative spaces. The former part is a brief review of noncommutative gauge theories focusing on ADHM construction of noncommutative instantons. The latter part is a report on recent results of existence of infinite conserved quantities and exact multi-soliton solutions for noncommutative Gelfand-Dickey hierarchies. Some examples of noncommutative Ward's conjecture are also presented. Finally, we discuss future directions on noncommutative Sato's theories, twistor theories and so on.
We describe a class of exact solutions of super Yang-Mills theory on even-dimensional noncommutative tori. These solutions generalize the solitons on a noncommutative plane introduced in hep-th/0009142 that are conjectured to describe unstable D2p-D0 systems. We show that the spectrum of quadratic fluctuations around our solutions correctly reproduces the string spectrum of the D2p-D0 system in the Seiberg-Witten decoupling limit. In particular the fluctuations correctly reproduce the 0-0 string winding modes. For p=1 and p=2 we match the differences between the soliton energy and the energy o…
We construct exact solitons on noncommutative tori for the type of actions arising from open string field theory. Given any projector that describes an extremum of the tachyon potential, we interpret the remaining gauge degrees of freedom as a gauge theory on the projective module determined by the tachyon. Whenever this module admits a constant curvature connection, it solves exactly the equations of motion of the effective string field theory. We describe in detail such a construction on the noncommutative tori. Whereas our exact solution relies on the coupling to a gauge theory, we comment …
