arXiv subjects

Masashi Hamanaka

Publications and source records attributed to Masashi Hamanaka.

27 records · Page 2Linked to original sources

arXiv2005

Noncommutative Solitons and Integrable Systems

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.

hep-th↗

arXiv2005

Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.

hep-th↗

arXiv2003

Towards Noncommutative Integrable Equations

We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving various noncommutative hierarchy equations in the framework of the Sato theory, which is brand-new. The existence of the hierarchy would suggest a hidden infinite-dimensional symmetry in the noncommutative Lax equations. We finally show that a noncommutative version of Burgers equation is completely integrable because it is linearizable via noncommutative Cole-Hopf transformation. These results are expected to lead to the completion of the noncommutative Sato theory.

hep-th↗

arXiv2003

Noncommutative Burgers Equation

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.

hep-th↗

arXiv2003

Towards Noncommutative Integrable Systems

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.

hep-th↗

arXiv2002

Gauge Fields on Tori and T-duality

We discuss gauge fields on tori in diverse dimensions, mainly in two and four dimensions. We construct various explicit gauge fields which have some topological charges and find the Dirac zero modes in the background of the gauge fields. By using the zero mode, we give new gauge fields on the dual torus, which is a gauge theoretical description of T-duality transformation of the corresponding D-brane systems including DDbar systems. From the transformation, we can easily see the duality expected from the index theorem. It is also mentioned that, for each topological charges, the corresponding constant curvature bundle can be constructed and their duality transformation can be performed in terms of Heisenberg modules.

hep-th↗

arXiv2002

ADHM/Nahm Construction of Localized Solitons in Noncommutative Gauge Theories

We study the relationship between ADHM/Nahm construction and ``solution generating technique'' of BPS solitons in noncommutative gauge theories. ADHM/Nahm construction and ``solution generating technique'' are the most strong ways to construct exact BPS solitons. Localized solitons are the solitons which are generated by the ``solution generating technique.'' The shift operators which play crucial roles in ``solution generating technique'' naturally appear in ADHM/Nahm construction and we can construct various exact localized solitons including new solitons: localized periodic instantons (=localized calorons) and localized doubly-periodic instantons. Nahm construction also gives rise to BPS fluxons straightforwardly from the appropriate input Nahm data which is expected from the D-brane picture of BPS fluxons. We also show that the Fourier-transformed soliton of the localized caloron in the zero-period limit exactly coincides with the BPS fluxon.

hep-th↗

arXiv2002

Moduli Space and Scattering of D0-Branes in Noncommutative Super Yang-Mills Theory

We study the moduli space of the D0-brane system on Dp-branes realized in the noncommutative super Yang-Mills theory. By examining the fluctuations around the solitonic solutions generated by solution generating technique, we confirm the interpretation of the moduli as the positions of D0-branes on Dp-branes. Low-energy scattering process is also examined for two D0-branes. We find that the D0-branes scatter at right angle for head-on collision in the D0-D4 system. For D0-D6 and D0-D8 systems we find special solutions which reduce to the D0-D4 case, giving the same behavior. This suggests that the scattering at right angle for head-on collision is a universal behavior of this kind of soliton scatterings.

hep-th↗

arXiv2001

On Exact Noncommutative BPS Solitons

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.

hep-th↗