Akifumi Sako

Instanton number of noncommutative U(n) gauge theory

We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative 4. Here the dimension of the vector space V that appear in the ADHM construction is called Instanton number. The calculation is done in operator formalism and the first Pontrjagin class is defined by converge series. The origin of the instanton number is investigated closely, too.

Elongated U(1) instantons on noncommutative Bbb R4

We found an exact solution of elongated U(1) instanton on noncommutative 4 for general instanton number k. The deformed ADHM equation was solved with general k and the gauge connection and the curvature were given explicitly. We also checked our solutions and evaluated the instanton charge by a numerical calculation.

Ring Structure of SUSY ∗ Product and 1/2 SUSY Wess-Zumino Model

There are two types of non(anti-)commutative deformation of D=4, N=1 supersymmetric field theories and D=2, N=2 theories. One is based on the non-supersymmetric star product and the other is based on the supersymmetric star product . These deformations cause partial breaking of supersymmetry in general. In case of supersymmetric star product, the chirality is broken by the effect of the supersymmetric star product, then it is not clear that lagrangian or observables including F-terms preserve part of supersymmetry. In this article, we investigate the ring structure whose product is defined by the supersymmetric star product. We find the ring whose elements correspond to 1/2 SUSY F-terms. Using this, the 1/2 SUSY invariance of the Wess-Zumino model is shown easily and directly.

Noncommutative cohomological field theory and GMS soliton

We show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter θμν. This is realized in a noncommutative cohomological field theory. As an example, a noncommutative cohomological scalar field theory is constructed, and its partition function is calculated. The partition function is the Euler number of Gopakumar, Minwalla, and Strominger (GMS) soliton space.

Explicit formulas for noncommutative deformations of CPN and CHN

We give explicit expressions of a deformation quantization with separation of variables for CP^N and CH^N. This quantization method is one of the ways to perform a deformation quantization of Kahler manifolds, which is introduced by Karabegov. Star products are obtained as explicit formulae in all order in the noncommutative parameter. We also give the Fock representations of the noncommutative CP^N and CH^N.

Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

A method to construct noncommutative instantons as deformations from commutative instantons was provided by Maeda and Sako [J. Geom. Phys. 58, 1784 (2008)]10.1016/j.geomphys.2008.08.006. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative R4, and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Greens function associated with instantons on noncommutative R4, as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Greens function derives noncommutative ADHM (Atiyah-Drinfeld-Hitchin-Manin) equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative R4 and ADHM data. An example of a...

Are vortex numbers preserved

Abstract We study noncommutative vortex solutions that minimize the action functional of the Abelian Higgs model in 2-dimensional noncommutative Euclidean space. We first consider vortex solutions which are deformed from solutions defined on commutative Euclidean space to the noncommutative one. We construct solutions whose vortex numbers are unchanged under the noncommutative deformation. Another class of noncommutative vortex solutions via a Fock space representation is also studied.

Recovery of full N=1 supersymmetry in non (anti-) commutative superspace

We investigate SUSY of Wess-Zumino models in non(anti-)commutative euclidean superspaces. Non(anti-)commutative deformations break 1/2 SUSY, then non(anti-)commutative Wess-Zumino models do not have full SUSY in general. However, we can recover full SUSY at specific coupling constants satisfying some relations. We give a general way to construct full SUSY non(anti-)commutative Wess-Zumino models. For a some example, we investigate quantum corrections and ?-functions behavior.

Euler number of instanton moduli space and Seiberg-Witten invariants

We show that a partition function of topological twisted N=4 Yang–Mills theory is given by Seiberg–Witten invariants on a Riemannian four manifolds under the condition that the sum of the Euler number and the signature of the four manifolds vanishes. The partition function is the sum of the Euler number of instanton moduli space when it is possible to apply the vanishing theorem. Also we obtain a relation of the Euler number labeled by the instanton number k with Seiberg–Witten invariants. All calculations in this article are done without assuming duality.

Noncommutative CPN and CHN and their physics

We study noncommutative deformation of manifolds by constructing star products. We start from a noncommutative d and discuss more genaral noncommutative manifolds. In general, star products can not be described in concrete expressions without some exceptions. In this article we introduce new examples of noncommutative manifolds with explicit star products. Karabegovs deformation quantization of PN and HN with separation of variables gives explicit calulable star products represented by gamma functions. Using the results of star products between inhomogeneous coordinates, we find creation and anihilation operators and obtain the Fock representation of the noncommutative PN and HN.