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 ${\bf R^4}$. 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.Comment: 6 color figures, 27 pages, some comments and references are added,typos fixe

Comments on the U(2) Noncommutative Instanton

We discuss the 't Hoof ansatz for instanton solutions in noncommutative U(2) Yang-Mills theory. We show that the extension of the ansatz leading to singular solutions in the commutative case, yields to non self-dual (or self-antidual) configurations in noncommutative space-time. A proposal leading to selfdual solutions with Q=1 topological charge (the equivalent of the regular BPST ansatz) can be engineered, but in that case the gauge field and the curvature are not Hermitian (although the resulting Lagrangian is real).Comment: Latex file, no figure

Large N Reductions and Holography

The large $N$ reductions in gauge theories are identified with dimensional reductions with homogeneous distribution of the eigenvalues of the gauge field, and it is used to identify the corresponding closed string descriptions in the Maldacena duality. When one does not take the zero-radii limit, the large $N$ reductions are naturally extended to the equivalences between the gauge theories and the "generalized" reduced models, which naturally contain the notion of T-dual equivalence. In the dual gravitational description, T-duality relates two type IIB supergravity solutions, the near horizon geometry of D3-branes, and the near horizon geometry of D-instantons densely and homogeneously distributing on the dual torus. This is the holographic description of the generalized large $N$ reductions. A new technique for calculating correlation functions of local gauge invariant single trace operators from the reduced models is also given.Comment: REVTeX4 v2: simple mistake in eq.(1) corrected. footnote 4 in v1 expanded in the main body. conflicting notations for the dilaton below eq.(6) (in v1) fixed. 6 pages, 5 figures v3: corrected typo v4: presentation improved with explanations and clarifications. results unchanged. refs added. 8page

Notes on Noncommutative Instantons

We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time R_{NC}^4 and noncommutative space R_{NC}^2 x R^2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R_{NC}^4 and on R_{NC}^2 x R^2. We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.Comment: Regular instanton solutions are now explicitly constructed also for the case of space-space noncommutativit

Non-Commutative Instantons and the Seiberg-Witten Map

We present several results concerning non-commutative instantons and the Seiberg-Witten map. Using a simple ansatz we find a large new class of instanton solutions in arbitrary even dimensional non-commutative Yang-Mills theory. These include the two dimensional ``shift operator'' solutions and the four dimensional Nekrasov-Schwarz instantons as special cases. We also study how the Seiberg-Witten map acts on these instanton solutions. The infinitesimal Seiberg-Witten map is shown to take a very simple form in operator language, and this result is used to give a commutative description of non-commutative instantons. The instanton is found to be singular in commutative variables.Comment: 26 pages, AMS-LaTeX. v2: the formula for the commutative description of the Nekrasov-Schwarz instanton corrected (sec. 4). v3: minor correction

Instanton Number Calculus on Noncommutative R^4

In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define ``Instanton number'' by the size of $B_{\alpha}$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes' theorem for the number operator representation. The Stokes' theorem on the noncommutative space shows that instanton charge is given by some boundary sum. Using the ASD conditions, we conclude that the instanton charge is equivalent to the instanton number.Comment: 29 pages, 7 figures, some statements in Sec.4.3 correcte

Confined Phase In The Real Time Formalism And The Fate Of The World Behind The Horizon

In the real time formulation of finite temperature field theories, one introduces an additional set of fields (type-2 fields) associated to each field in the original theory (type-1 field). In hep-th/0106112, in the context of the AdS-CFT correspondence, Maldacena interpreted type-2 fields as living on a boundary behind the black hole horizon. However, below the Hawking-Page transition temperature, the thermodynamically preferred configuration is the thermal AdS without a black hole, and hence there are no horizon and boundary behind it. This means that when the dual gauge theory is in confined phase, the type-2 fields cannot be associated with the degrees of freedom behind the black hole horizon. I argue that in this case the role of the type-2 fields is to make up bulk type-2 fields of classical closed string field theory on AdS at finite temperature in the real time formalism.Comment: v2: cases divided into sections with more detailed explanations. considerably enlarged with examples and a lot of figures. sec 4.1.2 for general closed cut-out circuits and appendix A for a sample calculation newly added. many minor corrections and clarifying comments. refs added. v3: refs and related discussion added. 1+46 pages, 26 figures. published versio

More on the Nambu-Poisson M5-brane Theory: Scaling limit, background independence and an all order solution to the Seiberg-Witten map

We continue our investigation on the Nambu-Poisson description of M5-brane in a large constant C-field background (NP M5-brane theory) constructed in Refs.[1, 2]. In this paper, the low energy limit where the NP M5-brane theory is applicable is clarified. The background independence of the NP M5-brane theory is made manifest using the variables in the BLG model of multiple M2-branes. An all order solution to the Seiberg-Witten map is also constructed.Comment: expanded explanations, minor corrections and typos correcte

Noncommutative U(1) Instantons in Eight Dimensional Yang-Mills Theory

We study the noncommutative version of the extended ADHM construction in the eight dimensional U(1) Yang-Mills theory. This construction gives rise to the solutions of the BPS equations in the Yang-Mills theory, and these solutions preserve at least 3/16 of supersymmetries. In a wide subspace of the extended ADHM data, we show that the integer $k$ which appears in the extended ADHM construction should be interpreted as the $D4$-brane charge rather than the $D0$-brane charge by explicitly calculating the topological charges in the case that the noncommutativity parameter is anti-self-dual. We also find the relationship with the solution generating technique and show that the integer $k$ can be interpreted as the charge of the $D0$-brane bound to the $D8$-brane with the $B$-field in the case that the noncommutativity parameter is self-dual.Comment: 22 page

Propagators in Noncommutative Instantons

We explicitly construct Green functions for a field in an arbitrary representation of gauge group propagating in noncommutative instanton backgrounds based on the ADHM construction. The propagators for spinor and vector fields can be constructed in terms of those for the scalar field in noncommutative instanton background. We show that the propagators in the adjoint representation are deformed by noncommutativity while those in the fundamental representation have exactly the same form as the commutative case.Comment: 28 pages, Latex, v2: A few typos correcte