Recent Developments in Instantons in Noncommutative ℝ^𝟒

Abstract

We review recent developments in noncommutative deformations of instantons in ℝ⁴. In the operator formalism, we study how to make noncommutative instantons by using the ADHM method, and we review the relation between topological charges and noncommutativity. In the ADHM methods, there exist instantons whose commutative limits are singular. We review smooth noncommutative deformations of instantons, spinor zero-modes, the Green's functions, and the ADHM constructions from commutative ones that have no singularities. It is found that the instanton charges of these noncommutative instanton solutions coincide with the instanton charges of commutative instantons before noncommutative deformation. These smooth deformations are the latest developments in noncommutative gauge theories, and we can extend the procedure to other types of solitons. As an example, vortex deformations are studied.

1. Introduction

Instantons in commutative space are one of the most important objects for nonperturbative analysis. We can overview them for example in [1] from the physicist's view points or in [2] from mathematical view points. See for example [3] for recent developments of them. Noncommutative (NC for short) instantons were discovered by Nekrasov and Schwarz [4]. After [4], NC instantons have been investigated by many physicists and mathematicians. However, many enigmas are left until now. Let us focus into instantons of 𝑈(𝑁) gauge theories in NC ℝ⁴ and understand what is clarified and what is unknown.

Instanton connections in the 4-dim Yang-Mills theory are defined by𝐹⁺=12(1+∗)𝐹=0, (1.1) where 𝐹 is a curvature 2-form and ∗ is the Hodge star operator. This condition says that curvature is anti-self-dual. In this paper, we call anti-self-dual connections instantons. The choice of anti-self-dual connection or self-dual connection to define instantons is not important to mathematics but just a habit.

NC instanton solutions were discovered by Nekrasov and Schwartz by using the ADHM method [4]. (See also [5] for the original ADHM method.) The ADHM construction which generates the instanton 𝑈(𝑁) gauge field requires a pair of the two complex vector spaces 𝑉=ℂ^𝑘 and 𝑊=ℂ^𝑁. Here −𝑘 is an integer called instanton number. Introduce 𝐵₁,𝐵₂∈Hom(𝑉,𝑉), 𝐼∈Hom(𝑊,𝑉), and 𝐽∈Hom(𝑉,𝑊) which are called ADHM data that satisfy the ADHM equations that we will see soon. In other words, 𝐵₁ and 𝐵₂ are complex-valued 𝑘×𝑘 matrices, and 𝐼 and 𝐽^† are complex-valued 𝑘×𝑁 matrices that satisfy (2.13) and (2.14) in Section 2.2. Using these ADHM data, we can construct instanton [6–17]. We call it NC ADHM instanton in the following. The NC ADHM construction is a strong method. A lot of instanton solutions are constructed by using the NC ADHM construction [6–17]. The NC ADHM method also clarifies some important features, for example, topological charge, index theorems, Green's functions, and so on. As a characteristic feature of NC ADHM construction, the NC ADHM instantons can be instantons that have singularities in the commutative limit. On the other hand, we can study NC instantons from a point of view of deformation quantization. Recently, NC instanton that is smoothly deformed from commutative instanton is constructed [18]. The method in [18] makes success in analysis for topological charges, index theorems, and the method derives the ADHM equations from NC instanton and proves a one-to-one correspondence between the ADHM data and NC instantons [19]. We review them in this article.

This paper is organized as follows. In Section 2, we review the NC ADHM instanton and their natures (For example, we investigate topological charges of instantons. We distinguish the terms “instanton number" from “instanton charge". In this article, we define the instanton number by the dimension of some vector space 𝑉; on the other hand, the instanton charge is defined by integral of the 2nd Chern class. We will soon see more details.). In Section 3, we construct an NC instanton solution which is a smooth deformation of the commutative instanton [18]. We study the NC instanton charge, an index theorem, and the correspondence relation with the ADHM construction for the smooth NC deformations of instantons [19]. In Section 4, we apply the method in Section 3 to a gauge theory in ℝ², and we make NC vortex solutions which are smooth deformations of commutative vortex solutions [20, 21].

2. Noncommutative ADHM Instantons

In this section, we review the NC ADHM instanton that may have singularities in commutative limit. An NC 𝑈(1) instanton is a typical example that has a singularity in commutative limit.

2.1. Notations for the Fock Space Formalism

Let us consider coordinate operators 𝑥^𝜇(𝜇=1,2,3,4) satisfying [𝑥^𝜇,𝑥^𝜈]=𝑖𝜃^𝜇𝜈, where 𝜃 is a skew symmetric real valued matrix and we call 𝜃^𝜇𝜈 NC parameter. We set the noncommutativity of the space to the self-dual case of 𝜃¹²=−𝜁₁, 𝜃³⁴=−𝜁₂, and the other 𝜃^𝜇𝜈=0 for convenience. By transformations of coordinates 𝑥^𝜇, the NC parameters are possible to be put in this form in general. Here we introduce complex coordinate operators𝑧₁=1√2(𝑥¹+𝑖𝑥²),𝑧₂=1√2(𝑥³+𝑖𝑥⁴). (2.1) Then the commutation relations become[𝑧₁,𝑧₁]=−𝜁₁,[𝑧₂,𝑧₂]=−𝜁₂,othersarezero. (2.2) We define creation and annihilation operators by𝑐^†_𝛼=𝑧_𝛼𝜁_𝛼,𝑐_𝛼=𝑧_𝛼𝜁_𝛼,(𝛼=1,2); (2.3) then they satisfy[𝑐_𝛼,𝑐^†_𝛼]=1,[𝑐_𝛼,𝑐_𝛽]=[𝑐^†_𝛼,𝑐^†_𝛽]=0(𝛼,𝛽=1,2). (2.4) The Fock space ℋ on which the creation and annihilation operators (2.4) act is spanned by the Fock state|𝑛₁,𝑛₂⟩=(𝑐^†₁)^𝑛1(𝑐^†₂)^𝑛2𝑛₁!𝑛₂!|0,0⟩, (2.5) with𝑐₁|𝑛₁,𝑛₂⟩=√𝑛₁|𝑛₁−1,𝑛₂⟩,𝑐^†₁|𝑛₁,𝑛₂⟩=𝑛₁+1|𝑛₁+1,𝑛₂𝑐⟩,₂|𝑛₁,𝑛₂⟩=√𝑛₂|𝑛₁,𝑛₂−1⟩,𝑐^†₂|𝑛₁,𝑛₂⟩=𝑛₂+1|𝑛₁,𝑛₂+1⟩, (2.6) where 𝑛₁ and 𝑛₂ are the occupation number. The number operators are also defined by𝑛_𝛼=𝑐^†_𝛼𝑐_𝛼,𝑁=𝑛₁+𝑛₂, (2.7) which act on the Fock states as𝑛_𝛼|𝑛₁,𝑛₂⟩=𝑛_𝛼|𝑛₁,𝑛₂⟩,𝑁|𝑛₁,𝑛₂⟩=(𝑛₁+𝑛₂)|𝑛₁,𝑛₂⟩. (2.8) In the operator representation, derivatives of a function 𝑓 are defined by𝜕_𝛼𝑓(𝑧)=[𝜕_𝛼,𝑓(𝑧)],𝜕_𝛼𝑓(𝑧)=[𝜕_𝛼,𝑓(𝑧)], (2.9) where 𝜕_𝛼=𝑧_𝛼/𝜁_𝛼 and 𝜕_𝛼=−𝑧_𝛼/𝜁_𝛼 which satisfy [𝜕_𝛼,𝜕_𝛼]=−1/𝜁_𝛼. The integral on NC ℝ⁴ is defined by the standard trace in the operator representation,𝑑⁴𝑥=𝑑⁴𝑧=(2𝜋)²𝜁₁𝜁₂Tr_ℋ. (2.10) Note that Tr_ℋ represents the trace over the Fock space whereas the trace over the gauge group is denoted by tr_𝑈(𝑁).

2.2. Noncommutative ADHM Instantons

Let us consider the 𝑈(𝑁) Yang-Mills theory on NC ℝ⁴. Let 𝑀 be a projective module over the algebra that is generated by the operator 𝑥_𝜇.

In the NC space, the Yang-Mills connection is defined by 𝐷_𝜇𝜓=−𝜓𝜕_𝜇+𝐷_𝜇𝜓, where 𝜓 is a matter field in fundamental representation type and 𝐷_𝜇∈End(𝑀) are anti-Hermitian gauge fields [22–24]. The relation between 𝐷_𝜇 and usual gauge connection 𝐴_𝜇 is 𝐷_𝜇=−𝑖𝜃_𝜇𝜈𝑥^𝜈+𝐴_𝜇, where 𝜃_𝜇𝜈 is an inverse matrix of 𝜃^𝜇𝜈. In our notation of the complex coordinates (2.1) and (2.2), the curvature is given as𝐹_𝛼𝛼=1𝜁_𝛼+[𝐷_𝛼,𝐷_𝛼],𝐹_𝛼𝛽=[𝐷_𝛼,𝐷_𝛽](𝛼≠𝛽). (2.11) Note that there is a constant term originated with the noncommutativity in 𝐹_𝛼𝛼. Instanton solutions satisfy the antiself-duality condition 𝐹=−∗𝐹. These conditions are rewritten in the complex coordinates as𝐹₁₁=−𝐹₂₂,𝐹₁₂=𝐹₁₂=0. (2.12) In the commutative spaces, instantons are classified by the topological charge 𝑄=(1/8𝜋²)∫tr_𝑈(𝑁)𝐹∧𝐹, which is always integer −𝑘 and coincide with the opposite sign of dimension of the vector space 𝑉 in the ADHM methods, and −𝑘 is called instanton number. In the NC spaces, the same statement is conjectured, and some partial proofs are given. (See Section 2.4 and see also [18, 25–30].)

In the commutative spaces, the ADHM construction is proposed by Atiyah et al. [5] to construct instantons. Nekrasov and Schwarz first extended this method to NC cases [4]. Here we review briefly on the ADHM construction of 𝑈(𝑁) instantons [22, 23].

The first step of ADHM construction on NC ℝ⁴ is looking for 𝐵₁,𝐵₂∈End(ℂ^𝑘), 𝐼∈Hom(ℂ^𝑛,ℂ^𝑘), and 𝐽∈Hom(ℂ^𝑘,ℂ^𝑛) which satisfy the deformed ADHM equations+[𝐵₂,𝐵^†₂]+𝐼𝐼^†−𝐽^†𝐽=𝜁₁+𝜁₂,+𝐼𝐽=0. (2.13)(2.14) We call −𝑘 “instanton number” in this article. In the previous section, we denote V as the vector space ℂ^𝑘. Note that the right-hand side of (2.13) is caused by the noncommutativity of the space ℝ⁴. The set of 𝐵₁,𝐵₂,𝐼, and 𝐽 satisfying (2.13) and (2.14) is called ADHM data. Using this ADHM data, we define operator 𝒟∶ℂ^𝑘⊕ℂ^𝑘⊕ℂ^𝑛→ℂ^𝑘⊕ℂ^𝑘 by𝒟^†=𝜏𝜎^†,𝜏=(𝐵₂−𝑧₂,𝐵₁−𝑧₁,𝐼)=(𝐵₂−𝜁₂𝑐^†₂,𝐵₁−𝜁₁𝑐^†₁𝜎,𝐼),_†=(−𝐵^†₁+𝑧₁,𝐵^†₂−𝑧₂,𝐽^†)=(−𝐵^†₁+𝜁₁𝑐₁,𝐵^†₂−𝜁₂𝑐₂,𝐽^†). (2.15) The ADHM equations (2.13) and (2.14) are replaced by𝜏𝜏^†=𝜎^†𝜎≡□,𝜏𝜎=0. (2.16)

Let us denote by Ψ∶ℂ^𝑛→ℂ^𝑘⊕ℂ^𝑘⊕ℂ^𝑛 the solution to the following equation:𝒟^†Ψ^𝑎=0(𝑎=1,…,𝑛),Ψ^†𝑎Ψ^𝑏=𝛿^𝑎𝑏. (2.17)

Theorem 2.1. Let Ψ^𝑎 be orthonormal zero-modes defined in (2.17). Then NC 𝑈(𝑁) instanton 𝐴_𝜇 with instanton number −𝑘 is obtained by 𝐴_𝜇=Ψ^†𝜕_𝜇Ψ=−𝑖Ψ^†𝜃_𝜇𝜈[𝑥^𝜈,Ψ]. (2.18) Here 𝜃_𝜇𝜈 is inverse of 𝜃^𝜇𝜈, that is, 𝜃_𝜇𝜈𝜃^𝜈𝜌=𝛿^𝜌_𝜇.

Proof. The curvature two-form determined by this connection is given as follows. 𝐹=𝑑𝐴+𝐴∧𝐴=𝑑(Ψ^†𝑑Ψ)+(Ψ^†𝑑Ψ)∧(Ψ^†𝑑Ψ)=𝑑Ψ^†∧𝑑Ψ−(𝑑Ψ)ΨΨ^†∧𝑑Ψ=𝑑Ψ^†(1−ΨΨ^†)∧𝑑Ψ. (2.19) Here we use 𝑑Ψ^†Ψ+Ψ^†𝑑Ψ=0 that follows from the differentiating of (2.17). Note that ΨΨ^†1=𝐼−𝒟𝒟^†𝒟𝒟^†, (2.20) since 𝐼=𝒟Ψ𝒟Ψ⁻¹(𝒟Ψ^†)⁻¹𝒟Ψ^†=𝒟𝒟Ψ^†𝒟001⁻¹𝒟Ψ^†1=𝒟𝒟^†𝒟𝒟^†+ΨΨ^†. (2.21) From (2.19) and (2.20), 𝐹=𝑑Ψ^†1(𝒟𝒟^†𝒟𝒟^†)∧𝑑Ψ=Ψ^†1(𝑑𝒟)∧𝒟^†𝒟(𝑑𝒟^†)Ψ, (2.22) where we use (𝑑𝒟^†)Ψ+𝒟^†𝑑Ψ=0 that follows from differentiating 𝒟^†Ψ=0. If the coordinates (𝑥₁,𝑥₂,𝑥₃,𝑥₄) are renamed (𝑥₂,𝑥₁,𝑥₄,𝑥₃) for convenience, we obtain 𝜕_𝜇𝒟^†=1√2(−𝜎_𝜇0),𝜕_𝜇1𝒟=√2−𝜎_𝜇0. (2.23) Here, we define 𝜎_𝜇 and 𝜎_𝜇 by (𝜎₁,𝜎₂,𝜎₃,𝜎₄)∶=(−𝑖𝜏₁,−𝑖𝜏₂,−𝑖𝜏₃,1_2×2(),𝜎₁,𝜎₂,𝜎₃,𝜎₄)∶=(𝑖𝜏₁,𝑖𝜏₂,𝑖𝜏₃,1_2×2), (2.24) where 𝜏_𝑖 are the Pauli matrices and 1_2×2 is an identity matrix of degree 2. Note that 𝒟^†𝒟=(^□0_0□) owing to (2.16), and 𝒟^†𝒟 and its inverse commute with 𝜎_𝜇. Then we find (2.22) is in proportion to 𝜎_𝜇𝜎_𝜈𝑑𝑥^𝜇∧𝑑𝑥^𝜈. (2.25)𝜎_𝜇𝜎_𝜈−𝜎_𝜈𝜎_𝜇 is a component of anti-self-dual two-form, that is easily checked by direct calculations. This fact and (2.22) show that the curvature 𝐹 is anti-self-dual and the connections given by (2.18) are instantons.

With the complex coordinate 𝑧_𝛼, NC instanton connections are given by𝐷_𝛼=1𝜁_𝛼Ψ^†𝑧_𝛼Ψ,𝐷_𝛼1=−𝜁_𝛼Ψ^†𝑧_𝛼Ψ. (2.26)

One of the most important feature to understand the origin of the instanton charges is existence of zero-modes of ΨΨ^†.

Theorem 2.2 (Zero-mode of ΨΨ^†). Suppose that Ψ and Ψ^† are given as above. The vector |𝑣⟩∈(ℂ^𝑘⊕ℂ^𝑘⊕ℂ^𝑛)⊗ℋ satisfying ΨΨ^†|𝑣⟩=⟨𝑣|ΨΨ^†=0,|𝑣⟩≠0 (2.27) is said to be a zero-mode of ΨΨ^†. The zero-modes are given by following three types: |𝑣₁⟩=(−𝐵₁+𝜁₁𝑐₁)|𝑢⟩(𝐵₂−𝜁₂𝑐₂)|𝑢⟩𝐽|𝑢⟩,|𝑣₂⟩=(𝐵^†₂−𝜁₂𝑐^†₂)|𝑢^′⟩(𝐵^†₁−𝜁₁𝑐^†₁)|𝑢^′⟩𝐼^†|𝑢^′⟩,|𝑣₀⟩=(exp_𝛼𝐵^†_𝛼𝑐^†_𝛼)|0,0⟩𝑣^𝑖₀(exp_𝛼𝐵^†_𝛼𝑐^†_𝛼)|0,0⟩𝑣^𝑖₀0. (2.28) Here |𝑢⟩ (|𝑢^′⟩) is some element of ℂ^𝑘⊗ℋ (i.e., |𝑢⟩ is expressed with the coefficients 𝑢_𝑖^𝑛𝑚∈ℂ as |𝑢⟩=∑_𝑖∑_𝑛,𝑚𝑢_𝑖^𝑛𝑚|𝑛,𝑚⟩𝑒_𝑖, where 𝑒_𝑖 is a base of 𝑘-dim vector space). 𝑣^𝑖₀ is a element of 𝑘-dim vector.

The proof is given in [25]. We will see the fact that zero-modes |𝑣₀⟩ play an essential role, in the following subsections.

2.3. 𝑈(1) N.C. ADHM Multi-Instanton

One of the most characteristic features of NC instantons is found in regularizations of the singularities. In commutative ℝ⁴, we cannot construct a nonsingular 𝑈(1) instanton. On the other hand, there exist in NC ℝ⁴. Let us see how to construct them as typical NC ADHM instantons.

At the beginning, we review the methods in [23]. Let 𝐵₁,𝐵₂,𝐼,𝐽 be constant matrices satisfying (2.13) and (2.14). We consider 𝜁=𝜁₁+𝜁₂>0; then we can put 𝐽=0 in general by using a symmetry. 𝐵₁ and 𝐵₂ are 𝑘×𝑘 matrices, and 𝐼 is 𝑘×1 matrices:𝐵₁=1⋮𝑘𝐵1⋯𝑘₁₁₁⋯𝐵¹_1𝑘𝐵⋮⋱⋮_1𝑘1⋯𝐵¹_𝑘𝑘,𝐵₂=1⋮𝑘𝐵1⋯𝑘₂₁₁⋯𝐵²_1𝑘𝐵⋮⋱⋮_2𝑘1⋯𝐵²_𝑘𝑘,112⋮𝑘𝐼𝐼=𝐼¹₂⋮𝐼_𝑘,𝐽=0. (2.29) We define 𝛽_𝛼, 𝑐^†_𝛼 and 𝑐_𝛼 by𝐵_𝛼=𝜁_𝛼𝛽_𝛼(𝛼=1,2). (2.30) We introduce Δ asΔ=_𝛼𝜁_𝛼(𝛽_𝛼−𝑐^†_𝛼)(𝛽^†_𝛼−𝑐_𝛼), (2.31) and we define a projection operator 𝑃 as a projection onto 0-eigenstates of Δ by𝑃=𝐼^†𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩𝐺⁻¹⟨0,0|𝑒^{∑𝛼𝛽𝛼𝑐𝛼}𝐼, (2.32) where𝐺=⟨0,0|𝑒^{∑𝛼𝛽𝛼𝑐𝛼}𝐼𝐼^†𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩. (2.33) We define shift operators 𝑆 and 𝑆^† and a operator Λ by𝑆𝑆^†=1,𝑆^†𝑆=1−𝑃,Λ=1+𝐼^†1Δ𝐼. (2.34)

Theorem 2.3 (Nekrasov). 𝑈(1) instantons are given by 𝐷_𝛼=−1𝜁_𝛼𝑆Λ^−1/2𝑐_𝛼Λ^1/2𝑆^†,𝐷_𝛼=1𝜁_𝛼𝑆Λ^1/2𝑐^†_𝛼Λ^−1/2𝑆^†. (2.35)

Proof. At first, we check that the inverse of Δ in (2.34) is well defined. Δ has 𝑘 zero-modes: 𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩⊗𝑒_𝑖(𝑖=1,…,𝑘) (2.36) which satisfy Δ𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩⊗𝑒_𝑖=0. Here 𝑒_𝑖=(𝛿_1𝑖,𝛿_2,𝑖,…,𝛿_𝑘𝑖)^𝑡 is a base of 𝑉. Note that 𝑆⋯𝑆^†=𝑆(1−𝑃)⋯𝑆^†. This implies that 𝑆 removes the zero-modes, and Hilbert spaces ℋ is projected on to a space that does not include the zero-modes. Therefore, the inverse of Λ exists if it is sandwiched between 𝑆 and 𝑆^† and (2.35) is well defined.
Next, we check that (2.35) is an instanton. Let us see how the equation 𝒟^†Ψ=0 is solved under orthonormalization condition Ψ^†Ψ=1. 𝜓_± and 𝜉 are introduced as 𝜓Ψ=𝜓⁺₋𝜉,𝜓_±∈𝑉⊗ℋ,𝜉∈ℋ. (2.37) The orthonormalization condition is expressed as 𝜓^†₊𝜓₊+𝜓^†₋𝜓₋+𝜉^†𝜉=1. (2.38) We put anzats for the solution by 𝜓₊=−𝜁₂(𝛽^†₂−𝑐₂)𝑣,𝜓₋=𝜁₁(𝛽^†₁−𝑐₁)𝑣, (2.39) and substitute them into 𝒟^†Ψ=0; then we get Δ𝑣+𝐼𝜉=0. (2.40) The orthonormalization condition is rewritten as 𝑣^†Δ𝑣+𝜉^†𝜉=1. (2.41) If there exist the inverse of Δ, 1𝑣=−Δ𝐼𝜉. (2.42) 0-eigenstates of Δ are (2.36) and we define the projection operator to project out the 0-eigenstates by 𝑃=I^†𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩𝐺⁻¹⟨0,0|𝑒^{∑𝛼𝛽𝛼𝑐𝛼}𝐼. (2.43) Shift operators 𝑆,𝑆^† satisfying 𝑆𝑆^†=1,𝑆^†𝑆=1−𝑃 (2.44) are determined by the definition of 𝑃. Then the inverse of Δ is well defined at the left side of 𝑆^† or the right side of 𝑆.
Using the orthonormalization condition, we obtain 𝜉=Λ^−1/2𝑆^†,Λ=1+𝐼^†1Δ𝐼. (2.45) Through these processes, Ψ is determined by the ADHM data, and after substituting this Ψ into (2.26) we obtain the instantons. 𝐷_𝛼1=−𝜉_𝛼𝜓^†𝑧_𝛼1𝜓=−𝜉_𝛼𝑆Λ^−1/2(𝐼^†1ΔΔ𝑧_𝛼1Δ𝐼−𝑧_𝛼)Λ^−1/2𝑆^†1=−𝜉_𝛼𝑆Λ^−1/2𝑐_𝛼Λ^1/2𝑆^†. (2.46)𝐷_𝛼 is given similarly.

This expression (2.35) is useful, but there exist other issues to get concrete expression of instantons. For example, it is not easy to obtain the explicit expression of Δ⁻¹.

As an example, let us construct an NC 𝑈(1) multi-instanton having concrete expressions with the instanton number −𝑘 [31, 32], which is made from the ADHM data:𝐵₁=^𝑘−1_𝑙=1𝑙𝜁𝑒_𝑙𝑒^†_𝑙+1=𝜁0√√10⋯⋯000√20⋯0⋮⋱⋱⋱⋮0⋯⋯0√𝑘−200⋯⋯0𝑘−10⋯⋯0,𝐵₂=0,𝐼=𝑘𝜁𝑒_𝑘=𝜁0⋮0√𝑘,𝐽=0, (2.47) where 𝜁=𝜁₁+𝜁₂. It is easy to check that this data satisfies the ADHM equations (2.13) and (2.14), and substituting them into definition of 𝑃 derives𝑃=^𝑘−1_𝑛1=0|𝑛₁,0⟩⟨𝑛₁,0|. (2.48) To construct an instanton, it is necessary to obtain Δ or Λ. By definition,Δ(𝑘)=𝜁₁𝑛₁+𝜁₂𝑛₂+𝜁^𝑘−1_𝑖=1𝑖𝑒_𝑖𝑒^†_𝑖−𝜁₁𝜁^𝑘−1_𝑖=1√𝑖{𝑐₁𝑒_𝑖𝑒^†_𝑖+1+𝑐^†₁𝑒_𝑖+1𝑒^†_𝑖Δ},Λ(𝑘)=1+𝜁𝑘⁻¹_𝑘𝑘(𝑘). (2.49)

Δ and Λ depend on 𝑘, so we denote them Δ(𝑘) and Λ(𝑘), respectively. Δ⁻¹_𝑘𝑘(𝑘) is (𝑘,𝑘) entry of matrix Δ⁻¹(𝑘). To obtain Δ⁻¹_𝑘𝑘(𝑘), it is enough to calculate the 𝑘th row vector of Δ⁻¹(𝑘). The 𝑘th row vector of Δ⁻¹(𝑘) is determined by Δ⁻¹Δ=1. We denote the 𝑘th row vector of Δ⁻¹(𝑘) by (𝑢₁,…,𝑢_𝑘), that is, Δ⁻¹_𝑘𝑖(𝑘)=𝑢_𝑖. Then, we obtain the following recurrence equation from the 𝑘th row of Δ⁻¹Δ=1𝑢₂𝑐^†₁−1𝜃𝑢₁(1+𝜃𝑛₁+(1−𝜃)𝑛₂√)=0,𝑖𝑢_𝑖+1𝑐^†₁−1𝜃𝑢_𝑖(𝑖+𝜃𝑛₁+(1−𝜃)𝑛₂)+√𝑖−1𝑢_𝑖−1𝑐₁=0(1≤𝑖≤𝑘−2), (2.50) where 𝜃=𝜁₁/(𝜁₁+𝜁₂). We change variables as 𝑢_𝑖=𝑤_𝑖−1𝑐₁^{†(𝑘−𝑖)};(𝑖−1)! (2.51) then we can rewrite the above recurrence relation by 𝑤_𝑖 as𝑤₁−1𝜃(1+𝜃(𝑛₁−𝑘+1)−(1−𝜃)𝑛₂)𝑤₀𝑤=0,_𝑖+11−{𝑖(𝜃+1𝜃)+𝜃(𝜃(𝑛₁−𝑘)+(1−𝜃)𝑛₂1)+(𝜃+𝜃)}𝑤_𝑖+𝑖((𝑖−1)𝑛₁−𝑘+2)𝑤_𝑖−1=0,(2≤𝑖≤𝑘−1). (2.52)

Note that 𝑛₁ and 𝑛₂ are commutative to each other, so we can treat them like C-numbers in the following. We introduce an anzats for the generating function 𝐹(𝑡;𝑘) by𝐹(𝑡;𝑘)=𝑒^𝑓(𝑡)(1−𝑎𝑡)^𝛼=^∞_𝑖=0𝑤_𝑖𝑡𝑖!^𝑖,𝑓(𝑡)=𝑑𝑡𝑐𝑡=𝑐(1−𝑎𝑡)(1−𝑏𝑡)2𝑎𝑏{ln(1−(𝑎+𝑏)𝑡+𝑎𝑏𝑡²)+𝑎+𝑏√𝐷√ln|2𝑎𝑏𝑡−(𝑎+𝑏)−𝐷2𝑎𝑏𝑡−(𝑎+𝑏)+√𝐷|}, (2.53) where 𝑎, 𝑏, 𝑐, and 𝛼 are real parameter determined by the request that 𝑤_𝑖 satisfy (2.52), and 𝐷=(𝑎−𝑏)². From the differentiation of 𝐹(𝑡;𝑘), we obtain(1−𝑎𝑡)(1−𝑏𝑡)^∞_𝑖=1𝑤₁𝑡(𝑖−1)!^𝑖−1={−𝑎𝛼+(𝑐+𝑎𝑏𝛼)𝑡}^∞_𝑖=0𝑤_𝑖𝑡𝑖!^𝑖, (2.54) and we find that 𝑤_𝑖 satisfy the following relation:𝑤_𝑖+1−(𝑖(𝑎+𝑏)−𝑎𝛼)𝑤_𝑖+𝑖(𝑎𝑏(𝑖−1)−𝑐−𝑎𝑏𝛼)𝑤_𝑖−1=0. (2.55) From (2.52) and (2.55), we obtain𝑎=𝜃or1𝜃1,𝑏=𝑎,𝛼=−ℎ(𝜃,𝑛₁,𝑛₂)𝑎,𝑐=−𝑛₁+𝑘−2+ℎ(𝜃,𝑛₁,𝑛₂)𝑎, (2.56) whereℎ(𝜃,𝑛₁,𝑛₂1)=𝜃{𝜃(𝑛₁−𝑘)+(1−𝜃)𝑛₂1}+𝜃+𝜃. (2.57) Thus the generating function 𝐹(𝑡;𝑘) is determined as an elementary function for each instanton number −𝑘. Using this 𝐹(𝑡;𝑘), we obtain 𝑤_𝑖, and Δ⁻¹_𝑘𝑘(𝑘) is determined as Δ⁻¹_𝑘𝑘(𝑘)=𝑢_𝑘=(1+𝜁₁𝜁√𝑘−1𝑤(𝑘−2)!^𝑘−2𝑛₁)(𝜁₁𝑛₁+𝜁₂𝑛₂)⁻¹. (2.58) Using them, 𝐺, 𝑃, 𝑆, and Λ are determined as𝐺=𝜁𝑘!^𝑘_𝑖=1{𝑖!(𝑘−𝑖)!𝜃^𝑘−𝑖}⁻¹𝑒_𝑖𝑒^†_𝑖,𝑃=𝐼^†𝑒^{∑𝛼𝛽†𝛼𝑐†𝛼}|0,0⟩𝐺⁻¹⟨0,0|𝑒^{∑𝛼𝛽𝛼𝑐𝛼}𝐼=^𝑘−1_𝑛1=0|𝑛₁,0⟩⟨𝑛₁𝑆,0|,_†=^∞_𝑛1=0|𝑛₁+𝑘,0⟩⟨𝑛₁,0|+^∞_𝑛1^∞_=0𝑛2=1|𝑛₁,𝑛₂⟩⟨𝑛₁,𝑛₂|,Λ=1+𝜁𝑘𝑢_𝑘. (2.59) Finally we obtain instanton gauge fields with instanton number −𝑘 as𝐷₁=1𝜁₁^∞_𝑛1^∞_=0𝑛2=0|𝑛₁,𝑛₂⟩⟨𝑛₁+1,𝑛₂|𝑑₁(𝑛₁,𝑛₂𝐷;𝑘),₂=1𝜁₂{^∞_𝑛1=0|𝑛₁,0⟩⟨𝑛₁+𝑘,1|𝑑₂(𝑛₁,0;𝑘)+^∞_𝑛1^∞_=0𝑛2=1|𝑛₁,𝑛₂⟩⟨𝑛₁,𝑛₂+1|𝑑₂(𝑛₁,𝑛₂;𝑘)}, (2.60) where𝑑₁(𝑛₁,𝑛₂;𝑘)=𝑛₁+𝑘+1[Λ(𝑛₁+𝑘+1,0)Λ(𝑛₁]+𝑘,0)^1/2,(𝑛₂=0),𝑛₁+1[Λ(𝑛₁+1,𝑛₂)Λ(𝑛₁,𝑛₂)]^1/2,(𝑛₂𝑑≠0),₂(𝑛₁,𝑛₂[;𝑘)=Λ(𝑛₁+𝑘,1)Λ(𝑛₁]+𝑘,0)^1/2,(𝑛₂=0),𝑛₂+1[Λ(𝑛₁,𝑛₂+1)Λ(𝑛₁,𝑛₂)]^1/2,(𝑛₂≠0). (2.61) Therefore, we obtain NC multi-instanton solutions expressed completely by elementary functions. This solution is one of the examples of the many kinds of the NC multi-instantons discovered until now [6–17].

2.4. Some Aspects

In this section, we overview some facts and important aspects of NC instantons without detailed derivations.

2.4.1. Instanton Charges and Instanton Numbers

Let us see a rough sketch of how to define instanton charges by using characteristic classes. The instanton charge in commutative space is determined (1/8𝜋²)∫tr𝐹∧⋆𝐹 and coincides with the instanton number defined by the dimension of the vector space 𝑉 in the ADHM construction. A naive definition of the instanton charges in NC ℝ⁴ is given by replacement of ∫𝑑⁴𝑥 by (2𝜋)²𝜁₁𝜁₂Tr_ℋ, but it is conditionally convergent in general. In [25, 26], we introduce cut-off 𝑁_𝐶 for the Fock space and make the instanton charge be a converge series. The region of the initial and final state of the Fock space with the boundary is|𝑛₁,𝑛₂⟩(𝑛₁=0,…,𝑁₁(𝑛₂),𝑛₂=0,…,𝑁₂(𝑛₁)), (2.62) where 𝑁₁(𝑛₂)(𝑁₂(𝑛₁)) is a function of 𝑛₂(𝑛₁) and we suppose that the length of the boundary is order 𝑁_𝐶≫𝑘, that is, 𝑁₁(𝑛₂)≈𝑁₂(𝑛₁)≈𝑁_𝐶≫𝑘.

Using this cut-off (boundary), we define the instanton charge by𝑄=lim_{𝑁𝐶→∞}𝑄_𝑁𝐶,𝑄_𝑁𝐶=𝜁²_𝑛1^𝑁₌₀^1(𝑛1)_𝑛2=0⟨𝑛₁,𝑛₂|(𝐹₁₁𝐹₂₂−𝐹₁₂𝐹₂₁)|𝑛₁,𝑛₂⟩. (2.63) As described in [25, 26], the regions for summations of intermediate states are shifted. This phenomenon is caused by the existence of the ΨΨ^† zero-mode ⟨𝑣₀|.

The following terms appear in the instanton charge 𝑄_𝑁𝐶:−tr_𝑈(𝑁)Tr_𝑁𝐶(12[Ψ^†𝑐^†₂Ψ,Ψ^†𝑐₂1Ψ]+2[Ψ^†𝑐^†₁Ψ,Ψ^†𝑐₁Ψ]). (2.64) We denote Tr_𝑁𝐶 as trace over some finite domain of Fock space characterized by 𝑁_𝐶 which is the length of the Fock space boundary. Using the Stokes' like theorem in [25], only trace over the boundary is left, then Tr_𝑁𝐶[Ψ^†𝑐^†₂Ψ,Ψ^†𝑐₂Ψ] becomes−tr_{𝑈(𝑁)boundary}(𝑁₂(𝑛₁)+1)=−tr_𝑈(𝑁)Tr_𝑁𝐶1−𝑘. (2.65) The same value is obtained from Tr_𝑁[Ψ^†𝑐^†₁Ψ,Ψ^†𝑐₁Ψ], too. The first term in (2.65) and the term from the constant curvature in (2.11) cancel out. The second term −𝑘 is occurred by zero-modes |𝑣₀⟩. Finally the second term of (2.65) is understood as the source of the instanton charge. The origin of the instanton charge is shift of intermediate states caused by 𝑘 zero-modes |𝑣₀⟩. After all, we get𝑄_𝑁=−𝑘+𝑂(𝑁^−1/2),𝑄=lim_𝑁→∞𝑄_𝑁=−𝑘. (2.66)

Theorem 2.4 (Instanton number). Consider U(N) gauge theory on NC ℝ⁴ with self-dual 𝜃^𝜇𝜈. The instanton charge 𝑄 is possible to be defined by limit of converge series and it is identified with the dimension 𝑘 that appears in the ADHM construction and is called “instanton number".

The strict proof is given in [25].

Note that the proof of the equivalence between the topological charge defined as the integral of the second Chern class and the instanton number given by the dimension of the vector space in the ADHM construction is not completed in NC space. In [27], Furuuchi shows how to appear zero-modes in the NC ADHM construction, and he shows that zero-modes project out some states in Fock space. In [28, 29], the geometrical origin of the instanton number for NC 𝑈(1) gauge theory is clarified. In [25], the identification between the topological charge and the dimension of the vector space in the ADHM construction is shown for a 𝑈(1) gauge theory. In [26], this identification is shown when the NC parameter is self-dual for a 𝑈(𝑁) gauge theory. In [30], the equivalence between the instanton numbers and instanton charges is shown with no restrictions on the NC parameters, but an NC version of the Osborn's identity (Corrigan's identity) is assumed. Until now, the relation between the instanton numbers and the topological charges in NC spaces had not been clarified completely. Moreover, the calculation in [25, 26] shows that the origin of the instanton number is deeply related to the noncommutativity. These results make us feel anomalous, because the instanton number of course exists for the instanton in the commutative space but |𝑣₀⟩ zero-modes or some counterparts of them do not exist in the commutative space. From these observations, we might wonder if there is a deep disconnection between commutative instantons and NC instantons. To clarify the connection between the NC instantons and commutative instantons, let us consider the smooth NC deformation from the commutative instanton in the next section.

Propagators and the Index Theorems
The zero-modes of the Dirac operator in the ADHM instanton background are studied in [33]. They show that the Atiyah-Singer index of the Dirac operator is equal to the instanton number. In [34], Green functions are constructed for a field in an arbitrary representation of gauge group propagating in NC ADHM instanton backgrounds.

Other Kinds of Solutions
We have reviewed the ADHM method. There are some other methods to construct NC instantons.

In [35], Lechtenfeld and Popov study the NC generalization of 't Hooft's multi-instanton configurations for the 𝑈(2) gauge group. They solve the problem in the naive application of Nekrasov and Schwarz method to the 't Hooft instanton solution. The problem originates from the appearance of a source term in the equation in the Corrigan-Fairlie-'t Hooft-Wilczek ansatz. They generalize the method of [36] to naive NC multi-instantons.

In [37], Horváth et al. use the method of dressing transformations, an iterative procedure for generating solutions from a given solution, and they generalize Belavin and Zakharov method to the NC case.

In [38], Hamanaka and Terashima construct NC instantons by using the solution generating technique introduced by Harvey et al. [39].

More details and an embracive list including other kinds of NC space and other kinds of BPS states are found in [40] for example.

Another approach that is smooth deformation of commutative instanton is given in the last few years. We will see it in the next section.

3. Smooth NC Deformation of Instantons

In this section, we construct NC instantons deformed smoothly from commutative instantons, and we study their natures.

We define NC deformations by formal expansions in a deformation parameter ℏ. So, let us pay attention to the mathematical meaning of the formal expansion. We introduce our star products by using formal expansions in ℏ, as we will see soon. Such products are not closed in the set of all smooth functions in general, so one of the simple ways to define the star products is using formal expansion. The star product is defined by putting some conditions on each order of ℏ expansion to be a smooth bounded function or a square integrable function and so on. Therefore, we have to check their conditions for all quantities represented by using the star product. Someone might wonder how can we manage such difficulties when the Fock space formalism is used. The Fock space formalism itself is regarded as a formal expansion by complex coordinates of ℂ²≅ℝ⁴. For example, an integrable condition of a function in the star product formulation is replaced by a convergence of the corresponding series. Space integrations are replaced by the trace operations (2.10). When we estimate topological charges like instanton charges by mathematically rigorous calculation, we have to use the Stokes' like theorem in the Fock space, as mentioned in Section 2.4. Therefore, the complexities of calculations are essentially same as the ones in star product formalism. One of the merits of using the star product formalism is that it does not require some specific representation. In calculations in the operator formalism, we have to introduce some basis like the Fock basis, but in the star product formalism, we can obtain physical values without introducing any representation.

3.1. Smooth NC Deformations

In this section, to easy understand that NC instantons smoothly connect into commutative instantons, we use a star product formulation. In the previous section, we use an operator formalism. Formally, there is a one-to-one correspondence between the operator formalism and the star product formalism, and the Weyl-transformation connects them with each other. Commutation relations of coordinates are given by[𝑥^𝜇,𝑥^𝜈]_⋆=𝑥^𝜇⋆𝑥^𝜈−𝑥^𝜈⋆𝑥^𝜇=𝑖𝜃^𝜇𝜈,𝜇,𝜈=1,2,…,4, (3.1) where (𝜃^𝜇𝜈) are a real, 𝑥-independent, skew-symmetric matrix entries, called the NC parameters. ⋆ is known as the Moyal product [41]. The Moyal product (or star product) is defined on functions by𝑖𝑓(𝑥)⋆𝑔(𝑥)∶=𝑓(𝑥)exp(2^⟵𝜕_𝜇𝜃^𝜇𝜈𝜕_𝜈)𝑔(𝑥). (3.2) Here ^⟵𝜕_𝜇 and 𝜕_𝜈 are partial derivatives with respect to 𝑥^𝜇 for 𝑓(𝑥) and to 𝑥^𝜈 for 𝑔(𝑥), respectively.

The curvature two form 𝐹 is defined by 𝐹∶=(1/2)𝐹_𝜇𝜈𝑑𝑥^𝜇∧⋆𝑑𝑥^𝜈=𝑑𝐴+𝐴∧⋆𝐴, where ∧⋆ is defined by 𝐴∧⋆𝐴∶=(1/2)(𝐴_𝜇⋆𝐴_𝜈)𝑑𝑥^𝜇∧𝑑𝑥^𝜈.

To consider smooth NC deformations, we introduce a parameter ℏ and a fixed constant 𝜃₀^𝜇𝜈<∞ with 𝜃^𝜇𝜈=ℏ𝜃₀^𝜇𝜈. We define the commutative limit by letting ℏ→0.

Formally we expand the connection as𝐴_𝜇=^∞_𝑙=0𝐴_𝜇^(𝑙)ℏ^𝑙. (3.3) Then,𝐴_𝜇⋆𝐴_𝜈=^∞_{𝑙,𝑚,𝑛=0}ℏ^{𝑙+𝑚+𝑛}1𝐴𝑙!^𝜇(𝑚)(Δ)^𝑙𝐴_𝜇^(𝑛),𝑖Δ≡2^⟵𝜕_𝜇𝜃₀^𝜇𝜈𝜕_𝑣. (3.4) We introduce the self-dual projection operator 𝑃 by𝑃∶=1+∗2;𝑃_{𝜇𝜈,𝜌𝜏}=14(𝛿_𝜇𝜌𝛿_𝜈𝜏−𝛿_𝜈𝜌𝛿_𝜇𝜏+𝜖_{𝜇𝜈𝜌𝜏}). (3.5) Then the instanton equation is given as𝑃_{𝜇𝜈,𝜌𝜏}𝐹^𝜌𝜏=0. (3.6) In the NC case, the 𝑙th order equation of (3.6) is given by𝑃^{𝜇𝜈,𝜌𝜏}(𝜕_𝜌𝐴_𝜏^(𝑙)−𝜕_𝜏𝐴_𝜌^(𝑙)+𝑖[𝐴_𝜌^(𝑙),𝐴_𝜏⁽⁰⁾]+𝑖[𝐴_𝜌⁽⁰⁾,𝐴_𝜏^(𝑙)]+𝐶^(𝑙)_𝜌𝜏𝐶)=0,_{(𝑙)𝜌𝜏}∶=_{(𝑝;𝑚,𝑛)∈𝐼(𝑙)}ℏ^{𝑝+𝑚+𝑛}1𝑝!(𝐴_𝜌^(𝑚)(Δ)^𝑝𝐴_𝜏^(𝑛)−𝐴_𝜏^(𝑚)(Δ)^𝑝𝐴_𝜌^(𝑛)),𝐼(𝑙)≡{(𝑝;𝑚,𝑛)∈ℤ³∣𝑝+𝑚+𝑛=𝑙,𝑝,𝑚,𝑛≥0,𝑚≠𝑙,𝑛≠𝑙}. (3.7) Note that the 0th order is the commutative instanton equation with solution 𝐴_𝜇⁽⁰⁾ being a commutative instanton. The asymptotic behavior of commutative instanton 𝐴_𝜇⁽⁰⁾ is given by𝐴_𝜇⁽⁰⁾=𝑔𝑑𝑔⁻¹+𝑂(|𝑥|⁻²),𝑔𝑑𝑔⁻¹=𝑂(|𝑥|⁻¹), (3.8) where 𝑔∈𝐺 and 𝐺 is a gauge group. (See, e.g., [2].) We introduce covariant derivatives associated to the commutative instanton connection by𝐷_𝜇⁽⁰⁾𝑓∶=𝜕_𝜇𝑓+𝑖[𝐴_𝜇⁽⁰⁾,𝑓],𝐷_𝐴⁽⁰⁾𝑓∶=𝑑𝑓+𝐴⁽⁰⁾∧𝑓. (3.9) Using this, (3.7) is given by𝑃^{𝜇𝜈,𝜌𝜏}(𝐷_𝜌⁽⁰⁾𝐴_𝜏^(𝑙)−𝐷_𝜏⁽⁰⁾𝐴_𝜌^(𝑙)+𝐶^(𝑙)_𝜌𝜏)=0. (3.10)

In the following, we fix a commutative instanton connection 𝐴⁽⁰⁾. We impose the following gauge fixing condition for 𝐴^(𝑙)(𝑙≥1) [18, 42]𝐴−𝐴⁽⁰⁾=𝐷^∗_𝐴(0)𝐵,𝐵∈Ω²₊, (3.11) where 𝐷^∗_𝐴(0) is defined by(𝐷^∗_𝐴(0))_𝜌^𝜇𝜈𝐵_𝜇𝜈=𝛿^𝜈_𝜌𝜕^𝜇𝐵_𝜇𝜈−𝛿^𝜇_𝜌𝜕^𝜈𝐵_𝜇𝜈+𝑖𝛿^𝜈_𝜌[𝐴^𝜇,𝐵_𝜇𝜈]−𝛿^𝜇_𝜌[𝐴^𝜈,𝐵_𝜇𝜈]=𝛿^𝜈_𝜌𝐷^(0)𝜇𝐵_𝜇𝜈−𝛿^𝜇_𝜌𝐷^(0)𝜈𝐵_𝜇𝜈. (3.12) We expand 𝐵 in ℏ as we did with 𝐴. Then 𝐴^(𝑙)=𝐷^∗_𝐴(0)𝐵^(𝑙). In this gauge, using the fact that the 𝐴⁽⁰⁾ is an anti-self-dual connection, (3.10) simplified to2𝐷²₍₀₎𝐵^{(𝑙)𝜇𝜈}+𝑃^{𝜇𝜈,𝜌𝜏}𝐶^(𝑙)_𝜌𝜏=0, (3.13) where 𝐷²₍₀₎≡𝐷^𝜌_𝐴(0)𝐷_𝐴⁽⁰⁾_𝜌. (3.14)

We consider the Green's function for 𝐷²₍₀₎: 𝐷²₍₀₎𝐺₀(𝑥,𝑦)=𝛿(𝑥−𝑦), (3.15) where 𝛿(𝑥−𝑦) is a four-dimensional delta function. 𝐺₀(𝑥,𝑦) has been constructed in [43] (see also [44, 45]). Using the Green's function, we solve (3.13) as𝐵^{(𝑙)𝜇𝜈}1=−2_ℝ⁴𝐺₀(𝑥,𝑦)𝑃^{𝜇𝜈,𝜌𝜏}𝐶^(𝑙)_𝜌𝜏(𝑦)𝑑⁴𝑦 (3.16) and the NC instanton 𝐴=∑𝐴^(𝑙)ℏ^𝑙 is given by𝐴^(𝑙)=𝐷^∗_𝐴(0)𝐵^(𝑙). (3.17) In the following, we call NC instantons smoothly deformed from commutative instantons SNCD instantons. The asymptotic behavior of Green's function of 𝐷²₍₀₎ is important, which is given by𝐺₀(𝑥,𝑦)=𝑂(|𝑥−𝑦|⁻²). (3.18)

We introduce the notation 𝑂^′(|𝑥|^−𝑚) as in [2]. If 𝑠 is a function of ℝ⁴ which is 𝑂(|𝑥|^−𝑚) as |𝑥|→∞ and |𝐷^𝑘₍₀₎𝑠|=𝑂(|𝑥|^{−𝑚−𝑘}), then we denote this natural growth condition by 𝑠=𝑂^′(|𝑥|^−𝑚).

Theorem 3.1. If 𝐶^(𝑙)=𝑂^′(|𝑥|⁻⁴), then 𝐵^(𝑘)=𝑂^′(|𝑥|⁻²).

We gave a proof of this theorem in [18].

In our case, 𝐶⁽¹⁾_𝜌𝜏=𝑂^′(𝑥⁻⁴) by (3.8), and so 𝐵⁽¹⁾=𝑂^′(|𝑥|⁻²), 𝐴⁽¹⁾=𝑂^′(|𝑥|⁻³) as 𝐴^(𝑙)=𝐷^∗_𝐴(0)𝐵^(𝑙). Repeating the argument 𝑙 times, we get|𝐴^(𝑙)|<𝑂^′(|𝑥|^−3+𝜖),∀𝜖>0. (3.19)

3.2. Instanton Charge

The instanton charge is defined by𝑄_ℏ1∶=8𝜋²tr_𝑈(𝑁)𝐹∧⋆𝐹. (3.20) We rewrite (3.20) as18𝜋²tr_𝑈(𝑁)2𝑑(𝐴∧⋆𝑑𝐴+31𝐴∧⋆𝐴∧⋆𝐴)+8𝜋²tr_𝑈(𝑁)𝑃_⋆, (3.21) where𝑃_⋆=13{𝐹∧⋆𝐴∧⋆𝐴+2𝐴∧⋆𝐹∧⋆𝐴+𝐴∧⋆𝐴∧⋆𝐹+𝐴∧⋆𝐴∧⋆𝐴∧⋆𝐴}. (3.22)

∫tr_𝑈(𝑁)𝑃_⋆ is 0 in the commutative limit, but it does not vanish in NC space, because the cyclic symmetry of trace operation is broken by the NC deformation.

The terms in ∫tr_𝑈(𝑁)𝑃_⋆ are typically written as_ℝ𝑑tr_𝑈(𝑁)(𝑃∧⋆𝑅−(−1)^{𝑛(4−𝑛)}𝑅∧⋆𝑃), (3.23) where 𝑃 and 𝑅 are some 𝑛-form and (4−𝑛)-form (𝑛=0,…,4), respectively, and let 𝑃∧𝑅 be 𝑂(ℏ^𝑘). The lowest order term in ℏ vanishes because of the cyclic symmetry of the trace, that is, ∫tr_𝑈(𝑁)(𝑃∧𝑅−(−1)^{𝑛(4−𝑛)}𝑅∧𝑃)=0. The term of order ℏ is given by𝑖2_ℝ⁴tr_𝑈(𝑁){ℏ𝜃₀^𝜇𝜈(𝜕_𝜇𝑃∧𝜕_𝜈=𝑖𝑅)}2_ℝ⁴(𝑛!(4−𝑛)!)𝜖^{𝜇1𝜇2𝜇3𝜇4}tr_𝑈(𝑁)𝑑{(∗𝜃)∧(𝑃_{𝜇1⋯𝜇𝑛}𝑑𝑅_{𝜇𝑛+1⋯𝜇4})}, (3.24) where ∗𝜃=𝜖_{𝜇𝜈𝜌𝜏}𝜃^𝜌𝜏𝑑𝑥^𝜇∧𝑑𝑥^𝜈/4. These integrals are zero if 𝑃_{𝜇1⋯𝜇𝑛}𝑑𝑅_{𝜇𝑛+1⋯𝜇4} is 𝑂^′(|𝑥|^{−(4−1+𝜖)})(𝜖>0) and this condition is satisfied for SNCD instantons. Similarly, higher-order terms in ℏ in (3.23) can be written as total divergences and hence vanish under the decay hypothesis. This fact and (3.19) imply that ∫tr_𝑈(𝑁)𝑃_⋆=0.

Because of the similar estimation, we found the other terms of ∫tr_𝑈(𝑁)𝐹∧⋆𝐹−∫tr_𝑈(𝑁)𝐹⁽⁰⁾∧𝐹⁽⁰⁾ vanish, where 𝐹⁽⁰⁾ is the curvature two form associated to 𝐴⁽⁰⁾.

Summarizing the above discussions, we get following theorems [18].

Theorem 3.2. Let 𝐴_𝜇⁽⁰⁾ be a commutative instanton solution in ℝ⁴. There exists a formal NC instanton solution 𝐴_𝜇=∑^∞_𝑙=0𝐴_𝜇^(𝑙)ℏ^𝑙 (SNCD instanton) such that the instanton number 𝑄_ℏ defined by (3.20) is independent of the NC parameter ℏ: 18𝜋²tr_𝑈(𝑁)1𝐹∧⋆𝐹=8𝜋²tr_𝑈(𝑁)𝐹⁽⁰⁾∧𝐹⁽⁰⁾. (3.25)

3.3. Index of the Dirac Operator and Green's Function

Dirac(-Weyl) operators 𝒟_𝐴∶Γ(𝑆⁺⊗𝐸)→Γ(𝑆⁻⊗𝐸) and 𝒟_𝐴∶Γ(𝑆⁻⊗𝐸)→Γ(𝑆⁺⊗𝐸) are defined as𝒟_𝐴∶=𝜎^𝜇𝐷_𝜇,𝒟_𝐴∶=𝜎^𝜇𝐷^†_𝜇. (3.26) Here, 𝜎_𝜇 and 𝜎_𝜇 are defined by (2.24). Consider ℏ expansion of 𝜓∈Γ(𝑆⁺⊗𝐸) and 𝜓∈Γ(𝑆⁻⊗𝐸) as𝜓=^∞_𝑛=0ℏ^𝑛𝜓^(𝑛),𝜓=^∞_𝑛=0ℏ^𝑛𝜓^(𝑛). (3.27) In [19], the zero-modes of 𝒟_𝐴 and 𝒟_𝐴, which are defined by𝒟_𝐴⋆𝜓=0,𝒟_𝐴⋆𝜓=0, (3.28) are investigated, and the following theorem is obtained.

Theorem 3.3. Let 𝒟_𝐴 and 𝒟_𝐴 be the Dirac(-Weyl) operators for an SNCD instanton background with its instanton number −𝑘. There is no zero-mode for 𝒟_𝐴⋆𝜓=0, and there are 𝑘 zero-modes for 𝒟_𝐴⋆𝜓_𝑖=0(𝑖=1,…,𝑘) that are given as 𝜓_𝑖=^∞_𝑛=0(^𝑘_𝑗=1𝑎^𝑗_𝑛,𝑖𝜂_𝑗)ℏ^𝑛+𝑂^′(|𝑥|^−5+𝜖),𝜂_𝑗=𝑂^′(|𝑥|⁻³), (3.29) where 𝑎^𝑗_𝑛,𝑖 is a constant matrix and 𝜂_𝑗 is a base of the zero mode of 𝒟_𝐴⁽⁰⁾.

Note that it is a well-known fact as an index theorem in commutative space that the dimension of ker𝒟_𝐴⁽⁰⁾ is equal to 𝑘 the instanton number (of opposite sign), and there exists 𝑘 zero-mode 𝜂_𝑖(𝑖=1,2,…,𝑘). Theorem 3.3 says that zero-modes deformed from the ones in commutative space are obtained, but there is no new zero-mode appearing. Then we get the following theorem [19].

Theorem 3.4. If Ind𝐷⁰∶=dimker𝒟_𝐴⁽⁰⁾−dimker𝒟_𝐴⁽⁰⁾=−𝑘, then Ind𝐷_&^&∶=dimker𝒟_𝐴−dimker𝒟_𝐴=−𝑘.

Next, we construct the Green's function of Δ_𝐴≡𝐷_𝜇⋆𝐷^𝜇,Δ_𝐴⋆𝐺_𝐴(𝑥,𝑦)=𝛿(𝑥−𝑦). (3.30)

We expand (3.18) by ℏ, for 𝑛>0, then ℏ^𝑛 order equation is given asΔ_𝐴⁽⁰⁾𝐺_𝐴^(𝑛)(𝑥,𝑦)+[Δ_{𝐴0≤𝑘<𝑛}ℏ^𝑘𝐺_𝐴^(𝑘)(𝑥,𝑦)]^(𝑛)=0, (3.31) where 𝐺_𝐴^(𝑛) is defined by 𝐺_𝐴(𝑥,𝑦)=∑^∞_𝑘=0𝐺_𝐴^(𝑘)ℏ^𝑘. We solve them recursively𝐺_𝐴^(𝑛)(𝑥,𝑦)=𝑑⁴𝑤𝐺_𝐴⁽⁰⁾(𝑥,𝑤)[Δ_{𝐴0≤𝑘<𝑛}ℏ^𝑘𝐺_𝐴^(𝑘)(𝑤,𝑦)]^(𝑛). (3.32) Note that 𝐺_𝐴⁽⁰⁾(𝑥,𝑤) was constructed in [43–45]. Using property of 𝐺_𝐴⁽⁰⁾(𝑥,𝑤) and 𝐴^(𝑛), we obtain the following decay condition in [19]:𝐺_𝐴^(𝑛)(𝑥,𝑦)=𝑂^′(|𝑥|⁻³). (3.33)

3.4. From an Instanton to the ADHM Equations

Let us see how to derive the ADHM equations from an SNCD instanton.

Let 𝜓_𝑖(𝑖=1,…,𝑘) be orthonormal zero-modes of 𝒟_𝐴 and 𝜓=(𝜓_𝑖), which are introduced in Section 3.3.

At first we define 𝑇^𝜇 by𝑇^𝜇∶=_ℝ4𝑑⁴𝑥12(𝑥^𝜇⋆𝜓^†⋆𝜓+𝜓^†⋆𝜓⋆𝑥^𝜇). (3.34)

Next we introduce an asymptotically parallel section 𝑔⁻¹𝑆 of 𝑆⁺⊗𝐸 by𝑔𝜓=−⁻¹𝑆𝑥^†|𝑥|⁴+𝑂^′(|𝑥|⁻⁴), (3.35) where 𝑥^†∶=𝜎_𝜇𝑥^𝜇 and 𝜓∶=^𝑡𝜓𝜎₂. This 𝑡 means transposing spinor suffixes.

Using various properties and decay conditions of 𝐴^(𝑛),𝐺_𝐴^(𝑛),𝜓^(𝑛), and theorems in the previous subsections, we finally obtain the following theorem.

Theorem 3.5. Let 𝐴^𝜇 be an SNCD instanton and 𝜓 the zero-mode of 𝒟_𝐴 determined by 𝐴^𝜇 as in Section 3.3. Let 𝑇^𝜇,𝑆 be constant matrices defined by (3.34) and (3.35), respectively. Then, they satisfy the ADHM equations: [𝑇^𝜇,𝑇^𝜈]⁺=12tr(𝑆^†𝑆𝜎^𝜇𝜈)−𝑖𝜃^𝜇𝜈+1_𝑘×𝑘. (3.36) Here 𝜎_𝜇𝜈∶=(1/4)(𝜎_𝜇𝜎_𝜈−𝜎_𝜈𝜎_𝜇) and 1_𝑘×𝑘 is an identity matrix.

Rough sketch of the proof
Let us see the essence of the proof. Let us introduce ⋆_𝑥 as ⋆ associated with variable 𝑥. The completeness of 𝜓(𝑥) is written as ⋆_𝑥𝜓(𝑥)𝜓^†(𝑦)⋆_𝑦=⋆_𝑥𝛿(𝑥−𝑦)⋆_𝑦−⋆_𝑥𝒟_𝐴⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦^⟵𝒟_𝐴⋆_𝑦. (3.37) From the definition of the 𝑇^𝜇, 𝑇^𝜇𝑇^𝜈=_ℝ4𝑑⁴𝑥_ℝ4𝑑⁴𝑦(𝑥^𝜇⋆_𝑥𝜓^†(𝑥)⋆_𝑥𝜓(𝑥))(𝜓^†(𝑦)⋆_𝑦𝜓(𝑦)⋆_𝑦𝑦^𝜈). (3.38) Using Theorem 3.3, (3.37), and integration by parts, (3.38) becomes 𝑇^𝜇𝑇^𝜈=_ℝ4𝑑⁴𝑥𝑥^𝜇⋆𝜓^†⋆𝜓⋆𝑥^𝜈+_𝑆3𝑑𝑆^𝜌_𝑥ℝ4𝑑⁴𝑦(𝑥^𝜇⋆_𝑥𝜓^†(𝑥)𝜎_𝜌)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦^⟵𝒟_𝐴⋆_𝑦(𝜓(𝑦)⋆_𝑦𝑦^𝜈)−_ℝ4𝑑⁴𝑥_ℝ4𝑑⁴𝑦(𝜓^†(𝑥)𝜎^𝜇)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦^⟵𝒟_𝐴⋆_𝑦(𝜓(𝑦)⋆_𝑦𝑦^𝜈), (3.39) where 𝑑𝑆^𝜇_𝑥=|𝑥|²𝑥^𝜇𝑑Ω and 𝑑Ω is the solid angle. The first term is deformed as follows. _ℝ4𝑑⁴𝑥𝑥^𝜇⋆𝜓^†⋆𝜓⋆𝑥^𝜈=_ℝ4𝑑⁴𝑥(𝜓^†⋆𝜓⋆𝑥^𝜈⋆𝑥^𝜇+[𝑥^𝜇,𝜓^†⋆𝜓]_⋆⋆𝑥^𝜈+𝜓^†⋆𝜓⋆[𝑥^𝜇,𝑥^𝜈]_⋆)=_ℝ4𝑑⁴𝑥(𝜓^†⋆𝜓⋆𝑥^𝜈⋆𝑥^𝜇+𝑖𝜃^𝜇𝜌𝜕_𝜌(𝜓^†⋆𝜓)⋆𝑥^𝜈+𝑖𝜃^𝜇𝜈𝜓^†⋆=𝜓)_ℝ4𝑑⁴𝑥𝜓^†⋆𝜓⋆𝑥^𝜈⋆𝑥^𝜇. (3.40) Here 𝜓=𝑂^′(|𝑥|⁻³) given in Theorem 3.3 is used in the third equality. By integration by parts again, we get 𝑇^𝜇𝑇^𝜈=_ℝ4𝑑⁴𝑥𝜓^†⋆𝜓⋆𝑥^𝜈⋆𝑥^𝜇+_𝑆3𝑑𝑆^𝜌_𝑥𝑆3𝑑𝑆^𝜏_𝑦(𝑥^𝜇⋆_𝑥𝜓^†(𝑥)𝜎_𝜌)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦(𝜎_𝜏𝜓(𝑦)⋆_𝑦𝑦^𝜈)−_𝑆3𝑑𝑆^𝜌_𝑥ℝ4𝑑⁴𝑦(𝑥^𝜇⋆_𝑥𝜓^†(𝑥)𝜎_𝜌)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦(𝜎^𝜈𝜓(𝑦))−_ℝ4𝑑⁴𝑥_𝑆3𝑑𝑆^𝜏_𝑦(𝜓^†(𝑥)𝜎^𝜇)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦(𝜎^𝜏𝜓(𝑦)⋆_𝑦𝑦^𝜈)+_ℝ4𝑑⁴𝑥_ℝ4𝑑⁴𝑦(𝜓^†(𝑥)𝜎^𝜇)⋆_𝑥𝐺_𝐴(𝑥,𝑦)⋆_𝑦(𝜎^𝜈𝜓(𝑦)). (3.41)(3.42)(3.43)(3.44)(3.45)
Equations (3.42) and (3.44) vanish when 𝑅_𝑦→∞, where 𝑅_𝑦 is a radius of 𝑆³_𝑦. Equation (3.45) will vanish on the self-dual projection [𝑇^𝜇,𝑇^𝜈]⁺∶=𝑃^{𝜇𝜈,𝜌𝜏}[𝑇_𝜌,𝑇_𝜏], because 𝜎^𝜇𝜎^𝜈−𝜎^𝜈𝜎^𝜇 is anti-self-dual with respect to the 𝜇,𝜈. Thus only (3.41) and (3.43) remain. By the asymptotic behaviors of 𝜓 and some calculations, we can prove that (3.43) becomes 18tr(𝑆^†𝑆𝜎^𝜇𝜎^𝜈), (3.46) where the trace tr is taken with respect to the spinor indices. In the [𝑇^𝜇,𝑇^𝜈]⁺ combination, (3.41) becomes −𝑖𝜃^𝜇𝜈+=−𝑖𝑃^{𝜇𝜈,𝜌𝜏}𝜃^𝜌𝜏. Therefore, we get (3.36). The complete proof is given in [19].

These ADHM equations (3.36) are coincident with the ones provided by Nekrasov and Schwarz [4]. After identification of𝑆^†=𝐼𝐽^†,𝑇^𝜇𝜎_𝜇=−𝐵₂−𝐵₁𝐵^†₁−𝐵^†₂, (3.47) and setting the NC parameter as in (2.2), we find that (3.36) is identified with (2.13) and (2.14).

Similar to the commutative case, we obtain the following theorem.

Theorem 3.6. There is a one-to-one correspondence between ADHM data satisfying (3.36) and SNCD instantons in NC ℝ⁴.

The proof is given in [19].

4. Smooth NC Deformation of Vortexes

In the previous section, we investigate the smooth deformation of instantons. This method is applicable to gauge theories in other dimensions. In this section we study NC deformation of the vortex solutions [46, 47]. We consider the Abelian-Higgs model in commutative ℝ² and deform the Taubes' vortex solutions into NC vortexes [48].

Let coordinates of NC Euclidean space ℝ² be 𝑥^𝜇,𝜇=1,2, with commutation relations[𝑥^𝜇,𝑥^𝜈]_⋆=𝑖ℏ𝜖^𝜇𝜈(𝜇,𝜈=1,2), (4.1) where 𝜖^𝜇𝜈=−𝜖^𝜈𝜇(𝜖¹²=1) is an antisymmetric tensor.

The curvature components of the connection 𝐴 are given by𝐹_𝑧𝑧=𝐹_𝑧𝑧𝐹=0,_𝑧𝑧=𝑖𝐹₁₂=𝜕_𝑧𝐴_𝑧−𝜕_𝑧𝐴_𝑧−𝑖[𝐴_𝑧,𝐴_𝑧]_⋆=∶𝑖𝐵. (4.2)

Using these complex coordinates, the covariant derivatives of the Higgs fields are𝐷⋆𝜙=(𝜕−𝑖𝐴)⋆𝜙,𝐷⋆𝜙=(𝜕−𝑖𝐴)⋆𝜙,𝐷⋆𝜙=𝜕𝜙+𝑖𝜙⋆𝐴,𝐷⋆𝜙=𝜕𝜙+𝑖𝜙⋆𝐴. (4.3)

The vortex equations are defined by𝐷⋆𝜙=(𝜕−𝑖𝐴)⋆𝜙=0,𝐵+𝜙⋆𝜙−1=0. (4.4) We call solutions of these equations NC vortexes.

The formal expansions of the fields are𝜙=^∞_𝑛=0ℏ^𝑛𝜙_𝑛(𝑧,𝑧),𝐴=^∞_𝑛=0ℏ^𝑛𝐴_𝑛(𝑧,𝑧). (4.5) The 𝑘th order equations for (4.4) are−𝑖(𝜕𝐴_𝑘+𝜕𝐴_𝑘)+𝜙_𝑘𝜙₀+𝜙₀𝜙_𝑘−𝛿_𝑘0+𝐶_𝑘(𝑧,𝑧)=0,𝜕𝜙_𝑘−𝑖𝐴_𝑘𝜙₀−𝑖𝐴₀𝜙_𝑘+𝐷_𝑘(𝑧,𝑧)=0. (4.6)(4.7) Here 𝐶_𝑘(𝑧,𝑧) is the coefficient of ℏ^𝑘 in −[𝐴,𝐴]_⋆+𝜙⋆𝜙−(𝜙_𝑘𝜙₀+𝜙₀𝜙_𝑘), so 𝐶_𝑘(𝑧,𝑧) is a function of {𝐴_𝑖,𝐴_𝑗,𝜙_𝑚,𝜙_𝑛∣0≤𝑖,𝑗,𝑚,𝑛≤𝑘−1}. Similarly, 𝐷_𝑘(𝑧,𝑧) is the coefficient of ℏ^𝑘 in −𝑖𝐴⋆𝜙−(−𝑖𝐴_𝑘𝜙₀−𝑖𝐴₀𝜙_𝑘) and a function of {𝐴_𝑖,𝐴_𝑗,𝜙_𝑚,𝜙_𝑛∣0≤𝑖,𝑗,𝑚,𝑛≤𝑘−1}.

In the case of 𝑘=0, (4.6) and (4.7) coincide with the commutative 𝑈(1) vortex equations 𝐷𝜙₀=(𝜕−𝑖𝐴₀)𝜙₀=0 and 𝐵₀+𝜙₀𝜙₀−1=0, where 𝐵₀=−𝑖(𝜕𝐴₀−𝜕𝐴₀). In the following, we consider the case that 𝐴₀,𝐴₀, and 𝜙₀ are smooth finite vortex solutions. We call it Taubes' vortex solution.

In the region 𝜙₀≠0, substituting (4.7) into (4.6) for 𝐴_𝑘 and 𝐴_𝑘, we get{𝜕𝜙₀𝜙²₀(𝜕𝜙_𝑘−𝑖𝐴₀𝜙_𝑘+𝐷_𝑘1)−𝜙₀(Δ𝜙_𝑘−𝑖𝜕𝐴₀𝜙_𝑘−𝑖𝐴₀𝜕𝜙_𝑘+𝜕𝐷_𝑘)}+{c.c.}+𝜙_𝑘𝜙₀+𝜙₀𝜙₀−𝛿_𝑘0+𝐶_𝑘=0. (4.8) Here {c.c.} is the complex conjugate of preceding terms and Δ=𝜕𝜕.

Setting𝜑_𝑘=𝜙_𝑘𝜙₀+𝜙_𝑘𝜙₀𝜙=2Re(^𝑘𝜙₀),𝑑_𝑘=𝐷_𝑘𝜙₀. (4.9) Equation (4.8) is simplified to(−Δ+|𝜙₀|²)𝜑_𝑘=𝐸_𝑘, (4.10) where𝐸_𝑘∶=−𝐶_𝑘+𝜕𝑑_𝑘−𝜕𝑑_𝑘. (4.11)

To show that there exists a unique NC vortex solution deformed from the Taubes' vortex solution, we consider the stationary Schrödinger equation(−Δ+𝑉(𝑥))𝑢(𝑥)=𝑓(𝑥) (4.12) in ℝ², where 𝑉(𝑥) is a real-valued 𝐶^∞ function. We impose the following assumptions for 𝑉(𝑥).(𝑎1)𝑉(𝑥)≥0, for all 𝑥⊂ℝ².(𝑎2) There exist 𝐾⊂ℝ² and ∃𝑐>0 such that 𝐾 is a compact set and for 𝑥∈ℝ²⧵𝐾, 𝑉(𝑥)≥𝑐.(𝑎3) There exist 𝑥₁,…,𝑥_𝑁∈ℝ² such that 𝑉(𝑥_𝑖)=0 and 𝑉(𝑥)>0 for 𝑥∉{𝑥₁,…,𝑥_𝑁}.(𝑎4) For any 𝛼=(𝛼₁,𝛼₂)∈ℤ²₊, There exists a positive constant 𝑐 such that |𝜕^𝛼_𝑥(𝑉−𝑐)|≤𝑐for any 𝑥∈ℝ².

Note that the system (4.10) satisfies the assumptions (𝑎1)–(𝑎4). We set𝐻_𝑙(𝑛)∶={𝑓∣‖𝑓‖∶=sup_𝑥∈ℝ2(1+|𝑥|^𝑛)|𝜕^𝛼_𝑥𝑓(𝑥)|<∞forany|𝛼|≤𝑙] (4.13) for 𝑛∈ℤ₊. Then we obtain the following theorem.

Theorem 4.1. Under the assumptions (𝑎1)–(𝑎4), there exists a unique solution 𝑢∈𝐻_𝑙(𝑛) of (4.12) for any 𝑓∈𝐻_𝑙(𝑛).

This theorem's proof is given by using standard techniques of Green's function [20].

Equation (4.10) is a particular example of (4.12). Theorem 4.1 and some asymptotic analysis derive the following theorem.

Theorem 4.2. Let 𝐴₀ and 𝜙₀ be a Taubes' vortex solution, in other words, (𝐴₀,𝜙₀) is a finite and smooth solution of the commutative vortex equations. Then there exists a unique solution (𝐴,𝜙) of the NC vortex equations (4.4) with 𝐴|_ℏ=0=𝐴₀,𝜙|_ℏ=0=𝜙₀, and its vortex number is preserved 1𝑑2𝜋²1𝑥𝐵=𝑑2𝜋²𝑥𝐵₀. (4.14)

The proof is given in [20].

5. Conclusions

We have reviewed developments for the last dozen years in NC instantons in ℝ⁴. The ADHM methods made great progress and broke ground to make strict solutions of the NC soliton equations. A lot of kinds of NC instanton solutions have been made by the ADHM method. Using the solutions and ADHM data, many aspects have been investigated. For example, topological charges, Dirac zero-modes, index theorems, and Green's functions in the NC ADHM instanton backgrounds. However, we could not understand the relation with commutative instantons and how instantons deform from commutative ones rigorously. In recent few years, the smooth NC deformation method has been investigated. For the smooth NC deformed instantons, many features are clarified. For example, the instanton charge, the number of the spinor zero-modes, and the index of the Dirac operator in the NC deformed instanton backgrounds coincide with the ones in commutative instanton backgrounds. The ADHM equations are derived from the NC deformed instantons and we find the ADHM equations coincide with the ones by Nekrasov and Schwarz. A one-to-one correspondence between smooth NC deformed instantons and the ADHM data are also obtained. Thus, about instantons in NC ℝ⁴, a lot of features have been investigated. The smooth NC deformation method is useful for other dimensional gauge theories. As an example, smooth deformations of vortexes are studied similarly. Their vortex numbers also coincide with the ones in commutative ℝ².

We have considered gauge theories in ℝ^𝑛. One of the essences to prove some theorems of NC instantons or NC vortexes is in infinity of size of the space. So, some of the theorems are changed when we consider finite size spaces. For example, topological charges are deformed under NC deformations of the spaces and they depend on the NC parameters in general [18]. The generic investigations of such changes from the point of view of smooth NC deformations are left for future subjects. Most NC instantons or some other solitons in gauge theories are still in deep mist.

Acknowledgment

The author was supported by KAKENHI (20740049, Grant-in-Aid for Young Scientists (B)).