# Recent Developments in Instantons in Noncommutativeℝ4

Authors: Akifumi Sako
Journal: Advances in Mathematical Physics
 (2010)

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## Abstract

We review recent developments in noncommutative deformations of instantons
inℝ4. 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.


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## Body

## 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 by(1.1)F+=12(1+*)F=0,
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=ℂk and W=ℂN. 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 Section2, 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 NCU(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 operatorsxμ(μ=1,2,3,4) satisfying [xμ,xν]=iθμν, where θ is a skew symmetric real valued matrix and we call θμν NC parameter. We set the noncommutativity of the space to the self-dual case of θ12=-ζ1, θ34=-ζ2, and the other θμν=0 for convenience. By transformations of coordinates xμ, the NC parameters are possible to be put in this form in general. Here we introduce complex coordinate operators(2.1)z1=12(x1+ix2),z2=12(x3+ix4).
Then the commutation relations become(2.2)[z1,z¯1]=-ζ1,[z2,z¯2]=-ζ2,othersarezero.
We define creation and annihilation operators by(2.3)cα†=zαζα,cα=z¯αζα,(α=1,2);
then they satisfy(2.4)[cα,cα†]=1,[cα,cβ]=[cα†,cβ†]=0(α,β=1,2).
The Fock space ℋ on which the creation and annihilation operators (2.4) act is spanned by the Fock state(2.5)|n1,n2〉=(c1†)n1(c2†)n2n1!n2!|0,0〉,
with(2.6)c1|n1,n2〉=n1|n1-1,n2〉,c1†|n1,n2〉=n1+1|n1+1,n2〉,c2|n1,n2〉=n2|n1,n2-1〉,c2†|n1,n2〉=n2+1|n1,n2+1〉,
where n1 and n2 are the occupation number. The number operators are also defined by(2.7)n̂α=cα†cα,N̂=n̂1+n̂2,
which act on the Fock states as(2.8)n̂α|n1,n2〉=nα|n1,n2〉,N̂|n1,n2〉=(n1+n2)|n1,n2〉.
In the operator representation, derivatives of a function f are defined by(2.9)∂αf(z)=[∂̂α,f(z)],∂α¯f(z)=[∂̂α¯,f(z)],
where ∂̂α=z¯α/ζα and ∂̂α¯=-zα/ζα which satisfy [∂̂α,∂̂α¯]=-1/ζα. The integral on NC ℝ4 is defined by the standard trace in the operator representation,(2.10)∫d4x=∫d4z=(2π)2ζ1ζ2Trℋ.
Note that Trℋ represents the trace over the Fock space whereas the trace over the gauge group is denoted by trU(N).


### 2.2. Noncommutative ADHM Instantons

Let us consider theU(N) Yang-Mills theory on NC ℝ4. Let M be a projective module over the algebra that is generated by the operator xμ.In the NC space, the Yang-Mills connection is defined byDμψ=-ψ∂̂μ+D̂μψ, where ψ is a matter field in fundamental representation type and D̂μ∈End(M) are anti-Hermitian gauge fields [22–24]. The relation between D̂μ and usual gauge connection Aμ is D̂μ=-iθμνxν+Aμ, where θμν is an inverse matrix of θμν. In our notation of the complex coordinates (2.1) and (2.2), the curvature is given as(2.11)Fαα¯=1ζα+[D̂α,D̂α¯],Fαβ¯=[D̂α,D̂β¯](α≠β).
Note that there is a constant term originated with the noncommutativity in Fαα¯. Instanton solutions satisfy the antiself-duality condition F=-*F. These conditions are rewritten in the complex coordinates as(2.12)F11¯=-F22¯,F12=F1¯2¯=0.
In the commutative spaces, instantons are classified by the topological charge Q=(1/8π2)∫trU(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 NCℝ4 is looking for B1,B2∈End(ℂk), I∈Hom(ℂn,ℂk), and J∈Hom(ℂk,ℂn) which satisfy the deformed ADHM equations(2.13)[B1,B1†]+[B2,B2†]+II†-J†J=ζ1+ζ2,(2.14)[B1,B2]+IJ=0.
We call -k “instanton number” in this article. In the previous section, we denote V as the vector space ℂk. 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:ℂk⊕ℂk⊕ℂn→ℂk⊕ℂk by(2.15)𝒟†=(τσ†),τ=(B2-z2,B1-z1,I)=(B2-ζ2c2†,B1-ζ1c1†,I),σ†=(-B1†+z¯1,B2†-z¯2,J†)=(-B1†+ζ1c1,B2†-ζ2c2,J†).
The ADHM equations (2.13) and (2.14) are replaced by(2.16)ττ†=σ†σ≡□,τσ=0.Let us denote byΨ:ℂn→ℂk⊕ℂk⊕ℂn the solution to the following equation:(2.17)𝒟†Ψa=0(a=1,…,n),Ψ†aΨb=δab.Theorem 2.1.
LetΨa be orthonormal zero-modes defined in (2.17). Then NC U(N) instanton Aμ with instanton number -k is obtained by
(2.18)Aμ=Ψ†∂μΨ=-iΨ†θμν[xν,Ψ].
Here θμν is inverse of θμν, that is, θμνθνρ=δμρ.Proof.
The curvature two-form determined by this connection is given as follows.(2.19)F=dA+A∧A=d(Ψ†dΨ)+(Ψ†dΨ)∧(Ψ†dΨ)=dΨ†∧dΨ-(dΨ)ΨΨ†∧dΨ=dΨ†(1-ΨΨ†)∧dΨ.
Here we use dΨ†Ψ+Ψ†dΨ=0 that follows from the differentiating of (2.17). Note that
(2.20)ΨΨ†=I-𝒟1𝒟†𝒟𝒟†,
since
(2.21)I=(𝒟Ψ)(𝒟Ψ)-1((𝒟Ψ)†)-1(𝒟Ψ)†=(𝒟Ψ)(𝒟†𝒟001)-1(𝒟Ψ)†=𝒟1𝒟†𝒟𝒟†+ΨΨ†.
From (2.19) and (2.20),
(2.22)F=dΨ†(𝒟1𝒟†𝒟𝒟†)∧dΨ=Ψ†(d𝒟)∧1𝒟†𝒟(d𝒟†)Ψ,
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
(2.23)∂μ𝒟†=12(-σ¯μ0),∂μ𝒟=12(-σμ0).
Here, we define σμ and σ¯μ by
(2.24)(σ1,σ2,σ3,σ4):=(-iτ1,-iτ2,-iτ3,12×2),(σ¯1,σ¯2,σ¯3,σ¯4):=(iτ1,iτ2,iτ3,12×2),
where τi are the Pauli matrices and 12×2 is an identity matrix of degree 2. Note that 𝒟†𝒟=(□00□) owing to (2.16), and 𝒟†𝒟 and its inverse commute with σμ. Then we find (2.22) is in proportion to
(2.25)σμσ¯νdxμ∧dxν.σμσ¯ν-σνσ¯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 coordinatezα, NC instanton connections are given by(2.26)D̂α=1ζαΨ†z¯αΨ,D̂α¯=-1ζαΨ†zαΨ.One of the most important feature to understand the origin of the instanton charges is existence of zero-modes ofΨΨ†.Theorem 2.2 (Zero-mode ofΨΨ†).
Suppose thatΨ and Ψ† are given as above. The vector |v〉∈(ℂk⊕ℂk⊕ℂn)⊗ℋ satisfying
(2.27)ΨΨ†|v〉=〈v|ΨΨ†=0,|v〉≠0
is said to be a zero-mode of ΨΨ†. The zero-modes are given by following three types:
(2.28)|v1〉=((-B1+ζ1c1)|u〉(B2-ζ2c2)|u〉J|u〉),|v2〉=((B2†-ζ2c2†)|u′〉(B1†-ζ1c1†)|u′〉I†|u′〉),|v0〉=((exp∑αBα†cα†)|0,0〉v0i(exp∑αBα†cα†)|0,0〉v0i0).
Here |u〉 (|u′〉) is some element of ℂk⊗ℋ (i.e., |u〉 is expressed with the coefficients uinm∈ℂ as |u〉=∑i∑n,muinm|n,m〉ei, where ei is a base of k-dim vector space). v0i 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 commutativeℝ4, 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:(2.29)B1=1⋯k1⋮k(B111⋯B1k1⋮⋱⋮Bk11⋯Bkk1),B2=1⋯k1⋮k(B112⋯B1k2⋮⋱⋮Bk12⋯Bkk2),I=112⋮k(I1I2⋮Ik),J=0.
We define βα, cα† and cα by(2.30)Bα=ζαβα(α=1,2).
We introduce Δ̂ as(2.31)Δ̂=∑αζα(βα-cα†)(βα†-cα),
and we define a projection operator P as a projection onto 0-eigenstates of Δ̂ by(2.32)P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI,
where(2.33)G=〈0,0|e∑αβαcαII†e∑αβα†cα†|0,0〉.
We define shift operators S and S† and a operator Λ by(2.34)SS†=1,S†S=1-P,Λ=1+I†1Δ̂I.Theorem 2.3 (Nekrasov).
U(1) instantons are given by
(2.35)Dα=-1ζαSΛ-1/2cαΛ1/2S†,Dα¯=1ζαSΛ1/2cα†Λ-1/2S†.Proof.
At first, we check that the inverse ofΔ̂ in (2.34) is well defined. Δ̂ has k zero-modes:
(2.36)e∑αβα†cα†|0,0〉⊗ei(i=1,…,k)
which satisfy Δ̂e∑αβα†cα†|0,0〉⊗ei=0. Here ei=(δ1i,δ2,i,…,δki)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
(2.37)Ψ=(ψ+ψ-ξ),ψ±∈V⊗ℋ,ξ∈ℋ.
The orthonormalization condition is expressed as
(2.38)ψ+†ψ++ψ-†ψ-+ξ†ξ=1.
We put anzats for the solution by
(2.39)ψ+=-ζ2(β2†-c2)v,ψ-=ζ1(β1†-c1)v,
and substitute them into 𝒟†Ψ=0; then we get
(2.40)Δ̂v+Iξ=0.
The orthonormalization condition is rewritten as
(2.41)v†Δ̂v+ξ†ξ=1.
If there exist the inverse of Δ̂,
(2.42)v=-1Δ̂Iξ.
0-eigenstates of Δ̂ are (2.36) and we define the projection operator to project out the 0-eigenstates by
(2.43)P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI.
Shift operators S,S† satisfying
(2.44)SS†=1,S†S=1-P
are determined by the definition of P. Then the inverse of Δ̂ is well defined at the left side of S† or the right side of S.
Using the orthonormalization condition, we obtain(2.45)ξ=Λ-1/2S†,Λ=1+I†1Δ̂I.
Through these processes, Ψ is determined by the ADHM data, and after substituting this Ψ into (2.26) we obtain the instantons.
(2.46)Dα=-1ξαψ†z¯̂αψ=-1ξαSΛ-1/2(I†1Δ̂Δ̂z¯̂α1Δ̂I-z¯̂α)Λ-1/2S†=-1ξαSΛ-1/2cαΛ1/2S†.Dα¯ 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 Δ̂-1.As an example, let us construct an NCU(1) multi-instanton having concrete expressions with the instanton number -k [31, 32], which is made from the ADHM data:(2.47)B1=∑l=1k-1lζelel+1†=ζ(010⋯⋯00020⋯0⋮⋱⋱⋱⋮0⋯⋯0k-200⋯⋯0k-10⋯⋯0),B2=0,I=kζek=ζ(0⋮0k),J=0,
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 derives(2.48)P=∑n1=0k-1|n1,0〉〈n1,0|.
To construct an instanton, it is necessary to obtain Δ̂ or Λ. By definition,(2.49)Δ̂(k)=ζ1n̂1+ζ2n̂2+ζ∑i=1k-1ieiei†-ζ1ζ∑i=1k-1i{c1eiei+1†+c1†ei+1ei†},Λ(k)=1+ζkΔ̂kk-1(k).Δ̂ and Λ depend on k, so we denote them Δ̂(k) and Λ(k), respectively. Δ̂kk-1(k) is (k,k) entry of matrix Δ̂-1(k). To obtain Δ̂kk-1(k), it is enough to calculate the kth row vector of Δ̂-1(k). The kth row vector of Δ̂-1(k) is determined by Δ̂-1Δ̂=1. We denote the kth row vector of Δ̂-1(k) by (u1,…,uk), that is, Δ̂ki-1(k)=ui. Then, we obtain the following recurrence equation from the kth row of Δ̂-1Δ̂=1(2.50)u2c1†-1θ̃u1(1+θ̃n̂1+(1-θ̃)n̂2)=0,iui+1c1†-1θ̃ui(i+θ̃n̂1+(1-θ̃)n̂2)+i-1ui-1c1=0(1≤i≤k-2),
where θ̃=ζ1/(ζ1+ζ2). We change variables as (2.51)ui=wi-1c1†(k-i)(i-1)!;
then we can rewrite the above recurrence relation by wi as(2.52)w1-1θ̃(1+θ̃(n̂1-k+1)-(1-θ̃)n̂2)w0=0,wi+1-{i(1θ̃+θ̃)+1θ̃(θ̃(n̂1-k)+(1-θ̃)n̂2)+(1θ̃+θ̃)}wi+i((i-1)n̂1-k+2)wi-1=0,(2≤i≤k-1).Note thatn̂1 and n̂2 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) by(2.53)F(t;k)=ef(t)(1-at)α=∑i=0∞wii!ti,f(t)=∫dtct(1-at)(1-bt)=c2ab{ln(1-(a+b)t+abt2)+a+bDln|2abt-(a+b)-D2abt-(a+b)+D|},
where a, b, c, and α are real parameter determined by the request that wi satisfy (2.52), and D=(a-b)2. From the differentiation of F(t;k), we obtain(2.54)(1-at)(1-bt)∑i=1∞w1(i-1)!ti-1={-aα+(c+abα)t}∑i=0∞wii!ti,
and we find that wi satisfy the following relation:(2.55)wi+1-(i(a+b)-aα)wi+i(ab(i-1)-c-abα)wi-1=0.
From (2.52) and (2.55), we obtain(2.56)a=θ̃or1θ̃,b=1a,α=-h(θ̃,n1,n2)a,c=-n1+k-2+h(θ̃,n1,n2)a,
where(2.57)h(θ̃,n1,n2)=1θ̃{θ̃(n1-k)+(1-θ̃)n2}+θ̃+1θ̃.
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 wi, and Δkk-1(k) is determined as (2.58)Δkk-1(k)=uk=(1+ζ1ζk-1(k-2)!wk-2n̂1)(ζ1n̂1+ζ2n̂2)-1.
Using them, G, P, S, and Λ are determined as(2.59)G=ζk!∑i=1k{i!(k-i)!θ̃k-i}-1eiei†,P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI=∑n1=0k-1|n1,0〉〈n1,0|,S†=∑n1=0∞|n1+k,0〉〈n1,0|+∑n1=0∞∑n2=1∞|n1,n2〉〈n1,n2|,Λ=1+ζkuk.
Finally we obtain instanton gauge fields with instanton number -k as(2.60)D1=1ζ1∑n1=0∞∑n2=0∞|n1,n2〉〈n1+1,n2|d1(n1,n2;k),D2=1ζ2{∑n1=0∞|n1,0〉〈n1+k,1|d2(n1,0;k)+∑n1=0∞∑n2=1∞|n1,n2〉〈n1,n2+1|d2(n1,n2;k)},
where(2.61)d1(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,(n2≠0),d2(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).
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)2ζ1ζ2Trℋ, but it is conditionally convergent in general. In [25, 26], we introduce cut-off NC 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(2.62)|n1,n2〉(n1=0,…,N1(n2),n2=0,…,N2(n1)),
where N1(n2)(N2(n1)) is a function of n2(n1) and we suppose that the length of the boundary is order NC≫k, that is, N1(n2)≈N2(n1)≈NC≫k.Using this cut-off (boundary), we define the instanton charge by(2.63)Q=limNC→∞QNC,QNC=ζ2∑n1=0∑n2=0N1(n1)〈n1,n2|(F11¯F22¯-F12¯F21¯)|n1,n2〉.
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 chargeQNC:(2.64)-trU(N)TrNC(12[Ψ†c2†Ψ,Ψ†c2Ψ]+12[Ψ†c1†Ψ,Ψ†c1Ψ]).
We denote TrNC as trace over some finite domain of Fock space characterized by NC which is the length of the Fock space boundary. Using the Stokes' like theorem in [25], only trace over the boundary is left, then TrNC[Ψ†c2†Ψ,Ψ†c2Ψ] becomes(2.65)-trU(N)∑boundary(N2(n1)+1)=-trU(N)TrNC1-k.
The same value is obtained from TrN[Ψ†c1†Ψ,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 get(2.66)QN=-k+O(N-1/2),Q=limN→∞QN=-k.Theorem 2.4 (Instanton number).
Consider U(N) gauge theory on NCℝ4 with self-dual θμν. 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.


## 2.1. Notations for the Fock Space Formalism

Let us consider coordinate operatorsxμ(μ=1,2,3,4) satisfying [xμ,xν]=iθμν, where θ is a skew symmetric real valued matrix and we call θμν NC parameter. We set the noncommutativity of the space to the self-dual case of θ12=-ζ1, θ34=-ζ2, and the other θμν=0 for convenience. By transformations of coordinates xμ, the NC parameters are possible to be put in this form in general. Here we introduce complex coordinate operators(2.1)z1=12(x1+ix2),z2=12(x3+ix4).
Then the commutation relations become(2.2)[z1,z¯1]=-ζ1,[z2,z¯2]=-ζ2,othersarezero.
We define creation and annihilation operators by(2.3)cα†=zαζα,cα=z¯αζα,(α=1,2);
then they satisfy(2.4)[cα,cα†]=1,[cα,cβ]=[cα†,cβ†]=0(α,β=1,2).
The Fock space ℋ on which the creation and annihilation operators (2.4) act is spanned by the Fock state(2.5)|n1,n2〉=(c1†)n1(c2†)n2n1!n2!|0,0〉,
with(2.6)c1|n1,n2〉=n1|n1-1,n2〉,c1†|n1,n2〉=n1+1|n1+1,n2〉,c2|n1,n2〉=n2|n1,n2-1〉,c2†|n1,n2〉=n2+1|n1,n2+1〉,
where n1 and n2 are the occupation number. The number operators are also defined by(2.7)n̂α=cα†cα,N̂=n̂1+n̂2,
which act on the Fock states as(2.8)n̂α|n1,n2〉=nα|n1,n2〉,N̂|n1,n2〉=(n1+n2)|n1,n2〉.
In the operator representation, derivatives of a function f are defined by(2.9)∂αf(z)=[∂̂α,f(z)],∂α¯f(z)=[∂̂α¯,f(z)],
where ∂̂α=z¯α/ζα and ∂̂α¯=-zα/ζα which satisfy [∂̂α,∂̂α¯]=-1/ζα. The integral on NC ℝ4 is defined by the standard trace in the operator representation,(2.10)∫d4x=∫d4z=(2π)2ζ1ζ2Trℋ.
Note that Trℋ represents the trace over the Fock space whereas the trace over the gauge group is denoted by trU(N).


## 2.2. Noncommutative ADHM Instantons

Let us consider theU(N) Yang-Mills theory on NC ℝ4. Let M be a projective module over the algebra that is generated by the operator xμ.In the NC space, the Yang-Mills connection is defined byDμψ=-ψ∂̂μ+D̂μψ, where ψ is a matter field in fundamental representation type and D̂μ∈End(M) are anti-Hermitian gauge fields [22–24]. The relation between D̂μ and usual gauge connection Aμ is D̂μ=-iθμνxν+Aμ, where θμν is an inverse matrix of θμν. In our notation of the complex coordinates (2.1) and (2.2), the curvature is given as(2.11)Fαα¯=1ζα+[D̂α,D̂α¯],Fαβ¯=[D̂α,D̂β¯](α≠β).
Note that there is a constant term originated with the noncommutativity in Fαα¯. Instanton solutions satisfy the antiself-duality condition F=-*F. These conditions are rewritten in the complex coordinates as(2.12)F11¯=-F22¯,F12=F1¯2¯=0.
In the commutative spaces, instantons are classified by the topological charge Q=(1/8π2)∫trU(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 NCℝ4 is looking for B1,B2∈End(ℂk), I∈Hom(ℂn,ℂk), and J∈Hom(ℂk,ℂn) which satisfy the deformed ADHM equations(2.13)[B1,B1†]+[B2,B2†]+II†-J†J=ζ1+ζ2,(2.14)[B1,B2]+IJ=0.
We call -k “instanton number” in this article. In the previous section, we denote V as the vector space ℂk. 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:ℂk⊕ℂk⊕ℂn→ℂk⊕ℂk by(2.15)𝒟†=(τσ†),τ=(B2-z2,B1-z1,I)=(B2-ζ2c2†,B1-ζ1c1†,I),σ†=(-B1†+z¯1,B2†-z¯2,J†)=(-B1†+ζ1c1,B2†-ζ2c2,J†).
The ADHM equations (2.13) and (2.14) are replaced by(2.16)ττ†=σ†σ≡□,τσ=0.Let us denote byΨ:ℂn→ℂk⊕ℂk⊕ℂn the solution to the following equation:(2.17)𝒟†Ψa=0(a=1,…,n),Ψ†aΨb=δab.Theorem 2.1.
LetΨa be orthonormal zero-modes defined in (2.17). Then NC U(N) instanton Aμ with instanton number -k is obtained by
(2.18)Aμ=Ψ†∂μΨ=-iΨ†θμν[xν,Ψ].
Here θμν is inverse of θμν, that is, θμνθνρ=δμρ.Proof.
The curvature two-form determined by this connection is given as follows.(2.19)F=dA+A∧A=d(Ψ†dΨ)+(Ψ†dΨ)∧(Ψ†dΨ)=dΨ†∧dΨ-(dΨ)ΨΨ†∧dΨ=dΨ†(1-ΨΨ†)∧dΨ.
Here we use dΨ†Ψ+Ψ†dΨ=0 that follows from the differentiating of (2.17). Note that
(2.20)ΨΨ†=I-𝒟1𝒟†𝒟𝒟†,
since
(2.21)I=(𝒟Ψ)(𝒟Ψ)-1((𝒟Ψ)†)-1(𝒟Ψ)†=(𝒟Ψ)(𝒟†𝒟001)-1(𝒟Ψ)†=𝒟1𝒟†𝒟𝒟†+ΨΨ†.
From (2.19) and (2.20),
(2.22)F=dΨ†(𝒟1𝒟†𝒟𝒟†)∧dΨ=Ψ†(d𝒟)∧1𝒟†𝒟(d𝒟†)Ψ,
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
(2.23)∂μ𝒟†=12(-σ¯μ0),∂μ𝒟=12(-σμ0).
Here, we define σμ and σ¯μ by
(2.24)(σ1,σ2,σ3,σ4):=(-iτ1,-iτ2,-iτ3,12×2),(σ¯1,σ¯2,σ¯3,σ¯4):=(iτ1,iτ2,iτ3,12×2),
where τi are the Pauli matrices and 12×2 is an identity matrix of degree 2. Note that 𝒟†𝒟=(□00□) owing to (2.16), and 𝒟†𝒟 and its inverse commute with σμ. Then we find (2.22) is in proportion to
(2.25)σμσ¯νdxμ∧dxν.σμσ¯ν-σνσ¯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 coordinatezα, NC instanton connections are given by(2.26)D̂α=1ζαΨ†z¯αΨ,D̂α¯=-1ζαΨ†zαΨ.One of the most important feature to understand the origin of the instanton charges is existence of zero-modes ofΨΨ†.Theorem 2.2 (Zero-mode ofΨΨ†).
Suppose thatΨ and Ψ† are given as above. The vector |v〉∈(ℂk⊕ℂk⊕ℂn)⊗ℋ satisfying
(2.27)ΨΨ†|v〉=〈v|ΨΨ†=0,|v〉≠0
is said to be a zero-mode of ΨΨ†. The zero-modes are given by following three types:
(2.28)|v1〉=((-B1+ζ1c1)|u〉(B2-ζ2c2)|u〉J|u〉),|v2〉=((B2†-ζ2c2†)|u′〉(B1†-ζ1c1†)|u′〉I†|u′〉),|v0〉=((exp∑αBα†cα†)|0,0〉v0i(exp∑αBα†cα†)|0,0〉v0i0).
Here |u〉 (|u′〉) is some element of ℂk⊗ℋ (i.e., |u〉 is expressed with the coefficients uinm∈ℂ as |u〉=∑i∑n,muinm|n,m〉ei, where ei is a base of k-dim vector space). v0i 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 commutativeℝ4, 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:(2.29)B1=1⋯k1⋮k(B111⋯B1k1⋮⋱⋮Bk11⋯Bkk1),B2=1⋯k1⋮k(B112⋯B1k2⋮⋱⋮Bk12⋯Bkk2),I=112⋮k(I1I2⋮Ik),J=0.
We define βα, cα† and cα by(2.30)Bα=ζαβα(α=1,2).
We introduce Δ̂ as(2.31)Δ̂=∑αζα(βα-cα†)(βα†-cα),
and we define a projection operator P as a projection onto 0-eigenstates of Δ̂ by(2.32)P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI,
where(2.33)G=〈0,0|e∑αβαcαII†e∑αβα†cα†|0,0〉.
We define shift operators S and S† and a operator Λ by(2.34)SS†=1,S†S=1-P,Λ=1+I†1Δ̂I.Theorem 2.3 (Nekrasov).
U(1) instantons are given by
(2.35)Dα=-1ζαSΛ-1/2cαΛ1/2S†,Dα¯=1ζαSΛ1/2cα†Λ-1/2S†.Proof.
At first, we check that the inverse ofΔ̂ in (2.34) is well defined. Δ̂ has k zero-modes:
(2.36)e∑αβα†cα†|0,0〉⊗ei(i=1,…,k)
which satisfy Δ̂e∑αβα†cα†|0,0〉⊗ei=0. Here ei=(δ1i,δ2,i,…,δki)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
(2.37)Ψ=(ψ+ψ-ξ),ψ±∈V⊗ℋ,ξ∈ℋ.
The orthonormalization condition is expressed as
(2.38)ψ+†ψ++ψ-†ψ-+ξ†ξ=1.
We put anzats for the solution by
(2.39)ψ+=-ζ2(β2†-c2)v,ψ-=ζ1(β1†-c1)v,
and substitute them into 𝒟†Ψ=0; then we get
(2.40)Δ̂v+Iξ=0.
The orthonormalization condition is rewritten as
(2.41)v†Δ̂v+ξ†ξ=1.
If there exist the inverse of Δ̂,
(2.42)v=-1Δ̂Iξ.
0-eigenstates of Δ̂ are (2.36) and we define the projection operator to project out the 0-eigenstates by
(2.43)P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI.
Shift operators S,S† satisfying
(2.44)SS†=1,S†S=1-P
are determined by the definition of P. Then the inverse of Δ̂ is well defined at the left side of S† or the right side of S.
Using the orthonormalization condition, we obtain(2.45)ξ=Λ-1/2S†,Λ=1+I†1Δ̂I.
Through these processes, Ψ is determined by the ADHM data, and after substituting this Ψ into (2.26) we obtain the instantons.
(2.46)Dα=-1ξαψ†z¯̂αψ=-1ξαSΛ-1/2(I†1Δ̂Δ̂z¯̂α1Δ̂I-z¯̂α)Λ-1/2S†=-1ξαSΛ-1/2cαΛ1/2S†.Dα¯ 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 Δ̂-1.As an example, let us construct an NCU(1) multi-instanton having concrete expressions with the instanton number -k [31, 32], which is made from the ADHM data:(2.47)B1=∑l=1k-1lζelel+1†=ζ(010⋯⋯00020⋯0⋮⋱⋱⋱⋮0⋯⋯0k-200⋯⋯0k-10⋯⋯0),B2=0,I=kζek=ζ(0⋮0k),J=0,
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 derives(2.48)P=∑n1=0k-1|n1,0〉〈n1,0|.
To construct an instanton, it is necessary to obtain Δ̂ or Λ. By definition,(2.49)Δ̂(k)=ζ1n̂1+ζ2n̂2+ζ∑i=1k-1ieiei†-ζ1ζ∑i=1k-1i{c1eiei+1†+c1†ei+1ei†},Λ(k)=1+ζkΔ̂kk-1(k).Δ̂ and Λ depend on k, so we denote them Δ̂(k) and Λ(k), respectively. Δ̂kk-1(k) is (k,k) entry of matrix Δ̂-1(k). To obtain Δ̂kk-1(k), it is enough to calculate the kth row vector of Δ̂-1(k). The kth row vector of Δ̂-1(k) is determined by Δ̂-1Δ̂=1. We denote the kth row vector of Δ̂-1(k) by (u1,…,uk), that is, Δ̂ki-1(k)=ui. Then, we obtain the following recurrence equation from the kth row of Δ̂-1Δ̂=1(2.50)u2c1†-1θ̃u1(1+θ̃n̂1+(1-θ̃)n̂2)=0,iui+1c1†-1θ̃ui(i+θ̃n̂1+(1-θ̃)n̂2)+i-1ui-1c1=0(1≤i≤k-2),
where θ̃=ζ1/(ζ1+ζ2). We change variables as (2.51)ui=wi-1c1†(k-i)(i-1)!;
then we can rewrite the above recurrence relation by wi as(2.52)w1-1θ̃(1+θ̃(n̂1-k+1)-(1-θ̃)n̂2)w0=0,wi+1-{i(1θ̃+θ̃)+1θ̃(θ̃(n̂1-k)+(1-θ̃)n̂2)+(1θ̃+θ̃)}wi+i((i-1)n̂1-k+2)wi-1=0,(2≤i≤k-1).Note thatn̂1 and n̂2 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) by(2.53)F(t;k)=ef(t)(1-at)α=∑i=0∞wii!ti,f(t)=∫dtct(1-at)(1-bt)=c2ab{ln(1-(a+b)t+abt2)+a+bDln|2abt-(a+b)-D2abt-(a+b)+D|},
where a, b, c, and α are real parameter determined by the request that wi satisfy (2.52), and D=(a-b)2. From the differentiation of F(t;k), we obtain(2.54)(1-at)(1-bt)∑i=1∞w1(i-1)!ti-1={-aα+(c+abα)t}∑i=0∞wii!ti,
and we find that wi satisfy the following relation:(2.55)wi+1-(i(a+b)-aα)wi+i(ab(i-1)-c-abα)wi-1=0.
From (2.52) and (2.55), we obtain(2.56)a=θ̃or1θ̃,b=1a,α=-h(θ̃,n1,n2)a,c=-n1+k-2+h(θ̃,n1,n2)a,
where(2.57)h(θ̃,n1,n2)=1θ̃{θ̃(n1-k)+(1-θ̃)n2}+θ̃+1θ̃.
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 wi, and Δkk-1(k) is determined as (2.58)Δkk-1(k)=uk=(1+ζ1ζk-1(k-2)!wk-2n̂1)(ζ1n̂1+ζ2n̂2)-1.
Using them, G, P, S, and Λ are determined as(2.59)G=ζk!∑i=1k{i!(k-i)!θ̃k-i}-1eiei†,P=I†e∑αβα†cα†|0,0〉G-1〈0,0|e∑αβαcαI=∑n1=0k-1|n1,0〉〈n1,0|,S†=∑n1=0∞|n1+k,0〉〈n1,0|+∑n1=0∞∑n2=1∞|n1,n2〉〈n1,n2|,Λ=1+ζkuk.
Finally we obtain instanton gauge fields with instanton number -k as(2.60)D1=1ζ1∑n1=0∞∑n2=0∞|n1,n2〉〈n1+1,n2|d1(n1,n2;k),D2=1ζ2{∑n1=0∞|n1,0〉〈n1+k,1|d2(n1,0;k)+∑n1=0∞∑n2=1∞|n1,n2〉〈n1,n2+1|d2(n1,n2;k)},
where(2.61)d1(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,(n2≠0),d2(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).
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)2ζ1ζ2Trℋ, but it is conditionally convergent in general. In [25, 26], we introduce cut-off NC 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(2.62)|n1,n2〉(n1=0,…,N1(n2),n2=0,…,N2(n1)),
where N1(n2)(N2(n1)) is a function of n2(n1) and we suppose that the length of the boundary is order NC≫k, that is, N1(n2)≈N2(n1)≈NC≫k.Using this cut-off (boundary), we define the instanton charge by(2.63)Q=limNC→∞QNC,QNC=ζ2∑n1=0∑n2=0N1(n1)〈n1,n2|(F11¯F22¯-F12¯F21¯)|n1,n2〉.
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 chargeQNC:(2.64)-trU(N)TrNC(12[Ψ†c2†Ψ,Ψ†c2Ψ]+12[Ψ†c1†Ψ,Ψ†c1Ψ]).
We denote TrNC as trace over some finite domain of Fock space characterized by NC which is the length of the Fock space boundary. Using the Stokes' like theorem in [25], only trace over the boundary is left, then TrNC[Ψ†c2†Ψ,Ψ†c2Ψ] becomes(2.65)-trU(N)∑boundary(N2(n1)+1)=-trU(N)TrNC1-k.
The same value is obtained from TrN[Ψ†c1†Ψ,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 get(2.66)QN=-k+O(N-1/2),Q=limN→∞QN=-k.Theorem 2.4 (Instanton number).
Consider U(N) gauge theory on NCℝ4 with self-dual θμν. 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.


## 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)2ζ1ζ2Trℋ, but it is conditionally convergent in general. In [25, 26], we introduce cut-off NC 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(2.62)|n1,n2〉(n1=0,…,N1(n2),n2=0,…,N2(n1)),
where N1(n2)(N2(n1)) is a function of n2(n1) and we suppose that the length of the boundary is order NC≫k, that is, N1(n2)≈N2(n1)≈NC≫k.Using this cut-off (boundary), we define the instanton charge by(2.63)Q=limNC→∞QNC,QNC=ζ2∑n1=0∑n2=0N1(n1)〈n1,n2|(F11¯F22¯-F12¯F21¯)|n1,n2〉.
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 chargeQNC:(2.64)-trU(N)TrNC(12[Ψ†c2†Ψ,Ψ†c2Ψ]+12[Ψ†c1†Ψ,Ψ†c1Ψ]).
We denote TrNC as trace over some finite domain of Fock space characterized by NC which is the length of the Fock space boundary. Using the Stokes' like theorem in [25], only trace over the boundary is left, then TrNC[Ψ†c2†Ψ,Ψ†c2Ψ] becomes(2.65)-trU(N)∑boundary(N2(n1)+1)=-trU(N)TrNC1-k.
The same value is obtained from TrN[Ψ†c1†Ψ,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 get(2.66)QN=-k+O(N-1/2),Q=limN→∞QN=-k.Theorem 2.4 (Instanton number).
Consider U(N) gauge theory on NCℝ4 with self-dual θμν. 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 parameterℏ. 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(3.1)[xμ,xν]⋆=xμ⋆xν-xν⋆xμ=iθμν,μ,ν=1,2,…,4,
where (θμν) 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 by(3.2)f(x)⋆g(x):=f(x)exp(i2∂⟵μθμν∂⃗ν)g(x).
Here ∂⟵μ and ∂⃗ν are partial derivatives with respect to xμ for f(x) and to xν for g(x), respectively.The curvature two formF is defined by F:=(1/2)Fμνdxμ∧⋆dxν=dA+A∧⋆A, where ∧⋆ is defined by A∧⋆A:=(1/2)(Aμ⋆Aν)dxμ∧dxν.To consider smooth NC deformations, we introduce a parameterℏ and a fixed constant θ0μν<∞ with θμν=ℏθ0μν. We define the commutative limit by letting ℏ→0.Formally we expand the connection as(3.3)Aμ=∑l=0∞Aμ(l)ℏl.
Then,(3.4)Aμ⋆Aν=∑l,m,n=0∞ℏl+m+n1l!Aμ(m)(Δ¯)lAμ(n),Δ¯≡i2∂⟵μθ0μν∂⃗v.
We introduce the self-dual projection operator P by(3.5)P:=1+*2;Pμν,ρτ=14(δμρδντ-δνρδμτ+ϵμνρτ).
Then the instanton equation is given as(3.6)Pμν,ρτFρτ=0.
In the NC case, the lth order equation of (3.6) is given by(3.7)Pμν,ρτ(∂ρAτ(l)-∂τAρ(l)+i[Aρ(l),Aτ(0)]+i[Aρ(0),Aτ(l)]+Cρτ(l))=0,Cρτ(l):=∑(p;m,n)∈I(l)ℏp+m+n1p!(Aρ(m)(Δ¯)pAτ(n)-Aτ(m)(Δ¯)pAρ(n)),I(l)≡{(p;m,n)∈ℤ3∣p+m+n=l,p,m,n≥0,m≠l,n≠l}.
Note that the 0th order is the commutative instanton equation with solution Aμ(0) being a commutative instanton. The asymptotic behavior of commutative instanton Aμ(0) is given by(3.8)Aμ(0)=gdg-1+O(|x|-2),gdg-1=O(|x|-1),
where g∈G and G is a gauge group. (See, e.g., [2].) We introduce covariant derivatives associated to the commutative instanton connection by(3.9)Dμ(0)f:=∂μf+i[Aμ(0),f],DA(0)f:=df+A(0)∧f.
Using this, (3.7) is given by(3.10)Pμν,ρτ(Dρ(0)Aτ(l)-Dτ(0)Aρ(l)+Cρτ(l))=0.In the following, we fix a commutative instanton connectionA(0). We impose the following gauge fixing condition for A(l)(l≥1) [18, 42](3.11)A-A(0)=DA(0)*B,B∈Ω+2,
where DA(0)* is defined by(3.12)(DA(0)*)ρμνBμν=δρν∂μBμν-δρμ∂νBμν+iδρν[Aμ,Bμν]-δρμ[Aν,Bμν]=δρνD(0)μBμν-δρμD(0)νBμν.
We expand B in ℏ as we did with A. Then A(l)=DA(0)*B(l). In this gauge, using the fact that the A(0) is an anti-self-dual connection, (3.10) simplified to(3.13)2D(0)2B(l)μν+Pμν,ρτCρτ(l)=0,
where (3.14)D(0)2≡DA(0)ρDA(0)ρ.We consider the Green's function forD(0)2: (3.15)D(0)2G0(x,y)=δ(x-y),
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) as(3.16)B(l)μν=-12∫ℝ4G0(x,y)Pμν,ρτCρτ(l)(y)d4y
and the NC instanton A=∑A(l)ℏl is given by(3.17)A(l)=DA(0)*B(l).
In the following, we call NC instantons smoothly deformed from commutative instantons SNCD instantons. The asymptotic behavior of Green's function of D(0)2 is important, which is given by(3.18)G0(x,y)=O(|x-y|-2).We introduce the notationO′(|x|-m) as in [2]. If s is a function of ℝ4 which is O(|x|-m) as |x|→∞ and |D(0)ks|=O(|x|-m-k), then we denote this natural growth condition by s=O'(|x|-m).Theorem 3.1.
IfC(l)=O'(|x|-4), then B(k)=O'(|x|-2).We gave a proof of this theorem in [18].In our case,Cρτ(1)=O'(x-4) by (3.8), and so B(1)=O′(|x|-2), A(1)=O'(|x|-3) as A(l)=DA(0)*B(l). Repeating the argument l times, we get(3.19)|A(l)|<O′(|x|-3+ϵ),∀ϵ>0.


### 3.2. Instanton Charge

The instanton charge is defined by(3.20)Qℏ:=18π2∫trU(N)F∧⋆F.
We rewrite (3.20) as(3.21)18π2∫trU(N)d(A∧⋆dA+23A∧⋆A∧⋆A)+18π2∫trU(N)P⋆,
where(3.22)P⋆=13{F∧⋆A∧⋆A+2A∧⋆F∧⋆A+A∧⋆A∧⋆F+A∧⋆A∧⋆A∧⋆A}.∫trU(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∫trU(N)P⋆ are typically written as(3.23)∫ℝdtrU(N)(P∧⋆R-(-1)n(4-n)R∧⋆P),
where P and R are some n-form and (4-n)-form (n=0,…,4), respectively, and let P∧R be O(ℏk). The lowest order term in ℏ vanishes because of the cyclic symmetry of the trace, that is, ∫trU(N)(P∧R-(-1)n(4-n)R∧P)=0. The term of order ℏ is given by(3.24)i2∫ℝ4trU(N){ℏθ0μν(∂μP∧∂νR)}=i2∫ℝ4(n!(4-n)!)ϵμ1μ2μ3μ4trU(N)d{(*θ)∧(Pμ1⋯μndRμn+1⋯μ4)},
where *θ=ϵμνρτθρτdxμ∧dxν/4. These integrals are zero if Pμ1⋯μndRμn+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 ∫trU(N)P⋆=0.Because of the similar estimation, we found the other terms of∫trU(N)F∧⋆F-∫trU(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.
LetAμ(0) be a commutative instanton solution in ℝ4. There exists a formal NC instanton solution Aμ=∑l=0∞Aμ(l)ℏl (SNCD instanton) such that the instanton number Qℏ defined by (3.20) is independent of the NC parameter ℏ:
(3.25)18π2∫trU(N)F∧⋆F=18π2∫trU(N)F(0)∧F(0).


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

Dirac(-Weyl) operators𝒟A:Γ(S+⊗E)→Γ(S-⊗E) and 𝒟¯A:Γ(S-⊗E)→Γ(S+⊗E) are defined as(3.26)𝒟A:=σμDμ,𝒟¯A:=σ¯μDμ†.
Here, σμ and σ¯μ are defined by (2.24). Consider ℏ expansion of ψ∈Γ(S+⊗E) and ψ¯∈Γ(S-⊗E) as(3.27)ψ=∑n=0∞ℏnψ(n),ψ¯=∑n=0∞ℏnψ¯(n).
In [19], the zero-modes of 𝒟A and 𝒟¯A, which are defined by(3.28)𝒟A⋆ψ=0,𝒟¯A⋆ψ¯=0,
are investigated, and the following theorem is obtained.Theorem 3.3.
Let𝒟A and 𝒟¯A be the Dirac(-Weyl) operators for an SNCD instanton background with its instanton number -k. There is no zero-mode for 𝒟A⋆ψ=0, and there are k zero-modes for 𝒟¯A⋆ψ¯i=0(i=1,…,k) that are given as
(3.29)ψ¯i=∑n=0∞(∑j=1kan,ijηj)ℏn+O′(|x|-5+ϵ),ηj=O′(|x|-3),
where an,ij is a constant matrix and ηj 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 ofker𝒟¯A(0) is equal to k the instanton number (of opposite sign), and there exists k zero-mode ηi(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.
IfIndD0:=dimker𝒟A(0)-dimker𝒟¯A(0)=-k, then IndD:=dimker𝒟A-dimker𝒟¯A=-k.Next, we construct the Green's function ofΔA≡Dμ⋆Dμ,(3.30)ΔA⋆GA(x,y)=δ(x-y).We expand (3.18) by ℏ, for n>0, then ℏn order equation is given as(3.31)ΔA(0)GA(n)(x,y)+[ΔA∑0≤k<nℏkGA(k)(x,y)](n)=0,
where GA(n) is defined by GA(x,y)=∑k=0∞GA(k)ℏk. We solve them recursively(3.32)GA(n)(x,y)=∫d4wGA(0)(x,w)[ΔA∑0≤k<nℏkGA(k)(w,y)](n).
Note that GA(0)(x,w) was constructed in [43–45]. Using property of GA(0)(x,w) and A(n), we obtain the following decay condition in [19]:(3.33)GA(n)(x,y)=O′(|x|-3).


### 3.4. From an Instanton to the ADHM Equations

Let us see how to derive the ADHM equations from an SNCD instanton.Letψ¯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 defineTμ by(3.34)Tμ:=∫ℝ4d4x12(xμ⋆ψ¯†⋆ψ¯+ψ¯†⋆ψ¯⋆xμ).Next we introduce an asymptotically parallel sectiong-1S of S+⊗E by(3.35)ψ̃=-g-1Sx†|x|4+O′(|x|-4),
where x†:=σ¯μxμ and ψ̃:=tψ¯σ2. This t means transposing spinor suffixes.Using various properties and decay conditions ofA(n),GA(n),ψ¯(n), and theorems in the previous subsections, we finally obtain the following theorem.Theorem 3.5.
LetAμ be an SNCD instanton and ψ¯ the zero-mode of 𝒟¯A determined by Aμ as in Section 3.3. Let Tμ,S be constant matrices defined by (3.34) and (3.35), respectively. Then, they satisfy the ADHM equations:
(3.36)[Tμ,Tν]+=12tr(S†Sσ¯μν)-iθμν+1k×k.
Here σ¯μν:=(1/4)(σ¯μσν-σ¯νσμ) and 1k×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
(3.37)⋆xψ¯(x)ψ¯†(y)⋆y=⋆xδ(x-y)⋆y-⋆x𝒟A⋆xGA(x,y)⋆y𝒟¯⟵A⋆y.
From the definition of the Tμ,
(3.38)TμTν=∫ℝ4d4x∫ℝ4d4y(xμ⋆xψ¯†(x)⋆xψ¯(x))(ψ¯†(y)⋆yψ¯(y)⋆yyν).
Using Theorem 3.3, (3.37), and integration by parts, (3.38) becomes
(3.39)TμTν=∫ℝ4d4xxμ⋆ψ¯†⋆ψ¯⋆xν+∫S3dSxρ∫ℝ4d4y(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y𝒟¯⟵A⋆y(ψ¯(y)⋆yyν)-∫ℝ4d4x∫ℝ4d4y(ψ¯†(x)σμ)⋆xGA(x,y)⋆y𝒟¯⟵A⋆y(ψ¯(y)⋆yyν),
where dSxμ=|x|2xμdΩ and dΩ is the solid angle. The first term is deformed as follows.
(3.40)∫ℝ4d4xxμ⋆ψ¯†⋆ψ¯⋆xν=∫ℝ4d4x(ψ¯†⋆ψ¯⋆xν⋆xμ+[xμ,ψ¯†⋆ψ¯]⋆⋆xν+ψ¯†⋆ψ¯⋆[xμ,xν]⋆)=∫ℝ4d4x(ψ¯†⋆ψ¯⋆xν⋆xμ+iθμρ∂ρ(ψ¯†⋆ψ¯)⋆xν+iθμνψ¯†⋆ψ¯)=∫ℝ4d4xψ¯†⋆ψ¯⋆xν⋆xμ.
Here ψ¯=O′(|x|-3) given in Theorem 3.3 is used in the third equality. By integration by parts again, we get
(3.41)TμTν=∫ℝ4d4xψ¯†⋆ψ¯⋆xν⋆xμ(3.42)+∫S3dSxρ∫S3dSyτ(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y(σ¯τψ¯(y)⋆yyν)(3.43)-∫S3dSxρ∫ℝ4d4y(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y(σ¯νψ¯(y))(3.44)-∫ℝ4d4x∫S3dSyτ(ψ¯†(x)σμ)⋆xGA(x,y)⋆y(σ¯τψ¯(y)⋆yyν)(3.45)+∫ℝ4d4x∫ℝ4d4y(ψ¯†(x)σμ)⋆xGA(x,y)⋆y(σ¯νψ¯(y)).
Equations (3.42) and (3.44) vanish when Ry→∞, where Ry is a radius of Sy3. Equation (3.45) will vanish on the self-dual projection [Tμ,Tν]+:=Pμν,ρτ[Tρ,Tτ], because σμσ¯ν-σνσ¯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
(3.46)18tr(S†Sσ¯μσν),
where the trace tr is taken with respect to the spinor indices. In the [Tμ,Tν]+ combination, (3.41) becomes -iθμν+=-iPμν,ρτθρτ. 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 of(3.47)S†=(IJ†),Tμσ¯μ=(-B2-B1B1†-B2†),
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].


## 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(3.1)[xμ,xν]⋆=xμ⋆xν-xν⋆xμ=iθμν,μ,ν=1,2,…,4,
where (θμν) 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 by(3.2)f(x)⋆g(x):=f(x)exp(i2∂⟵μθμν∂⃗ν)g(x).
Here ∂⟵μ and ∂⃗ν are partial derivatives with respect to xμ for f(x) and to xν for g(x), respectively.The curvature two formF is defined by F:=(1/2)Fμνdxμ∧⋆dxν=dA+A∧⋆A, where ∧⋆ is defined by A∧⋆A:=(1/2)(Aμ⋆Aν)dxμ∧dxν.To consider smooth NC deformations, we introduce a parameterℏ and a fixed constant θ0μν<∞ with θμν=ℏθ0μν. We define the commutative limit by letting ℏ→0.Formally we expand the connection as(3.3)Aμ=∑l=0∞Aμ(l)ℏl.
Then,(3.4)Aμ⋆Aν=∑l,m,n=0∞ℏl+m+n1l!Aμ(m)(Δ¯)lAμ(n),Δ¯≡i2∂⟵μθ0μν∂⃗v.
We introduce the self-dual projection operator P by(3.5)P:=1+*2;Pμν,ρτ=14(δμρδντ-δνρδμτ+ϵμνρτ).
Then the instanton equation is given as(3.6)Pμν,ρτFρτ=0.
In the NC case, the lth order equation of (3.6) is given by(3.7)Pμν,ρτ(∂ρAτ(l)-∂τAρ(l)+i[Aρ(l),Aτ(0)]+i[Aρ(0),Aτ(l)]+Cρτ(l))=0,Cρτ(l):=∑(p;m,n)∈I(l)ℏp+m+n1p!(Aρ(m)(Δ¯)pAτ(n)-Aτ(m)(Δ¯)pAρ(n)),I(l)≡{(p;m,n)∈ℤ3∣p+m+n=l,p,m,n≥0,m≠l,n≠l}.
Note that the 0th order is the commutative instanton equation with solution Aμ(0) being a commutative instanton. The asymptotic behavior of commutative instanton Aμ(0) is given by(3.8)Aμ(0)=gdg-1+O(|x|-2),gdg-1=O(|x|-1),
where g∈G and G is a gauge group. (See, e.g., [2].) We introduce covariant derivatives associated to the commutative instanton connection by(3.9)Dμ(0)f:=∂μf+i[Aμ(0),f],DA(0)f:=df+A(0)∧f.
Using this, (3.7) is given by(3.10)Pμν,ρτ(Dρ(0)Aτ(l)-Dτ(0)Aρ(l)+Cρτ(l))=0.In the following, we fix a commutative instanton connectionA(0). We impose the following gauge fixing condition for A(l)(l≥1) [18, 42](3.11)A-A(0)=DA(0)*B,B∈Ω+2,
where DA(0)* is defined by(3.12)(DA(0)*)ρμνBμν=δρν∂μBμν-δρμ∂νBμν+iδρν[Aμ,Bμν]-δρμ[Aν,Bμν]=δρνD(0)μBμν-δρμD(0)νBμν.
We expand B in ℏ as we did with A. Then A(l)=DA(0)*B(l). In this gauge, using the fact that the A(0) is an anti-self-dual connection, (3.10) simplified to(3.13)2D(0)2B(l)μν+Pμν,ρτCρτ(l)=0,
where (3.14)D(0)2≡DA(0)ρDA(0)ρ.We consider the Green's function forD(0)2: (3.15)D(0)2G0(x,y)=δ(x-y),
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) as(3.16)B(l)μν=-12∫ℝ4G0(x,y)Pμν,ρτCρτ(l)(y)d4y
and the NC instanton A=∑A(l)ℏl is given by(3.17)A(l)=DA(0)*B(l).
In the following, we call NC instantons smoothly deformed from commutative instantons SNCD instantons. The asymptotic behavior of Green's function of D(0)2 is important, which is given by(3.18)G0(x,y)=O(|x-y|-2).We introduce the notationO′(|x|-m) as in [2]. If s is a function of ℝ4 which is O(|x|-m) as |x|→∞ and |D(0)ks|=O(|x|-m-k), then we denote this natural growth condition by s=O'(|x|-m).Theorem 3.1.
IfC(l)=O'(|x|-4), then B(k)=O'(|x|-2).We gave a proof of this theorem in [18].In our case,Cρτ(1)=O'(x-4) by (3.8), and so B(1)=O′(|x|-2), A(1)=O'(|x|-3) as A(l)=DA(0)*B(l). Repeating the argument l times, we get(3.19)|A(l)|<O′(|x|-3+ϵ),∀ϵ>0.


## 3.2. Instanton Charge

The instanton charge is defined by(3.20)Qℏ:=18π2∫trU(N)F∧⋆F.
We rewrite (3.20) as(3.21)18π2∫trU(N)d(A∧⋆dA+23A∧⋆A∧⋆A)+18π2∫trU(N)P⋆,
where(3.22)P⋆=13{F∧⋆A∧⋆A+2A∧⋆F∧⋆A+A∧⋆A∧⋆F+A∧⋆A∧⋆A∧⋆A}.∫trU(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∫trU(N)P⋆ are typically written as(3.23)∫ℝdtrU(N)(P∧⋆R-(-1)n(4-n)R∧⋆P),
where P and R are some n-form and (4-n)-form (n=0,…,4), respectively, and let P∧R be O(ℏk). The lowest order term in ℏ vanishes because of the cyclic symmetry of the trace, that is, ∫trU(N)(P∧R-(-1)n(4-n)R∧P)=0. The term of order ℏ is given by(3.24)i2∫ℝ4trU(N){ℏθ0μν(∂μP∧∂νR)}=i2∫ℝ4(n!(4-n)!)ϵμ1μ2μ3μ4trU(N)d{(*θ)∧(Pμ1⋯μndRμn+1⋯μ4)},
where *θ=ϵμνρτθρτdxμ∧dxν/4. These integrals are zero if Pμ1⋯μndRμn+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 ∫trU(N)P⋆=0.Because of the similar estimation, we found the other terms of∫trU(N)F∧⋆F-∫trU(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.
LetAμ(0) be a commutative instanton solution in ℝ4. There exists a formal NC instanton solution Aμ=∑l=0∞Aμ(l)ℏl (SNCD instanton) such that the instanton number Qℏ defined by (3.20) is independent of the NC parameter ℏ:
(3.25)18π2∫trU(N)F∧⋆F=18π2∫trU(N)F(0)∧F(0).


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

Dirac(-Weyl) operators𝒟A:Γ(S+⊗E)→Γ(S-⊗E) and 𝒟¯A:Γ(S-⊗E)→Γ(S+⊗E) are defined as(3.26)𝒟A:=σμDμ,𝒟¯A:=σ¯μDμ†.
Here, σμ and σ¯μ are defined by (2.24). Consider ℏ expansion of ψ∈Γ(S+⊗E) and ψ¯∈Γ(S-⊗E) as(3.27)ψ=∑n=0∞ℏnψ(n),ψ¯=∑n=0∞ℏnψ¯(n).
In [19], the zero-modes of 𝒟A and 𝒟¯A, which are defined by(3.28)𝒟A⋆ψ=0,𝒟¯A⋆ψ¯=0,
are investigated, and the following theorem is obtained.Theorem 3.3.
Let𝒟A and 𝒟¯A be the Dirac(-Weyl) operators for an SNCD instanton background with its instanton number -k. There is no zero-mode for 𝒟A⋆ψ=0, and there are k zero-modes for 𝒟¯A⋆ψ¯i=0(i=1,…,k) that are given as
(3.29)ψ¯i=∑n=0∞(∑j=1kan,ijηj)ℏn+O′(|x|-5+ϵ),ηj=O′(|x|-3),
where an,ij is a constant matrix and ηj 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 ofker𝒟¯A(0) is equal to k the instanton number (of opposite sign), and there exists k zero-mode ηi(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.
IfIndD0:=dimker𝒟A(0)-dimker𝒟¯A(0)=-k, then IndD:=dimker𝒟A-dimker𝒟¯A=-k.Next, we construct the Green's function ofΔA≡Dμ⋆Dμ,(3.30)ΔA⋆GA(x,y)=δ(x-y).We expand (3.18) by ℏ, for n>0, then ℏn order equation is given as(3.31)ΔA(0)GA(n)(x,y)+[ΔA∑0≤k<nℏkGA(k)(x,y)](n)=0,
where GA(n) is defined by GA(x,y)=∑k=0∞GA(k)ℏk. We solve them recursively(3.32)GA(n)(x,y)=∫d4wGA(0)(x,w)[ΔA∑0≤k<nℏkGA(k)(w,y)](n).
Note that GA(0)(x,w) was constructed in [43–45]. Using property of GA(0)(x,w) and A(n), we obtain the following decay condition in [19]:(3.33)GA(n)(x,y)=O′(|x|-3).


## 3.4. From an Instanton to the ADHM Equations

Let us see how to derive the ADHM equations from an SNCD instanton.Letψ¯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 defineTμ by(3.34)Tμ:=∫ℝ4d4x12(xμ⋆ψ¯†⋆ψ¯+ψ¯†⋆ψ¯⋆xμ).Next we introduce an asymptotically parallel sectiong-1S of S+⊗E by(3.35)ψ̃=-g-1Sx†|x|4+O′(|x|-4),
where x†:=σ¯μxμ and ψ̃:=tψ¯σ2. This t means transposing spinor suffixes.Using various properties and decay conditions ofA(n),GA(n),ψ¯(n), and theorems in the previous subsections, we finally obtain the following theorem.Theorem 3.5.
LetAμ be an SNCD instanton and ψ¯ the zero-mode of 𝒟¯A determined by Aμ as in Section 3.3. Let Tμ,S be constant matrices defined by (3.34) and (3.35), respectively. Then, they satisfy the ADHM equations:
(3.36)[Tμ,Tν]+=12tr(S†Sσ¯μν)-iθμν+1k×k.
Here σ¯μν:=(1/4)(σ¯μσν-σ¯νσμ) and 1k×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
(3.37)⋆xψ¯(x)ψ¯†(y)⋆y=⋆xδ(x-y)⋆y-⋆x𝒟A⋆xGA(x,y)⋆y𝒟¯⟵A⋆y.
From the definition of the Tμ,
(3.38)TμTν=∫ℝ4d4x∫ℝ4d4y(xμ⋆xψ¯†(x)⋆xψ¯(x))(ψ¯†(y)⋆yψ¯(y)⋆yyν).
Using Theorem 3.3, (3.37), and integration by parts, (3.38) becomes
(3.39)TμTν=∫ℝ4d4xxμ⋆ψ¯†⋆ψ¯⋆xν+∫S3dSxρ∫ℝ4d4y(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y𝒟¯⟵A⋆y(ψ¯(y)⋆yyν)-∫ℝ4d4x∫ℝ4d4y(ψ¯†(x)σμ)⋆xGA(x,y)⋆y𝒟¯⟵A⋆y(ψ¯(y)⋆yyν),
where dSxμ=|x|2xμdΩ and dΩ is the solid angle. The first term is deformed as follows.
(3.40)∫ℝ4d4xxμ⋆ψ¯†⋆ψ¯⋆xν=∫ℝ4d4x(ψ¯†⋆ψ¯⋆xν⋆xμ+[xμ,ψ¯†⋆ψ¯]⋆⋆xν+ψ¯†⋆ψ¯⋆[xμ,xν]⋆)=∫ℝ4d4x(ψ¯†⋆ψ¯⋆xν⋆xμ+iθμρ∂ρ(ψ¯†⋆ψ¯)⋆xν+iθμνψ¯†⋆ψ¯)=∫ℝ4d4xψ¯†⋆ψ¯⋆xν⋆xμ.
Here ψ¯=O′(|x|-3) given in Theorem 3.3 is used in the third equality. By integration by parts again, we get
(3.41)TμTν=∫ℝ4d4xψ¯†⋆ψ¯⋆xν⋆xμ(3.42)+∫S3dSxρ∫S3dSyτ(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y(σ¯τψ¯(y)⋆yyν)(3.43)-∫S3dSxρ∫ℝ4d4y(xμ⋆xψ¯†(x)σρ)⋆xGA(x,y)⋆y(σ¯νψ¯(y))(3.44)-∫ℝ4d4x∫S3dSyτ(ψ¯†(x)σμ)⋆xGA(x,y)⋆y(σ¯τψ¯(y)⋆yyν)(3.45)+∫ℝ4d4x∫ℝ4d4y(ψ¯†(x)σμ)⋆xGA(x,y)⋆y(σ¯νψ¯(y)).
Equations (3.42) and (3.44) vanish when Ry→∞, where Ry is a radius of Sy3. Equation (3.45) will vanish on the self-dual projection [Tμ,Tν]+:=Pμν,ρτ[Tρ,Tτ], because σμσ¯ν-σνσ¯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
(3.46)18tr(S†Sσ¯μσν),
where the trace tr is taken with respect to the spinor indices. In the [Tμ,Tν]+ combination, (3.41) becomes -iθμν+=-iPμν,ρτθρτ. 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 of(3.47)S†=(IJ†),Tμσ¯μ=(-B2-B1B1†-B2†),
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 spaceℝ2 be xμ,μ=1,2, with commutation relations(4.1)[xμ,xν]⋆=iℏϵμν(μ,ν=1,2),
where ϵμν=-ϵνμ(ϵ12=1) is an antisymmetric tensor.The curvature components of the connectionA are given by(4.2)Fzz=Fz¯z¯=0,Fzz¯=iF12=∂zAz¯-∂z¯Az-i[Az,Az¯]⋆=:iB.Using these complex coordinates, the covariant derivatives of the Higgs fields are(4.3)D⋆ϕ=(∂-iA)⋆ϕ,D¯⋆ϕ=(∂¯-iA¯)⋆ϕ,D⋆ϕ¯=∂ϕ¯+iϕ¯⋆A,D¯⋆ϕ¯=∂¯ϕ¯+iϕ¯⋆A¯.The vortex equations are defined by(4.4)D¯⋆ϕ=(∂¯-iA¯)⋆ϕ=0,B+ϕ⋆ϕ¯-1=0.
We call solutions of these equations NC vortexes.The formal expansions of the fields are(4.5)ϕ=∑n=0∞ℏnϕn(z,z¯),A=∑n=0∞ℏnAn(z,z¯).
The kth order equations for (4.4) are(4.6)-i(∂A¯k+∂¯Ak)+ϕkϕ¯0+ϕ0ϕ¯k-δk0+Ck(z,z¯)=0,(4.7)∂¯ϕk-iA¯kϕ0-iA¯0ϕk+Dk(z,z¯)=0.
Here Ck(z,z¯) is the coefficient of ℏk in -[A,A¯]⋆+ϕ⋆ϕ¯-(ϕkϕ¯0+ϕ0ϕ¯k), so Ck(z,z¯) is a function of {Ai,A¯j,ϕm,ϕ¯n∣0≤i,j,m,n≤k-1}. Similarly, Dk(z,z¯) is the coefficient of ℏk in -iA¯⋆ϕ-(-iA¯kϕ0-iA¯0ϕk) and a function of {Ai,A¯j,ϕm,ϕ¯n∣0≤i,j,m,n≤k-1}.In the case ofk=0, (4.6) and (4.7) coincide with the commutative U(1) vortex equations D¯ϕ0=(∂¯-iA¯0)ϕ0=0 and B0+ϕ0ϕ¯0-1=0, where B0=-i(∂A¯0-∂¯A0). In the following, we consider the case that A0,A¯0, and ϕ0 are smooth finite vortex solutions. We call it Taubes' vortex solution.In the regionϕ0≠0, substituting (4.7) into (4.6) for Ak and A¯k, we get(4.8){∂ϕ0ϕ02(∂¯ϕk-iA¯0ϕk+Dk)-1ϕ0(Δϕk-i∂A¯0ϕk-iA¯0∂ϕk+∂Dk)}+{c.c.}+ϕkϕ¯0+ϕ0ϕ¯0-δk0+Ck=0.
Here {c.c.} is the complex conjugate of preceding terms and Δ=∂∂¯.Setting(4.9)φk=ϕkϕ0+ϕ¯kϕ¯0=2Re(ϕkϕ0),dk=Dkϕ0.
Equation (4.8) is simplified to(4.10)(-Δ+|ϕ0|2)φk=Ek,
where(4.11)Ek:=-Ck+∂dk-∂¯d¯k.To show that there exists a unique NC vortex solution deformed from the Taubes' vortex solution, we consider the stationary Schrödinger equation(4.12)(-Δ+V(x))u(x)=f(x)
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 existK⊂ℝ2 and ∃c>0 such that K is a compact set and for x∈ℝ2∖K, V(x)≥c.(a3)
There existx1,…,xN∈ℝ2 such that V(xi)=0 and V(x)>0 for x∉{x1,…,xN}.(a4)
For anyα=(α1,α2)∈ℤ+2, There exists a positive constant c such that |∂xα(V-c)|≤cfor any x∈ℝ2.Note that the system (4.10) satisfies the assumptions (a1)–(a4). We set(4.13)Hl(n):={f∣∥f∥:=supx∈ℝ2(1+|x|n)|∂xαf(x)|<∞forany|α|≤l]
for n∈ℤ+. Then we obtain the following theorem.Theorem 4.1.
Under the assumptions (a1)–(a4), there exists a unique solution u∈Hl(n) of (4.12) for any f∈Hl(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.
LetA0 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
(4.14)12π∫d2xB=12π∫d2xB0.The proof is given in [20].


## 5. Conclusions

We have reviewed developments for the last dozen years in NC instantons inℝ4. 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 inℝn. 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.



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