#### Abstract

We review recent developments in noncommutative deformations of instantons in . 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 gauge theories in NC and understand what is clarified and what is unknown.

Instanton connections in the 4-dim Yang-Mills theory are defined by where 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 gauge field requires a pair of the two complex vector spaces and . Here is an integer called instanton number. Introduce , , and which are called ADHM data that satisfy the ADHM equations that we will see soon. In other words, and are complex-valued matrices, and and are complex-valued matrices that satisfy (2.13) and (2.14) in Section 2.2. Using these ADHM data, we can construct instanton [6β17]. 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 [6β17]. 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 βinstanton number" from βinstanton charge". In this article, we define the instanton number by the dimension of some vector space ; 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 , 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 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 satisfying , where is a skew symmetric real valued matrix and we call NC parameter. We set the noncommutativity of the space to the self-dual case of , , and the other for convenience. By transformations of coordinates , the NC parameters are possible to be put in this form in general. Here we introduce complex coordinate operators Then the commutation relations become We define creation and annihilation operators by then they satisfy The Fock space on which the creation and annihilation operators (2.4) act is spanned by the Fock state with where and are the occupation number. The number operators are also defined by which act on the Fock states as In the operator representation, derivatives of a function are defined by where and which satisfy . The integral on NC is defined by the standard trace in the operator representation, Note that represents the trace over the Fock space whereas the trace over the gauge group is denoted by .

##### 2.2. Noncommutative ADHM Instantons

Let us consider the Yang-Mills theory on NC . Let be a projective module over the algebra that is generated by the operator .

In the NC space, the Yang-Mills connection is defined by where is a matter field in fundamental representation type and are anti-Hermitian gauge fields [22β24]. The relation between and usual gauge connection is , where is an inverse matrix of . In our notation of the complex coordinates (2.1) and (2.2), the curvature is given as Note that there is a constant term originated with the noncommutativity in . Instanton solutions satisfy the antiself-duality condition These conditions are rewritten in the complex coordinates as In the commutative spaces, instantons are classified by the topological charge which is always integer and coincide with the opposite sign of dimension of the vector space in the ADHM methods, and 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, 25β30].)

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 instantons [22, 23].

The first step of ADHM construction on NC is looking for , , and which satisfy the deformed ADHM equations We call βinstanton numberβ in this article. In the previous section, we denote as the vector space . Note that the right-hand side of (2.13) is caused by the noncommutativity of the space . The set of , and satisfying (2.13) and (2.14) is called ADHM data. Using this ADHM data, we define operator by The ADHM equations (2.13) and (2.14) are replaced by

Let us denote by the solution to the following equation:

Theorem 2.1. *Let be orthonormal zero-modes defined in (2.17). Then NC instanton with instanton number is obtained by
**
Here is inverse of , that is, .*

*Proof. *The curvature two-form determined by this connection is given as follows.
Here we use that follows from the differentiating of (2.17). Note that
since
From (2.19) and (2.20),
where we use that follows from differentiating . If the coordinates are renamed for convenience, we obtain
Here, we define and by
where are the Pauli matrices and is an identity matrix of degree . Note that owing to (2.16), and and its inverse commute with . Then we find (2.22) is in proportion to
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 is anti-self-dual and the connections given by (2.18) are instantons.

With the complex coordinate , NC instanton connections are given by

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

Theorem 2.2 (Zero-mode of ). *Suppose that and are given as above. The vector satisfying
**
is said to be a zero-mode of . The zero-modes are given by following three types:
**
Here () is some element of (i.e., is expressed with the coefficients as , where is a base of -dim vector space). is a element of -dim vector. *

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

##### 2.3. N.C. ADHM Multi-Instanton

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

At the beginning, we review the methods in [23]. Let be constant matrices satisfying (2.13) and (2.14). We consider ; then we can put in general by using a symmetry. and are matrices, and is matrices: We define , and by We introduce as and we define a projection operator as a projection onto 0-eigenstates of by where We define shift operators and and a operator by

Theorem 2.3 (Nekrasov). * instantons are given by
*

*Proof. *At first, we check that the inverse of in (2.34) is well defined. has zero-modes:
which satisfy . Here is a base of . Note that . This implies that removes the zero-modes, and Hilbert spaces is projected on to a space that does not include the zero-modes. Therefore, the inverse of exists if it is sandwiched between and and (2.35) is well defined.

Next, we check that (2.35) is an instanton. Let us see how the equation is solved under orthonormalization condition . and are introduced as
The orthonormalization condition is expressed as
We put anzats for the solution by
and substitute them into ; then we get
The orthonormalization condition is rewritten as
If there exist the inverse of ,
0-eigenstates of are (2.36) and we define the projection operator to project out the 0-eigenstates by
Shift operators satisfying
are determined by the definition of . Then the inverse of is well defined at the left side of or the right side of .

Using the orthonormalization condition, we obtain
Through these processes, is determined by the ADHM data, and after substituting this into (2.26) we obtain the instantons.
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 .

As an example, let us construct an NC multi-instanton having concrete expressions with the instanton number [31, 32], which is made from the ADHM data: where . It is easy to check that this data satisfies the ADHM equations (2.13) and (2.14), and substituting them into definition of derives To construct an instanton, it is necessary to obtain or . By definition,

and depend on , so we denote them and , respectively. is entry of matrix . To obtain , it is enough to calculate the th row vector of . The th row vector of is determined by . We denote the th row vector of by , that is, . Then, we obtain the following recurrence equation from the th row of where . We change variables as then we can rewrite the above recurrence relation by as

Note that and are commutative to each other, so we can treat them like C-numbers in the following. We introduce an anzats for the generating function by where , , , and are real parameter determined by the request that satisfy (2.52), and . From the differentiation of , we obtain and we find that satisfy the following relation: From (2.52) and (2.55), we obtain where Thus the generating function is determined as an elementary function for each instanton number . Using this , we obtain , and is determined as Using them, , , , and are determined as Finally we obtain instanton gauge fields with instanton number as where 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 [6β17].

##### 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 and coincides with the instanton number defined by the dimension of the vector space in the ADHM construction. A naive definition of the instanton charges in NC is given by replacement of by , but it is conditionally convergent in general. In [25, 26], we introduce cut-off 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 where is a function of and we suppose that the length of the boundary is order , that is, .

Using this cut-off (boundary), we define the instanton charge by As described in [25, 26], the regions for summations of intermediate states are shifted. This phenomenon is caused by the existence of the zero-mode .

The following terms appear in the instanton charge : We denote as trace over some finite domain of Fock space characterized by which is the length of the Fock space boundary. Using the Stokes' like theorem in [25], only trace over the boundary is left, then becomes The same value is obtained from , too. The first term in (2.65) and the term from the constant curvature in (2.11) cancel out. The second term is occurred by zero-modes . 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 zero-modes . After all, we get

Theorem 2.4 (Instanton number). * Consider U(N) gauge theory on NC with self-dual . The instanton charge is possible to be defined by limit of converge series and it is identified with the dimension that appears in the ADHM construction and is called βinstanton 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 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 gauge theory. In [26], this identification is shown when the NC parameter is self-dual for a 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 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 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], HorvΓ‘th 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 . So, let us pay attention to the mathematical meaning of the formal expansion. We introduce our star products by using formal expansions in , 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 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 . 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 where are a real, -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 Here and are partial derivatives with respect to for and to for , respectively.

The curvature two form is defined by , where is defined by

To consider smooth NC deformations, we introduce a parameter and a fixed constant with We define the commutative limit by letting .

Formally we expand the connection as Then, We introduce the self-dual projection operator by Then the instanton equation is given as In the NC case, the th order equation of (3.6) is given by Note that the th order is the commutative instanton equation with solution being a commutative instanton. The asymptotic behavior of commutative instanton is given by where and is a gauge group. (See, e.g., [2].) We introduce covariant derivatives associated to the commutative instanton connection by Using this, (3.7) is given by

In the following, we fix a commutative instanton connection . We impose the following gauge fixing condition for [18, 42] where is defined by We expand in as we did with . Then . In this gauge, using the fact that the is an anti-self-dual connection, (3.10) simplified to where

We consider the Green's function for : where is a four-dimensional delta function. has been constructed in [43] (see also [44, 45]). Using the Green's function, we solve (3.13) as and the NC instanton is given by In the following, we call NC instantons smoothly deformed from commutative instantons SNCD instantons. The asymptotic behavior of Green's function of is important, which is given by

We introduce the notation as in [2]. If is a function of which is as and , then we denote this natural growth condition by .

Theorem 3.1. *If , then .*

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

In our case, by (3.8), and so , as . Repeating the argument times, we get

##### 3.2. Instanton Charge

The instanton charge is defined by We rewrite (3.20) as where

is 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 are typically written as where and are some -form and -form , respectively, and let be . The lowest order term in vanishes because of the cyclic symmetry of the trace, that is, The term of order is given by where . These integrals are zero if is and this condition is satisfied for SNCD instantons. Similarly, higher-order terms in in (3.23) can be written as total divergences and hence vanish under the decay hypothesis. This fact and (3.19) imply that .

Because of the similar estimation, we found the other terms of vanish, where is the curvature two form associated to .

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

Theorem 3.2. *Let be a commutative instanton solution in . There exists a formal NC instanton solution (SNCD instanton) such that the instanton number defined by (3.20) is independent of the NC parameter :
*

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

Dirac(-Weyl) operators and are defined as Here, and are defined by (2.24). Consider expansion of and as In [19], the zero-modes of and , which are defined by are investigated, and the following theorem is obtained.

Theorem 3.3. *Let and be the Dirac(-Weyl) operators for an SNCD instanton background with its instanton number . There is no zero-mode for , and there are zero-modes for that are given as
**
where is a constant matrix and is a base of the zero mode of .*

Note that it is a well-known fact as an index theorem in commutative space that the dimension of is equal to the instanton number (of opposite sign), and there exists zero-mode . 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 , then .*

Next, we construct the Green's function of ,

We expand (3.18) by , for , then order equation is given as where is defined by . We solve them recursively Note that was constructed in [43β45]. Using property of and , we obtain the following decay condition in [19]:

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

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

Let be orthonormal zero-modes of and , which are introduced in Section 3.3.

At first we define by

Next we introduce an asymptotically parallel section of by where and . This means transposing spinor suffixes.

Using various properties and decay conditions of , and theorems in the previous subsections, we finally obtain the following theorem.

Theorem 3.5. *Let be an SNCD instanton and the zero-mode of determined by as in Section 3.3. Let be constant matrices defined by (3.34) and (3.35), respectively. Then, they satisfy the ADHM equations:
**
Here and is an identity matrix.*

*Rough sketch of the proof*

Let us see the essence of the proof. Let us introduce as associated with variable . The completeness of is written as
From the definition of the ,
Using Theorem 3.3, (3.37), and integration by parts, (3.38) becomes
where and is the solid angle. The first term is deformed as follows.
Here given in Theorem 3.3 is used in the third equality. By integration by parts again, we get

Equations (3.42) and (3.44) vanish when , where is a radius of . Equation (3.45) will vanish on the self-dual projection , because is anti-self-dual with respect to the . Thus only (3.41) and (3.43) remain. By the asymptotic behaviors of and some calculations, we can prove that (3.43) becomes