# Yang–Mills theory for semidirect products \(\mathrm{G}\ltimes \mathfrak {g}^*\) and its instantons

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## Abstract

Yang–Mills theory with a symmetry algebra that is the semidirect product \(\mathfrak {h}\ltimes \mathfrak {h}^*\) defined by the coadjoint action of a Lie algebra \(\mathfrak {h}\) on its dual \(\mathfrak {h}^*\) is studied. The gauge group is the semidirect product \(\mathrm{G}_{\mathfrak {h}}\ltimes {\mathfrak {h}^*}\), a noncompact group given by the coadjoint action on \(\mathfrak {h}^*\) of the Lie group \(\mathrm{G}_{\mathfrak {h}}\) of \(\mathfrak {h}\). For \(\mathfrak {h}\) simple, a method to construct the self–antiself dual instantons of the theory and their gauge nonequivalent deformations is presented. Every \(\mathrm{G}_{\mathfrak {h}}\ltimes {\mathfrak {h}^*}\) instanton has an embedded \(\mathrm{G}_{\mathfrak {h}}\) instanton with the same instanton charge, in terms of which the construction is realized. As an example, \(\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)\) and instanton charge one is considered. The gauge group is in this case \(SU(2)\ltimes \mathbf{R}^3\). Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given.

## 1 Introduction

Motivated by an interest in finding new gauge configurations, we consider Yang–Mills theory with a symmetry algebra that is the classical double of a real Lie algebra and study its self–antiself dual solutions. By the classical double of a real Lie algebra \(\mathfrak {h}\) we understand in this paper the semidirect product \(\mathfrak {h}\ltimes \mathfrak {h}^*\) defined by the action of \(\mathfrak {h}\) on its dual \(\mathfrak {h}^*\) via the coadjoint representation. Our concern here is Yang–Mills theory with gauge group the simply connected Lie group \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) obtained from \(\mathfrak {h}\ltimes \mathfrak {h}^*\) by exponentiation.

The group \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) admits several descriptions. From a geometric point of view, it is the cotangent bundle of the Lie group \(\mathrm{G}_\mathfrak {h}\) of \(\mathfrak {h}\). Algebraically, it can be regarded as the semidirect product \(\mathrm{G}_\mathfrak {h}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) of \(\mathrm{G}_\mathfrak {h}\) with the Lie group \(\mathrm{G}_{\mathfrak {h}^*}\) of \(\mathfrak {h}^*\!\). The cotangent bundle construction is standard in symplectic mechanics. The semidirect product approach is not new either in the physics literature. The Chern–Simons formulation of three-dimensional gravity [1, 2] is probably the most celebrated example of a gauge theory with a gauge group of this type. In that case, \(\mathfrak {h}\) is the Lorentz algebra in three dimensions, \(\mathfrak {h}^*\) is the algebra of three-dimensional translations, \(\mathfrak {h}\ltimes \mathfrak {h}^*\) is the algebra of isometries \(\mathfrak {i}\mathfrak {s}\mathfrak {o}(1,2)\), and \(\mathrm{G}_\mathfrak {h}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) is the isometry group \(\text {ISO}(1,2)\). Other forms of semidirect products, some involving finite groups, have been employed in various scenarios, including quantization of monopoles with nonabelian magnetic charges [3], neutrino mixing [4, 5], and hypercharge quantization [6, 7].

An important property of \(\mathfrak {h}\ltimes \mathfrak {h}^*\) is that it is a metric Lie algebra. This means that it admits an invariant, nondegenerate, symmetric, bilinear form, called metric, that takes values in \(\mathbf{R}\). The relevance of this property comes from the observation that if \(\mathfrak {g}\) is a metric Lie algebra and \({\Omega }\) is a metric on it, it is possible to formulate Yang–Mills theory with gauge group the Lie group \(\mathrm{G}_\mathfrak {g}\) of \(\mathfrak {g}\). To do this on a *d*-dimensional spacetime manifold, introduce a one-form gauge field \({\kappa }\) and its two-form field strength \(K=\text {d}{\kappa }+ {\kappa }\wedge {\kappa }\), both valued in \(\mathfrak {g}\), and consider the Yang–Mills *d*-form \(\mathcal{L}_{\text {ym}}={\Omega }(K,\star K)\). Nondegeneracy of \({\Omega }\) ensures that \(\mathcal{L}_{\text {ym}}\) contains a kinetic term for the gauge field \({\kappa }\), while invariance of \({\Omega }\) guarantees that \(\mathcal{L}_{\text {ym}}\) is invariant under \(\mathrm{G}_\mathfrak {g}\) gauge transformations. By considering the classical double \(\mathfrak {h}\ltimes \mathfrak {h}^*\), it is thus possible to define a gauge theory even if \(\mathfrak {h}\) is not metric. Similarly, four-dimensional topological field theory and three-dimensional Chern–Simons theory can be considered, with Lagrangians given by \({\Omega }(K,K)\) and \({\Omega }\left( {\kappa },\text {d}{\kappa }+\tfrac{2}{3}{\kappa }\wedge {\kappa }\right) \).

In view of this, it seems natural to ask how many different real metric Lie algebras there are. The list of them is exhausted by (i) reductive algebras, (ii) classical doubles, and (iii) double extensions. Reductive algebras are direct sums of semisimple Lie algebras and the Abelian algebra. They are the Lie algebras of the compact Lie groups, and their gauge theories have been the subject of continuous study over the last 40 years. Less is known about the gauge theories for algebras of type (ii) and (iii). Yang–Mills theory for classical doubles is the object of this paper. As regards double extensions, they are obtained by a nontrivial generalization [8] due to Medina and Revoy of the semidirect product that defines the classical double. In fact, a classical double can be regarded as a double extension of the trivial algebra. These authors proved a structure theorem that states (a) that every real metric Lie algebra is an orthogonal sum of indecomposable real metric Lie algebras, and (b) that every indecomposable real metric Lie algebra is simple, one-dimensional or the double extension of a metric Lie algebra by either a simple or a one-dimensional Lie algebra. A discussion of the theorem can be found in Ref. [9]. Some Wess–Zumino-Witten models and gauge theories for double extensions have been considered in Refs. [9, 10, 11, 12].

Let us center on the case of interest here, gauge theories with symmetry algebra \(\mathfrak {h}\ltimes \mathfrak {h}^*\). In these theories, the gauge field \({\kappa }\) takes values in \(\mathfrak {h}\ltimes \mathfrak {h}^*\) and has nonzero projections onto \(\mathfrak {h}\) and \(\mathfrak {h}^{*\!}\). New degrees of freedom are thus introduced when \(\mathfrak {h}\) is replaced with \(\mathfrak {h}\ltimes \mathfrak {h}^*\). In Sect. 2, it is shown, however, that the homology and homotopy invariants for the group \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) are the same as for \(\mathrm{G}_\mathfrak {h}\). This has two implications. Homotopically nontrivial solutions for \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) gauge theory exist if they do for \(\mathrm{G}_\mathfrak {h}\) gauge theory, and the \(\mathfrak {h}^*\)-component of the gauge field \({\kappa }\) does not contribute to the theory’s invariants. Here we study these questions. It will be shown that \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) instantons indeed have the same instanton charge as their embedded \(\mathrm{G}_{\mathfrak {h}}\) instantons, but larger moduli spaces. A method to construct \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*\!}\cong T^{*}\mathrm{G}_\mathfrak {h}\cong \mathrm{G}_\mathfrak {h}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) instantons and their moduli spaces from those of \(\mathrm{G}_\mathfrak {h}\) instantons will be presented.

This paper is organized as follows. Section 2 is dedicated to review the definition and basic properties of \(\mathfrak {h}\ltimes \mathfrak {h}^*\) and its Lie group \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\). The Lagrangian and field content of \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) Yang–Mills theory are discussed in Sect. 3. The construction of self–antiself dual \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) instantons in terms of the embedded \(\mathrm{G}_{\mathfrak {h}}\) instantons is presented in Sect. 4. This construction is explicitly realized for \(\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)\) and instanton charge one in Sect. 5, where expressions for the gauge field, the zero modes and the metric and complex structures of the moduli space are presented. In Sect. 6 we collect our final comments.

## 2 The classical double of a Lie algebra and its Lie group

*n*with basis \(\{T_i\}\) satisfying \([T_i,T_j]=f_{ij}{}^kT_k\). Denote by \(\mathfrak {h}^*\) its dual vector space, and take for \(\mathfrak {h}^*\) the canonical dual basis \(\{Z^i\}\), defined by \(Z^i(T_j)={\delta }^i{}_j\). Form the vector space \(\mathfrak {h}\oplus \mathfrak {h}^*\). Its elements are pairs (

*T*,

*Z*), with

*T*in \(\mathfrak {h}\) and

*Z*in \(\mathfrak {h}^*\), and as a basis on it one may take \(\{(0,T_i),(0,Z^j)\}\). Consider the semidirect product \(\mathfrak {h}\ltimes \mathfrak {h}^*\) that results from acting with \(\mathfrak {h}\) on \(\mathfrak {h}^*\) via the coadjoint representation. For

*T*in \(\mathfrak {h}\), the coadjoint representation \(\text {ad}_T^*:\mathfrak {h}^*\rightarrow \mathfrak {h}^*\) associates \(Z\mapsto \text {ad}^*_T Z\), with action on \(T^{\prime }\) in \(\mathfrak {h}\) given by \(\text {ad}^*_T Z(T^{\prime }) = Z(\text {ad}_T T^{\prime }) = Z([T,T^{\prime }])\). This results in a Lie algebra of dimension 2

*n*with Lie bracket

*C*in the algebra, it satisfies

*possibly degenerate*, invariant, bilinear form \({\omega }\) on \(\mathfrak {h}\). Hence \(\mathfrak {h}\ltimes \mathfrak {h}^*\) is a real metric Lie algebra, even if \(\mathfrak {h}\) is not, and \({\Omega }\) is a metric on it.

The algebras \(\mathfrak {h}\), \(\mathfrak {h}^*\) and \(\mathfrak {h}\ltimes \mathfrak {h}^*\) define through exponentiation simply connected Lie groups that we denote by \(\mathrm{G}_\mathfrak {h}, \mathrm{G}_{\mathfrak {h}^*}\) and \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\). From a geometric point of view, \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) is the cotangent bundle \(T^*\mathrm{G}_\mathfrak {h}\) of \(\mathrm{G}_\mathfrak {h}\), a standard construction in geometry. \(T^*\mathrm{G}_\mathfrak {h}\) is in turn isomorphic to the semidirect product \(\mathrm{G}_\mathfrak {h}\ltimes \mathfrak {h}^*\), where \(\mathrm{G}_\mathfrak {h}\) acts on \(\mathfrak {h}^*\) by the coadjoint action. For *h* in \(\mathrm{G}_\mathfrak {h}\), the coadjoint representation \(\mathrm{Ad}^*_h\!:\mathfrak {h}^*\rightarrow \mathfrak {h}^*\) maps *Z* to \(\mathrm{Ad}_h^*Z\), whose action on \(T'\) in \(\mathfrak {h}\) is given by \(\mathrm{Ad}_h^*Z(T^\prime )=Z(\mathrm{Ad}_hT^\prime ) = Z(h^{-1}T^\prime h)\). The elements of \(\mathrm{G}_\mathfrak {h}\ltimes \mathfrak {h}^*\) are pairs (*h*, *Z*) with product law \((h_1,Z_1)\,(h_2,Z_2)=(h_1h_2, \mathrm{Ad}^*_{h_2\!}Z_1\!+Z_2)\). Since *h* in \(\mathrm{G}_\mathfrak {h}\) can be uniquely written as \(h=e^T\!\), with *T* in \(\mathfrak {h}\), the derivative of \(\mathrm{Ad}_h^*\) is the coadjoint action \(\mathrm{ad}_T^*\) used to construct the semidirect product \(\mathfrak {h}\ltimes \mathfrak {h}^*\). As a group, \(\mathfrak {h}^*\) is Abelian, noncompact and homeomorphic to \(\mathbf{R}^{n}\), and \(\{0\}\times \mathfrak {h}^*\) is a normal subgroup. For example, for \(\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)\), this gives \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\cong SU(2)\ltimes \mathbf{R}^3\).

One may also adopt the following approach to \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\). Consider the Cartesian product \(\mathrm{G}_{\mathfrak {h}}\times \mathrm{G}_{\mathfrak {h}^*}\), whose elements are pairs (*h*, *n*) that can be uniquely written as \((e^T\!,e^Z)\), for some *T* in \(\mathfrak {h}\) and some *Z* in \(\mathfrak {h}^*\!\). The homomorphism \(\varphi \!: \mathrm{G}_{\mathfrak {h}}\!\rightarrow \mathrm{Aut}(\mathrm{G}_{\mathfrak {h}^*})\), where \(\varphi (h)=\varphi _h\) acts on \(\mathrm{G}_{\mathfrak {h}^*}\) by conjugation, \(\varphi _h(n)=h^{-1}nh\), defines a group structure on \(\mathrm{G}_{\mathfrak {h}}\times \mathrm{G}_{\mathfrak {h}^*}\). This results in the semidirect product \(\mathrm{G}_{\mathfrak {h}}\ltimes \mathrm{G}_{\mathfrak {h}^*}\), with group law \((h_1,n_1)\,(h_2,n_2)= (h_1h_2,\,(h_2^{-1}n_1\,h_2)\,n_2)\) and Lie algebra \(\mathfrak {h}\ltimes \mathfrak {h}^*\!\). As a group, \(\mathrm{G}_{\mathfrak {h}^*}\) is Abelian, noncompact, and homeomorphic to \(\mathbf{R}^n_+\). The map \([0,1]\times (\mathrm{G}_{\mathfrak {h}}\ltimes \mathrm{G}_{\mathfrak {h}^*}) \rightarrow \mathrm{G}_{\mathfrak {h}}\times \{0\}\), given by \(\left( t,(h,n)\right) \mapsto (h,tn)\), is then a homotopy. This means that \(\mathrm{G}_{\mathfrak {h}}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) and \(\mathrm{G}_{\mathfrak {h}}\times \{0\}\) are homotopically equivalent, hence have the same homology and homotopy invariants. In particular, they have the same third homotopy group. For the elements of \(\mathrm{G}_{\mathfrak {h}}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) we will use the notation \(g=hn=(h,n)\). It is clear that \(\mathrm{G}_{\mathfrak {h}}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) and \(\mathrm{G}_{\mathfrak {h}}\ltimes \mathfrak {h}^*\) are isomorphic.

We finish this section with two comments, one on representations and one on deformations.

*p*-dimensional matrix representation of \(\mathfrak {h}\) that associates to its basis \(\{T_i\}\) matrices \(\{\mathbf{M}_i\}\) with \([\mathbf{M}_i,\mathbf{M}_j]=f_{ij}{}^k\,\mathbf{M}_k\), it is very easy to see that

*p*-dimensional matrix representation of \(\mathfrak {h}\ltimes \mathfrak {h}^*\). In the adjoint representation of \(\mathfrak {h}\), the matrices \(\{\mathbf{M}_i\}\) are \(n\times n\) and have entries \((\mathbf{M}_i^\mathrm{ad})_j{}^k\!=\!-f_{ij}{}^k\). It is straightforward to check that \(\rho \) above is then the adjoint representation of \(\mathfrak {h}\ltimes \mathfrak {h}^*\). Representations other than (2.6) are possible. An example is the following. Let \(\mathbf{e}_i\) be the unit column vector in \(\mathbf{R}^n\), with components \((\mathbf{e}_i)_j\!={\delta }_{ij}\). Some simple algebra shows that the matrices

*s*in \([Z_i,Z_j]\) is an arbitrary real parameter. These commutators satisfy the Jacobi identity for all

*s*and reduce to the Lie bracket (2.9) of the classical double when \(s\rightarrow 0\). The vector space \(\mathfrak {h}\oplus \mathfrak {h}^*\) with the Lie bracket (2.10) is thus a Lie algebra, call it \(\mathfrak {h}\ltimes _{\!s}\mathfrak {h}^{*\!}\), and a deformation of \(\mathfrak {h}\ltimes \mathfrak {h}^*\) with deformation parameter

*s*. The algebra \(\mathfrak {h}\ltimes _{\!s}\mathfrak {h}^*\) is metric since it admits the metric

## 3 The gauge theory and its field content

Our interest here is Yang–Mills theory with gauge group \(G_{\mathfrak {h}\ltimes \mathfrak {h}^*}\). Consider a spacetime manifold \(M_d\) of dimension *d* equipped with a metric \({\gamma }\). Greek letters \({\mu },{\nu },\ldots \) will label coordinate indices \(1,2, \ldots , d\) in a local chart \(\{x^{\mu }\}\). In such a chart, \({\gamma }_{{\mu }{\nu }}\) will denote the metric components and \({\gamma }^{{\mu }{\nu }}\) the components of the inverse metric. For an *r*-form \(\zeta \) we will adopt the normalization \(\zeta =\frac{1}{r!}\,\zeta _{{\mu }_1\cdots {\mu }_r}\,\text {d}x^{{\mu }_1\!} \wedge \cdots \wedge \text {d}x^{{\mu }_r}\). Indices will be raised and lowered using \({\gamma }^{{\mu }{\nu }}\) and \({\gamma }_{{\mu }{\nu }}\) metric. For the commutator of an *r*-form \(\zeta \) with an *s*-form \(\xi \), both taking values in \(\mathfrak {h}\ltimes \mathfrak {h}^*\), we will use \([\zeta ,\xi ]=\zeta \wedge \xi -(-)^{rs}\,\xi \wedge \zeta \).

*r*-form \(\zeta \) is given by \(\text {d}_{\displaystyle {\kappa }}\,\zeta =\text {d}\zeta +[{\kappa },\zeta ]\), and a curvature two-form or field strength

*g*(

*x*). Under such transformations, the curvature changes as

*T*and

*Z*and keeping terms up to order one. With \(\Lambda \!:=T+Z\), they read

*d*-form \({\Omega }(K,\star K)\), where \(\star K\) is the Hodge dual of

*K*and \({\Omega }\) is an invariant metric on \(\mathfrak {h}\ltimes \mathfrak {h}^*\). The transformation law (3.3) for

*K*, the observation that any

*g*can be written as \(g=e^Te^Z\), and the invariance condition (2.4) imply that \({\Omega }(K,\star K)\) remains unchanged under gauge transformations. The functional

*K*can be expanded in the Lie algebra basis \(\{T_i,Z^j\}\) as

*F*has the same dependence on \({\alpha }\) as results from gauging the algebra \(\mathfrak {h}\). It is

*B*that mixes \({\alpha }\) with \({\beta }\). Secondly, the Lagrangian \({\Omega }(K,\star K)\) has a kinetic term for all the field components \({\alpha }^i\) and \({\beta }_i\) of the gauge field \({\kappa }\). Note in this regard that, for \({\omega }\) degenerate, \({\omega }(F,\star F)\) does not define a Yang–Mills Lagrangian since it does not contain a kinetic term for all the \({\alpha }^i\). Thirdly, the field strength

*B*, its Bianchi identity (3.20) and its field equation (3.22) are linear in \({\beta }\). And lastly, the field equations (3.21) and (3.22) do not depend on \({\omega }\).

## 4 Semidirect instantons: general analysis

*h*in \(\mathrm{G}_\mathfrak {h}\). Note that no boundary condition for \({\beta }_{\text {s}}\) is needed. These arguments can be made more explicit by noting that \(S_{\text {p}}[{\kappa }]\) is the integral over \(\text {S}^3_\infty \) of the Chern–Simons three-form \(\mathcal{L}_{\text {cs}}({\kappa })\) in Eq. (3.24). For a connection \({\kappa }=({\alpha },{\beta })\) that approaches \(({\alpha }_\infty \!=h^{-1}\text {d}h,\,{\beta }_{\infty })\) at \(\text {S}^3_\infty \), with \({\beta }_{\infty }\) arbitrary, Eq. (3.24) and \(F_{\infty }\!=0\) imply that \(S_{\text {p}}[\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*};{\kappa }] = S_{\text {p}}[\mathrm{G}_\mathfrak {h}; {\alpha }]\).

*N*, and the boundary conditions for a self–antiself dual \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) instanton \({\kappa }_{\text {s}}=({{\alpha }}_{\text {s}},{{\beta }}_{\text {s}})\) are specified by those of the embedded \(\mathrm{G}_\mathfrak {h}\) instanton,

Take \(\mathfrak {h}\) to be simple and \({\omega }_{ij}\) in Eq. (2.5) a metric on \(\mathfrak {h}\). This is the case of all self–antiself dual \(\mathrm{G}_\mathfrak {h}\) instantons known to date [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Introduce generators \(Z_i={\omega }_{ij}Z^j\). The commutation relations for \(\{T_i,Z_j\}\) and the metric \({\Omega }\) take the form (2.9) and (2.11). Since any gauge field \({\kappa }^{\,\prime \!}=({\alpha }^{\,\prime \!},{\beta }^{\,\prime })\) obtained from a solution \({\kappa }_{\text {s}}=({\alpha }_{\text {s}},{\beta }_{\text {s}})\) by a \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) gauge transformation is trivially a solution, we restrict our attention to gauge nonequivalent solutions. The space of all such solutions with instanton charge *N* is the moduli space \(\mathcal{M}_N(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*})\).

*N*self–antiself dual \(\mathrm{G}_\mathfrak {h}\) instantons and their zero modes. Given its solution \(\{{\alpha }_{\text {s}}, {\delta }{\alpha }_{\text {s}}\}\), we want to solve Eqs. (4.5), (4.11), and (4.12) for \({\beta }\) and \({\delta }{\beta }\). Let us first understand the solution to the \(\mathrm{G}_\mathfrak {h}\) problem. A solution \({\alpha }_{\text {s}}\) to Eq. (4.4) depends on a set of free parameters \(\{u^a\}\) that describe instanton degrees of freedom and that occur in the differential problem as integration constants [19, 20, 21, 22, 23, 24, 25, 26]. In the ADHM approach, the \(\{u^a\}\) appear as free parameters in the quaternion matrices in terms of which \({\alpha }_{\text {s}}\) is constructed. Using the fact that partial derivatives \({\partial }/{\partial }u^a\) commute with the exterior differential \(\text {d}\) and noting the Jacobi identity for the generators \(\{T_i\}\) of \(\mathfrak {h}\), it is trivial to check that (i) derivatives \({\partial }{\alpha }_{\text {s}}/{\partial }u^a\) of \({\alpha }_{\text {s}}\) along \(u^a\) and (ii) rotations \([{\alpha }_{\text {s}},T_i]\) of \({\alpha }_{\text {s}}\) about \(T_i\) solve the moduli equation (4.9). The problem is that they may not satisfy the gauge-fixing condition (4.10). To correct this, one includes infinitesimal local \(\mathrm{G}_\mathfrak {h}\) transformations and writes for the zero modes

*N*self–antiself dual \(\mathrm{G}_\mathfrak {h}\) instantons \(\mathcal{M}_N(\mathrm{G}_\mathfrak {h})\).

### 4.1 The connection

*F*, so that

*N*the charge of the \(\mathrm{G}_\mathfrak {h}\) instanton specified by \({\alpha }_{\text {s}}\), the derivatives on the right hand side vanish and Eq. (4.3) is reproduced.

Once we have \(({\alpha }_{\text {s}},{\beta }_{\text {s}})\), we look for the solutions \({\delta }{\beta }\) to Eqs. (4.11) and (4.12). There are two types of solutions: those with \({\delta }{\alpha }={\delta }{\alpha }_{\text {s}}\ne 0\), and those with \({\delta }{\alpha }=0\).

### 4.2 Zero modes with \({\delta }{\alpha }\ne 0\)

### 4.3 Zero modes with \({\delta }{\alpha }=0\)

To summarize, the gauge field \(({\alpha }_{\text {s}},{\beta }_{\text {s}})\), with \({\alpha }_{\text {s}}\) the connection of a charge *N* self–antiself dual \(\mathrm{G}_\mathfrak {h}\) instanton and \({\beta }_{\text {s}}\) as in Eq. (4.15), specifies a self–antiself dual \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) instanton with the same charge. The dimension of its moduli space \(\mathcal{M}_N(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*\!})\) is twice the dimension of \(\mathcal{M}_N(\mathrm{G}_\mathfrak {h})\). As local coordinates on \(\mathcal{M}_N(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*})\), one may take \(\{u^a,\tau ^j,\tilde{u}^a,\tilde{\tau }^j\}\), where \(u^a\) and \(\tau ^i\) are local coordinates on \(\mathcal{M}_N(\mathrm{G}_\mathfrak {h})\), and \(\tilde{u}^a\) and \(\tilde{\tau }^i\) are kind of dual coordinates. If the zero modes of the \(\mathrm{G}_\mathfrak {h}\) instanton \({\alpha }_{\text {s}}\) are given by Eqs. (4.13) and (4.14), the zero modes of the \(({\alpha }_{\text {s}},{\beta }_{\text {s}})\) instanton take the form in Eqs. (4.13)–(4.14), (4.19)–(4.20), and (4.25)–(4.26). We may call these instantons cotangent \(T^*\mathrm{G}_\mathfrak {h}\), or semidirect \(\mathrm{G}_\mathfrak {h}\ltimes \mathrm{G}_{\mathfrak {h}^*}\) instantons.

*U*and

*V*stand for two arbitrary moduli coordinates, the moduli space metric coefficients are given by

*H*the metric on \(\mathcal{M}_N(\mathrm{G}_\mathfrak {h})\), with components

*p*and

*q*.

In the next section we explicitly realize this construction for \(\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)\) and instanton charge one.

## 5 The semidirect extension BPST instanton and its moduli

On \(\mathbf{R}^4\) take coordinates \(x^{\mu }\!=(x^1,x^2,x^3,x^4)\) and Euclidean metric \({\delta }_{{\mu }{\nu }}\). Set \(\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)\), with basis \([T_i,T_j]={\epsilon }_{ijk}\,T_k\). The most general invariant bilinear form \({\omega }\) that can be defined on \(\mathfrak {s}\mathfrak {u}(2)\) is \({\omega }_{ij}\!={\omega }_0{\delta }_{ij}\), with \({\omega }_0\) an arbitrary constant that is conventionally set equal to \({1/2g^2}\).

*K*has components \(F^{\,i}\) and \(B^j\), given by

*SU*(2) selfdual instantons. Take as solution the BPST instanton [14], whose connection \({\alpha }^{\,i}_{\text {{s}}}\) and curvature \(F^{\,i}_{\text {{s}}}\) are given in singular gauge by

*t*is the function

### 5.1 The semidirect BPST instanton and its zero modes

*t*as in Eq. (5.12).

### 5.2 The moduli space metric

*f*and \(\tilde{f}\) are positive functions of \({\sigma }\) and \(\tilde{{\sigma }}\). This shows that the metric has signature (8, 8).

*SO*(4) rotations in \(\mathbf{R}^4\), and under \(SU(2)\ltimes \mathbf{R}^3\) gauge transformations. These symmetries go into isometries of the moduli metric. Indeed, \(\mathbf{R}^4\) translations give rise to translations in \(b^{\mu }\) and \(\tilde{b}^{\mu }\), generated by \({\partial }/{\partial }b^{\mu }\) and \({\partial }/{\partial }\tilde{b}^{\mu }\). Rotations become \(SO(4)\cong SU(2)_{+\!}\times SU(2)_-\) rotations in \(b^{\mu }\) and \(\tilde{b}^{\mu }\), generated by

### 5.3 Complex structures

We finish by studying the compatibility of the isometries of the moduli metric with the complex structures. Recall that for an isometry generated by a Killing vector \(\xi \) to be compatible with a tensor *A*, the Lie derivative \(\mathcal{L}_\xi A\) of *A* along \(\xi \) must vanish. For an isometry given in a chart \(\{u^a\}\) by \(u^{\,a\!}\rightarrow u^{\prime \,a}\!=u^a +\varepsilon \,\xi ^a(u)\), we use for the Lie derivative the convention \(\mathcal{L}_{\xi \!}{A} = \lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon }\, \left[ A^{\,\prime }(u)-A(u)\right] \). With this convention, one may check that the isometries generated by \(\xi ={\partial }_{b^{{\mu }}},\, {\partial }_{\,\tilde{b}^{\mu }}, \chi ^{i}_+,\,{\partial }_{\tau ^i}\) and \({\partial }_{\tilde{\tau }^i}\) are compatible with the complex structures \(J^i\). However, for \(\xi =\chi ^{i}_-\), one has \(\mathcal{L}_{\chi ^{i}_-\!}J^j\!={\epsilon }^{ijk}J^k\). The complex structures are thus rotated by \(SU(2)_-\) rotations, but they remain unchanged by the other isometries.

## 6 Outlook

In this paper we have proposed a method to obtain the self–antiself dual solutions for a gauge group \(\mathrm{G}_{\mathfrak {h}\ltimes \mathfrak {h}^*}\) from those for \(\mathrm{G}_{\mathfrak {h}}\). This hints to using Medina and Revoy’s theorem [8] to find structure results for the self–antiself dual instantons of the Lie groups with metric Lie algebras. One may advance a few ideas on the subject. According to the theorem, it would suffice to consider three cases: (1) simple Lie algebras, (2) Abelian algebras, and (3) double extensions of a metric Lie algebra by either a simple or a one-dimensional Lie algebra.

Simple real Lie algebras are the Lie algebras of real simple Lie groups, whose instantons would be regarded as the basic objects in terms of which state structure results. Next on the list is the Abelian Lie algebra. This case is trivial, since on \(\mathbf{R}^4\) there are no Abelian instantons. One is left with the Lie groups of double extensions.

The double extension \(\mathfrak {d}(\mathfrak {m},\mathfrak {h})\) of a metric Lie algebra \(\mathfrak {m}\) by a Lie algebra \(\mathfrak {h}\) is obtained [8, 9] by forming the classical double \(\mathfrak {h}\ltimes \mathfrak {h}^*\) and, then, by acting with \(\mathfrak {m}\) on \(\mathfrak {h}\) via antisymmetric derivations. Since \(\mathfrak {m}\) needs to be metric, three possibilities must be considered for \(\mathfrak {m}\). The first one is that \(\mathfrak {m}\) is a simple real Lie algebra. In this case [9], the algebra of antisymmetric derivations of \(\mathfrak {m}\) is \(\mathfrak {m}\) itself and the double extension is isomorphic to the direct product \(\mathfrak {m}\times (\mathfrak {m}\ltimes \mathfrak {m}^*)\). The corresponding Lie group is then the direct product \(\mathrm{G}_\mathfrak {m}\times \mathrm{G}_{\mathfrak {m}\ltimes \mathfrak {m}^*}\) and its instantons are determined in terms of the \(\mathrm{G}_\mathfrak {m}\) instantons using the construction presented here. The second possibility is that \(\mathfrak {m}\) is Abelian, of dimension *m*. Being Abelian, any nondegenerate, symmetric bilinear form on \(\mathfrak {m}\) is a metric, and this can always be brought to a diagonal form with all its eigenvalues equal to either \(+1\) or \(-1\). If there are *p* positive and *q* negative eigenvalues, the algebra \(\mathfrak {h}\) of antisymmetric derivations is any subalgebra of \(\mathfrak {s}\mathfrak {o}(p,q)\). In this case, by extending the arguments at the beginning of Section 4, it can be shown that the third homotopy group of \(\mathrm{G}_{\mathfrak {d}(\mathfrak {m},\mathfrak {h})}\) is equal to the third homotopy group of \(\mathrm{G}_\mathfrak {h}\). This motivates studying the self–antiself dual solutions of such theories in detail. The third option, that \(\mathfrak {m}\) is a double extension, takes us back to the starting point.

One would also like to include matter fields in the analysis. Their coupling to an \(\mathfrak {h}\ltimes \mathfrak {h}^*\) gauge field requires additional matter fields components, which introduce additional field equations that may lead to new nontrivial configurations.

## Notes

### Acknowledgments

This work was partially funded by the Spanish Ministry of Education and Science through Grant FPA2011-24568.

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