Yang-Mills instantons and dyons on homogeneous G_2-manifolds

We consider Lie G-valued Yang-Mills fields on the space R x G/H, where G/H is a compact nearly K"ahler six-dimensional homogeneous space, and the manifold R x G/H carries a G_2-structure. After imposing a general G-invariance condition, Yang-Mills theory with torsion on R x G/H is reduced to Newtonian mechanics of a particle moving in R^6, R^4 or R^2 under the influence of an inverted double-well-type potential for the cases G/H = SU(3)/U(1)xU(1), Sp(2)/Sp(1)xU(1) or G_2/SU(3), respectively. We analyze all critical points and present analytical and numerical kink- and bounce-type solutions, which yield G-invariant instanton configurations on those cosets. Periodic solutions on S^1 x G/H and dyons on iR x G/H are also given.

Initial choices for the internal manifold X 6 in string theory were Kähler coset spaces and Calabi-Yau manifolds, as well as manifolds with exceptional holonomy group G 2 for d=7 and Spin (7) for d=8. However, it was realized recently that the internal manifold should allow non-trivial p-form fluxes whose back reaction deforms its geometry. In particular, a three-form flux background implies a nonzero torsion whose components are given by the structure constants of the holonomy group, T a bc = κ f a bc , with a real parameter κ. String vacua with p-form fields along the extra dimensions ('flux compactifications') have been intensively studied in recent years (see e.g. [25][26][27] for reviews and references). Flux compactifications have been investigated primarily for type II strings and to a lesser extent in the heterotic theories, despite their long history [28][29][30][31][32]. The number of torsionful geometries that can serve as a background for heterotic string compactifications seems rather limited. Among them there are six-dimensional nilmanifolds, solvmanifolds, nearly Kähler and nearly Calabi-Yau coset spaces. The last two kinds of manifolds carry a natural almost complex structure which is not integrable (for their geometry see e.g. [33][34][35][36][37] and references therein).
In the present paper, we solve the torsionful Yang-Mills equations on G 2 -manifolds of topology R×X 6 with nearly Kähler cosets X 6 . The allowed gauge bundle is restricted by the G 2 -instanton equations [13,14]. For each coset X 6 = G/H, we parametrize the general G-invariant connection by a set of complex scalars φ i , which depend on the coordinate τ of the R factor. The Yang-Mills equations then descend to Newton's equations for the coordinates φ i (τ ) of a point particle under the influence of an inverted double-well-type potential, whose shape depends on κ. For this potential we derive the critical points of zero energy, which correspond to the τ →±∞ asymptotic configurations of the finite-action Yang-Mills solutions. We then present a variety of zero-energy solutions φ i (τ ), of kink and of bounce type, analytically as well as numerically. The kinks translate to instantons for the gauge fields.
Furthermore, by replacing the factor R with S 1 , we obtain periodic solutions with a sphaleron interpretation. Finally, in the Lorentzian case iR×G/H, the double-well-type potential gets flipped back, and there exist bounce solutions with a dyonic interpretation, some of which have finite action. The different types of finite-action Yang-Mills solutions on R×G/H or iR×G/H occur in the following ranges of the parameter κ: 2 Yang-Mills fields on R × G/H

Yang-Mills equations with torsion
Instantons [38] play an important role in modern gauge theories [39,40]. They are nonperturbative BPS configurations in four Euclidean dimensions solving the first-order antiself-duality equations and forming a subset of solutions to the full Yang-Mills equations. In dimensions higher than four, BPS configurations can still be found as solutions to first-order equations, known as generalized anti-self-duality equations [2][3][4][5][6][7][8][9][10] or Σ-anti-selfduality [11][12][13][14]. These appear in superstring compactifications as conditions of survival of at least one supersymmetry [1]. Various solutions to these first-order equations were found e.g. in [17][18][19][20][21][22][23][24], mostly on flat space R d and various cosets. The BPS-type instanton equations in d > 4 dimensions can be introduced as follows. Let Σ be a (d−4)-form on a d-dimensional Riemannian manifold M . Consider a complex vector bundle E over M endowed with a connection A. The Σ-anti-self-dual gauge equations are defined [11,12] as the first-order equations, where the torsion three-form H is defined by the formula * H := dΣ The torsion term in (2. 2) naturally appears in string theory [25][26][27]  We normalize the generators such that the Killing-Cartan metric on the Lie algebra g of G coincides with the Kronecker symbol, More general left-invariant metrics can be obtained by rescaling the generators. The Lie algebra g of G can be decomposed as g = h ⊕ m, where m is the orthogonal complement of the Lie algebra h of H in g. Then, the generators of G can be divided into two sets, For reductive homogeneous spaces we have the following commutation relations: For the metric (2.6) on g we have

Torsionful spin connection on G/H
The metric (2.8) on m lifts to a G-invariant metric on G/H. A local expression for this can be obtained by introducing an orthonormal frame as follows. The basis elements I A of the Lie algebra g can be represented by left-invariant vector fieldsÊ A on the Lie group G, and the dual basisê A is a set of left-invariant one-forms. The space G/H consists of left cosets gH and the natural projection g → gH is denoted π : G → G/H. Over a small contractible open subset U of G/H, one can choose a map L : U → G such that π • L is the identity, i.e. L is a local section of the principal bundle G → G/H. The pull-backs ofê A by L are denoted e A . Among these, the e a form an orthonormal frame for T * (G/H) over U , and for the remaining forms we can write e i = e i a e a with real functions e i a . The dual frame for T (G/H) will be denoted E a . By the group action we can transport e a and E a from inside U to everywhere in G/H. The forms e A obey the Maurer-Cartan equations, (2.10)

JHEP10(2010)044
The local expression for the G-invariant metric then is Recall that a linear connection is a matrix of one-forms Γ = (Γ a b ) = (Γ a cb e c ). The connection is metric compatible if g ac Γ c b is anti-symmetric, and its torsion is a vector of two-forms T a determined by the structure equations We choose the torsion tensor components on G/H proportional to the structure constants f a bc , where κ is an arbitrary real parameter. Then the torsionful spin connection on G/H becomes Γ a b = f a ib e i + 1 2 (κ+1) f a cb e c =: Γ a cb e c . (2.14)

Yang-Mills equations on R × G/H
Consider the space R × G/H with a coordinate τ on R, a one-form e 0 := dτ and the Euclidean metric g = (e 0 ) 2 + δ ab e a e b . (2.15) The torsionful spin connection Γ on R × G/H is given by (2.14), with Γ a cb = e i c f a ib + 1 2 (κ+1) f a cb and Γ 0 0b = Γ a 0b = Γ 0 cb = 0 . Consider the trivial principal bundle P (R×G/H, G) = (R×G/H)×G over R×G/H with the structure group G, the associated trivial complex vector bundle E over R×G/H and a g-valued connection one-form A on E with the curvature F = dA + A ∧ A. In the basis of one-forms {e 0 , e a } on R×G/H, we have In the following we choose a 'temporal' gauge in which A 0 ≡ A τ = 0. The Yang-Mills equations with torsion (2.2) on R×G/H are equivalent to

G-invariant gauge fields
Let us associate our complex vector bundle E → R × G/H with the adjoint representation adj(G) of the structure group G. Then the generators of G are realized as dim G×dim G matrices According to [51] (see also [52][53][54][55]), G-invariant connections on E are determined by linear maps Λ : m → g which commute with the adjoint action of H: Such a linear map is represented by a matrix (X B a ), appearing in For the cases we will consider one can always choose X i a = 0. In local coordinates the connection is written and its G-invariance imposes the condition The curvature F of the invariant connection (2.24) reads   (3)-structure manifolds used in flux compactifications of string theories (see e.g. [35][36][37][48][49][50] and references therein). Their geometry is fairly rigid and features a 3-symmetry, which generalizes the reflection symmetry of symmetric spaces. This allows for a very explicit description of their structure and a complete parametrization of G-invariant Yang-Mills fields, which we present in this section.

Nearly Kähler six-manifolds
An SU(3)-structure on a six-manifold is by definition a reduction of the structure group of the tangent bundle from SO(6) to SU(3). Manifolds of dimension six with SU(3)-structure admit a set of canonical objects, consisting of an almost complex structure J, a Riemannian metric g, a real two-form ω and a complex three-form Ω. With respect to J, the forms ω and Ω are of type (1,1) and (3,0), respectively, and there is a compatibility condition, g(J·, ·) = ω(·, ·). With respect to the volume form V g of g, the forms ω and Ω are normalized so that ω ∧ ω ∧ ω = 6V g and Ω ∧Ω = −8iV g .
Then, a nearly Kähler six-manifold is an SU(3)-structure manifold with the differentials dω = 3ρ ImΩ and dΩ = 2ρ ω ∧ ω (3.2) for some real non-zero constant ρ (if ρ was zero, the manifold would be Calabi-Yau). More generally, six-manifolds with SU(3)-structure are classified by their intrinsic torsion [56], and nearly Kähler manifolds form one particular intrinsic torsion class. There are only four known examples of compact nearly Kähler six-manifolds, and they are all coset spaces [33,34]: Here Sp(1)×U(1) is chosen to be a non-maximal subgroup of Sp(2): if the elements of Sp(2) are written as 2 × 2 quaternionic matrices, then the elements of Sp(1)×U(1) have the form diag(p, q), with p ∈Sp(1) and q ∈U(1). Also, SU(2) is the diagonal subgroup of SU(2) 3 . These coset spaces are all 3-symmetric, because the subgroup H is the fixed point set of an automorphism s of G satisfying s 3 = Id [33,34].
The 3-symmetry actually plays a fundamental role in defining the canonical structures on the coset spaces. The automorphism s induces an automorphism S of the Lie algebra g = h ⊕ m of G which acts trivially on h and non-trivially on m; one can define a map The map J satisfies J 2 = −1 and provides the almost complex structure on G/H. The components J a b of the almost complex structure J are defined via J(

Yang-Mills equations and action functional
In the previous subsection we described the geometry of nearly Kähler six-manifolds. Now we would like to consider the Yang-Mills theory on seven-manifolds R×G/H, where G/H is a nearly Kähler coset space. Note that on such manifolds one can introduce three-forms and Each of the two, Σ as well as Σ ′ , defines a G 2 -structure on R×G/H, i.e. a reduction of the holonomy group SO (7) to a subgroup G 2 ⊂ SO (7). From (3.18) [X a ,Ẋ a ] = 0 (sum over a) (3.21) after using the identities (3.8). We notice that the equations (3.20) and (3.21) are the equation of motion and the Gauss constraint for the action (3.23) The Euler-Lagrange equations for this matrix-model action are (3.20).

Solution of the G-invariance condition
The G-invariance condition (2.25), says that the X a must transform in the six-dimensional representation R of H which arises in the decomposition (2.21), It is real but reducible and decomposes into complex irreducible parts as with q p=1 dim R p = 3. This is the same H-representation as furnished by the I a . Hence, for each irrep R p one can find complex linear combinations I close among themselves for each p. In the absence of a condition on [X a , X b ], the X a appear linearly and thus may always be multiplied by a common factor φ p inside each irrep R p . By Schur's lemma this is in fact the only freedom, i.e.
is the unique solution to the G-invariance condition inside R p . The six antihermitian matrices X a are then easily reconstructed via and will depend on q complex functions φ p (τ ). The same holds for any smaller Grepresentation D instead of adj(G). For computations, we choose a basis in g such that the first dim(R 1 ) generators I α 1 span R 1 , the next dim(R 2 ) generators I α 2 span R 2 etc., and the last dim(H) generators span h. Such a basis decomposes R into the said blocks. Fusing all irreducible blocks and adj(H) together again, we obtain a realization of I i , I a and X a as matrices in adj(G). Since G is the gauge group, these matrices enter in the action (3.23). However, for calculations it is more convenient to take a smaller G-representation D. This affects only the normalization of the trace, tr D ( where the (2nd-order) Dynkin index χ D depends on the representation used. We normalize our generators such that χ adj(G) = 1, and choose D in all cases (see below) such that χ D = 1 6 . With this, the constant term in the action (3.23) computes to the solution to the SU(3)-invariance equation (3.24) then reads

Equations of motion
Substituting (4.4) into the action (3.23), we obtain the Lagrangian whose quartic terms may be rewritten as as well as The equations (4.7) are the Euler-Lagrange equations for the Lagrangian (4.5) obtained from (3.22) after fixing the gauge A 0 = 0.

Zero-energy critical points
Writing the equations of motion (4.7) as we see that they describe the motion of a particle on C 3 under the influence of the inverted quartic potential −V , where 10) or, alternatively, the dynamics of three identical particles on the complex plane, with an external potential given by the (negative of) the first line in (4.10) and two-and three-body interactions in the second line. The potential (4.10) is invariant under permutations of the φ i as well as under the U(1)×U(1) transformations which include the 3-symmetry, φ i → e 2πi/3 φ i . Such a transformation may be used to align the phases of the φ i , i.e. arg(φ 1 ) = arg(φ 2 ) = arg(φ 3 ). These phases only enter in the cubic term of the potential, which is proportional to cos( i arg φ i ). Therefore, the extrema of V are attained at i arg φ i = 0 or π, and so, employing (4.11), we may take φ i ∈ R in our search for them. 2 Furthermore, the Noether charges of the U(1)×U(1) symmetry (4.11) are just the differences ℓ i − ℓ j of the 'angular momenta' Hence, the constraints (4.8) may be interpreted as putting these charges to zero. Note, however, that the individual angular momenta are not conserved, sincė Finite-action solutions φ i (τ ) must interpolate between critical points with zero potential, (4.14) Modulo the symmetry (4.11) and permutations, the complete list of such critical points reads: where γ ± = −(1+ √ 3)±2 2( √ 3−1) takes the numerical values of −0.31 and −5.15. The zero modes of V ′′ are enforced by the symmetries; their number indicates the dimension of the critical manifold in C 3 . A critical point is marginally stable only when V ′′ has no positive eigenvalues. At the critical pointsl i = 0 is guaranteed, hence the product φ 1 φ 2 φ 3 has to be real unless κ = −3. The latter value is special because all phase dependence disappears, and the symmetry (4.11) is enhanced to U(1) 3 . We will not consider this special situation (type A') further. Appendix A proves that the list below is complete.

Some solutions
Finite-action trajectories φ i (τ ) require the conserved Newtonian energy to vanish, They can be of two types: . Since this choice occurs for each value of i = 1, 2, 3, mixed solutions are possible. We now present some special cases.
Transverse kinks at −3<κ<+3. The two-dimensional type A critical manifold exists for any value of κ, so one may try to find trajectories connecting two critical points of type A. As a particularly symmetric choice we wish to interpolate The three independent conserved quantities (E, ℓ i −ℓ j ) do not suffice to integrate the equations of motion (4.7), so generically one has to resort to numerical methods. With a little effort, zero-energy 'transverse' kinks can be found in the range κ ∈ (−3, +3). We display the trajectory (φ i (τ )) ∈ C 3 as three curves φ i (τ ) ∈ C in figure 1 for κ = −2, −1, 0, +1, +2. Apparently, the 3-symmetry effects a permutation since φ 2 (τ ) = e 2πi/3 φ 1 (τ ) = e −2πi/3 φ 3 (τ ). This relation takes care of the constraint (4.8). Of course, acting with the transformations (4.11) generates a two-parameter family of such 'transverse' kinks. At the magical value of κ=−1 the trajectories become straight, and the solution analytic:

(4.17)
Radial kinks at κ = 3. For this value of κ the critial point at the origin is degenerate with (1, 1, 1) and its symmetry orbits. Therefore, we can connect any type A critical point to the unique type B point via 'radial kinks', such as in a 3-symmetric fashion and is also marked in the lower right plot of figure 1. It is the limiting case of the transverse kinks for κ → +3. In the other limit, κ → −3, the particles move infinitely slowly on the degenerate unit circle, |φ| = 1. where f κ (τ ) is a real function, so the trajectories are straight. It is easy to find it numerically. Figure 2 shows the trajectories for κ = −4 and κ = +4.

JHEP10(2010)044
We choose the generators of the subgroup Sp(1)×U(1) of Sp(2) in the form Then solutions of the Sp(2)-invariance conditions (2.25) are given by matrices

Equations of motion
The equations of motion for Sp (2)

Some solutions
Clearly, the solutions to (5.4) and (5.5) form a subset of the solutions to (4.7) and (4.8), namely those where two functions coincide. Since in all examples of the previous section this can be arranged by applying a U(1)×U(1) transformation (4.11), one gets ϕ(τ ) = χ(τ ) equal to any of the functions appearing on the right-hand sides of (4.17) and (4.18) or depicted in figure 1, after dialling the corresponding κ value. In addition, (4.22) translates to a solution with ϕ ≡ 0 and a kink χ.
Let us for a moment investigate the possibility of straight-trajectory solutions φ(τ ) ∈ C to (5.12). With a 3-symmetry transformation, any such solution can be brought into a form where either Reφ(τ ) = const or Imφ(τ ) = const. Then, the vanishing of the left-hand side of Re (5.12) yields two conditions on Reφ and κ, whose solutions follow a Hamiltonian flow [24]:
Remark. Note that a nearly Kähler structure exists also on the space S 3 × S 3 . However, we do not consider the Yang-Mills equations on R × S 3 × S 3 since this was already done in [21].

Instanton-anti-instanton chains and dyons
If we replace R × G/H with S 1 × G/H, the time interval will be of finite length, namely the circle circumference L, and we are after solutions periodic in τ . In this case, the action is always finite, and the E=0 requirement gets replaced by φ i (τ +L) = φ i (τ ). The physical interpretation of such configurations is one of instanton-anti-instanton chains.

Periodic solutions
As the simplest case we take G/H = G 2 /SU(3) and consider the magical κ values which admit analytic solutions for φ(τ ) ∈ C. Switching from τ ∈ R to τ ∈ S 1 , we must impose the periodicity conditions φ(τ +L) = φ(τ )  At finite L, we obtain a different kind of solution (sphalerons), namely
The non-BPS solutions (6.3) can be embedded into the other cosets G/H, where they are special solutions, with ϕ = χ or φ 1 = φ 2 = φ 3 , respectively. Their degeneracy may be lifted by applying a symmetry transformation (5.9) or (4.11), respectively. Substituting our non-BPS solutions into (4.4) or (5.3) and then into (2.24), we obtain a finite-action Yang-Mills configuration which is interpreted as a chain of n instanton-anti-instanton pairs sitting on S 1 × G/H with six-dimensional nearly Kähler coset space G/H. Away from the magical κ values, such chains are to be found numerically.

Dyonic solutions
Let us finally change the signature of the metric on R × G/H from Euclidean to Lorentzian by choosing on R a coordinate t = −iτ so thatẽ 0 = dt = −idτ . Then as metric on R×G/H we have The G-invariant solutions (4.4) and (5.3) for the matrices X a are not changed. After substituting them into the Yang-Mills equations on R×G/H, we arrive at the same secondorder differential equations as in the Euclidean case, except for the replacemenẗ In particular, this implies a sign change of the left-hand side relative to the right-hand side in (4.7), (5.4) and (5.12). Thus, in the Lagrangians we effectively have a sign flip of the potential V , so that the analog Newtonian dynamics for (φ i (t)) is based on +V .

JHEP10(2010)044
Let us again for simplicity look at the case of G/H = G 2 /SU (3). Although the Lorentzian variant of (5.12), with V from (5.13), does not follow from first-order equations for any of the magical values κ = −1, −3, −7, +3 or +9, it can still be explicitly integrated in those cases, with (κ; β, γ) = + 3; 1 2 , 1 √ 3 , + 9; 1, 2 √ 3 . (6.10) The 3-symmetry action maps these solutions to rotated ones. Any such configuration is a bounce in our double-well-type potential, which most of the time hovers around a saddle point. For other values of κ, such bounce solutions may be found numerically. Inserting (6.10) into the gauge potential, we arrive at dyon-type configurations with smooth nonvanishing 'electric' and 'magnetic' field strength F 0a and F ab , respectively. The total energy − tr (2F 0a F 0a + F ab F ab ) × Vol(G/H) (6.11) for these configurations is finite, but their action diverges unless φ(±∞) = e 2πik/3 . These are saddle points for κ < −3 and κ > +5. Thus, for |κ−1| > 4 the potential (5.13) admits pairs φ ± (t) of finite-action dyons, with φ ± (±∞) = 1 and φ ± (0) = 1 6 κ−3 ± κ 2 −9 for κ > +5  and investigate the solution space of dV =0=V , i.e. and reproduce type C in the table. 3 It remains to study the situation where all φ i are nonzero. Multiplying (A.2) with φ i and taking the difference of any two of the resulting three equations, we obtain the three conditions Likewise, multiplying (A.2) with φ j φ k and taking the difference of any two of those three equations, we find three more conditions, A little thought reveals that there are only two options. The first one is The potential on this subspace becomes and its critical zeros on the positive real axis are The Jacobi elliptic functions arise from the inversion of the elliptic integral of the first kind, where k = mod u is the elliptic modulus and ξ = am(u, k) = am(u) is the Jacobi amplitude, giving ξ = F −1 (u, k) = am(u, k) .