Yang-Mills Solutions and Dyons on Cylinders over Coset Spaces with Sasakian Structure

We present solutions of the Yang-Mills equation on cylinders $\mathbb R\times G/H$ over coset spaces with Sasakian structure and odd dimension $2m+1$. The gauge potential is assumed to be $SU(m)$-equivariant, parametrized by two real, scalar-valued functions. Yang-Mills theory with torsion in this setup reduces to the Newtonian mechanics of a point particle moving in $\mathbb R^2$ under the influence of an inverted potential. We analyze the critical points of this potential and present an analytic as well as several numerical finite-action solutions. Apart from the Yang-Mills solutions that constitute $SU(m)$-equivariant instanton configurations, we construct periodic sphaleron solutions on $S^1\times G/H$ and dyon solutions on $i\mathbb R\times G/H$.


Introduction
Higher-dimensional Super-Yang-Mills theory appears in the context of string theory for example as the low-energy limit of the heterotic superstring. In this limit, heterotic string theory yields ten-dimensional supergravity coupled to N = 1 supersymmetric Yang-Mills theory [1,2]. In four dimensions, the full Yang-Mills equation is implied by the instanton equation, a first-order anti-self-duality equation. This fact generalizes to dimensions greater than four. The higher-dimensional instanton equation is particularly interesting for string compactifications on manifolds of the form M 10−d × X d with compact part X d and maximally symmetric M 10−d . Requiring the gauge field on the compact manifold to satisfy the instanton equation ensures the preservation of N = 1 supersymmetry on the non-compact spacetime part. Instantons in higher dimensions were first studied in [3], and solutions to the generalized anti-self-duality equation have been constructed, for example, in [4][5][6][7][8][9][10].
The requirement of supersymmetry preservation in string compactifications of the above type translates to a condition on the geometry of the compact internal manifold: imposing the instanton equation on the gauge field on X d is equivalent to requiring reduced holonomy on the compact space. In heterotic compactifications, Calabi-Yau 3-folds have been the preferred choice, leading to phenomenologically interesting models with N = 1 supersymmetry. Furthermore, G 2 -holonomy 7-manifolds as well as 8-manifolds with Spin(7)-holonomy have been of interest in this context. The problem of heterotic Calabi-Yau compactifications is that they come with a number of scalar fields with undetermined vacuum expectation value. These moduli can be fixed by allowing for nonvanishing p-forms, so-called fluxes, to exist on the internal compact manifold. Flux compactifications do solve the moduli problem but enlarge the number of possible string backgrounds significantly, leading to the string landscape problem. For a review of flux compactifications, see for example [11,12].
Nontrivial background fluxes on the internal compact manifold imply a backreaction on the geometry, relaxing the condition on the holonomy of the manifold. X d is no longer required to have reduced holonomy but to admit a G-structure, i.e. a reduction of the tangent bundle structure group from SO(d) to some subgroup G ⊂ SO(d). Such manifolds are equipped with a connection with totally antisymmetric torsion. In the following, the torsion will be determined up to a real scaling parameter κ. We will consider Sasakian manifolds of dimension 2m + 1 with structure group G = SU (m). For a particular choice of metric, these manifolds are in addition Einstein. Sasaki-Einstein manifolds have been studied in the context of non-compact flux backgrounds as AdS/CFT duals of confining gauge theories or, more precisely, as type IIB AdS vacua that lead to dual N = 1 Super Yang-Mills theories coupled to matter [11,13,14].
In this paper, we concentrate on cylinders R × G/H over coset spaces with Sasakian structure. We start by repeating the basics of Yang-Mills equations, G-structure and in particular Sasakian manifolds in chapter 2. We use an SU (m)-equivariant ansatz for the gauge connection, parametrized by two real scalar functions, to write out the Yang-Mills equation in components in chapter 3. This leads to a system of two coupled second-order ordinary differential equations, reducing Yang-Mills theory with torsion to the Newtonian mechanics of a point particle moving in R 2 under the influence of a potential. The shape of this potential depends on the torsion parameter κ. We derive the corresponding particle action in chapter 4 and discuss the critical points of zero energy. For a special value of κ, the second-order equations can be solved analytically and yield a tanh-kink-type solution, similar to solutions discussed before [15,16]. We construct further finite-action solutions numerically. When S 1 × G/H instead of R × G/H is considered, we obtain periodic solutions, so-called sphalerons, which are discussed in section 4.2. Considering the product space iR × G/H instead of R × G/H leads to a sign flip in the potential. Solutions to this case are known as dyons and can be constructed numerically. We present some of them in section 4.3.
In this note, we consider Sasakian manifolds of dimension 2m + 1, where 1 ≤ m ∈ N. A detailed introduction to Sasakian geometry can be found in [18] and [19]. Let us review the most important facts here.
Sasakian manifolds are special types of contact manifolds. According to [20,21], an almost contact structure (Φ, η, ξ) on an odd-dimensional Riemannian manifold (M, g) is characterized by a nowhere vanishing vector field ξ and a one-form η, satisfying η(ξ) = 1, plus a (1, 1)-tensor Φ such that Φ 2 = −1 + ξ ⊗ η. Such a structure is called contact if in addition the one-form satisfies η ∧ (dη) m = 0. (2.1) In this case, η is called contact form, and ξ is referred to as Reeb vector field. Contact structures are normal if for their Nijenhuis tensor N the relation N = −dη ⊗ ξ holds. 1 When the Riemannian metric g on an almost contact manifold (M, g) satisfies for any vector fields X, Y ∈ T M , the structure is referred to as almost contact metric. It is called contact metric if in addition with a two-form ω(X, Y ) := g(X, ΦY ), is satisfied.
A Sasakian manifold is defined to be a manifold with contact metric structure. Such manifolds admit a reduction of the tangent bundle structure group from SO(2m + 1) to U (m), which allows (apart from the existence of the one-form η ∈ Ω 1 (M ) and two-form ω ∈ Ω 2 (M )) for the introduction of forms P M ∈ Ω 3 (M ) and Q M ∈ Ω 4 (M ) that satisfy the following relations: The contraction is defined as η ω = * M (η ∧ * M ω) by use of the Hodge star operator on (M, g M ) (see for example [8]). All these forms are parallel with respect to the canonical connection introduced below. In addition to equation (2.3), the forms satisfy the relations (2.7) Condition (2.3) can be generalized. If the structure satisfies dη = αω for some real α, it is referred to as α-Sasakian. We will see below that in our case the α-Sasakian structure can be transformed into a Sasakian structure by rescaling of basis elements.
If the metric on a Sasakian manifold is proportional to the Ricci tensor, Ric ∝ g, we have a Sasaki-Einstein manifold. Note that a Sasakian structure need not necessarily be Einstein.

Canonical connection
Connections on the tangent bundle T M over a manifold M are locally determined by matrix-valued one-forms Γ ν µ = Γ ν σµ e σ , using Greek indices {µ, ν} = {1, 2, ...2m + 1} to label directions on M and a basis {e σ } of non-holonomic one-forms. A connection on the tangent bundle of a G-structure manifold M is called canonical if it has holonomy group G and totally antisymmetric torsion with respect to some G-compatible metric. We denote such a connection by ∇ P , or, locally, by P Γ ν µ . All of the above listed Gstructure manifolds come equipped with canonical 3-and 4-forms P M and Q M , and in all these cases the torsion of the canonical connection is proportional to the 3-form P M . The canonical connection can therefore be constructed as a sum of the (torsion-free) Levi-Civita connection ∇ LC and the 3-form P M , and is in the Sasakian case given by Here and in the following, we use the shorthand notation e a ∧ e b = e ab . The canonical connection ∇ P is compatible with the following family of metrics, all of which are Sasakian up to homothety: This can be seen by rescaling the metric with a real parameter γ, g h,γ = γ 2 (e 1 e 1 + e 2h δ ab e a e b ), (2.12) and introducing new basis forms e 1 = γe 1 , e a = γe h e a , such that the metric takes the form Remember that the original basis one-forms satisfy the Sasaki relation de 1 = 2ω. For still being Sasakian after rescaling, the new structure constants have to satisfy an analogous condition. We find (2.14) The structure is therefore α-Sasakian for all α(h, γ) (hence also for all scaling factors γ) and Sasakian (i.e. d e 1 = 2 ω) for the special value α = 2, or, equivalently, γ = e −2h .
For h = 0, the structure becomes Einstein. The value e 2h = 2m m+1 is special as it makes the torsion of the canonical connection totally antisymmetric. We will restrict our consideration to the latter case from now on and not study the Einstein case in detail.

Lie algebra structure and connection on Sasakian manifolds
We will consider Yang-Mills theory on the product manifold R × M over a Sasakian manifold with metric where e 0 := dτ denotes the coordinate in R direction. The canonical connection on M lifts to a connection on the tangent bundle over R × M with structure group SU (m). Note that in this case SU (m) is the holonomy group of the canonical connection. We will however use a more general connection, namely a perturbation of ∇ P by parallel sections (cf. [22] for details). The gauge group of this perturbed connection is SU (m+1). The corresponding Lie algebras split according to su(m + 1) = su(m) ⊕ m, where SU (m) acts irreducibly on su(m) and m denotes the (2m+1)-dimensional orthogonal complement to su(m). Representation theoretic arguments and the requirement of SU (m)-equivariance then allow for a connection of the following form, where χ(t) and ψ(τ ) are real functions of the variable τ parametrizing the R direction: We denote the generators of su(m) as {I i }, the dual one-forms by {e i } and the generators of m as {I 1 , I a }. They are chosen to be dual to the above introduced one-forms {e 1 , e a }.
The frame {e i } on su(m) can be expressed as a linear combination of the one-forms {e µ } as e i = e i µ e µ , where the e i µ are real functions. Written as matrices, the generators have the following nonvanishing entries: In this basis, the SU (m) structure constants satisfy Note that these identites differ from the corresponding equations in [22] by a sign. The sign here is in agreement with equations (2.17).

Yang-Mills equation on Sasakian manifolds
Before specifying to Sasakian coset spaces, let us review some general facts about the Yang-Mills equation. Consider a Riemannian manifold (N, g N ) of dimension d > 4, and let E be a complex vector bundle over N with connection A. We denote by F = dA+A∧A the curvature of this connection and by * the Hodge operator with respect to the metric g N . The generalized anti-self-duality equation is well-defined on any such manifold once it is equipped with a 4-form Q ∈ Ω 4 (N ), and given by * F = − * Q ∧ F.
Written out in components, (3.2) turns into the following set of 2m + 3 equations, with the free index B running from 0 to 2m + 2: For further specification, we have to compute the components of the 3-form H. According to [22], the 4-form on the cylinder over a Sasakian manifold is given by Using the definition H := * d * Q and decomposition rules for antisymmetric tensor indices, we find At this point, we have to distinguish between indices in 0-,1-and all other directions and find that the following components vanish for all m: The remaining components depend on the value of m. We demonstrate this by writing out H 231 explicitly, using equations (2.18) and (3.12). The other nonvanishing components of H behave in a similar way.
The B = 0 equation is identically satisfied. Let us take a look at the cases with B > 0.
, as well as from [A A , F AB ], we obtain terms proportional to the functions e i µ . These terms add up to zero by use of the Jacobi identity and SU (m)-equivariance of the connection and will therefore be omitted in the following computation. We evaluate the remaining terms explicitly. The connection coefficients are derived from the Maurer-Cartan structure equation where κ ∈ R is a real parameter, they take the form The value κ = 1 describes the instanton case discussed in [22]. For the derivation of explicit second-order equations, we will use along with equations (2.18). Using equation (2.16) and omitting the τ -dependence of the functions χ and ψ, we obtain the following curvature: Inserting F, T A , ω C AB as above and using The proof for equation (3.18) can be found in Appendix A.

Action functional and potential
The second-order equations (3.19) are equations of motion for the action where * M denotes the Hodge star operator on the Sasakian mainfold M with respect to the metric g M = e 1 e 1 + 2m m+1 δ ab e a e b , * denotes the Hodge operator on the cylinder, and V ol(M ) = |g M |e 1,2,··· ,2m+1 is the volume form on M . This can be verified by a direct computation, presented in Appendix B. Equations (3.19) form a gradient system of the form The potential V is symmetric with respect to sign changes of ψ and has the following critical points (i.e.χ =ψ = 0) for arbitrary m, κ: where the abbreviation Eigenvalues of Jacobian , 0, 0, − (m + 1) 2 2m 2 (0, ± 2(m + 1)) Table 2: Values of κ for which more than two critical points lie on the same axis is used. Finite-action Yang-Mills solutions χ(τ ), ψ(τ ) must interpolate between zero potential critical points. With κ arbitrary, the potential vanishes for the second critical point ( for the first critical point (χ 1 , ψ 1 ), which vanishes only for κ = 1, as well as lengthy nonzero expressions for V (χ 3 , ψ 3 ) and V (χ 4 , ψ 4 ). The critical points are listed in Table 1, together with the κvalues for which their potential becomes zero. For the values of κ listed in Table 2, more than two critical points are located on the same axis, and hence the system may admit analytic solutions. In addition, we note that at κ = m−2 m−1 , five of the seven critical points coincide at (0, 0), at κ = , (χ 4 , ψ 4 ) coincides with (χ 2 , ψ 2 ) and (χ 3 , ψ 3 ) becomes imaginary.

Analytic Yang-Mills solutions
The case κ = 1 1−m admits an analytic solution to the Yang-Mills equation, interpolating between the critical points (1, √ 2m) and (1, − √ 2m) for arbitrary m. All other critical points are located on the χ-axis and have potential V = 1 2 (m + 1) 2 . The zero-potential critical points are therefore minima of V , and we expect to find interpolating finite-action Yang-Mills solutions. With χ = 1, equations (3.19) take the form Equation (4.6) can be integrated to the first-order equatioṅ which is solved by This is a kink solution with finite energy and finite action. A plot of this solution in the χ, ψ-plane can be found in Figure 3.
For κ = 3 1−m , there are three critical points on the χ = 0 axis. However, none of them has zero potential, and we do not find any analytic solutions.

Periodic solutions
A different kind of solutions is obtained by changing from R × M to S 1 × M , i.e. when the additional direction is not a real line but a unit circle with circumference L. In this case, periodic boundary conditions have to be imposed: (4.9) We restrict the consideration to the analytically solvable case (4.6), which has the periodic solution This solution is known as a sphaleron [23]. sn[u, k] with 0 ≤ k ≤ 1 is a Jacobi elliptic function, details of which can be found for example in Appendix B of [16] or in [24]. The Jacobi elliptic function has a period of 4K(k), where K(k) denotes the complete elliptic integral of the first kind. The boundary condition (4.10) therefore turns into 4K(k)n = m + 1 fixing k = k(L, n) and φ(τ ; k(L, n)) =: φ (n) (τ ). Solutions (4.10) exist if L ≥ 2 3 2 πn (cf. [15,16]). The topological charge of the sphaleron φ(n) is zero due to the periodic boundary conditions. This solution is interpreted as a configuration of n kinks and n antikinks, alternating and equally spaced around the circle. The tanh-solution from chapter 4.1 arises from the Jacobi elliptic function in the limit k → 1. In the limit k → 0, the elliptic function approaches sin m+1 √ m(1+k 2 ) τ . In analogy to results in [25], our solution (4.10) with positive sign has the following total energy, with E(k) denoting the complete elliptic integral of the second kind:

Dyons
Replacing the coordinate τ in R direction by iτ changes the signature of the metric from Riemannian to Lorentzian: g = −e 0 e 0 + e 1 e 1 + e 2h δ ab e ab . (4.13) The Yang-Mills equations (3.19) remain unchanged, except for the fact that the secondorder derivatives now come with a minus sign: (χ,ψ) → (−χ, −ψ) (4.14) This corresponds to a sign flip of the potential, so that we have to study V instead of −V . Dyons are finite-energy solutions to the second-order equations obtained by this sign flip. Just as Yang-Mills solutions, they can interpolate between two critical points (kink), or start and end at the same point (bounce). Solutions that oscillate around a minimum can exist as well, but they do not lead to finite energy and hence will not be considered in the following.

Discussion and summary of solutions
Recall that in our sign convention, instanton solutions interpolate between minima and dyon solutions between maxima of V . In both cases, solutions that start or end at a saddle point are possible as well. With this in mind, we can expect the following solutions: • κ arbitrary: there exist at least two zero-potential critical points at (0, ± √ 2m) for all κ. According to Appendix C, they can be minima or saddle points of V , κ Eigenvalues of Jacobian (χ 1 , ψ 1 ) = (0, 0) 1 9, 9 4 (χ 2 , ψ 2 ) = (1, ±2) any 9 4 5 + κ + √ 5(1 + κ) , (1, 2) depending on the value of κ. This means that we can always find interpolating solutions, either of dyon or Yang-Mills type. These solutions have to be constructed numerically unless κ = 1 1−m . • κ = 1: this is the instanton case. Yang-Mills solutions exist between (0, 0) and (1, ± √ 2m) (cf. [22]). We do not expect to find finite-action dyon solutions, as the zero-potential critical points of V are minima.
• κ = 1 1−m (κ = −1 for m = 2): in this case, we find three nonzero critical points along the χ axis. An analytic Yang-Mills solution interpolates between the two remaining zero-potential critical points, which are minima for all m. This solution for arbitrary m is presented in chapter 4.1.
• κ = 3 1−m (κ = −3 for m = 2): we find four zero-potential critical points. Two of them are located at the lines with χ = 1 and χ = −1, respectively. We do not find any analytic solutions along the χ = ±1 and χ = 0 axes. There should, however, be a number of numerical solutions interpolating between various pairs of critical points.
We do not expect any analytic dyon solutions, as the zero-potential critical points are minima in the analytically solvable cases. For a better understanding, we present the case m = 2 as an example. The potential for various interesting values of κ is shown in Figure  1, and further dyon and Yang-Mills solutions for this example are presented in Figures 2  and 4. The list of zero-potential critical points can be found in Table 3.

Conclusion and outlook
Using a special ansatz for the gauge connection, we have derived a system of explicit second-order Yang-Mills equations on the cylinder over a class of Sasakian manifolds. We have constructed the corresponding action and potential, discussed the behaviour of the critical zero-potential points and found analytic as well as numerical solutions of Yang-Mills, dyon and sphaleron type.
A similar discussion for cylinders over certain SU (3)-structure manifolds can be found in [16]. A comparison with our results illustrates that Sasakian and SU (3)-structures are fundamentally different. The perhaps most striking fact is that the 3-symmetry of the SU (3)-structure manifold is recovered in the shape of the potential, whereas the potential in the Sasakian case is symmetric only under sign changes of the variable ψ. Furthermore, the Sasakian potential does not admit as many solutions with straight trajectories in the (χ, ψ)-plane as the SU (3)-structure potential does. In the latter case, the distribution of κ-dependent and κ-independent zero-potential critical points allows to systematically associate certain types of solutions (kinks, bounces) to intervals of the deformation parameter κ. In particular, there are always three critical points on the real axis. The Sasakian potential admits fewer κ-independent zero-potential critical points, and they are not as regularly distributed as in the SU (3)-case. The range and type of our solutions is therefore significantly different.
In spite of these differences, we have found that Sasakian manifolds do admit various interesting solutions. This, and in particular the fact that we have found an analytic kinktype solution of the Yang-Mills equation, makes them potentially interesting for string compactifications. It may be worth studying the instanton solution (4.8) in the context of the AdS/CFT duality mentioned in the introduction.
To complete the discussion, it would be interesting to consider Yang-Mills and dyon solutions on cylinders over G 2 -structure manifolds, i.e. 8-dimensional manifolds with Spin(7)-structure. We are planning to present results for this case in the near future. In addition, the analysis for the 3-Sasakian case is still missing. Another open question is how the Yang-Mills and dyon solutions change when considering cones and sine-cones instead of cylinders. On conical manifolds, the second-order equations acquire a first-order friction term, hence the analysis might have to be done numerically.
We can expect Yang-Mills solutions when these extrema are maxima, i.e. for κ − > κ > κ + , or saddle points, and dyon solutions when they are saddle points. As λ 1 and λ 2 do not simultaneously become smaller than zero, the critical points never become minima.