Journal of Mathematical Physics Volume 32, No. 5 (1991), 1263-1268.

Duality and Yang-Mills Fields on Quaternionic Kähler Manifolds

Krzysztof Galicki and Yat Sun Poon

Abstract: The concept of a self-dual connection on a four dimensional Riemannian manifold is generalized to the 4n-dimensional case of any quaternionic Kähler manifold. The generalized self-dual connections are minima of a modified Yang-Mills functional. It is shown that our definitions give a correct framework for a mapping theory of quaternionic Kähler manifolds. The mapping theory is closely related to the construction of Yang-Mills fields on such manifolds. We also discuss some monopole-like equations.

1. Introduction

A quaternionic Kähler manifold is a Riemannian manifold whose holonomy group can be reduced to a subgroup of Sp(n)·Sp(1), n > 1 [1, 2]. By definition, such manifold has dimension 4n. As demonstrated by Salamon [2, 3] it can be also viewed as a higher dimensional analogy of the anti-self-dual Einstein 4-manifold. The bundle of 2forms on a quaternionic Kähler manifold M has the following irreducible decomposition as representation of Sp(n)·Sp(1)

Λ^{2}T^{∗}M = S^{2}ℍ ⊕ S^{2}𝔼 ⊕ (S^{2}ℍ ⊕ S^{2}𝔼)^{⊥},  (1.1)

where ℍ and 𝔼 are vector bundles associated to the standard representations of Sp(n) and Sp(1) respectively. This decomposition resembles the decomposition of Λ^{2}T^{∗}M into the direct sum of self-dual and anti-self-dual 2-forms when M is four dimensional. Just as in the four dimensional case we are able to interpret the decomposition (1.1) in terms of the Hodge ∗-operator.

During the preparation of this work both authors was supported by NSF grant DMS-8906806.

If the curvature of a connection ∇ is in either the S^{2}ℍ or the S^{2}𝔼 part of (1.1) then ∇ is a minimum of the Yang-Mills functional and if the curvature is in the orthogonal complement of S^{2}ℍ ⊕ S^{2}𝔼 then ∇ is most likely a saddle point. We have found that the Yang-Mills functional can be modified so that whenever the curvature of ∇ is in one and only one component of (1.1) the connection is its minimum.

We demonstrate that our definitions are compatible with the description of Yang-Mills fields on 4-manifolds and that it gives a correct framework for mapping theory of quaternionic Kähler manifolds. On the other hand, when the energy functional is interpreted as a classical Lagrangian, our quaternionic mapping theory yields many new examples of quantum field theories with SU(2) (or SO(3)) gauge symmetry and composite gauge fields: 4-dimensional sigma models. We show that some fundamental properties of the well-known 4-dimensional σ-models on the quaternionic projective spaces are shared by such models on arbitrary quaternionic Kähler manifolds. Finally, we demonstrate that our formalism provides a global picture for the generalized monopole equation of Pedersen and Poon [4].

2. Duality

Let M be a 4n-dimensional Riemannian manifold whose holonomy group is contained in Sp(n)·Sp(1) ⊂ SO(4n). Then the cotangent bundle of M can be identified with

T^{∗}M = 𝔼 ⊗ ℍ,

where 𝔼 and ℍ are the standard representations of Sp(n) and Sp(1) respectively. Then S^{2}ℍ is a real rank 3 subbundle of EndT M. Locally, at each x ∈ M, S^{2}ℍ has a basis {I, J, K} satisfying

I^{2} = J^{2} = −1I, IJ = −JI = K.  (2.1)

The metric g on M is compatible with the bundle S^{2}ℍ in the sense that for each A ∈ S^{2}ℍ_{x}, g is Hermitian with respect to A, i.e., g(AX, AY) = g(X, Y) for all X, Y ∈ T_{x}M. One can use the metric to define an isomorphism

EndT M ≅ T^{∗}M ⊗ T^{∗}M

under which S^{2}ℍ is isometrically embedded in Λ^{2}T^{∗}M. Explicitly, any element A ∈ S^{2}ℍ_{x} is mapped into ω_{A} by

ω_{A}(X, Y) = g(AX, Y), X, Y ∈ T_{x}M.

Let {ω_{1}, ω_{2}, ω_{3}} be a local orthogonal frame of S^{2}ℍ ⊂ Λ^{2}T^{∗}M. For convenience of further computations let us normalize {ω_{1}, ω_{2}, ω_{3}} to have length 2n and then define

Ω = ω_{1} ∧ ω_{1} + ω_{2} ∧ ω_{2} + ω_{3} ∧ ω_{3}.  (2.2)

This Ω is a globally defined, non-degenerate 4-form on M and it is parallel. It is usually called the fundamental 4-form or the quaternionic structure on M as its parallelism determines reduction of the structure group on M. The condition ∇Ω = 0 can be used to define quaternionic Kähler geometry in dimension bigger than 4. In dimension 4 we shall say that M is quaternionic Kähler if it is self-dual and Einstein. The parallelism of Ω immediately implies that dΩ = 0. Recently, Swann [5] showed that the converse is also true provided dimM ≥ 12.

Pointwisely, Ω can be described as follows. At any point x ∈ M, T_{x}^{∗}M = 𝔼_{x} ⊗ ℍ_{x}, where 𝔼_{x} is the 2n-dimensional complex representation of Sp(n) and ℍ_{x} is the 2dimensional complex representation of Sp(1). Let ω_{E} and ω_{H} be the symplectic forms on 𝔼_{x} and ℍ_{x} respectively and j_{E} and j the quaternionic structures. Then the metric g on T_{x}^{∗}M can be expressed as

g = ω_{E} ⊗ ω_{H}.  (2.3)

Let {e^{j}, j_{E}e^{j} : j = 1, ..., n} be a symplectic basis on 𝔼_{x} and {h, jh} a symplectic basis on ℍ_{x}. We define

\begin{matrix}ω_{0}^{j} & ≐ \frac{1}{\sqrt{2}}(e^{j} ⊗ h + j_{E}e^{j} ⊗ jh) \\ ω_{1}^{j} & ≐ \frac{i}{\sqrt{2}}(e^{j} ⊗ h − j_{E}e^{j} ⊗ jh) \\ ω_{2}^{j} & ≐ \frac{1}{\sqrt{2}}(j_{E}e^{j} ⊗ h − e^{j} ⊗ jh) \\ ω_{3}^{j} & ≐ \frac{i}{\sqrt{2}}(j_{E}e^{j} ⊗ h + e^{j} ⊗ jh). \end{matrix}  (2.4)

Now {ω_{0}^{j}, ω_{1}^{j}, ω_{2}^{j}, ω_{3}^{j}, j = 1, ..., n} forms an orthonormal basis on T_{x}^{∗}M. Let

ω_{1} ≐ ∑_{j=1}^{n}(ω_{0}^{j} ∧ ω_{1}^{j} + ω_{2}^{j} ∧ ω_{3}^{j})

ω_{2} ≐ ∑_{j=1}^{n}(ω_{0}^{j} ∧ ω_{2}^{j} − ω_{1}^{j} ∧ ω_{3}^{j})

ω_{3} ≐ ∑_{j=1}^{n}(ω_{0}^{j} ∧ ω_{3}^{j} + ω_{1}^{j} ∧ ω_{2}^{j}).  (2.5)

Then {ω_{1}, ω_{2}, ω_{3}} forms an orthogonal basis on S^{2}ℍ_{x}. We shall choose Ω as in (2.2). The orthogonal basis for S^{2}𝔼_{x} can be written as

\{ \begin{matrix}Σ_{0}^{ij} & ≐ (ω_{0}^{i} ∧ ω_{0}^{j} + ω_{1}^{i} ∧ ω_{1}^{j}) + (ω_{2}^{i} ∧ ω_{2}^{j} + ω_{3}^{i} ∧ ω_{3}^{j}), 1 ≤ i < j ≤ n, \\ Σ_{1}^{ij} & ≐ (ω_{0}^{i} ∧ ω_{1}^{j} − ω_{2}^{i} ∧ ω_{3}^{j}) + (ω_{0}^{j} ∧ ω_{1}^{i} − ω_{2}^{j} ∧ ω_{3}^{i}), 1 ≤ i ≤ j ≤ n, \\ Σ_{2}^{ij} & ≐ (ω_{0}^{i} ∧ ω_{2}^{j} + ω_{1}^{i} ∧ ω_{3}^{j}) + (ω_{0}^{j} ∧ ω_{2}^{i} + ω_{1}^{j} ∧ ω_{3}^{i}), 1 ≤ i ≤ j ≤ n, \\ Σ_{3}^{ij} & ≐ (ω_{0}^{i} ∧ ω_{3}^{j} − ω_{1}^{i} ∧ ω_{2}^{j}) + (ω_{0}^{j} ∧ ω_{3}^{i} − ω_{1}^{j} ∧ ω_{2}^{i}), 1 ≤ i ≤ j ≤ n. \end{matrix}  (2.6)

Σ_{0}^{ij} give n(n − 1)/2 basis elements and Σij_{A}, A = 1, 2, 3, give n(n + 1)/2 basis elements respectively. One can easily check that

vol(M) = \frac{1}{(2n + 1)!}Ω^{n}  (2.7)

vol(M) = \frac{1}{12n(2n + 1)}Ω ∧ ∗Ω,  (2.8)

where vol(M) is the volume form of M and “∗” is the Hodge ∗-operator. As a consequence we have

∗Ω = \frac{6}{(2n − 1)!}Ω^{n−1}.  (2.9)

Note that all these equations are valid even when n is equal to 1.

Definition 2.1: A 2-form ω on M is c-self-dual if

∗ω = cω ∧ Ω^{n−1}.  (2.10)

When n = 1 then c^{2} = 1, because ∗^{2} = 1, and the above equation is reduced to the conformally invariant self-dual or anti-self-dual equations on a 4-dimensional oriented Riemannian manifold. Notice that the above definition depends on the choice of both the fundamental 4-form Ω and the constant c. In dimension higher than 4, as we shall now see, there are three different constants c that give non-trivial solutions to (2.10). Similar equations were studied in [6].

Theorem 2.2: Let ω be a non-zero c-self-dual 2-form. Then c = c_{i}, i = 1, 2, 3, where

c_{1} = \frac{6n}{(2n + 1)!}, c_{2} = \frac{−1}{(2n − 1)!}, c_{3} = \frac{3}{(2n − 1)!}.  (2.11)

Moreover, when c = c_{1} then ω ∈ S^{2}ℍ, when c = c_{2} then ω ∈ S^{2}𝔼, and when c = c_{3} then ω is in the orthogonal complement of S^{2}ℍ ⊕ S^{2}𝔼 in Λ^{2}T^{∗}M.

Proof: As the basis for S^{2}ℍ is given in (2.5) and the basis for S^{2}𝔼 in (2.6) the proof is an easy exercise in linear algebra. Therefore we only spell out the constraints on the coefficients of the 2-form ω. Using the orthonormal basis {ω_{0}^{j}, ω_{1}^{j}, ω_{2}^{j}, ω_{3}^{j} : j = 1, ..., n} any 2-form ω can be written as

ω = ∑_{i,j,α,β} ω\binom{i}{α}\binom{j}{β}ω_{α}^{i} ∧ ω_{β}^{j}.

Then ∗ω = c_{1}ω ∧ Ω^{n−1} if and only if

\{ \begin{matrix}ω\binom{i}{0}\binom{i}{1} & = ω\binom{i}{2}\binom{i}{3} = ω\binom{j}{0}\binom{j}{1} = ω\binom{j}{2}\binom{j}{3} \\ ω\binom{i}{0}\binom{i}{2} & = −ω\binom{i}{1}\binom{i}{3} = ω\binom{j}{0}\binom{j}{2} = −ω\binom{j}{1}\binom{j}{3} \\ ω\binom{i}{0}\binom{i}{3} & = ω\binom{i}{1}\binom{i}{2} = ω\binom{j}{0}\binom{j}{3} = ω\binom{j}{1}\binom{j}{2} \end{matrix}  (2.12)

for all i, j

ω\binom{i}{0}\binom{j}{0} = ω\binom{i}{1}\binom{j}{1} = ω\binom{i}{0}\binom{j}{2} = ω\binom{i}{3}\binom{j}{3} = 0 ∀i, j  (2.13)

and

ω\binom{i}{α}\binom{j}{β} = 0 ∀i ≠ j ∀α, β.  (2.14)

Similarly, ∗ω = c_{2}ω ∧ Ω^{n−1} if and only if

\{ \begin{matrix}ω\binom{i}{0}\binom{j}{1} & = −ω\binom{i}{2}\binom{j}{3}, ω\binom{i}{0}\binom{j}{2} = ω\binom{i}{1}\binom{j}{3}, ω\binom{i}{0}\binom{j}{3} = −ω\binom{i}{1}\binom{j}{2} ∀i, j \\ ω\binom{i}{α}\binom{j}{α} & = ω\binom{i}{β}\binom{j}{β}, ∀i, j, α, β \\ ω\binom{i}{α}\binom{j}{β} & = ω\binom{j}{α}\binom{i}{β}, ∀i, j, α, β, α ≠ β. \end{matrix}  (2.15)

Finally, ∗ω = c_{3}ω ∧ Ω^{n−1} if and only if

\{ \begin{matrix}∑_{i=1}^{n} ω\binom{i}{0}\binom{i}{1} = ∑_{i=1}^{n} ω\binom{i}{0}\binom{i}{2} = ∑_{i=1}^{n} ω\binom{i}{0}\binom{i}{3} = 0 \\ ∑_{α=0}^{3} ω\binom{i}{α}\binom{j}{α} = 0 ∀ i, j \\ ω\binom{i}{0}\binom{j}{1} + ω\binom{j}{0}\binom{i}{1} = ω\binom{i}{2}\binom{j}{3} + ω\binom{j}{2}\binom{i}{3} ∀i, j \\ ω\binom{i}{0}\binom{j}{2} + ω\binom{j}{0}\binom{i}{2} = −(ω\binom{i}{1}\binom{j}{3} + ω\binom{j}{1}\binom{i}{3}) ∀i, j \\ ω\binom{i}{0}\binom{j}{3} + ω\binom{j}{0}\binom{i}{3} = ω\binom{i}{1}\binom{j}{2} + ω\binom{j}{1}\binom{i}{2} ∀i, j. \end{matrix}  (2.16)

Definition 2.3: Let P be a principal bundle on M with connection ∇. This connection is c-self-dual if its curvature 2-form is c-self-dual.

Definition 2.4: For any real constant c , a generalized “Yang-Mills” functional on the space of connections on P is defined by

Y M_{c}(∇) ≐ \frac{1}{2} ∫_{M}[∥F ∥^{2} + c^{2}∥F ∧ Ω^{n−1}∥^{2}]vol(M),  (2.17)

where F is the curvature of the connection.

Y M_{c}(∇) has the following Euler-Lagrange equations

d ∗ F + c^{2}(d ∗ (F ∧ Ω^{n−1})) ∧ Ω^{n−1} = 0.  (2.18)

Notice that

0 ≤ ∥ ∗F − cF ∧ Ω^{n−1}∥^{2}
= ∥ ∗F ∥^{2} − 2⟨∗F, cF ∧ Ω^{n−1}⟩ + c^{2}∥F ∧ Ω^{n−1}∥^{2}
= ∥F ∥^{2} − 2c(trF ∧ F) ∧ Ω^{n−1} + c^{2}∥F ∧ Ω^{n−1}∥^{2}
= ∥F ∥^{2} − 16cπ^{2}p_{1}(P) ∧ Ω^{n−1} + c^{2}∥F ∧ Ω^{n−1}∥^{2}

or

c(8π^{2})p_{1}(P) ∧ Ω^{n−1} ≤ \frac{1}{2}[∥F ∥^{2} + c^{2}∥F ∧ Ω^{n−1}∥^{2}],

where p_{1}(P) is the first Pontrjagin class of the bundle P on M. Hence, after integrating over M, we get

8π^{2}c ∫_{M} p_{1}(P) ∧ Ω^{n−1}vol(M) ≤ Y M_{c}(∇).  (2.19)

The equality holds if and only if

∗F = cF ∧ Ω^{n−1},

i.e., if F is c-self-dual. In such case we shall call the connection ∇ itself a c-self-dual-connection. As p_{1}(P) is a topological invariant of the bundle P, we define

Q(P) ≐ 8π^{2} ∫_{M} p_{1}(P) ∧ Ω^{n−1}vol(M)  (2.20)

and call it a topological charge of the bundle P. We have just demonstrated that

Proposition 2.5: Any c-self-dual connection is a minimum of the Yang-Mills energy functional Y M_{c}(∇).

The following result is due to [7].

Proposition 2.6: Any c-self-dual connection is an extremum of the Yang-Mills energy functional Y M(∇). Moreover, c_{1}- and c_{2}-self-dual connections are minimizing.

Proof: Suppose ∇ is a c-self-dual connection. Then

d ∗ F = cd ∗ (F ∧ Ω^{n−1}) = 0

as dF = dΩ = 0. Hence, d ∗ F = 0 or ∇ is a Yang-Mills connection.

Let us write F (∇) ∈ Λ^{2}T^{∗}M as
F (∇) = F_{1} + F_{2} + F_{3},

where F_{1} ∈ S^{2}ℍ, F_{2} ∈ S^{2}𝔼, and F_{3} ∈ (S^{2}ℍ ⊕ S^{2}𝔼)^{⊥}. Then

Y M(∇) = \frac{1}{2} ∫_{M}(∥F_{1}∥^{2} + ∥F_{2}∥^{2} + ∥F_{3}∥^{2})vol(M)

because (1.1) is an orthogonal decomposition with respect to the usual norm ∥ · ∥ on Λ^{2}T^{∗}M. Notice that the topological charge of P can be written in terms of the components of F (∇)

Q(P) = ∫_{M} tr(F ∧ F) ∧ Ω^{n−1}vol(M) = ∫_{M}(\frac{1}{c_{1}} ∥F_{1}∥^{2} + \frac{1}{c_{2}} ∥F_{2}∥^{2} + \frac{1}{c_{3}} ∥F_{3}∥^{2})vol(M).

Hence, we can write Y M(∇) as

\begin{aligned}2Y M(∇) & = c_{1}Q(P) + ∫_{M}((1 − \frac{c_{1}}{c_{2}} )∥F_{2}∥^{2} + (1 − \frac{c_{1}}{c_{3}} ∥F_{3}∥^{2})vol(M) = \\ & = c_{1}Q(P) + ∫_{M}((1+ \frac{3}{2n+1})∥F_{2}∥^{2} + (1− \frac{1}{2n+1})∥F_{3}∥^{2})vol(M) \end{aligned}  (2.21)

or

\begin{aligned}2Y M(∇) & = c_{2}Q(P) + ∫_{M}((1 − \frac{c_{2}}{c_{1}} )∥F_{1}∥^{2} + (1 − \frac{c_{2}}{c_{3}} )∥F_{3}∥^{2})vol(M) = \\ & = c_{2}Q(P) + ∫_{M}((1+ \frac{2n+1}{3})∥F_{1}∥^{2} + \frac{4}{3}∥F_{3}∥^{2})vol(M) \end{aligned}  (2.22)

or

\begin{aligned}2Y M(∇) & = c_{3}Q(P) + ∫_{M}((1 − \frac{c_{3}}{c_{1}} )∥F_{1}∥^{2} + (1 − \frac{c_{3}}{c_{2}} )∥F_{2}∥^{2})vol(M) = \\ & = c_{3}Q(P) + ∫_{M}((−2n)∥F_{1}∥^{2} + 4∥F_{2}∥^{2})vol(M) \end{aligned}  (2.23)

It follows now from (2.21), (2.22) and Theorem 2.2 that c_{1}- and c_{2}-self-dual connections are minima of Y M(∇).

We do not know of any examples of c_{3}-self-dual connections but (2.23) seems to indicate that, if they exist, they will be unstable.

3. Quaternionic maps and sigma models

In this chapter we introduce a new concept of quaternionic maps. We shall do it in such a way that it generalizes the theory of holomorphic mappings between Kähler manifolds. On the other hand we shall see that it is also very natural in studying instantons on 4manifolds and 4-dimensional σ-models with composite SU(2) (or SO(3)) gauge fields and Yang-Mills fields on quaternionic Kähler manifolds.

It is well-known that, if one defines a quaternionic Kähler submanifold to be a submanifold with a quaternionic structure given by restrictions, then it is automatically a totally geodesic submanifold [4]. We shall therefore not insist that the whole quaternionic structure be preserved by such mappings. Instead we adopt a weaker definition.

Definition 3.1: Let M, N be quaternionic Kähler manifolds. A map f from M to N is called quaternionic if f^{∗}S^{2}ℍ_{N} ⊂ S^{2}ℍ_{M}.

The following theorem is in an obvious analogy to the well-known result stating that holomorphic maps between Kähler manifolds are energy minimizing.

Theorem 3.2: On the space of differentiable mappings between two compact oriented quaternionic Kähler manifolds M and N define the following functional

E(f) ≐ \frac{1}{2} ∑_{i=1}^{3} ∫_{M}(∥f^{∗}ω_{i}∥^{2} + c^{2}∥f^{∗}ω_{i} ∧ Ω^{m−1}∥^{2})vol(M),  (3.1)

where c = c_{1} = \frac{6m}{(2m+1)!}, 4m = dimM, and

Q(f) ≐ ∫_{M} f^{∗}Ω_{N} ∧ Ωm−1_{M}.  (3.2)

Then cQ(f) ≤ E(f) and equality holds if and only if the map f is quaternionic.

Proof: Let Ω_{M}, Ω_{N} be the fundamental 4-forms on M and N respectively. Once they are fixed Q(f) is a homotopy invariant. As usual, we shall call it the degree or the topological charge of f.

Let {ω_{1}, ω_{2}, ω_{3}} be a local orthogonal frame on S^{2}ℍ_{N} such that
Ω_{N} = ω_{1} ∧ ω_{1} + ω_{2} ∧ ω_{2} + ω_{3} ∧ ω_{3}.

We have to show that E(f) is well-defined. If ω_{i} = ∑_{j} ϕ_{ij}μ_{j} is an SO(3)-rotation of the frame field on S^{2}ℍ_{N} then pointwisely

f^{∗}ω_{i} = ∑_{i=1}^{3}(ϕ_{ij})f^{∗}μ_{j}.

Furthermore,

∑_{i=1}^{3} ∥f^{∗}ω_{i}∥^{2} = ∑_{i=1}^{3} f^{∗}ω_{i} ∧ ∗f^{∗}ω_{i} = ∑_{i=1}^{3} ∑_{j,k}(ϕ_{ij}f^{∗}μ_{j}) ∧ ∗(ϕ_{ik}f^{∗}μ_{k}) =

= ∑_{i=1}^{3} ∑_{j,k}(ϕ_{ij}ϕ_{ik})(f^{∗}μ_{j} ∧ ∗f^{∗}μ_{k}) = ∑_{i=1}^{3} ∑_{j,k}(ϕ_{ji}^{−1}ϕ_{ik})(f^{∗}μ_{j} ∧ ∗f^{∗}μ_{k}) =

= ∑_{j,k} δ_{jk}f^{∗}μ_{j} ∧ ∗f^{∗}μ_{k} = ∑_{j=1}^{3} f^{∗}μ_{j} ∧ ∗f^{∗}μ_{j} = ∑_{j=1}^{3} ∥f^{∗}μ_{j}∥^{2}.

Similarly,

∑_{i=1}^{3} ∥f^{∗}ω_{i} ∧ Ω^{m−1}∥^{2} = ∑_{i=1}^{3}(f^{∗}ω_{i} ∧ Ω^{m−1}) ∧ ∗(f^{∗}ω_{i} ∧ Ω^{m−1}) =

= ∑_{i=1}^{3} ∑_{j,k}(ϕ_{ij}ϕ_{ik})(f^{∗}μ_{j} ∧ Ω^{m−1}) ∧ ∗(f^{∗}μ_{k} ∧ Ω^{m−1}) =

= ∑_{j=1}^{3}(f^{∗}μ_{j} ∧ Ω^{m−1}) ∧ ∗(f^{∗}μ_{j} ∧ Ω^{m−1}) = ∑_{j=1}^{3} ∥f^{∗}μ_{j} ∧ Ω^{m−1}∥^{2}.

Hence, E(f) is independent of the choice of any normalized frame on S^{2}ℍ_{N} and therefore well defined. Now the inequality cQ(f) ≤ E(f) follows from

0 ≤ ∥ ∗ f^{∗}ω_{i} − cf^{∗}ω_{i} ∧ Ω^{m−1}∥^{2}

which can be written as

c⟨∗f^{∗}ω_{i}, f^{∗}ω_{i} ∧ Ω^{m−1}⟩ ≤ \frac{1}{2}(∥f^{∗}ω_{i}∥^{2} + c^{2}∥f^{∗}ω_{i} ∧ Ω^{m−1}∥^{2}).  (3.3)

Since

⟨∗f^{∗}ω_{i}, f^{∗}ω_{i} ∧ Ω^{m−1}⟩ = f^{∗}ω_{i} ∧ f^{∗}ω_{i} ∧ Ω^{m−1}

and

f^{∗}Ω_{N} = ∑_{i=1}^{3} f^{∗}ω_{i} ∧ f^{∗}ω_{i},

the inequality cQ(f) ≤ E(f) is simply obtained by summation of (3.3) over i and integration over M.

Finally, when c = \frac{6m}{(2m+1)!}, the assertion that cQ(f) = E(f) is equivalent to the requirement that

∗f^{∗}ω = cf^{∗}ω ∧ Ωm−1_{M}

holds for all ω ∈ S^{2}ℍ_{N}, or that f^{∗}ω ∈ S^{2}ℍ_{M} by Theorem 2.2., i.e., f is quaternionic.

Example 3.3: If dimM = 4, S^{2}ℍ_{M} ≅ Λ_{+}^{2}. As the Hodge ∗-operator is conformally invariant, any orientation preserving conformal automorphism is a quaternionic map in our sense.

In [8] Atiyah gave a geometric construction for all basic SU(2)-instantons, i.e., antiself-dual Yang-Mills fields on the Euclidean 4-sphere with topological charge −1, as follows: The Euclidean 4-sphere is viewed as the quaternionic projective line ℍℙ^{1}. The tautological bundle is the bundle ℍ with charge −1. The natural connection ∇ of ℍ is anti-self-dual. Let f be an orientation preserving conformal automorphism which is not an isometry. Then f^{∗}∇, the pull-back connection on f^{∗}ℍ, is a new anti-self-dual connection.

Example 3.4: The above example can be easily generalized as follows: The quaternionic projective space ℍℙ^{n} has a tautological bundle ℍ. By definition, any element of GL^{+}(n + 1, ℍ) is an orientation preserving quaternionic linear map. In other words, if f ∈ GL^{+}(n+ 1, ℍ) is considered as an automorphism of ℍℙ^{n}, then f^{∗}ℍ is isomorphic to ℍ. It follows that f^{∗}S^{2}ℍ ≡ S^{2}ℍ and hence f is a quaternionic map. As the natural connection ∇ on ℍ is c_{1}-self-dual, so is f^{∗}∇. Besides, as long as f is not an isometry, f^{∗}∇ is not gauge equivalent to ∇. We do not know if these are all c_{1}-self-dual connections on ℍℙ^{n}.

Example 3.5: Another well-known example of a mapping which in our language is quaternionic is a general SU(2)-instanton over 4-sphere with the topological charge k [8, 9]. The S^{2}ℍ bundle on the quaternionic projective space ℍℙ^{k} has a canonical Sp(1)-connection and all instantons over S^{4} are induced by an appropriate choice of f : S^{4} ⟶ ℍℙ^{k}. In fact f can be described explicitly as follows: If u ∈ ℍℙ^{k} is a local (Fubini-Study) quaternionic coordinate on the quaternionic projective space and x ∈ S^{4} is a local quaternionic coordinate on the 4-sphere identified with the quaternionic projective line ℍℙ^{1} then

u(x) = [λ⃗ · (𝔹 − x1I)]^{†}  (3.4)

where λ⃗ = (λ_{1}, ..., λ_{k}) is a quaternionic row vector, u is a quaternionic column vector, 𝔹 is a symmetric quaternionic n × n matrix, † denotes quaternionic conjugation and transposition, and (λ⃗, 𝔹) are subject to the following two conditions

\{ \begin{matrix}Im(𝔹^{†}𝔹 + λ⃗^{†}λ⃗) = 0 \\ (∀x ∈ ℍℙ^{1} (𝔹 − x1I)ξ⃗ = 0, λ⃗ · ξ⃗ = 0 where ξ⃗ ∈ ℍ^{k}) ⇒ ξ⃗ = 0. \end{matrix}  (3.5)

In the same way k-instantons over the complex projective plane can be generated by quaternionic maps from bbcℙ^{2} ⟶ ℍℙ^{2k} [10, 11].

The energy functional (3.1) may also be interpreted as an SO(3) locally gauge invariant lagrangian of an interesting class of nonlinear field theories called σ-models. In particular, if dimM = 4, one can think of M as a physical, possibly curved, space-time and f(x), x ∈ M, becomes an N-valued classical field with the action functional given by E(f). E(f) is manifestly invariant with respect to the global coordinate transformation on M (diffeomorphisms of M) as well as it is gauge invariant under the following gauge transformations

(f^{∗}ω_{i})_{x} ⟶ ∑_{j} Φ_{ij}(x)(f^{∗}ω_{j})_{x},  (3.6)

where Φ_{ij}(x) is a local SO(3) transformation and (f^{∗}ω_{i}) is the curvature 2-form of a gauge field A_{j} on M defined as follows

d(f^{∗}ω_{i}) = ∑_{j,k} ϵ_{ijk}A_{j} ∧ f^{∗}ω_{k}.  (3.7)

The gauge potential 1-form on A_{j} transforms in the usual way

δ(ϵ_{ijk}A_{k}) = −dΦ_{ij}(x).  (3.8)

A_{j}(f) depends on the choice of f(x), i.e., it is a composite gauge field. If N = ℍℙ^{n} and u ∈ ℍℙ^{n} as before then

A(u) = −\frac{1}{2} \frac{u^{†} · du − du^{†} · u}{1 + u^{†} · u} = iA_{1} + jA_{2} + kA_{3}.

This particular example was introduced and extensively studied by Gürsey and Tze [12]. Here we see that many interesting global and local properties of ℍℙ^{n}-model are common for a large class of field theoretical models based on E(f). All of them have duality equations built in and all possess global topological invariants.

4. Generalized Bogomolny equations

In this section we discuss some special solutions of the c-self-duality equations. If M = ℝ^{4} ∋ (x_{0}, x_{1}, x_{2}, x_{3}) and P is a principal bundle over M then one can study x_{0}-invariant solutions to the usual self-dual equations. They are called time invariant instantons or monopoles. In our case, let M = ℝ^{4n} ≃ ℝ^{4} ⊗ ℝ^{n} ∋ {x_{α}^{i}}_{α=0,...,3}^{i=1,...,n}, P be a principal bundle over M, and let Y M_{c}(∇) be our Yang-Mills functional. In an obvious analogy to the 4dimensional case we can study x_{0}^{i} invariant c-self-dual connections on M or “c-monopoles” on ℝ^{3} ⊗ ℝ^{n}. Let us start with the following observation.

Proposition 4.1: Let M = ℝ^{4} ⊗ ℝ^{n} be the 4n-dimensional Euclidean flat space with global linear coordinates x_{α}^{i}, α = 0, 1, 2, 3; i = 1, ..., n. For any (x_{1}, ..., x_{n}) in ℝ^{n} we define

p : ℝ^{4} ⟶ ℝ^{4} ⊗ ℝ^{n}

by

(x_{0}, x_{1}, x_{2}, x_{3}) ⟶ x_{α}^{i} = x_{α}x^{i}.  (4.1)

Suppose P is a principal bundle over M with connection ∇ and curvature F. Then p^{∗}∇ is an anti-self-dual connection on p^{∗}P if

∗F = −\frac{1}{(2n − 1)!}F ∧ Ω^{n−1},  (4.2)

i.e., F is c_{2}-self-dual.

Proof: In the x_{α}^{i}-coordinates dx_{α}^{i} is exactly the 1-form ω_{α}^{i} of (2.4). Now a 2-form ω satisfies the equation

∗F = −\frac{1}{(2n − 1)!}F ∧ Ω^{n−1}

if and only if

F = −\frac{1}{6} ∗ (F ∧ ∗Ω).  (4.3)

Using Theorem 2.2 we get the following solutions

\{ \begin{matrix}F\binom{i}{0}\binom{j}{1} & = −F\binom{i}{2}\binom{j}{3}, F\binom{i}{0}\binom{j}{2} = F\binom{i}{1}\binom{j}{3}, F\binom{i}{0}\binom{j}{3} = −F\binom{i}{1}\binom{j}{2} ∀i, j \\ F\binom{i}{α}\binom{j}{α} & = F\binom{i}{β}\binom{j}{β}, ∀i, j, α, β \\ F\binom{i}{α}\binom{j}{β} & = F\binom{j}{α}\binom{i}{β}, ∀i, j, α, β, α ≠ β. \end{matrix}  (4.4)

Let us denote the components of p^{∗}F by F_{αβ}. As a consequence of the chain rule we get

F_{αβ} = ∑_{i,j} x^{i}x^{j}F\binom{i}{α}\binom{j}{β}  (4.5)

and therefore

F_{01} = ∑_{i,j} x^{i}x^{j}F\binom{i}{0}\binom{j}{1} = −F_{23}

F_{02} = ∑_{i,j} x^{i}x^{j}F\binom{i}{0}\binom{j}{2} = F_{13}

F_{03} = ∑_{i,j} x^{i}x^{j}F\binom{i}{0}\binom{j}{3} = −F_{12}.  (4.6)

In other words, p^{∗}∇ is an anti-self-dual connection.

Recently, Pedersen and Poon used twistorial approach to find a generalization of the Bogomolny equations [5]. They introduced Yang-Mills-Higgs equations on ℝ^{3} ⊗ ℝ^{n}. If one considers monopoles on ℝ^{3} as time invariant instantons on ℝ^{4} the following simple geometric description of generalized monopoles comes with no surprise.

Proposition 4.2: Let x_{μ}^{i}, μ = 0, 1, 2, 3; i = 1, ..., n be a global linear coordinate on ℝ^{4} ⊗ ℝ^{n} and let

p : ℝ^{4} ⊗ ℝ^{n} ⟶ ℝ^{3} ⊗ ℝ^{n}

be a projection

(x_{0}^{i}, x_{1}^{i}, x_{2}^{i}, x_{3}^{i}) ⟶ (x_{1}^{i}, x_{2}^{i}, x_{3}^{i}).

If (∇, Φ^{i}) is a generalized monopole then

∇′ ≐ p^{∗}∇ + ∑_{i} Φ^{i}dx_{0}^{i}  (4.7)

is a c_{2}-self-dual connection. Conversely, any c_{2}-self-dual connection that is x_{0}^{i}-invariant determines a solution of the generalized monopole equation.

Proof: The curvature F′ of the connection ∇′ is given by

F′ = p^{∗}F + ∑_{j}(∇Φ^{j}) ∧ dx_{0}^{j} + \frac{1}{2} ∑_{i<j}[Φ^{i}, Φ^{j}]dx_{0}^{i} ∧ dx_{0}^{j},  (4.8)

where F is the curvature 2-form of ∇. Now, using equations (4.4), we get

\{ \begin{matrix}∇\binom{j}{1}Φ^{i} & = F\binom{i}{2}\binom{j}{3}, ∇\binom{j}{2}Φ^{i} = −F\binom{i}{1}\binom{j}{3}, ∇\binom{j}{3}Φ^{i} = F\binom{i}{1}\binom{j}{2} ∀i, j \\ F\binom{i}{α}\binom{j}{α} & = \frac{1}{2}[Φ^{i}, Φ^{j}], ∀i, j; α = 1, 2, 3 \\ ∇\binom{i}{α}Φ^{j} & = ∇\binom{j}{α}Φ^{i} ∀i, j; α = 1, 2, 3 \end{matrix}  (4.9)

which can be written as

\{ \begin{matrix}F\binom{i}{α}\binom{j}{β} & = ∑_{γ} ϵ_{αβγ}∇\binom{i}{γ}Φ^{j} + \frac{1}{2}δ_{αβ}[Φ^{i}, Φ^{j}] ∀i, j; ∀α, β = 1, 2, 3 \\ ∇\binom{i}{α}Φ^{j} & = ∇\binom{j}{α}Φ^{i} ∀i, j; α = 1, 2, 3. \end{matrix}  (4.10)

The converse is obvious.

We can also obtain “monopole” analogues of c-self duality equations in the c_{1} and c_{3} cases. The first one is not interesting, however, because it yields n decoupled self-dual Bogomolny equations. In the second case we can explicitly write down the set of equations

\{ \begin{matrix}F\binom{i}{α}\binom{j}{β} + F\binom{j}{α}\binom{i}{β} & = ∑_{γ} ϵ_{αβγ}(∇\binom{i}{γ}Φ^{j} + ∇\binom{j}{γ}Φ^{i}), ∀i, j; ∀α, β \\ ∑_{i=1}^{n} ∇\binom{i}{α}Φ^{i} = 0 ∀α, & [Φ^{i}, Φ^{j}] = − ∑_{α=1}^{3} F\binom{i}{α}\binom{j}{α} ∀i, j. \end{matrix}  (4.11)

For n = 1 these are just the usual Bogomolny equations with the reversed orientation. We do not known any non-trivial solutions of (4.11) for n > 1 at the moment. Finally, let us remark that we could introduce additional invariance and reduce the c-self-duality equation to 2n dimensions, assuming that the c-self-dual equations on ℝ^{4} ⊗ ℝ^{n} be both x_{0}^{i} and x_{1}^{i}-invariant. Then we obtain an analogue of the well-known vortex equation of the two-dimensional Yang-Mills-Higgs theory. Again c_{2} case is the most natural generalization and we shall address this problem in a future work.

References

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[2] S. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67, 143-171(1982).

[3] S. Salamon, Differential geometry of quaternionic manifolds, Ann. Scient. Éc. Norm. Sup. 19, 31-55(1986).

[4] H. Pedersen and Y. S. Poon, Hyper-Kähler Metrics and a Generalization of the Bogomolny Equations, Commun. Math. Phys. 117, 569-580(1989).

[5] A. Swann, On the Fundamental 4-Form of a Quaternion-Hermitean Manifold, Oxford University preprint (1990).

[6] E. Corrigan, P. Goddard and A. Kent, Some comments on the ADHM construction in 4k dimensions, Commun. Math. Phys. 100, 1-13(1985).

[7] M. M. Capria and S. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1, 517-530(1988).

[8] M. F. Atiyah, Geometry of Yang-Mills Fields, Lezioni Fermiane, Pisa 1979.

[9] M. F. Atiyah, V. G. Drin’feld, N. J. Hitchin, and Yu. I. Manin,Construction of instantons,

Phys. Lett., 65A, 185-187(1978).

[10] N. Buchdal, Instantons on ℂℙ^{2}, J. Differential Geometry 24, 19-52(1986).

[11] S. K. Donaldson, Vector bundles on flag manifolds and the Ward correspondence, Geometry Today, Progress in Math., No. 60, Birkhauser, Boston, 1985.

[12] F. Gürsey and C. H. Tze, Complex and quaternionic analyticity and gauge theories I, Annals of Physics 128, 29-130(1980).

Department of Mathmatics July 1990 Rice University P.O. Box 1892 Houston, Texas 77251