# Gauge Theory on Projective Surfaces and Anti-self-dual Einstein Metrics in Dimension Four

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## Abstract

Given a projective structure on a surface \(N\), we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space *M* of a certain rank 2 affine bundle \(M \rightarrow N\). The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on \(\mathbb {RP}^2\) is the non-compact real form of the Fubini–Study metric on \(M=\mathrm {SL}(3, \mathbb {R})/\mathrm {GL}(2, \mathbb {R})\). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

## Keywords

Projective structures Anti-self-dual metrics Einstein metrics Cartan geometry Gauge theory## Mathematics Subject Classification

Primary 53A20 Secondary 53B30 53C25 70S15## 1 Introduction

A projective structure on a smooth surface *N* is an equivalence class \([\nabla ]\) of torsion-free connections on *TN* having the same unparametrised geodesics. Canonically associated to a projective surface \((N,[\nabla ])\) is a rank 2 affine bundle \(M \rightarrow N\) which is modelled on \(T^*N\) and which arises as the complement of a certain \(\mathbb {RP}^1\)-subbundle of the projectivised cotractor bundle \(\mathbb {P}(E)\rightarrow N\) of \((N,[\nabla ])\). The aim of this paper is to canonically construct a pair \((g,\Omega )\) on *M*, consisting of a neutral signature anti-self-dual (ASD) Einstein metric *g*, as well as a symplectic form \(\Omega \). The pair \((g,\Omega )\) is related by an endomorphism \(I : TM \rightarrow TM\) whose square is the identity and hence it defines what is known as a bi-Lagrangian structure or almost para-Kähler structure on *M*. We construct the pair \((g,\Omega )\) by taking a \(\mathrm {GL}(2,\mathbb {R}\))-quotient of the Cartan geometry associated to \((N,[\nabla ])\) and in doing so, establish a one-to-one correspondence between projective vector fields on \((N,[\nabla ])\) and symplectic Killing vector fields on \((M,g,\Omega )\). In addition, we observe that every Killing vector field of (*M*, *g*) is symplectic with respect to \(\Omega \) and hence the lift of a projective vector field on \((N,[\nabla ])\).

*N*. The metric \(g_{\nabla }\) is thus part of a one-parameter family \(g_{\nabla ,\Lambda }\) of metrics on \(T^*N\) defined by

In the final part of the article we relate the metric *g* to a certain gauge-theoretic equation introduced by Calderbank in [7]. We also discuss some examples.

This paper mainly concerns itself with the two-dimensional case, but there are obvious higher dimensional generalisations which we briefly discuss in Appendix A.

## 2 Preliminaries

### 2.1 Algebraic Preliminaries

As usual, we let \(\mathbb {R}^n\) denote the space of column vectors of height *n* with real entries and \(\mathbb {R}_n\) the space of row vectors of length *n* with real entries. Matrix multiplication \(\mathbb {R}_n \times \mathbb {R}^n \rightarrow \mathbb {R}\) is a non-degenerate pairing identifying \(\mathbb {R}_n\) with the dual vector space of \(\mathbb {R}^n\).

Let \(\mathbb {RP}^2=(\mathbb {R}^3\setminus \{0\})/\mathbb {R}^*\) denote the space of lines in \(\mathbb {R}^3\) through the origin, i.e. two-dimensional real projective space. For any non-zero \(x \in \mathbb {R}^3\) let [*x*] denote its corresponding point in \(\mathbb {RP}^2\). Let \(\mathbb {RP}_2=(\mathbb {R}_3\setminus \{0\})/\mathbb {R}^*\) denote the dual projective space and likewise for any non-zero \(\xi \in \mathbb {R}_3\) we denote by \([\xi ]\) its corresponding point in \(\mathbb {RP}_2\).

*H*, so that \(\mathbb {RP}^2\simeq \mathrm {SL}(3,\mathbb {R})/H\). The elements of \(H\subset \mathrm {SL}(3,\mathbb {R})\) are matrices of the form

*H*acts faithfully from the left by affine transformations on the affine 2-space \(\mathbb {A}_2=\mathbb {RP}_2\setminus \mathbb {RP}_1\). Indeed, if we represent an element in \(\mathbb {A}_2\) by a vector \((1,\xi ) \in \mathbb {R}_3\) with \(\xi \in \mathbb {R}_2\), we obtain

*H*as the two-dimensional real affine group.

### 2.2 Projective Structures

*N*we mean a torsion-free connection on its tangent bundle

*TN*. The set of torsion-free connections on

*TN*is an affine space modelled on the smooth sections of the vector bundle \(V=TN\otimes S^2(T^*N)\). We have a canonical trace mapping \(V \rightarrow T^*N\) and an inclusion

*V*decomposes into a direct sum \(V\simeq V_0\oplus T^*N\), where \(V_0\) denotes the trace-free part of

*V*.

*X*,

*Y*,

*Z*on

*N*. We define the Ricci curvature of \(\nabla \) to be

*X*,

*Y*on

*N*.

^{1}The Ricci curvature need not be symmetric and we denote by \(\mathrm {Ric}^{\pm }(\nabla )\) its symmetric and anti-symmetric part.

*K*defined on some open set \(U\subset N\) is called

*affine*for the torsion-free connection \(\nabla \) on

*TN*if its local flow \(\phi _t\) preserves the geodesics of \(\nabla \). The set of such vector fields on

*U*is a Lie subalgebra of the Lie algebra of vector fields on

*U*which we will denote by \(\mathcal {A}_\nabla (U)\). Clearly, \(K \in \mathcal {A}_{\nabla }(U)\) if and only if

*U*. A straightforward computation yields that (2.1) is equivalent to the vanishing of the symmetric part of \(\nabla ^2 K\). By definition, the map \(K \mapsto \mathcal {L}_{K}\nabla \) takes values in \(\Gamma (V)\) and hence defines a second-order linear differential operator \(\mathcal {L}^{\nabla } : \Gamma (TN) \rightarrow \Gamma (V)\).

*projectively equivalent*if they share the same unparametrised geodesics. By a classical result of Weyl [45] this is equivalent to \(\hat{\nabla }-\nabla \) being pure trace, that is, the existence of a 1-form \(\Upsilon \) on

*N*such that

*X*,

*Y*on

*N*. Consequently, the set of projective structures on

*N*is an affine space modelled on the smooth sections of \(V_0\).

*K*defined on some open set \(U\subset N\) is said to be

*projective*for \([\nabla ]\) if its local flow \(\phi _t\) preserves the unparametrised geodesics of \([\nabla ]\). The set of such vector fields on

*U*is a Lie subalgebra of the Lie algebra of vector fields on

*U*which we will denote by \(\mathcal {P}_{[\nabla ]}(U)\). A vector field

*K*belongs to \(\mathcal {P}_{[\nabla ]}(U)\) if and only if

*U*, where \(\nabla \in [\nabla ]\), and the explicit expression for \(\mathcal {L}_{K}\nabla \) is given by (3.11). By definition, the right-hand side of (2.4) is a smooth section of \(V_0\) so that the map \(K \mapsto \mathcal {L}_{K}[\nabla ]\) defines a second-order linear differential operator \(\mathcal {L}^{[\nabla ]} : \Gamma (TN) \rightarrow \Gamma (V_0)\).

If *N* is orientable, we may restrict attention to connections in \([\nabla ]\) which preserve an area form \(\epsilon \) on *N*, so that \(\nabla \epsilon =0\). We shall refer to such connections as *special* [21]. Note that special connections always exist globally. For special connections the Schouten tensor is symmetric, that is \(\mathrm {P}_{[ij]}=0\). The residual freedom in special connections within a given projective class is given by (2.2) where \(\Upsilon =\mathrm{d} f\) for some smooth real-valued function *f* on *N*. The special condition is preserved if \(\hat{\epsilon }=e^{3f}\epsilon \).

### 2.3 The Cartan Geometry of a Projective Surface

*H*-bundle \(\pi : P_{[\nabla ]}\rightarrow N\) together with a Cartan connection \(\theta \in \Omega ^1(P_{[\nabla ]},\mathfrak {sl}(3,\mathbb {R}))\) having the following properties:

- (i)
\(\theta (X_v)=v\) for every fundamental vector field \(X_v\) on \(P_{[\nabla ]}\);

- (ii)
\(\theta _u : T_uP_{[\nabla ]}\rightarrow \mathfrak {sl}(3,\mathbb {R})\) is an isomorphism for all \(u \in P_{[\nabla ]}\);

- (iii)
\(R_h^*\theta =\mathrm {Ad}(h^{-1})\theta =h^{-1}\theta h\) for all \( h\in H\);

- (iv)Writefor an \(\mathbb {R}^2\)-valued 1-form \(\omega =(\omega ^i)\), an \(\mathbb {R}_2\)-valued 1-form \(\eta =(\eta _i)\) and a \(\mathfrak {gl}(2,\mathbb {R})\)-valued 1-form \(\phi =(\phi ^i_j)\). If \(X_{x}\) is a vector field on \(P_{[\nabla ]}\) having the property that$$\begin{aligned} \theta =\left( \begin{array}{cc} -{\text {tr}}\phi &{} \eta \\ \omega &{} \phi \end{array}\right) \end{aligned}$$for some non-zero \(x \!\in \mathbb {R}^2\), then the integral curves of \(X_{x}\), when projected to$$\begin{aligned} \omega (X_{x})=x, \quad \eta (X_{x})=0, \quad \phi (X_{x})=0, \end{aligned}$$
*N*, become geodesics of \([\nabla ]\) and conversely every geodesic of \([\nabla ]\) arises in this way; - (v)The curvature 2-form \(\Theta \) satisfiesfor a smooth curvature function \(L : P_{[\nabla ]}\rightarrow \mathrm {Hom}\left( \mathbb {R}^2\wedge \mathbb {R}^2,\mathbb {R}_2\right) \).$$\begin{aligned} \Theta =\mathrm{d} \theta +\theta \wedge \theta =\left( \begin{array}{cc} 0 &{} L(\omega \wedge \omega ) \\ 0 &{} 0\end{array}\right) , \end{aligned}$$(2.5)

*flat*if locally \([\nabla ]\) is defined by a flat connection. A consequence of Cartan’s construction is that a projective structure is flat if and only if

*L*vanishes identically.

### Remark 2.1

Cartan’s bundle is unique in the following sense: If \((\hat{\pi } : \hat{P}_{[\nabla ]} \rightarrow N,\hat{\theta })\) is another Cartan geometry of type \((\mathrm {SL}(3,\mathbb {R}),H)\) satisfying the properties (iii),(iv),(v), then there exists an *H*-bundle isomorphism \(\psi : P_{[\nabla ]}\rightarrow \hat{P}_{[\nabla ]}\) so that \(\psi ^*\hat{\theta }=\theta \).

### Remark 2.2

Let *w* be any real number. The line bundle associated to \(P_{[\nabla ]}\) via the *H*-representation \(\chi _w : H \rightarrow \mathrm {GL}^+(1,\mathbb {R})\), \(b\rtimes a \mapsto |\det a|^w\) will be denoted by \(\mathcal {E}(w)\). Following [2], we call its sections *densities of projective weight* *w*. In particular, nowhere vanishing sections of \(\mathcal {E}(1)\) are known as *projective scales*.

### 2.4 The Choice of a Representative Connection

For what follows it is necessary to have an explicit construction of the Cartan geometry \((\pi : P_{[\nabla ]}\rightarrow N,\theta )\) of a projective surface \((N,[\nabla ])\). This can be achieved conveniently by fixing a representative connection \(\nabla \in [\nabla ]\). To this end let \(\upsilon : F \rightarrow N\) denote the coframe bundle of \(N\) whose fibre at a point \(p \in N\) consists of the linear isomorphisms \(u : T_pN\rightarrow \mathbb {R}^2\). The group \(\mathrm {GL}(2,\mathbb {R})\) acts transitively from the right on each \(\upsilon \)-fibre by the rule \(R_a(u)=u \cdot a=a^{-1}\circ u\) for all \(a \in \mathrm {GL}(2,\mathbb {R})\). This action turns \(\upsilon : F \rightarrow N\) into a principal right \(\mathrm {GL}(2,\mathbb {R})\)-bundle. The bundle \(F \rightarrow N\) is equipped with a tautological \(\mathbb {R}^2\)-valued 1-form \(\omega =(\omega ^i)\) satisfying the equivariance property \((R_a)^*\omega =a^{-1}\omega \), where the 1-form \(\omega \) is defined by \(\omega _u=u\circ \upsilon ^{\prime }_{u}\).

*H*-action on \(F\times \mathbb {R}_2\) by the rule

*H*-bundle over \(N\). On \(F\times \mathbb {R}_2\) we define the \(\mathfrak {sl}(3,\mathbb {R})\)-valued 1-form

### 2.5 The Patterson–Walker Metric

*N*. As before, let \(\upsilon : F \rightarrow N\) denote the coframe bundle of

*N*with tautological 1-form \(\omega \) and let \(\varphi \) denote the connection form of \(\nabla \). The cotangent bundle \(\nu : T^*N \rightarrow N\) is the bundle associated to the \(\mathrm {GL}(2,\mathbb {R})\)-representation \(\chi \) on \(\mathbb {R}_2\) defined by the rule \( \chi (a)\xi =\xi a^{-1} \) for all \(a \in \mathrm {GL}(2,\mathbb {R})\) and \(\xi \in \mathbb {R}_2\). The 1-forms on \(F\times \mathbb {R}_2\) that are semi-basic for the projection \(\zeta : F\times \mathbb {R}_2 \rightarrow T^*N\simeq (F\times \mathbb {R}_2)/\sim _{\chi }\) are spanned by the components of \(\omega \) and \(\mathrm{d} \xi -\xi \varphi \). In particular, the equivariance properties of \(\omega ,\theta \) and \(\xi \) imply that the tensor field \(\left( \mathrm d\xi -\xi \varphi \right) \omega =\left( d\xi _i-\xi _k\varphi ^k_i\right) \otimes \omega ^i\) is invariant under the \(\mathrm {GL}(2,\mathbb {R})\)-right action,

*Patterson–Walker metric*or the

*Riemannian extension*of \(\nabla \). In canonical local coordinates \((x^i,\xi _i)\) on an open subset of the cotangent bundle, it takes the form

### 2.6 Anti-self-duality

*M*be an oriented four-dimensional manifold with a metric

*g*of signature (2, 2). The Hodge \(*\) operator is an involution on two-forms, and induces a decomposition

*g*. The Riemann tensor of

*g*has the symmetry \(R_{abcd}=R_{[ab][cd]}\) so can be thought of as a map \(\mathcal {R}: \Lambda ^{2}(T^*M) \rightarrow \Lambda ^{2}(T^*M)\) which admits a decomposition under (2.11):

*g*is ASD if \(C_{+}=0\). It is ASD and Einstein if \(C_{+}=0\) and \(\phi =0\). Finally it is ASD Ricci-flat (or equivalently hyper-symplectic) if \(C_{+}=\phi =\Lambda =0\). In this case the Riemann tensor is also anti-self-dual.

*M*equipped with parallel symplectic structures \({\varepsilon }, {\varepsilon }'\) such that

*M*iff \(C_+=0\).

## 3 From Projective to Bi-Lagrangian Structures

*H*also acts faithfully on \(\mathbb {R}_2\) by affine transformations defined by the rule

*H*-action is a rank-2 affine bundle \(M\rightarrow N\). We will refer to

*M*as the

*canonical affine bundle*of \((N,[\nabla ])\).

*M*is an equivalence class \([u,\xi ]\) with \(u\in P_{[\nabla ]}\) and \(\xi \in \mathbb {R}_2\) subject to the equivalence relation

*M*has a representative (

*u*, 0), unique up to a \(\mathrm {GL}(2,\mathbb {R})\) transformation, where here \(\mathrm {GL}(2,\mathbb {R})\subset H\) consists of those elements \(b\rtimes a \in H\) satisfying \(b=0\). For simplicity of notation, we will henceforth write

*a*instead of \(0\rtimes a\) for the elements of \(\mathrm {GL}(2,\mathbb {R})\subset H\). It follows that as a smooth manifold

*M*is canonically diffeomorphic to the quotient \(P_{[\nabla ]}/\mathrm {GL}(2,\mathbb {R})\) and we let \(\mu : P_{[\nabla ]}\rightarrow M\) denote the quotient projection.

### Remark 3.1

It can be shown that the sections of \(M \rightarrow N\) are in one-to-one correspondence with the \([\nabla ]\)-representative connections. The submanifold geometry in *M* of representative connections is studied in depth in two articles by the second author [32, 33].

*N*is the bundle associated to \(P_{[\nabla ]}\) via the natural

*H*-action on \(\mathfrak {sl}(3,\mathbb {R})/\mathfrak {h}\) induced by the adjoint representation of

*H*on its Lie algebra \(\mathfrak {h}\). An element in the Lie algebra \(\mathfrak {sl}(3,\mathbb {R})\) of \(\mathrm {SL}(3,\mathbb {R})\) can be written as

*H*is

Since \(\chi \) is precisely the linear part of the affine *H*-action (3.1), we see that the affine bundle \(M \rightarrow N\) is modelled on the cotangent bundle of *N*.

### 3.1 A Bundle Embedding

It turns out that we can embed \(P_{[\nabla ]}\rightarrow M\) as subbundle of the coframe bundle \(F\rightarrow M\) of *M*. Here, we define a coframe at \(p \in M\) to be a linear isomorphism \(T_pM \rightarrow \mathbb {R}_2\oplus \mathbb {R}^2\) and we denote the tautological \(\mathbb {R}_2\oplus \mathbb {R}^2\)-valued 1-form on *F* by \(\zeta \).

*M*, a vector field \(X\) on

*M*is represented by a unique \((\mathbb {R}_2\oplus \mathbb {R}^2)\)-valued function \((X_+,X_-)\) on \(P_{[\nabla ]}\) satisfying the equivariance condition

*M*and for all \(u \in P_{[\nabla ]}\)

*M*whose structure group is isomorphic to \(\mathrm {GL}(2,\mathbb {R})\). Furthermore, unravelling the definition of \(\zeta \), it follows that we have

*M*defined by the reduction of the coframe bundle of

*M*is a bi-Lagrangian structure, so we will study these structures next.

### 3.2 Bi-Lagrangian Structures

*bi-Lagrangian*structure on a smooth 4-manifold

*M*(or more generally an even dimensional manifold) consists of a symplectic structure \(\Omega \) together with a splitting of the tangent bundle of

*M*into a direct sum of \(\Omega \)-Lagrangian subbundles \(E_{\pm }\)

*I*is the unique endomorphism of the tangent bundle having these properties and, therefore, we may equivalently think of a bi-Lagrangian structure as a pair \((\Omega ,I)\) consisting of a symplectic structure \(\Omega \) and a \(\Omega \)-skew-symmetric endomorphism \(I : TM \rightarrow TM\) whose square is the identity.

*I*is skew-symmetric. Of course, a bi-Lagrangian structure is also equivalently described in terms of the pair (

*g*,

*I*) or the pair \((g,\Omega )\).

### Remark 3.2

What we call a bi-Lagrangian structure is also referred to as an *almost para-Kähler structure* and a *para-Kähler structure* provided \(E_{\pm }\) are both Frobenius integrable. Note that in [4] the term bi-Lagrangian structure is reserved for the case where both \(E_{\pm }\) are Frobenius integrable.

### Remark 3.3

We call a vector field defined on some open subset \(U\subset (M,\Omega ,I)\) *bi-Lagrangian* if its (local) flow preserves both \(\Omega \) and *I*. The set of such vector fields on *V* is a Lie subalgebra of the Lie algebra of vector fields on *V* which we will denote by \(\mathcal {B}_{(\Omega ,I)}(U)\).

*M*. To this end consider the symmetric bilinear form of signature (2, 2) on \(\mathbb {R}_2\oplus \mathbb {R}^2\)

*M*we say that a coframe

*u*at \(p \in M\) is

*adapted*to \((\Omega ,I)\) if for all \(v, w\in T_pM\)

*M*adapted to \((\Omega ,I)\) defines a reduction \(\lambda : B_{(\Omega ,I)}\rightarrow M\) of the coframe bundle \(F \rightarrow M\) of

*M*with structure group \(\mathrm {GL}(2,\mathbb {R})\). Conversely, every reduction of the coframe bundle of

*M*with structure group \(\mathrm {GL}(2,\mathbb {R})\) defines a unique pair \((\Omega ,I)\), consisting of a non-degenerate 2-form on

*M*and a \(\Omega \)-skew-symmetric endomorphism \(I : TM \rightarrow TM\) whose square is the identity. Note however that \(\Omega \) need not be closed.

*F*is said to be adapted to \((\Omega ,I)\) if it pulls back to \(B_{(\Omega ,I)}\) to become a principal \(\mathrm {GL}(2,\mathbb {R})\)-connection on \(B_{(\Omega ,I)}\). An adapted connection is given by a \(\mathfrak {gl}(2,\mathbb {R})\)-valued equivariant 1-form \(\nu \) on \(B_{(\Omega ,I)}\) such that

*M*in the usual way. By construction, the induced connection \({}^{\nu }\nabla \) on

*TM*is the unique (affine) connection with torsion \(\tau \) satisfying

*Libermann connection*. Of course, if \(\tau \) vanishes identically, then \({}^{\nu }\nabla \) is just the Levi-Civita connection of

*g*.

### 3.3 From Projective to Bi-Lagrangian Structures

Denoting by \(B_{(\Omega ,I)}\) the bundle of adapted coframes of a bi-Lagrangian structure \((\Omega ,I)\) and by \(P_{[\nabla ]}\) the Cartan bundle of a projective structure \([\nabla ]\), we obtain

### Theorem 3.4

### Proof

*M*. Furthermore, \(\psi \) satisfies

*M*and a unique \(\Omega \)-skew-symmetric endomorphism \(I : TM \rightarrow TM\) whose square is the identity. The 2-form \(\Omega \) pulled back to \(B_{(\Omega ,I)}\) becomes \(-\eta \wedge \omega \). The structure equations (2.5) imply that we have

*M*. The equivariance properties of \(\theta \) and (3.6) imply that the \(\psi \)-pushforward of \(\phi +\mathrm {I}{\text {tr}}\phi \) is a principal right \(\mathrm {GL}(2,\mathbb {R})\)-connection on \(B_{(\Omega ,I)}\) which satisfies (3.5) with \(T_-\equiv 0\) and \(T_+=L\circ \psi ^{-1}\). In particular, \(E_-\) is always integrable and \(E_+\) is integrable if and only if

*L*vanishes identically, that is, \([\nabla ]\) is flat. Denoting by \(\nu \) the Libermann connection of \((\Omega ,I)\), we obtain from its uniqueness that

### Remark 3.5

*N*, become geodesics of \([\nabla ]\). Conversely every geodesic of \([\nabla ]\) arises in this way. Likewise, a geodesic of the Libermann connection arises as the projection of an integral curve of a horizontal vector field on \(B_{(\Omega ,I)}\) which is constant on the canonical 1-form. It follows that the geodesics on \((N,[\nabla ])\) correspond to the geodesics of the Libermann connection on \((M,\Omega ,I)\) that are everywhere tangent to \(E_-\).

### 3.4 A Local Coordinate Descripition

*H*-bundle isomorphism \(P_{[\nabla ]}\simeq F \times \mathbb {R}_2\). In particular, we obtain a diffeomorphism \(\psi _{\nabla } : (F\times \mathbb {R}_2)/\mathrm {GL}(2,\mathbb {R}) \rightarrow M\). By construction, the quotient \((F\times \mathbb {R}_2)/\mathrm {GL}(2,\mathbb {R})\) is the cotangent bundle of \(N\). Denoting the projection \( F\times \mathbb {R}_2 \rightarrow T^*N\) by \(\mu \) as well, we obtain

*F*represents the Schouten tensor of \(\nabla \) and \(\varphi \) the connection form of \(\nabla \). Using (3.8), we see that in terms of the Patterson–Walker metric \(h_{\nabla }\) of \(\nabla \) and the Liouville 1-form \(\lambda \) of \(T^*N\), the metric can be expressed as

### Remark 3.6

Besides taking the quotient of the Cartan bundle by \(\mathrm {GL}(2,\mathbb {R})\), one might also consider the quotient by \(\mathbb {R}^2\rtimes H\), where *H* is the connected non-abelian real Lie group of dimension two. This quotient—which is a formal analogue to the construction of the conformal Fefferman metrics [22]—was studied in [35]. We also refer the reader to [24] for a generalisation of this construction to higher dimensions and its relation to the classical Patterson–Walker metrics [36].

### 3.5 Lift of Projective Vector Fields

Denoting by \(\rho : M \rightarrow N\) the basepoint projection, an immediate consequence of Theorem 3.4 is

### Corollary 3.7

For every open set \(U\subset N\) the Lie algebra of projective vector fields \(\mathcal {P}_{[\nabla ]}(U)\) is isomorphic to the Lie algebra of bi-Lagrangian vector fields \(\mathcal {B}_{(\Omega ,I)}(\rho ^{-1}(U))\).

### Proof

By standard results about Cartan geometries (c.f. [10]), the projective vector fields on \(U\subset (N,[\nabla ])\) are in one-to-one correspondence with the vector fields on \(\pi ^{-1}(U)\subset P_{[\nabla ]}\) whose flow preserves the Cartan connection \(\theta \) and which are equivariant for the principal right action. Theorem 3.4 implies that such a vector field corresponds to a vector field on \(\psi (\pi ^{-1}(U))\subset B_{(\Omega ,I)}\) preserving both the tautological form \((\eta ,\omega )\) and the Libermann connection. Again, by standard results about *G*-structures [10], such vector fields are in one-to-one correspondence with vector fields on \(\rho ^{-1}(U)\) preserving both \(\Omega \) and *I*. \(\square \)

*M*,

*g*) is also symplectic with respect to \(\Omega \) and hence the lift of a projective vector field on \((N,[\nabla ])\). As a warm up, we first consider a correspondence between affine vector fields and Killing vector fields for the associated Patterson–Walker metric (2.10). Let \(\nabla \) be an affine connection on \(N\). Recall that a vector field

*K*on \(N\) is affine with respect to \(\nabla \) if and only if

### Proposition 3.8

Let *K* be an affine vector field for a connection \(\nabla \) on \(U\subset N\). Then its complete lift (3.12) is a Killing vector field for the Patterson–Walker metric (2.10).

### Proof

*K*is projective for \(\nabla \) if and only if \((\mathcal {L}_K\nabla )_0=0\), that is, there exists a 1-form \(\rho \) on

*N*such that

### Proposition 3.9

*K*be a projective vector field with \(\rho _i=\nabla _i f\). Then

### Proof

*K*satisfies (3.14) with \(\rho =\mathrm{d} f\). \(\square \)

Finally we give the main result of this Section, and establish a one-to-one correspondence between projective vector fields on \((N, [\nabla ])\) and Killing vector fields on the Einstein lift on *M*.

### Theorem 3.10

*K*be a projective vector field on \((U, [\nabla ])\), where \(U\subset N\). Then

### Proof

To prove the converse, consider a general vector field \({\mathcal K}=K^i\partial /\partial x^i +Q_i\partial /\partial \xi _i\) on *M*, and impose the Killing equations. The \(\mathrm{d}\xi _i\odot \mathrm{d}\xi _j\) components of these equations imply that \(K^j=K^j(x^1, x^2)\). The \(\mathrm{d}\xi _i\odot d x^j\) components yield the general form (3.16), where \(\rho _i\) are some unspecified functions on *N*. Finally the \(\mathrm{d}x^i\odot \mathrm{d}x^j\) components imply that the vector field \(K^i\partial /\partial x^i\) on *N* is projective. \(\square \)

## 4 Local Characterization of the Metric

*g*constructed on the canonical affine bundle of a projective surface \((N,[\nabla ])\) is isometric to the metric

*N*, where \(\nabla \in [\nabla ]\) is any representative connection. The metric (4.1) has previously appeared in [9] as a member of a one-parameter family \(g_{\nabla ,\Lambda }\) of split-signature metrics on \(T^*N\) that one can associate to a torsion-free connection on

*N*. The metrics take the form

^{2}and Einstein with scalar curvature \(24\Lambda \), as can easily be verified by direct computation. Moreover, under the assumption that \(\nabla \) is non-flat, the metrics \(g_{\nabla ,\Lambda }\) are locally characterised as the neutral signature four-dimensional type II Osserman metrics whose Jacobi operators have non-zero eigenvalues. We refer the reader to [9, Thm. 7.3] for details. Here we provide another characterisation. Recall [3, 13, 43] that a distribution \({\mathcal D}\subset TM\) on a Riemannian manifold (

*M*,

*g*) is called parallel if \({{}^g\nabla }_X Y\in \Gamma ({\mathcal D})\) if \(Y\in \Gamma ({\mathcal D})\), where \({{}^g\nabla }\) is the Levi-Civita connection of

*g*. Thus, if \({\mathcal D}\) is parallel, then it is necessarily Frobenius integrable as \([X, Y]={{}^g\nabla }_X Y - {{}^g\nabla }_Y X \in \Gamma ({\mathcal D})\) if \(X, Y\in \Gamma ({\mathcal D})\).

### Theorem 4.1

Let (*M*, *g*) be an ASD Einstein manifold with scalar curvature 24 admitting a parallel ASD totally null distribution. Then (*M*, *g*) is conformally flat, or it is locally isometric to \((T^*N,g_{\nabla })\) for some torsion-free connection \(\nabla \) on *N*.

### Proof

*M*such that \(\text{ Ker }(\Theta )=\text{ span }\{\partial /\partial \xi _1, \partial /\partial \xi _2\}\). We can rescale \(\iota \) so that the corresponding two-form is closed, and proportional to \(\mathrm{d}x^1\wedge \mathrm{d}x^2\) for some functions \((x^1, x^2)\) which are constant on each \(\beta \)-surface in the two-parameter family. The functions \((\xi _1, \xi _2)\) are then the coordinates on the \(\beta \)-surface. The corresponding metric takes the form

### Remark 4.2

*N*with skew-symmetric Ricci tensor, then (4.2) simplifies to become

### Remark 4.3

*charged symplectic form*

^{3}

*L*is the pull-back to

*M*of the Liouville curvature \(\epsilon ^{ij}\nabla _{i}\mathrm {P}_{jk}\mathrm{d}x^k\otimes (\mathrm{d}x^1\wedge \mathrm{d}x^2)\) of \([\nabla ]\), which vanishes if and only if \(\nabla \) is projectively flat.

### Remark 4.4

### Remark 4.5

*g*with a two-plane distribution imposes topological restrictions on

*M*. If

*M*is compact then [1, 25]

*g*.

^{4}We may assume that \(\mathcal {D}\) is the graph of an isomorphism \(\mathcal {V}\rightarrow \mathcal {V}^{\prime }\), where \(TM=\mathcal {V}\oplus \mathcal {V}^{\prime }\) is an orthogonal decomposition into time-like and space-like subbundles with respect to some chosen background metric

*h*on

*M*. After possibly passing to a double cover, we can assume \(\mathcal {V}\) and \(\mathcal {V}^{\prime }\) to be orientable. Moreover, we may fix orientations so that the isomorphism \(\mathcal {V}\rightarrow \mathcal {V}^{\prime }\) is orientation reversing, thus equipping

*M*with an orientation so that \(\mathcal {D}\) is anti-self-dual. By rotating clockwise in \(\mathcal {V}\) and \(\mathcal {V}^{\prime }\) with respect to

*h*, we obtain an almost complex structure on

*M*such that \(\mathcal {V}\) becomes a complex line subbundle

*L*, and so that \(\mathcal {V}^{\prime }\) becomes its dual bundle \(L^*\). Consequently,

*M*admits an almost complex structure

*J*such that the canonical bundle of (

*M*,

*J*) is trivial. After possibly passing to a double cover, it therefore follows that

*M*is oriented and spin and—assuming

*M*is compact—that

*M*to be orientable hence (4.3) still holds true (assuming our choice of orientation) without passing to the cover as \(\chi \) and \(\tau \) are both doubled when passing to a double cover.

## 5 Gauge Theory of Tractor Connection

In this Section we shall present a gauge-theoretic construction of the metric (1.1). We shall introduce a projectively invariant equation on a connection, and a pair of Higgs fields on an auxiliary vector bundle \(E\rightarrow N\). In the special case when *E* is a rank-3 cotractor bundle (see Sect. 5.2) and the gauge group is \(SL(3, \mathbb {R})\), the horizontal lifts of the geodesic spray of \(\nabla \) and the Higgs field will give rise to an integrable \(\alpha \)-plane (twistor) distribution on *TM*, where \(M=\mathbb {P}(E)\) with a projective line removed from each fibre.

*A*, where \(\mathfrak {g}\) is some Lie algebra. Let \(\phi \) be a one-form on

*N*, called the Higgs pair, with values in the Lie algebra \(\mathfrak {g}\). In an open set \(U\subset N\) we shall write \(\phi =\phi _i \mathrm{d}x^i\) and regard \(\phi \) and

*A*as \(\mathfrak {g}\) valued one-forms on

*N*transforming as

*G*is the gauge group with the Lie algebra \(\mathfrak {g}\).

*A*is a connection on a principal (rather than a vector) bundle. While our construction below is self-contained, and does not rely on the results of [7, 8], we shall nevertheless refer to (5.1) as the Calderbank equations.

### 5.1 The Calderbank Equations

In Sect. 5.2 we shall show how the Calderbank equations with the gauge group \(\mathrm {SL}(3, \mathbb {R})\)—regarded as a subgroup of the group of diffeomorphisms of \(\mathbb {RP}^2\)—leads to the neutral signature anti-self-dual Einstein metric (3.10). We shall first list some other (implicit) occurrences of these equations for other gauge groups.

#### 5.1.1 Null Reductions of Anti-self-dual Yang–Mills Equations

If the projective structure is flat, then (5.1) is the symmetry reduction of the anti-self-dual Yang–Mills (ASDYM) equation on \(\mathbb {R}^{2, 2}\) by two null translations and such that the (2, 2) metric *g* restricted to the two-dimensional space of orbits \(N=\mathbb {R}^2\) is totally isotropic, and the bi-vector generated by the null translations is anti-self-dual.

*A*on \(\mathbb {R}^{2, 2}\), and set \(F=dA+A\wedge A\). In local coordinates adapted to \(\mathbb {R}^{2, 2}=TN\) with \(x^i\) the coordinates on \(N\), the null isometries are \(\partial /\partial \xi _i\), and the metric is

*A*. In [40] these equations have been solved completely for the gauge group \(\mathrm {SL}(2)\).

#### 5.1.2 Prolongation of the Calderbank Equations

*A*. The system is now closed. Commuting the covariant derivatives on \(\mu \) leads to an integrability condition

#### 5.1.3 Killing Equations

If the connection *A* is flat, and \(\mathfrak {g}=\mathbb {R}\), then the Calderbank equations become the projectively invariant Killing equations.

#### 5.1.4 Anti-self-dual Conformal Structures with Null Conformal Killing Vectors

*K*on \(\Sigma \). Let \(M\rightarrow N\) be a surface bundle over \(N\), with two-dimensional fibres \(\Sigma \). In this case the Calderbank equations are solvable by quadrature and the two-dimensional distribution

*M*is the twistor distribution for the most general ASD (2, 2) conformal structure which admits a null conformal Killing vector

*K*[7, 20, 34].

#### 5.1.5 The Patterson–Walker Riemannian Extension

*A*and the Higgs field given by

*g*on \(T^*N\). We shall restrict our discussion to

*special connections*in \([\nabla ]\) which preserve some volume. Consider the effect of transformation (2.2) with \(\Upsilon _i=\nabla _i f\), together with rescaling the fibres of \(TN\rightarrow N\)

^{5}(2.10). A straightforward calculation yields

### 5.2 Tractor Connection and ASD Einstein Metrics

In this Section we shall consider the Calderbank equations, where the gauge group is \(\mathrm {SL}(3, \mathbb {R})\), and *E* is the standard cotractor bundle for the projective structure \([\nabla ]\). Recall the Cartan bundle \(P_{[\nabla ]}\) from section (2.3). We may think of the left action of \(H\subset \mathrm {SL}(3,\mathbb {R})\) on \(\mathbb {R}_3\) by matrix multiplication as a (linear) *H*-representation and, consequently, we obtain an associated rank-3 vector bundle *E* for every projective surface \((N,[\nabla ])\). The vector bundle *E* is commonly referred to as the *cotractor bundle* of \((N,[\nabla ])\). Interest in *E* stems from the fact that it comes canonically equipped with an \(\mathrm {SL}(3,\mathbb {R})\) connection which is flat if and only if \((N,[\nabla ])\) is, see [2].

*E*. These generators descend to eight vector fields (which we shall also denote \({\mathbf{t}_\alpha }^\beta \)) which generate the action of \(\mathrm {SL}(3, \mathbb {R})\) on the fibres of the projective cotractor bundle \(\mathbb {P}(E)\) which is a quotient of

*E*by the Euler vector field \(\sum _{\alpha =0}^2 {\mathbf{t}_\alpha }^\alpha \). Setting \(\xi _i=\psi _i/\psi _0\) yields

*M*be a complement of a projective line in the total space of the bundle \(\mathbb {P}(E)\). The corresponding contravariant metric on

*M*is constructed by demanding that the leaves of the rank-2 distribution (5.4) \({\mathcal D}\subset T(M\times \mathbb {RP}^1)\) project down to self-dual two-surfaces on

*M*. This gives \( \epsilon ^{ij}(\partial /\partial x^i -A_i)\odot \phi _j, \) or, in the covariant form,

### Theorem 5.1

Formula (5.9) defines a metric which does not depend on a choice of a connection in a projective class.

### Proof

The metric is anti-self-dual, and Einstein with scalar curvature equal to 24. The anti-self-duality is a consequence of the fact that the connection *A* and the Higgs field \(\phi _i\pi ^i\) satisfy the Calderbank equations [7].

## 6 Examples

### 6.1 Homogeneous Model \(M = \mathrm{SL}(n + 1,\mathbb {R})/\mathrm{GL}(n)\)

To finish the proof we need to argue that \(\mathbb {R}^{n+1}\times \mathbb {R}_{n+1}\setminus \Delta \) projects down to a complement of an \(\mathbb {RP}_{n-1}\) subbundle in \(\mathbb {P}(E)\). This subbundle is just \(\mathbb {P}(T^*N)\) and it has an injection into \(\mathbb {P}(E)\) given by \(f\rightarrow (0, f)\). A point in \(N\) with homogeneous coordinates [1, 0, ..., 0] (corresponding to our choice of an affine chart) is not incident with any cotractor in \(\mathbb {P}(E)/\mathbb {RP}_{n-1}\), so removing a diagonal is equivalent to looking at the complement of this subbundle.

### 6.2 Ricci-Flat Limits

*N*and a connection \(\nabla \in [\nabla ]\) such that [46]

^{6}. In this case

### 6.3 Cohomogeneity: One Examples

*N*can be 8, 3, 2, 1 or 0 (see [30], and also [5, 17, 39]). If the dimension is maximal and equal to 8, then \(\mathfrak {g}=\mathfrak {sl}(3, \mathbb {R})\), and the projective structure is flat. We have shown that in this case the resulting metric (1.1) is given by (6.1), and admits 8 Killing vectors in agreement with Theorem 3.10. We shall now consider the submaximal case, where \(\mathfrak {g}=\mathfrak {sl}(2, \mathbb {R})\). There are two one-parameter families of non-flat projective structures with this symmetry. Their unparametrised geodesics are integral curves of a second-order ODE

*r*given by \(r^2\equiv {(x^1\xi _1+x^2\xi _2)}\) which is constant on the orbits. Let \(\sigma ^{\alpha }\) be right-invariant one-forms on \({\text {SL}}(2, \mathbb {R})\) such that

*r*, in choosing these one-forms. If we chose \(\Lambda <0\), and take

## Footnotes

- 1.
This definition is common in projective differential geometry, but differs from the more standard definition, where the Ricci curvature is defined as \(\mathrm {Ric}(\nabla )(X,Y)={\text {tr}}\left( Z\rightarrow R^{\nabla }(Z,Y)X\right) \).

- 2.
self-dual with respect to the orientation convention of [9].

- 3.
This terminology is motivated by the Hamiltonian description of a charged particle moving on a manifold, where the canonical symplectic structure on the cotangent bundle needs to be modified by a pull-back of a closed two-form (magnetic field) from the base manifold. In our case the two-form is the skew-symmetric part of the Schouten tensor, and the inverse of the Ricci scalar plays a role of electric charge. This magnetic term can always be set to zero by an appropriate choice of a connection in a projective class—here we find it convenient not to make any choices at this stage.

- 4.
Private communication, March 2016.

- 5.In [18] (see also [6, 23, 24] for other applications of this lift) it was proven that a ‘similar’ metric constructed out of the Thomas symbols \(\Pi _{ij}^k=\Gamma _{ij}^k-\frac{1}{3}\Gamma _{il}^l\delta ^k_j -\frac{1}{3}\Gamma _{jl}^l\delta ^k_i\), is anti-self-dual and null-Kähler (with ASD null-Kähler two-form) for any choice of \(\Gamma _{ij}^k\). The Patterson–Walker lift (2.10) is conformally equivalent (up to a diffeomorphism) to the projective Patterson–Walker lift (5.6) only if \(\Gamma _{ij}^j=\nabla _i F\) for some function
*F*on*N*. - 6.

## Notes

### Acknowledgements

The authors wish to thank Andreas Čap, Andrzej Derdziński, Nigel Hitchin and Claude LeBrun for helpful discussions regarding the contents of this paper. TM is grateful for travel support via the grant SNF 200020_144438 of the Swiss National Science Foundation. MD has been partially supported by STFC consolidated grant ST/P000681/1.

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